Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T13:45:21.125Z Has data issue: false hasContentIssue false
  • English
  • Français

Elastic wave fields generated by scalar wave functions

Published online by Cambridge University Press:  24 October 2008

P. Chadwick  and
E. A. Trowbridge
P. Chadwick
Affiliation:
School of Mathematics and Physics, University of East Anglia
E. A. Trowbridge
Affiliation:
Department of Mathematics, Lanchester College of Technology, Coventry
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we obtain a representation of elastic wave fields (i.e. solutions of the equations of motion of classical elastokinetics) in terms of three functions satisfying scalar wave equations. Although the form of the representation implies no restriction upon the choice of coordinate system or upon the shape of the elastic body, it is found that the result can only be applied advantageously to initial-boundary-value problems having spherical polar coordinates as a natural frame of reference. The representation is shown to be complete in the sense that every (sufficiently smooth) elastic wave field in a homogeneous, isotropic body bounded by two concentric spheres can be expressed in the given form.

Under conditions of axial symmetry the representation generates wave fields which can be decomposed into poloidal and toroidal constituents, the former arising from two scalar wave functions and comprising both dilatational and rotational waves, and the latter being associated with a single scalar wave function and a state of pure shear of the elastic solid. Finally, the representation is used to obtain a formal solution describing the elastic pulse generated in an infinite body by the application of time-dependent tractions to the surface of a spherical cavity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 4 , October 1967 , pp. 1177 - 1187
Copyright
Copyright © Cambridge Philosophical Society 1967

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Sternberg, E.Arch. Rational Mech. Anal. 6 (1960), 34–50.CrossRefGoogle Scholar
(2)Sternberg, E. and Gurtin, M. E.Proc. 4th U.S. Nat. Congr. Appl. Mech. (1962), vol. 2, 793–797. Amer. Soc. Mech. Engrs., New York.Google Scholar
(3)Morse, P. M. and Feshbach, H.Methods of theoretical physics, Part 2 (McGraw-Hill, New York, etc., 1953).Google Scholar
(4)Love, A. E. H.A treatise on the mathematical theory of elasticity, 4th edn. (Cambridge University Press, 1927).Google Scholar
(5)Einspruch, N. G., Witterholt, E. J. and Truell, R.J. Appl. Phys. 31 (1960), 806–818.CrossRefGoogle Scholar
(6)Allen, D. E. and Robinson, A. R.Technical Report. Contract Nonr 1834 (03), Office of Naval Research. University of Illinois, Urbana, 1966.Google Scholar
(7)Hook, J. F.J. Acoust. Soc. Amer. 33 (1961), 302–313, 967.CrossRefGoogle Scholar
(8)Hook, J. F.J. Acoust. Soc. Amer. 34 (1962), 354–355, 946–953.CrossRefGoogle Scholar
(9)Backus, G.Ann. Physics 4 (1958), 372–447.CrossRefGoogle Scholar
(10)Chadwick, P. and Trowbridge, E. A.Proc. Cambridge Philos. Soc. 63 (1967), 1189–1227.CrossRefGoogle Scholar
(11)Trowbridge, E. A. Thesis, University of Sheffield, 1964.Google Scholar
(12)Eringen, A. C.Quart. J. Mech. Appl. Math. 10 (1957), 257–270.CrossRefGoogle Scholar
(13)Chadwick, P. and Powdrill, B.Int. J. Engng. Sci 3 (1965), 561–595.CrossRefGoogle Scholar
(14)Bers, L., John, F. and Schechter, M.Partial differential equations (Interscience, New York, etc., 1964).Google Scholar
(15)Hellwig, G.Partial differential equations (Blaisdell, New York, etc., 1964).Google Scholar
(16)Todhunter, I. and Pearson, K.A history of the theory of elasticity and of the strength of materials, vol. 1 (Cambridge University Press, 1886).Google Scholar
(17)Hopkins, H. G.Progress in solid mechanics (eds. Sneddon, I. N. and Hill, R.), vol. 1, pp. 83–164 (North-Holland, Amsterdam, 1960).Google Scholar
(18)Jeffreys, H.Mon. Not. Roy. Astron. Soc. Geophys. Suppl. 2 (1931), 407–416.CrossRefGoogle Scholar
(19)Tupholme, G. E. Thesis, University of East Anglia, 1967.Google Scholar
(20)Tupholme, G. E.Proc. Cambridge Philos. Soc. Forthcoming.Google Scholar