Influence Tensors for the Analytical Mechanics of Anisotropic Eigenstressed Composites with Inclusions of Various Shapes and Orientations

Abstract

The Mori-Tanaka-Benveniste scheme is very popular for the homogenization of the elastic stiffness of microheterogeneous composites consisting of one matrix phase and any number of inclusion phases. In addition, the scheme allows for homogenization of eigenstresses/eigenstrains, e.g. in the fields of poroelasticity, thermoelasticity, drying shrinkage, and elastoplasticity. Still, the Mori-Tanaka-Benveniste scheme cannot appropriately represent matrix-inclusion composites with non-aligned ellipsoidal inclusion phases, because (i) the respective homogenized stiffness estimate becomes non-symmetrical, and (ii) the eigenstrain influence tensors do not satisfy the elastic reciprocal theorem. This problem has been recently solved by direct symmetrization of the homogenized Mori-Tanaka-Benveniste stiffness estimate, with corresponding modification of the matrix strains leading to improved macro-to-micro strain concentration tensor estimates. The present contribution extends these recent progress towards eigenstressed media, in terms of novel estimates for microscopic eigenstress-to-strain influence tensors which are (i) kinematically compatible, in the sense of satisfying the strain average rule, (ii) statistically admissible, in the sense that the stress average rule delivers the same macroscopic stress state as Levin’s theorem, and (iii) energetically consistent, in the sense of satisfying the elastic reciprocal theorem.