A criterion for the strong ellipticity of the equilibrium equations of an isotropic non-linearly elastic material

Abstract

A new criterion for satisfying the condition of strong ellipticity of the equilibrium equations of a homogeneous isotropic elastic medium is obtained, consisting of twelve unconditional and the same number of conditional inequalities. The criterion differs from previously known criteria in the fact that it is more effective for the analytical verification of the condition of strong ellipticity. A number of sufficient criteria of the strong ellipticity of the equations of the non-linear statics of an elastic body is presented. It is shown that the method used in this paper is also effective in the case of an incompressible material. A new constitutive inequality in the non-linear theory of elasticity is proposed that, in particular, ensures the strong ellipticity of the equilibrium equations and the stability of an elastic body in a state of hydrostatic compression. A mechanical interpretation of some implications of the above mentioned inequality is given. Using the criterion obtained, the condition of strong ellipticity is verified for a number of specific models of compressible elastic materials and examples of the violation of this condition are also considered.