Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations

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 consi...

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Bibliographic Details
Published in:Journal of nonlinear science 2019-10, Vol.29 (5), p.2055-2101
Main Authors: Lee, Min-Gi, Katsaounis, Theodoros, Tzavaras, Athanasios E.
Format: Article
Language:English
Subjects:
Analysis
Classical Mechanics
Competition
Dynamical systems
Economic Theory/Quantitative Economics/Mathematical Methods
Edge dislocations
Hardening rate
Localization
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Perturbation theory
Self-similarity
Shear bands
Shear flow
Singular perturbation
Strain rate
Theoretical
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Summary: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.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-019-09538-3