Existence and uniqueness of global weak solutions to strain-limiting viscoelasticity with Dirichlet boundary data

Bulicek, Miroslav, Patel, Victoria, Suli, Endre and Sengul, Yasemin 2022. Existence and uniqueness of global weak solutions to strain-limiting viscoelasticity with Dirichlet boundary data. SIAM Journal on Mathematical Analysis 54 (6) , pp. 6186-6222. 10.1137/21M1455322

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Abstract

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of these implicit constitutive relations, we establish the existence and uniqueness of a global-in-time large-data weak solution. Then we focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises. The Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfil natural compatibility conditions.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Funders:	GACR, EPSRC
Date of First Compliant Deposit:	3 October 2022
Date of Acceptance:	5 August 2022
Last Modified:	27 Nov 2024 08:45
URI:	https://orca.cardiff.ac.uk/id/eprint/153007

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