Conductivity for random polycrystals of laminates

For random polycrystals (see the earlier discussion of the basic model in the second section), it is most convenient to define a new canonical function:  

$$\Sigma_X(s) = \left[\frac{1}{3}\left(\frac{1}{\sigma_H+2s} + \frac{2}{\sigma_M+2s}\right)\right]^{-1} - 2s,$$ (36)

where the mean $\sigma_M = \sum_{i=1}^J v_i\sigma_i$ and harmonic mean $\sigma_H = \left[\sum_{i=1}^J \frac{v_i}{\sigma_i}\right]^{-1}$of the layer constituents are the pertinent conductivities (off-axis and on-axis of symmetry, respectively) in each layered grain. Then, the Hashin-Shtrikman bounds for the conductivity of the random polycrystal are  

$$\sigma_{HSX}^\pm = \Sigma_X(\sigma_\pm),$$ (37)

where $\sigma_+ = \sigma_M$ and $\sigma_- = \sigma_H$.These bounds are known not to be the most general ones since they rely on an implicit assumption that the grains are equiaxed. A more general lower bound that is known to be optimal is due to Schulgasser (1983) and Avellaneda et al. (1988):  

$$\sigma_{ACLMX}^- = \Sigma_X(\sigma_{ACLMX}^-/4).$$ (38)

Helsing and Helte (1991) have reviewed the state of the art for conductivity bounds for polycrystals, and in particular have noted that the self-consistent [or CPA (i.e., coherent potential approximation)] for the random polycrystal conductivity is given by  

$$\sigma_{CPAX}^* = \Sigma_X(\sigma_{CPAX}^*).$$ (39)

It is easy to show (39) always lies between the two rigorous bounds $\sigma_{ACLMX}^-$ and $\sigma_{HSX}^+$, and also between $\sigma_{HSX}^-$ and $\sigma_{HSX}^+$. Note that $\sigma_{ACLMX}^-$and $\sigma_{HSX}^-$ cross when $\sigma_m/\sigma_H = 10$, with $\sigma_{ACLMX}^-$ becoming the superior lower bound for mean/harmonic-mean contrast ratios greater than 10.