Elastic-plastic bending of plates with a central hole in a three-dimensional setting
https://doi.org/10.26907/2541-7746.2025.4.744-758
Abstract
This article presents a mathematical model and algorithm for analyzing the stress–strain state (SSS) of elastic-plastic plates with a central circular hole in a three-dimensional setting. The developed algorithm can be applied to any boundary conditions, dependencies, and materials for which experimental stress–strain diagrams are available. The model is based on the deformation theory of plasticity and was implemented using a combination of the finite element method (FEM) and the method of I.A. Birger’s variable elasticity parameters. To obtain reliable results, the type finite elements (FE) and their number in a three-dimensional setting were investigated, along with the convergence of the solutions on a mesh of tetrahedral and hexahedral FE for a plate with and without a hole in the center. The hexahedral FE was found to be the most optimal. Computational examples for a rectangular plate clamped along the contour and subjected to a constant load were provided. The plate material considered is pure aluminum described by the stress–strain diagram developed by Y. Ohashi and S. Murakami.
Keywords
deformation theory of plasticity,
method of variable elasticity parameters,
finite element method,
three-dimensional problem setting
Supporting agencies: This study was supported by the Russian Science Foundation (project no. 22-11-00160-P).
About the Authors
A. E. Makseev
Yuri Gagarin State Technical University of Saratov
Russian Federation
Russian Federation
Anton E. Makseev, Postgradute Student
K. S. Bodyagina
Yuri Gagarin State Technical University of Saratov
Russian Federation
Russian Federation
Kseniya S. Bodyagina, Cand. Sci. (Physics and Mathematics), Senior Lecturer
M. V. Zhigalov
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences
Russian Federation
Russian Federation
Maksim V. Zhigalov, Dr. Sci. (Physics and Mathematics), Professor
V. A. Krysko
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences
Russian Federation
Russian Federation
Vadim A. Krysko, Dr. Sci. (Engineering), Full Professor
References
1. Wagoner R.H., Lim H., Lee M.-G. Advanced issues in springback. Int. J. Plast., 2013, vol. 45. pp. 3–20. https://doi.org/10.1016/j.ijplas.2012.08.006.
2. Geka T., Asakura M., Kiso T., Sugiyama T., Takamura M., Asakawa M. Reduction of springback in hat channel with high-strength steel sheet by stroke returning deep drawing. Key Eng. Mater., 2013, nos. 554–557, pp. 1320–1330. https://doi.org/10.4028/www.scientific.net/KEM.554-557.1320.
3. Hama T., Matsudai R., Kuchinomachi Y., Fujimoto H., Takuda H. Non-linear deformation behavior during unloading of various metal sheets. ISIJ Int., 2015, vol. 55, no. 5, pp. 1067–1075. https://doi.org/10.2355/isijinternational.55.1067.
4. Fedotova D.V., Khamidullin R.M. Mixed-mode crack growth analysis using models of cyclic plasticity. XIII Vserossiisk. s”ezd po teor. i prikl. mekhan. Sankt-Peterburg, 21–25 avg. 2023 g. [Proc. XIII All-Russian Congr. on Theoretical and Applied Mechanics. St. Petersburg, August 21–15, 2023]. Vol. 3. St. Petersburg, 2023, pp. 724–726. (In Russian)
5. Shlyannikov V., Fedotova D., Khamidullin R. Mixed-mode crack growth analysis using a cyclic plasticity model. Theor. Appl. Fract. Mech., 2023, vol. 128, art. 104136. https://doi.org/10.1016/j.tafmec.2023.104136.
6. Nadezhkin M.V., Barannikova S.A. The localization of plastic deformation during stretching of strips by a stress concentrator. Fizicheskaya mezomekhanika. Materialy s mnogourovnevoi ierarkhicheski organizovannoi strukturoi i intellektual’nye proizvodstvennye tekhnologii: tez. dokl. Mezhdunarodn. konf., Tomsk, 11–14 sent. 2023 g. [Physical Mesomechanics. Materials with a Multilevel Hierarchical Structures and Intelligent Manufacturing Technologies: Proc. Int. Conf., Tomsk, September 11–14, 2023]. Tomsk, 2023. p. 166. (In Russian)
7. Barannikova S.A., Nadezhkin M.V. Kinetics of localization of plastic deformation zones in polycrystalline nickel. Metals, 2021, vol. 11, no. 9, art. 1440. https://doi.org/10.3390/met11091440
8. Shlyannikov V.N., Il’chenko B.V., Boichenko N.V., Tartygasheva A.M. The plate with a hole in a state of elasticity, plasticity, and creep. Izv. Vuzov. Probl. Energ., 2004, nos. 1–2, pp. 107–116. (In Russian)
9. Mir-Salim-zade M.V. Elastic-plastic problem for a stringer plate with a circular hole. J. Mech. Eng., 2021, vol. 24, no. 3, pp. 61–69. https://doi.org/10.15407/pmach2021.03.061.
10. Mamatova N. Quasistatic elastic-plastic loading and unloading of rods with Coulomb dry friction. E3S Web Conf., 2021, vol. 264, art. 01040. https://doi.org/10.1051/e3sconf/202126401040.
11. Awrejcewicz J., Krysko V.A., Zhigalov M.V., Krysko A.V. Contact interaction of two rectangular plates made from different materials with an account of physical nonlinearity. Nonlinear Dyn., 2018, vol. 91, no. 2, pp. 1191–1211. https://doi.org/10.1007/s11071-017-3939-6.
12. Birger I.A. Some general methods for solving problems in the theory of plasticity. Prikl. Mat. Mekh., 1951, vol. 15, no. 6, pp. 765–770. (In Russian)
13. Krysko A.V., Awrejcewicz J., Bodyagina K.S., Zhigalov M.V., Krysko V.A. Mathematical modeling of physically nonlinear 3D beams and plates made of multimodulus materials. Acta Mech., 2021, vol. 232, no. 9, pp. 3441–3469. https://doi.org/10.1007/s00707-021-03010-8.
14. Krysko V.A., Papkova I.V., Krysko A.V. Nonlinear dynamics of contact interaction porous sizedependent Euler-Bernoulli beams resonators with clearance: Numerical analysis of the stability problem. Commun. Nonlinear Sci. Numer. Model., 2024, vol. 135, art. 108038. https://doi.org/10.1016/j.cnsns.2024.108038.
15. Krysko V.A., Papkova I.V., Yakovleva T.V., Krysko A.V. Chaotic, hyperchaotic vibrations and stability of porous Euler–Bernoulli beams considering physical and geometrical nonlinearities. Dokl. Ross. Akad. Nauk. Fiz., Tekh. Nauki., 2025, vol. 521, no. 1, pp. 50–58. https://doi.org/10.31857/S2686740025020059. (In Russian)
16. Makseev A., Yakovleva T.V., Krysko A.V., Zhigalov M.V., Krysko V.A. Identification of inclusions of arbitrary geometry with different physical properties of materials in 3D structures. Int. J. Mech. Mater. Des., 2025, vol. 21, no. 1, pp. 53–79. https://doi.org/10.1007/s10999-024-09727-3.
17. Petryk V., Trubachev S., Kolodezhnyi V. Determination of stresses in a cylindrical shell taking into account holes. Young Sci., 2022, no. 5 (105), pp. 13–16. https://doi.org/10.32839/2304-5809/2022-5-105-3.
18. Astapov N.S., Kurguzov V.D. Simulation of elastoplastic fracture of a center cracked plate. PNRPU Mech. Bull., 2023, no. 1, pp. 12–25. https://doi.org/10.15593/perm.mech/2023.1.02. (In Russian)
19. Rudakov K., Dyfuchyn Y., Bakhtovarshoiev T. Concentration of stresses near the hole in contact with rigid cylinder in composite plate, taking into account lateral clearances. Mech. Adv. Technol., 2021, vol. 5, no. 2, pp. 183–192. https://doi.org/10.20535/2521-1943.2021.5.2.2243744. (In Ukrainian)
20. Khosravani M.R., Reinicke T. Mechanical strength of 3D-printed open hole polymer plates. Procedia Struct. Integr., 2022, vol. 41, pp. 664–669. https://doi.org/10.1016/j.prostr.2022.05.075.
21. Mirsalimov V.M. Elastoplastic tension problem for a plate with a circular hole with account for crack nucleation in an elastic deformation region. J. Appl. Mech. Tech. Phys., 2020, vol. 61, no. 4, pp. 641–651. https://doi.org/10.1134/S0021894420040185.
22. Silva B.F., de Souza L.F.G., de Aguair R.A.A., Pacheco P.M.C.L. Analysis of stress concentration in pseudoelastic plates using digital image correlation (DIC) and finite element method. Lat. Am. J. Solids Struct., 2025, vol. 22, no. 10, art. e870. https://doi.org/10.1590/1679-7825/e8703.
23. Sococol I., Mihai P., Petrescu T.-C., Nedeff F., Nedeff V., Agop M., Luca B.-I. Numerical study regarding the seismic response of a moment-resisting (MR) reinforced concrete (RC) frame structure with reduced cross-sections of the RC slabs. Buildings, 2022, vol. 12, no. 10, art. 1525. https://doi.org/10.3390/buildings12101525.
24. Wang Ch., Gonz´alez Ure˜na A., Afifi M., Rudman A., Tremblay R., Rogers C.A. Conventional construction steel braces with bearing plate energy dissipation. Proc. 17th World Conf. on Earthquake Engineering (17WCEE). Sendai, 2020, art. 2i-0070.
25. Hung L.-W., Chao C.-K., Huang J.-R., Lin J. Screw head plugs increase the fatigue strength of stainless steel, but not of titanium, locking plates. Bone Jt. Res., 2019, vol. 7, no. 12, pp. 629–635. https://doi.org/10.1302/2046-3758.712.BJR-2018-0083.R1.
26. Ohashi Y., Murakami S. The elasto-plastic bending of a clamped thin circular plate. In: G¨ortler H. (Ed.) Applied Mechanics. Berlin, Heidelberg, Springer. 1966. pp. 212–223. https://doi.org/10.1007/978-3-662-29364-5_25.
Review
For citations:
Makseev A.E., Bodyagina K.S., Zhigalov M.V., Krysko V.A. Elastic-plastic bending of plates with a central hole in a three-dimensional setting. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(4):744-758. (In Russ.) https://doi.org/10.26907/2541-7746.2025.4.744-758
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