{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/existence-of-localizing-solutions-in","title":"Existence of localizing solutions in plasticity via geometric singular perturbation theory","arxiv_id":"1608.00198","date":"2016-07-31","proceeding":null,"authors":["Min-Gi Lee","Athanasios Tzavaras"],"abstract":"Shear 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 developed 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 de-singularize 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. The geometric singular perturbation theory is applied to this problem to achieve an invariant surface. The flow on the invariant surface is analyzed via the Poincar\\'{e}-Bendixson theorem to construct a heteroclinic orbit.","url_abs":"http://arxiv.org/abs/1608.00198v2","url_pdf":"http://arxiv.org/pdf/1608.00198v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"existence-of-localizing-solutions-in","repo_url":"https://github.com/cebumactan/ming-lee","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}