An extension of the Eshelby conjecture to domains of general shape in anti-plane elasticity

TL;DR

This paper extends the Eshelby conjecture in anti-plane elasticity, characterizing shapes that produce uniform interior strain under polynomial loadings using conformal maps and Grunsky coefficients.

Contribution

It generalizes the Eshelby conjecture to arbitrary shapes and polynomial loadings, providing explicit shape characterizations and solutions.

Findings

Inclusion induces uniform strain iff conformal map is a Laurent series of degree N.

Shape characterized by first-degree polynomial loading in isotropic case.

Explicit interior potential solution using Grunsky coefficients.

Abstract

According to the Eshelby conjecture, an ellipse or ellipsoid is the only shape that induces an interior uniform strain under a uniform far-field loading. We extend the Eshelby conjecture to domains of general shape for anti-plane elasticity. Specifically, we show that for each positive integer $N$, an inclusion induces an interior uniform strain under a polynomial loading of degree $N$ if and only if the exterior conformal map of the inclusion is a Laurent series of degree $N$. Furthermore, for the isotropic case, we characterize the shape of an inclusion by only using the first-degree polynomial loading and explicitly solve the interior potential of the inclusion in terms of the Grunsky coefficients.