Existence of localizing solutions in plasticity via the geometric singular perturbation theory

Handle URI:
http://hdl.handle.net/10754/621820
Title:
Existence of localizing solutions in plasticity via the geometric singular perturbation theory
Authors:
Lee, Min-Gi ( 0000-0002-1953-2547 ) ; Tzavaras, Athanasios ( 0000-0002-1896-2270 )
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 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.
KAUST Department:
Computer, Electrical, Mathematical Sciences & Engineering Division
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Applied Dynamical Systems
Issue Date:
31-Jan-2017
DOI:
10.1137/16M1087308
ARXIV:
1608.00198
Type:
Article
Sponsors:
This research was supported by King Abdullah University of Science and Technology (KAUST).
Additional Links:
https://arxiv.org/abs/1608.00198; http://epubs.siam.org/doi/abs/10.1137/16M1087308
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorLee, Min-Gien
dc.contributor.authorTzavaras, Athanasiosen
dc.date.accessioned2017-02-15T07:25:05Z-
dc.date.available2016-11-13T08:22:10Z-
dc.date.available2017-02-15T07:25:05Z-
dc.date.issued2017-01-31-
dc.identifier.doi10.1137/16M1087308-
dc.identifier.urihttp://hdl.handle.net/10754/621820-
dc.description.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.en
dc.description.sponsorshipThis research was supported by King Abdullah University of Science and Technology (KAUST).en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttps://arxiv.org/abs/1608.00198en
dc.relation.urlhttp://epubs.siam.org/doi/abs/10.1137/16M1087308en
dc.rightsArchived with thanks to SIAM Journal on Applied Dynamical Systems.en
dc.titleExistence of localizing solutions in plasticity via the geometric singular perturbation theoryen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical, Mathematical Sciences & Engineering Divisionen
dc.identifier.journalSIAM Journal on Applied Dynamical Systemsen
dc.eprint.versionPost-printen
dc.identifier.arxivid1608.00198-
kaust.authorLee, Min-Gien
kaust.authorTzavaras, Athanasiosen

Version History

VersionItem Editor Date Summary
2 10754/621820grenzdm2017-02-15 07:20:30.368Article published with DOI and accepted manuscript received from Prof. Tzavaras.
1 10754/621820.1grenzdm2016-11-13 08:22:10.0
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