This research deals with problems arising in the mathematical
modeling of strongly heterogeneous materials, primarily composites.
A composite is a mixture of several different constituent materials or
phases so that it combines the most useful features of each.
For example, a high thermal conductivity composite consists of ceramic
filler beads in a polymeric matrix. The goal is to maximize thermal conductivity.
Ceramic is a very good thermal conductor while the polymer
is not. It is not practical to use pure ceramic since it is too brittle.
However, the polymer/ceramic composite is a good thermal conductor and it
has desirable mechanical properties. The problem in this context is
to optimize strength and thermal conductivity by the correct selection of
the filler identity, size, shape, and the random size distribution. Tools from
the mathematical theory of homogenization will be used to attack these questions.
The three problem areas of the proposed work are the optimization of dielectric and
mechanical properties of epoxy/ceramic composites, with applications to the
design of capacitors; an investigation of dynamical problems and frequency
dependent effects in transducer materials, with applications to sensors and
transmitters of acoustic signals; and a study of problems from superconductivity,
superfluidity, and liquid crystals, with the goal of understanding mathematical
issues caused by the presence of vortices, nonlinearity, and nonstandard
The planned work lies at the frontier between mathematics and materials science,
and close collaborations with materials scientists from universities and private industry
will be carried out. Mathematics can help to develop new materials with superior
properties for various industrials needs, verify the reliability of
experimental data, and devuise efficient ways to compute properties of
designer composites. The results of the work in the first of the three areas
(epoxy/ceramic composites) will provide guidance for future directions in
manufacturing and materials development in various important industrial applications.
Two typical examples are the design of new 'packages' for integrated circuits
which remove heat from the electronics more efficiently and significant enhancement
in functionality of capacitors which are used in various electronics products.
The planned work on transducer materials has immediate applications in underwater
sonar and medical imaging. The work in the third area (superconductivity) will help
to clarify which of the existing physical models provide the best way of calculating
effective properties of superconducting thin films with a large number of vortices,
and it will lead to better models for these phenomena.
|Effective start/end date
|7/1/99 → 6/30/03
- National Science Foundation: $108,160.00