A study of a hamiltonian model for martensitic phase transformations including microkinetic energy

How can a system in a macroscopically stable state explore energetically mo re favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations, the authors derive a Boussin esq-type equation p(u) double over dot = partial derivative (x)sigma(partial derivative (x)u) + beta partial derivative (2)(x)(u) double over dot, x is an element of (0 , 1), beta > 0, which models the dynamics of martensitic phase transformations. Here rho > 0 is the mass density, beta partial derivative (2)(x)(u) double over dot is a regularization term that models the inertial forces of oscillations with in a representative volume of length root beta, and sigma is a nonmonotone stress-strain relation. The solutions of the system, which the authors refe r to as the microkinetically regularized wave equation, exhibit strong osci llations after times of order root beta and relaxation of spatial averages can be confirmed. This means that macroscopic fluctuations of the solutions decay, to the benefit of microscopic fluctuations. From the macroscopic po int of view, this can be interpreted as a transformation of macroscopic kin etic energy into heat, i.e., as energy dissipation (despite the fact the au thors consider a conservative system). The mathematical analysis for the mi crokinetically regularized wave equation consists of two parts. First, the authors present some analytical and numerical results on the propagation of phase boundaries and relaxation effects. Despite the fact that the model i s conservative, it exhibits hysteretic behavior. Such behavior is usually i nterpreted in macroscopic models in terms of a dissipative threshold, which the driving force must overcome to ensure that the phase transformation pr oceeds. The threshold value depends on the volume of the transformed phase as observed in known experiments. Second, the authors investigate the dynam ics of oscillatory solutions. Their mathematical tools are Young measures, which describe the one-point statistics of the fluctuations. They present a formalism that allows them to describe the effective dynamics of rapidly f luctuating solutions. The extended system has nontrivial equilibria that ar e only visible when oscillatory solutions are considered. The new method en ables them to derive a numerical scheme for oscillatory solutions based on particle methods.