Advertisement

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

  • Oleksandr DemydovEmail author
  • Vsevolod Popov
Conference paper
Part of the Structural Integrity book series (STIN, volume 8)

Abstract

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).

Keywords

Stress intensity factor Ring-shaped crack Finite cylinder Finite differences Non-stationary torsion 

References

  1. 1.
    Shindo, Y., Li, W.: Torsional impact response of a thick-walled cylinder with a circumferential edge crack. J. Press. Vessel Technol. 112, 367–373 (1990)CrossRefGoogle Scholar
  2. 2.
    Bai, H., Shah, A.H., Popplewell, N., Datta, S.K.: Scattering of guided waves by circumferential cracks in composite cylinders. Int. J. Solids Struct. 39, 4583–4603 (2002)CrossRefGoogle Scholar
  3. 3.
    Dimarogonas, A., Massouros, G.: Torsional vibration of a shaft with a circumferential crack. Eng. Fract. Mech. 15, 439–444 (1981).  https://doi.org/10.1016/0013-7944(81)90069-2CrossRefGoogle Scholar
  4. 4.
    Andreikiv, O.E., Boiko, V.M., Kovchyk, S.E., Khodan, I.V.: Dynamic tension of a cylindrical specimen with circumferential crack. Mater. Sci. 36, 382–391 (2000).  https://doi.org/10.1007/BF02769599CrossRefGoogle Scholar
  5. 5.
    Ivanyts’kyi, Y.L., Boiko, V.M., Khodan’, I.V., Shtayura, S.T.: Stressed state of a cylinder with external circular crack under dynamic torsion. Mater. Sci. 43, 203–214 (2007).  https://doi.org/10.1007/s11003-007-0023-2CrossRefGoogle Scholar
  6. 6.
    Savruk, M.P.: New method for the solution of dynamic problems of the theory of elasticity and fracture mechanics. Mater. Sci. 39, 465–471 (2003).  https://doi.org/10.1023/B:MASC.0000010922.84603.8dCrossRefGoogle Scholar
  7. 7.
    Popov, P.V.: The problem of the torsion of a finite cylinder with a ring-shaped crack. Mashynoznavstvo. 9, 15–18 (2005). [in Ukrainian]Google Scholar
  8. 8.
    Popov, V.H.: Torsional oscillations of a finite elastic cylinder containing an outer circular crack. Mater. Sci. 47, 746–756 (2012).  https://doi.org/10.1007/s11003-012-9452-7CrossRefGoogle Scholar
  9. 9.
    Vaisfeld, N.D.: A nonstationary dynamic problem of torsion of a hollow elastic cylinder. Phys. Math. Sci. 6, 95–99 (2001) (in Russian)Google Scholar
  10. 10.
    Demydov, O.V., Popov, V.H.: Nonstationary torsion of the finite cylinder with circular crack. Phys. Math. Sci. 1, 131–142 (2017) (in Ukrainian)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.National University “Odesa Maritime Academy”OdesaUkraine

Personalised recommendations