Small volume asymptotics for anisotropic elastic inclusions

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

Research output: Contribution to journalArticle

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 diérence 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 M that encodes the eéct of the inclusions. We also derive some basic properties of this tensor. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for 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 in this setting and recover the formula previously obtained by Beretta and Francini.

Original languageEnglish (US)
Pages (from-to)1-23
Number of pages23
JournalInverse Problems and Imaging
Volume6
Issue number1
DOIs
StatePublished - Feb 1 2012

Fingerprint

Inclusion
Tensors
Tensor
Inhomogeneity
Asymptotic Expansion
First-order
Measurable set
Elastic body
Neumann Boundary Conditions
Conductivity
Strip
Elasticity
Polarization
Uniqueness
Boundary conditions
Moment
Curve
Modeling

Keywords

  • Anisotropic elasticity
  • Homogenization
  • Inclusions
  • Polarization tensor
  • Small volume asymptotics

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

Cite this

Small volume asymptotics for anisotropic elastic inclusions. / Beretta, Elena; Bonnetier, Eric; Francini, Elisa; Mazzucato, Anna L.

In: Inverse Problems and Imaging, Vol. 6, No. 1, 01.02.2012, p. 1-23.

Research output: Contribution to journalArticle

Beretta, Elena ; Bonnetier, Eric ; Francini, Elisa ; Mazzucato, Anna L. / Small volume asymptotics for anisotropic elastic inclusions. In: Inverse Problems and Imaging. 2012 ; Vol. 6, No. 1. pp. 1-23.
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