This project focuses on development and analysis of physically-motivated models of materials undergoing martensitic phase transition, a diffusionless deformation of a crystal lattice from the high-symmetry parent austenite phase to the low-symmetry martensite phase, which can exist in several symmetry-related twin variants. Under mechanical or thermal loading, these materials form finely layered twinning microstructures. The unique properties of martensites, such as their ability to accommodate large deformations and the marked hysteresis they exhibit under cyclic loading, are determined by the kinetics of phase and twin boundaries. Understanding how dynamics of the interfaces depends on their orientation, microstructural configuration, and properties of the crystal lattice involves challenging open problems in modeling of martensites. The project seeks to advance the understanding of these phenomena from the perspective of mesoscopic lattice models. Building on her prior work, the investigator focuses on several prototypical discrete models with the goal of capturing the essential features of interface kinetics and microstructure evolution and an emphasis on higher-dimensional phenomena and rate effects. The main ingredients of the models are nonconvex interactions between nearest neighbors allowing for the existence of two stable homogeneous states and the long-range interactions. Among the outcomes of this project are prediction of nucleated microstructural patterns and derivation of interfacial kinetic laws that can be used to solve problems involving temporal and spatial inhomogeneities. The results are compared to experimental observations and molecular dynamics simulations.

By focusing on the interface kinetics, this project helps determine how the dissipative properties of martensites and related active materials depend on the material structure and the loading conditions. This is important in emerging civil, aerospace and industrial applications that require significant passive damping, such as damage and vibration control in composite structures and attenuation of earthquake- and wind-induced vibrations in buildings and bridges. Results of this project may also help design new materials with desired properties. Mathematical methods developed during the project can be used to solve similar problems encountered in dislocation theory, fracture mechanics, DNA modeling, image recognition, and numerical analysis. The broader impacts of this program are also achieved through training of graduate and undergraduate students in an interdisciplinary research program, as well as educational and high-school outreach activities.

Project Report

Many physical and biological systems are both spatially discrete and nonlinear. The interplay of the discrete structure and nonlinearity in such systems, which include crystal lattices and biopolymers, plays a fundamental role in kinetics of phase transitions and emergence and propagation of nonlinear waves and microstructural patterns. This project provided insight into the effect of lattice structure and nonlinearity on the macroscopic kinetics of of phase boundaries and twinning dislocations in crystalline solids, focusing on the role of lattice heterogeneity and anisotropy, long-range interactions, additional degrees of freedom and the nonconvexity of the interaction potentials. It also contributed to a better understanding of these effects on the macroscopic properties of localized nonlinear waves in crystal lattices. In particular, dynamics of such waves in a novel class of locally resonant granular materials, which can be tuned to localize, transmit or reflect acoustic energy, was investigated. In addition, the work provided insight into the mechanism behind the ovestretching phase transition in DNA observed in recent experiments, which involved a large deformation under a nearly constant force, and clarified the role of stacking interactions between the base pairs in this phenomenon. The methods developed during this project may be applied to a variety of spatially discrete systems encountered in dislocation and fracture mechanics, neuroscience and biophysics and other research areas. The obtained exact and semi-analytical solutions for nonlinear waves in lattices are of interest in applied analysis since there are no general existence results for the problems under consideration. The work on nonlinear waves in locally resonant granular crystals provided an important initial step that could potentially lead to developing new applications of these materials in energy harvesting, shock absorbtion and vibration mitigation devices. More generally, understanding the effects of lattice structure on the macroscopic properties of the material may help design new materials with desired properties. The results of this research were made available to an interdisciplinary audience through journal publications and conference presentations. In addition, the PI facilitated exchange of ideas in mathematical modeling of materials and nonlinear lattice dynamics by co-organizing three minisymposia and a workshop. Student-oriented lectures about the work contributed to the educational impact of the project, in addition to research training of graduate and undergraduate students in this interdisciplinary field. An outreach component of the project involved organizing an annual Integration Bee competition for local high school students, aimed at encouraging the students to consider careers in mathematical sciences.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Eugene Gartland
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University of Pittsburgh
United States
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