Path independent integrals to identify localized plastic events in two dimensions

TL;DR

This paper develops a mathematical framework using path independent integrals to identify and analyze localized plastic events in two-dimensional elastic materials, aiding understanding of plasticity mechanisms.

Contribution

It introduces a novel method employing complex potentials and Cauchy integrals to detect and quantify localized plastic deformations in 2D materials.

Findings

Analytical expressions for elastic fields around plastic events.

Numerical validation using finite element data.

Potential for identifying structural reorganizations in amorphous materials.

Abstract

We use a power expansion representation of plane elasticity complex potentials due to Kolossov and Muskhelishvili, to compute the elastic fields induced by a localized plastic deformation event. Far from its center, the dominant contributions correspond to first order singularities of quadrupolar and dipolar symmetry which can be associated respectively to pure deviatoric and pure volumetric plastic strain of an equivalent circular inclusion. Constructing holomorphic functions from the displacement field and its derivatives, it is possible to define path independent Cauchy integrals which capture the amplitudes of these singularities. Analytical expressions and numerical tests on simple finite element data are presented. The development of such numerical tools is of direct interest for the identification of local structural reorganizations which are believed to be the key mechanisms for plasticity of amorphous materials.