Abstract
Historically, the vector Navier equation governing the dynamic response of an elastic, homogeneous, isotropic sphere has been solved using the Helmholtz decomposition of the displacement vector. Further, many of the problems in the literature have been restricted to ones involving axisymmetric geometry. In this presentation, the time-dependent Navier equation is solved using a set of vector spherical harmonics which, previously, has been used primarily in quantum mechanics studies but which seems particularly useful in solving asymmetric problems with nonconservative body forces. Expressions for the displacements, strains, and stresses and a discussion of the vibrations of an elastic sphere are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H.B. McClung, The elastic sphere under nonsymmetric loading.J. Elasticity 21 (1989) 1–26.
Cemal A. Eringen and S. Suhubi,Elastodynamics. Vol. II, Linear Theory. Academic Press, New York, N.Y. (1975).
A.H. Shah, C.V. Ramkrishnan and S.K. Datta, Three-dimensional and shell-theory analysis of elastic waves in a hollow sphere. Part 1.Journal of Applied Mechanics (September 1969) pp. 431–439.
H.A. Antosiewicz, Bessel functions of fractional order. In:Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Appl. Math. Ser. No. 55 (M. Abramowitz and I. Stegun, eds.), Sect. 10, U.S. Govt. Printing Office, Washington, D.C. (1964).
Y.H. Pao and C.C. Mow,Diffraction of Elastic Waves and Dynamic Stress Concentrations. Crane, Russak, and Co., New York, N.Y. (1973).
E.W. Hobson,The Theory of Spherical and Ellipsoidal Harmonics. Chelsea, New York, N.Y. (1965).
G.N. Watson,A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge Univ. Press, London and New York (1966).
E.L. Hill, The theory of vector spherical harmonics.Am. J. of Physics 22 (1954) 211–214.
J.M. Blatt, and V.F. Weisskopf,Theoretical Nuclear Physics. New York, N.Y. Wiley (1952).
R.M. Grey and A.C. Eringen, The elastic sphere under dynamic and impact loads. ONR Tech. Rep. No. 8 (1955) Purdue Univ., Lafayette, Indiana.
P.M. Morse and H. Feshbach,Methods of Mathematical Physics, Vol II. New York, N.Y. McGraw-Hill (1953).
George Arfken,Mathematical Methods for Physicists. New York, N.Y., Academic Press (1966).
G.L. Hill and R. Lanshoff, Dirac electron theory,Revs. Modern Physics, 10, 87 (1938) 118–123.
Larry C. Andrews,Special Functions for Engineers and Applied Mathematicians. New York, N.Y., Macmillan Publishing Company (1985).
Herbert Reismann, On the forced motion of elastic solids.Appl. Sci. Res. 18 (Sept. 1967) 156–165.
Author information
Authors and Affiliations
Additional information
Part of the material presented here was developed while the author was on a Developmental Leave at the University of Texas at Austin.