Peselnick-Meister-Watt (PMW) Bounds for Hexagonal Symmetry

Hashin-Shtrikman-style bounds (Hashin and Shtrikman, 1962a,b; 1963) on the bulk and shear moduli of isotropic random polycrystals composed of grains having hexagonal symmetry have been derived by Peselnick and Meister (1965), with corrections made later by Watt and Peselnick (1980). We will term these the PMW (for Peselnick-Meister-Watt) or the HS (Hashin-Shtrikman) bounds interchangably. The PMW notation was similar to that in the original Hashin-Shtrikman paper on random polycrystals of grains having cubic symmetry (Hashin and Shtrikman, 1962b). We will use a slightly modified notation here, taking into account the product formulas (Berryman, 2004b) in order to simplify the statement of the results. Derivations are found in the references, and therefore not repeated here.

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 ($\pm$) isotropic comparison materials. G₊,K₊ are the values used in the formulas for the upper bounds, and G_-,K_- for the lower bounds. Simplified formulas for the bulk modulus bounds are:  

$$K_{PMW}^\pm \equiv K_{HS}^\pm = \frac{K_V(G_{\rm eff}^r + \zeta_\pm)} {(G_{\rm eff}^v + \zeta_\pm)},$$ (27)

where  

$$\zeta_\pm = \frac{G_\pm}{6}\left(\frac{9K_\pm+8G_\pm}{K_\pm+2G_\pm}\right).$$ (28)

In (28), the values of $G_\pm$ and $K_\pm$ are those defined algorithmically according to:  

$$K_\pm = \frac{K_V(G_{\rm eff}^r-G_\pm)}{(G_{\rm eff}^v - G_\pm)},$$ (29)

where, for K_-,  

$$0 \le G_- \le \min(C_{44},G_{\rm eff}^r,C_{66}),$$ (30)

and, similarly, for the K₊ formula,  

$$\max(C_{44},G_{\rm eff}^v,C_{66}) \le G_+ \le \infty.$$ (31)

The corresponding formulas for shear modulus bounds $G^\pm_{hex}$ are  

$$\frac{1}{G_{\rm hex}^\pm + \zeta_\pm} = \frac{1}{5}\left[\f... ...rac{2}{C_{44}+\zeta_\pm} + \frac{2}{C_{66}+\zeta_\pm}\right],$$ (32)

where the constants $\alpha_\pm$ and $\beta_\pm$ are defined by  

$$\alpha_\pm = \frac{-1}{K_\pm + 4G_\pm/3}, \qquad \beta_\pm = \frac{2\alpha_\pm}{15} - \frac{1}{5G_\pm}.$$ (33)

Peselnick and Meister (1965) had originally obtained all the results for hexagonal symmetry, except for an additional condition that permits C₄₄ to be replaced in some circumstances by $G_{\rm eff}^r$. This condition was added later by Watt and Peselnick (1980).