Existence of localizing solutions in plasticity via geometric singular perturbation theory

TL;DR

This paper uses geometric singular perturbation theory to analyze the formation of shear bands in metals during plastic deformation, revealing conditions for localization onset through a mathematical model.

Contribution

It introduces a novel application of geometric singular perturbation theory to construct self-similar solutions for shear band formation in a viscoplasticity model.

Findings

Constructed a family of self-similar focusing solutions.

Reduced the problem to finding heteroclinic orbits in a dynamical system.

Applied geometric singular perturbation theory to analyze shear localization.

Abstract

Shear bands are narrow zones of intense shear observed during plastic deformations of metals at high strain rates. Because they often precede rupture, their study attracted attention as a mechanism of material failure. Here, we aim to reveal the onset of localization into shear bands using a simple model developed from viscoplasticity. We exploit the properties of scale invariance of the model to construct a family of self-similar focusing solutions that capture the nonlinear mechanism of shear band formation. The key step is to de-singularize a reduced system of singular ordinary differential equations and reduce the problem into the construction of a heteroclinic orbit for an autonomous system of three first-order equations. The associated dynamical system has fast and slow time scales, forming a singularly perturbed problem. The geometric singular perturbation theory is applied to this problem to achieve an invariant surface. The flow on the invariant surface is analyzed via the Poincar\'{e}-Bendixson theorem to construct a heteroclinic orbit.