Optimal microstructures
                  for two-component elastic composites
                        in two space dimensions

                                   by
                             Yury Grabovsky

                    JWB 335 3:20pm, Monday, October 9, 1995


We model elastic composites by elliptic equations of linear elasticity
with rapidly varying Hooke's law tensor:

$$\nabla \cdot C(x/\eps) e(v^\eps) = f$$

where

$$e(v) = \frac{1}{2} ( \nabla v + (\nabla v)^t ).$$

In this lecture we will consider two-component periodic composites,
which means that the function C(y) takes just two values Co and Ct and
it is assumed to be periodic.

The homogenization theory provides a tool for describing the effective
behavior of a composite by characterizing the limit of solutions to the
elastic composite equation. This limit also solves an equation of linear
elasticity but with a different Hooke's law tensor. This new tensor is
called the effective Hooke's law of a composite medium. It depends
on the elastic properties of the constituent materials Co and Ct, as
well as the microgeometry --- the geometric arrangement of the two
component materials.

The problem of describing all effective tensors that can be obtained via
the homogenization process from the given set of component materials is
called the G-closure problem. Tensors lying on the boundary of the
G-closure are called extremal. We will try to describe some interesting
microgeometries corresponding to extremal effective Hooke's laws. The
questions of finding all possible optimal geometries leads us to the
theories of quasi-convexity and Young measures and beyond.

Significant contributions to this circle of problems are due to several
of the Department's faculty: A. Cherkaev, K. Golden and G. Milton. You
can find references to their work in my reprints and preprints.

Order reprints via email to yuri@math.utah.edu.