Skip to main content
SpringerLink
Log in

The equivalent through-crack model for the surface part-through crack

  • Contributed Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

The three-dimensional problem of a surface part-through cracked plate is reduced to a two-dimensional problem by introducing an equivalent through-crack model, loaded by a convenient tension and bending stress distribution along the lips of the crack. The principle of the model is based on a similar idea as in theline-spring model, introduced by Rice and Levy [1]. In this model the singularities existing at the extremities of the surface lips of the part-through crack on the near face of the plate, indicated by experiments, were taken into account, whereas, in the line-spring model these extremities were considered as free. According to this model the effect of the surface part-through crack was replaced by a continuous distribution of forcesN(x, 0) and momentsM(x, 0) along the near-face crack lips applied to an equivalent through crack. These forces and moments were represented by power functions completely determining the boundary conditions along the crack. In this way the two-dimensional problem, derived from the model, can be readily reduced to a well-known Hilbert problem yielding integral equations containing the functions expressing theN(x, 0)- andM(x, 0)-distributions along the length of the crack. These equations were solved by an approximate numerical procedure. From this solution the expressions of stress- and displacement-fields for the surface part-through crack may be evaluated, expressed in terms of the distribution of the stress intensity factor function calculated along the whole front of the equivalent through crack.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Rice, J. R., Levy, N.: The part-through surface crack in an elastic plate. Jnl. Appl. Mech., Trans. ASME39, 2, 185–194 (1972).

    Google Scholar 

  2. Irwin, G. R.: Crack extension force for a part-through crack in a plate. Jnl. Appl. Mech., Trans. ASME33, 4, 651–654 (1966).

    Google Scholar 

  3. Green, A. E., Sneddon, I. N.: The distribution of stress in the neighbourhood of a flat elliptical crack in an elastic solid. Proc. Cambridge Phil. Soc.46, 159–163 (1960).

    Google Scholar 

  4. Smith, F. W.: The elastic analysis of the part-circular surface flow problem by the alternating method, in: The surface crack: Physical problems and computational solutions (Swedlow, J. L., editor) ASME Publ., New York, pp. 125–152 (1972).

    Google Scholar 

  5. Smith, F. W., Alavi, M. J.: Stress intensity factors for a penny-shaped crack in a half-space. Engrg. Fract. Mech.3, 3, 241–254 (1971).

    Google Scholar 

  6. Smith, F. W., Alavi, M. J.: Stress intensity factors for a part-circular surface flaw. Proc. First Intern. Conf. Pressure Vessel Technology, ASME, New York, pp. 793–800 (1969).

  7. Thresher, R. W., Smith, F. W.: Stress intensity factors for a surface crack in a finite solid. Jnl. Appl. Mech., Trans. ASME,39, 1, 195–200 (1972).

    Google Scholar 

  8. Shah, R. C., Kobayashi, A. S.: Stress intensity factor for elliptical crack under arbitrary normal loading. Engrg. Fract. Mech.3, 1, 71–96 (1971).

    Google Scholar 

  9. Shah, R. C., Kobayashi, A. S.: Stress intensity factors for an elliptical crack approaching the surface of a semi-infinite solid. Intern. Jnl. Fracture9, 2, 133–146 (1973).

    Google Scholar 

  10. Shah, R. C., Kobayashi, A. S.: On the surface flaw problem, in: The surface crack: Physical problems and computational solutions (Swedlow, J. L., editor) ASME Publ., New York, pp. 79–124 (1972).

    Google Scholar 

  11. Hartranft, R. J., Sih, G. C.: Alternating method applied to edge and surface crack problems. Methods of analysis and solutions, of crack problems (Sih, G. C., editor) pp. 179–238, The Netherlands: Noordhoff Intern. Publ. 1973.

    Google Scholar 

  12. Segedin, C. M.: A note on geometric discontinuities in elastostatics. Intern. Jnl. Engrg. Sci.6, 309–312 (1968).

    Google Scholar 

  13. Miyamoto, H., Miyoshi, T.: Analysis of stress intensity factor for surface-flawed tension plate. High-speed computing of elastic structures, Proc. of IUTAM Symposium, Univ. of Liege, pp. 137–155, 1971.

  14. Heliot, J., Labbens, R. C., Pellisier-Tanon, A.: Semi-elliptic cracks in a cylinder subjected to stress gradients. Fracture mechanics, ASTM Spec. Techn. Publ.677, 341–364 (1979).

    Google Scholar 

  15. Cruse, T. A.: Numerical solutional in three-dimensional elastostatics. Intern. Jnl. Sol. Struct.5, 1259–1274 (1969).

    Google Scholar 

  16. Cruse, T. A.: Numerical evaluation of elastic stress intensity factors by the boundary integral equation method. The surface crack: Physical problems and computational solutions (Swedlow, J. L., editor), ASME Publ., New York, pp. 153–170 (1972).

    Google Scholar 

  17. Tada, H., Paris, P. C., Irwin, G. R.: The stress analysis of cracks. Handbook, Hellertown, Pennsylvania: Del Res. Corp. 1973.

    Google Scholar 

  18. Delale, F., Erdogan, F.: Application of the line-spring model to a cylindrical shell containing a circumferential or axial part-through crack. Jnl. Appl. Mech., Trans. ASME49 [1], 97–102 (1982).

    Google Scholar 

  19. Parks, D. M.: Inelastic analysis of surface flaws using the line-spring model. Proc. Fifth Intern. Conf. on Fracture, Cannes, France (François, D., editor). Pergamon Press London Publ.5, 2589–2598 (1981).

  20. Benthem, J. P.: State of stress at the vertex of a quarter-infinite crack in a half-space. Int. J. Solids and Structures13, 470–492 (1977); see also: Bazant, Z. P., Estenssoro, L. F.: General numerical method for three three-dimensional sigularities in cracked or notched elastic solids, ICF 4, Waterloo, Canada, June 19–24. Fracture3, 371–385 (1979).

    Google Scholar 

  21. Francis, P. H., Davidson, D. L.: Experimental characterization of yield induced by surface flaws. The surface crack: Physical problems and computational solutions (Swedlow, J. R., editor), ASME Publ., New York, pp. 63–67 (1972).

    Google Scholar 

  22. Paris, P. C., Sih, G. G.: Stress analysis of crack, fracture toughness testing and it applications. ASTM STP.381, 30–83 (1964).

    Google Scholar 

  23. Smith, C. W., Peters, W. H., Kirby, G. C.: Crack-tip measurements in photoelastic models. Experimental Mechanics22, 448–451 (1982).

    Google Scholar 

  24. Theocaris, P. S.: The method of reflected caustics for the study of part-through circular cracks. Proc. Nat. Acad. Athens, Vol. X, No. I, 210–215 (1983).

    Google Scholar 

  25. Muskhelishvili, N. I.: Some basic problems of the mathematical theory of elasticity. Groningen: Noordhoff 1953.

    Google Scholar 

  26. Berger, G., Keller, H. B., Munz, D.: Determination of fracture toughness with linear elastic and elastic-plastic methods, elastic-plastic fracture, ASTM STP 668 (Landes, J. D., Begley, J. A., Clarke, G. A., eds.). Amer. Soc. Test. Mat. Publ. pp. 378–405, 1979.

  27. Knott, J. F.: Macroscopic aspects of crack extensions. Advances in elasto-plastic fracture mechanics (Larsson, L. H., ed.). Applied Science Publ., London, pp. 1–20, 1980.

    Google Scholar 

  28. Nishioka, T., Atluri, S. N.: Analytical solution for embedded elliptical cracks and finite-element alternating method for elliptical surface cracks subject to arbitrary loadings. Engrg. Fract. Mech.17 [3], 247–268 (1983).

    Google Scholar 

  29. Vajiyakumar, K., Atluri, S. N.: An embedded elliptical flaw in an infinite solid subject to arbitrary crack-face tractions. Jnl. Appl. Mech., Trans. ASME48, 88–96 (1981).

    Google Scholar 

  30. Delale, F., Erdogan, F.: Line spring model for surface cracks in a Reissner plate. Intern. Jnl. Engrg. Sci.19, 1331–1340 (1981).

    Google Scholar 

  31. Newman Jr., J. C., Raju, I. S.: Analysis of surface cracks in finite plate under tension or bending loads. NASA Technical Note No. 1578, 1979.

  32. Hartranft, R. I., Sih, G. C.: An approximate three-dimensional theory of plates with application to crack problems. Intern. Jnl. Engrg. Sci.8, 711–729 (1970).

    Google Scholar 

  33. Sih, G. C.: A review of the three-dimensional stress problem for a cracked plate. Intern. Jnl. Fract. Mech.7, 39–61 (1971).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Athens National Technical University, P. O. Box 77230, GR-17 510, Pal-Phaleron, Greece

    P. S. Theocaris (Fellow of the Academy, Professor) & D. L. Wu (Visiting Research Associate)

  2. Chinese Academy of Space Technology, Beijing, China

    D. L. Wu (Visiting Research Associate)

Authors
  1. P. S. Theocaris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. L. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

With 7 Figures

Rights and permissions

Reprints and permissions

About this article

Cite this article

Theocaris, P.S., Wu, D.L. The equivalent through-crack model for the surface part-through crack. Acta Mechanica 59, 157–181 (1986). https://doi.org/10.1007/BF01181662

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01181662

Keywords

Search

Navigation