Boundary perturbations due to the presence of small linear cracks in an elastic body

2012/06/27 by Habib Ammari, Ammari, Habib, Hyeonbae Kang +5
Mathematics · #35B30 #74B05 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B30 #msc:74B05

paper · pdf · doi:10.48550/arxiv.1206.6315

arxiv created 2012/06/27 · arxiv updated 2012/06/28

Abstract

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear crack. The formula reveals that the leading order term is ε² where εis the length of the crack, and the ε³-term vanishes. We obtain an asymptotic expansion of the elastic potential energy as an immediate consequence of the boundary perturbation formula. The derivation is based on layer potential techniques. It is expected that the formula would lead to very effective direct approaches for locating a collection of small elastic cracks and estimating their sizes and orientations.

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