Thermal stresses in a long cylinder containing a penny-shaped crack

Abstract

The paper contains an analysis of the state of stress in a long circular cylinder of elastic material when the inner surfaces of a concentric penny-shaped crack are subjected to a prescribed temperature distribution. The plane of the crack is perpendicular to the axis of the cylinder. It is assumed that the lateral surface of the cylinder is maintained at a constant zero temperature and is supported in such a way that the shear component of the stress tensor and the radial component of the displacement vector vanish on the boundary. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Then, using a suitable biharmonic function as the potential of thermoelastic displacements, the problem is reduced to the solution of a similar Fredholm integral equation in which the solution of the earlier integral equation arising from the heat conduction problem occurs as a known function. Iterative solutions of the integral equations are found which are valid for a certain chosen value of the ratio of the radius of the crack to that of the cylinder. The integral equations are also solved numerically by using a high speed computer with the same value of the ratio. The stress intensity factor is calculated for both of these solutions and the results are found to be in complete agreement.