Analytic Nonlinear Theory of Shear Banding in Amorphous Solids

TL;DR

This paper develops an analytic nonlinear theory for shear banding in amorphous solids under athermal quasi-static shear, revealing the mechanisms and critical conditions for shear band formation.

Contribution

It introduces a nonlinear elastic framework incorporating plastic deformation effects, predicting shear band formation and width based on elastic moduli.

Findings

Derived nonlinear equations for displacement fields in amorphous solids.

Identified an instability leading to shear band formation.

Provided analytic expressions for shear band width and critical stress.

Abstract

The aim of this paper is to offer an analytic theory of the shear banding instability in amorphous solids that are subjected to athermal quasi-static shear. To this aim we derive nonlinear equations for the displacement field, including the consequences of plastic deformation on the mechanical response of amorphous solids. The plastic events collectively induce distributed dipoles that are responsible for screening effects and the creation of typical length-scales that are absent in classical elasticity theory. The nonlinear theory exposes an instability that results in the creation of shear bands. By solving the weakly nonlinear amplitude equation we present analytic expressions for the displacement fields that is associated with shear bands, explaining the role of the elastic moduli that determine the width of a shear band from ductile to brittle characteristics. We derive an energy functional whose Hessian possesses an eigenvalue that goes to zero at the shear-banding instability, providing a prediction for the critical value of the accumulated stress that results in an instability.