On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations

We present two applications of the integro-differential volume equation for the eigenstrain, building on Eshelby's inclusion method [15,16], in the contexts of both static and dynamic linear elasticity. The primary objective is to address the inverse problem of recovering the elastic moduli of the inhomogeneity using a limited number of incident fields. In the static case, we adopt an efficient reformulation of Eshelby's equation proposed by Bonnet [7]. By employing a first-order approximation in addition with a limited number of incident loadings and measurements, we numerically determine the material coefficients of the inclusion. In elastodynamics, we focus on the inverse scattering problem, utilizing the Lippmann-Schwinger integral equation to reconstruct the elastic properties of the inclusion through a Newton-type iterative scheme. We construct the Frechet derivative and we formulate the linearized far-field equation. Additionally, the corresponding plane strain problems are analyzed in both static and dynamic elasticity.