Acta Mechanica

, Volume 224, Issue 3, pp 597–618 | Cite as

Dynamic behavior of a finite-sized elastic solid with multiple cavities and inclusions using BIEM

  • S. L. Parvanova
  • P. S. Dineva
  • G. D. ManolisEmail author


The 2D elastodynamic problem is solved for a finite-size solid containing multiple cavities and/or elastic inclusions of any shape that are arranged in an arbitrary geometrical configuration. The dynamic load is a tensile traction field imposed along the sides of the finite-size solid matrix and under time-harmonic conditions. Furthermore, the cavity surfaces are either traction-free or internally pressurized, while the inclusions have elastic properties ranging from very weak to nearly rigid. The presence of all these heterogeneities within the elastic matrix gives rise to both wave scattering and stress concentration phenomena. Computation of the underlying kinematic and stress fields is carried out using the boundary integral equation method built on the frequency-dependent fundamental solutions of elastodynamics for a point load in an unbounded continuum. As a first step, a detailed validation study is performed by comparing the present results with existing analytical solutions and with numerical results reported in the literature. Following this, extensive numerical simulations reveal the dependence of the scattered wave fields and of the resulting dynamic stress concentration factors (SCF) on the shape, size, number and geometrical configuration of multiple cavities and/or inclusions in the finite elastic solid. The pronounced SCF values invariably (but not always) observed are attributed to multiple dynamic interactions between these heterogeneities that may either weaken or strengthen the background elastic matrix.


Radial Stress Hoop Stress Circular Inclusion Observer Point Boundary Integral Equation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Boundary integral equation method


Finite element method


Stress concentration factor


Boundary value problem


Boundary integral equation


Boundary element




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  1. 1.
    Mow C.C., Mente L.J.: Dynamic stresses and displacements around cylindrical discontinuities due to plane harmonic shear waves. J. Appl. Mech. 30, 598–604 (1963)CrossRefGoogle Scholar
  2. 2.
    Kung, G.C.S.: Dynamical stress concentration in an elastic plate. M. Sci. Thesis, Cornell University, Ithaca, New York (1964)Google Scholar
  3. 3.
    Pao Y.H., Mow C.C.: Diffraction of Elastic Waves and Dynamic Stress Concentration. Crane Russak, New York (1971)Google Scholar
  4. 4.
    Achenbach J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  5. 5.
    Miklowitz J.: Elastic Waves and Waveguides. North-Holland, Amsterdam (1984)Google Scholar
  6. 6.
    Hirose S.: Scattering from an elliptic crack by the time-domain boundary integral equation method. In: Brebbia, C.A, Connor, J.J. (eds) Advances in Boundary Elements: Stress Analysis, pp. 99–110. Springer, Berlin (1989)Google Scholar
  7. 7.
    Manolis G.D.: Elastic wave scattering around cavities in inhomogeneous continua by the BEM. J. Sound Vib. 266, 281–305 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Meguid S.A., Wang X.D.: The dynamic interaction of a crack with a circular cavity under anti-plane loading. J. Mech. Phys. Solids 43, 1857–1874 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nakasone Y., Nishiyama H., Nojiri T.: Numerical equivalent inclusion method: a new computational method for analyzing stress fields in and around inclusions of various shapes. Mater. Sci. Eng. A285, 229–238 (2000)Google Scholar
  10. 10.
    Zhang J., Katsube N.: A hybrid finite element method for heterogeneous materials with randomly dispersed elastic inclusions. Finite Elem. Anal. Des. 19, 45–55 (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gong S.X., Meguid S.A.: Interacting circular inhomogeneities in plane elastostatics. Acta Mech. 99, 49–60 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ting K., Chen K.T., Yang W.S.: Applied alternating method to analyze the stress concentration around interacting multiple circular holess in an infinite domain. Int. J. Solids Struct. 36, 533–556 (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Squire V.A., Dixon T.W.: Scattering of flexural waves from a coated cylindrical anomaly in a thin plate. J. Sound Vib. 236, 367–373 (2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kratochvil, J., Becker, W.: Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite number of holes. Arch. Appl. Mech. doi: 10.1007/s00419-011-0587-z (2012)
  15. 15.
    Kushch V., Shmegera S., Buryachenko V.: Elastic equilibrium of a half plane containing a finite array of elliptic inclusions. Int. J. Solids Struct. 43, 3459–3483 (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Wang J., Crouch S.L., Mogilevskaya S.G.: A complex boundary integral method for multiple circular holes in an infinite plane. Eng. Anal. Bound. Elem. 27, 789–802 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hu C., Fang X.Q., Huang W.H.: Multiple scattering of flexural waves in a semi-infinite thin plate with a cut-out. Int. J. Solids Struct. 44, 436–446 (2007)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lee W.M., Chen J.T.: Scattering of flexural wave in thin plate with multiple holes by using the null-field integral equation approach. Comput. Model. Eng. Sci. 37, 243–273 (2008)MathSciNetGoogle Scholar
  19. 19.
    Ayatollahi M., Fariborz S.J., Ahmadi N.: Anti-plane elastodynamic analysis of planes with multiple defects. Appl. Math. Model. 33, 663–676 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Liu D., Gai B., Tao G.: Applications of the method of complex functions to dynamic stress concentrations. Wave Motion 4, 293–304 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Manolis G.D., Beskos D.E.: Dynamic stress concentration studies by boundary integrals and Laplace transform. Int. J. Numer. Methods Eng. 17, 573–599 (1981)CrossRefzbMATHGoogle Scholar
  22. 22.
    Niwa Y., Hirose S., Kitahara M.: Application of the boundary integral equation method to transient response analysis of inclusions in a half-space. Wave Motion 8, 77–91 (1986)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kitahara M., Nakagawa K., Achenbach J.D.: Boundary-integral equation method for elastodynamic scattering. Comput. Mech. 5, 129–144 (1989)CrossRefzbMATHGoogle Scholar
  24. 24.
    Providakis C.P., Sotiropoulos D.A., Beskos D.E.: BEM analysis of reduced dynamic stress concentration by multiple holes. Commun. Numer. Methods Eng. 9, 917–924 (1993)CrossRefzbMATHGoogle Scholar
  25. 25.
    Greengard L., Helsing J.: On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites. J. Mech. Phys. Solids 46, 1441–1462 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gao S.W., Wang B.L., Ma X.R.: Scattering of elastic wave and dynamic stress concentrations in thin plate with a circular cavity. Eng. Mech. 18, 14–20 (2001)Google Scholar
  27. 27.
    Yao Z., Kong F., Zheng X.: Simulation of 2D elastic bodies with randomly distributed circular inclusions using the BEM. Electron. J. Bound. Elem. 1, 270–282 (2003)MathSciNetGoogle Scholar
  28. 28.
    Rus G., Gallego R.: Boundary integral equation for inclusion and cavity shape sensitivity in harmonic elastodynamics. Eng. Anal. Bound. Elem. 29, 77–91 (2005)CrossRefzbMATHGoogle Scholar
  29. 29.
    Leite L.G.S., Venturini W.S.: Accurate modelling of rigid and soft inclusions in 2D elastic solids by the boundary element method. Comput. Struct. 84, 1874–1881 (2006)CrossRefGoogle Scholar
  30. 30.
    Dravinski M., Yu M.C.: Scattering of plane harmonic SH waves by multiple inclusions. Geophys. J. Int. 186, 1331–1346 (2011)CrossRefGoogle Scholar
  31. 31.
    Zienkiewicz O.C., Kelly D.W., Bettess P.: The coupling of the finite element method and boundary solution problems. Int. J. Numer. Methods Eng. 11, 355–375 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Beer, G.: Implementation of combined boundary element finite element analysis with application in geomechanics. In: Banerjee, P.K., Watson, J.O. (eds.) Developments in Boundary Element Methods 4, Chap. 7, pp. 191–225. Applied Science Publishers, London (1986)Google Scholar
  33. 33.
    Mogilevskaya S.G., Crouch S.L.: A Galerkin boundary integral method for multiple circular elastic inclusions. Int. J. Numer. Methods Eng. 52, 1069–1106 (2001)CrossRefzbMATHGoogle Scholar
  34. 34.
    Mogilevskaya S.G., Crouch S.L.: A Galerkin boundary integral method for multiple circular elastic inclusions with homogeneously imperfect interfaces. Int. J. Solids Struct. 39, 4723–4746 (2002)CrossRefGoogle Scholar
  35. 35.
    Kong F., Yao Z., Zheng X.: BEM for simulation of a 2D elastic body with randomly distributed circular inclusions. Acta Mechanica Solida Sinika 15, 81–88 (2002)Google Scholar
  36. 36.
    Tan C.L., Gao Y.L., Afagh F.F.: Anisotropic stress analysis of inclusion problems using the boundary integral equation method. J Strain Anal. 27, 67–76 (1992)CrossRefGoogle Scholar
  37. 37.
    Dong C.Y., Lo S.H., Cheung Y.K.: Stress analysis of inclusion problems of various shapes in an infinite anisotropic elastic medium. Comput. Methods Appl. Mech. Eng. 192, 683–696 (2003)CrossRefzbMATHGoogle Scholar
  38. 38.
    Dong C.Y.: The integral equation formulations of an infinite elastic medium containing inclusions, cracks and rigid lines. Eng. Fract. Mech. 75, 3952–3965 (2008)CrossRefGoogle Scholar
  39. 39.
    Venturini W.S.: Alternative formulations of the boundary element method for potential and elastic zoned problems. Eng. Anal. Bound. Elem. 9, 203–207 (1992)CrossRefGoogle Scholar
  40. 40.
    Dominguez J.: Boundary Elements in Dynamics. Elsevier, New York (1993)zbMATHGoogle Scholar
  41. 41.
    MATLAB: The Language of Technical Computing, Version 7.9. The MathWorks, Inc., Natick, Massachusetts (2009)Google Scholar
  42. 42.
    SAP 2000: Integrated Finite Element Analysis and Design of Structures, Version 14.0. Computers and Structures, Inc., Berkeley, California (2008)Google Scholar
  43. 43.
    ANSYS Release 10.0. Structural Mechanics Package, Canonsburg, Pennsylvania (2009)Google Scholar
  44. 44.
    Tsui C.P., Chen D.Z., Tang C.Y., Uskokovic P.S., Fan J.P., Xie X.L.: Prediction for debonding damage process and effective elastic properties of glass-bead filled modified polyphenylene oxide. Compos. Sci. Technol. 66, 1521–1531 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  • S. L. Parvanova
    • 1
  • P. S. Dineva
    • 2
  • G. D. Manolis
    • 3
    Email author
  1. 1.Department of Civil EngineeringUniversity of Architecture, Civil Engineering and Geodesy (UACEG)SofiaBulgaria
  2. 2.Institute of Mechanics, Bulgarian Academy of Mechanics (BAS)SofiaBulgaria
  3. 3.Department of Civil EngineeringAristotle University of Thessaloniki (AUTH)ThessalonikiGreece

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