Abstract
The Galerkin and Papkovich-Neuber solutions have the advantage of presenting a general solution to the problem of elasticity in a suitably compact notation, but they are not always the most convenient starting point for the solution of particular three-dimensional problems. If the problem has a plane of symmetry or particularly simple boundary conditions, it is often possible to develop a special solution of sufficient generality in one or two harmonic functions, which may or may not be components or linear combinations of components of the Papkovich-Neuber solution. For this reason, it is convenient to record detailed expressions for the displacement and stress components arising from the several terms separately and from certain related displacement potentials.
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Notes
J. Boussinesq, Application des potentiels à l’étude it l’équilbre ei du mouvement des solides élastiques, Gauthier-Villars, Paris (1885).
A.E. Green and W. Zerna, Theoretical Elasticity, 2nd.edn., Clarendon Press, Oxford, (1968), 165–176.
Other relations between the Boussinesq potentials are demonstrated by J.P. Bentham, Note on the Boussinesq-Papkovich stress functions, J.Elasticity, Vol. 9 (1979), 201–206.
A.E.H.Love, loc. cit.
A.E.Green and W.Zerna, loc. cit. §§5.8-5.10.
Notice that we denote the function of the complex conjugate ω¯(ξ¯) with a bar over the function as well as the argument. This is a redundant notation, but it permits us to drop the argument for brevity in lone derivations.
The complex variable method for plane strain problems in elasticity was developed simultaneously by Muskhelishvili in the Soviet Union and Stevenson and Green in England during the 1940s. The classical texts on this early work are N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, (English translation by J.R.M.Radok, Noordhoff, (1953) and A.E.Green and W.Zerna, Theoretical Elasticity, 2nd.edn., Clarendon Press, Oxford, (1968), Chapter 8. More approachable treatments for the engineering reader are given by A.H.England, Complex Variable Methods in Elasticity, John Wiley, London, (1971), S.P.Timoshenko and J.N.Goodier, loc. cit. Chapter 6 and D.S.Dugdale and C.Ruiz, Elasticity for Engineers, McGraw-Hill, London (1971), Chapter 2.
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© 1992 Springer Science+Business Media Dordrecht
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Barber, J.R. (1992). The Boussinesq Potentials. In: Elasticity. Solid Mechanics and Its Applications, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2454-6_16
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