Existence of localizing solutions in plasticity via the geometric singular perturbation theory
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Online Publication Date2017-01-31
Print Publication Date2017-01
Permanent link to this recordhttp://hdl.handle.net/10754/621820
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AbstractShear 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.
CitationLee, M.-G., & Tzavaras, A. (2017). Existence of Localizing Solutions in Plasticity via Geometric Singular Perturbation Theory. SIAM Journal on Applied Dynamical Systems, 16(1), 337–360. doi:10.1137/16m1087308
SponsorsThis research was supported by King Abdullah University of Science and Technology (KAUST).