Spatially periodic solutions in a 1D model of phase transitions with order parameter

A model of phase transitions with convex strain energy is investigated within the limits of ID nonlinear bar theory. The model is a special case of a coupled field theory using an order parameter that has been developed by Fried and Gurtin to study the nucleation and propagation of phase boundaries. The system of governing equations studied here consists of a wave equation coupled to a nonlinear reaction-diffusion equation. Using phase plane methods, the equilibria of the system have been constructed in order to obtain the macroscopic response of the bar. It is demonstrated that a large number of coexisting spatially periodic, inhomogeneous solutions can occur, with the number of these solutions being inversely proportional to the diffusion coefficient in the reaction-diffusion subsystem. A stability analysis of the equilibria is presented, and the bifurcation diagram is discussed in the context of quasistatic loading. It is shown that, though solutions with more than one interface are unstable, they are only weakly so, and can thus persist for a long time. The nucleation and propagation of phase boundaries are illustrated via a numerical study, which shows how nucleation relates to the loss of stability of the homogeneous equilibria.