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

Elasticity $\mathscr{M}$-tensors and the Strong Ellipticity Condition