Localization in adiabatic shear flow via geometric theory of singular perturbations

TL;DR

This paper models the formation of shear bands in metals during high-speed shear deformations using geometric singular perturbation theory to analyze the competition between instability and stabilization mechanisms.

Contribution

It introduces a novel application of geometric singular perturbation theory to construct self-similar solutions describing localization in shear flow.

Findings

Existence of self-similar localized solutions in shear deformation models

Application of geometric singular perturbation theory to a four-dimensional dynamical system

Construction of heteroclinic orbits representing shear band formation

Abstract

We study localization occurring during high speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the competition among Hadamard instability (caused by softening response) and the stabilizing effects of strain-rate hardening. We consider a hyperbolic-parabolic system that expresses the above mechanism and construct self-similar solutions of localizing type that arise as the outcome of the above competition. The existence of self-similar solutions is turned, via a series of transformations, into a problem of constructing a heteroclinic orbit for an induced dynamical system. The dynamical system is four dimensional but has a fast-slow structure with respect to a small parameter capturing the strength of strain-rate hardening. Geometric singular perturbation theory is applied to construct the heteroclinic orbit as a transversal intersection of two invariant manifolds in the phase space.