Advertisement

Acta Mechanica Solida Sinica

, Volume 20, Issue 2, pp 149–162 | Cite as

Computation of stress intensity factors by the sub-region mixed finite element method of lines

  • Si Yuan
  • Yongjun Xu
  • F. W. Williams
Article

Abstract

Based on the sub-region generalized variational principle, a sub-region mixed version of the newly-developed semi-analytical ‘finite element method of lines’ (FEMOL) is proposed in this paper for accurate and efficient computation of stress intensity factors (SIFs) of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations (ODEs) and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.

Key words

stress intensity factors finite element method of lines sub-region generalized variational principle ordinary differential equation solver 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Yuan, S., A new semi-discrete method-the finite element method of lines//Proceedings of 1st National Conference on Analytical and Numerical Combined Methods, Hunan, 1990, 132–136 (in Chinese).Google Scholar
  2. [2]
    Yuan, S. and Gao, J.L., A new computational tool in structural analysis: the finite element method of lines (FEMOL)//Proceedings of International Conference on EPMESC, Macau, 1990, 3(1–3): 517–526.Google Scholar
  3. [3]
    Yuan, S., The finite element method of lines. Chinese Journal of Numerical Mathematics and Applications, 1993, 15(1): 45–59.MathSciNetGoogle Scholar
  4. [4]
    Yuan, S., The Finite Element Method of Lines: Theory and Applications. Beijing-New York: Science Press, 1993.Google Scholar
  5. [5]
    Ascher, U., Christiansen, J. and Russell, R.D., Collocation software for boundary-value ODEs. ACM Transaction of Mathematical Software, 1981, 7(2): 209–222.CrossRefGoogle Scholar
  6. [6]
    Ascher, U., Christiansen, J. and Russell, R.D., Algorithm 569, COLSYS: collocation software for boundary-value ODEs [D2]. ACM Transaction of Mathematical Software, 1981, 7(2): 223–229.CrossRefGoogle Scholar
  7. [7]
    Yuan, S., A general-purpose FEMOL program-FEMOL92. Computational Structural Mechanics and Applications, 1993, 10(1): 118–122 (in Chinese).MathSciNetGoogle Scholar
  8. [8]
    Yuan, S., The ‘triangular’ elements in the finite element method of lines. Science in China, Series A, 1993, 23(5): 552–560 (in Chinese).Google Scholar
  9. [9]
    Xu, Y.J. and Yuan, S., Complete eigen-solutions for plane notches with multi-materials by the imbedding method. International Journal of Fracture, 1996, 81(4): 373–381.CrossRefGoogle Scholar
  10. [10]
    Xu, Y.J. and Yuan, S., Complete eigensolutions for anti-plane notches with multi-materials by super-inverse iteration. Acta Mechanica Solida Sinica, 1997, 10(2): 157–166.Google Scholar
  11. [11]
    Xu, Y.J., Yuan, S. and Liu, C.T., The progress on complete engen-solution of two dimensional notch problems. Advances in Mechanics, 2000, 30(2): 216–226 (in Chinese).Google Scholar
  12. [12]
    Xu, Y.J., Yuan, S. and Liu, C.T., Possible multiple roots for fracture problems. Acta Mechanica Sinica, 1999, 31(5): 618–624 (in Chinese).Google Scholar
  13. [13]
    Xu, Y.J., Eigenproblem in Fracture Mechanics for Reissner plate. Acta Mechanica Solida Sinica, 2004, 25(2): 225–228 (in Chinese).Google Scholar
  14. [14]
    Xu, Y.J. and Liu, C.T., The eigenvalues and eigenfunctions in shallow shell fracture analysis. Acta Mechanica Solida Sinica, 2000, 21(3): 256–260 (in Chinese).Google Scholar
  15. [15]
    Long, Y.Q., Sub-region generalized variational principles in elasticity. Shanghai Journal of Mechanics, 1981, 2(2): 1–9 (in Chinese).Google Scholar
  16. [16]
    Long, Y.Q., Zhi, B.C. and Yuan, S., Sub-region, sub-item and sub-layer generalized variational principles in elasticity. Proceedings of International Conference on Finite Element Methods, (ed. He Guangqian and Y. K. Cheung), Shanghai, China 1982, pp607–609.Google Scholar
  17. [17]
    Long, Y.Q. Zhi, B.C., Kuang, W.Q. and Shan, J., Sub-region mixed finite element analysis of stress intensity factors. Sinica Mechanica, 1982, 4: 341–353.zbMATHGoogle Scholar
  18. [18]
    Long, Y.Q. and Zhao, Y.Q., Calculation of stress intensity factors in plane problems by the sub-region mixed finite element method. Advances in Engineering Software, 1985, 7(1): 32–35.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Fan, Z. and Long, Y.Q., Sub-region mixed finite element analysis of V-notched plates. International Journal of Fracture, 1992, 56: 333–344.CrossRefGoogle Scholar
  20. [20]
    Xu, Y.J. and Yuan, S., Stress intensity factors calculation in anti-plane fracture problem by orthogonal integral extraction method based on FEMOL. Acta Mechanica Solida Sinica, 2007, 20(1): 87–94.CrossRefGoogle Scholar
  21. [21]
    Stern, M., Becker, E.B. and Dunham, R.S., A contour integral computation of mixed-mode stress intensity factors. International Journal of Fracture, 1976, 12: 359–368.Google Scholar
  22. [22]
    Gross, B. and Mendelson, A., Plane elastic analysis of V-notched plates. International Journal of Fracture Mechanics, 1972, 8: 267–276.CrossRefGoogle Scholar
  23. [23]
    Long, Y.Q., Introduction of New Finite Element Methods. Beijing: Tsinghua University Press, 1992.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2007

Authors and Affiliations

  • Si Yuan
    • 1
  • Yongjun Xu
    • 2
  • F. W. Williams
    • 3
  1. 1.Department of Civil EngineeringTsinghua UniversityBeijingChina
  2. 2.Chinese Academy of SciencesInstitute of MechanicsBeijingChina
  3. 3.Structural Engineering DivisionCardiff UniversityCardiffUK

Personalised recommendations