The Eshelby problem in amorphous solids

TL;DR

This paper revises the classical Eshelby problem for amorphous solids by incorporating screening effects and realistic boundaries, providing more accurate models of stress redistribution after plastic events.

Contribution

It presents a direct solution to the Eshelby problem in amorphous solids, highlighting significant modifications to classical elastic solutions for better modeling.

Findings

Classical Eshelby kernel is modified in amorphous solids.

Screening effects significantly alter stress redistribution.

Realistic boundary conditions are crucial for accurate modeling.

Abstract

The ``Eshelby problem" refers to the response of a 2-dimensional elastic sheet to cutting away a circle, deforming it into an ellipse, and pushing it back. The resulting response is dominated by the so-called ``Eshelby Kernel" which was derived for purely elastic (infinite) material, but has been employed extensively to model the redistribution of stress after plastic events in amorphous solids with finite boundaries. Here we discuss and solve the Eshelby problem directly for amorphous solids, taking into account possible screening effects and realistic boundary conditions. We find major modifications compared to the classical Eshelby solution. These modification are needed for modeling correctly the spatial responses to plastic events in amorphous solids.