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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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Elasticity Equations with High Contrast Parameters

Elasticity Equations with High Contrast Parameters

Chapter:
(p.33) Chapter Two Elasticity Equations with High Contrast Parameters
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/9780691165318.003.0003

This chapter presents some recent results on the elasticity equations with high contrast coefficients. It first sets up the problems for finite and extreme moduli before discussing the incompressible limit of elasticity equations. It then provides a complete asymptotic expansion with respect to the compressional modulus and considers the limiting cases of holes and hard inclusions. It proves that the energy functional is uniformly bounded and demonstrates that the potentials on the boundary of the inclusion are also uniformly bounded. It also shows that these potentials converge as the bulk and shear moduli tend to their extreme values and that similar boundedness and convergence result holds true for the boundary value problem.

Keywords:   elasticity equation, high contrast coefficient, incompressible limit, asymptotic expansion, compressional modulus, hole, hard inclusion, energy functional, boundary value problem, boundedness

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