Rigorous derivation of active plate models for thin sheets of nematic elastomers

Abstract: In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension-reduction techniques, based on the theory of Gamma-convergence. The two-dimensional models are nonlinear plate theories in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. The following three nematic textures are considered: splay-bend and twisted orientation of the nematic director, and uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable one with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.