Interplay Between Interfacial Patterns, Singular Perturbations and Heterogeneous Medium

Purdue University, West Lafayette IN

Investigators

Abstract

Yip DMS-1009102 The investigator and his colleagues study the interaction between the defect structures in materials (such as phase and grain boundaries) and the background heterogeneous medium. The heterogeneity can come from intrinsic atomic structures, externally imposed forcing, or even numerical approximations. The emphasis is on understanding the structure and propagation of interfaces. Due to the highly nonlinear interaction between the materials interface and background inhomogeneity, many new features arise, most notably boundaries between patterns instead of pure phases, and pinning and de-pinning transitions. In terms of mathematical disciplines, the project is related to the concept of homogenization of singularly perturbed problems in which multiple scales coexist. To tackle nonlinear problems as in the current setting, new techniques and interpretations beyond traditional approaches are required. The project investigates the structure and propagation of boundaries between patterns, and the connection between pulsating waves and singular perturbations. Both the discrete and continuum regimes are analyzed. The project is highly interdisciplinary and is well-motivated by problems coming from applied science, in particular materials science. It brings together rigorous applied analysis, numerical simulations and modeling in a variety of physical phenomena. It emphasizes the connections between multiple scales, as practical materials often are composed of structures that have greatly different length scales. Results can lead to better understanding of how applied forcing and changes of external environments can affect the behavior of materials responses. This in turn can lead to the manufacturing of stronger and better materials.