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Effective bulk modulus estimates for polycrystals

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
   \begin{eqnarray}
{{1}\over{K_R}} = 2S_{11} + 2S_{12} + S_{33} + 4S_{13}.
 \end{eqnarray} (20)
It is straightforward to show that this produces exactly the same result as either (18) or (20). So KR = K, which should be interpreted as meaning that the lower bound on the bulk modulus in the polycrystalline system is equal to the bulk modulus of the simple layered system.



 
next up previous print clean
Next: Voigt average Up: BULK MODULUS K AND Keff Previous: Bulk modulus K
Stanford Exploration Project
7/8/2003