A geometrical picture of anisotropic elastic tensors

TLDR: In this paper, a geometrical picture of fourth-order, three-dimensional elastic tensors in terms of Maxwell multipoles is developed and used to obtain the elastic tensor appropriate to various crystal symmetry groups.

Abstract: A geometrical picture of fourth-order, three-dimensional elastic tensors in terms of Maxwell multipoles is developed and used to obtain the elastic tensors appropriate to various crystal symmetry groups. Simply examining the picture shows whether an elastic tensor, described by its 21 independent components relative to an ill-chosen coordinate system, has an axis of symmetry of any order. The picture also facilitates obtaining the elastic tensor from the observed dependence of the three body-wave phase velocities on the direction of the propagation vector κ. In particular, for q = 2, 4, and 6, is a linear combination of the surface spherical harmonics of even orders up to and including q. Since determine uniquely, all the body-wave phase-velocity dependence on can be summarized by the 6 coefficients of spherical harmonics in P(2), the 15 coefficients in P(4), and the 28 coefficients in P(6). For nearly isotropic media, the anisotropy in the P velocity determines 15 of the 21 elastic coefficients, whereas determines the other 6 elastic coefficients. Our description of elastic tensors is generalized to all fourth-order tensors in three dimensions and certain fourth-order tensors in higher dimensions. The problem in higher dimensions produces simple examples of unitary representations of the rotation group ON+ with N ≥ 4 which contain no harmonic irreducible components.