Mathematical modeling of nonlinear thermodynamics of nanoplates

V.A. Krysko-, J. Awrejcewicz, E.Yu. Krylova and I.V. Papkova

Chaos, Solitons & Fractals, 2022, vol. 158, issue C

Abstract: The theory of nonlinear statics and dynamics of flexible plates, taking into account the modified couple stress theory and temperature field, is developed. The theory is based on the classical Kirchhoff model for an isotropic elastic body. Geometric nonlinearity is taken into account according to the von Kármán model. Variational differential equations are yielded by the Hamilton principle, and partial differential equation (PDE's) related to displacements of the middle surface and deflection are derived. The hypotheses of the modified couple stress theory implied an increase of the order of the system of partial differential equations due to the fact that moments of higher order appeared. The resolved PDE's are reduced to the Cauchy problem by the method of finite differences of the second order accuracy, which is solved by methods of the Runge-Kutta type (from the fourth to the eighth order of accuracy) and the Newmark method. No restrictions are imposed on the temperature field, and it determined from the solution of the three-dimensional heat equation using the finite element method (FEM). The convergence of the proposed algorithms is investigated depending on the number of finite elements and the time step. Static problems are solved by the dynamic approach. Nonlinear dynamics is analyzed based on (Fourier spectrum, wavelets of different types and phase portraits). The analysis of Lyapunov exponents obtained by different methods (Kantz, Wolf, Rosenstein, Sano-Sawada and the authors method) is carried out, which validated occurrence of chaotic vibrations. Hyperchaotic vibrations have been detected and studied. It is also illustrated how increase of the temperature field influences localization of the chaotic zones for size-dependent plates.

Keywords: Couple stress theory; Hyperchaotic vibrations; Mathematical modeling; Kantz, Wolf, Rosenstein and Sano-Sawada methods; Flexible plates (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0960077922002375
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:158:y:2022:i:c:s0960077922002375

DOI: 10.1016/j.chaos.2022.112027

Access Statistics for this article

Chaos, Solitons & Fractals is currently edited by Stefano Boccaletti and Stelios Bekiros

More articles in Chaos, Solitons & Fractals from Elsevier
Bibliographic data for series maintained by Thayer, Thomas R. ().