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Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians

Bevan, Jonathan and Kabisch, Sandra (2019) Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians Proceedings of the Royal Society of Edinburgh Section A: Mathematics.

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In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian det∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit \emph{explicit} twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer u σ :Ω→R 2 in a model, two-dimensional case. The shear map minimizer has the properties that (i) det∇u σ is strictly positive on one part of the domain Ω , (ii) det∇u σ =0 necessarily holds on the rest of Ω , and (iii) properties (i) and (ii) combine to ensure that ∇u σ is not continuous on the whole domain.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Kabisch, Sandra
Date : 23 January 2019
DOI : 10.1017/prm.2018.90
Copyright Disclaimer : © Royal Society of Edinburgh 2019
Uncontrolled Keywords : : energy minimiser; Jacobian constraints; uniqueness; regularity
Depositing User : Clive Harris
Date Deposited : 10 Nov 2017 08:19
Last Modified : 18 Mar 2019 15:19

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