David Aristoff's homepage Title: Modeling grain boundaries in metals with optimal transportation theory, calculus of variations, and the phase field method

Abstract: In crystalline solids such as metals, grain boundaries (GBs) form between regions of con- sistent crystallographic orientation. GBs are a unique type of material defect that affect many bulk material properties such as plasticity and fracture toughness; however their energetic dependence on crystallographic orientation (referred to as ”anisotropy”) is unknown for many GB types. In this work, a formalism is constructed to derive the energetic cost of grain boundary creation as the solution to an optimal transportation problem. It is shown that this method captures the effects of grain boundary anisotropy reasonably well for multiple GB types. The model is extended to account for non-planar GB types, specifically, GBs that form facetted structures. Using the recently-developed techniques of the modern calculus of variations, it is shown that these structure can be understood in the context of a minimizing sequence that approaches the infimum of the nonconvex energy grain boundary energy func- tional. This process induces an algorithm for predicting convexified energy and the corresponding facet morphology. Finally, the resulting model is implemented using a phase-field framework to investigate the effects of GB anisotropy on microstructural evolution.