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C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices
1. | School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
C-eigenvalues of piezoelectric-type tensors play an crucial role in piezoelectric effect and converse piezoelectric effect. In this paper, by the partial symmetry property of piezoelectric-type tensors, we present three intervals to locate all C-eigenvalues of a given piezoelectric-type tensor. Numerical examples show that our results are better than the existing ones.
References:
[1] |
H. T. Che, H. B. Chen and Y. J. Wang,
C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.
doi: 10.1016/j.aml.2018.09.014. |
[2] |
Y. Chen, A. Jákli and L. Qi, Spectral analysis of piezoelectric tensors, preprint, arXiv: 1703.07937v1. Google Scholar |
[3] |
Y. N. Chen, L. Q. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20 pp.
doi: 10.1088/1751-8121/aa98a8. |
[4] |
J. Curie and P. Curie,
Développement, par compression de l'éctricité polaire dans les cristaux hémièdres à faces inclinées, Bulletin de Minéralogie, 3, 4 (1880), 90-93.
doi: 10.3406/bulmi.1880.1564. |
[5] |
M. De Jong, W. Chen, H. Geerlings, M. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Scientific Data, 2 (2015), 150053.
doi: 10.1038/sdata.2015.53. |
[6] |
S. Haussl, Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag, Weinheim, 2007.
doi: 10.1002/9783527621156. |
[7] |
A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008. Google Scholar |
[8] |
C. Q. Li, Y. J. Liu and Y. T. Li,
C-eigenvalues intervals for piezoelectric-type tensors, Applied Mathematics and Computation, 358 (2019), 244-250.
doi: 10.1016/j.amc.2019.04.036. |
[9] |
D. Lovett, Tensor Properties of Crystals, 2$^{nd}$ edition, Institute of Physics Publishing, Bristol, 1989. Google Scholar |
[10] |
J. F. Nye, Physical properties of crystals: Their representation by tensors and matrices, Physics Today, 10 (1957), 26 pp.
doi: 10.1063/1.3060200. |
[11] |
L. Qi, Transposes, L-eigenvalues and invariants of third order tensors, preprint, (2017), arXiv: 1704.01327. Google Scholar |
[12] |
L. Q. Qi and Z. Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017.
doi: 10.1137/1.9781611974751.ch1. |
[13] |
W. J. Wang, H. B. Chen and Y. J. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035, 6 pp.
doi: 10.1016/j.aml.2019.106035. |
[14] |
W.-N. Zou, C.-X. Tang and E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469 (2013), 20120755.
doi: 10.1098/rspa.2012.0755. |
show all references
References:
[1] |
H. T. Che, H. B. Chen and Y. J. Wang,
C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.
doi: 10.1016/j.aml.2018.09.014. |
[2] |
Y. Chen, A. Jákli and L. Qi, Spectral analysis of piezoelectric tensors, preprint, arXiv: 1703.07937v1. Google Scholar |
[3] |
Y. N. Chen, L. Q. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20 pp.
doi: 10.1088/1751-8121/aa98a8. |
[4] |
J. Curie and P. Curie,
Développement, par compression de l'éctricité polaire dans les cristaux hémièdres à faces inclinées, Bulletin de Minéralogie, 3, 4 (1880), 90-93.
doi: 10.3406/bulmi.1880.1564. |
[5] |
M. De Jong, W. Chen, H. Geerlings, M. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Scientific Data, 2 (2015), 150053.
doi: 10.1038/sdata.2015.53. |
[6] |
S. Haussl, Physical Properties of Crystals: An Introduction, Wiley-VCH Verlag, Weinheim, 2007.
doi: 10.1002/9783527621156. |
[7] |
A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008. Google Scholar |
[8] |
C. Q. Li, Y. J. Liu and Y. T. Li,
C-eigenvalues intervals for piezoelectric-type tensors, Applied Mathematics and Computation, 358 (2019), 244-250.
doi: 10.1016/j.amc.2019.04.036. |
[9] |
D. Lovett, Tensor Properties of Crystals, 2$^{nd}$ edition, Institute of Physics Publishing, Bristol, 1989. Google Scholar |
[10] |
J. F. Nye, Physical properties of crystals: Their representation by tensors and matrices, Physics Today, 10 (1957), 26 pp.
doi: 10.1063/1.3060200. |
[11] |
L. Qi, Transposes, L-eigenvalues and invariants of third order tensors, preprint, (2017), arXiv: 1704.01327. Google Scholar |
[12] |
L. Q. Qi and Z. Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017.
doi: 10.1137/1.9781611974751.ch1. |
[13] |
W. J. Wang, H. B. Chen and Y. J. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035, 6 pp.
doi: 10.1016/j.aml.2019.106035. |
[14] |
W.-N. Zou, C.-X. Tang and E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469 (2013), 20120755.
doi: 10.1098/rspa.2012.0755. |
4.2514 | 0.1375 | 2.6258 | 12.4234 | 11.6674 | 7.7376 | 13.5021 | 27.4628 | |
7.3636 | 0.2834 | 5.6606 | 23.5377 | 16.8548 | 12.3206 | 20.2351 | 27.5396 | |
7.3636 | 0.2834 | 5.6606 | 23.5377 | 16.8548 | 12.3206 | 20.2351 | 27.5396 | |
7.3636 | 0.2744 | 4.8058 | 23.5377 | 16.5640 | 11.0127 | 18.8793 | 27.5109 | |
7.3636 | 0.2834 | 4.7861 | 23.5377 | 16.8464 | 11.0038 | 19.8830 | 27.5013 | |
7.3636 | 0.2737 | 3.3543 | 21.9667 | 16.0233 | 9.4595 | 16.7483 | 27.5012 | |
7.3636 | 0.2393 | 4.6717 | 22.7163 | 14.5723 | 12.1694 | 18.7025 | 27.5396 | |
6.3771 | 0.1943 | 3.7242 | 16.0259 | 11.9319 | 7.7540 | 13.5113 | 27.4629 | |
5.2069 | 0.1938 | 3.7025 | 14.9344 | 11.9319 | 7.7523 | 13.5054 | 27.4629 |
4.2514 | 0.1375 | 2.6258 | 12.4234 | 11.6674 | 7.7376 | 13.5021 | 27.4628 | |
7.3636 | 0.2834 | 5.6606 | 23.5377 | 16.8548 | 12.3206 | 20.2351 | 27.5396 | |
7.3636 | 0.2834 | 5.6606 | 23.5377 | 16.8548 | 12.3206 | 20.2351 | 27.5396 | |
7.3636 | 0.2744 | 4.8058 | 23.5377 | 16.5640 | 11.0127 | 18.8793 | 27.5109 | |
7.3636 | 0.2834 | 4.7861 | 23.5377 | 16.8464 | 11.0038 | 19.8830 | 27.5013 | |
7.3636 | 0.2737 | 3.3543 | 21.9667 | 16.0233 | 9.4595 | 16.7483 | 27.5012 | |
7.3636 | 0.2393 | 4.6717 | 22.7163 | 14.5723 | 12.1694 | 18.7025 | 27.5396 | |
6.3771 | 0.1943 | 3.7242 | 16.0259 | 11.9319 | 7.7540 | 13.5113 | 27.4629 | |
5.2069 | 0.1938 | 3.7025 | 14.9344 | 11.9319 | 7.7523 | 13.5054 | 27.4629 |
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