On a consistent finite-strain plate model of nematic liquid crystal elastomers

Nematic liquid crystal elastomer, abbreviated as NLCE, combines many excellent features of liquid crystal and elastomer, which promote its potential applications in many areas. In most reported situations, the thickness of an NLCE is relatively small compared with the other two dimensions. In particular, an NLCE can undergo large elastic deformation subjected to various stimuli. It is therefore of fundamental importance to derive a plate model describing nonlinear behaviors of an NLCE. This paper develops such a consistent plate theory for an NLCE incorporating both the hyperelasticity and the anisotropy. The 3D governing system, which is composed of the deformational momentum balance and the orientational momentum balance, is presented within the framework of nonlinear elasticity using a variational approach. Series expansions for all independent unknowns in terms of the thickness variable are conducted on the bottom surface of the plate. Furthermore, systematic manipulations of the expanded governing system generate two 2D vector plate equations containing five unknowns. Meanwhile, the associated edge boundary conditions are proposed. It turns out that the derived plate theory guarantees a required asymptotic order for each term in the variation of a generalized potential energy functional. In order to verify the accuracy of the obtained 2D plate system, we specify an exact form of the strain-energy function for an NLCE and study the finite pure bending of an NLCE-substrate structure where the substrate is assumed to be composed of an incompressible neoHookean material. It is found that the plate model can offer second-order correct results through comparisons between approximate and exact solutions. Remarkably, we find that for this benchmark problem the current plate model still works for a thick substrate plate and extremely large bending angles.