Collaborative Research; Mathematical Sciences, Transitions and Defects in Ordered Materials

DMS-9505077 Luskin and James This interdisciplinary group will investigate nonlinear material behavior and related issues in analysis and computation. Its focus is materials which undergo martensitic and magnetic phase transitions, with particular application to shape memory and giant magnetostrictive materials. The theories and computational methods under development by the authors will be applied to the design of new actuator materials and composites. The authors propose to design a composite that exhibits a strong two way shape memory effect, design small scale, large-force actuators with low hysteresis, and formulate a strategy for finding a magneto-memory material, a material which would combine the most useful aspects of a shape memory (large strain) and magnetostrictive (actuated by a magnetic field) material. The quantitative description of these phenomena rests in the framework of nonlinear analysis, in particular within the context of non-convex potentials whose equilibria are highly oscillatory across widely varying length scales. Innovative numerical methods and codes will be developed to compute the dynamics of complex microstructure in response to a variety of environmental changes. Active materials, that is, materials whose structure transforms dramatically when subjected to external stimuli like heat or electromagnetic fields, have enormous potential for use as actuators and sensors across the spectrum of technology, from aerospace to communications to medicine. This project addresses basic research directed toward the improvement and optimization of these materials and the discovery of new ones. The interplay between their microstructure and macroscopic properties is fundamental to their exploitation and the study of these properties leads to the frontiers of nonlinear theory, nonlinear analysis, and scientific computation. The interdisciplinary training of graduate students and postdoctoral associates is an integra l part of this project.