Shape derivative of the elastic moment tensor in two dimensions

Abstract

An elastic inclusion that has different material properties from the background elastic body induces a perturbation for a given far-field loading. This perturbation admits a multipole expansion whose coefficients are called the Elastic moment tensors (EMTs), which can be obtained from the exterior measurements of the field perturbation The EMTs contain information on the material and geometric properties of the inclusion, and they have been used as building blocks in the inverse problems of recovering the elastic inclusions from the exterior measurements. By the asymptotic analysis, the shape derivative of the EMTs can be expressed as a boundary integral in terms of the shape deformation of the inclusion. Based on this integral formula, an iterative optimization method to reconstruct the inclusion was derived. In this paper, we consider the shape derivative of the EMTs assuming that the inclusion is a small deformation of a disk, which can be approximated by the lower order terms of the EMTs. In particular, by employing the complex formulation for the solution to the linear elasticity problem in two dimensions, we explicitly express the shape derivative in terms of the Fourier coefficients of the shape deformation function from the disk. This expression provides us an analytic shape recovery formula for the elastic inclusion.