Stochastic stability of a magnetically affected single-layer graphene sheet resting on a viscoelastic foundation

Abstract

In this paper, we analyzed the stochastic stability of a single-layer graphene sheet resting on a viscoelastic foundation and influenced by the in-plane magnetic field. The mechanical model of a graphene sheet is given as an orthotropic and viscoelastic nanoplate while the viscoelastic foundation is of the Kelvin-Voigt type. We assume that the graphene sheet is influenced by the in-plane random forces variable with time and exerted in-plane magnetic field. Based on the Eringen's nonlocal continuum theory and Kirchhoff – Love plate theory, the governing equation of motion is derived by considering the influence of the Lorentz forces obtained from the classical Maxwell relations. In order to investigate the stochastic stability of such system, the maximal and moment Lyapunov exponents are considered by using the perturbation method. The predicted approximated analytical results for the p-th moment Lyapunov exponents are validated by the Monte Carlo method. Moreover, the boundaries of almost-sure and moment stability of the viscoelastic nanoplate are determined as functions of different system parameters. The influences of the nonlocal and magnetic field parameters, stiffness and damping coefficients and spectral density on the moment Lyapunov exponents are investigated through several numerical examples. Presented results reveal that the applied in-plane magnetic field could be successfully used to improve stability performances of nano-electromechanical systems based on graphene sheets.