For definiteness we also assume that this stiffness matrix in (1) arises from the lamination of N isotropic constituents having bulk and shear moduli Kn, , in the N > 1 layers present in each building block. It is important that the thicknesses dn always be in the same proportion in each of these laminated blocks, so that . But the order in which layers were added to the blocks in unimportant, as Backus's formulas (Backus, 1962) for the constants show. For the overall behavior for the quasistatic (long wavelength) behavior of the system we are studying, Backus's results [also see Postma (1955), Berryman (1998,2004a,b), Milton (2002)] state that
The bulk modulus for each laminated grain is that given by the compressional Reuss average KR of the corresponding compliance matrix sij [the inverse of the usual stiffness matrix cij, whose nonzero components are shown in (1)]. The well-known result is given by , where .When is constant in a layered grain, the definition of KR implies Hill's equation (Hill, 1963, 1964; Milton, 2002), which is given by
Even though is the same for every grain, since the grains themselves are not isotropic, the overall bulk modulus K* of the random polycrystal does not necessarily have the same value as KR for the individual grains (Hill, 1952). Hashin-Shtrikman bounds on K* for random polycrystals whose grains have hexagonal symmetry (Peselnick and Meister, 1965; Watt and Peselnick, 1980) show in fact that the KR value lies outside the bounds in many situations (Berryman, 2004b).