SpringerLink
Log in

Sharp Bounds for the Smallest M-eigenvalue of an Elasticity Z-tensor and Its Application

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

The smallest M-eigenvalue \(\tau _M ({\mathcal {A}})\) of a fourth-order partial symmetric tensor \({\mathcal {A}}\) plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if \(\tau _M ({\mathcal {A}})>0\), then the SE-condition of \({\mathcal {A}}\) holds. In this paper, we establish lower and upper bounds of \(\tau _M ({\mathcal {A}})\) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of \({\mathcal {A}}\). In addition, when \({\mathcal {A}}\) is an elasticity Z-tensor, we establish lower bounds for \(\tau _M ({\mathcal {A}})\) via the extreme C-eigenvalues of piezoelectric-type tensors. Finally, numerical examples show the efficiency of our proposed bounds in judging the SE-condition of \({\mathcal {A}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

No data was used for the research described in the article.

References

  1. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  Google Scholar 

  2. Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)

    Book  Google Scholar 

  3. Chen, Y., Jákli, A., Qi, L.: The \(C\)-eigenvalue of third order tensors and its application in crystals. J. Ind. Manag. Optim. 19(1), 265–281 (2023)

    Article  MathSciNet  Google Scholar 

  4. Dahl, G., Leinaas, J., Myrheim, J.: A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl. 420(2–3), 711–725 (2007)

    Article  MathSciNet  Google Scholar 

  5. Chirita, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27 (2007)

    Article  MathSciNet  Google Scholar 

  6. Walton, J., Wilber, J.: Sufficient conditions for strong ellipticity for a class of anisotropic materials. Int. J. Non-Linear Mech. 38, 44–455 (2003)

    Article  MathSciNet  Google Scholar 

  7. Zubov, L., Rudev, A.: On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials. Z. Angew. Math. Mech. 96, 1096–1102 (2016)

    Article  MathSciNet  Google Scholar 

  8. Han, D., Dai, H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97, 1–13 (2009)

    Article  MathSciNet  Google Scholar 

  9. Qi, L., Dai, H., Han, D.: Conditions for strong ellipticity and \(M\)-eigenvalues. Front. Math. China 4, 349–364 (2009)

    Article  MathSciNet  Google Scholar 

  10. Liu, X., Mo, C.: Calculating \(C\)-eigenpairs of piezoelectric-type tensors via a \(Z\)-eigenpair method. Appl. Math. Comput. 426, 127124 (2022)

    MathSciNet  Google Scholar 

  11. Liu, X., Yin, S., Li, H.: \(C\)-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices. J. Ind. Manag. Optim. 17(6), 3349–3356 (2021)

    Article  MathSciNet  Google Scholar 

  12. Liu, X., Liu, D., Shi, Y.: Perturbation bounds for the largest \(C\)-eigenvalue of piezoelectric-type tensors. Bull. Malays. Math. Sci. Soc. 46, 194 (2023)

    Article  MathSciNet  Google Scholar 

  13. Ding, W., Liu, J., Qi, L., Yan, H.: Elasticity \(M\)-tensors and the strong ellipticity condition. Appl. Math. Comput. 373, 124982 (2020)

    MathSciNet  Google Scholar 

  14. Huang, Z., Qi, L.: Positive definiteness of paired symmetric tensors and elasticity tensors. J. Comput. Appl. Math. 388, 22–43 (2018)

    Article  MathSciNet  Google Scholar 

  15. He, J., Xu, G., Liu, Y.: Some inequalities for the minimum \(M\)-eigenvalue of elasticity \(M\)-tensors. J. Ind. Manag. Optim. 16(6), 3035–3045 (2020)

    Article  MathSciNet  Google Scholar 

  16. He, J., Wei, Y., Li, C.: \(M\)-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity. Appl. Math. Lett. 102, 106137 (2020)

    Article  MathSciNet  Google Scholar 

  17. Li, S., Li, C., Li, Y.: \(M\)-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor. J. Comput. Appl. Math. 356, 391–401 (2019)

    Article  MathSciNet  Google Scholar 

  18. Li, S., Chen, Z., Liu, Q., Lu, L.: Bounds of \(M\)-eigenvalues and strong ellipticity conditions for elasticity tensors. Linear Multilinear Algebra 70(19), 4544–4557 (2022)

    Article  MathSciNet  Google Scholar 

  19. Wang, G., Sun, L., Liu, L.: \(M\)-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors. Complexity 2020, 2474278 (2020)

    Google Scholar 

  20. Wang, Y., Qi, L., Zhang, X.: A practical method for computing the largest \(M\)-eigenvalue of a fourth-order partially symmetric tensor. Numer. Linear Algebra Appl. 16, 589–601 (2009)

    Article  MathSciNet  Google Scholar 

  21. Liu, K., Che, H., Chen, H., Li, M.: Parameterized \(S\)-type \(M\)-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications. J. Ind. Manag. Optim. 19, 3060–3074 (2023)

    Article  MathSciNet  Google Scholar 

  22. Zhao, J.: Conditions of strong ellipticity and calculations of \(M\)-eigenvalues for a partially symmetric tensor. Appl. Math. Comput. 458, 128245 (2023)

    MathSciNet  Google Scholar 

  23. Wang, G., Sun, L., Wang, X.: Sharp bounds on the minimum \(M\)-eigenvalue of elasticity \(Z\)-tensors and identifying strong ellipticity. J. Appl. Anal. Comput. 11(4), 2114–2130 (2021)

    MathSciNet  Google Scholar 

  24. Wang, G., Wang, C., Liu, L.: Identifying strong ellipticity via bounds on the minimum \(M\)-eigenvalue of elasticity \(Z\)-tensors. J. Appl. Anal. Comput. 13(2), 609–622 (2023)

    MathSciNet  Google Scholar 

  25. Wang, C., Wang, G., Liu, L.: Sharp bounds on the minimum \(M\)-eigenvalue and strong ellipticity condition of elasticity \(Z\)-tensors. J. Ind. Manag. Optim. 19, 760–772 (2023)

    Article  MathSciNet  Google Scholar 

  26. Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  27. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)

    Book  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for their careful reading of the manuscript and helpful suggestions.

Funding

The first author was funded by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202200512), the Natural Science Foundation Project of Chongqing (Grant No. CSTB2022NSCQ-MSX0896). The second author was funded by Guizhou Provincial Science and Technology Projects (Grant No. QKHJC-ZK[2022]YB215).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, China

    **fu Liu

  2. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, China

    Jianxing Zhao

Authors
  1. **fu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianxing Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to **fu Liu.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Ethical Approval and Consent to participate

Not applicable.

Human and Animal Rights

Not applicable.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Zhao, J. Sharp Bounds for the Smallest M-eigenvalue of an Elasticity Z-tensor and Its Application. Bull. Malays. Math. Sci. Soc. 47, 119 (2024). https://doi.org/10.1007/s40840-024-01698-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-024-01698-0

Keywords

Mathematics Subject Classification

Search

Navigation