On the Nonlinear Eshelby Inclusion Problem and its Isomorphic Growth Limit

TL;DR

This paper develops a semi-inverse nonlinear elastic solution for growing spheroidal inclusions in soft materials, exploring their asymptotic shapes and pressures, extending Eshelby's classical linear results to the nonlinear regime.

Contribution

It introduces the first semi-inverse nonlinear solution for isotropic growth of spheroidal inclusions in neo-Hookean materials and analyzes their asymptotic behavior.

Findings

Existence of a non-spherical asymptotic shape for large inclusions

Identification of an isomorphic pressure associated with the asymptotic limit

Extension of classical linear Eshelby solutions to nonlinear elastic regime

Abstract

In the late 1950's, Eshelby's linear solutions for the deformation field inside an ellipsoidal inclusion and, subsequently, the infinite matrix in which it is embedded were published. The solutions' ability to capture the behavior of an orthotropically symmetric shaped inclusion made it invaluable in efforts to understand the behavior of defects within, and the micromechanics of, metals and other stiff materials throughout the rest of the 20th century. Over half a century later, we wish to understand the analogous effects of microstructure on the behavior of soft materials; both organic and synthetic; but in order to do so, we must venture beyond the linear limit, far into the nonlinear regime. However, no solutions to these analogous problems currently exist for non-spherical inclusions. In this work, we present an accurate semi-inverse solution for the elastic field in an isotropically growing spheroidal inclusion embedded in an infinite matrix, both made of the same incompressible neo-Hookean material. We also investigate the behavior of such an inclusion as it grows infinitely large, demonstrating the existence of a non-spherical asymptotic shape and an associated asymptotic pressure. We call this the isomorphic limit, and the associated pressure the isomorphic pressure.