Theoretical and numerical modeling of shape memory alloys

This work is concerned with the development of theoretical formulations and the necessary computational techniques required for analysis of the thermomechanical behavior exhibited by shape memory alloys. Based on consideration of the microstructure which forms during the solid-solid phase transformations known to occur in such materials, a multi-well energy structure comprised of free energy functions assigned to each of the minimally representative set of variants associated with the phase change in single crystals is adopted. Using this as the underlying framework, two physically motivated models for predicting phase transformation are investigated at length. The resulting constitutive models are designed to be appropriate for solving large scale initial boundary value problems at the macroscopic scale.; The first approach investigated derives the kinetics of phase transformation using concepts from statistical physics as applied to a thermally activated process. This involves the estimation of the energy barriers which hinder transformation between the variants as encapsulated in the multi-well framework. The idea is first developed in a one dimensional setting, then generalized to the full single crystal problem. Doing so required the creation of an integration strategy to improve the numerical efficiency of the method, as well as the extension of the notion of an energetic barrier to the multi-dimensional case. Numerical simulations demonstrate that the model qualitatively reproduces the unique behaviors observed in shape memory alloys.; The second group of models presented are formulated directly from energy minimization principles. Again using the same underlying energetic framework, it is initially assumed that among all possible microstructural configurations, the material will evolve to the one which minimizes the energy. Such an approach leads to a dissipation free model which successfully predicts the essential features of single crystal experimental data with relatively low numerical expense. Further generalizations allow for the inclusion of an anisotropic dissipation function, both in rate dependent and rate independent form, while retaining the optimal microstructure model as the limit state. Through simulation it is demonstrated that the theory provides a means to replicate both the qualitative and quantitative features of single crystal shape memory alloys.