Phase-field modeling of stress-induced instabilities - art. no. 036117

A phase-field approach describing the dynamics of a strained solid in conta ct with its melt is developed. Using a formulation that is independent of t he 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 independe ntly constructed models on which brief accounts have been given recently [J . Muller 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 appro ach recovers the sharp-interface limit corresponding to the continuum model equations describing the Asaro-Tiller-Grinfeld instability. Moreover, we u se our model to derive hitherto unknown sharp-interface equations for a sit uation including a field of body forces. The numerical utility of the phase -field approach is demonstrated by reproducing some known results and by co mparison with a sharp-interface simulation. We then proceed to investigate the dynamics of extended systems within the phase-field model which contain s an inherent lower length cutoff, thus avoiding cusp singularities. It is found that a periodic array of grooves generically evolves into a superstru cture which arises from a series of imperfect period doublings. For wave nu mbers close to the fastest-growing mode of the linear instability, the firs t period doubling can be obtained analytically. Both the dynamics of an ini tially periodic array and a random initial structure can be described as a coarsening process with winning grooves temporarily accelerating whereas lo sing ones decelerate and even reverse their direction of motion. In the abs ence of gravity, the end state of a laterally finite system is a single gro ove growing at constant velocity, as longs as no secondary instabilities ar ise (that we have not been able to see with our code). With gravity, severa l grooves are possible, all of which are bound to stop eventually. A latera lly infinite system approaches a scaling state in the absence of gravity an d probably with gravity, too.