Analysis of stress singularities at the corner point of lozenge hole and rigid lozenge inclusion in elastic plates by conformal mapping

Abstract

The plane elastostatic problems of a lozenge hole and a rigid lozenge inclusion in the infinite plate under tension loading condition are examined by the conformal mapping and Goursat stress functions. A new method is developed for the calculation of the orders of singularity and the stress intensity factors of a singular point, such as the corners of a lozenge hole and rigid lozenge inclusion in the elastic plates submitted to an overall tension at infinity. The rotation of rigid lozenge inclusion by the overall tension at infinity is also considered. Coefficients of Goursat stress functions are determined respectively, for the number m of terms of the finite series of the function for the conformal mapping. Then, the two types of the orders of stress singularities λ I and λ II which correspond to Mode I (opening mode) and Mode II (in-plane shear mode) deformations, respectively, are determined and their numerical results are in good agreement with the exact solutions by Williams for various corner angles of a lozenge. Also the stress intensity factors K 1 and K II of the singular stress field of the corner are determined for arbitrary equivalent Poisson's ratio k , and it is confirmed that K I and K II converge the exact solutions of crack and rigid line inclusion problems when a corner angle of a lozenge tends to zero. Moreover, the angle of the rotation ϵ converges the exact solutions of a rigid line inclusion problems when a corner angle of a lozenge becomes zero. Singular stress and displacement fields corresponding to Mode I and II deformations around the vicinity of the corner of a lozenge hole and a rigid lozenge inclusion are also obtained.