A Mathematical Theory for Nematic Elastomers with Non-uniform Prolate Spheroids

We address some problems arising in the mathematical modeling of nematic elastomers with variable step length tensors and directors with non-uniform length (which represents, in turn, the nonuniform prolate spheroid shapes). Using a similar idea of the Ericksen theory for liquid crystals with variable degree of orientation, we introduce a non-unit vector, called the shape parameter, to describe the step length and the director simultaneously. In addition to the usually used uniaxial elastic energy, we provide a model for the energy contribution due to change of the shape parameters associated with the elastic deformation of the elastomer. This energy is a generalization of the Oseen-Frank free energy for nematic orientational distortions of liquid crystals, but it incorporates coherently the elastic deformation of the network. The total free energy we obtain for the elastomer is suitable for the existence of energy minimizers and other mathematical studies of the elastomers and provides a concrete example that is consistent with both the continuum mechanics theory and the molecular-statistical theory.