  Polydomain liquid crystalline (nematic) elastomers have highly unusual
mechanical properties, dominated by the dramatically non-linear stress-strain
response that reflects stress-induced evolution of domain patterns. Here, we
study the classical Hertz indentation problem in such a material.
Experimentally, we find that polydomain nematic elastomers display a smaller
exponent than the classical 3/2 in the load vs. indentation depth response.
This is puzzling: asymptotically a softer stress-strain response requires a
larger exponent at small loads. We resolve this by theory where three regimes
are identified -- an initial elastic regime for shallow indentation that is
obscured in experiment, an intermediate regime where local domain pattern
evolution leads to a smaller scaling in agreement with experiments, and a final
stiffening regime where the completion of local domain evolution returns the
response to elastic. This three-regime structure is universal, but the
intermediate exponent is not. We discuss how our work reveals a new mechanism
of enhanced adhesion for pressure-sensitive adhesion of nematic elastomers.
