The Lokhin–Sedov table revisited: Tensor form of anisotropic elastic moduli for all symmetry groups

Abstract

Tensors that are invariant with respect to subgroups of three-dimensional group of rotations were constructed first by Lokhin and Sedov (J Appl Math Mech (PMM), 1963, 27, 597–629). In this paper, we do one more step and derive formulas for the elastic modulus tensors for all symmetry groups from the Lokhin–Sedov results. The tensor basis obtained in this way has an advantage of being built upon the same tensors that are used in construction of non-linear tensor functions. One consequence of the formulas derived is that the number of types of anisotropic elastic behaviors is ten. There is a long-standing controversy concerning this number; many authors claim that the correct number is eight. We discuss this controversy in detail, and add several points to the Fedorov–Khatkevich argument that initiated this controversy. We formulate the tensor relationship which is the underlying cause of the Fedorov–Khatkevich argument. This relationship yields several generalizations (extension to cubic symmetries, the inversed argument). We argue that for crystals, ten types of elastic response cannot be reduced to eight due to built-in intrinsic structure determining the crystal symmetry. For polycrystals with a priori unknown elastic symmetry, the problem of determining the symmetry by elastic moduli is shown to be ill-posed.