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1 J T AND J Q CHARACTERIZATION OF SURFACE CRACK TIP FI ELDS IN METALLIC LINERS UNDER LARGE SCALE YIELDING By SHAWN A. ENGLISH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLME NT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011
2 2011 Shawn A. English
3 Whatever you do, work at it with all your heart as working for the Lord, not for men Colossians 3:23
4 ACKNOWLEDGMENTS The a uthor expresses his gratitude to Dr. Nagaraj Arakere for his guidance, support and friendship as the graduate advisor on this project. Thanks also to the graduate committee members: Dr. John Mecholsky, Dr. Peter Ifju, Dr. Ashok Kumar and Dr. Ghatu Subhash for reviewing the work. Special thanks goes to Dr. Phillip Allen and others at the NASA Marshall Space Flight Center (MSFC). Ongoin g, unpublished research at MSFC laid the foundation for many of the analytical tools and techniques used in this study. A selflessness and unwavering support. T he author also thanks his fellow colleagues: Dr. Nathan Branch and Dr. George Levesque for their challenging discussions and willingness to review the work
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATIONS ................................ ................................ ........................... 13 NOMENCLATURE ................................ ................................ ................................ ........ 14 ABSTRACT ................................ ................................ ................................ ................... 16 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 18 Composite Overwrapped Pressure Vessels ................................ ........................... 20 NDE, Proof Test Logi ................................ ...................... 21 Coupon and Subscale Testing ................................ ................................ ......... 22 Two Parameter Fracture Mechanics ................................ ................................ ....... 23 J T and J Q Characterization of Near Tip Fields ................................ .............. 25 Fracture Prediction ................................ ................................ ........................... 26 Deformation Limits ................................ ................................ ........................... 27 2 BACKGROUND ................................ ................................ ................................ ...... 34 Linear Elastic Fracture Mechanics ................................ ................................ .......... 34 Elastic Plastic Fracture Mechanics ................................ ................................ ......... 36 Two Parameter Characterization ................................ ................................ ............ 37 J T Theory ................................ ................................ ................................ ........ 39 J Q Theory ................................ ................................ ................................ ....... 41 Modified Boundary Layer Formulation ................................ .............................. 43 Near Tip Stress Field Analysis and Plastic Collapse ................................ ........ 47 3 FINITE ELEMENT MODELING AND TEST GEOMETRIES ................................ ... 53 Geometries ................................ ................................ ................................ ............. 53 Finite Element Models ................................ ................................ ............................ 53 Mesh Configuration ................................ ................................ .......................... 53 Mesh Convergence ................................ ................................ .......................... 54 Element Form ulation ................................ ................................ ........................ 55
6 4 DEFORMATION LIMITS AND PLASTIC COLLAPSE IN A BONDED GEOMETRY ................................ ................................ ................................ ........... 61 Model Definitions ................................ ................................ ................................ .... 62 Materials ................................ ................................ ................................ ........... 62 Finite Element Models ................................ ................................ ...................... 63 J T Stress Field Characterization and Deformation Limit s ................................ ...... 63 J Integral and T Stress ................................ ................................ ..................... 63 Opening Stress Fields ................................ ................................ ...................... 65 Deforma tion Limits ................................ ................................ ........................... 67 Concluding Remarks ................................ ................................ ............................... 68 5 STRAIN HARDENING EFFECTS ON TWO PARAMETER CHARACTERIZATION ................................ ................................ ........................... 79 Model Definitions ................................ ................................ ................................ .... 81 Materials ................................ ................................ ................................ ........... 81 Finite Element Models ................................ ................................ ...................... 82 Results and Discussion ................................ ................................ ........................... 83 J Integral and Q ................................ ................................ ................................ 83 Crack Tip Deformation ................................ ................................ ...................... 85 Near Tip Opening Stress Fields ................................ ................................ ....... 88 Concluding Remarks ................................ ................................ ............................... 93 6 SURFACE CRACKED METALLIC LINERS IN COMPOSITE OVERWRAPPED PRESSURE VESSELS ................................ ................................ ......................... 104 Models Definitions ................................ ................................ ................................ 106 Materials ................................ ................................ ................................ ......... 106 Geometries and Loads ................................ ................................ ................... 107 Results and Discussion ................................ ................................ ......................... 111 J Q and Critical Crack Angle ................................ ................................ .......... 111 Near Tip Field Characterization Limits ................................ ........................... 115 Constraint Loss and Triaxiality in the Full Scale COPV ................................ .. 117 Concluding Remarks ................................ ................................ ............................. 121 7 FRACTURE PREDICTION WITH TWO PARAMETERS ................................ ...... 138 Model Definitions and Experimental Methods ................................ ....................... 139 Material ................................ ................................ ................................ ........... 139 Experimental Methods ................................ ................................ .................... 139 J integral and Constrai nt Factors ................................ ................................ .......... 140 Two Parameter Stress Field Characterization ................................ ...................... 141 Concluding Remarks ................................ ................................ ............................. 141 8 SUMMARY ................................ ................................ ................................ ........... 146
7 APPENDIX A POST PROCESSING PSEUDO CODE ................................ ............................... 149 B MATERIAL PROPERTY INPUTS ................................ ................................ ......... 152 Ramberg Osgood ................................ ................................ ................................ 152 Linear Plus Power Law ................................ ................................ ......................... 152 6061 T6 Tensile Data ................................ ................................ ........................... 153 C LINEAR PLUS POWER LAW DERIVATION ................................ ........................ 155 D INTERPOLATION METHODS ................................ ................................ .............. 159 E COMPOS ITE MATERIAL PROPERTIES ................................ ............................. 162 LIST OF REFERENCES ................................ ................................ ............................. 168 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 173
8 LIST OF TABLES Table page 4 1 Dimensions of models used in the analysis ( a = 0.889 mm (0.035 in)) .............. 70 5 1 Surface crack model dimensions ................................ ................................ ........ 95 6 1 Dimensions of the metallic liner models ................................ ........................... 123 6 2 Dimensions of the COPV models ................................ ................................ ..... 123 6 3 J Q dominance limit loads at = 24 ................................ ................................ 123 6 4 Points of large scale yielding ( J/J EL = 1.10) ................................ ...................... 123 6 5 Q at o ................................ ................................ ................................ ...... 123 7 1 Crack dimensions, FEA measured J and Q values and critical angles ............. 143 7 2 J Max and Q Max values and angles ( MBL ( T = 0 r = 2 o ) = 2.75) ....................... 143 B 1 Initial and final stress and plastic strain inputs for a LPPL model ..................... 154 B 2 Stress and plastic strain inputs 6061 T6 aluminum ................................ .......... 154
9 LIST OF FIGURES Figure page 1 1 Examples of common COPV geometries ................................ ........................... 31 1 2 Stress strain curve for the auto frettage cycle of a COPV ................................ .. 31 1 3 Picture of a COPV used for fracture testing ................................ ........................ 32 1 4 Surface crack face from a subscale COPV liner t = 2.29 mm (0.09 in.) .............. 32 1 5 Outer strain gage measurements from COPV test specimen with FEA comparisons ................................ ................................ ................................ ....... 33 2 1 Diagram showing the relationship between constraint and deformation for various fracture characterizations ................................ ................................ ....... 49 2 2 Plastic zone shapes diagram for high ( o = 1), zero ( o = 0) and low ( o = 1) T stresses ................................ ................................ ................................ .. 49 2 3 Modified boundary layer (MBL) model mesh and displacement boundary conditions. ................................ ................................ ................................ .......... 50 2 4 MBL generated near tip op ening stresses as a function of normalized radial distance r/ ( o ) for a series of T stresses ranging from T / o = 0.4 to 0.9. ........ 50 2 5 Relationship between normalized T stress and Q ................................ .............. 51 2 6 R eference opening stress as a function of o (A) and Q (B) for normalized radial distances r/ ( o ) = 2, 4, 6, and 8. ................................ ............................. 51 2 7 Examples of the near tip opening stress fields at a constant far f ield stress = o as a function of normalized radial distance r/ ( o ) ................................ ..... 52 3 1 General uniaxial surf ace crack model configuration ................................ ........... 57 3 2 Example of crack front co llapsed face nodal configuratio n ................................ 58 3 3 Finite element surface cracked mesh refinement detail. ................................ ..... 59 3 4 Data obtained from a single edge notched tensile (SEN(T)) specimen demonstrating the effects of using linear (small strain) and non linear (large strain) kinematic element formulations ................................ ............................... 60 4 1 Ramberg Osgood material models for a low ( n = 40) strain hardening material and 6061 T6 aluminum test data ................................ .......................... 71 4 2 General bonded surface crack model configuration ................................ ........... 72
10 4 3 J integral as a function angle at a constant nominal load = o .................... 73 4 4 T stress scaling factor as a function of angle ................................ .................. 73 4 5 Opening stress normalized by MBL reference solutions as a function of normalized T stress ( o ) at r/ ( o ) = 2 at = 90 (A) and 30 (B). .................. 74 4 6 Opening stresses normalized by MBL reference solutions as a function of normalized nominal stress ( ) at r/ ( o ) = 2 at = 90 (A) and 30 (B). ........... 7 5 4 7 Opening stresses normalized by MBL reference solutions as a function of deformation factor o /J at r/ ( o ) = 2 at = 90 (A) and 30 (B). ..................... 76 4 8 Opening stresses normalized by MBL reference solutions as a function of angle at r /( o ) = 2 and o = 0.95 (A) and o = 1.0 (B) .......................... 77 4 9 Normalized nominal load o at opening stresses corresponding to plastic collapse ( yy / MBL = 0.95) at various crack angles ................................ .......... 78 4 10 N ormalized J integral ( J/ ( o )) at opening stresses corresponding to plastic collapse ( yy / MBL = 0.95) at various crack angles ................................ ........... 78 5 1 Linear plus power law material models for the high ( n = 3) and low ( n = 20) strain hardening materials used in the surface crack analysis ............................ 96 5 2 General surface crack model configuration ................................ ........................ 97 5 3 J integrals as a function of crack angle at a constant average net section stress Net = o ................................ ................................ ................................ .... 98 5 4 Q as a function of crack angle at a constant average net section stress Net = o for a/t = 0.70 and 0.35 and aspect ratios a/c = 1.0 (A) and 0.40 (B). .......... 98 5 5 Local deformation factor o /J at crack angle = 30 as a function of normalized average net section stress Net o ................................ .................... 99 5 6 Schematic representation of the relationship between the crack tip blunting displacement ( u y ) and CTOD in a typical crack tip from this analysis. ................ 99 5 7 Crack tip blun ting displacement u y verses J integral at crack angle = 30 for a/t = 0.70 and 0.35 for aspect ratios a/c = 1.0 (A) and 0.40 (B). ....................... 100 5 8 Opening stresses normalized by the MBL predicted stresses at a constant normalized radial distance r/ ( o ) = 4 and crack angle = 30 as a function of normalized average net section stress Net o ................................ .............. 100 5 9 Opening stresses normalized by the MBL predicted stresses at a constant normalized radial d istance r/ ( o ) = 4 and crack angle = 30 as a function of local deformation factor o /J ................................ ................................ ....... 101
11 5 10 Opening stresses at an angle = 30 as a function of normalized radial distance r/ ( o ) for a load o /J = 40 (large deformation) ................................ 102 5 11 Opening stresses normalized by the MBL predicted stresses at a constant normalized radial distance r /( o ) = 4 as a function of crack angle ............... 103 6 1 Uniaxial stress strain data for 6061 T6 aluminum liner material and Abaqus material inputs for incremental plasticity ................................ ........................... 124 6 2 Surface crack from sub scale COPV ................................ ................................ 124 6 3 General COPV liner section surface crack model configuration ....................... 125 6 4 Strain da ta from the sub scale COPV specimen and 3D FEA results .............. 126 6 5 FEA visualization of the surface cracked liner sub model ................................ 127 6 6 J (A) and Q (B) crack front distributions for the sub scale COPV and liner models at a nominal equivalent stress = o ................................ ................ 128 6 7 J (A) and Q (B) crack front distributions for the full scale C OPV and liner models at a nominal equivalent stress = o ................................ ................ 129 6 8 The product of J ( ) and opening stress at a radial distance ( r = 2 o ) normalized by the maximum ................................ ................................ ............. 130 6 9 J normalized by J IC as a function of nominal equivalent liner stress ( ) for angles = 24 (A) and 90 (B) in the sub scale COPV and liner models. ....... 131 6 10 J normalized by J IC as a function of nominal equivalent liner stress ( ) for angles = 24 (A) and 90 (B) in the full scale COPV and liner models. ........ 131 6 11 Opening stress ( y y ) normalized by J Q MBL predicted stress at r = 4 o as a function of nominal equivalent liner stress (A) and local deformation factor o /J (B) at = 24 in the sub scale models. ................................ .......... 132 6 12 Opening stress ( yy ) normalized by J Q MBL predicted stress at r = 4 o as a function of nominal equivalent liner stress (A) and local deformation factor o /J (B) at = 90 in the sub scale models. ................................ .......... 132 6 13 Opening stress ( yy ) normalized by J Q MBL predicted stress at r = 4 o as a function of nominal equivalent liner stress (A) and local deformation factor o /J (B) at = 24 in the full scale models. ................................ ........... 133 6 14 Opening stress ( yy ) normalized by J Q MBL predicted stress at r = 4 o as a function of nominal equivalent liner stress (A) and local deformation factor o /J (B) at = 90 in the full scale models. ................................ ........... 133 6 15 T stress scaling factors ( h ) ................................ ................................ ........... 134
12 6 16 Q and Q(T) as a function of local deformation o /J. ................................ ........ 135 6 17 Q and Q(T) as a function nominal equivalent stress ................................ ... 136 6 18 The ratio of mean stress to von Mises stress at a radial distance r = 2 o verses Q for the full liner and COPV models ................................ .................... 137 7 1 J Q and Q crack front distributions normalized by their respective maximums. ................................ ................................ ................................ ....... 144 7 2 Opening stresses at the experimental critical crack angle ( Crit ) as a function of normalized radial distance r/ ( o ) ................................ ................................ 145 C 1 Schematic of the transition region in a linear plus power law material model. .. 158 C 2 Material constant K 2 uniquely defined as a function of strain hardening exponent n in the linear plus power law model. ................................ ................ 158 D 1 Example of cubic spline interpolation of nodal points for a surface crack geometry and a linear plus power law material. ................................ ............... 161 E 1 Example of angle ply orthotropic elasti c stiffnesses as a function of transformation angle ................................ ................................ ..................... 166 E 2 Example of angle transformation angle ................................ ................................ ..................... 166 E 3 Example of angle ply orthotropic shear moduli as a function of transformation angle ................................ ................................ ................................ ............. 167
13 LIST OF ABBREVIATION S ASTM American Society for Testing and Materials CT Compact Tension COPV Comp osite Overwrapped Pressure Vessel CMOD Crack Mouth Opening Displacement CTOD Crack Tip Opening Displacement EDM Electrical Discharge Machining FAD Failure Assessment Diagram LARC Langley Research Center LSY Large Scale Yielding LEFM Linear Elastic Fractur e Mechanics MEOP Maximum Expected Operating Pressure MBL Modified Boundary Layer NESC NASA Engineering & Safety Center NASA National Aeronautics and Space Administration NDE Non Destructive Evaluation SENB Single Edge Notched Bend SENT Single Edge Notched Tensile SSY Small Scale Yielding SCB Surface Cracked Bend SCT Surface Cracked Tensile
14 NOMENCLATURE t L thickness of the metallic section t b thickness of the bonded section for uniaxial specimens t Hel thickness of the helical overwrap layer t Hoop thickness of the hoop overwrap layer L half length w half width a crack depth c half crack width r radial distance from crack tip d i inner pressure vessel diameter d o outer pressure vessel diameter wrap angle measured from the longitudinal direction on the outer pressure vessel surface J J integral T T stress Q near crack stress difference constraint parameter Q normalized opening stress ( y y / o ) in the crack plane at r = 2 J / o y 0.2% offset yiel d strengt h o reference yield stress ( commonly o = y ) o yield strain n strain hardening exponent nominal stress of metallic section (Uniaxial: opening direction, Biaxial: Von Mises) Net average stress in the opening direction over the remaining area in the crack plane
15 far field displacement u y crack tip blunting displacement angle in plane perpendicular to crack front, = 0 is in the growth direction angle measured along the crack front in the crack plane
16 Abstract of Dissertation Prese nted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy J T AND J Q CHARACTERIZATION OF SURFACE CRACK TIP FI ELDS IN METALLIC LINERS UNDER LARGE SCALE YIELDI NG By Shawn A. English May 2011 Chair: Nagaraj K. Arakere Major: Mechanical Engineering C omposite over wrapped pressure vessels (COPV) are widely used in aerospace applications Understanding surface crack behavior in COPV metallic liners is challenging b ecause, (i) the liner experience s initial plastic deformation during manufacturing processes, (ii) the liner may experience cyclic plastic strains during normal operation, and (iii) surface crack behavior may be strongly influenced by the constraint effe c ts of the bonding. Methodology is developed to assess the constraint effects imposed by the backing on the surface crack tip stress fields in the liner. Near tip stress fields in uniaxial tensile loaded metallic liner specimens are developed using J T characterization and modified boundary layer (MBL) solutions where J measures the asymptotic field and T measures constraint. The increased elastic constraint imposed by the ba cking on the liner results in enhanced validity of J T characterization. In ad dition, presented is a rigorous study of strain hardening e ffects on near tip stress fields in surface crack s using J Q MBL reference fields and near tip deformation estimates where Q is a stress differe nce constraint parameter At moderate loads t he ra dial independence of Q cannot be assured for a low strain hardening material. I n
17 ASTM geometric standard development, this effect must be considered before a conservative limit is appli ed These methodologies are applied to sub scale and full scale COPV m odels. J Q fracture prediction of surface cracks in COPV metallic liners is found to be poss ible up to large deformations. Surface cracks in bonded geometries produce higher near tip triaxiality with respect to constraint level for large d eformations T his indicates that while J Q and J T predicted stress fields are valid at large loads in these region s the nonlinearity may hinder Q as a ductile fracture parameter. The results from this study aim to address the feasibility of using a two parameter fract ure theory to predict failure and characterize limits of both surface crack specimens and COPV liners These results and methodologies are of practical value for establishing testing standards for surface crack geometries, de velopment of proof test logic for COPVs and broader establishing of methodologies for assessing near tip dominance, parameterization deformation limits and plastic material property effects in fracture prediction.
18 CHAPTER 1 INTRODUCTION Fracture prediction has been studied extensively using both single parameter ( K for brittle fracture and J for ductile fracture) and two parameter ( J T and J Q ) characterizations. Both methods of fracture prediction are engineering approximations and thus have quantifiable limitations. A detailed unde rstanding of the meaning behind the fracture parameters is necessary to determine the distinct limitations for their use. The linear elastic stress intensity factor ( K ) is only valid when the crack tip experiences only moderate plastic deformations. Once the crack tip deformations exceed the applicable limits of linear elastic fracture mechanics (LEFM), an elastic plastic fracture parameter, the J integral, is introduced to predict failure. Under geometric conditions in which the stress state is not dete rmined by the single term K or J a constraint parameter can be introduced to help accurately predict the stress state and therefore the fracture process. The use of a single asymptotic term is limited to geometries that maintain high constraint, such as the compact tension (C(T)) specimen where the in plane stresses are dominated by bending. The two parameter appr oach also has its limitations. Many factors contribute to the validity range for two parameter characterization. Geometric and loading conditi ons, material properties and bi material interactions for structures with material discontinuities play important roles in the fracture process and in choosing viable fracture parameters. In this study, novel techniques are implemented to assess the geome tric and material intricacies that a complex structure has on two parameter characterizations. In particular, a liner/overwrap bi material structure with a liner surface crack is investigated. Surface cracks are among the most common flaws
19 present in str uctural components and frequently the most critical in terms of limiting structural life, for both ductile and brittle materials (Leach et al., 2007; Evans and Riley, 1983) Methodologies for evaluating the effects of the elastic backing on the two parame ter characterization are developed using elastic plastic finite element modeling. N ew techniques in constraint corrected stress field analysis are developed to evaluate hardening and stiffness material properties, loading conditions, and geomet ric constra ints. Homogeneous uniaxial surface crack specimens are generated for the investigation thickness effects and comparison to more complex bonded models. The homogeneous uniaxial models are also used in a detailed study on the effects of plastic material pr operties namely the strain hardening exponent in a power law hardening material, on two parameter near tip field characterization. A preliminary study is implemented to investigate the effects of a structure containing a through thickness elastic plastic to fully elastic material discontinuity in a uniaxial isotropic analysis. The elastic modulus ratio of the elastic plastic cracked body and the fully elastic backing is also varied in this analysis giving a range of possible stiffnesses seen in a more co mplex geometry such as a COPV. The culmination of methodologies and material/geometrical investigations is the sub scale and full scale COPV models containing liner surface flaws and actual material properties. This study fully investigates two parameter near tip stress field parameteriz ation. Results from th ese studies will facilitate the implementation of geometric limits in testing standards and proof test logic for surface cracked specimens and COPV liners
20 Composite Overwrapped Pressure Vessels The characterization of tip fields and fracture prediction of metal liners with surface cracks bonded to an elastically loaded structural backing has not been well established. The fracture prediction of such structures is of intrinsic and practical interest with wide application in bonded pressure vessel technology such as composite overwrapped pressure vessels (COPV). COPVs, used primarily in aerospace applications, generally consist of a thin (0.76+ mm) metallic (typically aluminum, titanium or stainless s teel) inner liner wrapped with a comparatively thick (greater than three times the liner thickness) composite shell, commonly made from carbon/epoxy. The shell/wrap serves as the primary structural component, with the liner supporting very little load, bu t acts as a barrier for leakage of propellants and other pressurants. During the manufacturing process, COPVs undergo an auto frettage pressure cycle in which the liner ex periences plastic deformation in order to compensate for thermal strains induced from curing and the mismatch in thermal expansion between the liner and overwrap resulting in residual compressive stresses in the liner An additional benefit of auto frettage is that the plastic strains accumulated in the liner will effectively extend the e lastic range under operating conditions. Figure 1 2 gives an example of the stresses and strains during an auto frettage cycle for an isotropic hardening material. The compressive loading comes from the overwrap elastically returning to its original shap e, deforming very little from the stresses in the liner. The result is an ope rating range that can handle greater elastic strains. Under some extreme situations, the liner may undergo post manufacturing proof/operating plastic strain, but these loads wil l typically be for very low cycle operation.
21 Understanding the behavior of surface cracks in COPV liners is challenging because, (i) the liner may experience initial plastic deformation during manufacturing processes, (ii) the liner may experience cyclic p lastic strains during normal operation, and (iii) surface crack behavior may be strongly influenced by the constraint effects resulting from the bonding. From a fundamental fracture mechanics perspective, the objective is to understand the constraint effe cts that arise in a thin liner bonded to a stiff composite and its role in characterizing crack tip fields. NDE, Proof Test Logic Non destructive evaluation (NDE) is performed on all fracture critical hardware NDE methods are used to detect existing flaws. The existence of a detectable flaw deems the hardware unusable. The NASA standard NASA STD 50 0 9 ( NASA, 2008) requires that all fracture critical parts shall be subjected to NDE and/or proof testing to screen fo r internal and extern al cracks. NDE methods are selected for all part or component life cycles, including but not limited to manufacturing, maintenance, and operations. NDE detection techniques for cracks in metals include eddy current, p enetrant magnetic p article r adiogra phic and ultrasonic. The minimum detectable crack size from these methods is assumed present in the structure at the most critical location. Simulated service life is then evaluated for fracture and fatigue. This method must demonstrate that the larges t flaw not detectable by NDE can survive at least four mission lives. Proof testing and if necessary post proof NDE are implemented to test hardware for service life given a verifiable proof safety factor from the above analysis. Proof testing begins wi th crack specimen testing and single parameter critical values or two parameter failure locus. The limit of applicability and level of conservativeness can be assessed by experimental repeatability of test specimens,
22 ensured us ing ASTM standards, and a de tail ed numerical and analytical analysis of specimen and structure near tip stress field characterizations. Aerospace pressurized structures do not lend themselves to simplified techniques such as LEFM, due to load levels up to net section yield. Additio nally, a luminum and its alloys often exhibit the ductile tearing mechanism during crack growth. This mechanism yields large plastic zones and crack tip deformation making numerical and experimental predictions difficult. Therefore, proof test logic must be supported by a rigorous simulated service test program using detailed numerical simulations and experimental coupons to evaluate proof factors, failure loci and parameterization limits. t progresses to unacceptable levels during normal operation. The smart flaw falls below critical limits achieved during proof testing yet extends and fails during normal operation. Comple x geometric constraints resulting in highly non linear crack respon se with growth, such as a surface flaw in the liner of a bonded geometry, near tip field analysis can aid in preventing the existe Coupon and Subscale Testing osite overwrap pressure vessel initiative includes the cyclic loadings of uniaxial surface cracked plate specimens and sub scale COPVs containing liner surface flaws. Figure 1 3 is an example of one of the COPVs tested at the NASA Langley Research Center (LARC). Figure 1 4 shows one of many crack surfaces from a sub scale COPV tested at NASA. The material is 6061 T6 aluminum. In addition, crack surfaces from many homogeneous uniaxial surface cracked coupons are available for analysis.
23 Figure 1 5 shows data from a COPV test specimen loaded under pneumatic pressure loadings. The strain measurements are taken from the center of the cylinder shown in Figure 1 3. The FEA results from a 3D elastic plastic FEA model with anisotropic overwrap are shown and match well with the experiment. Fracture test specimens validate the more complex composite p ressure vessel modeling implemented in a subsequent section Using the global strain results and the fracture surface measurements, a numerical model can be constructed with confidence to verify the fracture parameterization of the liner crack. Further COPV sub scale testing will be discussed in the final analysis section, Chapter 6. Two Parameter Fracture Mechanics Plastic deformation precedes fracture in metals and their alloys and resistance to fracture is therefore directly related to the development of the plastic zone at the crack tip. Among the major causes of metallic structural failure is the nucleation and propagation of cracks from regions of high stress concentration such as notches and surface flaws, due to both monotonic and fatigue loading. Understanding the evolution of plasticity in notches and cracks is therefore import ant for predicting fracture behavior of critical load bearing structures in many engineering applications. Consequently, development of crack tip elastic plastic fields, as a function of load configuration, geometry, monotonic and cyclic strain hardening behavior, and constraint effects have been subject of intensive study. Behavior of the crack tip in the plastic zone for strain hardening materials under small scale yielding for symmetric (Mode I) or antisymmetric (Mode II) two dimensional (2D) stress d istribution has been presented in the widely referenced HRR articles by Hutchinson (1968a, 1968b) and Rice and Rosengren (1968). Extensions to
24 combinations of Mode I and Mode II loadings were presented by Shih (1973, 1974). Comprehensive literature revie ws of plane strain fracture mechanics and crack tip field characterization are found in Anderson (2004), Irvin and Pari s (1971), McClintock (1971), Panayotounakos and Markakis (1991), and Rice and Rosengren (1968). The aforementioned works refer to fract ure prediction using single parameter ( K for brittle fracture and J for ductile fracture) crack tip field characterizations. The J integral alone does not uniquely and accurately describe the crack tip fields and resistance against initiation of ductile c rack growth when constraint effects arising from geometry and load configuration are considered. The constraint can be thought of as a structural obstacle against plastic deformation induced mainly by the geometrical and physical boundary conditions (Yuan and Brocks, 1991). Constraint effects can also arise from mismatch of material properties in a heterogeneous joint (English et al., 2010). Under these conditions, a second parameter, in addition to the J or K is introduced to quantify the crack tip con straint. Crack tip triaxiality, which is defined as the ratio of hydrostatic pressure to von Mises stress, or a parameter that maintains a linear relationship with triaxiality, can be used as a constraint parameter for predicting ductile crack growth. Th e in plane constraint is influenced by the specimen dimension in the direction of crack growth or the length of the uncracked ligament, and by global load configuration (bending or tension), whereas the out of plane constraint is controlled by the specimen dimension parallel to the crack front or the specimen thickness, for thorough thickness cracks (Giner et al., 2010 ). Out of plane constraint is typically denoted as a plane strain (high triaxiality) or plane stress (low triaxiality) state. In finite thi ckness fracture geometries, where the points are embedded entirely in material, there
25 exists a state of plane strain near the crack tip, the exception being very thin film s 1993; Wang, 2009). J T and J Q Characterization of Near Tip F ields Williams (1957) showed the existence of a non singular in plane normal stress component ( T stress) for linear elastic material. The significance of the T stress on the size and shape of the plastic zone under small scale yielding (SSY) conditi ons was shown by Larson and Carlsson (1973) and Rice and Tracey (1991, 1992) and Sharma and Aravas (1991) show the important role played by higher order terms in asymptotic solutions of crack tip fields and demonstrate that a two p arameter characterization of the crack tip fields involving J and a triaxiality or constraint parameter Q is necessary to satisfactorily describe the configuration dependence of fracture response of isotropic plastic solids, particularly under large scale yielding conditions. The use of a single parameter characterization ( K or J ) is limited to geometries that maintain a high constraint, such as the compact tension (C(T)) specimen, where the in plane stresses are dominated by bending. The Q stress differe nce factor has been found to maintain a material dependent linear relationship with near tip triaxiality independent of geometry, dimensions and deformation level. Moreover, the Q factor is shown to accurately predict near tip stress and strain fields, pa rticularly plastic strains, and therefore can be used as a ductile fracture parameter (Henry and Luxmoore, 1997; Kikuchi, 1995). A significant body of work has focused on two parameter characterization of through thickness cracks while limited work has bee n done on surface crack geometries. Wang (1993) established the J T characterization with modified boundary layer (MBL) reference solutions to topographical planes perpendicular to the crack front
26 in surface crack tension (SC(T)) and surface crack bend (S C(B)) geometries. Wang (2009) investigated Q as a function of load and radial distance from the crack tip for surface crack uniaxial and biaxial tension models. A J T or J Q family of fields generated with MBL formulation is used as a comparison for ass essing near tip dominance in actual structures. A near tip region is said to have J T or J Q dominance if the stress and strain distributions match the MBL reference field when the length scale is normalized by o where o is the yield stress. in addressing the limitations of plane strain references applied to three dimensional geometries, suggested that as the distance from the crack tip becomes small, the out of plane ( parallel to the crack front) strains become negligible when compared to the in plane singular fields. Therefore, plane strain MBL prediction of surface crack front stress fields should be accurate as the distance from the crack tip approaches zero. Fractu re P rediction Two parameter fracture mechanics is important in engineering applications because it provides a more adjusted assessment of failure limits (Cicero et al., 2010; Silva et al., 2006 ). The primary source of enhancement stems from the constraint effects on near tip plastic deformations and subsequent effect on critical loads. Iwamoto and Tsuta (2002) and Li and Chandra (2003), in their studies of crack growth characteristics in ductile materials, describe the direct influence of plastic strain a nd strain induced material transformations at the crack tip on apparent fracture toughness. Faleskog (1995) showed experimentally how an increase in the constraint factor Q results in a decrease in the critical value of J Chao and Zhu (2000) and MacLenn an and Hancock (1995) numerically and experimentally verified this effect on J resistance
27 ( J R) curves and failure assessment diagrams (FAD) respectively, with the highest constraint geometries having the lowest resistance. Leach et al. (2007), in a study using J and an average opening stress constraint parameter to predict critical crack growth angles in surface crack tension SC(T) and bend SC(B) specimens, verified the enhanced accuracy of a constraint modified fracture criterion. Similar results can be found throughout literature and it is now generally accepted that the increase in energy dissipation through plastic deformation due to loss of constraint can account for the increase in apparent fracture toughness. Deformation L imits Many investigators h ave examined single and two parameter characterizations limits (Kim et al., 2003; Larsson and Carlsson, 1973; McMeeking and Parks, 1979; and Shih, 1991, 1992; 1995; Varias and Shih, 1993; Wang, 1993; Wang and Parks, 1995; Wang, 2009; Zhu and Chao 2000 ). In single parameter fracture mechanics, the dominance limit of the asymptotic term is controlled by geometry and load/deformation. With the introduction of a non singular geometry dependent constraint term, such as T or Q the limits of cha racterization can be expressed by a load or deformation term. Wang and Parks (1995) considered the parametric limits of the constraint corrected asymptotically singular near tip stress fields based on crack tip deformations measured by o relative to a characteristic length l such as the remaining ligament or crack depth. The limiting relative local deformation factor o /J cr where J cr is the J integral at which two parameter dominance of near tip fields cannot be assured, has been suggested by many investigators and is a requirement in ASTM E1820 for compact tension (C(T)) and single edge notched bend (SEN(B)) specimens (ASTM, 2006). This factor is dependent on loading conditions and
28 material parameters, but given a proper definit ion of l and methodology to determine field dominance, a conservative limit locus can be found. Geometries with deformation limit loads below critical fracture load will often have inflated toughness values with increased scatter. However, stress fields p redicted using a micromechanical model within two parameter deformation limits can be used to construct a constraint modified locus for cleavage fracture (Faleskog, 1995). In addition to experimentally establishing J and Q as accurate predictors of cleava ge fracture in surface crack tensile (SC(T)) specimens, Faleskog (1995) provides a reasonable verification for the J Q approach. He observes that the hoop stress ( ) deviator of the Q defined difference fields is relatively consistent along the crack front for loads causing cleavage fracture, providing verification that Q is a proper measure of triaxiality. In addition, he uses MBL predicted opening stress fields to prove the existence of J Q dominance in the crack plane at the cleavage fracture initiation positions along the crack front. The preliminary limitations analysis by MacLennan and Hancock (1995) of opening stress fields compared to MBL predictions is u sed to validate their experimentally constructed modified FAD approach. To achie ve a proper verification of two parameter deformation limits, micro scale deformation data and detailed near tip stresses are necessary. Therefore, finite element analysis is used to extrapolate experimental data in order to verify the existence of J Q fields, as in MacLennan and Hancock (1995). Furthermore, the loss of two parameter near tip field dominance can only be predicted using numerical methods and can be observed sec ondarily from various experimental results.
29 Paul and Khan (1998) in studying finite element models of a centrally cracked thin circular disk show that the inability to maintain hydrostatic pressure accounts for the reduced stress state at the crack front e ven with small strain approximations, but uncontained hydrostatic pressure very close to the crack tip results in high triaxiality and a maximum opening stress hump. This region, denoted as the process zone, is where damage from void coalescence and crack growth commonly takes place (Xia and Shih, 1995) This phenomenon can help to explain the near tip stress field variances from MBL predicted stresses, namely the rapid opening stress reduction at distances after the hump for large levels of deformation. The importance of including effects of hydrostatic stress and the third invariant of the stress deviator on stress state computation has been shown by Bai and Wierzbicki (2008) and Gao et al. (2009, 2010) since it can play an important role on both the pla stic response and ductile fracture behavior of certain aluminum alloys. Ductile fracture analysis can be used subsequent to or in conjunction with traditional fracture parameterizations and has been the topic of much interest (Li et al., 2010; Mirone and C orallo, 2010; Wei and Xu, 2005; Xue et al., 2010) Li et al. (2010) give a thorough analysis of ductile fracture computational techniques and experimental results for a power law material in which the undamaged response follows J 2 plasticity theory. Alth ough this initial isotropic material model is widely used for numerical evaluation of fracture parameters and near tip stress field analysis, ductile fracture techniques using void growth, shear band and anisotropy are necessary for accurate prediction of critical failure beyond limit deformations (Huespe et al., 2009; Sun et al., 2009). These types of analyses are important for the failure assessment of structures
30 undergoing plastic deformations during normal operation in which the combination of geometry and plastic material properties prevents single and two parameter fracture prediction but add a greater level of complexity.
31 Figure 1 1. Examples of common COPV geometries Figure 1 2. Stress strain curve for the auto frettage cycle of a COPV
32 Figu re 1 3. Picture of a COPV used for fracture testing Figure 1 4. Surface crack face from a subscale COPV liner t = 2.29 mm (0.09 in.)
33 Figure 1 5. Outer strain gage measurements from COPV test specimen with FEA comparisons
34 CHAPTER 2 BACKGROUND Linea r Elastic Fracture Mechanics In modern fracture mechanics, crack sizes determined by the resolution of non destructive evaluation techniques are often assumed to be in a structure. The stability of an existing crack is the primary focus of fracture mechan ics. For sharp cracks in an isotropic linear elastic material, the stress state approaching the crack tip produces a singularity. Linear elastic fracture mechanics (LEFM) is the practice of design in such a situation. Understanding the basics of LEFM an d its limitations is necessary to assess the fracture problem of more complex materials and structures. The loading of a sharp crack in an isotropic linear elastic material produces a singularity. This stress state is proportional t o the stress intensity factors K I K II and K III for mode I, II and III (opening, sliding and tearing modes) respectively. Th e stress state is defined by (2 1) (2 2) (2 3) w here and are dimensionless functions of that are representative of the angular variation in stress for mode I, II and III load ings respectively. Williams (1957) showed that the above stress state is a series, where the first two terms under mode I loading are
35 (2 4) w her e T called T stress, is an elastically scaled parameter that plays an important role in elastic plastic fracture mechanics and will be discussed in the subsequent section. The linearly scalable stress intensity factor can be calculated in closed form for simple configurations. However, numerically determined fits and experimental methods are needed for geometries that are more complicated. The use of a single parameter to fully describe the stress state suggests that a material dependent critical stress intensity factor exists that will define the failure state of any crack in a structure. This practice has been used for many decades and is verified by extensive testing and standards, but has quantifiable limitations. In a real material, the infinite st ress prediction at the crack tip will not exist. The result is crack tip blunting due to plastic deformation. For moderate levels of plastic deformation, corrections are needed to use LEFM, but the K predicted stress state is still valid for regions suff iciently far from the plastic zone. Therefore, under certain conditions the stress intensity factor K can be used as a single term to predict failure. The size of the plastic zone compared to relevant structural dimensions must be small. Past emphasis h a s been placed on plane strain fracture for test specimen geometries. It is commonly accepted that the lower bound critical fracture toughness is found under plane plane strain ( K Ic ), which is consid ered a unique material property. The actual state of lower bound critical fracture is more closely related to the triaxiality or constraint at the crack tip, where the apparent fracture toughness is greater for lower constraint geometries and loadings. S ince the state of plane strain has high crack tip triaxiality the resulting
36 toughness values would appear smaller. Constraint will be discussed in a subsequent section. Elastic Plastic Fracture Mechanics For high toughness, low strength materials (high K c and low y ) the criterion for LEFM are difficult to satisfy. Large plastic zones before fracture invalidate LEFM assumptions. A different approach to fracture mechanics is necessary to account for the larger plastic deformations at the crack tip. Elast ic plastic fracture mechanics assumes crack growth occurs under stable ductile tearing and terminates by tearing instability. A new term that does not rely on the crack tip singularity and accounts for the plastic deformation is ne cessary to predict fracture. The J integral is crack tip: (2 5) where: w = strain energy density T i = components of the traction vector u i = displacement vector components ds = length incre (2 6) (2 7) where n j The units for J are energy per unit area; therefore, it is defined in an elastic material as the energy relea se rate (the energy released for a unit area of crack growth). Much like the stress intensity
37 factor K the J integral can be used as a single parameter for characterizing crack front fields. The benefit of a J integral approach verses other elastic plas tic methods (such as crack tip opening displacement or CTOD) is that under small scale yielding (SSY), the J integral simplifies correlatively to K A critical value of J exists to characterize fracture if the asymptotic fields about the crack tip are char acterized by this single parameter. This only happens when constraint is zero and SSY conditions apply. As large scale plasticity becomes prevalent, the stress field begins to deviate from the SSY curve. For a given crack tip with plasticity, annuli mea suring the extent of asymptotic dominance can be expressed. The annulus near the tip within the area affected by crack tip blunting is represented by plastically dominated large strains, then extending radially the J dominated region and finally the K dom fracture parameters accurately represent the near crack stress state for a given geometry and loading condition. Two Parameter Characterization Under large scale yielding geomet ry and size considerations in fracture become necessary. Much research has been done on the geometric effects on ne ar crack stress fields ( Leach et al., 2007; Wang and Parks Sharma and Aravas, 1991; Rice and Rosengren, 1968; Betegn and Hancock 1991; Wang 1993; Wang and Parks, 1995 ; Newman et al. 1993; Shih et al., 1993; 1995). Two parameter characterization methods exist to help predict failure by addressing the geometrically determined constraint of a struct ure. Figure 2 1 demonstrates the relationship between constraint, level of deformation and fracture parameterization necessary to characterize failure. Constraint is denoted
38 by and can be either T or Q ( o corresponds to T = Q = 0). The loading curves and critical points are shown for various testing standards. Examples from the Linear Elastic Fracture Toughness (E399), Elastic Plastic Fracture Toughness (E1820) and the Surfa ce Crack (E740) standards are shown. For a constraint sensitive material, the measured J c will be greater for low constraint test specimens such as middle crack tension (M(T)) or single edge notched tension (SEN(T)). Lower bound fracture toughness is ofte n found using a high lower bound fracture, the C(T) specimen is often used, but for a more adjusted design, a constraint corrected failure criteria is needed. Many invest igators have proposed the existence of a single failure measurement that is a function of both asymptotic and constraint parameters to predict crack growth and fracture (Leach et al., 2007; Wang and Parks 1995). For actual geometries fracture prediction s using standard test specimens are reliant upon the ability to predict the stress field within the damage zone around the crack tip. The limit of stress field prediction can be expressed as a region (annulus) around the crack tip or a load for a specific geometry and loading condition. Within this limit, stress fields can be compared directly, and damage prediction can be made from single (high constraint) and two parameter (low constraint) measurements. (1991) demonstrate that a two parameter characterization of the crack tip fields involving J and a constraint parameter is necessary to describe the stress fields under large scal e yielding conditions. Rice and Rosengren (1968) also notes the strong conf iguration dependence
39 of the near tip deformation fields, and that the asymptotic solution may not be applicable under large scale yielding conditions to low constraint fracture geometries such as center crack tensile (CC(T)) or single edge notch tensile (S EN(T)) specimens, where the stresses around the crack tip may be much lower. Two parameter characterization uses a measure of the asymptotic fields, such as the stress intensity factor K crack tip opening displacement (CTOD) or the J integral with an add itional constraint parameter, such as the T stress or stress difference Q to predict crack front stress fields. For loads causing deformation beyond the dominance limit, the near crack stress fields are considered plastically collapsed. Determining two parameter dominance relies on the ability to model idealized stress fields within the characterized range. This is done by use of modified boundary layer (MBL) solutions. an elastic plastic material. J T Theory The Williams expansion of a linear elastic stress field is a series with the leading term having a singularity and the second term being constant with r The second term is defined for mode I crack opening as the T s tress. Equation 2 4 shows the first two terms of the Williams expansion for mode I loading. T stress is defined as a uniform stress parallel to the crack plane and perpendicular to the crack front. Low crack tip constraint is ch aracterized by negative T stresses. Geometries with negative T stresses do not maintain single parameter J dominance because constraint loss is associated with a lowering of the stress state. However, negative T stress geometries can be described by the J integral and T stress up to and exceeding net section yield
40 ( Betegn and Hancock 1991) T stress is not defined under fully yielded conditions; nevertheless, elastically scaled T stresses calculated from far field loads in elastic plastic models are a relatively accurate constraint measurement for tw o parameter characterization (Wang, 1993) T stress is a measure of crack tip constraint (triaxiality); therefore, it changes the crack tip stress field conditions such as the opening stress (stress perpen dicular to the crack plane ) distributions an d plastic zone shape and size. strong constraint dependency on plastic strains near the crack tip for various power law hardening materials. Constraint or triaxiality directly influence s the size and shape of the plastic zone. Figure 2 2 demonstrates how normalized T stress ( o ) changes the plastic zone shape. Low T ( o = 1) results in a tendency away from the crack face while high T ( o = +1) shifts the zone towards the open ing face. Zero o results in a plastic zone common of th plane strain o is defined as the reference yield stress; this is typically taken as the 0.2% offset yield ( o = y ) unless otherwise noted. The applicable limits of J T two parameter characterization of elastic plastic crack tip fields using modified boundary layer (MBL) solutions have been explored by many investigators ( Newman et al. 1993; Wang and Parks, 1995) for th rough crack specimens. Wang (1993) extended the two parameter characterization with MBL reference solutions to topographical planes perpendicular to the crack front in surface crack tension (SC(T)) and surface crack bend (SC(B)) geometries. In addition to verifying the applicability of MBL solutio ns for J T limit determination in surface cracked plates, this study quantitatively explains the short falls of plane strain characterization.
41 The general conclusion was that a completely consistent two parameter description of crack tip stress fields is not possible beyond certain loads but is effective in determining the deformation limits. E lastically scaled T stress factors are readily available in literature for a number of geometries making it a valuable measure of constraint for engineering applica tions However, in large scale yielding conditions, this measure of constraint loses its physical meaning. J Q Theory The easily applied e lastically scaled T stress factors are useful in engineering applications; however, a direct measure of the stress fi eld within the plastic zone ( Q ) is a more accurate measure of constraint for geometries undergoing large scale plasticity J Q theory has been implemented for the characterization of near tip stress fields for 2D and 3D geometries by many investigators. Wang (2009) shows that the Q parameter as a measure of the difference between the actual opening stress and the small scale yielding solution is accurate for a range of normalized radial distances ( r/ ( o )) for loads exceeding yield, where the ratio o scales with the blunting zone diameter or crack tip opening displacement. The constraint parameter Q is defined as the difference between the actual hoop stress ( ) and a reference hoop stress within a cracked body. The hoop stres s ( ) is defined as the normal stress in the direction tangent to a circular region enclosing the crack tip. Figure 2 3 gives a definition of the and Shih (1991, 1992) showed that the following equation can be use d within a range of r/ ( o represent the near crack fields:
42 for 2 (2 8 ) where the first term is the SSY solution and ij is the Kronecker delta. The S SY solution In the analyses presented by Silva et al. (2006), plane strain results produce relatively unchanged Q values when the normalized radial distance fr om the crack tip is r/ ( o in three d imensional crack front fields of surface crack specimens, where the points are embedded entirely in material, there exists a state of plane strain near the crack tip. Therefore, va riations of the normalized radial distance within this range for J dominant surface crack fields should not change the general trends presented in the analysis or the conclusions derived from them. However, Bass et al. (1999) shows that under biaxial load ing, the opening stress difference fields do not correspond to the hydrostatic shift pro Shih (1991, 1992) indicated by the dependencies of Q on radial distanc e. Similarly, Wang (2009) shows that for deep surface cracks ( a/t = 0.6) unde r biaxial and uniaxial tension, Q fails to remain radially independent for angles close to the free surface at large deformations. These considerations make a thorough analysis of the applicable limits of Q as a ductile fracture parameter in pressure vess el geometries vital to predict crack growth and failure. The Q parameter is best defined as a fracture parameter when the normalized radial distance equals 2 J / 0 because this marks the location where cleavage mechanism is triggered ahead of the crack tip ( Silva et al., 2006). Therefore, in order to
43 be used in two parameter fracture characterization, Q is defined as the hoop stress difference in the cracked plan e (opening stress or yy ) at a radial distance equal to 2 o : at 0 and r = 2 J / 0 (2 9 ) A single value of Q represents a material dependent near crack stress field as a function of r/ ( o ). Using this definition of constrai nt, a plane strain J Q family of fields is constructed using the MBL finite element formulation. Nevertheless, the relative changes in Q with radial distance ( r ) when measured at larger deformations, necessitates the investigation of opening stress as a f unction of normalized radial distance r /( J/ o ). The deformation limit of J Q dominance may therefore be determined by the relative change of opening stress compared to those predicted at r = 2 J / 0 or when the radial independence of Q cannot be assured (Sharma et al., 1995). Modified Boundary Layer Formulation Determining two parameter dominance of a test specimen or structure relies on the ability to model idealized stress fields within the characterized range. Where analytical solutions exist to compare stress states in fully elastic cracked bodi es, the elastic plastic fields defined for an arbitrary material must be formulated using MBL finite element models. The MBL stress field solutions are derived from the existence of an asymptotic plane strain elastic stress field outside crack tip plastic zone under contained plasticity. This field is uniquely defined by the elastic parameters ( K and T ). Contained plasticity is defined as the state in which the plastic zone is on a length scale that is small compared to relevant dimensions. Under these conditions, the J integral simplifies correlatively to
44 K and therefore a region exists within the plastic zone that must be defined by the elastic plastic parameter J and a measure of constraint ( T or Q ). Moreover, the unique fields described by this sta te are representative of the stress state in any crack geometry and load configuration that is defined by these parameters. SSY conditions are a special case in the MBL formulation where the stress field measured outside the plastic zone but far from the geometry boundaries is characterized by the first singular term of the Williams Eigen expansion (Williams, 1957; Wang, 1993): (2 10 ) where K I is the mode I stress intensity factor. A material flow curve ( verses ), typically defined by uniaxial tension data, and the Poisson ratio are the only variables needed to produce a unique set of MBL reference fields. The MBL solution is therefore a derived material property. An accurate MBL reference field shoul d be reproducible when it is independent of K applied and mesh configuration as long as the formulation follows certain criteria defined later in this section The remote tractions for MBL formulation are given by the first two terms of the Williams Eigen expansion (Williams, 1957): (2 11 ) where K I is the mode I stress intensity factor and T is the T stress. Figure 2 3 shows the half symmetric plane strain FE mesh used for MBL analysis. The first two terms of the Williams series are applied as displacement boundary conditions to the r = r max outer surface. The corresponding in plane displacements are given by:
45 (2 12 ) where E and are material constants and g i and h i are the angular variations in displ acement caused by elastic singularity fields and T stresses respectively (Wang and Parks 1995). The angular variations are defined in plane strain as: (2 13 ) (2 14 ) (2 15 ) (2 16 ) Plane strain MBL solutions for elastic plastic crack tip fields can be obtained by applying these displacements to the outer boundary of the circular crack tip region and varying the T stress while K I remains constant. Th e J integral is related to the stress intensity factor K I in plane strain by: (2 17 ) Abaqus version 6.7 is used to compute the MBL solutions (Dassault Systmes, 2007 b ). Figure 2 4 shows the near crack openin g stress fields pre dicted by the MBL model for a 6061 T6 aluminum Wang (1993) showed that these MBL solutions provided accurate prediction of stress fields in surface crack tension (SC(T)) specimens as characterized by J and T and that the crack tip fields of the MBL sol ution far from the outer boundary and outside the crack tip blunting zone should represent those of any crack with the same K I and T
46 While an elastically scaled T stress can be used to accurately predict near tip stress fields up to and exceeding net se ction yield, the physical meaning of T is lost with the onset of large scale plastic deformation. Therefore, the Q factor, which is calculable at loads exceeding yield, is often used in stress field prediction Figure 2 5 shows the unique relationship bet ween T and Q in the MBL solutions. The near tip reference stress fields are often formula ted as a function of constraint for constant normalize d radial distances. Figures 2 6 (A) and ( B ) gives these curves for r/ ( o ) = 2, 4, 6, and 8 as functions of T and Q respectively Modified boundary layer solutions can only be formulated within certain constraint levels. As the compressive constraint stresses (negative T stress) approach the yield stress, crack tip deformations exceed the contained plasticity lim its and no longer provide an accurate representation of the J predicted field. Literature has shown the applied T stress limits to be approximately T = o but the actual limits can be determined by close examination of the near tip conditions. When the crack deformations exceed levels necessary for MBL formulation, the elastic plastic J integral ( J ) will deviate from the elastically predicted value ( J E l ), given by Equation 2 17 by a considerable amount. This can be formulated as an arbitrary limit where J must not deviate from J El by greater than 10%, as shown below: (2 18 ) One must also consider the limitation of MBL formul ation as a function of applied K With increasing applied K and the onset of large scale plasticity at a low level of constraint ( T o ), the J integral deviation, as defined by Equation 2 18 increases exponentially. Nevertheless, it is necessary to apply a sufficiently large K to achieve
47 radial data points close to the crack tip. Since K is normalized out for the final reference solutions, repeatability remains solely on adherence to the requirements of Equation 2 18 which also ensures deformations below large scale plasticity. Therefore, the MBL o 0.9) and a maximum K is applied such that the error of J at the minimum applied T stress ( T / o = 0.9) is within the above limits. Near Tip Stress Field Analysis and Plastic C ollapse The near tip stress fields are analyzed in the cracked plane along the vector normal to the cr ack front in the growth direction. The radial distance r from the crack front is normalized by o which gives a length scale comparable with MBL solutions. The nodal opening stress values are used along with cubic spline interpolation to determine the near tip stress distribution. From this curve, the value of Q is determined with Equation 2 9 The unique relationship of Q to a specific near tip field provides the reference solution for comparison. Figures 2 7 (A) and (B ) give examples of near tip o pening stress fields verses normalized radial distance ( o ) ) for high ( n = 3) strain hardening and low ( n = 20) strain hardening materials respectively as defined in a linear plus power law material model The J Q MBL predicted stress field and the neutral constraint (SSY) stresses are given as references for the actual stress state in a surface crack model with a/c = 1.0 and a/t = 0.35 at crack depth ( = 90) for a far field load = o Two processes exist by which near tip stress field parameterization is lost. First, single parameter dominance is l imited in low constraint geometries by a loss of constraint or triaxiality near the tip resulting in significant deviation from the SSY solution, even under contained plasticity. Secondly, two parameter dominance is limited by the amount of
48 deformation at the crack tip and the subsequent shape change and/or rapid relaxation of the fields compared to those predicted by MBL solutions. This process is referred to as plastic field collapse. Because the stress fields in J Q characterization are referenced to a point near the crack tip ( r = 2 o ), field collapse is measured as difference at larger radial distances or slope variation from the predicted field.
49 Figure 2 1. Diagram showing the relationship between constraint and deformation for various fractur e characterizations Figur e 2 2 Plastic zone shapes diagram for high ( o = 1), zero ( o = 0) and low ( o = 1) T stresses
50 Figure 2 3 Modified boundary layer (MBL) model mesh and displacement boundary conditions. Figure 2 4 MBL generate d near tip opening stresses as a function of normalized radial distance r/ ( o ) for a series of T stresses ranging from T / o = 0.4 to 0.9.
51 Figure 2 5 Relationship between normalized T stress and Q A B Figure 2 6 Reference opening stress as a funct ion of o (A) and Q (B) for normalized radial distances r/ ( o ) = 2, 4, 6, and 8.
52 A B Figure 2 7 Examples of the near tip opening stress fields at a constant far field stress = o as a function of normalized radial distance r/ ( o ) for a semicir cular surface crack ( a/c = 1.0 and a/t = 0.35) with the zero constraint small scale yielding (SSY) and the constraint corrected J Q predicted fields shown as so lid points for materials with n = 3 (A ) and n = 20 (B).
53 CHAPTER 3 FINITE ELEMENT MODEL ING AND TEST GEOMETRIES Geometr ies Surface cracked plates under monotonic tension are the primary focus of this study. Thr ee configurations are used to model this geometry. The first and the most general model tested is a n uniaxial surface cracked specimen with far field displacement ( ) or stress ( ) boundary conditions acting normal to the crack face, shown in Figure 3 1. Similar to this configuration, t he second model tested is the uniaxial surface cracked plate specimen bonded to an elastic plate on the fa ce opposite crack opening (Chapter 4) Lastly, a surface cracked sub model section is implemented from a global metallic pressure vessel or COPV geometry. The crack is oriented such that it experiences mode I opening in the direction of maximum normal st ress in the liner (hoop in the cylindrical portion) considered to be the most critical in terms of ductile crack growth The combination of specimen geometries and complex biaxial structures facilitates a complete understanding of the various geometric a nd material effects. The finite element crack geometries contain two planes of symmetry. The first is the xz plane at the crack in which the out of plane displacements in the remaining section are set to zero ( u yy = 0) and the second is t he central yz pla ne bisecting the c rack mouth in which u xx = 0. A more detailed description of geometries and boundary conditions will be given in the following chapters. Finite Element Models Mesh Configuration FEA Crack version 3.1.14 ( Quest Reliability, 2007) was used to generate the crack mesh and Abaqus version 6.7 (2007 b ) was used for the analysis. Both an elastic
54 plastic model and a fully elastic model for each of the geometries tested are necessary when using T as the constraint parameter The elastic pla stic mod el is used to calculate opening stress fields and J integrals. The fully elastic model is used to find the T stress scaling factors ( T ( )), which are normalized by the nominal s tress in the opening direction of the metallic section. The J integrals along the crack front are calculated using the domain integral method applied to three dimensions intrinsic to Abaqus. The T stresses are c alculated using the interaction integral method intrinsic to Abaqus in the fully elastic models ( Dassault Systmes 2007 a ) The finite element analyses are made using C3D20R 20 node isoparametric brick elements. At the crack tip, 20 node collapsed face prismatic elements are used. To allow for crack tip blunting in the elastic plastic models, the initially coincident nodes at the crack tip are left unconstrained. Figure 3 2 schematically illustrates the collapsed face element nodal configuration and bo undary conditions for elements with a face in the crack plane in the crack growth direction ( = 0). In order to maintain sufficient data points at the desired radial distances, the crack region contains a highly refined mesh consisting of 31 contours ar ound the crack tip and 72 collapsed face elements along the crack front. Figure 3 3 shows the crack face mesh with these refinements. Mesh Convergence A preliminary study of mesh refinement was conducted by comparing domain integral ( J integral) values, w hich tend to be sensitive to mesh refinement, from v arious models of mesh density. With increasing mesh refinement, the mean J integrals at each crack angle ( J ( )) calculated by averaging the values from the outer most contours in which the J integral varied by a negligible amount (< 0.10 %), converged to stable values and continued through the mesh refinement used for this analysis. The
55 chosen mesh refinement is beyond the level needed for accurate stress and domain integral calculations, but was necessary to achieve sufficient radial data points to assess the opening stress field using interpolation methods. Increasing the number of nodes in the radial directi on from the crack tip will decrease the maximum possible error in interpolation within numerical limitations. The near tip stress measurements obtained by a less refined mesh compared accurately to the results from the f inal mesh refinement ( Figure 3 3 ) w hen the number of radial points in the coarse mesh used in interpolation is maximized by limiting the range of deformations observed. In other words, an accurate comparison can be made between meshes when the radius of near tip mesh refinement is approxim ately equal to 10 o resulting in the maximum number of stress measurements. Outside this range, the coarse mesh does not contain sufficient data points for a meaningful comparison. Element Formulation Crack tip blunting effects can typically be neglected at distances g reater than roughly one and a half to twice the cr ack tip opening displacement (approximately 1.5 o or 2 o ) Inside this distance from the crack tip, it would be necessary to use large strain analysis (Betegn and Hancock 1991; Wang, 2009). Linear kinematic element formulation (small st r ain) simplifies the analysis because it does not require a finite radius crack tip and often converges with a relatively coarse mesh. Non linear kinematic element formulation (large strain) req uires a finite radius crack tip and therefore mesh convergence is often difficult to achieve. For this investigation, small strain approximation is used to analyze stress fields outside the crack tip blunting zone in the surface cracked models. Figure 3 4 (A ) gives an example of how the small strain approximation of opening stresses differ from those calculated using large strain
56 analysis with a finite radius key hole mesh configuration at the crack tip. This figure gives opening stress data from a singl e edge notched tension (SEN(T)) model ( crack depth to width ratio: a/W = 0.20) with plane strain boundary conditions at a load o /J = 40 using the two linear plus power law material models with n = 3 and 20 Figure 3 4 (B ) gives the percent difference between the large and small strain results. P ercent difference is defined as (3 1 ) Both linear plus p ower law low and high strain hardening materials, n = 20 and 3 respectively, achieve 95% opening stress accuracy within the radial distance r = 2 o Additionally, the higher strain hardening material achieves 95% accuracy at a smaller radial distance ( r = 1.5 o ) compared to the model with the low strain hardening material.
57 Figure 3 1 General uniaxial surface crack model configuration The applied loads, and are the average far field (nominal) stresses and displacements respectively in the y direction
58 Figure 3 2 Example of crack front collapsed face nodal configuration. For elements with a face in the = 0 plane, nodes 2, 3, 5, 6, and 7 are free while displacement u yy = 0 for nodes 1, 4 and 8. All other initially coincident nodes ar e free for remaining elements in the direction
59 Figure 3 3 Finite element surface cracked mesh refinement detail.
60 A B Figure 3 4 Data obtained from a single edge notched tensile (SEN(T)) specimen demonstrating the effects of using linear (small strain) and non linear (large strain) kinematic element formulations. (A) Opening stress ( yy ) and (B) percentage difference verses normalized radial distance ( o )) is given for a constant load o /J = 40.
61 CHAPTER 4 DEFORMATION LIMITS AND PLASTIC C OLLAPSE IN A BONDED GEOMETRY This chapter will investigate if a SC(T) geometry that is not well characterized by J and T at larger loads will maintain J T predicted stress fields for loads beyond yield when bonded to an elastic structure. This will assess the applicability of the J and T parameters for predicting fracture at loads seen in a typical COPV. Although the liner loading in a COPV is likely to be more complicated, assessing the effects of a uniaxial structure will aid in standardized specimen dev elopment that can be used to characterize this situation. Surface crack tip stress fields in a tensile loaded metallic liner bonded to a structural backing are developed using a two parameter J T characterization and elastic plastic modified boundary laye r (MBL) finite element solutions. The constraint effects that arise from the elastic backing on the thin metallic liner and the extent to which J T two parameter solutions characterize the crack tip fields are explored in detail. The goal of this study i s to assess how the increased elastic constraint imposed by the backing on the liner may result in an enhanced validity range of J T characterization by increasing the effective liner thickness from a fracture standpoint. The deformation limit determinati on methodology using two parameters can be used for any geometry, loading conditions, materials and fracture parameters. Results from this study will facilitate the implementation of geometric limits in testing standards for surface cracked tension specim ens bonded to a structural backing.
62 Model Definitions Materials The Ramberg Osgood power law hardening material model intrinsic to the deformation plasticity theory with Abaqus was used for this analysis. This model is de fined by the following equation ( Ramberg and Osgood, 1943) : (4 1) (4 2) where E o is the reference yield strength, n is the strain hardening exponent, and is the fitting coefficient or strain offset. A low strain hardening material, with a true stress strain curve seen in Figure 4 1, was used for our analysis with the following pa rameters: o = 250, = 0.5, n = 40 and the Poisson ratio = 0.33. When using the Ramberg Osgood material model, a low strain hardening material will minimize the nonlinear effects of scaling T stresses with fully elastic models. This material model was studied because of its simplicity and similarities to the 6061 T6 aluminum currently being tested for use tensile data for a t = 1.27 mm (0.05 inch) thick plate specimen is also given in Figure 4 1 to demonstrate t he similarities. COPVs consist of a comparatively thick composite shell relative to the liner This shell consists of multiple overlapping carbon fiber epoxy layers wound at different angles relative to the vessels axis comprising the layup. For the uniaxial tension specimen of interest, the strain distributions and constraint in the thickness direction will be significantly different from the complex loading seen in a COPV. Thus, the
63 anisotropic material effects would be difficult to isolate and sig nificantly complicate the analysis. Therefore, an isotropic linear elastic material with uniaxial stiffnesses ranging from three times the liner modulus, typical of a carbon epoxy in the fiber direction, to equal to the liner modulus ( E Backing /E Liner = 1, 2, 3 = 0.33 was used. This model will assess the effects of maintaining partial elasticity within the layered geometry while the cracked portion undergoes large scale plastic deformation. Finite Element Models The FE cracked plate implemented in the bonded geometry contains a semicircular surface crack with a depth of a = 0.889 mm (0.035 in) central in a w l t = 25.4 50.8 1.27 mm (1 2 0.05 in) plate. The thickness of the backing is chosen to be three times the cra cked plate thickness ( t Backing t Liner ) which is a good approximation for a COPV overwrap. Figure 4 2 provides a schematic representation of the bonded geometry with displacement boundary conditions. In order to compare the relative accuracy of crack tip characterizations between th e bonded model and unbonded models (SC ( T ) ), various thicknesses t = 1.27, 1.78, 2.54, 5.08 mm ( t = 0.05, 0.07, 0.10, 0.20 in) are implemented for the homogeneous surface cracked models with a constant crack size a = 0.889 mm (0.035 in), shown in Table 4 1. J T Stress Field Characterization and Deformation Limits J I ntegral and T Stress Figure 4 3 shows the J integral calculations at a constant load of = o The decreasing trend due to the thickness effects on the crack opening is typical of J integral calcu lations. The bonded cases show significant reduction in J when compared to the case of equal total thickness ( a/t = 0.175). This discrepancy is simply due to the differences in strain distribution on the plane of the bond and the equivalent plane in the
64 homogeneous models caused by the continued partial elasticity of the structure. The bonded structures will have a more uniformly distributed str ain field on this plane resulting in lower opening displacements near the crack for a given load causing large plastic deformations. By comparing the bonded model with E Backing /E Liner = 1 to the homogeneous model of equivalent total thickness ( a/t = 0.17 5), which under elastic loading are identical, it is shown that the effects of continued partial elasticity on this structure is the primary cause for lowered J values at loads causing sufficient plastic deformation. Similarly, the effects of modulus rati o in the bonded models on the J integral, although not significant at these loads, can explain the decrease in J with the increase in modulus. Observing that the J values from the bonded case are closer in magnitude to those of the equivalent thickness mo del ( a/t = 0.175), supports the claim that the effective thickness (characteristic length) of the liner increases when bonded to an elastic structure from a fracture standpoint. In fact, this effective thickness is most likely greater than the total compo site/liner model thickness. This postulate is supported further when looking at J T dominance of opening stress fields. Figure 4 4 gives the T stress scaling factors ( ) at various angles calculated from each of the models with linear elastic material properties. The homogeneous models show a trend typical of T stresses for increasing thickness. The bonded models however show a much stronger trend and are highly dependent upon modulus ratio. When the modulus ratio is unity, the elastic geometry is equivalent to the a/t = 0.175 homogeneous case. The T stresses near the bonded surface ( > 40) for the bonded models with modulus ratios of 2 and 3 are higher when c ompared to the homogeneous models. Conversely, for angles near the free surface ( < 40) the discrepancies seem
65 insignificant. These trends of course become heavily dependent on material properties like the ratio of moduli and anisotropy and are not cle arly representative of the extent of two parameter dominance before plastic collapse. Opening Stress Fields The most obvious and significant comparison with regard to the fracture driving force is of the opening stresses in the crack plane ( = 0) as func tions of constraint ( T s tress), the applied load level ( ) and a measure of deformation. The opening stress normalized by the plane strain MBL reference solutions at T stresses corresponding to those scaled by the nominal load in the surface cracked mode l and r/ ( o ) = 2 are given as a function of normalized T stress for angle = 90 in Figure 4 5 These curves show the effect of increasing T stress and consequently increasing nominal load on two parameter opening stress characterization. Due to the v arying T stress scaling factors and the fact that T stress alone is not sufficient to determine two parameter dominance; it is more beneficial to look at opening stresses as a function of nominal load and deformation on the metallic section. Figures 4 6 an d 4 7 give the opening stress normalized by the plane strain MBL reference solutions and o ) = 2 as a function of nominal load and a measure of deformation respectively for angle = 90. The level of deformation is shown as the inverse J integral multiplied by the characteristic length of the specimen, in this case crack depth a There fore, larger deformation is characterized by lower values of o /J The bonded cases show a greater adherence to the MBL predicted opening stresses for higher load levels and consequently higher levels of deformation when compared to those of the homogene ous models. Increasing thickness corresponds to an increase level of deformation in which J T dominance is sustained. The modulus ratios in the
66 bonded models do not have a significant effect on the dominance limit load, but has a strong effect at higher crack angles near the bonded surface on the level of allowable deformation. Normalized opening stresses are shown as a function of crack angle at constant normalized far field loads / o = 0.95 and 1.0 in Figures 4 8(A) and 4 8 (B), respectively. These curves give insight into how two parameter dominance varies with crack angle. It becomes necessary when becomes small (near the free surface ) to ignore angles < 5 because HRR singularity will only be present at near crack points em bedded entirely in material (Wang, 2009) Therefore, Figures 4 8(A) and (B) show opening stresses for angles > 5. The following observations are made when co mparing the bonded cases to the four homogeneous models. At both the loads examined, the bonded cases show greater adherence to J T characterized fields. At a nominal load of / o = 0.95 only the bonded cases and the thickest homogeneous case ( a/t = 0.1 75) have not yet shown signs of plastic collapse for angles > 21. At a load equal to far field yield ( = o ) all the homogeneous models have lost J T dominance, which is arbitrarily characterized by a 5% deviation from the MBL solution. For both loa ds, when the thickness of the homogeneous specimen is doubled ( a/t = 0.35 to a/t = 0.175) the opening stresses are only slightly closer to those predicted by the MBL solutions. This is in contrast to the significant difference that can be seen in the tran sitions between a/t = 0.70, 0.50 and 0.35. The asymptotic function of stress field size dependence with thickness is well established as t approaches infinity and is important when comparing results to a non homogeneous structure and the implications of a
67 Deformation Limits Figure 4 9 gives the nominal stresses in the opening direction at which the opening stresses at a constant normalized radial distance o ) = 2 deviate from the predicted plane strain MBL stresses by 5%. This point is arbitrarily called the point of field collapse, in which J and T no longer characterize the stress state within tolerable limits due to excessive plastic deformation. The bonded geometries exhibit higher J T dominance for l oads exceeding far field yielding when compared to the other homogeneous models. The homogeneous surface cracked models are approaching a limiting thickness in which further increase will not add to the limit load for J T characterization. This is most c learly seen in a/t = 0.35 and a/t = 0.175. Though the thickness is doubled, there is only a small change in limit load between these two geometries. This limitation of thickness effects may prevent the design of an cture that would have the same fracture characteristics as the bonded model, but will provide quantifiable limits to using homogeneous structures to represent bonded configurations. Similar to Figure 4 9, Figure 4 10 gives the normalized J integral or de formation factor at which the opening stresses at a constant normalized radial distance o ) = 2 deviate from the predicted plane strain MBL stresses by 5%. The bonded geometries exhibit higher J T dominance at angles nearer to the bonded surface for loads exceeding far field yielding when compared to the other homogeneous models. In the bonded models, at lower crack angles, the resistance to crack opening imposed by the continued elasticity of the backing dominates the near tip conditions resulting i n lower allowable J integrals before collapse. The most significant findings from this plot are the various points of variation in the homogeneous equivalent thickness model and the
68 bonded models in which the effects of the backing can be observed. At an gles of greater than > 25, the homogeneous model of equivalent total thickness begins to deviate from the bonded models. This angle represents the crack depth in which the continued linearity of the backing is influential in two parameter characterizat ions. Secondly, at angles greater than > 55, the deformation limit begins to deviate in the bonded models as a function of modulus ratio. For the range tested, this point represents the angle at which the material properties of the backing become infl uential. Conclu ding Remarks This section describes the feasibility of using J and T fracture parameters in characterizing crack tip stress fields in a surface cracked liner bonded to an elastic structure for loads typical to a COPV and establishes methodol ogy for determining the limits of such characterizations. The loads seen in a COPV liner exceed the yield strength in both manufacturing and normal operation; therefore, fracture characterization at these loads is critical. The bonded curves in Figures 4 9 and 4 10 show that a simple two parameter model for opening stresses in a metallic composite uniaxial structure can be used for loads seen in COPVs. From this analysis, it can be concluded that J T characterization is applicable for the uniaxial tensio n of bi material bonded specimens and may extend to pressure loadings of COPVs, but simple homogeneous uniaxial tension specimens do not provide a fracture equivalent geometry that maintains the level of J T dominance seen in the composite geometries. The difference of continued J T characterization in homogeneous models at large loads stems from the inability to maintain the elastic constraint effects of the backing. For bi material uniaxial test specimens, the effect of the backing to liner modulus rati o greatly affects the allowable level of deformation but not the far field load within the tested
69 range. From a loading standpoint, the most significant influence of the backing is its ability to impose an elastic constraint on the liner even at very high levels of far field loading.
70 Table 4 1. Dimensions of models used in the analysis ( a = 0.889 mm (0.035 in)) Model Type t L iner (mm) t B acking (mm) t Total (mm) a/t L iner a/t Total 1 Bonded 1.27 3.81 5.08 0.7 0.175 2 Homogeneous 1.27 N/A 1.27 0.7 0.7 3 Homo geneous 1.778 N/A 1.778 0.5 0.5 4 Homogeneous 2.54 N/A 2.54 0.35 0.35 5 Homogeneous 5.08 N/A 5.08 0.175 0.175
71 Figure 4 1 Ramberg Osgood material models for a low ( n = 40) strain hardening material and 6061 T6 aluminum test data from a t = 1.27 mm (0.05 inch) thick plate under uniaxial tension
72 Figure 4 2 General bonded surface crack model configuration
73 Figure 4 3 J integral as a function angl e at a constant nominal load = o Figure 4 4 T stress scaling factor as a function of angle
74 A B Figure 4 5 Opening stress normalized by MBL reference solutions as a functio n of normalized T stress ( o ) at r/ ( o ) = 2 at = 90 (A) and 30 (B).
75 A B Figure 4 6 Opening stresses normalized by MBL reference solutions as a function of n ormalized nominal stress ( ) at r/ ( o ) = 2 at = 90 (A) and 30 (B).
76 A B Figure 4 7 Opening stresses normalized by MBL reference solutions as a function of deformation factor o /J at r/ ( o ) = 2 at = 90 (A) and 30 (B)
77 A B Figure 4 8 Op ening stresses normalized by MBL reference solutions as a function of angle at r /( o ) = 2 and / o = 0.95 ( A) and / o = 1.0 ( B)
78 Figure 4 9 Normalized nominal load / o at opening stresses correspo nding to plastic collapse ( yy / MBL = 0.95) at v arious crack angles Figure 4 10 Normalized J integral ( J/ ( o )) at opening stresses corresponding to plastic collapse ( yy / MBL = 0.95) at various crack angles
79 CHAPTER 5 STRAIN HARDENING EFFECTS ON TWO PARAMETER CHARACTERI ZATION The geometry and loading conditions of surface cracks in actual structures will often be more complex than in laboratory specimens. Specifically, the near tip constraint conditions of a surface crack for both specimens and structures must be accounted for in order to comp are accurately the failure conditions. Without a two dimensional failure locus predicting values of the critical crack driving force as a function of near tip constraint, the application of a critical value calculated from a test specimen may not match wi th the actual point of structural failure. Two concerns of primary interest when implementing a two dimensional approach are, (i) evaluation of the conservativeness of the two parameter failure locus, and (ii) the geometric limits on test specimens in ord er to achieve a repeatable failure limit in testing. Similar to single parameter fracture prediction, stress fields surrounding the crack tip will not maintain two parameter dominance for loads exceeding deformation limits, and hence this measurement is u seful to evaluate limits for test specimens. It has been proposed in recent ASTM standard development that a material dependent limit is practical for surface crack specimens. However, a material based criterion is useful when a minimal number of measura ble input parameters are used. For example, a criterion dependent only on the ratio of elastic modulus to yield strength would be desirable if it can be shown that a conservative limit exists as a function of this ratio. The study presented in this chapt er aims to assess an additional degree of freedom, namely the level of strain hardening, keeping the elastic modulus to yield strength ratio fixed. Many investigators have addressed the differences in near tip fields for various strain hardening materials, but this literature does not provide a clear description of
80 these fields with respect to the two parameter fracture predictions for surface crack strains near the crack tip for various power law hardening materials. Constraint or triaxiality directly influences the size and shape of the plastic zone. Lower or negative constraint results in a large plastic zone shifted in the crack growth direction. In an owd and Shih (1991) showed that as plastic flow propagates through the remaining ligament of a cracked body of finite thickness, the material must have sufficient strain hardening for accurate J Q formulation. The strain hardening effects of crack tip def ormations with respect to the J integral directly influence the continued existence of a J Q dominated field with very little influence from constraint and elastic properties. This study investigates the influence of the strain hardening exponent, define d for a linear plus power law material model, on two parameter ( J Q ) fracture characterization and the associated limitations by presenting near tip stress fields for surface crack tensile geometries. The near tip stresses are referenced by the material s pecific J Q family of fields generated with MBL formulations for linear plus power law materials with strain hardening exponents of n = 3 (high strain hardening) and n = 20 (low strain hardening) with o = 250. The results from this study are of immedia te relevance towards establishing testing standards for surface crack geometries. Methodology for determining the limits of two parameter characterization for surface cracks are established and multiple techniques are presented.
81 Model Definitions Material s The focus of this analysis is to study the effects of strain hardening on crack tip stress fields relative to MBL solutions. Due to the fully plastic nature of the loadings, the strain hardening is expected to have a significant effect on the stress fie lds. Material dependent reference solutions are created for each of the elastic plastic materials tested. Therefore, the predicted stresses should take into account the material strain hardening for loads up to the deformation limits of the geometry. Fo r this study, two linear plus power law material models shown in Figure 5 1 are used to simulate low strain hardening ( n = 20) and high st r ain hardening ( n = 3). This material model correctly defines many actual material flow curve shapes often measured e xperimentally by a uniaxial tension specimen. The linear plu s power law model is defined by for (5 1 ) for (5 2 ) ( 5 3 ) where E is the elastic modulus, n is the strain hardening exponent, o is the reference yield stress and o is the yield strain For this investigation E = 68.95 GPa, o = 275.8 MPa and = 0.33 ( o = 250). In order to ensure gradual transition from elastic to plastic strains, a circular transition arc region is includ ed ( Wang, 1991 ). Equations 5 1 and 5 2 become
82 for (5 4 ) for (5 5 ) for (5 6 ) where ( Nc Nc ) is the center and r Nc is the radius of the circular arc in normalized space ( o o ). K 1 is the lower limit of the transition region, which is set at K 1 = 0.95, and K 2 is upper limit. Therefore, K 1 o defines the elastic limit of the materia l. By ensuring the slopes at the points of transition ( K 1 K 2 ) are continuous, the remaining variables can be calculated ( Nc Nc r Nc K 2 ). The flow curve smooth transition region is primarily implemented to aid in the convergence of the incremental pl asticity solver. Additionally, the curve defined by the latter equations represents more accurately the inherently smooth experimental curve from uniaxial specimens. The existence of the transition region has little effect on the results and is merely a numerical artifact. The material curve is discretized into stress and plastic strain inputs for the FEA model with a high concentration of points in the region of initial plastic strains. The material inputs are shown as solid points in Figure 5 1 Incr emental theory plasticity, which follows the von Mises yield criterion ( J 2 flow rule), is used for this analysis. Finite Element Models The surface crack tensio n SC(T) model, shown in Figure 5 2 has dimensions w L t = 19 mm 19 mm t with two thick ness values of t = 1.27 mm and 2.54 mm The crack geometries examined are a/c = 1.0 (semi circular) and 0.40 (semi elliptical). Table 5 1 gives a complete list of the four models tested. The crack depth remains
83 constant throughout the analysis at a = 0. 889mm. The uniaxial tensile load ( ) is applied to the xz face opposite the crack using displacement boundary conditions in the y direction Results and D iscussion J I ntegral and Q Figures 5 3 (A) and (B ) show the J integral as a function of crack angle at a constant average net section stress Net o for the semi circular and semi elliptical cracks respectively. Net section stress is defined as the far field stress divided by the area fraction or Net = ( A o / A N et ), where A o is the nominal section a rea and A N et is the remaining a rea at the cracked section ( Table 5 1). Geometries with lower strain hardening materials exhibit greater redistribution of stresses near the crack tip resulting in higher J integrals and crack tip deformations. Similarly, h igher strain hardening results in smaller plastic zones because the material inside the plastic zone is capable of supporting higher stresses, therefore less stress redistr ibution is necessary. Figure 5 3 gives a clear example of these trends at a constan t average net section stress for all the surface crack models tested. It is necessary when becomes small (near the free surface) to ignore angles < 5 because a J dominated field will only be present at near crack points embedded entirely in material (Wang 2009). Therefore, figures that are a function of show data for angles > 5. Figures 5 4 (A) and (B ) give Q defined in Equation 2 9 as a function of crack angle at a constant average net section stress Net o for the semi circular and s emi elliptical cracks respectively. The models with the lower strain hardening material have lower (more negative) Q factors for a given far field load. Near tip stress field reduction associated with constraint loss (more negative Q factors) is a result of redistributions in
84 the plastic zone which is amplified by the lack of strain hardening. The combination of stress redistribution and relaxation associated with constraint loss results in a greater degree of crack tip deformations in the low strain har dening cases compared to the high strain hardening cases. This postulate will be further supported in near tip stress field analysis. Much of the analysis is done for the case of = 30 for both aspect ratios a/c = 1.0 and 0.40. This angle is close to t hose predicted to be the most critical based on the criteri a proposed by Leach et al. (2007 ) in which the critical crack extension angle corresponds to the maximum product of the average opening stress over the plastic zone and the J integral. This criter ion roughly corresponds to the angles for which J and Q are maximum, which can be estimated from Figures 5 3 and 5 4 respectively. Experimentally, the critical angle analysis shows considerable scatter. Therefore, the angles shown here will not necessar ily be the most critical in terms of crack growth, but should represent the general trends within the critical region. Other angles show similar trends but the detailed analyses are not shown here. Figures 5 5 (A) and (B ) show the local deformation factor l o /J at crack angle = 30 as a function of normalized average net section stress Net o for the semi circular and semi elliptical cracks respectively. The local deformation parameter has been used by many investigators as a measure of load. This param eter is valuable in geometric limit analysis for fracture parameterization because it includes a measure of characteristic length ( l ) and the J integral. A material dependent limit, based on the local deformation factor, may exist that relates the J integ ral at field collapse to the lengths of the specimen. For this study, crack depth ( a ) is used to represent the
85 characteristic length of the specimen ( o o /J ), and therefore cannot be directly related to a geometrically independent limiting factor, but can be used to evaluate the relative characteristic lengths between specimens. All subsequent references in this chapter to the local deformation factor will be in the form o /J. At loads below average net section yield, the near tip conditions show only moderate plasticity and the differences in J are only due to geometric effects on crack mouth opening. The apparent drop in o /J on a log scale at an average net stress approximately yield ( Net o ) is a result of the non linear increase in J as the near tip fields proceed to large scale yielding. At a local deformation of o /J = 40 (used in following sections) the near tip fields can be chara cterized by either large scale yielding or plastic field collapse. Crack Tip D eformation Shih (1991) showed that the relationship between CTOD and J depends strongly on the strain hardening exponent and weakly on elastic properties and Q Simil arly, Chiodo and Ruggieri (2010), in a study on surface cracks in pipes under bending, showed this relationship is independent of geometry within easily achievable limits. In a low strain hardening material, the crack tip deformations are dominated by lar ge plastic strains. Conversely, near tip field with a high strain hardening material has greater resistance to deformation because the material is capable of carrying higher stresses in the plastic zone resulting in tip deformations with a larger portion attributed to elastic strains. The finite element models used in this analysis contain collapsed face prismatic elements at the crack tip. In order to simulate crack tip blunting the unconstrained nodes are allowed to separate during loading. The out o f plane displacement of this
86 blunting is used to estimate the level of crack tip de formation. Referring to Figure 5 6 u y is the out of plane (opening) displacement of the crack face measured from the vector perpendicular to the crack front in the crack p lane. In our analysis, u y will be used to denote crack tip blunting, calculated as the maximum out of plane nodal displacement in the initially coincident nodes at the crack tip. For models with the low strain hardening material, this measurement of crac k tip opening is similar to CTOD taken at the intersection of a 90 vertex with the crack flanks, as shown in Figure 5 6 (Anderson, 2004 ). In high strain hardening material, the crack tip blunting displacement ( u y ) will underestimate CTOD because tip defo rmations are more distributed along the crack face. In low strain hardening material, crack opening is more localized at the tip, resulting in 2 u y u y can be used to isolate the crack tip deformation/blunting with a higher dependency on plastic strains. Figures 5 7 (A) and (B ) show the crack tip blunting displacement ( u y ) as a function of normalized J integral ( J/ ( a o ) ) at crack an gle = 30. Similar to the abovementioned references, the slope ( u y o ) ) of the relationship is more or less constant for each material and seemingly independent of geometry. The models with the low strain hardening material have significantly higher values of u y o ) compared to the models with the high strain hardening material. This data not only shows that there is an increase in near tip deformation with lower strain hardening but also that the proportion of tip deformation to J is significant ly influenced by strain hardening. The higher ratio will have significant implications for near tip stress field analysis with MBL solutions which are functions of the J scaled length parameter o ) The direct proportionality
87 of J to the physical lo cation of measurement r for a constant o ) makes the proportion of J to near tip deformations important. The ratio of crack tip deformations to J has a direct effect on the physical meaning of the normalized radial distance and the local deformation f actor. For example, a constant normalized radial distance o ) = 4 corresponds to a physical distance that is four times the crack tip deformation predicted by o while a constant local deformation factor o /J = 80 corresponds to a state where the crack depth a is eighty times the crack tip deformation predicted by o Wang and Parks (1995) estimated this factor to be o u y for a linear plus power law material with n = 10. For our materials, o u y and o u y for n = 3 an d 20 respectively. MBL solutions have the same relationship between crack tip deformation and J but by definition, crack tip blunting has no effect on the near tip fields in MBL formulation. It is only when crack tip blunting ( u y ) is large compared to the characteristic length of the specimen that the effects on the near tip fields are significant. It is often useful to summarize crack tip deformation effects by examining the near tip fields at a constant normalized radial distance from the crack tip. Given these material dependent slopes, stress fields will be measured closer to the crack tip proportional to the blunting effected zone for models with the low strain hardening material. However, this should not be interpreted to mean that the actual l imits of fracture parameterization are better represented by the physical crack tip deformation such as u y or CTOD. It is entirely necessary to provide load/deformation limits as a function of the measured fracture parameters, in this case J
88 Near T i p Opening Stress F ields The most obvious and significant comparison with regard to the crack driving force is of the opening stresses in the crack plane ( = 0). This section presents the near tip opening stresses from the surface crack models as functio ns of net section loads and local deformations. All the stresses are normalized by the stress predicted by the modified boundary layer solutions at the corresponding Q factor ( MBL ). Figures 5 8 (A), (B), (C) and (D ) show the opening stresses normalized by the MBL J Q predicted stresses at a radial distance o as a function of average net section stress for surface crack specimen dimensions a/c = 1.0 and 0.40 and a/t = 0.70 and 0.35. The influence of far field and net section loads and crack tip deformations for limit methodology, which looks at quantifiable deviations from the predicted curve, can be effectively summarized by examining the near tip fields at a constant normalized radial distance from the crack tip. As discussed in the finite ele ment mesh section, the data presented at a constant normalized radial distance is limited to the radius of mesh refinement. When J increases, the physical point of measurement for a constant r = o increases. The radial independence of Q r/ ( o hydrostatic shift in stresses, given by Equation 2 8 may mark the limit of the usefulness of Q as a ductile fracture parameter. Given the definition of Q by Equation 2 9, t he analysis of opening stresses at a greater radial distance, say o compared to MBL predicted stresses is essentially a measur e of the radial independence Q providing the MBL solutions represent a radial independent measure of Q (Sharma et al., 1995).
89 In structures experiencing large loads during normal operations, a load based limit analysis is useful. This type of methodology is restricted to specific geometries and materials combinations and lack the robust features desired in a deformation limit methodology. English et al. (2010) showed the validity of two parameter characterization in bonded structures experiencing far field plasticity and the importance of load based limit determinations. Wang and Parks (1995) also addr ess the importance of load based limit determination for engineering applications. Although the calculation of a specific stress load at collapse for limit determination is not relevant when comparing t w o different strain hardening materials, the trends i n relative opening stress at net section yield is useful in determining the dependence on plastic flow through the remaining ligament. All the low strain hardening models, except the semi elliptical crack in the thin plate ( a/c = 0.40, a/t = 0.70), show a clear decrease in near tip opening stresses relative to those predicted by MBL solutions at average net section yield. The deviation from the pr edicted field at loads less than yield in the semi elliptical crack in the thin plate ( a/c = 0.40, a/t = 0.70) with the low strain hardening material is associated with the rapid loss of constraint at load s approaching yield ( Figure 5 4 (B ) ) Similarly, this geometry with the high strain hardening material is the only one to exhibit plastic collaps e at the loads tested where all other models maintain the predicted field through net section yield. Similar to the results shown in the previously referenced work, the deeply cracked structures ( a/t = 0.70) exhibit less accurate fields for lower loads in both materials tested. In addition, the elliptical crack ( a/c = 0.40) also demonstrates lower loads for accurate characterization for both materials. The results based on geometric
90 differences stem from a change in characteristic length of the specimen. Essentially, a small crack ( a/c = 1.0) in a thick specimen ( a/t = 0.35) should exhibit the highest level of allowable load before initiation of stress field collapse due to excessive deformation. These observations will be further supported by viewing opening stresses as function of the local deformation parameter. Figures 5 9 (A), (B), (C) and (D ) show the opening stress normalized by the J Q predicted stresses at a radial distance o as a function of local deformation factor o /J for surface crack specimen dimensions a/c = 1.0 and 0.40 and a/t = 0.70 and 0.35. Similar to crack tip opening displacement (CTOD), the parameter o /J gives insight into the physical deformations n ear the crack tip. From Figure 5 7 it is shown that the ratio of near tip deformation to the normalizing parameter o is material dependent. The lower strain hardening materials are shown to have larger near tip deformations for a constant J integral Figure 5 9 demonstrates that the increase in relative near tip deformations influenc es the stress fields. In Fig ure 5 9 (C ) ( a/c = 0.40, a/t = 0.70) the precise influence on geometric limit determination is clearly demonstrated in the case with lowest characteristic length (the lowest remaining net section area). The data shows the stress fields for both high and low strain hardening materials deviating from the predicted stress fields by greater than 5%. All geometries demonstrate this trend, but onl y this model gives a clear limit. The high and low strain hardening models show deviation from the predicted stresses by greater than 5% at o /J = 25 and 50 respectively. The geometric limits are not necessarily defined at a constant radial distance of r o ) = 4 with a 5% deviation. A criterion for the deformation limits of near tip stress
91 fields based on MBL formulation is debatable and may be better determined experimentally. Nevertheless, for limitation methodology in which o is used to normali ze both radial distance ( o ) ) and deformation ( o /J ), the plastic properties of the material, in this case the strain hardening exponent, must be accounted for. For determining relative geometric effects on prediction limits as a function of local de formation, it is important to address differences in net section areas. The ratios of net section area of the remaining ligament ( A N et ) to total nominal area ( A o ) are given in Table 5 1. The relative differences in net section area correlate with the lev el of local deformation at field collapse in the low strain hardening models. With lower remaining sectional area, the allowable local deformation at collapse decreases (higher limiting o /J ). Although crack tip deformation in the high strain hardening models is not sufficient to cause significant deviation from the predicted field at this radial distance, the trend of decreasing relative stress is still seen as a function of reducing n et section area. The accuracy of J Q characterization and the radial independence of Q are measured by the relative difference between a reference state and the field within a near tip zone where both cleavage and ductile fracture mechanisms are active and r/ ( o ) (Wang, 2009). For the previous plots, Q is defined at r = 2 o which serves as the reference state, and measurements of the relative differences are taken at r = 4 o This type of analysis is useful for deformation limit methodology, but in order to address the radial extent of J Q dominance, the opening stresses as a function of normalized radial distance are plotted at a cons tant load. Figures 5 10 (A), (B ), (C) and (D ) show
92 the opening stress as a function of normalized radial distance o ) for a load o /J = 40 (large deformation) on a logarithmic scale for all the surface cracked models implemented. The axes are typically shown as logarithms to enha nce resolution and simplify slope analysis. The reference solutions and 5% deviation curves are shown as solid points and dotted lines respectively. For this method of analyzing near tip fields, the limit of J Q dominance can be defined by the load at wh ich part of the stress field within the range specified above (1.5 o ) predicted curve by greater than 5%. Note that the points that fall o ) the load sho wn here all the models with the low strain hardening material have lost J Q dominance while all the models with the high strain hardening material maintain dominance for radial distances o ) Figure 5 10 (B ) shows a slight increase of relative nea r tip opening stresses at large normalized radial distances. This phenomenon is most likely due to far field effects. The far field stresses are significant in the n = 3 cases because a higher load is needed to achieve the constant local deformation fact or ( Figure 5 5 ). Similarly, the thicker models have greater resistance to crack opening as a function of far field loads. These factors only partially affect near tip stress field analysis and are typically negligible because even at large deformations t he near tip fields are still dominated by J and Q within o ) < 8. Fig ures 5 11 (A) and (B ) show the opening stress normalized by the J Q predicted stresses at a radial distance o and local deformation factor o /J = 4 0 as a function of crack angle for surface crack specimen dimensions a/c = 1.0 and 0.40
93 respectively. In general, the opening stress fields in the geometries with the high strain hardening material show two parameter dominance, measured as the percent deviation from the perfect ly predicted line ( yy MBL = 1 ), equal to or greater than the low strain hardening material. However, for angles closer to crack depth ( > 40), the differences between material models are diminished. The actual deformation limits at plastic collapse a re not clearly shown for these angles; therefore, postulates about strain hardening effects on J Q dominance limits are not possible. Nevertheless, the following observation can be made for loads resulting in a local deformation factor ( o /J Geo metries with the low strain hardening material exhibit plastic field collapse at levels of deformation lower than or equal to those found with the high strain hardening material in the surface cracked uniaxial tension geometry. Concluding Remarks The near tip constraint conditions of a surface crack must be accounted for in order to evaluate failure conditions. Recent ASTM standard development for surface cracked specimens has favored a material dependent limit that leads to conservative estimates. This c hapter assesses the level of strain hardening as a material limit while keeping the elastic modulus to yield strength ratio constant. The near tip stress fields in a surface cracked metallic plate loaded in uniaxial tension for loads exceeding net section yield are referenced to material specific J Q predicted plane strain modified boundary layer stresses for materials with strain hardening exponents from n = 3 (high strain hardening) to n = 20 (low strain hardening). The results from this study are of im mediate relevance towards establishing testing standards for surface crack geometries. Methodology for determining the limits of two parameter characterization
94 for surface cracks are established and multiple techniques are presented. Important conclusion s are enumerated below: a) Strain hardening level has to increase when plastic flow progresses through the remaining ligament for surface cracks under tension, to maintain the predicted field. High strain hardening materials maintained J Q predicted fields f or considerably higher levels of deformation compared to low strain hardening material. b) A radially independent Q parameter, that is applicable to ductile fracture prediction, cannot be measured for the low strain hardening material at larger deformations o /J mechanisms are present while the high strain hardening material maintained dominance at radial distances including and exceeding the damage zone. c) For the low strain hardening material the limit of J Q dominance and a radial independent measure of Q was a function of remaining net section area. d) The local deformation parameter, o /J is a relative measure of near tip deformation limits as a function of both geometry and material properties. e) Excessive relaxation at the crack tip results in a nonlinear decrease in the measure of constraint Q Hence, geometries with a low strain hardening material can exceed the MBL predictable limit due to plastic stress field collapse, leading to a loss of f racture parameterization. f) The geometric deformation limit of near tip stress field characterization is proportional to the level of stress the material is capable of carrying within the plastic zone. A simplified deformation limit criteria may be applicabl e if the strain hardening exponent is restricted within a range. The low strain hardening material may be used to provide a clear upper limit to the exponent. It is not recommended however that a deformation limit standard be implemented without addressi ng the significant effects of strain hardening in standard development or explicitly stated in the standard.
95 Table 5 1. Surface crack model dimensions Description Dimensions ( w L t ) a/t a/c A N et /A o Semi Circular Deep 19 19 1.27 mm 0.70 1.0 0.974 Semi Circular Shallow 19 19 2.54 mm 0.35 1.0 0.987 Semi Elliptical Deep 19 19 1.27 mm 0.70 0.4 0.936 Semi Elliptical Shallow 19 19 2.54 mm 0.35 0.4 0.968
96 Figure 5 1 Linear plus power law material models for the high ( n = 3) and low ( n = 20) strain hardening materials used in the surface crack analysis ( o = 250 and = 0.33)
97 Figure 5 2 General surface crack model configuration
98 A B Figure 5 3 J integrals as a function of crack angle at a constant average net section stress Net = o for a/t = 0.70 and 0.35 and aspect ratios a/c = 1.0 (A) a nd 0.40 (B). A B Figure 5 4 Q as a function of crack angle at a constant average net section stress Net = o for a/t = 0.70 and 0.35 and aspect ratios a/c = 1.0 (A) and 0.40 (B).
99 A B Figure 5 5. Local deformation factor o /J at crack angle = 30 as a function of normalized average net section stress Net o for a/t = 0.70 and 0.35 and aspect ratios a/c = 1.0 (A) and 0.40 (B). Figure 5 6. Schematic representation of the relationship between the crack tip blunting displacement ( u y ) and CTO D in a typical crack tip from this analysis.
100 A B Figure 5 7. Crack tip blunting displacement u y verses J integral at crack angle = 30 for a/t = 0.7 0 and 0.35 for aspect ratios a/c = 1.0 (A) and 0.40 (B ) A B C D Figure 5 8 Opening stresses norm alized by the MBL predicted stresses at a constant normalized radial distance r/ ( o ) = 4 and crack angle = 30 as a function of normalized average net section stress Net o for all the surface cracked models implemented.
101 A B C D Figure 5 9 Openin g stresses normalized by the MBL predicted stresses at a constant normalized radial distance r/ ( o ) = 4 and crack angle = 30 as a function of local deformation factor o /J for all the surface cracked models implemented.
102 A B C D Figure 5 10 Opening stresses at an angle = 30 as a function of normalized radial distance r/ ( o ) for a load o /J = 40 (large deformation) shown on a logarithmic scale for all the surface cracked models implemented. The reference solutions and 5% deviation curv es are shown as solid points and dotted lines respectively.
103 A B Figure 5 11 Opening stresses normalized by the MBL predicted stresses at a constant normalized radial distance r /( o ) = 4 as a function of crack angle for load o /J = 40 (large d eformation) for cracks with aspect ratios a/c = 1.0 ( A ) and 0.40 ( B )
104 CHAPTER 6 SURFACE CRACKED METALLIC LIN ERS IN COMPOSITE OVE RWRAPPED PRESSURE VESSELS Methodology for two parameter limits of surface cracks in a metallic liner of a uniaxial tension met allic/composite bonded geometry is presented in Chapter 4 The results presented in that study demonstrate the strong dependency of near tip fields referenced to J T predictions on the elastic plastic discontinuities of a bonded structure. Indeed, J and T along with MBL solutions predicted stresses for loads beyond full yielding where unbonded homogeneous specimens failed to maintain a predicted field. As stated in previous chapters the fracture prediction of a bonded structure is of intrinsic and practi cal interest with application in composite overwr apped pressure vessels (COPV). A through flaw in the metallic liner of a COPV is assumed detectable by leakage in proof testing due to the porous structure of the filament wound composite overwrap. Therefo re, surface cracks are the most critical flaws in terms of limiting structural life for liners during normal operation when proof cycles and crack detection techniques have been exhausted. Proof test logic for COPVs is not well established and is primaril y focused on the failure of the overwrap characterized by burst or the leakage of a through crack in the liner, while the growth of a surface crack is not well understood. Understanding the behavior of surface cracks in COPV liners for the development of proof test logic is challenging because the partial elasticity of the structure imposes complex constraint conditions significantly affecting the near tip fields. Additionally, plastic deformation of the liners during manufacturing processes and normal op eration facilitate the plastic collapse of near tip fields and a transition to the ductile tearing growth mechanism, especially in aluminum liners.
105 In Chapter 4 the applicable limits of J T two parameter characterization of elastic plastic crack tip fields using modified boundary layer (MBL) solutions were investigated for the bonded un iaxial surface crack geometry. Elastically scaled T stresses calculated from far field loads in elastic plastic models are a relatively accurate constraint measurement for t wo parameter characterization (Wang and Parks, 1995). T stress however is not defined under fully yielded conditions and therefore cannot provide a clear definition of field collapse under large deformations where the near tip field s are characteristic of J dominance but exceed elastic limits of the constraint measure. The construction of a two parameter failure locus, commonly the critical J value as a function of Q is necessary for engineering applications because it provides a more adjusted assessment of the failure limits (Cicero et al., 2010; Silva et al., 2006 ). Multiple experimental verifications of the constraint effects on critical crack driving force can be found in literature. Faleskog (1995), Chao and Zhu (2000) and MacLennan and Hancock (19 95) demonstrate the constraint effects on critical J values and resistance curves. In general, it has been shown that an increase in constraint results in a decrease in the critical value of J and crack growth resistance. For a fully adjusted surface cr ack failure criterion of COPV liners, it is necessary to construct a two parameter failure locus that covers a large range of possible constraint levels especially when the biaxial stress fields produce a slightly positive constraint near the free surface to highly negative at crack depth. This study will investigate the overwrap effects on constraint and crack driving force, near tip opening stress field characterizations limits, constraint loss and near tip triaxiality along with a comparison of T predict ed fields in homogeneous pressure
106 vessels (liner only models) and COPVs. This will assess the applicability of the J and Q parameters for predicting fracture with complex constraint conditions and high loads seen in a sub scale COPV specimen and a typical full scale COPV. Although many geometries and material combinations exist, assessing these characteristics will aid in standardizing methodology for COPV failure prediction and development of proof test logic. Models Definitions Materials The pressure ve ssel data and material properties were obtained through Carleton Technologies Inc. under the guidance of the NASA Engineering & Safety Center (NESC) Composite Pressure Vessels Working Group with the NASA Langley Research Center (LARC). All material proper ties are normalized for proprietary reasons. A common material for COPV liners is 6061 T6 aluminum. Load displacement data from a uniaxial tensile plate ( t = 2.29 mm) specimen of the liner material is used to construct the stress strain material flow cu rve. The material properties were calculated as follows: y = 267.5 and = 0.33, where E y is the 0.2% offset yield strength and length scale estimations, the reference stress will be defined as the 0.2% offset yield strength ( o y ). The resultant curve is discretized into stress and plastic strain inputs for the FEA model with a high concentration of points in the region of initial plastic strains. The stress strain specimen data and the discretized A baqus inputs are shown in Figure 6 1 as a solid line and hollow points respectively. Incremental theory plasticity, which follows the von Mises yield criterion ( J 2 flow r ule), is used for this analysis
107 A COPV overwrap consists of multiple overlapping carbon fiber e poxy layers filament wound at different angles relative to the vessels axis comprising the layup. In order to define the elastic nature of the overwrap, only material constants defining an orthotropic material are necessary. Assuming there is sufficient individual fiber windings for each layer defined by angle and the filament wound process produces a symmetric layup; the resultant material definition can be approximated as a symmetric angle ply laminate, essentially making an axisymmetric material mode l about the cylindrical axis (Gra y and Moser 2004). Engineering constants for each angle ply layer are calculated by transforming from the original properties ( = 0), where the wind angle is defined from the cylindrical axis of the pressure vessel. Therefore, the longitudinal direction is at = 0 and the hoop direction is at = 90. The Abaqus Wound Composite Modeler 6.7 3 plug in is used for the material calculations ( Dassault Systmes, 2008 ). The material properties of the carbon fiber epoxy T7 00 24K 50C are used in the finite element analysis. This material is used in the sub scale COPV fracture specimens tested for NASA. The elastic engineering constants for the T700 24K 50C carbon fiber epoxy material calculated based on 65% fiber volume no rmalized by the aluminum yield strength ( o ) are as follows: E 1 o = 527, E 2 o = E 3 o = 78.9, 12 = 13 = 0.326, 23 = 0.514, G 12 o = G 13 o = 27.1, G 23 o = 26.0. Where E 1 E 2 and E 3 are the moduli in the fiber, transverse and thickness directions respectively, G ij are the shear moduli and ij Geometries and Loads Figure 6 2 shows a surface crack from the sub scale COPV test specimens. The initial laser notch was placed in the center of the cylindrical se ction such that the crack opens i n the hoop direction. In or der to simplify experimental techniques, this crack is
108 located on the outer liner surface Pre cracking was done with cyclic pneumatic loading of the liner before wrapping. After wrapping, the test conditions consisted of a single over pressure load to s imulate autofrettage at 26.23 MPa (3804 psi) and 200 cycles at 16.96 MPa (2460 psi) to simulate operational life. During manufacturing, the autofrettage load is applied to compensate for thermal strains induced from curing by joining the liner to the over wrap with residual compressive stresses from plastic deformation. The initial and final crack fronts are shown in Figure 6 2 The pre cracks in the COPV specimens did not produce a through flaw during cyclic loading and crack growth from the outer surfac e was minimal. For purposes of finite element analysis, a crack opening on the interior liner surface is of greater interest (Figure 6 3). This placement is considered more critical in terms of crack mouth opening because crack opening is directly restri cted by the overwrap for a crack located on the outer liner surface. Crack growth is not directly modeled in this study; therefore, these experimental results will only serve as a base to compare the global material response of our finite element models. The sub scale COPV specimen i s designed to achieve an autofrettage and maximum operating pressure to ensure plasticity in the liner without burst. Due to the small cylindrical radius, the relative thickness of the applied overwrap is not typical. The ove rwrap to liner thickness ratio would typically be higher ( t Wrap t Liner ). Nevertheless, the effects of the overwrap on near tip fields should still be quantifiable. Figure 6 4 gives the average hoop and longitudinal surface strain data taken from eight gages (four gages oriented in the hoop direction and four in the longitudinal) adhered to the outer surface of the overwrap of an uncracked sub scale COPV
109 specimen at the center of the cylindrical section separated by 90 about the cylindrical axis. This specimen is identical to the initial COPV geometries before laser notching. This data will be simulated using a 3D finite element model to be described in a later section. The inability to acquire crack mouth opening data from this specimen inherently le ads to inaccuracies in near tip measurements (Faleskog, 1995), but the high accuracy in outer strain data provides confidence in the numerical estimations. The geometry, materials, and loads will be used in preliminary analyses before implementing additio nal COPV geometries. In this study, two liner and COPV geometries are implemented; a sub scale, matching the dimensions and loads described above and a full scale, in which typical COPV dimensions are used. The full scale COPV, with a three to one overwr ap to liner thickness ratio ( t Wrap t Liner ), is similar to those typically used in aerospace applications. Liner only models (homogeneous pressure vessels) are studied as a basis for comparison. Tables 6 1 and 6 2 provide the dimensions for the global liner and COPV models respectively Two semicircular crack sizes are inserted into each of the sub and full scale liners. The sub scale liner contains a crack with a/t = 0.267 corresponding to the maximum depth achieved in experimentation and a/t = 0.5 i mplemented to assess the effects of greater crack penetration. The full scale liner contains cracks of a/t = 0.5 and 0.7. The later depth ratio is typical for minimum detectable crack sizes for various non destructive evaluation (NDE) techniques. A glob al finite element model is implemented for each of the four geometries tested. Semicircular cracks of various depths are inserted in a sub model section of the liners at the center of the cylindrical portion. The longitudinal and hoop edge lengths of
110 thi s section are 38.1 38.1mm (1.5 1.5 in). Figure 6 3 is a schematic representation of the liner model with the crack section description and placement. Quarter symmetry boundary conditions are placed at half the axial length and the half circle cylindr ical cross section, such that only a quarter of the crack face placed at the center of the cylindrical section opening in the hoop direction is modeled. A typical single inlet COPV is not strictly quarter symmetric. However, the crack is assumed sufficie ntly far from end caps to be unaffected by the different geometries (inlet or dome). Only the quarter containing the inlet is modeled for the global COPVs. Figure 6 5 is a visualization of the mesh configuration. Sub model boundary conditions are then u sed to apply displacements to the cracked models. COPV sub models are implemented complete with anisotropic overwrap from the cut section of the global COPV. Greater mesh density is given to the region of the overwrap sec tion opposite the liner crack. A finite thickness adhesive layer is generally present between the overwrap and the liner. However, the layer thickness is small compared to overwrap and liner dimensions; therefore, it is ignored for this analysis. The COPV models use Abaqus tie constrai nts nodal displacements of the slave surface, in this case the liner, to the displacement field of the master surface (overwrap). Therefore, the model assumes a perfec t bond with zero adhesive thickness. Pressure loads are applied to the global and sub model inner surfaces using distributed loads. A simulation of the autofrettage load at 26.23 MPa (3804 psi) is applied to the sub scale COPV model. The maximum expected operating pressure (MEOP) of 16.96 MPa (2460 psi) is isolated for analysis before autofrettage load is achieved. The residual
111 stresses and strain hardening effects cause by autofrettage will be ignored for the MEOP cycle. A MEOP load of 20.68 MPa (3000 psi) is applied to the full scale COPV model. This load provides sufficient liner plasticity for this analysis; therefore, an autofrettage load will not be applied. Pressures of 8.76 MPa (1271 psi) and 1.671 MPa (242.9 psi) are applied to the liner sub s cale and full scale liner only models respectively, to achieve a nominal equivalent (von Mises) stress approximately equal to the their respective COPV liner load at MEOP Results and Discussion J Q and Critical Crack Angle Figures 6 6 and 6 7 show the J i ntegral (A ) and Q (B ) as a function of crack angle at a constant nominal equivalent stress = o for the sub scale (Fig ure 6 6) and full scale (Fig ure 6 7 ) models. The deeper cracks consistently show greater normalized J values even when normalized by crack depth. Similarly, due to the differences in strain distribution on the bonded surface caused by the continued partial elasticity of the structure, the overwrapped geometries (COPV) show significant reduction in J when compared to all the homogeneous geometries (liner only), for both deep and shallo w cracks. This indicates that the relative local deformation, measured as the ratio of J to relevant specimen dimensions, is significantly higher in deeper cracks and geometries without overwrap. Since this effect is a similar function of both overwrap a nd relative crack depth, the overwrap essentially increases the effective thickness of the geomet ry from a fracture standpoint. The uniaxial bonded structure investigated in Chapter 4, demonstrate a strong influence of the elastic backing on T stress const raint measurement. The greatest effect was shown at crack depth, where the in plane stress acting in the T direction was
112 directly influenced by the dimpling resistance caused by the backing, where dimpling is defined as displacement normal to the plate o n the face opposite crack opening. This study cannot be entirely applied to a biaxial structure with Q as the con straint measurement. Figures 6 6 (B) and 6 7 (B ) show the strong increase in Q near the free surface cause by the biaxial component of stress (longitudinal stress). In addition, since Q is measured here under large scale yielding conditions, the correlation of Q and elastically scaled T cannot be assured. Nevertheless, some similarities exist in Q measured here and the elastic T stress from t he uniaxial case. The Q values computed in the COPV geometries are greater compared to liner only geometries, with the greatest differences found at crack depth, caused by the overwrap, and near the angle of maximum Q caused by differences in the biaxial ratio. The existence of the overwrap often decreases the ratio of hoop stress to longitudinal stress (biaxial ratio) compared to homogeneous pressure vessels resulting in a higher longitudinal component. The shallow crack in the sub COPV does not demons trate any significant effect of the overwrap at crack depth because the crack is sufficiently far from the bonded layer. Crack depth has only a minor effect on Q measured in the COPV geometries, indicating the increased effective thickness caused by the ov erwrap is sufficient to maintain a value of Q unaffected by small increases in crack depth. However, crack depth in the homogeneous geomet ries has a significant effect on Q with deeper cracks showing greater constraint loss (lower Q ) for a given nominal load.
113 The presence of the overwrap also has a small effect of the angle of maximum J and Q values. In both cases, the maximum is found at an angle closer to the free surface in the COPV geometries Figures 6 8 (A) and (B ) give the product of the J integr al and opening stress ( yy ) at a radial distance ( r = 2 o ) normalized by the maximum product as a function of crack angle for o in the sub scale and full scale models respectively. The open ing stress value is used as a measure of constraint magnitude. Therefore, the product of J and yy ( r = 2 o ) can be used in a simi lar manner to Leach et al. (2007 ) to predict the critical crack growth angle. Assuming radial independence of Q this methodology should produce relatively accurate critical crack growth angles. The as sumption that the opening stress distribution will produce a radially independent measure of Q is shown to be invalid as the collapse or over relaxation of near tip fields is no longer represented by the J Q MBL predicted state. Consequently, the shape of this curve is moderately influenced by load and the actual crack growth mechanism may or may not be controlled by J and Q Therefore, the critical crack growth angles taken from this methodology, where the actual crack extension load is not known, should only be taken as an engineering approximation. The angle of interest corresponding to the COPV maxima is = 24. The crack angle of = 90 will also be analyzed in subsequent sections because growth in the crack depth direction is critical in terms of developing a through crack in the liner. A clear shift towards crack depth occurs in the liner only models. This corresponds to the angular shift towards crack depth in both maximum J and Q shown in Figures 6 6 and 6 7 For simplicity, comparisons will only be made at angles = 24 and 90 in both the COPV and liner only models
114 Typically, crack tip opening displacement (CTOD) and driving force ( J integral) measurements are taken from FEA coupon or structure models to compare with critical values. For two parameter fracture prediction, this is done with a failure locus as a function of constraint, where lower constraint yields higher toughness or resistance to crack growth. A conservative comparison between measured J and a critical J can be made with elastic plane strain fr acture toughness ( K IC ) of 6061 T6 aluminum converted to J IC using Equation 2 17 This value does not account for the increases in apparent fracture toughness with constraint loss and strain energy dissipation associated with plastic strains near the crack tip and therefore is not necessarily associated with cr itical crack growth. Figures 6 9 and 6 10 (A) and (B ) give J normalized by a critical value for angles = 24 and 90 in the sub and full scale models as a function of nominal equivalent stress in the liner ( ). The critical J integral ( J IC ) from plane strain fracture toughness K IC used to normalize Figures 6 9 and 6 10 is J IC MacMaster et al. 2000 ). These figures are shown only to the maximum load shared in each group. These loads are = 1.02 o and 1.10 o for the sub scale and full scale models respectively. By looking only at the driving force and ignoring constraint/triaxial effects on apparent toughness, the overwrap in a COPV significantly increases the resistance to crack failure as a function of load. Both the sub scale and full scale models show this trend in loading trajectory. However, the full scale model, with its higher overwrap to liner thickness ratio, experiences a stronger effect of the overwrap compared to the sub sc ale model of equivalent crack penetration ( a/t = 0.50)
115 Near Tip Field Characterization L imits Two general conclusions can be made by comparing near tip fields to J Q MBL predicted fields. Firstly, the adherence of near tip fields at a radial distance of r = 4 o indicates how the level of far field loads and crack tip deformations affect the near tip opening fields compared to a J Q prediction, providing a simple and clear measure of J Q dominance. Secondly, because Q, as defined in MBL formulation (Equ ation 2 9 ), is an offset measure of opening stress at r = o a comparison at a greater radial distance, here r = 4 o provides a relative measure of radial independence of Q The accurate prediction of stress fields in this range, where both cleavag e and ductile fracture mechanisms are present, provides a valid engineering measure of J Q dominance. Note that the ductile fracture mechanism is most prevalent in crack growth for aluminum alloys. The most obvious and significant comparison with regard t o the fracture driving force is of the opening stresses ( yy ) in the crack plane ( = 0) as functions of nominal load and near tip deformation. Engineering geometric standards for crack specimens use a factor that is directly proportional to relevant specimen dimensions and inversely proportional to crack driv ing force to specify minimum dimensional requirements in order to ensure a size independent measure of toughness. Therefore, the local deformation factor o /J is an inverse measure of local deformation and can be used as a relative measure of specimen ch aracteristic length requirements. Figures 6 11 through 6 14 give the opening stress ( yy ) normalized by J Q MBL predicted stress ( MBL ) at r = 4 o as a function of nominal equivalent liner stress (A ) and local deformation factor o /J (B ) at = 24 a nd 90 in all the geometries tested. In some cases, the homogeneous deep cracks at angles near crack depth yielded Q
116 values that were lower than the extent of MBL predictions; Q is therefore not extrapolated and these points are typically omitted. These trends will be analyzed in the subsequent section when Q is given as a function of load. Chapter 4 demonstrated that the allowable far field load before field collapse, designated by a greater than 5% difference from J T MBL solutions, is greater for bonde d g eometries It was also found that for these uniaxial geometries, the allowable local deformation was significantly greater at crack depth for the bonded case; however, in the region of maximum J ( r allowable deformations before J T collapse. J Q analysis of the biaxial fields in the pressure vessel geometries produces similar results. It can be observed that the sub scale and full scale models produce nearly identical trends. Although some differ ences exist for the near tip fields at crack depth ( = 90), the general trends in the data do not seem to demonstrate a clear effect of the overwrap at this angle. However, the fields remain within 5% of the MBL solutions for the loads tested. Therefore, the fields are said to maintain J Q dominance an d Q radial independence for the loads and deformations shown at = 90 and radial distances 2 o o A clearer picture of near tip trends and plastic collapse can be observed for the angle in the critical region ( = 24). Table 6 3 gives the values of nominal equivalent liner stress and corresponding local deformation factors for = 24 at the points of 5% deviation ( yy / MBL = 0.95) in the MBL normalized opening stress field This table alo ng with Figures 6 11 and 6 13 show the trends in n ear tip field collapse points as a function of overwrap. Similar to the results presented in Chapter 4 the allowable nominal stresses and local deformation
117 factors are higher (higher stress and smaller deformation) for the COPV models. This indicates th at the increased resistance to crack opening cau sed by the overwrap, ( Fig ures 6 11 and 6 13 ) dominates the near tip fields resistance to plastic collapse. This results in high stresses with limited crack opening at the point of field collapse Constraint Loss and Triaxiality in the Full S cale COPV The results from both the sub scale and full scale models have shown similar trends in J Q and near tip stress field measurements. In addition, the trends observed in the full scale model are stronger and ther efore more helpful in drawing conclusions from the results. The following sections deal only with the full scale COPV. Fully elastic models are used to calculate T stress scaling factors, which are normalized by the nominal hoop stress ( h ) of the liner. Figure 6 15 gives these factors for the full scale COPV and liner geometries. The elastic biaxial ratio (hoop over longitudinal stress) is significantly different in the wrapped verses un wrapped geometries. Nominal hoop verses longitu dinal stress in the liner is 2.0 without overwrap and 1.6 with overwrap. The higher longitudinal stresses in the COPV account for the increase in T stress at lower crack angles. Stresses acting in the crack plane are added to the T stress. In this case, the longitudinal stress is added to the T stress as a cosine function of crack angle. At crack depth, where no biaxial effects are present, the deep crack has a slightly higher T stress in the COPV compared to the liner only model. While this difference is not noticeable for a/t = 0.50, the deeper crack shows greater tensile stresses generated by the overwrap stiffness The MBL generated Q verses T stress curve, shown in Figure 2 5 is the difference in the J T predicted opening stress and the SSY soluti on at r = 2 o Therefore, comparing Q and Q predicted from T or Q ( T ) as a function of load is
118 essentially the comparison of J Q and J T opening stress fields. With increasing load and the onset of large scale yielding, near tip stresses will drop nonlinearly for low constraint geometries. Additionally, as plastic flow proceeds through the remaining ligament, T stress looses it meaning as a constraint parameter and Q deviates significantly from Q ( T ). Figures 6 16 and 6 17 give Q and Q ( T ) as a function of local de formation o /J and nominal equivalent stress respectively. Large scale yielding occurs when the near tip stress conditions deviate significantly from the elastic solution. The existence of a K scaled asymptotic field is not assured. The ratio of elastic plasti c J to elastic J EL estimate using Equation 2 17 and a linearly scalable K I can be used to determine the onset of large scale yielding. An arbitrary limit of 10% deviation is placed on this ratio and Table 6 4 gives the loads computed at this point for th e full scale models Therefore, Figures 6 16 and 6 17 primarily show only loads after the onset of large scale yielding and up to and including plastic collapse. Even under biaxial stresses Q remains relatively close to Q ( T ), prior to the onset of large scale yielding. However, the near surface ( 24) curves significantly deviate after this point At crack angle 24, the increase in crack penetration ( a/t both global geometries does not have a significant effect on Q Conversely increased crack penetration results in a more rapid decrease in Q for larger loads and deformations at 90 in the liner geometry. As plastic flow progresses through the remaining ligament at crack depth, stress field relaxation is a result of loss o f near tip triaxiality. This effect is enhanced with the decrease in remaining ligament. Interestingly, this same phenomenon has the opposite effect in the COPV geometry.
119 The increase in crack depth results in the fields being measured at a distance clo ser to the bonded surface for 90, resulting in higher constraint values for larger deformations compared to the shallower crack. Because MBL formulations rely on high triaxiality (plane strain), the Q maintains a closer relationship to Q ( T ) at large loads for the deeper crack. A t crack depth, Q maintains approximately a linear relationship with load up to very large deformations making Q ( T ) a more accurate approximation. However, for both angles tested Q deviates significantly from Q ( T ) at larger loads. The general conclusion i s that T fails to predict accurate stress fields for moderate levels of deformation and nominal equivalent stress levels in a biaxial configuration Ductile fracture occurs by void nucleation, growth and coalescence in metals (Rice and Tracey, 1974; Tver gaard 1982; Garrison and Moody, 1987; Tvergaard and Hutchinson, 2002) Near tip triaxiality, defined as the ratio of hydrostatic stress to von Mises stress, directly controls void nucleation and growth, therefore a constraint parameter should accurately represent triaxiality. Henry and Luxmoore (1997) in their study on the relationship between triaxiality and Q for low constraint geometries, indicate that there is a unique linear relationship between Q measured at r = 2 o and triaxiality. This relationship is independent of geometry and level of deformation. Therefore, one would expect the near tip triaxiality to maintain a linear relationship with Q for the homogeneous (liner only) models for all levels of deformat ion tested. This is not necessarily true for the COPV models, because the material is discontinuous through the thickness. The ratio of mean stress to von Mises stress is given by :
120 (6 1 ) where 1 2 and 3 are the principal stres ses. Figures 6 18 (A) and (B ) give the ratio of mean stress to von Mises stress verses Q at a radial distance r = 2 o for the full liner and COPV models at angles = 24 and 90 respectively. In the liner only models, for the near surface angle ( = 24 ), the triaxiality is lower for a given value of Q compared to those measured at crack depth ( = 90), indicating that the relationship between triaxiality and constraint loss is weakly affected by the addition of a biaxial stress. Nevertheless, triaxia lity and Q maintain a linear relationship that is a moderate function of Figures 6 12 and 6 14 indicate the relative independence of Q on radial distance for = 90. If deformation levels and geometry facilitate a hydrostatic shift in stress fields measured by Q opening stress must inherently agree with J Q predictions in the process zone. However, out of plane and hydrostatic stress components may determine whether the trends in Q maintain a linear relationship with triaxiality, which is desirable for Q as a ductile fracture parameter. It is important to indicate the level of Q experienced when plasticity prevails over the entire cylindrical portion of the liner in order to evaluate trends. Table 6 5 provides Q measured at a nominal equivalent str ess of yield ( o ) Triaxiality more or less maintains an equivalent relationship to Q for a given angle up to the region of nominal (far field) yield. When this load is exceeded, the triaxiality as a function of Q increases in the COPVs compared to the liner only, which maintains linearity through the loads shown. Except for the deep crack ( a/t = 0.70) at crack depth ( = 90), the increased
121 trend in the COPV models is gradual. However, an increase in triaxiality with a decrease in Q is seen for a small region corresponding with nominal yield for the deep crack ( a/t = 0.70) at crack depth ( = 90) in the COPV model. In this region, Q remai ns relatively unchanged ( Figure 6 17 (D )), while the far field load and therefore J and triaxiality continue t o increase. The increase in near tip triaxiality and radial distance of measure (linear with J ), results in a stronger dependency of the bonded relationship. Correspondingly, all bonded geometries demonstrate a greater influence on the material discontin uity for loads causing full scale yielding. Essentially, the continued elasticity in the structure maintains higher levels of stress for loads causing yield in the liner portion Concluding Remarks Fracture characterization in geometries where loads excee d net section yield is difficult and requires a thorough analysis of the near tip fields. The COPV geometry further complicates the analysis because of its material discontinuity and anisotropy. An engineering approach to the parameterization of surface cracks in COPV liners is given, both from a deformation limit and conservativeness perspective. J and Q crack front distributions, and the corresponding limits, and near tip triaxiality and constraint loss trends are obtained and the effects of elastic pl astic material discontinuity of the COPV and the biaxiality of stresses are evaluated. The following conclusions are summarized: a) Crack mouth opening and subsequent near tip crack intensity ( J integral) is significantly reduced as a function of far field l oad with the addition of the overwrap. Therefore, from a perspective of single parameter fracture prediction, COPVs cannot be treated as a homogeneous structure. Incorporating the overwrap in fracture measurements will achieve a more adjusted and less co nservative estimate of failure. b) J Q predicted fields maintain accuracy ( Q is radially independent) for higher far field loads and lower near tip deformations compared to the liner only models for angles near the free surface. This indicates that the resi stance to crack opening
122 (lower near tip deformations) outweighs the effects of near tip field relaxation in the COPV geometry at these angles. c) J Q fracture prediction of surface cracks in COPV metallic liners is possible up to large deformations. However, for the critical crack growth region ( semicircular crack, Q does not maintain a radially independent measure of constraint for loads seen in a typical COPV; therefore, these fracture predictors may not be applicable. d) The triaxiality and Q re lationship becomes non linear with large deformations for the COPV models, while homogeneous models maintain a linear relationship independent of deformation. At crack depth for the deeply crack full scale COPV, where the influence of the overwrap is most prevalent, large scale yielding marks a transition where triaxiality increases with loss of constraint. As a ductile fracture parameter, Q is to maintain a linear relationship with triaxiality; therefore, this effect will have consequences on the predict ion of ductile crack growth. e) The trends in near tip stress fields referenced to MBL solutions seen in the COPV geometries are similar to uniaxial cases studied in Chapter 4 The Q from elastic T stress ( Q ( T )) maintains relatively accurate predictions of Q for crack depth angles where the overwrap has the greatest effect, indicating J T fields are accurate in this region. However, the addition of the biaxial stress drastically reduces J T predictability for angles near the free surface.
123 Table 6 1. Dimensi ons of the metallic liner models Model Material t L (mm) di (mm) L (mm) Sub Liner 6061 T6 Al 2.29 157 504 Full Liner 6061 T6 Al 1.27 498 1149 Table 6 2 Dimensions of the COPV models Model Liner Material t Hel (mm, Deg) t Hoop (mm, Deg) Sub COPV Su b Liner T700 24K 50C 0.665, 12.2 0.686, 90 Full COPV Full Liner T700 24K 50C 1.91, 11.8 1.91, 90 Table 6 3 J Q dominance limit loads at = 24 Model a/t o /J o Sub COPV 0.267 71.4 0.985 Sub Liner 0.267 64.7 0.980 Sub COPV 0.50 73.4 0.964 Sub Liner 0.50 65.9 0.951 Full COPV 0.50 73.5 0.971 Full Liner 0.50 64.0 0.938 Full COPV 0.70 72.1 0.966 Full Liner 0.70 63.9 0.911 Table 6 4 Points of large scale yielding ( J/J EL = 1.10) Model a/t o /J o Full Liner 0.50 24 186 0.750 F ull COPV 0.50 24 176 0.820 Full Liner 0.50 90 174 0.847 Full COPV 0.50 90 174 0.883 Full Liner 0.70 24 219 0.661 Full COPV 0.70 24 177 0.798 Full Liner 0.70 90 197 0.792 Full COPV 0.70 90 173 0.872 Table 6 5 Q at o Model a/t Q Full Liner 0.50 24 0.577 Full COPV 0.50 24 0.335 Full Liner 0.50 90 1.13 Full COPV 0.50 90 0.974 Full Liner 0.70 24 0.703 Full COPV 0.70 24 0.342 Full Liner 0.70 90 1.23 Full COPV 0.70 90 0.985
124 Figure 6 1 Uniaxial stress strain dat a for 6061 T6 aluminum liner material and Abaqus material inputs for incremental plasticity Figure 6 2 Surface crack from sub scale COPV
125 Figure 6 3 General COPV liner section surface crack model configuration
126 Figure 6 4 Strain data from the sub scale COPV specimen and 3D FEA results
127 Figure 6 5 FEA visualization of the surface cracked liner sub model
128 A B Figure 6 6 J (A ) and Q (B ) crack front distributions for the sub scale COPV and liner models at a nominal equivalent stress = o
129 A B Figure 6 7 J (A ) and Q (B ) crack front distributions for the full scale COPV and liner models at a nominal equivalent stress = o
130 A B Figure 6 8 The product of J ( ) and opening stress at a radial distance ( r = 2 o ) normalized by the maximum for the sub scale ( A ) and full scale ( B ) models at a constant nominal equivalent stress = o
131 A B Figure 6 9 J normalized by J IC as a function of nominal equivalent liner stress ( ) for angles = 24 (A ) and 90 (B ) in the sub scale COPV and liner models. A B Figure 6 10 J normalized by J IC as a function of nominal equivalent liner stress ( ) for angles = 24 (A ) and 90 (B ) in the f ull scale COPV and liner models.
132 A B Figure 6 11 Opening stress ( yy ) normalized by J Q MBL predicted str ess at r = 4 o as a function of nomin al equivalent liner stress (A ) and l ocal deformation factor o /J (B ) at = 24 in the sub scale models. A B Figure 6 12 Opening stress ( yy ) normalized by J Q MBL predicted stress at r = 4 o as a function of nominal equivalent liner stress ( A ) and local de formation factor o /J (B) at = 90 in the sub scale models.
133 A B Figure 6 13 Opening stress ( yy ) normalized by J Q MBL predicted stress at r = 4 o as a function of nomin al equivalent liner stress (A ) and l ocal deformation factor o /J (B) at = 24 in the full scale models. A B Figure 6 14 Opening stress ( yy ) normalized by J Q MBL predicted stress at r = 4 o as a function of nominal equivalent liner stress (A ) and local deformation fa ctor o /J (B ) at = 90 in the full scale models.
134 Figure 6 15 T stress scaling factors ( h )
135 A B C D Figure 6 16 Q and Q(T) as a function of local deformation o /J
136 A B C D Figure 6 17 Q and Q(T) as a function nominal equivalent stress
137 A B Figure 6 18 The ratio of mean stress to von Mises stress at a radial distance r = 2 o verses Q for the full liner and COPV models at angles = 24 (A) and 90 (B ).
138 CHAPTER 7 FRACTURE PREDICTION WITH TWO PARAMETERS As a verification of the fracture prediction and limitation analysis methodologies presented in the previous chapters experimental data found in literature is applied Fracture prediction using a two parameter approach provides a more adjusted assessment of failure limit s (Cicero et al., 2010; Silva et al., 2006 ). Applying a fracture r tip stress and strain fields. Near tip plastic energy dissipation due to variations in geometric constraint will direct ly affect the apparent toughness of a crack. Experimental verifications of constraint effect on critical J values are presented throughout literature. For example, Faleskog (1995) showed experimentally how an increase in the constraint factor Q results i n a decrease in the critical value of J Crack growth is therefore a function of both driving force ( J integral) and loss of constraint. Apparent toughness of high or positive constraint geometries is not significantly affected by changes in constraint. Leach et al. (2007) used the product of a hyper local constraint parameter ( h ) and the J integral to predict initiation points of ductile crack growth in surface cracks compared to experimental specimens. Hyper local constraint is defined as the opening stress in the crack plane averaged over the plastic zone. In general, the magnitude of this parameter will be similar to the Q stress difference parameter taken at r = 2 o given in Equation 2 9, providing there is radial independence of Q and o scal es with plastic zone size. This chapter investigates the similarities and differences when using J Q fracture prediction in a similar manner to the methodology in Leach et al. (2007) for critical crack angle studies and the subsequent J Q near tip stress f ield predictions. It is not the
139 purpose of this study to reproduce the results presented in the abovementioned work. The most important aspect of this study is to verify the existence of a two parameter predicted field at the angles and loads where ducti le fracture is initiated. Model Definitions and Experimental Methods Material The material used in the experimental study by Leach et al. (2007) is a D6AC steel that in finite elements is modeled using t he Ramberg Osgood power law hardening material model intrinsic to the deformati on plasticity theory with Abaqus ( Ramberg and Osgood, 1943) This model is de fined by the equations 4 1 and 4 2 with the following parameters: E = 209.7 GPa, o = 1330 MPa = 0.315 n = 5 0 and the Poisson ratio = 0.3. A modified boundary layer J Q reference solution is generated for this material using the methodology presented in the Background chapter. Experimental Methods A complete description of the test methods can be found in Leach et al. (2007). Only summaries of methods and geometries are gi ven here. The schematic of the geometry is shown in Figure 3 1, where the load is applied as far field tensile stresses ( ) and w l t = 50.8 50.8 6 .35 mm (1 2 0.25 in). Electrical discharge machined (EDM) notches are used to facilitate pre cracking. Fatigue pre cracking is complete under remote bending loads. Crack extension during monotonic loading is detected by a 5% potential drop across t he cracked section at which point the load is removed. The crack face is marked using low load fatigue cycles. The critical location is assumed where maximum crack extension occurs. This relies on the assumption that maximum crack extension correspondin g to a 5% potential drop corresponds to the location of initiation. Four specimens are modeled with crack
140 dimensions shown in Table 7 1. These are denoted in the abovementioned paper as AT4, AT5, AT6 and AT7. Finite element modeling of these crack geome tries uses a mesh configu ration that is identical to those presented in the previous chapter. J integral and Constraint Factors The normalized crack front angle 90 is used to correspond with the angular definition found in the abovementioned paper. Table 7 1 gives the J integral and Q values at the loads and locations ( Crit ) associated with crack growth, denoted J Crit and Q Crit respectively. Additionally, th e experimentally measured crack growth angle ( Crit / 90 ) initiation load ( / o ) and the angle associated with maximum product of J and Q ( JQ / 90 ) are provided. In order to determine a relative maximum of this product and to retain similarities to the a bovementioned paper, the Q is offset by the SSY solution and thus the opening stress at r = 2 o denoted as Q is used to measure constraint magnitude for critical angle determina tion. Specimens AT4 through AT6 give relatively consistent critical J int egrals at crack extension, indicating the growth mechanism is the same. AT7 gives a slightly higher J Crit value, which is consistent with the larger crack size. Table 7 2 gives the maximum J and Q values calculated from the finite element models. The nor mal ized opening stress values are calculated as Q = Q + MBL ( T = 0 r = 2 J/ o ) Figures 7 1 A through D give the J Q and J Q crack front distributions normalized by their respective maxima. The normalized view of this data provides insight into the relative intensity at various points along the cra ck front. A 5% error corresponding to 0.95 on the vertical axis is generally an acceptable variation on this relative intensity. Therefore, values within 5% deviation of the max are considered viable initiation points. For many of the geometries the pre dicted range is very broad
141 and cannot pinpoint a single initiation point. However, the adjusted range observed in two parameter prediction provides a slightly narrower range of possible initiation points encompassing the actual point of initiations in all the models tested, while the J integral and opening stress values alone only provide a very broad range of possible points. Although the trends are consistent with observations in Leach et al. (2007), the difference in the measure of constraint gives sli ghtly different results. At angles near crack depth, J Q is lower, providing a narrower range of possible initiation points compared to those predicted in the abovementioned work. Two Parameter Stress Field Characterization The most significant comparison with regard to the fracture driving force is of the op ening stresses in the crack plane for radial distances where both cleavage and ductile fracture mechanisms are active and outside the area where finite strain effect s are r/ ( o ) Figures 7 2 A through D give opening stress as a function of normalized radial distance ( r/ ( o ) ) at loads and initiation points corresponding to experimental growth data. The range 1 r/ ( o ) is given on a logarithmic scale providing maximum resolution in the region des cribed above. Since, t he ductile fracture mechanism associated with this range of characterization is consistent in the specimens; the corresponding opening stress fields should be accurate when the measured critical values ( J Crit ) predict fracture. All the specimens modeled maintain a predicted field and the associated radially independent measure of Q for the full range shown in these plots Concluding Remarks The following conclusions regarding additions to the experimental and numerical studies presen ted in Leach et al. (2007) are enumerated below:
142 a) J and Q where the magnitude of Q is measured as Q or the opening stress at radial distance r = 2 o provid e similar trends in predicted critical crack growth angle as J and hyper local constraint present ed in Leach et al. (2007) for uniaxial tensile surface crack specimens. b) A first order constraint corrected fracture criterion, namely the J Q predictor, is more accurate at predicting the critical crack growth angle than J alone. c) For crack extension associ ated with cleavage or ductile fracture in a well constrained geometry, the J Q MBL predicted stress fields are accurate for a radial range where these mechanisms are active. The above conclusions give insight into the function of near tip stress field anal ysis in providing experimental verification of fracture parameterization in specimen and structural geometries.
143 Table 7 1 Crack dimensions, FEA meas ured J and Q values and critical angles Model a/c a/ t / o J Crit Q Crit Crit / 90 JQ / 90 AT4 0.72 0.50 0.866 55.3 0.505 0.43 0.337 AT5 0.70 0.52 0.836 53.5 0.504 0.28 0.356 AT6 0.72 0.52 0.857 57.0 0.499 0.45 0.322 AT7 0.61 0.65 0.757 70.2 0.417 0.34 0.315 Table 7 2 J Max and Q Max values and angles ( MBL ( T = 0 r = 2 J/ o ) = 2.75 ) Model J Max Q Max J / 90 Q / 90 JQ / 90 AT4 56.0 0.515 0.245 0.4 40 0.337 AT5 53. 6 0.48 7 0.23 3 0.444 0.356 AT6 58.1 0.50 9 0.2 30 0.4 40 0.322 AT7 70. 9 0.436 0.23 4 0.39 2 0.315
144 A B C D Figure 7 1. J Q and Q crack front distributions normalized by their respective maximums.
145 A B C D Figure 7 2. Opening stresses at the experimental critical crack angle ( Crit ) as a function of normalized radial dis tance r/ ( o ) for load s corresponding to crack growth on a logarithmic scale for all the models implemented. The reference solutions and 5% deviation curves are shown as solid points and dotted lines respectively.
146 CHAPTER 8 SUMMARY The results and concl usions from this study aim to address the feasibility of using a two parameter fracture theory to predict failure and characterize limits of both surface crack specimens and sur face cracked liners with a structural backing. These results and methodologies are of pract ical and intrinsic value for establishing testing standards for surface crack geometries development of proof test logic and failure assessment of COPVs and broader establishing of methodologies for assessing near tip dominance, parameterizat ion deformation limits and plastic material property effects in fracture prediction. Clear load limits and trends are given for specific material/geometry combinations. A brief summary of conclusions are enumerated below: a) Using a practical approach to ne ar tip characterization using MBL formulations, J T deformation limits are found to be a strong function of the partial elasticity of a bonded geometry. The resistance to crack opening combined with the increased near tip triaxiality and constraint for su rface crack angles near crack depth in a bonded geometry results in significant increase in applicable loads before plastic collapse. Similarly, allowable near tip deformations (scaled with J ) at crack depth are greater for the bonded geometries, even whe n the near tip deformations are greatly reduced as a function of load due to the increased resistance to crack opening imposed by the backing. The effect of the backing on near tip fields is significantly reduced near the free surface and the resistance t o crack opening dominates deformation limits resulting in slightly less allowable local deformations in the bonded geometry. A similar effect can be observed with the increase of backing to liner modulus ratio where the increased resistance to crack tip o pening for greater ratios causes near tip fields to collapse at lower local deformations. b) T stress provides a remarkably accurate constraint measure for the characterization of near tip fields at loads exceeding net section yield. However, T stress is not defined in fully yield ed conditions; therefore, cannot provide a clear definition of near tip field characterization limits for these loads. The near tip stress difference parameter Q is adopted to serve this purpose. J Q deformation limits are observed as function of material st r ain hardening for the uniaxial surface cracked geometry. Strain hardening level has to increase when plastic flow progresses through the remaining ligament for surface cracks under tension, to maintain the predicted field. Sim ilarly, the radial independence of Q necessary for ductile fracture prediction, cannot be assured for a low strain hardening material even for moderate loads.
147 c) Characteristic length measurements can be used to scale deformations limits as a function of mat erial properties within a range of applicable strain hardening levels. The remaining net section area or thickness of a surface crack specimen can be used along with a material dependent limit locus to determine applicability of surface crack tension spec imens. The local deformation parameter, o /J is a relative measure of near tip deformation limits as a function of both geometry and material properties, where l is the characteristic length of the specimen. d) Recent ASTM standard development for surface cracked specimens has favored a material dependent limit that leads to conservative estimates. A simplified deformation limit criteria may be applicable if the strain hardening exponent is restricted within a range. The low strain hardening material may be used to provide a clear upper limit to the exponent. e) The COPV geometry further complicates the analysis because of its material discontinuity and anisotropy of the overwrap. Crack mouth opening and subsequent near tip crack intensity (J integral) in a COPV liner is significantly reduced as a function of far field load compared to a homogeneous pressure vessel of equal liner thickness. f) In a COPV geometry, J Q predicted fields maintain accuracy for higher far field loads and lower near tip deformations c ompared to the liner only models for angles near the free surface. These results are similar to those presented for J T characterization of a uniaxial bonded specimen. g) J Q fracture prediction of surface cracks in COPV metallic liners is possible up to lar ge deformations. However, for the critical crack growth region ( semicircular crack, Q does not maintain a radially independent measure of constraint for loads seen in a typical COPV; therefore, these fracture predictors may not be applicable h) Crack tip triaxiality within the process zone plays an important role in ductile fracture prediction and a constraint parameter is a valid predictor only if it maintains a unique deformation and geometrically independent relationship. The triaxiality an d Q relationship becomes non linear with large deformations for the COPV models, while homogeneous models maintain a linear relationship independent of deformation. At crack depth for the deeply crack ed full scale COPV, where the influence of the overwrap is most prevalent, large scale yielding marks a transition where triaxiality increases with loss of constraint. This indicates that while J Q and J T predicted opening stress fields are valid at large deformation in a region with high triaxiality, the in fluence may hinder Q as a ductile fracture parameter for deep cracks in the liner of a COPV. i) An example of fracture prediction and near tip stress field characterization is given for surface crack tensile specimens at the point of ductile crack extension. A constraint corrected fracture prediction, in this case the product of J and Q (where Q is an offset measure of Q ) provides a more adjusted (narrower band) prediction
148 of critical crack growth angles. Additionally, at the point of crack extension the near tip opening stress fields are accurately predicted by J Q MBL solutions. Although this example provides assurance to the two parameter approach, the limits of this methodology can only be substantiated by further testing with geometries and loads th at facilitate plastic collapse before ductile tearing. Although this study is directed toward application in COPVs, the results and conclusions given for the bonded configuration is of use for any liner penetrating surface flaw, where the outer structura l layer can fail due to exposure to the contents. These structures include nuclear pressure vessels and hydrogen storage tanks where a liner material is often used to protect the structural steel f rom the effects of hydrogen. In general, the types of ana lysis presented in this study can and should be used for any surface cracked structure, where the combination of loads and complex geometries make characterization difficult, but where engineering applications necessitate a parameterized failure criterion
149 APPENDIX A POST PROCESSING PSEUDO CODE Data Extraction Extract data from FEA input file o Crack front nodes and associated crack angles written to array: [Node Number, x n y n z n Angle] o Q Vectors (defined as the unit vector normal to the crack front in the crack plane in the direction of crack growth) for crack front nodes written to array: [ q x q y q z ] o All nodal points associated with the part of interest written to array: [Node Number, x y z ] Find node numbers for points in the crack plane normal to the crack front in the growth direction o Find all nodes in set that fall on the crack plane, ( z = 0) for most cases o Plot line for a given crack angle at the crack front node location using q vectors, with z n = 0 and q z = 0: (A 1 ) o Becau se truncation error does not allow us to use the above equation directly, point line distance form with a low tolerance is necessary to find the nodal points that fall on this line. If the distance from a nodal point to the line is within the tolerance an d its position is in the growth direction it is of interest and is written to an array. (A 2 ) w here : o The confirmed radial points are written to an array and the ra dial distance is calculated and sorted (A 3 )
150 Using the node numbers of the radial points, the output data (stress, strain, The J integral values are extracted from the FEA *. dat file and written to a 3 dimensional array: [Contour Number, Crack Angle, Load Step] o The average J integral from the desired contours is calculated and written to a 2 dimensional array: [Crack Angle, Load Step] Similarly the T stresses are extracted fro m the FEA *.dat file from the elastic model All the extracted data is saved to various output files for further post processing Post Processing and Plotting Load the extracted data: J integrals (Crack Angle, Load Step), Normalized T Stress ( T Stress divide d by the Nominal Stress), Stress values (Crack Angle, r Load Step), Nominal Load Values (Load Step), Crack Parameters ( a c t ), Material Yield Stress Load the MBL reference data: ( T / o ) MBL MBL Calculate r / ( J/ o ) for all angles, r values and load steps : Calculate T EP / o for all angles and load steps: (A 4 ) Reduce data for r / ( J/ o ) values and load steps that are within nodal resolution, providing enough points for accurate interpolation The following oper ations are completed for each crack angle and specified r / ( J/ o ) value (typically r / ( J/ o ) = 2) o Using cubic spline interpolation, calculate stress values at the specified r / ( J/ o ) value for each load step o Using cubic spline interpolation, calculate MBL p redicted stress values at the specified r / ( J/ o ) value for each value of normalized T Stress from The following data has been extracted and calculated and can be used to form many combinations of plots: Normalized T stress from MBL reference data
151 Normalized opening stress from MBL reference data Crack angle Nominal elastic plastic load at each time step J Integral at crack angle and n ominal elastic plastic load Normalized T stress from data set Normalized radial distance at crack angle, nominal load and r Opening stress at crack angle, nominal load and r value T he above code is nearly identical for J Q characterization, except that the constraint values T MBL and T are replaced with Q MBL and Q which are calculated from the opening stress curves.
152 APPENDIX B MATERIAL PROPERTY IN PUTS Ramberg Osgood The Ramberg Osgoo d material model is defined as (B 1) (B 2) where E o is the reference yield strength, n is the strain hardening exponent, and is the fitting coefficient or strain offset ( Ramberg and Osgood, 1943) 6061 T6 aluminum (Chapter 4): E = 69.0 GPa o = 27 6 MPa n = 40, and = 0.33 D6AC steel (Chapte r 7 ): E = 209.7 GPa, o = 1330 MPa, n = 5 0, 0.315 and = 0.3 Linear Plus Power L aw The linear plus power law (LPPL) material model is defined as: for (B 3 ) for (B 4 ) for (B 5 ) (B 6 )
153 where E is the elastic modulus, n is the strain hardening exponent, o i s the reference yield stress, o is the yield strain ( Nc Nc ) is the center and r Nc is the radius of the circular arc in normalized space ( o o ). K 1 is the lower limit of the transition region, which is set at K 1 = 0.95, and K 2 is upper limit (He aly et al., 2009). Chapter 6 inputs: E = 69.0 GPa, o = 276 MPa n = 20 (low strain hardening) or 3 (high strain hardening) and = 0.33 n = 20 (low strain hardening): Nc = 1.079, Nc = 0.821 r Nc = 0.183 K 1 = 0.95 and K 2 = 1.003 n = 3 (low strain hardening): Nc = 1.159, Nc = 0.741, r Nc = 0. 295 K 1 = 0.95 and K 2 = 1.023 The material curve is discretized into stress and plastic strain inputs for the FEA model and is given in Table B 1. 6061 T6 Tensile Data Table B 2 provides data calculated from 6061 T6 tensile test data, that is used in the FEA models for Chapter 6
154 Table B 1. Initial and final stress and plastic strain inputs for a LPPL model n = 20 n = 3 Plastic Strain ( p ) (MPa) Plastic Strain ( p ) 262.0 0 262.0 0 268.7 2.37E 05 269.2 1.54E 05 273.1 8.28E 05 274.7 5.59E 05 275.7 0.00017 278.9 0.000116 276.7 0.00028 282.0 0.000192 ----339.3 0.24737 1097.8 0.236367 339.6 0.25237 11 05.0 0.241263 340.0 0.25736 1112.1 0.24616 340.3 0.26236 1119.1 0.251058 Table B 2 S tress and plastic strain inputs 6061 T6 aluminum Plastic Strain ( p ) 222.14 0 229.41 1E 04 233.36 0.0002 238.27 0.0002 244.25 0.0004 249.87 0.0008 254.7 5 0.0012 259.14 0.0018 263.02 0.0027 266.34 0.0038 269.32 0.0051 272.21 0.0068 275.23 0.009 278.62 0.0116 282.46 0.0148 287.03 0.0186 292.41 0.0233 298.63 0.029 305.76 0.0359 313.61 0.0442 322.09 0.0543 330.88 0.0665 339.80 0.0812 348.51 0 .0989 355.74 0.1202
155 APPENDIX C LINEAR PLUS POWER LAW DERIVATION The following equations define the linear plus power law material model: for ( C 1 ) for ( C 2 ) for ( C 3 ) Five constants are used to define the circular transition arc : , Conditions: 1. Set : Good approximation for first plastic strains 2. At point the Equations C 1 and C 2 are equal: (C 4 ) 3. At point the Equations C 2 and C 3 are equal : (C 5 ) 4. The line between and is normal to line defined by Equation C 1: (C 6 ) 5. The line between and is normal to curve defined by Equation C 3: First derivative of Equation C 3: Normal slope at :
156 (C 7 ) In order to ensure a valid solution to the above equations the following method is employed: 1. The point is formulated as a function of radial distance from denoted as where the connecting line is defined by Equation C 6 : (C 8 ) and (C 9 ) 2. The length of a radius along a line perpendicular to the power law portion and passing through point defined by Equation C 7, is determined using a numerical solver and the following rel ationship: (C 10 ) where is a point on the curve defined by Equation C 3. 3. The length of the line between and is determined and set equal to used in step 1. The result is an set of equations that are a function of only K 1 and Th e value of is found using a numerical solver. The resulting is equal to K 2 as defined in F igure C 1. The resulting solution s fully satisfy all the condit ions, however this method is time consuming. In his PhD dissertation, Wang (1991) derived a similar solution based on a small angle assumption for the initial slope of the power law portion re sulting in K 2 being an inverse function of n The resulting equation is: (C 11 )
157 The following plot gives the values of K 2 as a function of the strain hardening exponents n This method for K 2 prod uces almost identical results.
158 Figure C 1 Schematic of the transition region in a linear plus power law material model Figure C 2 Material constant K 2 uniquely defined as a function of strain hardening exponent n in the linear plus power law model
159 APPENDIX D INTERPOLATION METHOD S Nodal data is obtained from direct interpolation of Gauss point values in the surrounding elements, intrinsic to Abaqus CAE post processing code. Specific radial distances and constant load da ta is obtained from these points using cubic spline interpolation in MatLab (Mathworks, 2008) A highly refined mesh and small load steps are used in finite element analysis to ensure mini mal error in analysis. Figure D 1 gives an example of near tip str ess field nodal values and cubic spline interpolation curve. The following is a short review of the spine interpolation method and the order of error associated with this estimation. The text Chapra and Canale ( 2006 ) is used for this review and MatLab fu nctions are used for computations. Spline interpolation uses a subset of data points to construct low order connecting polynomials. For cubic spline interpolation, a third order curves a constructed between each pair of data points. Due to the condition s of evaluation, c onnecting cubic spline interpolation curves are visually smooth A spline interpolation method is preferable to high order methods or other curve fit functions because the nodal data is firstly assumed accurate and secondly distances of interpolation are small for many of our applications. In fact, a piecewise linear approximation provides nearly identical results in near tip stress field analysis. Sufficient data points are available for cubic interpolation, therefore this method was e mployed for nearly all estimations. A cubic polynomial is derived between every two data points in a set. The conditions for n cubic spline interpolation functions for n + 1 data points are as follows: 1) Function must be equal at interior points. The inte rpolation function must equal the function value at x i 1 and must equal the data point at x i
160 2) The first and last functions must pass through the end points of a set : The interpolation function must equal the function value and must equal the data point 3) First derivatives at the interior points must be equal : 4) Second derivatives at the interior points must be equal : 5) Second derivatives at the end points must are zero : The drafting spline. W here x i are independent variables (typically radial distance or load) for nodal values. A detailed derivation of the numerical implementation with derivative estimation using a first order Lagrange interpolating polynomial is given in Chapra and Canale (2006). The O ( h 2 ), where h is the distance between sequential data points. However, derivative estimates become more accurate further from the end points and the error becomes much less.
161 Figure D 1 Example of cubic spline interpolation of nodal points for a surface crack geometry and a linear plus power law material
162 APPENDIX E COMPOSITE MATERIAL P ROPERTIES The composit e material properties for the global part models are calculated using Wound Composite Modeler plug in for Abaqus ( Dassault Systmes, 2008). A detailed description of how this program calculates the material properties at the dome can be found in Gray and M oser (2004). This section focuses a general transformation of material constants for an angle ply orthotropic laminate for an arbitrary angle This method is implemented to determine the material properties for the COPV sub model cracked section backing. First, the constitutive relationship for an orthotropic material is given in the material coordinate system as: (E 1 ) or, (E 2 ) where, the 1, 2 and 3 directions are the fiber, transverse and normal directions respectively and S m is the compliance matrix. The stiffness matrix is the inverse of the compliance matrix:
163 (E 3 ) B ecause this matrix is symmetric, the following relationship ensues: (E 4 ) The following stress transformation is used for rotation of the material about the normal axis: (E 5 ) For simplicity, such that: (E 6 ) The transformed constitutive relationship in the problem coordinate system can be written as: (E 7 ) The inverse of the strain transformation matrix where is equal to ( ). Therefore, the transformed stiffness matrix in the problem coordinate system (x, y, z) is: (E 8 ) or,
164 (E 9 ) For a given layer composed of plies the in plane strains ( x y xy ) and the out of plane stresses ( z yz xz ) will be equal between each positive and negative angle ply. The stiffness matrix is rearranged such that the continuous terms are grouped. (E 10 ) Group the stresses, strains and quadrants as follows: (E 11 ) Solve the above relationship such that the constant terms are on the right. (E 12 )
165 This matrix can now be averaged for positive and negative (E 13 ) In order to obtain the stiffness and compliance matrices, the above relationship is reformulated as: (E 14 ) The compliance matrix can be found from the above stiffness matrix with Equation E 3 The resulting matrix takes on the form of Equation E 2 and the new elastic constants can be extracted.
166 Figure E 1 Example of angle ply orthotropic elastic stiffnesses as a function of transformation angle Figure E 2 Example of angle ply orthotropic transformation angle
167 Figure E 3 Example of angle ply orthotropic shear moduli as a function of transformation angle
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173 BIOGRAPHICAL SKETCH Shawn English received his Bachelor and Master of Science degree s in m echanical e ngineering from the University of Florida, Gainesville in 2007 and 2009. Shawn was admitted to the direct PhD progr am with the University of Florida Mechanical and Aerospace Engineeri ng Department in June 2007 He is currently completing the requirements for a Doctor of Philosophy degree in m echanical e ngineering under the advising of Dr Nagaraj Arakere Shortly aft er admission Shawn received the NASA Graduate Student Research Program Fellowship from the Marshall Space Fl ight Center in Huntsville, AL. s dissertation research was in conjunction with the NASA Engineering & Safety Center (NESC) Composite Pressur e Vessels Working Group which aims to characterize failure and develop proof test logic for composite overwrapped pressure vessels. Shawn has been happily married to Lisa Michelle since May 2008. He is a member of Creekside Commun ity Church and is an avi d angler photographer, and lover of the outdoors