Nematic Elastomers: Gamma-Limits for Large Bodies and Small Particles

We compute the large-body and the small-particle Gamma-limit of a family of energies for nematic elastomers. We work under the assumption of small deformations (linearized kinematics) and consider both compressible and incompressible materials. In the large-body asymptotics, even if we describe the local orientation of the liquid crystal molecules according to the model of perfect order (Frank theory), we prove that we obtain a fully biaxial nematic texture (the one of the de Gennes theory) as a byproduct of the relaxation phenomenon connected to Gamma-convergence. In the case of small particles, we show that formation of new microstructure is not possible and we describe the map of minimizers of the Gamma-limit as the phase diagram of the mechanical model.