This paper is concerned with an efficient dual boundary element method for 2d crack problems under antiplane shear loading. The dual equations are the displacement and the traction boundary integral equations. When the displacement equation is applied on the outer boundary and the traction equation on one of the crack surfaces, general crack problems with anti-plane shear loading can be solved with a single region formulation. The outer boundary is discretised with continuous quadratic elements; however, only one of the crack surfaces needs to be discretised with discontinuous quadratic elements. Highly accurate results are obtained, when the stress intensity factor is evaluated with the discontinuous quarter point element method. Numerical examples are provided to demonstrate the accuracy and efficiency of the present formulation.

The problem of a cracked body subjected to an antiplane shear loading had been studied extensively. Sih [

Several numerical solutions had been devised for antiplane crack problems. Wallentin et al. [

To overcome this shortcoming, we provide an efficient numerical procedure, based on the dual boundary element method (DBEM), for antiplane shear loading problems. The dual boundary element method seems to have certain apparent advantages for in-plane loading problems with a single region formulation. This method incorporates two independent boundary integral equations, the displacement and traction equations. Portela et al. [

Consider a finite domain subjected to an arbitrary antiplane shear loading, where the only nonzero displacement component

The stress components

The displacement integral equation (

Near tip displacement extrapolation is used to evaluate the numerical values of the stress intensity factor. The relative displacements of the crack surfaces are calculated using the DBEM and are used in the near crack tip stress field equations to obtain the stress intensity factor. Due to the singular behaviour of the stress around the crack tip, it is reasonable to expect a better approximation by replacing the normal discontinuous quadratic element with a transition element possessing the same order of singularity at the crack tip. The discontinuous quarter point element method is used in the present formulation [

In order to demonstrate the accuracy and efficiency of the technique previously described, and to illustrate possible applications, we now consider several examples. In all the numerical tests, the outer boundary is modelled by 24 continuous quadratic elements, and each crack discretization is carried out with three different meshes of 6, 8, and 10 discontinuous quadratic elements, respectively. The best accuracy is achieved with 6 elements, in which the crack discretization is graded, towards the tip, with ratios 0.25, 0.15, and 0.1. The plate is subjected to a uniform antiplane shear loading

Firstly, consider a rectangular plate containing a central slant crack as shown in Figure

Normalised mode III stress intensity factor for a straight central crack.

Present | 1.909 | 1.724 | 1.689 | 1.688 | 1.661 | |

Reference [ | 1.897 | 1.723 | 1.689 | 1.686 | 1.686 | |

Reference [ | 1.900 | 1.725 | 1.691 | 1.689 | 1.689 | |

Present | 1.796 | 1.467 | 1.371 | 1.361 | 1.361 | |

Reference [ | 1.780 | 1.460 | 1.369 | 1.359 | 1.358 | |

Reference [ | 1.782 | 1.463 | 1.370 | 1.361 | 1.360 | |

Present | 1.784 | 1.405 | 1.257 | 1.236 | 1.236 | |

Reference [ | 1.771 | 1.399 | 1.254 | 1.233 | 1.233 | |

Reference [ | 1.773 | 1.401 | 1.256 | 1.235 | 1.235 | |

Present | 1.792 | 1.384 | 1.179 | 1.131 | 1.129 | |

Reference [ | 1.770 | 1.377 | 1.176 | 1.127 | 1.126 | |

Reference [ | 1.772 | 1.379 | 1.178 | 1.130 | 1.128 |

Rectangular plate with a central slant crack.

The rectangular plate with a central slant crack at (a)

For the case where

The infinite plate with a central slant crack: (a) the analytical solutions and (b) the present method.

As shown in Figure

The rectangular plate with two identical collinear cracks. Normalised mode III stress intensity factors versus

The third example is an infinite plate (

The rectangular plate with two parallel cracks. Normalised mode III stress intensity factors versus

An efficient and accurate dual boundary element technique has been successfully developed for the analysis of two dimensional cracks subjected to an antiplane shear loading. The dual boundary equations are the usual displacement boundary integral equation and the traction boundary integral equation. When the displacement equation is applied on the outer boundary and the traction equation is applied on one of the crack surfaces, a general crack problem can be solved in a single region formulation. The discontinuous quarter point elements are used for evaluating the mode III stress intensity factor, which correctly describes the