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Nonstationary distributed axisymmetric load on an elastic half-space

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Abstract

This paper presents analytical and numerical-analytic approaches to solving the problem of the action of an arbitrarily distributed axisymmetric load applied instantly to the surface of an isotropic elastic half-space. The first approach is built around the Laplace and Hankel integral transforms whose inversion is performed jointly with Cagniard’s technique, and as a result, exact analytical expressions are obtained for computing stresses along an axis of symmetry. The second approach uses the Laplace integral transform and the expansion of sought for values into the Fourier–Bessel series to reduce the problem to a numerical solution of a series of Volterra integral equations. Concrete numerical analysis was performed for cases where the domain of application of a distributed load is fixed or expands in time with both constant and variable velocity.

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References

  1. De A, Roy A (2012) Transient response of an elastic half-space to normal pressure acting over a circular area on an inclined plane. J Eng Math 74:119–141

    Article  MathSciNet  MATH  Google Scholar 

  2. Gorshkov AG, Tarlakovskii DV (1995) Dynamic contact problems with moving boundaries. FIZMATLIT, Moscow, 352 p [in Russian]

  3. Kubenko VD, Marchenko TA (2004) Axisymmetric collision problem for two identical elastic solids of revolution. Int Appl Mech 40:766–775

    Article  Google Scholar 

  4. Kubenko VD, Ocharovich G, Ayzenberg-Stepanenko MV (2011) Impact indentation of a rigid body into an elastic layer; axisymmetric problem. J Math Sci 176:670–687

    Article  Google Scholar 

  5. Alexandrov VM, Vorovich II (2001) Mechanics of contact interactions. FIZMATLIT, Moscow, 672 p [in Russian]

  6. Lin X, Ballmann J (1995) A numerical scheme for axisymmetric elastic waves in solids. Wave Motion 21:115–126

    Article  MATH  Google Scholar 

  7. Laturelle FG (1989) Finite element analysis of wave propagation in an elastic half-space under step loading. Comput Struct 32:721–735

    Article  MATH  Google Scholar 

  8. Molotkov LA (1967) On the vibrations of a homogeneous elastic half-space under the action of a source applied to a uniformly expanding circular region. J Appl Math Mech 31:232–243

    Article  MATH  Google Scholar 

  9. Roy A (1979) Response of an elastic solid to non-uniformly expending surface loads. Int J Eng Sci 17:1023–1038

    Article  MATH  Google Scholar 

  10. Kutzenko AG, Ulitko AF, Oliynik VN (2001) Displacements of the elastic half-space surface caused by instantaneous axisymmetric loading. Int J Fluid Mech Res 28:258–273

    Article  Google Scholar 

  11. Ghosh SC (1970) Disturbance produced in an elastic half-space by impulsive normal pressure. Pure Appl Geophys 80:71–83

    Article  ADS  Google Scholar 

  12. Mitra M (1964) Disturbance produced in an elastic half-space by impulsive normal pressure. Math Proc Camb Philos Soc 60:683–696

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Duffy DG (2004) Transform methods for solving partial differential equations. Chapman & Hall/CRC Press, New York, 728 p

  14. Laturelle FG (1991) The stresses produced in an elastic half-space by a pressure pulse applied uniformly over a circular area: role of the pulse duration. Wave Motion 14:1–9

    Article  Google Scholar 

  15. Bresse LF, Hutchins DA (1989) Transient generation of elastic waves in solids by a disk-shaped normal force source. J Acoust Soc Am 86:810–817

    Article  ADS  Google Scholar 

  16. Singh SK, Kuo JT (1970) Response of an elastic half-space to uniformly moving circular surface load. Trans ASME J Appl Mech 37:109–115

    Article  MATH  Google Scholar 

  17. Kubenko VD (2004) Impact of blunted bodies on a liquid or elastic medium. Int Appl Mech 40:1185–1225

    Article  Google Scholar 

  18. Guz’ AN, Kubenko VD, Cherevko MA (1978) Diffraction of elastic waves. Sov Appl Mech 14:789–798

    Article  ADS  Google Scholar 

  19. Bateman H, Erdélyi A (1954) Tables of integral transforms [in two volumes]. McGraw-Hill, New York

  20. Wang Y-C, Murti V, Valliappan S (1992) Assessment of the accuracy of the Newmark method in transient analysis of wave propagation problems. Earthq Eng Struct Dyn 21:987–1004

    Article  Google Scholar 

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Authors and Affiliations

  1. S.P. Tymoshenko Institute of Mechanics of National Academy of Science of Ukraine, 3 Nesterova Str., Kiev, 03057, Ukraine

    Veniamin D. Kubenko & Igor V. Yanchevsky

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  1. Veniamin D. Kubenko
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  2. Igor V. Yanchevsky
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Correspondence to Igor V. Yanchevsky.

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Kubenko, V.D., Yanchevsky, I.V. Nonstationary distributed axisymmetric load on an elastic half-space. J Eng Math 96, 57–71 (2016). https://doi.org/10.1007/s10665-014-9778-2

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  • DOI: https://doi.org/10.1007/s10665-014-9778-2

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