CHAOTIC VIBRATIONS OF GEOMETRICALLY NONLINEAR NANO-DIMENSIONAL SHALLOW AXISYMMETRIC SHELLS
Abstract
The Hamilton - Ostrogradsky principle is used in the paper to construct a mathematical model of vibrations of geometrically nonlinear nano-dimensional shallow axisymmetric spherical shells. The model is based on the following relations and assumptions: the body of the shell is elastic, homogeneous and isotropic, the Kirchhoff - Love hypothesis, a modified momentary theory of elasticity is used to account for the dependence of elastic behavior of the shell on the dimension-dependent parameter; the shallowness of the shell is determined based on the hypotheses of V.Z. Vlasov, the geometrical nonlinearity is determined using T. Karman's hypothesis. An approach for determining the “true” solution is proposed. The partial differential equation is reduced to a Cauchy problem with the help of the second-order accuracy finite difference method, which is solved using several Runge - Kutta-type methods: the 4th- and 2nd-order Runge - Kutta methods, the 4th-order Runge - Kutta - Fehlberg method, the 4th-order Cash - Karp method, the 8th-order Runge - Kutta - Prince - Dormand method, the implicit 2nd- and 4th-order Runge - Kutta method. An algorithm and a software complex for obtaining numerical results are developed. The convergence of the above methods for a space and time coordinate is studied. The study is based on the qualitative theory of nonlinear dynamics: signals, 2D and 3D phase-plane portraits, power spectrums, Morley's wavelets, deflection curves, Poincare sections and autocorrelation functions are analyzed, the sign of Lyapunov's index is analyzed using several methods: Wolf's, Kantz's, Rosenstein's. An example of analyzing plates and shallow shells is given. The analysis of the obtained results shows that with the increase of the dimension-dependent parameter the vibration character changes from chaotic to harmonic, and the value of the dynamic critical load increases.
References
2. Nix W.D. Mechanical properties of thin films. Metallurgical Transactions A. 1989. Vol. 20. No 11. P. 2217-2245.
3. Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids. 2003. Vol. 51. Iss. 8. P. 1477-1508.
4. Scheible D.V., Erbe A., Blick R.H. Evidence of a nanomechanical resonator being driven into chaotic response via the Ruelle-Takens route. Applied Physics Letters. 2002. Vol. 81. No 2. P. 1884-1886.
5. Mindlin R.D., Tiersten H.F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis. 1962. Vol. 11. P. 415-448.
6. Toupin R. A. Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis. 1962. Vol. 11. P. 385-414.
7. Salajeghe S., Khadem S.E., Rasekh M. Nonlinear analysis of thermoelastic damping in axisymmetric vibration of micro circular thinplate resonators. Applied Mathematical Modelling. 2012. Vol. 36. No 12. P. 5991-6000.
8. Asghari M., Ahmadian M.T., Kahrobaiyan M.H., Rahaeifard M. On the size-dependent behavior of functionally graded micro-beams. Materials & Design. 2010. Vol. 31. No 5. P. 2324-2329.
9. Nayfeh A.H., Younis M.I. Modeling and simulations of thermoelastic damping in microplates. Journal of Micromechanics and Microengineering. 2004. Vol. 14. No 12. P. 1711-1717.
10. Ke L.-L., Yang J., Kitipornchai S. Dynamic stability of functionally graded carbon nanotube-reinforced composite beams. Mechanics of Advanced Materials and Structures. 2013. Vol. 20. No 1. P. 28-37.
11. Lifshitz R., Roukes M.L. Thermoelastic damping in micro- and nanomechanical systems. Physical Review B. 2000. Vol. 61. No 8. P. 5600-5609.
12. Reddy J.N. Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids. 2011. Vol. 59. No 11. P. 2382-2399.
13. Ma H., Gao X., Reddy J. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids. 2008. Vol. 56. No 12. P. 3379-3391.
14. Yin L., Qian Q., Wang L., Xia W. Vibration analysis of microscale plates based on modified couple stress theory. Acta Mechanica Solida Sinica. 2010. Vol. 23. No 5. P. 386-393.
15. Lazopoulos K.A. On bending of strain gradient elastic micro-plates. Mechanics Research Communications. 2009. Vol. 36. No 7. P. 777-783.
16. Reddy J.N., Kim J. A nonlinear modified couple stress-based third-order theory of functionally graded plates. Composite Structures. 2012. Vol. 94. No 3. P. 1128-1143.
17. Sheremet'yev M.P., Pelekh B.L. K postroeniyu utochnennoy teorii plastin [On the construction of a refined theory of plates]. Inzhenernyy zhurnal [Engineering Magazine]. 1964. Vol. 4. Iss. 3. P. 34-41 (In Russian).
18. Yang F., Chong A.C.M., Lam D.C.C., Tong P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures. 2002. Vol. 39. No 10. P. 2731-2743.
19. Krysko-jr V.A., Awrejcewicz J., Papkova I.V., Krysko V.A. General theory of geometrically nonlinear size dependent shells taking into account contact interaction. Part 1, 2. Chaotic dynamics of geometrically nonlinear axially symmetric onelayer shells. Mathematical and Numerical Aspects of Dynamical System Analysis: Proceedings of the 14th Conference on Dynamical Systems - Theory and Applications. Eds. J. Awrejcewicz, M. Kazўmierczak, P. Olejnik, J. Mrozowski. 11-14 Dec. 2017. Loўdzў. Poland. P. 289-300.
20. Volmir A.S. Nelineynaya dinamika plastin i obolochek [Nonlinear Dynamic of Plates and Shells]. Moscow. Nauka Publ. 1972. 432 p. (In Russian).
21. Fehlberg E. Low-order classical Runge - Kutta formulas with step size control and their application to some heat transfer problems. NASA Technical Report R-315. 1969.
22. Fehlberg E. Klassische Runge - Kutta - Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wa..rmeleitungsprobleme. Computing (Arch. Elektron. Rechnen.) Vol. 6. Iss. 1-2. P. 61-71. DOI: 10.1007/BF02241732.
23. Cash J.R., Karp A.H. A variable order Runge - Kutta method for initial value problems with rapidly varying right-hand sides. ACM Transactions on Mathematical Software. 1990. Vol. 16. No 3. P. 201-222. DOI:10.1145/79505.79507.
24. Dormand J.R., Prince P.J. A family of embedded Runge - Kutta formulae. Journal of Computational and Applied Mathematics. 1980. Vol. 6. No 1. P. 19-26. DOI: 10.1016/0771-050X(80)90013-3.
25. Lozi R. Can we trust in numerical computations of chaotic solutions of dynamical systems? World Scientific Series on Nonlinear Science. Topology and Dynamics of Chaos. In: Celebration of Robert Gilmore's 70th Birthday. 2013. Vol. 84. P. 63-98.
26. Devaney R.L. An Introduction to Chaotic Dynamical Systems. Second Edition. Addison-Wesley Publishing Company. 1989. 352 p.
27. Banks B.J., Davis G., Stacey P. On Devaney's Definition of Chaos. American Mathematical Monthly. Vol. 99. No 4. 1992. P. 332-334.
28. Knudsen C. Chaos without Periodicity. American Mathematical Monthly. Vol. 101. 1994. P. 563-565.
29. Gulick D. Encounters with Chaos. New York. McGraw-Hill. 1992.
30. Wolf A., Swift J.B., Swinney H.L., Vastano J.A. Determining Lyapunov exponents from a time series. Physica D. 1985. Vol. 16. No 3. P. 285-317.
31. Kantz H. A robust method to estimate the maximum Lyapunov exponent of a time series. Physics Letters A. 1994. Vol. 185. No 1. P. 77-87. DOI:10.1016/0375-9601(94)90991-1.
32. Rosenstein M.T., Collins J.J., De Luca C.J. A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets. Neuro Muscular Research Center and Department of Biomedical Engineering. Boston University. 1992.
