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Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor

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  • Zhao, Jianxing

Abstract

Strong ellipticity condition (SE-condition) of the equilibrium equations for general nonlinear elastic materials can be equivalently transformed into the SE-condition of a partially symmetric tensor. Qi et al. in 2009 proved that the SE-condition of a partially symmetric tensor holds if and only if its smallest M-eigenvalue is positive. In this paper, a criterion for the SE-condition of a partially symmetric tensor is first given. And then, an alternative method to compute all M-eigentriples of a partially symmetric tensor is presented. Finally, two numerical examples show the effectiveness of the proposed criterion and the calculation of M-eigenvalues in judging the SE-condition of the equilibrium equations.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:458:y:2023:i:c:s0096300323004149
    DOI: 10.1016/j.amc.2023.128245
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    References listed on IDEAS

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    1. Li, Suhua & Li, Yaotang, 2021. "Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    2. Gang Wang & Linxuan Sun & Lixia Liu, 2020. "M -Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors," Complexity, Hindawi, vol. 2020, pages 1-8, January.
    3. He, Jun & Liu, Yanmin & Xu, Guangjun, 2021. "New S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor with applications," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    4. 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).
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