ELASTICITY OF LAYERED MATERIALS

We assume that a typical building block of the random system is a small grain of laminate material whose elastic response for such a transversely isotropic (hexagonal) system can be described by:  

$$\left(\begin{array} {c} \sigma_{11} \\ \sigma_{22} \\ \sig... ...} \\ e_{33} \\ e_{23} \\ e_{31} \\ e_{12}\end{array}\right),$$ (2)

where $\sigma_{ij}$ are the usual stress components for i,j=1-3 in Cartesian coordinates, with 3 (or z) being the axis of symmetry (the lamination direction for such a layered material). Displacement u_i is then related to strain component e_ij by $e_{ij} = (\partial u_i/\partial x_j + \partial u_j/\partial x_i)/2$.This choice of definition introduces some convenient factors of two into the 44,55,66 components of the stiffness matrix shown in (2).

Although some of the results presented here are more general, we will assume for definiteness that this stiffness matrix in (2) arises from the lamination of N isotropic constituents having bulk and shear moduli K_n, $\mu_n$, in the N > 1 layers present in each building block. It is important that the thicknesses d_n always be in the same proportion in each of these laminated blocks, so that $f_n = d_n/\sum_{n'} d_{n'}$. But it is not important what order the layers were added to the blocks, as Backus's formulas (Backus, 1962) for the constants show. For the overall behavior for the quasistatic (long wavelength) behavior of the system we are studying, Backus's results [also see Postma (1955) and Milton (2002)] state that  

$$\begin{array} {ll} c_{33} = \left<\frac{1}{K+4\mu/3}\right\g... ...2}{K+4\mu/3}\right\gt, & c_{12} = c_{11} - 2c_{66}.\end{array}$$ (3)

This bracket notation can be correctly viewed: (a) as a volume average, (b) as a line integral along the symmetry axis x₃, or (c) as a weighted summation $\left<Q\right\gt = \sum_n f_nQ_n$ over any relevant physical quantity Q taking a constant value Q_n in the n-th layer.

The bulk modulus for each such building block (or crystalline grain if you like) is that given by the compressional Reuss average K_R of the corresponding compliance matrix s_ij [the inverse of the usual stiffness matrix c_ij, whose nonzero components are shown in (2)]. The well-known result is $e = e_{11}+e_{22}+e_{33} = \sigma/K_{\rm eff}$, where $1/K_{\rm eff} = 1/K_R = 2s_{11} + 2s_{12} + 4s_{13} + s_{33}$.This quantity can be expressed in terms of the stiffness elements as  

$$\frac{1}{K_R - c_{13}} = \frac{1}{c_{11} - c_{66} - c_{13}} + \frac{1}{c_{33} - c_{13}}.$$ (4)

When $\mu_n =$const, it is easy to show that (4) implies (1).

Even though $K_{\rm eff}$ is the same for every grain, since the grains themselves are not isotropic, the overall bulk modulus K^* of the random polycrystal is not necessarily the same as $K_{\rm eff}$ for the individual grains (Hill, 1952). Hashin-Shtrikman bounds on K^* for random polycrystals whose grains have hexagonal symmetry (Peselnick and Meister, 1965; Watt and Peselnick, 1980) show in fact that the value K_R lies outside the bounds in many situations. We will say more about this in the fourth section.

In general an upper bound on the overall shear modulus of an isotropic polycrystal (Hill, 1952) is given by the Voigt average over shear of the stiffness matrix, which may be written as  

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

This expression can be taken as the definition of $G_{\rm eff}^V$.Eq. (5) implies that $G_{\rm eff}^V = (c_{11} + c_{33} - 2c_{13} - c_{66})/3$.$G_{\rm eff}^V$ 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).