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Summary: Complete Asymptotic Expansions of Solutions of the
System of Elastostatics in the Presence of an Inclusion of
Small Diameter and Detection of an Inclusion
Habib Ammari Hyeonbae Kang y Gen Nakamura z
Kazumi Tanuma x
November 15, 2002
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 Lame constants of the imperfection are di erent from those of
the background medium. We establish a complete asymptotic formula for the
displacement vector in terms of the reference Lame 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 mo-
ment tensors (Polya-Szego tensor) and prove that the rst order elastic moment
tensor is symmetric and positive(negative)-de nite. We also obtain estimation
of it's eigenvalue. We then apply these asymptotic formulas for the purpose of
identifying with high precision the order of magnitude of the diameter of the
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