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

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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.

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Abbreviations

\( E \) :

Young’s modulus

\( \sigma^{\prime} \) :

The classical stress tensor

\( \sigma_{xx} , \sigma_{yy} , \sigma_{xy} \) :

Normal stress components

\( \varepsilon_{xx} , \varepsilon_{yy} , \varepsilon_{xy} \) :

Strain vector

\( \sigma_{ij,j} \) :

Stress

\( f_{i} \) :

Body forces

\( u_{i}^{ \pm } \) :

Surface displacement

\( \tau_{i\alpha }^{ \pm } \) :

Surface stresses

\( \sigma \) :

Detuning parameter

\( e_{0} a \) :

Nonlocal parameter

\( \beta \) :

Nonlinearity coefficient

\( \eta \) :

Phase angle

\( F \) :

Force amplitude

\( D \) :

Bending stiffness

\( t \) :

Time

\( \rho \) :

Mass density

\( \omega_{0} \) :

Natural frequency

\( u_{1} , u_{2} \) :

In-plane displacements

\( u_{3} \) :

Transverse displacement

\( u_{0} , v_{0} \) :

Middle surface displacements

\( N_{ij} \) :

Resultant forces

\( M_{ij} \) :

Resultant moments

\( \psi \) :

Time-dependent function

\( l \) :

Length of the nanoplate

\( b \) :

Width of the nanoplate

\( E^{s} \) :

Surface elastic modulus

\( \tau_{0} \) :

Residual surface stress

\( \sigma_{i3}^{ \pm } \) :

Bulk stresses

\( \rho_{0}^{ \pm } \) :

Surface density

\( h \) :

Thickness of the nanoplate

\( c_{11} , c_{12} , c_{21} c_{22} , c_{66} \) :

Bulk elastic constants

\( T_{0} , T_{1} \) :

Time scales

\( c_{d} \) :

Viscous damping coefficient

\( \delta \) :

Parametric excitation amplitude

\( I_{0} , I_{2} \) :

The moment of inertia

\( I_{00} , I_{22} \) :

The moment of inertia with surface effect

\( k_{w} \) :

Winkler foundation coefficient

\( k_{G} \) :

Pasternak foundation coefficient

\( f \) :

Transverse load

\( \varOmega \) :

Frequency excitation

\( \varepsilon \) :

Scaling parameter

\( w \) :

Nanoplate deflection

\( \mu \) :

Damping coefficient

\( \sigma_{\alpha \beta }^{s} \) :

Surface stresses

\( \sigma_{\alpha \beta }^{0} \) :

Residual surface stresses without applied strain

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, 3414896818, Iran

    Farzad Ebrahimi & S. Hamed S. Hosseini

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  1. Farzad Ebrahimi
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  2. S. Hamed S. Hosseini
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Correspondence to Farzad Ebrahimi.

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Ebrahimi, F., Hosseini, S.H.S. Nonlinear dynamics and stability of viscoelastic nanoplates considering residual surface stress and surface elasticity effects: a parametric excitation analysis. Engineering with Computers 37, 1709–1722 (2021). https://doi.org/10.1007/s00366-019-00906-x

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