# Elastic-Plastic Problem for a Stringer Plate with a Circular Hole

 DOI https://doi.org/10.15407/pmach2021.03.061 Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia Publisher A. Pidhornyi Institute for Mechanical Engineering Problems National Academy of Science of Ukraine ISSN 2709-2984 (Print), 2709-2992 (Online) Issue Vol. 24, no. 3, 2021 (September) Pages 61-69 Cited by J. of Mech. Eng., 2021, vol. 24, no. 3, pp. 61-69

Author

Minavar V. Mir-Salim-zade, Institute of Mathematics and Mechanics of the NAS of Azerbaijan (9, Vahabzade str., Baku, AZ1141, Azerbaijan), e-mail: minavar.mirsalimzade@imm.az, ORCID: 0000-0003-4237-0352

Abstract

When calculating the strength of machines, structures and buildings with technological holes, it is important to take into account the plastic zones that emerge around the holes. However, the unknown shape and size of the plastic zone complicate the solution of elastic-plastic problems. This paper gives an approximate method and solution of the plane elastic-plastic problem of the distribution of stresses in a thin plate, reinforced with a regular system of stiffeners (stringers). The stringer plate under consideration has a circular hole, which is completely surrounded by the zone of plastic deformation. At infinity, the plate is subjected to a uniform tension along the stiffeners. A constant normal load is applied to the contour of the hole. The plate and stringer materials are assumed to be isotropic. The loading conditions are assumed to be quasi-static. It is assumed that the plate is in the plane-stressed state. Taken as the plasticity condition in the plastic zone is the Tresca-Saint-Venant plasticity condition. Methods of perturbation theory, analytic function theory, and the least squares method are used. The solution to the stated elastic-plastic problem consists of two stages. At the first stage, the stress-strain state for the elastic zone is found, and then the unknown interface between the elastic and plastic zones is determined using the least squares method. A closed system of algebraic equations has been constructed in each approximation, the numerical solution of which makes it possible to study the stress-strain state of a stringer plate, with the hole entirely surrounded by the plastic zone, as well as to determine the magnitudes of the concentrated forces that replace the action of the stringers. The interface between the elastic and plastic deformations has been found. The presented solution technique can be developed to solve other elastic-plastic problems. The solution obtained in this paper makes it possible to consider elastic-plastic problems for a stringer plate with other plasticity criteria.

Keywords: plate, stringers, elastic-plastic problem, interface between elastic and plastic deformations.

References

1. Mirsalimov, V. M. (2015). Ob odnom sposobe resheniya ploskikh uprugoplasticheskikh zadach [On one method for solving plane elastoplastic problems]. Mekhanika predelnogo sostoyaniya i smezhnyye voprosy Mechanics of a limit state and related issues. Cheboksary: Chuvash State Pedagogical University, pp. 31–36 (in Russian).
2. Protosenya, A. G., Karasev, M. A., & Belyakov, N. A. (2016). Elastoplastic problem for noncircular openings under Coulomb’s criterion. Journal of Mining Science, vol. 52, no. 1, pp. 53–61. https://doi.org/10.1134/S1062739116010125.
3. Abashidze, Z. (2017). Elastoplastic problem for a plate with partially unknown boundary. Transactions of A. Razmadze Mathematical Institute, vol. 171, iss. 1, pp. 1–9. https://doi.org/10.1016/j.trmi.2017.01.004.
4. Senashov, S. I. & Gomonova, O. V. (2019). Construction of elastoplastic boundary in problem of tension of a plate weakened by holes. International Journal of Non-Linear Mechanics, vol. 108, pp. 7–10. https://doi.org/10.1016/j.ijnonlinmec.2018.09.009.
5. Mirsalimov, V. M. (2020). Elastoplastic tension problem for a plate with a circular hole with account for crack nucleation in an elastic deformation region. Journal of Applied Mechanics and Technical Physics, vol. 61, iss. 4, pp. 641–651. https://doi.org/10.1134/S0021894420040185.
6. Ma, Y., Lu, A., & Cai, H. (2020). Analytical method for determining the elastoplastic interface of a circular hole subjected to biaxial tension-compression loads. Mechanics Based Design of Structures and Machines. https://doi.org/10.1080/15397734.2020.1801461.
7. Senashov, S. I. & Savostyanova, I. L. (2020). Uprugoplasticheskaya zadacha v usloviyakh slozhnogo sdviga [Elastic-plastic problem under complex shear conditions]. Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I. Ya. Yakovleva. Seriya: Mekhanika predelnogo sostoyaniyaBulletin of the Yakovlev Chuvash State Pedagogical University. Series: Mechanics of Limit State, no. 1 (43), pp. 66–72 (in Russian). https://doi.org/10.37972/chgpu.2020.43.1.007.
8. Mirsalimov, V. M. & Kalantarly, N. M. (2021). Resheniye uprugoplasticheskoy zadachi dlya massiva, oslablennogo krugovoy vyrabotkoy pri deystvii tektonicheskikh i gravitatsionnykh sil [Solution of an elastoplastic problem for rock, weakened by a circular working under action of tectonic and gravitational forces]. Izvestiya TulGU. Nauki o ZemleProceedings of the Tula states university. Sciences of Earth, no. 1, pp. 207–216 (in Russian).
9. Ma, Y., Lu, A., Cai, H., & Zeng, X. (2021). An analytical method for determining the plastic regions around two circular holes in an infinite medium. Applied Mathematical Modelling, vol. 89, part 1, pp. 636–653. https://doi.org/10.1016/j.apm.2020.07.033.
10. Gomonova, O. V. & Senashov, S. I. (2021). Determining elastic and plastic deformation regions in a problem of uniaxial tension of a plate weakened by holes. Journal of Applied Mechanics and Technical Physics, vol. 62, pp. 157–163. https://doi.org/10.1134/S0021894421010193.
11. Mirsalimov, V. M. (2021). Elastic–plastic problem for a circular hole plate with regard to crack initiation in elastic zone. Archive of Applied Mechanics, vol. 91, pp. 1329–1342. https://doi.org/10.1007/s00419-020-01825-w.
12. Galin, L. A. (1984). Uprugo-plasticheskiye zadachi [Elastic-plastic problems]. Moscow: Nauka, 304 p. (in Russian).
13. Annin, B. D. & Cherepanov, G. P. (1983). Uprugoplasticheskaya zadacha [Elastoplastic problem]. Novosibirsk: Nauka, 239 p. (in Russian).
14. Mirsalimov, V. M. (1987). Neodnomernyye uprugoplasticheskiye zadachi [Non-one-dimensional elastoplastic problems]. Moscow: Nauka, 255 p. (in Russian).
15. Ostrosablin, N. I. (1984). Ploskoye uprugoplasticheskoye raspredeleniye napryazheniy okolo krugovykh otverstiy [Plane elastoplastic stress distribution around circular holes]. Novosibirsk: Nauka, 113 p. (in Russian).
16. Sokolovskiy, V. V. (1969). Teoriya plastichnosti [Theory of plasticity]. Moscow: Vysshaya shkola, 608 p. (in Russian).
17. Muskhelishvili, N. I. (1977). Some basic problem of mathematical theory of elasticity. Dordrecht: Springer, 732 p. https://doi.org/10.1007/978-94-017-3034-1.
18. Mirsalimov, V. M. (1986). Some problems of structural arrest of cracks. Soviet materials science, vol. 22, iss. 1, pp. 81–85. https://doi.org/10.1007/BF00720871.
19. Ivlev, D. D. & Yershov, L. V. (1978). Metod vozmushcheniy v teorii uprugoplasticheskogo tela [Perturbation method in the theory of elastoplastic solid]. Moscow: Nauka, 208 p. (in Russian).
20. Lomakin, V. A. (1976). Teoriya uprugosti neodnorodnykh tel [Theory of elasticity of inhomogeneous solids]. Moscow: Moscow State University, 368 p. (in Russian).