**Mathematical
Modeling of Fiber Reinforced Composites with Linear Grading**

**Participants:
K. L. Kreider (Theo and Applied Math), W. Binienda (Civil Eng)**

Many
materials are fabricated using fiber reinforced composite materials, which
consist of a set of aligned stiff fibers embedded in a softer matrix. A common
example is the automobile tire. It is becoming increasingly common for composites
manufacturers to add a thin linearly graded coating to improve the bulk
properties of the composite, but very little modeling of linearly graded
materials has been done. The linearly graded zone can also be used to model
damage and fiber-matrix separation. The goal of this project is to develop a
micro-mechanical mathematical model to determine the local displacement, stress
and strain fields surrounding a single fiber when there is a linearly graded
transition zone between the fiber and the matrix. Both fiber and matrix are
assumed to be isotropic or transversely isotropic. The composite is homogenized
to determine its effective mechanical properties (elastic moduli, Poisson ratios, and bulk modulus).

__Publications__

- Determination of the
effective properties of a fiber reinforced composite with a functionally
graded transition zone, C. F. Childers, K. L. Kreider, W. K. Binienda, submitted to
*Composites Science and Technology.*