Skip to main content
SpringerLink
Log in

A direct method for calculating M-eigenvalues of an elasticity tensor

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

The strong ellipticity condition (abbr. SE-condition) of the displacement equations of equilibrium for general nonlinearly elastic materials plays an important role in nonlinear elasticity and materials. Existing literature shows that it can be equivalently transformed into the SE-condition of an elasticity tensor, and that the SE-condition of an elasticity tensor holds if and only if all of its M-eigenvalues are positive. In order to judge the strong ellipticity of an elasticity tensor, we in this paper propose a direct method for calculating all M-eigenvalues of an elasticity tensor and show that the direct method is workable by a numerical example.

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

Access this article

Log in via an institution

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

References

  1. Che, H., Chen, H., Wang, Y.: M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors. J. Inequal. Appl. 2019, 32 (2019). https://doi.org/10.1186/s13660-019-1986-x

    Article  MathSciNet  Google Scholar 

  2. Che, H., Chen, H., Wang, Y.: On the M-eigenvalue estimation of fourth-order partially symmetric tensors. J. Ind. Manag. Optim. 16(1), 309–324 (2020). https://doi.org/10.3934/jimo.2018153

    Article  MathSciNet  Google Scholar 

  3. Che, H., Chen, H., Xu, N., Zhu, Q.: New lower bounds for the minimum M-eigenvalue of elasticity M-tensors and applications. J. Inequal. Appl. 2020, 188 (2020). https://doi.org/10.1186/s13660-020-02454-1

    Article  MathSciNet  Google Scholar 

  4. Che, H., Chen, H., Zhou, G.: New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors. J. Ind. Manag. Optim. 17(6), 3685–3694 (2021). https://doi.org/10.3934/jimo.2020139

    Article  MathSciNet  Google Scholar 

  5. Chiritǎ, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27 (2007). https://doi.org/10.1007/s10659-006-9096-7

    Article  MathSciNet  Google Scholar 

  6. Ciarlet, P.G.: Mathematical Elasticity, Three-Dimensional Elasticity, Vol. I. Society for Industrial and Applied Mathematics, Philadelphia (2022). ISBN: 9781611976786. https://epubs.siam.org/doi/pdf/10.1137/1.9781611976786.fm

  7. Cox, D.A., Little, J., O’shea, D.: Using Algebraic Geometry, 2nd edn. Springer, New York (2005). ISBN: 0-387-20706-6 (hardcover); ISBN: 0-387-20733-3 (softcover)

  8. Ding, W., Liu, J., Qi, L., Yan, H.: Elasticity M-tensors and the strong ellipticity condition. Appl. Math. Comput. 373, 124982 (2020). https://doi.org/10.1016/j.amc.2019.124982

    Article  MathSciNet  Google Scholar 

  9. Fichera, G.: Existence theorems in elasticity. In: Handbuch der Physik, VIa/2, pp. 347–389. Springer, Berlin (1973). https://linkspringer.53yu.com/chapter/10.1007/978-3-662-39776-3_3

  10. Han, D., Dai, H.H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97, 1–13 (2009). https://doi.org/10.1007/s10659-009-9205-5

    Article  MathSciNet  Google Scholar 

  11. Han, D., Qi, L.: A successive approximation method for quantum separability. Front. Math. China 8(6), 1275–1293 (2013). https://doi.org/10.1007/s11464-013-0274-1

    Article  MathSciNet  Google Scholar 

  12. He, J., Li, C., Wei, Y.: M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity. Appl. Math. Lett. 102, 106137 (2020). https://doi.org/10.1016/j.aml.2019.106137

    Article  MathSciNet  Google Scholar 

  13. He, J., Liu, Y., Xu, G.: New S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor with applications. Appl. Math. Comput. 398, 125992 (2021). https://doi.org/10.1016/j.amc.2021.125992

    Article  MathSciNet  Google Scholar 

  14. He, J., Liu, Y., Xu, G.: New M-eigenvalue inclusion sets for fourth-order partially symmetric tensors with applications. Bull. Malays. Math. Sci. Soc. 44(6), 3929–3947 (2021). https://doi.org/10.1007/s40840-021-01152-5

    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). https://doi.org/10.3934/jimo.2019092

    Article  MathSciNet  Google Scholar 

  16. Jia, L.: Study on spectral radius estimation of nonnegative matrices and M-eigenvalues of elasticity tensors. Master’s thesis, Tianjin University, Tianjin (2013). https://search.cn-ki.net/doc_detail?uid=4b0d6760-4640-4ae7-9d24-fbeb585dfca3

  17. Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast. 5(3–4), 341–361 (1975). https://doi.org/10.1007/BF00126996

    Article  MathSciNet  Google Scholar 

  18. Knowles, J.K., Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal. 63(4), 321–336 (1976). https://doi.org/10.1007/BF00279991

    Article  MathSciNet  Google Scholar 

  19. Li, M., Chen, H., Zhou, G.: Fourth-order partially symmetric tensors: theory and algorithm. Pac. J. Optim. 18(1), 233–264 (2022). http://yokohamapublishers.jp/online2/oppjo/vol18/p233.html

  20. 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). https://doi.org/10.1080/03081087.2021.1885600

    Article  MathSciNet  Google Scholar 

  21. 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). https://doi.org/10.1016/j.cam.2019.01.013

    Article  MathSciNet  Google Scholar 

  22. Li, S., Li, Y.: Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors. Appl. Math. Comput. 403, 126134 (2021). https://doi.org/10.1016/j.amc.2021.126134

    Article  MathSciNet  Google Scholar 

  23. Li, S., Li, Y.: Checkable criteria for the M-positive definiteness of fourth-order partially symmetric tensors. Bull. Iran. Math. Soc. 46, 1455–1463 (2020). https://doi.org/10.1007/s41980-019-00335-y

    Article  MathSciNet  Google Scholar 

  24. Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20(3), 1286–1310 (2009). https://doi.org/10.1137/080729104

    Article  MathSciNet  Google Scholar 

  25. 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(5), 3060–3074 (2023). https://doi.org/10.3934/jimo.2022077

    Article  MathSciNet  Google Scholar 

  26. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018). ISBN: 978-981-10-8057-9 (hardcover); ISBN: 978-981-13-4050-5 (softcover); ISBN: 978-981-10-8058-6 (eBook)

  27. Qi, L., Dai, H.H., Han, D.: Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4(2), 349–364 (2009). https://doi.org/10.1007/s11464-009-0016-6

    Article  MathSciNet  Google Scholar 

  28. Rosakis, P.: Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109, 1–37 (1990). https://doi.org/10.1007/BF00377977

    Article  MathSciNet  Google Scholar 

  29. Sang, C., Zhao, J.: Criteria for the strong ellipticity condition of a partially symmetric tensor. Acta Appl. Math. 180, 10 (2022). https://doi.org/10.1007/s10440-022-00513-x

    Article  MathSciNet  Google Scholar 

  30. Wang, C., Chen, H., Wang, Y., Yan, H.: An alternating shifted inverse power method for the extremal eigenvalues of fourth-order partially symmetric tensors. Appl. Math. Lett. 141, 108601 (2023). https://doi.org/10.1016/j.aml.2023.108601

    Article  MathSciNet  Google Scholar 

  31. 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(1), 760–772 (2023). https://doi.org/10.1007/10.3934/jimo.2021205

    Article  MathSciNet  Google Scholar 

  32. 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). https://doi.org/10.1155/2020/2474278

    Article  Google Scholar 

  33. 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). https://doi.org/10.11948/20200344

    Article  MathSciNet  Google Scholar 

  34. Wang, W., Li, M., Che, H.: A tighter M-eigenvalue localization set for fourth-order partially symmetric tensors. Pac. J. Optim. 16(4), 687–698 (2020). http://www.yokohamapublishers.jp/online2/pjov16-4.html

  35. Wang, X., Che, M., Wei, Y.: Best rank-one approximation of fourth-order partially symmetric tensors by neural network. Numer. Math. Theor. Meth. Appl. 11(4), 673–700 (2018). https://doi.org/10.4208/nmtma.2018.s01

    Article  MathSciNet  Google Scholar 

  36. 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(7), 589–601 (2009). https://doi.org/10.1002/nla.633

    Article  MathSciNet  Google Scholar 

  37. Zubov, L.M., Rudev, A.N.: On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials. Z. Angew. Math. Mech. 96(9), 1096–1102 (2016). https://doi.org/10.1002/zamm.201500167

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous referees for their insightful comments and constructive suggestions, which considerably improve our manuscript.

Funding

This work is supported by Guizhou Provincial Science and Technology Projects (Grant Nos. QKHJC-ZK[2022]YB215; QKHJC-ZK[2021]YB013), and Natural Science Research Project of Department of Education of Guizhou Province (Grant Nos. QJJ[2022]015; QJJ[2022]047).

Author information

Authors and Affiliations

  1. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, People’s Republic of China

    Jianxing Zhao, Yanyan Luo & Caili Sang

Authors
  1. Jianxing Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanyan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Caili Sang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Caili Sang.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests. That is, there is no financial or non-financial interests that are directly or indirectly related to this work submitted for publication.

Additional information

Publisher's Note

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

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, J., Luo, Y. & Sang, C. A direct method for calculating M-eigenvalues of an elasticity tensor. Japan J. Indust. Appl. Math. 41, 317–357 (2024). https://doi.org/10.1007/s13160-023-00598-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-023-00598-3

Keywords

Mathematics Subject Classification

Search

Navigation