Semi-Elliptical Cracks in a Cylinder Subjected to Stress Gradients

The calculation of stress intensity factors in three-dimensional situations, under any stresses, is an engineering necessity. The authors presented at the 9th National Symposium on Fracture Mechanics, Pittsburgh, 1975, a method for calculating three-dimensional weight functions by finite elements. But the computer time was found to be too long for engineering applications.

In this study the three-dimensional problem is limited to symmetrical problems with applied stresses expressed by a polynomial in one coordinate. Calculations are performed on semi-elliptical cracks in the meridional plane of a cylinder, and the applied stress is expressed by a polynomial of the fourth degree in the coordinate in the radial direction (see nomenclature and Fig. 1). The method could be extended to other symmetrical geometries and loads.

So-called “polynomial influence functions” are defined and correspond to the terms of the polynomial. These functions depend on the radii ratio, the shape, and the depth of the crack; since these parameters are fixed, they are functions ofthe eccentric angle that defines a point on the crack front.

The polynomial influence functions are computed by the boundary integral equation method. The method was first tested on a penny-shaped crack for which the known weight function allowed a direct computation of the polynomial influence functions. The accuracy was found sufficient to apply the method to more difficult problems. These functions were then calculated for semi-elliptical cracks in cylinders.

The results are presented in the form of curves; they are discussed and compared with the results published by other authors.