Volume 25, 2019
|Number of page(s)||27|
|Published online||23 July 2019|
Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers
Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid,
2 Department of Mathematics, Faculty of Sciences, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
* Corresponding author: firstname.lastname@example.org
Accepted: 13 January 2018
We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen–Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space W1,p (with p > n − 1 and n the dimension of the space) and does not present cavitation.
Mathematics Subject Classification: 49J45 / 49Q99 / 74B20 / 74F99
Key words: Nonlinear elasticity / nematic elastomers / relaxation / deformed configuration
© EDP Sciences, SMAI 2019
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