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Conditions for strong ellipticity and M-eigenvalues

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Abstract

The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive definite. The elasticity tensor is rank-one positive definite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive definite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for finding M-eigenvalues are presented.

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References

  1. Cardoso J F. High-order contrasts for independent component analysis. Neural Computation, 1999, 11: 157–192

    Article  Google Scholar 

  2. Chang K C, Pearson K, Zhang T. Perron-Frobenius theorem for nonnegative tensors. Commu Math Sci, 2008, 6: 507–520

    MATH  MathSciNet  Google Scholar 

  3. Chang K C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. Journal of Mathematical Analysis and Applications, 2009, 350: 416–422

    Article  MATH  MathSciNet  Google Scholar 

  4. Basser P J, Pajevic S. Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI. Signal Processing, 2007, 87: 220–236

    Article  Google Scholar 

  5. Cox D, Little J, O’shea D. Using Algebraic Geometry. New York: Springer-Verlag, 1998

    MATH  Google Scholar 

  6. De Lathauwer L, De Moor B, Vandewalle J. On the best rank-1 and rank-(R 1,R 2,…,R N) approximation of higher-order tensor. SIAM J Matrix Anal Appl, 2000, 21: 1324–1342

    Article  MATH  MathSciNet  Google Scholar 

  7. Knowles J K, Sternberg E. On the ellipticity of the equations of non-linear elastostatics for a special material. J Elasticity, 1975, 5: 341–361

    Article  MATH  MathSciNet  Google Scholar 

  8. Knowles J K, Sternberg E. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch Ration Mech Anal, 1977, 63: 321–336

    Article  MATH  MathSciNet  Google Scholar 

  9. Kofidis E, Regalia P A. On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J Matrix Anal Appl, 2002, 23: 863–884

    Article  MATH  MathSciNet  Google Scholar 

  10. Lasserre J B. Global optimization with polynomials and the problems of moments. SIAM Journal on Optimization, 2001, 11: 796–817

    Article  MATH  MathSciNet  Google Scholar 

  11. Lim L -H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP’ 05), Vol 1. 2005, 129–132

    Google Scholar 

  12. Ling C, Nie J, Qi L, Ye Y. SDP and SOS relaxations for bi-quadratic optimization over unit spheres. Department of Applied Mathematics, The Hong Kong Polytechnic University, July 2008. Manuscript

  13. Morse PM, Feschbach H. Methods of Theoretic Physics, Vol 1. New York: McGraw-Hill, 1979, 519

    Google Scholar 

  14. Ni G, Qi L, Wang F, Wang Y. The degree of the E-characteristic polynomial of an even order tensor. J Math Anal Appl, 2007, 329: 1218–1229

    Article  MATH  MathSciNet  Google Scholar 

  15. Parrilo P A. Semidefinite programming relaxation for semialgebraic Problems. Mathematical Programming, 2003, 96: 293–320

    Article  MATH  MathSciNet  Google Scholar 

  16. Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Computation, 2005, 40: 1302–1324

    Article  MATH  Google Scholar 

  17. Qi L. Rank and eigenvalues of a supersymmetric tensor, a multivariate homogeneous polynomial and an algebraic surface defined by them. J Symbolic Computation, 2006, 41: 1309–1327

    Article  MATH  Google Scholar 

  18. Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363–1377

    Article  MATH  MathSciNet  Google Scholar 

  19. Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Frontiers of Mathematics in China, 2007, 2(4): 501–526

    Article  MATH  MathSciNet  Google Scholar 

  20. Qi L, Wang F, Wang Y. Z-eigenvalue methods for a global polynomial optimization problem. Mathematical Programming, 2009, 118: 301–316

    Article  MathSciNet  MATH  Google Scholar 

  21. Qi L, Wang Y, Wu E X. D-eigenvalues of diffusion kurtosis tensor. Journal of Computational and Applied Mathematics, 2008, 221: 150–157

    Article  MATH  MathSciNet  Google Scholar 

  22. Rosakis P. Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics. Arch Ration Mech Anal, 1990, 109: 1–37

    Article  MATH  MathSciNet  Google Scholar 

  23. Simpson H C, Spector S J. On copositive matrices and strong ellipticity for isotropic elastic materials. Arch Rational Mech Anal, 1983, 84: 55–68

    Article  MATH  MathSciNet  Google Scholar 

  24. Thomson W (Lord Kelvin). Elements of a mathematical theory of elasticity. Philos Trans R Soc, 1856, 166: 481

    Article  Google Scholar 

  25. Thomson W (Lord Kelvin). Elasticity. Encyclopedia Briannica, Vol 7. 9th Ed. London, Edingburgh: Adam and Charles Black, 1878, 796–825

    Google Scholar 

  26. Wang Y, Aron M. A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. Journal of Elasticity, 1996, 44: 89–96

    Article  MATH  MathSciNet  Google Scholar 

  27. Wang Y, Qi L, Zhang X. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numerical Linear Algebra with Applications (to appear)

  28. Zhang T, Golub G H. Rank-1 approximation of higher-order tensors. SIAM J Matrix Anal Appl, 2001, 23: 534–550

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Liqun Qi

  2. Department of Mathematics, The City University of Hong Kong, Hong Kong, China

    Hui-Hui Dai

  3. School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing, 210097, China

    Deren Han

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  1. Liqun Qi
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  2. Hui-Hui Dai
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  3. Deren Han
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Correspondence to Liqun Qi.

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

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  • DOI: https://doi.org/10.1007/s11464-009-0016-6

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