Determination by Iterative Method of Diffraction Field at the Interaction Longitudinal Shear Wave with the System of Thin Rigid Inclusions

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


The problem of the diffraction field determination is arising as a result of the longitudinal shear wave interaction with the thin rigid inclusions system arbitrarily situated in an infinity body was solved. Inclusions are considered to be fully coupled to the elastic medium and are moving. Unknown amplitudes of inclusions are determined from the equations of motion. The solution method is based on the submission diffraction field displacement as sum of discontinuous solutions to the Helmholtz equation, the constructed for each inclusion. As result the original problem is reduced to the system of the singular integral equations for unknown jumps of stresses on the inclusions surface, The iterative method of this system solving, where the zero approximation are the solutions of the integral equations for the single inclusions, is proposed. This integral equation for single inclusions are numerical solved the mechanical quadrature method. The final result is the approximate formulas for calculating stress intensity factors and the amplitudes of the oscillations.


Thin ridged inclusion Wave interaction Integral equations Iterative method 


  1. 1.
    Jain, D.L., Kanval, R.P.: Diffraction of elastic waves by two coplanar parallel rigid strips. Int. J. Solids Struct. 10(11), 925–937 (1972). Scholar
  2. 2.
    Nazarenko, O.M., Lozhkin, O.M.: Plane problem of diffraction of elastic harmonic waves on periodic curvilinear inserts. Mater. Sci. 43(2), 249–255 (2007). Scholar
  3. 3.
    Popov, V.G.: Diffraction of elastic shear waves on radially distributed rigid inclusions. Int. Appl. Mech. 32(8), 624–630 (1996). Scholar
  4. 4.
    Popov, V.G.: Interaction of plane elastic waves with systems of radial defects [in Russian]. Izv. Ross. Akad. Nauk. Mekh. Tverdogo Tela. 4, 118–129 (1999)Google Scholar
  5. 5.
    Liu, E., Zhang, Z.: Numerical study of elastic wave scattering by cracks or inclusions using the boundary integral equation method. J. Comput. Acoust. 9(3), 1039–1054 (2001). Scholar
  6. 6.
    Lei, J., Yang, Q., Wang, Y.-S., Zhang, C.: An investigation of dynamic interaction between multiple cracks and inclusions by TDBEM. Compos. Sci. Technol. 69(7–8), 1279–1285 (2009). Scholar
  7. 7.
    Popov, V.G.: Comparison of displacement fields and stresses in the diffraction of elastic shear waves at various defects: crack and thin rigid inclusion [in Russian]. Dyn. syst. 12, 35–41 (1993)Google Scholar
  8. 8.
    Popov, V.G.: Investigation of the fields of displacements and stresses in the case of diffraction of shear elastic waves on thin rigid exfoliated inclusions [in Russian]. Izv. Ross. Akad. Nauk. Mekh. Tverdogo Tela. 3, 139–146 (1992)Google Scholar
  9. 9.
    Belotserkovskii, S.M., Lifanov, I.K.: Numerical methods in singular integral equations and their use in ferrodynamics, the theory of elasticity and electrodynamics [in Russian]. Nauka, Moscow (1985)Google Scholar
  10. 10.
    Sulym, H.T.: Foundations of the Mathematical theory of thermoelastic equilibrium of deformed bodies with thin inclusions [in Ukrainian]. Doslidno-Vydavnychyi Tsentr NTSh, Lviv (2007)Google Scholar

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Authors and Affiliations

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

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