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

We develop a general procedure for solving the first and secondfundamental problems of the theory of elasticity for cases whereboundary conditions are prescribed on a spherical surface, usingLove's general solution of the elastostatic equilibrium equationsin terms of three scalar harmonic functions. It is shown thatthis general solution combined with a methodology by Brennerpaves an elegant way to determine the three harmonic functionsin terms of the boundary data. Thus, with this general scheme,solution of any such boundary-value problem is reducible toa routine exercise thereby providing some `economy of effort'.Furthermore, we develop a similar general scheme for thermoelasticproblems for cases when temperature type boundary conditionsare prescribed on a spherical surface. We then illustrate theapplication of the procedure by solving a number of problemsconcerning rigid spherical inclusions and spherical cavities.In particular, apart from furnishing alternative solutions tothe known problems, we demonstrate the use of this general procedurein solving the problem of interaction of a rigid spherical inclusionwith a concentrated moment and that of a concentrated heat sourcesituated at an arbitrary point outside the inclusion. We alsoderive closed-form expressions for the net force and the nettorque acting on a rigid spherical inclusion embedded into aninfinite elastic solid under an ambient displacement field characterizedby an arbitrary-order polynomial in the Cartesian coordinates.To the best of our knowledge, these results are new.