Bending of Plates with Complex Shape Made from Materials that Differently Resist to Tension and Compression

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DOI https://doi.org/10.15407/pmach2023.02.016
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 26, no. 2, 2023 (June)
Pages 16-23
Cited by J. of Mech. Eng., 2023, vol. 26, no. 2, pp. 16-23

 

Author

Serhii M. Sklepus, Anatolii Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine (2/10, Pozharskyi str., Kharkiv, 61046, Ukraine), e-mail: snsklepus@ukr.net, ORCID: 0000-0002-4119-4310

 

Abstract

A new numerical-analytical method for solving physically nonlinear bending problems of thin plates with complex shape made from materials that differently resist to tension and compression is developed. The uninterrupted parameter continuation method is used to formulate and linearize the problem of physically nonlinear bending. For the linearized problem, a functional in the Lagrange form, given on the kinematically possible displacement rates, is constructed. The main unknown problems (displacements, strains, stresses) were found from the solution of the initial problem, which was solved by the Runge-Kutta-Merson method with automatic step selection, by the parameter related to the load. The initial conditions are found from the solution of the problem of linear elastic deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta-Merson scheme are found from the solution of the variational problem for the functional in the Lagrange form. Variational problems are solved using the Ritz method in combination with the R-function method, which allows to submit an approximate solution in the form of a formula – a solution structure that exactly satisfies the boundary conditions and is invariant with respect to the shape of the domain where the approximate solution is sought. The test problem for the nonlinear elastic bending of a square hinged plate is solved. Satisfactory agreement with the three-dimensional solution is obtained. The bending problem of the plate of complex shape with combined fixation conditions is solved. The influence of the geometric shape and fixation conditions on the stress-strain state is studied. It is shown that failure to take into account the different behavior of the material under tensile and compression can lead to significant errors in the calculations of the stress-strain state parameters.

 

Keywords: thin plate, physically nonlinear bending, complex shape, R-function method

 

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References

  1. Zhukov, A. M. (1986). Soprotivleniye nekotorykh materialov chistomu rastyazheniyu i szhatiyu [Resistance of some materials to pure tension and compression]. Izvestiya AN SSSR. Mekhanika tverdogo tela – Mechanics of Solids, no. 4, pp. 197–202 (in Russian).
  2. Miklyayev, P. G. & Fridman, Ya. B. (1986). Anizotropiya mekhanicheskikh svoystv metallov [Anisotropy of mechanical properties of metals]. Moscow: Metallurgiya, 224 p. (in Russian).
  3. Sarrak, V. I. & Filippov, G. A. (1977). Effekt raznogo soprotivleniya deformatsii pri rastyazhenii i szhatii martensita zakalennoy stali [The effect of different resistance to deformation in tension and compression of hardened steel martensite]. Fizika metallov i metallovedeniye – Physics of Metals and Metallography, vol. 44, iss. 4, pp. 858–863 (in Russian).
  4. Zolochevskiy, A. A. (1994). Razrabotka matematicheskikh modeley uprugosti, plastichnosti, polzuchesti izotropnykh i anizotropnykh tel s kharakteristikami, zavisyashchimi ot vida nagruzheniya [Development of mathematical models of elasticity, plasticity, creep of isotropic and anisotropic bodies with characteristics depending on the type of loading]: D. Sci. (Eng.) dissertation. Kharkov Polytechnic Institute, Kharkiv, 521 p. (in Russian).
  5. Zolochevskiy, A. A., Sklepus, A. N., & Sklepus, S. N. (2011). Nelineynaya mekhanika deformiruyemogo tverdogo tela [Nonlinear mechanics of a deformable solid body]. Kharkiv: «Biznes Investor Grupp», 720 p. (in Russian).
  6. Zolochevskiy, A. A. & Zolochevskiy, Yu. A. (1989). K metodike rascheta nelineynogo deformirovaniya tel vrashcheniya pri neosesimmetrichnom nagruzhenii [On the method for calculating nonlinear deformation of rotation bodies under nonaxisymmetric loading]. Prochnost tonkostennykh aviatsionnykh konstruktsiy – Strength of thin-walled aircraft structures, pp. 9–12 (in Russian).
  7. Zolochevskiy, A. A., Kozmin, Yu.S., & Konkin, V. N. (1990). Nelineynyye zadachi teorii tolstostennykh obolochek iz anizotropnykh materialov, raznosoprotivlyayushchikhsya rastyazheniyu i szhatiyu [Nonlinear problems of the theory of thick-walled shells made of anisotropic materials with different resistance to tension and compression]: Proceedings of the XV All-Union Conference on the Theory of Shells and Plates. Kazan: Kazan State University, vol. 1, pp. 286–290 (in Russian).
  8. Zolochevskiy, A. A. & Damasevich, S. V. (1990). Metodika rascheta nelineyno-uprugogo deformirovaniya obolochek iz materialov, raznosoprotivlyayushchikhsya rastyazheniyu i szhatiyu [Method for calculating nonlinear elastic deformation of shells made of materials differently resistant to tension and compression]. Izvestiya vuzov. Mashinostroyeniye – Bauman Moscow State Technical University Journal of Mechanical Engineering, no. 5, pp. 30–34 (in Russian).
  9. Zolochevskiy, A. A. & Kozmin, Yu. S. (1991). Metodika rascheta v prostranstvennoy postanovke pryamougol’nykh plastin iz materialov, raznosoprotivlyayushchikhsya rastyazheniyu i szhatiyu [Method of calculation in spatial formulation of rectangular plates made of materials with different resistance to tension and compression]. Izvestiye vuzov. Mashinostroyeniye – Bauman Moscow State Technical University Journal of Mechanical Engineering, no. 1–3, pp. 9–14 (in Russian).
  10. Oden, J. T. (2006). Finite Elements of Nonlinear Continua. USA, New York: Dover Publications, 464 p.
  11. Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotoryye yeye prilozheniya [The R-functions theory and some of its applications]. Kyiv: Naukova Dumka, 552 p. (in Russian).
  12. Rvachev, V. L. & Kurpa, L. V. (1987). R-funktsii v zadachakh teorii plastin [R-functions in problems of plate theory]. Kyiv: Naukova Dumka, 175 p. (in Russian).
  13. Smetankina, N., Merkulova, A., Merkulov, D., & Postnyi, O. (2021). Dynamic response of laminate composite shells with complex shape under low-velocity impact. In: Nechyporuk M., Pavlikov V., Kritskiy D. (eds.) Integrated Computer Technologies in Mechanical Engineering-2020. ICTM 2020. Lecture Notes in Networks and Systems. Cham: Springer, vol. 188, pp. 267–276. https://doi.org/10.1007/978-3-030-66717-7_22.
  14. Grigolyuk, E. I. & Shalashilin, V. I. (1988). Problemy nelineynogo deformirovaniya: metod prodolzheniya resheniya po parametru v nelineynykh zadachakh mekhaniki tverdogo deformiruyemogo tela [Problems of nonlinear deformation: method of continuing the solution with a parameter in nonlinear problems of mechanics of a solid deformable body]. Moscow: Nauka, 232 p. (in Russian).
  15. Washizu, K. (1982). Variational methods in elasticity and plasticity. Oxford: Pergamon Press, 630 p.
  16. Krylov, V. I, Bobkov, V. V., & Monastyrnyy, P. I. (1977). Vychislitelnyye metody [Computational methods]. Moscow: Nauka, 399 p. (in Russian).
  17. Chernilevskiy, D. V., Lavrova, Ye. V., & Romanov, V. A. Tekhnicheskaya mekhanika [Technical mechanics]. Moscow: Nauka, 544 p. (in Russian).
  18. Zolochevskii, A. A. (1990). Determining equations of nonlinear deformation with three stress-state invariants. Soviet Applied Mechanics, vol. 26, iss. 3, pp. 277–282. https://doi.org/10.1007/BF00937216.

 

Received 18 May 2023

Published 30 June 2023