含形变构型的非线性弹性能松弛及其在弹性体中的应用
Relaxation of nonlinear elastic energies involving deformed configuration and applications to nematic elastomers
摘要：
我们从一个变分模型的弹性体，涉及两个能量：机械和摩擦。第一个由非线性弹性能量组成，该能量受弹性体分子取向的影响。在变形组态中，能量是Oseen-Frank能量。对变形梯度的行列式的正性进行了约束。泛函不被假定为具有通常的多凸性或拟凸性假设是下连续的。相反，我们计算它的松弛，也就是下连续包络，结果是力学项的拟凸化加上非线性项的切向拟凸化。主要假设是力学项的拟凸化是多凸的，变形是在索伯列夫空间$W^{1,p}$（其中$p>n-1$和$n$是空间的维数）中，并且不存在空化。
英文摘要：
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 $W^{1,p}$ (with $p>n-1$ and $n$ the dimension of the space) and does not present cavitation.
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