Relation between Third-Order Elastic Constants of Single Crystals and Polycrystals

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Abstract
The approximations by Voigt, Reuss, and Hill for calculating the elastic constants of polycrystals from the elastic constants of single crystals are extended to the third-order elastic constants. The general relations including the expressions for the third-order elastic compliances are presented and given explicitly for cubic symmetry. They are used to calculate the polycrystalline third-order elastic constants of eleven cubic materials. For cubic symmetry, relations for the pressure derivatives of the second-order elastic constants in the approximations of Voigt, Reuss, and Hill are also presented, and the anisotropy of the third-order elastic constants is discussed. It is found that for all materials considered, the anisotropy for the third-order elastic constants is much larger than the anisotropy of the second-order elastic constants, and that a weak correlation exists between the anisotropy of the third-order elastic constants and of the second-order elastic constants.