International Applied Mechanics

, Volume 48, Issue 5, pp 487–551 | Cite as

Nonstationary contact of a rigid body with an elastic medium: plane problem (review)


The developed approaches to and results of studying the impact of rigid and deformable blunt bodies on an elastic half-space or a cavity in an elastic medium are stated using a plane problem of linear elasticity


elasticity plane problem impact blunt body stress state wave process 


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  1. 1.
    M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (1972).MATHGoogle Scholar
  2. 2.
    H. Bateman and A. Erdelyi, Higher Transcendental Functions, 2. Bessel functions, Parabolic Cylinder Functions, Orthogonal Polynomials, McGraw-Hill, New York (1953).Google Scholar
  3. 3.
    H. Bateman and A. Erdelyi, Fourier, Laplace, and Mellin Transforms, Vol. 1 of the two-volume series Tables of Integral Transforms [Russian translation], Nauka, GIFML, Moscow (1969).Google Scholar
  4. 4.
    H. Bateman and A. Erdelyi, Hankel Transform, Vol. 2 of the two-volume series Tables of Integral Transforms [Russian translation], Nauka, GIFML, Moscow (1969).Google Scholar
  5. 5.
    W. Goldsmith, Impact: The Theory and Physical Behavior of Colliding Solids, E. Arnold (Publ.) Ltd., London (1960).Google Scholar
  6. 6.
    W. Goldsmith, “Impact at intermediate velocities involving contact phenomena,” in: K. Vollrath and G. Thomer (eds.), High-Speed Physics, Springer-Verlag, New York (1967).Google Scholar
  7. 7.
    A. G. Gorshkov and D. V. Tarlakovskii, Dynamic Contact Problems with Moving Boundaries [in Russian], Nauka. Fizmatlit, Moscow (1995).Google Scholar
  8. 8.
    A. N. Guz, V. D. Kubenko, and M. A. Cherevko, Diffraction of Elastic Waves [in Russian], Naukova Dumka, Kyiv (1978).Google Scholar
  9. 9.
    V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, Pergamon Press, Now York (1965).MATHGoogle Scholar
  10. 10.
    V. A. Ditkin and A. P. Prudnikov, Handbook of Operational Calculus [in Russian], Vysshaya Shkola, Moscow (1965).Google Scholar
  11. 11.
    K. L. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge (1985).MATHGoogle Scholar
  12. 12.
    J. A. Zukas, T. Nicholas, H. F. Swift, L. B. Greszczuk, and D. R. Curran, Impact Dynamics, Wiley, New York (1982).Google Scholar
  13. 13.
    N. A. Kil’chevskii, Dynamic Contact Compression of Solids: An Impact [in Russian], Naukova Dumka, Kyiv (1976).Google Scholar
  14. 14.
    G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, and Formulas for Reference and Review, McGraw-Hill, New York (1984).MATHGoogle Scholar
  15. 15.
    B. V. Kostrov, “Indentation of a rigid punch into an elastic half-space: Self-similar mixed dynamic problems,” Izv. AN SSSR, OTN, Mekh. Mashinostr., No. 4, 54–62 (1964).Google Scholar
  16. 16.
    V. D. Kubenko, Nonstationary Interaction of Structural Members with a Medium [in Russian], Naukova Dumka, Kyiv (1979).Google Scholar
  17. 17.
    V. D. Kubenko, Penetration of Elastic Shells into a Compressible Fluid [in Russian], Naukova Dumka, Kyiv (1981).Google Scholar
  18. 18.
    V. D. Kubenko, “An approach to studying the stress distribution over the surface of a blunt body penetrating an elastic medium,” Teor. Prikl. Mekh., 17, 14–21 (1986).Google Scholar
  19. 19.
    V. D. Kubenko and S. N. Popov, “Two-dimensional problem of the impact of a rigid blunt body onto the surface of an elastic halfspace,” Int. Appl. Mech., 24, No. 7, 693–700 (1988).ADSMATHGoogle Scholar
  20. 20.
    V. D. Kubenko, “Impact interaction of bodies with a medium (survey),” Int. Appl. Mech., 33, No. 8, 933–957 (1997).MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    V. D. Kubenko, “Local wave theory of elastic-body collision: Plane problem in the ideal-liquid approximation,” Int. Appl. Mech., 34, No. 10, 997–1006 (1998).ADSCrossRefGoogle Scholar
  22. 22.
    V. D. Kubenko and T. A. Marchenko, “Collision of blunt elastic bodies made of the same material: A plane problem,” Vest. Donetsk. Univ., Ser. A, Estestv. Nauki, No. 1, 102–108 (2002).Google Scholar
  23. 23.
    V. D. Kubenko and T. A. Marchenko, “Plane collision problem for two identical elastic parabolic bodies—Direct central impact,” Int. Appl. Mech., 39, No. 7, 812–821 (2003).MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    V. D. Kubenko, “Impact of blunted bodies on a liquid or elastic medium,” Int. Appl. Mech., 40, No. 11, 1185–1225 (2004).ADSCrossRefGoogle Scholar
  25. 25.
    V. D. Kubenko and T. A. Marchenko, “Direct central collision of elastic cylindrical bodies,” Teor. Prikl. Mekh., 40, 161–168 (2005).Google Scholar
  26. 26.
    V. D. Kubenko, “Impact of a long thin body on a cylindrical cavity in liquid: A plane problem,” Int. Appl. Mech., 42, No. 6, 636–654 (2006).ADSCrossRefGoogle Scholar
  27. 27.
    V. D. Kubenko, T. A. Marchenko, and E. I. Starovoitov, “Determining the stress state of a plane elastic layer struck by a blunt rigid body,” Dokl. NAN Ukrainy, No. 8, 47–57 (2006).Google Scholar
  28. 28.
    V. D. Kubenko, “Wave processes in a perfect liquid layer impacted on by a blunted solid,” Int. Appl. Mech., 43, No. 3, 272–281 (2007).MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    V. D. Kubenko and T. A. Marchenko, “Nonstationary indentation of a rigid blunt indenter into an elastic layer: A plane problem,” Int. Appl. Mech., 44, No. 3, 286–295 (2008).MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    V. D. Kubenko and T. A. Marchenko, “Collision of dissimilar blunt elastic bodies: A plane problem,” Int. Appl. Mech., 45, No. 2, 139–147 (2009).MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    V. D. Kubenko, V. V. Gavrilenko, and A. Ya. Zhuk, “Nonstationary contact problem of elasticity (matched and unmatched surfaces),” Metody Rozv’yaz. Prikl. Zadach Mekh. Tverd. Tela, No. 10, 162–178 (2009).Google Scholar
  32. 32.
    V. D. Kubenko, “Impact of blunt bodies on a liquid or an elastic medium,” in: Vol. 5 of the six-volume series Advances in Mechanics [in Russian], Litera, Kyiv (2009), pp. 566–607.Google Scholar
  33. 33.
    V. D. Kubenko, “Wave processes in an elastic half-plane struck by a blunt solid,” Mekh. Tverd. Tela, No. 2, 118–129 (2011).Google Scholar
  34. 34.
    C. Lanczos, Applied Analysis, Englewood Cliffs, Prentice Hall (1956).Google Scholar
  35. 35.
    G. V. Logvinovich, Hydrodynamics of Free-Boundary Flows, Israel Program for Scientific Translations, Jerusalem (1972).MATHGoogle Scholar
  36. 36.
    P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, McGraw-Hill, New York (1953).Google Scholar
  37. 37.
    V. B. Poruchikov, Methods of Dynamic Elasticity [in Russian], Nauka, Moscow (1986).Google Scholar
  38. 38.
    A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 2: Special Functions, Gordon and Breach, New York (1986).Google Scholar
  39. 39.
    Yu. N. Savchenko, Yu. D. Vlasenko, and V. N. Semenenko, “Experimental research into high-speed cavitation flows,” Gidromekh., 72, 103–111 (1998).MATHGoogle Scholar
  40. 40.
    Yu. N. Savchenko, V. N. Semenenko, and S. I. Putilin, “Nonstationary processes in the supercavitation of bodies,” Prikl. Gidromekh. No. 1, 79–97 (1999).Google Scholar
  41. 41.
    A. Ya. Sagomonyan, Penetration [in Russian], MGU, Moscow (1974).Google Scholar
  42. 42.
    R. W. Hamming, Numerical Methods for Scientists and Engineers, McGraw-Hill, New-York (1962).MATHGoogle Scholar
  43. 43.
    L. I. Slepyan, Nonstationary Elastic Waves [in Russian], Sudostroenie, Leningrad (1981).Google Scholar
  44. 44.
    I. I. Vorovich and V. M. Alexandrov (eds.), Contact Mechanics [in Russian], Fizmatlit, Moscow (2001).Google Scholar
  45. 45.
    S. Ashley, “Warp drive underwater,” Scientific American, No. 5, 50–60 (2001).Google Scholar
  46. 46.
    A. E. Babaev and I. V. Yanchevskii, “Determining the shock load on an electroelastic bimorph beam with split conductive coatings,” Int. Appl. Mech., 46, No. 9, 1019–1026 (2010).CrossRefGoogle Scholar
  47. 47.
    L. Cagniard, Reflection and Refraction of Progressive Seismic Waves, McGraw-Hill, New York (1962).MATHGoogle Scholar
  48. 48.
    S. D. Gartsman, A. A. Zhukov, and I. I. Karpukhin, “Determining the parameters of the transverse impact of a ball on an elastic beam in the Timoshenko problem,” Int. Appl. Mech., 46, No. 9, 1050–1055 (2010).CrossRefGoogle Scholar
  49. 49.
    B. Kebli, G. Ya. Popov, and N. D. Vaisfel’d, “Dynamics of a truncated elastic cone,” Int. Appl. Mech., 46, No. 11, 1284–1291 (2010).CrossRefGoogle Scholar
  50. 50.
    V. D. Kubenko and M. V. Ayzenberg-Stepanenko, “Impact indentation of a rigid body into an elastic layer. Analytical and numerical approaches,” Mat. Metody Fiz.-Mekh. Polya, 51, No. 2, 61–74 (2008).MATHGoogle Scholar
  51. 51.
    V. D. Kubenko and O. V. Gavrilenko, “Impact interaction of cylindrical body with a surface of cavity during supercavitation motion in compressible liquid,” J. Fluids Struct., 25, 794–814 (2009).ADSCrossRefGoogle Scholar
  52. 52.
    E. L. Paryshev, “The plane problem of immersion of an expanding cylinder through a cylindrical free surface of variable radius,” in: Proc. Int. Summer Sci. School on High Speed Hydrodynamics, Cheboksary (Russia), June (2002), pp. 277–283.Google Scholar
  53. 53.
    A. R. Robinson and J. S. Thompson, “Transient stresses in an elastic half space resulting from frictionless indentation of a rigid wedge-shaped die,” Z. Angew. Math. Mech., 54, No. 3, 139–144 (1974).MATHCrossRefGoogle Scholar
  54. 54.
    Y. N. Savchenko, “High-speed body motion at supercavitating flow,” in: Proc. 3rd Int. Symp. on Cavitation, Grenoble, France (1998), pp. 9–14.Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine

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