Consider the process of formation of a laminate composite. Suppose that a portion of material is added to the composite with the effective tensor and that the periodic cell has volume . The material is added in a infinitesimal thin periodic layers with the normal . The resulting composite cell has the volume and an effective properties tensor denoted by .
Let us compute
by using 8, where we
As tends to zero, we obtain
the differential equation
This equation shows the rate of change of effective properties. It is integrated with respect to . The functions and determine the structure of the composite. We assume that different materials with properties are added to the composite in different ``times'' . We also assume that : The direction of laminates is generally changed during formation of the composite.
This scheme is effectively used in our Workshop to the Workshop to compute effective properties of the Hedgehogs, Pipe-brushes, end similar structures. It is flexible and robust.