Inspired by Cardano's method for solving cubic scalar equations, the additive decomposition of spherical/deviatoric tensor (DSDT) is revisited from a new viewpoint. This decomposition simplifies the cubic tensor equation, decouples the spherical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.
Haoyuan GUO, Liyuan ZHANG, YajunYIN, Yongxin GAO
. Relations between cubic equation, stress tensor decomposition, and von Mises yield criterion[J]. Applied Mathematics and Mechanics, 2015
, 36(10)
: 1359
-1370
.
DOI: 10.1007/s10483-015-1988-9
[1] Hill, R. The Mathematical Theory of Plasticity, Oxford University Press, New York, 15-23(1998)
[2] Xu, B. Y. and Liu X. S. Applied Elastic-Plastic Mechanics (in Chinese), Tsinghua University Press, Beijin, 88-97(1995)
[3] Han, W. and Reddy, B. D. Plasticity:Mathematical Theory and Numerical Analysis, Springer, New York, 61-62(2012)
[4] Huang, K. Z., Xue, M. D., and Lu, M. W. Tensor Analysis (in Chinese), Tsinghua University Press, Beijing, 101-102(2003)
[5] Lu, M. W. and Luo, X. F. Basis of Elastic Theory (in Chinese), Tsinghua University Press, Beijing, 226-228(2001)
[6] Fung, Y. C. Foundations of Solid Mechanics, Prentice-Hall, New Jersey, 80-81(1965)
[7] Fung, Y. C. A First Course in Continuum Mechanics, Prentice-Hall, New Jersey, 99-101(1994)
[8] Karasudhi, P. Foundations of Solid Mechanics, Kluwer Academic Publishers, The Netherlands, 59-61(1991)
[9] Asaro, R. and Lubarda, V. Mechanics of Solids and Materials, Cambridge University Press, New York, 463-465(2006)
[10] Guilbeau, L. The history of the solution of the cubic equation. Mathematics News Letter, 5, 8-12(1930)
[11] Nickalls, R. W. D. A new approach to solving the cubic:Cardan's solution revealed. The Mathematical Gazette, 77, 354-359(1993)
[12] Nahin, P. J. An imaginary tale:the story of √-1. Notices of the AMS, 46, 1233-1236(1998)
[13] Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra, Westview Press, New York, 27-30(1969)
[14] Friedberg, S. H. and Insel, A. J. Linear Algebra, Prentice Hall, New Jersey, 258-259(2003)
[15] Bhatia, R. Matrix Analysis, Springer-Verlag, New York, 7-9(1997)
Email Alert
RSS