Returning to the general problem for arbitrary *K*_{n},
suppose we construct a random polycrystal by packing small bits of
this laminate material into a large container in a way so that the
axis of symmetry appears randomly over all possible orientations and
also such that no empty volume (porosity) is left in the resulting composite.
If the ratio of grain size to overall composite is small enough so
the usual implicit assumption of scale separation applies to the
composite -- but not so small that we are violating the continuum
hypothesis -- then we have an example of the type of material we want to
study.

For each individual grain in this polycrystal, Eqs. (3)
are valid locally (*i.e.*, for locally defined coordinates),
and the grain bulk modulus *K*_{R} is given by (4) for all the
grains. The factors 3*K*_{R} and are not necessarily
eigenvalues of elastic stiffness for individual grains. The Voigt
average for shear is again given by (5),
which is an upper bound on the isotropic shear modulus of the random
polycrystal (Hill, 1952).

K4bounds
Various bulk modulus bounds:
The outer most bounds (blue
dot-dash lines) are the standard Hashin-Shtrikman bounds (HS) based only on
information about the layer constituents and their volume fractions.
The black solid lines are the Voigt and Reuss bounds (XV,XR) obtained from
appropriate averages of laminate constants in (3). The inner most
bounds (also blue dot-dash lines) are the Peselnick-Meister bounds (PM) for
hexagonal polycrystals. For contrast, the Dederich-Zeller bounds (DZ)
(see Appendix B) are also shown (dashed red lines).
Figure 1 |

mu4bounds
Same as Figure 1 for the various shear
modulus bounds.
Figure 2 |

The advantage of studying polycrystals of laminates is that we have
available an array of theoretical results from which to choose. For
example, since each grain is composed of isotropic constitutents,
standard Voigt and Reuss bounds (Hill, 1952), as well as the more
restrictive Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962; 1963)
on composites made up of isotropic constituents are all available.
Then, we can instead, or in addition, consider Voigt and Reuss bounds
on the laminated grain materials. Formulas for these bounds have
already been given here in Eqs. (4), (5),
and (8), respectively for *K*_{R}, , and .The remaining formula is well-known to be

(9) |

(10) |

(11) |

(12) |

We see in Figures 1 and 2 that these bounds (XR and XV) for the polycrystalline case are fairly substantial improvements over the uncorrelated Hashin-Shtrikman bounds (HS), which are themselves substantial improvements over the uncorrelated version of the Voigt and Reuss bounds (the Voigt bound is not shown here, but is just a straight line in each plot between the end points of these curves).

A correlated version of the Hashin-Shtrikman bounds
can be computed also, as has been shown by Peselnick and Meister (1965)
and Watt and Peselnick (1980) (see Appendix A for the details
of these formulas, but not their derivation). We see that these
bounds are very tight indeed in comparison to all the others
considered here.
In particular, note that *K*_{R} computed from (4) falls
outside the correlated Voigt and Reuss bounds (curves XV and XR) of
Figure 1.

For contrast, Figures 1 and 2 also plot another set of bounds derived by Dederichs and Zeller (1973) that is also intended for use in uncorrelated systems (see Appendix B for the formulas and a brief discussion). The DZ bounds behave quite differently from those of the correlated bounds XR, XV, PM.It is easy to see why this is so. In the laminates, as the volume fractions become small for one constituent at one end of the curves and for the other constituent at the other end, the low volume fraction constituent is approaching a flat disc-like geometry. It is well-known (Milton, 1981) that in this circumstance results for disc-like inclusions tend to dominate the behavior and, therefore, tend to hug the upper Hashin-Shtrikman bound in the lower left-hand limit, and then to hug the lower Hashin-Shtrikman bound in the upper right-hand limit of the Figures. We see that this is so for the correlated bounds XR, XV, PM. But the DZ bounds are uncorrelated and do not show this type of behavior at all.

PMW
Illustrating the graphical construction leading to the
optimum parameters for the comparison material of the lower and upper
Peselnick-Meister-Watt bounds: (Figure 3 G,_{-}K), (_{-}G,_{+}K),
shown as red circles. The case shown is for the middle point of the
examples shown in Figures 1 and 2 (volume fraction of 0.50).
Values of the constants entering the expressions (see Appendix A) are:
_{+}K_{V} = 30.2162, c = 7.2727, _{44}c = 22.0000,
, and , in units of GPa.
The two parts of the blue solid curve are determined by (14) and (16).
_{66} |

The best and also most relevant bounds here are obviously the Peselnick-Meister-Watt bounds (Peselnick and Meister, 1965; Watt and Peselnick, 1980), which are presented and briefly discussed in Appendix A. Figure 3 [following some similar figures in Watt and Peselnick (1980)] shows how the parameter sets for the elastic comparison materials are determined. The allowed regions in Figure 3 are the bounded area in the upper right-hand corner, and the similarly bounded area in the lower left-hand corner. The red circles are therefore the points in the (,)-plane that produce the optimum bounds. It is clear that the value of plays a very dominant role in the structure of this Figure as the singularity in the blue solid curve occurs exactly at this value.

For the case of constant bulk modulus *K*_{n} = *K*, Figure 3 should be contrasted
with Figure 4. Obviously, the structure is much simpler, as the
singularities in (22) and (24) have disappeared
through direct cancelation with the numerator.
It is still the case however that the allowed regions in Figure 4 are the
bounded areas in upper right-hand corner, and the lower left-hand
corner. Again the red circles are the points in the (,)-plane
that produce the optimum bounds. However, it is no longer clear from
this Figure whether is playing any role in the
analysis or not.

PMWconK
As in Figure 3 for the case of constant bulk
modulus, in which case Figure 4 K_{V} = K_{R} = K, and .Values of the constants entering the expressions (see Appendix A) are:
K = 50.0000, c = 7.2727, _{44}c = 22.0000,
, in units of GPa.
_{66} |

While attempting to find an answer to this question, the author has spent
some effort manipulating the form of the equations for the shear
modulus bounds and has found what may be a more enlightening
form of these equations. (The derivation will not be given here as it is
rather straightforward to find the result again, once the final
expression is known.)
The resulting simplified formula for the Peselnick-Meister-Watt
bounds on overall shear modulus of a polycrystal of laminates
when *K*_{n} = *K* is:

(13) |

(14) |

We see that still plays a dominant role here -- in the company
of *c _{44}* and

10/23/2004