  In this paper, we establish two sufficient conditions for the strong
ellipticity of any fourth-order elasticity tensor and investigate a class of
tensors satisfying the strong ellipticity condition, the elasticity
$\mathscr{M}$-tensor. The first sufficient condition is that the strong
ellipticity holds if the unfolding matrix of this fourth-order elasticity
tensor can be modified into a positive definite one by preserving the
summations of some corresponding entries. Second, an alternating projection
algorithm is proposed to verify whether an elasticity tensor satisfies the
first condition or not. Besides, the elasticity $\mathscr{M}$-tensor is defined
with respect to the M-eigenvalues of elasticity tensors. We prove that any
nonsingular elasticity $\mathscr{M}$-tensor satisfies the strong ellipticity
condition by employing a Perron-Frobenius-type theorem for M-spectral radii of
nonnegative elasticity tensors. Other equivalent definitions of nonsingular
elasticity $\mathscr{M}$-tensors are also established.
