Materials Science-inspired problems in the Calculus of Variations: Rigidity of shape memory alloys and multi-phase mean curvature flow

Simon, T.M.

Zusammenfassung

This thesis is concerned with two problems in the Calculus of Variations touching on two central aspects of Materials Science: the structure of solid matter and its dynamic behavior. The problem pertaining to the first aspect is the analysis of the rigidity properties of possibly branched microstructures formed by shape memory alloys undergoing cubic-to-tetragonal transformations. On the basis of a variational model in the framework of linearized elasticity, we derive a non-convex and non-discrete valued differential inclusion describing the local volume fractions of such structures. Our main result shows the inclusion to be rigid without additional regularity assumptions and provides a list of all possible solutions. We give constructions ensuring that the various types of solutions indeed arise from the variational model and quantitatively describe their rigidity via H-measures. Our contribution to the second aspect is a conditional result on the convergence of the Allen-Cahn Equations to multi-phase mean curvature flow, which is a popular model for grain growth in polychrystalline metals. The proof relies on the gradient flow structure of both models and borrows ideas from certain convergence proofs for minimizing movement schemes.