Topological Sensitivity Analysis of Thermo-elastic Constitutive Operator

Abstract

A simple analytical expression for the sensitivity of the two-dimensional macroscopic thermo-elastic constitutive tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework based on the Representative Volume Element (RVE) concept. These mathematical concepts allow the closed form calculation of the sensitivity, whose value depends on the solution of a set of equations over the original domain, of a given shape functional with respect to an infinitesimal domain perturbation. Their use in the context of multi-scale material design is reported in a number of recent publications. In the present context, the variational setting in which the underlying multi-scale theory is cast, is found to be particularly well-suited for the application of the topological derivative formalism. The derived sensitivity – a symmetric second order tensor field over the RVE domain – measures how the macroscopic thermo-elastic parameters estimated within the multiscale framework changes when a small circular inclusion is introduced at the micro-scale level.The final format of the proposed analytical formula is strikingly simple and can be potentially used in applications such as the synthesis and optimal design of microstructures to meet a specified macroscopic behavior.