Nonlinear dynamics and stability of viscoelastic nanoplates considering residual surface stress and surface elasticity effects: a parametric excitation analysis

Abstract

The present study mainly investigates surface effect on nonlinear dynamic instability of viscoelastic nanoplates under parametric excitation. In fact, great attention is given to the influence of residual surface stress on nonlinear dynamic behavior of the system. To achieve this goal, the governing equation of motion is derived by modeling a nanoplate embedded on a visco-Pasternak foundation and then, applying surface effect relations, nonlocal elasticity and nonlinear von Karman theories and Hamilton’s principle, respectively. Galerkin technique and multiple time scales method are also used to solve the equation. A class of nonlinear Mathieu–Hill equation is established to determine the bifurcations and the regions of nonlinear dynamic instability. The numerical results are performed, while the emphasis is placed on investigating the effect of residual surface stress, visco-Pasternak foundation coefficients, and parametric excitation. It is shown how residual surface stress leads to high values of amplitude response. Finally, stable and unstable regions in dynamic instability of viscoelastic nanoplates are addressed.