PSI - Issue 50

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000 – 000 Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000 – 000

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Procedia Structural Integrity 50 (2023) 131–136

© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Abstract The paper is devoted to the formulation and solution of the boundary value problem for a hollow circular cylinder whose properties continuously depend on the radial coordinate. It is shown that the mathematical model of such a problem is a homogeneous second-order differential equation with variable coefficients and two boundary conditions. In the case where the Poisson's ratio is constant and the Young's modulus logarithmically depends on the radial coordinate, the resulting equation can be reduced to the Whittaker equation, the solution of which is a linear combination of confluent hypergeometric functions. Based on this, an analytical solution of the boundary value problem is constructed in this particular case. Functions describing displacements, strains, and stresses occurring in the cylinder are found. © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Keywords: Boundary value problem ; inhomogeneity; Lamé problem; ordinary differential equation; confluent hypergeometric function 1. Introduction The formulas for determining the stress distribution in a hollow cylinder subjected to uniform internal and external pressures, obtained by Lam é in 1852, have found wide application in many engineering problems. These results, for example, helped to estimate the average values of the elastic modulus of some composites. Solutions for cylindrical anisotropic elastic pipes and rods were also obtained. The development of new production technologies, 16th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures (MRDMS 2022) Influence of logarithmic radial inhomogeneity of the elastic modulus on the stress-strain state of a hollow cylinder Kirill E. Kazakov* Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526, Russia Abstract The paper is devoted to the formulation and solution of the boundary value problem for a hollow circular cylinder whose properties continuously depend on the radial coordinate. It is shown that the mathematical model of such a problem is a homogeneous second-order differential equation with variable coefficients and two boundary conditions. In the case where the Poisson's ratio is constant and the Young's modulus logarithmically depends on the radial coordinate, the resulting equation can be reduced to the Whittaker equation, the solution of which is a linear combination of confluent hypergeometric functions. Based on this, an analytical solution of the boundary value problem is constructed in this particular case. Functions describing displacements, strains, and stresses occurring in the cylinder are found. © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers Keywords: Boundary value problem ; inhomogeneity; Lamé problem; ordinary differential equation; confluent hypergeometric function 1. Introduction The formulas for determining the stress distribution in a hollow cylinder subjected to uniform internal and external pressures, obtained by Lam é in 1852, have found wide application in many engineering problems. These results, for example, helped to estimate the average values of the elastic modulus of some composites. Solutions for cylindrical anisotropic elastic pipes and rods were also obtained. The development of new production technologies, 16th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures (MRDMS 2022) Influence of logarithmic radial inhomogeneity of the elastic modulus on the stress-strain state of a hollow cylinder Kirill E. Kazakov* Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526, Russia

* Corresponding author. Tel.: +7-926-401-26-47; fax: +7-495-434-00-17. E-mail address: kazakov@ipmnet.ru * Corresponding author. Tel.: +7-926-401-26-47; fax: +7-495-434-00-17. E-mail address: kazakov@ipmnet.ru

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers 2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the MRDMS 2022 organizers 10.1016/j.prostr.2023.10.032