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## Material Information- Title:
- Hygrothermal effects on complex moduli of composite laminates
- Creator:
- Bouadi, Hacene, 1954-
- Publication Date:
- 1988
- Language:
- English
- Physical Description:
- xvii, 137 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Composite materials ( jstor )
Damping ( jstor ) Diffusion coefficient ( jstor ) Experimental data ( jstor ) Laminates ( jstor ) Moduli of elasticity ( jstor ) Moisture content ( jstor ) Poisson ratio ( jstor ) Stiffness ( jstor ) Temperature ratio ( jstor ) Composite materials -- Testing ( lcsh ) Dissertations, Academic -- Engineering Sciences -- UF Engineering Sciences thesis Ph.D Hygrothermoelasticity ( lcsh ) Laminated materials -- Testing ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1988.
- Bibliography:
- Includes bibliographical references.
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- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Hacene Bouadi.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 024137148 ( ALEPH )
19755822 ( OCLC ) AFJ8829 ( NOTIS ) AA00004791_00001 ( sobekcm )
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HYGROTHERMAL EFFECTS ON COMPLEX MODULI OF COMPOSITE LAMINATES BY HACENE BOUADI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ACKNOWLEDGEMENTS I would like to express my gratitude to Professor Chang-T. Sun, the chairman of my doctoral committee, for his guidance, time, and encouragement during this research. Many thanks are owed to Professor Lawrence E. Malvern and Professor Martin A. Eisenberg for their teaching and financial support. I also want to thank the other members of my doctoral committee, Dr. Charles E. Taylor and Dr. Robert E. Reed-Hill for their helpful commen ts, critique, and advic e. In addition, I gratefully recognize the assistance of Dr. David A. Jenkins for teaching me how to operate the material testing equipment that was indispensable for my work. Finally, I appreciate Ms. Patricia Campbells help in typing this manuscript. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS i i LIST OF TABLES vi LIST OF FIGURES vii NOMENCLATURE xiii ABSTRACT xvi CHAPTERS 1 INTRODUCTION 1 1.1 General Introduction 1 1.2 Moisture Diffusion 2 1.3 Hygrothermal Effects 2 1.4 Scope and Methodology 3 1.5 Dissertation Lay-Out 4 2 DIFFUSION OF MOISTURE 6 2.1 Introduction 6 2.2 Fickian Diffusion 6 2.3 Fickian Absorption in a Plate 8 2.3.1 Infinite Plate 8 2.3.2 Semi-Inf inite Plate 10 2.3.3 Experimental Measurement of Moisture Content 11 2.3.4 Approximate Solutions of Moisture Content 12 2.3.5 Edge Effects Corrections in a Finite Laminated Plate 13 2.4 Diffusivity and Maximum Moisture Content . 15 3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES ... 21 3.1 Introduction 21 3.2 General Theory 21 3.3 Micromechanics Formulation of elastic Moduli 22 3.4 Complex Moduli 23 i i i 4 DAMPING 29 4.1 Damping Mechanisms 29 4.1.1 Nonmaterial Damping 29 4.1.2 Material Damping 30 4.2 Characterization of Damping 30 4.2.1 Free Vibration '. 30 4.2.2 Steady State Vibration 31 4.2.3 Complex Modulus Approach 32 5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES . 36 5.1 Introduction 36 5.2 Laminated Plate Theory Approach 36 5.3 Energy Method Approach 37 6 EXPERIMENTAL PROCEDURES 40 6.1 Introduction 40 6.2 Test Specimen 40 6.3 Environmental Conditioning 41 6.4 Four-Point Flexure Test Method 41 6.5 Impulse Hammer Technique 42 7 HYGR0THERMAL EXPANSION 48 7.1 Introduction 48 7.2 Coefficients of Thermal Expansion 49 7.3 Coefficients of Moisture Expansion . . 50 7.4 Experimental Data 51 7.4.1 Previous Investigations 51 7.4.2 Present Investigation 52 8 HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI 58 8.1 Literature Survey 58 8.2 Theoretical and Experimental Assumptions. . 59 8.3 Modeling of Epoxy Properties 61 8.4 Results 63 8.4.1 Epoxy Complex Moduli 63 8.4.2 Composite Complex Moduli 65 9 HYGROTHERMAL EFFECTS ON STRESS FIELD 79 9.1 Introduction 79 9.2 Description of Study Cases 80 9.3 Numerical Results and Discussion 81 9.3.1 Glass/Epoxy 81 9.3.2 Graphi te/Epoyx 82 9.3.3 Summary 82 i v 10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES 95 10.1 Introduction 95 10.2 Numerical Results and Discussion 95 10.2.1 Glass/Epoxy 95 10.2.2 Graphite/Epoxy 96 10.2.3 Summary 97 11 CONCLUSION Ill APPENDICES A COMPLEX STIFFNESSES OF COMPOSITES 115 A.l Elastic Stiffnesses 115 A.2 Complex Stiffnesses 117 B DEVELOPMENT OF THE FINITE ELEMENT METHOD .... 119 B.l Equilibrium Equations 119 B.2 Program Organization 122 B.3 Shape Functions, Jacobian and Strain Ma trix 123 B.4 Elasticity Matrix 125 B.5 Element stiffness Matrix 128 B.6 Equivalent Nodal Loadings 128 B.6.1 Element Edge Loadings 128 B.6.2 Hygrothermal Loadings 129 B.7 Element Stresses 130 REFERENCES 133 BIOGRAPHICAL SKETCH 137 v LIST OF TABLES T abIes Page 6.1 Initial properties of Magnolia 2026 epoxy, 3M Scotchply Glass/Epoxy, and a typical Graphite/epoxy composite 45 7.1 Coefficients of moisture and thermal expansion of epoxy and graphite and glassfibers 54 7.2 Properties of Glass and Graphite Fibers . 54 9.1 Description of cases in Figure 9.2 84 9.2 Typical strengths of Glass/Epoxy and Graphite/Epoxy 84 LIST OF FIGURES Figures Page 2.1 Plate subjected to a constant humid 18 environment on both sides. 2.2 Moisture distribution across a plate. The numbers on the curves are the values of (c c.)/(c c.) 18 v i y v oo i y 2.3 Semi-infinite plate in a humid environment 19 2.4 Comparison of the exact specific moisture concentration equation with some approximate so 1 u t ions 19 2.5 Geometry of a plate 20 2.6 Moisture content versus square root of time. On the curve Vt < Vt^< Vt^ and the slope is constant for Vt" < Vt^ 20 4.1 Schematic drawing of a free-clamped beam under free vibration and plot of its deflection versus time 35 4.2 Schematic drawing of a free-clamped beam under forced vibration and plots of the deflection versus time and deflection amplitude versus frequency 35 6.1 Schematic drawing of environmental and testing chambers 46 6.2 Loading configuration of the 4-point f 1 exu retest 46 v i i 6.3Schematic drawing of the impulse hammer technique apparatus and a typical display of the Fourier Transform 47 7.1 Transverse moisture strain of Magnolia epoxy and 3M Scotchply Glass/Epoxy 55 7.2 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry S Glassf iber/Epoxy at 20C 56 7.3 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20C 56 7.4 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry S Glassf iber/Epoxy at 20C 57 7.5 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20C 57 8.1 Schematic variation of the storage modulus of epoxy with temperature 67 8.2 Schematic variation of Poissons ratio of epoxy with temperature 67 8.3 Schematic variation of damping of epoxy with temperature 6S 8.4 Glass transition temperature of epoxy. From Delasi and Whiteside [6] 68 8.5 Experimental data of the storage modulus of epoxy as a function of temperature at diverse constant moisture contents 69 8.6 Experimental data of the storage modulus of epoxy as a function of moisture content at diverse constant temperatures 69 8.7 Experimental data of the storage modulus of epoxy as a function of normalized temperature (T Tq)/(T Tq) 70 8.8 Experimental data of damping of epoxy as a function of temperature at diverse constant moisture contents 71 v i i i 8.9 Experimental data of damping of epoxy as a function of moisture content at diverse constant temperatures 71 8.10 Experimental data of the storage modulus of epoxy as a function of normalized temperature (T Tq)/(T Tq) 72 8.11 Experimental data of Poissons ratio of epoxy in term of temperature 73 8.12 Experimental data of Poissons ratio of epoxy in term of moisture content 73 8.13 Experimental data of Poissons ratio in term of the normalized temperature T = (T T )/(T T ) 74 n v o g o' 8.14 Longitudinal storage modulus (Ej^) of Glass/Epoxy versus = (T Tq)/(T^ Tq). . 75 8.15 Transverse (E^) and shear (Gj^) storage moduli of Glass/epoxy versus T = (T T )/(T T ) 75 n o g o 8.16 Longitudinal transverse (rj 22^ and shear () damping of Glass/Epoxy versus T = (T T )/(T T ) 76 n v o g o' 8.17 Poissons ratio () of Glass/Epoxy versus T = (T T )/(T T ) 76 n o v g o 8.18 Longitudinal storage modulus (Ej^) of Graphite/Epoxy versus = (T Tq)/(T Tq) 77 8.19 Transverse (E^) an<3 shear (G^) storage moduli of Graphite/epoxy versus T = (T T ) / (T T ) 77 n v o' g o' 8.20 Longitudinal (77 ^ ^ ) transverse (1722) and shear (p^) damping of Graphite/Epoxy versus T = (T T )/(T T ) 78 n v o' v g o' 8.21 Poissons ratio (v ^ ) of Graphite/Epoxy versus T = (T T )/(T T ) 78 n o g o 9.1 Geometry of a laminate and finite mesh of a 1/4 cross-section area 85 9.2 Description of the applied moisture grad ients 86 9.3Profile of the hygrothermal stress a across a [(90/0)^^ Glass/Epoxy laminate at y/b = 0.472 87 9.4 Profile of the hygrothermal stress across a [(SO/O^lg Glass/Epoxy laminate at y/b = 0.472 88 9.5 Profile of the hygrothermal stress across a [(90/0)2^ Glass/Epoxy laminate at y/b = 0.472 89 9.6 Profile of the hygrothermal stress o across a [(OO/OJ^lg Glass/Epoxy laminate at y/b = 0.993 90 9.7 Profile of the hygrothermal stress a across a [(OO/OJ^jg Graphite/Epoxy laminate at y/b = 0.472 91 9.8 Profile of the hygrothermal stress across a [^O/OJ^jg Graphite/Epoxy laminate at y/b = 0.472 92 9.9 Profile of the hygrothermal stress across a [(QO/O^jg Graphite/Epoxy laminate at y/b = 0.472 93 9.10 Profile of the hygrothermal stress yX across a [(OO/O^lg Graphite/Epoxy laminate at y/b = 0.993 94 10.1 Line style legend of Figures 10.2-13 98 x 10.2 Complex in-plane stiffness A^ of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 99 10.3 Complex in-plane stiffness of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 100 10.4 Complex in-plane stiffness Agg of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 101 10.5 Complex bending stiffness of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 102 10.6 Complex bending stiffness of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 103 10.7 Complex bending stiffness Dgg of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 104 10.8 Complex in-plane stiffness A^ of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 105 10.9 Complex in-plane stiffness A^ of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 106 10.10 Complex in-plane stiffness Agg of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 107 10.11 Complex bending stiffness of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 108 a 10.12 Complex bending stiffness of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 109 10.13 Complex bending stiffness Dgg of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 110 B.l Organization of the F.E.M. program 131 B.2 Local axes f and rÂ¡, Gauss point numbers and local node numbers of an eight-node isoparametric element 132 x i i NOMENCLATURE B * B * B" c comp 1 ex in-p1ane stiff ne s s comp 1 ex coup 1ing stiff ne s s comp 1 ex modu1u s s torage modu1u s loss modulus moisture concentration c average specific moisture cm equilibrium moisture concentration D* . i J D D X XX [D] E11 E22 G12 K spec ific heat complex bending stiffness moisture d i ffusivities diffusivity matrix, elasticity matrix longitudinal Young modulus transverse Young modulus in-plane shear modulus thermal conductivity m weight of absorbed moisture M percent moisture content x i i i initial percent moisture content M . i equilibrium percent moisture content Q. transformed stiffness i J s t T v m w a . i T7 12 0 . J complex transformed stiffness specific gravity time temperature fiber volume fraction matrix volume fraction we igh t coefficient of thermal expansion coefficient of moisture expansion s t rain damping or loss factor major Poissons ratio fiber orientation of j-th layer density stress Subsc rip t s 1, 2, 3 principal directions of the fibers f fiber i initial j layer number x i v L longitudinal direction m ma t rix x, y, z Cartesian coordinates 00 maximum or equilibrium Superscripts H mo is ture o initial T transpose, thermal * complex value real part imaginary part xv Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillement of the Requirements for the Degree of Doctor of Philosophy HYGROTHERMAL EFFECTS ON THE COMPLEX MODULI OF COMPOSITE LAMINATES By Hacene Bouadi Apr i 1 1988 Chairman: Dr. Chang-T. Sun Major Department: Engineering Sciences The effects of absorbed moisture and temperature on the complex moduli of composite laminates are investigated and the mechanisms of moisture diffusion in a lamina are also analyzed. First, the variation of the complex moduli of epoxy in terms of temperature and moisture content are experimentally determined. Then, the hygrothermal effects on the complex moduli of composites are derived by using the complex moduli of the matrix, micromechanical formulas and experimental data. Only the hygrothermal effects on the complex moduli of pure epoxy need to be experimentally determined since these effects on the fibers properties are neg1 igib1e. xv i In addition, the effects of hygrothermal environment on the stress field and material damping of general laminated composite plates are analyzed. It is shown that hygrothermal stresses induced directly by moisture and indirectly by material property changes can be very high, but the effects on damping are less pronounced. xv i i CHAPTER 1 INTRODUCTION 1 1 General Introduction The introduction of advanced composites in aerospace applications has led to an extensive study of their mechanical behavior. The amount of experimental and theoretical findings of composite material researchers made during the 1960s was so vast that Broutman and Krock [1] needed eight volumes to edit a summary of the resulting know 1 edge. The interest in composite materials arose from their ideal characteristics for aerospace structures. Replacement of the commonly used aircraft material, aluminum, by high strength/density ratio and versatile composites can lead to a theoretical 60% weight reduction [2, p. 22], Due to such benefits, lower costs and better understanding of their mechanisms, the use of composite materials has been increasing slowly but steadily. Exposure of aircraft structures to high temperature and humidity in the environment and the tendency of composites to absorb moisture gave rise to concern about their performances under adverse operating conditions. 1 2 Therefore, considerable work has been done to understand the effects of hygrothermal environment on the mechanical behavior of composite materials. 1.2 Moisture Diffusion In a 1967 study on the effects of water on glass reinforced composites. Fried hypothesized that water can penetrate the resin phase by two general processes, by diffusion through the resin and by capillary or Poiseuille type of flow through cracks and pinholes [3], But no mathematical theory was presented. Later, investigators established that the primary mechanism for the transfer of moisture through composites is a diffusion process and adapted the general theory of mass diffusion in a solid medium to moisture diffusion in composite materials. The transfer of moisture through cracks is a secondary effect [4, 5], Experimental data indicate that for most composite materials, the diffusion of moisture can be adequately described by a concentration dependent form of Ficks law [4-10]. 1.3 Hygrothermal Effects The degradation of mechanical properties of glass reinforced plastics exposed to water has long been 3 recognized by marine engineers who use "wet" strengths in the design of naval structures [3]. Requirements in aircraft structures are more stringent. The mechanical properties of materials used in aerospace applications must be completely characterized. Therefore, the effects of hygrothermal environment on the elastic, dynamic, and viscoelastic responses of composites have been studied. To date, the effects of moisture and/or temperature on the following performances have been investigated: tensile strength, shear strength, elastic moduli [3, 11-14], fatigue behavior [15-17], creep, relaxation, viscoelastic responses [18-20], dimensional changes [21], dynamic behavior [22], glass transition temperature [23], etc. Only tensile and shear strengths and elastic moduli have been thoroughly studied by many researchers. But data on the other properties are more limited and hence inadequate to constitute a good design data-base. 1.4 Scope and Methodology The present investigation is a combined theoretical and experimental work and is concerned with predicting the hygrothermal effects (below the glass transition temperature) on the complex moduli of composite materials. This program is undertaken by carrying out the foil owing s teps: 4 i)The complex moduli of epoxy matrix in terms of temperature and moisture concentration are obtained by# using experimental tests and theoretical expressions. ii)The effects of temperature and moisture on the complex moduli of unidirectional composites can be derived by using the complex moduli of the matrix, micromechanics formulas, and experimental observations. In addition, we neglect the hygrothermal effects on the fibers. iii)The effects of hygrothermal conditions on the stress field and the material damping of some general laminated composite plates undergoing simple hygrothermal loadings are analysed. 1.5 Dissertation Lay-Out Right af ter the in t roduc tion , the mechanism o f moisture diffusion i s described in Chap ter 2, whe r e the absorption of moisture through thin compos ite 1aminas i s analyzed in detail. The complex moduli of unidirectional composites are defined in Chapter 3. Sections 3.3 and 3.4 give the microm echanics formulations of the elastic and complex moduli in terms of the constituent material properties. The damping of composites based on the dynamic and complex modulus approaches is characterized and the equivalence of both approaches is proven in Chapter 4. In Chapter 5. the damping and complex stiffnesses of general laminates are 5 derived by using the laminated plate theory, the energy approach, and the preceding derivations. The complex stiffnesses are completely expressed in Appendix A. The environmental conditioning of the test specimens, the static flexure test, and the impulse hammer techniques are presented in Chapter 6. These experimental methods, although simple, are very versatile and are adequate in determining the necessary data for the purpose of this investigation. The theoretical and experimental results are given in Chapters 7-10. The moisture and thermal expansions of composites are quantified in Chapter 7. The current experimental results and data and conclusions of previous investigators are used in Chapter 8 to model the complex moduli of epoxy as functions of temperature and moisture content. In Chapters 9 and 10, the hygrothermal effects on the stress field across laminates and on damping of composites are investigated with the help of the results in the preceding Chapters. The Finite Element Method (F.E.M.) used in determining the stresses is summarized in Append ix B. CHAPTER 2 DIFFUSION OF MOISTURE 2.1 Introduc tion The mechanism of moisture absorption and desorption in most fiber reinforced composites is adequately described by Ficks law [4]. Fick recognized that heat transfer by conduction is analogous to the diffusion process. Therefore, he adopted a mathematical formulation similar to Fouriers heat equation to quantify the diffusion process [24, 25]. 2.2 Fickian Diffusion The Fourier and Ficks equations, describing the one-dimens iona1 temperature and moisture concentration, are respectively given by (2.1) dc d ~ Sc_ 3t dx x dx (2.2) where p is the density of the material, is the 6 7 specific heat, T is the temperature, t and x are the time and spatial coordinates, respectively, is the thermal conductivity, c is the moisture concentration, and is the moisture diffusivity. The moisture diffusivity, D and the thermal diffusivity, Kx/(pCv), are the rate of change of the moisture concentration and the temperature, respectively. In general, both parameters depend on temperature and moisture concentration. But experimental data show that, for most composites, moisture diffusivity does not depend strongly on moisture concentration [4], Hence, Eq (2.2) becomes dc 3t D a2 a c x 2 dx (2.3) and is solved independently of Eq (2.1). The three-dimensional diffusion in an anisotropic medium is obtained by generalization of Eq (2.2) as follows dc mi * \ JTT = v. ( [D] vc) where the diffusivity ma t rix i s D D D XX xy xz [D] - D D D yz yy yz D D D zx zy zx (2.4) (2.5) 8 Expansion of Eq (2.4) results in an equation of the form 2 2 2 2 dc d c d c d c d c Â§7- = D + d 2_Â£ + d 2_Â£ + (D + D ). g-- 3t xxg^2 yyay2 K yz xy'ay ox ~2 .2 + (D + D + (D + D ).g g zx xz ox oz xy yx'ox oy (2.6) if the coefficients D. ,s are considered to be constant. i J 2.3 Fickian Diffusion in a Plate Laminated plates are widely used in the experimental characterization of composites. Hence, being of practical interest, the problem of moisture absorption in a plate is thoroughly discussed in this section. 2.3.1 Infinite Plate The case of moisture absorption through a material bounded by two parallel planes is considered. The initial and boundary conditions of an infinite plate exposed on both sides to the same constant environment (Figure 2.1) are given by T c for 0 < z < h and t < 0 (2.7) 9 T = T. ] 1 f for z = 0, z = h and t > 0 C = C<* J where T. is a constant temperature, is the initial moisture concentration inside the material, and c is the CO maximum moisture concentration. It is assumed that the moisture concentration on the exposed sides of the plate reaches cra instantly. The solution of Eq (2.3) in conjunction with the conditions of Eqs. (2.7) is given by Jost [25] c - c oo c . 1 c . 1 00 4 V 1 2 ,j + 1 W 1 (2jTTTsln ~Sz exp j=o ^iii)2!72D t ,2 z n (2.8) Equation (2.8) is plotted in Figure 2.2. The average moisture concentration is given by c 'll c dz ^ o (2.9) Substitution of Eq. (2.8) into Eq (2.9) and integration result in c c 00 c . 1 c . 1 8_ ir I j=o 1 (2j + l) exp .2_2 Dzt -(2j + 1) U s- (2.10) This analysis can be applied to the case of diffusion of 10 moisture into a laminated composite plate so thin that moisture enters predominantly through the plane faces. 2.3.2 Semi-Infinite Medium In the early stages of moisture diffusion into a plate, there is no interaction between moisture entering through different faces. Therefore, the solution of moisture absorption into a semi-infinite half-plane is applicable to a plate for short time. The initial and boundary conditions of a semi infinite plane (Figure 2.3) exposed to a constant moist environment are T c for 0 < z < 00 and t < 0 0 and t > o (2.11) The solution of Eq (2.3) in this case is [24, 25] erf z 2 / D t L v z J (2.12) The rate at which the total specific mass of moisture, m, is diffusing into the half-plane is de dz _ z =0 (2.13) 1 1 dm d t D Thus, the total mass of moisture entering through an area A in time t is n t m = - pAD dc dz dt = 2pA(c - z=0 D t z 17 (2.14) Equation (2.14) shows that the mass of diffusing substance is proportional to the square root of time. 2.3.3 Experimental Measurement of Moisture Content In the case of a finite plate, the total moisture con tent is m = pVc (2.15) where V is the volume of the test piece. The total moisture content is experimentally measured by subtracting the dry weight, w^, from the current weight, w, of the plate, i.e. m = w w (2.16) A parameter of practical interest is the percent moisture content defined as 12 M 100 (2.17) Since M M c c . n = 1 : M = 100c (2.18) M M. c c v oo i co i the experimentally measured M of Eq (2.18) can be compared to the analytical value given by Eq (2.10). 2.3.4 Approximate Solution of Moisture Content Approximate solutions of the specific moisture distribution in a plate subjected to the conditions given by Eqs (2.7) are useful, since the difficulty of dealing with infinite series can be avoided. Sma11 time. As discussed in section 2.3.2, Eq (2.14) can be applied during the early stages of absorption. It yields c c M M. i i Large time. Tsai and Hahn [26, p. 338] suggest that, for sufficiently large t, Eq. (2.10) can be approximated by using the first term of the series, i.e., D t z n h (2.19) 13 c c 00 C . 1 C . 1 8 -9-exp UZ IT D ti z (2.20) Shen and Springer formulation. These researchers have derived in Ref. [4] the following approximation c c 00 c . 1 c . 1 exp 7.3 D t z 0.75' 1 h2 j (2.20) Figure 2.4 shows a comparison of Eqs (2.19-21) with the exac t so 1u tion. 2.3.5 Edge-Effect Corrections in a Finite Laminated Plate A plate exposed to a humid environment absorbs moisture through all its six sides. At small time, the interaction of moisture entering through different sides is negligible. Therefore, Eq. (2.19) can be applied to such cases. It yields m = 4P(C oo_ ci) bL /d + bh nr + hL rr rm V z vx v y J v (2.22) where D D and D are the diffus ivities in the x, y, and x y z z directions, respectively. The geometry of the plate is shown in Figure 2.5. Rewriting Eq. (2.22) in terms of the percent moisture content gives 14 M 4M Dt > nh2 where the effective diffusivity D is (2.23a) D D 1 h 1 + L x h _ i y D b D > z >1 z (2.23b) The micromechanics formulation for diffus ivities proposed by Shen and Springer [4] and modified by Hahn [26] for impermeable, circular cross-section, fiber-reinforced compos i tes is D L D m D T (2.24) where D D_ and D, are the matrix, transverse, and m T L longitudinal diffus ivities, respectively. Equation (2.23b) for a unidirectional lamina with all fibers parallel to the x-direction can be written as D (2.25) 15 For a general laminated plate consisting of N layers with fiber orientations 0., the diffusivities are J D z D T D x D y N N D. y h .cos^0 + D~ T h.sin^0. L L j j TZ.J j J = i ii N 2", j = l N N D, y h.sin^0. + D y h.cos^0. L L j j 7 L j j 1 = 1 j=l N (2.26) where h. is the thickness of J diffusivity of a general substituting Eqs. (2.26) into the j-th layer. 1amina t e is Eq (2.23b). The effective obtained by 2.4 Diffusivity and Maximum Moisture Content The diffusivity and the maximum moisture content Mro must be experimentally determined in order to predict the moisture content and distribution in a lamina. These parameters are obtained by the following procedures: - a thin, unidirectional composite plate is completely dried and its weight is recorded, 16 - the specimen is then placed in a constant temperature and constant relative humidity environment, and its weight as function of time is recorded, - the moisture content, M, versus the square root of time, is plotted as shown in Figure 2.6. The maximum moisture content is determined from the plot and the diffusivity from the following equation m2 Ml = 4M D I7h (2.27) The subscripts 1 and 2 are defined in Figure 2.6. The diffusivity depends only on the material and temperature as follows D z D exp o (2.28) where R is the gas constant, D and E, are the o a pre-exponential factor and the activation energy, respec tively. Experimental research has shown that the maximum moisture content depends on environment humidity content and material. For a material exposed to humid air [4], the equilibrium moisture content can be expressed as M 00 (2.26) 17 where 0 is the relative humidity, a aad b are material cons tan t s. IS > > Moisture > 5 Fig. 2.1 Plate subjected to a constant humid environment on both sides. Z =* 0 is the center of the crosssection of the plate z/h Fig. 2.2 Moisture distribution across numbers on the curves are a plate. The the values of (c - c.)/(c - J v oo ci} (M M,) 19 Fig. 2.3 Semi-infinite plate in a humid environment. Exact Oneterm Shen and Springer Fig. 2.4 Comparison of the exact specific moisture concentration equation with some approximate solutions. Moisture content (%) 20 Fig. 2.5 Geometry of a plate i Fig. 2.6 Moisture content versus square root of time. On the curve J t ^ < J t^ < J t^ and the slope is for/T < cons tan t CHAPTER 3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES 3.1 In t roduction Composite materials, such as Glass/Epoxy and Graphite/Epoxy, have a polymeric matrix. Therefore, they display viscoelastic behavior. Some of the effects of this time-dependent phenomenon are: stress relaxation under constant deformation, creep under constant load, damping of dynamic response, etc. This chapter is an introduction to the dynamic behavior of viscoelastic composites in terms of complex modu 1 i . 3.2 General Theory A usual representation of the one-dimens iona1 stress-strain relation of a viscoelastic material subjected to a harmonic strain history of the form ( t) = e1(Jt (3.1) 21 22 is given by a(t) = B (iw)e*Wt = B (icj)G(t) (3.2) The complex modulus B can be decomposed into its real and imaginary parts as follows B*(iu) = B ((i)) + i B" ( gj ) (3.3) The terms B' and B are called the storage and loss moduli, respectively, and the ratio of the loss over the storage modulus D * V = g-r (3.4) is referred to as either the loss factor or damping. The loss modulus is a measure of the energy dissipated or lost as heat per cycle of harmonic deformation. 3.3 Micromechanics Formulation of Elastic Moduli The longitudinal modulus E^, the transverse modulus E22 the in-plane shear modulus G^, anc* the major Poissons ratio v 12 can be obtained by using the rule of mixtures and the Halpin-Tsai equations, viz., + v E m m E = v E 11 f f 11 (3.5) 23 E 22 E m 1 + 2rijV 1 n ^ v (3-6) 1 + n9v G10 = G t ^-3- 12 ml- n2vf (3.7) wher e The subscripts f respectively, and v n, = n~ = = VfUf12 + V V m m (3.8) (Ef22/Em> - i (3.9) (Ef22/Em) + 2 (Gfl2/Gm> - 1 (3.10) (Gfl2/Gm) + 1 m stand for fiber and ma t rix, v are the m volume fractions Como lex Moduli The micromechanics formulations of the complex moduli are obtained by applying the e1astic-viscoe1astic correspondence principle [27-29], i.e., by undertaking the following steps: i) determining the elastic moduli of composites in terms of the constituent material properties, ii) replacing the elastic moduli of fibers and matrix by corresponding complex expressions. For a viscoelastic composite. the properties of the constituent materials are Efll f11 + lEfll "Â£22 ~ Ef22 + lEf22 G, = G1 + iG" E = E' + iE m m (3-11) G = m G1 + iG" m m v = V + IB m m J f12 L f 12 The bulk modulus of eDoxy matrix, K is real and m independent of frequency [2S]. It is given by - %m 3V 1 2v ) v m' (3.12) while the viscoelastic bulk modulus is obtained from the correspondence principle 25 E' + iE" m m K* = m 3[1 2(u' + iu")] m m (3.13) Separation of the real and imaginary parts of Eq. (3.13) yields K * m (1 2d')E m m 3 [( 1 2u"E" + i[2E"d" + E'( 1 m m L mm m 2u')2 + 4d"2 J m m J 2d )] m'J (3.14) Since the dilatation bulk modulus is real, the imaginary part of Eq. (3.14) is equal to zero; hence 2E"d" + E(1 2d) = 0 (3.15) mmmv m' v ' Equation (3.15) results in m, d = ^(d 0.5) m E v m (3.16) m The shear modulus of the matrix is given by G* = m m 3K E* m m 2(1 + d*) 9K E* v m' m m (3.17a) Separating the real and imaginary parts and neglecting the 2 terms of the form (^) yield 26 3K E' P* nn m m 9K E m 1 + i 9K E" m m 9K E' E1 m mm Introduction of the material properties rj = ft m E 'f 1 1 f 1 1 E f 11 f 22 f 22 E f22 m 9K m 'Gm G 1 9K E 'm m mm into Eq s. (3.11) r e su 1t s m i n E f 11 ( 1 + 1T?f 11 Ef22^ + 1T?f22 G f 12 ( 1 + 1T?fl2 E'(1 + ip ) m v m' G 1 ( 1 + i r\r ) mv Gmy ) ) ) (3.17b) (3.18) (3.19) 27 * v m v + m T7 ( v ' m m V Â£ 12 ~ V Â£ 12 ~ v Â£ 12 There are no satisfactory data on the shear and transverse damping of fibers. Fibers have damping with a magnitude order ten times smaller than epoxy. The dampings ^fii T7f22> and q^.^ are assumed to be equal and are replaced by t]j. in subsequent equations. Since the fiber damping, q^. , is much smaller than the matrix damping, q the m imaginary part of the fiber Poissons ratio is neglected. The preceding assumptions have a negligible effect on the complex moduli of composites. Application of the e1astic-viscoelast i c correspondence principle to Eqs. (3.5)-(3.10) and substituting them into Eqs (3.19) yield the following complex material properties E. = v.E' ,(1 + i T7r) + v E'(l + iq ) 11 ffllv f mmv my E x 22 E ( 1 + i q ) mv m 1 + 2njVj. * 1 nivf (3.20) 12 Gm(1 + iTW 1 + n2Vf - n2vf 1 whe r e Ef22<1 + V Em( lflm> Ef22(l + "fl + 2Em(1 iT>m> Cfl2*1 lr|f) ~ Gm*1 1 ^Gm ^ Gf]2(l + iut) + Gm(l + lGm) The elastic moduli given by Eqs. (3.5)-(3.8) model experimental results with a good accuracy [2], Therefore, they are used instead of mathematically exact micromechanics formulas, such as those derived by Hashin [27, 29]. CHAPTER 4 DAMPING 4.1 Damping Mechanisms Any vibrational energy introduced in a structure tends to decay in time. This phenomenon is called damping. There are two types of damping mechanisms, external or nonmaterial and internal or material. 4.1.1 Nonmaterial Damping. Two common types of external damping are - Accoustic damping: a vibrating structure always interacts with the surrounding fluid medium (air, water, etc.). This effect can lead to noise emission and even to changes of the natural frequencies and mode shapes. Thus, mechanical responses might be modified. - Coulomb friction damping: two contacting surfaces in relative motion dissipate energy through frictional forces. 29 30 4.1.2 Material Damping There are many damping mechanisms that dissipate vibrational energy inside the volume of a material. Damping phenomena include thermal effects, magnetic effects, stress relaxation, phase processes in solid solutions [30, p. 61], etc. The internal damping of polymeric matrix composites, such as Glass/Epoxy and Graphite/Epoxy, is dominated by viscoelastic damping. 4.2 Characterization of Damping 4.2.1 Free Vibration A cantilever under free vibration oscillates regularly with an amplitude that decreases from one oscillation to the next one (Figure 4.1). A measure of damping is the logarithmic decrement defined as 6 1 n (4.1) whe r e A^ = amplitude of the n-th cycle ^n + N = amPlituc*e the (n+N)-th cycle The damping defined in Eq (4.1) is applicable to viscous 31 damping and for hysteretic damping that is represented by a complex modulus approach. 4.2.2 Steady State Vibration Damping also influences the dynamic equilibrium amplitude of structures (e.g. beams) that undergo harmonic oscillation. A resonance usually occurs (Figure 4.2). The following measure of damping is used V (j. - (jj. CJ (4.2) whe r e resonant frequency "l- w2 = frequencies on either sides of such that the amp 1itude is 1/y 2 times the resonant amp 1itude. I n the case of a vibration induced by the force f(t) = Fsin(ut) the response (deflection), w(t), is out of phase with f(t) by an angle e such that w( t) Wsinfut + e) 32 The work done per cycle is 217/u D f ( t)^- dt = I7WF sin(e) (4.3) J o The strain energy stored in the system at the maximum displacement is half the product of the maximum displacement by the corresponding value of the force, i.e., U = ^FW cos(e) (4.4) There is no damping if the work done per cycle is zero, i.e, if sin(e) = 0. The ratio of energy dissipated in a cycle to energy stored at the maximum displacement is another measure of damping. Therefore, the damping is (4.5) The definitions of damping given by Eqs. (4.2) and (4.5) are equivalent [33]. 4.2.3 Complex Modulus Approach The one-dimensional stress-strain relation of a viscoelastic material undergoing harmonic motion has been shown to be (Eq. (3.2)) 33 * ( t) = EH((o)o eUt = (E'(g)) + iE"())o eiwt (4.6) Noting that i |u | = d/dt, Eq. (4.6) can be written as ^ 1(Jt E iutc ct ( t ) = e + -ir ue o co o (4.7) The real part is given by (after algebraic manipulation) a(t) = E e^sinfut + e) i + rj + e ) -11 + r/2 (4.8) where q = tan(e) = E"/E ' The energy dissipated during a cycle per unit volume is D = () a d = x 217/(0 d x d t d t = UqE'e2 o (4.9) The maximum energy stored is 1 2 u = E* ~ 2 o (4.10) The r e f o r e, V (4.11) 34 Hence, the definitions of damping given by Eq. (3.4) and Eq. (4.5) are equivalent. This conclusion is also valid for general cases of structural vibration. 35 Fig. 4.1 Schematic drawing of free vibration and versus time. a free-clamped beam under plot of its deflection 4.2 Schematic drawing forced vibration versus time and f requency. of a free-clamped beam under and plots of the deflection deflection amplitude versus Fig . CHAPTER 5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES 5.1 Introduction Both the laminated plate theory and the energy method approaches for analyzing the damping and the stiffnesses of general laminates are presented in this chapter. 5.2 Laminate Plate Theory Approach Four independent parameters are needed to determine completely the damping of a unidirectional composite. But, the analysis of the material damping of a general laminated composite requires the use of eighteen parameters. These quantities are the ratios of the imaginary over the real parts of the complex in-plane stiffnesses A. ,s, the i J and the complex bending The terms A. ,s, B. ,'s, and 1 J i J and D. ,s are defined as i J +h/2 A. . i J dz J -h/2 36 37 r+h/2 -h/2 (5.1) 0 p+h/2 J-h/2 2tt* 2 Qij dz where the complex transformed stiffness Q_s depend on * * Ell E22 G12 the 1amina t e. 12 and the orientation of each layer of The in-plane, coupling, and flexural material damping are defined as 1*1 j A 7 . i J a: . i j C^ij B7 . -U. b: . i j (5.2) r-7? F i J D7 . i i d: . 1J respec tively. 5.3 Energy Method Approach The energy method can be used to damping of laminated composite materials determine the under certain loading and boundary conditions. The damping of a laminated 38 composite material in the first mode of vibration can be defined as N ^ ^ k^d ^ cyc. V = ^ (5.3) I 2n ks k= 1 where N is the total number of layers, (. U.) is the v k deye. energy dissipated in the k-th layer during a cycle, and ^_Us is the maximum energy stored in the k-th layer. The storage and the dissipated energy are given by k U s 1_ 2 C . V J J1 1 k dV kUd n e c v J J'1 1 k dV (5.4) where i and j are C'.' are the real J i stiffnesses and V. k total" damping of the material principal axes, Cj and and imaginary parts of the complex is the volume of each layer. Hence, the an N-layered laminate is given by V (5.5) 39 The maximum strain vector {} can be determined by the finite element method first. Then, the damping can be deduced. Equation (5.5) is used to determine the damping of a beam with variable thickness or of more general s t rue tures. CHAPTER 6 EXPERIMENTAL PROCEDURES 6 1 Introduction A description of the test specimens and the experimental procedures of the present investigation is given in this chapter. 6.2 Test Specimens The test specimens used to determine the complex moduli of epoxy and of composite materials are thin strips of approximate dimensions, 150mm by 25mm by 2mm. The only materials tested are Magnolia 2026 laminating epoxy and 3M Scotchply Glass/Epoxy. The curing temperatures of the epoxy and the Glass/Epoxy are 175C and 170C, respectively. The initial properties of these materials (at 20 C and without moisture), as well as those of a typical Graphite/Epoxy, are given in Table 6.1. 40 41 6.3 Environmental Conditioning The specimens are conditioned in a Thermotron environment chamber at a constant temperature and constant relative humidity. The weight gain of the test pieces as a function of time is monitored. Right after moisture equilibrium is reached, the specimens undergo all tests at diverse temperatures inside a testing chamber connected to the environment chamber (Figure 6.1). The range of temperature achieved inside the environment chamber is 4C to 90C and the range of relative humidity is 4% to 99% for temperatures below 75C. As temperature increases, the highest relative humidity that can be obtained decreases steadily to 75% at 90C. 6.4 Four-Point Flexure Test Method The Youngs modulus and the Poissons ratio can be determined with the four-point flexure test method. The loading configuration of this test is shown in Figure 6.2. The elastic flexural analysis yields [31] E PI' Sbh^w (6.1) where E is the effective modulus, P is the applied load, 1 is the length of the specimen, b is the specimen width, h 42 is the thickness, and w is the deflection at quarter-point. Poissons ratio is expressed as (6.2) where the transverse strain is measured y transverse strain gage cemented in the middle specimen. with a of the 6.5 Impulse Hammer Technique The material damping and the storage modulus of a one-dimensional thin beam are determined with the impulse hammer technique. This technique was pioneered by Halvorsen and Brown [32]. The equipment set-up is shown in Figure 6.3. The specimen is clamped inside the testing chamber. A force impulse is applied to the test piece by a force transducer. The end displacement of the specimen is recorded with a non-contacting motion transducer. Both responses from the force and motion transducers go through signal conditioning equipments (filters, amplifiers). These responses are digitized in a Fast Fourier Transform analyzer (FFT) to obtain the transfer function in terms of the frequency. The transfer function is defined as the ratio of the Fourier Transform of the output (displacement 43 v(t)) over the Fourier Transform of the input (force impulse u(t)); that is. H(f) _ mi ~ U(f) (6.3) where t = time f = f requency V(f) = Fourier Transform of v(t) U(f) = Fourier Transform of u(t) The real and imaginary parts of H(f) are displayed on the FFT analyser CRT (Figure 6.3). The material damping defined by Eq. (4.11) is experimentally obtained by the following expr e s sion (fa/fb>2 1 (6.4) where the frequencies f and f, are defined in Figure 6.3. a b The storage modulus is expressed as [33, p.464] 2 1 ^ E' = 38.32 f p^r (6.5) r li where f is the resonant frequency in Hz. p, is the material density, 1 is the length of the specimen and h is the thickness of the specimen. Equation (6.5) is valid only 44 for the case of the first mode free vibration of a clamped-free beam. A complete description and analysis of the impulse hammer technique are presented in Lees dissertation [34]. 45 Tab le 6.1 Initial properties of Magnolia 2026 epoxy, 3M Scotchply Glass/Epoxy, and a typical Graphite/Epoxy composite. Properties Epoxy G1as s/Epoxy Graphite/Epoxy Vf 0.50 0.70 p (g/cm3) 1.25 1.93 1.6 En (GPa ) 4.0 37.00 155.23 E22 (GPa) 4.0 11.54 10.81 G12 (GPa.) 1.52 3.46 4.35 U 1 2 0.32 0.285 0.217 *11 0.018 0.0023 0.0019 V22 0.018 0.015 0.0078 46 Fig. 6.1 Schematic drawing of environmental and testing chamber s. Loading configuration of f1exure test. Fig. 6.2 the 4-poin t 47 Fourier Transform Real part Im. part Schematic drawing technique apparatus the Transfer Fourier of the impulse hammer and a typical display of Trans form. Fig. 6.3 CHAPTER 7 HYGROTHERMAL EXPANSION 7.1 Introduction When a metallic or composite structure is subjected to a change of temperature, there are dimensional variations and there may be stress development. For a one-dimensional case, it is assumed that the thermal strain is given by T = a.(T T ) = a.AT i i o 1 where = coefficient of thermal expansion T = actual temperature Tq = reference temperature. A polymer matrix composite exposed to a humid environment absorbs moisture. Hence, it increases in weight and dimensions. This situation produces a moisture strain that varies linearly with moisture concentration [26]. In the one-dimensional case, the hygros train is given by (7.1) 48 49 H = p (c c ) = P.Ac (7.2) i iv o' l v where c is the initial moisture concentration and 6. is o 1 the coefficient of moisture expansion. 7.2 Coefficients of Thermal Expansion In the case of laminated composite plates, three coefficients of thermal expansion are used in determining the thermal strains. These parameters can be written in terms of fiber and matrix properties. The micromechanics formulas for a unidirectional orthotropic lamina are given by (see Refs. [35, p. 24], and [36, p. 405] for a detailed de riva tion) l = vParEr + v a E f f f m m m En a2 = vfaf + v a m m vfufaf v v a - m m m U12al (7.3) where the subscripts 1 and 2 represent the fiber and the transverse directions. The thermal expansion coefficients of an orthotropic lamina whose fibers make an angle 0 with the x-direction (Figure 2.5) are given by 50 2 2 a = a, cos 0 + o' sin 0 x 1 2 2 2 a = a, sin 0 + a_ cos 0 y 1 2 (7.4) a = 2(a, a~) cos 0 sin 0 xy v 1 2' 7.3 Coefficients of Moisture Expansion Similarly, the coefficients of moisture expansions of an orthotropic lamina with impermeable fibers can be expressed as [36, p. 406] sE p = Â£p 1 s E, m m 11 P2 = f-d + vjpm v12px (7.5) m *12 = where s and s^ are the specific gravities of the composite ma terial and the ma trix. The moisture expansion coefficients expressed in an axis system such that the x-direction makes an angle 0 with the fibers are given by Eqs. (7.4) after replacement of a's ^7 P^'s- 51 7.4 Experimental Data 7.4.1 Previous Investigations Hahn and coworkers investigations [21, 26, 37] of swelling of composites are outlined in this section. Some of the typical results of the transverse strain versus percent moisture gain are obtained by conducting the following tests: absorption is conducted in moisture saturated air such that Eqs (2.2) and (2.7) are satisfied; while desorption takes place in vacuum at the same temperature. Their data show a hysteretic nature of swelling in this case. But, when swelling of composites is given in terms of moisture concentration, the average behavior of S2-G1ass/Epoxy, Kevlar 49/Epoxy and Graphite/Epoxy can be approximated by 0.43c = P2c (7.6) Since the data presented in their publications display a wide scatter, Hahn et al. suggest that Eq. (7.6) can be used to estimate the moisture strains for most composite materials. 52 7.4.2 Present Investigation Epoxy and Glass/Epoxy specimens are conditioned at a constant relative humidity until the absorbed moisture reaches equilibrium. The changes in transverse dimensions are measured. This procedure is repeated at diverse values of relative humidity. The results are plotted in Figure 7.1. The longitudinal swelling strains could not be measured since the micrometer calipers used were not suf ficiently accurate. These data yield the foil ow i ng expe rimen ta1 values Pm(epoxy) = 0.25 (7.7) P2(G1ass/Epoxy) = 0.48 (7.8) Substitution of Eq. (7.7) and the parameters given in Table 6.1 into Eqs. (7.5) yields the following empirical values j3 = 0.042 (7.9a) P2 = 0.47 (7.9b) for the 3M Glass/Epoxy composite. The experimental and empirical values of P^ are practically equal. Hence, the 53 present results differ slightly from the approximation given by Eq. (7.6). The above coefficients and the typical coefficients of expansion of graphite are quantified in Table 7.1, while the storage moduli and the density of glass and graphite fibers are listed in Table 7.2. These properties are used to plot the thermal and moisture expansion coefficients versus the fiber volume fraction of Glass/Epoxy and Graphite/Epoxy in Figures 7.2 through 7.5. The values in these plots are valid for dry composites at room temperature. Since the storage modulus of epoxy varies with temperature and moisture content, this additional effect is investigated in Chapter 8. In general, the thermal expansion coefficients are functions of temperature, but this temperature effect is negligible below 100C. Therefore, in the subsequent chapters, the thermal expansion coefficients are assumed to be independent of temperature. 54 Table 7.1 Coefficients of moisture and thermal expansion of epoxy and graphite and glass fibers. Epoxy Glass Graphite a (pm/m)/K 54.0 5.0 0.2 P 0.25 0.0 0.0 Table 7.2 Proper ties of Glass and graphite Fibers. Glass Graphite En(GPa) 70.0 220.0 E-22(Gpa) 70.0 16.6 Gf12(GPa) 28.7 8.27 71 f 0.0015 0.0015 VÂ£ 12 0.22 0.16 p (g/cm3) 2.60 1.75 55 Moisture concentration (%) Fig. 7.1 Transverse moisture strain of Magnolia and 3M Scotchply Glass/epoxy. epoxy 56 Longitudinal transverse Fig. 7.2 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry S Glassfiber/Epoxy at 20C. Longitudinal Transverse Fiber volume fraction Fig. 7.3 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20C. Coefficient of moisture expansion ^ Coefficient of moisture expansion 57 Longitudinal Transverse Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry S G1assfiber/Epoxy at 20C. Fiber volume fraction Longitudinal Transverse Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20C. Fig. 7.5 CHAPTER 8 HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI S.1 Literature Survey The storage moduli (real parts of Eqs. (3.11)) of composites are usually determined by dynamic testings, such as the technique described in section 6.5. They can be approximated by using static tests [38]. Shen and Springer [12] investigated the environmental effects on the elastic moduli of a Graphite/Epoxy composite and made a survey of existing data showing the effects of temperature and moisture on the elastic modulus of several composites. Their conclusions are listed below. i) The hygrothermal effects on the 0 fiber direction laminates are negligible. ii) For 90 fiber direction laminates, the hygrothermal effects on the modulus are insignificant in the 200K to 300K temperature range. But, between 300K and 450K, the hygrothermal effects on the modulus are impor tan t. Putter et al. [38] investigated the influence of frequency and environmental conditions on the dynamic 58 59 behavior of Graphite/Epoxy composites. Their overall conclusions are i)The effects of frequency on the modulus and damping are quite small in all cases. ii)The effects of frequency on the modulus and damping are relatively greater for matrix-controlled laminates at higher frequencies (above 400 Hz.). iii)At the same temperature, damping increases with moisture saturation. But for dry laminates, damping decreases slightly as temperature increases. From all these experimental works, a general summary can be drawn: the influence of hygrothermal conditions on the elastic modulus, dynamic modulus and damping of composites is matrix dominated. 8.2 Theoretical and Experimental Assumptions Since the hygrothermal influence on composite properties is matrix controlled [12, 38], the fiber properties are assumed to be constant at any temperature below the glass transition temperature and at any moisture content. Therefore, to obtain the values of the complex moduli of composites, it is sufficient to know how temperature and moisture affect the complex moduli of the epoxy matrix, and then use the micromechanics formulations given by Eqs. (3.20). Thus, only the following functions 60 E ' m E 1 m (T.c) u = u (T c ) (8.1) mm v ' t] = r) (T c ) m mv ' need to be experimentally evaluated. The constant fiber properties are given in Table 7.2. The effects of frequency are negligible below 400 Hz. The results of this investigation are not accurate for higher frequencies since their effects have not been taken into account. The qualitative influence of temperature only on the storage modulus, real part of Poissons ratio, and damping of epoxy is illustrated in Figures 8.1-8.3. There are three distinct regions. At room temperature (in the glassy region), the storage modulus, Poissons ratio, and damping of epoxy are equal to about 4.0 GPa, 0.35, and 0.018, respectively. In the glassy region, the storage modulus decreases slowly, while Poissons ratio and the damping increase as temperature increases. In the next region (transition region), the storage modulus decreases rapidly, and both Poissons ratio and damping reach their maximum values. The last region is the rubbery region where the modulus takes a very low value, and all three parameters stay relatively constant. Typical values of the modulus in the rubbery region -2 could be 10 times the glassy modulus or lower. The damping 61 can reach a value of 1 or even 2 in the transition region [30, p. 90]. Poissons ratio reaches the 1 imiting value o f 0.5, wh i ch is approximated by incompressible rubber s [39, p. 293] The position of the transition region depends on the moisture concentration. The effects of moisture on the glass transition temperature, T^_, of six epoxy resins have been determined by Delasi and Whiteside [6]. These results are plotted in Figure 8.4. They are compatible with the data of McKague [40] and satisfy the theoretical relation derived in Ref. [41, p. 69]. 8.3 Modeling of Epoxy Properties The observations of the preceding section are used for modeling the material properties of epoxy that are given by Eq. (8.1). The glass transition region of epoxy resin is not broad [6], therefore, a glass transition temperature is used instead. The temperature T^ is usually obtained by measuring the expansion of a specimen as function of temperature. The point where the epoxy stops expanding as temperature increases corresponds to the first deviation from the glassy state and is termed T . According to the experimental data plotted in Figure 8.4, T^ is stongly dependent on absorbed moisture. These results show that, as the moisture content of epoxy 62 increases, the transition temperature moves to the left in Figures S. 1-8.3. Hence, the abrupt change of the material properties starts at a lower temperature as the moisture content increases. This fact and the conclusions reached by previous investigators [12] suggest that the following modelings of E, v', and n' are appropriate, mm m (8.2) (8.3) (8.4) where the temperature Tq is equal to 273K. The moisture concentration appears implicitly in T The glass transition temperature is represented by Tg = 210 exp(- 9c) (C) (8.5) where c is the moisture concentration. This modeling has been chosen so that it does not represent the material properties beyond T since the study of epoxy in the rubbery stage is not within the scope of this research. Equations (8.2)-(8.4) are valid only for 63 the continuous parts of the curves plotted in Figures 8.1-8.3. , 8.4 Results All test specimens are conditioned in a constant relative humidity environment until moisture equilibrium is reached. Then, the test pieces undergo the impulse hammer technique and the four-point flexure tests to determine the storage moduli, the material damping, and Poissons ratio at several temperature and moisture contents. 8.4.1 Complex Moduli of Epoxy Storage modulus. The experimental data on the storage modulus of epoxy in term of temperature at three different equilibrium moisture concentrations are plotted in Figure 8.5. It can be concluded tha t increase in either temperature or moisture content or both results in a decrease in the s t orage modu1u s. P1 o 11ing these data in terms of moisture content in Figure 8.6 does not lead to any additional insight. But, representing these results in term of the following normalized non-dimensional temperature (8.5) 64 in Figure 8.7 shows a clear trend. Experimental studies have shown that the modulus of polymer is very low at the glass transition temperature, therefore, adding the value E =0 for T = T to the data yields the following modeling m g E = 4.0(1 T ) (GPa) m v nJ y ' (8.6) Material damping. Similarly, the experimental data of the hygrothermal effects on the damping of epoxy are plotted in three Figures (8.8-8.10). There is very little change in damping for all the considered conditions. Therefore, it is proposed to let T7 = 0.018 (8.7) for temperatures up to 80C and moisture contents up to 5%. The conclusion that the hygrothermal effects on the damping of epoxy is negligible is qualitatively corroborated by Putter et al. [38]. A quantitative comparison cannot be made since these researchers have not included in their publication the values of the fiber volume fraction and moisture content of the test specimens. Poissons ratio. The experimental values of the Poissons ratio in terms of temperature at two different 65 moisture contents. are plotted in Figure 8.11. These results show that Poissons ratio increases at a negligible rate as temperature varies from 0 to SOC. Representing the same data in terms of the moisture content up to M = 4.5% (Figure 8.12) shows that the moisture effect is also negligible. Therefore, v = 0.32 (8.8) m v for temperatures up to 80C and moisture contents up to 5%. Since v equals 0.5 at the glass transition temperature m (T = 0), the plot of Poissons ratio versus the normalized temperature has been extrapolated as shown in Figure 8.13. The extrapolation displays a qualitative trend only. 8.4.2 Complex Moduli of Composites The complex moduli of Glass/Epoxy and Graphite/Epoxy in terms of moisture content and temperature can be determined by using the fibers properties given in Table 7.2, Eqs (8.6) through (8.8) and the micromechanics formulas (Eqs. (3.20)). This procedure is illustrated by determining the storage moduli and the damping of a Glass/Epoxy lamina with a fiber volume fraction of 0.5 and a Graphite/Epoxy lamina with a fiber volume fraction of 0.7. 66 G 1 a s s/Epoxy The parameters E^, E 22' ^12 'll T]22 and v 12 versus the normalized temperature are plotted in Figures 8.14-17. The experimental data substantiate the theoretical results. h. Graph i te/Epoxv Similarly, E^, ^22 *^12 11 T,22 and v'i2 versus the normalized temperature of Graphite/Epoxy are plotted in Figures 8.18-21. For both Glass/Epoxy and Graphite/Epoxy, the results show that the matrix-dominated parameters (E^ and G^) are strongly affected by moisture and temperature, while the fiber-dominated parameters (Ej^, s tay practically cons tan t. 67 Fig. 8.1 Schematic variation of the storage modulus of epoxy with temperature. Fig. 8.2 Schematic variation of Poissons ratio of epoxy with temperature. 68 Temperature Fig. 8.3 Schematic variation of damping of epoxy with temperature. O o Fig. 8.4 Glass De lasi transition temperature of epoxy. From and Whiteside [6]. 69 2.5 2 L_ i i I i i i 1 i 0 20 40 60 30 100 Temperature (C) M M M 0.0% 2.90% 3.70% Fig. 8.5 Experimental data of the storage modulus of epoxy as a function of temperature at diverse constant moisture contents. 20 C 50 C 70 C Fig. 8.6 Experimental data of the storage modulus of epoxy as a function of moisture content at diverse constant temperatures. Storage modulus (GPa) TO Normalized temperature Fig. 8.7 Experimental data of the storage epoxy as a function of normalized (T - T ) / (T o g " T ) o modulus of temperature o u 3.703 a. E a a 0.015 I 0.01 - 0.005 j- o r . i I > 0 20 40 SO 30 100 Temperature (C) Fig. 8.8 Experimental data of damping of epoxy as a function of temperature at diverse constant moisture contents. * T = 20 C O T = 50 C T = 70 C Fig. 8.9 Experimental data of damping of epoxy as a function of moisture content at diverse constant temperatures. 72 Normalized temperature * Experimental data Fig. 8.10 Experimental data of the storage epoxy as a function of normalized (T T )/(T o; g T ) o modulus of temperature 73 QZ 4.17S Fig. S.ll Experimental data of Poissons ratio of epoxy- in term of temperature * t = 20^: O T = 50 C a T => 75C Fig. 8.12 Experimental data of Poissons ratio of epoxy in term of moisture content. Poissons ratio 74 Fig ^ Experimental data Fit to data Extrapolation 8.13 Experimental term of the (T T )/(T o' g data of Poisson's ratio in normalized temperature = 75 Theoretical * Experimental data Fig. 8.14 Longitudinal s to rage modu1u s Glass/Epoxy versus = (T Tq)/(T (Eil) - v- o f Theoretical E 22 ^ Experimental EI,2 Theoretical G^2 Fig. 8.15 Transverse (E^) an<^ shear (G^) storage moduli of Glass/epoxy versus T = (T T )/(T T ). n o g o' 76 Theoretical *1,, ^ Experimental 7) n Theoretical T)^ O Experimental 7)^ Theoretical 71 s Fig. 8.16 Long itudi na 1 shear (le1 (Dll). damping n = (T - T ) / (T T ) . o g o transverse (^22 of Glass/Epoxy Theoretical * Experimental Fig. 8.17 Poisson s ra t i o T = (T T ) / (T n o g (i2) - v- of Glass/Epoxy ) and versus versus 180 a CL a 160 0) 3 3 T3 O 140 QJ cr a o -*-< tn 120 0 0.2 0.4 0.6 0.8 1 Normalized Temperature Fig. 8.IS Longitudinal storage modulus (E^) of Graphite/Epoxy versus T = (T T )/(T T ). n o g o Transversa Shear 0 0.2 0.4 0.6 0.8 1 Normalized Temperature (Gj2) storage versus T = n Fig. 8.19 T ransverse modu 1 i of (T T )/(T o' g (^2) anc* shear Graphite/epoxy - T ). o Poissons ratio ^ Damping 7 S 0.02 0.015 - 0.01 - 0.005 - Longitudinal damping Transverse damping Shear damping Normalized temperature (nn) ig. 8.20 Longitudinal shear (nG) T = (T T )/(T T ) . n v o' g o transverse damping of Graphite/Epoxy versus (ti22), and 0.4 Normalized temperature Theoretical Pois sons ratio ("2> o f versus T = (T T )/(T T ). n o g o Fig. 8.21 Graphite/Epoxy CHAPTER 9 HYGROTHERMAL EFFECTS ON STRESS FIELD 9.1 Introduction The hygrothermal effects on the stress field are investigated by considering an infinitely long, finite width and symmetric composite laminate undergoing hygrothermal loadings. The Finite Element Method is used in order to estimate the magnitude of hygrothermal stresses in laminated composites (see Appendix B). The geometry of a laminate and the finite mesh of a quarter cross-section are shown in Figure 9.1 and the boundary conditions are given by v = 0 for (y,z) = (0,z) (9.1) w = 0 f or (y,z) = (y 0) where v and w are the displacements in the y and z directions, respectively. The grid consists of 24 eight node isoparametric elements and 93 nodes. Only 24 elements 79 80 are used since increasing the number of elements to 48 results in a relatively small change in the stress magnitudes. The material properties in terms of temperature and moisture content have been derived in the preceding chapter. The constitutive equations are given by Eq. (B.12) and can be written in matrix form as {a} = [Q]({} {a}AT (P}c) (9.2) where {a} and {/3} are the vectors of thermal and moisture expansion coefficients. 9.2 Description of Study Cases The considered stacking sequence is the [^O/O^jg lay-up. The cross-ply laminate is preferred over other laminate since hygrothermal loadings induce very high stresses in this case. The volume fiber fractions of the Glass/epoxy and the Graphite/Epoxy are 0.5 and 0.7, respectively. The thickness and the width of the laminates are assumed to be 2 mm and 20 mm, respectively. Three cases of moisture gradients are applied. They are described in Figure 9.2 and Table 9.1. Cases A and C correspond to the dry and moisture saturated states, respectively. While the non-uniform moisture gradient (case B) corresponds to a moisture profile as derived in section 2.3. Two uniform temperatures (20C and 80C) are 81 used. All laminates are assumed to be initially (dry at 20C) free of stress. Hence, residual stresses are not taken into account. The elastic moduli used in computing the stresses are approximated by the real parts of the complex moduli. Therefore, the hygrothermal effects on the elastic properties can be deduced from the results given in Chapter 8 9.3 Numerical Results and Discussion For all considered cases, the following remarks can be drawn: at z/h = constant, the stresses away from the free edge stay constant and the shear stress (CTyz) zero, but, as y/b approaches 1, non-zero values and there are small variations in the a takes significant yz values of the other stresses. Hence, the stresses a a y z and a are plotted across the section of the laminate at x y/b = 0.472 and the shear stress a is plotted across yz the section at y/b = 0.993 (close to the free edge). The stresses are compared to typical strengths of Glass/Epoxy and Graphite/Epoxy that are provided in Table 9.2. 9.3.1 G1 as s/Epoxv The equilibrium moisture concentration, c of the Glass/Epoxy material is 0.025. 82 The stress a is plotted in Figure 9.3. It reaches y a maximum magnitude of 166 MPa. for case C at 20C. It is compressive for the 0 layer and tensile for the 90 layer. The stress a is shown in Figure 9.4. It is compressive everywhere and reach a magnitude of 288 MPa. for the case C at 20C. The stress is also compressive (Figure 9.5) and reaches a maximum of 245 MPa. The free edge shear stress yz (Figure 9.6) is very significant since its maximum magnitude is about 80 MPa.. 9.3.2 Graphite/Epoxy The equilibrium moisture concentration c^ for these cases is 0.015. The stresses a a o and a are y z x yz plotted in Figures 9.7-10. These results show the same trend as for the Glass/Epoxy cases. However, since the moisture concentration is lower and graphite fibers have stiffer moduli and lower coefficient of thermal expansion, the magnitude of the stresses is smaller. 9.3.3 Summary The hygrothermal conditions used in the preceding sections are practically achieved only under very adverse conditions. Hence, the induced stresses can be considered |

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COMPOSITE LAMINATES BY HACENE BOUADI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ACKNOWLEDGEMENTS I would like to express my gratitude to Professor Chang-T. Sun, the chairman of my doctoral committee, for his guidance, time, and encouragement during this research. Many thanks are owed to Professor Lawrence E. Malvern and Professor Martin A. Eisenberg for their teaching and financial support. I also want to thank the other members of my doctoral committee, Dr. Charles E. Taylor and Dr. Robert E. Reed-Hill for their helpful commen ts, critique, and advic e. In addition, I gratefully recognize the assistance of Dr. David A. Jenkins for teaching me how to operate the material testing equipment that was indispensable for my work. Finally, I appreciate Ms. Patricia Campbellâ€™s help in typing this manuscript. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS i i LIST OF TABLES vi LIST OF FIGURES vii NOMENCLATURE xiii ABSTRACT xvi CHAPTERS 1 INTRODUCTION 1 1.1 General Introduction 1 1.2 Moisture Diffusion 2 1.3 Hygrothermal Effects 2 1.4 Scope and Methodology 3 1.5 Dissertation Lay-Out 4 2 DIFFUSION OF MOISTURE 6 2.1 Introduction 6 2.2 Fickian Diffusion 6 2.3 Fickian Absorption in a Plate 8 2.3.1 Infinite Plate 8 2.3.2 Semi-Inf inite Plate 10 2.3.3 Experimental Measurement of Moisture Content 11 2.3.4 Approximate Solutions of Moisture Content 12 2.3.5 Edge Effects Corrections in a Finite Laminated Plate 13 2.4 Diffusivity and Maximum Moisture Content . . 15 3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES ... 21 3.1 Introduction 21 3.2 General Theory 21 3.3 Micromechanics Formulation of elastic Moduli 22 3.4 Complex Moduli 23 i i i 4 DAMPING 29 4.1 Damping Mechanisms 29 4.1.1 Nonmaterial Damping 29 4.1.2 Material Damping 30 4.2 Characterization of Damping 30 4.2.1 Free Vibration '. 30 4.2.2 Steady State Vibration 31 4.2.3 Complex Modulus Approach 32 5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES . . 36 5.1 Introduction 36 5.2 Laminated Plate Theory Approach 36 5.3 Energy Method Approach 37 6 EXPERIMENTAL PROCEDURES 40 6.1 Introduction 40 6.2 Test Specimen 40 6.3 Environmental Conditioning 41 6.4 Four-Point Flexure Test Method 41 6.5 Impulse Hammer Technique 42 7 HYGR0THERMAL EXPANSION 48 7.1 Introduction 48 7.2 Coefficients of Thermal Expansion 49 7.3 Coefficients of Moisture Expansion . . . . . 50 7.4 Experimental Data 51 7.4.1 Previous Investigations 51 7.4.2 Present Investigation 52 8 HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI 58 8.1 Literature Survey 58 8.2 Theoretical and Experimental Assumptions. . . 59 8.3 Modeling of Epoxy Properties 61 8.4 Results 63 8.4.1 Epoxy Complex Moduli 63 8.4.2 Composite Complex Moduli 65 9 HYGROTHERMAL EFFECTS ON STRESS FIELD 79 9.1 Introduction 79 9.2 Description of Study Cases 80 9.3 Numerical Results and Discussion 81 9.3.1 Glass/Epoxy 81 9.3.2 Graphi te/Epoyx 82 9.3.3 Summary 82 i v 10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES 95 10.1 Introduction 95 10.2 Numerical Results and Discussion 95 10.2.1 Glass/Epoxy 95 10.2.2 Graphite/Epoxy 96 10.2.3 Summary 97 11 CONCLUSION Ill APPENDICES A COMPLEX STIFFNESSES OF COMPOSITES 115 A.l Elastic Stiffnesses 115 A.2 Complex Stiffnesses 117 B DEVELOPMENT OF THE FINITE ELEMENT METHOD .... 119 B.l Equilibrium Equations 119 B.2 Program Organization 122 B.3 Shape Functions, Jacobian and Strain Ma trix 123 B.4 Elasticity Matrix 125 B.5 Element stiffness Matrix 128 B.6 Equivalent Nodal Loadings 128 B.6.1 Element Edge Loadings 128 B.6.2 Hygrothermal Loadings 129 B.7 Element Stresses 130 REFERENCES 133 BIOGRAPHICAL SKETCH 137 v LIST OF TABLES T abIes Page 6.1 Initial properties of Magnolia 2026 epoxy, 3M Scotchply Glass/Epoxy, and a typical Graphite/epoxy composite 45 7.1 Coefficients of moisture and thermal expansion of epoxy and graphite and glassfibers 54 7.2 Properties of Glass and Graphite Fibers . . . 54 9.1 Description of cases in Figure 9.2 84 9.2 Typical strengths of Glass/Epoxy and Graphite/Epoxy 84 LIST OF FIGURES Figures Page 2.1 Plate subjected to a constant humid 18 environment on both sides. 2.2 Moisture distribution across a plate. The numbers on the curves are the values of (c - c.)/(c - c.) 18 v i y v oo i y 2.3 Semi-infinite plate in a humid environment 19 2.4 Comparison of the exact specific moisture concentration equation with some approximate so 1 u t ions 19 2.5 Geometry of a plate 20 2.6 Moisture content versus square root of time. On the curve Vt < Vt^< Vt^ and the slope is constant for Vt" < Vt^ 20 4.1 Schematic drawing of a free-clamped beam under free vibration and plot of its deflection versus time 35 4.2 Schematic drawing of a free-clamped beam under forced vibration and plots of the deflection versus time and deflection amplitude versus frequency 35 6.1 Schematic drawing of environmental and testing chambers 46 6.2 Loading configuration of the 4-point f 1 exu retest 46 v i i 6.3Schematic drawing of the impulse hammer technique apparatus and a typical display of the Fourier Transform 47 7.1 Transverse moisture strain of Magnolia epoxy and 3M Scotchply Glass/Epoxy 55 7.2 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry S Glassf iber/Epoxy at 20Â°C 56 7.3 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20Â°C 56 7.4 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry S Glassf iber/Epoxy at 20Â°C 57 7.5 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20Â°C 57 8.1 Schematic variation of the storage modulus of epoxy with temperature 67 8.2 Schematic variation of Poissonâ€™s ratio of epoxy with temperature 67 8.3 Schematic variation of damping of epoxy with temperature 6S 8.4 Glass transition temperature of epoxy. From Delasi and Whiteside [6] 68 8.5 Experimental data of the storage modulus of epoxy as a function of temperature at diverse constant moisture contents 69 8.6 Experimental data of the storage modulus of epoxy as a function of moisture content at diverse constant temperatures 69 8.7 Experimental data of the storage modulus of epoxy as a function of normalized temperature (T - Tq)/(T - Tq) 70 8.8 Experimental data of damping of epoxy as a function of temperature at diverse constant moisture contents 71 v i i i 8.9 Experimental data of damping of epoxy as a function of moisture content at diverse constant temperatures 71 8.10 Experimental data of the storage modulus of epoxy as a function of normalized temperature (T - Tq)/(T - Tq) 72 8.11 Experimental data of Poissonâ€™s ratio of epoxy in term of temperature 73 8.12 Experimental data of Poissonâ€™s ratio of epoxy in term of moisture content 73 8.13 Experimental data of Poisson's ratio in term of the normalized temperature T = (T - T )/(T - T ) 74 n v o g o 8.14 Longitudinal storage modulus (Ej^) of Glass/Epoxy versus = (T - Tq)/(T^ - Tq). . . 75 8.15 Transverse (E^) and shear (Gj^) storage moduli of Glass/epoxy versus T = (T - T )/(T - T ) 75 n o g o 8.16 Longitudinal transverse ( rj22^â€™ and shear () damping of Glass/Epoxy versus T = (T - T )/(T - T ) 76 n v o g o' 8.17 Poissonâ€™s ratio () of Glass/Epoxy versus T = (T - T )/(T - T ) 76 n o v g o' 8.18 Longitudinal storage modulus (Ej^) of Graphite/Epoxy versus = (T - Tq)/(T - Tq) 77 8.19 Transverse (E^) an<3 shear (Gjg) storage moduli of Graphite/epoxy versus T = (T - T ) / (T - T ) 77 n v o' g o' 8.20 Longitudinal (77 ^ ^ ) , transverse (^22^ â€™ and shear (^q) damping of Graphite/Epoxy versus T = (T - T )/(T - T ) 78 n v o' v g o' 8.21 Poissonâ€™s ratio (v â€™ ^ ) of Graphite/Epoxy versus T = (T - T )/(T - T ) 78 n o g o 9.1 Geometry of a laminate and finite mesh of a 1/4 cross-section area 85 9.2 Description of the applied moisture grad ients 86 9.3Profile of the hygrothermal stress a across a [(90/0)^^ Glass/Epoxy laminate at y/b = 0.472 87 9.4 Profile of the hygrothermal stress across a [(OO/O^lg Glass/Epoxy laminate at y/b = 0.472 88 9.5 Profile of the hygrothermal stress across a [(90/0)2^ Glass/Epoxy laminate at y/b = 0.472 89 9.6 Profile of the hygrothermal stress a across a [(OO/OJ^lg Glass/Epoxy laminate at y/b = 0.993 90 9.7 Profile of the hygrothermal stress a across a [(OO/OJ^jg Graphite/Epoxy laminate at y/b = 0.472 91 9.8 Profile of the hygrothermal stress across a [^O/OJ^jg Graphite/Epoxy laminate at y/b = 0.472 92 9.9 Profile of the hygrothermal stress across a [(QO/O^jg Graphite/Epoxy laminate at y/b = 0.472 93 9.10 Profile of the hygrothermal stress Â°yX across a [(OO/O^lg Graphite/Epoxy laminate at y/b = 0.993 94 10.1 Line style legend of Figures 10.2-13 98 x 10.2 Complex in-plane stiffness A^ of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 99 10.3 Complex in-plane stiffness of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 100 10.4 Complex in-plane stiffness Agg of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 101 10.5 Complex bending stiffness of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 102 10.6 Complex bending stiffness of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 103 10.7 Complex bending stiffness Dgg of Glass/Epoxy. a) Non-dimensional Real part; b) corresponding damping 104 10.8 Complex in-plane stiffness A^ of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 105 10.9 Complex in-plane stiffness A^ of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 106 10.10 Complex in-plane stiffness Agg of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 107 10.11 Complex bending stiffness of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 108 a 10.12 Complex bending stiffness of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 109 10.13 Complex bending stiffness Dgg of Graphite/Epoxy. a) Non-dimensional Real part; b) corresponding damping 110 B.l Organization of the F.E.M. program 131 B.2 Local axes f and , Gauss point numbers and local node numbers of an eight-node isoparametric element 132 x i i NOMENCLATURE B * B * B" c comp 1 ex in-p1ane stiff ne s s comp 1 ex coup 1ing stiff ne s s comp 1 ex modu1u s s torage modu1u s loss modulus moisture concentration c average specific moisture cm equilibrium moisture concentration D* . i J D , D X XX [D] E11 E22 G12 K spec ific heat complex bending stiffness moisture d i ffusivities diffusivity matrix, elasticity matrix longitudinal Young modulus transverse Young modulus in-plane shear modulus thermal conductivity m weight of absorbed moisture M percent moisture content x i i i initial percent moisture content M . i equilibrium percent moisture content Q. . transformed stiffness i J s t T v m w a . i â‚¬ V 12 e . j complex transformed stiffness specific gravity time temperature fiber volume fraction matrix volume fraction we igh t coefficient of thermal expansion coefficient of moisture expansion s t rain damping or loss factor major Poissonâ€™s ratio fiber orientation of j-th layer density stress Subsc rip t s 1, 2, 3 principal directions of the fibers f fiber i initial j layer number x i v L longitudinal direction m ma t rix x, y, z Cartesian coordinates 00 maximum or equilibrium Superscripts H mo is ture o initial T transpose, thermal * complex value real part imaginary part xv Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillement of the Requirements for the Degree of Doctor of Philosophy HYGROTHERMAL EFFECTS ON THE COMPLEX MODULI OF COMPOSITE LAMINATES By Hacene Bouadi Apr i 1 1988 Chairman: Dr. Chang-T. Sun Major Department: Engineering Sciences The effects of absorbed moisture and temperature on the complex moduli of composite laminates are investigated and the mechanisms of moisture diffusion in a lamina are also analyzed. First, the variation of the complex moduli of epoxy in terms of temperature and moisture content are experimentally determined. Then, the hygrothermal effects on the complex moduli of composites are derived by using the complex moduli of the matrix, micromechanical formulas and experimental data. Only the hygrothermal effects on the complex moduli of pure epoxy need to be experimentally determined since these effects on the fibersâ€™ properties are neg1 igib1e. xv i In addition, the effects of hygrothermal environment on the stress field and material damping of general laminated composite plates are analyzed. It is shown that hygrothermal stresses induced directly by moisture and indirectly by material property changes can be very high, but the effects on damping are less pronounced. xv i i CHAPTER 1 INTRODUCTION 1 . 1 General Introduction The introduction of advanced composites in aerospace applications has led to an extensive study of their mechanical behavior. The amount of experimental and theoretical findings of composite material researchers made during the 1960â€™s was so vast that Broutman and Krock [1] needed eight volumes to edit a summary of the resulting know 1 edge. The interest in composite materials arose from their ideal characteristics for aerospace structures. Replacement of the commonly used aircraft material, aluminum, by high strength/density ratio and versatile composites can lead to a theoretical 60% weight reduction [2, p. 22]. Due to such benefits, lower costs and better understanding of their mechanisms, the use of composite materials has been increasing slowly but steadily. Exposure of aircraft structures to high temperature and humidity in the environment and the tendency of composites to absorb moisture gave rise to concern about their performances under adverse operating conditions. 1 2 Therefore, considerable work has been done to understand the effects of hygrothermal environment on the mechanical behavior of composite materials. 1.2 Moisture Diffusion In a 1967 study on the effects of water on glass reinforced composites. Fried hypothesized that water can penetrate the resin phase by two general processes, by diffusion through the resin and by capillary or Poiseuille type of flow through cracks and pinholes [3], But no mathematical theory was presented. Later, investigators established that the primary mechanism for the transfer of moisture through composites is a diffusion process and adapted the general theory of mass diffusion in a solid medium to moisture diffusion in composite materials. The transfer of moisture through cracks is a secondary effect [4, 5], Experimental data indicate that for most composite materials, the diffusion of moisture can be adequately described by a concentration dependent form of Fickâ€™s law [4-10]. 1.3 Hygrothermal Effects The degradation of mechanical properties of glass reinforced plastics exposed to water has long been 3 recognized by marine engineers who use "wet" strengths in the design of naval structures [3]. Requirements in aircraft structures are more stringent. The mechanical properties of materials used in aerospace applications must be completely characterized. Therefore, the effects of hygrothermal environment on the elastic, dynamic, and viscoelastic responses of composites have been studied. To date, the effects of moisture and/or temperature on the following performances have been investigated: tensile strength, shear strength, elastic moduli [3, 11-14], fatigue behavior [15-17], creep, relaxation, viscoelastic responses [18-20], dimensional changes [21], dynamic behavior [22], glass transition temperature [23], etc. Only tensile and shear strengths and elastic moduli have been thoroughly studied by many researchers. But data on the other properties are more limited and hence inadequate to constitute a good design data-base. 1.4 Scope and Methodology The present investigation is a combined theoretical and experimental work and is concerned with predicting the hygrothermal effects (below the glass transition temperature) on the complex moduli of composite materials. This program is undertaken by carrying out the foil owing s teps: 4 i)The complex moduli of epoxy matrix in terms of temperature and moisture concentration are obtained by# using experimental tests and theoretical expressions. ii)The effects of temperature and moisture on the complex moduli of unidirectional composites can be derived by using the complex moduli of the matrix, micromechanics formulas, and experimental observations. In addition, we neglect the hygrothermal effects on the fibers. iii)The effects of hygrothermal conditions on the stress field and the material damping of some general laminated composite plates undergoing simple hygrothermal loadings are analysed. 1.5 Dissertation Lay-Out Right af ter the in t roduc tion , the mechanism o f moisture diffusion i s described in Chap ter 2, whe r e the absorption of moisture through thin compos ite 1aminas i s analyzed in detail. The complex moduli of unidirectional composites are defined in Chapter 3. Sections 3.3 and 3.4 give the micromÂ¬ echanics formulations of the elastic and complex moduli in terms of the constituent material properties. The damping of composites based on the dynamic and complex modulus approaches is characterized and the equivalence of both approaches is proven in Chapter 4. In Chapter 5. the damping and complex stiffnesses of general laminates are 5 derived by using the laminated plate theory, the energy approach, and the preceding derivations. The complex stiffnesses are completely expressed in Appendix A. The environmental conditioning of the test specimens, the static flexure test, and the impulse hammer techniques are presented in Chapter 6. These experimental methods, although simple, are very versatile and are adequate in determining the necessary data for the purpose of this investigation. The theoretical and experimental results are given in Chapters 7-10. The moisture and thermal expansions of composites are quantified in Chapter 7. The current experimental results and data and conclusions of previous investigators are used in Chapter 8 to model the complex moduli of epoxy as functions of temperature and moisture content. In Chapters 9 and 10, the hygrothermal effects on the stress field across laminates and on damping of composites are investigated with the help of the results in the preceding Chapters. The Finite Element Method (F.E.M.) used in determining the stresses is summarized in Append ix B. CHAPTER 2 DIFFUSION OF MOISTURE 2.1 Introduc tion The mechanism of moisture absorption and desorption in most fiber reinforced composites is adequately described by Fickâ€™s law [4]. Fick recognized that heat transfer by conduction is analogous to the diffusion process. Therefore, he adopted a mathematical formulation similar to Fourierâ€™s heat equation to quantify the diffusion process [24, 25]. 2.2 Fickian Diffusion The Fourier and Fickâ€™s equations, describing the one-dimens iona1 temperature and moisture concentration, are respectively given by (2.1) 3c _ 3 ~ 3c 3t - 3x x dx (2.2) where p is the density of the material, is the 6 7 specific heat, T is the temperature, t and x are the time and apatial coordinates, respectively, is the thermal conductivity, c is the moisture concentration, and is the moisture diffusivity. The moisture diffusivity, D , and the thermal diffusivity, Kx/(pCv), are the rate of change of the moisture concentration and the temperature, respectively. In general, both parameters depend on temperature and moisture concentration. But experimental data show that, for most composites, moisture diffusivity does not depend strongly on moisture concentration [4], Hence, Eq (2.2) becomes 3c 3 t D a2 3 c x _ 2 dx (2.3) and is solved independently of Eq (2.1). The three-dimensional diffusion in an anisotropic medium is obtained by generalization of Eq (2.2) as follows 3c , mi â€”* \ 7TT = V. ( [D] . Vc) where the diffusivity ma t rix i s D D D XX xy xz [D] - D D D yz yy yz D D D zx zy zx (2.4) (2.5) 8 Expansion of Eq (2.4) results in an equation of the form 2 2 2 2 dc d c d c d c d c Â§7- = D + D 2_Â£ + d 2_Â£ + (D + D )_ â– g-- 3t xxg^2 yyay2 xx^^2 K yz xy'ay ox R2 R2 + (D + D + (D + D KÂ° Â§ zx xz ox oz xy yx'ox oy (2.6) if the coefficients D. ,â€™s are considered to be constant. i J 2.3 Fickian Diffusion in a Plate Laminated plates are widely used in the experimental characterization of composites. Hence, being of practical interest, the problem of moisture absorption in a plate is thoroughly discussed in this section. 2.3.1 Infinite Plate The case of moisture absorption through a material bounded by two parallel planes is considered. The initial and boundary conditions of an infinite plate exposed on both sides to the same constant environment (Figure 2.1) are given by T c for 0 < z < h and t < 0 (2.7) 9 T = T. j 1 f for z = 0, z = h and t > 0 C = C03 J where T. is a constant temperature, is the initial moisture concentration inside the material, and c is the CO maximum moisture concentration. It is assumed that the moisture concentration on the exposed sides of the plate reaches cra instantly. The solution of Eq (2.3) in conjunction with the conditions of Eqs. (2.7) is given by Jost [25] c - c oo c . 1 c . 1 oo 4 V 1 . 2 ,j + 1 â€ž W 1 (2jTTTsln ~Sz exp j=o ^iii)2!72D t v 2 z n (2.8) Equation (2.8) is plotted in Figure 2.2. The average moisture concentration is given by c â– 'h c dz ^ o (2.9) Substitution of Eq. (2.8) into Eq . (2.9) and integration result in c c 00 c . 1 c . 1 8_ ir I j=o 1 ( 2 j + 1 )' exp .2_2 Dzt -(2j + 1) U s- (2.10) This analysis can be applied to the case of diffusion of 10 moisture into a laminated composite plate so thin that moisture enters predominantly through the plane faces. 2.3.2 Semi-Infinite Medium In the early stages of moisture diffusion into a plate, there is no interaction between moisture entering through different faces. Therefore, the solution of moisture absorption into a semi-infinite half-plane is applicable to a plate for short time. The initial and boundary conditions of a semiÂ¬ infinite plane (Figure 2.3) exposed to a constant moist environment are T c for 0 < z < 00 and t < 0 0 and t > o (2.11) The solution of Eq (2.3) in this case is [24, 25] erf z 2 / D t L v z J (2.12) The rate at which the total specific mass of moisture, m, is diffusing into the half-plane is de dz _ z =0 (2.13) 1 1 dm d t D Thus, the total mass of moisture entering through an area A in time t is n t m = - pAD dc dz dt = 2pA(c - z=0 D t z 17 (2.14) Equation (2.14) shows that the mass of diffusing substance is proportional to the square root of time. 2.3.3 Experimental Measurement of Moisture Content In the case of a finite plate, the total moisture con tent is m = pVc (2.15) where V is the volume of the test piece. The total moisture content is experimentally measured by subtracting the dry weight, w^, from the current weight, w, of the plate, i.e. m = w w (2.16) A parameter of practical interest is the percent moisture content defined as 12 M 100 (2.17) Since M - M . c - c . n 7r~ = 1 : M = 100c (2.18) M - M. c - c . v â€™ 00 } CO i the experimentally measured M of Eq (2.18) can be compared to the analytical value given by Eq (2.10). 2.3.4 Approximate Solution of Moisture Content Approximate solutions of the specific moisture distribution in a plate subjected to the conditions given by Eqs (2.7) are useful, since the difficulty of dealing with infinite series can be avoided. Sma11 time. As discussed in section 2.3.2, Eq (2.14) can be applied during the early stages of absorption. It yields c - c . M - M. i i Large time. Tsai and Hahn [26, p. 338] suggest that, for sufficiently large t, Eq. (2.10) can be approximated by using the first term of the series, i.e., D t z TJh (2.19) 13 c c 00 C . 1 c . 1 8 -9-exp UZ IT D ti z (2.20) Shen and Springer formulation. These researchers have derived in Ref. [4] the following approximation c c 00 c . 1 c . 1 exp 7.3 D t z 0.75' 1 h2 j (2.20) Figure 2.4 shows a comparison of Eqs (2.19-21) with the exac t so 1u tion. 2.3.5 Edge-Effect Corrections in a Finite Laminated Plate A plate exposed to a humid environment absorbs moisture through all its six sides. At small time, the interaction of moisture entering through different sides is negligible. Therefore, Eq. (2.19) can be applied to such cases. It yields m = 4P(C oo_ ci) bL /d + bh nr + hL rr rm V z vx v y J v (2.22) where D , D , and D are the diffus ivities in the x, y, and x y z z directions, respectively. The geometry of the plate is shown in Figure 2.5. Rewriting Eq. (2.22) in terms of the percent moisture content gives 14 M 4M Dt ' nh2 where the effective diffusivity D is (2.23a) Â» D D 1 h 1 + L x h _ i y D b D > Z N z (2.23b) The micromechanics formulation for diffus ivities proposed by Shen and Springer [4] and modified by Hahn [26] for impermeable, circular cross-section, fiber-reinforced compos i tes is D L D m D T (2.24) where D , D_ and D, are the matrix, transverse, and m T L longitudinal diffus ivities, respectively. Equation (2.23b) for a unidirectional lamina with all fibers parallel to the x-direction can be written as D (2.25) 15 For a general laminated plate consisting of N layers with fiber orientations 0., the diffusivit i es are J D z D T D x D y N N D. y h .cos^0 . + D~ T h.sin^0. L L j j TZ.J j J = i iÃ¼i N 2", j = l N N D, y h.sin^0. + Dâ€ž y h.cos^0. L L j j 7 L j j â– 1 = 1 j=l N (2.26) where h. is the thickness of J diffusivity of a general substituting Eqs. (2.26) into the j-th layer. 1amina t e is Eq (2.23b). The effective obtained by 2.4 Diffusivity and Maximum Moisture Content The diffusivity and the maximum moisture content Mra must be experimentally determined in order to predict the moisture content and distribution in a lamina. These parameters are obtained by the following procedures: - a thin, unidirectional composite plate is completely dried and its weight is recorded, 16 - the specimen is then placed in a constant temperature and constant relative humidity environment, and its weight as function of time is recorded, - the moisture content, M, versus the square root of time, , is plotted as shown in Figure 2.6. The maximum moisture content is determined from the plot and the diffusivity from the following equation M2 - Ml = 4M D I7h (2.27) The subscripts 1 and 2 are defined in Figure 2.6. The diffusivity depends only on the material and temperature as follows D z D exp o (2.28) where R is the gas constant, D and E, are the o a pre-exponential factor and the activation energy, respec tively. Experimental research has shown that the maximum moisture content depends on environment humidity content and material. For a material exposed to humid air [4], the equilibrium moisture content can be expressed as M 00 (2.26) 17 where 0 is the relative humidity, a aad b are material cons tan t s. IS > > Moisture > 5Â» Fig. 2.1 Plate subjected to a constant humid environment on both sides. z =* 0 is the center of the crossâ€”section of the plate z/h Fig. 2.2 Moisture distribution across a plate. The numbers on the curves are the values of (c - c.)/(c - Ã J v co ci). (M - M,) 19 Fig. 2.3 Semi-infinite plate in a humid environment. Exact Oneâ€”term Shen and Springer Fig. 2.4 Comparison of the exact specific moisture concentration equation with some approximate solutions. Moisture content (%) 20 Fig. 2.5 Geometry of a plate i Fig. 2.6 Moisture content versus square root of time. On the curve J t ^ < J t^ < y t^ and the slope is fÂ°r/T < yr^. cons tan t CHAPTER 3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES 3.1 In t roduction Composite materials, such as Glass/Epoxy and Graphite/Epoxy, have a polymeric matrix. Therefore, they display viscoelastic behavior. Some of the effects of this time-dependent phenomenon are: stress relaxation under constant deformation, creep under constant load, damping of dynamic response, etc. This chapter is an introduction to the dynamic behavior of viscoelastic composites in terms of complex modu 1 i . 3.2 General Theory A usual representation of the one-dimens iona1 stress-strain relation of a viscoelastic material subjected to a harmonic strain history of the form â‚¬( t) = â‚¬Â°e1CJt (3.1) 21 22 is given by a(t) = B (iw)â‚¬Â°e^Wt = B (icj)G(t) (3.2) The complex modulus B can be decomposed into its real and imaginary parts as follows B*(iu) = B ' ((i)) + i B" ( gj ) (3.3) The terms B' and Bâ€ are called the storage and loss moduli, respectively, and the ratio of the loss over the storage modulus D Â»* V = g-r (3.4) is referred to as either the loss factor or damping. The loss modulus is a measure of the energy dissipated or lost as heat per cycle of harmonic deformation. 3.3 Micromechanics Formulation of Elastic Moduli The longitudinal modulus E^, the transverse modulus E22â€¢ the in-plane shear modulus G^, and the major Poissonâ€™s ratio v12 can be obtained by using the rule of mixtures and the Halpin-Tsai equations, viz., + v E m m E = v E 11 f f 11 (3.5) 23 E 22 E m 1 + 2rijV 1 - n ^ v (3-6) 1 + n9v G10 = G t ^-3- 12 ml- n2vf (3.7) wher e The subscripts f respectively, and v n, = n~ = = VfUf12 + V V m m (3.8) (Ef22/Em> - i (3.9) (Ef22/Em) + 2 (Gfl2/Gm> - 1 (3.10) (Gfl2/Gm) + 1 m stand for fiber and ma t rix, v are the m volume fractions Como lex Moduli The micromechanics formulations of the complex moduli are obtained by applying the e1astic-viscoe1astic correspondence principle [27-29], i.e., by undertaking the following steps: i) determining the elastic moduli of composites in terms of the constituent material properties, ii) replacing the elastic moduli of fibers and matrix by corresponding complex expressions. For a viscoelastic composite. the properties of the constituent materials are Efll ' â€œfll + lEfll ^f22 ~ Ef22 + lEf22 G, = G1 + iG" E = E' + iEâ€ m m (3-11) G = m G1 + iG" m m v = V + IB m m J f12 _ L f 12 The bulk modulus of epoxy matrix, K , is real and m independent of frequency [2S]. It is given by - â€¢%n _ 3V 1 - 2v ) v m' (3.12) while the viscoelastic bulk modulus is obtained from the correspondence principle 25 E' + iE" m m K* = m 3[1 - 2(u' + iu")] m m â€¢ (3.13) Separation of the real and imaginary parts of Eq. (3.13) yields K * m (1 2d ')Eâ€™ m m 3 1(1 2u"E" + i[2E"d" + E'( 1 m m L mm m 2u')2 + 4d"2 J m â€™ m J 2d ' )] m'J (3.14) Since the dilatation bulk modulus is real, the imaginary part of Eq. (3.14) is equal to zero; hence 2E"d" + Eâ€(1 - 2dâ€˜) = 0 (3.15) mmmv m' v ' Equation (3.15) results in m, d" = =^(d * - 0.5) m E v m â€™ (3.16) m The shear modulus of the matrix is given by G* = m m 3K E* m m 2(1 + d*) 9K - E* v m' mm (3.17a) Separating the real and imaginary parts and neglecting the 2 terms of the form (Eâ€) yield 26 3K E' P* nn m m â€œ 9K - E m 1 + i 9K E" m m 9K - Eâ€™ E* m mm Introduction of the material properties p = ft m E 'f 1 1 f 1 1 E â€™ f 11 f 22 f 22 E â€™ f22 m 9K m 'Gm " G 1 â€œ 9K - Eâ€™ 'm m mm into Eq s. (3.11) r e su 1t s i n E f 11 ( 1 + 1T7f 11 E f 22 ^ 1 + 1T?f 22 G f 12 ( 1 + 1T?fl2 E'(1 + ip ) m v m' G 1 ( 1 + ipp ) mv Gmy ) ) ) (3.17b) (3.18) (3.19) 27 * v m v ' + m ÃT7 ( v ' m m U f 12 â€œ U f 12 â€œ u f 1 2 There are no satisfactory data on the shear and transverse damping of fibers. Fibers have damping with a magnitude order ten times smaller than epoxy. The dampings ^fiiâ€™ T7f22> and q^.^ are assumed to be equal and are replaced by t]j. in subsequent equations. Since the fiber damping, q^. , is much smaller than the matrix damping, q , the m imaginary part of the fiber Poissonâ€™s ratio is neglected. The preceding assumptions have a negligible effect on the complex moduli of composites. Application of the e1astic-viscoelastic correspondence principle to Eqs. (3.5)-(3.10) and substituting them into Eqs (3.19) yield the following complex material properties E.. = V.E' ,(1 + iqr) + v E'(l + iq ) 11 ffllv fâ€™ mmv my E x 22 E'(1 + iq ) mv m l + 2njVj. ' * 1 - nivf (3.20) 12 = G'n + iTw) m , * 1 + n2Vf Gm; . ^ 1 ' n2Vf whe r e Ef22< 1 + â€˜V ~ Em(1 * lT|m> Ef22(I + â€˜"fl + 2Em(1 * iT>m> Cf 12^ 1 * lr|f ) ~ Gm*1 * 1â€™1Cni* Gf]2(l + iuf) + Gm(l + mGm) The elastic moduli given by Eqs. (3.5)-(3.8) model experimental results with a good accuracy [2], Therefore, they are used instead of mathematically exact micromechanics formulas, such as those derived by Hashin [27. 29], CHAPTER 4 DAMPING 4.1 Damping Mechanisms Any vibrational energy introduced in a structure tends to decay in time. This phenomenon is called damping. There are two types of damping mechanisms, external or nonmaterial and internal or material. 4.1.1 Nonmaterial Damping. Two common types of external damping are - Accoustic damping: a vibrating structure always interacts with the surrounding fluid medium (air, water, etc.). This effect can lead to noise emission and even to changes of the natural frequencies and mode shapes. Thus, mechanical responses might be modified. - Coulomb friction damping: two contacting surfaces in relative motion dissipate energy through frictional forces. 29 30 4.1.2 Material Damping There are many damping mechanisms that dissipate vibrational energy inside the volume of a material. Damping phenomena include thermal effects, magnetic effects, stress relaxation, phase processes in solid solutions [30, p. 61], etc. The internal damping of polymeric matrix composites, such as Glass/Epoxy and Graphite/Epoxy, is dominated by viscoelastic damping. 4.2 Characterization of Damping 4.2.1 Free Vibration A cantilever under free vibration oscillates regularly with an amplitude that decreases from one oscillation to the next one (Figure 4.1). A measure of damping is the logarithmic decrement defined as 5 1 n (4.1) whe r e A^ = amplitude of the n-th cycle ^n + N = amPlituc*e the (n+N)-th cycle The damping defined in Eq (4.1) is applicable to viscous 31 damping and for hysteretic damping that is represented by a complex modulus approach. 4.2.2 Steady State Vibration Damping also influences the dynamic equilibrium amplitude of structures (e.g. beams) that undergo harmonic oscillation. A resonance usually occurs (Figure 4.2). The following measure of damping is used V (j. - (jj. (i> o (4.2) whe r e resonant frequency "l- w2 = frequencies on either sides of such that the amp 1itude is 1/y 2 times the resonant amp 1itude. I n the case of a vibration induced by the force f(t) = Fsin(ut) the response (deflection), w(t), is out of phase with f(t) by an angle e such that w( t) Wsinfut + e) 32 The work done per cycle is â– 217/u D f ( t)^- dt = I7WF sin(e) (4.3) J o The strain energy stored in the system at the maximum displacement is half the product of the maximum displacement by the corresponding value of the force, i.e., U = ^FW cos(e) (4.4) There is no damping if the work done per cycle is zero, i.e, if sin(e) = 0. The ratio of energy dissipated in a cycle to energy stored at the maximum displacement is another measure of damping. Therefore, the damping is (4.5) The definitions of damping given by Eqs. (4.2) and (4.5) are equivalent [33]. 4.2.3 Complex Modulus Approach The one-dimensional stress-strain relation of a viscoelastic material undergoing harmonic motion has been shown to be (Eq. (3.2)) 33 Â°* ( t) = EH((o)â‚¬o ei(Jt = (E ' ((J) + iEâ€ (<>;)) â‚¬q eiwt (4.6) Noting that i |u | â‚¬ = dâ‚¬/dt, Eq. (4.6) can be written as ^ 1(Jt . E . l (J t ^ a (t) = â‚¬ e + -iâ€”r Ãue â‚¬ o co o (4.7) The real part is given by (after algebraic manipulation) a(t) = E' e^sinfut + t)'! i + rj + e ) -11 + r/2 (4.8) where 77 = tan(e) = E"/E' The energy dissipated during a cycle per unit volume is D = () a dâ‚¬ = x â– 217/(0 dâ‚¬ x d t d t = UqE'e2 o (4.9) The maximum energy stored is 1 2 u = E* â‚¬~ 2 o (4.10) The r e f o r e, V (4.11) 34 Hence, the definitions of damping given by Eq. (3.4) and Eq. (4.5) are equivalent. This conclusion is also valid for general cases of structural vibration. 35 Fig. 4.1 Schematic drawing of a free-clamped beam under free vibration and plot of its deflection versus time. 4.2 Schematic drawing forced vibration versus time and f requency. of a free-clamped beam under and plots of the deflection deflection amplitude versus Fig . CHAPTER 5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES 5.1 Introduction Both the laminated plate theory and the energy method approaches for analyzing the damping and the stiffnesses of general laminates are presented in this chapter. 5.2 Laminate Plate Theory Approach Four independent parameters are needed to determine completely the damping of a unidirectional composite. But, the analysis of the material damping of a general laminated composite requires the use of eighteen parameters. These quantities are the ratios of the imaginary over the real parts of the complex in-plane stiffnesses A. ,â€™s, the i J and the complex bending The terms A. ,â€™s, B. ,'s, and 1 J i J and D. ,â€™s are defined as i J â– +h/2 A. . i J dz J -h/2 36 37 r+h/2 -h/2 (5.1) 0 P+h/2 J-h/2 2tt* 2 Qij dz where the complex transformed stiffness Q_â€™s depend on * * * Ellâ€™ E22â€™ G12â€™ the 1amina t e. 12 and the orientation of each layer of The in-plane, coupling, and flexural material damping are defined as I77! j A 7 . i J a: . i j C^ij B7 . -U. b: . i j (5.2) r-7? â€¢ â€¢ F i J D7 . i i d: . 1J respec tively. 5.3 Energy Method Approach The energy method can be used to damping of laminated composite materials determine the under certain loading and boundary conditions. The damping of a laminated 38 composite material in the first mode of vibration can be defined as N ^ ^ k^d ^ cyc. V = ^ (5.3) I 2n kâ€œs k= 1 where N is the total number of layers, U.) is the vk d'cyc. energy dissipated in the k-th layer during a cycle, and ^_Us is the maximum energy stored in the k-th layer. The storage and the dissipated energy are given by k U s 2 â‚¬ . C '. . â‚¬ . V J J1 1 k dV kUd U â‚¬ C" â‚¬ V J J'1 1 k dV (5.4) where i and j are C'.' . are the real J i stiffnesses and V. k â€™â€™totalâ€ damping of the material principal axes, Cj . and and imaginary parts of the complex is the volume of each layer. Hence, the an N-layered laminate is given by (5.5) 39 The maximum strain vector {â‚¬} can be determined by the finite element method first. Then, the damping can be deduced. Equation (5.5) is used to determine the damping of a beam with variable thickness or of more general s t rue tures. CHAPTER 6 EXPERIMENTAL PROCEDURES 6 . 1 Introduction A description of the test specimens and the experimental procedures of the present investigation is given in this chapter. 6.2 Test Specimens The test specimens used to determine the complex moduli of epoxy and of composite materials are thin strips of approximate dimensions, 150mm by 25mm by 2mm. The only materials tested are Magnolia 2026 laminating epoxy and 3M Scotchply Glass/Epoxy. The curing temperatures of the epoxy and the Glass/Epoxy are 175Â°C and 170Â°C, respectively. The initial properties of these materials (at 20Â° C and without moisture), as well as those of a typical Graphite/Epoxy, are given in Table 6.1. 40 41 6.3 Environmental Conditioning The specimens are conditioned in a Thermotron environment chamber at a constant temperature and constant relative humidity. The weight gain of the test pieces as a function of time is monitored. Right after moisture equilibrium is reached, the specimens undergo all tests at diverse temperatures inside a testing chamber connected to the environment chamber (Figure 6.1). The range of temperature achieved inside the environment chamber is 4Â°C to 90Â°C and the range of relative humidity is 4% to 99% for temperatures below 75Â°C. As temperature increases, the highest relative humidity that can be obtained decreases steadily to 75% at 90Â°C. 6.4 Four-Point Flexure Test Method The Youngâ€™s modulus and the Poissonâ€™s ratio can be determined with the four-point flexure test method. The loading configuration of this test is shown in Figure 6.2. The elastic flexural analysis yields [31] E PI' Sbh^w (6.1) where E is the effective modulus, P is the applied load, 1 is the length of the specimen, b is the specimen width, h 42 is the thickness, and w is the deflection at quarter-point. Poissonâ€™s ratio is expressed as (6.2) where the transverse strain â‚¬ is measured y transverse strain gage cemented in the middle specimen. with a of the 6.5 Impulse Hammer Technique The material damping and the storage modulus of a one-dimensional thin beam are determined with the impulse hammer technique. This technique was pioneered by Halvorsen and Brown [32]. The equipment set-up is shown in Figure 6.3. The specimen is clamped inside the testing chamber. A force impulse is applied to the test piece by a force transducer. The end displacement of the specimen is recorded with a non-contacting motion transducer. Both responses from the force and motion transducers go through signal conditioning equipments (filters, amplifiers). These responses are digitized in a Fast Fourier Transform analyzer (FFT) to obtain the transfer function in terms of the frequency. The transfer function is defined as the ratio of the Fourier Transform of the output (displacement 43 v(t)) over the Fourier Transform of the input (force impulse u(t)); that is. H(f) _ mi ~ U(f) (6.3) where t = time f = f requency V(f) = Fourier Transform of v(t) U(f) = Fourier Transform of u(t) The real and imaginary parts of H(f) are displayed on the FFT analyser CRT (Figure 6.3). The material damping defined by Eq. (4.11) is experimentally obtained by the following expr e s sion T7 (fa/fb>2 - 1 (6.4) where the frequencies f and f, are defined in Figure 6.3. a b The storage modulus is expressed as [33, p.464] 2 1 ^ E' = 38.32 f p^r (6.5) r li where f is the resonant frequency in Hz. , p, is the material density, 1 is the length of the specimen and h is the thickness of the specimen. Equation (6.5) is valid only 44 for the case of the first mode free vibration of a clamped-free beam. A complete description and analysis of the impulse hammer technique are presented in Leeâ€™s dissertation [34]. 45 Tab le 6.1 Initial properties of Magnolia 2026 epoxy, 3M Scotchply Glass/Epoxy, and a typical Graphite/Epoxy composite. Proper ties Epoxy G1as s/Epoxy Graphite/Epoxy Vf 0.50 0.70 p (g/cm3) 1.25 1.93 1.6 En (GPa . ) 4.0 37.00 155.23 E22 (GPaâ€˜) 4.0 11.54 10.81 G12 (GPa.) 1.52 3.46 4.35 U 1 2 0.32 0.285 0.217 *11 0.018 0.0023 0.0019 *22 0.018 0.015 0.0078 46 Fig. 6.1 Schematic drawing of environmental and testing chamber s. Loading configuration of f1exure test. Fig. 6.2 the 4-poin t 47 Fourier Transform Real part Im. part Schematic drawing of the impulse hammer technique apparatus and a typical display of the Transfer Fourier Transform. Fig. 6.3 CHAPTER 7 HYGROTHERMAL EXPANSION 7.1 Introduction When a metallic or composite structure is subjected to a change of temperature, there are dimensional variations and there may be stress development. For a one-dimensional case, it is assumed that the thermal strain is given by â‚¬T = a.(T - T ) = a.AT i i o 1 where = coefficient of thermal expansion T = actual temperature Tq = reference temperature. A polymer matrix composite exposed to a humid environment absorbs moisture. Hence, it increases in weight and dimensions. This situation produces a moisture strain that varies linearly with moisture concentration [26]. In the one-dimensional case, the hygros train is given by (7.1) 48 49 â‚¬H = /3 . (c - c ) = 0 . Ac (7.2) i iv o' i v â€™ where c is the initial moisture concentration and 6. is o 1 the coefficient of moisture expansion. 7.2 Coefficients of Thermal Expansion In the case of laminated composite plates, three coefficients of thermal expansion are used in determining the thermal strains. These parameters can be written in terms of fiber and matrix properties. The micromechanics formulas for a unidirectional orthotropic lamina are given by (see Refs. [35, p. 24], and [36, p. 405] for a detailed de riva tion) â€œl = v Pa PE P + v a E iff m m m En a2 = vfaf + v a m m vfufaf v v a - m m m U12al (7.3) where the subscripts 1 and 2 represent the fiber and the transverse directions. The thermal expansion coefficients of an orthotropic lamina whose fibers make an angle 0 with the x-direction (Figure 2.5) are given by 50 2 2 a = a, cos 0 + o' sin 0 x 1 2 2 2 a = a, sin 0 + a_ cos 0 y 1 2 (7.4) a = 2(a, - a~) cos 0 sin 0 xy v 1 2' 7.3 Coefficients of Moisture Expansion Similarly, the coefficients of moisture expansions of an orthotropic lamina with impermeable fibers can be expressed as [36, p. 406] sE p 1 = â€”Â£â€”p 1 s E, , m m 11 P2 = f-O + vm)Pm - vl2Px (7.5) m * 12 = Â° where s and s^ are the specific gravities of the composite ma terial and the ma trix. The moisture expansion coefficients expressed in an axis system such that the x-direction makes an angle 0 with the fibers are given by Eqs. (7.4) after replacement of aÂ¿'s ^7 P^'s- 51 7.4 Experimental Data 7.4.1 Previous Investigations Hahn and coworkersâ€™ investigations [21, 26, 37] of swelling of composites are outlined in this section. Some of the typical results of the transverse strain versus percent moisture gain are obtained by conducting the following tests: absorption is conducted in moisture saturated air such that Eqs (2.2) and (2.7) are satisfied; while desorption takes place in vacuum at the same temperature. Their data show a hysteretic nature of swelling in this case. But, when swelling of composites is given in terms of moisture concentration, the average behavior of S2-G1ass/Epoxy, Kevlar 49/Epoxy and Graphite/Epoxy can be approximated by 0.43c = P2c (7.6) Since the data presented in their publications display a wide scatter, Hahn et al. suggest that Eq. (7.6) can be used to estimate the moisture strains for most composite materials. 52 7.4.2 Present Investigation Epoxy and Glass/Epoxy specimens are conditioned at a constant relative humidity until the absorbed moisture reaches equilibrium. The changes in transverse dimensions are measured. This procedure is repeated at diverse values of relative humidity. The results are plotted in Figure 7.1. The longitudinal swelling strains could not be measured since the micrometer calipers used were not suf ficiently accurate. These data yield the foil ow i ng expe rimen ta1 values Pm(epoxy) = 0.25 (7.7) ^2(G1ass/Epoxy) = 0.48 (7.8) Substitution of Eq. (7.7) and the parameters given in Table 6.1 into Eqs. (7.5) yields the following empirical values j3 = 0.042 (7.9a) P2 = 0.47 (7.9b) for the 3M Glass/Epoxy composite. The experimental and empirical values of 02 are practically equal. Hence, the 53 present results differ slightly from the approximation given by Eq. (7.6). The above coefficients and the typical coefficients of expansion of graphite are quantified in Table 7.1, while the storage moduli and the density of glass and graphite fibers are listed in Table 7.2. These properties are used to plot the thermal and moisture expansion coefficients versus the fiber volume fraction of Glass/Epoxy and Graphite/Epoxy in Figures 7.2 through 7.5. The values in these plots are valid for dry composites at room temperature. Since the storage modulus of epoxy varies with temperature and moisture content, this additional effect is investigated in Chapter 8. In general, the thermal expansion coefficients are functions of temperature, but this temperature effect is negligible below 100Â°C. Therefore, in the subsequent chapters, the thermal expansion coefficients are assumed to be independent of temperature. 54 Table 7.1 Coefficients of moisture and thermal expansion of epoxy and graphite and glass fibers. Epoxy Glass Graphite a (pm/m)/K 54.0 5.0 0.2 P 0.25 0.0 0.0 Table 7.2 Proper ties of Glass and graphite Fibers. Glass Graphite Efii(GPa) 70.0 220.0 E f 22 ^ GPa) 70.0 16.6 Gf12(GPa) 28.7 8.27 71 f 0.0015 0.0015 V{ 12 0.22 0.16 p (g/cm3) 2.60 1.75 55 Moisture concentration (%) Fig. 7.1 Transverse moisture strain of Magnolia and 3M Scotchply Glass/epoxy. epoxy Thermal expansion coefficient (/im/rn)/K ^ Thermal expansion coefficient Oim/m)/K 56 Longitudinal transverse 2 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry S Glassfiber/Epoxy at 20Â°C. Longitudinal Transverse 0 0.2 0.4 0.6 0.3 1 Fiber volume fraction Fig. 7.3 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20Â°C. Coefficient of moisture expansion ^ Coefficient of moisture expansion 57 Longitudinal Transverse Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry S G1assfiber/Epoxy at 20Â°C. Fiber volume fraction Longitudinal Transverse Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 20Â°C. Fig. 7.5 CHAPTER 8 HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI S.1 Literature Survey The storage moduli (real parts of Eqs. (3.11)) of composites are usually determined by dynamic testings, such as the technique described in section 6.5. They can be approximated by using static tests [38]. Shen and Springer [12] investigated the environmental effects on the elastic moduli of a Graphite/Epoxy composite and made a survey of existing data showing the effects of temperature and moisture on the elastic modulus of several composites. Their conclusions are listed below. i) The hygrothermal effects on the 0Â° fiber direction laminates are negligible. ii) For 90Â° fiber direction laminates, the hygrothermal effects on the modulus are insignificant in the 200K to 300K temperature range. But, between 300K and 450K, the hygrothermal effects on the modulus are impor tan t. Putter et al. [38] investigated the influence of frequency and environmental conditions on the dynamic 58 59 behavior of Graphite/Epoxy composites. Their overall conclusions are i)The effects of frequency on the modulus and damping are quite small in all cases. ii)The effects of frequency on the modulus and damping are relatively greater for matrix-controlled laminates at higher frequencies (above 400 Hz.). iii)At the same temperature, damping increases with moisture saturation. But for dry laminates, damping decreases slightly as temperature increases. From all these experimental works, a general summary can be drawn: the influence of hygrothermal conditions on the elastic modulus, dynamic modulus and damping of composites is matrix dominated. 8.2 Theoretical and Experimental Assumptions Since the hygrothermal influence on composite properties is matrix controlled [12, 38], the fiber properties are assumed to be constant at any temperature below the glass transition temperature and at any moisture content. Therefore, to obtain the values of the complex moduli of composites, it is sufficient to know how temperature and moisture affect the complex moduli of the epoxy matrix, and then use the micromechanics formulations given by Eqs. (3.20). Thus, only the following functions 60 E 1 m E 1 m (T.c) u ' = u â€™ (T . c ) (8.1) mm v â€™ t] = r) (T , c ) m mv ' need to be experimentally evaluated. The constant fiber properties are given in Table 7.2. The effects of frequency are negligible below 400 Hz. The results of this investigation are not accurate for higher frequencies since their effects have not been taken into account. The qualitative influence of temperature only on the storage modulus, real part of Poissonâ€™s ratio, and damping of epoxy is illustrated in Figures 8.1-8.3. There are three distinct regions. At room temperature (in the glassy region), the storage modulus, Poissonâ€™s ratio, and damping of epoxy are equal to about 4.0 GPa, 0.35, and 0.018, respectively. In the glassy region, the storage modulus decreases slowly, while Poissonâ€™s ratio and the damping increase as temperature increases. In the next region (transition region), the storage modulus decreases rapidly, and both Poissonâ€™s ratio and damping reach their maximum values. The last region is the rubbery region where the modulus takes a very low value, and all three parameters stay relatively constant. Typical values of the modulus in the rubbery region -2 could be 10 times the glassy modulus or lower. The damping 61 can reach a value of 1 or even 2 in the transition region [30, p. 90]. Poissonâ€™s ratio reaches the 1 imiting value o f 0.5, wh i ch is approximated by incompressible rubber s [39, p. 293] The position of the transition region depends on the moisture concentration. The effects of moisture on the glass transition temperature, T^_, of six epoxy resins have been determined by Delasi and Whiteside [6]. These results are plotted in Figure 8.4. They are compatible with the data of McKague [40] and satisfy the theoretical relation derived in Ref. [41, p. 69]. 8.3 Modeling of Epoxy Properties The observations of the preceding section are used for modeling the material properties of epoxy that are given by Eq. (8.1). The glass transition region of epoxy resin is not broad [6], therefore, a glass transition temperature is used instead. The temperature T^ is usually obtained by measuring the expansion of a specimen as function of temperature. The point where the epoxy stops expanding as temperature increases corresponds to the first deviation from the glassy state and is termed T . According to the experimental data plotted in Figure 8.4, T^ is stongly dependent on absorbed moisture. These results show that, as the moisture content of epoxy 62 increases, the transition temperature moves to the left in Figures S.1-8.3. Hence, the abrupt change of the material properties starts at a lower temperature as the moisture content increases. This fact and the conclusions reached by previous investigators [12] suggest that the following modelings of E', v', and n' are appropriate, mm m (8.2) (8.3) (8.4) where the temperature Tq is equal to 273K. The moisture concentration appears implicitly in T . The glass transition temperature is represented by Tg = 210 exp(- 9c) (Â°C) (8.5) where c is the moisture concentration. This modeling has been chosen so that it does not represent the material properties beyond T , since the study of epoxy in the rubbery stage is not within the scope of this research. Equations (8.2)-(8.4) are valid only for 63 the continuous parts of the curves plotted in Figures 8.1-8.3. , 8.4 Results All test specimens are conditioned in a constant relative humidity environment until moisture equilibrium is reached. Then, the test pieces undergo the impulse hammer technique and the four-point flexure tests to determine the storage moduli, the material damping, and Poissonâ€™s ratio at several temperature and moisture contents. 8.4.1 Complex Moduli of Epoxy Storage modulus. The experimental data on the storage modulus of epoxy in term of temperature at three different equilibrium moisture concentrations are plotted in Figure 8.5. It can be concluded tha t increase in either temperature or moisture content or both results in a decrease in the s t orage modu1u s. P1 o 11ing these data in terms of moisture content in Figure 8.6 does not lead to any additional insight. But, representing these results in term of the following normalized non-dimensional temperature (8.5) 64 in Figure 8.7 shows a clear trend. Experimental studies have shown that the modulus of polymer is very low at the glass transition temperature, therefore, adding the value E =0 for T = T to the data yields the following modeling m g Eâ€™ = 4.0(1 - T ) (GPa) m v nJ y ' (8.6) Material damping. Similarly, the experimental data of the hygrothermal effects on the damping of epoxy are plotted in three Figures (8.8-8.10). There is very little change in damping for all the considered conditions. Therefore, it is proposed to let r, = 0.018 (8.7) for temperatures up to 80Â°C and moisture contents up to 5%. The conclusion that the hygrothermal effects on the damping of epoxy is negligible is qualitatively corroborated by Putter et al. [38]. A quantitative comparison cannot be made since these researchers have not included in their publication the values of the fiber volume fraction and moisture content of the test specimens. Poissonâ€™s ratio. The experimental values of the Poissonâ€™s ratio in terms of temperature at two different 65 moisture contents. are plotted in Figure 8.11. These results show that Poissonâ€™s ratio increases at a negligible rate as temperature varies from 0 to 80Â°C. Representing the same data in terms of the moisture content up to M = 4.5% (Figure 8.12) shows that the moisture effect is also negligible. Therefore, vâ€™ = 0.32 (8.8) m v â€™ for temperatures up to 80Â°C and moisture contents up to 5%. Since u â€™ equals 0.5 at the glass transition temperature m (T = 0), the plot of Poissonâ€™s ratio versus the normalized temperature has been extrapolated as shown in Figure 8.13. The extrapolation displays a qualitative trend only. 8.4.2 Complex Moduli of Composites The complex moduli of Glass/Epoxy and Graphite/Epoxy in terms of moisture content and temperature can be determined by using the fibersâ€™ properties given in Table 7.2, Eqs . (8.6) through (8.8) and the micromechanics formulas (Eqs. (3.20)). This procedure is illustrated by determining the storage moduli and the damping of a Glass/Epoxy lamina with a fiber volume fraction of 0.5 and a Graphite/Epoxy lamina with a fiber volume fraction of 0.7. 66 G 1 a s s/Epoxy . The parameters Eâ€™^, E 22' ^12 'll 1 T]22â€¢ t7q . and v [2 versus the normalized temperature are plotted in Figures 8.14-17. The experimental data substantiate the theoretical results. Tli Graphi te/Epoxv . Similarly, Eâ€™^, ^â€™22â€™ *^12 11 â€™ T,22â€™ , and ujg versus the normalized temperature of Graphite/Epoxy are plotted in Figures 8.18-21. For both Glass/Epoxy and Graphite/Epoxy, the results show that the matrix-dominated parameters (E^ and Gâ€™g) are strongly affected by moisture and temperature, while the fiber-dominated parameters (E^, stay practically cons tan t. 67 Fig. 8.1 Schematic variation of the storage modulus of epoxy with temperature. Fig. 8.2 Schematic variation of Poissonâ€™s ratio of epoxy with temperature. Glass transition temperature, Tg (Â°C) ^ Damping 68 Tg Temperature .3 Schematic variation of damping of epoxy with temperature. 250 Fig. 8.4 Glass De lasi transition temperature of epoxy. From and Whiteside [6]. 69 0.0% 2.90% 3.70% Fig. 8.5 Experimental data of the storage modulus of epoxy as a function of temperature at diverse constant moisture contents. 20 Â°C 50 Â°C 70 Â°C Fig. 8.6 Experimental data of the storage modulus of epoxy as a function of moisture content at diverse constant temperatures. Storage modulus (GPa) TO Normalized temperature Fig. 8.7 Experimental data of the storage epoxy as a function of normalized (T - T ) / (T o g " T ) o modulus of temperature o u 3.703 a. E a a 0.015 I 0.01 - 0.005 j- o r . . . â– â– â– I â– â€¢ > 0 20 40 SO 30 100 Temperature (Â°C) Fig. 8.8 Experimental data of damping of epoxy as a function of temperature at diverse constant moisture contents. * T = 20 Â°C O T = 50 Â°C â–¡ T = 70 Â°C Fig. 8.9 Experimental data of damping of epoxy as a function of moisture content at diverse constant temperatures. 72 Normalized temperature * Experimental data Fig. 8.10 Experimental data of the storage epoxy as a function of normalized (T - T )/(T o; g T ) o modulus of temperature 73 QZ 4.17S Fig. S.ll Experimental data of Poissonâ€™s ratio of epoxy- in term of temperature * T =* 20^ O T = 50 Â°C d T => 75Â°C Fig. 8.12 Experimental data of Poissonâ€™s ratio of epoxy in term of moisture content. Poissonâ€™s ratio 74 Fig ^ Experimental data Fit to data â€¢ â€¢ â€¢ â€¢ Extrapolation 8.13 Experimental term of the (T - T )/(T o' g data of Poisson's ratio in normalized temperature = 75 Theoretical * Experimental data Fig. 8.14 Longitudinal s to rage modu1u s Glass/Epoxy versus = (T - Tq)/(T (Eil) - v- o f Theoretical Eâ€™â€ž 22 + Experimental EI,2 Theoretical G^2 Fig. 8.15 Transverse (E^) an<^ shear (G^) storage moduli of Glass/epoxy versus T = (T - T )/(T - T ). n o g oJ 76 Theoretical *1,, ^ Experimental 7) n â€¢ â€¢ â€¢ â€¢ Theoretical l]^ O Experimental 7)^ Theoretical 71 6 Fig. 8.16 Long itudi na 1 shear (vG) (Dll). damping n = (T - T ) / (T - T ) . o g o transverse (^22 of Glass/Epoxy Theoretical * Experimental Fig. 8.17 Poisson ' s ra t i o T = (T - T )/(T n o g (Â°i2) - v- of Glass/Epoxy ) , and versus versus 180 a CL a 160 0) 3 3 TJ O 140 QJ O' a o -*-< tn 120 0 0.2 0.4 0.6 0.8 1 Normalized Temperature Fig. 8.IS Longitudinal storage modulus (E^) of Graphite/Epoxy versus T = (T - T )/(T - T ). n o g o Transversa Shear 0 0.2 0.4 0.6 0.8 1 Normalized Temperature (Gj2) storage versus T = n Fig. 8.19 T ransverse modu 1 i of (T - T )/(T oJ g (^Â¿2) anc* shear Graphite/epoxy - T )â– o Poissonâ€™s ratio ^ Damping 0.02 â€” Longitudinal damping â€” - Transverse damping 0.015 - 0.01 - 0.0C5 - 0 L 0 0.2 0.4 0.6 0.3 1 Normalized temperature Shear damping ig. 8.20 Longitudinal ( p ^ ^ , transverse (^22 shear (p^) damping of Graphite/Epoxy T = (T - T )/(T - T ) . n v o' g oâ€™ 0.4 0.3 - 0.2 = 0.1 - 0 â– â€¢ â€¢ > â€˜ 0 0.2 0.4 0.6 0.3 1 Theoretical Normalized temperature Poissonâ€™s versus T n ratio (uj2) = (T - T )/(T - o' g of T o )â€¢ ) , and versus Fig. 8.21 Graphite/Epoxy CHAPTER 9 HYGROTHERMAL EFFECTS ON STRESS FIELD 9.1 Introduction The hygrothermal effects on the stress field are investigated by considering an infinitely long, finite width and symmetric composite laminate undergoing hygrothermal loadings. The Finite Element Method is used in order to estimate the magnitude of hygrothermal stresses in laminated composites (see Appendix B). The geometry of a laminate and the finite mesh of a quarter cross-section are shown in Figure 9.1 and the boundary conditions are given by v = 0 for (y,z) = (0,z) (9.1) w = 0 f or (y,z) = (y . 0) where v and w are the displacements in the y and z directions, respectively. The grid consists of 24 eight node isoparametric elements and 93 nodes. Only 24 elements 79 80 are used since increasing the number of elements to 48 results in a relatively small change in the stress magnitudes. The material properties in terms of temperature and moisture content have been derived in the preceding chapter. The constitutive equations are given by Eq. (B.12) and can be written in matrix form as {a} = [Q]({â‚¬} - {a}AT - (P}c) (9.2) where {a} and {/3} are the vectors of thermal and moisture expansion coefficients. 9.2 Description of Study Cases The considered stacking sequence is the [^O/OJgjg lay-up. The cross-ply laminate is preferred over other laminate since hygrothermal loadings induce very high stresses in this case. The volume fiber fractions of the Glass/epoxy and the Graphite/Epoxy are 0.5 and 0.7, respectively. The thickness and the width of the laminates are assumed to be 2 mm and 20 mm, respectively. Three cases of moisture gradients are applied. They are described in Figure 9.2 and Table 9.1. Cases A and C correspond to the dry and moisture saturated states, respectively. While the non-uniform moisture gradient (case B) corresponds to a moisture profile as derived in section 2.3. Two uniform temperatures (20Â°C and 80Â°C) are 81 used. All laminates are assumed to be initially (dry at 20Â°C) free of stress. Hence, residual stresses are not taken into account. The elastic moduli used in computing the stresses are approximated by the real parts of the complex moduli. Therefore, the hygrothermal effects on the elastic properties can be deduced from the results given in Chapter 8 9.3 Numerical Results and Discussion For all considered cases, the following remarks can be drawn: at z/h = constant, the stresses away from the free edge stay constant and the shear stress (CTyz) zero, but, as y/b approaches 1, non-zero values and there are small variations in the a takes significant yz values of the other stresses. Hence, the stresses a , a y z and a are plotted across the section of the laminate at x y/b = 0.472 and the shear stress a is plotted across yz the section at y/b = 0.993 (close to the free edge). The stresses are compared to typical strengths of Glass/Epoxy and Graphite/Epoxy that are provided in Table 9.2. 9.3.1 G1 as s/Epoxv The equilibrium moisture concentration, c , of the Glass/Epoxy material is 0.025. 82 The stress a is plotted in Figure 9.3. It reaches y a maximum magnitude of 166 MPa. for case C at 20Â°C. It is compressive for the 0Â° layer and tensile for the 90Â° layer. The stress a is shown in Figure 9.4. It is compressive everywhere and reach a magnitude of 2S8 MPa. for the case C at 20Â°C. The stress ct^ is also compressive (Figure 9.5) and reaches a maximum of 245 MPa. . The free edge shear stress Â°yz (Figure 9.6) is very significant since its maximum magnitude is about 80 MPa.. 9.3.2 Graphite/Epoxy The equilibrium moisture concentration cm for these cases is 0.015. The stresses o , o , a , and a are y z x yz plotted in Figures 9.7-10. These results show the same trend as for the Glass/Epoxy cases. However, since the moisture concentration is lower and graphite fibers have stiffer moduli and lower coefficient of thermal expansion, the magnitude of the stresses is smaller. 9.3.3 Summary The hygrothermal conditions used in the preceding sections are practically achieved only under very adverse conditions. Hence, the induced stresses can be considered S3 an upper bound for hygrothermal stresses. The results yield the following observations: 1) The stresses induced by temperature only (dry at 80Â°C) are much smaller than those induced by high moisture content. 2) The stresses due to a non-uniform moisture gradient can be as high as those induced by the saturated mo isture case. 3) Since the hygrothermal conditions degrade the modulus of the epoxy matrix, the stresses caused by the most severe hygrothermal condition (moisture case C at 80Â°C) are lower than for some of the other cases. 4) The hygrothermal stresses of the cross-ply laminates are very significant since their magnitude is of the same order of those of the strengths given in Table 9.2. S4 Table 9.1 Description of cases in Figure 9.2. Case Descrip tion A Dry B (M - M )/(Ho- M.) = 0.5 (absorption cycle) C (M - M.)/(Mffl- M.) = 1.0 (fully saturated) Table 9.2 Typical strengths Graphite/Epoxy of Glass/Epoxy S t reng ths G1as s/Epoxy Graphite/Epoxy (MPa.) Xt 1000.0 1200.0 X 1000.0 700.0 C Y t 30.0 o o Y 140.0 70.0 c S 40.0 70.0 and X (^c) = Longitudinal strength in tension (compression) Y^ (Yc) = Transverse strength in tension (compression) S Shear strength. S5 Finite element mesh of shaded area Fig. 9.1 Geometry of a laminate and finite mesh of a 1/4 cross-section area. 86 Moisture case A Moisture case B Moisture case C Fig. 9.2 Description of the applied moisture gradients. Stress (MPa.) 87 z/h â€”*â€” A at 80 Â°C --O- B at 20 Â°C â€¢â€¢â€¢Qâ€ B at 80 Â°C â€” Â¿r- C at 20 Â°C â€”â– â€” C at 80 Â°C Profile of the hygrothermal stress a across y a [(90/0)2] Glass/Epoxy laminate at y/b = 0.472. Fig. 9.3 Stress (MPa.) ss â€”A at 80 Â°C - -O- - B at 20 Â°C â€¢ â€¢ -a- â€¢ B at 80 Â°C â€”C at 20 Â°C â€”â€¢â€” C at 80 Â°C Fig. 9.4 Profile of the hygrothermal stress across a [(90/0)2]s Glass/Epoxy laminate at y/b = 0.472. 89 .. .0.. â€”Aâ€” A at 80 Â°C B at 20 Â°C B at 80 Â°C C at 20 Â°C C at 80 Â°C Fig. 9.5 Profile of the hygrothermal stress across a [(90/0)2] Glass/Epoxy laminate at y/b = 0.472. 90 â€”*â™¦â€” A at 80 Â°C --o-- B at 20 Â°C B at 80 Â°C â€”Aâ€” C at 20 Â°C â€”â€¢â€” C at 80 Â°C Fig. 9.6 Profile of the hygrothermal stress a across yz a [(90/0)2]s Glass/Epoxy laminate at y/b = 0.993. 91 A at 80 C --o-- B at 20 C â€¢â€¢â€¢&â€¢â€¢ B at 80 Â°C â€”Aâ€” C at 20 Â°C C at 80 Â°C Fig. 9.7 Profile of the hygrothermal stress aacross a [(90/0)2ls Graphite/Epoxy laminate at y/b = 0.472. 92 â€”+â€” A at 80 Â°C --o-- B at 20 Â°C â€¢â€¢â€¢a-- B at 80 Â°C -A- C at 20 C C at 80 C Fig. 9.8 Profile of the hygrothermal stress a across z a [(90/0)<-,]s Graphite/Epoxy laminate at y/b = 0.472. 93 --O-- â€”Aâ€” A at 80 B at 20 B at 80 C at 20 C at 80 Fig. 9.9 Profile of the hygrothermal stress across a [(90/0)2] Graphite/Epoxy laminate at y/b = 0.472. 94 - -O- - . . .Q. . â€”Aâ€” Fig. 9.10 Profile of the hygrothermal stress a [(90/0)2^ Graphi te/Epoxy y/b = 0.993. A at 80 Â°C B at 20 Â°C B at 80 Â°C C at 20 Â°C C at 80 Â°C a across yz laminate at CHAPTER 10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES 10.1 In t roduction The hygrothermal effects on the in-plane (A_) and the bending (D_) complex stiffnesses of Glass/Epoxy and Graphite/Epoxy angle-ply laminates are investigated. The applied moisture gradients are the cases A, B, and C that are given in section 9.2 and the uniform applied temperatures are 20Â°C and 80Â°C. The material properties in terms of moisture content and temperature have been determined in Chapter 8. The fibers properties are given in Table 7.2. The theoretical expressions of A and D. . in terms of the fibers and matrix properties have ij been developped in Chapter 5 and Appendix A. 10.2 Numerical Results and Discussion 10.2.1 G1 a s s/Epoxy The fiber volume fraction of the Glass/Epoxy is 0.5 and the equilibrium moisture content cm is 0.025. Since 95 96 the fibers do not absorb any moisture, the equilibrium moisture concentration of the matrix is 0.05. The thickness x x of the laminate is 2.0 mm and A. . and D. . are normalized i J i J with respect to 75.86 x 10 N/m and 25.29 N.m, respect ively. The real part of the longitudinal in-plane complex stiffness, Aj^, and its corresponding damping, are plotted in Figure 10.2. As the moisture gradients and the temperature change, the relative changes of Ajj vary from 6% (for Â±0 = 0Â°) to 33% (for Â±0 = 90Â°). The line style legend of the figures of this chapter is defined in Figure 10.1. Similarly, A^ and 1^12 are Plotted in Figure 10.3 and A â€™ and DO lV66 in Figure 10.4. All these cases show that the hygrothermal effects on A. . is matrix dominated. 1 J The real part of the bending stiffnesses, D ^ ^ . and their corresponding damping, -17. ., are plotted in Figures r 1J 10.5-7. These results yield similar conclusions to those of A* . . 1 J 10.2.2 Graphite/Epoxy The volume fraction of the Graphite/Epoxy laminate is 0.7 and its equilibrium moisture concentration cm is 0.015. Since all the moisture is absorbed by the matrix , cro of the epoxy is 0.050. The laminate is 2.0 mm thick. 97 The real parts of the complex stiffnesses (A^. an<^ Djj) and their corresponding damping (j^ij an<^ F^ij^ are plotted in Figures 10.8-13. The terms A! . and D; . are i j i j 0 normalized with respect to 311.3 x 10 N/m and 103.77 N.m, respectively. These results show the same tendency as those of Glass/Epoxy. Since the volume fraction and the longitudinal modulus of the graphite fibers are higher, the hygrothermal effects are less pronounced. 10.2.3 Summary The effects of moisture on the stiffnesses A.,â€™s and i J D. ,â€™s of composites at room temperature are negligible for all the considered cases. But, as temperature increases, the combined influence induces significant changes in the complex stiffnesses especially for the matrix dominated terms. 9S Fig. 10.1 Line Moisture gradient Moisture gradient Moisture gradient Moisture gradient Moisture gradient Moisture gradient style 1 egend o f case A at o o o CNj case B at 20 Â°C case C at 20Â°C case A at 80 Â°C case B at 80 Â°C case C at 80 Â°C Figures 10.2-13. Damping Real pQrÂ¿ Qf complex stiffness 99 (a) (b) Fig. 10.2 Complex in-plane stiffness of Glass/Epoxy. a) Non-dimensional Real part ; b) corresponding damping. Real part of complex stiffness 100 (a) Fig. 10.3 Complex in-plane stiffness of a) Non-dimensional Real part; b) damping, Glass/Epoxy. corresponding part of complex stiffness 101 (a) (b) 10.4 Complex in-plane stiffness Agg of a) Non-dimensional Real part; b) damping. Glass/Epoxy. co r r e sponding Fig . Real part of complex stiffness 102 (a) 0.02 cn 0.015 c 'cl a 0.01 a 0.005 0 0 15 30 45 60 75 90 (b) Fig. 10.5 Complex bending stiffness of a) Non-dimensional Real part; b) damping. Glass/Epoxy. c o r r e sponding Real part of complex stiffness 103 (a) (b) Fig. 10.6 Complex bending stiffness of a) Non-dimensional Real part; b) damping. Glass/Epoxy. corresponding Real part of complex stiffness 104 (a) Â±0 (b) Fig. 10.7 Complex bending stiffness D* of 66 a) Non-dimensional Real part; b) damping. Glass/Epoxy. corresponding Real part of complex stiffness 105 (a) (b) Fig. 10.8 Complex in-plane stiffness Graphite/Epoxy, a) Non-dimensional Real b) corresponding damping. of par t ; Real part of complex stiffness 106 (a) Fig. 10.9 Complex in-plane stiffness of Graphite/Epoxy, a) Non-dimensional Real part; b) corresponding damping. Real part of complex stiffness 107 (a) (b) Fig. 10.10 Complex in-plane stiffness of bo Graphite/Epoxy, a) Non-dimensional Real part; b) corresponding damping. Real part of complex stiffness 10S (a) (b) Fig. 10.11 Complex bending stiffness D*j of Graphite/Epoxy, a) Non-dimensional Real part; b) corresponding damping. Real part of complex stiffness 109 (a) (b) Fig. 10.12 Complex bending stiffness Graphite/Epoxy, a) Non-dimensional Real b) corresponding damping. o f par t; Real part of complex stiffness 110 (a) (b) Fig. 10.13 Complex bending stiffness D* bo Graphite/Epoxy, a) Non-dimensional Real b) corresponding damping. o f par t ; CHAPTER 11 CONCLUSION Theoretical and experimental methods have been incorporated in order to determine the effects of temperature and absorbed moisture on the complex moduli of composite materials. The effects of hygrothermal loadings and of the changes of the complex moduli on the stress field and on the structural damping of composite laminates are also analyzed. The Fickian theory of mass diffusion has been used for analyzing the diffusion of moisture. The Fickian diffusion is adequate for the experimental determination of moisture diffusion through unstressed test specimens. Therefore, theories that incorporate the coupling of moisture diffusion with stress, viscoelastic relaxation, entropy inequality, etc. [43-45], have not been included. The complex moduli of unidirectional composites are expressed in terms of the constituent material proper-ties in Chapter 3 by using the following steps: - first, the elastic moduli of unidirectional composites in terms of the fibers and epoxy properties are 112 obtained by using the rule of mixture and the Halpin-Tsai equa tions, - then, the correspondence principle is applied to determine the complex moduli. Equations (3.20) show that in order to determine the hygrothermal effects on the complex moduli of composites, the effects on eight distinct parameters need to be assessed. But, these effects on the fibers properties are negligible. Hence, only three terms, E^(T,c), tj^CT.c) and u^(T,c), have to be experimentally measured. The necessary test procedures are descibed in Chapter 6. The complex moduli of epoxy in terms of temperature and moisture content are presented in Chapter 8. The storage moduli, E^, is strongly dependent on temperature and moisture content. But, n and v' stay constant up to a moisture content M = 4.5% and a temperature T = 80Â°C. More severe hygrothermal conditions could not be reached in the environmental chamber due to operating temperature limitations of the chamber and of the motion and force transducers used in the impulse hammer technique. The experimental results of the epoxy properties in combination with the micromechanics formulas of Chapter 3 are used to determine the hygrothermal effects on the complex moduli of unidirectional Glass/Epoxy and Graphite/Epoxy laminates. It is shown that only the matrix 113 dominated terms (^-Â¿2 anc* ^12^ are strongly affected by temperature and/or moisture. Hygrothermal conditions and changes in the complex moduli influence the stress field and the structural damping of laminated composites. The stresses induced by hygrothermal conditions in Glass/Epoxy and Graphite/Epoxy cross-ply laminates are illustrated in Chapter 9. The [^O/O^jg lay-ups are chosen since they display the highest hygroscopic stresses. The induced stresses in the 90Â° layers are of the same order of magnitude as the strengths. Hence, high loadings induced by temperature and moisture content can lead to failure. The complex stiffnesses and structural damping of [(iGJ^jg Glass/Epoxy and Graphite/Epoxy laminates are analyzed in Chapter 10. These results display the same general trend as those of the complex moduli of unidirectional composites. That is, only the matrix dominated terms are strongly affected by moisture and/or temperature. The important conclusions of this investigations are: â€¢ hygrothermal effects are very significant for the matrix dominated properties only; â€¢ for certain laminates, severe hygrothermal stresses alone can lead to failure. Therefore, hygrothermal effects need to be taken into account when designing composite structures that are subjected to moisture and/or tempera ture. 114 Recommendations and additional remarks that arose from this investigation are listed below. â€¢ The results and conclusions could vary among different material systems. Hence, it might be necessary to repeat the tests and the methodology for different materials. â€¢ The experimental results are valid up to a 80Â°C temperature and a 4.5% moisture content. In order to determine the properties of epoxy through a wider range of temperature (from - 50Â°C to 200Â°C) and moisture content, an environmental chamber and transducers that can operate under more severe hygrothermal conditions need to be used. APPENDIX A COMPLEX STIFFNESSES OF COMPOSITES A.1 Elastic Stiffnesses The in-plane, coupling and bending stiffnesses of a general laminated elastic composite plate (Figure A.l) are given by A. . i J B. . i J D. . 1 J w-h/2 Q. . dz i J -h/2 p+h/2 zQ. . dz i J -h/2 P+h/2 z^Q. . dz 1 J J -h/2 (A.l) where the components of the transformed matrix [Q] ill ^12 ^16 Q12 ^22 5.26 Q16 Q26 Q66 (A.2) 115 116 are given by Q11 = Qllm + 2^Q12 + 2Q66^m n + Q22n Q12 = ^Qll + Q22 ~ 4Q66^m n + Q12^m + n ^ Q22 = Qnn4 + 2(Q12 + 2Q66)m2n2 + Q^m4 (A.3) Q16 = CQx x ~ Q12)m3n + (Q12 - Q22)mn3 + 2Q66(m2- n2)mn Q26 = (^n â€œ Q12)mn3 + (Q12 - Q22)m3n + 2Qg6(m2 - n2)mn Q66 = ^Qll + Q22 ~ 2Q12 ~ 2Q66^m n +-Q66^m â€ n ^ and m = cos 0 n = sin 0 The angle 0 represents the fibers orientation of the lamina under consideration. The parameters Q.^.â€™s can be expressed in terms of the longitudinal (E^) and transverse (E22) Young modulus, the shear modulus (Gj2) and the major Poissonâ€™s ratio (u^2) 117 Q11 = Ell/â€˜1 â€œ u12"21> Q12 - "l2Ell/(1 ' â€œ12U21> Q22 = E22/^1 - u12y21^ (A-4) â€œ21 â€ v12E22/E11 The parameters E^, ^22â€™ *^12 an<^ v 12 are obtained as functions of the properties of the constituent materials (fibers and matrix) by using Eqs. (3.5)-(3.10). A.2 Complex Stiffnesses The laminate complex stiffnesses are determined by carrying the following steps i)The elastic-viscoelastic correspondence principle is applied to Eqs. (3.5)-(3.10) ii)The resulting complex values replace their corresponding elastic modulus in Eqs. (A.4) iii)Consequently, the complex stiffnesses Q_â€™s are obtained and substituted for their correspondig elastic stiffnesses in Eqs. (A.3) 118 Qu Th i s iv) Finally, the complex transformed s are substituted in Eqs. (5.1). procedure is easily executed with a FORTRAN tiff ne s se s program. APPENDIX B DEVELOPMENT OF THE FINITE ELEMENT METHOD The finite element displacement method used to derive the stress field in the cases of Chapter 9 is presented in this Appendix. The F.E.M presented below is used only to give a first approximation of the hygrothermal stress magnitudes in laminated composites. In order to obtain more accurate results, another method that takes into account the zero-stress boundary conditions on the free surfaces should be applied. B. 1 Equilibrium Equations The stress-strain relation of a linear elastic solid continuum undergoing any type of loading can be written in the f orm W = [D]({â‚¬} - {â‚¬Â°}) + {aÂ°} (B.l) where [D] is the elasticity matrix, {â‚¬} is the strain vector, {â‚¬Â°} is the hygrothermal and/or initial strain vector, {ctÂ°} is any initial stress vector. 119 120 In this method, it is assumed that the displacements have unknown values only at the nodal points. The displacements are wher e n {6} = [N]{66} = l [N.]{6.} i = 1 ( B . 2 ) [N] = [N x, N2. ,Nn] [Ni] = N.[I] and n is the number of variables per node and [I] is the identity matrix. The strains in terms of the displacements are given by the following expression tr dv y dy J dvt Z dz nr dv dw yz dz + dy n = E B ] { <5e } = l [B.]{6.} ( B . 3 ) i = 1 where the strain matrix is defined as 121 [Bi] dN . i 3y 0 dN . i dz 0 dN dz dN 3 y (B.4) The set of functions are called the shape functions and are subsequently defined. The continuum is subdivided into a finite number of elements. Consider an element that is acted upon by nodal forces {F } and body forces {p}. Then, application of the virtual work principle to an element e yields [de]T{Fe} + [6]T{p}dV = J V e [â‚¬]T{a}dV J V e (B.5) Substitution of Eqs . (B.1)-(B.4) into Eq. (B.5) results in { F e } + [N]T{P}dV = [B]T[D][B]dV V e . V V e o r [B]T[D]{â‚¬Â°}dV + [B]T{aÂ°}dV (B.6) {Fe> + {Fp + {FeÂ£Q} + {F^o} = [Ke ] {6e} (B.7a) whe r e 122 [Ke] [B]T[D][B]dV J V e element stffness matrix { Fe } = P [N]â€˜{p}dV = equivalent nodal body force {FGo> [B]T[D]{eÂ°}dV J V e hygrothermal or initial strain 1oading J V e = initial stress loading These equations are valid for one element only. For the complete structure, Eq. (B.6) should be summed over all elements. It is noted that if an element is subjected to surface traction forces, {t}, then the additional equivalent nodal force {F*} = [N]â€˜{t}dS (B.7b) should be added to the left hand side of Eq. (B.7a) B.2 Program Organization Hinton and Owen [42] have developed a detailed F.E.M. code to solve isotropic beam, plane stress/strain and plate bending problems. Their procedure for the plane stress/strain has been modified so that it is adapted to 123 composite laminates that have nonconstant material properties and undergo thermal and moisture induced expansion. The F.E.M. operations are performed by modular subroutines. The general organization of these programs is shown in Figure B.l. B.3 Shape Functions. Jacobian and Strain Matrix Eight node isoparametric elements are used in the F.E.M. code. In this case, the shape functions are used to approximate both the geometry and the displacement field. The coordinates of any point of the element shown in Figure B.2 are 8 y(Â£ .t?) = ^ N . (f .T7)y . i = 1 (B.8) 8 z(f . T?) = J N . (f ,T7)z . i = l where (y. , z. ) are the coordinates of the node i and the w i i ' quadratic shape functions are defined as (1 - f)(l - T7) ( 1 + f + T7)/4 124 N2 = (1 ~f2)(l - i7)/2 N3 = (1 + f) ( 1 - v)(S - n - l)/4 N4 = (1 + f)(l - T72)/2 (B . 9) Ng = (1 + f) ( 1 + T7 )(f + T7 - l)/4 Ng = (1 - f2)(l + tj)/2 N? = (1 - f)(l + T7) (- f + tÃ - l)/4 Ng = (1 - f)(l - T72)/2 The direction of the local curvilinear coordinates f and 17 are given in Figure B.2. The displacements at any point are expressed as 8 v(f.ri) = ^ N.(f,T7)v. i = 1 (B.10) 8 w(f . T7) = ^ N. (f , T7)w. i = 1 The Jacobian matrix [J(f.h)] is necessary to derive the elemental volume dV, and area dS. It is defined as foilows 125 5N . i [J] = dy dÂ£ dz as 00 II as yi dy dz i = 1 3N . [ dr/ a T) an yi 3N . 3T*. <3N . dr) Z i (B.11) Then the elemental area dS is given by dS = dy dz = det[J] df dp Once the shape functions are chosen, the strains can be written in terms of and 6^ (Eq. (B.3)). B.4 Elasticity Matrix The stress-strain relation of an orthotropic lamina is given by â– * a X 'Qll ^12 Â°13 0 0 ^16 e X a AT - X Â¡3 c X a y Q12 ^22 ^23 0 0 ^26 e y a AT - y P c y o z Â« â€” Q13 Q23 Q33 0 0 Q36 â– â‚¬ z a AT - z /3 c ? z CT yz 0 0 0 q44 ^45 0 nr yz a xz 0 0 0 Q45 Q55 0 or xz a xy Â«16 ^26 ^36 0 0 ^66- 7 xy - a AT xy - /3 c xy J (Bâ€¢12) where the transformed stiffnesses that have not been defined in Appendix A are expressed as 126 Q13 = Q13m2 + Q23n2 G23 = Q13n2 + Q23m2 G33 = Q33 = (Q13 ' Q23)mn q44 = G23m2 + G12n2 G45 = (G12 - G23)mn G55 = G12m + G23n The stiffnesses Q. ,â€™s are i J Q13 = u21(1 + u12)E11/Q Q23 = u12 ^1 + U21^E22/Q Q33 = t1 " U12U21)E22/Q Q = 1 " 2v12v21^1 + 2v23^ ~ v23 wher e 127 r v, â€™23 V . + T) . V f 4 m f 12 m + ^4 Gâ€œ m ^4 = 3 - 4u + G /G10 m m 12 4(1 - v ) m' v 23 v m For the case of Chapter 9, Eq. (B.12) can be reduced to a y Q22 Q23 0 * * e y T & z Â» = ^23 ^33 0 < e ^ z a L yzJ 0 0 Q44 a L yzJ Q12 Q22 Q23 Q26 fa AT + X a AT + P c X Pc Ql3 ^23 ^33 ^36 < y a AT + z y pzc 0 0 0 0 a AT 1 xy + p c xy J or (B.15) W = [D1]{â‚¬> - [D2]{â‚¬Â°} The last equation is a reduced form of Eq. (B.l) with the initial stresses left out. The through the thickness stress is given by a = Q (- a AT - j3 c) + Q 10(â‚¬ - a AT - 0 c) x 1 1 v x 'x' 1 2 v y y Y + Q13(â‚¬z - azAT - pâ€žc) + Q1R(- a_,AT - /3_c) (B.16) 16v xy xy 1 28 B.5 Element Stiffness Matrix 0 A submatrix of the element stiffness matrix [K ] linking nodes i and j is evaluated from the expression â– >+1 p+ 1 J-lJ-1 [B.]T[D][B.] t de t[J ] df dp (B.17) where t is the thickness of the element under consideration. The integration is done by using a 3-point Gauss integration rule. B . 6 Equivalent Nodal Loadings B.6.1 Element Edge loadings An element edge might have both tangential and normal distributed load per unit length. Every edge of the isoparametric element has three nodal points. The values of the normal and tangential loads at each nodal point are called fp ). and (p ).. Then, the distributed loads at any point along the edge are given by * Pn 3 '(pn>i' = ) N- > Pt. L* 1 i = 1 .(Pn}i. (B.18) 129 It can be shown that the equivalent nodal forces are expressed as yi N . (p - p |f-)df iVit *n b (B.19) z . i N. (p ||- + p |r)df i v *n df *^t 1 if the loads are applied on an edge parallel to the curvilinear coordinate f. The Gaussian numerical integration is used to derive Eqs. (B.19) which are a form of Eq. (B.7b). B.6.2 Hygrothermal Loadings The equivalent nodal loadings due to hygrothermal strains are given in matrix form as [Bi]T([D ]{â‚¬Â°})dv J y 1 ^ e (B.20) where the matrices [D2] and {â‚¬Â°} are defined in Eq. (B.15). 130 B. 7 Element Displacements and Stresses The global stiffness matrix as well as the equivalent nodal loading matrix are assembled. 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Prentice-Hall, Englewood Cliff, New Jersey, 1969. 40. McKague, L., "Environmental Synergism and Simulation in Resin Matrix Composites," ASTM-STP 658, J. R. Vinson, Ed., 197S, p. 193-204. 41. Aklonis, J. J. .MacKnight, W. J. and Shen, M., INTRODUCTION TO POLYMER VISCOELASTICITY, John Wiley & Sons, New York, 1972. 42. Hinton. E. and Owen, D. R. J.. FINITE ELEMENT PROGRAMMING, Academic Press, London, 1977. 43. Aifantis, E. C. and Gerberich, W. W., "Gaseous Diffusion in a stressed-Thermoelastic Solid. I: The Thermomechanical Formulation," Acta Mechanica 28, 1977, p. 1-24. 44. Nowacki, W., "Certain Problems of Thermodiffusion in Solids," Archives of Mechanics 23, 6, 1971, p. 731-755. 45. Weitsman, Y., "Coupling of Moisture and Damage in Composites," in TWELFTH ANNUAL MECHANICS OF COMPOSITES REVIEW, Mueller. D. C.. Ed., Bal Harbour, Florida, October 1987, p 50-57. BIOGRAPHICAL SKETCH Hacene Bouadi was born in Algeria in August 12, 1954. After graduating from high school in 1972, he attended the Ecole Po1ytechni que dâ€™Alger (Polytechnic Institute of Algiers). He obtained a bachelorâ€™s degree in mechanical engineering in 1977. He was awarded a graduate scholarship by the Algerian Government which allowed him to obtain a Master of Science degree in 1979 and the Degree of Engineer in 1982 from the Aeronautics and Astronautics Department of Stanford University. Thereafter, he continued his studies toward the doctorate degree at the University of Florida. He completed his Ph.D. in December 1987 in the field of aerospace enginee ring. 137 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Charles E. Taylor x Professor of Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert E. Reed-Hill Professor Emeritus of Material Sciences and Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Apr i 1 1988 Dean, Graduate School I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Chang-T. Sun, Chairman Professor of Engineering Sciences I certify that opinion it conforms presentation and is as a dissertation for I have read this study and that in my to acceptable standards of scholarly fully adequate, in scope and quality, the degree of Doctor of Philosophy. Lawrence E. Malvern Professor of Engineering Sciences I certify tha t opinion it conforms presentation and is as a dissertation for I have read this study and that in my to acceptable standards of scholarly fully adequate, in scope and quality, the degree of Doctor of Philosophy. Martin A. Eisenberg Professor of Engineering Sciences UNIVERSITY OF FLORIDA 3 1262 08556 7880 xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EBTGX6WSB_LPAR00 INGEST_TIME 2011-11-08T18:05:01Z PACKAGE AA00004791_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |