An asymptotic formula for the displacement field in the presence of thin elastic inhomogeneities

We consider a plane isotropic homogeneous elastic body with thin elastic inhomogeneities in the form of small neighborhoods of simple smooth curves. We derive a rigorous asymptotic expansion of the boundary displacement field as the thickness of the neighborhoods goes to zero. © 2006 Society for Industrial and Applied Mathematics

Small volume asymptotics for anisotropic elastic inclusions

International audience; 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 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)

Scatter of elastic waves by a thin flat elliptical inhomogeneity

Elastodynamic fields of a single, flat, elliptical inhomogeneity embedded in an infinite elastic medium subjected to plane time harmonic waves are studied. Scattered displacement amplitudes and stress intensities are obtained in series form for an incident wave in an arbitrary direction. The cases of a penny shaped crack and an elliptical crack are given as examples. The analysis is valid for alpha a up to about two, where alpha is longitudinal wave number and a is a typical geometric parameter

Small Perturbations of an Interface for Elastostatic Problems

We consider solutions to the Lam\'e system in two dimensions. By using systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation), we establish a rigorous asymptotic expansion for the perturbations of the displacement field caused by small perturbations of the shape of an elastic inclusion with C2-boundary. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moments tensors (EMTs) due to the presence of small changes in the interface of the inclusion.Comment: 42pages,0 figures. arXiv admin note: text overlap with arXiv:1601.0677