  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.
