We assume that a typical building block of the random system is a small grain of laminate material whose elastic response for such a transversely isotropic (hexagonal) system can be described by:

(2) |

Although some of the results presented here are more general, we will
assume for definiteness that this stiffness matrix in
(2) arises from
the lamination of *N* isotropic constituents having bulk and shear
moduli *K*_{n}, , in the *N* > 1 layers present in each
building block. It is important that the thicknesses *d*_{n} always be
in the same proportion in each of these laminated blocks, so that
. But it is
not important what order the layers were added to the blocks, 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) and Milton
(2002)]
state that

(3) |

The bulk modulus for each such building block (or crystalline grain if
you like) is that given by the compressional Reuss average *K*_{R}
of the corresponding compliance matrix *s*_{ij} [the inverse of the
usual stiffness matrix *c*_{ij}, whose nonzero components are
shown in (2)]. The well-known result is
, where
.This quantity can be expressed in terms of the stiffness elements as

(4) |

Even though is the same for every grain, since the
grains themselves are not isotropic, the overall bulk modulus
*K ^{*}* of the random polycrystal is not necessarily the same as
for the individual grains (Hill, 1952). Hashin-Shtrikman bounds on

In general an upper bound on the overall shear modulus of an isotropic polycrystal (Hill, 1952) is given by the Voigt average over shear of the stiffness matrix, which may be written as

(5) |

10/23/2004