Acoustic waves in hypoelastic solids. I. Isotropic materials

Two models of hypoelastic isotropic materials based on the use of a nonholonomic strain measure, the generalized Yaumann derivative of which coincides with the strain rate tensor, are considered. The conditions under which elastic strain potential exists in such models are formulated. The elastic potential and the constitutive relations are written in terms of elastic eigen subspaces of an isotropic material. The models differ in the number of elastic constants. It is shown that the four-constant model satisfies the requirements of A.A.Ilyushin particular postulate of isotropy, and the five-constant model does not satisfy them. The equation of acoustic wave propagation in such materials is obtained.
The influence of the use of the particular isotropy postulate as a hypothesis on the results of dynamic problems solution is investigated. The phase velocities of acoustic wave propagation under various types of initial strains are determined for two models. At preliminary purely volumetric strains, calculations using five-constant and four-constant models give the same result. For strains located in the deviatoric subspace, the stress tensor has a component located in the first elastic eigen subspace, and its projection into the second subspace when using the five-constant model is misaligned to the strain deviator. At the same time, the initially isotropic material acquires anisotropy with respect to acoustic properties. The material model satisfying the particular postulate of isotropy in the case under consideration also describes the anisotropy of longitudinal waves propagation velocities.