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 W$$ (with p > n − 1 and n the dimension of the space) and does not present cavitation.
Mots-clés : Nonlinear elasticity, nematic elastomers, relaxation, deformed configuration
@article{COCV_2019__25__A19_0,
author = {Mora-Corral, Carlos and Oliva, Marcos},
title = {Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018005},
zbl = {1437.49029},
mrnumber = {3982966},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018005/}
}
TY - JOUR AU - Mora-Corral, Carlos AU - Oliva, Marcos TI - Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018005/ DO - 10.1051/cocv/2018005 LA - en ID - COCV_2019__25__A19_0 ER -
%0 Journal Article %A Mora-Corral, Carlos %A Oliva, Marcos %T Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018005/ %R 10.1051/cocv/2018005 %G en %F COCV_2019__25__A19_0
Mora-Corral, Carlos; Oliva, Marcos. Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 19. doi : 10.1051/cocv/2018005. http://www.numdam.org/articles/10.1051/cocv/2018005/
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