# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021205
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## Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors

 1 School of Management Science, Qufu Normal University, Rizhao, Shandong, China 2 School of Mathematics and Statistics, Xidian University, Xi'an, Shanxi, China

* Corresponding author: Gang Wang

Received  June 2021 Revised  September 2021 Early access November 2021

Fund Project: This work was supported by the Natural Science Foundation of Shandong Province (ZR2020MA025, ZR2019PA016), the Natural Science Foundation of China (12071250, 11801430, 11901343) and High Quality Curriculum of Postgraduate Education in Shandong Province (SDYKC20109)

In this paper, we establish sharp upper and lower bounds on the minimum M-eigenvalue via the extreme eigenvalue of the symmetric matrices extracted from elasticity Z-tensors without irreducible conditions. Based on the lower bound estimations for the minimum M-eigenvalue, we provide some checkable sufficient or necessary conditions for the strong ellipticity condition. Numerical examples are given to demonstrate the proposed results.

Citation: Chong Wang, Gang Wang, Lixia Liu. Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021205
 [1] H. Che, H. Chen and Y. Wang, On the M-eigenvalue estimation of fourth-order partially symmetric tensors, J. Ind. Manag. Optim., 16 (2020), 309-324.  doi: 10.3934/jimo.2018153. [2] S. Chirita, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7. [3] B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257. [4] W. Ding, L. Qi and Y. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038. [5] W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, Appl. Math. Comput., 373 (2020), 124982.  doi: 10.1016/j.amc.2019.124982. [6] M. E. Gurtin, The Linear Theory of Elasticity, Handbuch der Physik, Springer, Berlin, 1972. [7] D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5. [8] C. Hillar and L. H. Lim, Most tensor problems are NP hard, J. ACM, 60 (2013), 1-39.  doi: 10.1145/2512329. [9] J. He, Y. Wei and C. Li, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Appl. Math. Lett., 102 (2020), 106137.  doi: 10.1016/j.aml.2019.106137. [10] J. He, G. Xu and Y. Liu, Some inequalities for the minimum M-eigenvalue of elasticity $M$-tensors, J. Ind. Manag. Optim., 16 (2020), 3035-3045.  doi: 10.3934/jimo.2019092. [11] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994. [12] Z. Huang, X. Li and Y. Wang, Bi-block positive semidefiniteness of bi-block symmetric tensors, Front. Math. China, 16 (2021), 141-169.  doi: 10.1007/s11464-021-0874-0. [13] Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 388 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025. [14] S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, J. Comput. Appl. Math., 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013. [15] C. Ling, J. Nie, L. Qi and Y. Ye, Bi-quadratic optimization over unit spheres and semidefinite programming relaxations, SIAM J. Optim., 20 (2009), 1286-1310.  doi: 10.1137/080729104. [16] C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754. [17] L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6. [18] L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1. [19] C. Sang, A new Brauer-type Z-eigenvalue inclusion set for tensors, Numer. Algorithms, 80 (2019), 781-794.  doi: 10.1007/s11075-018-0506-2. [20] J. Walton and J. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Internat. J. Non-Linear Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X. [21] G. Wang, L. Sun and L. Liu, M-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors, Complexity, (2020), 247478. [22] G. Wang, L. Sun and X. Wang, Sharp bounds on the minimum M-eigenvalue of elasticity Z$-tensors and identifying strong ellipticity, J. Appl. Anal. Comput., 11 (2021), 2114-2130. doi: 10.11948/20200344. [23] G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear Multlinear Algebra, 68 (2020), 1817-1834. doi: 10.1080/03081087.2018.1561823. [24] G. Wang and Y. Zhang, Z-eigenvalue exclusion theorems for tensors, J. Ind. Manag. Optim., 16 (2020), 1987-1998. doi: 10.3934/jimo.2019039. [25] G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 187-198. doi: 10.3934/dcdsb.2017009. [26] K. Wang, J. Cao and H. Pei, Robust extreme learning machine in the presence of outliers by iterative reweighted algorithm, Appl. Math. Comput., 377 (2020), 125186. doi: 10.1016/j.amc.2020.125186. [27] Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601. doi: 10.1002/nla.633. [28] L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452. doi: 10.1137/130915339. [29] J. Zhao and C. Sang, New bounds for the minimum eigenvalue of M-tensors, Open Math., 15 (2017), 296-303. doi: 10.1515/math-2017-0018. [30] L. Zubov and A. Rudev, On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials, Z. Angew. Math. Mech., 96 (2016), 1096-1102. doi: 10.1002/zamm.201500167. show all references ##### References:  [1] H. Che, H. Chen and Y. Wang, On the M-eigenvalue estimation of fourth-order partially symmetric tensors, J. Ind. Manag. Optim., 16 (2020), 309-324. doi: 10.3934/jimo.2018153. [2] S. Chirita, A. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27. doi: 10.1007/s10659-006-9096-7. [3] B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 257-263. doi: 10.3934/dcdsb.2001.1.257. [4] W. Ding, L. Qi and Y. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl., 439 (2013), 3264-3278. doi: 10.1016/j.laa.2013.08.038. [5] W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, Appl. Math. Comput., 373 (2020), 124982. doi: 10.1016/j.amc.2019.124982. [6] M. E. Gurtin, The Linear Theory of Elasticity, Handbuch der Physik, Springer, Berlin, 1972. [7] D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13. doi: 10.1007/s10659-009-9205-5. [8] C. Hillar and L. H. Lim, Most tensor problems are NP hard, J. ACM, 60 (2013), 1-39. doi: 10.1145/2512329. [9] J. He, Y. Wei and C. Li, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Appl. Math. Lett., 102 (2020), 106137. doi: 10.1016/j.aml.2019.106137. [10] J. He, G. Xu and Y. Liu, Some inequalities for the minimum M-eigenvalue of elasticity$M$-tensors, J. Ind. Manag. Optim., 16 (2020), 3035-3045. doi: 10.3934/jimo.2019092. [11] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994. [12] Z. Huang, X. Li and Y. Wang, Bi-block positive semidefiniteness of bi-block symmetric tensors, Front. Math. China, 16 (2021), 141-169. doi: 10.1007/s11464-021-0874-0. [13] Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 388 (2018), 22-43. doi: 10.1016/j.cam.2018.01.025. [14] S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, J. Comput. Appl. Math., 356 (2019), 391-401. doi: 10.1016/j.cam.2019.01.013. [15] C. Ling, J. Nie, L. Qi and Y. Ye, Bi-quadratic optimization over unit spheres and semidefinite programming relaxations, SIAM J. Optim., 20 (2009), 1286-1310. doi: 10.1137/080729104. [16] C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525. doi: 10.1023/A:1020946506754. [17] L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364. doi: 10.1007/s11464-009-0016-6. [18] L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1. [19] C. Sang, A new Brauer-type Z-eigenvalue inclusion set for tensors, Numer. Algorithms, 80 (2019), 781-794. doi: 10.1007/s11075-018-0506-2. [20] J. Walton and J. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Internat. J. Non-Linear Mech., 38 (2003), 441-455. doi: 10.1016/S0020-7462(01)00066-X. [21] G. Wang, L. Sun and L. Liu, M-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors, Complexity, (2020), 247478. [22] G. Wang, L. Sun and X. Wang, Sharp bounds on the minimum M-eigenvalue of elasticity Z$-tensors and identifying strong ellipticity, J. Appl. Anal. Comput., 11 (2021), 2114-2130.  doi: 10.11948/20200344. [23] G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear Multlinear Algebra, 68 (2020), 1817-1834.  doi: 10.1080/03081087.2018.1561823. [24] G. Wang and Y. Zhang, Z-eigenvalue exclusion theorems for tensors, J. Ind. Manag. Optim., 16 (2020), 1987-1998.  doi: 10.3934/jimo.2019039. [25] G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009. [26] K. Wang, J. Cao and H. Pei, Robust extreme learning machine in the presence of outliers by iterative reweighted algorithm, Appl. Math. Comput., 377 (2020), 125186.  doi: 10.1016/j.amc.2020.125186. [27] Y. Wang, L. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633. [28] L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339. [29] J. Zhao and C. Sang, New bounds for the minimum eigenvalue of M-tensors, Open Math., 15 (2017), 296-303.  doi: 10.1515/math-2017-0018. [30] L. Zubov and A. Rudev, On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials, Z. Angew. Math. Mech., 96 (2016), 1096-1102.  doi: 10.1002/zamm.201500167.
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