Cracked semi-infinite cylinder and finite cylinder problems

This work considers the analysis of a cracked semi-infinite cylinder and a finite cylinder. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to axial tension. Solution of this problem can be obtained by superposition of solutions for an infinite cylinder subjected to uniformly distributed tensile load at infinity (I) and an infinite cylinder having a penny-shaped rigid inclusion at z = 0 and two penny-shaped cracks at z = ±L (II). General expressions for the perturbation problem (II) are obtained by solving Navier equations with Fourier and Hankel transforms. When the radius of the inclusion approaches the radius of the cylinder, the end at z = 0 becomes fixed and when the radius of the cracks approach the radius of the cylinder, the ends at z = ±L become cut and subject to uniform tensile load. Formulation of the problem is reduced to a system of three singular integral equations. By using Gauss–Lobatto and Gauss–Jacobi integration formulas, these three singular integral equations are converted to a system of linear algebraic equations which is solved numerically.