An asymptotic formula for the displacement field in the presence of small anisotropic elastic inclusions

TL;DR

This paper develops an asymptotic expansion for the displacement field in elastic bodies with small, possibly anisotropic, inhomogeneities, extending previous conductivity results to elasticity and providing explicit formulas for thin inclusions.

Contribution

It extends asymptotic analysis to anisotropic elastic inclusions without geometric restrictions, introducing an elastic moment tensor and explicit formulas for thin inhomogeneities.

Findings

Derived first-order asymptotic expansion for boundary displacement differences.

Established explicit formula for the elastic moment tensor in thin inhomogeneities.

Proved uniqueness of the elastic moment tensor in the case of planar inclusions.

Abstract

We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of CapdeBoscq and Vogelius ({\em Math. Modelling Num. Anal.} 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains an elastic moment tensor $\MM$ that encodes the effect of the inclusions. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for $\MM$ only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of $\MM$ in this setting and recover the formula previously obtained by Beretta and Francini ({\em SIAM J. Math. Anal.}, 38, 2006).