Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors

Abstract

The condition for strong ellipticity of the equilibrium equations plays a significant role in the theory of elasticity. For isotropic elastic materials and anisotropic linearly elastic materials, identification of the strong ellipticity conditions for the corresponding equilibrium equations has been discussed in many references and obtained some equivalent checkable criteria. But for general nonlinearly elastic materials, it is hardly possible to give checkable equivalent criteria for the strong ellipticity condition of the associated equilibrium equations. In 2009, Qi et al. pointed that the strong ellipticity condition of the equilibrium equations can be equivalently transformed into the strong ellipticity condition of partially symmetric tensors. In this paper, using the M-eigenvalues of partially symmetric tensors, we give some easily computable and verifiable sufficient conditions for the strong ellipticity of partially symmetric tensors. Based on these criteria, some direct algorithms for identifying the strong ellipticity condition of partially symmetric tensors are derived. Numerical examples show that the proposed criteria are efficient in identifying the strong ellipticity condition of the equilibrium equations, especially for general nonlinearly elastic materials.