Mathematical simulation of the strength of a pipe with three-dimensional defect

We develop a mathematical model of plastic fracture of a thin-walled pipe with three-dimensional part-through "smooth" defect and compute the degree of weakening caused by this defect. The defect is simulated by an elliptic (in plan) slot appearing in the external surface of the pipe. It is necessary to find the level of internal pressure under which the material in the zone of the defect passes into the limiting state. For sufficiently deep defects whose sizes are much smaller than the radius of the pipe, the problem of determination of the stress-strain state in the zone of the defect under given internal pressure is reduced to the solution of a nonlinear system of two one-dimensional integral equations. This system is solved by using an original algorithm and the limiting pressure is found within the framework of the proposed mathematical model. Under certain assumptions concerning the distribution of stresses in the zone of the defect, we deduce a formula which expresses the limiting pressure via the sizes of the defect and the strength characteristics of the material. The results of theoretical analysis are compared with numerical data obtained by the method of finite elements.