Journal Article

A general procedure for solving boundary-value problems of elastostatics for a spherical geometry based on Love's approach

M Rahman and T Michelitsch

in The Quarterly Journal of Mechanics and Applied Mathematics

Volume 60, issue 2, pages 139-160
Published in print May 2007 | ISSN: 0033-5614
Published online March 2007 | e-ISSN: 1464-3855 | DOI: https://dx.doi.org/10.1093/qjmam/hbm002

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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.

Journal Article.  5968 words. 

Subjects: Applied Mathematics ; Mathematical and Statistical Physics