  A phase-field approach describing the dynamics of a strained solid in contact
with its melt is developed. By rigorous asymptotic analysis we show that the
sharp-interface limit of this model recovers the continuum model equations for
the Grinfeld instability. Moreover, we use our approach to derive hitherto
unknown sharp-interface equations for a situation including a field of body
forces. The numerical utility of the phase-field approach is demonstrated by
comparison with a sharp-interface simulation. We then investigate the dynamics
of extended systems within the phase-field model which contains an inherent
lower length cutoff, thus avoiding cusp singularities. It is found that a
periodic array of grooves generically evolves into a superstructure which
arises from a series of imperfect period doublings. For wavenumbers close to
the fastest-growing mode of the linear instability, the first period doubling
can be obtained analytically. Both the dynamics of an initially periodic array
and a random initial structure can be described as a coarsening process with
winning grooves temporarily accelerating whereas losing ones decelerate and
even reverse their direction of motion.
