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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 1988 ACKNOWLEDGEMENTS I would like to express my gratitude to Professor ChangT. 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. ReedHill for their helpful comments, critique, and advice. 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 Campbell's help in typing this manuscript. TABLE OF CONTENTS ACKNOWLEDGEMENTS . . LIST OF TABLES . . . LIST OF FIGURES . . . NOMENCLATURE . . . ABSTRACT . . . CHAPTERS 1 INTRODUCTION . . 1.1 General Introduction . . 1.2 Moisture Diffusion . . 1.3 Hygrothermal Effects . . 1.4 Scope and Methodology . . 1.5 Dissertation LayOut . . 2 DIFFUSION OF MOISTURE . . 2.1 Introduction . . 2.2 Fickian Diffusion . . 2.3 Fickian Absorption in a Plate . 2.3.1 Infinite Plate . . 2.3.2 SemiInfinite Plate . 2.3.3 Experimental Measurement of Moisture Content . . 2.3.4 Approximate Solutions of Moisture Content . 2.3.5 Edge Effects Corrections in a Finite Laminated Plate . 2.4 Diffusivity and Maximum Moisture Content 3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES 3.1 Introduction . . 3.2 General Theory . . 3.3 Micromechanics Formulation of elastic Moduli . . . 3.4 Complex Moduli . . Page . ii S. vi . vii . xiii S. xvi 1 1 S 2 S 2 3 S 4 S 6 S 6 S 6 8 8 10 S. 11 12 13 S. 15 S. 21 21 21 22 23 iii 4 DAMPING . 4.1 Damping Mechanisms . . 4.1.1 Nonmaterial Damping . 4.1.2 Material Damping . . 4.2 Characterization of Damping . 4.2.1 Free Vibration . . 4.2.2 Steady State Vibration . 4.2.3 Complex Modulus Approach .. 5 DAMPING AND STIFFNESSES OF GENERAL LAMINATES 5.1 Introduction . . 5.2 Laminated Plate Theory Approach . 5.3 Energy Method Approach . . 6 EXPERIMENTAL PROCEDURES . . 6.1 Introduction . . 6.2 Test Specimen . . 6.3 Environmental Conditioning . 6.4 FourPoint Flexure Test Method . 6.5 Impulse Hammer Technique . . 29 . 29 . 29 . 30 . 30 30 . 31 . 32 7 HYGROTHERMAL 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 Graphite/Epoyx .. 82 9.3.3 Summary .... 82 10 HYGROTHERMAL EFFECTS ON COMPLEX STIFFNESSES 10.1 Introduction . . 10.2 Numerical Results and Discussion . 10.2.1 Glass/Epoxy . . 10.2.2 Graphite/Epoxy . 10.2.3 Summary . . . 95 . 95 . 95 . 95 . 96 . 97 11 CONCLUSION . . 111 APPENDICES A COMPLEX STIFFNESSES OF COMPOSITES . 115 115 117 A.1 Elastic Stiffnesses A.2 Complex Stiffnesses B DEVELOPMENT OF THE FINITE ELEMENT METHOD B.1 Equilibrium Equations . B.2 Program Organization . B.3 Shape Functions, Jacobian and Strain Matrix . . B.4 Elasticity Matrix . . B.5 Element stiffness Matrix . B.6 Equivalent Nodal Loadings . B.6.1 Element Edge Loadings . B.6.2 Hygrothermal Loadings . B.7 Element Stresses . . REFERENCES . BIOGRAPHICAL SKETCH . 119 . 119 . 122 . 123 . 125 . 128 . 128 . 128 . 129 . 130 . 133 137 LIST OF TABLES Page Initial properties of Magnolia 2026 epoxy, 3M Scotchply Glass/Epoxy, and a typical Graphite/epoxy composite .. 45 Coefficients of moisture and thermal expansion of epoxy and graphite and glass fibers. . ... 54 Properties of Glass and Graphite Fibers 54 Description of cases in Figure 9.2 .... 84 Typical strengths of Glass/Epoxy and Graphite/Epoxy. . .. 84 Tables 6.1 7.1 7.2 9.1 9.2 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 ci)/(c. ci). . ... 18 2.3 Semiinfinite plate in a humid environment. . . 19 2.4 Comparison of the exact specific moisture concentration equation with some approximate solutions. . .. 19 2.5 Geometry of a plate. . .. 20 2.6 Moisture content versus square root of time. On the curve v 7< VNT< vTL and the slope 1 2 L is constant for vt < . 20 4.1 Schematic drawing of a freeclamped beam under free vibration and plot of its deflection versus time .. 35 4.2 Schematic drawing of a freeclamped 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 4point flexure test. . . 46 vii 6.3 Schematic 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 Glassfiber/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 Glassfiber/Epoxy at 20C. .. 57 7.5 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 200C. . 57 8.1 Schematic variation of the storage modulus of epoxy with temperature .. 67 8.2 Schematic variation of Poisson's ratio of epoxy with temperature .. 67 8.3 Schematic variation of damping of epoxy with temperature . .. 68 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 T )/(T T ). 70 8.8 Experimental data of damping of epoxy as a function of temperature at diverse constant moisture contents .. 71 viii 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 T )/(T T ). 72 8.11 Experimental data of Poisson's ratio of epoxy in term of temperature ... 73 8.12 Experimental data of Poisson'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 o g o 8.14 Longitudinal storage modulus (E1 ) of Glass/Epoxy versus T = (T T )/(T T ). 75 n o g o 8.15 Transverse (E 2) and shear (Gi2) storage moduli of Glass/epoxy versus T = (T T )/(T T ) . .75 n o g o 8.16 Longitudinal (R711), transverse ('722). and shear (nG) damping of Glass/Epoxy versus T = (T T )/(T T ) .. 76 n o g o 8.17 Poisson's ratio (v12) of Glass/Epoxy versus T = (T T )/(T T ) .. 76 n o g o 8.18 Longitudinal storage modulus (E11) of Graphite/Epoxy versus T = (T T )/(T T ) 77 8.19 Transverse (E22) and shear (G'2) storage moduli of Graphite/epoxy versus T = (T T )/(T T ) . 77 n o g o 8.20 Longitudinal (TI11), transverse (7722)' and shear ('G) damping of Graphite/Epoxy versus T = (T T )/(T T ) .. 78 n o g o 8.21 Poisson's ratio (v12) of Graphite/Epoxy versus T = (T T )/(T T ) .. 78 9.1 Geometry of a laminate and finite mesh of a 1/4 crosssection area. .. 85 9.2 Description of the applied moisture gradients. . 86 9.3 Profile of the hygrothermal stress a y across a [(90/0)2]s Glass/Epoxy laminate at y/b = 0.472 . .. 87 9.4 Profile of the hygrothermal stress a Z across a [(90/0)2 s Glass/Epoxy laminate at y/b = 0.472 . .. 88 9.5 Profile of the hygrothermal stress a X across a [(90/0)2]s Glass/Epoxy laminate at y/b = 0.472 . .. 89 9.6 Profile of the hygrothermal stress a yz across a [(90/0)2]s Glass/Epoxy laminate at y/b = 0.993 . .. 90 9.7 Profile of the hygrothermal stress a y across a [(90/0)2]s Graphite/Epoxy laminate at y/b = 0.472 . .. 91 9.8 Profile of the hygrothermal stress a z across a [(90/0)2]s Graphite/Epoxy laminate at y/b = 0.472. .. . 92 9.9 Profile of the hygrothermal stress a x across a [(90/0)2]s Graphite/Epoxy laminate at y/b = 0.472 . .. 93 9.10 Profile of the hygrothermal stress a yz across a [(90/0)21s Graphite/Epoxy laminate at y/b = 0.993 . 94 10.1 Line style legend of Figures 10.213. ... 98 10.2 Complex inplane stiffness Al of Glass/Epoxy. a) Nondimensional Real part; b) corresponding damping. . 99 10.3 Complex inplane stiffness A2 of Glass/Epoxy. 12 a) Nondimensional Real part; b) corresponding damping .. 100 10.4 Complex inplane stiffness A66 of Glass/Epoxy. a) Nondimensional Real part; b) corresponding damping. .. 101 10.5 Complex bending stiffness D1 of Glass/Epoxy. a) Nondimensional Real part; b) corresponding damping .. 102 10.6 Complex bending stiffness D12 of Glass/Epoxy. a) Nondimensional Real part; b) corresponding damping . 103 10.7 Complex bending stiffness D66 of Glass/Epoxy. 66 a) Nondimensional Real part; b) corresponding damping .. 104 10.8 Complex inplane stiffness Al of Graphite/Epoxy. 11 a) Nondimensional Real part; b) corresponding damping .. 105 10.9 Complex inplane stiffness A2 of Graphite/Epoxy. 12 a) Nondimensional Real part; b) corresponding damping ... 106 10.10 Complex inplane stiffness A66 of Graphite/Epoxy. a) Nondimensional Real part; b) corresponding damping. . 107 10.11 Complex bending stiffness D1 of Graphite/Epoxy. 11 a) Nondimensional Real part; b) corresponding damping. . ... 108 10.12 Complex bending stiffness D12 of Graphite/Epoxy. 12 a) Nondimensional Real part; b) corresponding damping. . 109 10.13 Complex bending stiffness D66 of Graphite/Epoxy. a) Nondimensional Real part; b) corresponding damping .. 110 B.1 Organization of the F.E.M. program. B.2 Local axes f and n, Gauss point numbers and local node numbers of an eightnode isoparametric element . . 131 132 xii NOMENCLATURE A complex inplane stiffness. ij B.. complex coupling stiffness ij B complex modulus B' storage modulus B" loss modulus c moisture concentration c average specific moisture c equilibrium moisture concentration C specific heat v D.. complex bending stiffness ij D Dxx moisture diffusivities [D] diffusivity matrix, elasticity matrix E11 longitudinal Young modulus E22 transverse Young modulus G12 inplane shear modulus K thermal conductivity x m weight of absorbed moisture M percent moisture content xiii M. initial percent moisture content 1 M equilibrium percent moisture content Q.i transformed stiffness Qij complex transformed stiffness s specific gravity t time T temperature vf fiber volume fraction v matrix volume fraction m w weight a. coefficient of thermal expansion 1 .i coefficient of moisture expansion E strain T damping or loss factor v12 major Poisson's ratio 8. fiber orientation of jth layer p density a stress Subscripts 1. 2. 3 principal directions of the fibers f fiber i initial j layer number xiv L m x, y, z a longitudinal direction matrix Cartesian coordinates maximum or equilibrium Superscripts moisture initial transpose, thermal complex value real part imaginary part 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 April 1988 Chairman: Dr. ChangT. 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 negligible. xvi 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. xvii 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 1960's was so vast that Broutman and Krock [1] needed eight volumes to edit a summary of the resulting knowledge. 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. 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 Fick's law [410]. 1.3 Hygrothermal Effects The degradation of mechanical properties of glass reinforced plastics exposed to water has long been 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, 1114], fatigue behavior [1517], creep, relaxation, viscoelastic responses [1820], 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 database. 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 following steps: 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 LayOut Right after the introduction, the mechanism of moisture diffusion is described in Chapter 2, where the absorption of moisture through thin composite laminas is 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 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 710. 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 Appendix B. CHAPTER 2 DIFFUSION OF MOISTURE 2.1 Introduction The mechanism of moisture absorption and desorption in most fiber reinforced composites is adequately described by Fick's law [4]. Fick recognized that heat transfer by conduction is analogous to the diffusion process. Therefore, he adopted a mathematical formulation similar to Fourier's heat equation to quantify the diffusion process [24, 25]. 2.2 Fickian Diffusion The Fourier and Fick's equations, describing the onedimensional temperature and moisture concentration, are respectively given by BT 6 aT] C OT K, K (2.1) Pv at ax x jx ac [ c 1 O Dax (2.2) where is the dens t ia x is the where p is the density of the material, C is the v specific heat, T is the temperature, t and x are the time and spatial coordinates, respectively, K is the thermal x conductivity, c is the moisture concentration, and D is x the moisture diffusivity. The moisture diffusivity, Dx, 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 ac c t D a2c (2.3) at x 2 and is solved independently of Eq (2.1). The threedimensional diffusion in an anisotropic medium is obtained by generalization of Eq (2.2) as follows Oc at = v.([D].vc) (2.4) where the diffusivity matrix is D D D xx xy xz [D].= Dyz Dyy D (2.5) yz yy yz D D D zx zy zx Expansion of Eq (2.4) results in an equation of the form 2 2 2 8c 2c a2c c SD + D + D c + (D at xx ax2 yy 2 zz a2 yz 2 +(D + D ) z + (D x +(Dzx xz Ox oz xy 2 + D ) ac zy Oy 6z 2 + D ) a c yx ax oy if the coefficients D..'s are considered to be constant. 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 = T. c for 0 < z < h and t < 0 c = Ci (2.7) (2.6) 9 T = T. ST for z = 0, z = h and t > 0 c = c J where T. is a constant temperature, c. is the initial 1 1 moisture concentration inside the material, and c. is the maximum moisture concentration. It is assumed that the moisture concentration on the exposed sides of the plate reaches c instantly. The solution of Eq (2.3) in conjunction with the conditions of Eqs. (2.7) is given by Jost [25] c ce2 c c 4 V 1 1 i+l (2j+l) 2 = 1 1 }in21z exp 2j+1)U2D t co ci c (2j+1) h h2 IT z j=o (2.8) Equation (2.8) is plotted in Figure 2.2. The average moisture concentration is given by 1 c = c dz (2.9) h o Substitution of Eq. (2.8) into Eq. (2.9) and integration result in c c. D t 1 = 1 exp (j+)2 2 z c i j=0 (2j+1) h (2.10) This analysis can be applied to the case of diffusion of moisture into a laminated composite plate so thin that moisture enters predominantly through the plane faces. 2.3.2 SemiInfinite 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 semiinfinite halfplane 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 = T.] 1I for 0 < z < and t < 0 c = c. 1 (2.11) T = T i for z = O and t > o c = cJ The solution of Eq (2.3) in this case is [24, 25] c c. e r z ] 1 1 erf z (2.12) c ci The rate at which the total specific mass of moisture, m, is diffusing into the halfplane is dm [ac dt z z= (2.13) Thus, the total mass of moisture entering through an area A in time t is m= pADz dt = 2pA(c c. ) Jo z= 0 (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 content 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, wd. from the current weight, w, of the plate, i.e. m = w wd (2.16) A parameter of practical interest is the percent moisture content defined as w wd M = 100 (2.17) Wd Since M M. c. 1 1 M = lOOc (2.18) M0 M. c0 c. 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. Small 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. D t ci _=4 z (2.19) cw ci M0 M. 2 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., c c. D t 1 8 2 2z_ S1  ep I (2.20) c ci 1 2 h 2 Shen and Springer formulation. These researchers have derived in Ref. [4] the following approximation c C. D t 0.75 c 1 exp 7.3 (2.20) c ci h 2 Figure 2.4 shows a comparison of Eqs (2.1921) with the exact solution. 2.3.5 EdgeEffect 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 c ) bL /Dz + bh /DJx+ hL /JDy/T7i (2.22) where Dx. D and D are the diffusivities in the x, y, and 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 Dt M = 4MJ D2 (2.23a) 2h where the effective diffusivity D is D = D 1 + h + h (2.23b) The micromechanics formulation for diffusivities proposed by Shen and Springer [4] and modified by Hahn [26] for impermeable, circular crosssection, fiberreinforced composites is DL = Dm DL m (2.24) DT = 1 2 D T ~ v where D D and D are the matrix, transverse, and m T L longitudinal diffusivities, respectively. Equation (2.23b) for a unidirectional lamina with all fibers parallel to the xdirection can be written as D = D 1 + + m (2.25) z + 1 2 _f For a general laminated plate consisting of N layers with fiber orientations 8., the diffusivities are J Dz = DT N N D, h.cos2. + DT h.sin 2 DL Z j 2j jT Z J D= Nj=N j=l (2.26) h. j=l N N 2 2 DL 2 h.sin 9. + DT h cos 28 L ZJ J T J D j=l j=l y N hI j=l where h. is the thickness of the jth layer. The effective 3 diffusivity of a general laminate is obtained by substituting Eqs. (2.26) into Eq (2.23b). 2.4 Diffusivity and Maximum Moisture Content The diffusivity D and the maximum moisture content M 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, 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, q/ 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 M D = 4M (2.27) t/2 ./ t Hh2 The subscripts 1 and 2 are defined in Figure 2.6. The diffusivity depends only on the material and temperature as follows D =D exp (2.28) z 0 RT where R is the gas constant, Do and Ed are the preexponential factor and the activation energy, respectively. 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 = ab (2.26) 17 where is the relative humidity, a aad b are material constants. Moisture  h ^s^~ Moisture c 4 Fig. 2.1 Plate subjected to a constant humid environment on both sides. 1 I 8 . 0.8 0.4 I 01 0.2 .3 0.4 0~J z = 0 is the center of the crosssection of the plate z/h Fig. 2.2 Moisture distribution across numbers on the curves are (c ci)/(cm ci). a plate. the values The of x Moisture Dow Dow) Fig. 2.3 0.8 0. 0 Semiinfinite plate in a humid environment. Exact  Oneterm S* * Shen and Springer Vh2 Fig. 2.4 Comparison of the exact specific moisture concentration equation with some approximate solutions. Fiber direction Fig. 2.5 Geometry of a plate 0 M1 77 f Square root of time Fig. 2.6 Moisture content versus square root of time. On the curve /i < / < V and the slope is constant for,/T < /t. I CHAPTER 3 COMPLEX MODULI OF UNIDIRECTIONAL COMPOSITES 3.1 Introduction Composite materials, such as Glass/Epoxy and Graphite/Epoxy, have a polymeric matrix. Therefore, they display viscoelastic behavior. Some of the effects of this timedependent 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 moduli. 3.2 General Theory A usual representation of the onedimensional stressstrain relation of a viscoelastic material subjected to a harmonic strain history of the form E(t) = e0e (3.1) is given by a(t) = B(i)EOeit = B (ie)E(t) (3.2) The complex modulus B can be decomposed into its real and imaginary parts as follows B (iw) = B'(w) + iB"(w) (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 B" S B' (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 E11, the transverse modulus E22, the inplane shear modulus G and the major Poisson's ratio v12 can be obtained by using the rule of mixtures and the HalpinTsai equations, viz., E11 = fE + v mEm 11 f f1l m m (3.5) E = E 22 m G = G 12 m 1 + 2nlvf 1 n1vf 1 + n2vf 1 n2vf 12 = Vf f12 + Vm m (Ef22/E) 1 S= (Ef22/Em) + 2 (Gf12/G) 1 2 = (Gf12/Gm) + 1 The subscripts f and respectively, and vf and m stand for fiber and matrix, v are the volume fractions. m 3.4 Complex Moduli The micromechanics formulations of the complex moduli are obtained by applying the elasticviscoelastic correspondence principle [2729], i.e., by undertaking the following steps: i) determining the elastic moduli of composites in terms of the constituent material properties. (3.6) (3.7) (3.8) where (3.9) (3.10) ii) replacing the elastic moduli of fibers and matrix by corresponding coDplex expressions. For a viscoelastic composite, the properties of the constituent materials are = E 11 + iE l 11E1 E E' + iE" E22 = f22 22 G = G + iG" f f E = E' + iE" m* m G = G" + iG" m* m D = U' + iv" 1m m a Vf 12 = fl2 (3.11) The bulk modulus of epoxy independent of frequency [28]. matrix. K is real and m It is given by E K  m 3(1 2v ) (3.12) while the viscoelastic bulk modulus is obtained from the correspondence principle E' + iE" n m m m 3[1 2(v' + iv")] (3.13) m m Separation of the real and imaginary parts of Eq. (3.13) yields S(1 2v)E 2E"m + i[2E"v" + E'(1 2v')] m m mm mm m m S 3[(1 2v)2 + 4v2 J m m (3.14) Since the dilatation bulk modulus is real, the imaginary part of Eq. (3.14) is equal to zero; hence 2E"v" + E"(l 2v') = 0 (3.15) mm m m Equation (3.15) results in E" v" (v' 0.5) (3.16) m The shear modulus of the matrix is given by E 3K E n m mm G = m (3.17a) S2(1 + v) 9K  m m m Separating the real and imaginary parts and neglecting the 2 terms of the form (E") yield m Smm I m 9K E m Introduction of 9K E" m i 9K E' E' m m m the material properties E" m 7m E' m E' fl1 4f 11 ~ E ti=t E" f22 f22 E22 G" 9K ?m m Gm G' 9K E' m m m m into Eqs. (3.11) results in = E11 ( + if 1) = E 22(1 + i" 22) E fll1 E f22 G f12 E = m = G 12(1 + 'fl12) E'(1 + iTm) m im G = G'(1 + iT ) m m Gm (3.17b) (3.18) (3.19) n 1 v = U' + iTm (v'  m m m m 2 Vfl2 = 'fl2 = f12 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 rf11' "f22o and nfl2 are assumed to be equal and are replaced by Tf in subsequent equations. Since the fiber damping, 7f, is much smaller than the matrix damping, m, the imaginary part of the fiber Poisson's ratio is neglected. The preceding assumptions have a negligible effect on the complex moduli of composites. Application of the elasticviscoelastic correspondence principle to Eqs. (3.5)(3.10) and substituting them into Eqs (3.19) yield the following complex material properties E11 = fE11(l + inf) + vmE'(1 + inm) 1 + 2n vf E22 E(1 + inm) f 22 m m n 1 nlvf 1 (3.20) 1 + n2vf G12 = G(1 + inGm) 2 1 n2vf 1 1 v12 = (Vfvfl2 + v v') + im (v )v 12 fmm m m m where SEf22(1 + iqf) Em(1 + im) n1 E 22( + f) + 2E'(1 + im) n2 Gfl2(l + i) + Gm(1 + iGm) 2 = Gj1(l + inf) + G(l + in ) The elastic experimental they are micromechanic Hashin [27. 2 moduli given results with a used instead s formulas, by good of such Eqs. (3.5)(3.8) model accuracy [2]. Therefore, mathematically exact as those derived by 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. 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 N In[A (4.1) n+N where A = amplitude of the nth cycle An+N = amplitude of the (n+N)th cycle The damping defined in Eq (4.1) is applicable to viscous 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 "2 "1 n 2 (4.2) 0 o where W = resonant frequency o 1., W2 = frequencies on either sides of o such that the amplitude is 1//"2 times the resonant amplitude. In the case of a vibration induced by the force f(t) = Fsin(wt) the response (deflection), w(t), is out of phase with f(t) by an angle e such that w(t) = Wsin(wt + e) The work done per cycle is dw D = f(t) dt = TWF sin(e) (4.3) 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= 1FW cos(a) (4.4) There is no damping if the work done per cycle is zero, i.e, if sin(a) = 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 2IU = tan(e) (4.5) The definitions of damping given by Eqs. (4.2) and (4.5) are equivalent [33]. 4.2.3 Complex Modulus Approach The onedimensional stressstrain relation of a viscoelastic material undergoing harmonic motion has been shown to be (Eq. (3.2)) iWt iwt S(t) = E(w)E e0 = (E'(() + iE"(w))E et (4.6) Noting that i[il = dE/dt, Eq. (4.6) can be written as , i t E" i t a (t) = E e + i e E a 0Fo (4.7) The real part is given by (after algebraic manipulation) a(t) = E'E sin(wt + e)1J + n2 (4.8) where n = tan(e) = E"/E' The energy dissipated during a cycle per unit volume is dE x 2 D = 0 d = dE dt = lT1E'E x fdt o (4.9) The maximum energy stored is 1 2 U = E' E 2 o (4.10) Therefore, E" D 7 E' = 2 U (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. Vibrating beam Fig. 4.1 Envelope Equation w(t) = e t Response Schematic drawing of a freeclamped beam under free vibration and plot of its deflection versus time. Bem under forced vibration Beam under forced vibration 'A aI crI/ Deflection versus time aA Amplitude versus frequency Fig. 4.2 Schematic drawing of a freeclamped beam under forced vibration and plots of the deflection versus time and deflection amplitude versus frequency. MIM_ 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 inplane stiffnesses A. 's, the complex coupling stiffnesses B..'s, and the complex bending stiffnesses D. 's (see Appendix A). The terms A .'s, B .'s, and D .'s are defined as +h/2 A. = Q. dz ij ij h/2 .h/2 ij h/2 h/2 Dh/2 D. = h/ 1 h/2 zQi. dz 1J (5.1) z2 Q dz 1j where the complex transformed stiffness Q..'s depend on E1, E22, G12. v12 and the orientation of each layer of the laminate. The inplane, coupling, and flexural material damping are defined as Al. I ij A ij B" ij (5.2) D". Fij D:. ij respectively. 5.3 Energy Method Approach The energy method can be used to determine the damping of laminated composite materials under certain loading and boundary conditions. The damping of a laminated composite material in the first mode of vibration can be defined as N I kUdcyc. k=l S= N (5.3) 21 kUs k=l where N is the total number of layers. (kUd)cyc. is the energy dissipated in the kth layer during a cycle, and kU ks is the maximum energy stored in the kth layer. The storage and the dissipated energy are given by kUs = E .C'.. dV k 2 j j1 1 k (5.4) kUd = E.C".. dV k d JV k ji I k where i and j are the material principal axes, C. and Ji C". are the real and imaginary parts of the complex ji stiffnesses and Vk is the volume of each layer. Hence, the "total" damping of an Nlayered laminate is given by N I fV {}T[C"]{} dV k = k (5.5) kV {E}T[C']{} dV k=l k 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 structures. 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 1750C and 1700C. respectively. The initial properties of these materials (at 200 C and without moisture), as well as those of a typical Graphite/Epoxy, are given in Table 6.1. 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 4 C to 90 C and the range of relative humidity is 4% to 99% for temperatures below 75 C. As temperature increases, the highest relative humidity that can be obtained decreases steadily to 75% at 900C. 6.4 FourPoint Flexure Test Method The Young's modulus and the Poisson's ratio can be determined with the fourpoint flexure test method. The loading configuration of this test is shown in Figure 6.2. The elastic flexural analysis yields [31] 3 S= P3 (6.1) 8bh w where E is the effective modulus, P is fhe applied load, 1 is the length of the specimen, b is the specimen width, h is the thickness, and w is the deflection at quarterpoint. Poisson's ratio is expressed as E 2 S = E (6.2) 6hw y x where the transverse strain E is measured with a y transverse strain gage cemented in the middle of the specimen. 6.5 Impulse Hammer Technique The material damping and the storage modulus of a onedimensional thin beam are determined with the impulse hammer technique. This technique was pioneered by Halvorsen and Brown [32]. The equipment setup 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 noncontacting motion transducer. Both responses from the force and motion transducers go through signal conditioning equipment (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 v(t)) over the Fourier Transform of the input (force impulse u(t)); that is, H(f) = (f) (6.3) U(f) where t = time f = frequency 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 expression (fa/fb)2 1 S= 2 (6.4) (f/f b) + 1 where the frequencies fa and fb are defined in Figure 6.3. The storage modulus is expressed as [33, p.464] 2 1 E' = 38.32 fr p (6.5) 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 clampedfree beam. A complete description and analysis of the impulse hammer technique are presented in Lee's dissertation [34]. Table 6.1 Initial properties of Magnolia 2026 epoxy, 3M Scotchply Glass/Epoxy, and a typical Graphite/Epoxy composite. Properties Epoxy Glass/Epoxy Graphite/Epoxy vf 0.50 0.70 p (g/cm3) 1.25 1.93 1.6 E11 (GPa.) 4.0 37.00 155.23 E22 (GPa.) 4.0 11.54 10.81 G12 (GPa.) 1.52 3.46 4.35 v12 0.32 0.285 0.217 11 0.018 0.0023 0.0019 T22 0.018 0.015 0.0078 I I Testing chamber /Environn / mental chamber Fig. 6.1 Schematic drawing of environmental and testing chambers. Fig. 6.2 Loading configuration flexure test. of the 4point MTS Ic frame Fourier Transform Real part Im. part Motion transducer Testing chamber fr Frequency Fig. 6.3 Schematic drawing of the impulse hammer technique apparatus and a typical display of the Transfer Fourier Transform. 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 onedimensional case, it is assumed that the thermal strain is given by T = a.(T T ) = a.AT (7.1) 1 1 o 1 where a. = coefficient of thermal expansion T = actual temperature T = 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 onedimensional case, the hygrostrain is given by H EH = p.(c c ) = 1.Ac (7.2) 1 1 0o 1 where c is the initial moisture concentration and 3. is 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 derivation) VfafEf + vaE a EE a2 fa + mam + fa + vm am 12al (7.3) "12 = 0 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 6 with the xdirection (Figure 2.5) are given by 2 2 a = a1 cos 8 + a2 sin 8 2 2 a = a sin 8 + a2 cos 8 (7.4) axy = 2(al a2) cos 0 sin 8 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 m 1 sE m m 11 2 = s(1 + V)m v12 P (7.5) m 12 = where s and s are the specific gravities of the composite m material and the matrix. The moisture expansion coefficients expressed in an axis system such that the xdirection makes an angle 8 with the fibers are given by Eqs. (7.4) after replacement of a.'s by Pi's. 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 S2Glass/Epoxy, Kevlar 49/Epoxy and Graphite/Epoxy can be approximated by EH 0.43c = Pfc (7.6) 2 .2 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. 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 sufficiently accurate. These data yield the following experimental values Pm(epoxy) = 0.25 (7.7) P2(Glass/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 pi = 0.042 (7.9a) p2 = 0.47 (7.9b) for the 3M Glass/Epoxy composite. The experimental and empirical values of P2 are practically equal. Hence, the 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 1000C. Therefore, in the subsequent chapters, the thermal expansion coefficients are assumed to be independent of temperature. Table 7.1 a (Wm/m)/K P Table 7.2 Ef (GPa) E 22(GPa) G 12(GPa) If vf12 p (g/cm ) Coefficients of moisture and thermal expansion of epoxy and graphite and glass fibers. Epoxy 54.0 0.25 Glass 5.0 0.0 Graphite 0.2 0.0 Properties of Glass and graphite Fibers. Glass 70.0 70.0 28.7 0.0015 0.22 2.60 Graphite 220.0 16.6 8.27 0.0015 0.16 1.75  Glass/Epoxy  4 Epoxy Fig. 7.1 Moisture concentration (%) Transverse moisture strain of Magnolia epoxy and 3M Scotchply Glass/epoxy. (D c 1 'o 0) 11 L. I  Longitudinal  transverse 0 0.2 0.4 0.6 0.8 Fiber volume fraction Fig. 7.2 Plot of the thermal expansion coefficients in terms of fiber volume fraction of a dry S Glassfiber/Epoxy at 200C.  Longitudinal  Transverse 0 0.2 0.4 0.6 0.8 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 200C.  Longitudinal  . Transverse 0.4 0.6 0.8 Fiber volume fraction Fig. 7.4 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry S Glassfiber/Epoxy at 20C. ./ 3 0.2 0.4 0.6 0.8 Fiber volume fraction  Longitudinal  Transverse Fig. 7.5 Plot of the moisture expansion coefficients in terms of fiber volume fraction of a dry Graphite/Epoxy at 200C. 0.8 0.6 0 0.2 1 0.8 0.6 0.4 0.2 0 ......,....I......... CHAPTER 8 HYGROTHERMAL EFFECTS ON COMPOSITE COMPLEX MODULI 8.1 Literature Survey The storage moduli (real parts of Eqs. (3.11)) of composites are usually determined by dynamic testing, 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 00 fiber direction laminates are negligible. ii) For 900 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 important. Putter et al. [38] investigated the influence of frequency and environmental conditions on the dynamic 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 matrixcontrolled 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 E' = E'(T,c) m m v' = v'(T,c) (8.1) m m nm = 7m(T,c) 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 Poisson's ratio, and damping of epoxy is illustrated in Figures 8.18.3. There are three distinct regions. At room temperature (in the glassy region), the storage modulus, Poisson'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 Poisson's ratio and the damping increase as temperature increases. In the next region (transition region), the storage modulus decreases rapidly, and both Poisson'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 can reach a value of 1 or even 2 in the transition region [30, p. 90]. Poisson's ratio reaches the limiting value of 0.5, which is approximated by incompressible rubbers [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 g 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 g 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 strongly dependent on absorbed moisture. These results show that, as the moisture content of epoxy These results show that. as the moisture content of epoxy increases, the transition temperature moves to the left in Figures 8.18.3. 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 77 are appropriate, m m m E' E' f o (8.2) m o T T 0 g o v = g[ o] (8.3) m o T T g o T = i o h T T (8.4) g o where the temperature T is equal to 273K. The moisture concentration appears implicitly in T The glass g transition temperature is represented by T = 210 exp( 9c) (C) (8.5) g where c is the moisture concentration. This modeling has been chosen so that it does not represent the material properties beyond T since the g study of epoxy in the rubbery stage is not within the scope of this research. Equations (8.2)(8.4) are valid only for the continuous parts of the curves plotted in Figures 8.18.3. . 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 fourpoint flexure tests to determine the storage moduli, the material damping, and Poisson'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 that increase in either temperature or moisture content or both results in a decrease in the storage modulus. Plotting 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 nondimensional temperature T T T T T g o 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) (8.6) m n Material damping. Similarly, the experimental data of the hygrothermal effects on the damping of epoxy are plotted in three Figures (8.88.10). There is very little change in damping for all the considered conditions. Therefore, it is proposed to let Tm = 0.018 (8.7) for temperatures up to 80 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. Poisson's ratio. The experimental values of the Poisson's ratio in terms of temperature at two different moisture contents, are plotted in Figure 8.11. These results show that Poisson's ratio increases at a negligible rate as temperature varies from 0 to 80 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, v' = 0.32 (8.8) m for temperatures up to 80 C and moisture contents up to 5%. Since v' equals 0.5 at the glass transition temperature m (Tn = 0), the plot of Poisson'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 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. Glass/Epoxy. The parameters E'll E'22' G12' 11 T22' G, and v12 versus the normalized temperature are plotted in Figures 8.1417. The experimental data substantiate the theoretical results. Graphite/Epoxy. Similarly, E'22, GI2 711 T22' T), and vu2 versus the normalized temperature of Graphite/Epoxy are plotted in Figures 8.1821. For both Glass/Epoxy and Graphite/Epoxy, the results show that the matrixdominated parameters (E'2 and G12) are strongly affected by moisture and temperature, while the fiberdominated parameters (E11' v12) stay practically constant. Temperature Fig. 8.1 Schematic variation of the storage modulus of epoxy with temperature. _J Transition Glassy region region Rubbery region I I Temperature Fig. 8.2 Schematic variation of Poisson's ratio of epoxy with temperature. Transition region Glassy region I Rubbery region I "s T 1 I I m r S I Temperature Fig. 8.3 o 250 S200 E 150 100 50 : 50 c, 03 Fig. 8.4 Schematic variation of damping of epoxy with temperature. L 0 2 4 6 8 Moisture concentration x 100 (%) Glass transition temperature of epoxy. From Delasi and Whiteside [6]. 5 O4.5 a 4 0 0 3.5 O3 0 o, 2.5 2 SM = 0.0% * M = 2.90% 0 M = 3.70% Fig. 8.5 Experimental data of the storage modulus of epoxy as a function of temperature at diverse constant moisture contents. 5 a0 i3 ! 0 E 3.5 So 0' . 3 o 2.5 2 ) * T = 20 oC 0 T = 50 C O T = 70 C 1 2 3 4 Moisture content (%) Fig. 8.6 Experimental data of the storage epoxy as a function of moisture diverse constant temperatures. modulus of content at 0 0 20 40 60 80 1 Temperature (oC) . * 8 0* B I21 o 4 C0 E 03 1 0 0.2 0.4 0.6 0.8 Normalized temperature Fig. 8.7 SExperimental data  Fit to data Experimental data of the storage modulus of epoxy as a function of normalized temperature (T T )/(T T ) o g o  * 8 A  0 20 40 60 80 1 Temperature (oC) A M 0. SM U 2.90X 0 M 3.702 Fig. 8.8 Experimental data of damping of epoxy as a function of temperature at diverse constant moisture contents. 0.03 g, 0.025 S0.025 0.015 0.015 0.005 0 * T 20 OC 0 T 50 C 0 T 70 C 0 1 2 3 4 Moisture content (X) Fig. 8.9 Experimental data of damping of function of moisture content constant temperatures. epoxy as a at diverse 0.03 o 0.025 c S0.02 0.015 0.01 0.005 0 *  I. 0.03 0.025 0.02 0.015 0.01 0.005 0 I 0 0.2 0.4 0.6 0.8 1 Normalized temperature * Experimental data Fig. 8.10 Experimental data of the storage modulus of epoxy as a function of normalized temperature (T T )/(T T ) o g o 0 20 40 60 80 Temperature (OC) * U 0% 0 M 4.17% Fig. 8.11 Experimental data of Poisson's ratio of epoxy in term of temperature 0.2  0 1 2 3 4 Moisture content (%) * T = 20C 0 T = 50'C ST = 75C Fig. 8.12 Experimental data of Poisson's ratio of epoxy in term of moisture content. : * * 0 0.2 0.4 0.6 0.8 Normalized temperature * Experimental data  Fit to data *... Extrapolation Fig. 8.13 Experimental term of the data of normalized (T T )/(Tg T ). o vg o Poisson's ratio in temperature T = n m0 40 :3 a 35 0 E S3 25 0 U2 20  Theoretical * Experimental data Fig. 8.14 Longitudinal storage modulus (Ei1) Glass/Epoxy versus T = (T T )/(T T ). n o g o 0 0.2 0.4 0.6 0.8 Normalized Temperature Theoretical E'2 * Experimental E22 .... Theoretical G;2 Fig. 8.15 Transverse (E22) and shear (G12) storage moduli of Glass/epoxy versus T = (T T )/(T T ). n o g o  * ** 0 0.2 0.4 0.6 o0. Normalized Temperature t 0 Theoretical 31, Experimental 11, Theoretical 1122 Experimental 122 Theoretical a G 0 0.2 0.4 0.6 0.8 Normalized temperature Longitudinal shear (inG) T = (T T )/ n 0 (711), transverse (722), and damping of Glass/Epoxy versus (T T ). g o  Theoretical SExperimental 0.6 0.8 temperature Fig. 8.17 Poisson's ratio (v12) T = (T T )/(T T ). n o g o of Glass/Epoxy versus 0.03 0.025  0.02 0.015 0.01 0.005 0 :0 0 0 . *s *0 ' !... i,, i^ i*, , Fig. 8.16 * * * * I . 0 0.2 0.4 Normalized 0 0 0.2 0.4 0.6 0.8 Normalized Temperature Fig. 8.18 Longitudinal storage modulus (E11) of Graphite/Epoxy versus T = (T T )/(T T ). n o g o Transverie  Shear 0 0.2 0.4 0.6 0.8 Normalized Temperature Fig. 8.19 Transverse (E22) and shear moduli of Graphite/epoxy (T T )/(Tg T ). 1 ovg o' (G12) versus storage T = n 0.02 . 0.015 a0 E 0.01 0.005 0 0 0.2 0.4 0.6 0.8 Normalized temperature Fig. 8.20  Longitudinal damping  Transverse damping .... Shear damping Longitudinal (11 ), transverse (T22), and shear (7G) damping of Graphite/Epoxy versus T = (T T )/(T T ). n o g o 0 0.2 0.4 Normalized 0.6 0.8 temperature  Theoretical Fig. 8.21 Poisson's ratio (v12) of Graphite/Epoxy versus T = (T T )/(T T ). n o g o .. r ~' f * 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 crosssection are shown in Figure 9.1 and the boundary conditions are given by v = 0 for (y,z) = (Oz) (9.1) w = 0 for (y,z) = (y,O) 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 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 {o} = [Q]({E} {a}AT {j}c) (9.2) where {a} and {(} are the vectors of thermal and moisture expansion coefficients. 9.2 Description of Study Cases The considered stacking sequence is the [(90/0)2]S layup. The crossply 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 nonuniform moisture gradient (case B) corresponds to a moisture profile as derived in section 2.3. Two uniform temperatures (200C and 800C) are used. All laminates are assumed to be initially (dry at 20 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 (ayz) is zero, but, as y/b approaches 1, a takes significant yz nonzero values and there are small variations in the 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 Glass/Epoxv The equilibrium moisture concentration, cW, of the Glass/Epoxy material is 0.025. The stress a is plotted in Figure 9.3. It reaches a maximum magnitude of 166 MPa. for case C at 20 C. It is compressive for the 00 layer and tensile for the 900 layer. The stress a is shown in Figure 9.4. It is Z compressive everywhere and reach a magnitude of 288 MPa. for the case C at 200C. The stress a is also compressive (Figure 9.5) and reaches a maximum of 245 MPa.. The free edge shear stress a (Figure 9.6) is very significant yz 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 a and a are y z x yz plotted in Figures 9.710. 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 an upper bound for hygrothermal stresses. The results yield the following observations: 1) The stresses induced by temperature only (dry at 80 C) are much smaller than those induced by high moisture content. 2) The stresses due to a nonuniform moisture gradient can be as high as those induced by the saturated moisture 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 80 C) are lower than for some of the other cases. 4) The hygrothermal stresses of the crossply laminates are very significant since their magnitude is of the same order of those of the strengths given in Table 9.2. 