Geodesic
Stress state of elastic thick-walled ring with self-balanced pressures distributed on its internal and external borders
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2015), pp. 45-56.

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For the first time with the help of the theory of analytic functions and Kolosov–Muskhelishvili formulas the problem of the two-dimensional theory of elasticity for a thick-walled ring with the uneven pressures, acting on its borders, was solved. The pressure on the inner and outer boundaries is represented by Fourier series. The authors represent the two complex functions which solve boundary problem in the form of Laurent series. The logarithmic terms in these series are absent because the boundary problem has the self-balancing loads on each boundary of ring. The coefficients in the Laurent series are calculated by the boundary conditions. Firstly, the equations were obtained in the general form. But the hypothesis about even distributions of pressures at borders of ring was used for constructing an example. It leads to the fact that all coefficients of analytic functions represented in Laurent series have to be only real. As a solving example, the representation of pressures in equivalent hypotrochoids was used. The application of the computer algebra system Mathematica greatly simplifies the calculation of the distribution of stresses and displacements in ring. It does not require manual formal separation of real and imaginary parts in terms of Kolosov–Muskhelishvili to display the distribution of the physical parameters. It separates them only for calculated numbers with the help of built-in functions.
Keywords: analytic functions, Kolosov–Muskhelishvili formulas, complex numbers, elastic thick-walled ring, uneven pressure. 
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A. S. Kravchuk; A. I. Kravchuk. Stress state of elastic thick-walled ring with self-balanced pressures distributed on its internal and external borders. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2015), pp. 45-56. http://geodesic.mathdoc.fr/item/VVGUM_2015_4_a5/

[1] Yu.\;A. Amenzade, The Theory of Elasticity, Vysshaya shkola Publ., M., 1976, 272 pp. | MR

[2] N.\;I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Nauka Publ., M., 1966, 708 pp.