** Next:** BOUNDS ON ELASTIC CONSTANTS
** Up:** Berryman: Geomechanical constants of
** Previous:** INTRODUCTION

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 locally by:
| |
(1) |

where are the usual stress components for
*i*,*j*=1-3 in Cartesian coordinates, with 3 (or *z*) being
the axis of symmetry (the lamination direction for such a
layered material). Displacement *u*_{i} is then related to strain
component *e*_{ij} by
.This choice of definition introduces some convenient factors of two into the
44,55,66 components of the stiffness matrix shown in
(1).
For definiteness we also assume that this stiffness matrix in
(1) 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 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

| |
(2) |

This bracket notation can be correctly viewed: (a) as a volume
average, (b) as a line integral along the symmetry axis *x*_{3}, or
(c) as a weighted summation over any relevant
physical quantity *Q* taking a constant value *Q*_{n} in the *n*-th layer.
The bulk modulus for each laminated grain 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 (1)]. The well-known result is
given by
, where
.When is constant in a layered grain, the definition
of *K*_{R} implies Hill's equation (Hill, 1963, 1964; Milton, 2002),
which is given by

| |
(3) |

Here the bulk modulus of the *n*-th constituent is *K*_{n}, the shear
modulus takes the same value for all ,and the overall effective bulk modulus is *K*^{*}. The volume fractions *f*_{n} are all nonnegative, and sum to unity.
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 *K*_{R} 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 *K*_{R} value lies outside the bounds
in many situations (Berryman, 2004b).

** Next:** BOUNDS ON ELASTIC CONSTANTS
** Up:** Berryman: Geomechanical constants of
** Previous:** INTRODUCTION
Stanford Exploration Project

5/3/2005