Mathematical modeling of nonlinear thermodynamics of nanoplates

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.