We construct new basic solutions of equilibrium equations of a transversally isotropic paraboloid of rotation and obtain summation theorems that express them in terms of cylindrical solutions for a half-space and vice versa. The problem of the action of an axial concentrated force on an elastic transversally isotropic half-space with a fixed inclusion in the form of a paraboloid of rotation is investigated. The problem is solved by a generalized Fourier method and is reduced to a system of integral equations with a Fredholm operator on the condition that intersections of the boundaries of the half-space and inclusion are absent. We consider dependences of stresses on the shape of the paraboloidal inclusion and on the distance between the boundary surfaces and analyze the results of the calculation.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 52, No. 3, pp. 160–169, July–September, 2009.
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Nikolaev, A.G., Shcherbakova, Y.A. Apparatus and applications of a generalized Fourier method for transversally isotropic bodies bounded by a plane and a paraboloid of rotation. J Math Sci 171, 620–631 (2010). https://doi.org/10.1007/s10958-010-0162-0
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DOI: https://doi.org/10.1007/s10958-010-0162-0