In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.
Homogenization Elasticity Micromechanics
This is a preview of subscription content, log in to check access.
Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38(6):813–841CrossRefMathSciNetGoogle Scholar
Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40:3647–3679zbMATHCrossRefGoogle Scholar
Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) An approach to micro–macro modeling of heterogeneous materials. Comput Mech 27:37–48zbMATHCrossRefGoogle Scholar
Miehe C, Schotte J, Schröder J (1999) Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput Materials Sci 16:372–382CrossRefGoogle Scholar
Miehe C, Schröder J, Becker M (2002) Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction. Comput Methods Appl Mech Eng 191:4971–5005zbMATHCrossRefGoogle Scholar
Müller S (1987) Homogenization of nonconvex integral functionals and cellular elastic materials. Arch Ration Mech Anal 25:189–212Google Scholar
Mura T (1987) Micromechanics of defects in solids. Martinus Nijhoff, The HagueGoogle Scholar
Nemat-Nasser S (2004) Plasticity: a treatise on finite deformation of heterogeneous inelastic materials. Cambridge Press, Cambridge.zbMATHGoogle Scholar
Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials, 2nd edn. North-Holland, AmsterdamGoogle Scholar
Saiki I, Terada K, Ikeda K, Hori M (2002) Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling. Comput Methods Appl Mech Eng 191:2561–2585zbMATHCrossRefMathSciNetGoogle Scholar
Stroeven M, Askes H, Sluys LJ (2004) Numerical determination of representative volumes for granular materials. Comput Methods Appl Mech Eng 193:3221–3238zbMATHCrossRefGoogle Scholar
Sugimoto T (2001) Monodispersed particles. Elsevier, AmsterdamGoogle Scholar