Stress State in a Finite Cylinder with Outer Ring-Shaped Crack at Non-stationary Torsion

The axisymmetric dynamic problem of determining the stress state in the vicinity of a ring-shaped crack in a finite cylinder is solved. The source of the loading is the rigid circular plate, which is joined with one of the cylinder ends and loaded by the time-dependent torque. The proposed method consists in the difference approximation of only the time derivative. To do this, specially selected non-equidistant nodes and special representation of the solution in these nodes are used. Such an approach allows the original problem to be reduced to a sequence of boundary value problems for the homogeneous Helmholtz equation. Each such problem is solved by using integral Fourier and Hankel transforms, with their subsequent reversal. As a result, integral representations were obtained for the angular displacement through unknown tangential stresses in the plane of the crack. From boundary condition on a crack, an integral equation is obtained, which, as a result of using the Weber-Sonin integral operator and a series of transformations, is reduced to the Fredholm integral equation of the second kind. The numerical solution found made it possible to obtain an approximate formula for calculating the stress intensity factor (SIF).