Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations

Min-Gi Lee, Theodoros Katsaounis, Athanasios Tzavaras

Research output: Contribution to journalArticlepeer-review

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 (US)
Pages (from-to)2055-2101
Number of pages47
JournalJournal of Nonlinear Science
Volume29
Issue number5
DOIs
StatePublished - Mar 4 2019

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