Multi-inclusion method for finite deformations: exact results and applications

Abstract

For linearly elastic heterogeneous solids, averaging theorems are developed by Nemat-Nasser and Hori (S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids, Elsevier, Amsterdam, 1993; and S. Nemat-Nasser, M. Hori, J. Eng. Mater. Technol. 117 (1995) 412), using a multi-inclusion model. This model is based on the observation that the average strain and stress in any annulus of a nested sequence of ellipsoids embedded in an infinite homogeneous solid, can be computed exactly when each annulus undergoes a uniform transformation with an associated transformation strain different from the others; the central ellipsoid may have a non-uniform transformation strain. This result generalizes the pioneering observation of Tanaka and Mori (K. Tanaka, T. Mori, J. Elast. 2 (1972) 199; and T. Mori, K. Tanaka, Acta Met. 21 (1973) 571) who showed that the average strain and stress in the region between two similar and coaxial ellipsoids in an infinite uniform elastic solid, are zero for any transformation strain within the inner ellipsoid; see also Benveniste, and, and Mori and Wakashima (T. Benveniste, Mech. Mater. 6 (1987) 147; and T. Mori, K. Wakashima, Successive iteration method in the evaluation of average fields in elastically inhomogeneous materials, Micromechanics and Inhomogeneity — The T. Mura 65th Anniversary Volume, Springer, New York, 1990, pp. 269–282). Similar results apply to the finite deformation problems, provided that the nominal stress rate and the rate of change of the deformation gradient, (measured relative to any arbitrary state) are used as the dynamical and kinematical variables; see Nemat-Nasser (S. Nemat-Nasser, Mech. Mater. 31, (1999) 493) for a comprehensive account of a rigorous treatment of the transition from micro- to macro-variables of a representative volume element of a finitely deformed aggregate. An exact method for homogenization of an ellipsoidal heterogeneity in an unbounded finitely deformed homogeneous solid, is developed, using the generalized Eshelby tensor. It is shown that many results for single-, double-, and multi-inclusion problems in linear elasticity (see S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids, Elsevier, Amsterdam, 1993; and S. Nemat-Nasser, M. Hori, J. Eng. Mater. Technol. 117 (1995) 412), also apply to the finite-deformation rate problems, provided suitable kinematical and dynamical variables are used. The problem of the double inclusion is considered and exact expressions are given for the average field quantities, taken over the region between the two, as well as within each ellipsoidal domains, one containing the other, when arbitrary eigenvelocity gradients are prescribed within an arbitrary region contained in the inner ellipsoid. This generalizes to the fully nonlinear, finitely-deformed, elastoplastic case, the Tanaka–Mori (K. Tanaka, T. Mori, J. Elast. 2 (1972) 199) result, and the double inclusion result of Nemat-Nasser and Hori (S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids, Elsevier, Amsterdam, 1993; and S. Nemat-Nasser, M. Hori, J. Eng. Mater. Technol. 117 (1995) 412), which have been developed for linearly elastic solids. The application of the exact results to the problem of estimating the overall mechanical response of a finitely deformed heterogeneous representative volume element (RVE) is outlined and the overall effective pseudo-modulus tensor of the RVE is calculated for rate-independent elastoplastic materials.