Assuming that the simple layered material we have been studying is present locally at some small (micro-)scale in a heterogeneous (macro-)medium, and also assuming that the axis of symmetry of these local constituents is randomly distributed so the whole composite medium is isotropic, then we have a polycrystalline material. The well-known results of Reuss and Voigt provide simple and useful estimates for moduli of polycrystals [actually lower and upper bounds on the moduli as shown by Hill (1952)].

*Reuss average*

The Reuss average is obtained by assuming constant stress, which is
the same condition we applied already to estimate the bulk modulus *K*
for the simple layered material. The well-known result in terms of
compliances (*S* = *C ^{-1}*) is

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7/8/2003