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 locally by:  

$$\left(\begin{array} {c} \sigma_{11} \\ \sigma_{22} \\ \sig... ...} \\ e_{33} \\ e_{23} \\ e_{31} \\ e_{12}\end{array}\right),$$ (1)

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 (1).

For definiteness we also assume that this stiffness matrix in (1) 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 the order in which layers were added to the blocks in unimportant, 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), Berryman (1998,2004a,b), Milton (2002)] state that  

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

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 laminated grain 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 (1)]. The well-known result is given by $e = e_{11}+e_{22}+e_{33} = \sigma/K_{\rm eff}$, where $1/K_{\rm eff} = 1/K_R = 2s_{11} + 2s_{12} + 4s_{13} + s_{33}$.When $\mu_n = \mu$ is constant in a layered grain, the definition of K_R implies Hill's equation (Hill, 1963, 1964; Milton, 2002), which is given by  

$$K^*= \left[\sum_{n=1}^{N} \frac{f_n}{K_n+4\mu/3}\right]^{-1} - 4\mu/3.$$ (3)

Here the bulk modulus of the n-th constituent is K_n, the shear modulus takes the same value $\mu_n = \mu$ for all $n = 1,\ldots,N$,and the overall effective bulk modulus is K^*. The volume fractions f_n are all nonnegative, and sum to unity.

Even though $K_{\rm eff} = K_R$ is the same for every grain, since the grains themselves are not isotropic, the overall bulk modulus K^* of the random polycrystal does not necessarily have the same value as K_R 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 K_R value lies outside the bounds in many situations (Berryman, 2004b).