  Instabilities in thin elastic sheets, such as wrinkles, are of broad interest
both from a fundamental viewpoint and also because of their potential for
engineering applications. Nematic liquid crystal elastomers offer a new form of
control of these instabilities through direct coupling between microscopic
degrees of freedom, resulting from orientational ordering of rod-like
molecules, and macroscopic strain. By a standard method of dimensional
reduction, we construct a plate theory for thin sheets of nematic elastomer. We
then apply this theory to the study of the formation of wrinkles due to
compression of a thin sheet of nematic liquid crystal elastomer atop an elastic
or fluid substrate. We find the scaling of the wrinkle wavelength in terms of
material parameters and the applied compression. The wavelength of the wrinkles
is found to be non-monotonic in the compressive strain owing to the presence of
the nematic. Finally, due to soft modes, the critical stress for the appearance
of wrinkles can be much higher than in an isotropic elastomer and depends
nontrivially on the manner in which the elastomer was prepared.
