On the M-eigenvalues of elasticity tensor and the strong ellipticity condition

Strong ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong ellipticity condition holds if and only if all the M-eigenvalues of the elasticity tensor are positive. Thus, M-eigenvalues are some intrinsic parameters of the elasticity tensor. In this paper, we show that the M-eigenvalues of the elasticity tensor are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lam\'e's first parameter, the P-wave modulus, etc, and the positiveness of the M-eigenvalues are corresponding to some existing conditions for strong ellipticity in some special cases, such as the isotropic case, the cubic case, the polar anisotropic case and the tetragonal case. We also present new sufficient conditions for the strong ellipticity of the orthotropic case. These, in a certain sense, further reveal the physical meanings of M-eigenvalues.