# Abstract of Talk by Dr. James C. Turner Jr.

## Mathematical Problems Arising
in the Study of Smart Materials

Dr. James C. Turner Jr.
Department of Mathematics
Florida A & M University
Tallahassee, FL 32307

## Office Telephone Number:

904-599-3595
## Fax Number:

904-561-2155
## Electronic Mail Address:

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.

Go back to AARMS Menu