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Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

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Abstract

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lamé constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lamé constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Pólya–Szegö tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.

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Authors and Affiliations

  1. Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Habib Ammari

  2. School of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea

    Hyeonbae Kang

  3. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Gen Nakamura

  4. Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu, 376-8515, Japan

    Kazumi Tanuma

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  1. Habib Ammari
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  2. Hyeonbae Kang
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  3. Gen Nakamura
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  4. Kazumi Tanuma
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Ammari, H., Kang, H., Nakamura, G. et al. Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion. Journal of Elasticity 67, 97–129 (2002). https://doi.org/10.1023/A:1023940025757

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