A Landau-de Gennes theory of liquid crystal elastomers

In this article, we study minimization of the energy of a Landau-de Gennes liquid crystal elastomer. The total energy consists of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal. There are two related sources of anisotropy in the model, that of the rigid units represented by the traceless nematic order tensor Q, and the positive definite step-length tensor L characterizing the anisotropy of the 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. For this, we require sufficient regularity of the fields (φ,Q) of the problem, and that the deformation map satisfies the Ciarlet-Nečas injectivity condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. Alternatively, the approach of including the Rapini-Papoular surface energy for the pull-back tensor Q is also discussed. The regularity requirements also 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 of calculus of variations from isotropic nonlinear elasticity. Recovery of minimizing sequences of deformation gradients from the corresponding sequences of effective deformation tensors requires invertibility of the anisotropic shape tensor L. 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 -1/3. It turns out that L becomes singular as the minimum eigenvalue of Q reaches -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 [5].