## 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.

### Similar content being viewed by others

## References

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)Denda, M., Dong, Y.F.: Complex variable approach to the BEM for multiple crack problems. Comput. Methods Appl. Mech. Eng.

**141**, 247–264 (1997)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)Helsing, J.: A fast and stable solver for singular integral equations on piecewise smooth curved. SIAM J. Sci. Comput.

**33**, 153–174 (2011)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)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)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)Xing, J.: SIF-based fracture criterion for interface cracks. Acta Mech. Sin.

**32**, 491–496 (2016)Honggang, J., Yufeng, N.: Thermoelastic analysis of multiple defects with the extended finite element method. Acta Mech. Sin.

**32**, 1123–1137 (2016)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)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)Chen, Y.Z.: Various integral equations for a single crack problem of elastic half-plane. Eng. Fract. Mech.

**49**(6), 849–859 (1994)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)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)Kuo, C.H.: Contact stress analysis of an elastic half-plane containing multiple inclusions. Int. J. Solids Struct.

**45**, 4562–4573 (2008)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)Theocaris, P.S., Ioakimidis, N.I.: The V-notched elastic half-plane problem. Acta Mech.

**32**, 125–140 (1979)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)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)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)Ioakimidis, N.I., Theocaris, P.S.: A system of curvilinear cracks in an isotropic half-plane. Int. J. Fract.

**15**(4), 299–309 (1979)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)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)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)Parton, V.Z., Perlin, P.I.: New Integral Equation in Elasticity. Mir, Moscow (1982)

Chen, Y.Z.: Stress intensity factors for curved and kinked cracks in plane extension. Theor. Appl. Fract. Mech.

**31**, 223–232 (1999)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)Cheung, Y.K., Chen, Y.Z.: New integral equation for plane elasticity crack problems. Theor. Appl. Fract. Mech.

**7**, 177–184 (1987)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)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)Mogilevskaya, S.G.: Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks. Int. J. Fract.

**102**, 177–204 (2000)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)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)Datsyshin, A.P., Marchenko, G.P.: An edge curvilinear crack in an elastic half-plane. Fiz. Mekh. Mat.

**21**(1), 67–71 (1985)Jin, X., Keer, L.M.: Solution of multiple edge cracks in an elastic half plane. Int. J. Fract.

**137**, 121–137 (2006)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)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)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)

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

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

Lam, K.Y., Phua, S.P.: Multiple crack interaction and its effect on stress intensity factor. Eng. Fract. Mech.

**40**, 585–592 (1991)Sih, G.C.: Boundary problems for longitudinal shear cracks. In: Proceedings of 2nd conference on theoretical and applied mechanics,

**2**, 117–130 (1964)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)Chen, Y.Z.: Solution of integral equation in curve crack problem by using curve length coordinate. Eng. Anal. Bound. Elem.

**28**, 989–994 (2004)Chen, Y.Z.: Singular integral equation method for the solution of multiple curved crack problems. Int. J. Solids Struct.

**41**, 3505–3519 (2004)

## Acknowledgements

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

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### 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

Received:

Revised:

Accepted:

Published:

Issue Date:

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