  We derive the effective energy density of thin membranes of liquid crystal
elastomers as the Gamma-limit of a widely used bulk model. These membranes can
display fine-scale features both due to wrinkling that one expects in thin
elastic membranes and due to oscillations in the nematic director that one
expects in liquid crystal elastomers. We provide an explicit characterization
of the effective energy density of membranes and the effective state of stress
as a function of the planar deformation gradient. We also provide a
characterization of the fine-scale features. We show the existence of four
regimes: one where wrinkling and microstructure reduces the effective membrane
energy and stress to zero, a second where wrinkling leads to uniaxial tension,
a third where nematic oscillations lead to equi-biaxial tension and a fourth
with no fine scale features and biaxial tension. Importantly, we find a region
where one has shear strain but no shear stress and all the fine-scale features
are in-plane with no wrinkling.
