Abstract
Variational principles are derived for the analysis of dynamical phenomena associated with spherical inclusions embedded in homogeneous isotropic elastic solids. The starting point is Hamilton’s principle, with the material properties assumed to vary only with the radial distance r from the origin. Attention is restricted to disturbances that are symmetric about the polar (z) axis, such that the nonzero displacement components in spherical coordinates, u r and u θ, are independent of the polar coordinate φ. The symmetry allows for a decoupling of the polar components, the nth of which is described by U r, n (r, t)P n (cosθ) and U θ, n (r, t)dP n /dθ. A variational principle is subsequently derived for the field quantities U r, n and U θ, n . Concepts analogous to those of the theory of matched asymptotic expansions are used to embellish the principle in order to allow for the damping associated with the outward radiation of elastic waves. Examples illustrating the use of the variational principle for formulating plausible lumped-parameter models are given for the cases of n = 0 and n = 1.
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References
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 7: Theory of Elasticity, 3rd ed. (Nauka, Moscow, 1965; Pergamon, Oxford, 1970) (The result of interest is attributed to work in 1949 by M. A. Isakovich).
E. Meyer, K. Brendel, and K. Tamm, J. Acoust. Soc. Am. 30, 1116 (1958).
D. V. Sivukhin, Sov. Phys. Acoust. 1, 82 (1955).
C. F. Ying and R. Truell, J. Appl. Phys. 27, 1086 (1956).
K. F. Herzfeld, Philos. Mag. 9, 741 (1930).
L. Knopoff, Geophysics 24, 30 (1959).
N. G. Einspruch and R. Truell, J. Acoust. Soc. Am. 32, 214 (1960).
G. Johnson and R. Truell, J. Appl. Phys. 36, 3466 (1965).
C. C. Mow, J. Appl. Mech. 32, 637 (1965).
F. R. Norwood and J. Miklowitz, J. Appl. Mech. 34, 735 (1967).
G. C. Gaunaurd and H. Überall, J. Acoust. Soc. Am. 63, 1699 (1978).
V. N. Alekseev and S. A. Rybak, Acoust. Phys. 45, 535 (1999).
J. S. Allen and R. A. Roy, J. Acoust. Soc. Am. 107, 3167 (2000).
D. B. Khismatullin and A. Nadim, Phys. Fluids 14, 3534 (2002).
Yu. M. Zaslavskii, Acoust. Phys. 50, 46 (2004).
S. Y. Emelianov, M. F. Hamilton, Y. A. Ilinskii, and E. A. Zabolotskaya, J. Acoust. Soc. Am. 115, 581 (2004).
I. S. Sokolnikoff, Mathematical Theory of Elasticity (McGraw-Hill, New York, 1956), pp. 177–184.
A. D. Pierce, in Physical Acoustics: Principles and Methods, Ed. by W. P. Mason (Academic, San Diego, 1984; Mir, Moscow, 1984), Vol. 22, pp. 195–371.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge Univ. Press, Cambridge, 1927), p. 309.
M. van Dyke, Perturbation Methods in Fluid Mechanics (Academic, New York, 1964; Mir, Moscow, 1967), pp. 77–97.
J. N. Goodier, J. Appl. Mech., Paper APM-55-7, 39 (1933).
V. C. Anderson, J. Acoust. Soc. Am. 22, 426 (1950).
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From Akusticheski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Zhurnal, Vol. 51, No. 1, 2005, pp. 9–23.
Original English Text Copyright © 2005 by Pierce.
This article was submitted by the author in English.
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Pierce, A.D. Congratulations to Akusticheskii Zhurnal on the occasion of its 50th anniversary. Vibrations of spherical inclusions in elastic solids. Acoust. Phys. 51, 5–10 (2005). https://doi.org/10.1134/1.1851623
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DOI: https://doi.org/10.1134/1.1851623