Chaos-Type Identification in the Contact Interaction of Closed Cylindrical Nanoshells Embedded One into the Other with a Gap Between Them

Abstract

AbstractIn this work, a contact interaction theory of closed cylindrical nanoshells, embedded one into the other, is constructed. There is a small gap between the shells. Cylindrical nanoshells are elastic, homogeneous, isotropic, and subject to the kinematic hypothesis of the first approximation (Kirchhoff-Love); geometric nonlinearity is taken into account according to T. von Karman’s theory, and contact interaction is taken according to Winkler’s model and to B.Ya. Kantor’s theory. Size-dependent effects are taken into account according to the modified couple stress theory. The system of nonlinear partial differential equations is reduced to the Cauchy problem by the Faedo-Galerkin method in higher approximations for spatial coordinates. The method convergence, namely, the Faedo-Galerkin method, depending on the number of terms in the series is investigated. The Cauchy problem is solved by several methods: the Runge-Kutta-type methods from the second to eighth accuracy orders and the Newmark method for confirming reliable results. The results analysis is carried out by nonlinear dynamics with the construction of signals, phase portraits, Fourier power spectra, and wavelet spectra. This work is the first to analyze the chaotic oscillations type of closed cylindrical nanoshells during their contact interaction. The chaos-type analysis is carried out according to the Gulik criterion and on the basis of calculating the spectrum signs of Lyapunov exponents by the Sano-Sawada method. To confirm the reliability of the results obtained, the largest Lyapunov exponents are calculated by several methods: Wolf, Kantz, Rosenstein, and Sano-Sawada.KeywordsModified couple stress theoryFlexible closed cylindrical nanoshellsContact interactionChaotic dynamicsHyperchaosLyapunov exponents