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A Landau de Gennes theory of liquid crystal elastomers

In this article, we study minimization of the Landau-de Gennes energy for liquid crystal elastomer.The total energy, is of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal. Our model consider two sources of anisotropy represented by the traceless nematic order tensor $Q$ (rigid units), and the positive definite step-length tensor $L$ (network). This work is motivated by the study of cytoskeletal networks which can be regarded as consisting of rigid rod units crosslinked into a polymeric-type network. Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential that the deformation maps $φ$ be invertible. We require sufficient regularity of the fields $(φ, Q)$ of the problem, and that the deformation map satisfies the Ciarlet-Necas condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. The approach of including the Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also discussed. The regularity requirements lead us to consider powers of the gradient of the order tensor $Q$ higher than quadratic in the energy. We assume polyconvexity of the stored energy function with respect to the effective deformation tensor and apply methods from isotropic nonlinear elasticity. We formulate a necessary and sufficient condition to guarantee this invertibility property in terms of the growth to infinity of the bulk liquid crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular limit of $-\frac{1}{3}$. $L$ becomes singular as the minimum eigenvalue of $Q$ reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe of nematics (see Ball 2010).