Concentration tensors preserving elastic symmetry of multiphase composites

Abstract

The micromechanics of composites with multiple phases of different shapes embedded into a matrix phase, typically requires symmetrization strategies for their homogenized elasticity tensors, in particular so if the popular Mori–Tanaka estimate is employed. We here explore the implications of such symmetrization techniques, on the concentrations tensors, i.e. on the relations between macroscopic strains imposed onto a representative volume element of microelastic matter, and the microscopic phase strains developing across the materials’ microstructure. Thereby, we adopt the important idea of Mori and Tanaka to approximate the phase strains by the homogeneous strains inside an Eshelbian inhomogeneity embedded into an infinite matrix, together with the phase strains fulfilling the strain average rule; while we refrain from the identification of the strain in the matrix phase as the auxiliary strain imposed remotely at the infinite matrix of Eshelby's matrix-inhomogeneity problem. Instead of this identification, we allow for a general multilinear relation between auxiliary strains and RVE-related macroscopic strains, and we express the homogenized stiffness as (i) a function of the conversion tensor quantifying the aforementioned multilinear relation, and (ii) as the symmetrized Mori–Tanaka estimate. In this way, the conversion tensor and all phase concentration tensors can be determined in a way which allows the overall elastic stiffness to remain symmetric. This is illustrated by means of two benchmark examples concerning matrix–inclusion composites hosting both spherical and prolate inclusions, with the matrices being isotropic and transversely isotropic, respectively. These examples benefit from analytical expressions for the Hill tensor governing the underlying matrix-inhomogeneity problems.