Phenomenological Theory of Non-Uniform Nematic Elastomers: Free Energy of Deformations and Electric-Field Effects

The general phenomenological free energy is derived for a liquid crystalline elastomer under arbitrary strain and orientational distortions. Using the group representations method we obtained all invariants, describing the coupling of translational and orientational deformations and/or external electric field. It is shown that in centrosymmetric materials (nematic rubbers) the only contribution to the free energy which is linear in (small) gradients of the director, is the coupling with electric field and the strain tensor (16 independent terms analogous to the flexoelectric effect). In chiral materials (cholesterics) the electric field couples with the strain tensor for uniform nematic director (piezoelectric effect, 3 invariants) and, in the absence of an electric field, translational and linear orientational distortions interact with each other (couple-stress effects, 6 invariants). Experimental observations and previous models are critically analysed.