Skip to main content
SpringerLink
Log in

Numerical solutions for cracks in an elastic half-plane

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

The behavior of the stress intensity factor at the tips of cracks subjected to uniaxial tension \(\sigma ^{\infty }_{x}=p\) with traction-free boundary condition in half-plane elasticity is investigated. The problem is formulated into singular integral equations with the distribution dislocation function as unknown. In the formulation, we make used of a modified complex potential. Based on the appropriate quadrature formulas together with a suitable choice of collocation points, the singular integral equations are reduced to a system of linear equations for the unknown coefficients. Numerical examples show that the values of the stress intensity factor are influenced by the distance from the cracks to the boundary of the half-plane and the configuration of the cracks.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Panasyuk, V.V., Savruk, M.P., Datsyshyn, A.P.: A general method of solution of two-dimensional problems in the theory of cracks. Eng. Fract. Mech. 9, 481–497 (1977)

    Article  Google Scholar 

  2. Denda, M., Dong, Y.F.: Complex variable approach to the BEM for multiple crack problems. Comput. Methods Appl. Mech. Eng. 141, 247–264 (1997)

    Article  MATH  Google Scholar 

  3. Zozulya, V.V.: Regularization of hypersingular integrals in 3-D fracture mechanics: trigular BE, and piecewise-constant and piecewise-linear approximations. Eng. Anal. Bound. Elem. 34, 105–113 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Helsing, J.: A fast and stable solver for singular integral equations on piecewise smooth curved. SIAM J. Sci. Comput. 33, 153–174 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Nik Long, N.M.A., Eshkuvatov, Z.K.: Hypersingular integral equation for multiple curved cracks problem in plane elasticity. Int. J. Solids Struct. 46(13), 2611–2617 (2009)

    Article  MATH  Google Scholar 

  6. Rafar, R.A., Nik Long, N.M.A., Senu, N., et al.: Stress intensity factor for multiple inclined or curved cracks problem in circular positions in plane elasticity. ZAMM J. Appl. Math. Mech. 97, 1482–1494 (2017)

    MathSciNet  Google Scholar 

  7. Aridi, M.R., Nik Long, N.M.A., Eshkuvatov, Z.K.: Stress intensity factor for the interaction between a straight crack and a curved crack in plane elasticity. Acta Mech. Solida Sin. 29, 407–415 (2016)

    Article  Google Scholar 

  8. Xing, J.: SIF-based fracture criterion for interface cracks. Acta Mech. Sin. 32, 491–496 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Honggang, J., Yufeng, N.: Thermoelastic analysis of multiple defects with the extended finite element method. Acta Mech. Sin. 32, 1123–1137 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ji, X., Zhu, F., He, P.F.: Determination of stress intensity factor with direct stress approach using finite element analysis. Acta Mech. Sin. 33, 879–885 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, X., Ang, W.T., Fan, H.: Hypersingular integral and integro-differential micromechanical models for an imperfect interface between a thin orthotropic layer and orthotropic half-space under inplane elastostatic deformations. Eng. Anal. Bound. Elem. 52, 32–43 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Y.Z.: Various integral equations for a single crack problem of elastic half-plane. Eng. Fract. Mech. 49(6), 849–859 (1994)

    Article  Google Scholar 

  13. Li, Y.N., Hong, A.P., Bazant, Z.P.: Initiation of parallel cracks from surface of elastic half-plane. Int. J. Fract. 69, 357–369 (1995)

    Article  Google Scholar 

  14. Nik Long, N.M.A., Yaghobifar, M., Eshkuvatov, Z.K.: Stress analysis in a half plane elasticity. Int. J. Appl. Math. Stat. 20, 79–85 (2011)

    MathSciNet  Google Scholar 

  15. Kuo, C.H.: Contact stress analysis of an elastic half-plane containing multiple inclusions. Int. J. Solids Struct. 45, 4562–4573 (2008)

    Article  MATH  Google Scholar 

  16. Alexeyeva, L.A., Sarsenov, B.T.: Dynamics of elastic half-plane when resetting the vertical stress at the crack. IJPAM 107(3), 517–528 (2016)

    Google Scholar 

  17. Theocaris, P.S., Ioakimidis, N.I.: The V-notched elastic half-plane problem. Acta Mech. 32, 125–140 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, Y.Z.: Evaluation of the T-stress for multiple cracks in an elastic half-plane using singular integral equation and Green’s function method. Appl. Math. Comput. 228, 17–30 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Chen, Y.Z., Hasebe, N.: Solution of multiple-edge cracks problem of elastic half-plane by using singular integral equation approach. Commun. Numer. Methods Eng. 11, 607–617 (1995)

    Article  MATH  Google Scholar 

  20. Savruk, M.P., Kazberuk, A.: A plane periodic boundary-value problem of elasticity theory for half-plane with curvilinear edge. Mater. Sci. 44(4), 461–470 (2008)

    Article  Google Scholar 

  21. Ioakimidis, N.I., Theocaris, P.S.: A system of curvilinear cracks in an isotropic half-plane. Int. J. Fract. 15(4), 299–309 (1979)

    MathSciNet  Google Scholar 

  22. Panasyuk, V.V., Datsyshyn, O.P., Marchenko, H.P.: Contact problem for a half-plane with cracks subjected to the action of a rigid punch on its boundary. Mater. Sci. 31(6), 667–678 (1995)

    Article  Google Scholar 

  23. Panasyuk, V.V., Datsyshyn, O.P., Marchenko, H.P.: Stress state of half-plane with cracks under rigid punch action. Int. J. Fract. 101, 347–363 (2000)

    Article  Google Scholar 

  24. Monfared, M.M., Ayatollahi, M., Mousavi, S.M.: The mixed-mode analysis of a functionally graded orthotropic half-plane weakened by multiple curved cracks. Arch. Appl. Mech. 86, 713–728 (2016)

    Article  Google Scholar 

  25. Parton, V.Z., Perlin, P.I.: New Integral Equation in Elasticity. Mir, Moscow (1982)

    MATH  Google Scholar 

  26. Chen, Y.Z.: Stress intensity factors for curved and kinked cracks in plane extension. Theor. Appl. Fract. Mech. 31, 223–232 (1999)

    Article  Google Scholar 

  27. Mayrhofer, K., Fischer, F.D.: A Singular integral equation solution for the linear elastic crack opening displacement of an arbitrarily shaped plane crack: Part II regular integral solutions. FFEMS 20, 1497–1505 (1997)

    Google Scholar 

  28. Cheung, Y.K., Chen, Y.Z.: New integral equation for plane elasticity crack problems. Theor. Appl. Fract. Mech. 7, 177–184 (1987)

    Article  MathSciNet  Google Scholar 

  29. Chen, Y.Z., Cheung, Y.K.: New integral equation approach for the crack problem in elastic half-plane. Int. J. Fract. 46, 57–69 (1990)

    Article  Google Scholar 

  30. Chen, Y.Z., Lin, X.Y., Wang, X.Z.: Numerical solution for curved crack problem in elastic half-plane using hypersingular integral equation. Philos. Mag. 89(26), 2239–2253 (2009)

    Article  Google Scholar 

  31. Mogilevskaya, S.G.: Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks. Int. J. Fract. 102, 177–204 (2000)

    Article  Google Scholar 

  32. Dejoie, A., Mogilevskaya, S.G., Crouch, S.L.: A boundary integral method for multiple circular holes in an elastic half plane. Eng. Anal. Bound. Elem. 30(26), 450–464 (2006)

    Article  MATH  Google Scholar 

  33. Kratochvil, J., Becker, W.: Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite number of holes. Arch. Appl. Mech. 82, 743–754 (2012)

    Article  MATH  Google Scholar 

  34. Datsyshin, A.P., Marchenko, G.P.: An edge curvilinear crack in an elastic half-plane. Fiz. Mekh. Mat. 21(1), 67–71 (1985)

    Google Scholar 

  35. Jin, X., Keer, L.M.: Solution of multiple edge cracks in an elastic half plane. Int. J. Fract. 137, 121–137 (2006)

    Article  MATH  Google Scholar 

  36. Hejazi, A.A., Ayatollahi, M., Bagheri, R., et al.: Dislocation technique to obtain the dynamic stress intensity factors for multiple cracks in a half-plane under impact load. Arch. Appl. Mech. 84, 95–107 (2014)

    Article  MATH  Google Scholar 

  37. Hallback, M.W., Tofique, N.: Development of a distributed dislocation dipole technique for the analysis of multiple straight, kinked and branched cracks in an elastic half plane. Int. J. Solids Struct. 51, 2878–2892 (2014)

    Article  Google Scholar 

  38. Erdogan, F., Gupt, G.D., Cook, T.S.: Numerical solution of singular integral equations. In: Sih, G.C. (ed.) Mechanics of Fracture, vol. 1, pp. 368–425. Leyden Noordhoff, Berlin (1973)

    Google Scholar 

  39. Muskhelishvili, N.I.: Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff International Publishing, Leyden (1953)

    MATH  Google Scholar 

  40. Chen, Y.Z., Hasebe, N., Lee, K.Y.: Multiple Crack Problems in Elasticity. WIT Press, Southampton (2003)

    MATH  Google Scholar 

  41. Lam, K.Y., Phua, S.P.: Multiple crack interaction and its effect on stress intensity factor. Eng. Fract. Mech. 40, 585–592 (1991)

    Article  Google Scholar 

  42. Sih, G.C.: Boundary problems for longitudinal shear cracks. In: Proceedings of 2nd conference on theoretical and applied mechanics, 2, 117–130 (1964)

  43. Elfakhakhre, N.R.F., Nik Long, N.M.A., Eshkuvatov, Z.K.: Stress intensity factor for an elastic half plane weakened by multiple curved cracks. Appl. Math. Model. 60, 540–551 (2018)

    Article  MathSciNet  Google Scholar 

  44. Chen, Y.Z.: Solution of integral equation in curve crack problem by using curve length coordinate. Eng. Anal. Bound. Elem. 28, 989–994 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen, Y.Z.: Singular integral equation method for the solution of multiple curved crack problems. Int. J. Solids Struct. 41, 3505–3519 (2004)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank University Putra Malaysia for Putra Grant (9442300).

Author information

Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Universiti Putra Malaysia, 43400, Serdang, Selangor, Malaysia

    N. R. F. Elfakhakhre & N. M. A. Nik Long

  2. Institute for Mathematical Research, Universiti Putra Malaysia, 43400, Serdang, Selangor, Malaysia

    N. M. A. Nik Long & Z. K. Eshkuvatov

  3. Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800, Nilai, Negeri Sembilan, Malaysia

    Z. K. Eshkuvatov

Authors
  1. N. R. F. Elfakhakhre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. M. A. Nik Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. K. Eshkuvatov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. M. A. Nik Long.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elfakhakhre, N.R.F., Nik Long, N.M.A. & Eshkuvatov, Z.K. Numerical solutions for cracks in an elastic half-plane. Acta Mech. Sin. 35, 212–227 (2019). https://doi.org/10.1007/s10409-018-0803-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-018-0803-y

Keywords

Search

Navigation