Stressed state in a finite cylinder with a circular crack at non-stationary torsion
Keywords:
stress intensity coefficient (SIF), axially symmetric dynamic problem, finite differences, finite cylinder, circular crack, torque momentAbstract
This paper considers a solution to an axially symmetric dynamic problem of determining the stress-state in the vicinity of a circular crack in a finite cylinder. The cylinder lower base is rigidly fixed, and the upper one is loaded with time-dependent tangential stresses. In contrast to the traditional analytical methods based on the use of the integralLaplacetransform, the proposed one consists in the difference approximation of only the time derivative. To do this, specially selected unequally spaced nodes and a special representation of the solution in these nodes are used. Such an approach allows the initial problem to be reduced to a sequence of boundary problems for the homogeneous Helmholtz equation. Each such problem is solved by applying the finite Fourier and Hankel integral transforms with their subsequent inversion. As a result, an integral representation was obtained for the angular displacement through an unknown displacement jump in the crack plane. With regard to the derivative of this jump from the boundary condition on the crack, an integral equation was obtained which, as a result of the integral Weber-Sonin operator application and a series of transformations, was reduced to the Fredholm integral equation of the second kind regarding the unknown function associated with the jump. An approximate solution of this equation was carried out by the method of collocations, with the integrals being approximated by quadratic Gaussian-Legendre formulas. The numerical solution found made it possible to obtain an approximate formula for calculating the stress intensity factor (SIF). Using this formula, we studied the effect of the nature of the load and the geometric parameters of the cylinder on the time dependence of this factor. The analysis of the results showed that for all the types of loading considered, the maximum value of SIF can be observed during the transient process. When a sudden, constant load is applied, this maximum is 2-2.5 times higher than the static value. In the case of a sudden harmonic load, SIF maximum also significantly exceeds the values it acquires with steady-state oscillations, in the absence of resonance. Increasing the cylinder height and reducing the crack area result in an increase in the duration of the transient process and a decrease in the value of SIF maximum. The same effect can be observed when the crack plane approaches the stationary end of the cylinder.
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