A Short Introduction to Continuum Micromechanics

  • Helmut J. Böhm
Part of the International Centre for Mechanical Sciences book series (CISM, volume 464)


Basic issues in continuum mechanical modeling of microstructured materials are discussed and a number of physically based modeling approaches are presented, among them mean field and bounding methods as well as unit cell and embedding models. In addition, important aspects of multi-scale modeling strategies are addressed and a short introduction to the treatment of damage at the constituent level within micromechanical models is given.


Metal Matrix Composite Microstructured Material Inhomogeneous Material Unit Cell Model Eshelby Tensor 
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  1. J. Aboudi. Micromechanical analysis of composites by the method of cells. Appl. Mech. Rev., 42:193–221, 1989.Google Scholar
  2. J. Aboudi. Mechanics of Composite Materials. Elsevier, Amsterdam, 1991.MATHGoogle Scholar
  3. J. Aboudi, M.J. Pindera, and S.M. Arnold. Higher-order theory for periodic multiphase materials with inelastic phases. Int. J. Plast., 19:805–847, 2003.MATHGoogle Scholar
  4. A. Anthoine. Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int. J. Sol. Struct., 32:137–163, 1995.MATHGoogle Scholar
  5. M.S. Axelsen and R. Pyrz. Correlation between fracture toughness and the microstructure morphology in transversely loaded unidirectional composites. In R. Pyrz, editor, Micro structure-Property Interactions in Composite Materials, pages 15–26, Dordrecht, 1995. Kluwer Academic Publishers.Google Scholar
  6. Y. Benveniste. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mater., 6:147–157, 1987.Google Scholar
  7. Y. Benveniste and G.J. Dvorak. On a correspondence between mechanical and thermal effects in two-phase composites. In G.J. Weng, M. Taya, and H. Abé, editors, Micromechanics and Inhomogeneity, pages 65–82, New York, NY, 1990. Springer-Verlag.Google Scholar
  8. M.J. Beran and J. Molyneux. Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Quart. Appl. Math., 24:107–118, 1966.MATHGoogle Scholar
  9. H. Berns, A. Melander, D. Weichert, N. Asnafi, C. Broeckmann, and A. Gross-Weege. A new material for cold forging tools. Comput. Mater. Sci., 11:166–188, 1998.Google Scholar
  10. J.F.W. Bishop and R. Hill. A theory of the plastic distortion of a polycrystalline aggregate under combined stress. Phil. Mag., 42:414–427, 1951.MathSciNetMATHGoogle Scholar
  11. B. Bochenek and R. Pyrz. Reconstruction methodology for planar and spatial random microstructures. In R. Pyrz, J. Schjodt-Thomsen, J.C. Rauhe, T. Thomsen, and L.R. Jensen, editors, New Challenges in Mesomechanics, pages 565–572, Aalborg, Denmark, 2002. Aalborg University.Google Scholar
  12. H.J. Böhm. Numerical investigation of microplasticity effects in unidirectional longfiber reinforced metal matrix composites. Modell. Simul. Mater. Sci. Engng., 1:649–671, 1993.Google Scholar
  13. H.J. Böhm. Modeling the mechanical behavior of short fiber reinforced composites. This volume, pages 41–56, 2004.Google Scholar
  14. H.J. Böhm, A. Eckschlager, and W. Han. Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements. Corn-put. Mater. Sci., 25:42–53, 2002.Google Scholar
  15. H.J. Böhm and W. Han. Comparisons between three-dimensional and two-dimensional multi-particle unit cell models for particle reinforced metal matrix composites. Modell. Simul. Mater. Sei. Engng., 9:47–65, 2001.Google Scholar
  16. M. Bornert. Homogénéisation des milieux aléatoires: bornes et estimations. In M. Bornert, T. Bretheau, and P. Gilormini, editors, Homogénéisation en mécanique des materiaux 1. Matériaux aléatoires élastiques et milieux périodiques, pages 133–221, Paris, 2001. Editions Hermès.Google Scholar
  17. M. Bornert, T. Bretheau, and P. Gilormini, editors. Homogénéisation en mécanique des matériaux. Editions Hermès, Paris, 2001.Google Scholar
  18. M. Bornert, E. Hervé, C. Stolz, and A. Zaoui. Self consistent approaches and strain heterogeneities in two-phase elastoplastic materials. Appl. Mech. Rev., 47:66-S76, 1994.Google Scholar
  19. M. Bornert and P. Suquet. Propriétés non linéaires des composites: Approches par les potentiels. In M. Bornert, T. Bretheau, and P. Gilormini, editors, Homogénéisation en mécanique des matériaux 2. Comportements non linéaires et problèmes ouverts, pages 45–90, Paris, 2001. Editions Hermès.Google Scholar
  20. J.R. Brockenbrough and S. Suresh. Plastic deformation of continuous fiber-reinforced metal-matrix composites: Effects of fiber shape and distribution. Scr. metall. mater., 24:325–330, 1990.Google Scholar
  21. K.M. Brockmüller, O. Bernhardi, and M. Maier. Determination of fracture stress and strain of highly oriented oriented short fibre-reinforced composites using a fracture mechanics-based iterative finite-element method. J. Mater. Sci., 30:481–487, 1995.Google Scholar
  22. V.N. Bulsara, R. Talreja, and J. Qu. Damage initiation under transverse loading of unidirectional composites with arbitrarily distributed fibers. Compos. Sci. Technol., 59:673–682, 1999.Google Scholar
  23. V.A. Buryachenko. The overall elastoplastic behavior of multiphase materials with isotropic components. Acta Mech., 119:93–117, 1996.MATHGoogle Scholar
  24. C.M. Chimani, H.J. Böhm, and F.G. Rammerstorfer. On stress singularities at free edges of bimaterial junctions — A micromechanical study. Scr. mater., 36:943–947, 1997.Google Scholar
  25. R.M. Christensen and K.H. Lo. Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Sol., 27:315–330, 1979.MATHGoogle Scholar
  26. P.W. Chung, K.K. Tamma, and R.R. Namburu. Asymptotic expansion homogenization for heterogeneous media: Computational issues and applications. Composites A, 32A: 1291–1301, 2001.Google Scholar
  27. B. Clyne and P.J. Withers. An Introduction to Metal Matrix Composites. Cambridge University Press, Cambridge, 1993.Google Scholar
  28. F. Corvasce, P. Lipihski, and M. Berveiller. The effects of thermal, plastic and elastic stress concentrations on the overall behavior of metal matrix composites. In G.J. Dvorak, editor, Inelastic Deformation of Composite Materials, pages 389–408, New York, NY, 1991. Springer-Verlag.Google Scholar
  29. I. Doghri and A. Ouaar. Homogenization of two-phase elasto-plastic composite materials and structures. Int. J. Sol Struct, 40:1681–1712, 2003.MATHGoogle Scholar
  30. M. Dong and S. Schmauder. Modeling of metal matrix composites by a self-consistent embedded cell model. Acta mater., 44:2465–2478, 1996.Google Scholar
  31. W.J. Drugan and J.R. Willis. A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Sol., 44:497–524, 1996.MathSciNetMATHGoogle Scholar
  32. M.L. Dunn and H. Ledbetter. Elastic-plastic behavior of textured short-fiber composites. Acta mater., 45:3327–3340, 1997.Google Scholar
  33. G.J. Dvorak. Transformation field analysis of inelastic composite materials. Proc. Roy. Soc. London, A437:311–327, 1992.MathSciNetMATHGoogle Scholar
  34. A. Eckschlager. Simulation of Particle Failure in Particle Reinforced Ductile Matrix Composites. PhD thesis, TU Wien, Vienna, Austria, 2002.Google Scholar
  35. M. Elices, G.V. Guinea, J. Gomez, and J. Planas. The cohesive zone model: Advantages, limitations and challenges. Engng. Fract. Mech., 69:137–163, 2002.Google Scholar
  36. J.D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. London, A241:376–396, 1957.MathSciNetMATHGoogle Scholar
  37. J.D. Eshelby. The elastic field outside an ellipsoidal inclusion. Proc. Roy. Soc. London, A252:561–569, 1959.MathSciNetMATHGoogle Scholar
  38. F.D. Fischer, O. Kolednik, G.X. Shan, and F.G. Rammerstorfer. A note on calibration of ductile failure damage indicators. Int. J. Fract., 73:345–357, 1995.Google Scholar
  39. H.F. Fischmeister and B. Karlsson. Plastizitätseigenschaften grob-zweiphasiger Werkstoffe. Z. Metallkd., 68:311–327, 1977.Google Scholar
  40. C. Fond, A. Riccardi, R. Schirrer, and F. Montheillet. Mechanical interaction between spherical inhomogeneities: An assessment of a method based on the equivalent inclusion. Eur. J. Mech. A/Solids, 20:59–75, 2001.MATHGoogle Scholar
  41. A.C. Gavazzi and D.C. Lagoudas. On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Cornput. Mech., 7:12–19, 1990.Google Scholar
  42. M.G.D. Geers, R.A.B. Engelen, and R.J.M. Ubachs. On the numerical modelling of ductile damage with an implicit gradient-enhanced formulation. Rev. Eur. Elem. Fin., 10:173–191, 2001.MATHGoogle Scholar
  43. S. Ghosh, K.H. Lee, and S. Moorthy. Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model. Comput. Meth. Appl. Mech. Engng., 132:63–116, 1996.MATHGoogle Scholar
  44. S. Ghosh, K.H. Lee, and P. Raghavan. A multi-level computational model for multi-scale analysis in composite and porous materials. Int. J. Sol. Struct., 38:2335–2385, 2001.MATHGoogle Scholar
  45. S. Ghosh and S. Moorthy. Particle fracture simulation in non-uniform microstructures of metal-matrix composites. Acta mater., 46:965–982, 1998.Google Scholar
  46. C. Gonzalez and J. LLorca. A self-consistent approach to the elasto-plastic behavior of two-phase materials including damage. J. Mech. Phys. Sol., 48:675–692, 2000.MATHGoogle Scholar
  47. A.L. Gurson. Continuum theory of ductile rupture by void nucleation and growth: Part I — Yield criteria and flow rules for porous ductile media. J. Engng. Mater. Technol., 99:2–15, 1977.Google Scholar
  48. A.A. Gusev. Representative volume element size for elastic composites: A numerical study. J. Mech. Phys. Sol, 45:1449–1459, 1997.MATHGoogle Scholar
  49. Z. Hashin. Analysis of composite materials — A survey. J. Appl. Mech., 50:481–505, 1983.MATHGoogle Scholar
  50. Z. Hashin and S. Shtrikman. A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Sol, 11:127–140, 1963.MathSciNetMATHGoogle Scholar
  51. R. Hill. The elastic behavior of a crystalline aggregate. Proc. Phys. Soc. London, A65: 349–354, 1952.Google Scholar
  52. R. Hill. Continuum micro-mechanics of elastic-plastic polycrystals. J. Mech. Phys. Sol., 13:89–101, 1965a.MATHGoogle Scholar
  53. R. Hill. A self-consistent mechanics of composite materials. J. Mech. Phys. Sol., 13: 213–222, 1965b.Google Scholar
  54. R. Hill. The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Sol., 15:79–95, 1967.Google Scholar
  55. P.J. Hine, H.R. Lusti, and A.A. Gusev. The numerical simulation of the elastic and thermoelastic properties of short fibre composites. Compos. Sci. Technol, 62:1445–1453, 2002.Google Scholar
  56. G.K. Hu, G. Guo, and D. Baptiste. A micromechanical model of influence of particle fracture and particle cluster on mechanical properties of metal matrix composites. Comput. Mater. Sci., 9:420–430, 1998.Google Scholar
  57. C. Huet. Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Sol, 38:813–841, 1990.MathSciNetGoogle Scholar
  58. C. Huet. Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies. Mech. Mater., 31:787–829, 1999.Google Scholar
  59. H. Ismar and U. Reinert. Modelling and simulation of the macromechanical nonlinear behavior of fibre-reinforced ceramics on the basis of a micromechanical-statistical material description. Acta Mech., 120:47–60, 1997.MATHGoogle Scholar
  60. T. Iung and M. Grange. Mechanical behavior of two-phase materials investigated by the finite element method: Necessity of three-dimensional modeling. Mater. Sci. Engng., A201:L8–L11, 1995.Google Scholar
  61. D. Jeulin and M. Ostoja-Starzewski, editors. Mechanics of Random and Multiscale Microstructures. Springer-Verlag, Vienna, 2001.MATHGoogle Scholar
  62. M. Jiang, M. Ostoja-Starzewski, and I. Jasiuk. Scale-dependent bounds on effective elastoplastic response of random composites. J. Mech. Phys. Sol, 49:655–673, 2001.MATHGoogle Scholar
  63. M. Jirásek. Comparative study on finite elements with embedded discontinuities. Comput. Meth. Appl. Mech. Engng., 188:307–330, 2000.MATHGoogle Scholar
  64. J.W. Ju and L.Z. Sun. A novel formulation for the exterior point Eshelby’s tensor of an ellipsoidal inclusion. J. Appl. Mech., 66:570–574, 1999.Google Scholar
  65. J.W. Ju and L.Z. Sun. Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal inhomogeneities. Part I: Micromechanics-based formulation. Int. J. Sol Struct., 38:183–201, 2001.MATHGoogle Scholar
  66. M. Kailasam, N. Aravas, and P. Ponte Castaneda. Porous metals with developing anisotropy: Constitutive models, computational issues and applications to deformation processing. Comput. Model Engng. Sci., 1:105–118, 2000.Google Scholar
  67. T. Kanit, S. Forest, I. Gallier, V. Mounoury, and D. Jeulin. Determination of the size of the representative volume element for random composites: Statistical and numerical approach. Int. J. Sol. Struct., 40:3647–3679, 2003.MATHGoogle Scholar
  68. M. Karayaka and H. Sehitoglu. Thermomechanical deformation modeling of A12xxx-T4/SiCp composites. Acta metall. mater., 41:175–189, 1993.Google Scholar
  69. V.G. Kouznetsova, M.G.D. Geers, and W.A.M. Brekelmans. Multi-scale constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Num. Meth. Engng., 54:1235–1260, 2002.MATHGoogle Scholar
  70. K.H. Lee, S. Moorthy, and S. Ghosh. Multiple scale computational model for damage in composite materials. Comput. Meth. Appl. Mech. Engng., 172:175–201, 1999.MATHGoogle Scholar
  71. V.M. Levin. On the coefficients of thermal expansion of heterogeneous materials. Mech. Sol., 2:58–61, 1967.Google Scholar
  72. D.S. Li and M.R. Wisnom. Unidirectional tensile stress-strain response of BP-SiC fiber reinforced Ti-6A1–4V. J. Compos. Technol. Res., 16:225–233, 1994.Google Scholar
  73. M. Li, S. Ghosh, O. Richmond, H. Weiland, and T.N. Rouns. Three dimensional characterization and modeling of particle reinforced metal matrix composites, Part I: Quantitative description of microstructural morphology. Mater. Sci. Engng., A265: 153–173, 1999.Google Scholar
  74. J. LLorca. Deformation and damage in particle reinforced composites: Experiments and models. This volume, pages 87–124, 2004.Google Scholar
  75. K.Z. Markov and L. Preziosi. Heterogeneous Media: Micromechanics Modeling Methods and Simulations. Birkhäuser, Boston, MA, 2000.MATHGoogle Scholar
  76. R. Masson, M. Bornert, P. Suquet, and A. Zaoui. An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J. Mech. Phys. Sol., 48:1203–1227, 2000.MathSciNetMATHGoogle Scholar
  77. P.E. McHugh. Introduction to crystal plasticity theory. This volume, pages 125–172, 2004.Google Scholar
  78. P.E. McHugh and P. Connolly. Modelling the thermo-mechanical behavior of an Al alloy-SiCp composite. Effects of particle shape and microscale failure. Comput. Mater. Sci., 3:199–206, 1994.Google Scholar
  79. J.C. Michel, H. Moulinec, and P. Suquet. Effective properties of composite materials with periodic microstructure: A computational approach. Comput. Meth. Appl. Mech. Engng., 172:109–143, 1999.MathSciNetMATHGoogle Scholar
  80. J.C. Michel, H. Moulinec, and P. Suquet. Composites à microstructure périodique. In M. Bornert, T. Bretheau, and P. Gilormini, editors, Homogénéisation en mécanique des materiaux 1. Matériaux aléatoires élastiques et milieux périodiques, pages 57–94, Paris, 2001. Editions Hermès.Google Scholar
  81. J.C. Michel and P. Suquet. Nonuniform transformation field analysis. Int. J. Sol. Struct., 40:6937–6955, 2003.MathSciNetMATHGoogle Scholar
  82. C. Miehe, J. Schröder, and J. Schotte. Computational homogenization analysis in finite plasticity. simulation of texture development in polycrystalline materials. Comput. Meth. Appl. Mech. Engng., 171:387–418, 1999.MATHGoogle Scholar
  83. C.A. Miller and S. Torquato. Effective conductivity of hard sphere suspensions. J. Appl. Phys., 68:5486–5493, 1990.Google Scholar
  84. G.W. Milton. Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett., 46:542–545, 1981.Google Scholar
  85. G.W. Milton. The Theory of Composites. Cambridge University Press, Cambridge, 2002.MATHGoogle Scholar
  86. N. Moës, N. Cloirec, P. Cartraud, and J.F. Remacle. A computational approach to handle complex microstructure geometries. Comput. Meth. Appl. Mech. Engng., 192: 3163–3177, 2003.MATHGoogle Scholar
  87. A. Molinari, G.R. Canova, and S. Ahzi. A self-consistent approach for large deformation viscoplasticity. Acta metall., 35:2983–2984, 1987.Google Scholar
  88. H. Moulinec and P. Suquet. A fast numerical method for computing the linear and nonlinear mechanical properties of composites. C. R. Acad. Sci. Paris, série II, 318: 1417–1423, 1994.MATHGoogle Scholar
  89. W.H. Müller. Mathematical versus experimental stress analysis of inhomogeneities in solids. J. Phys. IV, 6:1–139-C1–148, 1996.Google Scholar
  90. T. Mura. Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht, 1987.Google Scholar
  91. A. Needleman. A continuum model for void nucleation by inclusion debonding. J. Appl. Mech., 54:525–531, 1987.MATHGoogle Scholar
  92. A. Needleman, S.R. Nutt, S. Suresh, and V. Tvergaard. Matrix, reinforcement, and interfacial failure. In S. Suresh, A. Mortensen, and A. Needleman, editors, Fundamentals of Metal Matrix Composites, pages 233–250, Boston, MA, 1993. Butterworth-Heinemann.Google Scholar
  93. S. Nemat-Nasser. Averaging theorems in finite deformation plasticity. Mech. Mater., 31: 493–523, 1999.Google Scholar
  94. S. Nemat-Nasser and M. Hori. Micromechanics: Overall Properties of Heterogeneous Solids. North-Holland, Amsterdam, 1993.MATHGoogle Scholar
  95. M. Ostoja-Starzewski. Random field models of heterogeneous materials. Int. J. Sol. Struct., 35:2429–2455, 1998.MATHGoogle Scholar
  96. R. Pandorf. Ein Beitrag zur FE-Simulation des Kriechens partikelverstärkter metallischer Werkstoffe. Reihe 5, Nr.585. VDI-Verlag, Düsseldorf, 2000.Google Scholar
  97. O.B. Pedersen. Thermoelasticity and plasticity of composites — I. Mean field theory. Acta metall., 31:1795–1808, 1983.Google Scholar
  98. R.H.J. Peerlings, M.G.D. Geers, R. de Borst, and W.A.M. Brekelmans. A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Sol. Struct., 38: 7723–7746, 2001.MATHGoogle Scholar
  99. H.E. Pettermann. Derivation and Finite Element Implementation of Constitutive Material Laws for Multiphase Composites Based on Mori-Tanaka Approaches. Reihe 18, Nr.217. VDI-Verlag, Düsseldorf, 1997.Google Scholar
  100. H.E. Pettermann, A.F. Plankensteiner, H.J. Böhm, and F.G. Rammerstorfer. A thermo- elasto-plastic constitutive law for inhomogeneous materials based on an incremental Mori-Tanaka approach. Comput. Struct., 71:197–214, 1999.Google Scholar
  101. H.E. Pettermann and S. Suresh. A comprehensive unit cell model: A study of coupled effects in piezoelectric 1–3 composites. Int. J. Sol. Struct., 37:5447–5464, 2000.MATHGoogle Scholar
  102. N. Phan-Thien and G.W. Milton. New third-order bounds on the effective moduli of n-phase composites. Quart. Appl. Math., 41:59–74, 1983.MathSciNetMATHGoogle Scholar
  103. P. Ponte Castaheda. Bounds and estimates for the properties on nonlinear inhomogeneous systems. Phil. Trans. Roy. Soc., A340:531–567, 1992.Google Scholar
  104. P. Ponte Castaneda and P. Suquet. Nonlinear composites. In E. van der Giessen and T.Y. Wu, editors, Advances in Applied Mechanics34, pages 171–302, New York, NY, 1998. Academic Press.Google Scholar
  105. P. Ponte Castaneda and J.R. Willis. The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Sol., 43:1919–1951, 1995.MathSciNetMATHGoogle Scholar
  106. R. Pyrz. Microstructural description of composites, statistical methods. This volume. pages 173–234, 2004.Google Scholar
  107. Y.P. Qiu and G.J. Weng. A theory of plasticity for porous materials and particle-reinforced composites. J. Appl. Mech., 59:261–268, 1992.MATHGoogle Scholar
  108. S. Quilici and G. Cailletaud. FE simulation of macro-, meso- and microscales in polycrystalline plasticity. Comput. Mater. Sci., 16:383–390, 1999.Google Scholar
  109. A. Reuss. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. ZAMM, 9:49–58, 1929.MATHGoogle Scholar
  110. M. Rintoul and S. Torquato. Reconstruction of the structure of dispersions. J. Colloid Interf. Sci., 186:467–476, 1997.Google Scholar
  111. A.P. Roberts and E.J. Garboczi. Elastic properties of a tungsten-silver composite by reconstruction and computation. J. Mech. Phys. Sol, 47:2029–2055, 1999.MATHGoogle Scholar
  112. M. Sautter, C. Dietrich, M.H. Poech, S. Schmauder, and H.F. Fischmeister. Finite element modelling of a transverse-loaded fibre composite: Effects of section size and net density. Comput. Mater. Sci., 1:225–233, 1993.Google Scholar
  113. R.A. Schapery. Thermal expansion coefficients of composite materials based on energy principles. J. Compos. Mater., 2:380–404, 1968.Google Scholar
  114. S. Schmauder, J. Wulf, T. Steinkopff, and H. Fischmeister. Micromechanics of plasticity and damage in an Al/SiC metal matrix composite. In A. Pineau and A. Zaoui, editors, Micromechanics of Plasticity and Damage of Multiphase Materials, pages 255–262, Dordrecht, 1996. Kluwer.Google Scholar
  115. J. Segurado, J. LLorca, and C. González. On the accuracy of mean-field approaches to simulate the plastic deformation of composites. Scr. mater., 46:525–529, 2002.Google Scholar
  116. H. Shen and C.J. Lissenden. 3D finite element analysis of particle-reinforced aluminum. Mater. Sci. Engng., A338:271–281, 2002.Google Scholar
  117. V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips, and M. Ortiz. Quasicontinuum models of interfacial structures and deformation. Phys. Rev. Lett., 80:742–745, 1998.Google Scholar
  118. J.A. Sherwood and H.M. Quimby. Micromechanical modeling of damage growth in titanium based metal-matrix composites. Comput. Struct., 56:505–54, 1995.Google Scholar
  119. T. Siegmund, R. Cipra, J. Liakus, B. Wang, M. LaForest, and A. Fatz. Processingmicrostructure-property relationships in short fiber reinforced carbon-carbon composite system. This volume, pages 235–258, 2004.Google Scholar
  120. R.J.M. Smit, W.A.M. Brekelmans, and H.E.H. Meijer. Prediction of the mechanical behavior of non-linear heterogeneous systems by multi-level finite element modeling. Comput. Meth. Appl. Mech. Engng., 155:181–192, 1998.MATHGoogle Scholar
  121. P. Suquet. Elements of homogenization for inelastic solid mechanics. In E. Sanchez-Palencia and A. Zaoui, editors, Homogenization Techniques in Composite Media, pages 194–278, Berlin, 1987. Springer-Verlag.Google Scholar
  122. P. Suquet, editor. Continuum Micromechanics. Springer-Verlag, Vienna, 1997a.MATHGoogle Scholar
  123. P. Suquet. Effective properties of nonlinear composites. In P. Suquet, editor, Continuum Micromechanics, pages 197–264, Vienna, 1997b. Springer-Verlag.Google Scholar
  124. T. Suzuki and P.K.L. Yu. Complex elastic wave band structures in three-dimensional periodic elastic media. J. Mech. Phys. Sol, 46:115–138, 1998.MATHGoogle Scholar
  125. N. Takano, M. Zako, and T. Okazaki. Efficient modeling of microscopic heterogeneity and local crack in composite materials by finite element mesh superposition method. JSME Int. J. Srs.A, 44:602–609, 2001.Google Scholar
  126. G.P. Tandon and G.J. Weng. The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites. Polym. Compos., 5:327–333, 1984.Google Scholar
  127. G.P. Tandon and G.J. Weng. A theory of particle-reinforced plasticity. J. Appl. Mech., 55:126–135, 1988.Google Scholar
  128. M. Taya, W.D. Armstrong, M.L. Dunn, and T. Mori. Analytical study on dimensional changes in thermally cycled metal matrix composites. Mater. Sci. Engng., A143: 143–154, 1991.Google Scholar
  129. K. Terada and N. Kikuchi. Nonlinear homogenization method for practical applications. In S. Ghosh and M. Ostoja-Starzewski, editors, Computational Methods in Micromechanics, pages 1–16, New York, NY, 1996. ASME.Google Scholar
  130. K. Terada, I. Saiki, K. Matsui, and Y. Yamakawa. Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain. Comput. Meth. Appl. Mech. Engng., 192:3531–3563, 2003.MathSciNetMATHGoogle Scholar
  131. J.F. Thovert, I.C. Kim, S. Torquato, and A. Acrivos. Bounds on the effective properties of poly dispersed suspensions of spheres: An evaluation of two relevant parameters. J. Appl. Phys., 67:6088–6098, 1990.Google Scholar
  132. S. Torquato. Random heterogeneous media: Microstructure and improved bounds on effective properties. Appl. Mech. Rev., 44:37–75, 1991.MathSciNetGoogle Scholar
  133. S. Torquato. Morphology and effective properties of disordered heterogeneous media. Int. J. Sol. Struct., 35:2385–2406, 1998.MATHGoogle Scholar
  134. S. Torquato. Random Heterogeneous Media. Springer-Verlag. New York, NY, 2001.Google Scholar
  135. S. Torquato, F. Lado, and P.A. Smith. Bulk properties of two-phase disordered media. IV. Mechanical properties of suspensions of penetrable spheres at nondilute concentrations. J. Chem. Phys., 86:6388–6392, 1987.Google Scholar
  136. V. Tvergaard. Analysis of tensile properties for a whisker-reinforced metal-matrix composite. Acta metall. mater., 38:185–194, 1990.Google Scholar
  137. V. Tvergaard. Fibre debonding and breakage in a whisker-reinforced metal. Mater. Sci. Engng., A190:215–222, 1994.Google Scholar
  138. V. Tvergaard and A. Needleman. Analysis of the cup-cone fracture in a round tensile bar. Acta metall, 32:157–169, 1984.Google Scholar
  139. E. van der Giessen. Creep rupture in polycrystalline materials. This volume, pages 283–306, 2004a.Google Scholar
  140. E. van der Giessen. Discrete dislocation plasticity. This volume, pages 259–282, 2004b.Google Scholar
  141. N. Vejen and R. Pyrz. Transverse crack growth in glass/epoxy composites with exactly positioned long fibers. Part II: Numerical. Composites B, 33B:279–290, 2002.Google Scholar
  142. W. Voigt. Über die Beziehung zwischen den beiden Elasticitäts-Constanten isotroper Körper. Ann. Phys., 38:573–587, 1889.MathSciNetGoogle Scholar
  143. W. Vonach. A General Solution to the Wrinkling Problem of Sandwiches. Reihe 18, Nr.268. VDI-Verlag, Düsseldorf, 2001.Google Scholar
  144. K. Wakashima, H. Tsukamoto, and B.H. Choi. Elastic and thermoelastic properties of metal matrix composites with discontinuous fibers or particles: Theoretical guidelines toward materials tailoring. In The Korea-Japan Metals Symposium on Composite Materials, pages 102–115, Seoul, 1988. The Korean Institute of Metals.Google Scholar
  145. L.J. Walpole. On bounds for the overall elastic moduli of inhomogeneous systems — I. J. Mech. Phys. Sol, 14:151–162, 1966.MATHGoogle Scholar
  146. S. Weihe and B.H. Kröplin. The fictitious crack concept in the mechanics of composites. In D.R.J. Owen, E. Ohate, and E. Hinton, editors, Computational Plasticity: Fundamentals and Applications, pages 1215–1226, Swansea, 1995. Pineridge Press.Google Scholar
  147. G.J. Weng. The theoretical connection between Mori-Tanaka theory and the Hashin-Shtrikman-Walpole bounds. Int. J. Engng. Sci., 28:1111–1120, 1990.MATHGoogle Scholar
  148. J.R. Willis. Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Sol, 25:185–202, 1977.MATHGoogle Scholar
  149. J.R. Willis. The overall response of nonlinear composite media. Eur. J. Mech. A/Solids, 19:S165–S184, 2000.Google Scholar
  150. P.J. Withers. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Phil Mag., A59: 759–781, 1989.Google Scholar
  151. J. Wulf, T. Steinkopff, and H. Fischmeister. FE-simulation of crack paths in the real microstructure of an Al(6061)/SiC composite. Acta mater., 44:1765–1779, 1996.Google Scholar
  152. S. Yang, A. Tewari, and A. Gokhale. Modeling of non-uniform spatial arrangement of fibers in a ceramic matrix composite. Acta mater., 45:3059–3069, 1997.Google Scholar
  153. A. Zaoui. Plasticité: Approches en champ moyen. In M. Bornert, T. Bretheau, and P. Gilormini, editors, Homogénéisation en mécanique des matériaux 2. Comportements non linéaires et problèmes ouverts, pages 17–44, Paris, 2001. Editions Hermès.Google Scholar
  154. A. Zaoui. Continuum micromechanics: Survey. J. Engng. Mech., 128:808–816, 2002.Google Scholar
  155. J. Zeman and M. Sejnoha. Numerical evaluation of effective elastic properties of graphite fiber tow impregnated by polymer matrix. J. Mech. Phys. Sol, 49:69–90, 2001.MATHGoogle Scholar
  156. Y.H. Zhao and G.J. Weng. A theory of inclusion debonding and its influence on the stress-strain relations of a ductile matrix composite. Int. J. Dam. Mech., 4:196–211, 1995.Google Scholar
  157. R.W. Zimmerman. Hashin-Shtrikman bounds on the Poisson ratio of a composite material. Mech. Res. Comm., 19:563–569, 1992.MathSciNetMATHGoogle Scholar
  158. T.I. Zohdi. Constrained inverse formulations in random material design. Comput. Meth. Appl. Mech. Engng., 192:3179–3194, 2003.MATHGoogle Scholar
  159. T.I. Zohdi and P. Wriggers. A model for simulating the deterioration of structural-scale material responses of microheterogeneous solids. Comput. Meth. Appl Mech. Engng., 190:2803–2823, 2001.MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2004

Authors and Affiliations

  • Helmut J. Böhm
    • 1
  1. 1.Institute of Lightweight Design and Structural BiomechanicsVienna University of TechnologyViennaAustria

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