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Abstract

How can a system in a macroscopically stable state explore energetically more favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations, the authors derive a Boussinesq-type equation [ILLEGIBLE] which models the dynamics of martensitic phase transformations. Here p > 0 is the mass density, 3O2fu is a regularization term that models the inertial forces of oscillations within a representative volume of length /, and of is a nonmonotone stress-strain relation. The solutions of the system, which the authors refer to as the micrmkinetically regularized wave equation, exhibit strong oscillations after times of order / 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 point of view, this can be interpreted as a transformation of macroscopic kinetic energy into heat, i.e., as energy dissipation (despite the fact the authors consider a conservative system). The mathematical analysis for the microkinetically 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 is conservative, it exhibits hysteretic behavior. Such behavior is usually interpreted in macroscopic models in terms of a dissipative threshold, which the driving force must overcome to ensure that the phase transformation proceeds. The threshold value depends on the volume of the transformed phase as observed in known experiments. Second, the authors investigate the dynamics 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 fluctuating solutions. The extended system has nontrivial equilibria that are only visible when oscillatory solutions are considered. The new method enables them to derive a numerical scheme for oscillatory solutions based on particle methods.

Keywords

Phase transformations oscillations Young measures Hamiltonian dynamics nonlinear partial differential equations

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A Study of a Hamiltonian Model for Martensitic Phase fransformations Including Microkinetic Energy

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