Nonlinear Composites

Aravas, 1995, Steady-state creep of fiber-reinforced composites: Constitutive equations and computational issues, Internat. J. Solids Structures, 32, 2219, 10.1016/0020-7683(94)00251-Q Avellaneda, 1987, Iterated homogenization, differential effective medium theory and applications, Comm. Pure Appl. Math., 40, 527, 10.1002/cpa.3160400502 Bao, 1991, Particle reinforcement of ductile matrices against plastic flow and creep, Acta Metall. Mater, 39, 1871, 10.1016/0956-7151(91)90156-U Bensoussan, 1978 Beran, 1965, Use of the variational approach to determine bounds for the effective permittivity of random media, Nuovo Cimento, 38, 771, 10.1007/BF02748596 Berveiller, 1979, An extension of the self-consistent scheme to plastically-flowing polycrystals, J. Mech. Phys. Solids, 26, 325, 10.1016/0022-5096(78)90003-0 Bhattacharya, 1997, Elastic energy minimization and recoverable strains of polycrystalline shape-memory materials, Arch. Rational Mech. Anal., 10.1007/s002050050049 Bishop, 1951, A theory of the plastic distortion of a polycrystalline aggregate under combined stresses, Philos. Mag., 42, 414, 10.1080/14786445108561065 Bishop, 1951, A theoretical derivation of the plastic properties of a polycrystalline face-center metal, Philos. Mag., 42, 1298, 10.1080/14786444108561385 Bornert, 1996 Bornert, 1996, Morphologically representative pattern-based bounding in elasticity, J. Mech. Phys. Solids, 44, 307, 10.1016/0022-5096(95)00083-6 Bouchitte, 1991, Homogenization, plasticity and yield design, 107 Brockenborough, 1991, Deformation of metal-matrix composites with continuous fibers: Geometrical effects of fiber distribution and shape, Acta Metall. Mater, 39, 735, 10.1016/0956-7151(91)90274-5 Budiansky, 1959, A reassessment of deformation theories of plasticity, J. Appl. Mech., 26, 259, 10.1115/1.4011992 Budiansky, 1965, On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids, 13, 223, 10.1016/0022-5096(65)90011-6 Budiansky, 1970, Thermal and thermoelastic properties of composites, J. Comp. Mater, 4, 286, 10.1177/002199837000400301 Budiansky, 1982, Void growth and collapse in viscous solids, 13 Buryachenko, 1996, The overall elastoplastic behavior of multiphase materials with isotropic components, Acta Mech., 119, 93, 10.1007/BF01274241 Buryachenko, 1989, Effective field method in ideal plasticity theory for composite materials, Prikl. Mekh. Tekhn. Fiz., 3, 149 Buryachenko, 1989, Prediction of nonlinear Row parameters for multi-component mixtures, Prikl. Mekh. Tekhn. Fiz., 4, 63 Christensen, 1979, Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, 27, 315, 10.1016/0022-5096(79)90032-2 Christman, 1989, An experimental and numerical study of deformation in metal-ceramic composites, Acta Metall. Mater., 37, 3029, 10.1016/0001-6160(89)90339-8 Chu, 1971, Plastic behavior of composites and porous media under isotropic stress, Internat. J. Engrg. Sci., 9, 971, 10.1016/0020-7225(71)90029-2 de Botton, 1995, The effective yield strength of fiber-reinforced composites, Internat. J. Solids Structures, 32, 1743, 10.1016/0020-7683(94)00203-9 de Botton, 1992, On the ductility of laminated materials, Internat. J. Solids Structures, 29, 2329, 10.1016/0020-7683(92)90219-J de Botton, 1993, Elastoplastic constitutive relations for fiber-reinforced solids, Internat. J. Solids Structures, 30, 1865, 10.1016/0020-7683(93)90222-S de Botton, 1995, Variational estimates for the creep behavior of polycrystals, Proc. Roy. Soc. London Ser. A, 448, 121, 10.1098/rspa.1995.0009 de Buhan, 1983, Homogénéisation en calcul à la rupture: le cas du matériau composite multi-couche, C. R. Acad. Sci. Paris Ser II, 296, 933 de Buhan, 1991, A homogenization approach to the yield strength of composites, European J. Mech. A Solids, 10, 129 Dendievel, 1991, Bounds for the creep behavior of polycrystalline materials, 175 Drucker, 1959, On minimum weight design and strength of non-homogeneous plastic bodies, 139 Drucker, 1966, The continuum theory of plasticity on the macroscale and the microscale, J. Mater., 1, 873 Duva, 1984, A self-consistent analysis of the stiffening effect of rigid inclusions on a power-law material, J. Engrg. Mater. Tech., 106, 317, 10.1115/1.3225723 Duva, 1984, Constitutive potentials for dilutely voided non-linear materials, Mech. Mater., 3, 41, 10.1016/0167-6636(84)90013-9 Dvorak, 1992, Transformation field analysis of inelastic composite materials, Proc. Roy. Soc. London Ser. A, 437, 311, 10.1098/rspa.1992.0063 Dvorak, 1987, A bimodal plasticity theory of fibrous composite material, Acta Mech., 69, 219, 10.1007/BF01175723 Dvorak, 1988, An experimental study of elastic-plastic behavior of a fibrous boron-aluminum composite, J. Mech. Phys. Solids, 36, 655, 10.1016/0022-5096(88)90003-8 Einstein, 1906, Eine Neue Bestimmung der Moleküldimensionen, Ann. Physik, 19, 289, 10.1002/andp.19063240204 Ekeland, 1974 Eshelby, 1957, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. Roy. Soc. London Ser. A, 241, 376, 10.1098/rspa.1957.0133 Fleck, 1997, Strain gradient plasticity, Adv. Appl. Mech., 33, 295, 10.1016/S0065-2156(08)70388-0 Francfort, 1986, Homogenization and optimal bounds in linear elasticity, Arch. Rational. Mech. Anal., 94, 307, 10.1007/BF00280908 Gărăjeu, 1996, Effective behaviour of porous viscoplastic materials containing axisymmetric prolate ellipsoidal cavities, C. R. Acad. Sci. Paris Ser. II, 323, 307 Gărăjeu, 1997, Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles, J. Mech. Phys. Solids, 45, 873, 10.1016/S0022-5096(96)00128-7 Germain, 1983, Continuum thermodynamics, J. Appl. Mech., 50, 1010, 10.1115/1.3167184 Gilormini, 1995, Insuffisance de l'extension classique du modèle auto-cohérent au comportement non-linéaire, C. R. Acad. Sci. Paris Ser. IIb, 320, 115 Golaganu, 1993, Approximate models for ductile metals containing non-spherical voids–+ase of axisymmetric prolate ellipsoidal cavities, J. Mech. Phys. Solids, 41, 1723, 10.1016/0022-5096(93)90029-F Golagdnu, 1994, Approximate models for ductile metals containing non-spherical voids+ ase of axisymmetric oblate ellipsoidal cavities, J. Engrg. Mater. Tech., 116, 290, 10.1115/1.2904290 Gurson, 1977, Continuum theory of ductile rupture by void nucleation and growth, J. Engrg. Mater. Tech., 99, 2, 10.1115/1.3443401 Hashin, 1962, The elastic moduli of heterogeneous materials, J. Appl. Mech., 29, 143, 10.1115/1.3636446 Hashin, 1980, Failure criteria for unidirectional fiber composites, J. Appl. Mech., 47, 329, 10.1115/1.3153664 Hashin, 1983, Analysis of composite materials: A survey, J. Appl. Mech., 50, 481, 10.1115/1.3167081 Hashin, 1962, On some variational principles in anisotropic and nonhomoge-neous elasticity, J. Mech. Phys. Solids, 10, 335, 10.1016/0022-5096(62)90004-2 Hashin, 1962, A variational approach to the theory of the elastic behavior of polycrystals, J. Mech. Phys. Solids, 10, 343, 10.1016/0022-5096(62)90005-4 Hashin, 1963, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids, 11, 127, 10.1016/0022-5096(63)90060-7 He, 1990, On the flow and creep strength of power-law materials containing rigid reinforcements, Mater Res. Soc. Symp. Proc., 194, 15, 10.1557/PROC-194-15 Hershey, 1954, The elasticity of an isotropic aggregate of anisotropic cubic crystals, ASME J. Appl. Mech., 21, 236, 10.1115/1.4010899 Hervé, 1991, A propos de l'assemhlage des sphères composites de Hashin, C. R. Acad. Sci. Paris Ser II, 313, 857 Hervé, 1993, N-layered inclusion-based micromechanical modelling, Int. J. Engrg. Sci., 31, 1, 10.1016/0020-7225(93)90059-4 Hill, 1963, Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, 11, 357, 10.1016/0022-5096(63)90036-X Hill, 1965, Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, 13, 89, 10.1016/0022-5096(65)90023-2 Hill, 1965, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 213, 10.1016/0022-5096(65)90010-4 Hill, 1967, The essential structure of constitutive laws for metal composites and polycrystals, J. Mech. Phys. Solids, 15, 79, 10.1016/0022-5096(67)90018-X Hom, 1992, Three-dimensional finite element analysis of plastic deformation in a whisker-reinforced metal matrix composite, J. Mech. Phys. Solids, 40, 991, 10.1016/0022-5096(92)90059-B Hu, 1996, A method of plasticity for general aligned spheroidal void of fiber-reinforced composites, Internat. J. Plasticity, 12, 439, 10.1016/S0749-6419(96)00015-0 Hutchinson, 1970, Elastic-plastic behavior of polycrystalline metals and composites, Proc. Roy. Soc. London Ser A, 319, 247, 10.1098/rspa.1970.0177 Hutchinson, 1976, Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. Roy. Soc. London Ser. A, 348, 101, 10.1098/rspa.1976.0027 Germain, 1983, Continuum thermodynamics, J. Appl. Mech., 50, 1010, 10.1115/1.3167184 Kachanov, 1971, 20 Kailasam, M., and Ponte Castñeda, P. (1997). A general constitutive theory for linear and nonlinear particulate media with microstructure evolution. Submitted for publication. Kailasam, 1997, The effect of particle size, shape, distribution and their evolution on the constitutive response of nonlinearly viscous composites. Part I. Theory, Philos. Trans. Roy. Soc. London Ser A, 355, 1883 Kailasam, 1997, The effect of particle size, shape, distribution and their evolution on the constitutive response of nonlinearly viscous composites. Part II. Examples, Philos. Trans. Roy. Soc. London Ser A, 355, 1901 Kohn, R. V., and Little, T. D. (1997). Some model problems of polycrystal plasticity with deficient basic crystals. Submitted for publication. Kohn, 1986, On bounding the effective conductivity of anisotropic composites, 97 Kreher, 1990, Residual stresses and stored elastic energy of composites and polycrystals, J. Mech. Phys. Solids, 38, 115, 10.1016/0022-5096(90)90023-W Kröner, 1958, Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Z, Physik, 151, 504, 10.1007/BF01337948 Kröner, 1977, Bounds for effective elastic moduli of disordered materials, J. Mech. Phys. Solids, 25, 137, 10.1016/0022-5096(77)90009-6 Laws, 1973, On the thermostatics of composite materials, J. Mech. Phys. Solids, 21, 9, 10.1016/0022-5096(73)90027-6 Laws, 1979, The effect of fibre length on the overall moduli of composite materials, J. Mech. Phys. Solids, 27, 1, 10.1016/0022-5096(79)90007-3 Leblond, 1994, Exact results and approximate models for porous viscoplastic solids, Internat. J. Plasticity, 10, 213, 10.1016/0749-6419(94)90001-9 Lee, 1991, Effect of inclusion shape on the stiffness of nonlinear two-phase composites, J. Mech. Phys. Solids, 39, 627, 10.1016/0022-5096(91)90044-O Lee, 1992, Axisymmetric deformation of power-law solids containing dilute concentration of spheroidal voids, J. Mech. Phys. Solids, 40, 1805, 10.1016/0022-5096(92)90052-4 Lee, 1992, Effective properties of power-law solids containing elliptical inhomogeneities. Part I. Rigid inclusions, Mech. Mater., 13, 313, 10.1016/0167-6636(92)90023-7 Levin, 1967, Thermal expansion coefficients of heterogeneous materials, Mekh. Tverd. Tela, 2, 83 Li, 1993, The effect of particle shape and stiffness on the constitutive behavior of metal-matrix composites, Internat. J. Solids Structures, 30, 3189, 10.1016/0020-7683(93)90109-K Li, 1994, Variational estimates for the elastoplastic response of particle-reinforced metal-matrix composites, Appl. Mech. Rev., 47, S77, 10.1115/1.3122825 Li, 1993, Constitutive models for ductile solids reinforced by rigid spheroidal inclusions, Mech. Mater, 15, 279, 10.1016/0167-6636(93)90004-B Lurie, 1986, Exact estimates of conductivity of a binary mixture of isotropic components, Proc. Roy. Soc. Edinburgh Sect. A, 104, 21, 10.1017/S0308210500019041 Mandel, 1972, Plasticité clussique et viscoplasticité, 97 McCoy, 1970, On the displacement field of an elastic medium: Random variations in material properties, 5, 235 McLaughlin, 1977, A study of the differential scheme for composite materials, Internat. J. Engrg. Sci., 15, 237, 10.1016/0020-7225(77)90058-1 Michel, 1994, A self-consistent estimate of the overall behavior of viscous and porous materials, C. R. Acad. Sci. Paris Ser. II, 319, 1439 Michel, 1996, A self-consistent estimate of the potential of a composite made of two power-law phases with the same exponent, C. R. Acad. Sci. Paris Ser. II, 322, 447 Michel, 1992, The constitutive law of nonlinear viscous and porous materials, J. Mech. Phys. Solids, 40, 783, 10.1016/0022-5096(92)90004-L Milton, 1982, Bounds on the elastic and transport properties of two-component composites, J. Mech. Phys. Solids, 30, 177, 10.1016/0022-5096(82)90022-9 Milton, 1985, The coherent potential approximation is a realizable effective medium scheme, Comm. Math. Phys., 99, 463, 10.1007/BF01215906 Milton, 1990, On characterizing the set of possible effective tensors: The variational method and the translation method, Comm. Pure Appl. Math., 43, 63, 10.1002/cpa.3160430104 Milton, 1988, Variational bounds of the effective moduli of anisotropic composites, J. Mech. Phys. Solids, 36, 597, 10.1016/0022-5096(88)90001-4 Milton, 1982, New bounds on the effective elastic moduli of two-component materials, Proc. Roy. Soc. London Ser. A, 380, 305, 10.1098/rspa.1982.0044 Molinari, 1987, A self-consistent approach of the large deformation polycrystal viscoplasticity, Acta Metall. Mater., 35, 2983, 10.1016/0001-6160(87)90297-5 Mori, 1973, Average stress in matrix and average elastic energy of materials with misfitting dislocations, Acta Metall. Mater., 21, 571, 10.1016/0001-6160(73)90064-3 Moulinec, 1995, A FTT-based numerical method for computing the mechanical properties of composites from images of their microstructure, 235 Nebozhyn, 1997, Second-order estimates for the effective behavior of nonlinear porous materials Nemat-Nasser, 1993 Olson, 1994, Improvements on Taylor's upper bound for rigid-plastic bodies, Mater. Sci. Engrg. A, 175, 15, 10.1016/0921-5093(94)91039-1 Ponte Castañeda, 1991, The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids, 39, 45, 10.1016/0022-5096(91)90030-R Ponte Castañeda, 1991, The effective properties of brittle/ductile incompressible composites, 215 Ponte Castañeda, 1991, Effective properties in power-law creep, 2, 218 Ponte Castañeda, 1992, New variational principles in plasticity and their application to composite materials, J. Mech. Phys. Solids, 40, 1757, 10.1016/0022-5096(92)90050-C Ponte Castañeda, 1992, Bounds and estimates for the properties of nonlinear heterogeneous systems, Philos. Trans. Roy. Soc. London Ser. A, 340, 531, 10.1098/rsta.1992.0079 Ponte Castañeda, 1992, A new variational principle and its application to nonlinear heterogeneous systems, SIAM J. Appl. Math., 52, 1321, 10.1137/0152076 Ponte Castañeda, 1996, Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. Mech. Phys. Solids, 44, 827, 10.1016/0022-5096(96)00015-4 Ponte Castañeda, 1996, Variational methods for estimating the effective behavior of nonlinear composite materials, 268 Ponte Castañeda, 1997, Corrigendum: Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. Mech. Phys. Solids, 45, 317, 10.1016/S0022-5096(97)85831-0 Ponte Castañeda, 1997, Nonlinear composite materials: Effective constitutive behavior and microstructure evolution, 131 Ponte Castañeda, 1992, On the homogenized yield strength of two-phase composites, Proc. Roy. Soc. London Ser. A, 438, 419, 10.1098/rspa.1992.0116 Ponte Castañeda, 1997, Nonlinear electrical conductivity in heterogeneous media, Proc. Roy. Soc. London Ser. A, 453, 793, 10.1098/rspa.1997.0044 Ponte Castañeda, 1997, Exact second-order estimates of the self-consistent type for nonlinear composite materials, Mech. Mater. Ponte Castañeda, P., and Nebozhyn, M. V. (1998). Variational estimates of the self-consistent type for the effective behavior of some model nonlinear polycrystals. Submitted Ponte Castañeda, 1995, On the effective mechanical behavior of weakly inhomo-geneous nonlinear materials, European J. Mech. A Solids, 14, 205 Ponte Castañeda, 1988, On the overall properties of nonlinearly viscous composites, Proc. Roy. Soc. London Ser. A, 416, 217, 10.1098/rspa.1988.0035 Ponte Castañeda, 1993, The effective behavior of nonlinear composites: A comparison between two methods, 351 Ponte Castañeda, 1995, The effect of spatial distribution on the effective behavior of composite materials and cracked media, J. Mech. Phys. Solids, 43, 1919, 10.1016/0022-5096(95)00058-Q Ponte Castañeda, 1994, Constitutive models for porous materials with evolving microstructure, J. Mech. Phys. Solids, 42, 1459, 10.1016/0022-5096(94)90005-1 Ponte Castañeda, 1996, The finite deformation of nonlinear composite materials. I. Instantaneous constitutive relations, Internat. J. Solids Structures, 33, 1271, 10.1016/0020-7683(95)00099-2 Qiu, 1992, A theory of plasticity for porous materials and particle-reinforced composites, J. Appl. Mech., 59, 261, 10.1115/1.2899515 Reuss, 1929, Calculation of the flow limits of mixed crystals on the basis of the plasticity of the monocrystals. Z, Angew. Math. Mech., 9, 49, 10.1002/zamm.19290090104 Rice, 1970, On the structure of of stress-strain relations for time-dependent plastic deformation in metals, J. Appl. Mech., 37, 728, 10.1115/1.3408603 Sachs, 1928, Zur Ableitung einer Fleissbedingun, Z. Ver Dtsch. Ing., 72, 734 Sanchez-Palencia, 1980, Non-homogeneous Media and Vibration Theory, 127 Smyshlyaev, 1995, Bounds and estimates for the overall plastic behavior of composites with strain gradients effects, Proc. Roy. Soc. London Ser A, 451, 795, 10.1098/rspa.1995.0156 Spitzig, 1988, The evolution of damage and fracture in iron compacts with various initial porosities, Acta Metall. Mater, 36, 1201, 10.1016/0001-6160(88)90273-8 Suquet, 1983, Analyse limite et homogénéisation, C. R. Acad. Sci. Paris Ser II, 296, 1355 Suquet, 1987, Elements of homogenization for inelastic solid mechanics, 272, 193 Suquet, 1992, On bounds for the overall potential of power-law materials containing voids with arbitrary shape, Mech. Res. Comm., 19, 51, 10.1016/0093-6413(92)90012-Y Suquet, 1993, Overall potentials and extremal surfaces of power law or ideally plastic materials, J. Mech. Phys. Solids, 41, 981, 10.1016/0022-5096(93)90051-G Suquet, 1993, Bounds and estimates for the overall properties of nonlinear composites, 361 Suquet, 1995, Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castaiieda's nonlinear variational procedure, C. R. Acad. Sci. Paris Ser II, 320, 563 Suquet, 1996, Overall properties of nonlinear composites: Secant moduli theories and variational bounds, 290 Suquet, 1996, Overall properties of nonlinear composites: Remarks on secant and incremental formulations, 149 Suquet, 1997, Effective properties of nonlinear composites, 377, 197 Suquet, P. (1997b). Work in progress. Suquet, 1997, A numerical method for computing the overall response of nonlinear composites with complex microstructure, 277 Suquet, 1993, Small-contrast perturbation expansions for the effective properties of nonlinear composites, C. R. Acad. Sci. Paris Ser. II, 317, 1515 Talbot, 1985, Variational principles for inhomogeneous nonlinear media, IMA J. Appl. Math., 35, 39, 10.1093/imamat/35.1.39 Talbot, 1991, The overall behavior of a nonlinear fiber reinforced composite, 527 Talbot, 1992, Some explicit bounds for the overall behavior of nonlinear composites, Internat. J. Solids Structures, 29, 1981, 10.1016/0020-7683(92)90188-Y Talbot, 1994, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry, Proc. Roy. Soc. London Ser. A, 447, 365, 10.1098/rspa.1994.0145 Talbot, 1995, Upper and lower bounds for the overall properties of a nonlinear elastic composite, 409 Talbot, 1997, Bounds of third order for the overall response of nonlinear composites, J. Mech. Phys. Solids, 45, 87, 10.1016/S0022-5096(96)00069-5 Tandon, 1988, A theory of particle-reinforced plasticity, J. Appl. Mech., 55, 126, 10.1115/1.3173618 Taylor, 1938, Plastic strains in metals, J. Inst. Metals, 62, 307 Tvergaard, 1990, Analysis of tensile properties for a whisker-reinforced metal-matrix composite, Acta Metall. Mater., 38, 185, 10.1016/0956-7151(90)90048-L Van der Giessen, 1995, Void growth due to creep and grain boundary diffusion at high triaxialities, J. Mech. Phys. Solids, 43, 123, 10.1016/0022-5096(94)00059-E Van Tiel, 1984 Voigt, 1889, Ueber die Bezienhung zwischen den beiden Elasticitlts-constanten isotroper, Ann. Physik, 38, 573, 10.1002/andp.18892741206 Walpole, 1969, On the overall elastic moduli of composite materials, J. Mech. Phys. Solids, 17, 235, 10.1016/0022-5096(69)90014-3 Walpole, 1981, Elastic behavior of composite materials: Theoretical foundations, Adv. Appl. Mech., 21, 169, 10.1016/S0065-2156(08)70332-6 Willis, 1977, Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. Mech. Phys. Solids, 25, 185, 10.1016/0022-5096(77)90022-9 Willis, 1978, Variational principles and bounds for the overall properties of composites, 185 Willis, 1981, Variational and related methods for the overall properties of composites, Adv Appl. Mech., 21, 1, 10.1016/S0065-2156(08)70330-2 Willis, 1983, The overall response of composite materials, ASME J. Appl. Mech., 50, 1202, 10.1115/1.3167202 Willis, 1986, Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, 247 Willis, 1989, The structure of overall constitutive relations for a class of nonlinear composites, IMA J. Appl. Math., 43, 231, 10.1093/imamat/43.3.231 Willis, 1989, Variational estimates for the overall behavior of a nonlinear matrix-inclusion composite, 581 Willis, 1991, On methods for bounding the overall properties of nonlinear composites, J. Mech. Phys. Solids, 39, 73, 10.1016/0022-5096(91)90031-I Willis, 1992, On method for bounding the overall properties of nonlinear composites: Correction and addition, J. Mech. Phy. Solids, 40, 441, 10.1016/S0022-5096(05)80020-1 Willis, 1994, Upper and lower bounds for nonlinear composite behavior, Mater. Sci. Engrg. A, 175, 7, 10.1016/0921-5093(94)91038-3 Zaidman, 1996, The finite deformation of nonlinear composite materials. II. Evolution of the microstructure, Internat. J. Solids Structures, 33, 1287, 10.1016/0020-7683(95)00100-X