An approach to nonlinear viscoelasticity via metric gradient flows

Mielke, Alexander, Ortner, Christoph and Sengul, Yasemin 2014. An approach to nonlinear viscoelasticity via metric gradient flows. SIAM Journal on Mathematical Analysis 46 (2) , 1317–1347. 10.1137/130927632

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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:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Date of Acceptance:	14 January 2014
Last Modified:	05 Jul 2022 11:07
URI:	https://orca.cardiff.ac.uk/id/eprint/145147

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