Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 1, 2019 (March)
Pages 16-24
Cited by J. of Mech. Eng., 2019, vol. 22, no. 1, pp. 16-24



Olga I. Kyrylova, National University “Odessa Maritime Academy” (8, Didrikhson St., Odesa, 65029, Ukraine), e-mail:, ORCID: 0000-0002-9221-182X

Vsevolod H. Popov, National University “Odessa Maritime Academy” (8, Didrikhson St., Odesa, 65029, Ukraine), e-mail:, ORCID: 0000-0003-2416-642X



This paper solves the problem of determining the stress state near cracks in an infinite hollow cylinder of arbitrary cross section during longitudinal shear oscillations. We propose an approach that allows us to separately satisfy conditions both on the cracks and boundaries of a cylinder. The problem reduces to the equations of motion in a flat domain with the defects bounded by arbitrary smooth closed curves under anti-plane deformation conditions. The solution scheme is based on the use of discontinuous solutions to the equations of motion of an elastic medium with displacement jumps on the surfaces of defects. Displacements in a cylinder with defects are represented both as a sum of discontinuous solutions constructed for each defect and an unknown specific function ensuring that the conditions of a harmonic load on the body boundaries are met. This function is sought as a linear combination of linearly independent solutions to the equations of the theory of elasticity in the frequency domain with unknown coefficients. The constructed representation makes it possible to separately satisfy the boundary conditions on the surfaces of defects, which results in a set of systems of integral equations that differ only in their right-hand sides and do not depend on the body boundary shape. The resulting systems of integral equations can be solved by the method of mechanical quadratures. After that, the conditions on the boundaries of the cylindrical body are satisfied, from which the unknown coefficients of the introduced specific function are determined by a collocation method. Using the approach proposed, the stress intensity factors in the vicinity of defects were calculated. With the help of those calculations, we investigated the effect of the frequency and location of the defects on the stress intensity coefficient values.


Keywords: hollow cylinder, harmonic oscillations, stress intensity factors, system of cracks.


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Received 11 September 2018

Published 30 March 2019