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An approach to nonlinear viscoelasticity via metric gradient flows
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Mielke, Alexander, Ortner, Christoph and Şengül, Yasemin (2014) An approach to nonlinear viscoelasticity via metric gradient flows. SIAM Journal on Mathematical Analysis, Volume 46 (Number 2). pp. 1317-1347. doi:10.1137/130927632
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Official URL: http://dx.doi.org/10.1137/130927632
Abstract
We formulate quasistatic nonlinear finite-strain viscoelasticity of rate type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can be obtained only by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous limit in a specific case.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Viscoelasticity | ||||||||
| Journal or Publication Title: | SIAM Journal on Mathematical Analysis | ||||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||||
| ISSN: | 0036-1410 | ||||||||
| Official Date: | 2014 | ||||||||
| Dates: |
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| Volume: | Volume 46 | ||||||||
| Number: | Number 2 | ||||||||
| Page Range: | pp. 1317-1347 | ||||||||
| DOI: | 10.1137/130927632 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Funder: | Deutsche Forschungsgemeinschaft (DFG), European Research Council (ERC), Leverhulme Trust (LT), Engineering and Physical Sciences Research Council (EPSRC), Türkiye Bilimsel ve Teknolojik Araştırma Kurumu [Scientific and Technological Research Council of Turkey] (TÜBİTAK) | ||||||||
| Grant number: | FOR797 MicroPlast (DFG), AdG 267802 (ERC), EP/D048400/1 (EPSRC), EP/E035027/1 (EPSRC), 2213(TÜBİTAK) |
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