Phase-field modeling of stress-induced instabilities

A phase-field approach describing the dynamics of a strained solid in contact with its melt is developed. Using a formulation that is independent of the state of reference chosen for the displacement field, we write down the elastic energy in an unambiguous fashion, thus obtaining an entire class of models. According to the choice of reference state, the particular model emerging from this class will become equivalent to one of the two independently constructed models on which brief accounts have been given recently J. M\"uller and M. Grant, Phys. Rev. Lett. 82, 1736 (1999); K. Kassner and C. Misbah, Europhys. Lett. 46, 217 (1999). We show that our phase-field approach recovers the sharp-interface limit corresponding to the continuum model equations describing the Asaro--Tiller--Grinfeld instability. Moreover, we use our model 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 reproducing some known results and by comparison with a sharp-interface simulation. We then proceed to 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 wave numbers 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. In the absence of gravity, the end state of a laterally finite system is a single groove growing at constant velocity, as long as no secondary instabilities arise (that we have not been able to see with our code). With gravity, several grooves are possible, all of which are bound to stop eventually. A laterally infinite system approaches a scaling state in the absence of gravity and probably with gravity, too.