  We model polydomain liquid-crystal elastomers by extending the neo-classical
soft and semi-soft free energies used successfully to describe monodomain
samples. We show that there is a significant difference between polydomains
cross-linked in homogeneous high symmetry states then cooled to low symmetry
polydomain states and those cross-linked directly in the low symmetry
polydomain state. For example, elastomers cross-linked in the isotropic state
then cooled to a nematic polydomain will, in the ideal limit, be perfectly
soft, and with the introduction of non-ideality, will deform at very low stress
until they are macroscopically aligned. The director patterns observed in them
will be disordered, characteristic of combinations of random deformations, and
not disclination patterns. We expect these samples to exhibit elasticity
significantly softer than monodomain samples. Polydomains cross-linked in the
nematic polydomain state will be mechanically harder and contain characteristic
schlieren director patterns. The models we use for polydomain elastomers are
spatially heterogeneous, so rather than solving them exactly we elucidate this
behavior by bounding the energies using Taylor-like (compatible test strain
fields) and Sachs (constant stress) limits extended to non-linear elasticity.
Good agreement is found with experiments that reveal the supersoft response of
some polydomains. We also analyze smectic polydomain elastomers and propose
that polydomain SmC* elastomers cross-linked in the SmA monodomain state are
promising candidates for low field electrical actuation.
