2017/07/17 by Min-Gi Lee, Lee, Min-Gi, Theodoros Katsaounis +3
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1707.05283
openalex publication_date 2017/07/17 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study localization occurring during high speed shear deformations of\nmetals leading to the formation of shear bands. The localization instability\nresults from the competition among Hadamard instability (caused by softening\nresponse) and the stabilizing effects of strain-rate hardening. We consider a\nhyperbolic-parabolic system that expresses the above mechanism and construct\nself-similar solutions of localizing type that arise as the outcome of the\nabove competition. The existence of self-similar solutions is turned, via a\nseries of transformations, into a problem of constructing a heteroclinic orbit\nfor an induced dynamical system. The dynamical system is four dimensional but\nhas a fast-slow structure with respect to a small parameter capturing the\nstrength of strain-rate hardening. Geometric singular perturbation theory is\napplied to construct the heteroclinic orbit as a transversal intersection of\ntwo invariant manifolds in the phase space.\n