Hashin-Shtrikman Bounds

It has been shown elsewhere (Berryman, 2004a,b) that the Peselnick-Meister-Watt bounds for bulk modulus of a random polycrystal composed of hexagonal (or transversely isotropic) grains are given by  

$$K_{PM}^\pm = \frac{K_V(G_{\rm eff}^r + \zeta_\pm)} {(G_{\rm ... ...K_RG_{\rm eff}^v + K_V\zeta_\pm} {G_{\rm eff}^v + \zeta_\pm},$$ (10)

where $G_{\rm eff}^v$ ($G_{\rm eff}^v$) is the uniaxial shear energy per unit volume for a unit applied shear strain (stress). The second equality follows directly from the product formula (9). Parameters $\zeta_\pm$ are defined by  

$$\zeta_{\pm} = \frac{G_\pm}{6}\left(\frac{9K_\pm+8G_\pm}{K_\pm+2G_\pm}\right).$$ (11)

In (11), values of $G_\pm$ (shear moduli of isotropic comparison materials) are determined by inequalities  

$$0 \le G_- \le \min(c_{44},G_{\rm eff}^r,c_{66}),$$ (12)

and  

$$\max(c_{44},G_{\rm eff}^v,c_{66}) \le G_+ \le \infty.$$ (13)

The values of $K_\pm$ (bulk moduli of isotropic comparison materials) are then determined by equalities  

$$K_\pm = \frac{K_V(G_{\rm eff}^r-G_\pm)}{(G_{\rm eff}^v - G_\pm)},$$ (14)

given by Peselnick and Meister (1965) and Watt and Peselnick (1980). Also see Berryman, 2004b).

Bounds on the shear moduli are then given by  

$$\begin{array} {r} \frac{1}{\mu_{\rm hex}^\pm + \zeta_\pm} = ... ..._{44}+\zeta_\pm} + \frac{2}{c_{66}+\zeta_\pm}\big],\end{array}$$ (15)

where $\gamma_\pm$ and $\delta_\pm$ are given by  

$$\gamma_\pm = \frac{-1}{K_\pm + 4G_\pm/3} \qquad\hbox{and}\qq... ... \left[\frac{4}{15} - \frac{2}{5G_\pm\gamma_\pm}\right]^{-1}.$$ (16)

K_V is the Voigt average of the bulk modulus as defined previously.