The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on March 21, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(2):139-160; doi:10.1093/qjmam/hbm002
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A general procedure for solving boundary-value problems of elastostatics for a spherical geometry based on Love's approach
( Daimler-Chrysler Corporation, Auburn Hills, MI 48326, USA )
( Department of Civil & Structural Engineering, University of Sheffield, Sheffield S1 3JD )
Corresponding author. Now at the Boeing Company, Seattle, Washington, USA mujibur.rahman{at}boeing.com
t.michelitsch{at}sheffield.ac.uk
Received 12 June 2006. Revise 12 December 2006. Accepted 3 January 2007.
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Abstract |
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We develop a general procedure for solving the first and second fundamental problems of the theory of elasticity for cases where boundary conditions are prescribed on a spherical surface, using Love's general solution of the elastostatic equilibrium equations in terms of three scalar harmonic functions. It is shown that this general solution combined with a methodology by Brenner paves an elegant way to determine the three harmonic functions in terms of the boundary data. Thus, with this general scheme, solution of any such boundary-value problem is reducible to a routine exercise thereby providing some `economy of effort'. Furthermore, we develop a similar general scheme for thermoelastic problems for cases when temperature type boundary conditions are prescribed on a spherical surface. We then illustrate the application of the procedure by solving a number of problems concerning rigid spherical inclusions and spherical cavities. In particular, apart from furnishing alternative solutions to the known problems, we demonstrate the use of this general procedure in solving the problem of interaction of a rigid spherical inclusion with a concentrated moment and that of a concentrated heat source situated at an arbitrary point outside the inclusion. We also derive closed-form expressions for the net force and the net torque acting on a rigid spherical inclusion embedded into an infinite elastic solid under an ambient displacement field characterized by an arbitrary-order polynomial in the Cartesian coordinates. To the best of our knowledge, these results are new.