Primary and Secondary Resonance Analyses of Viscoelastic Nanoplates Based on Strain Gradient Theory

Abstract

In this study, the nonstationary oscillation, secondary resonance and nonlinear dynamic behavior of viscoelastic nanoplates with linear damping are investigated based on the modified strain gradient theory extended for viscoelastic materials. The viscous component of the nonclassical and classical stress tensors are evaluated on the basis of the Leaderman viscoelastic model. Then, incorporating the size-dependent potential energy, kinetic energy and an external excitation force work, the governing equations of the oscillations are obtained based on the Hamilton’s principle. The governing formula is obtained as a nonlinear second-order integro-differential partial equation. This size-dependent viscoelastic formula is solved using analytical Harmonic balance method (HBM) and the fourth-order Runge–Kutta technique after applying the expansion theory. Additionally, the stability of the steady-state response is examined by means of HBM. Then, the secondary resonance conditions due to the super-harmonic motion are determined by performing frequency response, force response, Poincare map and phase portrait analyses. In addition, the nonstationary transient vibration of viscoelastic nanosystem is analyzed by performing Hilbert–Huang transform.

How does flexoelectricity affect static bending and nonlinear dynamic response of nanoscale lipid bilayers?

“…On the other hand, the constitutive equations for the viscoelastic nanostructures, according to the nonlinear Leaderman integral [51], considering the flexoelectric effects can be defined as [47]…”

“…The viscoelastic properties of the lipid bilayers were analyzed with atomistic simulation and experiment [42,43]. Baoukina and Mukhin used another approach based on continuum mechanics and modeled a flat lipid bilayer membrane as a thin plate and obtained its free energy functional [44] and recently, we developed the viscoelastic size-dependent models based on nonclassical elasticity theories for flat nanoplates [45][46][47] and are going to combine the same model with flexoelectricity, in this paper. The accuracy of this model was proved by Momeni Bashusqeh and Rastgoo who computed the elastic modulus and bending rigidity of a free-standing DDPC bilayer with coarse-grained molecular dynamics [48].…”