Analysis of variational models for nematic liquid crystal elastomers

The relaxation of a free-energy functional which describes the

order-strain interaction in nematic liquid crystal elastomers is obtained

explicitly. We work in the regime of small strains (linearized

kinematics). Adopting the uniaxial order tensor theory or Frank

model to describe the liquid crystal order, we prove that the

minima of the relaxed functional exhibit an effective biaxial

microstructure, as in de Gennes tensor model. In particular, this

implies that the response of the material is soft even if the

order of the system is assumed to be fixed. The relaxed energy

density satisfies a solenoidal quasiconvexification formula.