# Mathematical Models of Multilayer Flexible Orthotropic Shells Under a Temperature Field

• Jan Awrejcewicz
• Maxim V. Zhigalov
• Valeriy F. Kirichenko
• Anton V. Krysko
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 42)

## Abstract

In this chapter we present methods and algorithms for the numerical investigation of shell stability “in the large” of the following multilayer mathematical models: the Timoshenko second order of approximation (Volmir, Flexible plates and shells, Defense Technical Information Center, Gainesville, 1967, [1]), the Sheremetev–Pelekh–Reddy–Levinson third order of approximation (Levinson, J Sound Vib, 74:81–87, 1981, [2]; Reddy, J Appl Mech, 51:745–752, 1984, [3]; Sheremetev, Pelekh, Eng J 4(3):504–510, 1964, [4]), the Grigolyuk–Kulikov model (Grigolyuk, Kulikov, Multilayer reinforced shells: calculation of pneumatic tyres, Mashinostroenie, Moscow, 1988, [5]), and their modifications. We also construct novel mathematical models including a modified asymptotically compatible model obtained with the help of a stationary variant of the “projectional conditions” of the shell’s motion and a model with $$\varepsilon$$-regularization. In the latter case, a theorem on the existence of a general solution is formulated and proved. First a comparative analysis of the computational results in the framework of our mathematical models focused on stability estimation “in the large” of shallow multilayer orthotropic shells within the models and the Kirchhoff–Love first-order approximation model has been carried out.

## References

1. 1.
Volmir, A. S. (1967). Flexible Plates and Shells. Gainesville: Defense Technical Information Center.Google Scholar
2. 2.
Levinson, M. (1981). A new rectangular beam theory. Journal of Sound and Vibration, 74, 81–87.
3. 3.
Reddy, J. N. (1984). A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 51, 745–752.
4. 4.
Sheremetev, M. P., & Pelekh, B. L. (1964). On construction of improved theory of plates. Engineering Journal, 4(3), 504–510 (in Russian).Google Scholar
5. 5.
Grigolyuk, E. I., & Kulikov, G. M. (1988). Multilayer Reinforced Shells: Calculation of Pneumatic Tyres. Moscow: Mashinostroenie (in Russian).Google Scholar
6. 6.
Piskunov, V. G., Verijenko, V. E., & Prysyazhnyuk, V. K. (1987). Calculation of Inhomogeneous Shallow Shells and Plates by the Finite Element Method. Kiev: Vyscha SHKOLA (in Russian).Google Scholar
7. 7.
Awrejcewicz, J., Krysko, A. V., Kutepov, I., Zagniboroda, N., Zhigalov, M., & Krysko, V. A. (2013). Analysis of chaotic vibrations of flexible plates using Fast Fourier Transforms and wavelets. International Journal of Structural Stability and Dynamics, 13(7), 1340005-1–1340004-12.
8. 8.
Awrejcewicz, J., Krysko, V. A., Papkova, I. V., Yakovleva, T. V., Zagniboroda, N. A., Zhigalov, M. V. et al. (2014). Application of the Lyapunov exponents and wavelets to study and control of plates and shells. In J. Awrejcewicz (Ed.), Computational and Numerical Simulations (pp. 1–19). InTech.
9. 9.
Awrejcewicz, J., Krysko, A. V., Zhigalov, M. V., Saltykova, O. A., & Krysko, V. A. (2008). Chaotic vibrations in flexible multilayered Bernoulli–Euler and Timoshenko type beams. Latin American Jourmal of Solids and Structures, 5(4), 319–363.Google Scholar
10. 10.
Nowacki, W. (1970). Theory of Micropolar Elasticity. New York: Springer.
11. 11.
Rodionov, V. A. (1983). The Theory of Thin Anisotropic Shells Accounting for Transverse Shear and Compression. Leningrad: Publishing House Leningrad University (in Russian).Google Scholar
12. 12.
Ambartsumyan, S. A. (1990). Fragments of the Theory of Anisotropic Shells. Singapore: World Scientific.
13. 13.
Washizu, K. (1968). Variational Methods in Elasticity and Plasticity. Pergamon Press.Google Scholar
14. 14.
Volmir, A. S. (1972). The Nonlinear Dynamics of Plates and Shells. Moscow: Nauka (in Russian).Google Scholar
15. 15.
Pikul, V. V. (1982). Theory and Analysis of Shells of Revolution. Moscow: Nauka (in Russian).Google Scholar
16. 16.
Rasskazov, A. O., Sokolov, I. I., & Shul’ga, N. A. (1986). Theory and Calculation of Layered Orthotropic Plates and Shells. Kiev: Vishcha Shkola (in Russian).Google Scholar
17. 17.
Kirichenko, V.F., Bochkarev, V.V. (1989). Related problem of thermoelasticity for shallow shells within the framework of the generalized Timoshenko model. Saratov DEP. in VINITI, 6939-B89.Google Scholar
18. 18.
Lions, J.-L. (1969). Some Problems of Solving Non-Linear Boundary Value Problems. Paris: Dunod–Gauthier-Villars.Google Scholar
19. 19.
Mikhlin, S. G. (1970). Variational Methods in Mathematical Physics. Oxford: Pergamon Press.
20. 20.
Ladyzhenskaya, O. A. (1973). The Boundary Value Problems of Mathematical Physics. Berlin: Springer.Google Scholar
21. 21.
Kachurovskii, R. I. (1971). Nonlinear problems in the theory of plates and shells and their grid ap-proximation. Siberian Mathematical Journal, 12(2), 353–366 (in Russian).
22. 22.
Vorovich, I. I. (1999). Nonlinear Theory of Shallow Shells. New York: Springer.
23. 23.
Volmir, A. S. (1967). Stability of Deformable Systems. Moscow: Nauka (in Russian).Google Scholar
24. 24.
Samarskiy, A. A. (2001). Theory of Difference Schemes. New York: Marcel Dekker Inc.
25. 25.
Brucker, L. E. (1965). Some simplifications of the equations of bending of sandwich plates. Calculations of Elements of Aircraft Structures, 3, 74–99 (in Russian).Google Scholar
26. 26.
Krysko, V. A. (1976). Nonlinear Statics and Dynamics of Inhomogeneous Membranes. Saratov: Publishing House Saratov University Press (in Russian).Google Scholar

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## Authors and Affiliations

• 1
• Jan Awrejcewicz
• 2
Email author
• Maxim V. Zhigalov
• 1
• Valeriy F. Kirichenko
• 1
• Anton V. Krysko
• 3
1. 1.Department of Mathematics and ModelingSaratov State Technical UniversitySaratovRussia
2. 2.Department of Automation, Biomechanics and MechatronicsLodz University of TechnologyLodzPoland
3. 3.Department of Applied Mathematics and Systems AnalysisSaratov State Technical UniversitySaratovRussia
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