A phase-field model for the dynamics of a strained solid in contact with its melt is developed. The sharp-interface limit of this model reduces to the continuum equations describing the Asaro-Tiller-Grinfeld instability. In addition, it allows the derivation of generalizations of these equations. The dynamics of extended systems are investigated via the phase-field approach which contains an inherent lower length cutoff, thus avoiding cusp singularities. General initial conditions lead to coarsening. For periodic states, this arises via a series of approximate period doublings, the beginning of which can be demonstrated analytically.