Estimates of Nonlinear Elastic Constants and Acoustic Nonlinearity Parameters for Textured Polycrystals

Abstract

In this article, expressions are derived for the Voigt, Reuss, and Hill estimates of the third-order elastic constants for polycrystals with either cubic or hexagonal crystal symmetry and orthorhombic physical symmetry. General forms of the fourth- and sixth-rank elastic stiffness and compliance tensors for crystal and physical symmetries are given. Explicit expressions are reduced from these tensors for the case of polycrystals exhibiting orthorhombic sample symmetry with either cubic or hexagonal crystallites. The estimated third-order elastic constants of the textured polycrystal are obtained in terms of second- and third-order single-crystal elastic constants and orientation distribution coefficients (ODCs), which are used to account for anisotropic physical symmetry. The acoustic nonlinearity parameter, $\bar{\beta}$ , is defined through combinations of the second- and third-order Voigt, Reuss, and Hill estimates of the elastic constants for a textured polycrystal. The model predicts that $\bar{\beta}$ is dependent on the type of averaging scheme used and the texture-defining ODCs. The model is quantitatively evaluated for polycrystalline iron, aluminum, and titanium using second- and third-order single-crystal elastic constants and experimentally measured ODCs. The interrelation between $\bar{\beta}$ and polycrystalline anisotropy offers potential for techniques associated with quantitative texture analysis.