Small volume asymptotics for anisotropic elastic inclusions

Elena Beretta; Eric Bonnetier; Elisa Francini; Anna L. Mazzucato

doi:10.3934/ipi.2012.6.1

Article Contents

2012, Volume 6, Issue 1: 1-23. Doi: 10.3934/ipi.2012.6.1

This issue Previous Article Next Article On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

Small volume asymptotics for anisotropic elastic inclusions

1.
Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5 - 00185 Roma
2.
Laboratoire Jean Kuntzmann, Université de Joseph Fourier, CNRS, 38041 Grenoble Cedex 9
3.
Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67A - 50134 Firenze
4.
Mathematics Department, Penn State University, University Park, PA, 16802

Received: May 2011

Revised: November 2011

Published: February 2012

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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 (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).

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Mathematics Subject Classification: 35J57, 74B05 , 74Q05, 35C20.

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