2. Multiscale approaches for heterogeneous materials

The first homogenization methods developed for composite materials were based on the work of Eshelby. The principle of Eshelby's micromechanical theory (1957) is to replace a distribution of heterogeneities with a characteristic (disturbance-correcting) deformation field having equivalent homogeneous mechanical behavior. By applying various boundary conditions (imposed macroscopic stresses or strains) to infinity, several effective moduli can be deduced for the inclusion-laden medium under consideration. The fact that interactions between inclusions are not taken into account limits the use of this method to low volume fractions. However, it can be used to derive simple analytical expressions for specific geometries.

Later, Mori and Tanaka (1972) developed a theory that is one of the most widely used in composites. They noted the importance of taking into account...