Structure and evolution of martensitic phase boundaries

Abstract

This work examines two major aspects of martensitic phase boundaries. The first part studies numerically the deformation of thin films of shape memory alloys by using subdivision surfaces for discretization. These films have gained interest for their possible use as actuators in microscale electro-mechanical systems, specifically in a pyramid-shaped configuration. The study of such configurations requires adequate resolution of the regions of high strain gradient that emerge from the interplay of the multi-well strain energy and the penalization of the strain gradient through a surface energy term. This surface energy term also requires the spatial numerical discretization to be of higher regularity, i.e., it needs to be continuously differentiable. This excludes the use of a piecewise linear approximation. It is shown in this thesis that subdivision surfaces provide an attractive tool for the numerical examination of thin phase transforming structures. We also provide insight in the properties of such tent-like structures. The second part of this thesis examines the question of how the rate-independent hysteresis that is observed in martensitic phase transformations can be reconciled with the linear kinetic relation linking the evolution of domains with the thermodynamic driving force on a microscopic scale. A sharp interface model for the evolution of martensitic phase boundaries, including full elasticity, is proposed. The existence of a solution for this coupled problem of a free discontinuity evolution to an elliptic equation is proved. Numerical studies using this model show the pinning of a phase boundary by precipitates of non-transforming material. This pinning is the first step in a stick-slip behavior and therefore a rate-independent hysteresis. In an approximate model, the existence of a critical pinning force as well as the existence of solutions traveling with an average velocity are proved rigorously. For this shallow phase boundary approximation, the depinning behavior is studied numerically. We find a universal power-law linking the driving force to the average velocity of the interface. For a smooth local force due to an inhomogeneous but periodic environment we find a critical exponent of 1/2.