Application of boundary integral quadrature method to torsion problems of an isotropic bar containing edge cracks

Abstract

Regarding the Saint-Venant torsion problem of elastic cylindrical bar containing edge cracks, the boundary integral quadrature method (BIQM) in conjunction with the dual formula is employed to solve the stress function. In comparison with the conventional dual boundary element method (BEM), the present method is not required to generate the mesh. To achieve this advantage, the parametric representation for the boundary contour and the Gaussian quadrature for the boundary integral play important roles in the present method. When the collocation point is located on the ordinary boundary, the original adaptive exact solution is used to skillfully determine the singular integral in the Cauchy principal value sense. When the collocation point is located on the crack, the corresponding adaptive exact solution is rederived by using the linear combination of harmonic basis of elliptical coordinates. To realize the effect upon the elastic cylinder due to the crack, the torsional rigidity of the cross-section and the stress intensity factor (SIF) at the crack tip are considered to calculate. To conveniently determine the torsional rigidity, the formula is transformed into the form of the boundary quadrature of the boundary density. To the computation of the SIF, the corresponding boundary data on the boundary point that is the most nearly the crack tip is adopted. To check the validity of present results, the conventional dual BEM is adopted to examine those results. Finally, 3 elastic cylindrical bars of different cross-sections are considered. Two of them are circular cross-sections with a radial edge crack and a slant edge crack and the other is an elliptical cross-section with two edge cracks.