Relaxation of some transversally isotropic energies and applications to smectic A elastomers

We determine the relaxation of some transversally-isotropic energy densities, i.e. functions W : R^{3×3} → [0,∞] with the property W(QFR) = W(F) for all Q ∈ SO(3) and all R ∈ SO(3) such that Rn0 = n0, where n0 is a fixed unit vector. One physically relevant example is a model for smectic A elastomers. We discuss the implications of our result for the computation of macroscopic stress–strain curves for this material and compare with experiment.