A characteristic feature of materials undergoing martensitic phase transitions is hysteresis they exhibit in response to cyclic loading. The process involves formation and evolution of finely layered microstructures which depend in a crucial way on the orientation and history of the loading. The origin of energy dissipation, phase nucleation and kinetics of phase boundaries are important open problems in modeling of martensites and other active materials. This research program seeks to advance the understanding of these phenomena from the perspective of mesoscopic and microscopic frameworks. It will focus on a series of prototypical lattice models of increasing complexity with a goal of capturing the essential features of phase nucleation, interface kinetics and the associated hysteresis. The models have two main ingredients: (i) nonconvex interactions between nearest neighbors allowing for existence of two or more stable homogeneous states and (ii) the long-range interactions. Among the outcomes of this research will be derivation of an interface kinetic law and prediction of nucleated microstructural patterns. These results will be compared to experimental observations. In particular, the proposed work will relate the macroscopic observations such as the size and other features of a hysteresis loop to the microscopic characteristics of the material.
Results of this research will be important for emerging civil, aerospace and industrial applications which require significant passive damping exhibited by martensites. These include damage and vibration control in composite structures and attenuation of earthquake and wind-induced vibrations in buildings and bridges. The work will impact many other scientific areas where similar discrete systems are encountered, including dislocation theory, fracture mechanics, biology, image recognition and numerical analysis. The broader impact of this program will be also achieved through training of graduate and undergraduate students in an interdisciplinary research program and a variety of educational and outreach activities. These include developing new courses, mentoring women graduate students and running programs for middle and high school students. These activities will promote mathematics and science in general as an interesting intellectual pursuit and a prospective career path.
