Topological Derivative for Multi-Scale Linear Elasticity Models in Three Spatial Dimensions

Abstract

A remarkably simple analytical expression for the sensitivity of the three-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is obtained. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity, given by a symmetric fourth order tensor field over the microstructure domain, measures how the estimated three-dimensional macroscopic elasticity tensor changes when a small spherical void is introduced at the micro-scale level. The obtained result can be applied in the synthesis and optimal design of three-dimensional elastic microstructures.