Criterion for the strong ellipticity of the equations of motion of an anisotropic linear-elastic material

Two methods of verifying the strong ellipticity of the equations of motion (the SE-condition) for an arbitrary anisotropic linear-elastic material are proposed. A mechanical interpretation of a number of the corollaries of the SE-condition is given. Effective sufficient criteria for the realizability of the SE-condition and the weak version of it, the so-called Hadamard inequality, are found for materials of the monoclinic and 6-constant hexagonal systems. The case of a two-dimensional space is considered. Finite sets of elementary inequalities that are equivalent to the SE-condition and the Hadamard inequality are presented for materials of a 7-constant tetragonal system as well as a simple sufficient criterion for the ellipticity of the equilibrium equations of a uniform medium. Numerical examples are analysed and a comparison of the different methods of verifying the SE-condition is presented.