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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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Boundary Perturbations due to the Presence of Small Cracks

Boundary Perturbations due to the Presence of Small Cracks

Chapter:
(p.66) Chapter Four Boundary Perturbations due to the Presence of Small Cracks
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/9780691165318.003.0005

This chapter considers the perturbations of the displacement (or traction) vector that are due to the presence of a small crack with homogeneous Neumann boundary conditions in an elastic medium. It derives an asymptotic formula for the boundary perturbations of the displacement as the length of the crack tends to zero. Using analytical results for the finite Hilbert transform, the chapter derives an asymptotic expansion of the effect of a small Neumann crack on the boundary values of the solution. It also derives the topological derivative of the elastic potential energy functional and proves a useful representation formula for the Kelvin matrix of the fundamental solutions of Lamé system. Finally, it gives an asymptotic formula for the effect of a small linear crack in the time-harmonic regime.

Keywords:   small crack, Neumann boundary condition, asymptotic formula, boundary perturbation, displacement, asymptotic expansion, topological derivative, potential energy functional, Kelvin matrix, Lamé system

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