TY - JOUR

T1 - Localization in Adiabatic Shear Flow Via Geometric Theory of Singular Perturbations

AU - Lee, Min-Gi

AU - Katsaounis, Theodoros

AU - Tzavaras, Athanasios

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors thank Prof. Peter Szmolyan for valuable discussions on the use of geometric singular perturbation theory.

PY - 2019/3/4

Y1 - 2019/3/4

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/631130

UR - https://link.springer.com/article/10.1007%2Fs00332-019-09538-3

UR - http://www.scopus.com/inward/record.url?scp=85062715239&partnerID=8YFLogxK

U2 - 10.1007/s00332-019-09538-3

DO - 10.1007/s00332-019-09538-3

M3 - Article

SN - 0938-8974

VL - 29

SP - 2055

EP - 2101

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 5

ER -