Elasticity for random polycrystals of laminates

In order to have a more precise model for comparison purposes, and to get a better feeling for just how much difference it makes whether we model the microstructure very accurately or not, we will now consider a model material called a ``random polycrystal of laminates.'' Suppose we construct a random polycrystal by packing small bits of a laminate material (i.e., a composite layered along a symmetry axis) into a large container in a way so that the axis of symmetry of the grains appears randomly over all possible orientations and also such that no misfit of surfaces (and therefore porosity) is left in the resulting composite. If the ratio of laminate grain size to overall composite is small enough so the usual implicit assumption of scale separation applies to the composite -- but not so small that we are violating the continuum hypothesis -- then we have an example of a random polycrystal of laminates. See schematic in Figure 1.

The analytical advantage of this model is that the layers can be composed of the two elastic constituents in the composites discussed here previously. Furthermore, the elastic behavior of the laminate material itself can be predicted using well-known exact methods (Backus, 1962). We will not dwell on these details here, but just make use of the results to be found in many publications (Berryman, 2004b). The only results needed in the following are the Reuss and Voigt averages for the grains, which are 1/K_R = 2s₁₁ + 2s₁₂ + 4s₁₃ + s₃₃ for Reuss in terms of compliances, or  

$$\frac{1}{K_R - c_{13}} = \frac{1}{c_{11} - c_{66} - c_{13}} + \frac{1}{c_{33} - c_{13}},$$ (19)

in terms of stiffness, and  

$$K_V = \left[2(c_{11}+c_{12}) +4c_{13}+c_{33}\right]/9$$ (20)

for the Voigt average of bulk modulus. Similarly, the Voigt average for shear of the stiffness matrix may be written as  

$$\mu_V = \frac{1}{5}\left(G_{\rm eff}^v + 2c_{44} + 2c_{66}\right).$$ (21)

This expression can be taken as the definition of $G_{\rm eff}^v$.Eq.(21) implies that $G_{\rm eff}^v = (c_{11} + c_{33} - 2c_{13} - c_{66})/3$. In fact, $G_{\rm eff}^v$ is the energy per unit volume in a grain when a pure uniaxial shear strain of unit magnitude is applied to the grain along its axis of symmetry (Berryman, 2004a,b). Then, the Reuss average for shear is  

$$\mu_R = \left[\frac{1}{5}\left(\frac{1}{G_{\rm eff}^r} + \frac{2}{c_{44}} + \frac{2}{c_{66}}\right)\right]^{-1},$$ (22)

which is also a rigorous lower bound on the overall shear modulus of the polycrystal (Hill, 1952). Each laminated grain thus has hexagonal symmetry, so the product formulas $3K_RG_{\rm eff}^v = 3K_VG_{\rm eff}^r = \omega_+\omega_-/2 = c_{33}(c_{11}-c_{66})-c_{13}^2$ are valid (Berryman, 2004a). The symbols $\omega_\pm$ stand for the quasi-compressional and quasi-uniaxial shear eigenvalues for all the grains.

Once this notation has been established, then it is straightforward to express the Peselnick-Meister bounds (Peselnick and Meister, 1965) for hexagonal symmetry as  

$$K_{PM}^\pm = \frac{K_V(G_{\rm eff}^r + Y_\pm)} {(G_{\rm eff}^v + Y_\pm)}.$$ (23)

for effective bulk modulus K^* of the polycrystal, where  

$$Y_{\pm} = \frac{G_\pm}{6}\left(\frac{9K_\pm+8G_\pm}{K_\pm+2G_\pm}\right).$$ (24)

The precise values of the parameters $G_\pm$ and $K_\pm$ (being shear and bulk moduli of the HS isotropic comparison material) were given algorithmically by Watt and Peselnick (1980.) Similarly,  

$$\frac{1}{\mu_{PM}^\pm + Y_\pm} = \frac{1}{5}\big[\frac{1-A_... ..._\pm} + \frac{2}{c_{44}+Y_\pm} + \frac{2}{c_{66}+Y_\pm}\big],$$ (25)

for the effective shear modulus $\mu^*$ of the polycrystal. The meaning of $Y_\pm$ is the same in (23) and (25). Here $A_\pm = \frac{-1}{K_\pm + 4G_\pm/3}$,$B_\pm = \frac{2A_\pm}{15} - \frac{1}{5G_\pm}$,and $R_\pm = A_\pm/2B_\pm$. These bounds are of Hashin-Shtrikman type, but were first obtained for hexagonal symmetry by Peselnick and Meister (1965) with some corrections supplied later by Watt and Peselnick (1980).

Since we now have analytical forms for the bounds in (23)-(25), it seems it should be possible to arrive at self-consistent formulas (estimates related to the bounds) by making substitutions $K_\pm \to K^*$ and $\mu_\pm \to \mu^*$,as well as $K_{PM}^\pm \to K^*$ and $\mu_{PM}^\pm \to \mu^*$.This procedure can be followed without difficulty for the bulk modulus bounds in (23). However, for the shear modulus estimator, we need to take into account a step in the derivation of (25) that restricted its applicability to a certain curve in the ($G_\pm,K_\pm$)-plane. Since the self-consistent estimate will not normally lie on this curve, we need to back up in the analysis presented by Watt and Peselnick (1980) and take into account a correction term that vanishes along the curve in question but not in general. When we do this, and also make use of the self-consistent formula for the bulk modulus K^*, which is  

$$K^* = \frac{K_V(G_{\rm eff}^r + Y^*)} {(G_{\rm eff}^v + Y^*)},$$ (26)

we find that the self-consistent estimator for the shear modulus $\mu^*$ is  

$$\frac{1}{\mu^* + Y^*} = \frac{1}{5}\left[\frac{1-A^*(K_V-K^... ... + Y^*} + \frac{2}{c_{44}+Y^*} + \frac{2}{c_{66}+Y^*}\right].$$ (27)

The transform variable for these two formulas is just $Y^* = \Theta(K^*,\mu^*)$, with $\Theta$ defined as in (4).

From the derivation, it is expected that these self-consistent estimates based on the polycrystal bounds will always lie between the bounds. In fact, this feature is observed in all the results from calculations done using these formulas. It can also be shown that the self-consistent estimator obtained this way is the same as that found by Willis (1981) using different arguments. Furthermore, the results are also in agreement with the self-consistent formulas of Olson and Avellaneda (1992) for polycrystals composed of spherical grains when their results for orthorhombic symmetry are specialized to hexagonal symmetry.