Abstract
Based on the matrix unfolding technique of a tensor, three easily checkable sufficient conditions for the M-positive definiteness of fourth-order partially symmetric tensors are given. Numerical examples show that the proposed results are efficient.
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References
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)
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)
Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2. Springer, Heidelberg, New York, Berlin (1972)
Aron, M.: On the role of the strong ellipticity condition in nonlinear elasticity. Int. J. Eng. Sci. 21(11), 1359–1367 (1983)
Knowles, J.: On the representation of the elasticity tensor for isotropic materials. J. Elast. 39(2), 175–180 (1995)
Itskov, M.: On the theory of fourth-order tensors and their applications in computational mechanics. Comput. Methods Appl. Mech. Eng. 189(2), 419–438 (2000)
Walton, J.R., Wilber, J.P.: Sufficient conditions for strong ellipticity for a class of anisotropic materials. Int. J. Nonlinear Mech. 38(4), 411–455 (2003)
Chirita, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87(1), 1–27 (2007)
Han, D.R., Dai, H.H., Qi, L.Q.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97(1), 1–13 (2009)
Qi, L.Q., Dai, H.H., Han, D.R.: Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4(2), 349–364 (2009)
Horn, R.A., Johnson, C.R.: Matrix analysis, 1st edn. Cambridge University Press, Cambridge (2005)
Varga, R.S.: Gers̆gorin and His Circles. Springer, Berlin, Germany (2004)
Cvetković, L.: H-matrix theory vs. eigenvalue localization. Numerical Algorithms 42(3–4), 229–245 (2006)
Acknowledgements
Suhua Li’s work is supported in part by Yunnan University’s Research Innovation Fund for Graduate Students(Grant number 2018Z057); Yunnan Provincial Doctoral Graduate Academic Newcomer Award; China Scholarship Council (Grant number 201807030004).
Yaotang Li’s work is supported by National Natural Science Foundations of China (Grant number 11861077).
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Communicated by Abbas Salemi.
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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
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DOI: https://doi.org/10.1007/s41980-019-00335-y