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The forced two dimensional oscillations of a rigid strip in smooth contact with a semi-infinite elastic solid

Published online by Cambridge University Press:  24 October 2008

G. R. Wickham
G. R. Wickham
Affiliation:
University of Manchester
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Abstract

In this paper we investigate a time harmonic plane strain problem in a homogeneous isotropic semi-infinite elastic solid with mixed boundary conditions on its surface. The boundary is free from applied stress apart from on an infinite strip where the tangential components of stress are zero, and the normal component of displacement is a prescribed function. By reducing the mixed boundary value problem to the solution of a Fredholm integral equation of the second kind, an existence theorem is proved and a simple low frequency asymptotic formula relating the normal stress to the prescribed displacement is derived rigorously. The results are applied to the forced heaving and rocking motion of a rigid cylindrical body in smooth contact with an elastic half space. Some further applications to scattering problems are suggested.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 2 , March 1977 , pp. 291 - 311
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Awojobi, A. O.Harmonic rocking of a rigid rectangular body on a semi-infinite elastic medium. Jl. Appl. Mech. 33, Trans. ASME 88, Series E (1966), 547–552.Google Scholar
(2)Awojobl, A. O. and Grootenhuis, P.Vibration of rigid bodies on semi-infinite elastic media. Proc. Roy. Soc. 287 Series A (1965), 27–63.Google Scholar
(3)Bycroft, G. N.Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum. Phil. Trans. Roy. Soc. 287 Series A (1956), 327–368.Google Scholar
(4)Collins, W. D.The forced torsional oscillation of an elastic half-space and an elastic stratum. Proc. London Math. Soc. 12 No. 46 (1962), 226–244.Google Scholar
(5)Copson, E. T.Asymptotic expansions (Cambridge, 1965).CrossRefGoogle Scholar
(6)Gregory, R. D.The non-existence of standing modes in certain problems of linear elasticity. Proc. Cambridge Philos. Soc. 77 (1974), 385–404.Google Scholar
(7)Gregory, R. D.The propagation of waves in an elastic half space containing a cylindrical cavity. Proc. Cambridge Philos. Soc. 67 (1970), 689–710.CrossRefGoogle Scholar
(8)Karasudhi, P., Keer, L. M. and Lee, S. L.Vertical motion of a body on an elastic half plane. Jl. Appl. Mech. 35, Trans. ASME, 90, Series E (1968), 697–705.CrossRefGoogle Scholar
(9)Lamb, H.On the propagation of tremors over the surface of an elastic solid. Phil. Trans. Roy. Soc. 203 Series A (1904), 1–42.Google Scholar
(10)Lapwood, E. R.The disturbance due to a line source in a semi-infinite elastic medium. Phil. Trans. Roy. Soc. 242 Series A (1949), 63–100.Google Scholar
(11)Miller, G. F. and Pursey, H.The field and radiation impedance of mechanical radiators on the surface of a semi-infinite isotropic solid. Proc. R. Soc. 223 Series A (1954), 521–541.Google Scholar
(12)Noble, B.Electromagnetic waves, ed. Langer, R. E. (University of Wisconsin Press, 1962).Google Scholar
(13)Reissner, E.Stationäere axialsymmetrische, durch ire schüttelnde Masse erregte Schwingungen eines homogenen elastischen Halbraumes. Ingenieur-Archiv 7 (1936), 381–396.Google Scholar
(14)Robertson, I. A.Forced vertical vibration of a rigid circular disc on a semi-infinite elastic solid. Proc. Cambridge Philos. Soc. 62 (1966), 547–553.Google Scholar
(15)Sagoci, H. F.Forced torsional oscillations of an elastic half space. J. Appl. Phys. 15 (1944), 652–654.Google Scholar
(16)Shail, R. and Wickham, G. R.The torsional oscillations of a rigid spherical inclusion embedded in an elastic stratum. Journal of Elasticity 2 (1972), 323–333.Google Scholar
(17)Smithies, F.Integral equations (Cambridge, 1962).Google Scholar
(18)Sneddon, I. N.Fourier transforms (McGraw-Hill, 1951).Google Scholar
(19)Stallybrass, M. P.On the Reissner-Sagoci problem at high frequencies. Int. J. Engrg. Sci. 5 (1967), 689–703.Google Scholar
(20)Thomas, D. P.Torsional oscillations of an elastic half space. Quart. J. Mech. Appl. Math. 21 (1968), 51–65.CrossRefGoogle Scholar
(21)Ursell, F.Slender oscillating ships at zero forward speed. Journal of Fluid Mechanics 14 (1962), 496–516.CrossRefGoogle Scholar
(22)Watson, G. N.A treatise on the theory of Bessel functions, 2nd ed. (Cambridge, 1944).Google Scholar
(23)Wickham, G. R. Ph.D. Thesis, University of Surrey (1970).Google Scholar
(24)Williams, W. E.Slow torsional oscillations of a rigid inclusion in an elastic medium. Quart. J. Mech. Appl. Math. 24 (1971).Google Scholar