# The mathematical modeling of composite materials

• Grégoire Allaire
Part of the Applied Mathematical Sciences book series (AMS, volume 146)

## Abstract

This chapter is concerned with the application of the homogenization theory to the modeling of composite materials. Composite materials are heterogeneous materials obtained by mixing several phases or constituent materials on a very fine (or microscopic) scale. However, one is usually interested only in the large scale (or macroscopic) properties of such a composite. The main problem with composite materials is, therefore, to determine their effective properties without determining their fine scale structure. There is a huge mechanical literature on this topic, and the reader is referred to [1], [52], [81], [132], [133], [136], [145], [207], [216], [244], [246], [264], [287], [289], [290], and references therein. Mathematicians have been interested in composite materials since the 1970’s. Their first main contribution to this field was to give a firm theoretical basis for the notion of effective properties of a composite material. Indeed, homogenization theory permits one to properly define a composite material as a limit, in the sense of homogenization (an H-limit), of a sequence of increasingly finer mixtures of the constituent phases. Effective properties are now defined as homogenized coefficients. The application of homogenization to the modeling of composite materials has became a popular subject in applied mathematics. There is by now an extensive mathematical literature on this topic too.

## Keywords

Composite Material Nonlocal Term Fourth Order Tensor Optimal Microstructure Fourth Order Moment