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Mathematical Methods in Elasticity Imaging$
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Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, and Abdul Wahab

Print publication date: 2015

Print ISBN-13: 9780691165318

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691165318.001.0001

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Small-Volume Expansions of the Displacement Fields

Small-Volume Expansions of the Displacement Fields

Chapter:
(p.48) Chapter Three Small-Volume Expansions of the Displacement Fields
Source:
Mathematical Methods in Elasticity Imaging
Author(s):

Habib Ammari

Elie Bretin

Josselin Garnier

Hyeonbae Kang

Hyundae Lee

Abdul Wahab

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691165318.003.0004

This chapter deals with small-volume expansions of the displacement fields. It first introduces the notion of elastic moment tensor, a geometric quantity associated with a small-volume inclusion, before discussing some of its important properties such as symmetry and positive-definiteness. The asymptotic expansion of the displacement in the presence of a small-volume inclusion is expressed in terms of the elastic moment tensor. The chapter proceeds by deriving formulas for the elastic moment tensors under linear transformations and computes those associated with ellipses and balls. It also considers both the static and time-harmonic regimes and extends the small-volume asymptotic framework to anisotropic elasticity. Finally, it provides the leading-order terms in the asymptotic expansions of the solutions to the static and time-harmonic elasticity equations with respect to the size of a small inclusion.

Keywords:   small-volume expansion, asymptotic expansion, elastic moment tensor, linear transformation, ellipse, ball, displacement field, anisotropic elasticity, elasticity equation, small inclusion

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