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Strong ellipticity is an important property in the elasticity theory. In2009, M-eigenvalues were introduced for the elasticity tensor. It was shownthat M-eigenvalues are invariant under coordinate system choices, and thestrong ellipticity condition holds if and only if all the M-eigenvalues of theelasticity tensor are positive. Thus, M-eigenvalues are some intrinsicparameters of the elasticity tensor. In this paper, we show that theM-eigenvalues of the elasticity tensor are closely related with some elasticmoduli, 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 arecorresponding to some existing conditions for strong ellipticity in somespecial cases, such as the isotropic case, the cubic case, the polaranisotropic case and the tetragonal case. We also present new sufficientconditions for the strong ellipticity of the orthotropic case. These, in acertain sense, further reveal the physical meanings of M-eigenvalues.

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