Identification of a model of a nonlinear elastic anisotropic material with cubic symmetry of properties

The propagation of acoustic waves in nonlinear elastic anisotropic media with finite preliminary strains is considered. The media in the initial state are homogeneous with an elastic potential in which the first two nonzero terms of expansion in a series by degrees of the strain tensor are preserved. The dynamic equations are written as the equations of propagation of small displacement perturbations imposed on finite strains. The equations are concretized for the case of propagation of plane monochromatic waves.
An anisotropic material with the symmetry of properties inherent in crystals of cubic symmetry is considered. Constitutive relations of the nonlinear model are written in terms of the basis tensors of the eigen elastic subspaces of the fourth and sixth ranks. The relations include three second-order constants and six third-order constants. A program of experiments for determining the constants of elasticity of a cubic material is proposed.
To determine the elasticity constants of the second order, it is proposed to fulfill an experiment to measure the phase velocities of longitudinal and two transverse waves propagating along the edge of a prismatic sample. To determine the elasticity constants of the third order, the phase velocities of acoustic wave propagation are measured in two samples differing in the orientation of the main axes of anisotropy. In the samples preliminary tension-compression strains are created along the two edges.
The results of numerical simulation of the proposed experiments for niobium crystals, whose elastic properties are known from sources, are presented. Sections of the surfaces of the phase velocities of longitudinal (quasi-longitudinal) and transverse (quasi-transverse) waves found at different levels of preliminary deformations proposed in the experimental program are constructed. It is shown that not only the values of the wave propagation velocities depend on the level of strains, but also the shape of the cross sections of the phase velocity surfaces with different planes.