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Elasticity M-tensors and the strong ellipticity condition

Author

Listed:
  • Ding, Weiyang
  • Liu, Jinjie
  • Qi, Liqun
  • Yan, Hong

Abstract

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 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 M-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity 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 M-tensors are also established.

Suggested Citation

  • Ding, Weiyang & Liu, Jinjie & Qi, Liqun & Yan, Hong, 2020. "Elasticity M-tensors and the strong ellipticity condition," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    • Handle: RePEc:eee:apmaco:v:373:y:2020:i:c:s0096300319309749
    DOI: 10.1016/j.amc.2019.124982
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    Cited by:

    1. Zhao, Jianxing, 2023. "Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    2. Li, Suhua & Li, Yaotang, 2021. "Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors," Applied Mathematics and Computation, Elsevier, vol. 403(C).

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