Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations

Min Gi Lee, Theodoros Katsaounis, Athanasios E. Tzavaras

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 between 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 in four dimensions 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.

Original languageEnglish
Pages (from-to)2055-2101
Number of pages47
JournalJournal of Nonlinear Science
Volume29
Issue number5
DOIs
StatePublished - 1 Oct 2019

Keywords

  • Geometric theory of singular perturbations
  • Localization
  • Self-similarity
  • Shear bands

Fingerprint

Dive into the research topics of 'Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations'. Together they form a unique fingerprint.

Cite this