Dr. James C. Turner Jr. Department of Mathematics Florida A & M University Tallahassee, FL 32307
The smartness of a material is a consequence of its ability to form a flexible structure at one temperature or load while recognizing only a homogeneous equilibrium at a different temperature or load. These materials include shape-memory, magnetostrictive, and other active materials and are finding increased applications as actuators and sensors and inherently, they are highly nonlinear.
The mechanism leading to shape-memory behavior relies on a displacive transition or a temperature controlled change in shape. This is the hallmark of a martensitic transformation and it is frequently accompanied by a complicated microstructural pattern. A major challenge to materials science is the prediction and control of microstructure. In recent years a theory of microstructure has been development and put into practice that can describe the macroscopic properties of matensitic, shape-memory alloys. Mathematical models for the equilibrium of such materials give nonconvex variational problems that may fail to attain a minimum value for any admissible deformation.
It is my intention in this lecture to introduce several important nonlinear techniques developed in recent years for studying mathematical problems arising in materials science. We will discuss some of the first numerical simulations of equilibrium states for martensitic crystals. These numerical simulations have presented many new and challenging problems. In addition, we will describe a research program that has as its goal the development of a continuum theory and a computational model able to predict some of the behavior of shape-memory materials under the influence of loads, boundary displacements, and temperature changes.