International Applied Mechanics

, Volume 48, Issue 4, pp 430–437 | Cite as

Stress state of a finite elastic cylinder with a circular crack undergoing torsional vibrations

  • V. G. Popov

The stress intensity factors (SIF) for a plane circular crack in a finite cylinder undergoing torsional vibrations are determined. The vibrations are generated by a rigid circular plate attached to one end of the cylinder and subjected to a harmonic moment. The boundary-value problem is reduced to the Fredholm equation of the second kind. This equation is solved numerically, and the solution is used to derive a highly accurate approximate formula to calculate the SIFs. The calculated results are plotted and analyzed


finite cylinder plane circular crack stress intensity factor torsional vibrations 


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  1. 1.
    L. V. Vakhonina and V. G. Popov, “Stress concentration around a circular thin perfectly rigid inclusion interacting with a torsional wave,” Izv. RAN, Mekh. Tverd. Tela, No. 4, 70–76 (2004).Google Scholar
  2. 2.
    L. V. Vakhonina and V. G. Popov, “Interaction of elastic waves with a circular thin rigid inclusion in the case of smooth contact,” Teor. Prikl. Mekh., 38, 158–166 (2003).Google Scholar
  3. 3.
    I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Sums, Series and Products, Academic Press, New York (1980).Google Scholar
  4. 4.
    A. N. Guz and V. V. Zozulya, Brittle Fracture of Materials under Dynamic Loading [in Russian], Naukova Dumka, Kyiv (1993).Google Scholar
  5. 5.
    W. Kecs and P. P. Teodorescu, Introduction to the Theory of Distributions in Engineering [in Romanian], Editura Tehnica, Bucharest (1975).Google Scholar
  6. 6.
    A. V. Men’shikov and I. A. Guz, “Friction in harmonic loading of a circular crack,” Dop. NAN Ukrainy, No. 1, 77–82 (2007).Google Scholar
  7. 7.
    G. Ya. Popov, “On a method of solving mechanics problems for domain with slits or thin inclusions,” J. Appl. Mat. Mech., 42, No. 1, 125–139 (1978).CrossRefGoogle Scholar
  8. 8.
    G. Ya. Popov, S. A. Abdymanapov, and V. V. Efimov, Green Functions and Matrices of One-Dimensional Boundary-Value Problems [in Russian], Rauan, Almaty (1999).Google Scholar
  9. 9.
    G. Ya. Popov, V. V. Reut, and N. D. Vaisfel’d, Equations of Mathematical Physics. Integral Transform Method [in Ukrainian], Astroprint, Odessa (2005).Google Scholar
  10. 10.
    T. Akiyawa, T. Hara, and T. Shibua, “Torsion of an infinite cylinder with multiple parallel circular cracks,” Theor. Appl. Mech., 50, 137–143 (2001).Google Scholar
  11. 11.
    D.-S. Lee, “Penny shaped crack in a long circular cylinder subjected to a uniform shearing stress,“ Europ. J. Mech., 20(2), 227–239 (2001).MATHCrossRefGoogle Scholar
  12. 12.
    F. Narita, Y. Shindo, and S. Lin, “Impact response of a piezoelectric ceramic cylinder with a penny-shaped crack,” Theor. Appl. Mech., 52, 153–162 (2003).Google Scholar
  13. 13.
    A. N. Guz, V. V. Zozulya, and A. V. Men’shikov, “Three-dimensional dynamic contact problem for an elliptic crack interacting with a normally incident harmonic compression-expansion wave,” Int. Appl. Mech., 39, No. 12, 1425–1428 (2003).ADSCrossRefGoogle Scholar
  14. 14.
    A. N. Guz, V. V. Zozulya, and A. V. Men’shikov, “General spatial dynamic problem for an elliptic crack under the action of a normal shear wave, with consideration for the contact interaction of the crack faces,” Int. Appl. Mech., 40, No. 2, 156–159 (2004).ADSCrossRefGoogle Scholar
  15. 15.
    A. N. Guz, V. V. Zozulya, and A. V. Men’shikov, “Surface contact of elliptical crack under normally incident tension–compression wave,” Theor. Appl. Fract. Mech., 40, No. 3, 285–291 (2008).CrossRefGoogle Scholar
  16. 16.
    G.-Y. Huang, Y.-S. Wang, and S.-W. Yu, “Stress concentration at a penny-shaped crack in a nonhomogeneous medium under torsion,” Acta Mech., December, 180, No. 1–4, 107–115 (2005).MATHCrossRefGoogle Scholar
  17. 17.
    Z. H. Jia, D. I. Shippy, and F. I. Rizzo, “Three-dimensional crack analysis using singular boundary element,” Int. J. Numer. Meth. Eng., 28(10), 2257–2273 (2005).CrossRefGoogle Scholar
  18. 18.
    M. O. Kaman and R. G. Mehmet, “Cracked semi-infinite cylinder and finite cylinder problems,” Int. J. Eng. Sci., 44(20), 1534–1555 (2006).CrossRefGoogle Scholar
  19. 19.
    P. A. Martin and G. R. Wickham, “Diffraction of elastic waves by a penny-shaped crack: Analytical and numerical results,” Proc. Royal Soc. London, Ser. A, Math. Phys. Sci., 390, No. 1798, 91–129 (1983).MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    B. M. Singh, J. B. Haddow, J. Vibik, and T. B. Moodie, “Dynamic stress intensity factors for penny-shaped crack in twisted plate,” J. Appl. Mech., 47, 963–965 (1980).ADSMATHCrossRefGoogle Scholar
  21. 21.
    K. N. Srivastava, R. M. Palaiya, and O. P. Cupta, “Interection of elastic waves with a penny-shaped crack in an infinitely long cylinder,” J. Elast., 12, No. 1, 143–152 (1982).MATHCrossRefGoogle Scholar
  22. 22.
    W. Aishi, “A note on the crack-plane stress field method for analyzing SIFs and ITS to a concentric penny-shaped crack in a circular cylinder opened up by constant pressure,” Int. J. Fract., 66, 73–76 (1994).CrossRefGoogle Scholar

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© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Odessa National Maritime AcademyOdessaUkraine

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