" ‘13:. -..1rr.r:. ‘ 4'. -.r wv'v' 44’- v—..- I 1‘... . IH".'. -m.'( ‘1‘ cu"‘.". -‘. “59.. I a.“ ”I '1‘: ' .‘r. . 13?. .r. " xvi}; gk‘“ u‘ o’ '4‘»; " ~ My; ‘3' I' )‘(f'filc 4:") I; I} . v- A I I I! I ”an“! ' ”:14". -..' 3335A}. -.._ ,v MN. 29;; nun; \: 3 Of." ‘Lfib "fl Hrpl '4 _ . 'r 'Hl ‘.','. “Lg,“ .ILMo-Ocil ' I‘D‘. I‘l‘IT ‘0”,1‘34’; IA“ 7’ J h)?” - ,n 4 ‘ . - .Iu‘ Tn‘p/sx". 9”"; 2:!!! ‘33”??? Hr!» 5:23;“ 1"" ' . [‘33, h r’ S I}; ”3“??? m. ’3“ " 7:535}. .m f. 34+“. ’1'”; (1. -:; 7:15 4% “s ». .x*¥:::'q~”-xcv\;" : fbléf." ‘fi‘ 4 , In i av]; - u'; ' v: 3:112? ‘1453-‘7'2‘32'1'5'i". " t‘.’ a V» ‘ 3, . . A 4". 1}; I um ‘59: ‘3'; . "K“; s - x . r I .Ah.:{~)r.3..<‘ls r. '. - - , ' . I’m .‘ ~ ”19.155" ~v ' ' ,'” ‘l .C‘ ' b ::.::::"\A 4 . 'yL ‘1‘6 ,0‘9'. 4“" v' H“ c ”a “’5": ‘N §x5..~xi~vl. ”w I ‘ 4 1 ' . 4. y s 4 n W .’ I - o‘ , .- Eu‘m'f’z'14' . if I” " "'. ' .f" bl; vb ~ A I '4 \ .‘:‘f;‘:':‘-' 1 . ya. 1'. ( 4'. V I‘ll ' - “4:44 1"‘{-!" ' ‘ ' “II-‘41"?J' 1'. a z ' -’ - a I 1'; (‘l , ) . . . I? " (BEE? I. A . ‘ "4‘ ":"4’ .C‘ If» ’: try-taw- 5:4: -£l“'f" 2r.;'.1‘ kiwi; 441' 4, ’ 4’t\,\‘ 1““4'4 . ' if: ‘ C .J "\‘4i , A..g(y‘|‘; Vivaf ”3}; l 'lga‘z::.: ”:3 33):? ’lr';'+'if. $54, ’1’} 541?. .23 .‘f._'4¢'rf,'.f‘(;._. ’41" . ’ W £24912?» " ’. at > . ‘2‘” . ’ flu-hfra .II" I?! .‘ I “'.‘ ’FI'A "£3 1:4! 1'” H?" Dal,“- . It.“ A “(A . .ut“ 'IC-f-;;“I': nifihcmt If.“ 4.1179“ 3:. WWW r,” szgflg“ . .r W 0"}:- (A;‘L gu to“(thoer-l-fX-:.V~’3'mx,a'uh IIHIIlllllHIIIIHIIllllllllllllllllHlllllllllllllllllllllllll 3 1293 005613710 11468 LIBRARY Michigan State University This is to certify that the dissertation entitled DETERMINATION OF EFFECTIVE ELASTIC PROPERTIES AND THERMAL RESIDUAL STRESSES IN FIBER-REINFORCED COMPOSITES BY THE BOUNDARY ELEMENT METHOD presented by Younghan Youn has been accepted towards fulfillment of the requirements for Ph. D. degree in Mechanics Date ';/I/O/88 MS U i: an Aflinnative Action/Equal Opportunity Institution 0-12771 ‘IVISSIbJ RETURNING MATERIAL§z Place in book drop to LIBRARIES remove this checkout from “ your record. FINES will be charged if book is returned after the date stamped below. DETERMINATION OF EFFECTIVE ELASTIC PROPERTIES AND THERMAL RESIDUAL STRESSES IN FIBER-REINFORCED COMPOSITES BY THE BOUNDARY ELEMENT METHOD By Younghan Youn A DISSERTATION Submitted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Materials Science 1988 N'B \53 \3 LID ABSTRACT DETERMINATION OF EFFECTIVE ELASTIC PROPERTIES AND THERMAL RESIDUAL STRESSES IN FIBER-REINFORCED COMPOSITES BY THE BOUNDARY ELEMENT METHOD By Younghan Youn A boundary element formulation is developed for application to problems involving transverse-isotropic fiber-reinforced composites. Problems examined include determination of thermally-induced residual stresses due to the curing process and determination of effective transverse elastic properties in the presence of improper interfacial bonds. The boundary element method is shown to be an effective technique for the study of the fiber/matrix interface provided that the method is adapted to handle a very thin interphase region. In the estimation of effective elastic moduli for imprOper interface-bonded materials, two models are introduced, i.e. the interphase degradation model and the tangential displacement discontinuity model. These approaches have been incorporated into the boundary element model and representative numerical results are presented. Problems studied include a variety of fiber volume fractions, interphase volume fractions, and interfacial damage parameters. ACKNOHLEDGMENTS I would like to express appreciation to those who have helped and contributed to making this work a reality. This includes my advisor, Dr. Nicholas J. Altiero, for his great support, inspiration and valuable assistance through this period. I also would like to express appreciation to Dr. Ivvona Jasiuk for her priceless help and Drs. John F. Martin and David Yen, members of my guidance committee. A final note of appreciation must go to my family, my wife Younghee and my lovely daughters Marie, Miri and my parents for great support and encouragement . Financial support of this work by the State of Michigan through its Research Excellence Fund is gratefully acknowledged. iv TABLE OF CONTENTS PAGE LIST OF TABLES .................................................... vii LIST OF FIGURES .................................................. viii CHAPTER 1 INTRODUCTION AND BACKGROUND ........................ 1 CHAPTER 2 BOUNDARY ELEMENT METHOD ............................ 21 2.1 GOVERNING EQUATIONS FOR THERMOELASTICITY PROBLEMS ...................................... 21 2.2 DIRECT BEM FORMULATION FOR DISPLACEMENT ........ 23 2.3 TRANSFORMATION OF DOMAIN INTEGRALS TO BOUNDARY INTEGRALS ......................... 25 2.4 NUMERICAL IMPLEMENTATION OF BEM ................ 29 CHAPTER 3 THERMAL RESIDUAL STRESS DUE TO THE CURING PROCESS .. 33 3.1 MICRO-MECHANICS OF COMPOSITE MATERIALS ......... 33 3.2 STRESS ANALYSIS ................................ 33 3.3 BOUNDARY CONDITIONS ............................ 35 3.4 BEM APPLIED TO A NON-HOMOGENEOUS REGION ........ 37 3.5 INTERFACE CONDITIONS ........................... 39 CHAPTER 4 THE EFFECTIVE MECHANICAL BEHAVIOR .................. 40 4.1 BALANCE OF WORK AND ENERGY ..................... 40 4.2 STATISTICALLY TRANSVERSELY ISOTROPIC EFFECTIVE COMPOSITES ..................................... 43 4.3 DAMAGED INTERFACIAL BEHAVIOR ................... 47 4.3.1 INTERPHASE DEGRADATION MODEL ................ 51 4.3.2 TANGENTIAL DISPLACEMENT DISCONTINUITY MODEL . 57 CHAPTER 5 RESULTS AND DISCUSSION ............................. 58 5.1 THERMAL RESIDUAL STRESSES ...................... 58 5.2 EFFECTIVE ELASTIC CONSTANTS .................... 65 5.3 THE ELASTIC BEHAVIOR OF DAMAGED COMPOSITES ..... 73 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS .................... 86 BIBLIOGRAPHY ...................................................... 88 vi LIST OF TABLES TABLE PAGE 5.1 Material properties of S Glass/epoxy composite constituents (from ref. 71) ............................... 61 5.2 Comparison of thermal residual stresses at the fiber/matrix interface at room temperature (V;- 60 percent) ............ 61 5.3 Material properties of Glass/epoxy composite constituents (from ref. 16) ............................... 65 5.4 Comparison of transverse shear modulus G:;'with other analyses .................................................. 71 5.5 The thickness h and V1 of the interphase region ........... 71 5.6 Damaged effective elastic moduli for the interphase degradation and tangential displacement discontinuity models at VE- 0.6 ......................................... 85 vii LIST OF FIGURES PAGE Two phase model ........................................... 5 Modified two phase model .................................. 5 Three phase model ............................ i ............. 8 Rectangular array model ................................... 8 Unidirectional fiber-reinforced composite ................. 22 Discretized boundary ...................................... 30 Geometry of shrinkage ..................................... 36 Typical (a) boundary element grid and (b) finite element grid (from ref. 71) ............................... 38 Fundamental units of (a) a heterogeneous fiber/matrix composite and (b) a macroscopically homogeneous composite . 41 Boundary conditions for (a) biaxial tension and (b) uniaxial tension .......................................... 45 Nature of residual thermal stress, crack location, and fracture path (broken lines) .............................. 49 Schematic of the enhanced spherical particles. The location of cracks and fracture path (broken lines) ................ 50 Thin interphase BEM model ................................. 52 Line cracked material model ............................... 55 Normal thermal residual stress at the interface of a S Glass/epoxy composite (V;- 60 %) ...................... 59 viii FIGURE 5.2 Tangential thermal residual stress at the interface of a S Glass/epoxy composite (Vi- 6O %) ..................... Normal thermal residual stresses at points A, B and C for a square array of fibers in a S Glass/epoxy composite .... Tangential thermal residual stresses at points A, B and C for a square array of fibers in a S Glass/epoxy composite . Effective transverse shear modulus 0:3 of the composite ... Effective plane strain bulk modulus K:; of the composite .. Effective transverse Young's modulus E:; of the composite . * Effective transverse Poisson's ratio uzaof the composite .. * Effective transverse shear modulus ratio Cza/Cll for various fiber/matrix shear modulus ratios ................ Cracked interphase shear modulus ratio G /G due to cracked m micro-crack density k .................................... The relationship between micro-crack density k and interphase degradation factor DF ......................... * Effective transverse shear modulus 623 due to the degradation of interphase shear modulus Gll ............... Effective plane strain bulk modulus K:; due to the degradation of interphase shear modulus Gm ............... Influence of degradation factor DF on the effective shear modulus 6:; at 60 % fiber volume ......................... Influence of degradation factor DF on the effective plane strain bulk modulus K:3 at 60 % fiber volume ............. ix PAGE . 6O . 63 64 66 67 69 70 . 72 . 74 . 75 . 76 . 77 . 78 . 79 FIGURE 5.16 5.17 5.18 5.19 PAGE Effective transverse shear modulus 6:; due to a tangential displacement discontinuity ..................... 81 Effective plane strain bulk modulus K:; due to a tangential displacement discontinuity ..................... 82 Influence of tangential displacement discontinuity factor R on the effective transverse shear modulus 6:3 at 60 % fiber volume ...................................... 83 Influence of tangential displacement discontinuity factor R on the effective plane strain bulk modulus K:3 at 60 8 fiber volume ...................................... 84 CHAPTER 1 INTRODUCTION AND BACKGROUND The primary advantage of' " artificial materials ", such as composite materials, is that the thermal and mechanical behavior is controllable. In general, composite materials consist of two or more different materials that are formed from components large enough to be regarded as homogeneous continuous media. The search for materials with high specific strength and stiffness as well as good toughness has brought an enormous increase in research activity in the field of fiber-reinforced composite materials. The use of fiber-reinforced composite materials in engineering applications requires a fundamental understanding of the mechanical and thermal interactions between the fiber and matrix components as well as the effective material properties of the composite system. These material properties can be determined experimentally by testing actual composite specimens. However, during the last two decades, increased attention has been given to methods for analytical prediction of these basic composite material properties as a function of the prOperties of the constituent materials and their geometrical relationship to each other. The previous predictions of the effective elastic properties have been 'based on the assumption of perfect bonding between the constituents in the composite medium. However, the elastic moduli of a composite are not merely controlled by the elastic properties of the two phases. Cracks that can develop during cooling of the composite from it's fabrication temperature and cracks formed during stressing due to poor interfacial bonding lead to lower elastic moduli than are otherwise predicted. In this dissertation, a more realistic effective transverse behavior of unidirectional fiber-reinforced composite materials will be studied. Here the effect of imperfect interfacial bonding and thermally-induced residual stresses, which build-up during the fabrication process, will be considered. .1. C. Maxwell [1] in 1873 and Lord Rayleigh [2] in 1892 introduced the notion of analytical determination of properties of composite materials. They computed the effective (or average) electrical conductivity of composites consisting of a matrix containing a certain distribution of spherical particles [3]. In the field of elasticity, the first approximation of.effective elastic moduli of composite materials was postulated in 1889 by W. Voigt [4]. Voigt examined a composite material which consisted of a spherical inhomogeneity, with shear and bulk moduli ”2 and K2, imbedded in a matrix with moduli p1 and K1. The volume fraction of the inhomogeneity was denoted by Ca and that of the matrix by c1 and the composite was subjected to uniform shear strain. He concluded that the effective shear and bulk moduli, I; and R, can be expressed as: - + ” c1 ”1 c2 ”2 7" - c1 K.1 + c2 K2. (1) In 1929, Reuss [5] assumed that all of the elements of the composite material are subjected to a uniform stress equal to the average stress. According to Reuss's assumption, the effective moduli were obtained as: - c1 c2 -1 "-l + l #1 #2 c1 c2 -1 K - [ K + K } (2) 1 2 Hill [6], in. 1952, initiated. the notion. of ‘bounding, of the effective elastic moduli. He also showed on the basis of the classical extremum principles of elasticity that the Voigt approximation and the Reuss approximation are the upper and lower bounds, respectively, of the true effective elastic moduli. In 1957, Eshelby [7] suggested the "equivalent inclusion" method. He examined the change in the gross elastic constants of a material when a dilute dispersion of ellipsoidal inhomogeneities is introduced into an infinite homogeneous isotropic elastic matrix, see Figure 1.1. Eshelby simulated a composite material as a homogeneous material having the same elastic moduli as those of the matrix and containing inclusions with shear eigenstrain. £1: (stress-free strain). The effective shear modulus, by Eshelby's equivalent inclusion method, can be formulated for a spherical inclusion as: ”1[”1+281212(“z"‘1)] p - (3) 1 + clog-#2) where S12 - (4-v1)/15(l-v1) and v1 is Poisson's ratio of the matrix. 12 When the equivalent inclusion method is applied to obtain the stress disturbance of inhomogeneities, the existence of a matrix is assumed. However, the meaning of the matrix becomes rather vague when the volume fraction of the inclusion is increased, i.e. Cié 0. Furthermore, the interaction among inhomogeneities becomes more prominent as the number of particles increases. Krbner [8] and Budiansky and Wu [9], in 1958 and 1962 respectively, proposed the self-consistent method in order to avoid this difficulty. In 1964, Hill [10] employed. this technique to derive expressions for the elastic constants of aggregates of crystals. He modeled the composite as a single fiber embedded in an unbounded homogeneous medium which was macroscopically indistinguishable from the composite, i.e. of shear modulus E and Poisson's ratio 5, as shown in Figure 1.2. The effective shear modulus, I; and effective bulk modulus, R can be expressed by the self-consistent method approximation as : clog-no) It u - 111+ (48) u + 2512120540 Infinite matrix Inclusion Figure 1.1. Two phase model (a fiber is embedded in an infinite matrix medium) Infinite effective medium Inclusion Figure 1.2. Modified two phase model (a fiber is embedded in an infinite effective medium) 3c1(R1-Ro) 12 (4 b) 7:0 I 7‘: O 4. 3K 4» $11.13 “(1'10 In the case of a spherical inclusion, s - (4-5&)/15(1-&), s - 1212 11.1.1 (l+§)/(l-D) and 5- (3R -2fi)/2(fl+3R). The effective moduli L and R can be determined by solving equations (4 a) and (4 b) simultaneously. For inclusions much stiffer than the matrix, equations (4) overestimate the effective moduli while for inclusions much more compliant than the matrix, the effective moduli are underestimated. More generalized versions of self-consistent methods have been proposed by Hashin [11], 1968, and Christensen and Lo [12], in 1979, for the thermal conduction and elasticity problems respectively. In their generalized versions of self-consistent methods, a model consisting of an inclusion of radius a and a concentric matrix shell with radius b was embedded in an effective medium as shown Figure 1.3. The generalized self-consistent method appears to be a more realistic approximation than the first version of the method. However the ratio 0 - a/b is an unknown parameter and the choice of n for a spherical inclusion is not obvious. In early 1960's, Hashin and Shtrikman [13,14,15] obtained improved bounds for the effective elastic moduli .of a statistically isotropic composite material on the basis of the energy theorems of classical elasticity. The minimum complementary energy theorem yields the lower bound, while the minimum potential energy theorem yields the upper bound. In general their method gave better bounds than the Voigt and Reuss bounds and also provided good agreement with experimental data. Hashin and Shtrikman's variational method has been generalized and modified by Hill [16], Walpole [17,18], Wu [19] and others [20,21]. The bounds are of practical value for a fiber/matrix stiffness ratio up to approximately 10. Since the. only geometrical information is volume fraction, they can not provide good estimates for extreme stiffness ratios such as a rigid or empty inclusion. To improve the bounds of the effective moduli, it is necessary to incorporate additional geometrical information. Assuming rectangular packing and perfect bonding between the two constituents, Adams and Doner [22,23], in 1967, examined the effective shear and transverse Young's moduli of composite materials by a finite difference analysis. Because of the assumed periodicity, they were able to isolate a fundamental or repeating element as typical of the entire composite, see Figure 1.4. They formulated the effective moduli in terms of the average elastic stresses and strains obtained by solution of an elasticity boundary-value problem. From their results, Adams and Doner concluded that the predictions of shear and Young's moduli of a square array of fibers is in more reasonable agreement with experimental measurements than those obtained assuming a hexagonal array of fibers. Infinite effective medium Figure 1.3. Three phase model \ K) K1 //\ \ 2\3223 Matrix /\ \J F\ k/ Fundamental unit C \ r \ \J/ \4/ f K \J \J Figure 1.4. Rectangular array model Chen and Cheng [24] considered a hexagonal array of fibers and defined a triangular fundamental element. They assumed that the fiber and the matrix were homogeneous and isotropic on the microscopic level. Assuming perfect bonding of the interface of the fiber and matrix, they obtained a solution by a combination of the least squares method and the Fourier method. Springer and Tsai [25] approached the effective thermal conductivity problem based on the analogy beWeen the response of the unidirectional composite to shear loading [22] and heat transfer. Without considering thermal contact resistance, they considered a square array and modeled the fundamental element as rectangular. From their results, the transverse effective thermal conductivity was in reasonable agreement with experimental data. In 1977, similar to Springer and Tsai [25], Gogél and Furmanski [26] predicted the transverse thermal conductivity of a rectangular fundamental element by a finite element analysis. From their results, the prediction of the conductivity, «:22, was in good agreement with experimental results except for at high fiber/matrix thermal conductivity ratios (act/um > 100) and high fiber volume fractions. They also concluded that a finite element analysis gave accurate results up to 0.6 percent fiber volume fractions. The most commonly adopted premise in the theory of the mechanical and thermal behavior of composites is the requirement of continuity of tractions and displacements (in elasticity problems) or temperatures and normal heat flux (in heat transfer problems) at the constituent 10 interfaces. Such perfect interface conditions, assumed by previous investigators, are a convenient idealization which have given valid results in a large variety of situations [28]. However, internal micro-cracks, voids, debonding and imperfect contact are well known to exist in composite materials. Incorporation of such phenomena in the general theory of effective properties requires the modification and relaxation of the continuity conditions between the constituents. In 1982, Lane and Leguillon [27] examined the equivalent behavior of an elastic composite material when a tangential slip was allowed on the interface of the components. They assumed tangential slip to be a linear law in terms of a scalar coefficient k: _ u(1) -k[u(2) } t t. s s s 12 12 12 0' t where 812 is the interface between fiber and matrix and superscripts ”1" and "2" indicate that the relevant quantities are associated with the matrix and inclusion respectively. They used an asymptotic expansion method for the unstressed periodic-structure material and employed a finite element analysis. Their results showed the existence of two limit values (DIII and D") of a damage coefficient D expressed in terms of k (D - log {(lO‘Et/kH-l). For D < D“, the material behaved like the perfectly bonded composite. In the intermediate range, the elastic properties of the equivalent material rapidly decreased. For D > D", a damaged state of the material was reached. In 1985, Benveniste [28] made the same tangential slip assumptions at the fiber-matrix interface. Based on Christensen and Lo's [12] generalized version of the self-consistent scheme, he ll formulated the representative volume average of the elastic strain energy in terms of the surface tractions and displacements using the divergence theorem. From his results, he concluded that the effective shear modulus agreed with Christensen and Lo's predictions when the debonding parameter R (-1/k) was zero (perfect bonding) and, as the debonding parameter increased, the effective shear modulus asymptotically decreased. For periodically distributed inhomogeneities, Mura, Jasiuk, Kouris and Furuhashi [29] examined an extreme case of imperfect bonding of the interfaces. They employed the equivalent inclusion method to evaluate the effect of pure sliding of the interface on the overall elastic constants. They developed an iteration method and obtained an approximation formula for the average shear modulus. From their results, the sliding caused relaxation of the average shear modulus and the relaxation was higher for larger ratios of the inclusion shear and matrix moduli. In 1988, Jasiuk, Mura and Tsuchida [30] examined the thermal stresses and effective thermal expansion coefficients of a spheroidal fiber composite in which the fiber-matrix interface was allowed to slip. Neglecting the effect of friction, they employed the Boussinesq-Papkovich displacement potential method to solve the stress and displacement disturbances due to a spheroidal inhomogeneity embedded in the matrix. The Mori-Tanaka theorem [31] was used to predict the average thermal expansion coefficients of the composite containing a finite concentration of fibers. From their results, they 12 observed that sliding at the fiber-matrix interface caused higher stress concentrations and significantly affected the average coefficients of thermal expansion of the composite. Assuming an initial stress free condition, previous investigators have predicted the effective thermal and mechanical properties by various analytical and numerical schemes. However, high performance fiber composite materials are normally cured at elevated temperature and severe thermally-induced residual stresses may develop in the vicinity of the fibers because of the mismatch of the coefficients of thermal expansion and, elastic moduli between the fibers and the surrounding matrix. The presence of residual stresses in fiber composites can not be avoided because the most commonly used method of reducing their intensity (thermal stress relief) can not be applied to such. materials. ‘Because of the different thermal expansion characteristics of fiber and matrix, any stresses relieved at the annealing temperature will be re-generated during cooling to ambient temperature [35,37,45]. Some of the detrimental effects caused by such residual stresses are time-dependent dimensional changes, distortion on machining, and reduction of the elastic limit and yield strength of the composites. In a tensile test analysis, the existing state of residual stresses affects the stress-strain response and the performance of the composite [35,37]. For more comprehensive predictions of fiber- reinforced composite material performance, full knowledge of the thermally-induced residual stress distribution is required. 13 In 1964, Dundurs and Zienkiewicz [32] investigated the stresses in the vicinity of a long circular cylindrical elastic inclusion embedded in an infinite surrounding elastic material subjected to a uniform temperature gradient in a direction transverse to the axis of the cylinder. Their solution was based on an Airy stress function which they obtain by superposition of a "zero-stress“ solution and a second solution introduced in order to cancel multivalued and discontinuous displacements arising in the first solution. The problem could be solved in a direct way using the thermal potential function derived by Sternberg and McDowell [33] and Tauchert [34]. Instead of an infinite matrix with a single embedded fiber, Hecker, Hamilton and Ebert [35] examined a concentric cylindrical OFHC copper-4340 steel composite in 1969. They developed a modified Sachs [36] boring-out procedure to obtain an experimental measure of the intensity and distribution of residual stresses due to cool-down from final heating temperature. From their semi-empirical results, they concluded that the magnitudes of the residual stresses in the fabricated composite were very high, e.g. in the copper core region, they greatly exceeded the yield strength in uniaxial tension. In a later paper [37], they formulated an elastic solution to predict thermally-induced residual stresses for their previous concentric cylindrical composites. They assumed that the interfacial bond was perfect and the residual stresses resulted only from the difference in thermal expansion coefficients between homogeneous, isotropic elastic components. From their results, they concluded that the theoretical 14 predictions and the experimental results of residual stresses differed by less than 25 percent. During fabrication of metal matrix composites, thermally-induced local plastic deformation can occur. To account for these plastic strains Hoffman [38], in 1973, estimated the magnitude of elastic stresses and elastic-plastic stresses and strains in a tungsten fiber-reinforced 80 Ni+20 Cr matrix composite during heating and cooling between 80 and 2000 deg F. He developed approximate equations for high fiber volume and low fiber volume fractions of fibers in homogeneous, isotropic components. A hydrostatic stress state within each constituent had been assumed in the high volume fraction of fibers. However, in the low volume fraction of fibers, he used thick wall cylinder theory in the derivation and included the possibility of plastic strains. In his derivation, he did not consider either a metallurgical interphase or stresses due to differences in the Poisson's ratios of fiber and matrix. He concluded that large elastic stresses, sufficient to cause fracturing of either constituent or the interface, could quickly occur in cyclic temperature usage. In 1978, Miller and Adams [39] approached a more realistic behavior of composites. They modeled a micro-region, consisting of a square matrix containing a cylindrical fiber, by means of the finite element method. They assumed inelastic matrix behavior, an elastic fiber and a state of generalized plane strain. They calculated stresses due to cool down from curing temperature and moisture absorption in graphite/epoxy composites. The results indicated that, 15 for 60 percent fiber volume fraction, the maximum residual stresses were on the order of the yield strength of the matrix material. In 1979, Uemura, Iyama and Yamaguchi [40], assuming the matrix to be isotropic and the cylindrical fibers to be transversely isotropic in carbon reinforced composites, calculated thermally-induced residual stresses using effective thermal expansion coefficients and effective mechanical properties. Experiments on carbon-fiber/epoxy composite cylinders showed good agreement with the theoretically predicted values. They concluded that the residual stresses induced by curing could not be neglected when compared with the low tensile strength transverse to the fibers. In 1979, Weitsman [41] and Herrmann and Mattheck [42], rather than applying linear thermo-elastic theory, proposed a visco-elastic response of the matrix and accounted for temperature-dependence and stress sensitivity of the creep compliance. Comparisons with linear elastic solutions indicated that the viscoelastic relaxation may reduce the residual stresses by about 20 percent. In 1986, Rohwer and Jiu [43] studied the effects of free edge conditions during the curing process. They employed a three dimensional finite element model for calculation of residual stresses in a hexagonal array of fibers. Concentrations of radial and tangential stresses were found along the free edge interface. The matrix was predominantly endangered by high tangential and axial stresses arising at a certain distance from the free edge. 16 In 1986, Avery and Herakovich [44] utilized an elastic solution to analyze orthotropic fibers in an isotropic matrix. They showed that, for orthotropic fibers, stress distributions are radically different than those predicted assuming the fiber to be transversely isotropic. Fiber volume fraction greatly influences the stress distribution for transversely isotropic fibers, but has little effect on the distribution if the fibers are transversely orthotropic. In 1987, Adams [45] studied the influence of interface degradation, and thermal and moisture effects on the performance of various carbon fiber-reinforced unidirectional composite materials. For a square packing array of circular fibers, he calculated hygrothermal stresses using a two dimensional finite element model. The predicted thermal residual normal stress at the fiber-matrix interface for 0.6 volume fiber fraction of ASA/3501-6 was compressive, the highest values occurring in the regions of closest fiber spacing. At 16 MPa, it was approximately 25 percent of the tensile strength of the Hercules 3501-6 epoxy matrix. The addition of 2.9 weight percent moisture to the matrix induced residual stresses of magnitudes over 2.3 times those induced during cool down. In 1988, Vedula, Pangborn, and Queeney [46] examined the effect of anisotropy of the fiber thermal expansion properties by extending a thick-walled cylinder model that had been developed for a graphite/aluminum composite in which the fiber and matrix materials had isotropic elastic moduli. They showed that transverse thermal l7 stresses are 75 percent lower for the case of anisotropic thermal expansion coefficients than for the isotropic case. They also showed that, when these stresses were examined for incipient plasticity, anisotropy of the fiber thermal expansion can result in a large temperature range of elastic defamation during cooling from processing temperatures. In a later paper [47], they examined a three-component cylinder model of a unidirectional composites. The components consisted of the fiber, an interfacial coating region and a surrounding matrix. As in their previous paper, they assumed that all three components had homogeneous and isotropic elastic mechanical properties and the fiber had anisotropic thermal expansion coefficients. Under the assumption of perfect bonding of the three components, they concluded that the existence of the interfacial region contributed significantly to residual stress build-up. Furthermore, the use of a third coating component in the form of an interfacial region could control the intensity of the residual thermal stresses . Finally, Hahn and Kim [48], instead of an interfacial region model, proposed that the concentric fiber and matrix were surrounded by an infinite homogeneous medium that had the properties of the composite, as in Figure 1.3. They assumed that the interfacial bonds were perfect, the fiber and the composite were homogeneous and transversely isotropic and the matrix was isotropic. Their three-phase model predicted a compressive radial stress of 1.1 MPa at room temperature (25 0C) after cool-down from 60 0C cure temperature along the fiber/matrix interface of a Kevlar 49/epoxy composite. They also 18 performed a finite element analysis for a square array of fibers. The results indicated that the maximum compressive radial stress was in the region of closest fiber spacing. From their three-phase model and finite element analysis results, they concluded that the predicted radial stress, by the three-phase model, was comparable to the average stress over the fiber surface determined by the finite element analysis. A number of investigators [49,50] have discussed the presence of a thin interphase region 'between. fiber and matrix” of different material properties. Others [51,52,53] have discussed the existence of voids and micro-cracks in the interface and the possibility of imperfect bonding. Voids, formed during fabrication, and cracks, formed due to differential thermal contraction, significantly reduce the elastic moduli from the expected values [54]. The primary motivation of the present work is to develop techniques for better understanding how various types of interface conditions and thermally-induced residual stresses can influence the thermal and mechanical behavior of the composite material. Past investigations have been aimed primarily at evaluation of the effective elastic constants in composites with perfect bonding of the interface and results have been obtained in closed-form or by finite element analysis. In Chapter 2, the boundary element method is formulated for the plane strain thermo-elasticity problem. In the Navier's equations, l9 written in terms of displacement, thermal load is treated as an effective body force. As is usual in boundary element analysis, the body force is treated using a domain integration. However, in this formulation, the domain integration can be transformed to a boundary integration by defining the Galerkin tensor and employing the divergence theorem . In Chapter 3, two fundamentally important concepts are introduced for the calculation of thermally-induced residual stresses during the cool-down process. First, we assume the composite medium to be macroscopically homogeneous and isotropic or transversely isotropic. Instead of considering the entire composite medium, we isolate a single fiber and it's surrounding matrix region as a fundamental unit cell which represents behavior of the entire composite. Second, the macroscopically-homogeneous medium will be deformed unifomly during the cool-down process. Since the deformation of a fundamental unit must be consistent with the entire medium, we use the superposition technique with displacement-controlled boundary conditions. In Chapter 4, in order to estimate effective elastic properties of the composite material, we examine the ”balance of work and energy" theorem. Due to lack of a proper technique for prediction of crack development due to thermal residual stresses and phases of differing elastic properties, we introduce two different damage material models. These are the interphase degradation model and the tangential displacement discontinuity model. For each of these damaged models, we 20 will examine the elastic moduli of the damaged material. Finally, in Chapters 5 and 6, we will apply and discuss our boundary element technique to obtain results for thermally-induced residual stresses and effective elastic behavior of fiber-reinforced composites assuming both a perfect bond interface and a damaged interface. We will also examine our present method in comparison with other published results. CHAPTER 2 BOUNDARY ELEMENT METHOD 2.1 Governing equations for thermoelasticity problems Plane strain thermoelasticity problems are considered in this analysis. The cross-section of the composite body is assumed to lie in the xzz-x3 plane, which is perpendicular to the fiber direction x1 as shown in Figure 2.1. The first assumption made for this problem is that each lcomponent of the strain tensorj 613 can be additively (o) (r) decomposed into elastic and thermal parts, CL: and 6L3, respectively. Thus, 6 - e“) + END - 6“” + a T 6 ' (2.1) 13 13 13 13 id where a is the coefficient of thermal expansion, T is the change in temperature and 81‘1 is the Kronecker delta. The range of indices in equation (2.1) is i-2,3 ; j-2,3. The equations of equilibrium can be written in terms of displacements by combining them with the constitutive and kinematic equations. The governing equation, expressed in terms of displacements, for an elastic homogeneous isotropic medium under thermal loading is : G 1+u G “.... + I727 “.... " 'F. + 2“ {-1727 ) °‘ T,. (2-2) 21 22 l’ “ l ’ \ 0 ° I I \ U. \- ‘ -‘ " I ' I ‘ 1 I \ ' l s' \" \ Figure 2.1 Unidirectional fiber-reinforced composite 23 where G is the shear modulus, v is Poisson's ratio and F1 are the components of the body force vector per unit volume. The partial differential equations (2.2) must be solved subject to the appropriate boundary' conditions. The 'usual. mechanical ‘boundary' condition, i.e. displacement, traction, or combination, must be prescribed on the boundary F of the body. The components of the traction vector t1 at a point on the boundary 1‘ are given by 2y - 2(1+V) t. CH”... ”3.1)“; + ‘17? “... “3] T57 ““51 ”-3) where n1 are the components of the unit outward normal to 1‘ at that point. 2.2 Direct BEM formulation for displacement A boundary element method (BEM) formulation for the displacement field is obtained by following the method outlined by Danson [55]. Kelvin's fundamental solution, chosen here as the auxiliary solution, is [56] 1.11 - U“ eJ (2.4) a- where u1 satisfies the equation * _ 1 * A(¢>,E) ‘11,” + W uk,ki __G—- 61.1 e3“ (2'5) In equation (2.5), A is the Dirac delta function and eJ are unit orthogonal base vectors. The two point influence function U“ is the displacement in the 1 direction at a field point C due to a unit load 24 in the j direction at a source point 3. The function U“ for plane strain is available in many references [55,56,57,58,59]. The traction * vectors t1 in the homogeneous elasticity problem are related to the gradients of the displacement field according to the equation * * * 2y * t1 IIG [(u1,3+u1,3)nj+T2v—uk,kni]' (2.6) * Multiplying both sides of equation (2.2) by ui and integrating over the domain 0 of the body results in the equation * 6 n1 [G “1,13 + W uk,k1] d0 0 * 1+v . - 111 [ -Fi+ 26 W a T,i] d9 . (2.7) 0 Using the definitions of tractions from equations (2.3), (2.6), and the divergence theorem, equation (2.7) can be reduced to u1 (a) + 1130”?) u1(x) at; - uu(o,i) t1(i) ari r r _ E _ + UiJ,J(¢’=) [ m a T (3.3)] d0; 0 - lflJ(O,E) F1(E) d0: (2.8) 0 where the two point influence function'Iu(§,i) is the traction in the 1 direction at a field point 0 due to a unit load in the j direction at 25 a source point i, <1, and 5 denote points inside the body 0 and x and 1? denote points on the boundary I‘. Equation (2.8) is seen to contain the unknown temperature field inside both the domain integrals and the boundary integrals. The temperature field must be obtained by solving the diffusion equation with the appropriate thermal boundary conditions. The domain integrals which contain the body force 11(5) and thermal loads (effective body force) must be dealt with by dividing the domain 0 into integration cells, thus increasing the amount of data preparation required to run a particular problem, e.g. plasticity problems [60], viscoelasticity problems [61] or nonhomogeneous problems [62] . However Cruse [56] and Danson [55] have shown that for many of the commonly-encountered body forces such as gravity, rotational inertia or steady state thermal loads, the domain integrals in equation (2.8) may be reduced to a surface integral by a further application of the divergence theorem. In this analysis, the thermal load is considered to be the only non-zero domain integral. 2.3 Transformation of domain integrals to boundary integrals Let's denote the domain integral in equation (2.8) as an effective body force vector 81(6) ._ E. .. Bi(¢) "' uiJ,J(¢’-‘ [m a T(—)] d‘lE. (2.9) 0 In order to transform the domain integral, equation (2.9), to an equivalent boundary integral, it is useful to define a tensor (31.1 (QUE (the Galerkin tensor) such that the displacement kernel “1.) («5.3) is 26 given by G ($.53 a _ = _ is.“ 111301,...) G13,u(°*“ 2w”) . (2.10) Differentiating the Galerkin tensor 61301.5) and renaming dummy indices gives 1'2” c (as . (2.11) Humans) - l-v 11.1mm Thus, the domain integral B105) becomes .. E .— Bi(¢) "" U13,J(Q’= [T27 a T(:)] dag - G dams [Wyn T(=.)] “15- (2.12) For steady state heat conduction T .13 (E)-0 and one may write aE .. .. .. .. 810D) - T2; [813.Jkk(§'=) T(=) - Gid’J(¢,=) T’JJ(=)) d“;- 0 (2.13) Using the divergence theorem, equation (2.13) can be transformed to a boundary integral as : 31(6) - {1:ng [613.51 (65:) mi) - cudwjo T360] nkdI‘i. I‘ (2.14) The boundary integral expression for displacements for the 27 thermoelasticity equation (2.8) may now be obtained by taking the limit as an internal point C approaches a point x on boundary 1‘ c13(x) uJ(x) + T1J(x,i) 1116:) ari - owns?) :10?) (11‘; r r + P1(x,i) T(i) cu‘;I - 01(x,i) rho?) nk at; (2.15) r r where - (1+y) 1 7’8” “13(X,X) - m [{ (3'4V)1n(-r—) ' T } 61., + rirj] ru(x,i) - mini-’1), r [nkrkt(l-2u)613+2rirJ}-(l-2u)(rind-rjni) ] - 1+1! 2 1 613(3)!) " 4_1rE— 513 r 111(T) - a(l+u) 1 1 PM”) " m {[ WT) ‘ T J“. ' “.133. } 01(x,i) - % rir [ ln(+) - —%— ]. (2.16) and rZ-[(x1-i1)2+(x2-i )2] 2 ri-(xi -§1)/r. (2.17) The tensor C13 (x) is a function of the local geometry and the location of x. If the boundary is locally smooth at x, cij(x) - —%— 615. As 28 shall be seen, knowledge of cu(x) is not required. Once the displacement and traction fields have been obtained everywhere on the boundary using equations (2.15), the displacements and stresses for any internal point C of the body can be calculated from the equations ui(<1>) - Unani) 1:36;) (11‘; - ru(c,i) u3(i) dI‘i r r + Pi(o,i) ml) at; - 01mg?) 'r,k(i) nk cm5E r r 0130») - om(a,i) tk(i) dl‘i - [ smani) uk(i) (11‘; 1‘ Jr * - - * - - aE .- + Sid(¢,x) T(x) ari - Vid(¢,x) T’k(x) nq dFi - -T:§;—-T(x)6ij. F F (2.18) where the influence functions D , S , 5* and V* are derived from hi) kid 13 1J the displacement gradient and stress-displacement relations. 2.4 Numerical implementation of BEM The boundary of the body P is discretized into N straight segments, as shown in Figure 2.2. A discretized version of equation (2.15) is written as 29 n (n) (n) ' '- Ci.) (1:) u.1 (x) +§ 'I’ 1.1 (x,x) 11.1 (x) dl‘iIn 111-1 1‘ D n I! -E U 1.1 (x,x) tJ (x) (11'; «PE P1(x,x) T(x) (11‘; 111-1 1‘ ' n1 1‘ m II I N - E 01(x,x) T,k(x) nk dI‘i . (2.19) m-l F to II where uJ (x)m) are the displacements components at node n and n-1,2, ...... ,N, j-2,3. After having determined the temperature and heat flux distributions on the boundary 1‘, suitable shape functions must be chosen to represent the displacement, traction, temperature and heat flux on 1‘. Displacement components are assumed to be piecewise linear on the boundary segments 1". However, the traction components and temperature and heat flux are assumed to be constant on the boundary segments, i.e. 30 x(n+1) x(n) element n Figure 2.2. Discretized boundary element 31 - (m-l) (m) 113(1) - “1 111(6) + uJ N205) (m) td(:-t) - e-J T (i) _ TOM T’n(i) - 1"?) (2.20) where ‘ N103) - (Lo/2. N2(€) - (1+£)/2 i _ N1(£) x‘ln'l) + 112(5) x011) arm - [—;.4 sm - em.1 )] a: - ( Asa/2) as (2.21) and 6 is a local coordinate for the segment m with value -1 at node m-l, value 0 at the center of the segment, and value 1 at node m. Note that the order of traction tJ(x) in the interval is less than that of displacement uJ(x) on the boundary and it is consistent within elements, i.e. linear displacements and constant tractions on each element. If equations (2.18) and (2.19) are substituted into equations (2.17) the following is obtained (a) (n) (m-l) 2c” (1:) u.j (x) + Asia} I id(x,€) N1(€) d5 . u.j m-l P mfln+1 32 (In) (In) + Asa} I 13(x,£) N1(£) d6 . u.1 - Asll§ U 1J(x,£) d5 . t.1 m-I P III P mi‘n m m H N (In) (In) (II!) + Asa} P1(x,£) dé’ - T - [ls-E 01(x,£) d6 - T ’k n k m-l Pm 1rd. 1‘- m—l,2, ...... ,N (2.22) Equation (2.20) can also be written in matrix form as [v1-(u)-m-(E)+(b) <2.23> where [U ] and [I ] are coefficient matrices which can be calculated from the fundamental solutions (2.16). In equation (2.23) the column vectors ( u ) and ( t ) denote the nodal displacement and resultant segment force vectors, respectively, and ( b ) the column vector corresponding to the resultant segment temperature and heat flux distributions on the boundary. For more detail of the numerical derivations, see reference [63]. 33 CHAPTER 3 THERMAL RESIDUAL STRESSES DUE TO THE CURING PROCESS 3.1 Micro-mechanics of composite materials To treat the problem analytically, assumptions must 'be made regarding the fiber packing arrangement and the geometry of the individual fibers. Here, a rectangular fiber packing array has been assumed, as shown in Figure 1.4. A rectangular array, whidh permits the variation of fiber spacing in two coordinate directions independently, offers more versatility than a hexagonal array which has only one geometry spacing variable. The individual fiber cross-sections are assumed to be symmetrical about each of the coordinate axes, x2 and x3. Within this assumption, the fiber can be of arbitrary shape, i.e., circular, elliptical, diamond, square, rectangular, hexagonal, etc. Because of assumed periodicity, a fundamental or repeating unit, as indicated by the dashed lines of Figure 1.4, can be isolated and considered as typical of the entire composite. Because of the assumed fiber symmetry, only one quadrant of this fundamental unit need be considered. 3.2 Stress analysis Having established the fundamental unit by assuming rectangular 33 34 packing and doubly symmetric cylindrical fibers, the problem can be formulated exactly by the theory of chemo-elasticity. Let u1 denote the components of the displacement vector field. Then the components of the infinitesimal strain field are 1 6” - T a3, either hemispherical cracks will form around the particle when fracture energy 1’ > 7”. Cracks can also form at the particle/matrix interface when the fracture energy of the interface is lower than both that of the particle and matrix phase. This occurs for poor interfacial bonding. For the case a; > up, radical cracks develop regardless of the fracture energy of the two phases. Binns [67] and Evans [52] observed that of the cracks shown in Figure 4.3, the radial cracks were most detrimental since they may easily link together and form a network of interparticle cracks. The non-symmetric stress distribution due to the different elastic properties of the two phase must also be considered since, for most composite, the cracks associated with the particles form during stressing as shown in Figure 4.4. Unless the thermal expansion coefficients of both phases are equal, these more complex stress distributions must be superimposed on residual stress distributions to 49 From ref. 53 particle in tension \ 7" Q 1p I <:EEE;;:) (::;Ei::) Same as Figure 4.3. Nature of residual thermal stress, crack location, and fracture path (broken lines) 50 From ref. 53 Figure 4.4. Schematic of the enhanced tensile stresses around spherical particles. The location of cracks and fracture path (broken lines) 51 obtain crack locations and fracture modes. Although no direct technique has been developed to determine the crack size due to the thermal residual stresses and differential elastic moduli of the two phase, in this analysis we introduce the simple micro-crack density parameter to estimate the effective elastic moduli of the fiber-reinforced composite. In the estimation of effective elastic moduli for improper interface-bonding materials, we will introduce the interphase degradation model and tangential displacement discontinuity model. For simplicity, we assume that debonded region will occur only at the fiber/matrix interface. 4.3.1 Interphase degradation model Let's consider a third phase between the fiber and matrix as shown in Figure 4.5. The existence of an interphase layer has been demonstrated by infra-red spectroscopy, ESP, NMR, electron microscopy and other experimental methods [49,50] . It has also been shown that the thickness of this layer depends on the polymer cohesion energy, free surface energy of the solid and the flexibility of the polymer chains. However, the thickness of the interphase has been shown to be very small in comparison with the size of the two constituent materials. 52 matrix interphase fiber fiber Figure 4.5. Thin interphase BEM model 53 To apply the B.E.M subregional technique, the interphase layer has to be discretized into elements small enough to ensure that nodes on opposite interfaces are sufficiently distant from each other in comparison with the size of the boundary elements. Otherwise, numerical inaccuracies will arise. An alternative scheme to improve the boundary element formulation consists of finding a relationship between the points on opposite faces of the thin layer [68]. Consider the thin layer and apply one-dimensional stress-strain relationships for shear and compression in a finite difference manner. If the thickness of the layer element, 12, is small by comparison with the boundary element lengths, one can write (f) (f) ( ) Ei t t - ( u - um ) n 02' n n n h (f) (f) (m) Ginter t. - (ut - ut ) T. (4.15) where E and G are Young's and shear moduli respectively of the inter inter interphase material and the displacements are referred to the local system of coordinates indicated in Figure 4.5. Two other relationships are provided by the equilibrium conditions i.e. ,tit) 4» ti") - 0. In the interphase model, the effects of improper interfacial bonding strength can be easily considered as the degradation of elastic properties in the interphase region. When the elastic properties of the interphase is equal to those of the matrix, it is 54 considered as a perfect bond. However, when microcracks or voids are formed in the interphase due to the thermal residual stresses, one may expect that the interphase elastic properties will be less than those of the surrounding matrix. For simplicity, we assume that micro-cracks only occur in the interphase region and other two constituent materials perform their inherent elastic properties. Then the degradation of interphase elastic properties can be calculated from micro-crack density a: . We assume that the cracks are unifomly distributed with a characteristic density 5 and an average crack length L in the interphase region as shown in Figure 4.6. Thus, if A is any area which is large compared to L2, this area should contain Afl cracks regardless of the location or geometry of the region A. Since we assume the cracks are mechanically independent, this approximation is valid for a lightly cracked material. The model is defined by two geometric parameters: a length L and an inverse area a (- A”). Walsh [69] examined these parameters to describe certain lightly cracked rock materials. Using the known solution for displacements at a crack [70], we can calculate the interface work of equation (4.2) as AW -I [u]1t a; - 2mc(l-u )(é +2.: )2; (4.16) cracked where ac - (IL2 is the dimensionless measure of crack density and E is Young's modulus of an uncracked homogeneous and isotropic matrix body. 55 crack Figure 4.6. Line cracked material model 56 The average stress-strain relations corresponding to the strain energy of cracked material is [70] l 62 - T {02 - v(03 + 01)} -1{a-<+ >}+2<1-2)/E 63 -E— 3 V 02 01 IN V 03 l+v 2 ‘23- -E— 023+ 2xn(1-u’)aza/E. (4.17) They have the effect of making the overall mechanical response anisotropic, i.e. 1 £2 - E {02 ' "2303} 2 1 £3 - E {03 - "3202} 3 - .—1— a 23 G 23 23 where E - E 53 - wanna-VS) V -V "V 23 32 (:23 - G/(l+21nc(l-v2)) Due to the thermal (4.18) (4.19) and mechanical properties of two the constituents, more complex cracks will be formed. For simplicity, we assume the overall cracked material to be isotropic in the sense of an ave rage response . 57 4.3.2 Tangential displacement discontinuity model Due to several damage phenomena, some separation may exist between the constituents at their interface. This debonding is certainly non-uniform in the composite. In order to model the complicated effect of debonding on the thermo-mechanical behavior of the composite, it is 'necessary to introduce a relaxation. of' the tangential displacement continuity condition, which simulates separation at the interface [27,28]. The partial discontinuity model is that in.*which the continuity of the normal displacements and tractions prevail at the fiber/matrix interface. However, some degree of slip, reflected. by a jump in the tangential displacement is allowed. Thus [ 0“,] TB - 0 [ u1 ] n‘ - O on 1‘ (4.21) interface where [. ] denotes the jump of the bracketed expression across the interface, but [u t.] - R t . on P (4.22) 3 interface where ut and tt are the tangential displacement and shear traction vector respectively. R is a constant of proportionality, which represents the degree of bond or lubrication at the interface. Note that the usual case of perfect continuity is obtained by the limit condition R a 0, while the case of lubricated sliding at the interface is obtained by R " w. While, intermediate range, 0 < R < m, this model demonstrates the certain damage state of composite behaviors. Chapter 5 RESULTS AND DISCUSSION 5.1 Thermal residual stresses Here we employ the superposition method to obtain a numerical approximation of the thermally-induced residual stresses during the curing process in the fundamental unit of a fiber-reinforced composite as shown in Figure 3.1. The thermal residual stresses at room temperature (21°C) after cool down from cure temperature (177 °C) were calculated using the boundary element analysis in conjunction with the S Glass/Hercules 3501-6 resin composite material properties listed in Table 5.1. Although the matrix material has temperature dependent properties [45], for simplicity, we assume that the matrix properties are constant throughout the cure process. The predicted thermal residual stresses at the fiber/matrix interface for the S Glass/epoxy composite, assuming a fiber volume of 60 percent, are indicated in Figures 5.1 and 5.2. As can be seen in Figure 5.1, the normal stress on: is primarily compressive, the highest values (an- -35.5 Mpa) occurring in the regions of closest fiber spacing. In the resin rich region, the normal stress is tensile although its magnitude is small (9.6 Mpa). The shear stress distribution is indicated in Figure 5.2. The highest interfacial shear 58 59 om AN 00 uw>v mufimoasou hxoem\mmmao w m we wommumucw wnu um mmmuum Hmsvwmmu Hmauonu Hmahoz .H.m wuswfim m mootmfi: x_.;oE\..mn: of 0:02. on om may om mm . o _ _ _ _ Owl B_manoo >xoam\mmo_o m mE:_o> N ow 10¢! .. \. “o (de) 889.113 lonpgsea iDLweqi [DUJJON 60 AN oo um>v ouamomfiou mxoao\mmmaw m o no unawuwucw mnu um ammuum Hmsvwmmu awakens Hmaucmwcma .~.m muswam m mootBE xEoE\._mn: or: 0:02 om mm om we on m_ o T b p _ P _ ON] mtmanoo >xoam\mmo_o m 1 mE20> N om 109! o .f I s ro— e - _ _ L _ _ muN fiuoi 94D (de) seems lonplsea |DLUJ8LLL [onus 61 stress is 15.5 Mpa. It will be noted that highest shear stresses do not occur at the same locations around the interface as the normal stresses. The present boundary element analysis is in good agreement with F.E.M [71] as shown in Table 5.2. Table 5 . 1 Material properties of S Glass/epoxy composite constituents (Ref.7l) Materials Properties E: - 86 Gpa S Class Fiber ut - 0.22 alt-5x10“5 °/c E. - 4.3 Gpa Hercules 3501 - 6 ”. ' 0'3“ _6 0 matrix a. - 40 x 10 / C Table 5.2 Comparison of thermal residual stresses at the fiber/matrix interface at room temperature (Vt - 60 percent) ** a 7 Node rr Hex rr Min Max B.E.M ~35.5 9.6 15.4 70 F.E.M * -31 10 13.5 230 * Adams [71] ** see Figure'3.2 units - Mpa Figures 5.3 and 5.4 show the normal and tangential stress distributions at the three characteristic points A,B and C versus fiber volume fraction V:' It can be seen from these figures that, at 62 the minimum interfiber distance, the stresses are compressive normal (radial) stresses and tensile tangential (hoop) stresses. There is a region around point C which is in a state of biaxial tension. The state of stress at point C is always hydrostatic tension since this point is an isotropic point. It can also be deduced that an increase of V1 results in an increase of the compressive normal stress at point A. Koufopoulos and Theocaris [72] showed that this stress increases very rapidly for higher values of Er/En' It is therefore expected that for some combination of high V: and Et/En’ the compressive normal stress will exceed the strength of the matrix and initiate a crack. The tensile tangential stress at the point A decreases rapidly with increasing V:' This stress may become compressive at V! 2 70 percent creating a biaxial compression at the point A. A tensile normal stress is developed at point B when Vf z 55 percent. It is in fairly good agreement with a value of VI 2 54 percent in the case of rigid inclusion, given by Hsu [73] and VI 2 49 percent for Et/Em - 2.4 and Vi z 54 percent for Et/Em - 11.4 [72]. This tensile normal stress at the point B may initiate a bond crack at the interface. Since the tangential (hoop) stress distribution curve runs parallel to the normal stress distribution curve the tangential stress may also be large enough to initiate a crack in the radial direction. 63 ouwmoaaou zxoam\mmmao m m ecu muonam mo amuse mumsum m ca 0 use m .< mucuoe um mommmuum Hmsvfimou Hmahmnu Hmauoz .m.m musmwm 1> 0.0 _ c0500.: mE3_o> twp: md _ To Md Nd L .L --r__.__1,- utmanoo >xoam\mmo_o m om! 1.0.?! 6 “£03 030.302 I Am “50$ EJEES aria _ 2 “£03 E:E_xoz me -la..lc|.’.llul'\l'uo’ ‘lfluh: 1.4!!4-Il'." linens-IF .llI-l'... nil--.- om 888.118 [anlsea [DLUJGLLL |DUJJON 64 euamoeaoo mxoee\meeau m e you euenam mo heave whence e ca 0 use m .< mucaoa ue eemeeuue Heseaeeu Heahenu Heaueewcey .e.m euswfih c> c0500.; 0E3_0> 000E A tivltc.. I I1! L ..3 ll - ..‘li..0'||r‘l'o to}; 0..“ 4 1.) ..ELIIII’I-" .4 Q»: r £516-.--i 00.600800 >x00e\mm0_d w G “Eon; 200.502 ole W m dEOQ ale _ < argon. Glo ......i..i-.-..-z _ 2.. -..... :1: _: -...s.;.....:.. ..---_..i ...... .. ...-..L. strietnluli Ad dd mud ed Md m. E 990 830118 [Unplsag [01111314] [DHUSBUDL j CD eceuu 0>Huoemmm .m.m euswfim ... .> c0500.; 0E3_0> EQE dd md ed 1 _ .- k 1.: ..‘Ie II-.u. m1xm.d..hs\~ NdHS _ \\ 62 Fm¢F.HEO 62 mhmdnd 1 \ 7000:: 63.6530 3050.6 0.5200: \ .000E 00010 n whemceymtco 1W 0:300 .632 m.cEmoI .2005 <00 Magma: Sure BEE $2885 5.0m 1121:: who N. 104 Tm; Adflmmv _0D0E 0001.035 .2.m.m . _ “ rile--11.-.. u: L .5-..l sLil-.- p i 1....— .----:-shoe---::-.n.r.mu.mw smnpow 100143 SSJBASUDJi 0.11109113 67 eufimoaaou 000 m0 MN 0% M measeos xasn caeuue eceaa 0>Huuewwm .oom ouswfim _> 00:00: 0E3_0> 000$ ed M d T. ........... Li.- air-.----. ed Md N lb b L x. a ....... [IL KP \\ ensue: Nous \\\ \ -62 $3.10 02 meme-“.0 X. 600E “cedflmcoolrem PEI 0.0 0:300 0000: 0.5000: Z 609.: <00 030001 «10 8“.de .000E emocatefiz Edd elm. Adnmv _000E 00203,: .E.m.m $10 u---h-. . ...i 1:: i.—..- .. u. ....I.— u. -. . -. -..--... * 6- - - - ... -Iu. ..!-..I.!|L.. .d 10 Snmpow we was eumd SAD-38113 68 modulus K; is in an excellent agreement with Hashin's CCA model [65]for the entire range of volume fractions. For high Vt ( z 50% ), the present square array model gives an effective transverse shear modulus 6:3 which is higher than those predicted by other (cylindrical) array analyses. A comparison with other analyses for 6:3 is given in Table 5.4. Figures 5.7 - 8 show that the transverse effective moduli 15:3 and u* are functions of fiber volume fraction 23 V1' 51:3 and y; of the present square array model are also in good agreement for up to 50 percent fiber volume fraction in comparison with Hashin’s CCA model. According to Laws's investigation [75], the cylindrical array models under-estimate the transverse effective moduli 0:3 and E; for the high fiber volume fraction composites (VI 2 45%) in comparison with experiments, while 11:3 is over-estimated. Adams and Doner [23] and Springer and Tsai [25] also showed that for higher Vt the square array's transverse effective moduli and effective thermal conductivity are in better agreement than those of the cylindrical square array. It is also seen that the interface and interphase models are in excellent agreement with one another if elastic properties of the interphase are equal to those of the matrix (perfect bond). Since the interphase volume fraction is very small compared to that of the two constituents, the interphase region has little effect on the overall mechanical behavior . The values of the interphase thickness 11 and interphase volume fraction V1 for various fiber contents are given in Table 5.5 [50] . 69 00H000500 000 m0 mmm 03H30oe e.wcsow emu0>mceuu e>wuuemwm .m.m euawfim 0% .> :00.00: 6E3_0> 60E dd Md vd Md - : L _ b III-III. .‘|I.II|I‘|. 600:: 002.60:_ Edd 600E 0000060E ._2.m.m m 0:300 .626. 0.50001 5| 0:300 .6003 0.5000: - 3 .3 rub..- vial:- ..I. I '1. L .- 151 1).. III-VI...I.I_I 5| . ‘5 II. 'llfl'llii-flvl. .01 i. .9: .1. ..u'F.ll.ri4-. ‘: I I 1| Uh. 1!. MMngS mdni 03 63.0.4.6 03 £3.10 Nd eAmem (Isd 9QLX) 9:3 0001 agar-sum] pow 8.5 1 Sflifl 70 eufimomaoo 0:0 mo oaueu m.commH0m eeue>mceuu 0>Huoemmm .w.m euswfim .> :0500: 6E3_0> 60E —. m.O ®.O v.0 N.O 0.0 _l L L - L _ L - F P P LII-.-- 0.0 fl EDGE mmOLQLUHE .§.m.m 010 EDGE $00557: .§.w.m mT-m. r w. UCDOD 0005: m.c_£mOI I m 0:300 .630. 0.:EmOI 01:0 . a . MMstA NdHS 62 MNMcencd 7x/ 02 63.0.1.0 w. // _ ,. T a 0 F.11L..- -.-- 0-43.1.1.-. -L ......Li -..L --.-.--.!:...L:©d d .1 .J {in 01103 sluossgod aSJeAsumi sax-110011 71 Table 5.4 * Comparison of Transverse shear modulus 023 with other analyses units - 106 psi Inter- Inter- Hashin' s Hashin ' s Christensen Vt face phase CCA lower 3 phase 0.2 0.2125 0.209 0.2174 0.2043 0.2063 0.3 0.2787 0.2763 0.2765 0.2435 0.2505 0.4 0.3693 0.3656 0.3605 0.2946 0.3121 0 . 5 0 . 508 0 . 5036 0 . 4802 0 . 3642 0 . 4001 0.6 0.730 0.718 0.6525 0.4641 0.5293 Table 5.5 The thickness h and V1 for the interphase region (ref.50) V t 0.2 0.3 0.4 0.5 0.6 h 0.0109 0.0113 0.0177 0.0243 0.0315 V1 0.005 0.011 0.020 0.031 0.044 Figures 5.9 shows the transverse shear modulus ratios versus V1 for various ratios Gt/Gn' It is shown that in the low range of volume fractions, the higher 6/61. ratio composites do not improve the effective transverse shear modulus. CD T—r—F: I 72 T'"”"l"" ' T'- ””‘R"°' "Y“‘w'l— "'T.‘T \ T . N“. T woaumu msasuoa umwnm Nahuma\umnfiw msoaum> you EU\mmu moaumu msaavoa ummnm mmum>mamuu m>auommmm .m.m madman t> c0300: mE:_o> .60; 0.0 0.0 v.0 m0 0.0 L g b _ . p L . '7 IL “50 \ mimanoo xxoam\mmo_o - 10— \ \ , \ - -\ low amuso\.o I T mange-.0 «L .. Eneoxo min flange-so ole j ..- _ _ _ 2-..-- -_ .- _. - . _- L [on “9/29 03103:] .10qu SSJSASUDJi amoegg 73 5.3 The elastic behavior of damaged composites The damage characteristics at the fiber/matrix interface due to thermal residual stresses, voids, and inherent micro-cracks may be too complex to describe by simple mathematical tools. Due to localized stress concentrations and micro-crack initiation, the damage is not uniformly distributed. throughout the interfacial region. For simplicity, we here assume that the elastic moduli of the interphase region will be uniformly reduced due to the micro-crack density s. As shown in Figure 5.10, the shear modulus is dramatically decreased with an increase in micro-crack density n. In order to establish the shear modulus of the damaged interface, we represent the interphase condition using a degradation factor DF as shown in Figure 5.11. If there is no interphase damage (DF - 0), the shear modulus of the interphase will be the same as that of the matrix. The results for the effective elastic moduli of the damaged-interphase composite are plotted in Figures 5.12 - 13. For small values of DF. the damage effects are not significant. The reason is that the micro-crack density n increases steadily at low values of DF as shown in Figure 5.11. However, for high values of BF, the damage effects are severe. The effective elastic moduli, K; and 0:3, are dramatically decreased by an increase in DF. Figures 5.2.14 -15 also demonstrate that the damaged behavior of the effective shear and plane strain bulk moduli at 60 percent fiber volume fraction and various degradation factors DF. 74 x zuwmsmv xomuolouoaa ou wow 5 vexomuo 0\ 0 oaumu msaavoa Human momsdumuafi vmxomuu 2 36ch xoocolotoi .oe.m acumen Mw.mv mw.m0 .v.m0 Nugu Ru.mv L L _ H L L L h L F p — b F LL L b If 0.0 T iii/ 4 ,- fwd fl T .. rmd ”- j m Nr 830.: 20333 IN.— “ Eoum vmumHsuamo Ellirfy rli-PI.-—t-;+s..a-alLu ...... _.elt;! p:-:i:-;g:-zaf -_.?:Ln -_ ;I;;;@ttlnuillps dJSlUf wO/panooloo ogog sngnpow Joaqg panama asoq 75 we nouomm soaumvmuwmv mmmnduouafi vow x mufimcmv xomuolouoas somaumn nanmsofiumHmu och .0H.m ouswfim .5 L803 cozonocmon mmocaLBE 0.0 0.0 v.0 N0 0.0 fl P L F L P L iiilLinl! ..... .I.L!I.tlr . I O O \4 _ m . LI IN 0 _ .. 15 _ _ _ _ . W... 10.0 T 0.00 L--..-..i L-.. ,-a....L-, ...-3;...- , L --L‘ _.f ‘_- illnfio. _ 31 Misuap mama—0.43m 76 50 msasvoa ummnm mumsaumuca mo sofiumvmuwmv mm 0:“ OH 05‘ U mfiHflfiOE mer—m OMHO>QGMHU 0>HUUUHMN .1 L> c0500: mEBo> LmQE .Ns.m ouswfim 0.0 0.0 T0 0.0 m0 .rllxll.:z. :-.:.:.: _ .:--a-1,. _ .-2:i:.~.,-..a.||hli|c:.:,-. a :z--1|i|+|.mu.mw . j r T L. .. T ...?- m0 r ... QHH\..\..\..\ T‘s... .. a f T , _ o H... mam-one; Non: 1.0 w. a: 5:.qu a: mmmend . m . .. .h\ 0.01.00 I . WT .. GOULD : F©.O ,L.\ roams «La . f...“ Gonna mum _ . _ _ onus @Io . O CZ .*. (18d 90L X) snmpow .10qu SSJBASUDJi eAnoeHg 77 aw wadsvoe ummnm mmmnnumuaa mo coaumvmuwmv MN man 0» mav x maasvoa xasn cwmuum madam o>Huommmm a. .> 00:00: 0E3_o> 50C 0.0 p .m~.m muswam .v.mv mugu mfiflu L -.a F _ p 0.0 DOOHEA Ndfli _m_2 Fwt _..HEO _m_2 MNMJVIILO 1.mwgu mdnha odnmo TOHLQ Ndflha on“. ..0:. -- g-._-sy +_.___..._.. - ---..- TO.— m4 >1 CZ Xv (18d 90L X) sninpow mng ugmqs euold aAgpebg 78 muamoaaou manao> umnam N ow um mmo msaavoa * ummnm m>fiuumwum may co ma wouUMM coaumvmuwmv mo mummSHmcH .c~.m wuswfim ...Q .303 005000.000 30:05.32 wd md v.0 Nd L L Lillrllsll! L . _ . mmdne; Ndni . BE PwVFdHEO 6.2 mum-VH6 -/ mtmoarcoo mEEo> .00: Row Thu—“w --. T... --.... ... -- .1--. -5--51-52.55 5.55. _ L. 5;... . _ -_ 5.-.. :5...-L!s- in. ..---. - ..-- sir—55:3: 5- .L . 5 f/Iil... 0. o O. o 5.0.0 I 03 CD L.-_... O CZ -¥ (18d QOLX) smnpow 1:)qu esxaAsumi ampagg 79 0.0 _ .... mufimanou maaao> umnfim N 00 um mmx mafiavoa xaaa :Hmuum mamaa m>wuommmm mnu :0 ma u0u00m coaumvmuwmv mo muamsamaH .m~.m muswfim 000000000rka00000000 mmOLQLmfi: 0.0 . .. #0 N0 . L L i i ...‘l mmdusz N055 62 53.0“...0 .22 037.0 8600500 0E:_0> .60: N00 0.0 0.0 50.— ‘r IN; I F I I I 5 I L I I I I I I I I I I I 1 CZ 318 BSDId BAQOSHE IDJ (18d 90 IX) II‘IPOW Wing U 85’": 80 Introduction of a tangential displacement discontinuity at the interface is another approach for examining the mechanical behavior of the damaged composite as described in equations (4.21 - 22). This model allows a jump of the tangential displacements at the fiber/matrix interface. If R - 0, one has a perfect interface bond. For R -* no, one has pure sliding at the interface. In the intermediate range (0 < R < an), it is very difficult to interpret the physical meaning associated to the behavior of the interface. The results are shown in the Figures 5.16 -l9. The effective transverse shear modulus will be significantly decreased by an increase in R value. Specifically, for a certain R range (0 s R s 5), the reductions are dramatic. However, for R z 50, the decrease of the effective transverse shear modulus is steady. The plane strain bulk moduli remain the same as the perfect bond case for the entire range of R values. The reason may be that the shear stress at the fiber/matrix interface is much smaller than the normal stress due to the biaxial displacement boundary conditions, i.e. u2 - u3 - uo. In Table 5.6, the damaged effective plane strain bulk modulus and transversly shear modulus are listed for the interphase degradation model for the case of 60 percent fiber volume fraction. 81 zuaacauaoowwv uamamumadmfiv HafiuaowaMu wnu ou 0:0 0M0 msasvoa ummnm omum>mcmuu m>auoomwm .0H.0 muswam .> 00:00: mEJ_0> $0; 0.0 0.0 T0 0.0 m0 h L b I— p L h 0.0 - T ... \\ \. N.O “\m T 00.05.: N0": .40 .ms. .03.“...0 5: 0305.0 . 3.0 885... I oflmum 05... 50.0 0 any. a!» 0.05m «I... floum min . . .Amnc cum $50 .500 O CZ -# (13d 90L X) snmpow .50qu BSJeAsqui eAIpeHg 82 zuasaauaoomwv ucmamomanmwv Hmwusmwamu mnu ou osv mNM msasvoa xasn cwmuum mamad m>auoomwm .n~.m muawfim « .> 00000: 00.5.0». .607. 0.0 0.0 #0 0.0 N0 . L L P . 00.0“2: N.0HS 62 F0¢TH50 62 000.¢H.0 >I CZ ”It (18d 90L X) smnpow wing UIDJlS euold eAIpeHE ...l..I.I-5-.; .--il.i-i..---_- ...I-.......-I Lilli - --...L -5.-. ....I. III...- 5I..- -._.|---I..iI. r: .I.,I.||I_.l m. P 83 mufimoaaoo masao> umafim N 00 um mmo maasvoe umwnm 00u0>mamuu 0>fiuoowmw mnu co m uouomm xuwscfiuaoomav ucoamomadmav Hmfiucmwamu mo wonmsamcH .0~.0 muawfim 0 350000006 0008002020 60.000000 0.0 0... 0.0 0.0 0.. .0 _0I _ _ Ir F . h + L p -..-.-IIAWO.O 00.0....5 0.0.55 w a... 53.0"...0 .mz 00.0”.0 . T 000000000 9029 .00: x00 . _.||5I-I.. -_ .- .- II.~-I- -IIIT-.I-:.;. H -.---. .-.-LIII- _..II!-I1_ -- --I_-.....-I-.--.. . --IIIIHIO. _. ...E) CZ (18d SOLX) SfllnpOW JDeLiS eSJeASUDJi GAHOS‘LLB 84 mufimoaaoo wasao> umnwm N 00 um max msasvoa xasn cqmuum onwan 0>wuomwmw 050 so a. m nouomu %ua:cfiuaoomuv ucmamomaamfiv amauamwcmu mo mocmsamcH .0~.0 005000 1 350500020 000000003000 00.00000? W . m .r . . 0:0 0.0 ON 0.— L 0 _ - 0.0 0 LI L '40.— . 00.0“5: m.0HS 62 0.070320 62 0000......0 IN.— .5./...! 00009000 0E:_0> 000: N00 [04 w . . _.... 1--.-..0- ---. -. -_ .--...I. .. ..II_ I- -.-. - 75-35.. . 0-x-..- I--..r....I.----IL. .>I CZ .X (18d BOLX) . sngnpow >11fl8 UIDflS aUDId BAIWBJB Damaged effective plane strain bulk modulus and transversely shear 85 Table 5 . 6 modulus for the interphase degradation models at the V: - 0.6. unit - 106 psi * * * *- DF G K DF G K 23 23 23 23 0.0 0.718 1.30 0.5 0.632 1.156 0.1 0.7126 1.282 0.6 0.581 1.103 0.2 0.696 1.262 0.7 0.525 1.021 0.3 0.6772 1.235 0.8 0.4h2 0.893 0.4 0.653 1.203 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS A boundary element method has been presented for the analysis of thermally-induced residual stresses due to the cool-down process and effective transverse elastic properties for a transversely isotropic fiber-reinforced composite material. The square array of fibers as a fundamental unit has been applied to a number of problems and is in good agreement with other techniques. The main advantage of this approach over other analytical techniques is its ability to handle interfiber effects and various fiber cross-sectional shapes. Furthermore, only the, external boundary and interface needed to be discretized unlike the finite element technique. The numerical calculations were performed on a personal computer. In all problems, simple linear displacements and constant tractions on each element were used as shape functions for numerical treatment . The main advantage of this work is its potential for modelling mechanical behavior of damaged interfaces in fiber-reinforced composites. If one measures micro-crack density n at the fiber/matrix interface using non-destructive methods (e.g. acoustic emissions), the degradation of elastic properties of the interphase can be calculated. Then, with the present interphase degradation model, one can determine 86 87 the effective elastic properties of the fiber-reinforced composite material. ‘Unlike the tangential displacement discontinuity’ method, this method gives a more physical meaning for a damaged or improperly bonded interfacial region. Although axisymmetric and three-dimensional analyses have not been discussed here, it is felt that the method presented here could be extended to axisymmetric and three-dimensional micro-mechanics composite analysis without unreasonable difficilty. This extension will provide a full analysis of the effective elastic properties and thermally-induced residual stress distributions for composites. BIBLIOGRAPHY 10. ll 12 13 BIBLIOGRAPHY Maxwel, J. C., Treatise on Electricity and Magnetism, lst Ed., Clarendon Press, Oxford, 1873, p. 365 Lord Rayleigh, ”On the influence of obstacles arranged in rectangular order upon the properties of a medium", Philosophical Magazine, Vol. 34, 1892, p. 481 Mura, T., Micro mechanics of Defects in Solids, Martinus Ninhoff, Hague, 1982 Voigt, W., ”fiber die beziehung zwischen den beiden elastizitAtskonten isotroper kbrper", Wied. Ann., Vol. 38, 1889, p. 573 Reuss, A., ”Berechnung der fliessgrenze uon mischkristallen auf grund der plastizitatsbedingung fur einkristalle”, Z. angew. Math. Mech., Vol. 9, 1929, p. 49 Hill, R., "The elastic behaviour of a crystalline aggregate", Proc. Physical Socity, London, Vol. A65, 1952, p. 349 Eshelby, J. D., ”The determination of the elastic field of an ellipsoidal inclusion, and related problems", Proc. Roy. Soc., Vol. A241, 1957, p. 376 Krdner, E., "Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls", Z. Phys., Vol. 51, 1958, p. 504 Budiansky, B., and Wu, T.T., ”Theoretical prediction of plastic strains of polycrystals", Proc. 4th U.S. Nat. Congr. Appl. Mech., 1962, p. 1175 Hill, R., "Theory of mechanical properties of fiber-reinforced materials -'I[. elastic behaviour”, J. Mech. Phys. Solids, Vol. 12, 1964, p. 199 Hashin, 2., "Assessment of the self-constistent scheme approximation: conductivity of particulate composites”, J. Comp. Mater., Vol. 2, 1968, p. 284 Christensen, R. M. and Lo, K. H., "Solution for effective shear properties in three phase sphere and cylinder Models", J. Mech. Phys. Solids, Vol. 27, 1979, p. 315 Hashin, Z. and Shtrikman, ”On some variational principles in anisotropic and nonhomogeneous elasticity”, J. Mech. Phys. 88 14 15 16 17 18 19 20 21 22 23 24 25 26 89 Solids, Vol. 10, 1962, p. 335 Hashin, Z. and Shtrikman, "A variational approach to the elastic behaviour of multiphase materials", J. Mech. Phys. Solids, Vol. 11, 1963, p. 127 Hashin, Z. and Shtrikman, "A variational approach to the theory of the elastic behaviour of polycrystals”, J. Mech. Phys. Solids, Vol. 10, 1962, p. 343 Hill, R., “Elastic properties of reinforced solids; some theoretical principles", J. Mech. Phys. Solids, Vol. 11, 1963, p. 347 Walpole, L. J., ”On bounds for the overall elastic moduli of inhomogeneous systems - I“, J. Mech. Phys. Solids, Vol. 14, 1966, p. 151 Walpole , L. J . , "On bounds for the overall elastic moduli of inhomogeneous systems - II" , J . Mech- Phys . Solids , Vol . 14 , 1966, p. 289 Wu, T. T., "The effect of inclusion shape on the elastic moduli of a two-phase material”, Int. J. solids and structures, Vol. 2, 1966, p. 1 Willis, J. R., ”Bounds and self-consistent estimates for the overall properties of anisotropic composites", J. Mech. Phys. Solids, Vol. 25, 1977, p. 185 Krdner, E., ”Bounds for disordered materials", J. Mech. Phys. Solids, Vol. 25, 1977, p. 137 Adams, D. F. and Doner, 0. R., "Longitudinal shear loading of a unidirectional composite", J. Composite Materials, Vol. 1, 1967, p. 4 Adams, D. F. and Doner, O. R., ”Transverse normal loading of a unidirectional composite“, J. Composite Materials, Vol. 1, 1967, p. 152 Chen, C. H. and Cheng, 8., "Mechanical properties of fiber reinforced composites”, J. Composite Materials, Vol. 1, 1967, p. 30 Springer, 8. G. and Tsai, S. W., "Thermal conductivities of unidirectional materials", J. Composite Materials, Vol. 1, 1967, p. 166 Gogél, W. and Purmanski, P., ”Some inwastigations of effective thermal conductivity of unidirectional fiber-reinforced composite”, Enviromental Effects on Composite Materials, Vol. 2, Ed. Springer, Techomic Publishing, PA., 1984, p. 261 27 28 29 30 31 32 33 34 35 36 37 38 39 40 90 Lene, F. and Leguillon, D., ”Homogenized constitutive law for a partially cohesive composite material", Int. J. solids structures, Vol. 18, 1982, p. 443 Benveniste, Y., "The Effective mechanical behaviour of composite materials with imperfect contact between the constituents", Mech. Materials, Vol. 4, 1985, p. 197 Mura, T., Jasiuk, I., Kouris D. and Furuhashi "Recent results of the equivalent inclusion method applied to composite materials", Composites '86 : Recent Advances in Japan and the United States proce. Japan-U.S. CCM-III, Tokyo, 1986, p.169 Jasiuk, I., Mura, T. and Tsuchida, E., ”Thermal stresses and thermal expansion coefficients of short fiber composites with sliding interfaces”, J. Eng. Mater. Technology, Vol. 110, 1988, p. 96 Mori, T. and Tanaka, R., “Average stress in matrix and average energy of materials with misfitting inclusions", Act. meta11., Vol. 21, 1973, p. 517 Dundurs, J. and Zienkiewicz, O. C., ”Stresses around circular inclusions due to thermal gradients with particular reference to reinforced concrete”, J. Amer. Concrete Inst., 1964, p. 1523 Sternberg, E. and McDowell, E. L., ”On the steady-state thermoelastic problem for the half-space”, Quart. Appl. Math., Vol. 14, 1957, p. 381 Tauchert, T. R., "Thermal stress concentrations in the vicinity of cylindrical inclusions", J. Composite Materials, Vol. 5, 1970, p. 868 Hecker, S. 8., Hamilton, C. H. and Ebert, L. J., "Improved fiber composite tensile performance by mechanical residual stress relief", Transactions Quart. ASMQA, Vol. 62, 1969, p. 740 Sachs, G. and Espey, 6., ”Residual stresses in metals", Iron Age, Vol. 128, 1941, p. 63 Hecker, S. 8., Hamilton, C. H. and Ebert, L. J., "Elastoplastic analysis of residual stresses and axial loading in composite cylinders”, J. Materials, JMLSA, Vol. 5, 1970, p. 868 Hoffman, C. A., ”Effects of thermal loading on fiber-reinforced composites with constituents of differing thermal expansivities", J. Eng. Mater. Technology, Vol. 95, 1973, p. 55 Miller, A. K. and. Adams, D. P., "Inelastic micromechanical analysis of graphite/epoxy composites subjected to hygrothermal cycling", ASTM STP 658, 1978, p. 121 Uemura, M., Iyama, H. and Yamaguchi, Y., "Thermal residual 41 42 43 44 45 46 47 48 49 50 51 52 53 54 91 stresses in filament-wound carbon-fiber-reinforced composites", J. Thermal Stresses, Vol. 2, 1979, p. 393 ' Weitsman, Y., "Residual thermal stresses due to cool-down of epoxy-resin composite", J. Appl. Mech., Vol. 46, 1979, p. 563 Herrmann, K. and Matteck, R., ”Thermal stresses in the unit cell of a fiber-reinforced Material”, J. Thermal Stresses, Vol. 2, 1979, p. 15 Rohwer, K. and .Jiuq X., 'Micromechanical curing stresses in CFRP", Composite Science and Technology, Vol. 25, 1986, p. 169 Avery, W. B. and Herakovich, C. T., ”Effect of fiber anisotriopy on thermal stresses in fibrous composites", J. Appl. Mech., Vol. 53, 1986, p. 751 Adams, D. P., “A Micromechanics analysis of the influence of the interface on the performance of polymer-matrix composites”, J. Reinforced plastics and composites, Vol. 6, 1987, p. 66 Vedula, M., Pangborn, R. N. and Queeney, R. A., ”Fiber anisotropic thermal expansion and residual thermal stress in a graphite/aluminium composite", Composites, Vol. 19, 1988, p. 55 Vedula, M., Pangborn, R. N. and Queeney, R. A., "Modification of residual thermal stress in a metal-matrix composite with the use of a taileres interfacial region”, Composites, Vol. 19, 1988 Hahn, H. T. and Kim, K. S., "Hygroscopic effects in aramid fiber/epoxy composite". J. Eng. Mater. Technology, Vol. 110, 1988, p. 153 Rich, M» and Drzal, L. T., "Interfacial properties of some high-strain carbon fibers in an epoxy matrix”, J. Reinforced plastics and composites", Vol 7, 1988, p. 145 Sideridis, E., "The ineplane shear modulus of fiber reinforced composites as defined by concept of interphase", Composite Science and Technology, Vol. 31, 1988, p. 35 Davidge, R. W. and Green, T. T., "The strength of two-phase ceramic/glass materials", J. Materials Science, Vol. 3, 1968, p. 629 Evans, A. G. and Fu, Y., ”Some effects of microcracks on the mechanical properties of brittle solids - II", Acta Meta11., Vol. 33, 1985, p. 1525 Lange, F. P., "Fracture of brittle matrix, particulate composites", Composite Materials Vol.5, Ed. Broutman, Academic Press, 1974 Rubinstein, R. and Fitzgerald, J. E., "Further developments of a 55 56 57 58 59 60 61 62 63 64 65 66 67 68 92 damage theory for filled polymers", Polymer Eng. Science, Vol. 19, 1979, p. 151 Danson, D., "Linear isotripic elasticity with body forces", Progress in Boundary Element Method, Vol. 2, Ed. Brebbia, Pentech press, 1983 Cruse, T. A., "Boundary-integral equation method for three-dimensional elastic fracture mechanics analysis", AFOSR-TR-75-0813, 1975 Brebbia, C. A”, The Boundary Element Method for Engineers, Pentech press, London, 1978 Lachat, J. C. , "A further development of the boundary integral technique for elastostatics", Ph.D Thesis, University of Southampton, 1975 ‘ Rizzo, F. J. , "An integral equation approach to boundary value problems of classical elastostatics", Quart. Appl. Math. Vol. 25, 1967, p. 83 Riccardella, P. C., "An implementation of the boundary-integral technique for planar problems of elasticity and elastoplasticity", Report No. SM-73-10. Department of Mechanical Engineering, Carnegie Mellon University, 1973 Kumar, V. and. Mukherjee, 8., ”A. boundary integral equation formulation for time-dependent inelastic deformation in metals", Int. J. Mech. Sci., Vol. 19, 1977, p. 713 Mukherjee, S. and Ghosh, 8., "Boundary element method analysis of thermoelastic deformation in nonhomogeneous media", Int. J. solid structures, Vol. 20, 1984, p. 829 Sur, U., "A numerical method for the treatment of kanked cracks in finite bodies", Ph.D Thesis, Michigan State University, 1987 Haenerr, J. and .Ashbaugh, N., ”Three-dimensional stress distribution in a unidirectional composite", J. Composite Materials, Vol. 1, 1967, p. 54 Hashin, 2., ”Analysis of properties of fiber composites with anisotropic constituents”, J. Appl. Mech., Vol. 46, 1979, p. 543 Benedetto, A, T. and. Nicolais, L., " .Advances in composite materials", Ed. Piatti, Applied Science Publishers, p. 153 Binns, D. B., "Science of Ceramics”, Ed. Steward, Vol. 1, Academic press, NY, 1962, p. 315 Venturini, W. S. and Brebbia, C. A., "Application in geomechanics", Topics in Boundary Element Research, Ed. Brebbia, Springe-Verlag, Vol. 1, 1984 69 70 71 72 73 74 75 93 Walsh, J. M., J. Geophys. Res. Vol. 70, 1965, p. 399 Fitzgerald, J. E. and Rubinstein, R., "Metatropic material : a theory for permament memory effects", Report AFOSR-72-2322, 1976 Adams, D. F. , "Influence of the polymer matrix on the mechanical response of a unidirectional composite", Progress in Science and Engineering of Composites Vol. 1, ICCM-IV, 1982, p. 507 Koufopoulos, T. and Theocaris, P. 8., ”Shrinkage stresses in tao-phase materials”, J. Composite Materials, Vol. 3, 1969, p. 308 Hsu, T. T., "Mathematical analysis of shrinkage stresses in a model of hardened concrete", Proc. Amer. Concr. Inst., Vol. 60, 1963, p.371 Christensen, R. M., Mechanics of Composite Materials, 1979, John Wiley 6: Sons, New York Laws, N., ”Composite materials : Theory vs. Experiment", J. Composite Materials, Vol. 22, 1988, p. 396 "I11111111111101W