Abstract
A rigorous study of the sector problem is presented by using the Mellin transform technique. The stress function is obtained as an asymptotic expansion of the complex inversion integral. The number of terms of this expansion, as well as the differentiability of the stress function, depend on the differential properties of boundary conditions on the radial edges. If these boundary conditions belong toC ∞, this asymptotic expansion is transformed to a uniformly convergent infinite series. The coefficients of the series, which depend only on the boundary conditions along the circumferential edges, are calculated by applying a bi-orthogonality condition, or, by a technique based on the Betti formula.
Résumé
En utilisant la technique de la transformation de Mellin on présente une étude rigoureuse du problème d'un secteur. La fonction des contraintes est obtenue par un développement asymptotique de l'intégral complexe d'inversion. Le nombre de termes de ce développement, ainsi que la differentiabilité de la fonction des contraintes, dépendent de proprietés différentielles des conditions aux limites sur les bords radiaux. Si ces conditions aux limites sont dansC ∞, ce développement asymptotique est transformé en série infinie uniformement convergente. Les coefficients de cette série, qui dépendent uniquement de conditions aux limites sur le bord circulaire, sont calculés à l'aide d'une condition de bi-orthogonalité que nous démontrons, ou, à l'aide d'une technique basée sur la formule de Betti.
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Tsamasphyros, G., Theocaris, P.S. On the solution of the sector problem. J Elasticity 9, 271–281 (1979). https://doi.org/10.1007/BF00041099
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DOI: https://doi.org/10.1007/BF00041099