Reconstruction of shear modulus inclusions in elastostatics via divergence-free localization

Abstract

We formulate an improved version of the monotonicity method for shape reconstruction in the inverse problem of linear elastostatics. We show that this method can reconstruct inclusions in the second Lam\'e parameter, i.e., the shear modulus, independently of inhomogeneities in the first Lam\'e parameter. This improves on earlier methods that only recover inclusions in the bulk modulus under certain assumptions on the local interplay between the Lam\'e parameters. Our proofs are based on linearization and divergence-free localization of energy using harmonic vector fields.