The history of laminates and why they are important

The simplest conceivable composite is a stratified material, such as the one illustrated in figure 9.1 on the next page, where the material properties vary only in one direction, called the direction of lamination, represented by a unit vector n. Of course lamination in direction n is equivalent to lamination in direction –n. This chapter is devoted to obtaining formulas for the effective tensors of such laminates (called simple or rank-1 laminates), and also laminates of laminates (called multiple-rank laminates) where there is a large difference in the length scales of the successive laminations and the subsequent laminations are in different directions; see figure 9.2 on the following page. Laminates of laminates were introduced by Max well (1873), who provided a formula for the effective conductivity tensor of certain third-rank laminates, and later by Bruggeman (1930) for estimating the elastic moduli of polycrystalline aggregates. Beginning with the work of Schulgasser (1976), the effective tensors of appropriately designed multiple-rank laminates were found to attain many bounds on effective tensors. Numerous examples of such optimal laminate microstructures will be given in the latter part of this book. When one is trying to get some idea of the range of properties that composite materials can exhibit as the microstructure is varied, it is usually best to first examine the range of properties that multiple-rank laminate materials can exhibit.