Skip to main content
Log in

A Crack in the Form of a Three-Link Broken Line Under The Action of Longitudinal Shear Waves

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We solve the problem of determination of the dynamic stress intensity factors for a crack in the form of a three-link broken line. The crack is located in an infinite elastic medium with propagating harmonic longitudinal shear waves. The initial problem is reduced to a system of three singular integrodifferential equations with fixed singularities. A numerical method is proposed for the solution of this system with regard for the true asymptotics of the unknown functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. V. Andreev, “Direct numerical method for solving singular integral equations of the first kind with generalized kernels,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 126–146 (2005); English translation: Mech. Solids, 40, No. 1, 104–119 (2005).

  2. B. A. Afyan, “On the integral equations with fixed singularities in the theory of branching cracks,” Dokl. Akad. Nauk Arm. SSR, 79, No. 4, 177–181 (1984).

    MathSciNet  MATH  Google Scholar 

  3. R. V. Duduchava, Integral Equations of Convolution with Discontinuous Presymbols, Singular Integral Equations with Fixed Singularities, and Their Applications to Problems of Mechanics [in Russian], Proc. Razmadze Math. Inst., 60 (1979).

  4. V. I. Krylov, Approximate Calculation of Integrals, Dover, New York (2006).

    MATH  Google Scholar 

  5. P. N. Osiv and M. P. Savruk, “Determination of stresses in an infinite plate with broken or branching crack,” Prikl. Mekh. Tekh. Fiz., No. 2, 142–147 (1983); English translation: J. Appl. Mech. Tech. Phys., 24, No. 2, 266–271 (1983).

  6. V. G. Popov, “Diffraction of elastic shear waves on an inclusion of complex shape located in an unbounded elastic medium, in: Hydroaeromechanics and Elasticity Theory: Numerical and Analytic Methods of Solution of Problems of Hydroaerodynamics and Elasticity Theory [in Russian], Dnepropetrovsk State University, Dnepropetrovsk (1986), pp. 121–127.

  7. V. G. Popov, “Investigation of the fields of stresses and displacements in the case of diffraction of elastic shear waves on a thin rigid separated inclusion,” Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 139–146 (1992).

  8. V. G. Popov, “Stressed state near two cracks leaving the same point under harmonic longitudinal shear oscillations,” Visn. Kyiv. Shevchenko Nats. Univ. Ser. Fiz.-Mat. Nauk., Issue 3, 205–208 (2013).

  9. G. Ya. Popov, Concentration of Elastic Stresses near Punches, Notches, Thin Inclusions, and Reinforcements [in Russian], Nauka, Moscow (1982).

  10. M. P. Savruk, Two-Dimensional Problems of Elasticity for Bodies with Cracks [in Russian], Naukova Dumka, Kiev (1981).

  11. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., New York (1959).

  12. M. Isida and H. Noguchi, “Stress intensity factors at tips of branched cracks under various loadings,” Int. J. Fract., 54, No. 4, 293–316 (1992).

    Article  Google Scholar 

  13. V. Vitek, “Plane strain stress intensity factors for branched cracks,” Int. J. Fract., 13, No. 4, 481–501 (1977).

    Google Scholar 

  14. X. Yan, “Stress intensity factors for asymmetric branched cracks in plane extension by using crack-tip displacement discontinuity elements,” Mech. Res. Comm., 32, No. 4, 375–384 (2005).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 58, No. 1, pp. 112–120, January–March, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Popov, V.G. A Crack in the Form of a Three-Link Broken Line Under The Action of Longitudinal Shear Waves. J Math Sci 222, 143–154 (2017). https://doi.org/10.1007/s10958-017-3288-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3288-5

Navigation