Self-consistent Estimates for Hexagonal Symmetry

The results obtained for self-consistent estimates can be written in many different ways (Berryman, 2005). We take the self-consistent estimate for bulk modulus to be  

$$K^* = \frac{K_V(G_{\rm eff}^r + \zeta^*)} {(G_{\rm eff}^v + ... ...{(G_{\rm eff}^vK_R + \zeta^*K_V)}{(G_{\rm eff}^v + \zeta^*)},$$ (34)

where  

$$\zeta^* = \frac{G^*}{6}\left(\frac{9K^*+8G^*}{K^*+2G^*}\right).$$ (35)

In (35), K^* is determined by (34), depending also on G^*; G^* is determined by the self-consistent expression for the shear modulus to follow, also depending on K^*; and $\zeta^*$ is then determined by (35). The final result for G^* = G^*_hex in polycrystals having grains with hexagonal symmetry is  

$$\frac{1}{G_{\rm hex}^* + \zeta^*} = \frac{1}{5}\left[\frac{... ...+ \frac{2}{C_{44}+\zeta^*} + \frac{2}{C_{66}+\zeta^*}\right].$$ (36)

These formulas can be successfully solved by iteration, starting for example by using values corresponding to upper or lower bounds for the values of K^* and G^*. Some details of the derivation of these formulas can be found in Willis (1977, 1981) and Berryman (2005).