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## Material Information- Title:
- Characterizing the Fiber-Matrix Interface via Single Fiber Composite Tests
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Carbon fibers ( jstor ) Crystallites ( jstor ) Diameters ( jstor ) Fiber strength ( jstor ) Interphase ( jstor ) Modeling ( jstor ) Polymers ( jstor ) Shear stress ( jstor ) Simulations ( jstor )
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- 7/12/2007
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PAGE 1 CHARACTERIZING THE FIBER-MATRIX INTERPHASE VIA SINGLE FIBER COMPOSITE TESTS By DAVID MICHAEL BENNETT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 Copyright 2007 by David M. Bennett PAGE 3 To my wife, Kerri, on the first anniversary of our marriage. PAGE 4 ACKNOWLEDGMENTS I wish to thank my parents for their c ontinuous support throug hout this work, and Prof. Charles L. Beatty for his invaluable in sight, assistance, and s upport. I would also like to acknowledge the patience and willingn ess of my supervisory committee members to cooperate and contribute during this protracted and fractionated process. PAGE 5 v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................ix LIST OF FIGURES..........................................................................................................xii ABSTRACT.....................................................................................................................xx i CHAPTER 1 INTRODUCTION..........................................................................................................1 2 CARBON FIBER STRENGTH AND STATISTICS.....................................................4 Introduction................................................................................................................4 Experimental Methods...............................................................................................6 Theoretical Discussion...............................................................................................7 Fiber Strength Statistics....................................................................................7 Carbon fiber failure mechanisms/models..............................................18 Reynolds-Sharp failure mechanism......................................................18 Stewart-Feughelman model..................................................................31 Batdorf-Crose failure theory.................................................................33 Experimental Methods and Error Sources......................................................44 Results......................................................................................................................47 Fiber Diameters..............................................................................................47 Load-Deflection..............................................................................................51 Failure Stresses ...............................................................................................55 Discussion................................................................................................................58 Weibull Strength Statistics.............................................................................58 Weibull two-parameter distribution......................................................60 Concurrent distribution.........................................................................66 Exclusive and partially-c oncurrent distributions..................................74 Carbon Fiber MorphologyProperty Relationship..........................................75 Conclusion................................................................................................................78 RecommendationsÂ…................................................................................................81 PAGE 6 vi 3 CHARACTERIZING THE FIBER-MATRIX INTERFACE....................................170 Introduction............................................................................................................170 Historical Development...............................................................................170 Additional Testing Methods.........................................................................171 Discussion..............................................................................................................175 Early One-Dimensional Single Fibe r Composite Fragmenation (SFCF).....175 Yielded matrix assumption.................................................................178 Shear-lag assumption..........................................................................183 Yielded matrix assumption reconsidered............................................187 Ineffective length................................................................................190 Additional assumptions.......................................................................191 SFCF advanced interpretation......................................................................193 Multi-dimensional analysis.................................................................195 3-D continuum treatment....................................................................196 Numerical/Empirical evaluation.........................................................198 Interfacial Debonding...................................................................................203 General discussion..............................................................................203 Single fiber pull-out............................................................................207 SFCF Modelling Multi-Modal......................................................................215 Stochastic simulation..........................................................................217 Effective physical properties and continuum considerations..............221 Interface adhesion modeling...............................................................222 Ineffective length reconsidered...........................................................224 Interface/Interphase Influence......................................................................227 Interphase formation...........................................................................229 Interphase modeling............................................................................230 Interphase Chemistry/Morphology/Dynamics..............................................233 Interphase morphology.......................................................................233 Interphase reaction kinetics/products..................................................234 Interphase definition...........................................................................236 Thermodynamic and spectroscopic evaluation...................................237 Kinetic explanation.............................................................................246 Mechanistic explanation......................................................................250 Interphase composition/physical modeling.........................................254 Interphase non-linear behavior............................................................257 Surface energetics and dynamics........................................................264 Interphase fracture mechanics.............................................................266 Crack formation energetics.................................................................270 CF surface free energy........................................................................272 CF topology/microstructure................................................................273 Schematic representation of fracture mechanics.................................276 PAGE 7 vii Analytical Model Assessment.....................................................................279 Cox and Rosen one-dimensional analytical models............................279 Kelly-Tyson yielded matrix................................................................282 Bi-Modal load absorption models.......................................................292 Spring-layer SFCF modeling .............................................................301 Modeling Difficulties .........................................................................304 Summary .............................................................................................306 Conclusions............................................................................................................310 Recommendations..................................................................................................314 4 SIMULATING THE SINGLE FI BER COMPOSITE FRAGAMENTATION TESTÂ…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…Â…..394 Introduction............................................................................................................394 Theoretical Background.........................................................................................396 Modeling Uncertainties and Variability.......................................................396 Variabilities Â…Â…Â…Â…Â…....................................................................397 Response surface methodology (RSM)...............................................399 Non-probabilistic NDA ......................................................................401 Model Quality .............................................................................................403 Model Development...............................................................................................404 Bi-modal Model............................................................................................405 Interphase Control Diagram.........................................................................409 Property covariance Â…Â…Â…Â…Â….....................................................410 Model definition .................................................................................410 Debonding prediction..........................................................................410 Recovery and plateau length...............................................................413 Simulation Details...............................................................................415 Results....................................................................................................................418 Discussion..............................................................................................................419 Conclusions............................................................................................................421 Recommendations..................................................................................................422 APPENDIX A WEIBULL-GRIFFITH STATISTICS.......................................................................438 B COX LOAD TRANSFER..........................................................................................446 C ROSEN LOAD TRANSFER.....................................................................................448 D LAWRENCE PULL-OUT ANALYSIS....................................................................451 E GAO ENERGETICS..................................................................................................454 F HENSTENBURG AND PHOENIX SIMULATIONS...............................................459 PAGE 8 viii G BELTZER INEFFECTIVE LENGTH.......................................................................462 H SFCF SIMULATION PROGRAM............................................................................469 REFERENCES................................................................................................................481 BIOGRAPHICAL SKETCH...........................................................................................497 PAGE 9 ix LIST OF TABLES Table page 2.1 Exclusive and partially-concurrent di stribution parameters for Cases 1 through 6 (Figures 2.8 and 2.9).....................................................................................82 2.2 Exclusive and partially-concurrent di stribution parameters for Cases 1 and 2 (Figure 2.10)....................................................................................................82 2.3 Exclusive and partially-concurrent parameters for the Weibull probability plots (Figures 2.13 and 2.14)...........................................................................82 2.4 Statistical measures for a simulated two-parameter Weibull distribution.......83 2.5 PAN-based carbon fiber physic al properties as a func tion of heat treatment temperature (HTT)...........................................................................................84 2.6 Statistics for extremes of normal distributions with =0 and various standard deviations.........................................................................................................85 2.7 Set A carbon fiber diameters dete rmined by Fraunhoffer diffraction..............86 2.8 Set B carbon fiber diameters determined by Fraunhoffer diffraction..............87 2.9 Set C carbon fiber diameters determined by Fraunhoffer diffraction..............88 2.10 Set D carbon fiber diameters dete rmined by Fraunhoffer diffraction..............89 2.11 Set A carbon fiber (20 mm long, untreated) load-defl ection and stress-strain results...............................................................................................................90 2.12 Set B carbon fiber (50 mm long, untreated) load-defl ection and stress-strain results...............................................................................................................91 2.13 Set C carbon fiber (20 mm long, plasma treated) load-def lection and stressstrain results.....................................................................................................92 2.14 Set D carbon fiber (20 mm long, plasma treated) load-def lection and stressstrain results.....................................................................................................93 2.15 Set A through Set D two-parameter We ibull distribution parameters and GOF factors...............................................................................................................94 PAGE 10 x2.16 Two-population Weibull conc urrent simulation statisti cs for various scaling ratios.................................................................................................................95 2.17 Two-population Weibull conc urrent distribution para meters and GOF factors for Set A through D fibers...............................................................................96 2.18 Two-population exclusive and partially -concurrent distribution parameters and GOF factor for Set A and B carbon fibers................................................96 3.1 Fiber reinforcement efficiency, Eratio, for different fiber gage lengths..........316 3.2 Input parameters for fiber reinfor cement efficiency calculations (Eqn. 3.8, Figure 3.5)......................................................................................................317 3.3 Physical properties and range variables used in Eqns. 3.9-3.12 (Figures 3.93.12)...............................................................................................................318 3.4 Failure mechanisms in fibrous composites....................................................319 3.5 Typical interfacial properties and pull-out parameters (Eqns. 3.37-3.38).....320 3.6 Henstenburg and Phoenix (1989) bi-l inear stress recovery simulation parameters and results....................................................................................321 3.7 Glass transition temperatures and esti mated number of chain atoms involved in cooperative movement at Tg for various polymers....................................322 3.8 Tg, coefficient of thermal expansion ( ), and thermodynamic values for neat and filled polymer systems............................................................................323 3.9 Heat capacity vari ation (DSC) at Tg, Cp, and interphase characteristics ( i and ri :Eqns. 3.51-3.52) for iron-particulate composites....................324 3.10 Rutile material characteristics, ruti le-CPE composite damping capacity, and calculated interaction parameters ( B, and R) (Eqn. 3.55, and Figures 3.433.44)......................................................................................................325 3.11 Glass transition parameters for various neat and Aerosil filled polymer systems...........................................................................................................326 3.12 Work of adhesion (Wa) and interfacial fracture energy (Wf)values via inverted blister test for adhesive/s ubstrate couples at sub-Tg(adhesive) temperatures (Figure 3.53)..................................................................................................327 3.13 Selected surface energy properties, a nd the harmonic and geometric means of Wa x (x=h,g) for HM and HS carbon fiber-epoxy composites........................328 PAGE 11 xi3.14 Selected surface energy and physical properties of PAN (T-300, HS CF, Toray) and Pitch (P-55, HM CF, Un ion Carbide) carbon fibers, and pure graphite..........................................................................................................329 3.15 Physical properties and adhesion strengt h for different carbon fiber treatments in an epoxy matrix.........................................................................................330 3.16 Interfacial shear strength for coated and uncoated carbon fibers..................331 3.17 Laminate (0/904/0) void content and composite failure strain against fiber surface treatment and test temperature..........................................................332 4.1 Interphase control diagram indica ting interaction of major composite parameters on interphase properties...............................................................423 4.2 Simulation parameters for single fiber composite fragmentation NDA........424 4.3 Single-tail normal parameter uncerta inty NDA fragment length statistics....425 4.4 Fiber Weibull moduli and single-tail normal parameter uncertainty NDA fragment length statistics...............................................................................426 4.5 Interfacial shear stress and singletail normal parameter uncertainty NDA fragment length statistics...............................................................................427 PAGE 12 xii LIST OF FIGURES Figure page 2.1 Fiber tensile testing jig and load-aligning fixture............................................97 2.2 Laser diffraction system used to measure carbon fiber diameters...................97 2.3 Radio Frequency plasma volume and rota ting carousel used to treat discrete carbon fibers.....................................................................................................98 2.4 Two-dimensional failure state space, for a fiber of length L with theoretical tensile strength.................................................................................................98 2.5 Cumulative distribution functions for ideal two-parameter Weibull distribution.......................................................................................................99 2.6 Probability density functions for idea l two-parameter Weibull distribution.100 2.7 Concurrent two population Weibull CDFs....................................................101 2.8 Exclusive two population Weibull CDFs......................................................102 2.9 Partially-concurrent tw o population Weibull CDFs......................................103 2.10 Exclusive and partiallyconcurrent Weibull CDFs........................................104 2.11 Two-parameter Weibull di stribution probability plot....................................105 2.12 Concurrent Weibull dist ribution probability plot..........................................106 2.13 Exclusive Weibull distribution probability plot.............................................107 2.14 Partially-concurrent Weibull distribution probability plot............................108 2.15 Simulated two-parameter Weibull distribution probability plots..................109 2.16 Reynolds-Sharp carbon fiber tensile failure mechanism...............................110 2.17 Effective basal plane compliance versus basal plane misorientation from fiber axis.................................................................................................................111 PAGE 13 xiii2.18 Reynolds-Sharp and Griffith failure criteria as a function of fiber axis/crystallite misorientaion ......................................................................112 2.19 Small angle x-ray scattering patt erns for PAN-based carbon fibers..............113 2.20 The averaged pore axis angular dist ribution for two types of PAN-based carbon fibers...................................................................................................114 2.21 First asymptote maximum value distri bution and generated data for a normal distribution.....................................................................................................115 2.22 Weibull, third asymptote, maximum va lue distribution and generated data for a normal distribution......................................................................................116 2.23 First asymptote minimum value distri bution and generated data for a normal distribution.....................................................................................................117 2.24 Weibull, third asymptote, minimum valu e distribution and generated data for a normal distribution.........................................................................................118 2.25 First asymptote absolute extreme valu e distribution and generated data for a normal distribution.........................................................................................119 2.26 Weibull, third asymptote, absolute extreme value distribution and generated data for a normal distribution.........................................................................120 2.27 Effect of crystallite misorientati on dispersion and magnitude on simulated carbon fiber tensile failure strength...............................................................121 2.28 Effect of crytallite misorientaion ma gnitude on simulated carbon fiber tensile failure strength...............................................................................................122 2.29 Effect of numerical parameters on simulated carbon fiber tensile failure strength ........................................................................................................123 2.30 Ribbon-like morphology proposed for PAN-based carbon fibers....................124 2.31 Polar projections of effective stress acting on a crack subjected to simple uniaxial tension..............................................................................................125 2.32 The bounded solid angle ( cr) that contains the gra phitic crystallite basal plane orientations for which failure via transverse rupt ure is expected.........126 2.33 Uniaxial Batdorf-Crose failure predic tion utilizing the Reynolds-Sharp failure criteria ............................................................................................................127 2.34 Frequency histogram for flaw free lengths in acrylic fibers..........................128 PAGE 14 xiv2.35 Notional flaw size/failure strength distribution for surface and volume flaws..................................................................................................129 2.36 Thin beam load cell calibration curve............................................................130 2.37 Fraunhoffer diffraction patterns produ ced by a single carbon fiber and HeNe laser light........................................................................................................131 2.38 Fraunhoffer diffraction schematic indi cating object/viewing plane separation, L, and intensity node separation Zn.............................................................132 2.39 Pixel intensity along th e centerline of the Fra unhoffer diffraction pattern....133 2.40 Carbon fiber diameters determ ined by Fraunhoffer diffraction.....................134 2.41 Normal distributions for carbon fiber diameter measurements.....................135 2.42 Typical PAN-based carbon fiber cross section..............................................136 2.43 Typical tensile load-deflection cu rves for PAN-based carbon fibers............137 2.44 Failure load versus failure stress for ca rbon fibers with assumed constant fiber cross section...................................................................................................138 2.45 Cumulative failure probabilities fo r as-received PAN-based carbon fibers..139 2.46 Two-parameter Weibull probability pl ot for carbon fiber tensile tests..........140 2.47 Two-parameter PDFs for carbon fiber tensile tests.......................................141 2.48 Two-parameter Weibull probability plot for set A tensile tests.....................142 2.49 Prediction residuals for LLSQ regr ession of Set A and B carbon fibers.......143 2.50 Set A tensile strength two-parameter Weibull CDF determined by LLSQ regression.......................................................................................................145 2.51 Set B tensile strength two-parameter Weibull CDF determined by LLSQ regression.......................................................................................................146 2.52 Set A tensile strength two-parame ter Weibul CDF determined by MLE......147 2.53 Set B tensile strength two-parameter Weibull CDF determined by MLE.....148 2.54 Set A tensile strength two-paramete r Weibul CDF determined by non-linear estimation.......................................................................................................149 2.55 Set B tensile strength two-paramete r Weibul CDF determined by non-linear estimation.......................................................................................................150 PAGE 15 xv2.56 Two-parameter Weibull 95% lower c onfidence bounds for set A carbon fiber tensile tests.....................................................................................................151 2.57 Two-paramter Weibull 95% confidence intervals for set A carbon fiber tensile tests................................................................................................................152 2.58 Set A and B carbon fiber two-parameter Weibull PDFs................................155 2.59 Log-Log plot of average fiber fa ilure strength vs. gage length.....................156 2.60 Relationship between Weibull modulus scaling ratio, and ultimate strength, sb, for ideal two-parameter Weibull distributions..........................................157 2.61 Relationship between Weibull modulus, scaling ratio, and standard deviation, for ideal two-parameter Weibull distributions..........................................158 2.62 Weibull probability plot for si mulated two population concurrent distributions....................................................................................................159 2.63 Set A carbon fiber tensile streng th two population concurrent CDF.............160 2.64 Set B carbon fiber tensile streng th two population concurrent CDF.............161 2.65 Set A carbon fiber tensile strength two population concurrent Weibull probability plot...............................................................................................162 2.66 Set B carbon fiber tensile strength two population concurrent Weibull probability plot...............................................................................................163 2.67 Two population concurrent PDFs for fiber tensile tests of two different gage lengths .............................................................................................................. 164 2.68 Set A nd B scaling preditions for two population concurrent Weibull distributions....................................................................................................165 2.69 Failure strengths and two-parameter Weibull CDFs for set C and D carbon fibers..............................................................................................................166 2.70 Discrete CDFs for surface and volume flaw failure strength populations.....167 2.71 Discrete PDFs for surface and volume flaw failure strength populations.....169 3.1 Single fiber composite frag mentation (SFCF) schematic..............................333 3.2 Single fiber pull-out schematic i ndicating the embedded fiber length ( le) and a partially debonded section ( l ).........................................................................334 3.3 Fiber pull-out stress against em bedded length aspect ratio (l/d)eff.................335 PAGE 16 xvi3.4 Fiber strength variation and fiber axia l stress for an arbitrary length single fiber composite (SFCF) with perfect bonding (PB) and interacial shear yielding ( y)....................................................................................................336 3.5 Composite reinforcement modulus ratio, Eratio, against the normalized difference in fiber-matrix modulus ratio (Ef/Em)...........................................337 3.6 Three-dimensional representation of the composite reinforcement ratio......338 3.7 Stringer load diffusion into a semi-infinite half-plate....................................339 3.8 Generalized self-consistent scheme (G SCS) schematic for a multi-phase fiber and interphase embedded in an infinite effective medium............................340 3.9 Cox and Rosen FAS predictions ( f/ appl) against fiber axial position for several fiber aspect ratios...............................................................................341 3.10 Cox and Rosen interfacial shear stress predictions ( ifss/ appl) against fiber axial position for several fiber aspect ratios..................................................342 3.11 Comparison of shear stress prediction s and assumptions at the fiber tip......343 3.12 Comparison of FAS predictions for a discontinuous fiber...........................344 3.13 Initial pull-out stress and maximu m debond stress against embedded fiber length..............................................................................................................345 3.14 Load adsorption for a semi-infinite fiber in an infi nite continuum...............346 3.15 Normalized fiber axial stress, (z)/ ( ), against normalized axial position (z/a)..................................................................................................347 3.16 Load adsorption for a discontinuous, but contiguous, filament in an infinite continuum......................................................................................................348 3.17 Normalized load adsorption ( (z)/ ( ) against position (z/a) for a disjointed, but contiguous, infinite filame nt in an infinite continuum............................349 3.18 Effect of matrix poisson ratio and modul us ratio on the load transfer length [P(l/a)=0.9P( )] for a fractured infinite length fiber.....................................350 3.19 Matrix stress concentrati on near the fiber end (.06df) for different gap sizes and filler fraction............................................................................................351 3.20 Matrix shear and principal stress con centrations in an ideal five-fiber microcomposite with a central end-bonded discontinuous fiber...................352 3.21 Pull-out toughness predicti on against fiber length.........................................353 PAGE 17 xvii3.22 Maximum fiber pull-out load against length factor ( a l/2) for various interfacial frictional and shear debond strengths...........................................354 3.23 Schematic representation of fiber ax ial stress vs. displa cement for a single fiber pull-out experiment...............................................................................355 3.24 Remote fiber pull-out stress vs. displacement schematics for (a) and (b) q0> qth (c) q0 qth, and (d) a totally stable condition (zmax 0)....................................356 3.25 Single fiber pull-out schematic indi cating partial debond length (L-z) and shear stress interaction extent (rb ra)............................................................358 3.26 Partial debond stress against debond length using Gao's LFM prediction and Hsueh's strength criterion...............................................................................359 3.27 Bi-modal stress transfer across yielded terminal sections and a central shearlag approximation..........................................................................................360 3.28 Strengthening factor ( c/2 y) against ratio of yielded terminal section to fiber length for various fiber-matrix shear modulus...............................................361 3.29 Bi-linear stress recovery model fo r a broken fiber with a disbonded and yielded matrix section....................................................................................362 3.30 Hashin's collapsed two-dimensional interface...............................................363 3.31 Change in Tg of PMMA with Aerosil c ontent and testing frequency........364 3.32 Dielectric Relaxation Spectra (tan ) for Aerosil filled (vol. %) PMMA indicating dipole group and dipole segment mobility variation....................365 3.33 Variation in heat capacity jump at Tg ( Cp) with filler fraction...................366 3.34 Heat capacity variation of an ir on particulate-epoxy composite at Tg ( Cp) against filler size and fraction........................................................................367 3.35 Estimated boundary layer fraction ( bl) of an iron particulate-epoxy composite against filler fraction and particle size...........................................................368 3.36 Interphase thickness ( rbl) as a function of filler size and fraction ( f).........369 3.37 Normalized minimum particle-particle se paration of spherical elements in an ideal cubic packing configur ation vs. filler fraction......................................370 3.38 Heat capacity variation of an iron-particulate composite at Tg ( Cp) against filler size and minimum particle-particle separation.....................................371 PAGE 18 xviii3.39 Change in heat capacity variation with total filler su rface area and f iller size of an iron-particulate composite at Tg ( Cp).....................................................372 3.40 Glass transition temperature variat ion of phenoxy composites against filler concentration and type...................................................................................373 3.41 Tan max at Tg for chlorinated PE-rutile composites as a function of filler basicity, and filler concentration, f.........................................................374 3.42 Filler fraction correction against the ac id-base interaction parameter (W from IGC) for various chlorinated PE-rutile composites.......................................375 3.43 Bulk material flow vi a altering hole content..................................................376 3.44 Excess molar hole energy, h, and molar hole volume, Vh, of Aerosil composites against filler fraction...................................................................377 3.45 Variation in T as a function of molar cohesion energy, Wcg, for different Aerosil filled composites...........................................................................378 3.46 Molar cohesion energy, Wcg, and maximum Tg variation ( T) for different Aerosil composites relative to macromolecular rigidity, .......................379 3.47 Shift in temperature ( T by NMR) at tan max for a ED-20-Aerosil composite with different length alcohols (No. of C atoms in chain) grafted to the filler surface.............................................................................................380 3.48 Critical aspect ration, lc/d, against testing temper ature for a carbon fiber-epoxy single fiber composite....................................................................................381 3.49 Epoxy volume percent (Palmese predicti on and Skourlis disc retization) and a matrix shear modulus-stair-step model ag ainst radial position from the fiber surface............................................................................................................382 3.50 Unfolding matrix modulus relative to radial position (r) and fitting parameter ratio................................................................................................................383 3.51 Representative non-linear shear strain hardening ma terial (solid line) and idealized constitutive model for the composite interphase............................384 3.52 Work of Fracture, Wf, against work of adhesion, Wa, for various adhesives, substrates, and temperatures..........................................................................385 3.53 Schematic depicting a reorientation/st rain hardening stre ss-strain curve and the expected exponential growth for Wf versus Wa.......................................386 3.54 Interfacial shear strength, ifss, for HS and HM carbon fibers in epoxy resin against the geometric (x = g) and ha rmonic (x = h) work of adhesion, Wa x..387 PAGE 19 xix3.55 Apparent interfacial shear stress, ifss, versus embedded fiber length for various carbon fiber-matrix systems..............................................................388 3.56 Double-box fiber strength pr obability density function, f( ), for a fiber with distinct surface and volume flaws..................................................................389 3.57 Glass-coupling agent-matr ix interphase regions............................................390 3.58 Crack driving force, G( ), and continuum dissipation, D( ), versus load/displacement factor................................................................................391 3.59 Weibull failure probability, F, agai nst element/link fracture strain and the influence of improved wetting and bond strength.........................................392 3.60 Ideal Weibull failure distributions (Weibull modulus = ) and the influence of matrix ductility, defects, and constraints.......................................................392 3.61 Shear-lag model applied to a 0x/90y/0x composite laminate and normalized axial stress build-up in the 90 plies (t reated as a continuum) against distance from an existing crack....................................................................................393 4.1 Hypercubic approximations for an arbitrary two-dimensional (x1, x2) response solution...........................................................................................................428 4.2 Evolution of model prediction unc ertainty with increasing process understanding.................................................................................................429 4.3 Average debond lengths for AS-4 graph ite fibers in epoxy (new breaks only) as a function of applied strain........................................................................430 4.4 NDA treatment of covari ant property variation.............................................431 4.5 Bi-modal fragmentation debonding and load-uptake model.........................432 4.6 Fragment length histograms at satu ration for simulated parameter uncertainty......................................................................................................433 4.7 Normal distribution approximation and fragment length histogram at saturation for nominal (zero CV) SFCF simulations (n=25).........................434 4.8 Average saturation fragment length and maximum fragment length as a function of input parameter uncertainty.........................................................435 PAGE 20 xx4.9 Fragment length histogram and normal distribution approximation for singletail 10% CV SFCF simulations with various fiber Weibull strength moduli............................................................................................................436 4.10 SFCF fragment length histograms a nd normal distribution approximations (NDA simulation) for single-tail 0.01% CV parameter uncertainty and various interfacial shear stress values.........................................................................437 PAGE 21 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARACTERIZING THE FIBER-MATRIX INTERPHASE VIA SINGLE FIBER COMPOSITE TESTS By David Michael Bennett May 2007 Chair: Charles L. Beatty Major Department: Materials Science and Engineering ABSTRACT The assumptions of weakest-link theory and statistics were considered, especially as they relate to high-strength and high-m odulus carbon fibers. A critical review of proposed failure mechanisms and flaw populations was conducted. Numerical simulations and morphological evidence showed that the Reynolds-Sharp failure criteria for transverse rupture of misoriented turbostr atic graphite crystall ites are infrequently satisfied. Furthermore, the mean crystallite size in these carbon fibe rs is too small to cause catastrophic tensile failure, except in cooperation with other misoriented crystallites. A Reynolds-Sharp failure-strengt h model and a phenomenological/statistical model developed by Batdorf and Crose were used to predict failure statistics for nonuniformly oriented flaw populations e xposed to polyaxial stress states. PAGE 22 Ultimate tensile tests were conducted on carbon fibers of various gauge lengths and surface treatments. Results indicate two di stinct flaw populations in these PAN-based carbon fibers and that their failure statistic s are represented by a concurrent Weibull distribution with a broad, lower-strength surface flaw population, and a tighter, higherstrength volume distribution. Next, we explored the morphology, chemis try, and micromechanics of the fibermatrix interphase and their influence on fibe r-matrix load transfer. In particular, the development, utility, accuracy, and analytical interpretation methods for the single fiber composite fragmentation (SFCF) test are investigated. The complex nonlinear threedimensional stress state near fiber-matrix imperfections in conjunction with irregular interphase properties discounts simple shear -lag or yielded-matrix micromechanical analysis. It is shown, however, that interp retation approaches that include fiber-matrix debonding and fiber strength variation, while more representative of the physical fragmentation process, are una ble to accurately represent the stochastic fragmentation process. Additionally, it is argued that the interfacial shear strength microvariable ifss does not unambiguously define ma trix-fiber load absorption. A non-deterministic model and analysis (NDA) was developed for simulating the SFCF test and its attendant uncertaintie s, reducible and non-reducible. The NDA demonstrated that the accumulated uncer tainties in SFCF testing, modeling, and interpretation contribute to a larg e response prediction uncertainty for ifss. The utility and suitability of SFCF testing is therefore questioned, especially if the results of the tests are reported without confidence intervals. PAGE 23 CHAPTER 1 INTRODUCTION Natural and engineered composite material s and structures are essential to our everyday existence. We often identify com posite materials using only their components (i.e., graphite-epoxy, glass-polyester). This enables macrocomposite identification but neglects the role that the interface plays in macrocomposite performance. We cannot create a composite material without this inte rface, and a composite material would not be greater than the sum of its pa rts if not for the interface. Indeed, the interface and the interaction between the components is the m eans by which composite materials transfer load from the matrix material to the reinfo rcement. In continuum mechanics, this is called load absorption. Load absorption is a rela tively young micromechanics fi eld compared to load diffusion micromechanics, roughly coinciding with the introducti on of strong fiber reinforcements (metal fibers, 1955; carbon fibe rs, 1960s). The first, and still surviving, load absorption analysis (1952) considered the short-fiber rein forcement of paper composites. This seminal analysis was predicated on restrictive and faulted assumptions, yet provided a solid opening for continuing research into reinforcement-matrix micromechanics. More involved analyt ical treatments followed and a broad, interdisciplinary fiber-matrix adhesion and reinforcement micromechanics field developed. As we step away from the reinforcemen t we must consider an intermediate interphase layer that is morphologically, chem ically, and mechanically different from the PAGE 24 bulk matrix properties. We consider the mi cromechanics of the load transfer from the macrocomposite structure through the matr ix and across the interphase into the reinforcing fiber. In particular, we focus on a model microcomposite test: the single fiber composite fragmentation (SFCF) test. SFCF analysis/interpretation and relate d model microcomposite tests were reviewed and analyzed with particular at tention to the assumptions and analytical approaches used to predict th e physical properties of the in terphase (Chapter 3). The various influencing physical fact ors that affect fiber-matrix lo ad transfer were examined, especially their empirical ev aluations and attendant statis tical variability. We have described the numerical impact of this stat istical variability and suggested confidence intervals for the predictive quality of the prim ary result of the SFCF test (i.e., interfacial shear stress). We considered whether the SFC F test is representative of macrocomposite morphology and performance. We have also described the underlying physical and chemical causes for interphase formation in model microcomposites to understand how interphase formation and structure influen ces SFCF interpretation and macrocomposite performance. The earliest SFCF tests used a simple av erage fiber strength to enable a fibermatrix interfacial shear strength calculation. For carbon fibers and other brittle fibers, this simple strength statistic is far from representative of the act ual statistical strength variation. Ultimate tensile tests on uns upported single fibers, as-received and room temperature air plasma treatments, were conducted (Chapter 2). A laser diffraction apparatus/technique for measuring small diam eter carbon fibers is described. It was shown that the variation in fiber diameter and across a fiber tow (~2000 fibers in this PAGE 25 case) must be considered when measuring fiber ultimate tensile strength. The tensile strength statistics were desc ribed using Weibull strength st atistics for single and multiple populations. The strength variability is ascrib ed to two separate popul ations (i.e., volume and surface flaws). We developed a new bi-modal fiber load-u ptake model to simulate single fiber fragmentation testing. In the model deve loped, a debonded fiber-mat rix zone, initiated and energized by fiber fracture, and a sec ond region with perfect bonding but a yielded interphase material provided for the load-uptake in the fiber. The influences of the uncertainties in composite and component mechanical properties, as well as the consequences of modeling assumptions and techniques, on the response prediction are evaluated by a non-determinis tic Monte Carlo analysis. PAGE 26 CHAPTER 2 CARBON FIBER STRENGTH AND STATISTICS Introduction Early European mechanists knew that mech anical component strength was strongly related to specimen geometry and loading c onditions, but they coul d not account for all of the strength variations seen in real structures. Resear chers and engineers were also aware (Leonardo Da Vinci) that specimen vol ume differences gave rise to changes in failure strength. A more complete understand ing of fracture, its mechanisms and causes, was needed to satisfy the increasing demand a nd complexity of structures and machinery. The need was finally satisfied by Griffith's seminal treatise on fracture mechanics [1], which along with work by Irwin [2] became the basis for modern fracture mechanics. While Griffith's fracture mechanics went a long way toward explaining strength variations related to specim en geometry and loading conditions, it was not until Weibull introduced the concepts of probabilistic fr acture [3] that the causes for strength differences associated with specimen volum e changes first became apparent, at least qualitatively. Although root causes of volume/strength variations were not well understood before Weibull's landmark paper, engineers r outinely implemented corrections to systems and structures to account for these strengt h variations. For example, when Brooklyn Bridge construction engineer Washington Roeb ling learned that the quality of delivered steel suspension wire was subs tandard, he simply increased the number of steel wires per PAGE 27 bundle [4], effectively relying on the wire's strength variation to meet the mechanical requirements. Weibull's novel approach to probability view ed components as consisting of sets of sub-units or links, which themselves exhibit intrinsic variations: this approach is the underlying principle of probabilistic fracture mechanics and, here, is called WeibullGriffith (W-G) theory. Critical assump tions of W-G theory are as follows. Failure strength associated with a give n link is independent of the other links. The flaws in a volume are randomly distri buted and their distribution is best described by a Poisson point process. It follows from these assumptions that the component will necessarily break at its weakest link when exposed to a uniform stre ss field. Thus, the di stribution of weakestlinks (both in space and in strength, for a sa mple population) controls that population's strength statistics and explai ns why volume variations give rise to stre ngth differences. After W-G theory was introduced extensive research was conducted on volume/strength variations [5-14], confirmi ng the basic W-G assumptions about weakestlinks and their independence. Most of the research concerned supplementing or altering the form of the W-G cumulative flaw distribu tion to better fit empirical observations. The W-G cumulative flaw distribution, its asso ciated failure probabilities, and various modified W-G failure probabil ities are discussed next a nd in Appendix A. Our study considered the ultimate tensile strength (UTS ) of brittle materials, in particular High Strength (Type II) carbon fibers, and how Weibull statistics and statistical treatments in general can be used to characterize and pred ict failure strengths as a function of gage length and surface treatment. PAGE 28 Experimental Methods Untreated polyacrylonitrile (PAN) based carbon fibers from Hercules (HS AU4, Magna, Utah) were used in this work. Speci mens were extracted from the fiber tow and affixed to paper testing jigs with cyanoacrylate (Figure 2.1). Two lengths were tested: 20 and 50 mm. The specimens were taken from a tow with ~200 fibers. Extreme care was exercised to avoid contact with or damage to the gage length. The diameter for each fiber tested was found using a laser diffraction technique (Figure 2.2). The diffraction pattern was captured with a di gital video image-processing system installed in an 80386 personal com puter. A TargaM8 (AT&T, Bedminster, New Jersey) digital video board and a Pana sonic VW-1410 black and white CCD video camera were used to capture the image. Vide o analysis software from Jandel Scientific Co. (JAVA, San Rafael, California) and pol ynomial fitting software, Grapher (Golden Software Inc., Golden, Colorado), were us ed to determine node separations of the diffraction patterns. The individually selected fibers (manua lly drawn from a room temperature tow) were tensile tested on a servo-controlled hydraulically actuate d MTS-880 system. A MTS 418.91 microprofiler was used to contro l the strain rate at +0.1 mm/mm/min-1. A specially constructed fixture held the fiber jig and maintained axial loading. A small thin-beam load cell (maximum load ,113 grams) coupled to an electronic amplifier was incorporated into the fixture to measure the axial loading. The load and displacement signals were recorded at 200 Hz by a highspeed A/D data acqui sition board (DAS-50, Keithley Instruments, Cleveland, Ohio). Radio frequency (RF) plasma surface trea tments on individual carbon fibers were performed in a cylindrical volume, using a rotating carousel (Figur e 2.3). The working PAGE 29 volume was evacuated to approximately 200 mi crons for 1 h before all treatments. Treatment times and RF power se ttings for the sample sets we re C: 2 h, 130 watts; D: 1 h, 100 watts. All the treatments were conducted with room temperature air as the plasma gas. All fiber tensile tests were performed at room temperature and for treated fibers 24 h after treatment. Fiber diameter dimens ions before RF treatment were used. Theoretical Discussion Fiber Strength Statistics W-G theory predicts that fiber strength is controlled by a distribution of flaws in space and size. The assumption that the flaws occur randomly (Poisson point process) is critical to the usefulness of W-G theory; its appropriateness is disc ussed later. WeibullGriffith theory can often be e ffectively applied to fiber failure strength data to elucidate or differentiate flaw populati ons and their effects on fiber st rengths. Scaling laws that enable prediction of fiber strengths for leng ths not tested can also be derived. These scaling laws are discussed next a nd used in the results section. Henstenburg and Phoenix [6] described a flaw distribution along a fiber of length L where each individual flaw can be loca ted in a two-dimensional state space by its position and intrinsic strength ( x' s' ) (Figure 2.4). The state space is defined by [0, sint] x [0, L ] where sint is the maximum theoretical failure strength for a given material. Any subregion is described by the area [ x x+x ] x [ s s+s ]. Given that flaws in are described by a Poisson point process, the probability of containing a flaw is proportional to the area and is vanishingly small as x and s tend to zero. This is equivalent to the general statement for Poi sson point processes that as the number of PAGE 30 observations (area ) decreases, the probability of an event occurring becomes increasingly small. The two-dimensional state space (Figure 2.4) defined by a Poiss on point process in failure stress and length, coupled with a pow er law relation for the Poisson intensity function for that space gives the failur e probability for a fiber of length L at a stress s ] ) s s ( L L [1 = F(s)0 0exp (2.1) where is the shape factor or modulus and s0 is the scale parameter for reference length L0 [6]. Equation 2.1 describes a two-parame ter Weibull cumulative distribution function (CDF). It and other multi-parameter models are derived and considered in Appendix A. CDFs are generally s-shaped: the broader the dist ribution the lazier th e s. In particular, the two-parameter Weibull CDF (Figure 2.5) can assume all the slope possibilities from a vertical line ( = ) to a slowly changing curve ( approximately 1.0). The greater the modulus, the steeper the curve (t hat is, the tighter the distribut ion). The inflection points for two-parameter Weibull CDFs do not generally occur at F(s) = 0.5; that is, they are generally asymmetric. For = 3.25889 the Weibull distribution is very nearly symmetrical, or pseudosymmetrical, except at the extremes. For the two-parameter Weibull distribution, the mode precedes the median for < 3.25889, equals the median for = 3.25889, and exceeds the median when > 3.25889. In other words, for > 3.25889 the inflection point of the CDF is at a cumulative probability greater than 0.50. The converse holds for <3.25889. This asymmetry is more clearly seen by looking at the probability density functions (PDFs) associated with each CDF. A CDF is defined as the integr al of the PDF, CDF = PAGE 31 PDF ds or inversely PDF = d / ds (CDF). The PDF for Weibu ll's two-parameter model is given by ] ) s s ( L L [ s s L L = f(s)0 0 1 0 0 exp (2.1a) The PDFs corresponding to the CDFs in Fi gure 2.5 are given in Figure 2.6. The skew of each PDF is readily apparent. S caling with length ratios greater than one ( L/L0>1) shifts the distribution to lower values and tightens the dist ribution while scaling to shorter lengths shifts the curve to higher va lues and broadens the distribution. It is interesting to note that, while scaling shifts and alters the breadth of the distribution, it has little or no effect on th e skew of the distribution. The two-parameter Weibull model describe d above is only capable of describing a process that is controlled by a single population. If howev er, two or more populations control a distribution, then a modified approach is re quired. One possi ble distribution considers two distinct concurre nt subpopulations (both occurr ing within each sample) and results in Eqn. 2.2: ] ) s s ( L L ) s s ( L L [ 1 = F(s)B AB 0 0 A 0 0 exp (2.2) where the sub/superscripts A and B refer to concurrent subpop ulations of flaws in the total flaw population [7]. The scaling effects for concurrent CDFs (Figure 2.7) are more complicated than for the simple two-paramter Weibull distribution and will be addressed in the results section. Neglecting scaling effects, there is a tendency for the lower modulus population to predominate at lo wer stresses and for the higher modulus population to predominate at higher stresses pr ovided the two scale parameters are or the same order. PAGE 32 If the subpopulations occur exclusively within the samp le (either A or B but not simultaneously) then the failure probability is given by an exclusive distribution ] ) ) s s ( L L (1 [ + ] ) ) s s ( L L (1 [ = F(s)B AB 0 0 B A 0 0 A exp exp (2.3) where A and B are the fractions of samples with type A and type B flaws respectively (A +B =1). Again, just as for concurrent di stributions, Figure 2.8 shows that exclusive CDFs can assume a variety of s-shapes. If it were possible to identify the flaw origin as either type A or type B then these distinct distributions could be considered individually using a two-parameter Weibull distribution. Because of this, scaling and modulus changes to either population produce similar eff ects to the overall distribution. However, it is often difficult to identify each subpopulati ons contribution to the total distribution especially considering there are now five adjustable parameters. A partially-concurrent flaw di stribution is one that is a combination of exclusive, type A flaws and concurrent flaws A and B. The cumulative failure probability for partially-concurrent flaw di stributions is described by )] F )(1 F (1 [1 + F ) (1 = F(s)B A B A B (2.4) where ] ) s s ( L L [ 1 = Fii 0 0 iexp and B is the fraction of samples with concurrent A and B flaws. Concurrent failure probability, Eqn. 2.2, is a special case of Eqn. 2.4, where B is equal to one. Similarly the two-parameter Weibull model is a special case of exclusive flaw populations where A is approximately one. It is a pparent that partially-concurrent PAGE 33 CDFs are similar to exclusive CDFs (Figure 2.9). Indeed, with appropriate parameter adjustment the two types are largel y indistinguishable (Figure 2.10). Another common form used for describing fiber failure probability is the threeparameter model, ] ) s s s ( L L [ 1 = F(s)0 L 0exp (2.5) where sL is the failure strength lower bound; it is taken as zero in the two-parameter model. The three-parameter model can only de scribe a distribution arising from a single population just as for the two-parameter model. Changes to the scaling ratio, L/L0, and Weibul modulus, have the same effect as for the two-parameter Weibull distribution. The three parameter model, Eqn. 2.5, exhibi ts little difference from the two-parameter model described above, except that th e low-strength outliers are bounded by sL for the former, and zero for the latter. Discerning th e lower leg of the s-shape is also more difficult for the three parameter model. Furthermore, identifying the lower bound, sL, is typically very difficult [7]. An explicit three-parameter CDF will therefore not be used here (a two-parameter model implicitly assumes sL = 0 ). Eqns. 2.1.5 provide explicit forms for fa ilure probabilities; however, they are difficult to interpret visually. More often, these equations are rearranged (Appendix A) to give Weibull probability Eqns. 2.6.9 [9-14]. s s = F(s))] (1 L L [-0 0ln ln ln ln (2.6) ] ) s s ( + ) s s ( [ = F(s))] (1 L L [-B AB 0 A 0 0 ln ln ln (2.7) PAGE 34 )]] ) s s ( L L (+ ) ) s s ( L L ([ L L [= F(s))] (1 L L [-B AB 0 0 B A 0 0 A 0 0 exp exp ln ln ln ln (2.8) )]] ) s s ( L L (+ [1 L L ) s s [( = F(s))] (1 L L [-B AB 0 0 B 0 A 0 0 exp ln ln ln ln (2.9) Weibull probability curves (Eqns. 2.6.9 ) are given in Figures 2.11.14. The solid and dashed lines in these figures are gi ven by Eqns. 2.6.9, while the symbols are randomly generated data from their respectiv e strength populations. The characteristics for each of these curves will be discussed be low as they relate to empirical carbon fiber failure strengths and fiber morphology. Equation 2.6 describes a strai ght line with slope equal to when ln ( L0/L (ln ( 1 F ( s ))) is plotted against ln ( s ) (Figure 2.11). Data that f it the two-parameter Weibull CDF (Eqn. 2.1) will fall on the straight line (Fi gure 2.11). Empirical nonlinearity is evidence that either the fibers' stre ngth distribution is not accura tely described by Eqn. 2.1, suggesting that modifications such as Eqns. 2.2.5 may be necessary, or that there was an insufficient sampling of the total population. Still another compli cating possibility is that the population cannot be well described by Weibull-like CDF's at all, and some other type of CDF is required (i.e., normal or lognormal). The two-parameter Weibull CDF is widely used by researchers in many fields with considerable success [9-18]. Weibull probabilitys broad applicability stems from its relative simplicity, inherent asymmetry and as will be shown below its relati onship to extreme value distributions. PAGE 35 An interesting side note is that Weibu ll originally thought that the intensity functions, ( s ) (Appendix A), used in Weibull failure models Eqns. 2.1.5 were not invested with any theoretical basis for descri bing strength distributions but rather, they were simply convenient analytical forms to represent empirical evidence. In fact, Weibull thought it "utterly hopeless to expect a theoretical basis for distributions of random variables such as strength properties of materials"[15]. However, as will be shown below, the Weibull distribution (Eqn. 2. 1) is actually one of three extreme value distributions, and as such is a natural consequence of some very simple underlying statistical assumptions. The solid line in Figure 2.12 is the concurrent flaw distribution (Eqn. 2.7). It has a positive curvature with the low and high tails of the distribution approaching the low modulus, A, and high modulus, B, asymptotes (dashed lines). This asymptotic behavior is endemic to concurrent flaw popula tions. Johnson [7] noted that if A < B then Type A flaws will predominate at lower stresses a nd Type B flaws will predominate at higher stresses. The location of the knee will also depend on the loading configuration if Type A were surface flaws and Type B were volum e flaws [7]. For all of the multiple population strength distributions described above, off-axis fiber loading could bias the total strength distribution towa rd surface flaws. Off-axis loading increases the stress intensity factor (SIF) near the surface flaws more than for volume flaws. The apparent modulus and scale factor for Type A surface flaws may be decreased by this off-axis loading and the disparity in SIFs. In practice, it would be extremely difficu lt, if not impossible, to differentiate between a biased and non-biased distribut ion without knowing a priori the surface and PAGE 36 volume flaw distributions as well as the nature of the off-axis loading. It is therefore imperative that exacting contro l of fiber loading is main tained throughout the tensile loading of the fiber. Johnson [7] states that true concurrent popul ations are rare and cites Bansal et al. [19] as one possible example; however, the pr esent author thinks that concurrent flaw populations are more common, albeit difficult to identify. A concur rent population could be obscured if the two populat ions exhibit a large difference in scale parameters while having approximately equal shape factors. Likewise, if the shape and scale parameters are approximately equal the concurrent na ture of the flaws populations would be obscured. Both of these distributions would appear to be fit well with a single population two-parameter model. Lastly, obfuscation of one or more popul ations is possible depending on the geometry of the sample and/or its size [20-24]. The shape for an exclusive distribution function plotted on Weibull probability paper can vary from essentially a straight line to a curve with strong knees and asymptotic behavior. Figure 2.13 gives seve ral different examples of exact fits to exclusive populations (Table 2. 3). Several important features of exclusive distributions can be identified. The distribution approaches the low modulus asymptote at high and low strengths. The midrange values approach the high modulus asymptote. Separation and evaluation of the di stinct populations is possible. An exclusive distributio n might occur if two sets of ceram ic test bars are received from two different sources. If source A produced poorly sintered well machined bars with high porosity and source B produced well sint ered but poorly machined bars, then the total population could be consid ered exclusive. Test spec imens from a single source are PAGE 37 less likely to exhibit an ex clusive distribution as proces sing conditions would remain relatively constant. The processing conditions for carbon fibers are tightly controlled and the process is generally continuous. This ma kes an exclusive distribution for a batch of fibers an unlikely proposition, especially for fibers in the same tow. Several partially-concurrent Weibull dist ributions are plotted in Figure 2.14. Again, just as for an exclusive distribution a partially-concurrent di stribution can take on many shapes depending on the parameters. For example, if B = 1 then a concurrent distribution is recovered. A graph similar to an exclusive distribution can also be recovered with the proper parame ters. Johnson [7] notes that it can be very difficult to distinguish between an excl usive and partially-concurre nt distribution even for n = 100 200. This is troublesome because even t hough it is physically improbable that an exclusive distribution exists for carbon fibers, the actua l strength distribution may seem to be well represented by an exclusive distri bution. This elicits suspicion concerning the validity of multiple parameter distributions. In order to estimate the scale and shape factors for a distribution (Eqns. 2.1.5), it is necessary to first determine the failure probability F ( s ). This is easily accomplished by ranking the failure strengths for n tests from lowest to highest and calculating the probability according to the rankings, i 0.5) + (n i = F(s) (2.10) The nature of this equation is such that a pr obability of one can neve r be reached as this would imply an infinite number of tests. Note also that Eqn. 2.10 is only one variation of many forms available to calculate the empirical probability [25,26]. PAGE 38 Kamiya and Kamigaito [27] argue that om itting 2% of the distributions outliers from consideration gives Weibull estimators that are closer to the true values. While, at first, this approach seems ou tlandish there is credible th eoretical and empirical support for this approach [27]. More details on pa rameter estimation and accuracy will be given in the results section below. Still, the most critical co ncern in analyzing any distri bution is the sample size, n In particular, it can be shown that fo r a population precisely described by a twoparameter Weibull distribution, the norm of the correlation coefficient for Eqn. 2.6 decreases as the number of randomly genera ted data decreases, even though the data originated from an ideal two-parameter Wei bull population. Table 2.4 gives the norm of R for 10 separate randomly generated trials of n samples each. It is clear that as n decreases the norm of R decreases as well. This is a consequence of the strength population's distribution, its si gnificance should not be overlooked. Because only a finite number of tests can be practically performed it may appear that a certain distribution poorly describes the true population, howev er, the total population may be accurately described. Representative data used to generate Ta ble 2.4 is plotted on Weibull Probability paper in Figure 2.15. The Weibull probability plots in Figure 2.15ae show that for increasing n the empirical data more closely appr oximates a straight line with slope just as the norms of the correlation coeffici ents in Table 2.4 indicate. The disparity between the apparent Weibull modulus calcula ted from the randomly generated data and the modulus used in generating the data are also strongly affected by the sample size. The coefficient of variation (CV) for the modul us is much greater for small sample sizes PAGE 39 (53% for n = 10 and 2.3% for n = 500). It is obvious from this simple example that choosing the sample size is equally as important as the form of the CDF. The simulations represented in Table 2.4 suggest an appropria te sample size to reach a certain confidence in the empirical estimators and s0. To verify that populations are normally di stributed, 30 tests are usually considered adequate [25-27]. Once normality is established for a process, much smaller sample sets can be used, especially if censoring is used. Estimator confidence is certainly affected for small sample sets. A more detailed disc ussion of censored data and extreme-value statistics and their effect on parameter esti mation will be given in the results section. Larger sample sets are also required to establish good agreement for Weibull populations and subsequently less data is required for paramete r estimation. It is clear then, that some measure is necessary to qualify the empirical estimators and s0. Confidence intervals provide one means for qualifying estimators. These confidence intervals broaden as n decreases. For normal distributi ons, the confidence intervals are usually calculated from the t-distribution, which is an approximation to the normal distribution. Determining confidence in tervals for non-normal distributions (i.e., Weibull) is more involved. In fact, fiber strength data cannot be pr ecisely described by any of the equations noted above because of the vici ssitudes of materials, testing procedures and apparatus. However a goodness-of-fit (GOF) paramter can be defined that measures data/model correlation (Appendix A). A value of one fo r the GOF parameter indicates the model PDF and sample distribution are equivalent wh ile values less than one suggest reduced correlation between model and data. GOF pa rameters are particul arly amenable to PAGE 40 numerical procedures and are used iterativel y to obtain optimum fit parameters; whereas, confidence intervals are calculated after the fit parameters are determined using the sample size and the fitted CDF. Carbon fiber failure mechanisms/models Equation 2.1 describes a streng th distribution that arises from a single process or single flaw population. Usi ng a two-parameter Weibull mo del therefore precludes the distinction between multiple flaw populations/f ailure processes. While it seems unlikely that a single process or flaw population will completely cont rol the strength distribution of carbon fibers, which are known to have a complex and irregular morphology, a number of researchers have used the tw o-parameter Weibull model with reasonable success [5,13,14]. Reynolds-Sharp failure mechanism A single process that may control fiber st rength in carbon fibers has been proposed by Reynolds and Sharp [28]. The Reynolds a nd Sharp failure mechanism is based on previous work by Sharp and Burnay (ref. 5 of [2 9]) that asserts that the failure origins in carbon fibers are the turbostratic1 graphitic structures at in ternal void surfaces and not the voids (pores) themselves. Reynolds a nd Sharp [28] contend that misoriented crystallites cause tens ile failure in carbon fibers Sh ear stresses are produced on the misoriented basal planes causing transverse rupt ure of the planes and, ultimately, failure of the misoriented crystallite (Figure 2.16). 1 The term turbostratic graphite was first used by Biscoe and Warren in 1942 [30] to describe `true graphite layers arranged rou ghly parallel and equidistant, but otherwise completely random that exist in most carbon blacks. Layer plan e separation for turbostratic graphite d(002) is roughly 3.42 although this varies depending on the regularity of the structure, versus d(002)=3.3 5 for true graphite [40]. PAGE 41 Complete failure of the fiber occurs wh en either one or both of the following conditions hold: 1. The incipient crystallite is sufficiently c ontinuous with neighboring crystallites for crack propagation. 2. Either La (avg. turbostratic gra phite layer width) or Lc (avg. stack height normal to basal plane, c direction) is greater than the critical flaw size, C for tensile fiber failure. The Reynolds-Sharp (R-S) condi tion for tensile failure is C > 2 s a 2 (2.11) where sin cos sin cos_ 4 33 2 2 13 44 4s + ) s 2 + s ( + s = s is the effective compliance on the ba sal plane oriented at an angle to the fiber axis, a is the equilibrium surface energy, sii are the stiffness components for graphite, and C is the critical flaw size for tensile fiber failure [ 29]. Basal plane compliance versus basal plane orientation, is shown in Figure 2.17. Supposing that one or both of the R-S cond itions is true, the axial stress necessary for failure in carbon fibers is plotted in Figur e 2.18. The solid line is the locus of UTS, measured radially from the origin, for cr ystallites with basa l plane orientation to the fiber axis (positive vertical axis). A simple Griffith fracture criteria C 2E = a 2 GR is plotted in Figure 2.18 for co mparison. An infinite axial stress is predicted for perfect fiber and layer plane alignment while the minimum UTS is found at misorientation. In contrast, Griffith fracture is independent of crystallite orientation, assuming instead a homogeneous medium with randomly oriented line cracks. PAGE 42 Of course, perfectly aligned, flawless crys tals do not exist in carbon fibers, nor are these crystallites randomly oriented. Rather x-ray diffraction studi es indicate that the polycrystalline orientation varies with respect to the fiber ax is [31-41]. Not surprisingly, preferred basal plane orientati on along the fiber axis results in higher modulus fibers [3741]. Excessive crystallinity, how ever, can lead to lower strength fibers because both R-S conditions (1) and (2) are more readily satisfied as crystallinity increases. X-ray diffraction has shown that with in creasing two-dimensional order there is a concomitant increase in the Young's modulus for PAN and pitch based carbon fibers. Pitch based carbon fibers also show strong three-dimensi onal ordering in the skin producing an increase in both the Young's modul us and tensile strength [42]. However, for PAN-based carbon fibers, the high heat treatment temperatures and exposure times required for generating high crystallinity produces randomly oriented 3-dimensional graphite crystallites at the cost of highly or iented two-dimensional tu rbostratic graphite. This will generally reduce fiber UTS [43] a nd for extreme graphitization will render the fiber friable with little or no tensile strength. Assuming, for the moment, that R-S failure is the only failure mechanism operating during carbon fiber axial deformation and that the crystallite distribution, orientation and density, are known it should th en be possible to calcula te the cumulative failure probability F ( s ) using the R-S criteria, Eqn. 2.11. The nature of this crystallite orientation distribution is addressed below. Observations using x-ray, SEM, and TEM have shown there is a strong correlation between pore and crystallite angular distribution in T ype I and II PAN-based carbon fibers [34,35 ,40,41,44-49]. With proper ca re small angle x-ray measurements can PAGE 43 readily detect and measure the size, orientat ion and relative number of the pores. The pore dimensions in Type I and II PAN-base d carbon fibers are approximately 10 x 200 with the long axis oriented preferentially along the fibe r axis [40,41]. Perret and Ruland [34] and Johnson and Tyson [40] inde pendently concluded that the sharp density transition at the pore walls ( 0.3 ) indicates that the surf aces of the pores are graphite layer planes. Thus it can be logically concl uded that the pore axia l angular distribution relative to the fiber axis and the basal plane orientation ar e roughly equivalent. SEM and TEM have also been used to support this conclusion [44-48], although there is some evidence of even more highly misoriented crystallites at th e pore surface [29]. The small angle x-ray patterns in Type I and II PAN-based carbon fibers (Figure 2.19) [49] demonstrate the por e and crystallite orientation distribution. The bow-shaped scattering patterns are primarily th e result of thin crystallites ( Lc = 60 from 002 line width [40]) and narrow polyhedral pores possessing an orientation distribution gA() [50]. The graphite layer plane stack height, Lc, is also known to be a decreasing function of layer plane misorientation [ 29], which would tend to increase scattering density at low angles, emphasizing the bow-type distribution seen for Type II fibers and to a lesser degree for Type I fibers. A reduction in the angular width about the equator for these small angle x-ray patterns (F igure 2.19) therefore indicates increased basal plane preferred orientation along the fiber axis (v ertical in the figure) [41]. High Modulus Type I fibers clearly have greater basal plan e alignment with the fiber axis compared to High Strength Type II fibers. A more uniform scattering particle geometry (pore and/or crystallite) would also lessen scattering. Additionally, interparticle interference is al so especially strong and the PAGE 44 scattering effects of pore geometry and separa tion are sometimes difficult to distinguish [34]. All this notwithst anding, it is the particular nature of these scattering influences that allows such precise determination of crys tallite and pore morphology in carbon fibers. The angular width of the 002 x-ray diff raction arc at half peak density Z is the primary indicator for crystallite preferred or ientation. The angular width is a strong function of heat treatment temperature (HTT) as are the fiber's m odulus and strength. Table 2.5 presents Youn g's modulus, UTS, and Z as a function of HTT as reported by different investigators [32,33,37,38,51,52]. It is clear from Table 2.5 that with increasing HTT there is a corresponding in crease in modulus and preferred orientation, while fiber tensile strength initially increases, plateaus and eventually decreases. It is also evident from Table 2.5 that th ere is some disagreement on the quality of preferred orientation in carbon fibers, even for fibers of the same type. For Type II carbon fibers researchers have quoted Z values as large as 41 [49] and as small as 4 [39]. Much of this variation stems from the details of the stre ss graphitization/heat treatment process used by different carbon fi ber producers. In this regard, Johnson and Tyson [41] subjected PAN-based carbon fibers to HTT's from 1000CC and found there is a complex relationship between lp, Porod's `distance of he terogeneity, and UTS. The discontinuity of lp at 1900C appears to be related to crosslink destru ction between adjacent layer planes. This destruction allows for more labile reorganization of the layer planes above 2000C as exhibited by increased crystall ite alignment and size, especially with the application of stress during graphitization [39,41]. There is little doubt that a distribution of basal plane orient ation about the fiber axis exis ts. However, variations in the pore dimensions make it impossible to definitively determine the exact angular PAGE 45 distribution. That is, without first assuming an explicit pore geometry we cannot make an unambiguous determination of the angular distribution and vice versa [50]. In their effort to determine pore angu lar distribution Johnson and Tyson [41] assumed a size distribution of the form ) R / R ( R 1) + (n 2 1 R 2 = M(R)2 0 2 n 1 + n 0exp where R is the particle radius, and n and R0 are empirically determined. Unfortunately, this gives unsatisfying results and is also difficult to support as SEM resolution is insufficient. In a more rigorous analysis of low-angle scattering Perret and Ruland [35] assume that the scattering intensity Ip is roughly proportional to the angular distribution of layer plane normals. ) ( g ) const, = (s IN p This relation holds for large scattering vectors, s > 1 / L where L is the particle length. In other words, for these needle-lik e pores, the pore geometry has a negligible effect on the scattering distri bution, provided the proper inte rval is considered. More importantly, gN() can be determined directly from consideration of the scattering intensity without assuming a pore size distribution ( gA() is obtained from gN() with a change of variables [35]). Strictly speaking, disregardi ng pore size effect s is invalid; however, Perret and Ruland's [35] results are compelling. The pore axis angular distribution gA(), found by Perret and Ruland [35], for a carbon fiber with HTT = 2800 C is shown in Figure 2.20. A simple normal distribution ( = 0, = 9.5 ) is also plotted in Figure 2.20. Th e angular width at half-peak height, Z PAGE 46 for this pore axis angular distribution is 22 in general agreement with values obtained from the 002 diffraction arc. Given that for carbon fibers the pore axis angularity (viz. crystallite basal plane orientation) is reasonably approximated by a normal distribution and further assuming that either R-S Condition (1) or (2) is satisfie d, the R-S failure criteria can be directly applied to give an approximate CDF for carbon fiber tensile failure. In order to quantitatively evaluate the R-S failure crit eria, a Monte Carlo simulation was used to generate 50 ( N ) normal distributions with either 100 or 1000 ( n ) entries (misorientation angles) each. For all cases the population m ean was fixed at zero. The standard deviation varied as described in Table 2.6. The carbon fiber ultimate tensile strengths were predicted from the 50 normal extremes using Eqn. 2.11 and ranked in ascending order. The extremums for the misorientation a ngle can be treated using extreme value statistics provided the variables are statistically random and independent, and the original distribution from which the extremes are drawn remains constant throughout the test [53]. These conditions are strictly valid for the misorientation angle extremes, for positive and negative angles as well as the combined abso lute extreme distribution, but not necessarily for the predicted UTS distribution. There are three genera l types of extreme value dist ributions, the exponential, the Cauchy, and the Weibull type, or alternativ ely the first, second, and third asymptote distributions. The second asymptote distri bution is unbounded on the le ft and right while both the first and third distribu tions are bounded on the left, 0 < x < The three Extreme Value distributions are related by changes of va riables [53]. In cont rast to the first and PAGE 47 third asymptote distributions, the second asym ptote distribution is not well suited for practical applications owing to the form of its initial distri bution and lack of a left bound, it will not be considered in this discussion. The ubiquity and us efulness of the Weibull (third asymptote) distribution has been men tioned previously and, as noted, the first asymptote can be obtained with a relati vely simple change of variables. The fitted first and third asymptote distributions for the extremes of a normally distributed population with = 0, = 8.5 ( Z = 20), n = 1000, and N = 50 are given in Figures 2.21.24. The maximum extremes are plotted in Figures 2.21 and 2.22 while the minimum extremes are shown in Figures 2. 23 and 2.24. These figures suggest that there is very little difference in the quality of fit between the first and third asymptotes for these simulations. The correlation coefficien ts for the two asymptotes reinforce this notion (Table 2.6). The tendency for the mini mum angles to deviate more dramatically from the asymptotes can only be an artifact of the Monte Carlo simulation itself, its cause at present unknown, although anomalous ra ndom number generation is suspected. Although the physical situation does not specifically a llow for negative misorientation angles, there is no loss of gene rality in treating the absolute value extreme distribution with extreme value st atistics. It has been demonstr ated [54], that the absolute value extremes from a symme trical distribution of size n is equivalent to considering a single tail of the same distribution with 2 n entries. Extreme absolute value probability plots are given in Figures 2.25 and 2.26. In n early all cases, absolute extreme and simple extremes, the first asymptote appears to slightly outperform the Weibull asymptote (Table 2.6). PAGE 48 It should be noted that extremes derive d from a normal distribution will only slowly approach the first asymptote and even more slowly toward the third asymptote [54] as N increases. Tippet (Tippet (1925) of [5 4]) has shown that there is sensible deviation from the third asymptote prediction for normal distribution extremes, even for N = 1000. So the GOF for the first and third asymptotes seen in Figures 2.25 and 2.26 is not unexpected. The first and third asymptotes are better suited for initial distributions of the exponential and limited types respectively, which explains their usefulness in lifetime and strength statistics. While the angular extremes are reasonab ly predicted by both the first and third asymptotes, the failure strength extremes f ound via Eqn. 2.11 are not well represented by the Weibull distribution. Several Weibull pl ots for minimum failure strengths are given in Figures 2.27 and 2.28. It is obvious from these figures and the statistics in Table 2.6 that the failure stress distribution becomes distinctly non-Weibull as the angular width increases beyond 30 For angular distribut ion widths of 30 and especially 40 the simulated strength distribution is strongly biased toward the minimum failure stress predicted by R-S criteria (cf. Figure 2.27). A simulate d orientation angle near misorientation at minimum UTS, is increasingly probable as angular width increases resulting in strong low-strength biasing. As the angular width decreases the conditions required for extreme value statistical treatment are more valid and th e UTS distributions ar e more Weibull-like (Figure 2.28 and Table 2.6). As the number of entries in each no rmal distribution increases, greater misorientation angles will naturally be found. A shift to greater Weibull moduli and PAGE 49 lesser scale factors is ex pected (Figure 2.29). Fo r an angular width of 20 the predicted UTS Weibull modulus for 1000 samples is nearly double that for n = 100 entries. There is little effect on the UTS distribution however if the number of simulations is increased (e.g., 100 tests vs. 50 tests). For actual fibers the number of crysta llites is at least 109 times greater than simulated here. This w ould increase the predicted Weibull modulus at all angular widths for the case where R-S condi tions (1) and (2) are everywhere satisfied. If Conditions (1) and (2) are everywhere sa tisfied, and there are a large number of misoriented crystallites (Z~30 ), then misorientation angles close to will exist and the UTS distribution for carbon fibers would be even more biased than seen in Figure 2.27. This is contrary to previous research on carbon fibers [5,13,14] and results to be presented later. This contradiction is a good indication that simply assuming that R-S conditions (1) and (2) are universally satisf ied is invalid, even though carbon fibers are known to have high porosity (20% Type I, 25% Type II [34]) a nd high crystallinity (>95% Type I, 95% Type II [41], bulk carbon 90% at 3000 C [47]). The Reynolds-Sharp conditions for failure are expl ored more closely below. Considering Condition (2): The mean cr ystallite dimensions for Type I carbon fibers are approximately 70 x 70 x 60 ( La x La x Lc) [40], whereas the critical flaw sizes, C are generally an order of magnitude larger (Table 2.5). Additionally, the layer plane stack height Lc decreases with misorientation [29]. It is clear then that the mean graphitic crystallite is not causing catastro phic failure by itself. Fiber failure, if it is occuring at misoriented crystallites as R-S argue [cf. Condition (1)], must therefore involve more than a single mean crystallite. This collection of cooperating crystallites PAGE 50 must give rise to the strength limiting flaws that are on the order of the Griffith critical flaw size. Considering Condition (1): Condition (1) re quires that neighbor ing crystallites be sufficiently continuous to permit crack propagati on, a fairly vague physical requirement. The local stress field on the cr ystallite involves not only that crystallites or ientation and compliance but also the dynamics of the propa gating crack and other near field stress intensifiers, all of which will tend to amp lify the local stress field magnitude. The situation is far more complex than the elemen tary picture Reynolds-Sharp failure criteria suggests. The nearness of crystallites can be determined by considering the average pore volume, pore surface area and specific surface area Ssp, which can all be determined using small angle x-ray techniques. From geomet ry it follows that there are approximately 1.3x1012 voids/mm in a 7 micron diameter fiber. These voids are, on the average, separated by 60 [40]. The number of crystallites per millimeter is of the same order (<5 % amorphous carbon) and the crystallites are separated by pores approximately 10 in diameter. Additionally, n earest neighbors would tend to be like oriented and thus of comparable strength further weaken ing misoriented crystallite regions. SEM and TEM observations [44-48], subs equent to R-S theory introduction, indicate that layer plane dimensions, La and La are much larger than the 70 suggested by x-ray measurements [40]. The layer plan es are generally intertwined and folded among crystallites in a complicated and c onvoluted manner and cannot be considered simply as existing in isolated crystallites. PAGE 51 Crystallite interconnectivity in carbon fibers is further evidenced by poor mechanical dampening and an increasing s onic modulus with great er HTT [55]. The higher sonic modulus indicates that there are more continuous pathways (increased crystallite perfection and c ontinuity) in High Modulus fi bers. Small angle x-ray measurements for loaded and unloaded fibers [55] indicate that an increasing sonic modulus is closely associated with reduced basal dislocatio ns and increasing crystallite orientation. It is thus difficult to imagine, given all the collateral evidence, a crystallite that is not sufficiently continuous with its ne arest neighbor, at least in terms of shared layer planes, or separated from other crystallites by small pores (~10 ). The modified Griffith criterion (Eqn. 2.11) is based on proven micromechanical and fracture mechanics principles and as such is a valid represen tation for misoriented crystallite failure. Condition (2 ) must therefore be infrequent ly satisfied. In contrast, Condition (1) would seem to be, at first glance, generally satisfied. However, simulated CDFs with large n and Z 30 are not consistent with empirical results [5,13,14]. Also, the predicted Weibull moduli (cf. Table 2.6) ar e much larger than ac tual moduli. It is therefore concluded that Condition (1), despite morhologica l evidence to th e contrary, is not everywhere satisfied, especially for highly misoriented crystallites. The contradiction here between morphol ogical evidence supporting extended long range continuity amongst crystallites and predic ted failure strength arises from the quality of continuity between crystallites. Specifically the state of being su fficiently continuous is dependent on the testing method. Using an x-ray probe the tu rbostratic graphite crystallites appear to be disc rete [40], while with electron microscopy the complex nature of crystallite interconnectivity and overla pping is clear [4448]. When probed PAGE 52 macroscopically (i.e., tensile tests, sonic m odulus tests, conductiv ity) the continuity between adjoining crystallites is indicated by strong mechanical integrity (UTSavg = 2.5 GPa [43]), and high thermal and electrical condu ctivities [42] These material properties are strong functions of crystallite size, orient ation, spatial distribut ions, and crystallite continuity. Crystallite continuity is via sh ared layer planes, or intertwining, and is primarily along the fiber axis, not normal to the fiber axis as Condition (1) requires. The complex morphology of carbon fibers ma kes the direct application of R-S failure criteria for predicting fiber UTS intractable without first making unreasonable assumptions regarding crystallite continuity While these simulations, which assume Conditions (1) and (2) are everywhere tr ue, do not accurately predict genuine UTS distributions, they do indicate that the pres ence of misaligned crystallites near will strongly bias UTS distributions toward the minimum UTS given by the R-S failure criteria. The simulations also suggest that one of the primary benefits brought about by stress graphitization is increased crystallit e alignment which purges the fiber of highly misoriented crystallites and their deleterious ef fects. This crytallite alignment is in addition to increased crystallinity and crystalline perfection seen with stress graphitization [39]. Another interesting result supported by these simula tions (by induction via. disproving the assumptions) is that the layer plane stack height Lc for single crystallites and adjoining crystallites does not generally exceed the critical flaw size for fiber failure. Rather, it appears that Condition (1) and (2) are very infrequently satisfied considering the large number of crystallites and this is a primary reason carbon fibers exhibit such high ultimate tensile strengths. PAGE 53 Stewart-Feughelman model An alternative micromechanical model proposed by Stewart and Feughelman (alternatively referred to as ribbon model he re) [56] asserts that carbon fiber failure occurs when misoriented gr aphitic ribbon-like structures (Fordeux et al. (1971) [49]) within the fiber are overloaded (Figure 2.30). In this micromechanic al model, the ribbon geometry (thickness and curvature) controls the ribbon's effective modulus and strength, thus producing the strength variations seen in carbon fibers. According to the ribbon failure model the ribbon failure strength decreases with increasing ribbon thickness and/or decreasing ribbon curvature, a relati onship that is qualita tively supported by carbon fibers heat treated above 1200 C [41]. Stewart and Feughelman [56] contend that the volume flaws discussed above are, in general, `insufficiently abundant to fu lly account for observed carbon fiber fracture. While it is generally true that individual vol ume flaws (i.e., viz. misoriented crystallites) are not capable of causing catastrophic fiber failure, the interconnectivity of the ribbonlike crystallites and the close proximity of needle-like pores (approx. 70 [40]) leads to a cooperative failure mode involving multiple vo lume flaws that is contrary to Stewart and Feughelman's argument. It seems likely that both Reynolds-Sharp and Stewart-Feughelman failures are occurring simultaneously in PAN-based carbon fibers. This is probable because basal plane misorientation relative to the fiber axis, the primar y factor affecting failure strengths, is common to both failure mechanis ms. Another factor common to both failure mechanisms is ribbon/crystalli te thickness, which itself is affected by pore density and orientation [41]. PAGE 54 Once again the complex morphology of carbon fibers makes the direct application of the ribbon failure criteria difficult wit hout first making assumptions regarding ribbon macrostructure and crystallite continuity, ne ither of which are accessible via analytical evaluation. However, just as for the RS failure mechanism, the Stewart-Feughelman failure mechanism illuminates the primary f actors responsible for carbon fiber tensile failure. Specifically: (1) Crystallite basal pl ane misorientation relative to the fiber axis, and (2) Crystallite thickness and interconnectivit y. Parenthetically, bo th (1) and (2) could influence the local toughness positively, thus opposing their own flawed nature. Now it may be argued that surface flaws ar e simply intersections of volume flaws with the fiber surface. To a degree this is certainly true as the complex morphology of carbon fibers near the surface is largely re sponsible for the irregular, flawed surface morphology. However, there are many othe r factors that can produce surface flaws peculiar to carbon fiber surfaces. The mechan ics and equipment used for fiber spinning and stress graphitization will invariably da mage the fiber surface despite precautions. Also fiber-fiber contact in the tow duri ng production and handling generates additional surface flaws. Lastly, while a surface flaw and a volume flaw may owe their origin to similar morphology, the associated failure mechanisms may be different a nd therefore evidence themselves in distinct failure strength populations. The definitions/demarcations for flaws in carbon fibers are masked by the ve ry complexity of the morphology. One can only define a flaw relative to a perfect or 'unflawed' state, which, as ample investigation has shown, is difficult if not impossible to identify in carbon fibers. For instance, the PAGE 55 capricious voids in carbon fibers, while co nsistent enough to produ ce distinguishable xray scattering patterns, are nothing ( voids) and flaw boundaries simultaneously. Batdorf-Crose failure theory Another failure model/criteria that may help explain the tensile failure process for carbon fibers was developed by Batdorf and Crose [57] to "reduce the gap between physically based and weakest link theories." It is particualrly well suited for studying brittle fracture under polyaxial stress states. The assumpti ons made by Batdorf and Crose [57] are very similar to the basic tenets of W-G theory: The material is isotropic and contains ra ndomly oriented and located microcracks. Crack initiation is neglected. Each crack within the material will caus e specimen failure when the macroscopic normal stress on that crack exceeds some critical stress, cr. These statements are simply reformulat ions of W-G assump tions with physical interpretations. Batdorf and Heinisch [58] pr oved that these assumptions lead directly to Weibull's then-unproven polyaxial stress failure equation [3], which assumes that cracks act independently, are randomly oriented and that failure is shear insensitive. This last condition, shear insensitivity, is the critical drawback of B-C failure theory as it was first introduced [57]. Specifically, Batdorf [59] concluded that B-C failure prediction for polyaxial stress states is not conservative especially for materials that are sensitive to shear failure. Batdorf et al [57-60] did show, however, that failure for polyaxial stress states can be predicted from uniaxial Weibull statistics, at least fo r generally isotropic materials insensitive to shear induced failure. In a later paper, Batdorf and Heinisch [ 58] modified the failure criterion for B-C theory allowing for an arbitrary fracture criterion and specifically introduced shear sensitive failure criteria. This variation on th e original B-C failure theory predicts that PAGE 56 specimen failure will occur when the `effective normal stress e on the crack exceeds the critical normal stress cr on the crack. The effective st ress acting on a randomly oriented crack depends on the failure criterion chosen. In order of increasing shear sensitivity, the effective stress formulas suggested by Batdorf and Heinisch [58] are given in Eqns. 2.12a through 2.12e. n e = (2.12a) ] + + [ 0.5 = 2 2 n n e (GC) (2.12b) ] ) 0.5 /(1 + + [ 0.5 = 2 2 2 n n e (PSC) (2.12c) + = 2 2 n e (GC) (2.12d) ) 0.5 /(1 + = 2 2 2 n e (PSC) (2.12e) GC and PSC indicate that the effective st resses are derived from Griffith cracks and penny shaped cracks. These equations sweep out axisymmetric surfaces and are represented in Figure 2.31a using polar diagram projections All the crack normals for which e > cr are subtended by the solid angle (Figure 2.31b). With increas ing shear sensitivity, it is apparent that will increase, thus the conclusion th at Weibull's polyaxial stress failure criterion and the premiere B-C form ulation are non-conservative [58]. Unfortunately, the material conditions nece ssary to apply B-C failure theory are fairly restrictive. While B-C theory was su ccessfully applied to Poco graphite [57], a nominally isotropic graphite, direct applic ation of B-C theory to highly anisotropic carbon/graphite fibers is not possible. B-C th eory, however, is not without merit in this PAGE 57 regard as it can provide valu able insight into the micromechanics of carbon fiber tensile failure and may be modified to provide a uni que failure criteria for graphite fibers. In particular, it should be possible to use the R-S failure criteria as an alternative to Eqns. 2.12ae in the reformulated B-C failure theory, provided special consideration is given to the preferred crystallit e orientation in graphite fibe rs. This should permit failure envelope determination for highly anisotropic carbon fibers subjected to polyaxial stress states. As noted above B-C theory assumes that cracks are uniformly distributed in both location and orientation. Where B-C theory sp eaks of crack normals, graphite crystallite basal plane orientation can be readily subs tituted without affecting application or interpretation. Also, a uniform location distri bution for the crystallites is consistent with our previous discussions rega rding Poisson distributions. The crystallite orientation distribution, however, is cl early not uniform but instead approximately normally distributed about the fiber axis with = 0 and ~ 8.5 (Z = 20 ) [35] (c.f. Figure 2.20). The failure probability for a single randomly oriented crack is )/4 ( = Fc (2.13) In contrast for graphite basal planes no rmally distributed a bout the fiber axis ) (1 )/2 ( = Fcr cos (2.13a) where varies with the breadth of the orientation distri bution and desired F. Equation 2.13 is recovered from Eqn. 2.13a when = Following B-C formulation, the failure probabi lity for graphite fiber material with a volume V and subjected to stress is given by PAGE 58 . ] d d dN ) (1 2 ) ( V [ 1 = ) F(V,cr cr cr 0 cos exp (2.14) For Reynolds-Sharp type failure, cr (Eqn. 2.14) is the critical remote axial stress for shear induced transverse crysta llite rupture. Figure 2.18 indicates that R-S failure is strongly dependent on basal plane orientation. With increasing axial loading, the rate at which new orientations are incl uded in the failure envelope, dN(cr)/dcr, decreases. N(cr) and dN(cr)/dcr can be determined by combini ng the R-S failure condition and a normal approximation to the crystallite orientation. That is, cr cr cr crd d d ) dN( = d ) dN( where dN/d is the Normal PDF and d/dcr can be numerically determined from Eqn. 2.11. For the simple uniaxial case, = (,0,0), the solid angle ( ,cr) is graphically depicted in Figure 2.32. The R-S failure criteria at small misorientation angles produces bounded volumes (Figure 2.32). The solid angle limits min and max are determined from Eqn. 2.11 and the uniaxial tension The solid angle ( ,cr) is then calculated using the definition of a solid angle and the limits min and max. ( ,cr) = 4 (cosmin cosmax) where min and max are the half angles of the solid angles min and max. It is clear from Figure 2.18 that is zero for cr (all angles), only gradually builds to, and asymptotically approaches 4 as Equation 2.14 can now be numerically evaluated using ( ,cr) as above, along with the previously determined dN(cr)/dcr. The resulting cumulative failure probabilty and Weibull plot are shown in Figure 2.33 for axial loading and Z = 20 This prediction PAGE 59 assumes that the R-S failure conditions, (1) a nd (2), are everywhere satisfied in the body of the material, a condition that was dispr oved above, but is show n here for comparison and completeness. Figure 2.33a shows that th e cumulative failure probability only slowly approaches a magnitude of one, a direct cons equence of the highly anistropic R-S failure condition and the normally distri buted crystallite or ientations. Note also that the distribution is decidedly non-Weibull (Figure 2.33b). The modified B-C failure model developed he re will be used in the results section to predict failure strengths fo r biaxial and off-axis tensile loading. These cases are important in composite structures subjected to polyaxial stresses. Off-axis fiber loading in carbon fibers is characterized by a dramatic reduction in failure strength. This is primarily the result of strong preferential crystallite and pore alignment. Specifically, as loading becomes more transverse the crystallit es closely aligned with the fiber axis are exposed to large shearing stresses and c onsequently rupture through the crystallite thickness (Figure 2.16) thereby reducing failure strength relative to on-axis loading. For the combined R-S and B-C failure model, off-axis loading is roughly equivalent to the case where basal planes ar e normally distributed about an orientation with on-axis loading. The cumulative failure probability so derived is traced in Figure 2.33. As expected the CDF for off-axis load ing precedes the case for pure axial loading. Also, as for axial loading, the failure dist ribution is clearly non-Weibull. The lowstrength low probability tail seen for the o ff-axis case is primarily caused by the smaller stress increments used to numerically evalua te Eqn. 2.14 compared to the on-axis case. For small s, and dN/dcr are correspondingly small and th e integral area builds very PAGE 60 slowly, especially for misori ented basal planes at ~46 to the loading axis (cf. Figure 2.18). In addition to the increased likelihood for RS type failure as loading deviates from axial loading, it is also probable that addi tional failure modes will develop or become more pronounced. Specifically with increasing off-axis loading existing irregular shaped pores will be exposed to increasing normal and shearing forces making Griffith fracture more likely (Poco graphite has exhibited Griffith type fractur e [57]). Also, layer plane delamination is more probable as misorientation relative to the loading direction increases owing to the relatively weak Van Der Waals forces binding layer planes. Given the complexity and irregularity of High Strength and High Modulus carbon fibers, completing a list of all the probabl e failure modes seems unlikely and frankly unnecessary. However, for off-axis loading it seems imprudent to neglect Griffith fracture of the pores considering their number de nsity. In this regard it has been argued by Barnett (ref. 7 of [57]) and Freudenthal [61] that the survival probability, Fs, for polyaxial stress states can be approximated by ) ( F ) ( F ) ( F = ) , ( F3 s 2 s 1 s 3 2 1 s where 1, 2, and 3 are the principal stress es. In a like ma nner the same approach may be applied to two types of failure criteria, namely ) ( F ) ( F = ) , ( FGr s BC s 3 2 1 s The superscripts BC and Gr correspond to Batdorf-Crose (ribbon failure) and Griffith failure criteria respectively. The resulting survival probability Fs would be less than for R-S failure alone and especially so as polyaxial stress states are imposed. Once again it would be difficult to quantify this distribution without making a number of PAGE 61 assumptions regarding pore size and orientat ion distribution. Re gardless, the trends predicted by the modified B-C theory and competing failure modes (Reynolds-Sharp and Griffith) are consistent with experimental evidence (see discussions above regarding crystallite alignment Table 2.5, and Poco gr aphite [57]). Recall also that the ReynoldsSharp failure criteria is simply a modified Gr iffith criteria with a variable compliance and as such a failure probability combining R-S and Griffith seems quite natural. Early research also showed that carbon fi ber strength was strongly associated with impurity concentration in the PAN precursor [36,62,63]. PAN precursor fibers spun in clean room environments displayed marked improvements in ultimate tensile strength over control group fibers [62]. The impurities, which act as nucleating sites for graphite crystals, are generally burned off during graphitization leaving irregular voids. Additionally, the polymeric precursor fiber ma y also contain voids as a result of the spinning operation [64]. These voids are subsequently elongat ed during the stress graphitization process providing additiona l nucleating sites for crystallites. Finally, if the misoriented crystallites are the weakestlink in the carbon fiber then they alone might be considered the contro lling population for fiber strength and thus a two-parameter Weibull model would be appropriate. For this reason and because of its simplicity the two-parameter model will be used to describe the strength distributions of all failed carbon fibers in this work. Add itional theoretical support for the two-parameter Weibull model will be given below. A number of researchers ha ve associated the strength distributions of brittle ceramics with their incipient flaw populations [ 65] generally using an optical or scanning electron microscope to identify flaw type. Sometimes inferences from the strength PAGE 62 distributions and sample geomet ries were used to identify the presence of multiple populations. For instance, on the basis of empi rical strength distributi ons Own et al. [10] concluded that there were two flaw populati ons present in the PAN-based carbon fibers they tested, surface and internal flaws. Th ey found that the two flaw populations gave rise to a bimodal strength distribution that was best described by a bimodal lognormal model, Eqn. 2.15, (cf. Eqn. 4 of [10]): ) : (x f q + ) : (x f p = f(x)2 2 2 1 1 1 (2.15) where ] 2 ] (x) [ [ x 2 1 = ) : (x f2 2 i i i ln exp for x > 0 and p + q = 1. Equation 2.15 is similar to Eqn. 2.3 except that the form of the PDF's for f1 and f2 are lognormal in Eqn. 2.15 and Weibull in Eqn. 2.3. To use multiple population distributions (e.g., concurrent, exclusive) two distinct subpopulations must be identified in the sp ecimens. Substantia l evidence exists to identify misoriented graphitic crystallites in carbon fibers as a di stinct population of strength limiting defects [29-49] These crystallites are of va rying sizes and orientations with respect to the fiber axis. Their number density and locations throughout the fiber are controlled by the PAN precursor fiber homoge neity and the subsequent carbonizing and graphitization processes. A second possibl e defect population might be ascribed to surface defects, irregularities and scratches. Usually any defect that disrupts the morphology of the precursor fiber tends to decrease the fiber modulus and strength [63,66]. This is as expected for a homoge nous material and has also been amply proven by studies that consider the structure/strength relationshi p of carbon fibers [66-68]. PAGE 63 Controlling the flaw population and flaw seve rity is therefore a primary concern to fiber manufacturers [67], irresp ective of whether the internal and surface flaws are caused by polymeric precursor impurities, fiber spinning flaws, or mechanical handling effects. Protecting the carbon fiber during and after pr oduction usually invo lves applying some type of surface treatment (e.g., a liquid-phase treatment, plasma treatment). This has a twofold advantage. Sizing can be specifically tailored to en hance wetting out of the fiber and simultaneously provide mechanical protec tion from deleterious contact with other fibers and processing equipment. Sizing, however, is not completely effective and handling will invariably damage some of th e fiber surfaces. The severity and frequency of this damage is difficult to discern and control so extreme care must be taken when testing single carbon fibers to avoid a ny untoward damage to the fiber surface. It seems probable then, that at least tw o distinct flaw populations exist for carbon fibers. Establishing whether the populations are concurre nt, exclusive or partiallyconcurrent is more difficult. If we consider an acrylic precursor fiber, the difficulty in avoiding internal flaws in car bonized and graphitzed PAN-based fibers becomes readily apparent. Thorne noted [64] that internal defect locations are di stributed in a Poisson point process with Poisson in tensity equal to the mean fl aw concentration. With increasing mean flaw concentration there is an attendent drop in the probability of obtaining a flaw free gage length. This is clearly seen in Figure 2.34 where the probability of obtaining a flaw free length is plotted for various measured mean flaw concentrations and different gage lengths. Even for the moderate concentrations considered in Figure 2.34, there is virtually no chance of obtaining a flaw free length of 20 mm. PAGE 64 Of course one would expect the flaw free length in carbon fibers to be much shorter. Tensile failure tests, SEM, a nd TEM inspection overwhelmingly confirm this supposition. Bennett [29] found that the inci pient flaws, volume or surface, can not be identified on carbon fiber fractur e surfaces 20% of the time. Th is indicates that there are critical flaws on a scale smalle r than can be distinguished, even by SEM inspection. In addition, surface defects are not restricted to fiber sections devoid of internal flaws. Indeed, supposing that the surface and interior defect location distributions can each be described by a Poisson process, the possibili ty for a two-population exclusive distribution along the carbon fiber (i.e., no physical overlap of the flaws) becomes extremely remote. In a like manner, the existence of a partia lly-concurrent distribu tion is difficult to support considering the large number of intern al flaws that are known to exist in carbon fibers. If however, surface flaws occur only infrequently compared to the gage length then a partially-concurrent model is reasonabl e. Additionally, if both volume and surface flaw populations were describe d by a Poisson point process th en a partially-concurrent distribution would become more likely as the gage length decreases. Again, SEM and STM evidence has shown that carbon fiber surfaces are far from smooth and thus surface irregularities and flaws are comm on. In addition, physical cont act with other fibers in the tow (handling effects) can ne ver be completely avoided. Lastly, as noted above, the exclusive a nd partially-concurrent models can be confused with each other and are therefore of dubious practical value. An attempt to fit fiber Sets A and B with an exclusive and partially-concurrent distributions will be undertaken if only to prove their questionable va lue in this case. Wh ile the exclusive and partially-concurrent models will prove to be of little use here in describing carbon fiber PAGE 65 failure statistics, they can be used effectively in carefully co ntrolled situations. All this leads to the simple conclusion that the tw o flaw populations in PAN-based carbon fibers exist concurrently. Discrete flaws acting in conj unction with other flaws is also a strong possibility. That is there are a number of flaws simultane ously involved in the failure process. The discrete flaws for HM and HS carbon fibers are very small. Consequently, any single flaw is unlikely to cause fiber tensile failure by itself, rather numerous flaws combine, or interact, to produce a critical flaw that results in total fiber failure. Bennett et al. [29] failed 20 fibers and viewed 11 fracture surface pairs with high resolution SEM and TEM. Of the 11 pairs viewed two fibers failed at the surface first and the remaining nine in the fibers' interior. This suggests that eith er the number density (number per unit length) of surface flaw s is less than for volume flaws, NS < NV, or that the volume flaws are more severe (greater st ress concentration) or possibly larger than the surface flaws [69]. A combinat ion of both is also possible. Schematic strength distributions for severe volume flaws and surface flaws are given in Figure 2.35. The larger percentage of severe internal flaws (lower fiber strength) would naturally give rise to a greater number of fibers failing at in cipient volume flaws. The overlap of the two strength distributions (Figure 2.35) does not necessarily imply that there is a physical overlap of the flaw populat ions. Indeed, consider ing the scale of the flaws physical overlap seems unlikely. In th is vein, the mathematical approach can be much simplified by considering the flaws as point defects with no physical dimensions. In support of this simplification, Thorne [64] not ed that there is a neg ligible effect on the mean flaw free length in acrylic fibers when the flaws physical dimensions (~ 0.5 m PAGE 66 avg. axial dimension) are considered. In a like manner, the needle-like voids have dimensions on the order of 10 nm wide by 30 nm long [35,41] and can be neglected with respect to the overall gage le ngth. Surface defect dimensions are also negligible with respect to testing length and therefore will be considered point defects. Experimental Methods and Error Sources Measuring a carbon fiber's modulus and failur e stress requires accurate readings for load, displacement and fiber diameter. The thin beam load cell used here has a rated capacity of 113 gm. The output from the lo ad cell was amplified with an operational amplifier. Each point on the calibration curve for the load cell (Figure 2.36) represents the average of approximately 100 readings. The load/voltage conversion factor is 86.83 gm/volt with a correlation coefficient of 0.9999. The displacement or strain in the fiber was obtained directly from the integrally mounted LVDT on the MTS system. The conversion factor and correlation coefficient for linear displacement are 9.17 mm/volt and 0.9999. Measuring the carbon fiber diameter is more troublesome, especially considering the precision available for load and deflec tion measurements. Errors in diameter measurement and inherent fluctuations in the fi ber diameter will likely be the sources of the largest errors in repo rted Young's modulus and failure stress. The Fraunhoffer diffraction pattern used to determine the fi ber diameter is created by the wave-like interference of coherent laser light with the carbon fiber (Figure 2.2). A representative Fraunhoffer diffraction pattern produced by a sing le carbon fiber is shown in Figure 2.37. The dark region at the center of the diffract ion pattern is caused by a light stop that is necessary to avoid CCD array damage. The location of the first intensity minima is evident in Figure 2.37 and even more so when the digital image information is inspected. PAGE 67 As a first approximation to the fiber diameter, d, one can use n Z L 2 + = dn (2.16) where is the probe wavelength, L is the separation of the ob ject and viewing planes, and Zn is the distance separating the nth nodes [70] (Figure 2.38). Equation 2.16 is a one term approximation to Fraunhoffer diffraction, which itself is a 2-D approximation of 3D Fresnel diffraction. Provide d the impinging light is coll imated (e.g. HeNe laser), Fraunhoffer diffraction is a good approximation to the actual 3-D Fresne l diffraction. A more accurate diameter value is f ound when Eqn. 2.17 is used [71]. ] ) n Z 2L ( + 1 [ n = d2 n (2.17) The two critical measurements in these two equations are L and Zn. L is determined by inserting a diffraction grati ng of known spacing at the object plane and then using Eqn. 2.18 to back out the distance L [72]. tan 2 Z = L (2.18) where ) d ( = 1 sin is the wavelength (HeNe laser, = 632.8 nm), and d is the grating spacing (10 microns, 100 lines/mm). Once again, determination of L depends on the ability to determine Zn accurately. Zn was found by using a calibra ted digital black and white video imaging system with an eight bit (0) gray scal e. The imaging system was calibrated using a 20 mm grid PAGE 68 with circles located at the grid intersections. Coordinates and intensity of any pixel in the digital image (400 rows x 512 columns) could then be ascertained and recorded. Pixel intensity along a line co uld also be recorded and provide d a convenient method to record the Fraunhoffer diffraction pattern. Figure 2.39 gives the pixel inte nsity along the line in Figure 2.37. The flat intensity profile near the light stop (central minimum) is caused by imaging system limitations. Determining the first minima separation, Z1, in the diffraction pattern was accomplished by fitting the intensity curve minima regions with fourth-order polynomials and rounding to the nearest pixel. The pol ynomial fits to the minima are shown as solid lines in Figure 2.39. Diffraction patterns at two different positions were taken for each fiber. The average of the two diameter measurements was corrected for a slight Fraunhoffer technique over prediction. Corrected diamet ers are used in all subsequent modulus and stress calculations, except where indicated. Multiple measurements at the same location on a fiber showed excellent repeatability. The CV for fiber diameter m easurements at a fixed position was less than 2.0%. This suggests that if an error exists in the measur ement it is operating systematically, obviating any concern that mi ght be raised about noise in the digital image affecting diameter values. Noise in the digital image ma y be caused by a number of sources (e.g., video camera CCD array, poor electrical connections, electromagnetic interference, and A/D conversion). However, it appears that the la rge number of data points in the line intensity curves overwhelm a ny noise effects. The systematic errors in diameter measurement are expected to be the result of the digital image system x-y calibration and the 2-D Fraunhoffer approximation, Eqn. 2.17. PAGE 69 Clearly, the minimum visual resolution of a digital image is the pixel. XY resolution however depends on both the imag e magnification and the number of pixels for the system. Difficulty in identifying th e intersections of the calibration grid and possible errors in the grid itself could result in a small systematic error for pixel to pixel distance. In addition, the vi deo analysis software used ca nnot always equally divide the calibration length by the number of pixels. This gives rise to a series of pixel locations that are not equally spaced. The xy reso lution would then be 1 pixel dimension + modulo/(no. of pixels in series). More importantly, Li and Tietz [71] found that the Fr aunhoffer diffraction technique consistently over predicts the fi ber diameter by 5% when compared to the values obtained by careful SEM examination. According to Li and Tietz [71], this error is a direct result of the Fraunhoffer approximati on. A 5% error in diameter can lead to a 10.25% error in modulus and failure strength values because the error is compounded by squaring the diameter. In their work, Li and Ti etz [71] correct for this systematic error by dividing the diameter obtaine d by Fraunhoffer diffraction by 1.05. The same correction factor for the Fraunhoffer diffraction tec hnique will be used in this research. Results Fiber Diameters A representative example of the Fraunhoffer diffraction pattern created by the wave-like interference of the incident HeNe laser light and a carbon fiber is shown in Figure 2.37. The signal intensity along this diffraction pattern and the fourth order polynomial fits to the minima are given in Fi gure 2.39. The dark region at the center of the diffraction pattern is created by an opaque light stop and is necessary to avoid damage to the video camera's CCD array. The node se paration obtained from Figure 2.39 is used PAGE 70 in Eqn. 2.17 to find the fiber diameter. The object/viewing plane separation L used in Eqn. 2.17 was calculated using Eqn. 2.18 with 100 lines/mm and Z1 as determined by digital video analysis. The results were then corrected for the over prediction error introduced by the Fraunhoffer diffraction ap proximation (~ 5% [71]). The corrected carbon fiber diameters (Fraunhoffer diffraction) are given in Tables 2.7.10 and in Figure 2.40. The data in Figure 2.40 represen t the average of two measurements for each fiber. Considering the XY resolution of th e digital image and the laser wavelength there are 3 significant dig its (x.xx micrometers) for this Fraunhoffer diffraction technique. The diameters (avg. of 2 readings) for each sample set can be represented by a normal frequency distribution in accord with the Mean Value Theorem [25]. The normal distributions for the averaged measurements are shown in Figure 2.41. The sample means for Sets C and D do not lie within the 95% confidence intervals for either Set A or B. In other words, we are 95% confident that the difference in the means is not the result of random probability. There are two possible e xplanations for this difference in means: The variation arises naturally during fiber processing (i.e. slight variations in filament tension or cooling). The mean differences are the result of a systematic error associated with the determination of the object/viewing plan e separation (Tables 2.7 through 2.10). The apparent diameter variation is a comb ination of both factor s with the latter contributing more in this instance. Hughes [ 37] reported that for a batch of HM and HS carbon fiber diameters the CV was 4% and 6% respectively. The variation for each sample set tested here was within this CV a nd even the overall varia tion was near to these reported values supporting the contention that the variations within a sample set were indeed natural and not an artifact of the measuring process. However, the tighter groupings for sample Sets C and D suggest that at least some of the scatter for Set A and PAGE 71 B fiber diameters can be attributed to measurement error. The larger scatter seen for Sets A and B is the result of the fibers being di splaced out of the object plane either by air currents or because of fiber drooping. The tension applied to Set C and D fibers was greater than that for A and B fibers insuring that the fiber remained in the object plane and thus the tighter grou ping for C and D fibers. While the maximum difference between the sample means is 4.7%, for Sets A and D, this variation does not represent an unreas onable experimental un certainty considering reported diameter variations [37] and th e complex and irregular morphology of carbon fibers. More importantly, the difference betwee n the reported engineering stress (failure load divided by the original cr oss section) and the true stre ss (failure load divided by the instantaneous cross section at failure) is caused mainly by two necessary, but faulted assumptions: 1. The fiber has a constant circular cross s ection along the gage length and it remains constant throughout the test. 2. The longitudinal and transverse stress fields in the fibe r are uniform along the gage length. Considering assumption (1): Clearly, the fi bers, as the data bear out, do not have constant circular cross sections. The av erage difference between the two readings (difference = abs [( dia1dia2)/ diaavg]*100) for each fiber is 1.5%, a relatively small value. However, the standard deviation was nearly as large, = 1.4%. The maximum difference between the two readings for a single fiber was almost 8%. Multiple readings on single fibers confirmed the magnitude of th e diameter variation al ong fiber sections as short as 4 mm. Irregular carbon fiber geometry at a given cross section (non-uniform diameter) is also apparent with Fraunhoffer diffraction measurements (Figure 2.42) [73]. Note that the axes for Figure 2.42 are chosen to exaggerate the diameter variation. There PAGE 72 is also considerable SEM and STM evidence [29,42,68] that the surfaces of carbon fibers are irregular and the cross section is not circular. It might be possible to measure the cro ss sectional area of each failure surface by SEM, however this is neither expedient nor ne cessary. While it is cl ear that the fibers do not possess constant circular cr oss sections equating the fiber diameters to the average of their respective readings produces only a small error in the failure stress, CV 5% (from multiple measurements on a single fiber), whereas the variation in the fiber stress among fibers is much larger, CV = 25%. In other words, the large scatter seen for carbon fiber tensile strength in this instance obviates th e need for such precise SEM measurement of the failure surface. In this vein, it can also be argued that the small reduction in fiber diameter accompanying tensile loading can be neglected. A quick example bears this out. The difference between true stress and engineering stress for a 7.0 m diameter fiber with 2% ultimate strain is only 1%. It is therefore concluded th at assuming a constant circular cross section with a diameter equal to the average of two readings for that fiber is a reasonable and defensible assumption fo r fiber tensile strength determination. Considering assumption (2): There is general agreement that carbon fiber morphology is irregular and complex consis ting of turbostratic graphite regions, amorphous carbon, needlelike voids, irregular voids, small impurities and inclusions, and sometimes highly ordered 3-D graphitic re gions [32-37,39-49]. This morphology is schematically depicted in Figure 2.16. From this figure, it is obvious that a homogeneous uniform stress field cannot exist throughout the fiber during uniaxial tension tests. The mechanical anisotropy of graphite and the ot her inhomogeneities in the fiber will give rise to a non-uniform stress field. The surf ace and volume defects will intensify stress in PAGE 73 their vicinity making it extremely unlikely that a fiber will fail at its smallest cross section as would be expected for an isotr opic homogeneous material. However, without assuming a priori a uniform stress field it w ould otherwise be impo ssible to quantify the fiber failure stress. In addition, assuming a uniform stress field allows for comparisons among specimens that are known to have dis tinct microscopic differences but which otherwise behave similarly macroscopically. Finally, it is believed that any errors in troduced by the diameter measurement, a constant circular cross section assumpti on or a uniform stress field assumption are subsumed by the distribution and severity of surface and volume defects in the fiber. In other words, it is the nature of the defects and their distri bution throughout th e fiber that broadens the failure strength data. A ny distribution broadening introduced by the assumptions noted above or by diameter m easurement errors are overwhelmed by the statistical nature of brittle failure. Load-Deflection A selection of load-deflection curves is given in Figure 2.43 and the results for all fibers are tabulated in Tables 2.11 through 2.14. Several impo rtant features of the curves in Figure 2.43 and for all the load-deflecti on results can immediately be noted. For instance Tables 2.11 through 2.14 show that when the results for failu re load and failure stress are ranked by increasing magnitude the order of their rankings are not the same (Figure 2.44). The solid line in Figure 2.44 represents the lo cus of points for Set A fibers if a single diameter (average for all A fibers) is assumed. The scatter of each point about the solid line represents that fibers de viation from the sample set mean (7.25 m). The PAGE 74 need to measure each fiber's diameter is evident from this figure, as assuming a single constant diameter will lead to faulty statistics. This finding is consistent with work by Wa gner et al. [13] that notes that simply reporting the failure load for aramide fibers is misleading as fiber di ameter variation has a marked effect on the distribution of failure st ress or tenacity for a set of fibers. The results for carbon fibers here are no exception to this axiom as similar results for sample Sets B through D were also found. Before proceeding, a note on the electronic noise present in the load and stroke signal is in order. The noise in the load signal appeared irregular and random while the displacement signal noise was very regular for al l tests. Much of the noise in the load signal could be eliminated by proper gr ounding as evidenced by linear correlation coefficients (Table 2.11 through 2.14) for fibe rs A1 through A17 versus subsequent fiber tests for which grounding was improved. A re gular 20 Hz perturbation on the DC stroke signal can be readily identified using a FFT power spectrum plot. The 20 Hz perturbation has been eliminated from the signals in Figur e 2.43. To eliminate this small perturbation a sliding average of width 5 readings (20 Hz data acquisition rate = 100 Hz) was used. This sliding average was used in plotting the load deflection curves in Figure 2.43 as well as in the computation of all associated materi al properties. Note that the remaining noise seen in the load-deflection curves of Figure 2.43 is caused primarily by irregular fluctuations in the stroke signal. All of the fibers tested exhibited the sa me brittle behavior as depicted in Figure 2.43. The load-deflection curves were all linear with no discerni ble yielding or yield points. In addition, fiber fa ilure was always immediate and catastrophic. The nature of PAGE 75 these load-deflection curves is common to ceramic fibers and has been reported by numerous investigators [74-78]. As to whether load-deflection behavior for tensile deformed carbon fibers is strictly linear the answer is somewhat clouded by our inspecti on method. Using sonic modulus measurements Curtis [55] found that th ere was marked nonlinearity of the dynamic Young's modulus during initial load ing (cf. fig. 1 of [55]) with asymptotic linear behavior at higher loading. Curtis concluded via X-ray scattering that the linear region is associated with crystallographic reorientation during fiber loadi ng. It is unlikely that the sensitivity of the present load cell and disp lacement head would allow detection of this initial nonlinear region. This reorienting effect duri ng loading would be dependent on the initial preferred alignment of the graphite basal planes along the fiber axis. High Modulus Type I carbon fibers would likely exhibit less initial nonlinearity than HS fibers since the former are more preferentially aligned. The range over which this reorienting process takes place surely contributes to the incr eased failure strains seen fo r HS carbon fibers. It also suggests that there is a str ong relationship between the distribution of basal plane orientations and fiber strengt h/modulus, a conclusion that is supported by x-ray scattering [29-35,39-41]. We also know that as we reduce the gage length (effective volume) the fiber's ultimate strength tends toward the material's in trinsic strength. Practically, fiber tensile test gage lengths are limited by end-clamping effects and fiber alignment considerations (the shorter the gage length the more critical the alignment). Phoenix and Sexsmith [79] derived relations that account for the stress build-up in th e clamping matrix and possible PAGE 76 failure in the clamping matrix. They showed that for gage lengths of one inch (25.4 mm) or greater there is only a slight effect on th e predicted mean failure load. As the gage length decreases below this limit, there is a roll-off tendency toward under predicting the true failure load. No end-cl amping effect corrections were made in this study (minimum gage length = 20 mm) and in all reported cases fiber failure occurred in the free unclamped gage length. Figure 2.43 and Tables 2.11 through 2.14 i ndicate that there is a significant variation in stiffness (kg/mm ) for the fibers tested. So me, although not all, of the stiffness variations can be accounted for by cons idering fiber cross sec tions. Again, this underscores the need for measuring each fi ber diameter individually. The average modulus and standard deviation for each samp le set are given in their respective table (Tables 2.11 through 2.14). This modulus di stribution must be considered statistically significant, as the overall modulus CV is mu ch larger than the diameter CV, 16.5% and 5.5% respectively. This type of modulus variation will not generally be caused by handling or testing procedures (i.e. surface sc ratches, testing fixtures, humidity, testing temperature, etc.). These parameters would tend to adversely affect fiber strength and have little effect on fiber modulus, especially surface scratches [77] and elevated testing temperatures [80,81]. More probably, these modulus variations are the result of small fluctuations in processing conditions that can give rise to local density and morphology/orientation differences in the fibers [32-37,39-49]. Lastly, the ultimate load and deflection for the different fibers varies widely, which is indicative of a flaw-controlled failure process. For exampl e, in comparing the diameter and failure load CVs for sample Set A, 4.9% and 20% respectively, it is apparent that PAGE 77 diameter variation alone will not account for the observed fail ure load dispersion. Note also that within each sample set and for all fibers of the same gage length, the difference in deflection at failure indicates that the ultimate strain f is varying statistically. Once again, this supports th e conclusion that for axial loading a flaw-controlled failure process is operative for th ese carbon fibers. Failure Stresses The failure stresses for each sample set were ranked a nd the corresponding cumulative failure probability was assigned according to Eqn. 2.10. The cumulative failure probabilities for as -received 20 mm and 50 mm long carbon fibers are plotted in Figure 2.45. Recalling that the fiber diamet er varies along the fiber by an average of 1.5%, one could plot horizontal error bars fo r each fiber (error bars are left off Figure 2.45 for clarity). Technically, the width of the error bar for each fiber would be different as each fiber diameter varies independently of the others, however, for practical reasons the width of the error bar is set at +/1.5% (the average of the differences between the two diameter readings for all fibers, cf. Tabl e 2.7 through 2.10). We are concerned here with the statistical nature of the failure stresses and as such, th e possible measurement error in failure stress for any one fiber is overwhelmed by the much broader stress distribution for many fibers that is the result of a distribution of flaws in both space and severity. The cumulative probability curves in Figur e 2.45 for Set A and B fibers have the expected s shape. It is also evident that the cumulative failure probability F ( s ) at a stress state s is greater for the 50 mm long fi bers than for the 20 mm long fibers, F50( si) > F20( si), for all i These results are in accord with W-G theory, which may be stated as PAGE 78 ] (s)ds dV [ 1 = F(s)s 0 v exp (2.19) where ( s ) is the intensity function. With the restrictions that ( s ) be a positive nondecreasing boundless function of s it is obvious that the cumulative failure probability F ( s ) at any given stress will increase as VdV increases. As the fiber gage length increases, there is a concomitant increase in F ( si) when si is held constant. The cumulative failure probability F ( s ) for Set C plasma treated samples is also seen in Figure 2.45. The plasma treatment has adversely affected the fiber strength ( F20,plasma( si) > F20( si)) for nearly all s The larger failure strength scatter seen for the plasma treated fibers evidences varying degr ees of treatment for the individual fibers even though the fibers were continuously rota ted inside the plasma chamber (Figure 2.3). In addition, an increased numb er of treated fiber failure tests would likely weight the distribution to more moderate values (cf. both the untreat ed 20 and 50 mm long tests in Figure 2.45). Inspecting Eqn. 2.19 one can see th at the increase in failure probability for these plasma treated samples as compared to untreated 20 mm fibers is due to a change in the integral ( s )ds not in the volume integral as was the case for the 50 mm fibers. The oxygen component in the plasma physically etch es the fiber surface, altering the fiber's response to tensile loading. The other species in the plasma have little effect on carbon fibers at low temperatures. The physical chan ge on the fiber surface is manifested in the intensity function ( s ). The intensity and duration of the oxidation treatment determines the change in ( s ). Initially, an oxidation trea tment will strongly etch disorganized or higher energy carbon forms on the fiber surface [82]. This pr eliminary etching eliminates or lessens severe surface flaws resulting in improved te nsile strength [82-84]. However, with PAGE 79 increased duration or intensity oxidating trea tments will tend to increase the rugosity of the carbon fiber by producing micropores on th e fiber surface [85-87] While increased rugosity generally improves composite performance (except for fiber dominated properties, i.e. 0 tensile strength of a unidirectional laminate ) by providing additional fiber/matrix coupling sites, th e fiber's tensile strength is ma rkedly decreased (Tables 2.11, 2.13, and 2.14, and [88]). Independently, and utilizing different oxidative techniques, Donnet et al. (microwave plasma [86,87] vs. Marshal and Price liquid phase oxidation [82]) found that severe oxid ation treatments of PAN-based carbon fibers produce surface pits or slits that are transver se to the fiber axis. STM meas urements [82] showed nearly a doubling of the RMS roughness parallel and transv erse to the fiber axis for a severe oxidative treatment versus the standard oxidatio n treatment. Again, the etching occurred preferentially at high ener gy carbon sites (i.e., edges of graphitic-like planes, disorganized or amorphous car bon) exposing graphite platelets that are misaligned with the fiber surface [82,86,87]. A standard co mmercial oxidation treatment produced slits with dimensions on the order of 50 nm long x 3 nm wide x 10 nm deep [82]. Overtreating the fibers is known to increase these dimensions [82]. The resulting surface morphology is similar to the morphology Reynolds and Sharp [28] propose as the cause for carbon fiber failures In the case of over-treated fibers, the surface roughness and especially transverse slit s expose the misoriented graphite platelets to large local stress concentrations. For ex ample, a transverse slit with a rounded tip would expose the neighboring crys tallite to a stress field thr ee times that of the farfield stress, making transverse rupture via induced shear much more probable for the local misoriented basal planes. In fact, the actual transverse slit tip is not rounded and would PAGE 80 therefore produce an even larger stress con centrating effect. In contrast, the sharp needle-like interior voids ar e oriented primarily along the fiber axis [34,35,40,41] and will thus have only a slight effect on the local stress field. In the limit, a void of zero width aligned perfectly w ith the fiber axis and will have no effect on the fiber tensile strength or modulus. The net effect of over-treatment and incr eased surface rugosity in particular is therefore a considerable reduction in tensile st rength as there is an increased likelihood for satisfying the Reynolds-Sharp conditions on the fiber surface. This argument holds for moderate oxidative treatments as well, although the effect is not as pronounced, because as Marshall and Price [82] have not ed surface etching and pitting is not as severe. The fiber Set D RF plasma treatment is presumed to be relatively light while Set C fibers were subjected to a harsher (longer and more powerful) oxidative treatment. Failure statistics will be used be low to support this contention. Commercial oxidative treatme nts are generally more severe than Set D treatments owing to the manufacturer's desire to incr ease fiber wet-out and improve interfacial adhesion, which requires a more severe oxidative treatment. From the preceding discussion it should be apparent that surface treatments (moder ate and extreme) and fiber handling adversely affect fiber strength and that the severity of these introduced flaws will likely supersede existing interior flaws (cf. Figure 2.45). Discussion Weibull Strength Statistics In the above discussion, we have been speaking of ( s ) in general terms without specifying its form. To quantify the distributi on we must assume a form for the intensity PAGE 81 function. Choosing to use Eqn. 2.1, a modi fied form of the original CDF proposed by Weibull [3], we can begin a more quantitative discussion of the failu re stress. Utilizing Eqn. 2.19 as the general form for Eqn. 2.1, the intensity function is given by ) s s ( ) L s ( = (s)1 0 0 0 (2.20) Recall that Weibull believed that the power law form in Eqn. 2.20 did not possess any unique qualities that would make it appropriate for failure strength predictions. However, as noted above the Weibull distribution is one of three extreme value distributions, and is thus well suited for life statistics and failure strength data. In this work three different analytical tec hniques have been used to obtain the scale factor ( s0) and Weibull modulus () seen in Eqns. 2.1 and 2.20. The most straightforward approach involves fitt ing Eqn. 2.7 using a linear leas t square regression (LLSQ). This method has been discussed above with regard to the simulated two-parameter Weibull distributions of Table 2.4 and Fi gure 2.15. The second approach is via Maximum Likelihood Estimation (MLE). Specifi cs for the MLE technique can be found in Appendix A and references [89-92]. Th e final approach uses a standard nonlinear curve fitting procedure (TableCurve 2D, Jandel Scientific). All three techniques compare estimators, and s0' to the true parameters and s0. The techniques do not generally return the same estimators. However, for large n these estimators asymptotically approach the same true parameters. For clar ity, the primes are left off the estimators below and unless otherwise noted parameters cite d actually refer to estimators to the true parameters. PAGE 82 Weibull two-parameter distribution The Weibull moduli and scale factors for the failure st rengths of fiber Sets A through D are given in Table 2.15. It is imme diately apparent that for a given sample set the moduli obtained by MLE and LLSQ differ c onsiderably, while the scale factors are consistent. Generally, the MLE technique te nds to weight more h eavily large groupings of data lessening the influe nce of outliers, whereas LLSQ uses a more conservative approach by weighting each point evenly. Th e net effect in this case is that the moduli determined by MLE tend to be greater than the moduli found by LLSQ. In addition, the GOF for MLE is generally much less than that for LLSQ fits. The difference between the MLE and LLSQ estimators seen in Table 2.15 highlights the difficulty associated with dete rmining Weibull estimators. The question as to the most efficient Weibull parameter estim ators has been considered by a great many researchers [53,54,89-99]. An efficient esti mator is defined as one that rapidly and asymptotically approaches, for small n the true parameter. There are, in fact, a number of other a pproaches available for estimating Weibull statistics besides MLE and LLSQ. The most notable are best linear invariant estimate (BLIE) [93-95], best linear unbi ased estimate (BLUE) [96], and a simple linear estimate (SLE) proposed by Bain [97]. BLIE and BLUE, which themselv es are linear functions of each other, involve extensive use of tables and Monte Carlo simulations to determine appropriate weighting fa ctors. For example, ci,m,n in Eqn. 2.21, ) s ( c = 1/n i, n m, i, m =1 iln (2.21) where ( n-m ) is the number of censored data. PAGE 83 BLIE and BLUE estimators are realistically limited to data sets with n < 25 or those that are deeply censored, r = n-m <25. As n increases, it becomes increasingly difficult to determine precisely the weighting coefficients [95]. The SLE proposed by Bain [97] is easier to use than both BLIE and BLUE as it requires only a single table ( W, the 100 th of W ). An added advantage of SLE is that an associated approximate statistic varies as a chi-square variable thus allowing easy confidence interval prediction. It has been shown that MLE and BLIE ar e both equally asymptotically efficient [94]. The original stumbling block in using MLE was the iterative procedure required to optimize the fitting parameters, and s0. This is no longer a problem and noting that Thoman [91] showed that MLEs are unbiased for n 30, MLE is the clear choice between MLE and BLIE. MLE is efficient, easy to apply and does not suffer from any precision problems for n > 25. In contrast to BLIE, BLUE and SLE, the LLSQ approach uses only the sum of squared errors (SSE) to arrive at its estimators. For n paired observations, the slope of the LLSQ line is given by n ) X ( X n ) Y ( ) X ( Y X = slope2 i n 1 2 i n 1 i n 1 i n 1 i i n =1 i While fitting data with LLSQ regression is exceedingly simple it suffers from the incorrect assumption, in this ca se, that the data is distributed about the predicted value in a Gaussian manner. This biases the LLS Q prediction by weighting each data point equally. As a result, the LLSQ approach is less efficient than MLE in determining Weibull parameters and will generally under predict the Weibull modulus (Table 2.15). PAGE 84 The failure strengths for sample Sets A and B are plotted on Weibull probability paper in Figure 2.46. The solid and dashed lines in Figure 2.46 represent the LLSQ and MLE two-parameter Weibull fits respectively. The curves are straight lines with slopes equal to their respective Weibull moduli and th ey cross the abscissa at a value equal to the natural log of their scale f actors, which in this case are essentially equal. The PDFs, empirical and fitted, for fiber Sets A and B are displayed in Figure 2.47. In these figures, the solid lines are MLE predicted and the bars are the actual freque ncy histogram plots. It should be noted here that the 5% Fra unhoffer diffraction correction [71] used in fiber diameter calculations has no effect on the Weibull modulus. However, the scale factor is shifted upward by 10. 25%. This can be seen in Figure 2.48 where the Weibull two-parameter corrected and uncorrected cumu lative distribution curves for Set A fibers are identical except for a shift along the x-axis. The most obvious feature of Figures 2.46 and 2.47 is th at the data are by no means perfectly represented by the two-parameter Wei bull model, Eqn. 2.1. Graphically, this reaffirms the generally poor GOF parameters given in Table 2.15. The LLSQ residuals for the untreated 20 and 50 mm long fiber sets are given in Figure 2.49. There is considerable systematic scatter of the data about the predicted lines for each data set. The MLE and nonlinear approaches (not shown here) have similar residual patterns. Systematic scatter ordinarily indicates either th at the chosen model is invalid or that there was insufficient sampling or both. In this case, model inadequacies are suspected as n 30 usually assures sufficient sampling. The 95% confidence and prediction inte rvals, predicted assuming a normally distributed error, for fiber Sets A through D are quite narrow considering the complexity PAGE 85 of carbon fiber microstructure and the rela tively few samples measured. The LLSQ 95% confidence and prediction interv als for fiber Sets A and B ar e plotted in Figures 2.50 and 2.51. The intervals for Set A and B fibers are consistent with each other while the intervals for the plasma treated fibers, Sets C and D, are broader still, a direct result of varying plasma treatment effects and the sma ller sampling. The Gaussian confidence and prediction intervals for the MLE derived es timations (Figures 2.52 and 2.53) and the nonlinear estimations (Figures 2.54 and 2. 55), are similar to those for LLSQ fits. As noted above there has been considerable work on the subject of estimating Weibull parameters and their predictive qua lities [53,54,89-99]. For material strength and lifetime statistics the error, in the linear probabilistic model + x + = Y1 0 (compare Eqn. 2.7) is not generally normally distributed as Gaussian confidence interval prediction assumes. More likely the distribution of varies as a function of the predicted value (deterministic component ). The width of the error distribution shoul d broaden as the predicted value nears the tails of the We ibull PDF and narrow considerably at values near the mean of the Weibull distribution. Thoman [91] and Harter [98] devel oped methods for confidence interval predictions that consider the natural skew of Weibull distributions. Extensive sample simulations were undertaken by both investigat ors to generate the necessary tables for confidence interval estimation. The Set A 95% lower confidence bounds predicted by the methods of Thoman and Harter (MLE estimat ors) are presented in Figure 2.56. Figure 2.56 illustrates that both methods give nearly identical 95% lower confidence bounds. Consequently, unless otherwise noted, the c onfidence intervals used below are those PAGE 86 predicted by Harter's approach, the simpler and more straight-forward approach of the two. In contrast, the Gaussian confidence inte rvals are always narrower, except at the extremes, than those predicted by Harter's approach regardless of the parameter estimating technique. The Harter 95% c onfidence bands and Gaussian confidence intervals for Set A fibers are compared in Figure 2.57. Utilizing simple LLSQ confidence intervals suggests unwarranted confidence in the pred ictive qualities of Weibull distributions. A more realis tic consideration of the random error results in considerable broadening of the 100(1) confidence interval. A lthough Harter [98] did not give an explicit method for calculating the 100(1) prediction interval it would be broader still, further weakening the predictive quali ties of Weibull statistics for single fiber strength. Considering the complexity of carbon fibers and their inherent strength variation, the predictive qualities provided by the confidence and prediction intervals are still quite reasonable an d useful for carbon fibers with co mparable diameters and lengths. Now, if the distributions given in Ta ble 2.15 and Figures 2.50 through 2.55 are reliable for fibers tested under similar conditi ons, the question remains as to the validity of strength distributions found by scaling from the referen ce length. The scaled PDF predictions (Eqn. 2.1a) are compared to the MLE fitted curves (Figure 2.58). The solid lines in Figure 2.58 represent the MLE fits for fiber Sets A and B. The solid lines with symbols and dashed lines are the scaled pred ictions. The scaled distributions' maxima correspond well to the maxima for the base curv es. In contrast, the breadth of the base and scaled distributions are quite different. The dispar ity between actual and scaled distribution breadths underscores a particul ar problem associated with the scaling PAGE 87 concept. Which is: scaling to other gage le ngths involves th e critical assumption that the modulus remains constant over the lengths consid ered. In other words, there is one modulus that describes the failure pro cess for all specimen volumes and loading configurations. An alternate graphical method of evaluati ng Weibull modulus and the validity of a constant Weibull modulus over a range of gage lengths further illustrates this critical assumption. A straight line on a Log-Log plot of average failure strength versus gage length indicates that the Weibu ll modulus is constant over th at gage length range [9-11]. The average failure strengths and scaled predictions for Set A and B fibers are given in Figure 2.59. The modulus s uggested by this approach ( = 3.9) is only slightly larger than that predicted for 50 mm long fibers ( = 3.4, LLSQ) but it is considerably less than the MLE estimator for 20 mm long fibers ( = 6.6, MLE). A change in Weibull modulus with specimen volume often indicates th e presence of multiple flaw populations, especially when orders of magnitude di fferences in specimen volume are involved [7,20,21,23,100-103] (cf. Figure 2.58). The scatter in Figure 2.46 is not unexpected considering the simulated failure tests (Table 2.4), which indicate that there is considerab le scatter about the expected curves for n < 50, and that there is likely more than one flaw population controlling carbon fiber tensile failure. Indeed, the Weibull plot for 20 mm data appears to exhibit a distinct knee, suggesting that two populations are controlli ng the failure process [7]. Likewise, the Weibull plot for the 50 mm failure tests indicates multiple flaw populations. In this case, concurrent subpopulations are indicated b ecause of the concave-up Weibull plot. PAGE 88 Concurrent distribution The physical evidence (SEM, TEM and x -ray [30-49]) previously discussed indicates that there are at leas t two distinct flaw populations th at incite tensile failure in carbon fibers. The four-parameter concurrent distribution, Eqn. 2.2, seems a likely model to represent these two populations. Proving th at there are multiple flaw populations in carbon fibers by inspecting failure statistics is possible. However, effective specimen volume and loading conditions must be carefully considered as failure to do so can lead to erroneous statistical information. In this regard, it can be shown that for an ideal two-parameter Weibull CDF the predicted ultimate strength sb and the standard deviation in crease as the volume/sample space decreases [3] (Figures 2.60 and 2.61). For fiber Sets A and B the Weibull modulus increases (equivalent to a decrease in the st andard deviation) as the volume decreases contrary to expected behavior for a singl e population distribution. The decrease in standard deviation with decreasing specimen volume is a compelling statistical argument for the existence of multiple flaw populat ions in these PAN-based carbon fibers. In addition, Johnson [7] has shown that when there are large differences in effective volumes (e.g., 105x) it is possible to censor nearly an entire subpopulation of a concurrent distribution. Simulated data for concurrent distributions (Figure 2. 62) exhibit the strong self-censoring associated with effective volum e extremes and the more subtle censoring accompanying moderate effective volume differ ences. It is evident from Figure 2.62 and Table 2.16 that failures at high stresses in all cases asymptotically approach the high modulus asymptote and, in a ddition, are predominantly hi gh modulus type failures. Similarly, for L / L0 > ~1 the failure strengths fall a bout the low modulus asymptote and are primarily low modulus type failures. PAGE 89 The behavior for scali ng to lengths less than L0 is not as simple. For small volumes, both extremes of the failure di stribution are dominated by the high modulus population and naturally fall a bout the high modulus asympt ote. The total population quickly becomes dominated by high modulus type failures (Figure 2.62 and Table 2.16) as the scaling ratio drops below one. It s hould be noted that the severity of population censoring depends on both the absolute magnit udes of the moduli a nd their disparity. The difference in the scale factor also c ontributes to population censoring especially when they are widely disparate (little or no overlap of discrete populations). The curves in Figure 2.62 further indicate that negative ki nks (upward, cf. Figure 2.12) in concurrent distributions are possible despit e Johnson's [7] contention to the contrary. Finally, it is apparent that careful consideration is warrant ed when dealing with failure strength data that likely arises from multiple concurrent distributions. For these carbon fibers, the use of a multip le population distribution is evidently supported not only by strong physical/morphologi cal evidence but also by empirical evidence of a decreasing Weibull modulus with decreasing specimen volume and the inherent censoring effects for concurrent distributions dem onstrated in Figure 2.62. Additionally, a concurrent distribution is the next simple st model following the generally unstable three-parameter distribu tion. An exclusive distributi on will be considered later. The nonlinear fitted concurrent distribution parameters (Eqn. 2.2) for sample Sets AD are given in Table 2.17. The CDF predic tions for sample Sets A and B are shown in Figures 2.63 and 2.64. The corresponding Weib ull probability plots are given in Figures 2.65 and 2.66. The concurrent CDF traces a ppear very similar to the two-parameter CDFs in Figures 2.54 and 2.55. Indeed, de spite possessing both physical justification PAGE 90 (i.e. surface and volume flaws) and two additio nal fitting parameters the four-parameter concurrent model shows little improvement over the simpler two-parameter Weibull model (Eqn. 2.1 and GOF values in Tables 2.15.17). For Set A fibers two realistic distinct populations can not be identified by the nonlinear estimating procedure. The second modulus and scale factor, 0.16 and 9.1x1024 Pa (a physically unreasonable value) resp ectively, do not contri bute to the cumulative failure probability in the range considered This is most evident in the Weibull probability plot (Figure 2.65) where the low modulus population (represented by the dashed line) is clearly not contributing to the failure pr obability. The two and fourparameter predictions for Set A fibers are thus virtually indist inguishable. No new information has been obtained in this instance concerning the flaw populations, concurrent or otherwise, in carbon fibers by using the more complex and supposedly more appropriate conc urrent distribution. For Set B fibers, however, the concurrent distribution exhibits a slightly higher GOF than the previously considered Weibu ll two-parameter distribution, 0.985 and 0.977 respectively. The asymptotic approach of Set B failure strengths to high and low modulus asymptotes is evident in Figure 2. 66, a strong indicator for the presence of two distinct simultaneously o ccurring flaw populations. In this case, the fit improvement seen for a concurrent over a two-parameter distribution might be attributed to the larger volume being te sted. Larger volumes (larger sample space) will naturally bias strength distributions to lower modulus populations. More accurate failure prediction is therefor e possible with a concurrent distribution, PAGE 91 because it allows for a distin ction between two simultaneously occurring and contributing populations provided the sample space (specimen volume) is sufficient. The concurrent PDF traces for Set A and B fibers are given in Figure 2.67. These traces have essentially the same breadth at half maximum intensity, HWHM ~ 0.55x109 Pa. This is not unexpected for a two populat ion concurrent distri bution provided the two specimen volumes are of the same order. The low-strength tail and negative skew for the 50 mm long fibers are more pronounced than fo r the shorter 20 mm long fibers. For Set B fibers the skew and low-stre ngth tail are the resu lt of the low modulus contribution that is finite, albeit small, over a considerable range. The low modulus contribution for the 20 mm failure tests is non-existent in the ra nge considered here (Figures 2.65.66). The statistics for Set B fibers establish th at two distinct conc urrent subpopulations exist in these PAN-based carbon fibers. One population has a high modulus and moderate scale parameter (7.1, 2.9x109 Pa), the other population, a low modulus and high scale parameter (0.85, 2.7x1010 Pa). Scaling the 50 mm predictions to shorter ga ge lengths is accomplished by adjusting the scaling ratio, L / L0, in Eqn. 2.2. The best fit 20 mm prediction scaled from 50 mm fitted data is given in Figure 2.68. The prediction is conservative, but more importantly it suggests that greater than 90% of fibers tested at 20 mm will be asymptotically represented by the high modulus population. In other words, the cumulative failure distribution for a 20 mm gage length should be well repres ented by a single population failure model, a result confirmed by experiment here (cf. Figure 2.54). In light of the scaled predictions prov ided by the 50 mm failure statistics, the unreasonable four-parameter concurrent para meters obtained for Set A fibers must PAGE 92 therefore be an indication that the effective specimen volume has dropped below a critical value that would enable multiple population distinction. The predicted twoparameter Weibull PDFs for Set A and B fi bers (Figure 2.47) also have dissimilar breadths, which is contrary to single failure population behavior, instead indicating multiple flaw populations. As the scaling ratio L / L0 decreases below 0.4 (20 mm prediction) the resulting distribution will become increasingly biased toward the appearance of a single population distribution. The scaled pr ojection for 5 mm long fiber failu re strengths is given in Figure 2.68. Over 97% of the predicted failure s for 5 mm long fibers ar e expected to fall about the high modulus asymptote. Asse ssing the failure strength population in PANbased carbon fibers simply by failing 5 mm or even 20 mm long fibers alone would give an incomplete and incorrect pi cture of the true failure dist ribution. The lower modulus population would likely be overlooked altogether. In a like manner, scaling to lengths greater than 50 mm, L / L0 > 1, biases the prediction toward the low modul us asymptote. The scaled prediction for 100 mm long carbon fibers is shown in Figure 2.68. The bias toward the low modulus population is evident and it is apparent that at lengths not much greater than 50 mm the high modulus population is quickly becoming obscured. Once again, just as for shorter lengths, there exists a critical length (range) beyond whic h an entire population will be essentially overlooked because of the gage length. Clearly, care must be taken when choosing th e gage length. This is similar to our earlier discussion concerning the appropriate sample size and emphasizes the need for testing a range of fiber lengths, specimen vol umes or loading conditi ons to ensure that PAGE 93 multiple flaw populations, if they exist, will be identified. In fact by testing a range of fiber lengths it should be possibl e to determine, within statis tical certainty, the existence or nonexistence of multiple flaw populations. This excludes the case for distinct multiple populations with similar scale parameters a nd Weibull moduli, which by all statistical measures appear to be ju st one single population. We conclude that two distinct flaw populat ions control the tens ile failure strength distribution in these carbon fibers. That is, the GOF parameters found for a twoparameter distribution (Table 2.15) and GOF parameters fo r a concurrent distribution (Table 2.17) coupled with evidence from x-ray scattering, SEM, TEM and STM measurements indicate hat there are two distin ct flaw types in HS and HM fibers. The surface and volume flaw populations arise owing to the irregular graphite basal plane and amorphous carbon morphology generated primarily during the high temperature stress-graphitization process. Earlier it was suggested that surface flaw s would generally be more severe than volume flaws owing to the increased like lihood for satisfying the Reynolds-Sharp criteria. Now while we can confidently id entify two separate flaw populations in the carbon fibers tested, fiber Sets A and B, simply by close inspection of the failure statistics it is another matter still to associate the population paramete rs (Table 2.17) with their respective physical origins. SEM inspection could be used to associate the fitted parameters to their respective physical origins. The small fiber diameter s and the nature of the tensile failure (catastrophic and dynamic) make this process difficult [29,67]. This inspection approach was not attempted here. Instead, statistical information obtained from failure tests PAGE 94 conducted on RF plasma treated fibers, Sets C and D, will provide the necessary supporting evidence to confidently associat e the population parame ters with their respective physical origins. The effects of oxygen plasma and othe r oxidative treatments on carbon fiber surfaces have been previously discussed. Recall that the net effect of oxidative treatments is an incr ease in surface roughness characterized by transverse pits or slits that bound misaligned graphite platelets. Noting that an oxidative treatment will etch higher energy carbon sites (e.g.. amorphous carbon, plat elet edges, crystallite imperfections) and recognizing that for an untreated fiber thes e sites represent exis ting surface flaws or imperfections, we conclude that the surf ace flaw populations for plasma treated and untreated carbon fibers will begin with the same broad character (see additional discussion below and Chapter 3 regarding e ffects of surface treatments on carbon fibers). In addition, considering the vagaries of fiber handling and treatment procedures it is likely that the Weibull modulus for incipi ent surface flaw failures is less than the relatively large Weibull modulus for internal flaws. That is, preferred crystallite orientation in an otherwise complex and ir regular morphology predominates the incipient volume flaws population, which thereby pr oduces relatively tight groupings for macromechanical properties such as tensile strength and modulus. This is strongly evidenced by the relationship between pr eferred alignment (as measured by x-ray scattering) and fiber strength (Table 2.5) and the marked improvement in fiber strength seen with clean precursors and clean room spinning [36,62,63]. At first glance, the two-parameter Weibull distribution fits for Set C and D fibers (Table 2.15 and Figure 2.69) provide apparently contradictory informa tion as Set C fibers PAGE 95 have a low Weibull modulus ( = 3.25 : MLE) and Set D fibers a high modulus ( = 9.64 : MLE). Excepting the very remote possibility that the fiber samples chosen for Set C and D were somehow statistically different prior to plasma treatment, the resulting Weibull modulus disparity must therefore be related to RF plasma treatment severity. The gentler treatment (Set D) has apparent ly only lightly etched the carbon fiber surface lessening the severity of gross mechanical surface flaws. This allows the less severe internal flaws to dominate the failure population. However, the more aggressive treatment for Set C fibers has severely weakened the carbon fibers by producing transverse pits on the surface at the edges of misoriented graphite platelets [82]. Considering the effects of these two treatments on failure strength and our previous conclusion, morphological and st atistical, that there are two distinct flaw populations we conclude that the surface fl aw population in these carbon fibe rs is characterized by the smaller Weibull modulus and the internal fl aw population is desc ribed by the larger Weibull modulus (see Table 2.17, Set B f it parameters). These discrete failure populations are indicated in Figure 2.66 by the low and high modulus asymptotes. The individual CDFs and PDFs for the two discre te failure strength populations are given in Figures 2.70 and 2.71 respectively. From Figure 2.70a and Figure 2.70b it is obvi ous that the initial rise in failure probability for the low modulus distribution ( = 0.85) is much more rapid than for the high modulus distribution ( = 7.14). This further illustrates the low modulus asymptotic behavior for weak fibers. In addition, th e dominant high modulus asymptotic behavior for strong fibers is evident in Figure 2. 70a. Essentially 100% of the high modulus population failures are expected well before 20% of the low modulus population failures PAGE 96 are predicted. Finally, the PDF traces in Figure 2.71 s how that the two-parameter Weibull density function ha s exponential character when < 1. This does not however impact our argument for discrete failure populations. Exclusive and partially-concurrent distributions As noted above it is ofte n difficult to distinguish between an exclusive and partially-concurrent distribution (Figure 2.10). In order to ap ply an exclusive or partiallyconcurrent distribution the in cipient flaw type must first be identified and then a determination as to the exis tence or nonexisten ce of a second flaw type in the same specimen must be performed. Clearly, for car bon fibers this is an impossible task and even for simple homogeneous materials repres ents a very difficult undertaking. In addition, since there are now five fitting pa rameters nonlinear estimation becomes more sensitive to initial guesses. Finally, as the number of parameters increases each parameters individual cont ribution to the overall dist ribution decreases and thus confidence in the parameters suffers. Table 2.18 gives separate nonlinear estimates for exclusive and partially-concurrent distributions for Set A a nd B fibers. The factor b has been arbitrarily set at 0.7 for the exclusive distribution an d 0.85 for the partially-concurrent case. It is clear that large changes in fitting parameters have only a small effect on the GOF. Despite the high GOF values for the exclusive distribution, no c onfidence should be placed in the parameters listed in Table 2.18, as the pr ovisions for exclusive and partia lly-concurrent distributions do not hold for graphite fibers. PAGE 97 Carbon Fiber Morphology-Property Relationship The historical development and basic prin ciples of Weibull-Griffith weakest-link theory were discussed and in particular thei r relationships to flaw populations and failure mechanisms in High Modulus and High Stre ngth carbon fibers. St rong and substantial evidence garnered from previous investigators SEM, TEM, and small angle x-ray scattering work was used to support the conclu sion that crystallite orientation is nearly normally distributed about the fiber axis and the breadth of this distribution is intimately related to the physical prope rties of the carbon fiber. Failure simulations using a normal orient ation distribution, the Reynolds-Sharp failure criteria (Eqn. 2.11), and a Griffith fracture model with a va riable Young's modulus indicated that the mean graphitic crystallite is incapable of initiati ng catastrophic failure, unless it is involved in a cooperative pro cess with neighboring crystallites. The simulations also showed that the R-S failure conditions are very infrequently satisfied, especially considering the hi gh crystallinity (> 95% Type I, 95% Type II [41], bulk carbon 90% at 3000C [47]) and crystallite de nsity in carbon fibers. The simulated failure stre ngth distributions were d ecidedly non-Weibull and strongly biased toward low strengths especia lly as the angular width at half-maximum Z increased beyond 30. This low-strength bias contra dicts empirical be havior thereby providing theoretical evidence that misoriente d crystallites are the principal incipient volume flaw. Their preferred orientation, which is brought about mainly by stress graphitization at high temperatures (> 2000C, as evidenced by the discontinuity in lp, Porod's distance of heterogeneity [41]), is the fundamental reason that carbon fibers exhibit such excellent ax ial physical properties. PAGE 98 The Reynolds-Sharp criteria, especially c ondition (1) which requi res that there be sufficient continuity among neighboring crys tallites for crack propagation, is not everywhere satisfied in the complex car bon fiber morphology. Despite layer plane intertwining and folding between crystallites the mean crystallite thickness Lc (70 x 70 x 60 : La x La x Lc [40]) is still much smaller than the critical flaw size (measured normal to the loading axis) for tensile failure (500 ). Condition (1) is therefore only infrequently satisfied. Finally, crystallite thickness decreases as basal plane misorientation to the fiber axis increases [29], which will tend to increase the number of high-strength incipient flaws and lessen the number of low-strength flaws. This trend also explains the drop in failure strength s een with excessive crys tallinity, especially random three-dimensional ordering at the co st of preferred two-dimensional axial alignment. Batdorf and Crose's failure theory [57], or iginally developed fo r isotropic materials with randomly oriented and located cracks, was modified to consider the preferred graphite crystallite orientat ion (approximately Normal, = 0, cf. Figure 2.20) and expected Reynolds-Sharp failure mechanism for carbon fibers. For the simple uniaxial loading case = (,0,0), the solid angle containing th e basal plane orientations (Figure 2.32) for which transverse crystalline rupt ure is expected is bounded by the angles min and max, which are functions of and the R-S failure criteria. Likewise, the CDF for off-axis loading can be simulated using nor mally distributed basal plane orientations about a mean = 30 and axial loading = (,0,0). The predicted off-axis loading strength distribution is much tighter and pr ecedes the case for axial loading owing to the PAGE 99 increased possibility for larger basal plan e misorientations relative to the loading direction. In addition, there is an increasing chance fo r simple Griffith type fracture initiation at the irregular polyhedral shaped pores as the off-axis loading component increases. Basal plane continuity along the fiber axis reinforces this competing failure mode relative to the R-S failure. The combined failure proba bility for polyaxial lo ading is given by the product of the two failure modes ) ( F ) ( F = ) , ( FGr s BC s 3 2 1 s and is qualitatively consistent with empirical evidence. Morphological evidence and fa ilure strength statistics strongly supports the notion that both volume and surface flaws are randomly distributed in carbon fibers, a necessary condition for W-G weakest-link application. The simultaneous random distribution of these flaw types and their intensity precludes th e use of either an exclusive or a partiallyconcurrent distribution to model the carbon fiber failure population which leaves the four-parameter concurrent distribution as the most probable failure distribution. Numerous investigators have used Weibull statistics in failure strength and lifetime studies including carbon, gla ss, and ceramic fiber strengths [5-9,11-13,15,20-24]. However, there has been little consideration as to the validity or accuracy of the derived Weibull estimators. Fortunately, the Weibull CDF is one of three general extreme value distributions and is thus partic ularly well suited for strength and lifetime statistics. That is, the extreme value is the weakest link in a population. For Type I and II carbon fibers, Weibull statistics provide not only a convenient mathematical representation for the failure di stribution, they also suggest the underlying PAGE 100 nature of the failure process/population. Gi ven the high density of discrete volume flaws (specifically needle-lik e pores) and their very small extent (10 x 200 [40,41]) it is apparent that these pore flaws are not large enough to induce te nsile failure on their own. This is the essence of Reynolds-Sharp criter ia [28] for carbon fiber tensile failure. In fact, the statistics quoted here and in other articles on Type I and II carbon fibers describe an ensemble of flaws cooperating to form a macro-flaw population. The strength distribution arising from this macro-flaw population may be described by a twopopulation Weibull concu rrent distribution. Conclusions A broad low-strength surface flaw populati on and relatively narrow higher strength volume flaw population for tensile failure have been identified in Type II carbon fibers by testing at two different gage lengths (20 and 50 mm) and also by considering the effects of radio frequency plasma surface treatments. Despite the complex morphology in HS and HM carbon fibers, the failure distribution remains quite narrow. This is primarily a result of strong preferential graphite basal plane orientations about the fi ber axis (a direct a nd extremely beneficial consequence of stress graphitization above 2000 C). Considering the extensive evidence provided by early researchers on the relationship between por e/crystallite alignment and carbon fiber physical properties it can be concluded that the failure mechanism envisioned by Sharp and Burnay (ref. 5 of [29]) and formalized by Reynolds and Sharp [28] is consistent with morphological and failure strength statistical evidence. Note that Griffith fracture criteria requires a critical flaw size; and that, in the case of HS and HM carbon fibers, it is evident that R-S condition (1) is not satisfied at every misoriented PAGE 101 crystallite (compare R-S simulations, Figures 2.27.28 and actual carbon fiber failure plots Figure 2.46). The narrow distribution for the pore/crystallite orientation naturally gives rise to a tight failure strength dist ribution for uniaxial tensile loading, as the relative misorientation of the crystallite to the fibe r axis determines the crystallite's strength (Figure 2.18). This failure mechanism app lies to both volume and surface links although surface flaws tend to be more se vere (free surface effect) a nd their distribution is much broader. This was evidenced by the broad failure strength dist ribution seen for RF oxygen plasma over-treated fibers (Figure 2.69). A simple LLSQ approach, a MLE, and a non linear fitting procedure were used to estimate the two-parameter distributions seen in Table 2.15. There was considerable difference in the Weibull moduli predicted by the various methods while the scale factors were essentially equal. The LLSQ approach gave the shallowest CDFs while the more aggressive MLE method returned larger moduli. For the two-parameter Weibull distribution, Thoman [91] found that MLE provides an unbiased estimate to and s0 for n 30. In contrast, LLSQ estimation incorrectly assumes equal weighting for the data and consequently will under predict the populati on Weibull modulus. MLE is therefore a more efficient estimating procedure than th e simpler LLSQ approach and is preferred over a nonlinear approach as well. For hi gher order Weibull distributions MLE is intractable and LLSQ inapplicable leavi ng only the nonlinear estimating procedure. Scaling predictions that used the MLE determined two-parameter Weibull distribution parameters (Figure 2.58) unders cored the primary rest riction for scaling Weibull distributions. That is, the Weibull modulus is assumed constant over the PAGE 102 scaling range. For the fiber gage lengths tested here, 20 and 50 mm, the moduli are not the same, 6.0 and 3.3 respectively, a strong in dication that multiple failure processes are operating in these fibers and th at scaling these Weibull distri butions to longer or shorter lengths will lead to faulted predictions. Typically, for a single population fail ure process as the number of fitting parameters increases, the GOF increases as well, although confidence in the individual parameters is lessened. The case for multiple population failure processes is more complicated as one or more of the populati ons may be completely, or partially obscured depending on the volume/sample size tested This obfuscation was evidenced by fiber Sets A and B failure strength distributions and the concurrent distribution failure simulations. For Set A fibers (20 mm gage length) tw o distinct physically realistic failure populations could not be iden tified. The narrow high-stre ngth population dominates the failure process and obscures any contribution from the broader low-strength surface flaw population. Two failure populat ions are evident in the Set B (50 mm) fiber failure distribution just as one woul d expect from morphological evidence. The limiting sample space for multiple population detection apparently lies between a 20 and 50 mm gage length. Scaling the 50 mm concurrent distributi on to 20 mm indicates that greater than 90% of the expected failures will be asym ptotically represented by the high modulus population. This prediction is consistent with Set A fiber failure distribution. Furthermore, as the scaling ratio, L / L0, decreases the failure st rength distribution will become increasingly biased toward the hi gh modulus population. Conversely, as the PAGE 103 concurrent distributi on is scaled upwards, L / L0 > 1, the low modulus population will tend to dominate the failure process. The simple conclusion here is that testing a single gage length can lead to incomplete and faulty failu re statistics. Multiple gage lengths should therefore be failed to ensure that al l flaw populations will be identified. Recommendations If we are concerned with carbon fiber fr acture in an ideal microcomposite (e.g., single fiber composite) it is important to understand the basic fracture mechanics of brittle fibers and the influence of the co mposite processing and environment on carbon fiber fracture. Inspection of the carbon fibe r fracture surface woul d increase confidence in the assignment of multiple flaw (sur face and internal) populations. Additionally, information regarding carbon fiber in-situ (e.g., carbon fiber-epoxy composite) strength could conceivably be acquired by fragmenting the fiber in a single fiber composite, which allows for testing at much shorter gage lengt hs; however, improvements in the analytical interpretation of single fiber fragmentati on tests are necessary. Specifically, details regarding the matrix-fiber lo ad absorption and the observed carbon fiber raman frequency shift with fiber strain. PAGE 104 Table 2.1 Exclusive and partially-concurrent distribution parameters for Cases 1 through 6 (Figures 2.8 and 2.9). Exclusive Distribution Partia lly-concurrent Distribution Case No. s0 A, A, s0 A, B, L, L0, A, B s0 A, A, s0 A, B, L, L0, B 1 2e9, 3, 3e9, 4, 20, 20, 0.5, 0.5 2e9, 3, 3e9, 4, 20, 20, 0.5 2 2e9, 3, 3e9, 4, 20, 20, 0.0, 1.0 2e9, 3, 3e9, 4, 20, 20, 0.0 3 2e9, 3, 3e9, 4, 100, 20, 0.5, 0.5 2e9, 3, 3e9, 100, 20, 0.5 4 2e9, 3, 3e9, 4, 4, 20, 0.5, 0.5 2e9, 3, 3e9, 4, 4, 20, 0.5 5 2e9, 3, 3e9, 1, 20, 20, 0.5, 0.5 2e9, 3, 3e9, 1, 20, 20, 0.5 6 2e9, 3, 3e9, 4, 20, 20, 0.75, 0.25 2e9, 3, 4e9, 4, 20, 20, 0.5 Table 2.2 Exclusive and partially-concurrent distribution parameters for Cases 1 and 2 (Figure 2.10). Exclusive Distribution Partia lly-concurrent Distribution Case No. s0 A, A, s0 A, B, L, L0, A, B s0 A, A, s0 A, B, L, L0, B 1 2.3e9, 3, 3.1e9, 3.7, 20, 20, 0.3, 0.7 3e9, 3, 4e9, 4, 20, 20, 0.7 2 2.3e9, 3.3, 1.5e9, 3.7, 20, 20, 0.4, 0.6 1.8e9, 3.25, 3.8e9, 5, 20, 20, 0.5 Table 2.3 Exclusive and partially-concurrent parameters for the Weibull probability plots (Figures 2.13 and 2.14). Exclusive Distribution Partia lly-concurrent Distribution Case No. s0 A, A, s0 A, B, L, L0, A, B s0 A, A, s0 A, B, L, L0, B 1 2e9, 3, 3e9, 4, 20, 20, 0, 1 2e9, 3, 3e9, 4, 20, 20, 0.5 2 1.5e9, 3, 3e9, 1, 20, 20, 0.5, 0.5 1.5e9, 3, 3e9, 1, 20, 20, 0.5 3 2e9, 3, 3e9, 4, 20, 20, 0.5, 0.5 1.5e9, 3, 3e9, 20, 1000, 0.5 4 .5e9, 4, 2e9, 4, 20, 20, 0.5, 0.5 1.5e9, 1,13e9, 3, 20, 20, 0.5 Random 2e9, 3, 3e9, 4, 20, 20, 0.25, 0.75 2e9, 3, 3e9, 1, 20, 20, 0.5 PAGE 105 Table 2.4 Statistical measures for a simu lated two-parameter Weibull distribution ( =3.0, s0=3.0x109 Pa.). Weibull Modulus LLSQ Correlation Coefficient, R No. of Simulations Mean Std. Dev.CV Mean Std. Dev. CV 10 3.302 1.749 52.95 0.891 0.084 9.41 25 2.754 0.437 15.87 0.964 0.022 2.24 50 2.788 0.406 14.55 0.964 0.016 1.64 100 3.019 0.295 9.77 0.982 0.011 1.07 500 2.989 0.070 2.34 0.998 0.001 0.11 Measures are for 10 separate trials. PAGE 106 Table 2.5 PAN-based carbon fibe r physical properties as a function of heat treatment temperature (HTT). Young's Ultimate Angular HTT Modulus Strength Width, Z Refs ( C) (GPa.) (GPa.) (degrees) 350 28 [37] 600 26 [37] 1000 18.3 [33] 30/14 (1) [37] 1200 221 1.6 [51] 1400 234 2.8 [51] 1500 241 3.0 [51] 12 [37] 1600 255 3.1 [51] 1800 276 2.1 [51] 2000 310 2.6 [51] 345/359 (2) [52] 7-8 (3) [37] 2200 345 2.4 [51] 2500 414 2.2 [51] 400/490 (2) [52] 8 [37] 2600 414 1.7 [32] 6.9 [33] 2750 414/531 (2) [52] 3000 434/558 (2) [52] 3900 (4) 700 19.6 0 [38] (1) Amorphous/crystalline graphite. (2) Inert atmosphere in the presence of Boron. (3) From 2000-3000C very little change seen in Z. (4) DC arc spray technique produces nearly perfectly aligned graphite whisker rolls. End clamping effect s (ignored) lower absolute values. PAGE 107 Table 2.6 Statistics for extremes of normal distributions with =0 and standard deviations, PAGE 108 Table 2.7 Set A carbon fiber diameter s determined by Fraunhoffer diffraction. Fiber Diameter % Diff. No. (meters) two meas. 1 7.73E-06 3.18 2 7.56E-06 0.53 3 6.96E-06 0.40 4 7.08E-06 4.69 5 7.47E-06 1.80 6 7.45E-06 0.58 7 7.30E-06 1.72 8 7.20E-06 1.39 9 6.98E-06 0.37 10 7.36E-06 1.81 11 7.68E-06 0.11 12 7.67E-06 0.84 13 7.34E-06 1.64 14 7.32E-06 1.74 15 7.05E-06 0.55 16 6.47E-06 2.96 17 7.28E-06 1.46 18 7.30E-06 3.29 19 6.64E-06 1.37 20 6.95E-06 1.26 21 7.18E-06 0.77 22 7.92E-06 8.49 23 7.57E-06 0.40 24 7.56E-06 1.54 25 7.13E-06 1.04 26 7.47E-06 0.07 27 6.99E-06 1.45 28 7.78E-06 1.30 29 7.11E-06 1.79 30 6.46E-06 0.99 31 6.86E-06 2.44 mean 7.25E-06 1.68 std. dev. 3.58E-07 1.60 CV 4.92 95.5 Diameters are average of two readings. PAGE 109 Table 2.8 Set B carbon fiber diameter s determined by Fraunhoffer diffraction. Fiber No. Diameter % Diff. (meters) (two meas.) 1 7.80E-06 0.79 2 7.56E-06 1.56 3 7.59E-06 0.79 4 7.16E-06 2.57 5 7.85E-06 0.39 6 7.82E-06 1.59 7 7.37E-06 3.76 8 8.09E-06 3.70 9 7.48E-06 0.76 10 7.13E-06 3.99 11 7.31E-06 0.75 12 7.28E-06 0.00 13 7.25E-06 0.77 14 7.67E-06 1.16 15 7.26E-06 0.01 16 7.55E-06 0.38 17 7.15E-06 0.00 18 7.25E-06 0.00 19 7.19E-06 0.37 20 7.51E-06 0.39 21 7.20E-06 0.74 22 7.62E-06 0.00 23 7.81E-06 1.58 24 7.90E-06 1.61 25 7.68E-06 2.38 26 7.48E-06 3.04 27 8.23E-06 0.00 28 8.11E-06 0.41 29 7.16E-06 1.08 30 8.04E-06 0.40 31 7.25E-06 1.86 32 7.22E-06 0.73 33 6.92E-06 0.64 mean 7.51E-06 1.16 std. dev. 3.31E-07 1.13 CV 4.41 97.9 Diameters are average of two readings. PAGE 110 Table 2.9 Set C carbon fiber diameter s determined by Fraunhoffer diffraction. Fiber No. Diameter % Diff. (meters) (two meas.) 1 6.37E-06 2.66 2 6.95E-06 3.70 3 7.13E-06 1.26 4 6.93E-06 1.64 5 7.12E-06 2.98 6 7.02E-06 0.85 7 7.15E-06 1.27 8 6.73E-06 3.17 9 6.98E-06 0.41 10 6.94E-06 0.81 11 6.84E-06 1.22 12 7.31E-06 1.29 13 6.75E-06 0.41 14 6.68E-06 0.40 mean 6.92E-06 1.58 std. dev. 2.29E-07 1.06 CV 3.30 67.4 Diameters are average of two readings. PAGE 111 Table 2.10 Set D carbon fiber diameter s determined by Fraunhoffer diffraction. Fiber No. Diameter % Diff. (meters) (two meas.) 1 6.84E-06 2.84 2 6.94E-06 0.81 3 6.76E-06 0.41 4 6.78E-06 0.43 5 6.90E-06 0.82 6 7.33E-06 0.87 7 6.86E-06 2.85 8 6.75E-06 2.02 9 6.64E-06 0.00 10 6.62E-06 0.39 11 6.91E-06 0.01 12 6.69E-06 1.58 13 6.90E-06 1.63 14 6.85E-06 1.65 mean 6.84E-06 1.17 std. dev. 1.67E-07 0.92 CV 2.43 78.6 Diameters are average of two readings. PAGE 112 Table 2.11 Set A carbon fiber (20 mm long, untreated) load-deflecti on and stress-strain results. Fiber Load Deflection Diameter Stess Strain Load Stress Stiffness Modulus No. (gm) (mm) (meters) (Pa. ) Ranking Ranking (kg/m) (Pa.) 1 17.35 0.467 7.73E-06 3.62E+0 9 0.0233 27 24 37.18 155.3 2 12.29 0.331 7.56E-06 2.68E+0 9 0.0166 8 4 37.09 161.8 3 11.37 0.378 6.96E-06 2.92E+0 9 0.0189 6 6 30.08 154.8 4 7.92 0.239 7.08E-06 1.97E+0 9 0.0119 1 1 33.20 165.2 5 17.66 0.660 7.47E-06 3.95E+0 9 0.0330 29 28 26.76 119.8 6 8.83 0.288 7.45E-06 1.99E+0 9 0.0144 3 2 30.67 138.1 7 14.51 0.395 7.30E-06 3.40E+0 9 0.0198 20 21 36.73 172.2 8 16.63 0.501 7.20E-06 4.00E+0 9 0.0251 26 29 33.20 159.7 9 13.85 0.538 6.98E-06 3.55E+0 9 0.0269 16 23 25.76 131.9 10 13.48 0.397 7.36E-06 3.10E+0 9 0.0199 14 13 33.95 156.3 11 15.05 0.435 7.68E-06 3.18E+0 9 0.0218 22 17 34.56 146.0 12 16.06 0.507 7.67E-06 3.41E+0 9 0.0254 25 22 31.65 134.4 13 18.20 0.590 7.34E-06 4.22E+0 9 0.0295 31 30 30.83 143.0 14 13.30 0.510 7.32E-06 3.10E+0 9 0.0255 13 12 26.10 121.7 15 17.46 0.635 7.05E-06 4.39E+0 9 0.0318 28 31 27.48 138.1 16 12.54 0.659 6.47E-06 3.74E+0 9 0.0330 10 26 19.02 113.3 17 12.86 0.355 7.28E-06 3.02E+0 9 0.0178 12 8 36.23 170.4 18 14.00 0.404 7.30E-06 3.27E+0 9 0.0202 17 18 34.66 162.1 19 10.85 0.415 6.64E-06 3.07E+0 9 0.0208 5 10 26.13 147.8 20 14.07 0.551 6.95E-06 3.64E+0 9 0.0276 18 25 25.53 132.1 21 12.58 0.394 7.18E-06 3.04E+0 9 0.0197 11 9 31.92 154.5 22 15.95 0.592 7.92E-06 3.17E+0 9 0.0296 24 15 26.92 107.1 23 17.80 0.643 7.57E-06 3.88E+0 9 0.0322 30 27 27.67 120.5 24 14.36 0.448 7.56E-06 3.13E+0 9 0.0224 19 14 32.05 139.9 25 13.74 0.452 7.13E-06 3.37E+0 9 0.0226 15 20 30.39 149.2 26 14.68 0.438 7.47E-06 3.28E+0 9 0.0219 21 19 33.54 149.9 27 11.80 0.498 6.99E-06 3.01E+0 9 0.0249 7 7 23.67 120.8 28 15.41 0.413 7.78E-06 3.17E+0 9 0.0207 23 16 37.27 153.6 29 12.51 0.611 7.11E-06 3.09E+0 9 0.0306 9 11 20.48 101.2 30 9.48 0.411 6.46E-06 2.84E+0 9 0.0206 4 5 23.06 138.0 31 8.38 0.350 6.86E-06 2.22E+0 9 0.0175 2 3 23.95 126.9 mean 13.71 0.468 7.25E-06 3. 24E+09 0.0234 29.93 141.5 std. dev. 2.75 0.109 3.58E-07 5.55E+08 0.0055 5.01 18.3 CV 20.0 23.4 4.9 17.1 23.4 16.7 13.0 PAGE 113 Table 2.12 Set B carbon fiber (50 mm long, untreated) load-deflecti on and stress-strain results. Fiber Load Deflection Diameter Stess Strain Load Stress Stiffness Modulus No. (gm) (mm) (meters) (Pa.) Ranking Ranking (kg/m) (Pa.) 1 15.61 0.760 7.80E-06 3.20E+09 0.0152 32 29 20.54 210.6 2 12.38 0.724 7.56E-06 2.70E+09 0.0145 18 19 17.10 186.5 3 10.95 0.577 7.59E-06 2.37E+09 0.0115 12 9 18.99 205.6 4 10.53 0.641 7.16E-06 2.56E+09 0.0128 9 15 16.43 199.9 5 14.24 0.824 7.85E-06 2.89E+09 0.0165 26 25 17.28 175.1 6 12.34 0.760 7.82E-06 2.52E+09 0.0152 17 13 16.24 165.7 7 12.48 0.778 7.37E-06 2.86E+09 0.0156 19 24 16.05 184.2 8 8.19 0.418 8.09E-06 1.56E+09 0.0084 7 3 19.58 186.9 9 16.14 0.937 7.48E-06 3.60E+09 0.0187 33 33 17.23 192.3 10 11.91 0.830 7.13E-06 2.92E+09 0.0166 15 26 14.35 176.0 11 10.71 0.698 7.31E-06 2.50E+09 0.0140 10 12 15.35 179.0 12 12.08 0.835 7.28E-06 2.85E+09 0.0167 16 22 14.46 170.4 13 7.17 0.467 7.25E-06 1.70E+09 0.0093 3 4 15.36 182.5 14 14.67 0.881 7.67E-06 3.11E+09 0.0176 28 27 16.66 176.8 15 15.12 1.040 7.26E-06 3.58E+09 0.0208 30 31 14.54 172.2 16 12.55 0.738 7.55E-06 2.75E+09 0.0148 20 21 17.02 186.2 17 7.91 0.573 7.15E-06 1.93E+09 0.0115 6 7 13.80 168.3 18 9.77 0.714 7.25E-06 2.32E+09 0.0143 8 8 13.69 162.6 19 11.13 0.728 7.19E-06 2.69E+09 0.0146 13 18 15.29 184.5 20 12.91 0.869 7.51E-06 2.85E+09 0.0174 21 23 14.85 164.1 21 7.74 0.458 7.20E-06 1.86E+09 0.0092 5 6 16.89 203.3 22 15.25 0.798 7.62E-06 3.28E+09 0.0160 31 30 19.10 205.2 23 11.86 0.677 7.81E-06 2.43E+09 0.0135 14 11 17.52 179.3 24 13.23 0.816 7.90E-06 2.65E+09 0.0163 23 16 16.21 162.3 25 3.50 0.252 7.68E-06 7.42E+08 0.0050 1 1 13.92 147.4 26 10.78 0.683 7.48E-06 2.41E+09 0.0137 11 10 15.77 176.0 27 14.50 0.852 8.23E-06 2.67E+09 0.0170 27 17 17.02 156.8 28 13.34 0.733 8.11E-06 2.53E+09 0.0147 24 14 18.19 172.4 29 14.73 0.888 7.16E-06 3.58E+09 0.0178 29 32 16.58 201.6 30 14.18 0.768 8.04E-06 2.74E+09 0.0154 25 20 18.46 178.2 31 7.39 0.383 7.25E-06 1.76E+09 0.0077 4 5 19.32 229.4 32 13.08 0.847 7.22E-06 3.13E+09 0.0169 22 28 15.44 184.9 33 4.67 0.346 6.92E-06 1.22E+09 0.0069 2 2 13.50 175.7 mean 11.61 0.706 7.51E-06 2.56E+09 0.0141 16.45 181.9 std. dev. 3.08 0.179 3.31E-07 6.48E+08 0.0036 1.82 16.9 CV 26.6 25.4 4.4 25.3 25.4 11.0 9.3 PAGE 114 Table 2.13 Set C carbon fiber (20 mm long, plasma tr eated) load-deflection a nd stress-strain results. Fiber Load Deflection Diameter Stess Strain Load Stress Stiffness Modulus No. (gm) (mm) (meters) (Pa.) Ranking Ranking (kg/m) (Pa.) 1 10.56 0.320 6.37E-06 3.24E+09 0.0160 7 8 32.97 202.5 2 5.48 0.148 6.95E-06 1.42E+09 0.0074 3 3 37.11 191.7 3 10.17 0.251 7.13E-06 2.50E+09 0.0126 6 6 40.47 198.6 4 12.69 0.347 6.93E-06 3.29E+09 0.0174 9 9 36.56 189.8 5 14.80 0.341 7.12E-06 3.64E+09 0.0171 12 10 43.39 213.4 6 16.90 0.414 7.02E-06 4.28E+09 0.0207 14 14 40.78 206.5 7 12.25 0.430 7.15E-06 2.99E+09 0.0215 8 7 28.48 138.9 8 4.22 0.153 6.73E-06 1.16E+09 0.0077 1 1 27.55 151.8 9 15.01 0.586 6.98E-06 3.84E+09 0.0293 13 11 25.60 131.0 10 5.36 0.260 6.94E-06 1.39E+09 0.0130 2 2 20.62 106.9 11 8.28 0.250 6.84E-06 2.21E+09 0.0125 4 4 33.16 177.1 12 9.53 0.248 7.31E-06 2.23E+09 0.0124 5 5 38.45 179.7 13 14.65 0.509 6.75E-06 4.01E+09 0.0254 11 12 28.80 157.6 14 14.36 0.443 6.68E-06 4.01E+09 0.0222 10 13 32.41 181.2 mean 11.02 0.34 6.92E-06 2.87E+09 0.02 33.31 173.35 std. dev. 3.90 0.12 2.29E-07 1.02E+09 0.01 6.32 30.52 CV 35.4 37.0 3.3 35.6 37.0 19.0 17.6 PAGE 115 Table 2.14 Set D carbon fiber (20 mm long, plasma tr eated) load-deflection a nd stress-strain results. Fiber Load Deflection Diameter Stess Strain Load Stress Stiffness Modulus No. (gm) (mm) (meters) (Pa.) Ranking Ranking (kg/m) (Pa.) 1 10.80 0.365 6.84E-06 2.88E+09 0.0183 3 3 29.57 157.8 2 13.38 0.476 6.94E-06 3.47E+09 0.0238 12 10 28.09 145.6 3 10.19 0.408 6.76E-06 2.78E+09 0.0204 2 2 24.96 136.3 4 12.61 0.294 6.78E-06 3.42E+09 0.0147 8 8 42.94 233.1 5 12.48 0.430 6.90E-06 3.27E+09 0.0215 7 6 29.02 152.0 6 10.05 0.251 7.33E-06 2.34E+09 0.0125 1 1 40.09 186.4 7 14.71 0.456 6.86E-06 3.90E+09 0.0228 14 13 32.29 171.0 8 10.98 0.340 6.75E-06 3.01E+09 0.0170 4 4 32.29 176.8 9 12.28 0.368 6.64E-06 3.48E+09 0.0184 6 11 33.34 188.8 10 13.07 0.411 6.62E-06 3.72E+09 0.0206 10 12 31.81 181.0 11 12.73 0.411 6.91E-06 3.33E+09 0.0205 9 7 31.00 162.1 12 11.60 0.336 6.69E-06 3.24E+09 0.0168 5 5 34.55 192.9 13 13.14 0.413 6.90E-06 3.44E+09 0.0206 11 9 31.83 167.0 14 14.65 0.415 6.85E-06 3.90E+09 0.0207 13 14 35.33 187.9 mean 12.33 0.38 6.84E-06 3.30E+09 0.02 32.65 174.20 std. dev. 1.42 0.06 1.67E-07 4.18E+08 0.00 4.47 23.37 CV 11.5 15.6 2.4 12.7 15.6 13.7 13.4 PAGE 116 Table 2.15 Set A through Set D two-parameter Wei bull distribution parameters and GOF factors. 2-Parameter Weibull Distribution Estimating Procedure Fiber LLSQ MLE Nonlinear Set modulus scale gof modulusscale gof modulus scale gof A raw 5.98 3.15 corrected 5.98 3.47 0.940 6.61 3.47 0.956 7.58 3.41 0.965 B raw 3.39 2.58 corrected 3.39 2.85 0.908 4.76 2.80 0.973 5.47 2.81 0.977 C raw 2.41 2.89 corrected 2.41 3.19 0.943 3.25 3.22 0.799 2.56 3.27 0.961 D raw 7.61 3.14 corrected 7.61 3.46 0.960 9.64 3.47 0.873 8.89 3.46 0.971 PAGE 117 Table 2.16 Two-population Weibul l concurrent simulation stat istics for various scaling ratios. Scaling Simulation Bounds % Failed by Failure Stresses Ratio Type 1 Type 2 Type 1 Type 2 Type 1 Type 2 Overall (MPa) (MPa) (MPa) (MPa) (MPa) 10/10 0.86 0.66 81 19 1.74 1.46 1.68 mean (1) 2.60 8.70 0.45 0.47 0.47 std. dev 26 33 28 CV 100/10 0.62 0.20 66 34 1.25 0.77 1.09 mean (10) 2.00 4.20 0.40 0.33 0.44 std. dev 32 43 41 CV 1000/10 0.46 0.08 62 38 0.92 0.54 0.78 mean (100) 1.50 2.10 0.30 0.33 0.37 std. dev 33 63 47 CV 1/10 1.20 0.20 95 5 2.18 2.52 2.20 mean (.1) 3.40 4.20 0.65 0.44 0.64 std. dev 30 17 29 CV 0.1/10 1.60 6.30 100 0 3.00 0.00 3.00 mean (.01) 4.60 35.00 0.86 0.00 0.86 std. dev 29 29 CV PAGE 118 Table 2.17 Two population Weibull concurre nt distribution parameters and GOF factors for set A through D fibers as determined by nonlinear estimation. 4-Parameter Concurrent Weibull Distribution Nonlinear Estimation Fiber low scale high scale Set modulus GPa modulusGPa gof A 0.156 9.08E+21 7.62 3.41 0.9652 B 0.847 26.9 7.14 2.87 0.9853 C 1.95 3.59 16.7 4.10 0.9885 D 0.396 4360 0.396 4360 0.9731 Table 2.18 Two population exclusive and partia lly-concurrent distribution parameters and GOF factor for set A and B fi bers as determined by nonlinear estimation. 5-parameter Weibull Distributions Nonlinear Estimation Fiber scale scale Set modulus parameter modulusparametergof (a) (a) (b) (b) (b) GPa GPa Exclusive A(1) 14.2 3.16 0.3 14.4 3.96 0.9856 A(2) 45.9 3.14 0.3 6.07 3.57 0.9947 B(1) 9.39 2.79 0.3 1.49 2.76 0.9910 B(2) 19.7 2.77 0.3 3.50 2.81 0.9922 Par. Con. A(1) 5.00 100.0 0.85 5.26 2.71 0.2675 A(2) 16.2 4.16 0.85 2.81 3.44 0.6152 B(1) 1.00 .0261 0.85 0.689 3.07 0.3027 B(2) 16.2 4160 0.85 2.81 3.44 0.1561 PAGE 119 Paper Jig Cyanoacrylate Carbon Fiber Double Sided Ta p e Cardboard Tabs Thin Beam Load Cell Aligning Pins Specimen Grip Aligning Gimbal Figure 2.1 Fiber tensile testi ng jig and load-aligning fixture. Figure 2.2 Laser diffraction system used to measure carbon fiber diameters. PAGE 120 Figure 2.3 Radio Frequency plasma volume and rotating carousel used to treat discrete carbon fibers. Figure 2.4 Two-dimensiona l failure state space, for a fiber of length L and theoretical tensile strength Sth, after [6]. Rotating Carousel Air, gas or monomer Radio Frequency Energy Carbon Fibers supported on Paper Jigs Pressure Gage Vacuum Line 0 x x+dx x L x s s + ds s Sint d S PAGE 121 Figure 2.5 Cumulative distribution functi ons for ideal two-parameter Weibull distribution, Eqn. 2.1, with s0=3.0x109 Pa. and =(1.0, 1.5, 2.0, 5.0, 10.0, 20.0). Weibull Modulus = 1.0 = 1.5 = 2.0 = 5.0 = 10.0 = 20.0 Stress (Pa.) 1081091010 Cumulative Probability 0.0 0.2 0.4 0.6 0.8 1.0 = 1.0 = 1.5 = 2.0 = 5.0 = 10.0 = 20.0 Weibull Modulus PAGE 122 Figure 2.6 Probability density function for ideal two-parameter We ibull distribution, Eqn. 2.1a, with s0=3.0x109 Pa. and =(1.0, 1.5, 2.0, 5.0, 10.0, 20.0). Stress (Pa.) 1081091010 Frequency 0.0e+0 5.0e-10 1.0e-9 1.5e-9 2.0e-9 2.5e-9 3.0e-9 Weibull Modulus = 1.0 = 1.5 = 2.0 = 5.0 = 10.0 = 20.0 PAGE 123 `Figure 2.7 Concurrent two population We ibull CDFs, Eqn. 2.2, for various Weibull moduli and scaling ratios. Stress (Pa.) 1081091010 Cumulative Probability 0.0 0.2 0.4 0.6 0.8 1.0 Concurrent Distribution a, b L/L0 4.0, 2.0, 1.0 4.0, 2.0, 5.0 4.0, 2.0, 0.5 10.0, 1.0, 1.0 1.0, 10.0, 1.0 4.0, 10.0, 1.0 PAGE 124 Figure 2.8 Exclusive two poulation Wei bull CDFs, Eqn. 2.3. See Table 2.1 for particulars. Stress (GPa.) 1.02.03.04.05.06.07.0 Cumulative Probability 0.0 0.2 0.4 0.6 0.8 1.0 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Exclusive Distribution PAGE 125 Figure 2.9 Partially-concurrent two populat ion Weibull CDFs, Eqn. 2.4. See Table 2.1 for particulars. Stress (Pa.) 1.02.03.04.05.06.07.0 Cumulative Probability 0.0 0.2 0.4 0.6 0.8 1.0 Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Partially-Concurrent Distribution PAGE 126 Stress (GPa.) 1081091010 Cumulative Probability 0.0 0.2 0.4 0.6 0.8 1.0 Figure 2.10 Exclusive and partially-c oncurrent Weibull CDFs, Eqn. 2.3 and 2.4 respectively. See Table 2.2 for particulars. Par., 1 Par., 2 Exc., 1 Exc., 2 Distribution, Case PAGE 127 Figure 2.11 Two-parameter Weibull distributi on probability plot, Eqn. 2.1, with =3.0 and s0= 3x109 Pa.. ( ) simulated data (n=25), (solid line) Eqn. 2.6. ln(s) 18192021222324 ln(-ln(1-f(s))) -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 PAGE 128 Figure 2.12 Concurrent Weibull distri bution probability plot, Eqn. 2.2, with A=4.0, s0 A=2x109 Pa., B=2.0, and s0 B=3x109 Pa. (solid line) Eqn. 2.7; (dashed line) Eqn. 2.6, with =( A, B) and s0=(s0 A,s0 B). Low Modulus High Modulus Concurrent Distribution, Eqn. 18 19 20 21 22 23 4 2 0 -2 -4 -6 -8 -10 -12 ln (stress [Pa]) PAGE 129 Figure 2.13 Exclusive Weibull distribution probability plot, Eqn. 2.3, with various parameters. See Table 2.3 for cases. 18 19 20 21 22 23 4 2 0 -2 -4 -6 -8 -10 -12 ln (stress [Pa]) -14 Exclusive Distribution Case 1 Case 2 Case 3 Case 4 PAGE 130 Figure 2.14 Partially concurrent Weibull distribution probability plot, Eqn. 2.4, with various parameters. See Table 2.3 for cases. 13 15 17 19 21 23 4 2 0 -2 -4 -6 -8 -10 ln (stress [Pa]) Case 1 Case 2 Case 3 Case 4 Partially Concurrent Distributions PAGE 131 Figure 2.15 Simulated two-parameter Weibull distribution probability plots (N=10), Eqn. 2.1, with =3.0, s0=3.0x109 Pa., and (a) n=10; (b) n=25; (c) n=50; (d) n=100; (e) n=500. 18202224 ln(-ln(1-f(s))) -7 -6 -5 -4 -3 -2 -1 0 1 2 18202224 ln(-ln(1-f(s))) -7 -6 -5 -4 -3 -2 -1 0 1 2 ln(stress) 18202224 ln(-ln(1-f(s))) -7 -6 -5 -4 -3 -2 -1 0 1 2 ln(stress) 18202224 ln(-ln(1-f(s))) -7 -6 -5 -4 -3 -2 -1 0 1 2 ln(stress) 18202224 ln(-ln(1-f(s))) -7 -6 -5 -4 -3 -2 -1 0 1 2 (a) (b) (c) (d) (e) PAGE 133 Figure 2.17 Effective basal plane compliance versus basal plane misorientation from fiber axis, Eqn. 2.11. Misorientaion Angle (degrees) 020406080100120140160180 Basal Plane Compliance (m2/N) 2.0e-11 4.0e-11 6.0e-11 8.0e-11 1.0e-10 1.2e-10 1.4e-10 PAGE 134 Figure 2.18 Reynolds-Sharp and Griffith failure criteria as a function of fiber axis/crystallite misorientaio n UTS is measured ra dially from the origin. a=4.2 J/m2, C=50x10-9 m, sij (Tablexx); (solid line) R-S criteria, Eqn. 2.11; (dashed line) Griffith fracture cr=1x109 Pa. 0 2 4 6 8 10 12 Log Stress (Pa) 0 2 4 6 8 10 12 Log Stress (Pa) UTS Reynolds-Sharp Criteria Griffith Fracture cr = 1 Mpa PAGE 135 Figure 2.19 Small angle x-ray scattering patterns for PAN-based carbon fibers. Fiber axis is vertical in figures. (a) Type II high strength carbon fiber. (b) Type I high modulus carbon fiber. Reprinted wi th permission from Kluwer Academic Publishers. [ 49 ], fig.5. PAGE 136 Figure 2.20 The averaged pore axis angular distribution for two types of PAN-based carbon fibers as determined by small angle x-ray scattering, data from Perret and Ruland [ 35 ]. (thin line) WYB Thornel, Union Carbide; (thick line) WYD Thornel, Union Carbide; (open circle) Normal Dist. 0.0 0.5 1.0 -90-75-60-45-30-150153045607590Pore Axis Misorientation Angle ()Distribution Intensity (Normalized ) Union Carbide (WYD) Union Carbide (WYB) Normal Dist. PAGE 137 -2 -1 0 1 2 3 4 5 Reduced Variate 20 25 30 35 40 Angle (degrees) Figure 2.21 First asymptote maximum value di stribution and generate d data for a normal distribution with =0 =8.5 (Z=20 ), and n=1000. (open circle) maximum value, N=50; (solid line) pr edicted first asymptote distribution, LLSQ. PAGE 138 -2 -1 0 1 2 3 4 5 Reduced Variate 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Angle (degrees) Figure 2.22 Weibull, third asymptote, maximum value distribution and generated data for a normal distribution with =0 =8.5 (Z=20 ), and n=1000. (open circle) maximum value, N=50; (solid line) predicted Weibull distribution, LLSQ. PAGE 139 -3 -2 -1 0 1 2 3 4 Reduced Variate -36 -34 -32 -30 -28 -26 -24 -22 Angle (degrees) Figure 2.23 First asymptote minimum value distribution and generated data for a normal distribution with =0 =8.5 (Z=20 ), and n=1000. (open circle) minimum value, N=50; (so lid line) predicted first asymptote distribution, LLSQ. PAGE 140 -3 -2 -1 0 1 2 3 4 Reduced Variate 3.1 3.2 3.3 3.4 3.5 3.6 Angle (degrees) Figure 2.24 Weibull, third asym ptote, minimum value distribu tion and generated data for a normal distribution with =0, =8.5 (Z=20), and n=1000. (open circle) minimum value, N=50; (so lid line) predicted Weib ull distribution, LLSQ. PAGE 141 -2 -1 0 1 2 3 4 5 6 Reduced Variate 20 25 30 35 40 Angle (degrees) Figure 2.25 First asymptote abso lute extreme value distributi on and generate d data for a normal distribution with =0, =8.5 (Z=20), and n=1000. (open circle) absolute extreme value, N=50; (solid line) predicted first asymptote distribution, LLSQ. PAGE 142 -2 -1 0 1 2 3 4 5 Reduced Variate 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Angle (degrees) Figure 2.26 Weibull, third asymptote, absolu te extreme value distribution and generated data for a normal distribution with =0, =8.5 (Z=20), and n=1000. (open circle) absolute extreme value, N=50; (solid line) predicted Weibull distribution, LLSQ. PAGE 143 Figure 2.27 Effect of crystallite misorientati on dispersion and ma gnitude on simulated carbon fiber tensile failure strength. Failure strengths cal culated using Eqn. 2.11 and crystallite misorientation angles given by a normal distribution with =0, n=100, and N= 50. (square) =12.75 (Z=30); () =17 (Z=40); (solid line) predicted Weibull distributions, LLSQ. Z=40, =17 Z=30, =12.75 20.24 20.56 20.52 20.48 20.44 20.40 20.36 20.32 20.28 3 2 1 0 -1 -2 -3 -4 ln (stress [Pa]) ln(-ln(1-F(s))) PAGE 144 Figure 2.28 Effect of crystallite misorienta tion magnitude on si mulated carbon fiber tensile failure strength. Failure strengths calculated using Eqn. 2.11 and crystallite misorientation angles given by a normal distribution with =0, n=100, N=50, and () Z=5; (+) Z=10; () Z=20; () Z=30; () Z=40. 21.8 21.6 21.4 21.2 21.0 20.8 20.6 20.4 2 1 0 -1 -2 -3 -4 ln (stress [Pa]) 20.2 22.0 Z=40 Z=30 Z=20 Z=10 Z=5 ln(-ln(1-F(s))) PAGE 145 Figure 2.29 Effect of numerical parameters on simulated carbon fiber tensile strength. Failure strengths calcula ted using Eqn. 2.11 and crystallite misorientation angles given by a normal distribution with =0, Z=20, and () n=100, N=50; () n=100, N=100; (x) n=1000, N=50 ; (solid line) predicted Weibull distributions, LLSQ. 20.9 20.8 20.7 20.6 20.5 20.4 2 1 0 -1 -2 -3 -5 ln (stress [Pa]) 20.3 -4 3 100, 50 100, 100 1000, 50 n, # of simulations Z=20 ln(-ln(1-F(s))) PAGE 146 Figure 2.30 Ribbon-like morphology proposed for PAN-based carbon fibers -Fordeux et al. (1971) of [ 49 ]. Fiber axis is vertical. Reprinted with permission from Kluwer Academic Publishers. PAGE 147 0 2 4 6 8 10 12 Log (Effective Stress, Pa.) 0 2 4 6 8 10 12 Log (Effective Stress, Pa.) s = cr const. Eqn. 2.12d Eqn. 2.12b Eqn. 2.12a Eqn. 2.12c Eqn. 2.12e Figure 2.31 Polar projections of effective stress acting on a crack subjected to simple uniaxial tension. Eqn. 2.12a-Eqn. 2.12e; and Griffith criteria cr=const. PAGE 148 1E-14 1E-13 1E-12 1E-11 1E-10 1E-9 1E-8 1E-7 1E-6dN/ds, dtheta/ds0 0.01 0.02 0.03 0.04 0.05 dN/dtheta -45 -30 -15 0 Misorientation () d N/d theta d theta / d s dN/ds Figure 2.32 The bounded solid angle ( cr) that contains the gr aphitic crystallite basal plane orientations for which failure via transverse rupture is expected. Fiber loading is uniaxial, =( ,0,0), and the failure criteria is given by Eqn. 2.12. 1E-9 1E-8 1E-7 1E-6-dtheta/ds1E-35 1E-30 1E-25 1E-20 1E-15 1E-10-dN/ds -90 -75 -60 -45 Misorientation () PAGE 149 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 0.0E+00 5.0E+09 1.0E+10 1.5E+10 2.0E+10 Failure Stress (Pa.) -20 -15 -10 -5 0 5 ln(-ln(1-F)) 20.0 21.0 22.0 23.0 24.0 ln(stress) 0.01 0.02 0.05 0.1 0.2 0.5 0.9 0.999 (a) (b) Figure 2.33 Uniaxial Batdorf-Crose failure prediction u tilizing the Reynolds-Sharp failure criteria. (a) CDF and (b ) Weibull probability plot. Z=20 ; C=50x109m; =4.2 J/m2. PAGE 150 Figure 2.34 Frequency histogram for flaw free lengths in acrylic fibers. After [ 64 ]. 0 0.2 0.4 0.6 0.8 1 Probability 0 5 10 15 20 Gage Length (mm) 2.32 0.98 0.57 2.1 1.2 0.32 Mean Flaw Conc. (mm) PAGE 151 Intensity (Arb.) Surface Flaws Volume Flaws Increasing Flaw Size Increasing Strength 0 0 Figure 2.35 Notional flaw size/failure strength distribution for surface and volume flaws. PAGE 152 Load (grams) 010203040506070 D/A output (units @ +/5 V range) 0 40 80 120 160 200 240 280 Load Cell Output (volts) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 2.36 Thin beam load cell calibration curve. Each (closed circle) represents approximately 100 readings. (solid line) LLSQ regression. PAGE 153 (a) (b) Figure 2.37 Fraunhoffer diffraction patterns produced by a single carbon fiber and HeNe laser light (two positions on the same fiber: (a) and (b)). Dark region at the center of the pattern is caused by a light stop. See Figure 2.39 for pixel intensity along centerline. PAGE 154 LProjection PlaneCarbon Fiber1 Figure 2.38 Fraunhoffer diffraction schematic indicating object/viewing plane separation, L, and intensity node separation Zn. HeNe Laser Coherent Li g h t PAGE 155 Figure 2.39 Pixel intensity along the centerline of the Fraunhoffer di ffraction pattern. (a) and (b) taken at two diff erent positions on the same fiber. (dot) pixel intensity; (solid lin e) fourth-order polynomial f its of patterns in Figure 2.37. 0 50 100 150 200 250 050100150200250300350 Pixel Number (along line)Intensity (grey level, 0-255)(a) 0 50 100 150 200 250 050100150200250300350 Pixel Number (along line)Intensity (grey level, 0-255)(b) PAGE 156 Fiber No. 0102030405060708090 Corrected Fiber Diameter ( m) 6.0 6.5 7.0 7.5 8.0 8.5 Figure 2.40 Carbon fiber diameters determined by Fraunhoffer diffraction. Symbols are averages of two measurements. ( ) set A, ( ) set B, ( ) set C, ( ) set D, (solid line) set mean, (dashed line) overall mean. PAGE 157 Fiber Diameter ( m) 5.56.06.57.07.58.08.59.0 Frequency (arb.) Fiber Group Set A Set B Set C Set D Overall Normal Approximation Figure 2.41 Normal distributions for carbon fiber measurements.( ) set A, ( ) set B, ( ) set C, ( ) set D. PAGE 158 0 Figure 2.42 Typical PAN-based carbon fiber cross section, after Krucinska [ 73 ]. The diameter differences are exagerrated to emphasize non unif orm cross section. PAGE 159 Figure 2.43 Typical tensile load-deflection curves for PAN-based carbon fibers. ( ) set A, ( ) set B, ( ) set C, ( ) set D. Deflection (mm) 0.00.10.20.30.40.50.6 Load (gm) 0 2 4 6 8 10 12 14 16 Typical Fiber Set A Set B Set C Set D PAGE 160 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 791113151719 Failure Load (gm)Failure Stress (GPa) Solid line represents a common const. cross section assumption for all Set A (20 mm) fibers. Diameter = 7.25 m Figure 2.44 Failure load versus failure stress for set A ( ) carbon fibers. The solid line represents a constant cross section as sumption for all fibers (dia.= 7.25 m). PAGE 161 Figure 2.45 Cumulative failure probab ilities for as-received PANbased carbon fibers. (Strain rate, =0.1 min-1) ( ) set A fibers, 20 mm untreated; ( ) set B fibers, 50 mm untreated; ( ) set C fibers, 20 mm plasma treted. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.51.01.52.02.53.03.54.04.5 Failure Stress (GPa)Cumulative Probability (%) Set A Fibers Set A Fibers Set C Fibers PAGE 162 Figure 2.46 Two-parameter Weibull probab ility plot for carbon fibe r tensile tests. ( ) set A, 20 mm; ( ) set B, 50 mm; (so lid line) LLSQ regre ssion; (dashed line) MLE. -7 -6 -5 -4 -3 -2 -1 0 1 2 20.320.520.720.921.121.321.521.721.922.122.3 ln(stress [Pa])ln[(-L0/L) ln(1-F)] 20 mm 50 mm LLSQ MLE PAGE 163 Failure Stress (GPa) 0.01.02.03.04.05.06.0 Predicted Intensity (arb. Units) Failure Count 0 1 2 3 4 5 6 7 8 9 10 20 mm histogram 50 mm histogram 20 mm predicted 50 mm predicted Figure 2.47 Two-parameter PDFs fo r carbon fiber tensile tests. ( ) set A; ( ) set B; (solid line) set A, MLE; (dashed line) set B, MLE. PAGE 164 -4 -3 -2 -1 0 1 2 21.221.421.621.822.022.222.4 ln(stress [Pa])ln[(-L0/L) ln(1-F)] 20 mm (Dia. Corrected) : rho = 5.985, s0 = 3.15 GPa 20 mm (Dia. Corrected) : rho = 5.985, s0 = 3.47 GPa LLSQ Figure 2.48 Two-parameter Weibull probab ility plot for set A tens ile tests: corrected ( ) for Fraunhoffer diffra ction -Li and Tietz [ 71 ]and uncorrected (+). (solid line) LLSQ regression. PAGE 165 ln (stress [Pa.]) 21.221.421.621.822.022.222.4 Residuals -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 Zero (a) Figure 2.49 Prediction residuals for LLSQ regres sion. (a) Set A, 20 mm. (b) Set B, 50 mm. PAGE 166 ln (stress [Pa.]) 20.420.821.221.622.022.4 Residuals -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 zero (b) Figure 2.49 cont. PAGE 167 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) 20 mm LLSQ 95% conf. 95% pred. Figure 2.50 Set A tensile strength two-pa rameter Weibull CDF determined by LLSQ regression. ( ) tensile tests. (solid line) LLSQ regression, (dotted line) 95% confidence interval; (dashed line) 95% prediction interval. PAGE 168 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 5.0E+08 1.5E+09 2.5E+09 3.5E+09 Failure Stress (Pa.) 50 mm LLSQ 95% conf. 95% pred. Figure 2.51 Set B tensile strength two-pa rameter Weibull CDF determined by LLSQ regression. () tensile tests. (solid line) LLSQ regression, (dotted line) 95% confidence interval; (dashed line) 95% prediction interval. PAGE 169 Figure 2.52 Set A tensile strength two-para meter Weibul CDF determined by MLE. ( ) tensile tests. (solid line) MLE, (dotted line) 95% co nfidence interval; (dashed line) 95% prediction interval. 0 0.2 0.4 0.6 0.8 1 Cumulative Probability1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) 20 mm MLE pred. 95%conf 95%conf 95%pred 95%pred PAGE 170 Figure 2.53 Set B tensile strength two-para meter Weibull CDF determined by MLE. ( ) tensile tests. (solid line) MLE, (dotted line) 95% co nfidence interval; (dashed line) 95% prediction interval. 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 5.00E+08 1.50E+09 2.50E+09 3.50E+09 Failure Stress (Pa.) 50 mm MLE pred. 95%conf 95%conf 95%pred 95%pred PAGE 171 Figure 2.54 Set A tensile strength twoparameter Weibul CDF determined by nonlinear estimation. ( ) tensile tests. (solid line) non-linear estimation, (dotted line) 95% confidence interval; (dashed line) 95% prediction interval. 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) 20 mm Non-linear fit 95% conf. 95% pred. PAGE 172 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 5.0E+08 1.5E+09 2.5E+09 3.5E+09 Failure Stress (Pa.) 50 mm Non-linear fit 95% conf. 95% pred. Figure 2.55 Set B tensile strength twoparameter Weibul CDF determined by nonlinear estimation. ( ) tensile tests. (solid line) non-linear estimation, (dotted line) 95% confidence interval; (dashed line) 95% prediction interval. PAGE 173 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) MLE FIt Set A Fibers (Harter) MLE Lower Conf.(Thoman) LLSQ 95% Lower Conf. (Harter) nonlinear 95% Lower Conf. (Harter) 95% Lower Confidence Bounds Figure 2.56 Two-parameter Weibull 95% lower confidence bounds for set A tensile tests (solid line). Confidence estima tion; MLE(Harter); MLE(Thoman); LLSQ(Harter); and non-linear(Harter). PAGE 174 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) LLSQ (Gaussian) LLSQ (Harter) 95% Confidence Intervals Param. est. (conf. int) (a) Figure 2.57 Two-paramter Weibull 95% confidence intervals for set A carbon fiber tensile tests. Fit estimation (a) LLSQ, (b) MLE, (c) non-linear. Fit (dotted line); confidence estimati on, Gaussian (solid line); Harter (dashed line). PAGE 175 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) MLE (Gaussian) MLE (Harter) 95% Confidence Intervals Param. est. (conf. int.) (b) Figure 2.57 cont. PAGE 176 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) nonlinear (Gaussian) nonlinear (Harter) 95% Confidence Intervals Param. est. (conf. int.) (c) Figure 2.57 cont. PAGE 177 0.0E+00 2.0E-10 4.0E-10 6.0E-10 8.0E-10 1.0E-09 Frequency 1.0E+09 2.0E+09 3.0E+09 4.0E+09 5.0E+09 Failure Stress (Pa.) 20 mm 50 mm from 20 50 mm 20 mm from 50 Figure 2.58 Set A and B carbon fiber two-parameter Weibull PDFs. Reference length and scaling ratio, (L0,L/L0); (20,1); (20, 2.5) ; (50,1); (50, 0.4). PAGE 178 9.35 9.40 9.45 9.50 9.55 log (avg. Strength [Pa.]) 1.2 1.3 1.4 1.5 1.6 1.7 1.8 log (gage length [mm]) Actual, Weibull Modulus = 3.88 Scaled 'LLSQ prediction Scaled MLE prediction Weibull Scaling Predictions Figure 2.59 Log-Log plot of average failure strength vs. gage length. ( ) untreated carbon fibers; scaled predictions, (+) LLSQ, ( ) MLE. PAGE 179 0 1 2 3 4 0 4 8 12 16 4 8 12 16 20 24 28 32sb / s0W e i b u l l M o d u l u s L / L0 ( S c a l i n g r a t i o ) Figure 2.60 Relationship between Weibull modulus, scaling ratio (L/L0), and ultimate strength ratio (sb/s0) for ideal two-parameter Weibull distributions. PAGE 180 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 / s0W e i b u l l M o d u l u sL / L0 ( S c a l i n g R a t i o ) Figure 2.61 Relationship between Weibull modulus, scaling ratio, and standard deviation, for ideal two-parameter Weibull distributions. PAGE 181 Figure 2.62 Weibull probability plot for simulated two population concurrent distributions. See Figure for cases and Table 2.16 for staistical details. 18 19 20 21 22 23 Ln(stress) -5 -4 -3 -2 -1 0 1 2 3 Reduced Variate 0.020.050.10.20.30.40.50.60.70.80.90.950.99 1,type1 1,type2 10,type1 10,type2 100,type1 100,type2 .1,type1 .1,type2 .01,type1 .01,type2 high modulus asymptote low modulus asymptote .1 1 10 100 .01Cumulative Probability PAGE 182 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) 20 mm Concurrent 95% conf 95% pred Figure 2.63 Set A carbon fiber tensile stre ngth two population concurrent CDF. ( ) tensile tests. (solid lin e) non-linear estimation, ( dotted line) 95% confidence interval; (dashed line) 95 % prediction interval. PAGE 183 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 5.0E+08 1.5E+09 2.5E+09 3.5E+09 Failure Stress (Pa.) 50 mm Concurrent 95% conf 95% pred Figure 2.64 Set B carbon fiber tensile stre ngth two population concurrent CDF. ( ) tensile tests. (solid lin e) non-linear estimation, ( dotted line) 95% confidence interval; (dashed line) 95 % prediction interval. PAGE 184 -6 -4 -2 0 2 Reduced Variate21.2 21.4 21.6 21.8 22 22.2 22.4 ln (stress [Pa.]) low modulus asymptote high modulus asymptote Figure 2.65 Set A carbon fiber tensile stre ngth two population concurrent Weibull probability plot. ( ) tensile tests. (solid line) non-linea r estimation; (dashed line) low and hi gh modulus asymptotes. PAGE 185 -10 -8 -6 -4 -2 0 2 Reduced Variate20.2 20.6 21 21.4 21.8 22.2 ln (stress [Pa.]) low modulus asymptote high modulus asymptote Figure 2.66 Set B carbon fiber tensile stre ngth two population concurrent Weibull probability plot. ( ) tensile tests. (solid line) non-linea r estimation; (dashed line) low and hi gh modulus asymptotes. PAGE 186 0 2E-10 4E-10 6E-10 8E-10 1E-09 Frequency 5.0E+08 1.5E+09 2.5E+09 3.5E+09 4.5E+09 Failure Stress (Pa.) 50 mm 20 mm FWHM Figure 2.67 Two population concurrent PDFs for fiber tensile tests of two different gage lengths. (solid line) set A, 20 mm; (dashed line) set B, 50 mm. PAGE 187 -6 -4 -2 0 2 4 Reduced Variate Cumulative Probability20 20.5 21 21.5 22 22.5 ln (stress [Pa.]) 0.01 0.02 0.05 0.1 0.2 0.5 0.8 0.9 0.99 0.999 20 mm actual 20 pred 50 from 20 50 mm actual 50 pred 20 from 50 Concurrent Distributions and Scaling Predictions Figure 2.68 Set A nd B scaling preditions for two population concurrent Weibull distributions: set A ( ) tensile test s; set B ( ) tensile tests; probability plots (solid lines) and scaling predictions (dashed lines). Reference length and scaling ratio, (L0,L/L0); (20, 2.5); (50, 0.4). PAGE 188 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 5.0E+08 1.5E+09 2.5E+09 3.5E+09 4.5E+09 5.5E+09 Failure Stress (Pa.) Set C MLE fit Set D MLE fit Figure 2.69 Failure strengths and two-parame ter Weibull CDFs for set C and D fibers. See Table 2.15 for fit parameters. PAGE 189 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 0.0E+00 1.0E+09 2.0E+09 3.0E+09 4.0E+09 5.0E+09 Failure Stress (Pa.) High Modulus Low Modulus (a) Figure 2.70 Discrete CDFs for surface and volum e flaw failure strength populations as determined from set B fibers via a two population concurrent model (see Table 2.17 for fit parameters). (a) and (b) contain the same curves on different scales. PAGE 190 0 0.2 0.4 0.6 0.8 1 Cumulative Probability 0.0E+00 5.0E+10 1.0E+11 1.5E+11 2.0E+11 Failure Stress (Pa.) Low Modulus High Modulus (b) Figure 2.70 cont. PAGE 191 0 2E-10 4E-10 6E-10 8E-10 1E-09 Frequency 0.0E+00 2.5E+10 5.0E+10 7.5E+10 1.0E+11 Failure Stress (Pa.) High Modulus Low Modulus Figure 2.71 Discrete PDFs for surface and vol ume flaw failure strength populations as determined from set B fibers via a two population concurrent model. See Table 2.17 for fit parameters. PAGE 192 CHAPTER 3 CHARACTERIZING THE FIBER-MATRIX INTERFACE Introduction Historical Development The development of high strength (HS) a nd high modulus (HM) graphite fibers in the early 1960's sparked concurrent inte rdisciplinary research in interface micromechanics, surface science, and chemistr y in order to gain the full advantage of these new high performance graphite fibers1. In his seminal paper on fibrous reinforcement micromechanics Cox wrote: The elastic properties and st rength of any material de pend upon the detail of its structure, and analysis of this structure should enable the strength and stiffness of the whole to be correlated to that of its parts. [105] Indeed, this is a statement of the obvious, one with which mechanists have been grappling with for some time. The comple xity of even regula r, ordered composite structures and the sometimes-sweeping gene ralizations necessary for their evaluation belies the obviousness of such an aphorism. None-the-less with concerted effort and over time, there have been excellent advances in interface micromechan ics understanding. The earliest investigations on HS and HM fiber reinforcement were conducted by British and GE researchers in th eir efforts to utilize recently developed carbon fibers and advanced Metal Matrix Composite (MMC) t echnology, especially as pertaining to load transfer across the interface. A delayed flu rry of articles addressing fiber-matrix load transfer [106-109] refined and expanded C oxs passing mention of the same subject PAGE 193 [105]. Rosens 1965 article M echanics of Composite Streng thening [110] provides an excellent snapshot of the a ggregate knowledge and notes th e fine distinction between model prediction and inference from empirical evidence. Using molybdenum and tungsten wires, Kelly and others [111-113] conducted some of the earliest fiber fragmentation a nd pull-out tests based on the work of Cox, Sadowsky, and Dow [105,108,109]. The Single Fi ber Composite Fragmentation (SFCF) test (Figure 3.1) is a model microcomposite tensile test. Fo r SFCF tests a single continuous fiber is suspended in a dogbone sa mple that has a rectangular matrix cross section (e.g., graphite in epoxy). As the com posite tensile load increases, the fiber axial stress (FAS) will exceed the local fiber strength resulting in progressive fiber fragmentation. The resulting fragment leng th distribution (FLD) during loading, and at fragmentation saturation, characterizes the fiber-matrix interaction and the fibers strength/flaw population. The simplest SFCF analysis owing to Kelly and others [111-113] defines the critical fiber length, lc, as the minimum fiber length that will just allow the fibers ultimate axial strength to be reached. The SFCF test pioneered by Kelly and others has been widely reproduced, refined, and redresse d, and along with fiber pull-out tests will be the primary focus of this chapter. Additional Testing Methods Microdebonding2 tests are closely related to SFCF and pull-out tests in that they subject the fiber to tensil e loading across a shear stre ssed interface [114]. While 1 "There is currently some interest in the idea of using parallel bundles of strong fibers with large elastic moduli as structural materials" [112]. 2 The term microdebond has also been use by Narkis et al. [120] to describe compressive microindentation tests. PAGE 194 microdebond tests are easy to conduct, variatio ns in droplet geometry and vice-related stress fields influence bond strength calculations [115]. A similar microbundle pull-out test has also been suggested [116-119]. Samp le preparation and interpretation difficulties limit this test's utility. Another family of microcomposite tests is compressive in nature and includes the SFC compression test, the fiber push-out te st (microindentation), and the slice compression test. For SFC compression tests various sample geometries, regular and dogbone, allow for both shear and transverse te nsile debonding strength determinations. SFC compression tests, however, are little used owing to fiber alignment sensitivity and an inability to identify incipient debonding. The slice compression test is the most recently developed compressive model composite test. It has been used exclusiv ely for Ceramic Matrix Composites (CMCs). Empirical [121] and analytic al [121-122] developments are still ongoing. The test involves compressively loading a composite section between tw o plates (one harder than the fibers and the other softer). The residual impression depths in th e soft plate and fiber protrusion after relaxation are measures of interface properties, be they frictional or bonding. The fiber push-out or microindentation te st has seen much more widespread use than the alternative compressive microcomposite tests. Fiber push-out tests have proved useful for both polymer and ceramic matrix co mposites. In addition to the large body of empirical work, extensive theoretical and numerical micromechanical analyses have advanced interface understanding. Fiber push-out particulars are folded into the PAGE 195 discussion on pull-out and SFCF tests to illumi nate the mechanisms and parameters that affect SFCF interpretation. Whereas model composite tests enable di rect assessment of fiber-matrix interface mechanical properties, interlaminar strength tests require a macroscopic approach with the expectation that interface properties can be deduced from the test results. The interlaminar strength properties most comm only associated with interface quality are interlaminar shear strength (ILSS), transver se tensile strength, and in-plane shear strength. Provided sufficient care is taken to insure that the failure mechanisms are consistent with testing procedures, these strength properties are excellent comparative measures for exploring compos ite fabrication parameters. The Iosipescu shear, short-beam shear, and and off-axis tensile tests are all used for determining ILSS. Iosipesc u shear is the simplest, most common, and accepted. Descriptions of the testing procedur es, variations, and specifications are given in a review by Herrera-F ranco and Drzal [115]. Bulk and interphase molecular relaxation phenomena can also be utilized to analyze composite and interface /interphase properties. The relaxation processes studied range from extremely slow dilatometric re laxations to very high frequency spectra associated with Dielectric Relaxation Spectroscopy (DRS) and Nuclear Magnetic Resonance (NMR). Much of the early and important work in this area, especially calorimetric, can be attributed to Y. S. Lipatov [123]. Vari ous researchers have employed Dynamic Mechanical Analysis (DMA), DRS, Differential Scanning Calorimetry (DSC), Dilatometry, Fourier Transform Infrar ed Spectroscopy (FTIR), NMR, and Raman Spectroscopy to investigat e the interphase chemistry, morphology, and physical PAGE 196 properties in close detail. Ind eed, the extent of the research is astounding. Lipatov [123] offers an extensive review and bib liography on these subj ects (see [115]). Relaxation phenomena are used to inves tigate composite interface properties for several important reasons. Firstly, and most importantly, is that re laxation processes are sensitive to variations in ch emical and physical structure ac ross a broad scale (viz. Bulk polymer volumetric relaxations, Dilatometr y vs. proton spin resonance, NMR). Secondly, relaxation spectroscopic techniques are generally non-destructive and can be conducted on either ideal or practical compos ite structures. Finally, these tests can be both complimentary and unique in their analys is relative to the previously described mechanical tests. In Lipatovs view, the chemical a nd structural development of the interface/interphase derives from the composite systems colloidal nature3. As such, the chemical and structural development at th e interface is a comple x interaction between matrix/filler adsorption and adhesion, peculiar physicochemical reactions associated with the solid inclusion, matrix molecular rigidit y, and various surface re laxation processes. Utilizing a physical chemistry approach, it can be shown [123] that small changes in interface chemistry, morphology, or extent can produce large changes in composite relaxation phenomena. We will employ reasoning and results derived from relaxation spectra extensively in our discussion below, especially as they pertain to interface morphology and chemistry. 3 Lipatov prefers the colloid chemistry definition advanced by Reibinder as a division of physical chemistry in which there are considered the processes of formation and break-up of fine particle systems, and also their characteristic proper ties, which are connected mainly w ith the surface phenomena at the phase boundaries in these systems. (ref. (29) [123]) PAGE 197 Optical and electron microscopy can provide in some cases, definitive qualitative proof for failure mechanisms and they usua lly provide excellent corroborative evidence. Unfortunately, fractography can not provi de quantitative data on fiber-matrix micromechanics and, furthermore, it is ofte n difficult to discern small changes in interface properties by visual inspection. Un fortunately, macrocomposite testing such as transverse tensile strength and ILSS do not al low for direct evaluation of the complex fiber-matrix interaction. In particular, the irregular geom etric fiber distribution in macrocomposites, combined with mixed mode failures for both the ILSS and transverse tensile tests obscure the discre te contribution of the fiber-m atrix interfacial strength in macrocomposites. Other related techniques include ultras onic [124,125], infrared imaging [125], and eddy-current analysis [126]. These techni ques generally do not have the necessary resolution to conduct detailed anal ysis of the fiber-matrix interface, but have been used successfully to monitor fatigue and imp act damage on composite structures. Discussion We will now begin our discussion of the comp osite interface in earnest, especially regarding data provided by SFCF tests that were first advanced by Kelly and Tyson [111113]. The validity and conseque nces of the underlying assumptions and principles of the SFCF test will be considered in light of the considerable accumulated understanding of interface morphology, chemistry, and physical properties provided by alternative inspection techniques and analytical approaches. Early One Dimensional Single Fib er Composite Fragmentation (SFCF) The origin of single fiber tests can be tr aced to initial work by Cox [105], who considered the elasticity and st rength of paper and other fibr ous materials for a range of PAGE 198 fiber orientation distributions Cox concluded that modulus reinforcement efficiency depends on fiber geometry, orientation and spatial distributions, and the mechanical properties of both the fiber and the matrix. In his analysis, Cox a ssumed that the load was transferred to the fiber by shear stress alone, and that the transverse mechanical properties of the fiber and matrix were equi valent. This effectively generates a onedimensional problem along the reinforcement axis. Kelly and Tyson [111-113] further simplified Coxs load transfer description to enable evaluation for their SFCF test. Utilizing simple fiber pull-out tests (Figur e 3.2) that related the fiber pull-out or failure stress (f) to the imbedded fiber length aspect ratio ( l/df) Kelly and Tyson [112] identified a linear relationship of the form where k is a proportional constant For free-end specimens 0 tends to zero (Figure 3.3) and a constant slope implies uniform load transfer across the interface. A simple composite axial force balance across the interface yields: where is the interfacial shear stress (IFSS) and z is measured from the break point along the discontinuous fiber. Assuming furthe r "that the whole of the matrix yields plastically," [112] the key simplifying and lim iting plastic matrix assumption, yields the familiar where y is the matrix yield stress. ) / (0 f fd l k (3.1) z fdz r r z0 22 ) ( or z fdz r z02 ) ( (3.2) y r z zf2 ) ( (3.3) PAGE 199 Before the first fiber break in SFCF test s, the FAS increases uniformly along its length with increasing composite strain (Figure 3.4). The exception being the FAS near the loading clamps and any FAS perturbation as sociated with fiber diameter variations, imperfect bonding, or matrix/interphase inhom ogeneities. These discrepancies will be considered below. For now we hold that perfect bonding exists, th e fiber diameter is constant, and the matrix/inter phase is homogeneous, at leas t on the scale required here. After the first fiber break and w ith increasing composite strain, c, the FAS buildsup linearly on both sides of a fiber break until nom, the nominal fiber stress, is reached (nom = Ef c, Ef is fiber axial modulus) or the fiber may fracture if an additional weak cross section is overloaded (Fi gure 3.4). Each successive fi ber break necessarily occurs at higher composite strains (viz. FAS) until fragmentation saturation is achieved (dynamic stress amplification, te sting rate effects, and fatigue -life possibilitie s are, for the moment, neglected). At satura tion, Kelly and Tyson [112] argue that lc and 1/2 lc bound the fiber length distribution. This critical length is obtained by substituting the fibers ultimate strength ult,f into Eqn. 3.3. That is, the critical fiber length is that leng th that will just allow the fibers ultimate strength to be reached, any longer and fiber failu re is assured. Note that this definition involves implied linear stress build-up in the fiber, perfec t bonding between the fiber and matrix, localized matrix yielding, and the exis tence of a single, cons tant fiber strength statistic. y r lf ult c (3.4) PAGE 200 The critical fiber length [ 112] depends on the matrix yi eld stress, fiber diameter, and ultimate fiber strength. In more recent SFCF tests, the fiber length distribution and critical fiber length are the tacit objec tives. The interfacial shear strength, ifss, is derived according to where lc is an appropriate averaged statistic from the FLD. It might be argued that Eqn. 3.5 represents an improvement relative to Eqn. 3.4 in that matrix yielding at the interface is not proscribed, a major limiting assumption for Kelly and Tysons original analysis [112], and also because a more discrimi nating critical length is employed. The apparent simplicity of Eqns. 3.3.5 belies the statistical and experimental difficulty attendant and obscures the necessary assumptions. We will separately consider the influence and validity of the various a ssumptions required to obtain Eqns. 3.3.5. Yielded matrix assumption Kelly and Tysons yielded matrix assump tion [111-113]: We must first determine the actual interfacial stress pattern at a fiber break to assess the assumptions validity. The first attempt to characteri ze the matrix stress state near discontinuous fiber ends is due to Cox [105]. Coxs approach apparently4 followed work on box beams by Reissner [127], which related the lagging flange axial displacement (wrt the distance to the web) relative to web axial disp lacement (Sections 6.26.6 [128]), to load diffusion5 from the web, and by equilibrium considerations, a non-ze ro shear stress distribu tion in the flange. 4 The lack of citations in Coxs article [105] is curiou s, albeit excusable, considering the extensive body of research which precedes his work and s eems to have influenced his analysis. 5 In the literature [133], diffusion refers to load transfer from a loaded fiber/stringer to the matrix (e.g., fiber pull-out, fiber push-out) and absorption [138-139] refers to load transfer from the matrix to a fiber/stringer (e.g., SFCF tests, compression tests). c f ult ifssl r, (3.5) PAGE 201 This is the etymology for shear-lag a nd as Nairn [129] notes, shear-lag is phenomenological and not an analytical appr oach as it is often misidentified [130]. The FAS build-up described by Coxs shear -lag model assumes perfect bonding or, at least, a uniform ability to transfer load acr oss the matrix/fiber interface (viz. frictional forces with no slipping) via shear stresses. Additionally, the model requires that the matrix and fiber lateral stiffness must be equa l. This last condition is well satisfied for paper systems but it is often violated for hi gh modulus ratio system s, especially carbon fiber/epoxy combinations. Implications for th is transverse mechanical disparity have been considered by Nairn [129], Feillard [131], and Zhou [132] and will be further examined below. For now we keep the shear -lag model one-dimensional, as it was first envisioned by Cox. [105] Coxs shear-lag model provides that the fi ber axial loading, P(z), for a hexagonally ordered fiber in an isotro pic medium is (Appendix B) Here Ef-m is the modulus difference for fiber and matrix, Af is the fiber cross sectional area, is the matrix strain (assumed uniform), and l is the fiber length [0, l ]. In most research the distinction betw een the modulus difference ( Ef-m) and fiber modulus ( Ef) is ignored owing to large modulus ra tios. The shear-lag parameter cox is given by f m f coxA E HEqn. 3.7, where 0 1ln 2 r r G Hm is a geometric factor for hexagonally spaced circular fibers, Gm is the matrix shear modulus, and r1 and r0 are the mean fiber 2 cosh 2 cosh 1 ) ( l z l A E z Pcox cox f m f (3.6) PAGE 202 separation and fiber radius respectivel y. The correctness and accuracy of H a Cox model approximation and an experimentally diffi cult evaluation, is addressed below. The mean fiber axial load is then The bracketed term (Eqn. 3.8) predicts a decrease in the fibe rs apparent modulus (reinforcement efficiency) as fiber length is reduced. Shorter fibers cannot reinforce the matrix as efficiently as lo ng fibers. Modulus ratios ( Eratio), defined by Cox [105] as the percent modulus difference ( Ef-m) effectively maintained for a given configuration, are given in Table 3.1 for various ga ge lengths. Values obtained he re are slightly higher than values given by Cox [105]; approximation or truncation errors for tanh() and filler fraction are suspected. It is interesting to look at the rate of ch ange in fiber axial lo ad (Eqn. 3.8) with variations in Ef/ Em and f because the fiber modulus a nd fiber volume fraction are often modulated in practical composites. As th e fiber modulus tends toward the matrix modulus ( Ef/ Em 1), the fiber will carry less load and its reinforcement efficiency paradoxically increases. Conversely, as the fiber volume fraction increases ( r1 r0), both the modulus ratio and fiber load increase In the limit, thes e cases represent the composite as either all matrix or all fiber, respectively. Modulus ratios for Coxs original exampl e of a paper compos ite (Section 8 [105]) and a simple graphite fiber/epoxy system (Tab le 3.2) are compared in Figure 3.5. Ef/ Em values range from 65 to 250 for practical gr aphite/epoxy systems (highlighted region in Figure 3.5). The paper reinforcement is genera lly more efficient than the graphite fiber 2 2 tanh 1 l l A E Pcox cox f m f (3.8) PAGE 203 reinforcement, primarily because the relative magnitudes of the paper fiber and its matrix are closer (cf. Eqn. 3.8). Equation 3.8 al so predicts that the graphite/epoxy system modulus ratio, Eratio, varies much more quickly with relative changes in fiber modulus when compared to the paper composite system. With higher volume fractions ( r1 decreasing), the rate of change in Eratio for the graphite/epoxy syst em is similarly greater than that for the paper system. Combining the two curves const m f ratiofE E E and const E E f rationm fE while sweeping f and Ef/Em generates a three-dimensional Eratio surface. For a given system, this volume is bounded on all sides by the physical cons traints for the composite. Namely, the composite is all fiber 1 fm f ratioE E E the composite is all matrix 0 fm f ratioE E E, the fiber modulus is equal to the matrix modulus 1 m fE E f ratioE and by m fE E f ratioE which is zero for all f as l /2 tends to zero [105]. The general shape of this Eratio surface holds for practical composite systems. Eratio always increases along iso-f ( Ef/Em 1) and isomodulus contours (f 0) (cf. dashed lines in Figur es 3.5 and 3.6). The rate of change in Eratio with Ef/Em or f depends on the composite system particulars and the independent variable under consideration ( Ef/Em or f, Figure 3.5). The third key variable in Eqn. 3.8 is the fiber fragment length, l We can generate a series of Eratio surfaces with different fragment lengths and thereby produ ce a three-dimensional state equation where f, Ef/Em, and l are the substate variables. Here Eratio is path independent and depends PAGE 204 only on the substate variables magnitudes. We will show below that the fragmentation length distribution is not indepe ndent of path or process. This graphical volume demonstrates that with increasing fragment length, Coxs model predicts that reinforcement efficiency is everywhere equal or greater than for shorter lengths. At low f or high Ef/Em combinations, reinforcement efficiency improvement is very slight or non-existent. In addition, for a given ( Ef/Em, f)i there is a maximum reinforcement efficiency Ei,m ratio (to a certain confidence limit) which is obtained at a limiting aspect ratio ( l/d )i,m. Ei ratio increases above this limit are insignificant and below ( l/d )i,m, the efficiency approaches asymptotically. Lastly, Ei ratio does not change for any ( Ef/Em, f)i when the aspect ratio exceeds a certain limiting ratio, ( l /d )limit. Inspection of Eqn. 3.8 likewise shows that if coxl /2 is large, the reinforcement efficiency tends to one. In addition, if coxl /2 is small then Eratio tends to zero and there is no effective reinforcement. A small coxl /2 implies either a large modulus difference ( Efm), a very short ga ge length (although Eratio sensitivity to l variations is less severe), or that the geometric factor H is small (viz. small f). The data in Table 3.1 and graphical repr esentations in Figures 3.5 and 3.6 are derived from Coxs shear-lag analysis, Eqn. 3.6, and as such are accurate with respect to the constraints/assumptions described by Cox [105]. The controlli ng assumptions stated by Cox are6: 1. Fiber load absorption is described by cox, Eqn. 3.7, where H is a constant. 6 These are the expressed assumptions in Cox (1952) [105], they have been reordered here for discussion purposes. Nairn [129] has formalized the minimum a ssumptions required to obtain the putative shear-lag Eqn. 3.6. PAGE 205 2. The matrix strain is homogeneous excep t for the local perturbation caused by fiber load absorption. 3. The fiber-matrix bond is perfect. 4. The matrix and fiber lateral stiffness are equal. Shear-lag assumption Looking first at the shear-lag assumption: It is clear that Coxs shear-lag parameter is a direct consequence of the assumpti on that fiber load absorption is given by dP/dz = H(u-v) assumption (4) (Appendix B). In other words, cox, which controls the rate at which FAS (Eqn. 3.6) builds in the fiber, is dependent not on exact el asticity analysis, but rather critically depends on the first simplifying assumption. It is not clear if Cox considered this assumption to be wholly de fensible or simply sufficiently accurate and expedient for his quick, almost back-of-the-na pkin, analysis for short fiber reinforcement (section 8 of Cox (1952) [105]). Whatever the case, Coxs work [105] has been widely referenced and utilized. While Cox is generally credited with in itiating and laying the groundwork for fiber reinforcement micromechanics/elastostatics, the more general problem of load diffusion from an elastic stringer to an elastic sheet, which has broad applicab ility to structures, was first studied by Melan in 1932 [ref. Mela n (1932) 133]. Muki and Sternberg [133] provide a concise histor y of stringer load diffusion devel opment as well as a rigorous solution to load diffusion from a rigid string er to a semi-infinite plate (Figure 3.7). Similarly rigorous treatments for load ab sorption of a stringer/fiber in a semiinfinite or infinite elasti c medium have been consider ed by numerous authors [108110,134-139]. Sternberg and Muki [138] not ed that load absorption understanding tended to lag behind both load diffusion knowledg e and successful practic al applications. PAGE 206 Considering the current ubiquitous use of strong-fiber reinforced composites and the continuing research on load transfer micromech anics, it does not seem that research has yet come abreast of application technology. As evidence of the c ontinued interest and importance of the subject, McCartney [140] Nayfeh and Abdelrahman [141], and more recently Nairn (1997) [129], fully 45 years af ter Coxs original publ ication (1952) [105], reconsidered shear-lag models, this time using exact elasticity equations and with special attention paid to the requisite assumptions to reach cox. As a means of comparison and as an intr oduction to Generalized Self-Consistent (GSC) analysis we consider here the wo rk by Dow and Rosen [109,137]. Utilizing GSC and a variant of Coxs shear-lag a ssumption (Eqn. B.1) Dow and Rosen [109-110,137] developed an alternate shear-lag solution for FAS (Appendix C). In GSC analysis nconcentric cylinders are assigned n different physical properties ( Ei i)i=1,n with the outermost ring having the effec tive composite properties ( Ec c)i=n Figure 3.8 depicts an n-ring model and the three phase a pproximation employed by both Dow and Rosen [109-110,137]. The primary advantage for this GSC approach is that the interphase and effective composite extents and moduli can be uniquely selected which allows for considerable experimental fitting. Unfortunately, the same versatility yields a solution that is sensitive to parameter error propagation. To allow for comparison between fiber lo ad absorption models, we restate Coxs fiber axial load solution, Eqn. 3.6, by translat ing the origin to the fibers midpoint (Figure 3.9, Table 3.3). Fiber axia l stress is then given on [l / 2 + l / 2 ] as 2 cosh cosh 1 l z Ecox cox m f f (3.9) PAGE 207 Additionally, composite axial loading equilibr ium renders the matrix interfacial shear stress (Figure 3.10), 2 cosh sinh 2 1 l z E rcox cox cox m f f z (3.10) By comparison, the FAS and matrix shear stress from Rosens GSC approach [110] are (Appendix C) 2 cosh cosh 1 l z Erosen rosen c f f (3.11) and 2 cosh sinh 2 1 l z Erosen rosen rosen c c z (3.12) respectively, where 2 2 2 21 2b a f m f f f b f b rosenr r r E E r r r E G (3.13) or f f b f b rosenr r r E G 22 if ra >> rb, and c cE It is apparent from Eqns. 3.10 and 3.12, and Figures 3.9 and 3.10 that the shear-lag solutions obtained by Cox and Rosen shar e the same general form. The primary difference between the two solutions is in their shear-lag parameters (cox vs. rosen, Eqn. 3.7 & Eqn. 3.13 respectively) and to a lesser extent the strain and moduli, (Ef-m,)cox vs. (Ec,c)rosen, values employed. The assumptions for the two analyses are very similar except there is an additional simplifying assumption for Rosens approach, which PAGE 208 requires that ra >> rb (Appendix C). In addition, the first assumption noted for Coxs analysis is restated as rz f b f ar r w w ) ( which because exact shear strain is given by z u r wrz implies that z u r w an implicit assumption for Coxs shear-lag analysis. Nairn refers to this inequality as the fundamental sh ear-lag assumption [129] and argues that in the vicinity of the fibe r break this assumption is nowhere valid, although it is approached at some distance from the break. To be fair, the Cox and Rosen analyses are admitted approximations to the exact solution and this fundamental shear-lag assu mption is the most obvious evidence of their approximate nature. Curiously, an exact an alysis suggested by Sadowsky (1961) [108] has been largely ignored, generally incl uded only in order to conclude that an approximate analysis is sufficient. Figure 3.11 compares shear stress predictions for the Cox, Rosen, and Sadowsky analyses (Eqn. 3.10, Eqn. 3.12, and [108]). Ordinarily one would expect approximate solutions to sa tisfy, or at least approach, the boundary conditions of the exact solution or physical problem; however, in th is case, neither the Cox nor Rosen solutions satisfy the equ ilibrium and symmetry boundary conditions for zero shear stress at the fiber end. This departure from the exact solution at the crack tip, in many cases, only slightly influences FAS mid-segment prediction (Figure 3.12). The deviation from the actual shear st ress pattern at the fiber tip is far more significant. The FAS predictions in Figure 3.12 indi cate that the exact and approximate solutions differ. The exact curve reaches a plateaus more quickly (no further load transfer), load transfer for the exact prediction is more rapid. However, this conclusion is qualified by the facts that both the Cox and Rosen solutions are sensitive to parameter selection, especially matrix extent ( r1 and ra respectively), interphase extent and modulus PAGE 209 ( rb and Ei) and composite ef fective modulus ( Ea). Determining the extent and mechanical properties (e.g., modulus, Poisson ratio, therma l expansion coeff.) of the interphase is no small feat The task is subject to considerable interpretation and has been approached from various directions (FTIR, DSC, DMA, etc.). The subject is deferred to a later section where empirical and analytical argumen ts will be explored. For now we maintain that these parameters are well described and address simply their mathematical influence on prediction. Despite the difficulties associated with pr operty evaluation, we assert that the differences in FAS shapes seen here fo r the Cox, Rosen, Tyson, and Sadowsky approaches (Figure 3.12) are relevant and these differences will lead to erroneous single fiber fragmentation interpretation and prediction. That is, the FAS build-up is critical to fiber fragmentation evolution and influences alternative energy absorbing mechanisms near the fiber break. This seems obvious e nough, and yet the Kelly-Tyson yielded matrix assumption [112] and Cox/Rosen models have been widely utilized with little regard for the implications [142-145] Yielded matrix assumption reconsidered As for Kelly and Tysons yielded matrix assumption (Figure 3.1), it seems that it was primarily arrived at via empirical consid erations, which suggested that there are high shear stresses near the fiber ends [113,146] and linear fiber pull-out vs. embedded length results (Figure 3.3) [112]. Now, if we look only at the embedded fiber pull-out tests (Fig. 14 [112]) and conclude that th e fiber pull-out load is well described by Eqn. 3.1, then both perfect fiber-matrix bonding and matrix inte rface yielding appear to be reasonable, even accurate, assumptions. There are, howev er several problems with this inference. PAGE 210 The first: Fiber pull-out stress vs. embedded length is not characteristically linear as Kelly and Tyson concluded. The relations hip is generally non-linear as Figure 3.13 indicates [147,148,refs. (6,7,11,12) [148]). Mo reover, Kelly and Tysons data may not be linear either, as there is noticeable non-linearity and insufficient data at longer embedded lengths to confirm a linear relationship (F igure 3.3). Experiments and analytical arguments point toward residual radial fiber clamping stresses, a result of differential thermal contraction during curing, and large di fferences in Poisson contraction as causes for this non-linear behavior [132,149,150]. Ne ither residual radial stresses nor Poisson contraction differences are ac counted for in the three simple load-absorption models considered thus far. Secondly, Kelly and Tysons own results ca st doubt on the validity of a simple yielded matrix assumption. Recognizing that th e predicted yield stress (Eqn. 3.1) is more than 20% smaller than the independently meas ured matrix yield stress (23 ksi vs. 30 ksi) [112] we must conclude that stress build-up in the fiber is not accurately represented by the simple Kelly-Tyson yielded matrix assumption. Finally, continuity and symmet ry conditions require that rzz = 0 = 0, which is definitely violated by an assumption that the matrix is everywhere yielded (rz = y). For uniaxial composite loading, the matrix shear stress outboard of the fiber is zero and by continuity must be zero at the fiber end. Additionally, the interfacial shear stress at the fiber end must tend to zero both ra dially and at the fibers midpoint. For Kelly-Tyson modeling the extent of th e yielded region at a given composite strain/stress is given by the recovery length, PAGE 211 This approach obviously bounds the shear stre ssed interface along the fiber, in this case approximating it by the matrix yield stress y. By design, the Kelly-Tyson model is onedimensional (axial) so we cannot address the radial shear stress decay. This onedimensional restriction might be viewed as an extreme case of shear stress radial decay. Clearly, Kelly and Tysons yielded matrix assumption introduces approximating errors into the FAS profile re lative to the exact solution pr ovided by Sadowsky [108] (cf. Figure 3.12). FEA results and othe r analytical approaches indicat e that the shear stress at the fiber tip is far from consta nt. It has been shown that the shear stress distribution at the fiber tip is a function of the fiber diamet er, tip and nearest neighbor geometry as well as the matrix/fiber bonding, modulus ratio, and thermal mismatch. The situation may be further complicated by matrix cracking, po ssible debonding and/or the presence of an interphasial region with physic al properties distinct from the bulk matrix and fiber. Unfortunately, if we wish to determine the quality or nature of the fiber-matrix adhesion using a simple Kelly-Tyson model we have already dictated the interface behavior. Therefore, any conclusion will be, at best, misleading if the load absorption profile was not originally we ll described by a simple one-d imensional case with yielded interfaces and all the other at tendant assumptions for a Kelly-Tyson analysis. This point has been largely ignored for many SFCF e xperiments [142-145,151] and almost certainly leads to false confidence in the resulting critical length/ifss value and its interpretation. Simulated and actual SFCF tests will be re viewed below using various simple and y f f c T K recr E 2 (3.14) PAGE 212 advanced load absorption models to highlight the implications of utilizing different load absorption models. Ineffective length Single fiber composite fragmentation mode ls and interpretations, which employ a yielded matrix assumption, often exclude th e linear FAS build-up (Figure 3.1) regions from subsequent failures (cf. Fig. 6 [6]). The lengths of these excluded regions depend on the matrix yield stress7 and the remote composite stress magnitudes. The excluded lengths would be different for the various FAS profiles depicted in Figure 3.12. This notion of excluded length is more commonly, if not ambiguously, referred to as ineffective length ( ). Rosen [110,137] developed the ineffective leng th concept to enable statistical composite strength predicti ons from simple solutions to the interface stress distribution (Eqn. 3.11. 12 and Appendix A [110]). In effective length is defined as that portion of the fiber on which FAS < k FASmax, where k is variously defined on [0.9.99]. The assignment of k has been arbitrary and even the name seems inappropriate in that the ineffective length is clearly actively involve d in load transfer. Regardless of the load absorption model chosen, is a dependent function of FAS build-up and it consequently depends on the a ssumptions used to obtain the FAS. The arguments relating to the accuracy and valid ity of FAS models discussed previously apply equally to With this in mind, ineffec tive length does provide a convenient measure of reinforcement efficiency and is proportional to Coxs modulus ratio ( Eratio). That is, lf/ Eratio or %ineff. = / lf = 1/ Eratio. The preceding discussion regarding Eratio and the effects of Ef/ Em and f can be applied to the inverse of the ineffective length. 7 The matrix yield stress is usually independently dete rmined via matrix tensile or shear strength tests. PAGE 213 Additional assumptions We have seen that the Kelly-Tyson, C ox, and Rosen approximate SFC models differ from the exact analytical solution primarily because of the assumptions these models make regarding the nature of the load absorption: namely, Cox and Rosens fundamental shear-lag assumption and Ke lly-Tysons yielded matrix assumption. However, there are other physical factors (i gnoring for the moment any statistical or empirical difficulties) which contribute to model deficiencies. For instance, all three mode ls neglect nearest-neighbor fiber stress perturbation influence (cf. influence functions describe d by Hedgepeth and Van Dyke [152]) despite the practical knowledge that r eal composite fiber failures ma y cluster [153], especially when high fiber volume fractions, low toughness matrices and/or str ong adhesion exist. It might be possible, however, to insert the effects of neares t-neighbor fiber stress perturbations in real composite modeling while continuing to utilize simple load absorption approximations. For instance, clus tering might be modeled as a statistical tendency for near-neighbor fiber failures [154155]. Locally increas ing FAS (artificial mechanical load-sharing rule [156]) over and above that predicted by equal load sharing (ELS) [157], might also be considered [ 152]. Unfortunately, bot h these approaches require, a priori, some understanding of the influence of near-neighbor breaks and would therefore also bias and restrict the prediction. In addition, Coxs analysis does not c onsider the influence of remote fiber reinforcing segments unlike Rosens GSC sc heme, which employs effective composite properties. This is seemingly equivalent to restricting the solution to very dilute fiber PAGE 214 concentrations8 and yet in the same analysis Cox makes use of high volume concentrations to approximate H [105]. It has been shown that Coxs analysis is increasingly inaccurate as fiber fracti on decreases [158], a consequence of the approximate shear-lag parameter cox. Perfect bonding between fiber and matrix is another major physical assumption all three models employ. Assuming perfect bonding is surely a mathematical convenience; however, several investigations have proven perfect fiber-matrix bonding is far from realistic in many practical composites [131]. Much of the recent SFC and real composite modeling has been concerned with more accura tely representing the quality of the fibermatrix interface adhesion while maintaining simple approximations to the interfacial stress state. Lastly, all three models discussed above ne glect any matrix hoop or radial stress contributions. It is well know n that most fiber composites experience dissimilar fiber and matrix cure shrinkage, which leads to non-zero, non-negligible radial stresses [149-150]. As for fiber-matrix adhesion, the research ha s focused on including radial stresses that have been variously attribut ed to residual cure shrinka ge and differential thermal contraction [159], dissimilar matrix/fiber moduli/Poisson contraction [150], and fiber surface irregularities [160], in load absorption models. Hoop stress contributions to strain energy (constant with respect to angular positio n) are nearly always neglected in favor of simplified one and two-dimensional treatments The effects of these assumptions/simplifications will be addre ssed below with respect to simple load absorption models as well as more advanced modeling. 8 A dilute condition applies when individual particles in a composite body under homogeneous boundary conditions are not influenced by neighboring or distant particles. PAGE 215 SFCF Advanced Interpretation We have considered some of the earlies t and simplest single and ideal fiber composite models (Cox, Sadowsky, Dow, Rose n, Kelly-Tyson) to establish and isolate the factors that influence load absorpti on in ideal composites. The accompanying assumptions have been shown to be rest rictive and oversimplifying, although not completely without merit. We will now disc uss more advanced load absorption models, especially as they address the aforementioned assumptions. Following Coxs paper (1952) [105] the interest in fi ber reinforced composite modeling has continued unabated for fifty y ears owing to, at first, the newness and perceived benefits9 and later as an analysis tool to enable evolutionary understanding and practical development. Muki and Sternberg [139] noted that the general load diffusion and load absorption problems have interested continuum mechanists for an even longer period. The range and number of approach es to load transfer micromechanics is impressive. No attempt will be made here to recount all of this work, only the most significant and pertinent st udies will be addressed. For instance, Outwaters (1956) [146] conclusion that high shear stress concentrations at discontinuities precipitat ed localized debonding in resin/fiber systems initiated a more realistic treatment of th e load absorption problem. Subsequent researchers, although not all, recognized th e fallacy of assuming perfect fiber-matrix bonding except for limited cases. With partia l or complete debonding, it is evident that fiber-matrix relative translation is accompan ied by frictional forces; and furthermore, frictional forces imply normal (viz. radial) tractions, which have been ignored in the simplest analyses. PAGE 216 Kelly and Tyson [111] argue that the normal matrix stress, r m, can be determined from a pseudo-energy balance between the radi al cure shrinkage forces and the tensile forces required to create the small gap (2 t ) between fiber breaks10. Then replacing y in Eqn. 3.4 with r m ( is the coefficient of friction between fiber and matrix at the interface) gives Eqn. 3.16, a corollary of Eqn. 3.4 that neglec ts transverse stresses, explicitly considers radial compressive stresses. This contra diction, coupled with the difficulty in determining limits Eqn. 3.16 utility. However, recognizing the influence of residual radial stresses on load absorption is a key element to more accurate SFC and multi-fiber composite modeling. Note also that load transfer across the interface via frictional stresses will generally be slower than that for an intact yielded interface [112] or an elastically strain ed interface [161]. The development of transverse stress dur ing axial loading wa s ignored in Eqns. 3.9.12 and in Figures 3.11.12. However, a tr iaxial stress state is inevitable provided there is a mismatch in component elastic c onstants (Cox assumes equal lateral stiffness thus insuring a uniaxial one-dimensional problem). With dissimilar elastic constants lateral movement is constrained and a tria xial stress state ensues. Cox and others 9 The earliest investigations were predominan tly concerned with metal matrix composites. 10 Only matrix strain energy is used here [111]. Ne ither fiber or matrix fracture mechanics nor Poisson contraction stresses are included. f z m ult r md t, (3.15) f r m z f ult cd t l 2, (3.16) PAGE 217 [105,108-110,112] were aware of this but for thei r analyses reasoned that the influence of any triaxial stress was negligible. For paper composite systems equal lateral fiber and matrix stiffness is justified as the component s have essentially the same composition, just different configurations. Similarly, Kelly and Tyson were primarily interested in metal reinforced metal matrix composites for which equal lateral stiffness is also justified. Dow [109] and Rosen [110] make certain that the approximate nature of their solutions are understood. They use th eir solutions only for explor ing the implications of component property variations. The problem is not the analyses proffered by Cox, Dow, Rosen, and Kelly-Tyson, but the extension of these results to composite systems where assuming equal lateral stiffness is cl early not justified (e.g., [162-164]). Multi-dimensional analysis In contrast to the load diffusion problem which received considerable attention dating to Melan (1932) (refs. [133]), load absorption analysis remained one-dimensional until Mooney and McGarry (1956) [ref. (22) 165] and Islinger et al. (1960) [ref. (23) 165] considered the transverse radial response aris ing from Poisson contraction, albeit while still ignoring circumferentia l stress. Mooney and McGarry [ref. (22) 165] derived a radial stress equation for a single fiber com posite where the matrix cylinder is much greater than the fiber volume, ra>> rb (Figure 3.1). In this cas e, the radial stress at the matrix fiber interface is given by ) 1 ( ) 1 ( ) 1 ( ) ( ) ( ) (2 2 m f m m m b a f m z m rv E E v v r r v v z (3.17) PAGE 218 where m z( z ) is the matrix axial stress. The sense of r is opposite that of axial stress m z and is therefore, for most fiber composites (mf), compressive. By comparison, Islingers [ref. (23) 165] multi-fiber composite solution, for which displacement continuity at rb is required, is given by Eqn. 3.18 predicts tensile stre sses at the interface, suggesting that the SFC radial stress poorly represents the multi-fib er systems radial stress. The radial contraction/tens ion seen in Eqn. 3.17Eqn. 3.18 is proportional to the matrix axial stress m z and the Poisson difference (m-f). Subsequent numerical and FEA investigations have confirmed this f undamental relationship [166-168]. Ebert and Gadd (1965) [165] looked at radial and ta ngential stresses for the less common case where the matrix is stiffer and stronger than the fiber ( Em > Ef and Y,m > Y,f). When both components are elastic, the sense of the matrix radial stress is opposite both the axial stress and the transverse tangential stress. Th at is, compressive radial stresses and tensile hoop stresses, consistent with Mooney and McGa rrys single fiber composite model, Eqn. 3.17. Ebert and Gadd [165] considered only continuous long fibers so axial stresses in the fiber and matrix are constant. 3-D continuum treatment Sternberg and Muki [138] provided the fi rst rigorous three-dimensional treatment for a fully bonded (including term inal cross section) semi-inf inite fiber in an infinite matrix, Figure 3.14 (cf. Figure 3.1). This l eads to non-zero axial loading and an infinite ) 2 1 ( ) 1 ( ) 2 1 ( ) ( ) (f f f m f m m f m z m rv v E E v v v v v z (3.18) PAGE 219 slope at the fiber end (Figure 3.15), whic h according to Sternberg and Muki [138] suggests a tendency for fiber-matrix dis bonding or localized matrix yielding. Bonding across the fiber end may be represen tative for short fiber reinforcement at small strains but is not suitable for SFCF mode ling. Muki and Sternberg [139] advanced an analysis more compatible with SFCF phys ical conditions. The filament here has two disjointed but contiguous semi-infinite segmen ts (Figure 3.16). As for their preceding article [138], Muki and Sternbe rg [139] approximate the filament as a one-dimensional elastic continuum. Unfortunately, this di sregards the likely fiber-matrix Poisson contraction mismatch, although the three dime nsional matrix treatment does allow for radial and hoop stresses. As expected, in accord with the free surface at z/a = 0, the fiber load at the break is zero. In contrast to the solutions provided by Dow and Rosen, the fiber load absorption curves at the fiber brea k/end have infinite sl ope, again suggesting fiber-matrix debonding or matrix yielding. Muki and Ster nberg [139] acknowledge the three-dimensional Stress Concentration Factor (SCF) at the fiber end; however, owing to their one-dimensional fiber approximation, th ey argue that evaluation of the SCF would be neither accurate nor justified. As w ith both Cox and Rosens simpler approximations, this analysis predicts that load transfer efficiency decreases with in creasing modulus ratio, Ef/Em (Figure 3.17, cf. Figure 3.5). The rigorous continuum mechanics appro ach undertaken by Muki and Sternberg [139] represents an improvement relative to the simpler Cox and Rosen models, which make sweeping assumptions regarding load transfer resulting in inadmissible stress states. The main difficulties for Muki and Sternbergs analyses would seem to be the one-dimensional (isotropic) approximation for the fiber, especially considering the PAGE 220 appreciable affect seen for variations in matrix Poisson ratio (Figure 3.18) and the loading singularity at the fiber end. The singularity at the fiber tip predicted by Muki and Sternberg [139] (Figure 3.17) cannot exist in a real composite, so the acute stress concentration must be mitigated by one or several energy consuming processes. The most obvious are ma trix yielding, fibermatrix debonding, viscoelastic deformation, and matrix cohesive failure. Deviations from the ideal physical conditions (e.g., non-square fiber ends, inhomogeneous, anisotropic, or inelastic matr ix properties) are also possibl e. These conditions may be accretive or reduce the matrix stress along an intact fiber. To this point, we have seen a number of analyses that approximate the load absorption profile all of whic h produce slightly different re sults at the fiber end while maintaining the same general trends for variatio ns in modulus ratio, recovery lengths, and ineffective lengths. In particular, at the fiber end, we have seen non-zero shear stress with a large negative slope (C ox), constant non-zero shear strength (Kelly-Tyson), zero shear stress with a finite sl ope (Dow and Rosen), and fina lly zero shear stress with a positive infinite slope (Muki and Sternberg). Clearly, we have a crisis of confusion regarding load absorption at the fiber break. The difficulty and variation seen in this early work arises from anal ytical solution sensitivity to proscribed boundary conditions as well as the attendant assumptions requi red to simplify or enable the analyses. Numerical/Empirical evaluation The complex three-dimensional stress states at discontinuities or inhomogeneities are most readily analyzed utilizing numerical or empirical methods as opposed to closedform analytical solutions. The case for shor t and discontinuous fiber composites, where discontinuities and inhomogeneities are unavoid able, is no different. These alternative PAGE 221 methods were largely unavailable to early re searchers in this field (Cox, Sadowsky, and Dow) owing to the scarce use of computers, relative infancy of computer aided numerical evaluation (FEA, finite difference), and underutilized empirical approaches (e.g., photoelastic analysis or Moir e interference techniques). Following Outwaters [146] id entification of large shear stresses at the interface, possibly in excess of the matrix shear or matrix/fiber adhesive strength, Tyson and Davies [113] utilizing photoelastic techniques considered the simplest case for a perfect interface11 and dilute filler concentration (eliminating near-neighbor influences). Tyson and Davies [113] concluded that the theory significantly underpredicted the shear stresses at the fiber end (Eqn. 3.10, Eqn. 3.12, and Figure 3.11). Maclaughlin [169] also employed 2-D photoelastic methods to reach similar conclusions. These investigations were quickly followed by Iremonger and othe rs using both FEA and visual methods [(FEA) 170-174; (Photoelastic) 175; (Moir) 175 ]. Despite considering very different modulus ratios (2,3000) and volum e fractions (0.01,0.23) these early studies a ll identified maximum stress concentrations at the fi ber break/end and especially intense concentrations for smaller gaps between fiber breaks and for square-end fiber geometries. In addition, Carrara and McGarry [ref. (3) 174] found that gradually tapered tips produce the smallest shear stress concentration. Th e fiber end geometry affects the local stress concentration [146,169,175] and by nature of the load absorption profile, the composite reinforcement efficiency. 11 A perfect interface involves continuous tractions and displacement continuity. PAGE 222 Considering only dilute part icle concentrations, which isolates component property influences and allows fiber/fiber interactions to be neglected, Russel [176] formulated an equation for the composite longitudinal Youngs modulus12 ( Ec long) such that where the variables and the order of operator O( ) have their usual meanings and A = f ( l / df, Ef/ Em, f, m). Russels slender-body approxima tions, adapted from dilute fluid mechanics [refs. (4-7) 176], showed that A increases as the aspect ratio and Ef/Em increase (cf. fig. 2 [176]). Russel employs an ineffective length definition A/A0 = 0.9, where A0 is evaluated at l/df = Russel [176] also demonstrated that reinforcing efficiency for a given ( Ef/Em, f, m) combination increases in order for blunt/square, prolat e spheroidal, and double cone end geometries, which is consistent with earlie r FEA investigations [r ef. (3) 174]. It is also evident from Russels data that th ere exists, for a given aspect ratio ( l/df)i, a critical modulus ratio ( Ef/Em)cr above which any further increase does not increase the reinforcement level (Ec) (cf. discussion for Cox and Rose ns reinforcing efficiency). That is, there is a practical composite modulus limit for as pect ratios less than some arbitrarily large number. Barker and MacLaughlin [174] using FEA show ed that for square fibers in a square array the matrix maximum principal stress near a discontinuity is it self at a maximum and continually increases as the gap decreases (F igure 3.19). Concurrent work by the same authors [177] on an ideal 2-D five-fiber mode l indicates that the maximum shear stress concentration is even more acute than the maximum principal stress (Figure 3.20). The 12 Composite longitudinal modulus exhibited maximum sensitivity to property variations. )] ( 1 [2f f m l cO A E E (3.19) PAGE 223 stress concentration along the fiber-matri x interface predicted by early FEA [174-175] (defined here as max > 1.5comp) extends approximately 10 df inboard of the fiber end and even less into the unfilled matrix radially (5 df). This small extent has been generally confirmed by FEA and numerical technique s [166-167,178-181]. These analyses also compare favorably with ineffective length cal culations from Cox, Rosen, Muki and Sternberg, and Phoenix models [105,110,138,139, 142], although, as we will show below, the Cox and Rosen calculations be nefit from fortuitous coincidence. While the extent of the stress concentration region is limited, the severity (especially for lager fiber-matrix modulus ratios) of the SCF is in accord with the mathematical si ngularity predicted by Muki and Sternberg [139]. The high stra in energy near the fiber break cannot be elastically supported near the fiber break, suggesting matrix yielding, cohesive failure and/or disbonding. FEA by Iremonger and others [170,173] also indicates that the fiber end SCF adversely influences near-neighbor fibe r loading. Specifically, for gaps 2 df or larger, the nearest neighbor fiber e xperiences a 150% increase in surfac e loading. The extent of the affected region is small (~5 df), but considering high surface flaw populations, it does evince fiber fracture clustering. Lastly, near neighbor fibers appear to laterally constrain the matrix, thereby slightly increasing the fiber maximum principal stress and matrix shear stress (obvious shoulde r in Fig. 9b [173]), which is counter to many model assumptions that isolate the single fibe r in an expansive or infinite matrix. Specifically, Nayfeh [141] has shown anal ytically that the single fiber pull-out stress fields differ for laterally free (plane stress, z = 0) and laterally constrained (plane strain, z = 0) boundary conditions. Moreover, Nayfeh argues that SFCF tests are PAGE 224 consistent with laterally free BCs and laterally constrained BCs are more in-line with macro-composites. Radial stresses are compressi ve in the former and tensile in the latter case, consistent with the solutions provi ded by Eqn. 3.17 and Eqn. 3.18 previously discussed. Therfore we cannot assume an elastic-perfectly pl astic matrix without introducing uncertainty into our evaluation. Th e impact of this assumption may be small, however, compared to statistical and empirical difficulties for SFCF tests. The evidence supporting acute stress con centrations near fiber ends and fiber breaks is considerable and incontrovertible the question remains however, considering the relatively limited extent (~10 df) of the stress concentration region, what is its influence on ideal SFC and macr o-composite performance. As we have seen, the nature of the stress pattern at the fiber end can affect FAS buildup (cf. Figures 3.12 and 3.15) and consequently the composite reinforcement efficiency. Even still, many researchers have employed Coxs simplistic analysis to predict composite strength/modulus properties for which the physical confi guration/ analysis is incompatible If fiber loading via simple shear across a yielded interface is not the case, how exactly is the fiber loaded? Various investig ations (empirical, analytical, and numerical) have identified a number of mechanisms th at control fiber loading along with energy absorbing mechanisms that can delay or re duce FAS build-up. These mechanisms may or may not operate simultaneously or in all fi ber-matrix systems. The processes include interface debonding, matrix yielding, frictional fo rces and slipping, shear-lag loading, and transverse and collinear matrix microcracking. An intermediate interphasial region, if it exists, will also affect FAS build-up. PAGE 225 Interfacial Debonding We have considered elastic-perfectly plas tic matrix behavior via Kelly-Tyson type analyses wherein perfect bonding exists along th e intact fiber fragme nt; however, given the empirically observed and theoretical acute stress concentration at the fiber end, it is evident that fiber-matrix disbonding is probabl e and therefore must be examined for most composite systems. Failure to include interfacial debonding e ffects can lead to considerable interpretation errors, especially wi th regards to critical and recovery length calculations. The debonding criterion can be taken as either energetic (viz. strain energy/fracture mechanics) or as a simple strength criterion (ifss > debond). The two criteria while providing the same general pe rformance characteristics differ slightly with regards to FAS build-up. This is especi ally true for fiber pull-out and push-out predictions for which interfacial debonding foreseen by energetic means is preferred owing to its accuracy and also because it is more consistent with the actual failure mechanism [182]. General discussion Gershon [183] in an early review of com posite fracture toughness notes that while composite strengths are usually a function of the constituent material properties (i.e. expressed by a rule of mixtures), the fracture energies relate to the composite structur e and to the fracture mechanism. In particular, the principal possible fract ure mechanisms and corresponding equations available for fiber-matrix composites ar e given in Table 3.4 (Eqns. 3.20.25). The total work of fracture for a composite is the summation of the discrete processes in Table 3.4. Note that all mechanisms may not c ontribute significantly to the total work of fracture ( Wf) and furthermore Gershon and othe rs [183-185] assert that fiber PAGE 226 pull-out during failure is th e dominant energy absorbing process for glass, carbon, and aramide fiber-polymer matrix systems. The original form for pull-ou t work, attributed to Cottre ll (1964), is the work done against friction over the pull-out length lpo, d Rpo f f po 22 (3.20) Eqn. 3.20 is obtained when lpo = lc and by using Eqn. 3.22. For fragment lengths greater than the critical length, the pullout work decreases according to c c f f pod R 62 (3.21) These two equations for pull-out work along with that for the case where lpo < lc, d T Rf f po62 (3.22) are presented in Figure 3.21. The assertion that pull-out work dominates the total work of fracture presupposes that lpo = lc (dashed line in Figure 3.21), and clearly overestimates the pull-out work contribution in most cases. The debonding toughness given by Eqn. 3.23 (Table 3.4) is also questionable. In Eqn. 3.23, the debonding toughness is actually the fiber strain energy at failure in the debonded section divided by the co mposite cross sectional area. Besides the point that the descriptor debonding toughness is misleadi ng, Eqn. 3.23 is further in error because it assumes that the strain energy per cross se ction is constant across the debonded region, which as we have shown is not valid owi ng to non-linear FAS build-up. Additionally, the work of fracture contribution from de bonding should be the rela tive difference in strain energy between a bonded intact fi ber and a fractured disbonded fiber. An PAGE 227 alternative and more prec ise debonding toughness has been proposed by Kim and Mai [185]. 22 f d d f dE R (3.24) where d is the debonding stress. The stress redistribution work (Eqn. 3.25) is also confusing, as some authors believe that this work is implicitly counted in the debonding toughness [183,184]. Mathematically, the stress redistribution work is 2lc/ ld times debonding toughness; again intimating its ambiguous definition and furthe r suggesting that both stress redistribution and debonding toughness, Eqn. 3.24, should not be simultaneously considered. A less confusing approach developed by Marston et al. [186], termed Total Toughness, asserts that composite toughness is a combination of fiber pull-out ( Rpo, Eqn. 3.20Eqn. 3.22), new surface generation ( Rs), and stress redistribution ( Rr, Eqn. 3.25). r s po TR R R R (3.26) where Rs includes the energy required to create new fiber and matrix surfaces ( Rf and Rm), and the fiber-matrix interface Rif. if c f m f f f sR d R R R ) 1 ( (3.27) Taking Rif Rm and neglecting the fiber contribution, Rs becomes [185] m c f sR d l R 1 1 (3.28) Assuming Rif Rm is questionable, a point th at will be addressed later. The aforementioned toughness measures are all derived from linear elastic properties and are, in principle, reversible processes. Howeve r, we have noted that acute PAGE 228 stress concentrations near fiber ends are likely to lead to elastic-pl astic behavior (KellyTyson modeling assumes elastic-perfectly plastic) so an elastic treatment of the failure process can not fully account for all the en ergy consumed during composite fracture and in particular fiber-matrix de bonding. Indeed, fiber-matrix debonding in simple SFCF is decidedly inelastic just as for macrocomposites. It should also be noted that a propagati ng matrix crack intersecting a fiber-matrix bi-material interface at a right angle may either induce cohesive (transverse fiber fracture) or adhesive (longitudinally deflected along inte rface) failure, a result of the crack tip stress concentration and elastic constraints at the interface [187]. Transverse propagation implies fiber failure clustering and possi ble catastrophic composite failure while deflected or interfacial debondi ng suggests toughening (blunti ng the propagating crack). Inelastic deformation in this damage zone will also contribute to composite toughening/energy loss. Just as for composite structures, the de bonding and failure criteria for simple SFCF, fiber pull-out, and fiber push-out tests can be interpreted using either strength or energy based methods. Mullin and others [188,189] discussed and induced simple fracture modes in low and moderate f composites that indicated the effects of remote reinforcement, strain rate, fiber-matrix adhesion, matrix shear strength, and matrix tensile strength on fracture modes and composite ultima te strength. This work [188,189] clearly indicates that complex fracture pa tterns are possible even in low f composites and especially in moderate to high f composites, which further underscores the necessity to account for these energy consuming mechanisms to assure accurate strength and SFCF modeling. PAGE 229 Single fiber pull-out Lawrence [190] following a simple fibe r pull-out model suggested by Greszczuk13 (1969) [191] used shear-lag assumptions and a shear strength criterion (max s, s is the debonding shear strength) to predict the fibe r loading required for complete debonding and pull-out ( l /2 zmax). where Pf is the pull-out load for an infinitely long fiber with no frictional forces, f is the interfacial frictional stress (constant), zmax is the maximum bonded length before catastrophic debond failure, and other variab les are as described in Appendix C and Figure 3.22 As s/f or as the debond length ( l/2 zmax) 0 the frictional contribution (second term in Eqn. 3.29) can be neglected an d we are left with si mple shear-lag loading and catastrophic debond failure ( zmax l/2 ). This is equivalent to the original form (cf. Eqn. 3.6) given by Greszcz uk [191] (solid curve (s/f = ) in Figure 3.22). Note also that deviations from Greszczuks solution (solid line) occur at azmax for a particular shear debond stress and inte rfacial frictional stress (s/f) ratio. Most notably, Eqn. 3.29 predicts an unbounded pull-out load for s/f = 1 (~no debonding). Takaku and Arridge [192] extended the pul l-out analysis by in cluding resin-fiber cure/thermal mismatch as well as fiber Poiss on contraction during pull-out. These radial hydrostatic stresses oppose one another and give rise to a non-linear variation in debonding and pull-out stresses vs. embedded fi ber length (Figure 3.13). A rapid and substantial drop in fiber load during disp lacement controlled fibe r pull-out indicates 13 The Greszczuk model [192]assumes catastrophic de bonding and frictionless pull-out once the critical max max max2 tanh z l a az P Ps f f f (3.29) PAGE 230 complete debonding [192]. Additionally, pa rtial debonding is suggested by stepwise loading before maximum loading at the debonding stress. A key element supporting Takakus proposed failure mechanism is eviden ced by a linear drop in pull-out stress as fiber pre-load increases (Fig. 6 [192]). Hsueh (1990) [148] reconsidered Takaku and Arridges [192] simple shear-lag analysis, which employed a relatively crude fric tional contribution. This analysis [148] allowed for a distinction between the stress le vels during fiber pull-out, namely the initial debond stress14, partial debonding stress, complete debonding stress, and fiber pull-out stress (Figure 3.23). While Hsuehs analysis [148] is still primarily a shear-lag approach, the frictional contribution is more rigorously considered thus enabling good agreement with regards to debonding and pull-out stresses compared to Takaku a nd Arridges fit (cf. Figure 3.13 and Figs. 5 and 7 [148]). Note how ever, that while the shear strength based pull-out analysis provides fo r a monotonic increase in d and eventual plateau against embedded length (Fig. 4 [148]), a constant debond stress is employed in predicting d and po. Here Hsueh [148] argues that the ex act interfacial debond criterion is not well established. A curious conclusion and appro ach considering the analysis is predicated on a shear strength criterion. Finally, a st able debonding process is assumed by Hsueh, which should preclude its use for composite systems with unstable debonding. This restriction can be alleviated by using an appropriate energy criterion [182,193]. An alternative analytical treatment for fiber pull-out employs linear (elastic) fracture mechanics (LFM) to predict fiber-m atrix debond initiation and propagation, and fiber pull-out work against fric tion. As noted previously Co tterell [194] first described shear strength at the fiber exit s is reached. PAGE 231 fiber pull-out energy. Outwater and Mu rphy [ref. Outwater (1969) 195] adapted Cotterells formulation specifically for singl e fiber pull-out experiments and after some approximating they obtained for the critical debonding stress In Eqn. 3.30, G is the strain energy release rate and Pd is the critical debonding load. Gao et al. [149] refined Outwater-Murphy (Eqn. 3.30) using shear-lag type assumptions to account for frictional fiber loading and fiber Pois son contraction effects. In this case, the debond criteria is given by the integr o-differential equation (Appendix D). The first term is a restatement of the complia nce-strain energy release rate relationship15, with A = 2rl and C = -uf(0)/P The second term in Eqn. 3.31 accounts for the frictional contribution by creating a fricti onal interfacial shear stress, f = (q0-q*) q0 is the initial interfacial radial pressure and q* is the Poisson cont raction pressure. uf(0) and v(y), the axial displacement at the fiber end and radial displacement are separately derived from shear-lag considerations (Eqns. 2.14 and 2. 15 [149]). Equation 3.31 then reduces to 14 Kim and others [194] later referred to this as crack tip debond stress. It is also sometimes called the frictionless debond stress. 15 Note that Eqns. 3-30 and 3.31 implicitly assume that Gic is constant for different debond lengths (Figure 3.26). f f f d dr G E r P 22 (3.30) 0) ( 2 1 ) 0 ( 4 dy y v u r P Gf f (3.31) A C P Gd 22 (3.32) 2 3 2) 1 ( ) 2 1 ( ) 1 ( 4 Q P K G E rf ic r (3.33) PAGE 232 where Q and are defined in Appendix D. Furt her simplification provides the debonding load. where K Pd *, and P0 (frictionless debond load) are gi ven in Appendix D. From this debonding criteria an initial compressive pressure limit qth can be found, above which frictional loading contributes during debondi ng and, below which the fiber Poisson contraction renders zero fric tional loading during debonding. When the initial pressure is less than or equal to the theoretical pressure ( q0 qth) the frictionless debond load P0 equals Pd The remote debonding load (Eqn. 3.34) is independent of debond length ( ld), or equivalently P = P0. By contrast, Hsuehs [148] analysis suggests that the debonding load is a function of both the debonded length and matrix/fiber shear bonding strength. If however, q0 > qth, then Eqn. 3.34 predicts that P will increase and eventually plateau with increasing ld. When q0 > qth (Figure 3.24.a and Figure 3.24.b) partially stable debonding between points A and B is expected in a stepwise ma nner, culminating in complete debonding at position B ( Pd *). The magnitude of the load drop upon complete debonding ( Pd *-Ppo) depends upon the relative difference between s and f 16, and the embedded length. Frictionless debonding ( q0 > qth) is shown in Figure 3.24c. Here a constant partial debond loading ( P = Pd *) is seen from the initial de bond (A) to complete debonding (B) followed by a drop to the frictional pull-out load Ppo (C). Kim et al. [195] also identify a totally stable debonding condition (Figure 3. 24d), wherein the co mplete debonding load 16 Recall that f = (q0-q*) where q* is a non-linear function of positio n, Poisson contraction, modulus differences, volume fractions, and geometry. K e P P P Pd d 1 ) (0 * (3.34) PAGE 233 is equal to the frictional loading. This t ype of behavior is pos sible for glass-ceramic composites [193] where bonding is weak and residual compressive forces are high. Consideration of the maximum su stainable load in the fiber ( Pc) leads to a definition for the critic al fiber debonded length ldc, Gao [149] refers to those fibers where Pc > P* as strong and those with Pc < P* as weak. This definition leads to no fiber fractures in unidirect ional strong-fiber composites during fiber pull-out, and multiple ( lc/l ) failures in weak-fiber composites, depending on the location of the fracture pl ane relative to the fiber ends [196]. Gao also defines a characteristic debonded length, lT is a measure of the friction affected z one during debonding and is approximately the length where debonding stress plateaus. The characteristic debonded lengths for uncoated and coated wires are lT = 18 and 95 mm, respectively [192], which agree with experiment (cf. Fig. 6 [149]). Both Gaos (LFM) and Hsuehs (strengt h criterion) [148,149] predictions for frictionless debond stress, fric tional stress, and partial debond stress against debond length ( L-z Figure 3.25) are depicted in Figure 3.26. Both models can be reformulated to the general form c d c dcP P P P K* 01 ln 1 (3.35) ) (m f f Tr (3.36) PAGE 234 where d is the initial debond stress, L is the plateau debonding stress for long L, and is the reciprocal effectiv e shear transfer length ( = 2 k / a is the coefficient of friction, and k is a geometric and strength factor, Appendix E). Both approaches work well for composite systems w ith stable interfacial debondi ng. That is, the partial debonding stress d p continuously increases and the maximum debond stress d is reached at complete fibe r debonding (Figure 3.24.b, L > zmax, partially stable; Figure 3.24.d, zmax 0). Frictionless debond stre ss is independent of fi ber length for Gaos LFM analysis (an explicit assumption in [149]). Gaos LFM analysis is also unable to account for debond growth instability, which for Hs uehs analysis [148] is provided by the instability condition dd p/ dz 0. For Hsuehs approach, unstable debondi ng is initiated at the maximum debond stress (bonded lengt h at instability = zmax, see Figure 3.26) which in this case is necessarily greater than the stress at complete fiber debonding. Note also that the partial debond stress at complete debonding is equal to the in itial frictional stress (d p = po) as required by equilibrium conditions. Karbhari and Wilkins [ref. (24) 193] approximated the criterion zmax for totally unstable ( L zmax, A = B in Figure 3.24.a), partially stable ( L > zmax, Figure 3.24.a), and totally stable debonding ( zmax 0, Figure 3.24.d) by neglec ting Poisson contraction and assuming constant frictional traction. Ld L d p d exp 1 (3.37) 0 *dz d p d dp d 2 1 1 max1 cosh 1 f sz (3.38) PAGE 235 Despite disparate interfacial and composite properties, zmax can be quite small for composite systems (e.g., zmax(CF-epoxy) 0.15 mm and zmax(SiC-glass) 0.06 mm, Table 3.5, and [193]). While zmax tends to zero for both the CF-epoxy and SiC-glass systems, only the latter system displays cl assic totally stable behavior at room temperature (recall both Gao and Hsueh assume stable propagation). Consequently, for SiC-glass systems there is excellent agreemen t between theory and experiment. That is, for this SiC fiber-glass composite system the debonding and pull-out mechanisms are consistent with the proposed model, however for the CF-epoxy system tested Gaos model over predicts the de bonding stress, especially at short embedded lengths. For short embedded lengths, Lawrence argues that if s/f cosh2al/2 then debonding is catastrophic once it is initiated [ 190]. From this argument, it would seem that for very short embedded lengths (relative to transfer length and fiber diameter) perturbations and likely inelastic matrix de formation would influence apparent debonding strength. That is, while both Gao and Hsusehs models suggest stable debonding ( zmax(CF-epoxy) 0.15 mm) the actual debonding init iation and propagation is more stochastic than determinative. Data from Kim et al. [193] also reve als that Hsuehs model under predicts maximum debonding stress as L increases. To bring prediction in line with experiment s must be increased substan tially (e.g., 65 MPa vs. 43.5 MP a, steel-epoxy system) which means that s is very nearly equal to the epoxy ultimate tensile stress (ult 66.5 MPa). This is indicative of sizable nonlinear plastic deformation in the process zone and further emphasizes the need to take into account matr ix inelastic deformation, as the proposed PAGE 236 failure mechanism (elastic interfacial failure at s) is not consistent with the actual failure process. Similarly, Gaos artificial restriction that Gic remain constant as the debond length changes violates expected R-curve behavior That is, since model fiber composites approximate plane stress (laterally free) conditions [141], and for plane stress conditions17, crack resistance generally increases with crack growth [197], a constant Gic clearly oversimplifies the process. Interestingly, the interface/matrix stress field in macro composites approaches a plane strain (laterally constrained) c ondition [141], which implies a constant KIc, independent of crack growth ex tent. So, while a constant Gic assumption is apparently invalid for single fiber pull-ou t experiments, it may hold for fiber pull-out in practical composites. This raises the obvious question, if the fiber pull-out process differs for model and practical composites, is it even appropriate, assuming the model process is well described, to utilize mech anical properties derived from simple systems to predict behavior in more complex practical system s. Obviously, the accuracy and validity for any particular model relies on its ability to predict or simulate the genuine mechanical process to an extent such that any appr oximations can be neglected or sufficiently accounted for in the final prediction. Zhou and others [182,198] made modificat ions to Gaos [149] LFM based fiber pull-out model that removed the restrictiv e stable debonding and constant crack tip debond stress conditions. Zhous model include s the strain energy c ontributions for the bonded and debonded regions (fiber and matrix ) as well as the matrix shear strain 17 Plane stress, in this case, is equivalent to the stat ement that the extent of the plastic/process zone is no longer small compared to the matrix-fiber diameter ratio. PAGE 237 energy18. The resulting crack tip debond stress is sensitive to the debonding extent and this allows for possible debonding instability d d p/dz 0, similar to that seen in Hsuehs shear strength model. Additionally, this gr eatly improves the predictive quality for short embedded lengths (Fig. 4 [182]), especially w ith regards to Gaos LFM analysis [149]. The instability conditions with respect to embedded length and zmax are as described above. Zhous LFM analysis [182] is clearly superi or and more representative than that proposed by Gao [149]. However, it is difficu lt to definitively conclude that either an LFM or strength based criterion is more su itable for determini ng partial debond stress (initial debond, maximum debond, and initial fri ctional pull-out stresses) as both models require evaluating Gic, q0, f, and s, which are difficult to determine. Clearly, both approaches involve simplifying assumptions an d cannot realistically expect to represent the pull-out process in all its detail. Zhou et al. [182] argue that an LFM based criterion is more suitable because it deals with a more fundamental problem of macroscopic material behavior, whereas, the shear st rength criterion cannot account for debonding instability19. SFCF Modeling Multi-Modal We have previously discussed the early sing le fiber load transfer shear-lag models of Cox, Rosen, and Kelly-Tyson without re gards to fiber-matrix debonding. A great proportion of the work following these simple load transfer models has focused on the nature of the fiber-matrix de bonding criterion, and in partic ular the influence of the interface and/or the interphase. Over time the modeling has grown increasingly complex 18 Gaos model neglects both of these terms [149] 19 The instability condition for Hsuehs model arises from the debonding stress functional dependence on PAGE 238 allowing for imperfect adhesion, non-linear matrix behavior, a distinct interphase (chemical, morphological, and mechanical), stoc hastic processes, asperity interaction, complex failure processes, and dynamic effects. The body of work is substantial and convoluted, sometimes repetitive and simplistic. The empirical work has aimed to characterize and eluc idate the fundamental and critical physical properti es, which the analytical re searchers employ to predict practical and ideal composite performance. This relationship between theory and practical/empirical work seems obvious, but in this case, the interpla y between empiricist and theoretician is especially critical owing to the large number of variables involved in composite mechanics. In the authors view, the simplifying assumptions involved in some analyses obscure and obfuscate the na ture of the composites performance, and provide misleading or false confidence in the material and mechanical properties. The evolution of SFCF modeling20 followed a natural path from the simple Cox, Rosen, and Kelly-Tyson approaches to bi-m odal models which separate the fiber into two regions, generally bonded and disbonded. The Kelly-Tyson approach might be considered the first of this family with its yielded matrix fiber ends and zero shear central portion. Piggotts early work (1966) [161] refined Kelly -Tyson by considering both a plastically yielded terminal section and a cen tral region which loaded according to Coxs treatment (Figure 3.27). Piggo tt observed that elastic load transfer provides only limited composite strengthening and that some stress tr ansfer at least must occur through plastic deformation of the matrix [161]. Piggott [161] demonstrated that the composite strengthening attributed to pl astic loading increases with the extent of the plastic zone debond length. 20 We distinguish here between generalized load transfer modeling and that specifically derived for SFCF PAGE 239 (Figure 3.28) and also that reinforcement efficiency plateaus at both small and large aspect ratios (not shown). It is also apparent from Figure 3.28 that the rate of increase in reinforcement efficiency with an incremental pl astic zone increase is less for stiffer fibers than for weaker. Piggotts results did nothing to dispel the main tenet of Kelly-Tyson and likely gave subsequent researchers conf idence in the oversimplified Kelly-Tyson analysis. Now while numerous researchers recogni zed the likelihood for incipient debonding and propagation from the fiber end duri ng load transfer [183,187-189,199-202], and related fiber pull-out mode ling clearly suggested an avenue for evaluation [130,147,190,192], the first delineated bi-lin ear models for SFCF (1989) [5-6,203] lagged well behind generalized load transfer modeling and its pull-out counterpart [149,196,199,204-211]. This was due, in part, to a preference for the pull-out test over SFCF because of the pull-out tests perc eived simplicity, representative nature, and an apparent reluctance to complicate the simple SFCF interpretation (Eqn. 3.6). Still, considering that Cox first proffered his simple shear-lag approach in 1952, it took nearly 40 years to correct the Cox and Kelly-Tys on perfect bonding assumptions used for SFCF interpretation. Stochastic simulation Once bi-modal models became the norm the progress to more advanced and inclusive load transfer models was swift. Henstenburg and others [6,203] used a matrix failure strain criterion21 at the fiber-matrix interface to differentiate between bonded and disbonded sections. Following Kelly-Tys ons yielded matrix assumption and further interpretation. 21 Technically Henstenburg and Phoenix [6] limit the extent of the yielded matrix region to b (or maximum PAGE 240 assuming constant frictional stress over the debonded portion (Figure 3.29), the authors developed a Monte Carlo simulati on for the SFCF test. The si mulation utilized a Poisson Point Process to approximate the random spa tial distribution of stre ngth controlling flaws which themselves are assumed to follow a Weibull (volume scaleable) distribution. A simulated strength distribution (spa tial and severity) overcomes an obvious inadequacy of Eqn. 3.4 which adopts a consta nt fiber strength statis tic. CFs and other brittle fibers have real and s ubstantial strength variation (Ch. 2) that is associated with their morphology, chemistry, and proce ssing (e.g., diameter variation, sizing, graphitization, and handling dur ing composite fabrication). Simply choosing the mean failure strength introduces considerable uncer tainty into SFCF interpretation. Indeed, Watson and Smith [156] point to diameter vari ation as a major contributor to apparent fiber strength distribution, esp ecially across a fiber tow. A second advantage for the simulated SFC F tests conducted by Henstenburg and Phoenix [6] is that they provide a means to as sess the FLD for a large number of trials. Prior to these simulations the FLD was assu med to be either normally distributed, a consequence of work by Ohsawa [200], or a lternatively as a Weibu ll distribution [212]. Neither distribution can be em pirically or theoretically ju stified. While Handge [213] argues that Weibull statistics are a good approximation to w eakest-link scaling provided the number of links is great, the distri bution has a bounded tail, and the failure distribution on each link is independent and identically distributed, the FLD dependence on failure proximity and excluded leng ths renders the FLD non-conforming. Simple normal distributions are usually employed. For a normal distribution on [1/2 lc, lc] the mean fragment length is simply l = 3/4 lc or lc = 4/3 l. The problem with FAS, sb) before debonding at the fiber end begins. PAGE 241 this argument is twofold: First, the FLD is not normal [214], and second, since during fragmentation the fiber gage length varies, th e corresponding failure st ress at that gage length also varies according to Weibull scali ng laws. The combination of these factors leaves Eqn. 3.5 a not so obvious approximation and suspect at best. Henstenburg and Phoenix [6] utilizing simp le Weibull weakest link scaling, the Kelly-Tyson recovery length definition (Eqn. 14), and a non-dimensional mean fragment length obtain for the interfacial yield stress (Appendix E) or Here sl is the fibers Weibull scaled strength for gage length l and sf,l is the mean fiber strength for length l is the fibers Weibull modulus, and () is the gamma function. The appended terms in Eqn. 3.40 and Eqn. 3.41 (cf. Eqn. 3.5) provide stochastic corrections for the fibers inherent strength distribution during frag mentation and are not equivalent to the simple 4/3 multiplier propos ed by Ohsawa [200] (Figs. 9-10 [6]). Note that in this approach is a function of the fibers We ibull modulus and the composite system recovery length, which in the bilinear model includes a bonded and disbonded region; whereas, for the simple Kelly-Tyson approach, the recovery length has only a single linear loaded section (Figs. 4 and 5 [6]). rec (3.39) 1 *2 2 s dy (3.40) 1 1 2 21 f ys d (3.41) PAGE 242 It is evident (Figure 3.29) that the recovery length rec and (by definition) are functionally dependent on f and b (extent of bonded yielded matrix region). In this analysis, f and b must be assigned values based on external evidence or supposition. Simulated SFCF tests [6] indicated (Table 3.6) that the aggregate (multiple simulations) mean fragment length l decreases as b increases (viz. increasing debond). That is, increased load transfer over b raises the probability that the fiber will fracture somewhere on its plateau section22. The data also shows that the mean fragment length coefficient of variation (COV) increases as b increases. Again there is increased fiber failure probability, although th is time it is related to bot h the higher plateau stress and shorter exposed gage lengths, which increa ses low and high tail fi ber failure proportions with concomitant fragment length dist ribution broadening. Likewise, as f increases, mean fragment lengths decreases, except w ith reduced COV owing to higher plateau stress and unchanged gage lengths. The SFCF simulations in [6,203] intimate complex interactions that must be considered for realistic SFCF interpretation. These simulations, however, do not allow for matrix non-linearity or for that matter, a distinct interphase. As we have previously noted, a number of researchers have identifi ed a region at the fiber-matrix interface [123] that has distinct physiochemical properties and ex tent. The true nature of this interphase is complex and is dependent on the particular phases involved as we ll as the processing conditions. Moreover, characte rizing the physical, mechani cal, and chemical properties of the interphase is exceedingly difficult and is sensitive to the testing technique. 22 Henstenburg and Phoenix exclude the load transfer section, including the incr ementally loaded section (Fig. 6 [6]) during their simulations. Fatigue-life pred ictions suggest that this omission may not be valid. PAGE 243 Effective physical properties and continuum considerations With regards to Coxs analysis [105], the interphase extent is defined by the geometric factor r1/r0 that appears in the shear-lag parameter cox. Recall that r1 is the mean fiber separation, it defines the effective load transferring volum e. Subsequent work [129] has shown that cox is increasingly inaccurate as f decreases. Refinements to the shear-lag parameter provided by Nairn [215] and Nayfeh [ref. (23) 216] likewise require assigning an appropriate filler fraction (f) or effective load transfer volume. The effective physical properties of a continuum derived from a relevant micromechanical model must consider the geometric and physical properties of the constituent phases. The continuum, Represen tative Volume Element (RVE), and micro scales must also satisfy Hashins MMM pr inciple [217] (MICRO << MINI << MACRO). Therfore an accurate micromechanical mode l requires knowledge of both the extent of the RVE and the interaction particulars at the multiphase interface. Analysis by Direct23 approach requires that the averaged pha se microvariables satisfy the external macrocomposite BCs, the phase-phase in terface conditions (e.g., perfect bonding, no slipping, no tractions, etc.), a nd the phases constitutive differential equations. Since only the averaged microfields must satisfy the ex ternal BCs, disparat e constitutive phase properties and interactio ns may lead to effective propert ies that are indistinguishable. That is, while the effective properties deri ved from a micromechanics model and physical testing may agree, this does not guarantee that the model phase geometry and interactions accurately depict the true composite. More rigorously Hashin states, with re gard to elastic heterogeneous media: 23 A Direct Approach implies exact calculation of ef fective properties for some geometrical model of a composite material [218]. PAGE 244 The stress and strain fields in a large SH24 heterogeneous body subjected to homogeneous boundary conditions are SH except in a boundary layer near the external surface. [217] Hashins postulate might be vi ewed as a reverse application of St. Venants principle to heterogeneous media (cf. [218]). Paga no and Tandon [168] provide numerical and analytical arguments that make clear Ha shins fundamental postulate for elastic heterogeneous media. In particular, Paga no and Tandon [168] argue that mathematical (volume averaged) and physical (surface) co mposite strains provide different solutions for some composite moduli, especially when imperfect interfaces are involved [219]. Mathematical and physical strains are equal for perfect bonding and homogeneous BCs25. The implication here for SFCF interpretati on, analysis, and exte nsion to effective macrocomposite performance is that local st ress/strain field pert urbations may not be inconsequential. Single fiber RVEs do not sa tisfy the loose requirement that SH fields are statistically indistinguishable within di fferent RVE in a heterogeneous body [217]. This intimates that extension of SFCF and fiber pull-out tests to effective macrocomposite performance should be undertak en with appropriate reliability concerns. Interface adhesion modeling Perfect Bonding (PB) between fiber a nd matrix (continuous tractions and displacement continuity) is obviously an idea l condition and as such represents an upper bound with respect to composite moduli. Paga no concludes that volume averaged and surface strain measurements provide for identical composite moduli when perfect bonding exists.26 By contrast, Complete Separation (CS) is characterized by a traction 24 A composite is statistically homogeneous (SH) if all global geometrical characteristics are the same for all RVE regardless of position or axes orientation. 25 Homogeneous BCs produce homogeneous fields in a homogenous body. 26 Surface displacement and surf ace traction BCs produce coincide nt upper and lower bounds for E11, 12, G12, and K23 but differ for E22 (Table 1 [222]). PAGE 245 free interface and a Perfectly Smooth (PS) c ondition refers to continuous normal tractions and displacements, and zero shear stress at the interface [168]. CS and PS are idealized interface conditions that produce lower bounds for the composite moduli. An intermediate PS condition rela tes the tangential displacemen t to a non-zero interfacial shear traction with a propor tionality constant [219-222] This is commonly referred to as a spring la yer interface model, with R representing the interface stiffness. The spri ng layer model can depict a bond ed or disbonded condition. That is, R = 0 indicates PB and R implies PS conditions. Benveniste [219] showed that as R increases the effective recovery length (rec) increases leading to violation of St. Venants principle with respect to continuous unid irectional composite and consequently the inverse relationship between elastic m oduli and compliances ceases to be valid. Since Henstenburg and others [6,203] simp le bilinear stress recovery model, investigators have sought to account for th e observed fiber-matrix debonding, matrix cracking, and acute matrix non-linearity by addi ng layers/phases and complexity to SFCF test modeling and interpretation. The importa nt work here considers either a distinct interphase or a spring layer interface. Just as for simpler load ab sorption models (Cox, Rosen, Kelly-Tyson, etc.), various analyt ical, numerical, and empirical techniques have been employed to characterize the complex in terphasial interaction between fiber and matrix. Imperfect interface/interpha se interaction is analogous to discontinuous fiber reinforcement, at least with regards to lo ad absorption and simple Rule of Mixtures (ROM) approximations. It is known that shor t or discontinuous fi bers do not provide fR x u (3.42) PAGE 246 composite reinforcement efficiencies equiva lent to that for long/continuous fibers (Figures 3.5 and 3.17, [105,111]). In a like ma nner, imperfect interfaces are less efficient than perfect interfaces on equal length fibers. The degree and distribution of interface/interphase imperfection along an in tact fiber can be li nked to a number of factors. Namely: 1. Non-uniform fiber diameter 2. Surface irregularities (p hysical and chemical) 3. Imperfect bonding 4. Local matrix microstructural variations 5. Residual and stress concentrations 6. Matrix non-linearity and cohesive failure Ineffective length reconsidered Recall previous discussion c oncerning ineffective length, i, as that portion of the fiber on which FASi < k FASmax. For simple shear-lag models which assume PB, i is a function of the shear-lag parameter (cox or rosen) and can be quite small (i 2 Df). It is consequently of little practical si gnificance [223]. Additionally, owing to its functional dependence on i is restricted to idealized lo ad transfer conditions attendant to simplistic shear-lag models (e.g., PB, one -D restriction, perfect fibers, etc.). Beltzer et al. [223], aware of the limitations associated with heretofore ineffective length definitions, introduced an ineffective le ngth definition that util izes strain energy densities for an idealized intact fiber, V0(x,y,z), and a broken fiber, V(x,y,z). Leff is the length of an intact perfect fiber that is capable of carrying the same strain energy as a broken fiber of length L The ineffective length is simply i, = L-Leff. The advantage for this ineffective length definiti on is that it makes no assumptions regarding effL S L Sdx ds z y x V dx ds z y x V00 0, , (3.43) PAGE 247 the load transfer particulars. The definiti on is dependent only on the fiber strain energy, arguably a more appropriate averaged statistic as compared to localized stress concentrations. The trivial bounding conditions for PB ( V(x) V0) and PS ( V(x) 0) give i,b 0 and i,b L as required. Moreover, i,b is a convenient measure of the reinforcements overall load transfer effi ciency (including impe rfections and breaks), provided appropriate strain en ergy fields are available. Supposing that the fiber aspect ratio is high ( L/Df >> 1) then Eqn. 3.43 reduces to [223] where V0 is the maximum uniform strain energy density which insures 0 i,b L Further defining V F as the critical strain energy density provides27 i,b = 0 ( V0 < V F) and i,b = L ( V0 = VF). By definition then, all real fibers carry strain energy densities less than prescribed by the ideal ( V0) and are accordingly referred to as inferior or imperfect (in a strain energy density sense) [223]. The present ineffective length definition allows for the introduction of more complex imperfection distributions besides a simp le fiber break. The fiber break is seen (with respect to fiber strain energy density ) as a particular example of a periodic imperfection. Beltzer et al. [223] consider and derive strain energy density expressions for three basic imperfection distributions (Appendix F). 1. A periodic model (Deterministic). 2. A random model (Stochastic). 27 This criterion implies complete debonding and PS conditions, although this is clearly a mathematical contrivance as critical fiber loading would likely l ead to fiber fracture and incomplete debonding. L b idx V x V0 0 ,) ( 1 (3.44) PAGE 248 3. A distributed inferiority model (e.g., broken fibers, matrix inhomogeneities, etc.). To use Eqn. 3.44 a differentiable expression s for V(x) is required. Beltzer et. al. [223] suggest starting with a single term e xpansion of a simple shear-lag model (e.g., Eqn. 3.9) for the near-field stress at a fiber break. Here k is a simple shear-lag parameter (Appendix F) and x is the position along the fiber. Combining the near-field energy density and the far-field c ontribution (periodic imperfection strain energy density) gives the total strain energy density along a broken fiber with periodic imperfections28. Then substituting V(x) and V0 = 0 2/ 2 Ef into Eqn. 3.44 gives 2 is the inferiority strength/severity, d is the break separation ( xixi+1), and [ l ] is the unit dimensional length (function of L and L*, App. F). Note that Eqn. 3.46 holds only when L d L* rf. That is, the fiber length L is long compared to both the imperfection separation ( d ) and fragment length ( L*), which is itself longer than the fiber radius. In Eqn. 3.46 the first term arises from th e stress recovery regi on at the break while the second term reflects the influence of pe riodic imperfections. Increasing imperfection separation or reducing flaw seve rity decreases the ineffective length. However, with little effort i,b can quickly approach L reflecting the substantial influence periodic imperfections can have on composite reinforcement efficiency. 28 Note that the periodic imperfections are considered to have zero extent, enabled with the use of a dirac delta function. d L k rf b i2 ,2 3 (3.46) f fr kx x exp 1 ) (0 (3.45) PAGE 249 For a random inferiority distribution, the ineffective length must also be a random quantity (Appendix F). Followi ng a derivation similar to that for periodic imperfections [223], it can be shown that that the first moment ( ) for random inferiority distributions is: The severity ( g2) and density () of the random imperfections generally leads to a less inferior fiber than that for pe riodic imperfections. Again, the imperfections are localized using dirac delta-functions. By comparison, distributed infe riority allows for extended influence near random or periodic imperf ections; consequently, both the random and periodic cases described appear as particular examples of a general distributed inferiority strain energy density approxima tion (Appendix F). Note that the first terms in Eqn. 3.46 and Eqn. 3.47 are identical and a consequence of the shear-lag FAS fo rm of Eqn. 3.45. Choosing an alternative FAS bu ild-up will not affect the infl uence of the periodic or random perturbations. Certainly, fiber-matrix disbonding, matrix cracking (cf. Eqn.60 [223]), matrix yielding, and other types of imperfection distributions can be considered utilizing the ineffective length definition proposed by Beltzer et al, Eqn. 3.44. We will return to this discussion later, but first we will consider more closely the physical nature of the interface/interphase and, in particular, its effect on load transfer/absorption. Interface/Interphase Influence Now while it is abundantly clear from bot h analytical arguments and empirical evidence, that the interaction between the reinforcement and matrix is critical to composite performance, it is not as clear-cut as to the particulars of this interaction. The L g k rf b i 2 ,2 3 (3.47) PAGE 250 earliest work assumed PB and linear elastic matrix properties, which leads to unrealistically sustainable stre ss concentrations [133,138,139]. Yielded matrix behavior was the next logical step [111, 112] but unfortunately, this di sregarded the possibility of fiber-matrix disbonding. Subsequent improve ments allowed for disbonding according to strength [161] or energetic criteria [182]. Debonding and yi elding behavior were later combined in a bi-linear stress recovery mode l [6]. That is, linear stress build-up across both the unbonded and bonded fiber sections (F igure 3.29). This was an improvement relative to the orig inal PB analyses [105,110,111] but l acked refinement with regards to matrix non-linearity near the inclusion interface. Separate treatment of a distinct interphase proceeded in parallel with increasingly complex interface interaction interpretation. GSCS and other advanced continuum mechanics techniques [167,224,225] were utili zed to consider various phase properties (modulus, strength, thickness, viscoelasticity, conductance, etc.) a nd their influence on effective macrocomposite properties. Desp ite the large body of ongoing research that models the interface/interphase micromechanics, there has been relatively little work on direct characterization of the interpha se morphology and its physical properties, especially as those properties relate to the proposed microm echanical models. That is, the materials science at the in terface has been overlooked, for the most part, in favor of convenient and relatively simplistic microm echanical models. We will look at how morphology and physical properties of an interphase might give rise to the macrocomposite performance observed, but more importantly for this work, how or if this interphase is consistent with existing micromechanical models. PAGE 251 Interphase formation Addressing first the interpha sial region, its morphology, chemistry and mechanical properties: The physical and chemical propertie s of the interphase are different from the bulk matrix properties. It is difficult to imagine a scenario where the physical constraints imposed by the fiber surface29 and the fiber's surface chemistry have no effect on the local matrix chemistry and morphology. The incipient causes for interphase forma tion at the fiber-matrix interface are as noted above both physical and ch emical. The most often cited influences for interphase formation are (1) the fiber surface chemistr y (surface energy being a function of the chemistry), particularly the percent oxide c ontent, (2) matrix-sizi ng (or surface treatment) chemical interaction, and (3) macromolecula r entropic effects associated with limited mobility at the rigid fiber surface. However, there is by no means agreement on the relative importance of these factors on inter phase formation and performance. This is due primarily to the complex physical and ch emical interaction be tween the fiber and matrix during fabrication and performance, a nd the difficulty in probing the interphasial region with analytical or empi rical techniques (SFC tests be ing of particular interest here). It may not be possible to define th e interphase and its re lationship to composite performance using only a single techni que, empirical or analytical [226]. Still another complicating issue is related to the very limited extent of the interphase. The interphase ha s been variously estimated to extend anywhere from 50 nm to 500 nm [227,228] from the fiber surface. Some of the difference in estimated interphase extent is associated with identifying what constitutes an interphase. Whereas DSC measurements consider the interphase as the fraction of the composite that PAGE 252 influences the cooperative movement of the macromolecules, IR measurements consider only the averaged local thermal absorption ch aracteristics, a much smaller effective volume, and thus do not survey the same phys ical effects or interphase. The same argument holds for Dynamic Mechanical An alysis (DMA) which surveys both small segment/group mobility and much larger m acromolecular relaxations for anelastic polymeric matrices. Dielectric Response Spectrometry (DRS) examines much smaller effective volumes, but just as for DMA does so in a volumetric averaging sense. Unlike IR or Raman spectroscopy, which have small spot sizes, DMA and DRS require that the total composite structure be tested from which the interphase properties are inferred given an apriori physical composite model. The generally accepted definition for the inte rphase is that region near the fibermatrix interface, which differs morphologically and/or compositionally from the bulk matrix properties. The earliest models we discussed above are mechanical in nature. They do not refer to the underl ying phenomenological causes fo r the ascribed mechanical behavior. This enables treatment of ideali zed physical models (e.g., PB, PS, CS) which provide bounding solutions for effective com posite properties such as tensile modulus and strength. However, interface conditions in actual composites deviate from ideal PB and consequently require more detailed micromechanical modeling. Interphase modeling To understand how an interphase might in fluence composite performance we must delve into its morphology and physical propert ies, but we must also consider how the interphase is modeled. There are two mode ling approaches for deal ing with an interface in composite reinforcement micromechanics Spring-layer type models approximate 29 The fiber surface limits the conformations av ailable for relaxation processes [123,228]. PAGE 253 irregular/inhomogeneous load transfer with an all-inclusive interface parameter (e.g., Eqn. 3.42). They collapse the complicated three-dimensional stress state onto the twodimensional fiber-matrix interface. Three or four-phase GSCS modeling is the other analysis tool. The former is mathematically simpler, but because of this, the results are less transparent. In contrast, GSCS mode ling is much more involved, but its added complexity limits its reliability. That is, gi ven the large number of input parameters error propagation becomes important. In addition, there are concerns that even though the macro performance may be satisfied in the aver age sense, this does not necessarily insure that the micromechanical model represents the true phase-phase interaction [217]. The two different approaches to modeling an interphase require unique assessments of the interphase performance during load tr ansfer. For spring-layer application, all possible interphase irregularities and influe nces are collected and averaged over an appropriate fiber length to approximate thei r aggregate influence. A stress-displacement jump relationship is employed at the two-di mensional fiber-matrix interface [222,229] (Figure 3.30). m n f n n n n m nn f nnu u u u D : (3.48a) m s f s s s s m ns f nsu u u u D : (3.48b) m t f t t t t m nt f ntu u u u D : (3.48c) For the axisymmetric SFCF case, f nt = 0, so Dt can be ignored. With compressive radial stresses and no overlap between phases [ un] = 0 or equivalently, Dn Nairn argues that for SFCF tests the interfacial radial stresses is predominan tly compressive (Eqn. 3.18, [165]) except very near fiber breaks and thus the quality of the interface in the radial direction should have no effect on frag mentation results [229]. In light of the PAGE 254 results by Muki and others [133, ref. (23) 165], which show that interfaci al radial stress in macrocomposites is tensile, this calls into question the representative nature of SFCF with respect to macrocomposite behavior, a nd in particular, the validity of assuming Dn = If however, we do set Dn= and assume axisymmetric stresses, then we are left only with the axial sh ear interface parameter Ds, Eqn. 3.48b. This obviously simplifies SFCF interpretation, but despite Hashins assertion [222], is not more transparent as the interface parameter subsumes various morphol ogical and mechanical effects, which consequently cannot be disti nguished. The advantage for spring-layer models lies primarily with avoiding complications for de termining the interphases physical or mathematical boundaries. Recall that both th e Cox and Rosen models are sensitive to matrix extent, r1 and rb respectively, and that these radi i do not represent the actual fiber volume fraction, but rather the mean fibe r separation and the shear stress influence extent. Spring-layer models al so obviate the need to descri be the exact nature of the interface, especially as a f unction of axial position, allowing instead for an averaging approach. By contrast, generalized continuum mech anics approaches can quickly become very complex and involved, even with the ai d of advanced numeri cal techniques (e.g., FEA and Finite Difference) [230-233]. Conseque ntly, analysis is generally restricted to idealized or simplified interface conditions with limited fiber count. While this approach affords exceptional flexibility with regards to interphase morphology and load transfer characteristics, the considerable number of required interphase parameters (extent, modulus, non-linearity, etc.) makes application di fficult and can lead to substantial error PAGE 255 propagation [234]. In addition, with contin uing research the interphase extent and influence apparently continue to increase. Fisher [235] ar gues that any reasonable load transfer modeling must include a distinct inte rphase, especially with regards to effective viscoelastic properties (cf. Fig. 11 [235]). Interphase Chemistry/Morphology/Dynamics Postulating and modeling interface microm echanics can be, as described above, relatively complex, but empirical justificat ion of the particulars has proven far more difficult and contentious. Some of the earlie st efforts to establish the morphology and dynamics of interphases can be attributed to Lipatov and ot hers [123] and their physical chemistry work concerning filler addition. Lipat ov [123] refers to the interphase as either a boundary or surface laye r and stresses that defini ng the boundary layer depends on the property being investigated and also the means to id entify that property. The principle argument regarding th e interphase structure tender ed by Lipatov [123] is that macromolecular adsorption onto the solid surface is disti nguished by super-molecular (globule) or aggregate molecular adsorption, which depends upon, among other factors, macromolecular mobility, molecular cohesion, and filler surface energy. Furthermore, this adsorbed layer leads to the formati on of a boundary layer with its attendant morphology and physical propertie s. The boundary layer is ma nifested by a reduction of available macromolecular conformations a nd consequently distinct physiochemical properties relative to bulk-matrix properties. Interphase morphology Globule and aggregate macromolecular struct ures at the interf ace and in the bulk have been estimated by SEM. For crosslinked PUR globule size ranges from 30 nm, with aggregates possibly as la rge as 250 nm [ref. (94) 123]. The size and distribution of PAGE 256 globular structures at surfaces are related to the substrate's surface energy (5 nm 30 nm for graphite vs. 5 nm 10 nm for diamond, [r ef. (97) 123]). One would expect graphite fibers with different surface energies, whether introduced by oxidizing treatments, coupling agents, or stress graphitization sc hedules (Type I or Type II fibers), would likewise influence the supermolecular structur e and consequently the molecular mobility, crosslink chemistry, density, and thermal prope rties of the interphase region. Larger globule structures will certai nly influence nano-scale stre ss concentrations at the interface, and the globule size increase is al so in opposition to [236], which asserts that smaller more uniform globules improve mechani cal properties (imagine fine vs. coarse grain metals). For semicrystalline thermoplastic composites, paracrystalline (transcrystalline) structures may dominate at the interface. In cipient transcrystalline formation at the interface is related to the f iller surface energy and the presence of sufficient nucleation sites [214]. These thin cr ystallites extend well away fr om the interface and with sufficient fiber fraction may constitute the enti re matrix phase (Figure 1 [237]). In an analogous situation, film formation studies [ref. (94) 123] suggest that by a cascading process, the interface influence on structure may extend as far as 200 m [ref. (20) 238]. Finally, there is evidence that interphase st ructure and extent is affected by packing density in a complicated manner [123,236]. Interphase reaction kinetics/products If macromolecular mobility is affected by a solid surface, and it surely is [123,239], then it is reasonable to conc lude that reaction kinetics a nd possibly reaction products are similarly influenced. However, there is by no means agreement on this issue, especially PAGE 257 in the presence of sizing or surface treatments. Again, some of the earliest and most complete work on this subject can be attrib uted to Lipatov and others [refs. (104,105) 123] in which IR measurements indicate th at a solid interface increases PUR conversion relative to bulk conversion rate s. The argument here is that the termination reaction is reduced in the interphase owing to decr eased poly(oxypropylene) glycol mobility (cf. Figures 1.21 and 1.22 [123]). In support of this premise, raising the cure temperature reduces the kinetic difference between su rface and bulk reaction rates. Increasing mechanical agitation during PUR curing produces similar results as evidenced by hard and soft segment phase separation (Tg(soft) decreases and Tg(hard) increases) [240]. Lipatov [123] also argues that adsorpti on ordering affects reaction kinetics to varying degrees depending on the boundary la yer structure and its interaction with reacting species. This is particularly ev ident for branched network polymers for which limited mobility at the interf ace may slow reaction rates relative to the bulk. For linear macromolecules, the reaction rate is only affect ed after sufficient conversion; that is, with higher MW there are fewer conformations available and therefore less mobility. There is also the possibility that reacting or unreacting species will be selectively adsorbed at the interface, which can create an interphase/bulk either deficient or enriched in a particular reaction component. For inst ance, adsorbed water on aramide and graphite fibers can alter the crosslinking chemistr y in the first 200 nm adjacent to these idealized surfaces [241]. The adsorbed wa ter increases anhydride consumption and consequently reduces ester formation (% conve rsion). The alpha transition temperatures ( T) for composite systems with similar chemistries decrease, in comparison to neat PAGE 258 matrix and dry fiber composite s [241]. This indicates a re latively soft interphase with lower crosslink density. Unfortunately, the effect for different sizings, treatments, and fiber-matrix combinations is not always so clear-cut, which has lead to disparate results and conclusions. Clearly, it is unrealistic to expe ct a single universal in terphase description to be representative for the myriad filler-treatment-matrix systems. The controlling interphase parameters must be identified and appropriate micromechanical models developed so that informed conclusions can be drawn regarding interphase influence on fiber-matrix load transfer. Interphase definition Since we are concerned here with mechanical load transfer from the matrix to the fiber, it is understood that we are interested in mechanical pr operties of the interphase. We previously described the interphase as that region adjacent to the reinforcement that is morphologically and/or compositionally dist inct from the bulk matrix. However, for polymeric materials, precise demarcation betw een the interphase and the bulk matrix is not possible owing to the complex, possibly in terpenetrating, macromolecular structure. Polymer testing rate sensitivity also contributes to the difficulty in defining the interphase extent. Hence, just as for polycrystalline metals, where the effective moduli of an RVE30 enables representation of the microscopically heterogeneous material as a homogeneous continuum, we must choose an RVE for the in terphase which likewis e enables treatment of the interphase as a ho mogeneous continuum. Here homogeneous refers to a continuum which is piecewise continuous and not necessarily isotropic, orthotropic or otherwise simply described. The interphase RVE is obviously a sub region of the larger PAGE 259 composite RVE, which as noted above can aff ect effective composite moduli calculations [242]. Thermodynamic and spectroscopic evaluation While statistical thermodynamics and light scattering techniques may provide an end-to-end radial distribution function, W(r) and Radius of Gyration, Rg, information that characterizes the scale, rigidity, and mo lecular weight of the polymer, the actual macromolecular conformation in a relaxed bulk state is unknown. Furthermore, given the small scale of macromolecules ( Rg 10nm) [243] and their comple x intertangled structure, it is evident that cooperative movement mu st be considered to realize mechanical estimations for effective macromolecular properties. Cooperative movement and molecular weight distributions (MWD) lead to a distribution of relaxation times (i) for chain, segment, side chain, and group mobility. Collectively, dynamic spectroscopic technique s enable evaluation of the relaxation time distribution and hence are invaluable tools for studying morphology and physical properties. Macromolecular in teraction with the solid surf ace (via the adsorbed layer) will alter the relaxation time distribution. In some cases this contribution will be evident [228,241,refs. (10,11,13) 243,244-247], while in ot hers, the effect will not be discernible [247,refs. (9,12) 247]. This incongruity is a consequence of complex energetic and entropic influences at the solid surface-ma trix interface and will be explored further below. The apparent glass transition temperature for neat viscoelastic materials increases with testing frequency (~6 C/decade, WLF eqn. [248-249]). For an otherwise non30 Hashin [218] refers to this as a classical approximation PAGE 260 interacting filler and polymer matrix, rest ricted macromolecular mobility at the solid surface will increase Tg, and likely broaden the dispersion [250]. The change in Tg for non-interacting filled materials relative to the neat polymer ( Tg) is related to the testing frequency, filler volume fraction (Figure 3.31), local packing density near the interface, molecular cohesion energy, and filler surface ener gy. In accord with the relative testing volumes at the different probi ng frequencies, the greatest Tg is seen at the lowest frequencies (viz. largest effective testing volumes). The rate of change in Tg with filler concentratio n decreases at higher volume fractions and eventually tends to zero. Th e higher testing frequencies display smaller plateau concentrations (f at which Tg ~ const.). These platea u concentrations provide information regarding both the extent of the interface influence at a given testing frequency and the effective testing volume As the Aerosil content in PMMA increases, the total volume available for uni nhibited bulk relaxation declines while the boundary layer extent increases only gradually and with a limited extent31. That is, the boundary layer volume fraction (bl) at higher filler concentr ations does not increase in proportion to the filler fraction. By definition and effect, the boundary la yer extent limits the solid surface influence. As evidenced by plateau Tg concentrations seen in Figure 3.31 the effective boundary layer extent varies with testing fr equency. Higher frequency tests suggest thinner boundary layers while considerably larger volumes are involved in cooperative movement for lower testing rates (e.g., DSC, dilatometry). The testing volume involved 31 Re change in BL thk, irregular prop. Var. [lipatov.1979.1] PAGE 261 in cooperative movement at higher pr obing frequencies can be very small32 (NMR ~2 bond lengths, DRS ~2-bonds). Given the small testing volumes, the bulk of the relaxing dipole groups are well separated from the solid surface influence, at least on the scale of the effective testing volumes. Accordingly, onl y a very small fraction of the sample will see the solid surface. Thus, appreciable cha nges in dipole group relaxations must be attributed to packing density reduction fo r surface adsorbed polymer. This affords increased side group mobility (increase in tr ansition magnitude) and a consequent shift to lower temperatures (increased facility fo r relaxation) (Figure 3.32). Decreased dispersion, suggesting distinct matrix-int erphase separation, is also possible. For the same PMMA-Aerosil system as above [123], the dipole segment transition shifts to higher temperatures, Figur e 3.32, with increasing filler fraction. The solid surface reduces availabl e conformational options (magni tude of transition peak) and decreases facility for relaxa tion (increasing transition te mperature). This despite evidence derived from dipole group relaxations indicating decreased packing density at the interface [123,250]. Additionally, large group, segment, and long range cooperative relaxations are not possible at higher testing frequencies, so elevated packing densities will induce little change in the measured Tg at these frequencies (Figure 3.31). In contrast, for lower testing frequencies the volume of material involved in cooperative movement is relatively large (Table 3.7). Consequently a greater proportion of the sample is influenced by the solid surface, which in conjunction with constricted mobility at the interface heightens the Tg increase with filler concentration (Figure 3.31). 32 Here we distinguish between dipole group and dipole segment cooperative relaxation, where dipole segment implies chain conformational/cooperative relaxations. PAGE 262 Restricted mobility and decreased pack ing density near the interface are both consistent with molecular relaxation and Tg as an iso-free-volume process, as first posited by Fox and Flory [ref. (54) 249]. The WLF Equation, originally developed for linear macromolecular viscosity characterization, de scribes the relationship between relaxation times (I) and relative temperature differences (T-Tg). 0 0 0 0ln T T f T T f B Af T (3.49) For linear, uncrosslinked polymers considerab le evidence supports a constant universalfree volume ( f0 = 0.025) at Tg. However, with constricted mobility near an interface and/or crosslinking, it is clea r that a universal-free volume ( f0) is not defensible. Lipatov [123] argues that the WLF Eqn. (Theory of re duced Variables) rema ins valid for filled crosslinked polymers, albeit with a revised non-universal free volume ( ff 0.08). ff is not universal either, as it depends on molecu lar rigidity, molecular cohesive energy33, and, to a lesser extent, filler surface energy. An in crease in free volume with filler addition is usually indicated by an increase in thermal e xpansion (Table 3.8). Just as with the fractional free volume f0, Simha and Boyer [ref. (55) 249] argue that liqTg and (liq-g)Tg are constants for polymeric materials. Again, it appears that this is not entirely true for neat or filled polymers (cf. Table 3.8). Empirically, the simplest estimates for boundary layer thickness are obtained using calorimetric measurements. The ch ange in heat capacity at Tg, Cp, is associated with increased molecular motion, particularly segmental mobility. As filler content increases, there is a corresponding drop in segmental m obility of adsorbed/boundary layer polymer, 33 Molecular rigidity and cohe sive energy also affect f0 for linear polymers, although this is usually ignored in favor of the simpler, generally accepted, universal f0=0.025. PAGE 263 consequently Cp decreases [123,250-252]. Lipatov [123] attributes the entire Cp reduction to an increase in boundary layer fraction, The polymer in the boundary layer is effec tively excluded from the cooperative glass transition and the boundary layer extent is defined by the percent drop in Cp at Tg relative to a neat polymer tr ansition. With the further as sumption that the boundary layer is uniformly distributed around the filler/fi ber the boundary layer is estimated by [123] Eqn. 3.50 and Eqn. 3.51 assume a sharp bounda ry layer/bulk phase transition, which is obviously unrealistic and simplistic. This a ssumption is, however, convenient and can be elucidating. Equations 3.50 and 3.51 notwithstanding, f iller/fiber addition generally reduces the heat capacity difference for second-order phase transitions (Figure 3.33). Note that for filled polymers it is possible to observe simultaneously both a reduction in Cp at Tg and an increase in the magnitude of Cp relative to that for unfilled specimens at temperatures above and below transitions (Fig. 3.4 [123] ). This confirms DRS evidence, which indicates looser molecular packing near the interface, but does not preclude restricted macromolecular mobility near the interface. In a series of papers, Theocaris and ot hers [251,252] employed DSC measurements and Eqns. 3.50 and 3.51 to estimate boundary layer thickness34 for iron particulate filled 34 Theocaris generally uses the term mesophase, rather than the more ambiguous interphase, or more appropriate boundary layer preferred by Lipatov. 01p f p BLC C (3.50) f f BL f fr r r 1 13 (3.51) PAGE 264 epoxy systems. Selected results are repr oduced in Table 3.9 and Figures 3.34.36. From inspection of Figure 3.34, it is evident that, while increasing filler fraction reduces Cp at Tg, the sensitivity to filler fraction addition continuously decreases. This trend echoes that seen for Tg variation at higher filler con centrations (cf. Figure 3.31) and suggests boundary layer growth is not cont inuous or monotonic with filler volume fraction. Conversely, if the boundary layer growth is continuous then its properties cannot be uniform with respect to the thickness direction. Boundary layer fraction and th ickness estimates are obtai ned directly from Eqn. 3.50 and Eqn. 3.51 (Figure 3.35 Figure 3.36). A cubic relationship between bl and f (bl = Cf 3, C = const.) falls naturally out of Eqn. 3.51 [252], and although appealing, is difficult to justify, especially at high filler fractions. Figure 3.36 indicates quadratic interphase growth in proportion to the filler diameter rbl = ( rbl rf) = C( rf)f 2. Again, this is difficult to justify physically, in light of known adsorption and agglomeration behavior as well as steric p acking considerations, which w ould suggest looser packing on the larger diameter particles and conseque ntly greater mobility (viz. a thinner boundary layer). Recall that Eqns. 3.50 and 3.51 are a di rect result of an assumption that confines the change in Cp at Tg to polymer existing only within the boundary layer. This has the net effect of predicting an ever-increasi ng boundary layer fraction, despite the plateau behavior seen in Cp with f (Figure 3.33). If the rate of change in Cp decreases with f then it is reasonable to expect the nature of rbl vs. f to vary as well. The limiting behavior at high filler frac tions (Figure 3.33) suggests that the boundary layer thickness might be related to pa rticle separation. The minimum particle separation (viz. mean particle separation) for uniformly distributed pa rticles/fibers (e.g., PAGE 265 cubic cell, BCC, HCP) decreases non-uniformly as volume fraction increases. The initial separation is large and closes rapidly at lower concentratio ns, then gradually decreases for higher concentrations (Figure 3.37). Replotting Theocaris Cp vs. f data, assuming a simple cubic cell particle distribution, gives the Cp vs. minimum particle separation curves in Figure 3.38. Similar Cp values are achieved at sign ificantly different particle separations (f = 5,10%, Figure 3.38), which indi cates that with sufficient particle separation roughly equivalent volumes of mate rial are influenced by the solid surface, independent of both filler scale and surface ar ea (Figure 3.39). This implies that the fillers region of influence for glass transition relaxations is small compared to both filler size and separation. The straight lines in Figure 3.38 are for reference only and are not meant to imply expected or observed behavior Deviations from lineari ty do however hint at the possibility of non-ideal packing (particle aggl omeration, random packing) and that this phenomena is more acute at higher filler fracti ons. The deviation from linearity at higher filler fractions, and general insensitivity of Cp to minimum particle separation (i.e., 75 vs. 150 radius particles at the same filler fr action, Figure 3.38) and surface area, suggests that boundary layer development cannot be e xplained simply in terms of particle separation or filler surface area. Evidently, filler separation will influence cooperative movement only when it falls below approximately four times the effec tive boundary layer thickness. That is, boundary layers will not only exclude relaxa tions within their ex tent, but may also restrict/exclude relaxations between neighbor ing elements. Recall that the effective PAGE 266 boundary layer thickness depends also on the tes ting frequency so this influence zone is a moving target. Work by Droste (1969) [250] regarding changes in Tg with spherical and rod-like filler (AR~50, L~1 m) additions to a Phenoxy polymer, i ndicates that the increase in Tg is related to mechanical reinforcing influences and interaction/modification of the polymer phase near the interface. It can also argued that interfaci al area influences the glass transition temperature variation with fille r fraction, and that this influence is nonlinear with respect to filler c oncentration (Figure 3.40). The sh ape of this curve is similar to that seen for Tg vs. MW behavior (cf., Fig. 6.27 [249]). An empirical relationship between Tg and f is presented [250] that is similar to Eqn. 3.50. where Tg c and Tg 0 are the composite and neat polymer transition temperatures, Tg is the maximum Tg change for the filled system, f is the corrected fill er fraction (including interphase), and B is an empirical constant. Th e interphase fraction is given by where f = 1/1+ is the filler specific surface area, Vc is the total composite volume, and is the zone of influence. Using Eqn. 3.52 and Eqn. 3.53 Droste [250] estimates the zone of influence for the rod-like particles at ~35, approximately one monolayer. This is considerably smaller th an interphase thickness estimates provided by Theocaris [252] (0.25 m, Table 3.9). The disparity between the two interphase thickness predictions highlights the effect of model formulation on interphase extent estimation. )) exp( 1 (0 f g g c gB T T T (3.52) c f ipV ) 1 ( (3.53) PAGE 267 Mechanical damping capacity for filled polymers drops in proportion to specific filler/polymer interaction [241,245,253]. Fo r example, acid/base reactions between polymer (chlorinated PE, acidic) and increasing ly basic filler (rutile, acidic-basic series) decreases mechanical damping capacity at Tg [253]. Here again a linear drop in tan is attributed to an increase in in terphase thickness according to with B = (1+ rf / rf)3. The data for this system (PE and rutile) are reproduced in Table 3.10 and displayed in Figure 3.41 Figure 3.42. B increases with increasing filler basicity ( = ( Vg 0)a/( Vg 0)b-1, ( Vg 0)a and ( Vg 0)b are retention volumes for IGC, a = butanol, b = butylamine) and rf follows, by definition and model s upposition. Note also that the breadth of the tan transition will increase wit h network structure [254]. Deviations from ideality in Figure 3.41 suggest disproportionate particle-matrix interaction or an auxiliary c ontributing mechanism. As in Figure 3.33, where there is limiting behavior at higher filler fractions the acid-base reaction influence is limited (Figure 3.42) at higher values. Boluk and Schreiber [ 253] suggest irregular particle separation, interparticle fric tion, and preferential matr ix component adsorption as possible sources for the nonlineari ty seen in Figure 3.41 and plateau behavior evident in Figure 3.42. The interphase thickness deri ved from damping magnitude (Eqn. 3.54 and Eqn. 3.51) is similar to that given by Tg variation with filler concentratio n [250]. It is interesting and informative to note that Boluk and Schreiber find no discernible Tg change with filler addition, but inspection does suggest slightly increased dispersion. The three spectroscopic techniques describe d for boundary layer i nvestigation (i.e.; Cp, Eqn. 3.50Eqn. 3.51; Tg, Eqn. 3.52Eqn. 3.53; tan, Eqn. 3.54 and Eqn. 3.51), ) 1 ( tan tan0 f cB (3.54) PAGE 268 exhibit similar limiting characteristics at extreme filler/reaction conditions despite considering different thermodynamic propertie s. The data are somewhat conflicting however with regards to the effects of filler surface area. Lipatov and Theocaris [123,252] heat capacity data indicate no obvi ous surface area influence, while Droste [250] argues that high specific surface area, E qn. 3.53, induces substantial interphase thickness growth. It is more probable however, that the Tg variation seen in Figure 3.40 is caused by both mechanical and energetic (entropic and reaction based) mechanisms. That is, in this case, both the filler shape and material were changed (clay rods vs. glass spheres) leaving the possibility for comp eting interphase formation mechanisms. The limiting behavior for Cp, Tg, and tan at higher filler fractions suggests that variations in these fundamental ther modynamic properties at higher filler fractions might be related to scarce free-volume. As noted above, the free volume necessary for bulk relaxations ( f0 = 0.025) is different from that for filled composites or crosslinked systems ( f0 = 0.08). There is a natural monotonic increase in the free-volume necessary for relaxations for the series, neat dilute lightly packed densely packed materials. Furthermore, as filler concentration increases it will become increasingly unlikely that sufficient free-volume or energy will be av ailable for relaxations consistent with devitrification. Kinetic explanation A kinetic explanation for a heat capacity jump at Tg ( Cp) was developed by Eyring and others [refs. (67-70) 249]. In this approach the molar hole energy ( h) and molar cohesion energy ( Wcg) are employed in an activation energy formula (Eqn. 3.55 and Figure 3.43). PAGE 269 and Vh = h/( VgWcg ) with Wcg = WcgVh, where Vh is the molar hole volume, Vg is the molar volume at Tg, and R is the universal gas constant. Data provided by Lipatov [123] (Table 3.11) indicates a drop in heat capacity with increasing filer fraction as well as a corresponding increase in Vh and h (Figure 3.44). Increasing h is in accord with decreasing segm ental mobility attendant to evolving boundary layer influence. Likewise, increasi ng hole volume is required for devitrifying relaxations as filler fraction increases. The relationship between Wcg and Tg with filler addition is depicted in Figure 3.45. From inspection (Table 3.11), it is evident th at molecular rigidity and molar hole volume both contribute to this non-lin ear response. In this case, T exhibits similar functional dependence on h / Vh, which underscores the importance of free volume for second-order glass relaxation Flexible, isolated molecular chains (low Tg) do not insure facile cooperative macromolecular movement in the bo undary layer (cf. irregular correlation in Figure 3.46). Molecular interaction near the solid interface and boundary layer packing also affect T. At higher filler fractions Vh and Vh/ Vh 0 ( Vh 0 is the molar hole volume with no filler) asymptote (Figure 3.44). This is similar to the behavior described previously (Figure 3.31, Figure 3.33, and Figure 3. 42). It is a direct conse quence of the supposed kinetic formulation (Eqn. 3.55) and assertion that Wcg and Vg remain constant with filler addition. Lipatov [123] claims that the error associated with assuming Vg is constant is ~2%, and g h g h g cg pRT RT T W C exp (3.55) PAGE 270 likewise, Wcg, which depends on the molar volume, among other factors, varies only slightly with filler addition. Mechanical coupling between adsorbed/g rafted macromolecules and bulk polymer may also influence Tg and system mechanical performance. Wang and others [NMR: 255, refs. (16,32-36) 255] established that distinct motional rates of silane coupling agents adhered to silica can be distinguishe d. Whereas the first monolayer of the two coupling agents ( -aminopropyltriethoxysilane, APS; -aminobutyltriethoxysilane, ABS) have similar motional rates, as surveyed by NM R, the outer layers of ABS are ten times more mobile than APS. The extra carbon atom in ABS allows for increased silane flexibility relative to APS. DCB testing i ndicated superior APS performance. That is, failure occurred through the bulk matrix for APS compared to an interface/bulk failure for ABS, and an interface failu re for the untreated glass e poxy system. The differences are attributed to better load transfer ac ross the APS coupling layer owing to motional matching between the bulk macromolecules a nd the silane coupling agent [255]. The ABS silane is too flexible to dynamically c ouple with the bulk matrix and hence unable to efficiently transfer load across its influence zone. NMR sensitivity to small boundary layer va riation was also shown in [ref. (229) 123], which considered a homologous series of grafted alcohols a nd their plasticizing effect. With increasing graft molecule le ngth, there is a reduction in the dipole and segmental temperature shift re lative to an ungrafted system (Figure 3.47). This suggests that the interface influence is very short ra nge, and also that the shift in transition temperature is closely associated with adso rption/grafting onto the interface surface, packing density, and the adsorbed molecule s mobility at the boundary layer/matrix PAGE 271 interface. The plateau behavior for longer gr afted alcohols indicates that the adsorbed molecules mobility (conformational facility) enables increased packing density as evidenced by reduced dipole-group facility (i ncreasing temperature). The dipole-segment variation with graft chain length is both sma ller and comparatively slower than that for dipole-group relaxations. This reinforces th e argument that segmental mobility, and by extension, cooperative moveme nt, is entropically limited by the interface, i rrespective of bulk/filler energetic interaction. Lipatov and others [123,239] argue that in most filler-matrix systems, even those with interacting phases, the entropic effect is the primary contributor to observable changes in relaxation processes. The comp lex entropic and energe tic factors affecting boundary layer formation can lead to either a perceptible change (up or down) or no change in Tg. The presence or absence of a Tg change is therefore not an unambiguous indicator for a distinct interphase. While a boundary layer is a necessary condition for a change in Tg, the presence of a boundary layer does not insure a Tg shift. The changes in thermodynamic properties w ith filler/fiber addition are real and often, although not always, substantial and discernible. Ascertaining the underlying causes for these thermodynamic variations wit h filler fraction is, however, much more complicated. The interpretation models described by equations ( Cp) Eqn. 3.50Eqn. 3.51, ( Tg) Eqn. 3.52Eqn. 3.53, and ( tan) Eqn. 3.54 and Eqn. 3.51 provide several methods to evaluate the interphase/boundary layer influence. While the three methods consider different thermodynamic properties, they share a common physical model. They presuppose an evolving uniform (geome tric and physical) bounda ry layer that can be precisely demarcated from the bulk matrix properties. This allows for concise and PAGE 272 intuitive arguments regarding boundary layer ev olution, but given the physical models assumptions, the conclusions are virtually assured a priori. Equations 3.50 and 3.51 invariably pred ict an unbounded interphase fraction as filler fraction increases ( Cp at Tg drops, Figure 3.34 and Figure 3.35). The Cp interpretation [123] ignores th e filler surface ar ea influence, which results in the curious quadratic functional dependence seen in Figur e 3.36. For uniformly dispersed filler, suspended in an otherwise homogeneous matr ix, increasing boundary layer thickness is inconsistent with globule and agglomerat ion behavior seen for dilute matrix concentrations. Furthermore, boundary la yer formation is physically bounded, its properties are not uniform either. Evidence sup ports an interphase that varies chemically and mechanically with distance from the interface [123,227,228,243,252,255,256]. Mechanistic explanation Preferential species adsorption or accumula tion at the interface is often cited as a primary cause for inhomogeneous interphase fo rmation. Off-stoichiometric epoxy-amine ratios reduce crosslinking relative to the st oichiometric point (SP) Consequently, the glass transition temperature (DSC, DMA) drops on either side of the SP (cf. Figure 3 [256]), especially for aminepoor conditions. Likewise, the flexural modulus (DMA) will exhibit a maximum near the SP for measurements taken 30 C above Tg (cf., Figure 5 [256]). This local maximum holds in the rubbery plateau sugges ting that the highest crosslink density exists at the SP (molecular weight between crosslinks is a minimum, MWc = 3 RT / E ). However, in the glassy region (50 C below Tg, cf., Figure 6 [256]), the flexural modulus is a mi nimum at the SP, which indica tes that off-stoichiometric ratios will elevate the local modulus although this effect may be slight. PAGE 273 For graphite/epoxy-amine systems it is postulate d that oxidized graphite fibers will attract the more basic amine leaving the im mediate region amine-rich [228, refs. (1-2) 257]. Slightly enriched amine concentrations35 which are in close proximity to the solid interface will provide relatively moderate m odulus material (cf. Figures 4 and 6 [256]) with crosslink density lower than that seen in bulk experiments because of restricted macromolecular mobility during curing. The net effect at the interface would be a low Tg, low yield stress, and moderate to weak modulus interphase material [243]. Acute matrix agglomeration may further decr ease boundary layer modulus by decreasing molecular entanglement. The lowered boundary layer modulus does not contradict the reduced mobility (entropic) argument because we are distinguishing here between the boundary layer modulus as if it could be measur ed without the solid surface influence. In other words, the boundary layer modulus is de termined solely by its network structure and morphology, independent of th e constraining solid surface. In contrast with above, Mars hall et al. [257] conclude th at the amine concentration at the interface is depleted re lative to the SP. Although a w eak interface is indicated by SFCF and microdebond tests, it is not clear that the weak interf ace is attributable only to low amine hardener concentration. Amin e-poor conditions lead to reduced alpha transition temperatures and lo wer yield stresses [256] but th e data does not conclusively link observed cohesive interpha se failure with low amine concentrations. The authors [257] incorrectly associate maximum in terfacial shear stress (~70 MPa: SFCF, microdebond) at high surface oxidation levels to a bulk yield stress value which then suggests an amine-poor condi tion (cf. Figure 3 [257]). 35 Harris [35] argues that adsorbed am ine concentrations are insufficient to account for observed interphase physical performance. PAGE 274 Low amine concentration near the interface contradicts viscoelastic measurements taken during epoxy curing [254], which imply th at before gelation there is ample time, energy, and opportunity for the relatively sma ll and mobile diethytoluene diamine curing agent to equilibrate at the interface. Moreover, th e TGDDM epoxy will be increasingly stiffened by cure extent and entropic restrict ions near the interface leaving the interphase epoxy-lean. Marshall et al. [ 257] also fail to account for acute matrix non-linearity and strain hardening [213,231,238,255] that may arise within the bounda ry layer, which makes their maximum ifss-bulk yield stress association tenuous. As to the effects of adsorbed coupling agen ts and other species (sizings, oligomers, etc.), advances in predicting the (neg ative) molar Gibbs free energy of mixing36 [(Gmix)0.5] [258] points toward ready in terdiffusion of the two phases37 at the interface. For unreactive thermoplastic systems practical adhesion st rength increases with the (negative) molar Gibbs free energy of mixing fo r equal molar contributions (see also Zeta potential utility in this regard [259-262]). Deviations from the linear relationship between (Gmix)0.5 and adhesion strength is an indicator of either strong steric hindrance (not foreseen by UNIFAC calculations) or cohesive bonding (cf. Figures 3 [258]). The situation for thermosetting systems is further complicated by diffusion of unreacted species to the interface (i.e., crossl inking agents, catalysts, low MW mobile oligomers/polymers) and their reaction (crosslin king) with time and temperature. Miller [258] also recognizes possible embedding of co mpatible coupling agents in the matrix, which is similar to [255] with regards to the dynamic affinity of some silane coupling agents and epoxy. 36 Using UNIFAC calculations based on acid-base reactions and some steric considerations [260]. PAGE 275 Low MW self-segregation at the solid su rface has been observed for carbonaceous filled polyethylene blends (HDPE and UHMWPE) [263]. Williams et al. [243] surmise that this type of phase separation leads at fi rst to amine-rich regions at the filler surface and after some reaction, to an insoluble fract ion. Phase separation is further encouraged by the two-dimensional interface; consequently a network structure with elevated freevolume content is generated. This us hers lower crosslink density and low Tg properties relative to the bulk [243]. Again, the physic al properties are ascr ibed to the network structure in the absence of the solid surface. Microindentation tests conducted near a single embedded fiber (Figure 1 [228]) give results consistent w ith the physical model for th e interphase described above38. In particular, empirical fitti ng provides that the interpha se has limited extent (~0.5 m) and a modulus approximately equal to 0.5 Em. It is evident that inte rphase yielding, creep, and microdamage are possible even at low strains (~ 0.4% fiber strain). In these deformation mapping experiments [228] the interphase inel astic deformation was approximately five times greater than the elastic response. W illiams et al. [243] ascribe a large portion of this inelastic response to resi dual radial compressive forces developed during cure. In an attempt to relieve this stress before defo rmation mapping, the samples were relaxed by applying a creep or fati gue loading spectrum. Recall that compressive radial stresses are predicted for fibers in dilute concentrations, Eqn. 3.17, whereas tensile fo rces are expected for more densely packed fibers, Eqn. 3.18. The relaxation-loading sp ectrum in [243] was in tended to isolate the elastic and inelastic interphase responses, but it also had a second unintended benefit. By 37 Adsorbed species may include integral matrix species. 38 More precisely the proposed physical mode l fits the observed displacement profiles. PAGE 276 eliminating or alleviating radial compressive stresses at the interface, the stress state at the single fiber interface will more closely a pproximate the radial tensile forces expected in a macrocomposite and consequently the results and conclusions should be more representative. The very low strains required for inter phase inelastic deformation indicated by deformation mapping [228,243] and pull-ou t [182,193], which Williams and others [228,243] attribute mainly to residual compressi ve forces, also indicate that the boundary layer structure is highly faulted and hence una ble to sustain even small strains before yielding. Incipient interphase yielding is further encourag ed by irregular fiber surface geometry, fiber misalignment, fiber splitting, fi ber fracture, and near-n eighbor influences. Birefringence patterns during and after SFC F loading indicate that these same small strains are not sufficient to yield the bulk ma trix material [264-266], a further indication that interphase yielding is locali zed at the fiber-matrix interface.. Interphase composition/physical modeling An intriguing approach for estimating in terphase extent a nd modulus from SFCF data has been proposed by Skourlis and McCullough [227]. Combining a thermodynamic mixing model, which predicts the interphase st oichiometry (viz. interphase modulus), a therma l spectrum, and a simple three-phase CCA model, Skourlis and McCullough [227] estimate that IPthk 30nm and Gip 0.2Gm (glassy region). The interphase modulus is estimated by firs t applying the thermodynamic model [256], determining the epoxy content, and then testing a bulk sp ecimen at that epoxy ratio (DMA) to find its modulus-temperature profil e (Fig. 5 [227]). Finally, a three-phase CCA model [ref. (15) 227] is employed to predict the lc/d ratio. The results are PAGE 277 reproduced in Figure 3.48. The predicti on, while overestimating the change in lc/d with temperature, mimics the transitional behavior at ~60 C very well. This is not unexpected given the CCA models dependency on Gip and rip. It has been previously shown that the matrix axial and ra dial stresses decay rapidly from the interface. Because the bulk propertie s do not exhibit any transitional behavior at 60 C (Tg bulk 160 C), the transitional behavior for lc/d (Figure 3.48, and [200,212,267,268]) must be confined to th is small mechanical interaction zone surrounding the reinforcement. In this approach [227], an increasing stair-step variation in the modulus (interphase to matrix) follows from the variation in th e epoxy/hardener ratio with radial distance from the fiber surface (lattice layers) [256], Figure 3.49. The stair-step discretization seen in Figure 3.49 is generally consiste nt, although clearly an approximation, with evolving boundary layer morphology suggested by increasing dipole-group mobility with grafted chain length [123] and a physical pr operty-motional rate gradient from the interface outward [255]. This trend is in opposition with th e unfolding model employed by Theocaris and others [269,270], where th e local matrix modulus unfolds from Ef at the solid interface to Em at the mesophase/bulk matrix interf ace. Here the form is variously taken as exponential, power or linear. Lipatov and others [1 23,ref. (16) 243,271] entropic arguments regarding limited chain mobility at the interface also s uggest that the local modulus should be higher in the vicinity of the solid surface and decrease to the unperturbed matrix modulus at so me distance from the interface. f f ip ip m ip f f cE T G T G T G r r E C d1 12 2 12 1 1 1 (3.56) PAGE 278 The unfolding model used by Theocaris a nd others [269-270], despite recognizing the inhomogeneous nature of the interphase is not based on independent empirical evidence indicating a larger modulus near the interface, but is rather an assumed mathematical form that allows for extensive model-empirical fitting. One radial function for the unfolding interphase modulus is given by where 1 and 2 are equal for perfect bonding, and 1 0 for poor adhesion/load transfer across the interphase. The magnitude of 1 is estimated from Cp at Tg and Eqns. 3.50 Eqn. 3.51 [269]. with bl from Eqn. 3.50. This allows for Eip(r) vs. 1/2 determination (Figure 3.50). While convenient, this model cannot be theore tically justified and any empirical fit is attributable to the limited interphase extent and inherent numerical adjustability in the fitting procedure. Namely, 1 depends on bl, which is itself, paradoxically influenced by a uniform interphase assumption. Part of the apparent opposition/confusion between unfolding models (Figure 3.50) and a Skourlis-McCullough type model (Figure 3.49) arises owing to the distincti on between the actual physical pr operties of an RVE near the interface and the measured properties, whic h are strongly dependent on testing method and rate. As for entropic arguments, restricted macr omolecular mobility does not necessarily portend a higher interphase modulus as the restricted mobility may be accompanied by 2 1 r r E r r E E r Ef m f f m ip (3.57) f f bln n 1 1 5 255 21 PAGE 279 reduced density (steric hindrance), reduced entanglement (globules, agglomeration, or because of lower MW), or in the cas e of thermosetting materials reduced crosslinking/network formation. The eff ective interphase modulus may therefore be reduced, unaffected, or increased owing to competing entropic, mechanical, and energetic (chemical) effects. Entropic arguments do not preclude an interphase/boundary layer with an effective modulus less than the unrestricted matrix modulus ( Em). Interphase non-linear behavior A reduced interphase modulus in conjuncti on with lower yield stress and incipient SCFs will promote localized non-linear yield behavior. Pe nn and Defex [272] demonstrated that, even in the absence of common irreversible processes in the interphase (e.g., cohesive failure, molecular disentanglement, mechanical matrix/filler interlocking), the total work of fracture ( Wf) far exceeds the reversible work of adhesion ( Wa) for simple frozen adhesives (i.e., water, diodomethane, bromonapthalene, etc., cf. Table 1 [272]). In this case, the adhesives were frozen (brittle) and displayed no affinity for the surface. This eliminated interdi ffusion or chemical inte raction (irreversible processes), leaving only reversible molecular in teractions. Inverted blister tests (Figure 1 [272]) showed that Wf increased nearly exponentially with Wa (cf. Figs. 5 [272]). Following arguments by Wei and Hutchins on [273] and supported by clean interface failure along with carefully controlled e nvironmental conditions, Penn and Defex [272] attribute the exponential increase in Wf to orientation harden ing in the polymeric substrates and strain hardening for the metal substrates. The reoriented region (polymer, Lre 1.2nm; metal, Lre 0.75nm) is estimated using [273]. PAGE 280 where 0 is the characteristic Wf for the process zone (in [273] Wf is defined as approx. Wa, slow process). An alternative estimation for the inelastic zone by Gupta [274] gives Linelastic 10 m (CMC). There are, however, difficu lties with direct application of Eqn. 3.58. For instance, we know that Em varies with strain for most yielding polymers (Figure 3.51). Additionally, the interphase yi eld stress varies with position and is very likely less than Y [243,257]. There is also th e added complexity that the interphase/boundary layer modulus ( Eip) varies from the interf ace to its plateau value Em. These difficulties notwithstanding, the anal ysis suggested here for reoriented length in inverted blister tests (Eqn. 3.58, [272, 273]) is analogous to th e situation for load transfer at a fiber-matrix interface. In these experiments [272], the brittle adhe sive (assured by testi ng temperature) is unyielded (linear elastic behavior ) and remains adhered to the substrate until catastrophic Mode I failure. All the irreversible excess energy ( Wf Wa) is expended in the yielded substrate at the interface (Figure 3.52). Si nce we are dealing with the energy consumed in the yielded substrate, the mixed Mode I and II failure for axial load disbonding does not pose a problem. Both failure modes (inve rted blister and disbonding) will generate process zones prior to and during dynamic fract ure; the extent of this process is what Eqn. 3.58 tries to estimate. For the SFCF test we could imagine that the massive solid support, aluminum in the inverted blister test, is represented by th e fiber surface (i.e., graphite, glass, aramide), which is much stiffer than a polymeric matrix especially in the axial direction. In 2 2 01 3Y m reE L (3.58) PAGE 281 addition, a brittle unyielding adhesive joint is the very definition of a perfect bond. Consequently, the reorientation zone starts at the interface and extends into the interphase in our general load transfer problem. The in terphase extent previ ously defined as that region near the fiber-matrix interface wh ich differs morphologically and/or compositionally from the bulk matrix properties may be less or greater than the affected zone, Lre, predicted by Eqn. 3.58. Care is taken for this inverted blister test to exclude irreversible processes common at an interface. In particular, cohesive matrix failure, molecular disentangling, and mechanical matrix/filler interlocking are avoi ded by careful experimental configuration and environmental conditions. Th is concentrates and isolates the process zone into a very small region (est. ~ 1.2nm thick for polymers) Even with this small process zone, the total work of fracture is still one to two orde rs of magnitude greater than the reversible work of adhesion. Recall that the wo rk of adhesion is given by (Dupree Eqn.) and by the Young equation After substitution (Young-Dupree Eqn.) where is the surface free energy, subscripts s and l are for solid and liquid respectively, and is the solid-liquid contact angle. Selected Wa and Wf are reproduced in Table 3.12 [272]. Although there are some outliers (Figure 3.52) the trend for Wf vs. Wa is roughly exponential. From Eqn. 3.58 the pro cess length increases linearly with Wa. As Wa sl l s AW (3.59) sl s l cos (3.60) cos 1 l AW (3.61) PAGE 282 increases, greater separation forces will be resisted by the adhesive and this will precipitate increased yielding, beginning at th e interface. The process is similar to that for stable neck propagation in tensile dogbone samples, except here the yield zone is limited by an adhesion strength-orientation hardening trade-off at the interface. Compare also increasing interlaminar toughness and associated larg e matrix deformation with increasing debonding stress [195]. Large tensile reorientation capacity is supported by tests on thin (1 m) epoxy films [ref. (20) 238] that othe rwise exhibit 2% ultimate strain in bulk specimens [275]. The shear stress-strain diagram follows the typical non-linear strain hardening profile seen in Figure 3.51. Gulino et al. [238] approximate this non-lin ear profile in their modified shear-lag approach for predicti ng debonding and load transfer in model microcomposites (cf. dashed line in Figure 3.51). Adhesive failure may occur at any point along this stress-strain curve, al though joint failure prior to yielding ( Wf = Wa) was not observed in these tests [272]. Cohe sive matrix failure is possible if Wa is large or if the network structure limits deformat ion [213,238]. The excess energy ( Wf Wa) is, by definition, from an irreversible process and its exponential growth can arise only from adhesive failure during matrix strain harden ing (Figure 3.53). That is, for a process where the dependent variable in creases in direct proportion to the extent of the process itself, exponential curves are expected and, for mechanical beha vior in particular, orientation hardening is indicated. In this instance [272], the smooth substr ate and brittle non-interacting adhesives isolate the reorientation beha vior that would otherwise act in conjunction with other irreversible processes at the interface. The process volume will be larger for a typical PAGE 283 inhomogeneous interacting boundary layer, wh ere the yield strength and modulus are both reduced. Evidence [243] indicates that the modulus changes mo re slowly than the yield strength (cf. Eqn. 3.58). Irregular surface morphology and imperfect adhesion/wetting will create stress concentra tions that will be the incipient sites for reorientation/yielding be havior and disbonding. The total work of fracture for more comp lex higher MW adhesives, includes the various irreversible processes avoided by careful experimental design in [272]. Note that irreversible processes and/or cohesive inte rphase failure are no t possible without an adhesive joint capable of transferring suffici ent stress to induce deformation (substrate, interphase or matrix). The work of adhesion described above, Eqn. 3.61, is not strictly applicable for resin-fiber systems because of the aforementioned irreversible processes. Wa does however afford insight into the total work of fracture. In this regard, it is often easier to estimate Wa by its polar and dispersive compone nts. This is accomplished by determining the so-called harmonic (h) and geometric (g) means of Wa x ( x = h,g ), Eqn. 3.62 and Eqn. 3.63 respectively [ref. (14) 260]. The superscripts p and d denote polar and dispersive surface tension components, and l and s are as before. For slow (non-viscoelastic) testing rates, the total work of fracture is proportional to the work of adhesion ( Wf Wa) and, therefore, for com posite samples with a common matrix (viz. same VE contributions), the adhesion strength will be proportional to Wa p s p l p s p l d s d l d s d l h aW 4 (3.62) p s p l d s d l g aW 2 (3.63) PAGE 284 [260]. Utilizing this basic argument a nd modified Willhelmy surface tension measurements on untreated and basic/oxidized treated carbon fibers Ramanathan et al. [260] found a nearly lin ear relationship between the interf acial shear stress (fiber pullout) for HS carbon fibers and Wa x (Figure 3.54 and Table 3.13). Earlier fiber pull-out work by Mader et al. [259] on bi-functional epoxy resins provided results similar to that for simple brittle adhesives with acute reorientation hardening [272] (Figure 3.54). Here [259], in contrast to HS carbon fiber results [260], the relationship between Wa and debonding shear strength39 is approximately exponential, suggesting both non-linear VE processes and re orientation hardening in the process zone during fiber pull-out. The results for tetra-functional epoxy (similar Wa values to bifunctional epoxy) are not as st raight forward. Sizing-e poxy chemical interaction not evident for the bi-functional epoxy is indicate d (higher debond shear strength for similar fiber surface treatments), but wetting behavior (cf. Figur e 4 [259]) contravenes debond strength results for oxidized as compared to oxidized/sized fibers (Figure 3.54). Improved dynamic mechanical coupling betw een the sizing and epoxy or other physiochemical interactions must contribute to account for this tetra-f unctional epoxy behavior. By comparison, Zeta-potential and debondi ng shear strength tests on treated glass fibers in polyamide [259] reveal linear Wa vs. debonding strength be havior as seen in [260]. Linear elastic force-displacement curves (cf. Figure 9 [259]) support this behavior and imply uniform interphases with brittle characteristics It should be clear from above th at a linear relationship between Wa (predicted or measured) and adhesion strength occurs only for cases where the adhesion strength is 39 Mader [259] uses the term debonding shear strength instead of ifss. It is not clear, however, that the pullout behavior is strictly adhesive. PAGE 285 limited to simple interfacial adhesive failure only. This is a direct consequence of the definition for Wa, which is defined in [260] as the wo rk required to reversibly separate two phases from the equilibrium state at their common interface. That is, Wa is strictly two-dimensional and therefore cannot accoun t for any phase-phase interdiffusion or specific phase-phase irreve rsible interaction (chemi cal reaction, mechanical coupling/entanglement). Consequently, when adhesion strength is substantially greater than Wa simple interfacial adhesive failure is discounted in favor of a more complex, more energetic interphasial failure. Curiously, for single fiber pull-out tests, a lack of functional dependence for the apparent interfacial shear strengt h against fiber embedded length ( Le) implies ductile failure of the process zone [130,261, ref. (41) 262]. In contrast, if the apparent (averaged) interfacial shear strength (ifss) decreases with fiber embedded length (Figure 3.55) a more brittle failure is indicated. This interfacial failure is not restricted to the obvious fiber-matrix interface [259,276]. Adhesive failure between a fiber and its coating, which itself is cohesively bonded to the matrix, is also possible [262,277]. Cohesive failure between the transcrystalline morphology adhe red to the filler and bulk spherulitic crystallites has also been speculated40 [214, ref. (14) 214]. In each case, the weakest interface (link in the adhesive ch ain) fails giving rise to monotonically decreasing ifss vs. Le behavior (Figure 3.55). Note how ever, that adhesive failure does not preclude or disprove non-lin ear interphase mechanical pr operties; only that by this technique (single fiber pull-out) it is not evident. These va rious failure modes reveal the 40 This is analogous to hoar crystalline interfaces in snow packs, which when overloaded precipitate avalanches. PAGE 286 inadequacy of performing pullout test without visually, or otherwise, determining the locus of the failure path and failure mode. Surface energetics and dynamics The influence of filler we ttability is complicated by filler surface roughness, capillary wicking action between fibers, which acts to reduce fiber separation [278], and sizing physical barrier issues [279]. Fiber surface roughne ss generally in creases after oxidative and non-oxidative surf ace treatments [280]. This increased rugosity can lead to incomplete wetting despite strong energetic dr iving forces [281]. Increased wetting is usually accompanied by reduced void content [278], which increases the apparent interfacial shear strength by reducing the severity and density of surface stress concentrations. This trend is echoed in [279] where void cont ent increases as wettability decreases. Fisher [235] also argues that elevated cure temp eratures relative to the matrix Tg allows for increased molecular mobility at the interface and hence better bonding. For a spreading liquid (contact angle 0 cos 1), with known surface tension (i.e., (hexadecane) = 21mJ/m2), Eqn. 3.61 allows for perimeter estimation, Perim. = f / l ( f is the emersion force, Eqn. 6 [282]). Wh en incomplete wetting is caused by surface roughness or irregular perimeter extent [282], the perimeter will be underestimated (e.g., T-300 CF: f actual = 7.0 m, f surf = 5.7 m [281]) and therefore, by extension, geometric and harmonic surface energies Wa x will be artificially elevated. For example, the total surface energies for Toray T-300 carbon fibers s ubjected to heat treatments in air (15 min. at 400 C, 500 C, 600 C) increase in order (T = 50, 58, 62 mJ/m2), in general agreement with fiber surface roughness ( Rz = 8.4,16.4,15.5 nm) and XPS spectra, which indicate increasing O/C with heat treatment temperat ure [283]. The increased surface PAGE 287 energies in this case are attributable prim arily to the polar surface energy contribution (Fig. 3 [283]). However, just as [260] remark s that separating the pa rticular contributions to adhesion strength (i.e., increased mechani cal interlocking or chem ical interaction) is difficult, so to is distinguishing between r eal polar surface energy increases and apparent increases related to incomplete fiber wetti ng attendant to increas ed surface roughness. It is important to recognize that the C OV for surface energy measurements can be quite large (p: 34%, d: 28%, t: 4% [239,283]) owing to surface roughness, adsorbed species [284], and ma terial density fluctuations41. Empirical difficulties related to contact angle resolution ( 8 [259]) and non-flat surfaces, especially small diameter fibers (e.g., CF dia. ~7 m), also contribute to the considerable COV [83]. The large COV in Wa notwithstanding, the asserti on by Ramanathan and others [261, ref. (23) 261] that there is no direc t correlation between wettability, work of adhesion and the adhesion strength [261] is too narrow. While a simple single parameter functionality between Wf (viz. ifss) and Wa is not supported by the data for HM carbon fibers in epoxy [260] nor HS and HM carbon fibers in a thermoplastic composite (PPS) [261], the work of adhesion does influe nce the apparent interf acial shear strength in pull-out, push-out, and SFCF tests, as we ll as the failure path in fracture and delamination experiments. In fact, deviation from a simple Wf vs. Wa relationship indicates a process zone that is three-dimensional. That is the process zo ne involves the two-dimensional interface, the adjacent matr ix, and possibly, fiber surface layers (CF: [refs. (9,10) 260,276,284]; UHM WPE: [ref. (19) 285]). 41 The dispersive component, d, is proportional to the surface charge density and thus sensitive to material density fluctuations [228]. PAGE 288 Gupta et al. [274] considered an analogous situation with regards to an ideal mixed Mode I and II two-dimensional pu re cleavage fracture (3 J/m2, hard inorganic solid), and a more involved three-dimensi onal process/inelas tic region (60 6.85 J/m2, SiCpyroltic graphite system). As above, the en ergy consumed in the process zone is far greater than the simple two-dimensional cas e. According to [ 274] the total toughness ( Gc) scales with the intr insic interface toughness ( Gc0) and K where and Gp is the inelastic contribution. That is, the intrinsic interface toughness controls the total toughness, although the major contribution arises from the inelastic contribution. The same authors [274] argue that interface cohesive and shear strength are inherently difficult to measure because they are str ongly influenced by local imperfections. Moreover, composite failure is more likely en ergetically controlled and consequently, the work of separation is a more appropriate indicator than interf ace cohesive or shear strength. Interphase fracture mechanics It is evident that th e work of adhesion and Wf, while more fundamental measurements than the flawed interfacial shear strength, are physical properties of the substrate and adherent. As such, practical Wa and Wf values are subject to a range of mechanical, chemical, and empirical influe nces. Indeed, filler surface energetics, topology, and morphology are intr insic to composite performance. These factors do not ) 1 (0K G Gc c (3.64) 0 c pG G K PAGE 289 act independently but act colle ctively to provide a particul ar boundary layer and evolving interphase microstructure, which in turn play a critical role in m acrocomposite properties. The topology contribution is particularly difficult to isolate because topology and surface energetics are intimately related. Fu rthermore, surface treatments designed to increase/decrease fiber-matrix adhesion i nvariably affect the fiber topology. Consequently, the surface treatments alter th e fibers flaw and strength distributions. Accordingly, indiscriminate use of ph ysical models and equations for SFCF interpretation (e.g., Eqn. 3.4 and Eqn. 3.5) which necessarily include a fiber strength statistic will provide mislead ing information (all other fact ors understood and constant). In fact, the change in fiber strength with surface treatm ent has long been acknowledged [187,284], but is inexplicably ignored in mo st SFCF and microcomposite interpretations. Surface treatm ents affect the fiber strength distribution by modifying the surface defect distribution. Light surface trea tments for carbon fibers (oxidative and nonoxidative) first attack carbon and/or hetero atoms on the edges of surface micropores (width<2nm) [276,286,287,276]. These high-en ergy sites are the exposed edges of turbostratic42 graphite basal planes and amorphou s carbon, which intersect the fiber surface and micropores (Fig. 9 [289]). The initial surfaces attacked are sometimes re ferred to as weak outer layers [260]. This is somewhat of a misnomer as materi al removal is far from uniform [276,286,287]. The notion of layer removal is inaccurate. In addition, a weak outer layer usually refers to the condition where there is increased fibe r-matrix adhesion post-treatment. This does not necessarily imply weak fiber surface layers, but instead may be attributed to increased 42 Turbostratic graphite (first described for carbon black by Biscoe and Warren (1942) [288]) describes an imperfect graphitic structure where the basal plan es are separated more than for ideal graphite PAGE 290 fiber rugosity, improved wetting (more effec tive surface area), or modified surface energetics, all of which may lead to improved fiber-matrix adhesion. The particular fracture mechanisms c ontrolling post-treatment fiber strength statistics and in-situ fiber fracture are not well understood. The di stinction between asspun and treated fibers is often ignored and th e effect of the fibers in-situ environment has not been addressed. In this regard Rand and Robinson [ref. (8) 280] found that microporosity increased from 2x10-5 cm3/gm to 7x10-5 cm3/gm for High Modulus (Type II) carbon fibers with a commercial oxidat ive treatment (electrolytic). Increasing porosity and roughness portends decreasing m ean fiber strength [284,290]. However, this is not always borne out. Denison et al. [286] found th at carbon fiber strength was unaffected by electrolytic surface treatmen ts (ammonium bicarbonate) despite obvious improvements in ILSS with increasing trea tment aggressiveness, which suggests fiber surface modification. Similarly, the mean fi ber strength increased for low temperature plasma polymerization surface treatment (PS and PAN) on carbon fibers [285] and silane treated glass fibers [266], this is sometimes referred to as heali ng the surface flaws. DiBenedetto and Lex [266] contend that the fl aw density is unchanged with the silane treatments, but the flaw severity is some how reduced. No reasoning is provided and details regarding porosity, weight loss, or roughness were not provi ded. A carbon fiber surface is far more disorganized and pitted [289] than a glass fibe r surface. The glass fiber topology and morphology is more unifo rm, energetically and geometrically. Furthermore, silane treatments are mild [266], which promotes uniform surface deposition, possibly accompan ied by slight physical etch ing or chemical ablation. (d002(gr)=0.246nm, d002(turbo)=0.335nm), a consequence of random orientation about the layer normal. PAGE 291 The similar responses for the dissimilar fibers and treatments suggest a common mechanism for the observed trea tment-strength relationship [266,285]. Considering first the glass fiber silane treatment: DiBenede tto and Lexs [266] statistical approach, wherein two rectangular boxes are used to re present distinct surface and volume flaw populations in a classical proba bility density function (Figure 3.56), indicates that surface flaw severity is reduced (cf. Table 2 [266]) while the higher strength internal flaws are unaffected by the surface treatment. It is known that the most severe surface fl aws (transverse nicks and axial scratches) are attributable to processing and handling, hence the use of sizing agents. Uniform surface treatment with silane coupling agents is unlikely to preferen tially attack these flaws. The silane solution (5 silane, 5 water, 90 methanol by volume [266]) treatment does however cover the surface43 (such a dilute solution would not be expected to be volume filling). The silane treatment likely hydrolyzed the M-OH surface groups (polysiloxane formation) as evidenced by increased ifss relative to untreated fibers. Any chemisorbed (hydrogen bonding, dispersion bondi ng) or physisorbed silane [ref. Schrader (1970) 195] was removed by the post-treatment (100 C water for 30 min.). These three regions (polysiloxane formation, chemisorp tion, physisorption) are depicted in Figure 3.57. The resulting glass fiber surface is smoothed out by the conforming polysiloxane coating and, although not reported [266], the su rface energy is likely increased relative to the untreated glass fibers. 43 The authors [268] imply that the silane thickness varies by less than ~1-2 m. PAGE 292 Crack formation energetics To understand this surface treatment-strengt h relationship we must consider the energetics of crack formation in continuous bodies. The classical energy balance for crack growth (new surface A ) in a continuous body subjecte d to external tractions and body forces provides [290]. W is the external work done by tractions and body forces, T is the total kinetic energy, is the elastic strain energy, D is the continuum dissipation, and C is the separation work44. The net mechanical energy expended (LHS of Eqn. 3.65), commonly the driving force: G is opposed by the resistance to crack fo rmation (RHS of Eqn. 3.65). Recall that Griffiths original approach for brittle glass-like materials set C = 2, where is, again, the material surface energy. Simple substitution gives Both G and D generally depend on a displacement factor and its history (Figure 3.58). The critical condition for crack initiation ( Gc) by Linear (Elastic)45 Fracture Mechanics (LFM) is simply: Dc is depicted in Figure 3.58 at displacement c, where Gc(c)Dc(c) = 2. For crack initiation D is constant (material and fracture mode held const.), while for propagation it is a specific function of the crack tip translat ion as well as the materi al and fracture mode. 44 Separation work is the work done to generate new surface area (A ) per unit crack area, or for brittle fracture (2 ) [1]. 45 Unfortunately, often referred to as LEFM even though it includes the decidedly inelastic dissipation contribution. D C dA d dA dT W (3.65) D G 2 (3.66) c cD G 2 (3.67) PAGE 293 It is evident from Eqn. 3.67, variations in Gc (viz. KIc) may arise from changes in either surface energy () or continuum dissipation ( D ). It is known that water and other strongly polar species have considerable influence on the fracture behavior of glass and ceramic materials. The small positive ions, H+, in water can form strong hydrogen bonds with pol ar molecules in the material. This effectively reduces the materials surface energy, which naturally leads to reduced fracture toughness, as predicte d by Griffith Energy Failure criterion. That is, the moisture lowers the formation energy for new surfaces at the crack tip, which makes crack propagation more energetically favorab le and therefore fracture more likely. For instance, the drop in surface ener gy, 20%, for a silicate glass in the presence of water is consistent with an en ergy drop caused by sharing electrons with the Hydrogen proton (Hydrogen bond), ~6 kCal /mole. Accordingly, there is a proportional decrease in the fr acture toughness of this silicate glass [291] in the presence of water. For this type of system, the maximum crack growth rate is determined by the slowest process, diffusion of the water molecule into the interatomic width at the propagating crack tip. Embrittlement and plasticization is possible for some binary liquid metal-solid metal interactions as well. Unlike the case for ceramics, this phenomenon is not based on the Griffith Energy Failure criterion. Rather, it is postulated that the change in fracture toughness is associated with a lowering of the po tential at the surface. This allows nearly instantaneous brittle fracture propagating perpen dicular to the farfield stress. Once again the crack growth rate is determined by the slowest process of hydrodynamic and migration movement of the liquid in the cav ity of the crack [291] . Additionally the PAGE 294 incipient crack formation is independent of any local stress intensit y factor, although it is still dependent on the farfield stress magnitude which must be sufficient to allow liquid metal migration to the new surfaces. It would be similarly nave to believe that a fiber surface treatment or the proximity of an interphase has no effect on a fibers fa ilure strength. From above, an increase in Gc (viz. KIc) for silane treated glass fibers or plasma polymerized carbon fibers (e.g., [266,285]) may be attributable to either an increase in or Dc. For the carbon fiber treatment in [285], pref erential carbon and hetero atom ab lation during the early stages of plasma polymerization [285,292,293] will soft en the micropore edges. The effect on Dc is subtle and complex here. Recall that Dc is a function of both th e material and fracture mode. The fracture mode broadly includes th e geometry and loading conditions as well as the specimen surface topology. That is, surface irregularities and micropores, which act as stress concentrators, are reduced by the actions of the surface treatment. The surface treatment consequently a lters the driving force curve G () (dashed curve in Figure 3.58). The critical displacement factor increases and, as a result, Dc and Gc also increase (compare argument rega rding localized reorientation and Wa). CF surface free energy A plasma polymerized coating (PAN, 40nm thk.; PS, 100nm thk. [285]) will dissipate additional energy prior to fiber failure thus affecting D () directly. By contrast, embrittled or damaged coatings can markedly reduce fiber strength [167,207]. Finally, the plasma treatment raises the surface ener gy relative to an untreated carbon fiber (c PAN = 54, c PS = 40, c un = 32 mJ/m2 [285]). This surface energy increase may, however, be inconsequential with regards to fiber streng th because the new surface area for fracture is PAGE 295 transverse to the external surface. While there is some evidence that subsurface oxygen content may be slightly increased with high temperature oxidation (15 min. at 400 C, 500 C, 600 C [283]); X-ray Photoelectron Spectr oscopy (XPS) data, which averages through its penetration depth (~80nm [294]), evidences oxyge n content variation against treatment temperature (10 ) that is much smaller than the observed surface energy variation (25 Fig. 6 [283]). This implies very rapid oxygen content decline from the surface and consequently, transverse frac ture surfaces will be largely unaffected by this type of surface treatment. Moreover, low temperature plasma polymerization [285] is a low energy process, consequently, s ubsurface carbon fiber modification is not expected. An intriguing aside attends to the actua l surface energy of clean, untreated carbon fibers. Clean graphite surfaces have hi gh surface energies (est. HOPG, 90 mJ/m2 [239]) which are lowered r eadily by oxygen, nitrogen, CO2, and water adsorption [239,295] (Table 3.14). It therefore follows that newly created surfaces, as in Mode I fracture, will have surface energies gr eater than that measured by common surface techniques (e.g., Willhelmy Force, Sessile dr op). Furthermore, this surface energy will be unaffected by common surface treatments. CF topology/microstructure Carbon fiber microstructure and topology also affect surface energetics. For example, Vukov and Gray [239] determined that the dispersive energies for cleaned (100 h in Nitrogen at 160 C) HS and HM graphite fibers, 76.8 and 54.4 mJ/m2 respectively (Table 3.14), were related to the characteristic semicrystalline structure (10 and 50 respectively). That is, HS carbon fibers have more high-energy prismatic edges PAGE 296 intersecting the fiber surface a nd consequently a greater di spersive energy than HM carbon fibers. This holds despite material densities ( (HS) = 1.75 gm/cm3 and (HM) = 2.0 gm/cm3) which would otherwise suggest the opposite trend42. As suggested above, oxygen, CO2, and water adsorption lower the surf ace energy, leaving the HS and HM fibers energetically indistinguishable ( d (HS,untreated) = 38.8 mJ/m2; d (HM,untreated) = 39.4 mJ/m2 [239]). This natural equilibrium uptake in air is referred to as the protective antioxidant layer (PAOL) [293]. For high surface free energy graphite surfaces and carbon fibers, it is evident that a percentage of the high energy prismatic carbon edge sites adsorb chemical species from air (e.g., water and CO2). These physisorbed species can be removed by relatively mild surface treatments, 100 h in nitrogen at 160 C [239] and 20 h in helium at 105 C [ref. (35) 239]. The latt er treatment gives d(HS) = 50.4mJ/m2, indicative of an intermediate state between as received and cleaned HS carbon fibers (38.8 and 76.8mJ/m2 respectively [239]). This further suggests that the species are physisor bed or chemisorbed at most (Figure 3.57), which allows for their removal short of cohesive energies. Favre and Perrin [284] recognized the hom ogenizing effect of surface treatments for high surface energy fibers (Table 3.14), which are known to readily adsorb species (chemically and physically [239,285,293]). W ithout an ameliorating surface treatment, the irregular surface topology and energetics of as-received untreated CFs result in a large COV for interfacial shear strength. In contrast, the COV was considerably reduced and adhesion strength increased by various oxidative and non-oxida tive treatments (Table 3.15 and [284]). PAGE 297 Surface topology may also affect mechanic al interlocking between the fiber and bulk matrix [277,296,262], although proof of this is by inference as direct evidence is not available. Kim et al. [277] argue that a volume filling PVAL coating reduces the roughness seen by the matrix thereby lessening mechanical keying between the fiber, with its coating, and the epoxy matrix. This is a somewhat specious argument as the PVAL acts as both a physical and energetic ba rrier; it would be difficult to separate the two contributions (cf. [260]). In a similar vein, single fiber pull-ou t tests [262] for an electrocopolymerized carbon fi ber coating (acrylamide and carbazole) reveal a weak fiber-coating interface (strong coating-matrix adhesion) as compared to strong fibermatrix adhesion without the fiber coating. In this case [262], the i rregular coating (Fig. 1 [262]) is both energetically (IEP46(coated fiber) = 7.5 pH, IEP(uncoated fiber) = 4.2 pH, IEP(epoxy matrix) = 4.0pH) and mechanically beneficial fo r coating-epoxy adhesion. Untreated HS carbon fiber and the epoxy matrix have similar pot traces (Fig. 2 [262]), which would suggest poor adhesion (cf. Figure 3.42). However, the ifss results [262] indicate otherwise (Table 3.16); strong mech anical interlocking is therefore inferred [259]. Finally, with regard to topology and wettabili ty, it has been established that surface roughening increases wettability for wettable surfaces (contact angle: <90 ) and decreases wettability for non-wettable surfaces ( 90 ) [292]. Accordingly, for wettable systems (substrate and adherent) the void content should decrease as wettability increases. Peters [296] found that laminate void content drops with fiber pretreatment level (Table 3.17). The results in [296] underscore the interplay between topology, 46 The isoelectric point (IEP) is the pH value at pot=0. PAGE 298 wettability, and adhesion strength. In partic ular, interphase development is found to be dependent on the specific interaction between the fiber surface and matrix. Schematic representation of fracture mechanics The effects of fiber surface treatment (viz wettability and defects) and adhesion strength (various influences) on element failure probability are schematically represented in Figure 3.59. The schematic applies to 0/90x/0 transverse cracki ng experiments [296]. Its correlation with the more common SFCF te st will be discussed below. First, the significant features of Figure 3.59 and Figure 3.60 will be addressed. The curves in Figure 3.60 represent the locus of ordered failure probability ( Fi) against the element failure strain (f,i). For transverse ply cracking, the fa ilure strain distri bution arises owing to the disparity between axial load-bearing capacity and transver se failure strain, which is generally much smaller. Multiple transverse cracks are generated before ultimate failure across the axial plies. The loading condition is derived assuming shear-lag load transfer across the axial-transverse ply inte rface as depicted in Figure 3.61. Referring to Figure 3.59, Peters [296] argue s that improved fiber wetting shifts the early, defect dominated (Figure 3.60), failures to higher failure strains (Figs. 7 [296]). Increased fiber-matrix bond strength and a reduc tion in defect concentration and severity (Table 3.17) are given as explanation. As above (cf. Eqn. 3.64), the increased fibermatrix bond strength enables increased stra in to failure capacity in the compliant interphase [refs. (23,24) 296]. The 0y/90x/0y cross-ply stress state and that for SFCF testing are similar, which has been noted and used to good effect [158]. The primary difference being the constraining effect of the neighboring 90x layers and the added statistical influence of multiple 90 PAGE 299 core plies. Generally, the 90 core layers are treated as an effective medium, obviating the need to consider probable fiber misa lignment (through the thickness) and area density variations (Figure 3.61), this al so greatly simplifies the analysis. Delayed load transfer to the 90 transverse plies is similar to the situ ation for load transfer across a compliant interphase for aligned fibers [FEA: 178,181,297; Analytical: 179,211; Empirical: 227,255]. Peters [296] attributes shif ts in the upper end of the failure distribution (Figure 3.59) to improved bond strength and mechani cal influences, as opposed to the defect controlled lower strength failures. The particular influences of fiber Weibull modulus (), matrix ductility, and mechani cal constraints on element failure47 are indicated in Figure 3.60. The number of links is chosen to afford high failure probability estimation ( F ) for the ultimate link fracture via and the failure ordering condition Fj = j /(N+1). Here j is the ordered link failure number, N is the number of links, f,lo is the link failure strain, f,lo is the characteristic fracture strain, and is the Weibull modulus. According to [296] the ultimate link fracture ( j = N ) represents the strain at interphase failure for an ideal fiber ( = Figure 3.60, Fig. 3 [296]). Of course, no fiber is ideal and ultimate fragmentation48 does not necessarily occur at interphase failure as argued by [ 296]. More importantly, because there is an 47 In [299], an element is a link of the continuum 90 plies (L=105 mm, 100-200 elements). 48 Ultimate fragmentation indicates fragmentation satura tion. That is, no additional fragmentation with increasing testing stress/strain. 0 0, ,exp 1l f l fF (3.68) PAGE 300 obvious multi-modal failure process (cf. Figs 7 [296]), extrapolation from the defect controlled early failures to saturati on fragmentation cannot be supported. The failure distribution for SFCF tests (Simulated: [6,203]; Empirical: [298,299]; Analytical: [300]) can be dis tinguished by fiber defect and load transfer controlled regions (cf., Fig. 3 [299], Fig. 4 [300]). The transition in failure rate with displacement/loading indicates a change in load transfer micromechanics and/or statistical variation in fiber strength [158]. Simple matrix yielding is often argued [111113,296]; although, as we have detailed above th e nature of load transfer is decidedly more complex. The changing fiber-matrix micromechanics may derive from fiber-matrix debonding, random fiber-matrix imperfections, in terphase/matrix reorientation or strain hardening, and simple matrix pl astic yielding. Whatever the cause, it is clear that the fracture progress in SFCF and 0y /90x /0y experiments is process dependent. Simple inspection of the final FLD without considerat ion of the process and/or progression (e.g., Eqn. 3.5) will obscure important aspects of the composite performance, especially interphase behavior. Work by Ikuta et al. [298] clearly indicates the effect an imperfect interface conditions on fragmentation progression. The investigators [298] utilized SFCF tests with fibers surface treated on only one-half of th eir length. This mostly eliminates fiberfiber (within a bundle) diamet er variation [156] and reduces SFC specimen-specimen variation (cure shrinkage, matrix chemistry, handling, geometry, etc.). A two-stage progressive fragmentation proce ss is clearly indicated for the treated half-length, whereas for the untreated section only a single failure mechanism is indicated. Initial average fragmentation lengths (beg inning of the SFCF test) for the treated section l are PAGE 301 consistently shorter than t hose for the untreated section lb whereas, the average ultimate fragmentation lengths for both systems are roughly equivalent (Fig. 5 [298]). In light of these results [298] and other data Felliard et al. [ 299] conclude that: All of the results show that the interf acial reinforcement reflects the fracture progress rather than the final fracture state. This conclusion elicits serious concerns re garding SFCF interpretation, and reflects the oversimplification and idealized load adsorption assumptions attendant to Eqn. 3.5. For reasons enumerated above, perfect bonding [105] and/or localized matrix shear yielding [112,6] may not adequately de scribe the complex fiber-mat rix interaction for aligned uniaxial loading conditions. Analytical Model Assessment In the preceding discussion, we have e xplored the nature of the fiber-matrix interaction, primarily with regards to load ab sorption from the matrix to the fiber across a nebulously defined interphase, which may be chemically and morphologically different from the bulk matrix, and consequently ex hibit unique mechanical properties. Macromolecular conformational constraints adj acent to the solid filler surface may also affect the dynamic properties of the interphase In this section we are looking to assess analytical model accuracy, where model accuracy here refers to a sense that the model must provide an accurate prediction and fa ithfully represent the underlying physical process. Cox and Rosen One-Dimensional Analytical Models It is clear that the mechanical propertie s of the interphase (e.g., modulus, yield strength, failure strain, adhesion, and th ermodynamic properties) influence the load absorption across the interphase. The varia tion in these mechanical properties for a PAGE 302 particular composite system (e .g., high strength carbon fibers in an epoxy matrix) is both spatial49 (radial) and statistical (chemical com position, macromolecular conformational, surface energetics/topology, and morphological). The statistical variation in mechanical properties are often neglected to enable anal ytical (deterministic) evaluation. The onedimensional (axial) load absorption profiles pr ovided by Cox ( Eqn. 3.9) and Rosen (Eqn. 3.11) are examples of determin istic models. Not only do both of these analytical models neglect any statistical va riation in mechanical properties, they also preclude matrix radial property variation, matrix yi elding behavior, and hoop stress contributions. Subsequent to Coxs first attempt at short/broken fibe r load absorption [105], there was convincing evidence provided by Outwater [146] that inte rphase shear yielding occurs during load absorption. Obviously, Coxs analyses did not include this yieldi ng effect, nor would it likely have been included, as the scope of his shear-lag solution was narrow and limited to paper systems where the strains considered are small and thus a representative elastic solution is appropriate. Rosens analys is, however, clearly ignores the yielding condition, despite evidence to the contrary. A purely elastic one-dimensional analysis (e.g., Eqns. 3.9.12) cannot capture the actual physical process of shear load transfer across an in terphase when matrix shear yielding is inevitable (e.g., n ear fiber breaks/ends: Analyti cal [139]; FEA/numerical [166167,178-181]; Empirical [113,146]). This is es pecially true for SFCF evaluation where large matrix strains are required to re ach fiber fragmentation saturation (~3 x f ,ult [181]), the required ultimate condition for lc and ifss determination. Even for stress states where ideal fiber-matrix interaction is expected (i.e., no distinct interphase, perfect bonding, no 49 The axial mechanical response will vary from a fiber break/end, but the axial mechanical property variation is entirely statistical. PAGE 303 solid interface influence), the statistical variation in the mechanical properties of the interphase can lead to non-lin ear inelastic interphase behavi or [238] for areas of acute matrix stress/strain such as the terminus of fiber/matrix debonding, regions with irregular interphase chemistry or morphol ogy, and large fiber asperities. Yielding in the interphase is even more probable given the cons iderable empirical evidence that indicates a thin (~50 nm nm) interphasial re gion exists with mechanical properties that ar e different from the bulk matr ix properties [123,257]. In particular, the yield strength of the interphase is reduced relative to bulk matrix yield strength. Acute mechanical loading near th e fiber break, and other irregularities (fiber asperities, fiber waviness, etc.), in combina tion with a weak interphase will result in permanent yielding in the interphase. This of course, precludes elastic treatment regardless of the assumptions made in the elastic analysis. As for the elastic analysis, recall the assumptions that the Cox and Rosen analyses employ regarding radial stress variation, that, while in the limit are reasonable, require assigning radial influence parameters ( r1: Cox and rb: Rosen), which ul timately control the load absorption predictions (cox and rosen). Approximations for the shear interaction parameter (, Eqns. 3.7 and 3.13) also fail to hold for low volume fractions, which is vital for accurate interpretation of microcomposite tests such as SFCF and single fiber pullout. Consequently, and for reasons previously enumerated, we contend that the Cox and Rosen elastic analyses (Eqns. 3.10 and 3.12) fail to satisfy even the most basic model accuracy requirement; that is, faithfully re presenting the underlying physical process, and therefore should not be used to evaluate load absorption profiles, especially SFCF test interpretation or prediction PAGE 304 Kelly-Tyson Yielded Matrix Model Arguably the most common, if not simplest, analytical treatment for fiber-matrix load absorption (with particular usefulness for SFCF interpretation) was posited by Kelly and Tyson [111], Eqn. 3.5. The main premis e of the Kelly-Tyson approach is that the interphase region is everyw here yielded and its yield strength is constant (y, perfectly plastic). This naturally produces linear FAS build-up (Figure 3.1) from the fiber end. This approach to load abso rption analysis is invitingly simple, but unfortunately suffers, in a predictive sens e, because of that same simplicity. In the broadest sense, a predictive model is a mathematical function/formula with known parameters that is representative of the underlying physica l problem. The final mathematical formulation invariably involve s assumptions that simplify the model with the expectation that the error introduce d by the assumptions is understood and the magnitude of the error is acceptable. The assumptions that enable Equation 3.5 and ifss determination disregard mechanical propert y variation (interpha se yield strength, adhesion quality, etc.) and require single valu ed estimates for the stochastically varying lc and f,ult. The validity and accuracy of Kelly-T yson SFCF analysis is consequently dependent not only on the empirically determined FLD, but also, and more importantly, on the fiber strength variati on with testing volume (process fragment length distribution) and the inadequately/incorrectly defined fiber length distribution statistic lc. The fiber length distribution is not easily described, it is neither normal nor Weibull [6,200]. Efforts to estimate lc from the FLD typically invol ve simple (e.g., Kelly-Tyson) load absorption models [5,6]. The interpretation of the FLD and its defining statistic lc are necessarily biased by the load absorption model itself. We know that the statistical PAGE 305 variation in interphase and fiber mechanical properties al ong the fiber contribute to the actual FLD during SFCF tests; therefore, ch aracterizing the shape of the FAS build-up adjacent to fiber breaks apriori (e.g., shear-lag or linear) to enable interphase performance interpretation is clearly flawed, as it predetermines what is being investigated. The use of a constant value fiber strength statistic in SFCF interpretation is also flawed. As the single fiber fragments during a SFCF test (increasing fiber strain/stress), the effective fiber gage length for the fibe r segments is continuously reduced until fragmentation saturation. The effective gage length can also be affected by fiber matrix debonding, which reduces the portion of the fi ber subject to the maximum monotonically increasing fiber load (plateau length). Fibe r clamping can also di srupt the actual fiber axial stress for short gauge length fiber tests [79]. The failure probability for these fiber fragments is a function of their length/vol ume and the underlying fiber flaw/strength distribution. The ultimate fiber strength in Eqn. 3.5 should, at the very least, be scaled to a length consistent with the fragmented lengt hs. What length and, by extension, fiber strength to choose is a key problem for this deterministic approach. Recall that Weibull scaling requires that the flaw strength distribution be well characterized on the lengths under considera tion, and that large scaling ratios risk excluding or biasing the streng th prediction (Ch. 2). The in-situ fiber strength after composite fabrication may also be reduced relative to ex-situ measurements. This is especially true for macrocomposites where c ontact with near-neighbor fibers is certain during the fiber surface treatment process, for wet lay-up, and pre impregnated sheet/tape/tow fabrication. Dry fiber handli ng, required for SFCF specimen preparation, also risks damaging the fibers. Typically, fiber strength measurements are performed on PAGE 306 untreated fibers. The effects of fiber surface treatments and fiber handling during composite fabrication are not factored into the in-situ fiber strength predictions, even though the in-situ fiber strength is affected. Monte Carlo simulations may be used to derive FLD lc correlations that account for individual fiber strength-volume scaling effects [5,6]; however, the correlations can only narrowly be applied as the underlying phys ical model is predic ated on a particular fiber strength distribution and fiber-matrix inter action model. It is also important to note that the load absorption model is ideal for this approach. That is, there is no statistical variation in interphase propertie s or load absorption mechanism. We have noted the statistical difficult ies with applicati on of the Kelly-Tyson model, Eqn. 3.5; however, the preceding disc ussion does not address the main premise of the Kelly-Tyson formulation which is: The ma trix is everywhere yielded and load absorption occurs exclus ively through this yielded zone. The original formulation [111] also assumes perfect fiber-matrix adhesion. It is therefore understood, that when there is imperfect adhesion or the matrix response is not perfect plasticity (i.e., elastic deformation or non-linear pl astic strain hardening, Fi gure 3.51), the Kelly-Tyson formulation should not be used. The Kelly-Tyson assumption [111] that the whole of the matrix is brought to its yield stress is not strictly followed when the Kelly-Tyson model (Eqn. 3.5) is employed for SFCF interpretation. Whereas high matrix strains are required for SFCF tests to insure fragment length satu ration (e.g., graphite-epoxy, 7-8% [181]); typically, and this applies to SFCF tests in general regardless of the interpretation formula, a modified resin system is used to avoid transverse matrix mi crocracking that would otherwise occur if the PAGE 307 unmodified resin were used. The effect of this material substitution on SFCF interpretation is generally overlooked. If we are concerned only with matrix yielding, then this material substitution may not be critical; however, if resin chemistry changes measurably, there is a real possibility that fiber-matrix adhesion will be altered. One could easily argue that the SFCF specimen is not consistent with the macrocomposite material and consequently, the SFCF results are not representative for this reason alone. The assumption regarding matrix yielding fo r SFCF interpretation is that the large stress concentration near the fiber break yields the matrix locally to an extent that maximum fiber loading is achieved for the ap plied remote stress. All of the other statistical questions/di fficulties notwithstanding, we must still choose an appropriate yield strength (y) if we are looking to model load ab sorption using a Kelly-Tyson approach. Assigning the bulk yield strength, determined at an appropriate strain rate and temperature does not satisfy the accuracy tenet to faithfu lly represent the underlying physical process. Specifically, we know that interfacially confined polymeric materials exist in a non-equilibrium state [123,239, 302] and consequently, their mechanical properties differ from their equilibrium bulk properties. Recent nuclear magnetic resonance (2H NMR) measurements [303] and positron annihilation measurements [304,305] support the position that interphase mechanical properties are reduced relati ve to bulk properties [123,257]. Distinct mechanical properties in the interphase are inferred from : (1) glass transition temperature differences for adsorbed poly (methyl acrylate) on amor phous silica [303] relati ve to the bulk glass transition temperature, and (2 ) positronium (o-Ps) lifetim e variation with depth for polystyrene films on a silica wafer [304]. PAGE 308 The effective volume for mechanical respons e in a macromolecular material is, of course, much larger than the probing volum e for NMR and PAL experiments so we must be sure that the interphase properties (mat erial) identified with these techniques has sufficient extent to effect a change in the compoiste mechanical response. The problem is two-fold. First, there is an expected m echanical response adjacent to the fiber for an ideal isotropic matrix material and second, th e extent of any inter phase is not, nor can it be, clearly defined in a macromolecular matrix -fiber composite system. Recall that threedimensional continuum analyses [139] predict a stress singularity at the fiber tip and also that FEA elastic results indicate limited radial and axial influence. For graphite fibers ( f ~ 7 m) this would suggest a shear stress fi eld extending over 2000 nm into the matrix, which exceeds the expected interphase range. Therefore, utilizing distinct interphase properties seems warranted, although ther e remains the problem of assigning an appropriate yield strength. Assigning a proper interphase yield strength is complicat ed by the effects of an inhomogeneous radially-dependent interpha se chemistry and morphology [228,243,257] hydrostatic constraints to plas tic flow [181,306], and accounting for its finite, albeit fuzzily defined, extent. Finally, the influence of the abutting solid interface on interphase mechanical performance must be considered. As for the interphase inhomogeneous chem istry, it is highly dependent on the species (e.g., coupling agent, curing agen t, fiber surface treatment), reactions, and composite processing conditions. We will consider interphase chemistry evolution as part of our discussion on interface influence, as interphase chemistry cannot be explained without addressing interfacial e ffects. Indeed, an interphase no matter how we choose to PAGE 309 define it, exists only because there are co mpeting energetic and dynamic effects at the phase-phase boundary. Hysdrostatic confinement on the fiber rest ricts shear/axial plastic flow thereby increasing the effective interphasial yield shear strength. FEA results indicate a 60% increase in yield stress for a stress confined interphase system [304 ]. Empirical single fiber pull-out tests indicate 10 MPa radial compressive stre sses in ideal microcomposites [192], but analytical results indicate matrix radial tension among adjacent fibers in macrocomposites, Eqn. 3.19.50 Consequently, if we are considering ideal composite performance (e.g., SFCF tests) then hydrostatic confinement will increase the interphase yield strength, but in macrocomposites, there will be no expectation for interphase yield modification as a result of hydrostatic stress c onfinement. Here again, we must question whether ideal composite results faithfully represent macrocomposite performance if the initial stress states and functional responses are fundamentally different. Interfacial confinement of the interphase is critical to the chemical, morphological, and mechanical property development of the interphase. There is evidence indicating strongly scale-dependent interact ions between solid inclusions and polymer chains [307]. As the scale of the inclusion approaches the size of the polymer molecule (~100 nm) there is a reduction in molecular mobility (i .e., decrease in free vol ume/density increase, increase in Tg). This size-scale dependence holds also for molecular confinement in sufficiently small domains such as thin f ilms (<100 nm) [304,305], forced-assembly of immiscible nanolayers (<100 nm thick) [308] and spherical phase domains (dia. < 100 nm) [302]. 50 Williams [williams.1990.305] microindentation results suggest a statistical circumferential property variation (adhesion, matrix microcracking). PAGE 310 There is also the additional mechanical reinforcement advantage. Namely, as the reinforcement diameter decreases the relative ratio of interphase thickness to filler radius ( rf/ tip) increases. Consequently, the proportion of matrix material that might be considered part of the interphase increases in proportion to the square of the filler fraction (f). In particular, recent molecular dynamic simulations [ref. (8) 307] indicate that for single wall carbon nano-tube (SWNT, rswnt ~ 100 nm) reinforced-polyethylene composites, the matrix molecular structure is modified and the macromolecular mobility is restricted near the solid interface. The increa sed surface area/volume ratio afforded by nanotube and nanoparticle reinforcement ( Af/ Vf = 2x107 m-1), coupled with the scaledependent interfacial interaction be tween particle and polymer, produces disproportionately large reinfo rcing effects [309] relative to the reinforcing ratios for micro-scale reinforcements at the same filler fractions (e.g., carbon fibers; rf ~ 10 m, Af / Vf = 2x105 m-1). Microscale-dependent effects are al so seen for thin films (~ 25 m). That is, the mechanical behavior of these thin films is inconsistent with bulk mechanical properties. Here, strain to failure for the film is seen to be ~10 times that for bulk material failure strain [ref. (21) 310]. It app ears that mechanical confinemen t contributes to this change in yield behavior, although the exact cause wa s not identified. Likewise, a series of tensile test conducted on 1 m thick films [ref. (20) 238] indicated extensive plastic yield behavior and strain hardening prior to te nsile failure (Figure 3.51) for length scales on the order of 100 m, whereas brittle behavior is s een in large specimen testing (~100 mm). So it appears that scale-dependent behavior extends from the nanoscale to the macroscopic range for matrix/interphase yiel ding, although in the mi croscopic case, the PAGE 311 effect is an increase in matrix strain to failure whereas for nanoscale reinforcement the effect is enhanced composite modulus reinfor cement. Matrix/interphase behavior on this scale (100 nm 2000 nm) is, of course, critical to composite performance and is yet to be fully understood [310]. The effect of the interface on interphase formation in microcomposites is not as clearly defined. There have been various es timates of the interphase thickness (0 nm 500 nm), these dimensions might create a bounde d region that would effectively confine the interphase. It seems more likely, however, that the in terphase boundary can act only as a weak confinement interface, if at all. It is postulated inst ead, that there is preferential migration of low MW material and unreacted species toward the interface, and that this preferential migration is a function of thei r increased mobility near the interface (fewer conformational restrictions relative to higher MW species) and lower Tg during cure (increased facility for conformational moveme nt). The chemistry at the interface (fiber surface activity, migrated species, steric considerations, cure conditions) gives rise to a network and supermolecular structure that is different from the remainder of the interphase region. Monte Carlo lattice simu lation [ref. (4) 303] and 2H quadrupole echo NMR measurements confirm the conformational excl usion (actually low probability) of higher MW species at the interface. Instead, these high MW chains (tails and loops primarily) are located off the interface, leaving confor mational positions close to the interface for lower MW species. That is, reduced cro sslink density, lower de nsity, and lower MW; consequently, the interphase ha s a reduced modulus, yield st rength, and increased strain PAGE 312 to failure relative to the bulk. This conf ormationally graded interphase argument is consistent with empirical and modeling efforts that assert Gip/ Gbulk < 1. Microscale reinforcements ( f ~ 10 m) are clearly not generating nanoscale synergistic reinforcement effects, otherwise we would not be able to identify a scaledependent reinforcing effect as the reinfo rcement characteristic length drops below ~100 nm. In other words, there is no artificial molecular confinement interface that will produce a functionally conf ined interphase for microcomposite systems. Idealized macrocomposites (e.g., unifor mly spaced fiber arrays with 1.2 m graphite fiber-fiber separation, f = 0.35) are likewise not geometrically disposed to create a functionally confined interphase at the fiber-matrix interface. Practical macrocomposites, however, are far from uniform ly distributed. Fi ber-fiber separation can easily approach interfacial confinement dimensions (~ 100 nm) along finite lengths throughout the composite. However, this in termittent interfacial confinement condition will not produce reinforcement efficiencies on the level seen for nanoscale systems (<100 nm). The conditions at the in terface will also differ from ideal microcomposites as cure shrinkage and differential thermal expansion in macrocomposites will generate tensile stresses in the matrix in contrast to compressive radial stresses in sparse microcomposites. Furthermore, the indu ced radial tensile stresses created during manufacture of macrocomposites may be suffi cient to initiate fiber-matrix debonding. Additionally, ply orientation cure/shrinkage effects in macrocomposites can create significant, undesirable, and pe rmanent structural warpage. That is, microcomposites and macrocomposites will have different matrix resi dual stress fields near the fibers and their PAGE 313 interphasial properties (i.e., adhesion quality, chemical composition, physical properties) may also be different, at least on the average. In conclusion, the essential assumptions for Kelly-Tyson load absorption modeling require that the fibre-matrix adhesion is pe rfect and the matrix be haves as a perfectly plastic material. Neither of these assumpti ons are universally true, although localized matrix yielding is expected. The interphase st ress-strain behavior is instead characterized by initial yielding, a measure of near-perfect plasticity, and finally, acute non-linear strain hardening. It is a fact that, assuming pe rfect adhesion between the fiber and matrix during a SFCF test precludes any possibili ty for debonding. However, for SFCF tests when there is fiber-matrix debonding, the observed critical fiber length lc, regardless of the statistical treatment, will increase a nd the apparent interfacial shear strength prediction ifss will be reduced accordingly. This w ill happen despite the fact that there was no real change in the interphase shear st rength. This predictive error is simply a consequence of the Kelly-Tyson SFCF interpre tation scheme and is in addition to any empirical vagaries attached to the fiber diameter measurement ( rf) and tensile fiber strength (ult,f). The interfacial shear strength value pr ovided by a simple Kelly-Tyson analysis (Eqn. 3.5) or by including a FLDlc correlation (Eqn. 3.40), can, at best (ideal fibermatrix interaction, ideal Weibull fiber strength scaling, constant fibe r diameter, etc.), only be considered a measure of the effective inte rfacial shear strength. Specifically, in Eqn. 3.5 we must use an average d fiber strength value (ult,f ) and a statistical measure for the critical fiber length ( lc). In addition, the critical fiber length estimate is derived from the PAGE 314 fragment length distribution at fragmentati on saturation without any consideration for the shape of the fragment length distribution during the SFCF test. In general we can say that, even though the Kelly-Tyson load absorption model (Eqn. 3.5) includes estimators for the stochastic variables lc and ult,f the final formulation is still, ultimately deterministi c. Moreover, interpre ting a SFCF test using a Kelly-Tyson analytical scheme imposes certain artificial restrictions on fiber fragmentation interpretation that do not exist in the actual physical process. Specifically, (1) perfect adhesion between the fiber and matrix must be maintained throughout the fragmentation, and (2) the matrix behavior is assumed to be perfectly plastic. Failure to satisfy either of these two conditions shoul d disqualify analysis of SFCF tests using a Kelly-Tyson approach; however, the almost comically simplistic formulation (Eqn. 3.5) has broad appeal because of its simplicity and the mistaken belief that the load transfer into a broken/short fiber can be, or should be described using a single material property, ifss. Bi-Modal Load Absorption Models Bi-modal load absorption models disti nguish between two regions/mechanisms for load absorption along a reinforcing fiber. The first region is typically characterized by a very large stress concentration near the fiber end/break and the second region is distinguished by more modest stress/strain conditions. Gene rally, these two regions are bonded and unbonded [232], although Piggotts a pproach [161] considers a yielded region and an elastic region, respectively). There is a th ird, implicit region along the fiber where no load is transferred across the fiber-matrix interface. The extent of this third region is suggested by thought experiments where the length of perfectly aligned PAGE 315 short fibers (constant f) are increased until the compos ite modulus reaches a plateau value. Any further increase in fiber length will not increase the composite modulus. This third section of the fiber does not participate in absorbing lo ad from the matrix, although it still carries its share of the composite load. This condition is difficult to imagine in real composites. In real composite structures and in ideal microcomposites, there will be fiber diameter variation, irregular fiber surf ace topology, fiber waviness, near-neighbor influences, and interphase inhomogeneity th at will produce local st ress concentrations along the fiber length. As a consequence, th e macrocomposite system will reconfigure to minimize the composites total strain energy by a number of different mechanisms (e.g., matrix yielding, viscoelastic deformation, and/or adhesive or cohesive failure. Bi-modal load absorption models [6,158,161] are a natural evolution of the KellyTyson yielded matrix load absorption model. Allowing for fiber-matrix debonding overcomes an obvious inadequacy of a perfect bonding assumption, which is an integral part of both Coxs shear-lag elastic analys is and Kelly-Tysons yielded matrix load transfer model. For matrix systems where transverse matrix cracking occurs preferentially to fiber-matrix debonding or matrix yielding (e.g., silane treated glassepoxy systems that exhibit excellent adhesion [181,203,299]) a bi-modal load absorption mechanism (bonded/debonded) is inappropr iate; however, for most graphite-epoxy composite systems, where fiber-matrix adhesion is relatively weak (e.g., deb(graphiteepoxy) ~ 20-70 MPa (pull-out and SFCF [257 ]), a bi-modal load absorption model will be much more representative. PAGE 316 Piggotts [161] bi-modal model closely follo ws the Kelly-Tyson model, except that the matrix in the fiber-matrix interaction region is not uniformly yielded. Rather, there is progressive yielding from the fiber end with increasing matrix strain, inboard of this yielded region Piggott used Coxs shear-lag solution. Obviously, as a segmented combination of Cox and Kelly-Tyson models, this model will be subject to the same limitations and concerns already noted. An impor tant result of this plastic-elastic model [161] however, is that the overwhelming majority of load absorption from the matrix to the fiber is predicted to take place across th e yielded matrix/interphase region. In this regard, Piggotts model is not far off the mark. The very early evidence of acute birefri ngence near the terminal sections of discontinuous fibers in SFCF test s that is consistent with the expected high matrix strains is often accompanied by a semi-permanent birefringent sheath along the fiber length [311]. Some of the apparent birefringent sh eath may be attributed to evanescent wave scattering at debonded sections. The radial extent of this sh eath can be seen to decrease with time in some systems [311] (viscoelastic shear deformation followed by stress relaxation). The permanent pattern that remain s after load removal is a tell-tale indicator of interphase yielding. The ra dial and axial extent of the pe rmanent birefringent sheath is also suggestive of the interphase yield streng th and may also intimat e interphase extent. For well bonded fiber-matrix systems, the wi dth of the permanent birefringent zone is controlled by the external loading condition and a combination of factors in the matrix (interphase thickness, interphase modulus, inte rphase yield strength, interphase strain hardening). Here, we include the interpha se in the matrix phase and consider the resultant mechanical response. Unfortunately, it is not possible to precisely define the PAGE 317 interphase extent because the mechanical properties of polymeric materials involves cooperative macromolecular movement51. Couple this cooperative macromolecular movement with a gradient in molecular weight near the inte rface and mechanical property dispersion across the inte rphase is virtually guaranteed. For fiber-matrix systems that exhibit load absorption debonding during load transfer, the width of the permanent birefr ingent sheath is roughly uniform along the debonded fiber length. This sheath uniformity is partially associated with birefringence measurement resolution, but primarily i ndicates a debonding/propa gation condition that is a consequence of the stress-strain history at the fiber-matrix interf ace. That is, as the external composite loading is increased, the process zone, which is attached to the debonded fiber-bonded fiber interface, reac hes a critical debonding-propagation condition and the debond propagates. In the wake of the debonding is a nearly uniform birefringent sheath. Based on evidence for non-linear plastic stra in hardening of thin films [ref. (20) 238, ref. (21) 310] and scale-dependent inter phase behavior, ~100 nm or less (e.g., film thickness [304,305], phase domain size [302], na nolayer thickness [308]), it is claimed here that the intermediate interphase formed in micro-fiber reinforced polymeric composites would exhibit characteristics more li ke the micron scaled films. In particular, the interphase properties differ fr om the bulk matrix properties (Tg,ip < Tg,bulk, y,ip(r) < y,bulk) and for small scale testing (< 100 m), the ultimate shear strain far exceeds the ultimate bulk shear strain (ult,ip > ult,bulk). It is an interphase of this type that gives rise to the observed yielding/debondi ng behavior seen in m oderate to weakly bonded 51 Recall that the interphase definition and extent are dependent on the probing frequency and experimental PAGE 318 microcomposites (e.g., graphite fibers in an epoxy matrix). A bi-modal model that accounts for matrix yielding and debondi ng would therefore be well suited for understanding load absorption in many differe nt fiber-matrix composite systems. For instance, the bi-linear load absorption model proposed by Henstenburg and Phoenix [6] considers two linear stress bu ild-up sections, a yielded matr ix section (Kelly-Tyson), and for higher strains, a debonded sectio n with frictional stress build-up. Modeling fiber-matrix debonding requires establishing debonding/propagation criteria, which as we have noted involves the evaluation of still more matrix (interphase), fiber, and fiber-matrix interaction mechanic al properties. As we layer on model complexity, we generally expect that the pred iction error would decrease. This supposes that the revised model is more representative of the actual load absorption mechanism, and that the model parameters themselv es are well characterized. For SFCF interpretation, we are dealing with process variables that have inherent statistical variability (fiber tensile strength, interphase modulus/yield strength, debonding, etc.) and yet the predictive models (e.g. Eqns. 3.5, 3.9.12) are predominantly used in a deterministic manner without consideration fo r parameter statistical variability. Adding additional fitting parameters that are poorly characterized (i.e., inherent statistical variation, empirical interpretation difficulties) may leave the theoretician and empiricist little closer to understanding the process or, at the very least, with false confidence in the model prediction. There are numerous empirical difficulties associated with appl ication of bi-modal models that include fiber-matrix debonding. Recall our previous discussion on the nature of fiber-matrix debonding in single fiber pullout tests by shear failure (Mode II), and the technique. PAGE 319 requirement in that case to define both the load diffusion (shear-lag) and determine a suitable failure criterion. The debonding cr iterion can be either strength based, max>s, where s is the debonding shear strength, or st rain energy based, LFM (e.g., Eqns. 3.35 3.36). Establishing suitable criteria for fibermatrix debonding is complicated by mixedmode failure at the fiber-matrix interface that includes non-linear matrix yielding, mode I and II interface fracture, and transverse matrix microcracking. The trade-off amongst matrix microcracking, interphase yieldi ng, and fiber-matrix debonding is controlled, largely, by the interphase stiffn ess. A stiff interphase, either by virtue of its modulus or thickness, tends to promote transverse micr ocracks (Fig. 9 [181], Fig. 1 [5]) at the expense of interphase yielding or debondi ng. Henstenburg and Phoenix [6] utilize a simple stress criterion to identify inci pient fiber-interphase debonding (Figure 3.29). DiAnselmo et. al [181] meanwhile, make use of a critical strain en ergy release rate for the interphase Gipc. The important conclusion here is that ther e are a number of pathways that might be followed during load absorption, each of which is a response to the matrix strain energy. The composite system responds by absorbing the strain energy across the interphase into the fiber, or alternatively, reconfigures (interphase/matrix yielding, viscoelastic deformation, adhesive or cohesive failur e) to minimize the system strain energy. By comparison to the simpler single-mode load absorption approximations (Cox: elastic; Kelly-Tyson: plastic ), properly formulated bi-modal models can capture an important aspect of real composite loading (fiber matrix debonding/imperfect adhesion). These bi-modal models [6,161,181] are still rela tively simplistic in that they do not allow PAGE 320 for any statistical variation in interphase properties along the fibe r. The analyses by DiAnselmo et. al [181] and Piggott [161], to th eir detriment, also re quire defining a zone of influence, beyond which matrix axial stress is equal to the farfield axial stress. This artificial constraint enables closed form so lutions; but, unfortunately, has a large effect on the rate at which the FAS prediction is built-up (cf. Figure 3. 12), and is clearly inconsistent with the actual morphology a nd physical properties of the interphase. Ideally, one would like to determine the inte rphase properties as a function of radial position and be able to identify the interphase boundary. Unfortunate ly, the viscoelastic behavior and morphology of macromolecule s near solid interfaces precludes the determination of discrete values for mate rial properties that involve cooperative movement (e.g., modulus, yield strength, gl ass transition temperature). The limited extent of the interphase further limits mechanical property evaluation. SFM measurements on this scale and near the fibe r-matrix interface have been taken. The measurements indicate an interphase with di stinct mechanical pr operties [243], although the variability in the measured modulus near the interface makes it difficult to discern between real matrix modulu s changes and stiffness effects accompanying matrix microcracking and fiber-matrix debonding. Henstenburg and Phoenix load absorption model [6] establishes a strain based debonding criterion and employs a yielded matr ix assumption for the remainder of the shear loaded fiber fragment. An alternativ e statistical approach for critical length assessment lc (Eqn. 3.40) is also provided [6], as opposed to the more common, but flawed, normal distribution treatment. A uniform frictional shear stress, f, is used to account for load absorption on the debonded sections. Simulations (Table 5 [6]) PAGE 321 indicated that as the f/y ratio decreased the mean fragment length COV increased. Similarly, as the percentage of load ab sorbed through the yiel ded section increased (increasing y or yielded length), the mean fragment length COV again increased. These cases are noteworthy because: (1) Frictional shear loading in macrocomposites is very small relative to loading in microcomposites because of matrix radial tension in macrocomposites and compressive radial tract ions in microcomposites. That is, the SFCF test may not be representa tive of the underlying physical pr ocess in this regard. (2) The variance in fragment lengths for an ideally simulated SFCF test at saturation is sensitive to the yield stress paramete r. In other words, estimates of y from non-ideal SFCF tests would have even larger statistical variance and reduced predictive confidence. Finally, we have previously noted that for a Kelly-Tys on yielded matrix approach, the inclusion of a simple fiber strength statistic (e.g., mean fiber strength) and an averaged critical fiber length, lc, leads to SFCF interpretation ambiguity. These concerns apply also to the use of bi-modal schemes fo r SFCF interpretation. That is, the final model predictions for interfacial shear strengt h (Eqn. 6 [161], Eqn. 14 [6]) can provide, at best, only an averaged interf acial shear strength; although in this case, the averaged prediction also includes a statistical contribu tion associated with the reduced gage length attendant to fiber-matrix debonding. The disbonded fiber sections do not carry sufficient load to fail the fiber, this results in a decreasing effective gage length. The expected failure strength for the shorter fiber increases, a consequence of the reduced specimen volume and its flaw-strength distribution. Further fiber fragmentation or debonding is the response. The evolving SFCF fragmentation dist ribution subject to a bi-modal fa ilure mechanism is not normally PAGE 322 distributed, nor is it necessa rily well described by a Wei bull distribution [6]. The fragmentation distribution is st rongly dependent on the fiber st rength-scaling relationship. A smaller underlying fiber Weibull modulus in creases scaling sensit ivity and stretches the evolving FLD symmetrically; whereas, a fiber characterized by a large Weibull modulus will exhibit a tighter fr agmentation distribution skewed to shorter gage lengths. The predicted interfacial shear strength for a composite system that contains a fiber with a small Weibull strength modulus will be mo re susceptible to vari ability in the fiber strength. That is, the reliability of the av erage interfacial shear strength obtained for a debonding/fragmenting fiber-matrix system with a small fiber Weibull strength modulus is less than that for a composite system w ith a large fiber Weibull strength modulus. In contrast, a perfectly bonded system with ideal fibers (i.e., a constant diameter and a single strength with no length dependency) would pr oduce fiber fragments of equal lengths and consequently, an interfacial shear strength prediction with no ambi guity. Accordingly, fewer SFCF specimens would be required fo r high Weibull modulus composite systems to achieve a certain confiden ce in the interfacial strength prediction, provided that the fragmentation process is properly described by the bi-modal model and the other controlling parameters are also well described. One important parameter that may not be well understood is the role of the excluded length during fiber fragmentati on evolution. Single fiber composite fragmentation interpretation is further m uddied if the debond strength is small or variable, thereby generating long debonded lengths and short eff ective fiber gage lengths. There is increased fiber strength variation as the gage length decreases and there is also the possibility of flaw/strengt h population biasing when the effective gage length differs PAGE 323 by orders of magnitude from the tested gage length. Additionally, the bi-modal models discussed [6,161] assume that fiber matrix adhesion on the remaining bonded section is perfect; although it has been s hown that this assumption can not be supported along a fiber length. The net effect is that the fibe r axial stress build-up is delayed by imperfect adhesion (e.g., viscoelastic in terphase, irregular fiber mo rphology/surface chemistry). The effective gage length is reduced even further and confidence in the predicted interface strength is thereby reduced. Spring-Layer SFCF Modeling We have previously introduced bi-modal52 stress recovery models [6,161,181] that allow for more precise load transfer estima tion, at least for fiber-matrix systems where the models are consistent with the actual load transfer. However, as is evident in the preceding discussion, there is a range of mechanisms that exis t and control the interaction at the fiber-matrix interface. The complex interaction and trade-off be tween these several mechanisms renders Coxs original shear-la g model (typified by Eqn. 3.2, FB D; Eqn. 3.9, FAS; Eqn. 3.10, ifss), which assumes perfect bonding and other limiting conditions, simplistic and in most cases well off the mark. The more realis tic, although still highly flawed, Kelly-Tyson yielded matrix analysis (typified by Eqn. 3.4, critical fiber length ) likewise cannot be unequivocally utilized to interpret most SFCF tests53 owing to gross generalizations relating to perfect bonding, fiber strength, critical fiber length and an everywhere yielded matrix assumption. Recall al so that both the Cox and Kelly-Tyson analyses neglect hoop 52 See also multi-modal analysis by Chen (1990) [304] 53 Recall Hashins [218] admonition regarding model fit and physical actuality. PAGE 324 and radial stress contributions as well as any near neighbor influences (dilute assumption). In light of the multifaceted interaction at the fiber-matrix interface in SFCF tests and in macrocomposites, many investigators ha ve chosen to use a spring-layer interface modeling approach to avoid the difficulties in representing the actual fiber-matrix micromechanics. With this approach, th e various mechanisms are collected and averaged over the fiber lengt h to approximate their aggregate influence. Using stressdisplacement jump relations, Eqns. 3.48, the disc rete interface/interphase interactions are subsumed by the overarching interface parameters ( Ds, Dn, Dt). That is, we have replaced requirements for measuring and interpreting th e discrete fiber-matrix mechanisms and their contributions to overal l load absorption performance w ith three artificial interface parameters to facilitate analysis. This t ype of approach is not without precedent, although clearly one must be careful when comparing load absorption predictions amongst composite systems with different underlying load absorption mechanisms. If, by assumption (no radial displacement at interface, Dn = 0; minimal hoop influence, Dt = 0 [300]), we are left with ju st a single interf ace parameter, Ds, that considers axial displacements only, then it should be obvious that we have recovered a one-parameter model for load absorption. A model by any other name is still the same model. In this case, repl acing the actual physical paramete rs and mechanisms with a single fitting parameter ( Ds) allows for load absorption es timation but we cannot ascribe any physical meaning to the parameter. If we take the interphase parameter out of context, we are not able to identify the underlying mechanisms, nor can we use the interface parameter in other analytical expressi ons. Without contextual information (e.g., PAGE 325 elastic behavior, yielding, matrix microc racking, debonding) the interface parameter appears to be nothing more than an admissi on that the actual load profile is poorly represented by the chosen model. It does not inspire confidence in the de-facto loadabsorption model, whether it is a one-dimensi on elastic shear-lag model, a yielded matrix model, or a three-dimens ional continuum solution. The benefit of such an approach, which al lows for the accumulation of multiple and disparate mechanisms into a single parameter that varies on [0, ], on the other hand is buttressed by Benvenistes results [219] rega rding an increase in stress recovery length54 as R (interface imperfection increasing). Benven istes results are also in accord with Beltzers [223] ineffective length calculations for periodic (Eqn. 3.46) and distributed imperfections (Eqns. F.22 and F.23, increasing se verity and length, respectively). In a practical sense, one might wonder about the ab solute value (signifi cance) of a parameter whose value ranges from zero to infinity. SFCF modeling and interpretation using a tw o-dimensional spring-layer interface is relatively straight forward by comparison to the alternative multi-modal analyses (e.g., [6, 203]). While a spring-layer approach simp lifies the treatment fo r matrix-fiber load absorption arising from matrix imperfec tions such as microdebonding (imperfect adhesion along the fiber), composition and morphology gradients (interphase/boundary layer), matrix microcracking, and matrix nonlinear/yielding behavior the contribution of fiber-matrix disbonding and propaga tion at a fiber break must be considered separately, if it is considered at all. An accurate elastic load transfer fo rmulation for the case with no 54 Specifically, Benveniste considered the rate of decay for end effects relative to PB ( R =0) as R (figs. 3 and 4 [219]). PAGE 326 imperfections, excepting a possible fiber break, is still required, as is a correlation between fiber loading and fiber failure. Modeling Difficulties The Cox and Rosen solutions, Eqns. 3.9.12, have obvious inadequacies that have been detailed above. In partic ular, they are greatly influen ced by their shea r interaction parameters (cox, Eqn. 3.7 and rosen, Eqn. 3.13), which themselves are reliant on nebulous shear interaction extents ( r1 and rb respectively). Muki and Sternberg [133,138,139] provided more rigorous solutions for fiber lo ad absorption and diffusion at the fiber ends, which while satisfying boundary conditions, do not consider likely matrix (interphase) mechanical non-linearity or yielding. Herein lies the problem: load absorption near the fiber break quickly induces stress concentrations in the matrix well in excess of the probable matrix/interphase yiel d strength and/or fibermatrix adhesion strength. Hence, the yielded matrix assumption suggested by Kelly and Tyson [111-113] is not without merit; however, it is too broad and cannot be supported for low matrix strains away from the fiber break. It also discounts the possibility of alternative energy absorbing mechanisms besides matrix yielding. Matrix yielding by shear stress alone also neglects principal stress influences (fig. 9 [229]). Appropriate criteria for yi elding, fiber-matrix disbondi ng, and crack propagation must therefore be selected in order to employ spring-layer interface or any other multimodal analysis technique. The criteria ar e tested against the predicted property to determine the onset of matrix yielding fiber-matrix disbonding, and/or crack propagation. Note, however, that the particul ar approach/analysis used influences the evolution (prediction) of the pr operty in question. In this re gard, it is important to utilize PAGE 327 a micromechanics model that accurately mimics the actual physical conditions; otherwise, the modeling and conclusions are bias ed or possibly meaningless. To be clear, the results are biased with regards to the model chosen versus competing mechanisms that are not considered. Recognizing the shortcomings of Cox a nd Rosen shear-lag analyses and the over simplification of a Kelly-Tys on elastic-plastic approximati on, Nairn [158] developed a variational mechanics three-dimensional axisymmetric solution for load absorption at a fiber discontinuity. Following analogous55 cross-ply laminate [0y/90x/0y] analyses [301, refs. Liu and Nairn (1990, 1992) 158] Nairn modeled the microcomposite as three concentric cylinders (fiber rad. = r1, matrix rad. = r2, and shear extent r rc). The matrix axial stress for r rc is equivalent to the far-field stress in the absence of any fiber breaks. Although the closed-form solution provided by [158] assumes that both the fiber and matrix are linear elastic, as well as perf ect adhesion between the fiber and matrix, all of which are known to be invalid, the FAS soluti on is still informative. In particular, the predicted load uptake in the fiber by this analysis [158] is ve ry rapid (approx. 2 Df) compared to Cox shear-lag prediction (approx. 8 Df). Furthermore, the variational approach provides results for FAS, ifss, and radial stress (compressive) that agree well with axisymmetric finite element results (fig 6 [158]). Note that the finite element solution uses similar assumptions regarding load absorption mechanisms, so agreement between the analytical and numerical solutio ns is not surprising. More importantly, agreement does not confirm that the analytic al model accurately represents the actual fiber-matrix load absorption mechanism, but instead, the analytical formulation does a 55 See preceding arguments regarding [0y/90x/0y] cross-ply laminates and Figures 3.60-3.62. PAGE 328 good job of reproducing the numerical FEA re sults, both of which assume perfect adhesion and linear matrix behavior. The results of Nairn [158] are meaningful on regions where an elastic response dominates, although it is clear th at the majority of fiber-matrix load transfer occurs prim arily through yielded zones [113,146,161]. It should also be remarked that just as for the simple r shear-lag approaches (Cox and Rosen), a shear affected zone (i.e., r rc) must be demarcated for Nairns approach, and that the extent of this n ear-field region affects stress fiel d predictions (cf. variations in cox and rosen wrt r1 and ra respectively). In this inst ance, Nairn [158] argues that rc (i.e., V2, fiber volume fraction in th e article) can be backed-o ut using photoelastic or Raman experiments. However, using a singl e-fiber test and the model itself (perfect interface), to fit one of the m odels critical parameters empiri cally, is certainly flawed. Indeed, Nairn et al. [300] argue as much in a later article with regards to a perfect interface assumption. Summary We have identified and discussed a number of analytical load absorption models and model types (one-dimensional elastic: Cox, Rosen; one-dimensional perfectly plastic: Kelly-Tyson; one-dimensional elastic-plastic: Piggott; one-dimensional unbondedplastic: Henstenburg; two-dimensional el astic: Mooney & McGarry; axisymmetric: Nairn; three-dimensional continuum: Muki, Sternberg) with particular emphasis on model predictive accuracy and usefulness for SFCF interpretation. The one-dimensional elastic load absorption models (Cox, Rosen) are not suitable for SFCF interpretation. The primary reason these models should not be used for SFCF fragmentation is their supposition that during load transfer the matrix behavior is elastic, PAGE 329 this despite incontrovertib le evidence from empirical measurements [113,146,161], continuum mechanical analyses [133,138,139] and analytical co mputations [166-168] that the shear stress field/concentration at th e fiber tip far exceeds the bulk polymer yield stress. Moreover, it has been shown that the matrix adjacent to the fiber surface (interphase) has distinct mechanical prope rties [123,259,302]. Specifically, the yield strength, modulus, and glass transition te mperature are reduced relative to the bulk properties, all of which conspi re to promote yielding near th e fiber break and at positions of local stress perturbations (fiber diameter variation, fiber misalignment, fiber asperities, interphase inhomogeneity, fiber-matrix im perfect adhesion/unbonding) along the fiber length. Additionally, the high matrix strains required to reach fragmentation saturation (e.g., graphite-epoxy, 7-8% [181]) cannot be reached without localized interphase yielding and/or fiber-matrix debonding. A one-dimensional perfectly plastic load absorption model (e.g., Kelly-Tyson) is both enabled by and limited by its primary assump tion: the interphase is yielded over the load absorption region (ineffective length). The importance of the various mechanical parameters, which influence matrix-fiber lo ad absorption and SFCF interpretation, is made abundantly clear by the simplicity of the Kelly-Tyson load absorption scheme (Equation 3.5). It is seen that the predicted interfacial shear strength, ifss, is a nondeterministic function of stochastic variables. The nature of the estimating approach and statistical variability of the parameters infl uence the ultimate shear strength prediction. A Weibull strength-scaling re lationship has been establis hed for brittle fibers [1013] and for carbon fibers in particular [Ch. 2]. Often, this strength-volume relationship is ignored in favor of a simple average fiber strength statistic. At the very least however, PAGE 330 the ultimate fiber strength statistic (ult,f) should be scaled to a length roughly twice the average fragment length of the saturation FLD. Additionally, for a one-dimensional perfectly plastic load absorption model, the effective fiber length for each fragment is controlled by the perfect fiber-matrix adhesi on assumption, which precludes fiber failure along the stress build-up zones at the fiber ends (Figure 3.1). It should also be noted that the expected fiber strength (a stochastic variable) is continuously changing during the fiber fragmentation process. That is, as th e fiber gage length d ecreases, the expected fiber strength distribution is shifted to higher values because of both the excluded strengths and a smaller gage length (volume). Unfortunate ly, the Kelly-Tyson approach accommodates only a single fiber strength variable, even if it were stochastically consistent with the underlyi ng fiber strength variation. Fiber to fiber diameter varia tion in a tow (e.g., 2.5-5.0% in Ch. 2, greater than 5% for IM6 carbon fibers [312]) and diameter vari ation along a single fiber (1-2% in Ch. 2), must also be considered for accurate SFCF interpretation. Th is diameter variation affects both the numerator (ultimate strength, Fult,f / r2) and denominator (r) in Eqn. 3.5. A simple sensitivity analysis for Eqn. 3.5 indicates that a 5% diameter variation leads to a predicted interfacial stre ngth variation of +8.1% and -7.0%, respectively. The second major assumption for Kelly-Tyson load absorption analysis, the first being that the matrix in the mechanical interaction zone is yielded (perfectly plastic) during shear load transfer, is that there is perfect adhesion between the fiber and matrix. That is, there is no fiber-matrix debonding for a ny fiber-matrix interfaci al shear stress and furthermore, perfect adhesion implies that there are uniform prope rties along and around the fiber (e.g., no fiber surface irregulariti es, homogeneous radial and axial matrix PAGE 331 properties). Obviously, any fiber-matrix debond ing, a consequence of matrix-fiber load absorption, during a SFCF test should disc redit a simple Kelly-Tyson load absorption analysis. However, for well bonded fibe rglass-epoxy systems, a perfect adhesion assumption is essentially justified for SFCF testing. Perfect fiber-matrix adhesion for carbon fiber-epoxy systems, is not supported either by SFCF tests [203] or by surface chemistry considerations [262,283]. Carbon fi bers are far less reactive with an epoxy matrix, although there is improved adhesion (chemical, energetic, and physical) with appropriate fiber surface treatment (d ry, wet, or plasma) (Table 3.16). Strong, or essentially perfect, adhesi on however, does not guarantee load absorption behavior consiste nt with a Kelly-Tyson interpretation. Namely, strong adhesion may induce transverse matrix microcra cking during SFCF testing. This matrix microcracking will create localized matrix de formation/displacements and consequently alter the matrix stress distribution and there by, the fiber load-uptake. In an ideal microcomposite, the matrix microcracking will tend to shield the fiber in the vicinity of the microcrack, which increases the fiber ineffective length, The FLD will be shifted to longer lengths and as a result, ifss will decrease. Once again, it is clear that the SFCF test does not provide an accurate measur e of the interfacial shear strength, or alternatively, the presumed definition of th e interfacial shear strength (Eqn. 3.5) is unreliable with regards to matrix microcra cking or any other mechanism that would increase/decrease fiber load absorption rela tive to perfect adhesion and yielded matrix assumptions. More advanced biand multi-modal matrix-fiber load absorption models have been proposed to account for fiber-matrix debondi ng and non-uniform interface/interphase PAGE 332 properties. For instance, stress-displacement jump relationships avoid the complex threedimensional fiber load absorption behavior /response by replacing it with a much simplified, mechanistic and mathematical, formulation (Eqns. 3.48). This approach sidesteps the empirical and philosophical diffi culties of defining th e interphase extent (mechanical interaction) that is required for Cox and Rosen analysis, r1 and ra, respectively. Conclusions In general, we can say that the mechanic al response at the fiber-matrix interface during SFCF testing cannot be understood with out considering the small (nanomicro) scale response of the interphase. The respons e in the interphase is a consequence of the macromolecular forces (energetic and entropic) present during composite fabrication (i.e., driving forces for interphase formation) and the acute matrix strains in the vicinity of fiber-matrix irregularities (e.g ., fiber ends and fiber-matrix de bonding). This response is best understood/followed by considering ve ry small domains/scales. Fiber surface irregularities (e.g., crystalline graphite edges, surface pitting, and surface chemistry inhomogeneity), macromolecular conformationa l statistics, species chemistry, and macromolecular cooperative mechanical behavi or provide for a very localized response that forms the basis of the microand macr o-scale responses. While micro-variables are typically assigned to enable phase-phase interaction interpretation in micromechanics (e.g., ifss), it is unrealistic to expect that the complex mechanical response to exomechanical inputs/probes/tests (forcing func tion originates outsid e the interphase) in fiber-matrix systems can be adequately de fined with a single micro-variable (ifss). PAGE 333 The composite micro and macro performance hinge upon the small scale response at the interface; they are th e averaged response across a large number of intermingled macromolecular domains. These domains, individually and colle ctively, exhibit mechanical properties distinct from the bul k matrix properties. In particular, the interphase modulus and yield strength ar e generally reduced relative to the bulk ( Eip < Ebulk, y,IP < y,bulk), while the strain to failure is increased (ult,IP > ult,bulk). Scaledependent behavior of the interphase is indicated by the creation of permanent birefringent sheaths during SFCF testing and by thin film testing where failure strains far exceed (~10 times) bulk strains [ref. (21) 310]. These large matrix strains are the result of highly non-linear stressstrain behavior, which leads to early permanent matrix deformation and, as the interphase tangent modulus increases rapidly (Figure 3.51), cohesive matrix failure or matrix-fiber adhesive failure. Given irregular fiber surface topology, non-uniform fiber surface chemistry/energetics, and acute interphase non-linear mechanical behavior, localized situational matrix yielding o ccurs along the fiber length postmatrix vitrification, even without external loadi ng. For example, situational in terphase yielding may occur in macrocomposites during cool down from the cu re temperature. That is, continuum [ref. (23) 165] and FEA analysis [166-168] predict te nsile stresses at the fiber matrix interface in macrocomposites (Eqn. 3.18) as co mpared to compressive stresses for microcomposites (Eqn. 3.17). Furthermor e, as external loading increases in macrocomposites the instances of situational yielding in creases monotonically. An analogous condition to situational yielding in microcomposites, and a good indication of macrocomposite reconfigur ation (e.g., matrix yielding, tape-tape PAGE 334 delamination, fiber failure) in response to external lo ading occurs during hydroproof testing of filament wound gl ass and graphite-epoxy pressure vessels. There is audible crackling and popping throughout the composite case as the intern al pressure is increased and then held at a predetermined proof pre ssure. During this loading cycle, there is typically no indication of any decrease/increase in case stiffness, a clear indication that the load carrying capacity of the fibers is maintained via effective matrix-fiber load absorption along the continuous fibers, desp ite the audible evidence that suggests otherwise. The macrocomposite modulus, aligned with the ta pe and loading, is largely unaffected because any imperfect adhesion or mechanical irregularity is overwhelmed by the continuous fiber reinforcement ( lineff << leff). The matrix-fiber load absorption models discussed here have been developed for interpreting load transfer in SFCF tests and/or for evaluating load transfer in short fiber composites. The SFCF test has been variously used to assess the effects of fiber surface treatments, matrix chemistry variations, and environmental conditions on the nature of the fiber-matrix interaction, specifically the interfacial shear strength ifss. The simplest SFCF interpretation approaches (Cox, Rosen, and Kelly-Tyson) rely on the ubiquitous perfect fiber-matrix adhesion assumption, wh ich has been roundly refuted. Imperfect fiber-matrix adhesion (debonding) has been included in bi-modal and spring layer interface load absorption models, which is an improvement relative to perfect adhesion models, especially for weakly bonded systems where the deb onded length, if it were not considered, could lead to a gross underpre diction of the interf acial shear strength. What is more problematic for these analytical SFCF interpretation schemes is their inability to accurately capture the stochastic nature of the fiber fragmentation process. PAGE 335 That is, the predicted in terfacial shear strength (ifss) is generally reported as a single value as though it were a material property of the composite system when in fact the SFCF test involves statistical variations in the fiber strength, fiber diameter, and fibermatrix adhesion, which contribute to a sec ond order effect on the fragment length distribution. It should be cl ear that, the saturation FLD is a function of both the fiber strength distribution and the na ture of the load uptake in discontinuous fibers. We cannot simply assume that the FLD is normal or Weibull because the FLD is affected by a range of factors such as non-linear fiber load upt ake, length exclusion effects (debonding and ineffective length), clamp effects, fiber stre ngth variation (fiber di ameter variation and Weibull scaling, Eqn. 2.1a), and matrix /interphase inhomogeneity (e.g., non-uniform fiber surface energy, matrix-coupling agent inte raction, transverse matrix microcracking, and macromolecular-interface en tropic effects). Consequent ly, the saturation FLD and any statistical measure for the FLD can not uniquely identify the interfacial shear strength. Any numerical procedure for estim ating the critical fiber length lc from the FLD must consider the fragmentation process and not simply the saturation FLD. An estimating procedure for lc must include a micromechanics model for load absorption, which necessarily biases the estimation and therefore the interfaci al shear strength prediction. The underlying model for any num erical correction for the critical fiber length (e.g., Eqn. 3.40) must be carefully evaluated and compared against the observed fragmentation process. Add itionally, the FLD correction factor must implicitly account for the fiber strength distributi on and any length scaling effects. PAGE 336 In closing, we should call attention to some fundamen tal differences between the physical configuration and loading conditions of SFCF tests as compared to that for macrocomposites. The residual radial stress at the fiber surface is compressive in single fiber and other sparse microcomposites while it is tensile for macrocomposites. This can be important when there is fiber-matrix debondi ng. That is, compre ssive stresses will produce frictional shear loading whereas tensile stresses cannot. The magnitude of this frictional loading, in any case, is expect ed to be small by comparison to the load absorption by shear across the adhered interphase. The residual tensile loading in macrocomposites may contribute to fiber-mat rix debonding; however, th is contribution is absent in the microcomposite specimens. For SFCF specimens the matrix chemistry is sometimes modified slightly to enable higher matrix strains and fragmentation satura tion. The yield behavi or of the modified SFCF matrix material may be inconsistent w ith the macrocomposite matrix and therefore a poor representation of the macrocomposite performance. A more important concern with the SFCF test is that while th e SFCF test assays the load uptake for discontinuous/fragmented fibers (i.e., highly non-linear even singular behavior at the fiber tip), the load transfer in macrocomposites is more uniformly distributed with localized shear loading crea ted by situational yielding. Recommendations It is recommended that a detailed Mont e Carlo (non-deterministic) model be developed for load absorption that incl udes all known/understood variations in mechanical properties and mechanisms that define the load absorption and enable SFCF interpretation. Researchers should abandon simplistic SFCF test analysis (e.g., Cox/Rosen and Kelly-Tyson) in favor of more realistic multi-modal analysis that consider the non-deterministic nature of the load absorption mechanism and fragmentation process. PAGE 337 The unrealistic expectation that the complex mechanical response to exomechanical inputs/probes/tests (forcing f unction originates outs ide the interphase) can be adequately defined with a sing le/simple micro (phase-phase) variable across the entire phase-phase interface s hould also be abandoned, unless the associated assumptions and limitations are clearly defined and understood. A confidence interval must be reported w ith any measure of interfacial load absorption behavior as the pr ocess is non-deterministic. PAGE 338 Table 3.1 Fiber reinforcement efficiency, Eratio, according to Eqn. 8 (bracketed term) for different fiber gage lengths. Modulus Ratio Eratio Length (mm) Aspect Ratio Present WorkCox (1952) 0.1 10 0.580 0.518 0.2 20 0.786 0.750 0.3 30 0.857 0.833 0.4 40 0.893 0.875 0.5 50 0.914 0.900 1.0 100 0.957 0.950 2.0 200 0.979 0.975 3.0 300 0.986 0.983 4.0 400 0.989 0.988 5.0 500 0.991 0.990 Source: Table 3.3 [105]. PAGE 339 Table 3.2 Input parameters for fiber rein forcement efficiency calculations (Eqn. 3.8) displayed in Fig. 3.5. Property Units Graphite-EpoxyPaper Comp. (Ef)ref GPa 230 73.3 (Em)ref GPa 3 4.3 (Ef/Em)ref 76.7 17 (Ef-m)ref GPa 227 69 Gm GPa 1.07 1.72 cox @ (Ef/Em)ref m-1 33723 1187 (r0)ref m 3.0 5.0 (r1)ref m 7.55 12.5 (r0/r1)ref 0.4 0.4 Range Variables (Ef/Em)min 1.003 1.001 cox @ (Ef/Em)min m-1 5080859 118668 (Ef/Em)max 383 85 cox @ (Ef/Em)max m-1 15002 518 Fiber Aspect Ratio 50 50 (r1)min m 3.001 5.05 (r0/r1)min 0.9997 0.99 (r1)max m 65.2 55.6 (r0/r1)max 0.046 0.09 Paper composite values from [105 ], graphite-epoxy are generic. PAGE 340 Table 3.3 Physical properties and range variables used in Eqns, 9-12 (Figures 3.93.12). Property Units Cox (1952) Rosen (1965) Kelly-Tyson (1965) Ef (Carbon) GPa 230 230 Em (Epoxy) GPa 3.0 3.0 f 0.3 0.3 (Ec)ref GPa 71.1 71.1 y MPa 50 f 0.2 0.2 m 0.4 0.4 Gm GPa 1.07 1.07 r0 m 3.0 r1 mm 1.0 rf m 3.0 3.0 ra mm 1.0 rb m 4.2 Range Variables AR 1.5-1000 1.5-1000 Length m 9.0-6000 9.0-6000 f 0.3-0.9 Sources: [105, 110, 112] PAGE 341 Table 3.4 Failure mechanisms in fibrous composites. Toughening Mechanism Eqn. Origin Fiber pull-out c po po f f pol l d R ; 22 (3.20) Cotterell1 Fiber pull-out c c c f f pol l d R ; 62 (3.21) Cotterell1 Fiber pull-out c po f f pol l d T R ; 62 (3.22) Cotterell1 Interfacial debonding f d f f dE R 22 (3.23) [47] Interfacial debonding f d d f dE R 22 (3.24) Harris2 Stress redistribution f c f f RE l R 32 (3.25) Piggott3 Surface energy if c f m f f f sR d R R R ) 1 ( (3.27) Marston4 Surface energy 0 ) 27 ( ; 1 1 f s if m c f sR R R with R d l R (3.28) Marston4 1 A. H. Cotterell, Proc. Roy. Soc. Lond. A 282 (1964) 2-9. 2 B. Harris, Metal. Sci. 14 (1980) 351-362. 3 M. R. Piggott, J. Matl. Sci. 5 (1970) 669-675. 4 T.U. Marston et. al., J. Matl. Sci. 9 (1974) 447-455. PAGE 342 Table 3.5 Typical interfacial properties and pull-out para meters (see Eqns. 3.37 and 3.38). Fiber pull-out parameters Interfacial properties 0 L zmax Gic q0 s f Composite system Fiber surface treatment GPa mm-1 GPa mm J/m2 MPa MPa MPa CF-epoxy Untreateda nd oxidized 3.4 1.5 5.4 0.152 37.7 1.25 -9.97 72.7 12.2 Steelepoxy Uncoated 1.95 0.0142 2.41 7.8 1316 0.48 -8.85 43.5 5.0 Steelepoxy Release agent 0.316 .0065 1.98 6.5 34.7 0.22 -7.28 8.96 1.77 SiC-glass Untreated 0.149 0.0304 2.92 0.06 0.964 0.048 -64.5 3.18 3.11 SiC-glass Acid etched 0.235 .049 3.27 0.08 2.4 0.078 -72.3 5.83 5.6 Source: [193] PAGE 343 Table 3.6 Henstenburg and Phoenix (1989) bi -linear stress recovery simulation parame ters and results: normalized gage length s=200; table entries are normalized (where noted, norm.) and results are aggregates of multiple simulations. Norm. Debond Sress Weibull Modulus Norm. Frictional Stress No. of Fragments Norm. Effective Gage Length Norm. Mean Fragment Length Coeff. of Variation Confidence Interval Max. Peak Stress kb Tf nt slt COV( lt) CI c 0.7 3 0.3 71 197.7 2.784 42.9 2.660,2.909 1.684 0.7 5 0.3 69 197.3 2.868 36.4 2.757,2.978 1.881 0.7 10 0.3 72 199.1 2.765 29.1 2.681,2.849 1.293 0.7 15 0.3 73 198.0 2.712 25.3 2.642,2.783 1.251 0.7 0.3 87 198.8 2.296 21.1 2.250,2.341 1.000 0.5 5 0.3 59 196.8 3.336 32.7 3.211,3.461 1.610 0.6 5 0.3 63 197.6 3.147 34.7 3.025,3.268 1.495 0.7 5 0.3 69 197.3 2.868 36.4 2.757,2.978 1.881 0.8 5 0.3 77 198.6 2.573 37.4 2.477,2.669 1.625 0.9 5 0.3 83 198.2 2.377 36.9 2.292,2.461 1.591 1.0 5 0.3 97 198.1 2.033 38.2 1.964,2.103 1.589 0.7 5 0.3 69 197.3 2.868 36.4 2.757,2.978 1.881 0.7 5 0.5 90 198.6 2.217 33.0 2.149,2.285 1.661 0.7 5 0.7 100 198.3 1.983 29.1 1.932,2.034 1.770 See Eqns. 3.39-3.41, text, and Appendix F for definitions and descriptions, simulation results from [6]. PAGE 344 Table 3.7 The glass transition temperatures and estimated number of chain atoms involved in cooperative movement at Tg for various polymers. Polymer Tg ( C) No. of Chain Atoms Involved Polydimethylsiloxane -127 40 Polyisoprene -73 30-40 Poly(ethylene glycol) -41 30 Polystyrene +100 40-100 Sources: Ch. 6 (7a,13), [249]. PAGE 345 Table 3.8 Reproduced Tg, coefficient of thermal expansion ( ), and thermodynamic values for neat and filled polymer systems Polymer Tg ( K) g x 103 liq x 103 ( liqg)Tg liqTg Low MW PMMA Neat 335 .150 0.398 0.083 0.133 With 5% glass fiber 354 .110 0.312 0.071 0.110 With 20% glass fiber 370 .175 0.405 0.085 0.152 With 30% glass fiber 387 .180 0.550 0.143 0.213 With 5% Nitron 355 .250 0.525 0.098 0.186 With 20% Nitron 369 .280 0.550 0.098 0.199 High MW PMMA Neat 372 .150 0.475 0.121 0.177 With 5% Nitron 374 .325 0.650 0.122 0.243 With 20% Nitron 376 .330 0.675 0.129 0.254 PS Neat 364 .200 0.500 0.110 0.183 With 5% quartz 365 .200 0.505 0.110 0.183 With 20% quartz 370 .215 0.570 0.131 0.213 With 30% quartz 371 .220 0.620 0.148 0.232 With 5% Nitron 365 .250 0.505 0.101 0.180 With 20% Nitron 367 .250 0.625 0.138 0.229 Source: [123], Table 3.2. PAGE 346 Table 3.9 Heat capacity variation (DSC) at Tg, Cp, and interphase characteristics ( i and ri :Eqns. 50-51) for iron-particulate composites. f i m Cp rf ri Percent J/kg m m 5 0.048 94.95 276 0.185 75 75.24 10 0.386 89.61 220 0.352 75 75.96 15 1.27 83.73 172 0.494 75 77.12 20 2.828 77.17 138 0.593 75 78.53 25 5.098 69.9 117 0.654 75 80.1 5 0.052 94.48 272 0.198 150 150.518 10 0.413 89.59 211 0.377 150 152.0632 15 1.348 83.65 161 0.525 150 154.4935 20 2.698 77.03 128 0.623 150 157.4199 25 5.457 69.54 100 0.704 150 160.9148 5 0.05 94.95 274 0.191 200 200.6691 10 0.399 89.6 216 0.364 200 202.6621 15 1.333 83.67 163 0.519 200 205.9229 20 2.94 77.06 130 0.617 200 209.7998 25 5.28 69.72 109 0.679 200 214.0751 Source: [252]. PAGE 347 Table 3.10 Rutile material characteristics, rutile-CPE compos ite damping capacity, and calcula ted interaction parameters ( B, and R). See Eqn. 3.54 and Figs. 3.41-3.42. Material Surface Area (Vg 0)a (Vg 0)b 0 3 10 20 B R m2/gm ml/gm ml/gm Filler Percent CPE 9.1 24.7 1.021 -1.71 TiO2-1 7.32 688 677 0.989 0.916 0.810 0.02 1.03 10 TiO2-2 8.55 988 940 0.981 0.892 0.761 0.06 1.27 70 TiO2-3 7.55 285 106 0.967 0.844 0.665 1.69 1.74 195 TiO2-4 9.05 200 32 0.963 0.834 0.645 5.25 1.84 190 TiO2-5 8.52 343 151 0.968 0.848 0.674 1.27 1.7 170 TiO2-6 7.68 109 18 0.966 0.842 0.661 5.05 1.76 200 TiO2-7 8.7 1047 1071 0.978 0.881 0.741 -0.02 1.37 90 TiO2-8 8.98 96 136 0.982 0.896 0.769 -0.42 1.23 60 TiO2-9 8.64 38 57 0.982 0.896 0.769 -0.5 1.23 60 Source: [253]. PAGE 348 Table 3.11 Glass transition paramete rs for various neat and Aerosil filled polymer systems. Polymer Filler Fraction Tg Cp v Ea Vg h Vh Tg 0 Tmax Wcg %bm K J/kgK J/mol cm3/mol J/mol cm3/mol K K J/cm3 PS 0 368 26.17 30.6 100.5 5.15 16.9 368 0 304.8 2.22 1 368 23.45 0.105 5.76 18.9 5 368 18.74 0.27 7.1 23.5 10 368 12.98 0.505 9.04 29.7 15 368 14.6 0.52 9.17 30.1 PMMA 0 378 41.86 47.7 85.9 4.94 8.95 378 105 552.6 2.14 1 383 40.36 0.02 5.09 9.2 5 391 37.68 0.1 5.65 10.2 7 394 35.17 0.16 6.09 11 10 396 35.76 0.19 6.41 11.5 PUR 0 239 82.06 68.6 143 3.75 7.85 239 14 477.3 1.68 1 240 72.05 0.12 4.27 8.9 5 241 66.15 0.195 4.67 9.75 10 243 61.12 0.255 4.9 10.2 20 243 59.45 0.275 5.02 10.5 PDMS 0 148 30.14 20.9 65 2.91 9.05 148 7 322.4 1.47 10 149 27.23 0.095 3.16 9.85 30 150 24.36 0.19 3.37 10.5 50 150 22.31 0.26 3.54 10.95 PVAC 0 291 291 121 582 2.12 Source: Tables 3.5 and 3.7 [123]. PAGE 349 Table 3.12 Work of adhesion (Wa) and interfacial fracture energy (Wf)values via inverted blister test for adhe sive/substrate couples at sub Tg (adhesive) temperatures (Figure 3.52). Adhesive Substrate Temp. Wa Wf C mJ/m2 mJ/m2 water PTFE -23 44.8 564 water PS -23 70.1 1970 water PMMA -23 84.7 5820 water PC -23 101.7 1910 water STEEL -23 144.4 19300 water PTFE -10 44.8 120 water PS -10 70.1 730 water PMMA -10 84.7 3310 water PC -10 101.7 729 water STEEL -10 144.4 16300 diodomethane PTFE -23 39.7 218 diodomethane PS -23 86.5 a diodomethane PMMA -23 89.1 1490 diodomethane PC -23 86.8 621 diodomethane STEEL -23 101.6 2490 diodomethane PTFE -16 39.7 792 diodomethane PS -16 86.5 a diodomethane PMMA -16 89.1 1810 diodomethane PC -16 86.8 2246 diodomethane STEEL -16 101.6 4710 diodomethane PTFE -6 39.7 705 diodomethane PS -6 86.5 b diodomethane PMMA -6 89.1 1240 diodomethane PC -6 86.8 1540 diodomethane STEEL -6 101.6 a bromonapthalene PTFE -23 55 719 bromonapthalene PS -23 79.9 a bromonapthalene PMMA -23 88.4 655 bromonapthalene PC -23 87.3 a bromonapthalene STEEL -23 89.2 1120 bromonapthalene PTFE -10 55 a bromonapthalene PS -10 79.9 b bromonapthalene PMMA -10 88.4 a bromonapthalene PC -10 87.3 b bromonapthalene STEEL -10 89.2 a hexadecane PTFE -4 45 a hexadecane PS -4 55.2 a hexadecane PMMA -4 a hexadecane PC -4 54.9 a hexadecane STEEL -4 55.4 a hexadecane PTFE -5 45 287 hexadecane PS -5 55.2 a hexadecane PMMA -5 a hexadecane PC -5 54.9 a hexadecane STEEL -5 55.4 a a cohesive fracture b adhesive/substr ate interaction PTFE=polytetrafluorethy lene; PS=polystyrene; PMMA=polymethylmethacrylate; PC=polycarbonate Inverted blister test: Figure 1 and Table III [272] PAGE 350 Table 3.13 Selected surface energy propertie s and the harmonic and geometric means of Wa x (x=h,g) for HM and HS carbon fiber-epoxy composites. FiberMatrix p d Wa h Wa g O/C ifss mj/m2 mj/m2 mj/m2 mj/m2 mj/m2 MPa HMA 6.7 1.4 37.4 1.3 44.1 1.9 80.5 85.3 11.6 26.6 3.9 HMBox 3.8 0.5 41.0 1.7 44.8 1.8 76.1 81.3 4.1 54.7 3.1 HMB 14.4 1.9 31.2 1.0 45.6 2.2 87.6 87.9 21.9 63.25 4.6 HMEC 13.9 3.2 31.6 2.2 45.5 3.8 87.4 87.7 27.5 84.4 10.2 CA 10.0 1.6 27.5 2.3 37.5 2.3 78.5 79.3 28.6 61.4 5.9 CBox 8.1 1.4 39.0 1.5 47.1 2.1 84.3 86.9 20.3 71.9 4.3 CAA 39.8 4.3 22.0 13.8* 61.8 4.5 94.4 99.2 29.1 83.9 6.8 Epoxy 16.5 1.5 26.1 1.3 42.6 2.0 HMA=unmodified high modulus (HM) carbon fiber: HM48.00A SGL Sigri Carbon Group (Meitingen, Germany) HMBox=basic surface oxidized, HM CF HMB=unsized, industial modified, HM CF HMEC=electrochemichal oxidation (10 min. in KNO3/KOH soln.), HM CF CA=unmodified, unsized, high strength (HS) CF: C320.00A, SGL CBox=basic surface activated, HS CF CAA=thermally oxidized, HS CF Epoxy=DGEBA resin LY556, amine agent HY 932, Ciba-Geigy, Basel, Switzerland Large scatter present in original data Source: [260]. PAGE 351 Table 3.14 Selected surface energy and physical properties of PAN (T-300, HS CF, Toray) and Pitch (P-55, HM CF, Union Carbide) carbon fibers and pure graphite: AR=as received T= heat treated. Sources as indicated. Property Structure Surface Area Surface Area Alkane Length d Source (31,32) of [226] [226] [226] [226] [226] Technique TEM BET(Kr)GC calc. GC Material Units gm/cm3 m2/gm m2/gm # of C atoms mJ/m2 40C 45C 50C 60C 70C 75C T-300 AR 1.75 10 0.4 8 39.5 38.6 36.6 35.2 T-300 T 0.62 0.61 4.5 79.4 76.6 73.9 70.2 P-55 AR 2 50 0.59 7 41.5 39.9 39.3 37.0 P-55 T 0.74 0.75 5 54.0 50.9 48.3 47.1 Graphite AR 2.2 Graphite T Property d Source (35) of [226] Technique GC Material Units mJ/m2 mJ/m2 mJ/m2 T-300 AR 42.4, 41.8 T-300 T 50 P-55 AR P-55 T Graphite AR 32 Graphite T 40,54 90-130#, 165, 119, 151 Average crystallite size, estimated alkane length of entropic effect, [ 285], # [226], ref. (24) of [226], ref. (25) of [226], ref. (26) of [226]. PAGE 352 Table 3.15 Physical properties and adhe sion strength for different carbon fiber treatments in an epoxy (Araldite LY556 plus HT972, Ciba Geigy) matrix. Fiber Modulus Fiber Strength Adhesion Strength COV Fiber Type Fiber Treatment GPa GPa MPa Percent HS AR 270 2.56 9.7 38 Benzene wash 270 2.56 7.8 18 Oxidation (nitric acid) 2.14 32.5 Oxidation (hypochlorite) 273.5 2.46 40 Oxidation (hot air) 1.96 40.5 SiC coating 18 HM AR 357 1.73 1.8 Oxidation (hypochlorite) 270-325 1.0-1.8 up to 26 SiC coating 281.5 0.88 (70) Reduction by wet hydrogen 259 1.04 3.8 Modified microdebond pull-out (fig. 1 of [284]) Textual note [284] indicates all treated COV(treated) PAGE 353 Table 3.16 Interfacial shear strength for coated and uncoated carbon fibers. ifss Pull-out Test ifss Push-out Test Fiber/ Treatment MPa MPa CA 61.4 5.9 37.7 3.6 F15 38.8 6.7 34.7 3.1 HMox 84.4 10.2 28.6 5.5 F14 38.8 6.7 68.7 9.0 same value, similar mechanism fibers from SGL Sigri Carbon Group, Meitingen, Germany CA=HS PAN C320.00A F15=CA fiber plus electrocoat ed acryl amide and carbazole HMox=HM 48.00A plus electro chemical oxidation in KNO3/KOH F14=Hmox plus electrocoated acryl amide and carbazole Source: [262] PAGE 354 Table 3.17 Laminate (0/904/0) void content and composite failure strain against fiber surface treatment level and test temperature. Surface Treatment Void Content ult ult Percent Percent (100 C) (130 C) 0 1.2 0.5 0.85 10 0.75 1.6 50 0.75 1.5 100 0.5 0.6 1.85 Source: [296]. PAGE 355 (a) (b) Figure 3.1 Single fiber composite fragment ation (SFCF) schematic: (a) Dog-bone specimen with fiber breaks ; (b) Fiber axial stress ( f) and interfacial shear stress ( y) profiles for yielded matrix assumption [111]. f y Fiber Fragment Length C L Stress (arb) PAGE 356 L Z le Fixed Base Free Fiber End rf ra rb Figure 3.2 Single fiber pull-out schematic indicating the embe dded fiber length ( le) and a partially debonded section ( l ). PAGE 357 0.0 0.4 0.8 1.2 02468 (l/d)eff Pull-Out Fracture End Bonded End Free W Cu po (GPa) Figure 3.3 Fiber pull-out stress agains t embedded length aspect ratio (l/d)eff, tungsten wires in a copper matrix. After [112], fig. 14. PAGE 358 Figure 3.4 Fiber strength vari ation and fiber axial stress fo r an arbitrary length single fiber composite (SFCF) with Perfect Bonding (PB) and interacial shear yielding ( y). Position (Arb.)Stress (Arb.)Fiber StrengthIncreasing Composit e Increasing0 0 0 0 0 0 0 0 0 0 Fracture PAGE 359 Figure 3.5 Composite reinforcement modulus ratio, Eratio (from Eqn. 3.8), against the normalized difference in fi ber-matrix modulus ratio (Ef/Em: dashed lines, f = 0.05, 0.25, 0.50) and shear interaction extent (r1, symbols) for a paperpaper and graphite-epoxy composite See Table 3.2 for parameters. 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 00.511.522.533.544.55 Normalized Difference [1+(y-x)/(x)]E (%)ratioGr.-Epoxy Range Gr.-Epoxy Paper Comp., Cox (1952) Increasing rb, Ef f=0.25f=0.50f=0.05 PAGE 360 Figure 3.6 Three-dimensional representation of the composite reinforcement ratio, Eratio (Eqn. 3.8), as a function of fiber-matrix modulus ratio, Ef/Em, and fiber volume fraction, f. PAGE 361 P0x 1x 2 h E, vl E' 0 Figure 3.7 Stringer load diffusion into a se mi-infinite half-plate (h< PAGE 362 EFFECTIVE MEDIUM MATRIXPhase 1Phase 2: INTERPHASECOREPhase n r 0rn-1 Figure 3.8 Generalized Self-Consistent Sc heme (GSCS) schematic for a multi-phase fiber and interphase embedded in an infinite effective medium. PAGE 363 Figure 3.9 Cox and Rosen FAS predictions ( f/ appl: Eqns. 3.9 & 3.11) against fiber axial position for several fiber aspe ct ratios (AR=10, 100, and 1000). Only half-length is depicted, curves are sy mmetric about fiber midpoint, z/L=0. See Table 3.3 for parameters. 3.2350 1 2 3 0.00.20.40.60.81.0 10: Cox (1952) 10: Rosen (1965) 100: Cox 100: Rosen 1000: Cox 1000: Rosen Position (z/L)FAS (/ )f applf,max (L ) C L PAGE 364 Position (x/L) Figure 3.10 Cox and Rosen interf acial shear stress predictions ( ifss/ appl: Eqns. 3.10 & 3.12) against fiber axial position for se veral fiber aspect ratios (AR=10, 100, and 1000). See Table 3.3 for parameters. ifss / appl ifss / appl -0.10 -0.05 0.00 0.05 0.10 -1.20-0.80-0.400.000.400.801.20 10: Cox (1952) 100: Cox 1000: Cox -0.250 -0.125 0.000 0.125 0.250 -1.20-0.80-0.400.000.400.801.20 10: Rosen (1965) 100: Rosen 1000: Rosen Centerline Centerline PAGE 365 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.00.20.40.60.81.0 Position (z/L)/ifss applCox and RosenSadowsky and Kelly-Tyson ifss (arb.)Cox (1952) Rosen (1965) Sadowsky (1961) Kelly-Tyson (1965) Figure 3.11 Comparison of shear stress pred ictions and assumptions at the fiber tip (AR=100, see Table 3.3 for parameters): Cox and Rosen predictions share a common scale; Sadowsky, 'exact', and Kelly-Tyson yielded matrix assumption are on an arbitrary, but common scale (area under these two curves is equivalent). PAGE 366 Figure 3.12 Comparison of FAS predictions for a discontinuous fiber (AR=100, see Table 3.3 for parameters): Cox a nd Rosen predictions share a common scale; Sadowsky, 'exact', and Kelly-Tyson yielded matrix assumption are on an arbitrary, but common scale (area under these two curves is equivalent). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.00.20.40.60.81.0 Position (z/L)Cox and RosenSadowsky and Kelly-Tyson FAS (arb.)Cox (1952) Rosen (1965) Sadowsky (1961) Kelly-Tyson (1965) FAS (/ )f appl PAGE 367 Figure 3.13 Initial pull-out stress and maximum debond stre ss against embedded fiber length: (a) initial pull-out stress fo r uncoated steel wire in epoxy (open circles); coated steel wire in epoxy matr ix (filled circles), data from [192]; acid treated SiC fiber in glass matrix (f illed triangles), data from [148]; (b) maximum debond stress for the same data Solid lines are theoretical fits from respective references, see text. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 020406080 Embedded Length (mm)Initial pull-out stress (GPa) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 020406080 Embedded Length (mm)Maximum Debond Stress (GPa)(a) (b) PAGE 368 Figure 3.14 Load adsorption for a semi-infinite fiber in an infinite continuum. After [138]. x 1x 2 x 3Matrix B 1Region R 1E 1, v Filament B 2 Region R 2 E 2 v Radius a 0 0 Cross-section 2Cross-section 0 infi n z 0 PAGE 369 Figure 3.15 Normalized fiber axial stress ( (z)/ ( )) against normalized position (z/a). See Fig. 3.14 for geometry. Image courtesy of American Institute of P hysics. After [138], fig. 3. 1000 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1.0 E2 / E1 5 20 z/a =E2/E1 =100 =20 =5 =2 =1 ( z )/ ( ) PAGE 370 Figure 3.16 Load adsorption for a discontinuous, but contiguous, filament in an infinite continuum. After [139], fig. 1. x 1x 2x 3Matrix B 1 Region R 1E 1, v Discontinuous but contiguous Filament B 2Region R 2E 2 v Radius a 0 Cross-section 2Crack-region 0 z 0 infi n 0 infi n PAGE 371 02468101214161820 0 0.2 0.4 0.6 0.8 1.0 =100 =E2/E1z/a (z) / infin50 20 10 5 1 Figure 3.17 Normalized load adsorption ( (z)/ ( )) against position (z /a) for a disjointed, but conti guous, infinite filament in an infinite continuum, see Fig. 3.16 for geometry. Image courtesy of American Institute of Physics. After [139], fig. 2. =E2/E1 =100 ( z )/ ( ) PAGE 372 Figure 3.18 Effect of matrix Poi sson ratio and modulus ratio on the load transfer length [P(l/a)=0.9P( )] for a fractured infinite fiber (Figure 3.16). Dashed line is for semi-infinite end-bonded case. Image courtesy of American Institute of Physics. After [139], fig. 3. 02468101214161820 0 10 20 30 40 50 60 E2/E1load transfer length ( l/a )1/2 1/4 0 1/4 PAGE 373 0 5 10 15 20 25 30 35 02468101214 GAP/d infinity filler fraction = 0.16 filler fraction = 0.45 Figure 3.19 Matrix stress concentration near the fiber end (.06df) for different gap sizes and filler fr action. After [174], fig. 6. E m c E r f m PAGE 374 0 4 8 12 16 20 24 04080120160200Modulus Ratio (Ef/Em)Shear SCF,0 4 8 12 16 20 24Principal SCF, SCF shear FEA SCF shear Moire SCF shear Photoelastic SCF principal FEA SCF principal Moirem[1+(Ef/Em -1) f] c m[1+(Ef/Em -1) f] c FEA = 0.06d Moire = 0.05-1.0d Photo = 0.1d Figure 3.20 Matrix shear and pr incipal stress concentrations in an ideal five-fiber microcomposite with a central end-bonded discontinuous fiber. Data from [ 175]:photoelastic and [177]:FEA, Moire. PAGE 375 Rpo lcl E q n. 3. 201l l2Eqn. 3.21 Eqn. 3.22 Figure 3.21 Pull-out toughness prediction against fiber le ngth. After [195], fig.6.2. PAGE 376 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 01234s/f = s/f = 9 s/ f = 4 s/ f = 1 friction contributions s/ f = 9 s/ f = 4 a l /2Pf / P f max Figure 3.22 Maximum fiber pull-ou t load against length factor ( a l/2) for various interfacial frictional and shear debond strengths ( f and s respectively). Image courtesy of Chapman Ha ll ltd. After [191], fig. 5. PAGE 377 Figure 3.23 Schematic representation of remote fiber axial stress vs. displacement for a single fiber pull-out experi ment. After [148], fig. 1. DisplacementTensile Stres s Initial Debonding (d)Partial Debonding (p)Fiber pull-out (po)Complete Debonding (d) PAGE 378 DisplacementTensile Stres s B C A DisplacementTensile Stres s dpo A B C d d d (a) (b)po Figure 3.24 Remote fiber pull-out stress vs displacement schematics for (a) and (b) q0> qth (c) q0 qth, all after [149], fi g. 4 and (d) a totally stable condition (zmax 0), after [192], fig.4c. PAGE 379 DisplacementTensile Stres s B C A DisplacementTensile Stres s po A, B C *d d = d (c) (d)po d = Figure 3.24 cont. PAGE 380 rbraL z z =0 L-z Debonded Length d p d Figure 3.25 Single fiber pull-out schematic indicating partial de bond length (L-z) and shear stress interaction extent (rb ra). After [192], fig. 1. PAGE 381 0.0 1.0 2.0 3.0 4.0 5.0 0100200300400500Stress (GPa)Debond length, L-z ( m) (a) Frictional shear component fInitial debond stress dPartial debond stress ,p d Figure 3.26 Partial debond stress against debond length using Gao' s LFM prediction (a) and Hsueh's strength crit erion (b) (Eqns. 1-3 and 4-7 of [193]): solid lines (partial debond); squares (initial debond); circles (frictional shear stress); dashed line (locus of init ial frictional shear stress). 0 1 2 3 4 5 0100200300400500Stress (GPa)Debond length, L-z (mm) L=50 L=300 L=100 Z max (b) PAGE 382 0E+10 0E+09 0E+00 0E+09 0E+10 5E+10 0E+10 5E+10 0E+10 5E+10 0E+10 -2.0E+08 0.0E+00 2.0E+08 4.0E+08 6.0E+08 8.0E+08 1.0E+09 1.2E+09 0 L l y End bonded contribution IFSS per Cox (1952) Shear Stres s Fiber Axial Stres s FAS per Piggott (1966) Figure 3.27 Bi-modal stress tran sfer across yielded terminal sections and a central shearlag approximation (Cox, [105] ). After [162], fig. 1. PAGE 383 0 5 10 15 20 25 30 35 00.20.40.60.81Ef/G=100 Ef/G=10 Ef/G=1,000 Ef/G=5,000 Ef/G=50,000 Ef/G=1 ~97l / L c2 m Figure 3.28 Strengthening factor ( c/2 y) against ratio of yielded terminal section to fiber length for various fiber-matrix shear modulus ratios (fiber aspect ratio=100). After [162], fig. 3. l /L PAGE 384 0 200 400 600 800 0102030405060 0 20 40 60 80 100Position (arb.)FA S ( ar b.) ifss (arb.) fy Yielded Section Disbonded Section 0 Fiber End K b S b Figure 3.29 Bi-linear stress recovery model for a broken fiber with a disbonded and yielded matrix section: Fiber Axial Stre ss (FAS, solid line); Interfacial Shear Stress (IFSS, dashed line). After [6], fig. 5. PAGE 385 Figure 3.30 Hashin's collapsed tw o-dimensional interface schematic: s is axial; t is tangential; n is normal. After [222], fig. 1. Material 1 Material 2 S12 t n s PAGE 386 0246810 5 0 10 15 20 Tg (deg. C)Aerosil content (vol. %) 1 2 3 4 5 Figure 3.31 Change in Tg of PMMA with Aerosil cont ent and testing frequency: 1, DSC; 2, Dilatometry; 3, DMA; 4, DRS; 5 NMR. Image courtesy of Rubber and Plastic Research Assoc. of Great Britain. After [123], fig. 3.1. PAGE 387 020406080100120140 0 2 4 6 8 10 Temperature (C) x Neat PMMA PMMA + 1.32% Aerosil PMMA + 12.5% Aerosil PMMA + 23% Aerosil Figure 3.32 Dielectric Relaxation Spectra (tan ) for Aerosil filled (vol. %) PMMA indicating dipole group and dipole segmen t mobility variation. Image courtesy of Rubber and Plastic Research Assoc. of Great Britain. Data from [123], fig. 3.20. tan x 102 T emperature C PAGE 388 0 20 40 60 80 100 0102030405060 PUR PMMA PDMS PS Cp (J / kg K)Filler Fraction (% bm) Figure 3.33 Variation in heat capacity jump at Tg ( Cp) with filler fraction (Aerosil ) for: Polyurethane (PUR); Polymethylmethacrylate (PMMA); Polydimethylsiloxane (PDMS); Polystyr ene (PS). Data from [123], Table 3.5. PAGE 389 0 50 100 150 200 250 300 051015202530 Filler Fraction (%) Cp (J / kg K)rf = 75 mrf = 150 mrf = 200 m Figure 3.34 Heat Capacity variation of an iron partic ulate-epoxy composite at Tg ( Cp) against filler size and fraction: rf = 75, 150, and 200 mm. Data from [252], Table 1. PAGE 390 Figure 3.35 Estimated boundary layer fraction ( bl) of an iron particulate-epoxy composite against filler fraction and pa rticle size using DSC data and Eqns. 3.50-3.51: rf = 75, 150, and 200 m; lines are cubic fits. Data from [252], Table 1. 0 1 2 3 4 5 6 051015202530 (%)bl r f= 75 m r f= 150 m r f= 200 m bl ~ C f3 f (%) PAGE 391 Figure 3.36 Interphase thickness ( rbl) as a function of filler size and fraction ( f) utilizing DSC data and Eqns. 50-51: rf = 75, 150, and 200 ( m), lines are quadratic fit. Data from [252], Table 1. 0 4 8 12 16 051015202530f (%) rbl ~ C(rf) f2 rf = 75 mrf = 150 mrf = 200 m rbl (m) PAGE 392 0 0.2 0.4 0.6 0.8 1 01020304050Normalized min. Particle SeparationClosest packing = 52.4% f (%) Figure 3.37 Normalized minimum particle-particle se paration of sphe rical elements in an ideal cubic packing conf iguration vs. filler fraction. PAGE 393 0 50 100 150 200 250 300 0100200300400500 Min. Particle Separation (m ) Cp (J / kg K)rf = 75 mrf = 150 mrf = 200 m Increasing f Figure 3.38 Heat Capacity variation of an iron-pa rticulate composite at Tg ( Cp) against filler size and minimum particle-particle separation (cubic cell config.): rf = 75, 150, and 200 m; lines are for reference only (see text). Data from [252], Table 1. PAGE 394 Figure 3.39 Change in heat capacity variati on with total filler surf ace area and filler size of an iron-particulate composite at Tg ( Cp): rf = 75, 150, and 200 m; lines are for reference only. Data from [252], Table 1. 0 50 100 150 200 250 300 0.01.02.03.04.0rf = 75 mrf = 150 mrf = 200 m Total Filler Surface Area x 10 (m )3 2 Cp (J / kg K) PAGE 395 95 100 105 110 01020304050f (%)Tg ( C) Figure 3.40 Glass transition temperature va riation of phenoxy compos ites against filler concentration and type: Attapulgite Clay, open circle; Glass Beads, open square. Data from [250], fig. 1. PAGE 396 Figure 3.41 Tan max at Tg for chlorinated PE-rutile composites as a function of filler basicity, and filler concentration, f: see Table 3.10 for material specifics; Eqn. 3.54, dashed lines. Data from [253], fig. 4 and Table I. 0.6 0.7 0.8 0.9 1.0 1.1 0510152025 TiO 2 2 TiO 1 2 TiO 4 2 f (%)tan ma x Filler Basicity PAGE 397 Figure 3.42 Filler fraction correction parame ter (B in Eqn. 3.54) against the acid-base interaction parameter (W from IGC) for various chlorinated PE-rutile composites: See Table 3.10 for details; dashed line is for reference only. Data from [253], Tables I & II. 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 -10123456, Acid-Base Interaction Param.B, Filler Fraction Correction Param. PAGE 398 Figure 3.43 Bulk material flow via altering hole content: Vh = molar hole volume; j = activation energy to eliminate hole; h = excess molar hole energy relative to "no hole" condition. After [249], fig. 6.23. -4 -2 0 2 4 6 8 10 12 024681012141618 h j Volume Potential EnergyVh Reaction Coordinate PAGE 399 0 1 2 3 4 5 6 7 8 9 10 0102030405060 0 10 20 30 40 50h Excess Molar Hole energy (J/mole ) V h MolarHoleVolume(cm 3 /mole) h : PSh : PMMAh : PURh : PDMSVh : PSVh : PMMAVh : PURVh : PDMS f Figure 3.44 Excess molar hole energy, h, and molar hole volume, Vh, of Aerosil composites against filler fraction. Data from [123], Table 3.5. PAGE 400 Figure 3.45 Variation in T as a function of molar cohesion energy, Wcg, for different Aerosil filled composites: materials as indicated; line is for ref. only. Data from [123], Table 7. 0 20 40 60 80 100 120 140 300400500600 Wcg Molar Cohesion Energy (J/cm3) T Maximum Tg Diff. For Filled Composite ( C) PS PMMA PUR PDMS PVAC PAGE 401 0 100 200 300 400 500 600 700 11.522.5 0 20 40 60 80 100 120 140 PS PMMA PUR PDMS PVAC Macromolecular RigidityT, Maximum Tg Diff. For Filled Composites (C)Wcg Molar Cohesion Energy (J/cm )3 Figure 3.46 Molar cohesion Energy, Wcg, and maximum Tg variation ( T) for different Aerosil composites relative to macromolecular rigidity, : T, open symbols; Wcg, filled symbols. Data from [123], Table 3.7. PAGE 402 -150 -140 -130 -120 -110 -100 -90 02468101214 Segment-Dipole Group-Dipole -150 -140 -130 -120 -20 -10 0 No. of C Atoms in Grafted AlcoholT ( C) Figure 3.47 Shift in temperature ( T by NMR) at tan max for a ED-20-Aerosil composite with different length alcohols (No. of C atoms in chain) grafted to the filler surface. Da ta from [123], fig. 3.25. PAGE 403 Figure 3.48 Critical aspect ration, lc/d, against testing temper ature for a carbon fiberepoxy single fiber composite, experi ment and prediction (Eqn. 3.56 and Eqn. 3 of [227]. Data from [227], fig. 2. 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 20406080100120 lc/d, experiment Eqn. 3 of [230] Eqn. 3.56Temperature (C)lc/d PAGE 404 Figure 3.49 Epoxy volume percent (Pal mese prediction [256]) and Skourlis discretization [227] ) and a matrix shear modulus-stair-step model against radial position from the fiber su rface. Data from [227], fig. 4. 0.50 0.55 0.60 0.65 0.70 0.75 0.80 012345678910 0.0 0.3 0.6 0.9 1.2 1.5 1.8 Palmese Epoxy vol% Discretized Epoxy vol% Matrix Shear Modulus Radial Position from Fiber Surface (lattice layers)Epoxy Volume Fraction (%)Matrix Shear Modulus @25C (GPa) PAGE 405 0 1 2 3 4 5 6 7 8 9 10 101214161820222426 Radial Position, r ( m)Matrix Tensile Modulus, E (GPa) =0.1 =0.3 =0.5 =0.9 =0.2 Em fiber radius Figure 3.50 Unfolding matrix modulus (Eqn. 3. 57) relative to radial position (r) and fitting parameter ratio ( = 2/ 1): Em=3.2 GPa; Ef=69.9 GPa; rf=12 m; 1= 26. After [269], fig. 2. PAGE 406 0 2 4 6 024681012 p lastic debon d debon d y iel d Shear Stress, Extension Ratio, Figure 3.51 Representative non-linear shear st rain hardening material (solid line) and idealized constitutive model for the composite interphase (dashed line). After [238], fig. 3. PAGE 407 0 4000 8000 12000 16000 20000 020406080100120140160 Work of Adhesion (mJ/m2)Work of Fracture (mJ/m2) water -23 C water -10 C (b) Figure 3.52 Work of fracture, Wf, against work of adhesion, Wa, for various adhesives, substrates, and temperatures with (a) hydrophobic character, and (b) hydrophylic character. Data from [272], Table III. 0 1000 2000 3000 4000 5000 020406080100120140160 Work of Adhesion (mJ/m2)Work of Fracture (mJ/m2) diodomethane -23 C diodomethane -16 C diodomethane -6 C bromonapthalene -23 C hexadecane 5 C(a) PAGE 408 Exponentia l R2 = 0.993-2000 0 2000 4000 6000 8000 10000 12000 0123 -4 0 4 8 12 16 20 24 W Reorientaion / Strain Hardening W Plastic Flow W Elastic Region Stress-StrainStress (arb)Work of Fracture (arb) 0 0 Strain (arb) and Work of Adhesion (arb) 0Not zero but small relative to reorientation region Figure 3.53 Schematic depicti ng a reorientation/strain hard ening stress-strain curve and the expected exponential growth for Wf versus Wa seen for reorientation/strain hardening. f f f PAGE 409 Figure 3.54 Interfac ial shear strength, ifss, for HS and HM carbon fibers in epoxy resin against the geometric (x = g) and ha rmonic (x = h) work of adhesion, Wa x. Data from [260], Table 4. R2 = 0.991 R2 = 0.9870 20 40 60 80 100 708090100110 HS, Basic Oxidized, x=g HS, Basic Oxidized, x=h HM, Basic Oxidized, x=g HM, Basic Oxidized, x=h W (mJ/m )x a2 ( MPa ) IFSS PAGE 410 Figure 3.55 Apparent in terfacial shear stress, ifss, versus embedded fiber length for various carbon fiber-matrix systems: closed symbols from [262], HMox=High Modulus oxidized, Electr ocoated with Poly(carbazole-coacrylimide); open symbols from [261], QW=quenched PPS matrix, SC=slow cooled, ISO=1hr at max. crystallization temperature. R2 = 0.6303 R2 = 0.936 0 10 20 30 40 50 60 70 80 1030507090110130 HMox Electrocoated HS Electrocoated HM cleaned QW HM cleaned SC HM cleaned ISO Embedded Fiber Length (m) ( MPa ) IFSS PAGE 411 Fiber Strength, 124f( )p 1-p Figure 3.56 Double-box fiber stre ngth probability density function, f( ), for a fiber with distinct surface and volume flaws: 1 = min. strength for a continuous fiber, 2 = min. strength for a continuous fiber with no severe flaws, 4 = max. strength for a flaw free fiber, p = fraction of link popul ation containing a severe surface flaw. After [266], fig. 1. PAGE 412 Figure 3.57 Glass-coupling agen t-matrix interphase regions: (1) chemically reacted; (2) chemisorbed; (3) physisorbed; (4) in terdiffusion/IPN. See text for descriptions and sources. O M--O--Si--R O M--O--Si--R O OH H M--O O--Si--R H OH Matrix/Coupling Agent Interdiffusion / IPN Physisorbed Siloxanols Chemisorbed Siloxanol Chemically Reacted Poly(siloxane) R-Group/Matrix ReactedGlass PAGE 413 Gc(1) Gc(2) c(1)c(2) G ( 2 ) G ( 1 ) DcD ( ) 2 G D Figure 3.58 Crack driving force, G( ), and continuum dissipation, D( ), versus load/displacement factor A reduction in G( ) (or increase in D( )) increases c, where Gc-Dc=2 (Eqn. 3.66). After [290], fig. 4.2. PAGE 414 Figure 3.59 Weibull failure probability, F, against element/link fracture strain and the influence of improved wetting and b ond strength. After [296], fig. 12. Figure 3.60 Ideal Weibull failure distributions (Weibull Modulus = ) and the influence of matrix ductility, defects, and constraints. After [296], fig. 4. Element Fracture Strain (%)Element Failure Probability, F Improved Wetting Increased Bond Strength ln (Fracture Strain)Element Failure Probability, F ln ( ln (1-F)) Ideal Fracture ( f =const) Constraint Defects Matrix Ductilit Weibull Modulu PAGE 415 u1u1u2b a2a1a1X Y 0 plies 0 plies 90 plies 0 0.2 0.4 0.6 0.8 1 1.2Position Relative to Crack Y axis0 Element Length L0 (x) / (a) (b) Figure 3.61 Schematic of a shear-lag model applied to a 0x/90y/0x composite laminate: (a) u1 and u2 are displacements of the plies; a1 and a2 are the ply thicknesses; b is the shear stress transfer zone: (b ) normalized axial stress build-up in the 90 plies (treated as a continuum) agai nst distance from an existing crack; dashed line indicates 90% stre ss transfer and element length L0. After [296], figs. 1 & 2. PAGE 416 CHAPTER 4 SIMULATING THE SINGLE FIBER CO MPOSITE FRAGMENTATION TEST Introduction Continuum treatments for load absorp tion in a strong discontinuous fiber suspended in a loaded composite structure (effective continuum) apparently provide simple deterministic solutions (e.g., Eqns. 3.17-3.18) In fact, the actual physical process for load absorption into the fiber is not deterministic because of the complex and irregular matrix morphology at the fiber/matrix interface1, and also the non-uniform mechanical and surface properties of the fiber (viz. Chapters 2 and 3). A progression of increasingly complex micromechanical models were developed in an effort to estimate and characterize fiber/matrix performan ce (e.g., Cox (1952) [105], Rosen (1965) [110], Kelly and Tyson (1965) [112], Henstenburg ( 1989) [151], and Nair n (1996) [229]). These models, and others, used in conjunction with the Single Fiber Composite Fragmentation (SFCF) test assess interpha se mechanical and fiber/matrix adhesion strength, and, more generally, predict strong-fiber/matrix in teraction in regards to the composites modulus, strength, and other mechanical properties. A micromechanical models utility hinges on: (1) faithfully representing the underlying physical process, (2 ) the certainty of parameters and processes modeled (e.g., matrix mixing, lot-lot variation, and fiber strength), and (3) any modeling errors (e.g., discretization errors in nu merical techniques). Points (1) and (2) are commonly 1 This interaction region is variously referred to as a mesophase [251], boundary layer [123], and most commonly as an interphase. PAGE 417 distinguished as reducible (epistemic/s ubjective) and non-reducible (aleatory) uncertainties. Here, and in the literature, unc ertainty is defined as a potential deficiency in any phase or activity of the modeling proce ss that is due to lack of knowledge [313]. As the descriptors indicate, epistemic un certainty can be reduced by an increased understanding of the physical process; whereas, aleatory uncertainty is inherent in the system and cannot be objectively reduced. A further distinction of epistemic uncertainty is particularly appropriate for SFCF modeling. Specifically, reducible uncertainty in modeling attends to (a) vagueness, (b) nonspecificity, or (c) dissonance (ref. 16 of [313 ]. As we have noted previously for load absorption modeling, and SFCF mo deling in particular (viz. Chapter 3), there is acute nonspecificity (that is, numerous models re presenting the same process) and dissonance (incompatible evidence) with regards to describing the debonding criteria at the fiber/matrix interface and interphase yielding. Epistemic uncertainties that influence, either independently or collectively (second order effects or covariance), nondeterministic analysis (NDA) of SFCF expe riments will be considered below. The impact of the inherent uncertainty in fi ber strength estimation (range and distribution shape) on the NDA will also be investigate d. A probabilistic-based Monte Carlo model for the SFCF test will be presented that enables parameter gradient and sensitivity determination. That is, the un certainty in the in terfacial shear stre ngth (response), the objective2 measure of SFCF tests, will be evalua ted with regards to the inherent and epistemic uncertainties of the NDA. Additi onally, the NDA (response) will be compared to simple deterministic estimations of the fiber fragment distribution and the interfacial 2 Here, objective refers to the goal of the test (determining the interfacial shear strength, ifss) and not specifically an objective measure because, as we will see, the interpretation of a series of SFCF tests is subject to the method/assumptions used to analyze the results. PAGE 418 shear strength, ifss, prediction. Finally, a non-probabil istic NDA for SFCF test modeling will be described that might better handle the epistemic uncertainties in the physical process. Theoretical Background We will consider first, the sources and t ypes of uncertainties for micromechanical modeling and how they are modeled. That is, wh at type of non-deterministic analysis is appropriate for the different ki nds of uncertainty (epistemic and aleatory) and how do the uncertainties in the modeling processes/ parameters propagate through the model. A new micromechanical model (non-determi nistic software simulation) for the SFCF test will be described that is based on the deterministic models discussed in the previous chapter (e.g., Eqns. 3.x and 3.y). The uncertainties in the physical processes and model variables will be deri ved from available data a nd the modelers opinion. Modeling Uncertainties and Variability Not all uncertainties are created equal. Clearly, it is preferable to lessen the impact of epistemic uncertainty on model predictions That is, where practicable, the modeler should make use of available data and argum ents that explain the physical process and thereby reduce, or eliminate, the subjective uncertain ty. This is obvious, but in practice, the vagueness, nonspecificity, and dissonance at tached to an explanation of a process (e.g., fiber/matrix debonding) does not always allow for such clarity. Typically, micromechanical models, and load absorption models in particular have glossed over this modeling difficulty (fuzziness) by em ploying broad or unreasonable assumptions (e.g., perfect fiber/matrix bonding or interphase elastic-plas tic behavior), which, while enabling the analyses, introduce uncertaint y into the response (NDA) predictions. PAGE 419 If the subjective uncertainty in a system cannot be completely eliminated, it is incumbent on the analyst to model the physical process in a way that minimizes the epistemic uncertainty. Indeed, the complex m echanical interaction in the interphase for fiber load absorption and SFCF testing cannot be uniquely describe d by any one physical process (e.g., perfect fiber/matr ix bonding). When reducible uncertainty persists, it is imperative that the consequences of modeli ng choices, assumptions, and uncertainties be carefully communicated. A classic probabilistic m odeling approach for epistemic uncertainties in micromechanical models is not appropriate because, in this case, we likely know only the intervals or, perhaps a subject ive probability distribution (expert opinion), about the model parameters/physical pro cess. Typically, the subjectiv e uncertainty is obscured by modeling assumptions that are in adequately supported by the da ta, that is, the previously referenced vagueness, nonspecificity, and di ssonance. Assumptions of this type are recast as invariable uncertainties. To be clea r, they are recognized uncertainties in the modeling, but for practical and mathematical r easons these parameters are held constant (invariable). The constant shear stress assumption (iffs = y), made by Kelly and Tyson for their fiber load absorption studies (cf. Eqns. 3.4 and 3.5) [112.], is an example of invariable uncertainty. Variabilities With regards to non-reducible uncertainty, th ere are two types of variability, certain and uncertain. Both the range and distribution of values are known fo r certain variables, whereas, for uncertain variables, there is a lack of knowledge about either the range or PAGE 420 distribution3. Certain variables in micromechanical modeling are ideally suited for a frequentist interpretation via non-deterministic analysis. Of course, sufficient sampling must be undertaken to accurately represent the possible response (outcome) even when only certain variable s are involved [314]. The outcome for a properly posed probabi listic NDA of certain variables is representative of both the fr equency and range of possible outcomes. In contrast, the outcome for a probabilistic NDA involving uncer tain variability (unknown distributions) can only provide information on the ra nge of possible outcomes because subjective PDFs were necessarily assumed to handle the un certain variable distributions. Normal distributions are typically assumed for uncerta in variables without due consideration for other PDF possibilities that might otherwise improve predictive performance [313]. This bears reiterating, the outcome distributi on, which has only limited, if any, objective qualities [315], does not correspond to the actual response frequency. An alternative approach for handling un certain variables involves employing a Gibbs sampler to generate random variables from a distribution w ithout first knowing its density (PDF). This method involves first de fining a response prediction. For example, Yi = 0 +1 xi + i, where i ~ N(0, 2) and 0 +1 xi is the mean for a given value of x Two-stage Monte Carlo simulations are performed using mean-deviation pairs (0 +1 xi 2) that are obtained by first choosing 0 and from a bivariate normal distribution and then, iteratively determining the mean-d eviation pairs. This hybrid Bayesian4 (response prediction) probabilistic NDA a pproach requires matched i nput-response pairs, where the 3 The classic example of an uncertain variable is the case of an unknown distribution of actual dimensions bounded by a drawing tolerance for parts from a manufacturer with unknown process controls or equipment. 4 A reference to original work by mathematician Thomas Bayes (~1701-1761), but broadly applied to a non-frequentist consideration of probability. PAGE 421 input may be multivariate. Confidence in the response prediction depends on the quality of the data that provides th e empirical model estimators (0, 2). Additionally, it must be recognized that an NDA (viz., proba bilistic Monte Carl o) cannot overcome deterministic model deficiencies. Response surface methodology (RSM) For very complex and computer intens ive simulations (e.g., rocket flight trajectories and large finite element models ) that may involve only certain variables, straight-ahead Monte Carlo non-deterministic analysis may not be practical, or even necessary. Replacing the complex analysis with a response surface prediction, that is estimated from either actual input-response da ta or computer simulations, allows the analyst to determine responses for complex systems quickly. Response surface methodology (RSM) also provides a means to eval uate parameter partials and explore the possibilities of higher order pa rameter interactions. In the latter, the response surface prediction can be formulated, for instance, to account for known and/or postulated covariance (e.g., matrix interphase modul us and fiber-matrix debonding strength). In general, the response surface takes the form e x x f Yn i .. Eqn. 4.1 where e is the error (uncertainty) and f ( xi xn) is the response surface. A particular example of a second order response surface prediction is ij i j i j i i i ie x x x Y, 0 Eqn. 4.2 One advantage of RSM is that explicit depe ndence on parameters can be modeled a priori without a complete understanding of the variable s function in the physical process. That is, when there is sufficient data availabl e for response surface fitting, the modeler can PAGE 422 introduce explicit variable de pendence even in light of incomplete understanding (epistemic uncertainty) of that variables phenomenological influence or its physically based mathematical representation. While RS M may be able to represent the outcomes for a complex analysis, care must be exer cised near or beyond the response surface domain boundaries because large gradients often exist there. The subjective response surface function (e .g., Eqn. 4.2) is not necessarily based on physical process representation; however, its form may be inferred from an understanding of the process. In complex systems however, this inference may be obscured by the large number of parameters involved. RSM was developed to enable the exploration of inherent variability via Monte Carlo simula tion (draws from the respective variable distributions) or design of experiments, but it cannot model errors or uncertainties (e in Eqn. 4.1) because of the error assumption N(0, ). In fact, sparse or inconsistent data, which may be a consequence of uncertainty, wi ll be effectively ignored in favor of a smoothly changing response surface that is typi cally formulated (via reducing the sum of the squares of the differences) to include certain variables of the modelers choosing. The difference between the response su rface prediction and the actual response (again, e in Eqn. 4.1) contains c ontributions arising from: Limited sampling, simulated/actual (affects response surface shape and estimators) Uncertain variability and invariable uncertainty (affects simulated response predictions) Response surface mathematical form (affect s response surface shape and parameter sensitivity) Epistemic uncertainty (affects model prediction confidence) Response surface methodology (e.g., Eqn. 4.2) obviously limits the response prediction to the surface itself. The respons e prediction variation with respect to a PAGE 423 particular variable is consequently also controlled by the response surface shape (multidimensional). For example, j j jx x const x x f1 2 12 1 1. Eqn. 4.3 The point here is that, piecewise smooth re sponse surfaces are a necessary consequence of the chosen response surfaces. The sh ape of this multi-dimensional surface is controlled by the convenient n-dimensional func tional format and is not representative of the underlying physical processes. Partials, or computed sensitivities, for a singular variable may contain contributi ons from covariance and, in a ddition, are also influenced by the choice of the other model variables (v ector space), uncertainti es, and discretization errors. That is, a precise physical meaning for variable partials and sensitivities for response surfaces is clouded by the transformation of the actual physical process onto the response surface. Non-probabilistic NDA A distinct, and arguably more appropri ate, approach for handling subjective uncertainty is paradoxically referred to as non-pr obabilistic non-determin istic analysis. A Bayesian treatment of the available informa tion (possibly incomplete or dissonant) for a physical process assigns probabilities to all un certainties (objective and subjective) in a simulation. For subjective uncertainties, the probabilities are based on the available information or expert opinion. A collection of propositions, or interval estimations, and rules for combining the probabilities (e.g., De mpsters combining rule, eqn. (1) [316], and fuzzy logic membership functions, eqn. ( 10) [313]) allow for treatment of subjective uncertainty in complex systems. Again, as for uncertain variables in probabilistic NDA, a non-probabilistic NDA contains a subjective component that must be acknowledged. PAGE 424 The proper use of probabilistic and non-pr obabilistic NDA is consequently strongly dependent on an accurate assessment of th e model variables and problem uncertainties. For certain variables and uncertain variab les that exhibit no rmal-like behavior centered on an interval, a proba bilistic approach is clearly more appropriate than an interval (non-probabilistic) analysis [313]. However, a non-probabilistic (interval) NDA is more appropriate when uncertain variab les with inadequately supported PDFs and invariable uncertainties are part of the mode l definition. The exact solution set of a nonprobabilistic NDA, { y } = f ({ x }) Figure 4.1 , contains a ll the possible outcomes of the input parameter { x }, which itself is a subset of the interval vector {x}5. Typically, the analyst is interested in the extrema of the solution set, which is aptly represented by a hypercubic approximation ( n -dimensional for n variables, 2-D in Figure 4.1). Several basic strategies exist for determining the n -dimensional hypercubic approximation: Vertex analysis (ref. 33 of [313]) Global optimization (refs. 1 and 4 of [313]) Interval arithmetic strategy Hybrid interval finite element analysis [313] Vertex analysis, as its name implies, looks at the boundary solutions for the range of input intervals. However, vertex analysis is not able to identify local extrema on the response surface (cf. RSM limitations due to smoo thing) and it is consequently restricted to monotonic functions, otherwise conservati sm cannot be guaranteed. In contrast, interval arithmetic strategy, which reformats the deterministic formulation using interval operator definitions (addition, subtraction, multiplication, and division) is necessarily conservative. In fact, the degree of conser vatism, which is a consequence of successive 5 Typically, the tails of PDFs are reduced in inte rval analysis according to expert opinion (e.g., 2). PAGE 425 exact solution bounding (Figure 4.1), is not known and cannot be easily controlled for complex analyses. Global optimization of th e deterministic solution set quickly becomes numerically intensive as the number of un certain variables and model uncertainties increase. A hybrid interval finite element analysis [313] uses a combination of global optimization and interval arithmetic whereby the excessive conservatism of interval arithmetic is reduced, first by determining an intermediate global optimization result and then finishing with an inte rval arithmetic approach. The advantage for this hybrid technique is efficiency (less complex global optimization) and a less conservative, more representative, prediction because ther e are fewer hypercubic approximations. Model Quality Model certainty is a complex function of the variables i nvolved in the deterministic description, their variability (certain and uncertain), modeling assumptions (invariable uncertainty), and any modeling/discretization errors (e.g., hypercubic approximations and finite element mesh size). The domain of a deterministic solution for a nominal input vector is a single point (dotted line in Figure 4.2). This nominal vector { x } may itself change over time as the true population mean s are approached, the process/parameters are changed (e.g., new graphitization temper ature for graphite fi bers), or as our understanding of the process/system changes. The width of the predictive band in Figure 4.2 (exact solution set for the interval vector set {x}) is representative of the ndimensional hypercubic volume depicted in Figure 4.1. With an increasing understanding of the proce ss (increasing abscissa in Figure 4.2; e.g., imperfect fiber-matrix adhesion vs. pe rfect adhesion), we can reduce the subjective uncertainty but uncertain variable intervals s till exist. When we eliminate all reducible uncertainty (invariable uncertainty and uncerta in variables), we ha ve only the inherent PAGE 426 variability of the process/system dashed lines . Ideally, one would prefer to model a system in this objective regime using a strai ghtforward probabilistic NDA. Again, in this regime, both the response interval and its di stribution are significant while left of the demarcation (vertical line separating reduc ible and non-reducible uncertainty) only the interval is relevant. Model Development There have been numerous attempts to m odel matrix-fiber load absorption using deterministic formulas. Several representativ e equations and approach es were considered in Chapter 3, they are repeated here for convenience6. 2 cosh 2 cosh 1 ) ( l z l A E z Pcox cox f m f Cox shear-lag y r z zf2 ) ( Kelly-Tyson Yielded Matrix f b y bs s d s d s x 4 4 ) ( Bi-linear Model We have previously eliminated the Cox shear -lag approach for load absorption evaluation as overly simplistic and unrealistic. In pa rticular, it does not allow for fiber-matrix debonding and suggests infinite matrix stress cap acity at the fiber end. Likewise, the yielded matrix and perfect bonding assumptions employed in the Kelly-Tyson equation cannot be supported for most fiber-matrix systems, even though there is substantial evidence indicating that an interphase materi al exists adjacent to the fiber surface with mechanical properties different from the bulk ma trix. Specifically, there is contradictory (dissonant) evidence demonstrating that the interphase modulus/yield strength may be 6 The equations given here represent the essential na ture of the analysis indicat ed and not necessarily the model, per se, used to simula te load absorption or SFCF. PAGE 427 either greater than [123,250-252] or less than the bulk matrix [refs.]. The Henstenburg bi-linear approach does allow for fiber-mat rix debonding but require s linear load-uptake on the unbonded and bonded sections (Figure 5 of [6]), which cannot be completely justified and consequently introduces an additi onal uncertainty into any prediction. In this case, the uncertainty is both epistemi c (the debonding/propagati on criteria are not certain) and aleatory (e.g., inherent vari ations in carbon fiber surface energetics and topology naturally give rise to debonding uncertainty). Bi-modal Model We will use a bi-modal model as the basis for our SFCF simulation. In particular, there will be two distinct load absorption re gions (cf. Figure 3.27). The mechanisms for load-uptake on the two different sections will be discussed in more detail below as they relate to a non-deterministic analysis of a SFCF test. Note that the load-uptake in a discontinuous fiber is the mechan ical response to an external stimulus (matrix load/strain) and, as such, the Henstenburg bi-linear fibe r load-uptake model is essentially a response surface prediction inferred from previous models (e.g., Kelly-Tyson debonding models) and empirical evidence [111-113]. Although the exact mechanism for fiber lo ading may still be unclear [216], the confidence in the magnitude and shape of the load-uptake in the fiber can be increased with the use of Raman shif t spectroscopy [226,229,264] ther eby reducing the subjective uncertainty (i.e., defining the interval). That is, we may be able to follow the fiber load increase but we cannot be certain as to th e contributions of the frictional and residual radial compression components. The Raman shif t frequency is representative only of the surface fiber strain (probing depth <~ 2 m). In combination with possible fiber skin- PAGE 428 core morphology that may bias the Raman shift frequency; the surface strain may misrepresent the axial stress distribution inside the fiber. This is important because a non-uniform strain field may influence the measur ed fiber strength, espe cially if two flaw populations exist in the fiber (viz surface and volume flaws, Ch. 2). Matrix-fiber debonding To utilize a bi-modal model of this type for NDA we must quantify the variability in the factors affecting the load-uptake shape and magnitude (e.g., debond length, unbonded friction, residual stresses, and matr ix mechanical properties). Specifically, there is considerable disagreement (nonspecificity and dissonance) on the criteria for fiber-matrix debonding and debonding propa gation. There are two primary phenomenological approaches for evalua ting incipient fiber-matrix debonding: Strength limited debonding [6] Energy (fracture) crite rion debonding [317.] Empirical evidence does indicate that th e debond length increases with increasing macrocomposite strain [216,300]. Single fibe r composite results by Kim and Nairn [216] indicate that the debond lengt h for isolated fiber breaks increases monotonically with applied matrix strain and that there is signi ficant sample-to-sample variation (Figures 3 and 6 of [216]) even when the results are nor malized for fiber diameter variations (fiberto-fiber diameter variation). That is, there is inherent uncertainty layered on top of the imprecise understanding of the underlying debonding mechanism (i.e. frictional and residual stress contributions). We could us e the total variation in observed debond length at a given matrix strain (normali zed for fiber diameter variation7) to gauge the input vector (friction coefficient, residual stress magnitude, matrix modulus, fiber modulus) 7 This normalization is for an average fiber diam eter (no axial diameter variation is considered). PAGE 429 range. This would provide the interval vect or for the variables that affect the debond length, at least according to this data [216]. Again, the interval vector range, including any possible covariance, is also dependent on the response prediction (Eqn. 6 of [216]). In fact, the range of actual debond lengths is la rger than this set of data [216] suggests. If we simply replace the crisp deterministic variable s used for debonding evaluation with an interval range, the response frequency is not representative of the likelihood; only the range is significant. For the larger anal ysis (SFCF interpretation), knowledge of only the range of debonding lengths does not allo w for a meaningful simulation as the debonded length on any given fiber fragment in fluences the load uptake and eventually fragmentation saturation. In other words, the range of debonded lengths for a given composite system at a particular matrix (far field) strain depends on the interphase and matrix mechanical properties. That is, we must avoid specifying interphase mechanical properties without supporting evidence b ecause any covariance with our objective measure (interfacial shear st rength) will confuse the NDA. The results in [216] indicate a coefficient of variation (CV) in debonded length at a given strain (actually binned data) of great er than 30 percent, which appears to be increasing slightly as the matrix strain in creases (Figure 4.3). The variation in debond length for an AS4-epoxy system is apparently bi-modal (two populations), while that for an E glass-epoxy system appears more normally distributed (Figure 6 of [216]). The root cause of the underlying variability in debond leng th at a given matrix strain is not known nor is a possible explanation offered [216]. If we assume, however, that a particular set of data is representative of the larger population of debond lengths and further that we know its PDF (e.g., normal), then we can insert this information into our non- PAGE 430 deterministic analysis. Of course, we must be mindful of the assumptions involved here as they may limit our confidence and bias our predicted response. For example, the Kelly-Tyson yielded matrix assumption suppos es that there is perfect adhesion (no debonding) and that the matrix is elastic-perfectly plastic, wh ich when there is discernible fiber-matrix debonding leads to a gross overes timate of the interpha se load transfer capacity. Interphase Control Diagram The variability in fiber-matrix debonding is but one of many f actors affecting SFCF interpretation. In an ideal single fiber com posite, the fiber sits in an expansive matrix, one that is many times larger than required to transfer load across the interface into the fiber. The idea is, of course, to isolate the ma trix-fiber interface from external influences. That is, eliminate all farfield influences (e .g., clamp effects, free surfaces, and fractures on adjacent fibers) except the uniform matrix st rain. The local and farfield stresses in ideal single fiber composites are obviously fa r less complicated than in macrocomposites; however, the matrix-fiber physical-chemical in teraction is still complex and irregular. The various factors that aff ect interphase morphology, chemistry, and mechanical properties were discussed previously (Chapter 3). The major contributors to interphase form ation, recognizing that defining the extent and characteristics of an interphase is difficu lt, are tabulated in Table 4.1, which we will call an interphase control diagram8. Each row in the ICD repr esents a controllable input parameter (e.g., matrix chemistry and car bon fiber surface treatment) in composite fabrication. The list is not exhaustive, but we assert that it does include the factors that 8 This interphase control diagram echoes n2 charts used to track interf ace specifications and controlling authority in complex aerospace systems. PAGE 431 affect interphase formation, at least to an extent that can be sensibly measured. The columns in the ICD are the same as the rows thus allowing for covariance indications. The intersection of the various input parame ters (grid square) contains information regarding the trending and influence for that input parameter pair on the interphase. The diagonal of the chart provides information on the input parameter measurement method, parameter variability, and confidence. The t ypical ensemble effect of the various input parameters is given at the bottom of each column. Inspection of the ICD reveals that, in ge neral, the variability of interphase properties is high and co nfidence in these values is low (a consequence of dissonance and nonspecificity) as compared to the input pa rameters, which can be assessed much more confidently and readily. For example, we know, with high confidence, the matrix chemistry prior to composite processing, but the interphase chemistry post-processing is substantially affected by the carbon fi ber surface (i.e., surface energy and surface treatment) and also macromolecular entropic effects [123]. Conseque ntly, the interphase chemistry is less well understood and it is cl early dependent on composite system inputs. Note also that still another functional laye r exists for SFCF test evaluation that will further increase variability a nd decrease confidence in the de rived properties. That is, any empirical difficulties or an inability to simulate accurately the underlying physical process (epistemic uncertainty ) of the single fiber fragment ation process (i.e., the NDA) will increase prediction uncertainty over and above actual physical variation. Property covariance Recall that any interphase property covariance confounds a simple linear combination response surface prediction, and furthermore, an NDA that does not consider property covariance cannot claim to re present the actual physical problem. It is PAGE 432 clear then that any NDA parame ter covariance should consider ed. There are a number of options for handling property covariance in NDA. In one approach, property (A) covariance with properties B and C are identified. The range for variables B and C will be established by the relative relationship of the sample A draw to its mean, Ax, and standard deviation, A (Figure 4.4). Another approach would be simply to ignore any property covariance. The first approach is em ployed here for the cova riance of interfacial shear stress and frictional shear stress. The remaining parameters are treated independently, although modification of the mode l described below is certainly possible. Model Definition We have previously described the genera l approach for modeling load absorption with a bi-modal model. In this section, we will provide the equations and methodology for simulating the SFCF test. In particular we will define the fiber-matrix debonding and subsequent debond propagati on criteria along with the process for assigning fiber strength (Poisson point process) on the discrete fiber fragments. We will identify the assumptions (invariable and epistemic uncertainty) necessary to formulate a mathematical model of the fiber fragmentation process. Additionally, we will classify the variables in the model formulation as either certain or uncertain variables and provide (estimate when necessary) their ranges and distributions. Debonding prediction In light of considerable evidence (biref ringence and FEA), it is clear that a representative SFCF model must at the very l east allow for the possibility of fiber-matrix debonding. What is less clear, and here we must mark this up to epistemic uncertainty, is the nature of the fiber-matrix debonding proces s. In this work, we choose to model the PAGE 433 initial fiber-matrix debonding as arising from an energy balance of fiber fracture and the newly created fracture surf aces at the interface [216]. d d f f f L d f f fL r r dx x G r G rd 2 2 22 2 0 2 Eqn. 4.4 Gf is the energy release rate for an isolated fiber break, Gd (x) is the energy release rate per unit length for a fiber matrix debond, Ld is the debond length (on both sides of the fracture), and f and d are the fiber and interface fracture toughness, respectively. The critical assumption here is that this fiber fracture is isolated from other fractures and stress concentrators, which as Hedgepeth and others have shown [152-155] is not realistic for macrocomposites or near fibe r discontinuities. Note that transverse microcracking at the fiber break is not modeled here, alt hough this phenomenon is certainly common to numerous glass fiber-epoxy matrix systems that exhibit exceptional fiber-matrix cohesion. The energy balance equation, Eqn. 4.4, can be rearranged to provide the fibermatrix debonding toughness (d ) in terms of parameters that are known, or can be estimated (e.g., the effective friction coefficient on the unbonded section, f) (ref. 19 of [216]). d f f d f f d f f d f a f dL r L r r L r L E r2 3 1 2 1 1 122 Eqn. 4.5 where is the far-field stress on the fiber (residual and remote applied stress 0, Eqn. 4.8), Gf in Eqn. 4.4 is given by a f fE r G2 and PAGE 434 2 3 1 ln 1 1 2 1 4 1 1 22 f f f m a f f m f a m aG G E E E E Eqn. 4.6 Eqn. 4.7 Ga and Gm are the fiber axial shear and matrix shear moduli, Ea and Em are the fiber axial and matrix Youngs moduli, and f is the effective fi ber volume fraction. is the aforementioned shear-lag parameter as derived in [141]. Recall that the shear-lag parameter plays a criti cal role in the rate of loaduptake into the fiber and, as is evidenced by Eqn. 4.6, is a strong non-linear function of the effective fiber fraction. The Cox shear-lag parameter (Eqn. 3.7), by comparison, has been roundly discounted as inaccurate at low fiber fractions (i.e., si ngle fiber composites). The insertion of into Eqn. 4.5 introduces both epistemic and aleatory uncertainty into our modeling effort. For one, there are a number of formulae for the shear-lag parameter (nonspecificity) and dissonant empirical results (i.e., the apparent extent of the in terphase region is affected by the testing method). Additionally, the mechani cal response in the matrix at the fiber break extends farther into the matrix than many investigators ha ve determined the thickness of the interphase to be An appropriate value for the matrix properties is also in question as the matrix exhibits both radial variation, a consequence of energetic and entropic factors, and non-reduc ible uncertainty associated with matrix/fiber chemistry and fiber surface energy. In this model, is calculated by using a Bessel-Fourier series m m t t a a a m a m m t t m t a a m a m m t t a m aE E E T E E E E E E E E E 1 1 2 1 1 2 1 1 22 0 PAGE 435 to estimate f (0.16%) [229]. An alternative method employing back fitting using Raman shift spectroscopy has also been demonstrated [226]. The interfacial fracture toughness ( d) can be found from debonded length versus applied strain plots. Using an optical bi refringence method to identify the debonding extent ( Ld in Eqn. 4.5) for a given far-field stress (e.g., Figures 3-8 of [216]) one can determine d to minimize the predicted error. By this approach Kim and Nairn [317] estimate d to be 220 J/m2 for an epoxy/AS-4 graphite system. Once the interfacial fracture toughness is known, Eqn. 4.5 can be used to predict the fiber-matrix debond length Ld, the essential condition fo r fiber fragmentation in the SFCF test, at a given remote stress 0,. Obviously, the interface fractur e toughness is integral to the fragmentation process, its va lue in the modeling is corres pondingly important. Again, the uncertainty in the interfacial fracture t oughness is both reducible and non-reducible. A simple fitting estimate using the data in [ 216] indicates that the interfacial toughness would have to vary by +19%/-38% to cover the range of observed debond lengths if one assumed that the interfacial toughness was the only source of uncertainty in the energy balance model (Eqn. 4.5). Recovery and plateau length Once the interfacial fracture toughness is estimated, the length of the interfacial debond can be predicted at a fibe r break during the fragmentation process. This provides the length of the first zone in the load-uptak e bi-modal model (Figure 4.5). The length of the second load-uptake region (in this model, Lyield) is the difference between the far-field fiber stress and the average fiber stress afte r frictional loading on the debonded section. PAGE 436 f yield y f f debd L d L 4 4 Eqn. 4.8 In this approach, both th e initial yielded length Lyield,init and initial debonded length, Ldeb,init, at fiber fracture vary with the fractur e strength on a fiber segment. Here, the initial yielded length, Lyield,init at a fiber fracture is held c onstant throughout the simulation while the interface fracture front propagates in ward to satisfy the far-field stress. The debonded length for the ith load increment ( Ldeb,i) is determined from Eqn. 4.5 where Lyield = Lyield,init and is known (Eqn 4.7). The simula tion tracks the change in Lyield and Ldeb as the remote stress is increased. The total recovery length ( Lrec = Lyield + Ldeb) at each fiber break enables the determination of the plateau length (fiber axial stress is fully developed, on each fragment and on the overall gage length (typically 25 mm for dogbone samples [6]). The overall plateau length is an important measure for the simulation because the fiber strength prediction relies on a Weibull le ngth scaling approach tied to the plateau length. At each incremental loading step, th e recovery length on each fiber segment is determined, and the total plateau length is n i rec gage plati L i L L1 (n total fragments). Fragmentation saturation occu rs on a particular fiber fr agment when the predicted recovery length equals the fiber segment le ngth. Consequently, no further fragmentation is possible. The SFCF test simulation is co mplete when the total plateau length reaches zero (fragmentation saturation). PAGE 437 Simulation details Single fiber fragmentation tests are typical ly conducted incrementally to allow for fragment and debond length determination (e.g., via birefringence [317]) between loading steps. In this non-deterministic analysis, a looping routine allows for the incremental loading of the single fiber as well as a means; to check fo r fiber fragmentation (local Weibull strength scaling) and location, to predict fiber-matrix debonding (e.g., energetic fracture criterion, Eqn. 4.4), to track debond propagation with increasing remote stress, and to identify fragmentation saturation. The debonding criterion used in this analysis is one of many possible options (e.g., no debondi ng, an energetic criterion, or a strengthlimiting criterion). The debonding criteri on or energy absorbing mechanism (e.g., transverse matrix microcracking, or nonlinear matrix yielding) at th e fiber break in this simulation can be modified to examine the influence of the debonding criterion on SFCF testing and interpretation. The approach taken here for predicting de bonding length at fiber fragmentation is supported by empirical [317] and analytical work [229]. While the nature of the debonding criterion is important, its real influence for SFCF si mulation is evident in the debonded length prediction, Ldeb (Eqn. 4.5). The uncertainty in the debonding criterion is transmitted to the simulation in the in terface fracture toughness parameter d (-38% to +19%, Figure 4.3). The uncertainty in the in terface fracture toughness arises because of both the inherent variability in the mechan ical properties (matrix and fiber) and the assumptions used to obtain Eqns. 4.5.7. That is, possible invariab le uncertainty and nonspecificity in formulation of Eqns. 4.5.7. For this model, we will assume that the formulations are correct and therefore, in order to define the variability in d, we must PAGE 438 quantify the range and distribution of the mech anical properties. In our simulation we have bracketed the observed variation in mech anical properties of the fiber and matrix based on a review of the literature9. Note that the carbon fibers ultimate tensile strength is handled separately (W eibull strength scaling). Treatment of the debonding criterion and fibe r strength are, of course, critical to any SFCF simulation, as are the particulars of debond propagation afte r initial debonding. The details of the iterative procedure fo r determining the current debonded length, Ldeb, from the initial yielded recovery length, Lyield,init, and the current far-field stress were described previously. The key point here is that the recovery lengt h adjacent to a fiber fragment length increases with each su ccessive loading step, a strength-limited propagation criterion, until it inte rsects with the recovery le ngth from the opposite end of the segment. Consequently, the recovery leng ths on either end of a fiber segment are not likely to be the same. This debond propagation approach simplifie s the stress field near the fiber break and neglects any possible near-break influe nce. Furthermore, the uniform frictional loading assumption on the de bonded section (Figure 4.5) ca nnot be fully supported. Raman shift spectroscopy m easurements on fragmented graphite-epoxy single fiber composites [229] suggest a non-uniform load ing over this debonded length (non-linear load-uptake in the fiber). The disparity be tween the present simulation and the observed behavior, as well as the means to repres ent the process is suggestive of model nonspecificity. We could reduce this uncerta inty if we were cer tain of the actual mechanisms controlling fiber fragmentation; however, there is still disagreement as to 9 Restricted to High Modulus PAN based carbon fi bers and epoxy matrices for tractability and clarity. PAGE 439 whether the debonding process is controlled by a strength or fractur e energy crierion. In this model we will make use of the observati on that fiber fragmentation is a consequence of the cumulative fiber load and consequently the precise mechanism for load absorption is not as critical as the magn itude of the fiber load itself. Fiber strength predictions are enabled by the combination of a scaleable Weibull strength distribution and a Poisson dist ribution (after [hen stenburg..1989]), which determines the probability that a random even t will occur on a certain gage length or time interval. The Poisson probability density f unction (Eqn. 4.9) is defined by the Poisson parameter, M ( A ). ... 2 1 0 ) ( Pr k e k A M k A NA M k Eqn. 4.9 where Ads dx s x A M ) ( We treat the intensity, as a function only of the fiber stress, s When the intensity takes the form of the classic power-law relations hip (Eqn. 4.10), the We ibull scaling strength prediction (Eqn. 4.11) follows directly. 1 0 0 0 s s L s s Eqn. 4.10 0 0exp 1 s s L L L s F Eqn. 4.11 One advantage of this approach is that the program can adjust the stress increment automatically (simulation iteration) in accord with the likelihood that a flaw exists on the PAGE 440 plateau length. This quickens the simulati on time and lessens the discretization error associated with an otherwise analog flaw strength spectrum and loading sequence. The non-deterministic simulation (Matla b) for a single fiber composite fragmentation test (Appendix H) includes options for adjusting the COV for each parameter individually or collectively. The absolute magnitudes of the nominal fiber and interphase properties were chosen to ground the simulation to previous deterministic SFCF analyses (Table 4.2). Multiple simu lations with the same parameter conditions (mean and standard deviation) provides a means to assess the influence of parameter variability on the simulation response (fragment length distribution, FLD). Results A series of single fiber fragmentation nondeterministic simulations (Tables 4.34.5) were conducted using th e purpose written Matlab function (sfcf_sim.m). In one series of simulations, each of the mechanical parameters was assigned a unique average value and a common coefficient of variati on (CV), except for the interfacial fracture toughness, d, and the effective friction coefficient, f, which were held constant in an attempt to isolate the effects of the debonding criterion. Fragment length histograms (n = 25) for four single-tail parameter distributions (CV = 0.0001, 0.05, 0.10, 0.25) are presented in Figure 4.6 (see Table 4.3 for simulation statistics). The fitted normal distribution for the bin-averaged fragment length distribution of the nominal (CV = 0.0001) cas e is provided in Figure 4.7. Additional statistical information for this simulation set is provided in Figure 4.8. The histograms and fitted normal distributions for SFCF simulations using unique fiber strength populations (Table 4.4) are pl otted in Figure 4.9. Table 4.5 describes a series of non- PAGE 441 deterministic analyses for different average matrix yield strengths, y. Fragment length histograms and normal distribution approximati ons for these simulations are shown in Figure 4.10. Discussion Obviously, it is unrealistic to expect the m echanical parameters (Table 4.2) to share the same CV; however, it is a reasonable first test for the simulation program. These Monte Carlo simulations, which use normally di stributed certain variables, indicate the ensemble sensitivity of the response prediction. It is difficult to discern a visual difference between the histograms in Figure 4. 6. There is, however, increasing positive skewness as the input variability increases (Table 4.3), whic h is also crudely suggested by an increasing maximum fragment length Additionally, the fragment length CV for each simulation (n=25) generally tends to increase. The significant result of these simulations is that, even with large mechanical property variability (i.e., CV = 25%), there is virtually no change in the average fragment length (~ 3.5%). If we had conducted, for instance the CV = 0.25 experiments manually, and we were able to correctly identify the debonded length, a not-altogether simple task, we would not be able of identify the discrete contribution of the interf acial shear stress to the fragment length distribution, which is ultimately, the objective of the SFCF experiment. The small response variability also indicates that the SFCF experiment is not sensitive enough to the range of parameter va riability explored in the NDA to identify the real changes in interfacial mechanical properties. A second set of simulations (Table 4.4 and Fi gure 4.9) looked at th e effect of larger changes in the fibers Weibull modulus (i.e., =3, 5, 8). In these simulations, CV = 0.1. PAGE 442 It is seen in Figure 4.9 that as the fibers Weibull modulus increases (tig hter strength distribution), the average fragme nt length and standard deviat ion decrease. As the fiber strength distribution narrows, there are more fiber segments created on a given stress range. The result is that these segments have similar initial debond lengths and consequently at fragmentation saturation the dist ribution is less skewed. A real change in fiber statistics could very well be confused with a change in the interfacial shear strength, which underscores the need for accurate a nd reliable fiber strength statistics. The direct influence of the interfacial shear strength on the response prediction was probed using a set of simulations with fixe d input parameters (Table 4.2) and three different interfacial shear stress values (y = 25, 50, and 100 MPa). The magnitude of the interfacial shear strength cl early influences the response prediction in this analysis (Figure 4.10). The interfacial shear stress, y, simultaneously affects both the fiber axial stress and the segment plateau length (see dashed lines in Figure 5). The only nondeterministic variable in thes e particular simulations was the fiber strength (Weibull streng th scaling) and location (uniform distribution on plateau length). The fragment length average increases by 71% when the interfacial shear strength is doubled and decreases by 33% when the interfacial shear strength is halv ed. It is evident that as the interfacial shear stress decreases there is a marked increase in the recovery length on each segment and consequently, a d ecrease in the plateau length. The number of fragments naturally decreases as the recovery length increase s (Table 4.5). Finally, the FLD tightens (standard deviation decreases) as the load-uptake rate increases. This suggests that for any particular average inte rfacial shear strength, the breadth of the PAGE 443 distribution indicates the variab ility in the shear strength value itself, provided there are sufficient samples. Conclusions A probabilistic nondeterministic model has been created for analyzing the micromechanics and uncertainties associated with the SFCF test. The NDA indicates that the SFCF test is incapable of identifying larg e variations in mechan ical properties (singletail CV = 0.25) when the variations are ac ross the range of input variables, excluding fiber-matrix interfacial toughness. In particular, the fragme nt length distribution becomes very broad and positively skewed with increas ing input parameter variability (CV for the FLD also increases). In contrast, this bi-modal NDA predicts a di scernible decrease in fragment length averages as the fibers Weibull modulus increases (i.e., = 3, 5, 8). As the fibers Weibull modulus increases there is a sli ght decrease in the FLD breadth, although inclusion of certain variability (CV = .10 simulated) mitigates this response. The change in the fiber length distribution associated w ith large Weibull modulus variations can be mistaken for a change in the interfacial propert ies. It is therefore critical that accurate and reliable fiber strength statistics be availa ble that are appropriate for the gage lengths under consideration (e.g., ~1-2 mm). NDA simulations for various interfacial shear strengths (y = 25, 50, and 100 MPa) and no other parameter variation (CV = 0.01%) indicate that with decreasing interfacial shear strength there is the combined eff ect of an increasing recovery length and decreasing fiber axial loading. This combina tion causes rapid broadening of the fragment length distribution and a large upward sh ift in the fragment length average. PAGE 444 The NDA tool developed here can be used to track the influence of inherent mechanical property variability and micr omechanical modeling details (e.g., debonding criterion and load-uptake) on the possible outc ome of the decidedly stochastic SFCF test. The experimentalist is cauti oned with regards to SFCF interpretation as simulations indicate that disparate causes can produce similar response predictions. Recommendations The current bi-modal (debonding and yiel ding) probabilistic non-deterministic model should be modified to represent alternate fiber-matrix debonding criteria debond propagation schemes, and fibe r strength statistical treatments. A series of SFCF tests that systemati cally varied the primary fiber-matrix interphase influencing factors (Table 4.1) could be undertaken to bound the uncertainties in the interphase properties. These results could then be back-fitted with the current nondeterministic simulation to determine the range of the interphase properties. Simulation results clearly indicate that the experimentalist cannot rely solely on SFCF saturation fragmentation results because in some instances the test appears insensitive to parameter uncertainty a nd consequently unable to discern small changes in the composite fabrication proce ss. The current model could be used to predict the response (SFCF results) for a particular composite process variable beforehand, thus allowing the investigat or foreknowledge of the outcome and an interpretive tool to analyze th e results against expectations. PAGE 445 Table 4.1 Interphase control diagram indicating interaction of major composite parameters on interphase properties. M: Measurement method; V: Variability; C: Confidence. PAGE 446 Table 4.2 Simulation parameters for si ngle fiber composite fragmentation NDA. Parameter Symbol Units Mean Value Fiber Diameter df mm 5.7 Fiber Axial Modulus Ea GPa 231 Fiber Transverse Modulus Et GPa 40 Fiber Axial Shear Modulus Ga GPa 20 Fiber Axial Poisson Coefficient a 0.2 Fiber Transverse Poisson Coefficient t 0.25 Fiber Axial CTE a ppm/K -0.7 Fiber Transverse CTE t ppm/K 10 Fiber Weibull Modulus GPa 3 Fiber Weibull Scale Strength s0 GPa 6 Fiber Weibull Reference Length L0 mm 10 Matrix Modulus Em GPa 2.6 Matrix Shear Modulus Gm GPa 0.97 Matrix Poisson Coefficient m 0.34 Matrix CTE m ppm/K 40 SFCF Gage Length Lgage mm 25 Temperature Differential (Tstress free -Tcure) T K -100 Effective Fiber/Matrix Friction Coefficient f 0.01 Shear-Lag Parameter 0.06207 Fiber Fracture Toughness f J/m2 10 Fiber-Matrix Interfac ial Fracture Toughness d J/m2 90 Matrix Yield Strength y Mpa 50 Frictional Shear Stress f Mpa 16.7 Parameters extracted from [6] and [317]. PAGE 447 Table 4.3 SFCF simulation parameters and fragment length statistics from a NDA for single-tail normal parameter uncertainty. Simulation Designation Parameter Units sfcf_25_0.0001_3 sfcf_25_0.05_3 sfcf_2 5_0.10_3 sfcf_25_0.25_3 No. of Simulations 25 25 25 25 CV (positive single-tail) 0.0001 0.05 0.1 0.25 Weibull Strength Modulus 3 3 3 3 Results Average (n=25) mm 0.580 0.596 0.602 0.589 Standard Deviation (n=25) mm 0.464 0.473 0.493 0.502 CV 80.1 79.2 82.3 85.3 Maximum (all fragments) mm 1.99 2.155 2.288 2.322 Minimum (all fragments) mm 0.024 0.025 0.029 0.019 Skew (all fragments) 1.406 1.660 1.751 1.883 No. of Fragments (avg.) 43.7 42.3 42.0 43.0 Simulation designation: sfcf_xx_yy_zz; xx = no. of simulations; yy = para mater uncertainty, CV; zz = fiber Weibull strength modulus. PAGE 448 Table 4.4 SFCF simulation parameters and fr agment length statistics from a NDA for di fferent fiber Weibull moduli and single-ta il normal parameter uncertainty. Simulation Designation Parameter Units sfcf_25_0.10_3 sfcf _25_0.10_5 sfcf_2_0.10_8 No. of Simulations 25 25 2 CV (positive single-tail) 0.1 0.1 0.1 Weibull Strength Modulus 3 5 8 Results Average (n=25,25,2) mm 0.602 0.549 0.521 Standard Deviation (n=25,15,2) mm 0.493 0.45 0.408 CV 82.2 82.1 78.3 Maximum (all fragments) mm 2.288 1.987 1.760 Minimum (all fragments) mm .0292 .0173 .0065 Skew (all fragments) 1.75 1.55 1.07 No. of Fragments (avg.) 42.0 46.3 48.0 Simulation designation: sfcf_xx_yy_zz; xx = no. of simulations; yy = paramater uncertainty, CV; zz = fiber Weibull strength modulus. PAGE 449 Table 4.5 SFCF simulation parameters and fr agment length statistics from an NDA for different fiber Weibull moduli and single-t ail normal parameter uncertainty. Simulation Designation Parameter Units sfcf_5_0.0001_25 sfcf_2 5_0.0001_50sfcf_2_0.0001_100 No. of Simulations 5 5 5 CV (positive single-tail) 0.1 0.1 0.1 Weibull Strength Modulus 3 3 3 Interfacial Shear Strength MPa 25 50 100 Results Average (n=5) mm 1.107 0.645 0.429 Standard Deviation (n=5) mm 0.131 .0513 .0159 CV 74.8 76.4 68.4 Maximum (all fragments) Mm 3.97 2.709 1.313 Minimum (all fragments) mm 0.004 0.008 0.001 Skew (all fragments) .907 1.08 .627 No. of Fragments (avg.) 22.8 39.3 58.4 Simulation designation: sfcf_xx_yy_zz; xx = no. of simulations; yy = pa ramater uncertainty, CV; zz = matrix interfacial shear strength. PAGE 450 Figure 4.1 Hypercubic approximations for an arbitrary two-dimensional (x1, x2) response solution. x2x1 conservative hypercube (2-D) smallest hypercube (2-D) exact solution set conservatism PAGE 451 Figure 4.2 Evolution of model predicti on uncertainty with increasing process understanding. Epistemic bounds Aleatory bounds Process_1 Process_2 Average Increasing Process Understanding Process Change Subjective Uncertainty Eliminated Inerval Analysis Fuzzy/Subjective Probabilistic PAGE 452 0 2 4 6 8 10 12 14 16 18 0.00.51.01.52.02.53.0 Applied Strain (%)Debond Growth (fiber dia. ) AS-4 graphite in epoxy Bin average, bin size = 0.125% Std. Dev. of bin Figure 4.3 Average debond lengths for AS-4 graphite fibers in epoxy (new breaks only) as a function of applied strain. Data from [317]. PAGE 453 0 0.2 0.4 0.6 0.8 1 1.2 PropertyNormalized Property Valu e draw A range draw B range draw C range A draw B draw C draw Figure 4.4 NDA treatment of c ovariant property variation. PAGE 454 L y ieldLdebond Lrecover y Ld, init Ld, propa g ation Initial debond length from Eqns. 4.5-4.7 Propagation length required to build current fiber stress level Fiber break at ,break yield friction ,break Remote stress 0 Remote stress 0 Fiber axial stress (FAS) Lplateau Figure 4.5 Bi-modal fragmentation debonding and load-uptake model. PAGE 455 0 20 40 60 80 100 120 140 00.00050.0010.00150.0020.00250.0030.00350.0040.0045 Fragment Length (m)Frequncy (histogram) sfcf_25_0.0001 sfcf_25_0.05 sfcf_25_0.10 sfcf_25_0.25 Figure 4.6 Fragment length hi stogramsat saturation for simulated parameter uncertainty. See Tables 4.2 and 4.3 for NDA simu lation parameters and statistics (sfcf_xx_yy; xx = no. of simulations; yy = paramater uncertainty, coefficient of variation). PAGE 456 0 20 40 60 80 100 00.00050.0010.00150.0020.00250.0030.00350.0040.0045 Fragment Length (m)Frequncy (histogram)0 200 400 600 800 1000Frequency (distribution) sfcf_25_0.0001 Normal dist. Average standard deviation Figure 4.7 Normal distri bution approximation and frag ment length histogram at saturation for nominal (zero CV) SFCF si mulations (n=25). See Tables 4.2 and 4.3 for simulation parameters and fragment length statistics (sfcf_xx_yy; xx = no. of simulations; yy = paramater unc ertainty, coefficient of variation). PAGE 457 0.0 0.5 1.0 1.5 2.0 2.5 sfcf_25_0.0001_3sfcf_25_0.05_3sfcf_25_0.10_3sfcf_25_0.25_3Length (mm) Mean Fragment Length Maximum Fragment Length Figure 4.8 Average saturation fragment length and maximum fragment length as a function of input parameter uncertainty (sfcf_xx_yy; xx = no. of simulations; yy = paramater uncertainty, coefficient of variation). PAGE 458 0 20 40 60 80 100 00.00050.0010.00150. 0020.00250.0030.00350.004 Fragment Length (m)Frequncy (histogram)0 200 400 600 800 1000Frequency (distribution) sfcf_25_0.10_3 sfcf_25_0.10_5 sfcf_2_0.10_8 sfcf_25_0.10_3 sfcf_25_0.10_5 sfcf_2_0.10_8 Figure 4.9 Fragment length histogram and normal distribut ion approximation for singletail 10% CV SFCF simulations with va rious fiber Weibull strength moduli. (sfcf_xx_yy_zz; xx = no. of simulati ons; yy = paramater uncertainty, coefficient of variation; zz = fiber Weibull strength modulus) PAGE 459 0 2 4 6 8 10 12 14 16 18 00.00050.0010.00150.0020.00250.0030.00350.0040.0045 Fragment Length (m)Frequency (histogram)0 100 200 300 400 500 600 700 800 900 1000Frequency (distribution) sfcf_5_0.0001_25 sfcf_5_0.0001_50 sfcf_5_0.0001_100 Normal, sfcf_5_0.0001_25 Normal, sfcf_5_0.0001_50 Normal, sfcf_5_0.0001_100 Figure 4.10 SFCF fragment length histograms and normal distribution approximations (NDA simulation) for single-tail 0.01% CV parameter uncertainty and various interfacial shear stress va lues. See Table 4.5 for simulation parameters and statistics (sfcf _xx_yy_zz; xx = no. of simulations; yy = paramater uncertainty, coefficient of variation; zz = interfacial shear strength). PAGE 460 APPENDIX A WEIBULL-GRIFFITH STATISTICS Weibull first developed his distributio n function in 1939 [3]. The Weibull distribution function is easily developed from first principles in statistics. A concise derivation in English was given by Weibull himself in 1951 [15]. This derivation is reproduced here as well as a more rigorous derivation advanced by Taylor and Karlin (1980) and utilized by Henstenburg and Phoenix [6]. Weibull began with the general formula for all Cumulative Distribution functions (CDFs) (s)) (1 = F(s) exp A.1 where F(s) is the probability of choosing an individual S from the total population such that P(Ss)=F(s). The usefulness of Eqn. A.1 hinges on the relationships (1-P)n = exp (-n (s)) Eqn. A.2, and (1-Pn) = (1-P)n Eqn. A.3, where n can be thought of as the number of lengths and Pn is the failure probability for n links Now, if the CDFs for each link are described by Eqn. A.1 then combining A.1 through A.3 yields (s)) (-n 1 = Pn exp A.4 The conditions for (s) are that it be a positiv e, non-decreasing function and (s)|s<0 =0. Weibull chose the simple function s ) s (s = (s)0 u A.5 PAGE 461 where is the modulus or shape factor, s0 is the scale parameter and su is the lower bound (su0). Substituting A.5 into A.4 yields ] s ) s (s [ 1 = F(s)0 uexp A.6 This is the original form for the Weibull CDF [3]. A more rigorous derivation was advan ced by Taylor and Karlin (1980) and illuminated by Henstenburg and Phoenix [6]. Assuming that the flaws that cause carbon fiber failure are described by a Poisson point process, it can be shown that the mean number of points falling in area :[0,L]x[0,s] (see Figure 2. 4 and text) is given by s d x d ) s ( = M(A)A A.7 or A = M(A) A.8 where A is the area swept by [0,l]x[0xs ] in two dimensional state space and is a constant. (x',s') is the intensity function and M(A) is the Poisson parameter. A fiber will fail at a stress level s, if within the region :[0,l]x[0,s] N(A)1, where N(A) is the number of events in that space. Furthermor e the probability that the number of flaws in is equal to an integer k is given by (-M(A)) k! ) M(A = k] = Pr[N(A)kexp A.9 The probability that failure will occur in a fiber of length L at stress s (area ) is equal to one minus the probability of non-failure -N(A)=0or s d ) s ( L 1 = L) F(s,s 0] exp[ A.10 This is essentially Eqn. A.4. The restrictions on the intensity function are PAGE 462 . = s d ) s ( 0, ) s ( 0, < s when 0 = ) s (s 0 s snt i lim Literally, the intensity must be a positive, non-decreasing boundless function of s. One can assume a simple power-law relationship for the intensity, ) s s ( L s = (s)1 0 0 0 A.11 where L0 is the reference length. Co mbining Eqns. A.11 and A.10 gives ] ) s s )( L L ( [ 1 = L) F(s,0 0exp A.12 or ] ) s s ( L L [ 1 = (s) Fii 0 0 iexp A.12a where the subscript/superscript i refers to subpopulations i if there are multiple populations. Note that the Wei bull distribution is actually one of three general Extreme Value distributions (see discussion in text a nd reference [54]) and is not just simply a convenient analytical form, although it is that as well. The general form for the non failure of a specimen with two concurrent subpopulations is given by (s)) F (s))(1 F (1 = F(s)) (1B A A.13 where F is the total distributions failure probability. FA is the failure probability for subpopulation A if it alone were th e total population. Likewise FB is the failure probability for subpopulation B. Rearranging A.13 yields (s)) F (s))(1 F (1 1 = F(s)B A A14 PAGE 463 After substituting the Weibull dist ribution Eqn. (A.12) for both FA and FB in Eqn. (A.14) and with some algebra one arrives at ) ) s s ( L L () ) s s ( L L (1 = F(s)B AB 0 0 A 0 0 exp exp A.15 The failure probability for an exclusive di stribution is a weighted average of the individual populations: 1 = with (s) F = F(s)i i i i i A.16 For two distinct populations this gives (s) F + (s) F = F(s)B B A A A.17 Inserting Eqn. A.12 into A.17 gives the exclusive Weibull distribution )] ) s s ( L L ([1 + )] ) s s ( L L ([1 = F(s)B AB 0 0 B A 0 0 A exp exp A.18 Rearranging and recalling that A + B =1 one gets ) ) s s ( L L () ) s s ( L L (1 = F(s)B AB 0 0 B A 0 0 A exp exp A.19 A partially concurrent distri bution has the general form (s))] F (s))(1 F (1 [1 + (s) F ) (1 = F(s)B A B A B A.20 where the first term on the RHS is a weighted Weibull distribution with front factor (1B) and the second term is the previously di scussed concurrent distribution multiplied by the weighting factor B. Population A exists in all specimens and B<1. If B=1 then the concurrent distributi on is recovered as the first term of Eqn. A.20 disappears. In other words, Eqn. A.14 is a special case of a partially concurrent distribution with B=1. As before one can insert the We ibull distribution or any othe r function that satisfies the PAGE 464 conditions for F(s); that is a positive, non-decreasi ng function. After some algebra Eqn. A.21 is obtained. ] ) ) s s ( L L (1 [ ) ) s s ( L L (+ 1 = F(s)B AB 0 0 A 0 0 B exp exp A.21 Representative graphs for two-parameter, concurrent, exclusive, and partially concurrent distributions, Eqns A.12, A.15, A.19 and A.21 respectively, are plotted in Figures 2.5, and 2.7-2.9. Eqns. A.12, A.15, A.19 and A.21 are commonly referred to as Cumulative Distribution Functi ons (CDFs). A CDF by defin ition is the integral of a Probability Density Function (PDF). The PD Fs for Eqns. A.12, A.15, A.19 and A.21 can be found by taking their derivativ es with respect to s and ar e given by Eqns. A.22 A.25. ] ) s s ( L L [s s L L = f(s)0 0 1 0 0 exp A.22 ] ) s s ( L L [ ] ) s s ( L L [ ] s ) s ( L L + s ) s ( L L [ = ) f(sB A B B A AB 0 0 A 0 0 1 B B 0 0 1 A A 0 0 con exp exp A.23 f + f = ) f(sB B A A exc A.24 ) F (1 f + ) F (1 f = ) f(sA B B B B A par A.25 where Fi is given by Eqn. A.12a for subpopulations A and B, and fA and fB are the derivatives of FA and FB with respect to s see Eqn. A.22-. The PDFs described by Eqns A.22-A.25 are asymmetric about their maxima. The maxima in these PDF occur at the inflection points of the CDFs [F(s)=0.5]. The curves are skewed toward low strength values, as is the case for all Weibull-like distribution functions. In contrast, Normal distri butions are symmetric about their mean. PAGE 465 An alternate, and sometimes more enlighteni ng, way to plot a Weibull CDF is by taking the double natural logarithm of both sides and rearranging such that the equation's left hand side (ordinate of the graph) is ln[-L0/L(ln(1-F(s)))]. These types of graphs are called Weibull probability plots. The rearra nged, but equivalent forms, for Eqns. A.12, A.15, A.19 and A.21 are given in Eqns. A.26-A.29. Rearranging A.12 and taking the na tural log of both sides gives )] ) s s ( L L ([ = F(s)) (10 0exp ln ln or ) s s ( L L = F(s)) (1 -0 0ln Taking the natural log again yields s s + ) L L ( = F(s))] (1 [-0 0ln ln ln ln ln Finally, rearranging gives the desired form s s = F(s))] (1 L L [-0 0ln ln ln ln A.26 This is the equation for a straight line with slope and intercept -ln s0 when ln[L0/L(ln(1-F(s)))] is plotted against ln s. The corresponding equations for concurrent exclusive and partially concurrent distributions are be found in a similar manner. The equations are given in Eqns. A.27, A.28, and A.29 respectively. ] ) s s ( + ) s s ( [ = F(s))] (1 L L [-B AB 0 A 0 0 ln ln ln A.27 PAGE 466 )]] ) s s ( L L (+ ) ) s s ( L L ([ L L [= F(s))] (1 L L [-B AB 0 0 B A 0 0 A 0 0 exp exp ln ln ln ln A.28 )]] ) s s ( L L (+ [1 L L ) s s [( = F(s))] (1 L L [-B AB 0 0 B 0 A 0 0 exp ln ln ln ln A.29 Normal distributions are sy mmetric about their mean and their breadth is determined by the standard deviation of the population. The PDF for a normal distribution is given by 0 > x for ] 2 ) (s [ ) (2 1 = ) f(s;2 2 1/2 exp A.30 There is no closed form solution for the CDF of a normal distributi on, the CDF must be found by numerical methods. Confidence interv als for normal distributions are easily obtained by using t-dist ribution tables (1-) which provide a close approximation to the normal distribution confidence interval. As th e number of tests decreases the confidence interval broadens. A lognormal distribution is one, such that if it were plotte d as the frequency against the log of the independent variable, the di stribution would be normal. Equation A.31 gives the PDF for a lognormal distribution. ] 2 ) s ( [s ) (2 1 = ) : f(s2 2 1/2 ln exp A.31 where and are the lognormal mean and lognormal standard deviation. Multiple population lognormal distributions can be easily generated. The process is analogous to PAGE 467 that for multiple population Weibull distribut ions -Eqns. A.13, A.16 and A.21-. A two population exclusive lognormal distribution wa s used by Own et al. [10] to describe carbon fiber failure strength distributions. The PDF is ) : (s f + ) (s; f = f(s)B B B B A A A A A.32 where fA and fB are as in A.31 and A and B are the populations' proportions with A+B=1. Goodness-of-fit (GOF) parameters are ordinari ly used in iterative procedures as measures of fit correl ation. A GOF value of one indicat es an exact fit of equation and data. For linear least square fit the correlation coefficient, R, is given by ] ) ) y ( y (n ) ) x ( x (n [ y x)( ( xy n = R1/2 2 2 2 2 A similar parameter, Q, can be calculated using the equation ) s s ( ) s s ( 1 = Q2 i n =1 i 2 i i n =1 i where si is the measured value, si is the mean and si is the expected value. Given a PDF f(s) and interval limits asb one can use random numbers to generate data that fits the distribution f(s). The distribut ion f(s) can contain only one maxima on the interval chosen for the follo wing procedure to hold. First the maximum on the interval fmax must be identified A random number sR is then generated from a uniform distribution on the interval [a,b]. A uniform distribution is one for which choosing any individual in th e population is equally probable (viz. a uniform distribution is completely random). Another random number fR is generated on the interval (0,fmax). If fR PAGE 468 APPENDIX B COX LOAD TRANSFER The seminal work on load transfer across a fiber-matrix interface was authored by Cox (1952) [105]. In this work, Cox utili zed four major assumptions regarding load transfer from the matrix to a discontinuous fiber: 1. The fibers are embedded in a continu ous solid medium: perfect bonding between matrix and fiber. 2. The matrix strain is homogeneous except for local perturbations caused by fiber load absorption. 3. The matrix and fiber have equal latera l stiffness: one-dimensional assumption. 4. The fiber load, P, is given by dP/dz = H(u-v) where H is a constant, u is the fiber displacement, and v is the matrix displacement in the absence of the fiber. Accordingly; and from the stress-strain relationship (f = Ef f) where Ef-m= EfEm, and Af is the fiber cross sectional area. Rearranging Eqn. B.2 and substituting into Eqn. B.1, along with the strain definition, dv / dz =, gives v u H dz dP B.1 dz du A E Pf m f B.2 f m fA E P H dz P d2 2 B.3 PAGE 469 The general solution to Eqn. B.3 is where Evaluating Eqn. B.4 with th e boundary conditions P(x=0, =0 provides and Substituting Eqns. B.6 and B.7 into Eqn. B .4 and simplifying gives the fiber load on z=[0, In this analysis, the shear lag parameter, cox, controls the rate of load absorption into the fiber. cox itself is a function of the geometric constant H. To evaluate H, Cox [105] approximated the fiber mat spa tial distribution as hexagonal. Hence, where r1 is the mean fiber separation and r0 is the fiber diameter. z S z R A E Pcox cox f m f cosh sinh B.4 2 1 f m f coxA E H B.5 f m fA E S B.6 l l l A E Rcox cox cox f m f sinh sinh cosh B.7 2 cosh 2 cosh 1l z l A E z Pcox cox f m f B.8 0 1ln 2r r G H B.9 PAGE 470 APPENDIX C ROSEN LOAD TRANSFER Rosens (1965) [110] analysis roughly follows Cox orig inal approach (1952) [105] with the added complexity (improvement) of a distinct interphase and effective composite (cf. Figure 3.8). A simple free body diagram for a fiber element gives (cf. Eqn. 3.2) where is the matrix/interfacial shear stress and f is the fiber axial stress. Composite equilibrium in the axial direction likewise providesHere a is the axial stress in the effective medium and is the applied axial stress. The shear strain in the interphase (Rosen uses the term binder), assuming rb-rf is small and uniform shear across this region, is simplyIf we differentiate Eqn. C.3 twice with resp ect to z and make use of the stress-strain relationships, = E and = G we get Now differentiating Eqn. C.2 (solving for da/dz) and, substituting this result along with df/dz from Eqn. C.1, into Eqn. C.4, we get the familiar differential form 0 2 dz d rf f C.1 a a b a f a fr r r r r2 2 2 2 2 C.2 f b f ar r u u C.3 2 21 1 dz d G r r dz d E dz d Eb f b f f a a C.4 PAGE 471 where The solution to Eqn. C.5 is (cf. Eqn. B.4) and the boundary conditions are (0)=0 (zero shear stress at the fiber end) and f()=0 (no load transfer across fiber ends. Th e boundary coefficients are consequently Thus the interfacial shear stress is The fiber axial stress is obtained from Eqns. C.9 and C.1, When ra>>rb, Eqn. C.6 reduces to and Eqn. C.10 simplifies to In the limit, as l the maximum axial stress at the fiber midpoint (z=0) is 02 2 2 rosendz d C.5 2 2 2 21 2b a f a f f f b f b rosenr r r E E r r r E G C.6 z B z Arosen rosen cosh sinh C.7 l r r r r E r G A Bb a f b a rosen a b cosh ; 02 2 2 C.8 l r r r r E z r Gb a f b a rosen rosen a b cosh sinh2 2 2 C.9 1 cosh cosh2 2 2 2l z r E r r E E rrosen rosen f f b a a f a f C.10 2 2 2 21 2b a f a f f f b f b rosenr r r E E r r r E G C.11 1 cosh cosh l z E Erosen rosen a f f C.12 PAGE 472 Rosen [110] considered the stress carryi ng capacity (efficiency) of a short fiber relative to a very long fiber (Eqn. C.13) The fiber efficiency at a point from the fiber end is therefore This expression can be further simplified for large l because tanh(rosen l )=1. This leads to (after hyperbolic identity substitution and rearrangement): where f is the filler fraction. By setting the fiber efficiency (typically =[0.9-0.99]), the ineffective length can be determined from Eqn. C.15. a f l fE E 0 C.13 rosen rosen rosen l f fl l sinh tanh cosh 1 0 C.14 1 2 1 1 cosh 1 2 12 1 2 1 2 1 b f f fG E d C.15 PAGE 473 APPENDIX D LAWRENCE PULL-OUT ANALYSIS Lawrences pull-out analysis (1972) [190], like Cox origin al analysis (1952) [105], assumes that the interfacial shear stress is propo rtional to the difference between fiber displacement, u (supposing the matrix has the fiber pr operties), and matrix displacement, v (supposing the fiber has matrix properties). That is, where K is a constant. As above (App. B and C), employing a force balance over a fiber element of length dz gives where C is the fiber circumference. Combining Eqns. D.1 and D.2 where H = CK Differentiating Eqn. D.3 we get Equation D.4 can be reformulated ut ilizing the stress-strain relations f= Eff and m= Emm. which yields (cf. Eqns. B.3 and B.5) v u K D.1 dz z C dP D.2 v u H dz dP or v u K C dP D.3 m fH dz dv dz du H dz P d 2 2 D.4 m m f fE A E A P H dz P d1 12 2 D.5 02 2 2 P a dz P d D.6 PAGE 474 Here a2=H/R with 1/R= (1/AfEF 1/AmEm). The fiber load takes the familiar hyperbolic sine form after BC evaluation, The shear stress along the fiber can be obtained by recalling Eqn. D.1 and differentiating it directly to get: Substituting Eqn. D.7 for P(z) in Eqn. D.8, and integrating, we obtain The maximum shear stress occurs at z=l/2, and for an infinitely long fiber The shear stress along the fiber(Eqn. D.9) can consequently be normalized by Eqn. D.11. Lawrence considered fiber-matrix in terfacial failure at a distance (l/2-z) from the fiber end. The fiber load at this point is obtained from Eqn. D.2 and a force balance as 2 sinh sinhl a az P z Pf D.7 R z KP dz d D.8 2 sinh cosh l a az a R KPf D.9 2 cothmaxl a a R KPf D.10 a R KPfmax D.11 2 sinh coshmaxl a az D.12 z l C P Pf f f 2 D.13 PAGE 475 In Eqn. D.13, f is the frictional shear stress at the interface (assumed constant along length of fiber) and C is the fiber circumference. The shear stress at this point is given by Eqn. D.9 with Pf=Pf and l/2=z, Debonding at z continues without further load incr ease if Eqn. D.14 is equal to the interfacial debond stress s. Setting =s (debonding criteria) in Eqn. D.14, solving for Pf, and equating with Eqn. D.13 gives Differentiating Eqn. D.15 with respect to z and setting this equal to zero to find z=zmax at maximum Pf gives Eqn. D.16. Equation D.15 describes the critical length for catastrophic debonding. Note that it follows from Eqn. D.5 that if s/fcosh2(al/2), then zmax=l/2 and the debonding process is catastrophic from the onset. Complete debonding will occur at maximum loading Pf max according to Eqns. D.17a and D.17b. where Pf is the load required to debond an infini tely long fiber with no frictional forces. That is, az a R KPfcoth D.14 az K a R z l C PS f ftanh 2 D.15 f sa z z 1 maxcosh 1 D.16 2 2 tanhmax maxl z l a P Pf f D.17a 2 2 tanhmax max max maxl z z l a az P Ps f f f D.17b K a R Ps f D.18 PAGE 476 APPENDIX E GAO ENERGETICS Fiber-matrix (interphase) de bonding may be considered by using either strength (see Appendix D [190]) or energetic criterion [146,201, 149]. The energetic (LFM) approach described in the text (Eqns.3.30-3. 37, Figure 3.26) follows primarily Gao et al. [149]. The analysis begins with the compliance energy-debonding toughness relationship, where Pd is the critical debonding load, C is the body compliance (function of debond length l ), and A =2rl An exact solution for C is not available, but after approximation and rearrangement [146] gives, for the critical debonding stress1, Gao et al. [149] refined [146] utilizing shear-lag type assumptions to account for frictional fiber loading and fibe r Poisson contraction effects. An energy balance for a generalized cracked body with external trac tions and frictional forc es across the crack surface provides 1 In this analysis [146] neglects the effects of friction. The debonding work accounts only for fiber strain energy. A C P Gd 22 E.1 f f f d dr G E r P22 E.2 TFSS fdU ds dv dA g ds du T E.3 PAGE 477 where g is the specific work of fracture, the in tegral on the LHS is the external traction work, the integral on the RHS is the work of friction, and U is the strain energy of the continuum body. We also know that for an elastic system Combining Eqns. E.3 and E.4 gives for the specific work of fracture If we consider the traction T as n concentrated forces, P1Pn, with corresponding displacements 1n, then Eqns. E.5 can be rewritten as A force balance allows for g=G, A=2rl, ds=2rdy, and Pi=P. If we also substitute v(y) (Eqn. 2.15 of [149]) and i= -uf(0) (Eqn. 2.14 of [149]) into Eqn. E.6 we arrive at an integro-differential equation for the debonding criteria. Note that the first term on the RHS of Eqn. E.7 is equivalent to Eqn. E.1 because C = -uf(0)/P, while the second term represents th e frictional contribution to the debonding toughness. Further simplifying Eqn. E.7 gives where TFSS fds dv ds du T dU2 1 2 1 E.4 TFSS fds dv A ds du T A g2 1 2 1 E.5 FS f n i i ids A v A P g12 1 E.6 l f f fdy l y v l u r P G02 1 0 4 E.7 2 3 21 2 1 1 4Q P k G E rf f E.8 1* l m f f me P P l T Q E.8a f mk k 2 1 2 1 E.8b PAGE 478 with =Em/Ef, =f/m, f and m are the Poisson ratios, q0 is the residual cure shrinkage/thermal mismatch betw een the fiber and matrix, and is the coefficient of friction. We can recover the frictionless debonding load, P0, from Eqn. E.8 when Q=0, =0, and =0. This is not equivalent to Outwater and Murphys original solution Eqn. E.2, which ignored frictional contributions altogether and underscores the significant frictional contribution to fracture energy. The debonding criteria in Eqn. E.8 is more transparently described by where The above debonding criteria (Eqn. E.10) invol ves the frictional c ontribution across the debonded interface (second term on RHS of Eqn. E.7). The frictional shear stress is given by 2 1 10 2 m f f fq r P E.8c 2 1 1 2m v m f fr E.8d 2 1 2 3 02 1 1 2 f f fk G E r P E.9 K e K P P P Pl 10 * E.10 1f m fK 2 1 1 m f m fk E.8e PAGE 479 Here, -q* is the normal pressure contribution ar ising due to Poisson contraction and geometric factors, Tm and Tf are the tensile forces in the fiber and matrix respectively. Tm and Tf are given by: We now have the necessary equations to consider frictionl ess and friction controlled debonding. Frictionless debonding across the interface will occur when the frictionless debond load P0 (Eqn. E.9) is equal to P* (Eqn. E.8c). Then by E.10, P=P0. Accordingly, equating Eqns.E.8c and E.9 and solving for q0 gives the theoretical threshold value for frictionless debonding, qth. When q0 qth there is effectively no frictional co ntribution to the debonding load, hence P=P0 as in Eqns. E.9 (see Figure 3.24). Furt hermore, this debonding load is independent of the debonded length l. For example, [149] found, fo r a steel-epoxy composite system, qth=19 MPa (cf. q0=23 MPa [6]). Conversely, if q0 qth, the debonding load is given by Eqn. E.10, which has a strong func tional dependence on the debonded length l. For friction-controlled debonding (P* P P0) Gao et al. [149] argue that 0q qf E.11 2 1 1 12 m f m m f f fT T r q E.12 1* l m f f me P P T E.13 m fT P T E.14 2 12 1 1 2 1 1 2 f f f m f f thk r G E q E.15 PAGE 480 because for practical engineering composites 1, 1, f0.5, and m0.5. Equation E.10 can then be simplified to Furthermore, a characteristic debond length (lT) may be defined, which is a measure of the friction affect ed zone at the debonding front. Again for practical engineering composites (e.g., steel-epoxy, lT=8.3mm or ~4df) this characteristic debond length is relatively small, indicating th at the friction contri bution is significant only near the debonding front. Finally, it is possible to de fine a critical fiber debond length lc using Eqn. E.16 with P=Pc. where Pc is the critical external fiber tensile load at fiber failure. Strong fibers will continue to debond with incr easing external load (from P0 to a plateau value of P*) when Pc P*, and weak fibers (Pc P*) will break in proportion to the debond length with a ratio equal to (lc/l). m f fr le K P P P P 0 * E.16 m f f Tr l E.17 c c cP P P P K l* 01 ln 1 E.18 PAGE 481 APPENDIX F HENSTENBURG AND PHOENIX SIMULATIONS In an effort to more accurately determine and interpret the FLD for a SFCF test Hentstenburg and Phoenix [6] employed a bilinear stress recovery model (Figure 3.29) and Monte Carlo simulation. The Monte Ca rlo simulation allowe d for fiber strength variation (stochastic with gage length and position) during the frag mentation process as opposed to the single strength statistic employed by K-T analysis (ult,f, Eqn. 2.5, repeated here for convenience). The approach in [6] is straightforward and follows K-T analysis, except with regards to the statistics employed for fiber stre ngth and averaged critical length lc. In this analysis [6], stress recovery in the bonded and yielded section is limited by the strain criterion, b (reference b in Figure 3.29). Extern al conditions which produce fiber axial stresses (Sb) in excess of the cri tical strain level (b), result in fiber-matrix disbonding at the fibers terminal end. Th ere is progressive de bonding along the fiber as the stress level increases. However, the leng th of the yielded and bonded section remains constant and translates with the debonding fr ont. The lengths of these two regions are given here as a function of FAS(s). c f ult ifssl r, Eqn. 2.5 b yS s s d s : 4 F.1a b f b y bS s S s d S d s : 4 4 F.1b PAGE 482 where Henstenburg and Phoenix [6] nondimensionalized their analysis by making use of the Weibull two-parameter weakest-link scaling law and Eqn. F.1a. In Eqn. F.2, and s0 are the Weibull modulus and scale parameter respectively for reference length L0, and s1 is the scaled Weibull strength at length L1. Defining s* and *, where s* is both the peak fiber stress and scaled strength for a fiber with length 2* (*=(s*)). These definitions in combin ation with F.2 and F.1a provide and The following nondimensional variables can then be defined: where l is the saturation mean fragment length Utilizing these nondimensional variables Eqns. F.1a and F.1b can be rewritten as Combining the last of the nondimensional variables for mean fragment length l and d Sb y b 4 0 1 0 1 1s L L s F.2 1 1 0 0 *2 s d L sy F.3 1 0 1 1 0 *4 2 ys d L F.4 *s s x * s sL *s S sb b b b y f fT *l bs : F.5a b f b bs T s s : F.5b PAGE 483 gives Finally, we can replace s* in Eqn. F.6 using Weibull weakest link scaling (Eqn. F.2), Hence, In contrast with Eqn. 2.5, Eqn. F.8 uses the Weibull scaled strength statistic at the mean fragment length, sl, rather than the ultimate fiber strength sult,f. In addition, the mean fragment length l is used in F.8 as compared to the critical length lc in Eqn. 2.5. In a like manner, it is possible to reformulate Eqn. F.8 using the mean fiber strength sf,l (as determined from Weibull statistics) at l Namely, Note that the correction factor s (bracketed terms) in Eqns. F.8 and F.9 are not equivalent to the simple 4/3l (normally distributed FLD) propos ed by Ohsawa et al. [200]. The correction factors are also obvious functions of the inherent fibe r strength distribution (viz. Weibull modulus ) and fragment length distribu tion prior to saturation, which cannot be said for the K-T analysis provided by Eqn. 2.5. ys d 4* 2 2*l s dy F.6 l ls s l s 1 1 *2 2 F.7 12 2 l s dl y F.8 1 1 2 21 ,l s dl f y F.9 PAGE 484 APPENDIX G BELTZER INEFFECTIVE LENGTH Beltzer et al. (1992) [223], were aware of the limitations associated with heretofore ineffective length definitions. For example, [ 223] provides, for the in effective length of a fiber, where is given by and the shear parameter beltzer (cf., shear parameters in B.5 and B.6) is given by Variables in the above equation have th eir customary meanings. The value of is arbitrarily chosen on [0.9-0.99]. The obvious limita tions of Eqn. G.1 are: There is no standard value and more importantly, why Eqn. G.1? Equation G.1 is restricted to a uniaxial stress state. Perfect bonding between fiber and matrix is assumed despite considerable evidence to counter this assumption. The magnitude of is so small in practical composites as to be inconsequential (e.g., ~2-5 rf). To counter the limitations and concerns re garding Eqn. G.1, Beltzer et al. [223] introduced an ineffective length definition th at employs strain en ergy densities for an idealized intact fiber, V0( x,y,z ), and a broken fiber, V ( x,y,z ). 1 2 1 1 cosh 12 1 beltzer i G.1 L x xf f L lim G.2 21 2f f f f m beltzerr E G G.3 PAGE 485 Leff is the length of an intact perfect fiber that is capable of carrying the same strain energy as a broken fiber of length L The ineffective length is simply i,b= L-Leff. For the conditions, L / rf>>1 and uniaxial uniform stress on an in tact fiber, Eqn. G.4 simplifies to It is evident that Eqn. G.5 avoids the ambiguities of Eqn. G.1, and provides the trivial bounding conditions for Perfect Bonding ( V(x) V0 as i,b 0) and Perfectly Smooth (i,b L as V(x) 0). Beltzer et al. [223] determined that th e FAS for a broken, bu t otherwise perfect fiber is given by (cf. B.8 and C.12): The shear interaction parameter, k is given by a is the fiber radius and b (by geometry of HCP) is Therefore the strain energies V0 and V ( x ) are effL S L Sdx ds z y x V dx ds z y x V00 0, , G.4 L b idx V x V0 0 ,1 G.5 fE V 22 0 0 G.9 a kx fe x 10 G.6 a b E G kf mln 22 G.7 a bf 1 3 22 1 G.8 2 2 01 2a kx fe E x V G.10 PAGE 486 Substituting Eqns. G.9 and G.10 into Eqn. G.5 yields The two ineffective length estimations (R osen: Eqn. G.1 and Beltzer: Eqn. G.11) for perfect fibers are in relatively clos e agreement. For example, [223] finds for a carbide-glass composite with Gm/Ef=0.2, a =7.5 m, and f=0.5. Once again, these values are very small, calling into question th eir significance and utility. Beltzer et al. [223] argue that the discrepa ncy between the two estimators is insignificant in the context of statistical theories of strength. It is important to recognize that both ineffective length calculations (Eqns. G.1 and G.11) utilize shear-lag arguments, so the similarity is not surprising. It is also important to note that Eqn. G.5 makes no assumptions or restrictions regarding the na ture of the fiber-matrix interaction (save a uniform uniaxial stress state). More importa ntly, the integral equation Eqn. G.4 is applicable for any three-dimensional stress state ( V0 and V ( x ) evaluation may become very complicated). Given the very small ineffective length values for perfect fibers, it is informative to consider imperfect fiber-matrix interacti on. The present ineffective length definition allows for the introduction of more comp lex imperfection distributions besides the obvious fiber break. The fiber break is seen (wrt fiber strain energy density) as a particular example of a period ic imperfection. Beltzer et al. [223] consider and derive strain energy density expressions for three basic imperfection distributions. A periodic model (Deterministic). A random model (Stochastic). A distributed inferiority model (e.g., broken fibers, matrix inhomogeneities, etc.). k ab i2 3, G.11 f b ir m 72 1 91 12, f ir m 34 2 57 17 PAGE 487 The key to Eqn. G.5 application involves choosing an appropriate form the fiber strain energy density V ( x ). A periodic distribution of understressed (imperfect, Vim( x )) regions can be described by the strain energy density, where () is the dirac delta function and 2 (2 1) is the strength of the inferiority (e.g.,matrix inhomogeneities: Gm, agglomeration, incomple te wetting, compositional variation, etc.; fiber irre gularities: diameter variati on, surface roughness, surface treatment, etc.). Supposing that the inferiority spacing, d is much greater than the fiber radius ( a ) and the extent of stress perturbation, we can approximate the FAS near the periodic inferiority using Eqn. G.6 (near-field approx.). Furthermore, at distances far from the imperfection f( x ) 0 (far-field approx.). Combining the near-field (Eqn. G.6) and the far-field approximation, the complete strain energy density for a periodic inferiority distribution is Then inserting Eqns. G.13 and G.9 into Eqn. G.4 yields the effective length, Leff. where [ l ] is the unit length (arising from dirac delt a function integration). The ineffective length (i,b= L-Leff) is obtained from Eqn. G.14. For sufficiently large N the ineffective length is approximately ) ( 0 2 2 01 2 ) (x N n n f imx x E x V G.12 ) ( 1 2 2 2 01 1 2 ) (x N n n a kx fx x e E x V G.13 L N n a kdn L N n a kdn a kL a kL effe e d L l e k a e k a L L1 1 2 2 22 1 2 1 2 G.14 PAGE 488 The first term on the RHS recovers the fiber break contribution per Eqn. G.11, while the second term arises solely from the distributed imperfection di stribution. It is apparent that as [ l ]/d approaches unity, the ineffective length will quickly approach L Random inferiorities (includi ng strength and spacing) can be treated in much the same manner as above. That is, the strain energy density for understressed (imperfect, Vim( x )) unbroken fibers is given by, where Supposing that the random imperfections are sufficiently separated so that imperfectionimperfection interaction can be ignored and also th at the first moment of j is Combining the near-field and far-field solutio ns for the strain energy density of a broken randomly inferior fiber (cf. Eqn. G.13) Then, proceeding as above (Eqn. G.13 to Eqn. G.15) it is found that where is the average number of imperfections per unit length (Poi sson distribution). The similarity to Eqn. G.15 is obvious. Beltzer et al. [223] argue that random imperfections generally lead to a less inferior fiber than that for periodic imperfections. d L l k ab i2 ,2 3 G.15 x E x Vf im 1 22 0 G.16 x x j jjL x x x x00 ; G.17 2gj G.18 x e E x Va kx f 1 1 22 2 0 G.19 l L g k ab i 2 ,2 3 G.20 PAGE 489 The isolated imperfections in the peri odic and random distri butions, Eqns. G.15 and G.20 respectively, are derived assumi ng negligible interaction between the imperfections. However, it is also possible that imperfections may be distributed along the fiber-matrix interface (i.e., disbonding, localized matrix yielding, etc.) in closeenough proximity to influence the perturbation distribution. Distri buted imperfections may be considered by defining the di stributed inferiority distribution, (x). Then Eqn. G.5 becomes With this definition, Eqns. G.15 and G.20 can be reconsidered with the following distributed inferiority distributions. and Beltzer et al. [223] provide a particular example for random residual compressive stress (avg. density of length LR) acting on a fiber. From inspection, the strain energy in the volume element (LR s) is where s is the fiber cross section. From Eqns G.16 and G.17 the combined near and farfield strain energies are given by Combining Eqns. G.25 and G.26 and solving for j yields 1 0 ; 10 x V x V x G.21 L b idx x0 G.22 x n j jx x x k a x12 3 G.23 s L ER f R22 0 G.25 s l L Ej R f 22 0 G.26 PAGE 490 The first moment, PAGE 491 APPENDIX H MATLAB SFCF SIMULATION SCRIPT PAGE 492 function [trans_sorted_lrc, parameters] = sfcf_s im(random,COV,COV_mult) % enable for wrapper % SFCF simulation % David M Bennett % University of Florida clf('reset'); COV_mult = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1 ;1;1;1;1;1;1;1;1;1;1]; %always on %random = 1; %disable for wrapper % COV = 0.05; %disable for wrapper parameters = []; % get inputs from input fi le "sfc_sim_input.inp" fid = fopen('sfcf_sim_input.inp'); inVars = textscan(fid,'%s %*s %f','commentStyle','//'); file_status = fclose(fid); mode_flag = inVars{2} (strmatch('mode_flag',inVars{1})) % 1=kim, 2=henstenburg COV = inVars{2} (strmatch('COV',inVars{1})); %coeffeicient of variation rhoFiber = inVars{2} (strmatch('rhoFiber',inVars{1})); % fiber Weibull Modulus scale0 = inVars{2} (strmatch('scale0',inVars{1})); % fiber scale factor dia_F = inVars{2} (strmatch('dia_F',inVars{1})); %fiber diameter (m) mod_M = inVars{2} (strmatch('mod_M',inVars{1})); %matrix modulus (Pa) T_y = inVars{2} (strmatch('T_y',inVars{1})); %matrix yield (Pa) % Nominal mechanical properties (mean values) %Fiber properties % dia_F = 5.7e-6; rad_F = dia_F/2; mod_a_F = 278e9; mod_t_F = 21E9; axialshearMod_F = 20e9; transshearMod_F = 16e9; nu_a_F = 0.25; nu_t_F = 0.25; alpha_a_F = -0.7e-6; alpha_t_F = 10e-6; % rhoFiber = 2; % was 4.95 % scale0 = 5.762e9; %Matrix properties % mod_M = 1.62e9; axialshearMod_M = 0.605e9; nu_M = 0.34; PAGE 493 alpha_M = 40e-6; %Composite/test parameters sigma0 = 0.7e7; %initial stress guess Lgage0 = 10e-3; %weibull reference length Lgage = 25e-3; %composite test gage length dogbone parallel section Lplat = 25e-3; %fiber le ngth at max fiber plateau stress % adjusted M_p down to lower first stress level (was 0.0454) M_p = 0.0454; delT = -100; psi_f = 0.01; beta = .06207; gamma_F = 10.0; gamma_d_fit = 90.0; % T_y = 48.44e6; % 62.25 for phi =200 T_f = 0.3*T_y; % End of nominal values table if random == 1 %Fiber properties dia_F = rand_param(dia_F,COV*COV_mult(1)); %1 rad_F = dia_F/2*COV_mult(2); %2 mod_a_F = rand_param(mod_a_F,COV* COV_mult(3)); %3 mod_t_F = rand_param(mod_t_F,COV*COV_mult(4)); %4 axialshearMod_F = rand_param(axialshearMod_ F,COV*COV_mult(5)); %5 transshearMod_F = rand_param(transshearMod_F,COV*COV_mult(6)); %6 nu_a_F = rand_param(nu_a_F,COV*COV_mult(7)); %7 nu_t_F = rand_param(nu_t_F,COV*COV_mult(8)); %8 alpha_a_F = rand_param(alpha_a_F,COV*COV_mult(9)); %9 alpha_t_F = rand_param(alpha_t_F,COV*COV_mult(10)); %10 rhoFiber = rand_param(rhoFiber,COV*COV_mult(11)); %11 scale0 = rand_param(scale0,COV*COV_mult(12)); %12 %Matrix properties mod_M = rand_param(mod_M,COV*COV_mult(13)); %13 axialshearMod_M = rand_param(axialshearMod_M,COV*COV_mult(14)); %14 nu_M = rand_param(nu_M,COV*COV_mult(15)); %15 alpha_M = rand_param(alpha_M,COV*COV_mult(16)); %16 %Composite/test parameters sigma0 = 0.7e7; %initial stress guess %17 Lgage0 = 10e-3; %weibull reference length %18 PAGE 494 Lgage = 25e-3; %composite test ga ge length dogbone parallel section %19 Lplat = 25e-3; %fiber length at max fibe r plateau stress %20 M_p = M_p; % adjusted M_p down to lower first stress level (was 0.0454) %21 delT = -100; %22 psi_f = rand_param(psi_f,COV*COV_mult(23)); %23 beta = .06207*COV_mult(24); %24 gamma_F = gamma_F; %*COV_mult(25); %25 gamma_d_fit = gamma_d_fit; %*COV_mult(26); %26 T_y = rand_param(T_y,COV*COV_mult(27)); % 62.25 for phi =200 %27 T_f = 0.3*T_y*COV_mult(28); %28 parameters = [COV; dia_F; mod_a_F; mod_t_F; axialshearMod_F; transshearMod_F; nu_a_F ; nu_t_F; alpha_a_F; alpha_t_F; rhoFibe r;scale0]; parameters = [parameters; mod_M; axialshearMod_M; nu_M; alpha_M]; parameters = [parameters; sigma0; Lgage0; Lgage; Lp lat; M_p; delT; psi_f; beta; gamma_F; gamma_d_fit; T_y; T_f]; end %initializing LrecStar = 0; sigmaStar = 0; flaw =0; count =1; t_flaws =0; flaws =0; cdfFiber_gage = 0; cdfFiber_gage_old = 0; rand_pos(1) = 0; rand_pos(2) = 0; psi_infinity = sigma0; psi_infin_1 = 0; psi_infin_2 = 0; psi_infin_3 = 0; psi_infinityNon = 0; psi_infinityNon_old =0; del_psiNon = 0; array_full = []; len_guess = 0; lenDebondNon = 0; lenRecovery_current = 0; lrc_array = []; gamma_d =[]; fit_gamma_d = []; value_gamma_d = 0; cdf_array =[]; PAGE 495 rand_pos = []; simul = []; M_p_array = []; sigma0_frag = 0; first_flaw = 0; increment_flaw = 0; len_excluded = 0; len_excludedNon = 0; rowCount = 1; do_this = 1; excluded_fragments = 0; lgage_fragments = 25e-3; sim_done =0; temp = []; xx_excluded = []; plot_flag = 0; sorted_ln_ar = []; temp_prob = []; temp_sort_ln_ar =[]; plot_flag = 0; total_lrc = 0; total_frag = 0; closed_frags = 0; % dummy count variable for number of fragnments saturated on entire gage length discarded_flaws =0; % dummy count variable for flaws discarded because they are in excluded region print_flag = 0; % print flag for limiting screen print new_flaw = 0; % flag for new flaw on load increment count % parameters= []; for i=1:200 for j = 1:10 simul(i,j)= 0; end end for i=1:1 for j = 1:11 lrc_array(i,j)= 0; end end psi_infin_1 = (2*nu_a_F*nu_M/mod_a_F-(1nu_t_F)/mod_t_F-(1+nu_M)/mod_M); psi_infin_2 = (2*nu_a_F/mod_a_F*(alpha_t_F-a lpha_M)+ ((1-nu_t_F)/mod_t_F+(1+nu_M)/m od_M)*(alpha_a_F-al pha_M))*mod_a_F*delT; psi_infin_3 = (2*nu_a_F^2/mod_a_F-(1 -nu_t_F)/mod_t_F-(1+nu_M)/mod_M); % non-dimensionalizing PAGE 496 sigmaStar = (2*T_y*Lgage0/dia_F*s cale0^rhoFiber)^(1/(rhoFiber+1)); LrecStar = (Lgage0/2)^(1/(rhoFiber+1))*(dia_F *scale0/(4*T_y))^(rhoFiber/(rhoFiber+1)); sigma0Non = sigma0/sigmaStar; gageNon = Lgage/LrecStar; LplatNon = Lplat/LrecStar; Lgage0Non = Lgage0/LrecStar; scale0Non = scale0/sigmaStar; LgageNon = Lgage/LrecStar; Lgage_fragNon = lgage_fragments/LrecStar; LplatNon_old = LplatNon; disp(sprintf('\n sigmaStar, LrecSta \n %5.5s %5.5s ',sigmaStar,LrecStar)) disp(sprintf('\n sigma0Non, gageNon, LplatN on, Lgage0Non, Lgage_frags scale0Non \n %5.3s %5.3f %5.3f %5.3f %5.3f %5.3f',sigma0Non, gageNon, LplatNon, Lgage0Non, Lgage_fragNon, scale0Non)) pause(2); % choose sigma0 while (psi_infin_1*mod_a_F*sigma0/m od_M+psi_infin_2)/psi_infin_3 < 0 sigma0 = sigma0 + 0.01e7; disp(sprintf(' sigma0 = %4.3g and psi_ininfity = %5.3 s', sigma0, (psi_infin_1*mod_a_F*sig ma0/mod_M+psi_infin_2)/psi_infin_ 3)) end % 0.7e7 while (lgage_fragments > 10*dia_F) if (count == 1) % first step of simulation % first step of loading fiber % prior to this there is zero load in the fiber, external load is zero % SFC composite is virtually all matrix psi_infinity = (psi_inf in_1*mod_a_F*sigma0/mod_M+psi_infin_2)/psi_infin_3; psi_infinityNon = psi_infinity/sigmaStar; % fiber stress calculations (Tyson, Henstenberg) LplatNon = LgageNon-lenD ebondNon*t_flaws; %simple (no debond propagation) else psi_infinity = (psi_inf in_1*mod_a_F*sigma0/mod_M+psi_infin_2)/psi_infin_3; psi_infinityNon = psi_infinity/sigmaStar; % fiber stress calculations (Tyson, Henstenberg) for j=1:t_flaws lrc_array(j,1) = si mul(j,3); % random position on gage length (possibly excluded) % [xxx yyy] = lrc(simul(j,4)*sigmaStar, simul(j,6), dia_F, T_y, T_f, psi_infinity); %format for function [lenRecovery_current, lenYield, lenDebond_c urrent, debond_flag] = lrc(psi_infinity_break, lenDebond, dia_F T_y, T_f, psi_infinity_current, mode_flag) PAGE 497 [lenRecovery_curr, lenYield_cu rr, lenDebond_curr, debond_flag_curr] = lr c(simul(j,4)*sigmaStar, simul(j,6), dia_F, T_y, T_f, psi_infinity,mode_flag); lrc_array(j,2) = lenRecovery_curr; lrc_array(j,2) = lrc(simul(j,4)*sigmaStar, si mul(j,6), dia_F, T_y, T_f, psi_infinity); lrc_array(j,3) = simul(j,4); % psi_infinityNon lrc_array(j,4) = simul(j,1); % flaw number lrc_array(j,5)= lrc_array(j,1)*Lgage; %flaw position on fiber (meters) lrc_array(j,6) = lrc_array(j,5)/LrecStar; %flaw position normalized lrc_array(j,10)= simul(j,5); % cdf on gage length end % disp(sprintf(' lrc = %4.8f j = %3.0f', lrc_array(j,2), j)) if (first_flaw == 1) so rted_lrc_array = sort rows(lrc_array,5); first_flaw = 0; else sorted_lrc_array = sortrows(lrc_array,5); end % start of while loop do_this ==1 while (do_this) == 1 %section to check for excluded flaws if (rowCount == 1) sorted_lrc_array(1,7)=sorted_lrc_array(rowCount,5); if (t_flaws == 1) sorted_lrc_array(2,7) = Lgage sorted_lrc_array(rowCount,5); elseif (t_flaws == 2) sorted_l rc_array(2,7) = sorted_lrc_array(rowCount+1,5)-sorted_lrc_array(rowCount,5); end end if (rowCount == 2) sorted_lrc_arr ay(2,7)=sorted_lrc_array(rowCount,5 )-sorted_lrc_array(rowCount-1,5); end if (rowCount >1) if (0.5*sorted_lrc_arra y(rowCount-1,2)+0.5*sorted_lrc_array(ro wCount,2) > (sorted_lrc_array(rowCount,5) -sorted_lrc_array(rowCount-1,5))) check = -0.5*sorted_l rc_array(rowCount-1,2)-0.5*sorted_lrc_array(rowCount,2) + sorted_lrc_array(row Count,5)sorted_lrc_array(rowCount-1,5); closed_frags =closed_frags +1; % disp(sprintf('\n fragm ent %3.0f associated with flaw %3.0f %4.3s',rowCount, sorted_lrc_array(row Count,4), check)) % disp(sprintf('\n frag %3.0f closed, ass. w/ flaws %3.0f and %3.0f closed = %3.0f, count =%3.0f, t_flaws =%4.0f',rowCount, sorted_lrc_array(rowCount,4), sorted_lrc_array(rowCount+1,4),cl osed_frags, count, t_flaws)) if (s orted_lrc_array(rowCount,4) == t_ flaws) && (new_flaw == 1); discarded_flaws = discarded_flaws+1; PAGE 498 % disp(sprintf('frag %3.0f closed, ass. w/ flaws %3.0f and %3.0f closed = %3.0f, count =%3.0f t_flaws =%4.0f',rowCount, sorted_lrc_array(rowCount,4), sorted_lrc_array(rowCount+1,4),closed_frags, count, t_flaws)) % disp(sprintf('ex cluded flaw %3.0f in fragment %4.0f total discarded %4.0f',sorted_lrc_array (rowCount,4), rowCount, discarded_flaws)) lrc_array(sorted_lrc _array(rowCount,4),:) = []; % delete flaw from consideration sorte d_lrc_array(rowCount,5)sorted_lrc_array(rowCount-1,5) [t_flaws,n] = size(lrc_array); % determine lrc_array size sorted_lrc_array(rowCount,:) = []; % delete row in sorted_lrc_array for ex cluded flaw (current @ rowcount) new_flaw = 0; rowCount = rowCount-1; % increment rowCount end end if (rowCount == t_flaws) sorted_lrc_array(ro wCount,7)=Lgage-sorted_lrc_array(rowCount,5); % flaw separation else sorted_lrc_array(ro wCount+1,7)=sorted_lrc_array(rowCount+1,5)-s orted_lrc_array(rowCount,5); % fla w separation end end if rowCount == 1 sorted_lrc_array(r owCount,9) = sorted_lrc_array(rowC ount,7)-sorted_lrc_array(rowCount,2); if (sorted_lrc_array(rowCount,9) < 0) sorted_lrc_array(rowCount ,9) = 0; % limit recove ry length to distance between flaws, minimum gage length is zero (0) end lgage_fragments = sorted_lrc_array(rowCount,9); sorted_lrc_array(rowCount,11) = lgage_fragments; else sorted_lrc_array(rowCount ,9) = sorted_lrc_array(rowCount,7)-0.5 *sorted_lrc_array(rowCount,2)-0.5*sorted _lrc_array(rowCount-1,2); if (sorted_lrc_array(rowCount,9) < 0) sorted_lrc_array(rowCount ,9) = 0; % limit recove ry length to distance between flaws, minimum gage length is zero (0) end lgage_fra gments = lgage_fragments+sor ted_lrc_array(rowCount,9); sorted_lrc_array(rowCount,11) = lgage_fragments; end rowCount = rowCount+1; if (rowCount >= t_flaws) do_this = 0; end end % end while loop do_this <> 1 % conditional to get gage length of final fragment % (avoids matrix index problem) if (t_flaws > 1) sorted_lrc_array(rowCount ,9) = sorted_lrc_array(rowCount,7)-0.5 *sorted_lrc_array(rowCount,2)-0.5*sorted _lrc_array(rowCount-1,2); if (sorted_lrc_array(rowCount,9) < 0) PAGE 499 sorted_lrc_array(rowCount ,9) = 0; % limit recove ry length to distance between flaws, minimum gage length is zero (0) end lgage_fra gments = lgage_fragments+sor ted_lrc_array(rowCount,9); sorted_lrc_array(rowCount,11) = lgage_fragments; if (rowCount == t_flaws) sorted_lrc_arr ay(rowCount+1,7)=Lgage-sorted_lrc_array(ro wCount,5); % flaw separation sorted_lrc_a rray(rowCount+1,9) = sorted_lrc_array(rowCount+1,7)-sorted_lrc_array(rowCount,2); if (sorted_lrc_array(rowCount+1,9) < 0) sorted_lrc_array(rowCount+1,9) = 0; % limit recovery length to distance betwe en flaws, minimum gage length is zero (0) end lgage_fragments = lgage_fragments+sorted_lrc_array(rowCount+1,9); sorted_lrc_array(rowCount+1,11) = lgage_fragments; sorted_lrc_array(rowCount+1,5) = Lgage; end end new_flaw = 0; % table generated if (t_flaws ==0) | (t_flaws ==1) lgage_fragments = 0.025; end LplatNon = lgage_fragments/LrecStar; end if sim_done == 0 % just checknig poisson calculation prob_1 = M_p exp(-M_p); prob_2 = M_p^2/2 exp(-M_p); % calculations for psi_infini ty and delta psi_infinity % psi_infinityNon, realized average fiber stress del_psiNon = (M _p/LplatNon_old + psi_infinityNon^rhoFib er)^(1/rhoFiber) psi_infinityNon; psi_infinityNon = psi_infinityNon + del_psiNon; psi_infinity = psi_infinityNon*sigmaStar; sigma0 = (psi_infinit y*psi_infin_3-psi_infin_2)*m od_M/(psi_infin_1*mod_a_F); sigma0Non = sigma0/sigmaStar; strain0 = sigma0/mod_M; % interpolation of debond length for a given psi_infinity and gamma_d mult = (10:20:390)'; PAGE 500 len_guess = 0.01*mult*dia_F; for k = 1:20 fit_gamma_d(k) = rad_F *psi_in finity^2/(12*mod_a_F)*(1+(1-psi_f*len_gue ss(k,:)/rad_F)*(2-psi_f*len_guess(k, :)/rad_F)*(1+3*rad_F/(beta*len_guess(k,:))))gamma_F*rad_F/(2*l en_guess(k,:)); gamma_d(k,1) = fit_gamma_d(k); end lenDebond = inte rp1(gamma_d, len_guess, gamma_d_fit,'spline'); lenDebondAR = lenDebond/dia_F; lenDebondNon = lenDebond/LrecStar; % ****************************************************************** % disp(sprintf('M_p before =%3.7f, psi_Non_ol d = %5.4s, psi_Non = %5.4s, psi = %5.4s', M_p,psi_infinityNon_old, ps i_infinityNon, psi_infinity)) M_p = LplatNon*((psi_infin ityNon)^rhoFiber psi_infinityNon_old^rhoFiber); % disp(sprintf('\n M_p after =%3.7f count %4.0f, t_flaws %4.0f', M_p, count, t_flaws)) if M_p < 0.002 M_p = 0.02; end simul(count,9) = count; simul(count,10)= M_p; %CDF value for 2-parameter Weibull distribution cdfFiber_gage = 1 exp(-Lplat Non/Lgage0Non*(psi_infinity Non/scale0Non)^rhoFiber); p_param = LplatNon*(psi_infi nityNon^rhoFiber-psi_inf inityNon_old^rhoFiber); if p_param < 0.002 p_param = 0.02; end random_sample = poissrnd(p_param); % disp(sprintf('\n poisson section M_p =%4.3f No. flaws = %3.0f ',M_p, random_sample)) if random_sample >= 1 % disp(sprintf ('random flaw generated %3.0f %3.3f', random_sample,p_param)) % pause(1); for j=1:random_sample t_flaws = t_flaws + 1; if (t_flaws == 1) first_flaw = 1; end incr ement_flaw = increment_flaw +1; simul(t_flaws,1) = t_flaws; simul(t_flaws,2) = count; simul(t_flaws,3) = rand; simul(t_flaws,4) = psi_infinityNon; simul(t_flaws,5) = cdfFiber_gage; simul(t_flaws,6) = lenDebond; PAGE 501 % simul(t_flaws,6) = 10e-6; % NO DEBONDING new_flaw = 1; % disp(sprintf('%3.4f %3.4f %4.4f flaw %3.0f of%3.0f, Lplat %5.5f psi %5 .5g %5.0f %3.0f',p_param,M_p, cdf Fiber_gage, j,random_sample,LplatNon*LrecStar, psi_infinity, count, t_flaws)) end end print_flag = print_flag +1; plot_flag = plot_flag + 1; if (count <101) | (print_flag == 100) disp(sprintf(' %5.0f %4.4f %4.5f %4.5f %4.4s %1.9f %4.3f %3.0f',... count, p_param, random_sample ,del _psiNon, psi_infinity, cdfFiber_gage, lgage_fragments/LrecStar, t_flaws)) print_flag = 0; total_lrc = 0; total_frag = 0; end if plot_flag == 100; % [flag] = pl ottingsim(sorted_lrc_arra y,dia_F, pos1, pos2); [flag] = plottingsim(sorted_lrc_array,dia_F); plot_flag = 0; end if (lgage_fragments < 10*dia_F) disp(sprintf('\n fragmentation saturation')) sim_done = 1; disp(sprintf('\n rhoFiber = %4.5f and total flaws = %4.0f',rhoFiber, t_flaws)) % [flag] = pl ottingsim(sorted_lrc_arra y,dia_F, pos1, pos2); [flag] = plottingsim(sorted_lrc_array,dia_F); end % iteration counting and bookkeeping cdfFiber_gage_old = cdfFiber_gage; 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Vo l. 4 (A99-24601 05-39) (AIAA, Reston, Virginia, 1999) 3051. 315 K. Lyle, E. Fasanella, M. Me lis, K. Carney, and J Gabrys, 8th International LSDYNA Users Confernce, Dearborn, Mich igan, May 2004 (Livermore Software Technology Corporation, Liverm ore, California, 2004) 3-23. 316 H. R Bae, R. V. Granhi, and R. A. Canfield, in th Annual AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia, Sept. 2002 (AIAA, Reston, Virginia, 2002) AIAA 2002-5558. 317 B. W. Kim and J. A. Nairn, J. Mater. Sci. 37 (2002) 3965. PAGE 519 BIOGRAPHICAL SKETCH David Bennett did his undergraduate studies at University of California, Irvine where he graduated with a B.S. degree in M echanical Engineering. After graduation he spent a few years working in the aerospace field. Looking to further advance his career and knowledge of Materials Science, he went back to school at the Univ ersity of Florida. He received a Master of Science degree from the University of Florida in Materials Science and Engineering, with an emphasis on Polymers. David spent the next 6 years working in the golf industry where he applied his skills to a game, and in the intirim helped develop golf equipment for professionals and amateurs alike. Recently, David has completed his studies in the pursuit of a Ph.D in Material Science and Engineering with a particular interest in ch aracterizing the fiber-matrix interface and understanding its influence on macrocomposite performance. Most recently David has again found gainful employment in the Aerospace field. Hopefull y, his knowledge of fiber-matrix interfaces and materials will enable missiles systems a nd their effective, if not peaceful, use. |