  We consider a stochastic heat conduction model for solids composed by N
interacting atoms. The system is in contact with two heat baths at different
temperature $T_\ell$ and $T_r$. The bulk dynamics conserve two quantities: the
energy and the deformation between atoms. If $T_\ell \neq T_r$, a heat flux
takes place in the system. For large $N$, the system adopts a linear
temperature profile between $T_\ell$ and $T_r$. We establish the hydrodynamic
limit for the two conserved quantities. We introduce the fluctuations field of
the energy and of the deformation in the non-equilibrium steady state. As $N$
goes to infinity, we show that this field converges to a Gaussian field and we
compute the limiting covariance matrix. The main contribution of the paper is
the study of large deviations for the temperature profile in the
non-equilibrium stationary state. A variational formula for the rate function
is derived following the recent macroscopic fluctuation theory of Bertini et
al.
