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 & Management Optimization, doi: 10.3934/jimo.2021205
References:
[1]

H. CheH. 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.  Google Scholar

[2]

S. ChiritaA. 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.  Google Scholar

[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.  Google Scholar

[4]

W. DingL. 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.  Google Scholar

[5]

W. DingJ. LiuL. 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.  Google Scholar

[6]

M. E. Gurtin, The Linear Theory of Elasticity, Handbuch der Physik, Springer, Berlin, 1972. Google Scholar

[7]

D. HanH. 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.  Google Scholar

[8]

C. Hillar and L. H. Lim, Most tensor problems are NP hard, J. ACM, 60 (2013), 1-39.  doi: 10.1145/2512329.  Google Scholar

[9]

J. HeY. 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.  Google Scholar

[10]

J. HeG. 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.  Google Scholar

[11] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.   Google Scholar
[12]

Z. HuangX. 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.  Google Scholar

[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.  Google Scholar

[14]

S. LiC. 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.  Google Scholar

[15]

C. LingJ. NieL. 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.  Google Scholar

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar

[17]

L. QiH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

G. WangL. 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.  Google Scholar

[23]

G. WangY. 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.  Google Scholar

[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.  Google Scholar

[25]

G. WangG. 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.  Google Scholar

[26]

K. WangJ. 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.  Google Scholar

[27]

Y. WangL. 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.  Google Scholar

[28]

L. ZhangL. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. CheH. 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.  Google Scholar

[2]

S. ChiritaA. 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.  Google Scholar

[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.  Google Scholar

[4]

W. DingL. 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.  Google Scholar

[5]

W. DingJ. LiuL. 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.  Google Scholar

[6]

M. E. Gurtin, The Linear Theory of Elasticity, Handbuch der Physik, Springer, Berlin, 1972. Google Scholar

[7]

D. HanH. 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.  Google Scholar

[8]

C. Hillar and L. H. Lim, Most tensor problems are NP hard, J. ACM, 60 (2013), 1-39.  doi: 10.1145/2512329.  Google Scholar

[9]

J. HeY. 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.  Google Scholar

[10]

J. HeG. 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.  Google Scholar

[11] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.   Google Scholar
[12]

Z. HuangX. 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.  Google Scholar

[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.  Google Scholar

[14]

S. LiC. 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.  Google Scholar

[15]

C. LingJ. NieL. 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.  Google Scholar

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar

[17]

L. QiH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[22]

G. WangL. 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.  Google Scholar

[23]

G. WangY. 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.  Google Scholar

[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.  Google Scholar

[25]

G. WangG. 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.  Google Scholar

[26]

K. WangJ. 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.  Google Scholar

[27]

Y. WangL. 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.  Google Scholar

[28]

L. ZhangL. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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