  We propose a lattice statistical model to investigate the phase diagrams and
the soft responses of nematic liquid-crystal elastomers. Using suitably scaled
infinite-range interactions, we obtain exact self-consistent equations for the
tensor components of the nematic order parameter in terms of temperature, the
distortion and stress tensors, and the initial nematic order. These equations
are amenable to simple numerical calculations, which are used to characterize
the low-temperature soft regime. We find a peculiar phase diagram, in terms of
temperature and the diagonal component of the distortion tensor along the
stretching direction, with first- and second-order transitions to the soft
phase, and the prediction of tricritical points. This behavior is not
qualitatively changed if we use different values of the initial nematic order
parameter.
