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 ε 2 where ε is the length of the crack, and the ε 3-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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Authors would like to thank André Novotny for helpful comments on this paper.
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This work was supported by the ERC Advanced Grant Project MULTIMOD-267184 and NRF grants No. 2009-0085987, 2010-0004091, and 2010-0017532.
Appendix: Derivation of (3.3)
Appendix: Derivation of (3.3)
The Kelvin matrix (2.12) can be rewritten as
where
Using the operator T(∂) defined by (3.2) one can see that
or
We use the formulas
By (6.3), we have
Since Δlog|x−y|=0 for x≠y, we have
Since
we obtain
Similarly, we can compute
Since λ(μ′−λ′)+2μμ′=μ(λ′−μ′), we have
We also have
This proves (3.3).
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Ammari, H., Kang, H., Lee, H. et al. Boundary Perturbations Due to the Presence of Small Linear Cracks in an Elastic Body. J Elast 113, 75–91 (2013). https://doi.org/10.1007/s10659-012-9411-4
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DOI: https://doi.org/10.1007/s10659-012-9411-4