A general elastic-anisotropy measure

Abstract

We propose an elastic-anisotropy measure. Zener’s familiar anisotropy index A=2C44∕(C11−C12) applies only to cubic symmetry [Elasticity and Anelasticity of Metals (University of Chicago Press, Chicago, 1948), p. 16]. Its extension to hexagonal symmetry creates ambiguities. Extension to orthorhombic (or lower) symmetries becomes meaningless because C11−C12 loses physical meaning. We define elastic anisotropy as the squared ratio of the maximum/minimum shear-wave velocity. We compute the extrema velocities from the Christoffel equations [M. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970), p. 84]. The measure is unambiguous, applies to all crystal symmetries (cubic-triclinic), and reduces to Zener’s definition in the cubic-symmetry limit. The measure permits comparisons between and among different crystal symmetries, say, in allotropic transformations or in a homologous series. It gives meaning to previously unanswerable questions such as the following: is zinc (hexagonal) more or less anisotropic than copper (cubic)? is alpha-uranium (orthorhombic) more or less anisotropic than delta-plutonium (cubic)? The most interesting finding is that close-packed-hexagonal elements show an anisotropy near 1.3, about half that of their close-packed-cubic counterparts. A central-force near-neighbor model supports this finding.