One of the most important engineering tasks over the years has been the design and manufacture of increasingly sophisticated structural materials as a result of the requirements related to the technological progress. In the last decades, the growing needs for improved properties of products have been partially solved through the development of composite materials. A key to the success of many modern structural components is the tailored behavior of the material to given applications. Therefore, research efforts in material science engineering have been focused in the design of new materials either through the creation of new structures at the scale of single atoms and molecules or through the development of structural materials by changing the composition, size, arrangement and topology of the constituents at larger scales: the microscopic/mesoscopic level.

The development of new materials has been linked to the development of a new theoretical field within the mechanics of solids. This branch of the mechanical, known as Continuum Micromechanics, introduces a series of new concepts that are key to the definition of the macroscopic properties of composite materials on the basis of the definition of the characteristics of its components. Starting from the premise of separation of scales and the concept of Representative Volume Element, defined the so-called homogenization methods, whose number has been increasing as the Micromechanics is gone extend over the years. Such methods are many and varied, although especially there have been two that have been used and developed by the majority of authors: the so-called Mean-Homogenization techniques and the multi-scale based on Finite Element Approaches.

Mean-field homogenization schemes are an efficient way to predict the behavior of heterogeneous materials. They range from the simplest hypotheses of the stress or strain sharing among the phases which do not require analytical solution on the associated boundary-value problem to more involved geometric models based on the solution of a boundary-value problem involving a single or composite inclusion embedded in an equivalent homogenized medium whose elastic module become part of the solution procedure. In general, they are based on analytical solutions of the boundary value problem defined in the microstructure level of the inhomogeneous material and provide good predictions for the mean values over the RVE. Although originally designed for elastic materials, some approaches to deal with elastoplastic materials and even with viscoplastic materials have been developed over the years and compared with the results obtained using Finite Element Approaches. The comparison between different methods of homogenization allows the definition of a range of validity between the different methods, which helps to discover the limitations of the various methods and aspects to take into account for future developments and research.

The main goal of this work is, firstly, to present a general overview of the different techniques that have been developed in the last years in order to obtain a prediction of the behavior of elastoplastic composites by taking into account geometrical and mechanical aspects. Secondly, a comparison between the different approaches is carried out through a numerical implementation of such techniques. Both objectives will be carried out through eight different chapters. The first chapter serves as an introduction and historical review of the advances that have been made in the field of micromechanics. On the other hand, the second chapter deals with some important theoretical background that is important in the field of Continuum Micromechanics, as well as a short introduction of the different approaches that traditionally have been considered to solve the problem. One group of methods, based on analytical solutions - the so-called Mean Field Analysis - will be commented in chapter 3. Chapter 4 is devoted to the implementation and validation of a numerical tool that solves the mean-field homogenization using analytical schemes for elastoplastic materials. Subsequent chapters are devoted to the comparison of the results with the results given by the Finite Element Method. The general formulation of such method - applied to multi-scale problems - is presented in chapter 5 from a theoretical point of view, as well as the corresponding numerical examples. Finally, last chapter will be dedicated to enumerate some conclusions extracted from the present work, including some aspects that can be object of future works or improvements.

The current work presents some important aspects about the theoretical concepts and the numerical implementation of some key approaches for solving the mechanical problem regarding composite materials. There exist a large number of possibilities to approximate the response of such complex materials, based in different assumptions. This document shows the general efficiency of the so-called mean-field homogenization schemes to capture correctly the macroscopic behavior of composites. Although these techniques show some limitations, like the incapability to provide results for the distribution of the different variables over the microgeometry or the low accuracy in the case of complex microgeometries (like porous materials), they represent an efficient way to predict the main general behavior of a composite material spending low computational effort. They are specially indicated to be used in the previous steps of an analysis or as a tool to validate the results with more involved approaches.

[1] J. Aboudi. Micromechanical analysis of composites by the method of cells. Appl. Mech. Rev., 42:193-221, 1989.

[2] J. Aboudi, M.-J. Pindera and S. Arnold. Linear thermo-elastic higherorder theory for periodic multiphase materials. J. Appl. Mechanics, 68:697-707, 2001.

[3] J. Aboudi, M.-J. Pindera and S. Arnold. Higher-order theory for periodic multiphase materials with inelastic phases. Int. J. Plasticity, 19, 6:805-847, 2003.

[4] L. Bardella. An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials. Int. J. of Eng. Sci., 41:741768, 2003.

[5] B. Bednarcyk, S. Arnold, J. Aboudi and M.-J. Pindera. Local field effects in titanium matrix composites subject to fiber-matrix debonding. Int. J. Plasticity, 20:1707-1737, 2004.

[6] M. Bervellier and A. Zaoui. An extension of the sel-consistent scheme to plastically-owing polycrystals. J. Mech. Phys. Sol., 26:325-344.

[7] A. Bensoussan, J. Lions and G. Papanicolaou. Asymptotic analysis for periodic structures. North Holland, Amsterdam, 1978.

[8] Y. Benveniste. A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater., 6:147-157, 1987.

[9] 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. [10] H. J. Böhm. Continuum Micromechanics of Materials. Institut für Leichtbau und Biomechanik, TU Wien, 2011.

[11] M. Bornert. Homogénéisation des milieux aléatoires: bornes et estimations, vol. 1. Hermes Science, Paris, 5:133-221, 2001.

[12] I. ? Özdemir, W.A.M. Brekelmans and M.G.D. Geers. FE^{2} computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput. Methods Appl. Mech. Engrg., 198:602613, 2008.

[13] V. A. Buryachenko. The overall elastoplastic behavior of multiphase materials with isotropic components. Acta Mechanica, 119:93-117.

[14] C. R. Chiang. An extended Mori-Tanaka micromechanics model. 16th. International Conference of Composite Materials, 2007.

[15] P. Chung, K. Tamma and R. Namburu. Asymptotic expansion homogenization for heterogenous media: computational issues and applications. Composite part A, 32, 9:1291-1301, 2001.

[16] L.C. Davis. Flow rule for the plastic deformation of particulate metal matrix composites. Comp. Mater. Sci., 6:310-318, 1996.

[17] I. Doghri and A. Ouaar Homogenization of two-phase elasto-plastic composite materials and structures: study of tangent operators, cyclic plasticity and numerical algorithms. Int. J. Solids Struct., 40:1681-1712, 2003.

[18] I. Doghri and C. Friebel. Effective elasto-plastic properties of inclusionreinforced composites. Study of shape, orientation and cyclic response. Mech.Mater., 37:45-68, 2005.

[19] I. Doghri and L. Tinel. Micromechanical modeling and computation of elasto-plastic materials reinforced with distributed orientation fibers. Int. J. Plasticity, 21:1919-1940, 2005.

[20] I. Doghri, A. Ouaar, L. Delannay and J. F. Thimus. Micromechanics of the Deformation and Damage of Steel Fiber-reinforced Concrete. Int. J. of Damage Mech., 16, 2:227-260, 2007.

[21] W. Drugan and J. Willis. A micromechanics-based nonlocal constitutive equations and estimates of representative volume element size for elastic composites. J. Mech. Phys. Sol., 44:497-524, 1996.

[22] G. Dvorak. Transformation field analysis of inelastic composite materials. Proc. Roy. Soc. Lond., A 437:311-327, 1992.

[23] G. Dvorak, Y.A. Bahei-el Din and A. M. Wafa. The modelling of inelastic composite materials with the transformation field analysis. Modell. Simul. Mater. Sci. Engrg., 2:571-586, 1994.

[24] J. Eshelby. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. Lond., A 24:376-396, 1957.

[25] F. Feyel. A multilevel finite element method (FE^{2}) to describe the response of highly non-linear structures using generalized continua. Comput. Mech. Appl. Mech. Engrg., 192:3233-3244, 2003.

[26] C. Friebel. Mechanics and Acoustics of viscoelastic inclusion reinforced composites: micro-macro modeling of effective properties. PhD thesis, Université Catholique de Louvain, Belgium, 2007.

[27] A. Gavazzi and D. Lagoudas. On the numerical evaluation of Eshelbys tensor and its application to elastoplastic fibrous composites. Comput. Mech., 7:13-19, 1990.

[28] S. Ghosh, K. Lee and S. Moorthy. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method. Int. J. Solids Struct., 32:27-62, 1995.

[29] S. Ghosh, K. Lee and S. Moorthy. Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model. Comput. Methods Appl. Mech. Engrg., 132,1-2:63-116, 1996.

[30] S. Gong, Z. Li, Y.Y. Zhao. An extended MoriTanaka model for the elastic moduli of porous materials of finite size. Acta Materialia, 59:68206830, 2011.

[31] D. Gross and T. Seelig. Bruchmechanik, mit einer Einfhrung in die Micromechanik. Springer, Berlin, Heidelberg, 2001.

[32] Z. Hashin and S. Shtrikman. Note on a variational approach to the theory of composite elastic materials. J. Franklin Inst., 271:336-341, 1961.

[33] Z. Hashin. The differential scheme and its application to cracked materials. J. Mech. Phys. Sol., 36:719-733, 1988.

[34] J. A. Hernández, J. Oliver, A. E. Huespe and M. Caicedo. Highperformance Model Reduction Procedures in Multiscale Simulations. CIMNE Monograph, Barcelona 2012.

[35] R. Hill. Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Sol., 11:357-372, 1963.

[36] R. Hill. Continuum micro-mechanics of composite materials. J. Mech. Phys. Sol., 13, 4:89-101, 1965.

[37] J. W. Hutchinson. Elastic-plastic behavior of polycrystalline metals and composites. Proceedings of the Royal Society of London, A319, 247-272, 1970.

[38] 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.

[39] M. Jiang, M. Ostoja-Starzewski and I. Jasiuk. Apparent elastic and elastoplastic behavior of periodic composites. Int. J. Solids Struct., 39:199-212, 2002.

[40] T. Kanit, S. Forest, I. Galliet, V. Mounoury and D. Jeulin. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct., 40, 13-14:3647-3679, 2003.

[41] H. Khatam, M.-J. Pindera. Parametric finite-volume micromechanics of periodic materials with elastoplastic phases. Int. J. of Plasticity, 25:1386-1411, 2009.

[42] B. Klusemann and B. Svendsen. Homogenization methods for multiphase elastic composites: Comparisons and benchmarks. Technische Mechanik, 30, 4:274-286, 2009.

[43] V. Kouznetsova, M.G.D. Geers and W.A.M. Brekelmans. Advanced constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Num. Meth. Engrg., 54:1235-1260, 2002.

[44] V. Kouznetsova. Computational homogenization for the multi-scale analysis of multi-phase materials. PhD thesis, Technische Universiteit Eindhoven, The Nederlands, 2002.

[45] V. Kouznetsova. Computational homogenization. Micromechanics graduate course lecture notes, Eindhoven, 2004.

[46] W. Kreher. Residual stresses and stored elastic energy of composites and polycrystals. J. Mech. Phys. Sol., 38:115-128, 1990.

[47] E. Kr?öner. Zur plastischen Verformung des Vielkrystalls. Acta Metall. 9, 2:155-161, 1961.

[48] W. P. Kuykendall, W. D. Cash, D. M. Barnett, W. Cai. On the Existence of Eshelby's Equivalent Ellipsoidal Inclusion Solution. Mathematics and Mechanics of Solids, in press, 2011.

[49] S. Li, R. Sauer and G. Wang. A circular inclusion in a finite domain I. The Dirichlet-Eshelby problem. Acta Mechanica, 179:67-90, 2005.

[50] G. Lielens. Micro-Macro Modeling of Structured Materials. PhD thesis, Université Catholique de Louvain, Belgium, 1999.

[51] 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.

[52] J. C. Michel and P. Suquet. Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Comput. Meth. Appl. Mech. Engrg., 193:5477-5502, 2004.

[53] C. Miehe. Computational micro-to-macro transitions for discretized microstructures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Comput. Mech. Appl. Mech. Engrg., 192:559-591, 2003.

[54] L. Mishnaevsky. Micromechanics of hierarchical materials: a brief overview. Rev.Adv.Mater. Sci., 30:60-72, 2012.

[55] J. Moraleda, J. Segurado and J. Llorca Finite deformation of incompressible fiber-reinforced elastomers: a computational micromechanics approach. J. of the Mech. and Phys. of Solids, 57:1596-1613, 2009.

[56] H. Moulinec and P. Suquet. A fast numerical method for computing the linear and nonlinear properties of composites. C.R. Acad. Sci. Paris II, 318:1417-1423, 1994.

[57] H. Moulinec and P. Suquet. A FFT-based numerical method for computing the mechanical properties of composites from images of their microstructure. Kluwer Academic Publ., Dordrecht, :235-246, 1995.

[58] T. Mura. Micromechanics of defects in solids. Martinus Nijhoff Publishers, 1987.

[59] W. H. M?üller. Mathematical versus experimental stress analysis of inhomogeneities in solids. J. Phys. IV, 6:1-139-C1-148, 1996.

[60] S. Nemat-Nasser. Averaging theorems in finite deformation plasticity. Mech. Mater., 31:493-523, 1999.

[61] S. Nemat-Nasser and M. Hori. Micromechanics: Overall Properties of Heterogeneous Solids. North-Holland, Amsterdam, 1993.

[62] S. Nemat-Nasser and M. Hori. Micromechanics: Overall Properties of Heterogeneous Materials. 2nd Ed., Elsevier, Amsterdam, 1999.

[63] M. Ortiz and E. P. Popov. Accuracy and stability of integration algorithms for elastoplastic constitutive relations. Int. J. Numer. Methods, 21:571-574, 1985.

[64] M. Ostoja-Starzewski. Material spatial randomness: From statistical to representative volume element. Prob. Eng. Mech., 21:112-132, 2006.

[65] M. Paley and J. Aboudi. Micromechanical analyisis of composites by the generalized cells model. Mech. Materials, 14,2:127-139, 1992.

[66] H. E. Petterman, A. F. Planskensteiner, H. J. B?öhm, F. G. Rammerstorfer. A termo-elasto-plastic constitutive law for inhomogeneous materials based on an incremental Mori-Tanaka approach. Computers and Structures, 71:197-214, 2002.

[67] O. Pierard, C. Friebel, I. Doghri. Mean-field homogenization of multiphase thermo-elastic composites: a general framework and its validation. Composites Science and Technology, 64:15871603, 2004.

[68] O. Pierard. Micromechanics of inclusion-reinforced composites in elastoplasticity and elasto-viscoplasticity: modeling and computation. PhD thesis, Université Catholique de Louvain, Belgium, 2006.

[69] O. Pierard, C. González, J. Segurado, J. LLorca, I. Doghri. Micromechanics of elasto-plastic materials reinforced with ellipsoidal inclusions. J. Mech. Phys. Sol., 44:6945-6962, 2007.

[70] M. J. Pindera, H. Khatam, A. Drago and Y. Bansal. Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches. Composites: Part B, 40:349378, 2009.

[71] P. Ponte Casta~neda. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Sol., 39:45-71, 1991.

[72] P. Ponte Casta~neda 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.

[73] G. Povirk. Incorporation of microstructural information into models of twophase materials. Acta Metall. Mater., 43, 8:3199-3206, 1995.

[74] Y. P. Qiu and G. J. Weng. On the application of Mori-Tanaka's theory involving transversely isotropic spheroidal inclusions. Int. J. Engng. Sci., 28, 11:1121-1137, 1990.

[75] J. Qu and M. Cherkaoui. Fundamentals of micromechanics of solids. John Wiley & Sons, New Jersey, 2006.

[76] R. Roscoe. Isotropic composites with elastic or viscoelastic phases: General bounds for the moduli and solutions for special geometries. Rheol. Acta, 12:404-411, 1973.

[77] E. Sánchez-Palencia. Non homogeneous media and vibration theory - lecture notes in physics, 127. Springer, Berlin, 1980.

[78] M. Sautter, C. Dietrich, M. H. Poech, S. Schmauder and H. F. Fischmeister. Finite element modeling of a transverse-loaded fibre composite: Effects of section size and net density. Comput. Mater. Sci., 1:225-233, 1993.

[79] J. Segurado, J. Llorca, C. González. On the accuracy of mean-field approaches to simulate the plastic deformation of composites. Scripta Materialia, 46:525-529, 2002.

[80] J. Segurado, J. Llorca. A numerical approximation to the elastic properties of sphere-reinforced composites. J. Mech. Phys. Sol., 50:2107-2121, 2002.

[81] J. Segurado. Micromecánica computacional de materiales compuestos reforzados con partículas. PhD thesis, Universidad Politécnica de Madrid, Spain, 2004.

[82] J. Segurado, J. Llorca, C. González. Numerical simulation of elastoplastic deformation of composites: evolution of stress microfields and implications for homogenization models.J. Mech. Phys. Sol., 52:1573-1593, 2004.

[83] J. Segurado, J. Llorca. Computational micromechanics of composites: the effect of particle spatial distribution. Mechanics of Materials, 38:873-883, 2006.

[84] J. C. Simo and T. J. R. Hughes. Computational Inelasticity. Springer Verlag, New York, 1998.

[85] E. S. Perdahcoglu and H. J. M. Geijselaers. Constitutive modeling of two phase materials using the mean field method for homogenization. Int. J. Mater. Form. , 2010.

[86] E. A. de Souza Neto and R. A. Feijóo. Variational Foundations of Multi-Scale Constitutive Models of Solid: Small and Large Strain Kinematical Formulation. LNCC Research and Development. Report 16.

[87] P. Suquet. Local and global aspects in the mathematical theory of plasticity. Plasticity today: modelling, methods and applications, A. Sawczuk G. Bianchi. London. Elsevier Applied Science Publishers, 1985.

[88] P. Suquet. Overall properties of non-linear composites: a modified secant moduli theory and its link with Ponte Casta~neda's non-linear variational procedure. Comptes rendus de l'Academie des Sciences, Paris, Serie IIb, 320:563-571.

[89] P. Suquet. Effective properties of nonlinear composites. Continuum Micromechanics. CISM Course and Lecture Notes:197-264.

[90] 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.

[91] H. Tan, Y. Huang, C. Liu, P.H. Geubelle. The MoriTanaka method for composite materials with nonlinear interface debonding. Int. J. of Plasticity, 21:1890-1918, 2005.

[92] G. P. Tandon, G. J. Weng. A theory of particle-reinforced plasticity. Journal of Applied Mechanics, 55:126-135, 1988.

[93] M. Tane, T. Ichitsubo, M. Hirao, H. Nakajima. Extended mean-field method for predicting yield behaviors of porous materials. Mechanics of Materials, 39:5363, 2007.

[94] S. Torquato. Effective stiffness tensor of composite media: II. Applications to isotropic dispersions. J. Mech. Phys. Solids, 35-7:411-440, 1998.

[95] J. Vorel, J. Sýkora and M. Šejnoha. Two Step Homogenization of Effective Thermal Conductivity for Macroscopically Orthotropic C/C composites. Bulletin of applied mechanics, 4-14:48-53, 2008.

[96] C. Weinberger, W. Cai and D. Barnett Lecture Notes Elasticity of Microscopic Structures. ME340B Stanford University Winter 2004., 2005.

[97] P. Withers. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium and its relevance to composite materials. Philosophical Magazine A 59, 4:759-781, 1989.

[98] W. Yu and T. Tang. Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials. Int. J. Sol. Struct., 44:3738-3755, 2007.

[99] Q. S. Zheng and D. X. Du. An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution. J. Mech. Phys. Sol., 49, 11:2765-2788, 2001.

[100] Q. S. Zheng and D. X. Du. A further exploration of the interaction direct derivative (IDD) estimate for the effective properties of muhiphase composites taking into account inclusion distribution. Acta Mechanica, 157:61-80, 2002.

[101] T. Zohdi and P. Wriggers. Introduction to computational micromechanics. Springer Verlag, Berlin, 2005.

Back to Top

Published on 25/10/17

Accepted on 14/07/17

Submitted on 12/07/17

Licence: CC BY-NC-SA license

Are you one of the authors of this document?