Estimation of Influence Tensors for Eigenstressed Multiphase Elastic Media with Nonaligned Inclusion Phases of Arbitrary Ellipsoidal Shape

The analysis of microheterogeneous materials exhibiting eigenstressed and/or eigenstrained phases requires an estimation of eigenstrain influence tensors. Within the framework of continuum micromechanics, we here derive these tensors from extended Eshelby-Laws matrix-inclusion problems, considering, as a new feature, an auxiliary matrix eigenstress. The auxiliary matrix eigenstress is a function of all phase eigenstresses and, hence, accounts for eigenstress interaction. If all material phases are associated with one and the same Hill tensor, the proposed method degenerates to the well-accepted transformation field analysis. Hence, the proposed concept can be interpreted as an extension of the transformation field analysis toward consideration of arbitrarily many Hill tensors, i.e., as an extension toward heterogeneous elastic media comprising inclusion phases with an arbitrary ellipsoidal shape and with arbitrary spatial orientation. This is of particular interest when studying heterogeneous media consisting of constituents with nonspherical phase shapes, e.g., cement-based materials including concrete, or bone. As for polycrystals studied by means of the self-consistent scheme, the auxiliary matrix eigenstress turns out to be equal to the eigenstresses homogenized over the representative volume element (RVE), which is analogous to the self-consistent assumption that the auxiliary stiffness is the average stiffness of the RVE. The proposed method opens the door for micromechanics-based modeling of a great variety of composite phase behaviors characterized by eigenstresses or eigenstrains, e.g., thermoelasticity, poroelasticity, drying-shrinkage, as well as general forms of inelastic behavior, damage, fatigue, and fracture.