Acta Mechanica Solida Sinica

, Volume 20, Issue 1, pp 87–94 | Cite as

Stress intensity factors calculation in anti-plane fracture problem by orthogonal integral extraction method based on FEMOL

  • Yongjun Xu
  • Si Yuan
Article

Abstract

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions (or between the derivatives of eigenfunctions) of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors (SIFs), so any order SIF can be extracted based on a certain known solution of displacement (an analytic result or a numerical result). Many numerical examples based on the finite element method of lines (FEMOL) show that the present method is very powerful and efficient.

Key words

anti-plane problem Hilbert space eigenvalue eigenfunction orthogonal relationship stress intensity factor finite element method of lines 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Williams M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in tension. J. Appl. Mech., 1952, 14: 526–528.Google Scholar
  2. [2]
    Williams M. L., On the stress distribution at the base of a stationary crack. J. Appl. Mech. 1957, 24: 109–114.MathSciNetMATHGoogle Scholar
  3. [3]
    Williams M. L., Surface stress singularities resulting from various boundary conditions in angular coners of plates under bending. Proc. First U. S. Nat, Congress of Appl Mech., 1950: 325–329.Google Scholar
  4. [4]
    Hartranft R. J. and Sih G. C., The use of eigenfunction expansions in the general solution of three-dimensional crack problems. Journal of Mathematics and Mechanics, 1969, 19(2): 123–138.MathSciNetMATHGoogle Scholar
  5. [5]
    Liu Chuntu, Stress and deformation near the crack tip for bending plate. Acta Mechanica Solida Sinica, 1983, 3: 441–448 (in Chinese)Google Scholar
  6. [6]
    Gross B., Srawley J. E. and Brown W. F., Stress intensity factors for a single-edge-notch tension specimen by boundary collocation, NASA TN D-2395, 1964.Google Scholar
  7. [7]
    Kobayashi A. S., Cherepy R. B. and Kinsel W. C., A numerical procedure for estimating the stress intensity factor of a crack in a finite plate. Journal of Basic Engineering, 1964, 86: 681–684.CrossRefGoogle Scholar
  8. [8]
    Gross B. and Mendelson A., Plane elastic analysis of V-notched plates. Int. J. Fract. Mech., 1972, 8: 267–276.CrossRefGoogle Scholar
  9. [9]
    Wilson W. K., Numerical method for determining stress intensity factors of an interior crack in a finite plate. Journal of Basic Engineering, 1971: 685–690.CrossRefGoogle Scholar
  10. [10]
    Carpenter W. C., A collocation procedure for determining fracture mechanics parameters at a corner. International Journal of Fracture, 1984, 24: 255–266.CrossRefGoogle Scholar
  11. [11]
    Rzasnicki W., Mendelson A and Albers L. U., Application of boundary integral method to elastic analysis of V-notched beams. NASA TN-F-7424, 1973.Google Scholar
  12. [12]
    Rzasnicki W. and Mendelson A, Application of boundary integral method to elastoplastic analysis of V-notched beams. International Journal of Fracture, 1975, 11: 329–342.CrossRefGoogle Scholar
  13. [13]
    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(3): 359–368.Google Scholar
  14. [14]
    Mohan Lal Soni and M. Stern, On the computation of stress intensity factors in composite media using a contour integral method. Int. J. Fract., 1976, 12(3): 331–344.Google Scholar
  15. [15]
    Long Yuqiu, Zhi Bingchen, Kuang Wenqi, Shan Jian, Sub-region mixed finite element method for the calculation of stress intensity factor. Sinica Mechanica, 1982, 4: 341–353 (in Chinese).MATHGoogle Scholar
  16. [16]
    Long Yuqiu, Zhi Bingchen and Yuan Si, 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, 1982: 607–609.Google Scholar
  17. [17]
    Long Yuqiu and Zhao Yiqiang, Technical note: Calculation of stress intensity factors in plane problems by the sub-region mixed finite element method. Adv. Eng. Software, 1985, 7(1): 32–35.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Xu Yongjun, Yuan Si, Complete eigensolutions for plane notches with multi-materials by the imbedding method. International Journal of Fracture, 1996, 81: 373–381.CrossRefGoogle Scholar
  19. [19]
    Xu Yongjun, Yuan Si, Complete eigensolutions for anti-plane notches with multi-materials by super-inverse iteration. Acta Mechanica Solida Sinica, 1997, 10(2): 157–166.Google Scholar
  20. [20]
    Xu Yongjun, Yuan Si, Liu Chuntu, The progress on complete engen-solution of two dimensional notch problems. Advances in mechanics, 2000, 30(2): 216–226 (in Chinese).Google Scholar
  21. [21]
    Xu Yongjun, Yuan Si, Liu Chuntu, Possible multiple roots for fracture problems. Acta Mechanica Sinica, 1999, 31(5): 618–624 (in Chinese).Google Scholar
  22. [22]
    Xu Yongjun, Eigenproblem in fracture mechanics for reissner plate. Acta Mechanica Solida Sinica, 2004, 25(2): 225–228 (in Chinese).Google Scholar
  23. [23]
    Xu Yongjun, Liu Chuntu, The eigenvalues and eigenfunctions in shallow shell fracture analysis. Acta Mechanica Solida Sinica, 2000, 21(3): 256–260 (in Chinese).Google Scholar
  24. [24]
    Yuan Si, 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
  25. [25]
    Yuan Si and Gao Jianling, A new computational tool in structural analysis: the finite element method of lines (FEMOL)//Proceedings of International Conference on EPMESC, Macau, 3 (Aug. 1–3, 1990): 517–526.Google Scholar
  26. [26]
    Yuan Si, The finite element method of lines. Chinese Journal of Numerical Mathematics and Applications, 1993, 15(1): 45–59.MathSciNetGoogle Scholar
  27. [27]
    Yuan Si, The Finite Element Method of Lines: Theory and Applications. Beijing-New York: Science Press, 1993.Google Scholar
  28. [28]
    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
  29. [29]
    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
  30. [30]
    Yuan Si, A general-purpose FEMOL program—FEMOL92. Computational Structural Mechanics and Applications, 1993, 10(1): 118–122 (in Chinese).MathSciNetGoogle Scholar
  31. [31]
    Yuan Si, The finite element method of lines: theory and applications. Beijing-New York: Science Press, 1993.Google Scholar
  32. [32]
    Erwin Kreyszig, Introductory Functional Analysis with Application. JOHN WILEY & SONS, 1978.Google Scholar
  33. [33]
    Xanthis L. S., A pseudo-ODE modeling trick for the direct method of lines computation of important fracture mechanics parameters. ACM SIGNUM Newsletter, 1986, 21(1–2).CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Yongjun Xu
    • 1
  • Si Yuan
    • 2
  1. 1.Chinese Academy of SciencesInstitute of MechanicsBeijingChina
  2. 2.Department of Civil EngineeringTsinghua UniversityBeijingChina

Personalised recommendations