The Eshelby theorem and patch optimization problem

Abstract

Abstract. Let Ω 0 be an ellipsoidal inclusion in the Euclidean space R n . It is checked that if a solution of the homogeneous transmission problem for a formally selfadjoint elliptic system of second order differential equations with piecewise smooth coefficients grows linearly at infinity, then this solution is a linear vector-valued function in the interior of Ω 0 . This fact generalizes the classical Eshelby theorem in elasticity theory and makes it possible to indicate simple and explicit formulas for the polarization matrix of the inclusion in the composite space, as well as to solve a problem about optimal patching of an elliptical hole. §1. Setting of the transmission problem Let Ω 0 be a nonempty domain in the Euclidean space R n , n ≥ 2, with a smooth (of class C ∞ ) boundary, and let Ω 1 = R n \ Ω 0 be its exterior. Suppose it has compact closure:We place the origin of a Cartesian coordinate system x = (x 1 , . . . , x n ) in the interior of Ω 0 . Let D(∇ x ) be an (N ×k)-matrix of first order differential operators with constant and, in general, complex coefficients, and let A i , i = 0, 1, be numerical Hermitian positive definite matrices of size N × N . We introduce the formally selfadjoint (k × k)-matrices of second order differential operatorsand consider the following system of differential equations with transmission conditions on the interface Γ = ∂Ω 0 = ∂Ω 1 :

The topological gradient in anisotropic elasticity with an eye towards lightweight design

“…This result was deduced in [33] for the isotropic case and [50] for the general case of anisotropic materials. See [38] for a mathematically rigorous derivation. S is called Eshelby's tensor.…”

“… This result was deduced in for the isotropic case and for the general case of anisotropic materials. See for a mathematically rigorous derivation. is called Eshelby's tensor.…”

“…Summarizing our improvements rely upon the following facts: Eshelby's theorem holds for ellipsoids in an arbitrary anisotropic material, see for a mathematically rigorous treatment. Eshelby's tensor has been explicitly computed for spheroidal inhomogeneities in transversely isotropic materials in . Recent advances in the numerical evaluation of Eshelby's tensor (more precisely of the related Hill tensor, see ) in materials with arbitrary anisotropy enable a fast numerical evaluation of the corresponding EMT. The formulae are particularly simple for orthotropic symmetry.…”

“…Eshelby's theorem holds for ellipsoids in an arbitrary anisotropic material, see for a mathematically rigorous treatment.…”