Abstract :
Suppose that a material is composed of single crystals of three or more different phases of the same composition, which may have been in equilibrium at some time but the conditions now are such that the phases are not in equilibrium. The interface between any pair of crystals Ai(t) and Aj(t) at time t moves at a prescribed velocity υj→i. A least-time formulation is given for the motions of the interfaces, provided hypotheses are made on the velocities which are appropriate to this situation, and existence and uniqueness of solutions are proved by an optimization procedure which essentially constructs the solutions. A consistent formulation is also given, based on characteristics, which needs no hypotheses on the velocities. For arbitrary initial interface configurations consisting of three half-lines in the plane meeting at a point, or six planar pieces separating four phases in R3 and sharing a common point, an explicit construction of consistent solutions is given (except in one non-physical case where no solutions can exist) and these solutions are unique except in one other case. When infinitessimally thin layers form in the least-time solutions, least-time and consistent solutions differ. Additionally, a catalog is given and proved to be complete for which configurations of three half-lines in the two-dimensional plane can result in flat growth for constant velocities; analogous results could be proved in three dimensions.
