A refined model of viscoelastic-plastic deformation of flexible spatially-reinforced cylindrical shells

Cover Page

Cite item


A model of viscoelastic-plastic deformation of flexible circular cylindrical shells with spatial reinforcement structures is developed. The instant plastic behavior of the materials of the composition is determined by flow theory with isotropic hardening. The viscoelastic deformation of the components of the composition is described by the equations of the Maxwell–Boltzmann body model. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The relations used make it possible to calculate with varying degrees of accuracy the residual displacements of the points of the construction and the residual deformed state of the components of the composition. In this case, a possible weak resistance of the reinforced shell to transverse shear is simulated. In the first approximation, the equations used, the initial and boundary conditions, are reduced to the formulas of the nonclassical Ambardzumyan theory.

The numerical solution of the formulated initial boundary-value problem is constructed according to the explicit “cross” scheme. Elastoplastic and viscoelastic-plastic dynamic deformation of thin fiberglass shells under the influence of internal pressure of an explosive type is investigated. Two reinforcement structures are considered:

1) orthogonal reinforcement in the longitudinal and circumferential directions;
2) spatial reinforcement in four directions.

It is shown that even for relatively thin composite shells the Ambardzumyan theory is unacceptable to obtain adequate results of calculations of their viscoelastic-plastic dynamic deformation. It has been demonstrated that a calculation according to the theory of elastoplastic deformation of reinforced structures does not allow even an approximate estimate of the residual states of composite shells after their dynamic loading. It is shown that even for a relatively thin and long cylindrical shell, the replacement of the traditional “flat”-cross-reinforcement structure with a spatial structure can significantly reduce the residual strain of the binder material. In cases of relatively thick and especially short shells, the positive effect of such a replacement of the reinforcement structures is manifested to a much greater extent.

About the authors

Andrey P Yankovskii

Khristianovich Institute of Theoretical and Applied Mechanics,
Siberian Branch of the Russian Academy of Sciences

Author for correspondence.
Email: lab4nemir@rambler.ru
ORCID iD: 0000-0002-2602-8357
SPIN-code: 9972-3050
Scopus Author ID: 7003288442

(Dr. Sci. (Phys. & Math.)), Leading Research Scientist, Lab. of Fast Processes Physic

4/1, Institutskaya st., Novosibirsk, 630090, Russian Federation


  1. Mouritz A. P., Gellert E., Burchill P., Challis K. Review of advanced composite structures for naval ships and submarines, Compos. Struct., 2001, vol. 53, no. 1, pp. 21–42. https://doi.org/10.1016/S0263-8223(00)00175-6.
  2. Reddy J. N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. Boca Raton, CRC Press, 2003, xxiii+831 pp. https://doi.org/10.1201/b12409.
  3. Soutis C. Fibre reinforced composites in aircraft construction, Progress in Aerospace Sciences, 2005, vol. 41, no. 2, pp. 143–151. https://doi.org/10.1016/j.paerosci.2005.02.004.
  4. Gibson R. F. Principles of Composite Material Mechanics. Boca Raton, CRC Press, 2016, 700 pp. https://doi.org/10.1201/b19626.
  5. Vasiliev V. V., Morozov E. Advanced Mechanics of Composite Materials and Structural Elements. Elsever, Amsterdam, 2013, xii+412 pp. https://doi.org/10.1016/C2011-0-07135-1.
  6. Gill S. K., Gupta M., Satsangi P. S. Prediction of cutting forces in machining of unidirectional glass fiber reinforced plastics composite, Front. Mech. Eng., 2013, vol. 8, no. 2, pp. 187–200. https://doi.org/10.1007/s11465-013-0262-x.
  7. Solomonov Yu. S., Georgievskii V. P., Nedbai A. Ya., Andryushin V. A. Prikladnye zadachi mekhaniki kompozitnykh tsilindricheskikh obolochek [Applied Problems of Mechanics of Composite Cylindrical Shells]. Moscow, Fizmatlit, 2014, 408 pp. (In Russian)
  8. Kazanci Z. Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses, Int. J. Nonlin. Mech., 2011, vol. 46, no. 5, pp. 807–817. https://doi.org/10.1016/j.ijnonlinmec.2011.03.011.
  9. Tarnopol’skii Yu. M., Zhigun I. G., Polyakov V. A. Prostranstvenno-armirovannye kompozitsionnye materialy [Spatially Reinforced Composite Materials]. Moscow, Mashinostroenie, 1987, 224 pp. (In Russian)
  10. Grigorenko Ya. M. Izotropnye i anizotropnye sloistye obolochki vrashcheniia peremennoi zhestkosti [Isotropic and Anisotropic Layered Shells of Revolution of Variable Stiffness]. Kiev, Naukova dumka, 1973, 228 pp. (In Russian)
  11. Ambartsumyan S. A. Obshchaia teoriia anizotropnykh obolochek [General Theory of Anisotropic Shells]. Moscow, Nauka, 2002, 446 pp. (In Russian)
  12. Reissner E. On transverse vibrations of thin, shallow elastic shells, Quart. Appl. Math., 1955, vol. 13, no. 2, pp. 169–176. https://doi.org/10.1090/qam/69715.
  13. Bogdanovich A. E. Nelineinye zadachi dinamiki tsilindricheskikh kompozitnykh obolochek [Non-Linear Dynamic Problems for Composite Cylindrical Shells]. Riga, Zinatne, 1987, 295 pp. (In Russian)
  14. Abrosimov N. A., Bazhenov V. G. Nelineinye zadachi dinamiki kompozitnykh konstruktsii [Nonlinear Problems of Dynamics of Composite Structures]. Nizhni Novgorod, Nizhni Novgorod State Univ., 2002, 400 pp. (In Russian)
  15. Yankovskii A. P. Modelling the viscoelastic-plastic deformation of flexible reinforced plates with account of weak resistance to transverse shear, Computational Continuum Mechanics, 2019, vol. 12, no. 1, pp. 80–97 (In Russian). https://doi.org/10.7242/1999-6691/2019.12.1.8.
  16. Yankovskii A. P. Modeling of viscoelastoplastic deformation of flexible shallow shells with spatial-reinforcements structures, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2020, vol. 24, no. 3, pp. 506–527 (In Russian). https://doi.org/10.14498/vsgtu1709.
  17. Andreev A. N., Nemirovskii Yu. V. Mnogosloinye anizotropnye obolochki i plastiny. Izgib, ustoichivost’ i kolebaniia [Multilayer Anisotropic Shells and Plates. Bending, Stability, and Vibrations]. Novosibirsk, Nauka, 2001, 287 pp. (In Russian)
  18. Whitney J. M., Sun C. A higher order theory for extensional motion of laminated composites, J. Sound Vib., 1973, vol. 30, no. 1, pp. 85–97. https://doi.org/10.1016/s0022-460x(73)80052-5.
  19. Lo K. H., Christensen R. M., Wu E. M. A high-order theory of plate deformation. Part 2: Laminated plates, J. Appl. Mech., 1977, vol. 44, no. 4, pp. 669–676. https://doi.org/10.1115/1.3424155.
  20. Yankovskii A. P. The refined model of viscoelastic-plastic deformation of reinforced cylindrical shells, PNRPU Mechanics Bulletin, 2020, no. 1, pp. 138–149 (In Russian). https://doi.org/10.15593/perm.mech/2020.1.11.
  21. Kompozitsionnye materialy [Composite Materials], ed. D. M. Karpinos. Kiev, Naukova Dumka, 1985, 592 pp. (In Russian)
  22. Handbook of Composites, ed. G. Lubin. New York, Van Nostrand Reinhold Company Inc., 1982, xi+786 pp. https://doi.org/10.1007/978-1-4615-7139-1.
  23. Mohamed M. H., Bogdanovich A. E., Dickinson L. C., Singletary J. N., Lienhart R. R. A new generation of 3D woven fabric preforms and composites, Sampe J., 2001, vol. 37, no. 3, pp. 3–17.
  24. Schuster J., Heider D., Sharp K., Glowania M. Measuring and modeling the thermal conductivities of three-dimensionally woven fabric composites, Mech. Compos. Mater., 2009, vol. 45, no. 2, pp. 241–254. https://doi.org/10.1007/s11029-009-9072-y.
  25. Houlston R., DesRochers C. G. Nonlinear structural response of ship panels subjected to air blast loading, Comput. Struct., 1987, vol. 26, no. 1–2, pp. 1–15. https://doi.org/10.1016/0045-7949(87)90232-X.
  26. Khazhinskii G. M. Modeli deformirovaniia i razrusheniia metallov [Deformation and Long-Term Strength of Metals]. Moscow, Nauchnyi Mir, 2011, 231 pp. (In Russian)
  27. Komarov K. L., Nemirovskii Yu. V. Dinamika zhestkoplasticheskikh elementov konstruktsii [Dynamics of Rigid-Plastic Structural Elements]. Novosibirsk, Nauka, 1984, 234 pp. (In Russian)

Supplementary files

There are no supplementary files to display.

Copyright (c) 2021 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies