Small volume asymptotics for anisotropic elastic inclusions

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 (Math. Modeling 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 a moment or polarization tensor $\mathbb{M}$ that encodes the effect of the inclusions. We also derive some basic properties of this tensor $\mathbb{M}$. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for $\mathbb{M}$ 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 $\mathbb{M}$ in this setting and recover the formula previously obtained by Beretta and Francini (SIAM J. Math. Anal., 38, 2006).