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Mechanics of Composite Materials

, Volume 55, Issue 1, pp 13–28 | Cite as

Discontinuous Solutions of Axisymmetric Elasticity Theory for a Piecewise Homogeneous Layered Space with Periodical Interfacial Disk-Shape Defects

  • V. N. HakobyanEmail author
  • L. V. Hakobyan
  • L. L. Dashtoyan
Article
  • 22 Downloads

By the method of Hankel integral transformation, discontinuous solutions of equations of the axisymmetric elasticity theory are constructed for a piecewise homogeneous uniform layered space obtained by alternately joining two heterogeneous layers of equal thickness and whose junction contains a periodic system of parallel circular disk-shaped defects. On the basis of the solutions found, as examples, the governing systems of integral equations with Weber–Sonin kernels are presented for two cases: with defects in the form of absolutely rigid disk-shape inclusions and circular cracks. Using rotation operators, the governing systems of equations, in both cases, are reduced to a singular integral equation of the second kind, which is solved by the method of mechanical quadratures. Simple formulas for determining the rigid-body displacement of inclusions and crack opening are obtained.

Keywords

periodical mixed boundary-value problems disk-shape crack circular rigid inclusion 

Notes

Acknowledgements

This research was performed at a financial support by the GKN MON of Armenia Republic and FSI of Russian Federation within the framework of joint scientific projects SCS 18RF061 and RFBR 18-51-05012, respectively.

References

  1. 1.
    F. Erdogan, “Stress distribution in bonded dissimilar materials containing circular or ring-shaped cavities,” Trans ASME. Ser. E. J. Appl. Mech., 32, No. 4, 829-836 (1964).CrossRefGoogle Scholar
  2. 2.
    V. I. Mossakovskii and M. T. Rybka, “Generalization of the Griffith-Sneddon criterion for the case of a nonhomogeneous body,” J. Appl. Math. and Mech., 28, No. 6, 1061-1069 (1964).CrossRefGoogle Scholar
  3. 3.
    J. R. Willis, “The penny-shaped crack on an interface,” Quart J. Mech. and Appl. Math., 25, No. 3, 367-385 (1972).CrossRefGoogle Scholar
  4. 4.
    G. Ya. Popov, “About concentration of elastic stresses near thin delaminated inclusion,” Modern Problems of Mechanics and Aviation, 156-162 (1980).Google Scholar
  5. 5.
    V. N. Hakobyan and L. L. Dashtoyan, “An axisymmetric problem for a compound space weakened by a semi-infinite ring-shaped crack,” Izv. NAN RA, Mechanics, 59, No. 1, 3-10 (2006).Google Scholar
  6. 6.
    V. N. Hakobyan, “Stresses near an absolutely rigid penny-shaped inclusion in a piecewise homogeneous space,” Proc. of Int. Conf. “Topical Problems of the Mechanics of Continuous Media,” devoted to the 95th anniversary of birth of N. Kh. Arutyunyan, Yerevan, 45-51 (2007).Google Scholar
  7. 7.
    V. N. Hakobyan, S. E. Mirzoyan, and L. L. Dashtoyan, “Axisymmetric mixed problem for a compound space with a penny-shaped crack,” Bulletin of MGTU named by N. E. Bauman. Series Natural Sci., No. 3, 31-46 (2015).Google Scholar
  8. 8.
    V. N. Hakobyan, Mixed Boundary-Value Problems on the Interaction of Continuous Deformable Bodies with Stress Concentrators of Various Types [in Russian], Izd. Gitutyun NAN RA, Yerevan (2014).Google Scholar
  9. 9.
    S. M. Mkhitaryan, L. A. Shekyan, S. V. Verlinski, D. Aidun, and P. Marzocca, “Stationary theory of heat-conductivity for an axi-symmetrical piecehomogeneous space with circular inclusion,” J. Thermal Stresses, 35, No. 5, 424-447 (2012).CrossRefGoogle Scholar
  10. 10.
    G. Ya. Popov, Selected Works [in Russian], Vol. 1, Izd. Tipogr. “VBV” (2007).Google Scholar
  11. 11.
    G. Ya. Popov, Selected Works [in Russian], Vol. 2, Izd. “VBV” (2007).Google Scholar
  12. 12.
    V. N. Hakobyan, “Axisymmetric stress state of piecewise-homogeneous layered space with parallel penny-shaped cracks,” Proc. of XVIII Int. Conf. “Modern Problems of Mechanics of Continuous Media,” Rostov-on-Don, Vol. 1, 35-39 (2016).Google Scholar
  13. 13.
    B. Korenev, H. J. Glaeske, E. Moiseev, and M. Saigo, Bessel Functions and Their Applications, CRC Press, London, (2002).Google Scholar
  14. 14.
    F. D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow (1977).Google Scholar
  15. 15.
    A. A. Amirjanyan and A. V. Sahakyan, “Mechanical quadrature method as applied to singular integral equations with logarithmic singularity on the right-hand side,” Comput. Math. and Math. Phys., 57, No. 8, 1285-1293 (2017).CrossRefGoogle Scholar
  16. 16.
    I. S. Gradshtein, and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1962).Google Scholar
  17. 17.
    A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marychev, Integrals and Series [in Russian], Nauka, Moscow (1981).Google Scholar
  18. 18.
    M. K. Kassir and A. M. Bregman, “The stress-intensity factor for a penny-shaped crack between two dissimilar materials,” Trans ASME. Ser. E. J. Appl. Mech., 39, No. 1, 308-310 (1972).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. N. Hakobyan
    • 1
    Email author
  • L. V. Hakobyan
    • 1
  • L. L. Dashtoyan
    • 1
  1. 1.Institute of Mechanics of the National Academy of Sciences of Republic of ArmeniaYerevanArmenia

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