Acoustic waves in hypoelastic solids. II. Anisotropic materials

For hypoelastic anisotropic materials, the constitutive relations of nonlinear elasticity are written out. The relations establish a connection between the generalized Yaumann derivatives of the stress tensor and the nonholonomic strain measure introduced in the works of A. A. Markin. The relation is concretized for anisotropic materials with cubic symmetry of properties. The relations are written in projections into elastic eigen subspaces of a cubic material. The analysis of the mutual influence of finite deformation processes belonging to different eigen subspaces is carried out.
Deformation processes in which the main axes of strains coincide with the same material fibers, whose position relative to the main axes of anisotropy does not change, are considered. Elastic potentials for such processes are written out for two material models: a general one containing nine elastic constants, and a model satisfying the generalization of A. A. Ilyushin particular postulate to anisotropic materials and containing six constants.
The results of solution of the problem of acoustic wave propagation in hypoelastic anisotropic materials with cubic symmetry of properties are presented. As initial strains of a material, deformations located entirely in its elastic eigen subspaces are considered: purely volumetric strain, tension-compression in the main axes of anisotropy, pure shear in the plane of the main axes of anisotropy. Preliminary finite purely volumetric deformations do not affect the shape of the angular dependences of the phase velocities of wave propagation in the vertical plane, but only affect the values of the phase velocities. At preliminary shaping, the material acquires additional anisotropy, and the shape of the angular dependencies of the phase velocities changes. The results show that in some cases, the six-constant model of the material does not predict a change in the propagation velocities of transverse waves at initial finite strains.