APPENDIX A: Peselnick-Meister-Watt Bounds for Hexagonal Symmetry

Hashin-Shtrikman-style bounds (Hashin and Shtrikman, 1962; 1963) on the bulk and shear moduli of isotropic random polycrystals composed of hexagonal grains have been derived by Peselnick and Meister (1965), with later corrections by Watt and Peselnick (1980) The main results are presented here using notation consistent with that of our text, in order to emphasize the connections to the analysis presented. To keep this summary brief, we will merely quote the results and refer the reader to the original papers for the derivations.

Parameters used to optimize the Hashin-Shtrikman bounds are $K_\pm$ and $G_\pm$, which have the significance of being the bulk and shear moduli of two isotropic comparison materials. G₊,K₊ are the values used in the formulas for the upper bounds, and G_-,K_- for the lower bounds. Formulas for the bounds are:  

$$K_{PM}^{\pm} = K_\pm + \frac{K_V - K_\pm}{1 - 2\beta_\pm(G_{\rm eff}^V - G_\pm)},$$ (17)

and  

$$\mu_{PM}^{\pm} = G_\pm + \frac{B_2^\pm}{1+2\beta_\pm B_2^\pm},$$ (18)

where  

$$\alpha_\pm = \frac{-1}{K_\pm + 4G_\pm/3}, \qquad \beta_\pm =... ...pm},\qquad \gamma_\pm = \frac{1}{9}(\alpha_\pm - 3\beta_\pm),$$ (19)

and  

$$B_2^\pm = \frac{1}{5}\big[\frac{G_{\rm eff}^V - G_\pm}{{\cal... ...)} + \frac{2(c_{66}-G_\pm)}{1-2\beta_\pm(c_{66}-G_\pm)}\big],$$ (20)

with  

$${\cal D}_\pm = 1 - \beta_\pm(c_{11}+c_{12}+c_{33} - 3K_\pm-2G_\pm) - 9\gamma_\pm(K_V-K_\pm).$$ (21)

Optimum values of the moduli for the comparison materials have been shown to be (in our notation)  

$$K_- = \frac{K_V(G_{\rm eff}^R-G_-)}{(G_{\rm eff}^V - G_-)}$$ (22)

with  

$$0 \le G_- \le \min(c_{44},G_{\rm eff}^R,c_{66}),$$ (23)

and  

$$K_+ = \frac{K_V(G_+-G_{\rm eff}^R)}{(G_+-G_{\rm eff}^V)}$$ (24)

with  

$$\max(c_{44},G_{\rm eff}^V,c_{66}) \le G_+ \le \infty.$$ (25)

Note that, when G_- = 0, K_- = K_R, because $K_R = K_VG_{\rm eff}^R/G_{\rm eff}^V$from the product formulas (Berryman, 2004). Also, note that, if K_n = K is constant, then $K_\pm = K_V = K_R = K$ for any choice of $G_\pm$, since then we also have that $G_{\rm eff}^V = G_{\rm eff}^R$.

For the laminated materials considered here, the minimum condition in (23) will never be satisfied by c₆₆ except in the trivial case of constant shear modulus. Each of the other two arguments can possibly become the minimum under certain nontrivial circumstances. For the materials considered here, it follows from (7) that the maximum condition in (25) will always be uniquely satisfied by c₆₆, except again for the trivial case of constant shear modulus.

Peselnick and Meister (1965) had originally obtained all the results here except for the additional condition in (23) that permits c₄₄ to be replaced in certain circumstancs by $G_{\rm eff}^R$. This new condition was added later by Watt and Peselnick (1980).