Torsion problems of finite cylinders weakened by ring-shaped cracks

Abstract

Abstract: A finite elastic hollow and solid cylinders are considered. The bottom faces of the cylinders are fixed, the upper faces are free from stress. The tangent axisymmetric loading is applied along their cylindrical surfaces. This leads to the torsion axisymmetric deformation. A system of N ring – shaped cracks is situated inside the cylinders parallel to the cylinder's axis. It is supposed that the branches of the cracks are free from stress. It is necessary to construct the formulas for the stress intensity factor calculation and investigate the stress state of a solid. The initial boundary value problem is reduced with Fourier transformation to a system of integral singular equations with regard to the jumps of the displacements at the cracks' branches. The singularity of the equations kernels is extracted. The system of singular integral equations is solved with the orthogonal polynomial method. The solution of the system is searched as the series by Chebyshev polynomials with the weight function. The realisation of orthogonal polynomial method leads to an infinite system of linear algebraic equations with regard to the unknown coefficients of the series. The formulas for the stresses and displacements of an elastic finite cylinder are presented. The numerical realisation of the proposed method is demonstrated in cases with two and three cracks; the stress state is investigated dependingon the cracks' locations and sizes.