Abstract
The Erdogan fundamental solutions are derived from an infinite plane containing a crack. When they are used in the formulation of the boundary element method (BEM), the stress boundary conditions on the crack surface are automatically satisfied and the singular behavior at the crack tip can be naturally reflected. Using the multi-domain technique, the multi-crack problem can be transformed into a series of single-crack problems involving displacement continuity conditions along common boundaries. In this paper, the displacements which are expressed in terms of the integral of a complex function in the Erdogan fundamental solutions are derived in closed-form expressions. Then, the multi-domain spline fictitious boundary element method (SFBEM) based on the above fundamental solutions is proposed and formulated for analyzing multi-crack problems. The computational accuracy and stability of the proposed method are verified by comparing the stress intensity factor (SIF) results of a double-inner crack problem with different inclined angles and crack lengths against those calculated by the finite element method. Also, the SIF results of a double-edge crack problem with different crack lengths are compared with those obtained from studies. Finally, the proposed method is applied to the analysis of the triple-crack problem, in which the shielding effects of multi-crack and stress contours are studied with different crack lengths and locations.
Similar content being viewed by others
References
Fischer, G., Li, V.C.: Influence of matrix ductility on tension-stiffening behavior of steel reinforced engineered cementitious composites (ECC). ACI Struct. J. 99, 104–111 (2002)
Park, S., Ahmad, S., Yun, C.B., Roh, Y.: Multiple crack detection of concrete structures using impedance-based structural health monitoring techniques. Exp. Mech. 46, 609–618 (2006)
Li, S., Song, G., Kwakernaak, K., van der Zwaag, S., Sloof, W.G.: Multiple crack healing of a Ti 2 AlC ceramic. J. Eur. Ceram. Soc. 32, 1813–1820 (2012)
Astiz, M.: An incompatible singular elastic element for two-and three-dimensional crack problems. Int. J. Fract. 31, 105–124 (1986)
Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)
Banerjee, P.K., Butterfield, R.: Boundary Element Methods in Engineering Science, vol. 17. McGraw-Hill, London (1981)
Zhang, Y., Feng, W.: Investigation in weighted function and optimization of isoparametric singular boundary elements’ size in 3-D crack problem. Eng. Fract. Mech. 26, 611–617 (1987)
Luo, G., Zhang, Y.: Application of boundary element method with singular and isoparametric elements in three dimensional crack problems. Eng. Fract. Mech. 29, 97–106 (1988)
Cruse, T.A.: Boundary Element Analysis in Computational Fracture Mechanics, vol. 1. Springer Science & Business Media, New York (2012)
Crouch, S.: Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution. Int. J. Numer. Methods Eng. 10, 301–343 (1976)
Dong, C., De Pater, C.: Numerical implementation of displacement discontinuity method and its application in hydraulic fracturing. Comput. Method Appl. Mech. 191, 745–760 (2001)
Hong, H.K., Chen, J.T.: Derivations of integral equations of elasticity. J. Eng. Mech. 114, 1028–1044 (1988)
Rudolph, T., Rizzo, F.: Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering. J. Appl. Mech. 57, 404–414 (1990)
Portela, A., Aliabadi, M., Rooke, D.: The dual boundary element method: effective implementation for crack problems. Int. J. Numer. Methods Eng. 33, 1269–1287 (1992)
Vera-Tudela, C.A.R., Telles, J.C.F.: The dual reciprocity method and the numerical Gree’s function for BEM fracture mechanic problems. Acta Mech. 227, 3205–3212 (2016)
Erdogan, F.: On the stress distribution in plates with collinear cuts under arbitrary loads. In: Proceedings of the Fourth US National Congress of Applied Mechanics, pp. 547–553 (1962)
Tan, P.W., Raju, I.S., Newman, J.: Boundary force method for analyzing two-dimensional cracked bodies. NASA Langley Research Center, Hampton (1986)
Ang, W.T.: A boundary integral solution for the problem of multiple interacting cracks in an elastic material. Int. J. Fract. 31, 259–270 (1986)
Su, C., Zheng, C.: Probabilistic fracture mechanics analysis of linear-elastic cracked structures by spline fictitious boundary element method. Eng. Anal. Bound. Elem. 36, 1828–1837 (2012)
Su, C., Han, D.: Multidomain SFBEM and its application in elastic plane problems. J. Eng. Mech. 126, 1057–1063 (2000)
Su, C., Xu, J.: Reliability analysis of Reissner plate bending problems by stochastic spline fictitious boundary element method. Eng. Anal. Bound. Elem. 51, 37–43 (2015)
Su, C., Zhao, S., Ma, H.: Reliability analysis of plane elasticity problems by stochastic spline fictitious boundary element method. Eng. Anal. Bound. Elem. 36, 118–124 (2012)
Ji, M., Chen, W., Lin, J.: Crack analysis by using the enriched singular boundary method. Eng. Anal. Bound. Elem. 72, 55–64 (2016)
Gu, Y., Chen, W., Zhang, C.: Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method. Int. J. Solids Struct. 50, 3460–3471 (2013)
Prenter, P.M.: Splines and Variational Methods. Courier Corporation, New York (2008)
Della-Ventura, D., Smith, R.N.L.: Some applications of singular fields in the solution of crack problems. Int. J. Numer. Methods Eng. 42(5), 927–942 (1998)
Liu, J., Lin, G., Du, J.: SBFEM analysis of multiple crack problems based on SBFEM. J. Dalian Univ. Technol. 48(3), 392–397 (2008). in chinese
Ritchie, R.: Mechanisms of fatigue crack propagation in metals, ceramics and composites: role of crack tip shielding. Mat. Sci. Eng. A Struct. 103, 15–28 (1988)
Seyedi, M., Taheri, S., Hild, F.: Numerical modeling of crack propagation and shielding effects in a striping network. Nucl. Eng. Des. 236, 954–964 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Xu, Z., Su, C. & Guan, Z. Analysis of multi-crack problems by the spline fictitious boundary element method based on Erdogan fundamental solutions. Acta Mech 229, 3257–3278 (2018). https://doi.org/10.1007/s00707-018-2160-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2160-0