### 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 language | English (US) |
---|---|

Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Inverse Problems and Imaging |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2012 |

### Fingerprint

### 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

*Inverse Problems and Imaging*,

*6*(1), 1-23. https://doi.org/10.3934/ipi.2012.6.1

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

Research output: Contribution to journal › Article

*Inverse Problems and Imaging*, vol. 6, no. 1, pp. 1-23. https://doi.org/10.3934/ipi.2012.6.1

}

TY - JOUR

T1 - Small volume asymptotics for anisotropic elastic inclusions

AU - Beretta, Elena

AU - Bonnetier, Eric

AU - Francini, Elisa

AU - Mazzucato, Anna L.

PY - 2012/2/1

Y1 - 2012/2/1

N2 - 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.

AB - 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.

KW - Anisotropic elasticity

KW - Homogenization

KW - Inclusions

KW - Polarization tensor

KW - Small volume asymptotics

UR - http://www.scopus.com/inward/record.url?scp=84857184970&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857184970&partnerID=8YFLogxK

U2 - 10.3934/ipi.2012.6.1

DO - 10.3934/ipi.2012.6.1

M3 - Article

VL - 6

SP - 1

EP - 23

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

SN - 1930-8337

IS - 1

ER -