Nonlinear Deformation of Cylinders from Materials with Different Behavior in Tension and Compression

image_print
DOI https://doi.org/10.15407/pmach2024.01.026
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher Anatolii Pidhornyi Institute for Mechanical Engineering Problems
of National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 27, no. 1, 2024 (March)
Pages 26-34
Cited by J. of Mech. Eng., 2024, vol. 27, no. 1, pp. 26-34

 

Authors

Oleksandr Z. Galishin, S. P. Timoshenko Institute of Mechanics of NAS of Ukraine (3, Nesterova str., Kyiv, 03057, Ukraine), e-mail: plast@inmech.kiev.ua, ORCID: 0000-0003-0286-872X

Serhii M. Sklepus, S. P. Timoshenko Institute of Mechanics of NAS of Ukraine (3, Nesterova str., Kyiv, 03057, Ukraine), 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 deformation problems of axisymmetrically loaded cylinders made of materials with different behavior in tension and compression has been developed. To linearize the problem, the uninterrupted parameter continuation method was used. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible displacement rates, is constructed. To find the main unknowns of the problem of physically nonlinear cylinder deformation, the Cauchy problem for the system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta-Merson method with automatic step selection. The initial conditions were established by solving 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’s 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. The test problem for the nonlinear elastic deformation of a thin cylindrical shell is solved. Coincidence of the spatial solution with the shell solution was obtained. Physically nonlinear deformation of a thick-walled cylinder was studied. It is shown that failure to take into account the different behavior of the material under tension and compression leads to significant errors in the calculations of stress-strain state parameters.

 

Keywords: thick-walled cylinder, different behavior in tension and compression, physically nonlinear deformation, uninterrupted parameter continuation method.

 

Full text: Download in PDF

 

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. Shevchenko, Yu. N., Babeshko, M. Ye., Piskun, V. V., & Savchenko, V. G. (1980). Prostranstvennyye zadachi termoplastichnosti [Spatial problems of thermoplasticity]. Kyiv: Naukova dumka, 264 p. (in Russian).
  5. Podgornyy, A. N., Bortovoy, V. V., Gontarovskiy, P. P., Kolomak, V. D., Lvov, G. I., Matyukhin, Yu. I., & Morachkovskiy, O. K. (1984). Polzuchest elementov mashinostroitelnykh konstruktsiy [Creep of elements of machine-building structures]. Kyiv: Naukova dumka, 264 p. (in Russian).
  6. Shevchenko, Y. N., Piskun, V. V., & Kovalenko, V. A. (1992). Elastoplastic state of axisymmetrically loaded laminated bodies of revolution made of isotropic and orthotropic materials. International Applied Mechanics, vol. 28, iss. 1, pp. 25–32. https://doi.org/10.1007/BF00847326.
  7. Rvachev, V. L. & Sinkop, N. S. (1990). Metod R-funktsiy v zadachakh teorii uprugosti i plastichnosti [R-functions method in problems of the theory of elasticity and plasticity]. Kiyev: Naukova dumka, 216 p. (in Russian).
  8. Breslavs’kyi, D. V., Korytko, Y. M., & Morachkovs’kyi, O. K. (2011). Cyclic thermal creep model for the bodies of revolution. Strength of Materials, vol. 43, iss. 2, pp. 134–143. https://doi.org/10.1007/s11223-011-9279-8.
  9. Braam, H. & Haverkate, B. R. W. (1994). Creep lifetime prediction of pressurized tubes with continuum damage mechanics. Structural Integrity: Experiments – Models – Applications: Proceedings of the 10th Biennial European Conference on Fracture – ECF 10 (20–23 September 1994, Berlin, Federal Republic of Germany). Auspices of the European Structural Integrity Society, pp. 279–284.
  10. Galishin, A. Z., Zolochevsky, A. A., & Sklepus, S. M. (2018). Study of creep and damage for a hollow cylinder on the basis of space and refined shell models. Journal of Mathematical Sciences, vol. 231, iss. 5, pp. 629–640. https://doi.org/10.1007/s10958-018-3840-y.
  11. 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).
  12. 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).
  13. Zolochevsky, A., Sklepus, S., Galishin, A., Kühhorn, A., & Kober, M. (2014). A comparison between the 3D and the Kirchhoff-Love solutions for cylinders under creep-damage conditions. Technische Mechanik – European Journal of Engineering Mechanics, vol. 34, no. 2, pp. 104–113. https://doi.org/10.24352/UB.OVGU-2017-056.
  14. Podgornyy, A. N., Gontarovskiy, P. P., Kirkach, B. N., Matyukhin, Yu. I., & Khavin, G. L. (1989). Zadachi kontaktnogo vzaimodeystviya elementov konstruktsíy [Problems of contact interaction of structural elements]. Kyiv: Naukova dumka, 232 p. (in Russian).
  15. Voronin, V. Ye. & Shirshov, A. A. (1989). Neustanovivshayasya polzuchest’ tolstostennoy truby iz materiala s razlichnymi kharakteristikami pri rastyazhenii i szhati [Unsteady creep of a thick-walled pipe made of a material with different characteristics under tension and compression]. Izvestiya vuzov. Mashinostroyeniye – Bauman Moscow State Technical University Journal of Mechanical Engineering, no. 8, pp. 7–9 (in Russian).
  16. Gorev, B. V. & Tsvelodub, I. Yu. (1974). Primeneniye energeticheskikh uravneniy polzuchesti k raschetu tolstostennoy tsilindricheskoy truby [Application of energy creep equations to the calculation of a thick-walled cylindrical pipe]. Dinamika sploshnoy sredy – Continuum dynamics, iss. 17, pp. 99–105 (in Russian).
  17. Gao, J., Huang, P., & Yao, W. (2017). Analytical and numerical study of temperature stress in the bi-modulus thick cylinder. Structural Engineering and Mechanics, vol. 64, no. 1, pp. 81–92. https://doi.org/10.12989/sem.2017.64.1.081.
  18. He, X.-T., Wang, X.-G., & Sun, J.-Y. (2023). Application of the variational method to the large deformation problem of thin cylindrical shells with different moduli in tension and compression. Materials, vol. 16 (4), article 1686. https://doi.org/10.3390/ma16041686.
  19. He, X.-T., Chang, H., & Sun, J.-Y. (2023). Axisymmetric large deformation of thin shallow shells with different moduli in tension and compression. Thin-Walled Structures, vol. 182, part B, article 120297. https://doi.org/10.1016/j.tws.2022.110297.
  20. 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).
  21. Demidov S. P. (1979). Teoriya uprugosti [Theory of elasticity]. Moscow: Vysshaya shkola, 432 p. (in Russian).
  22. 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.
  23. Krylov, V. I, Bobkov, V. V., & Monastyrnyy, P. I. (1977). Vychislitelnyye metody [Computational methods]. Moscow: Nauka, 399 p. (in Russian).

 

Received 05 October 2023

Published 30 March 2024