Existence of localizing solutions in plasticity via the geometric singular perturbation theory

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 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 desingularize 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. Geometric singular perturbation theory is applied to this problem to achieve an invariant surface. The flow on the invariant surface is analyzed via the Poincaré--Bendixson theorem to construct a heteroclinic orbit.
Original languageEnglish (US)
Pages (from-to)337-360
Number of pages24
JournalSIAM Journal on Applied Dynamical Systems
Volume16
Issue number1
DOIs
StatePublished - Jan 31 2017

Fingerprint Dive into the research topics of 'Existence of localizing solutions in plasticity via the geometric singular perturbation theory'. Together they form a unique fingerprint.

Cite this