Singular Integral Equation Method For The Crack Problem Of Circular Region In Anti-Plane Elasticity

In this paper, the branch crack or the edge crack problem of circular region or infinite region containing a circular hole in elasticity antiplane shear are investigated systematically by using complex potentials of dislocation model and the singular integral equation approach.It is firstly studied that there is an arbitary branch crack in circular region when there are concentrated forces along the circular boundary. A point dislocation is placed at the branch point and the distributed dislocations are assumed along all the branches to simulate the branch crack. According to the elementary solution for a point dislocation of infinite region in antiplane elasticity, the elementary solution for branch crack problem of circular region in antiplane elasticity is obtained. By matching the traction along the cracks, Cauchy singular integral equations are gotten, where the point dislocation intensity and the distributed dislocation density served as the unknown function. A constraint equation can be formulated for displacement single value condition. And then by using a semi-open quadrature rule, the singular integral equations are solved. The results of the numerical solution directly relate to the stress intensity factors (SIF) at the branch tips, which are paid more attention to by engineers in practice.Based on the branch crack problems in circular region, the multiple edge-cracks problem of circular region or infinite region containing a circular hole in antiplane elasticity are also considered.The interactions and the reciprocal effects between cracks are exposed quantificationally,and the 'shadow effect' of multiple edge-cracks are observed. Meanwhile, the crack opening displacement of multiple edge-cracks for infinite region containing a circular hole is solved. Using the method proposed in this paper makes SIF values calculated more easily and more accurately. When the problems are reduced to cracks in infinite region,the calculate results using this method are found to be very similar to those by conformal mapping in the handbook of stress intensity factors. There are lots of numerical examples given. The numerical results and charts in this paper can be applied directly into practice. All the conclusions in this paper can be further used to investigate the complex crack problems in antiplane elasticity.