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Effect of residual surface stress on parametrically excited nonlinear dynamics and instability of viscoelastic piezoelectric nanoelectromechanical resonators

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Abstract

The present study mainly investigates the effect of the residual surface stress and the applied electric voltage on the nonlinear dynamic instability of the viscoelastic piezoelectric nanoresonators under parametric excitation. In fact, great attention is given to the influence of the residual surface stress on the nonlinear instability of the system. Numerical examples are treated which show various bifurcations. By means of a bifurcation analysis, it is shown that the instability of the system can be significantly affected by considering the residual surface effect. The results also show that a discontinuous bifurcation is always accompanied by a jump. Finally, stable and unstable regions in dynamic instability of viscoelastic piezoelectric 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

\(f\) :

Body forces

\(\Phi\) :

Electric potential

\(E_{z}\) :

Electric field

\(\sigma\) :

Detuning parameter

\(e_{0} a\) :

Nonlocal parameter

\(\beta\) :

Nonlinearity coefficient

\(\eta\) :

Phase angle

\(F\) :

Force amplitude

\(N_{\rm AB}^{*}\) :

Total resultant forces

\(M_{\rm AB}^{*}\) :

Total resultant moments

\(D_{11} , D_{12} ,D_{22} , D_{66}\) :

Bending stiffness

\(t\) :

Time

\(\rho\) :

Mass density

\(\omega_{0}\) :

Natural frequency

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

In-plane displacements

\(u_{3}\) :

Transverse displacement

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

Middle surface displacements

\(N_{AB}\) :

Resultant forces

\(M_{AB}\) :

Resultant moments

\(D_{{\Lambda }}^{s}\) :

Plane electric movements

\(D_{{\Lambda }}^{0}\) :

Residual plane electric movement

\(l\) :

Length of the nanoplate

\(b\) :

Width of the nanoplate

\(h\) :

Thickness of the nanoplate

\(\tau_{0}\) :

Residual surface stress

\(e_{31}^{s}\) :

Surface piezoelectric constant

\(q_{z}\) :

Lateral load

\(Q_{A} , Q_{B}\) :

Shear forces

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

Bulk elastic constants

\(c_{11}^{s} , c_{12}^{s} , c_{66}^{s}\) :

Surface 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

\(e_{31}\) :

Bulk piezoelectric constant

\(k_{33}\) :

Dielectric constants

\(k_{w}\) :

Winkler foundation coefficient

\(k_{G}\) :

Pasternak foundation coefficient

\(f\) :

Transverse load

\({\Omega }\) :

Frequency excitation

\(\in\) :

Scaling parameter

\(w\) :

Nanoplate deflection

\(\mu\) :

Damping coefficient

\(\sigma_{\rm AB}^{s}\) :

Surface stresses

\(\sigma_{\rm AB}^{0}\) :

Residual surface stresses without applied strain

\(\psi\) :

Time-dependent function

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ali Shariati

  2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ali Shariati

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

    S. Hamed S. Hosseini, S. Sedighi Bayrami & Farzad Ebrahimi

  4. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Ali Toghroli

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  1. Ali Shariati
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Shariati, A., Hosseini, S.H.S., Bayrami, S.S. et al. Effect of residual surface stress on parametrically excited nonlinear dynamics and instability of viscoelastic piezoelectric nanoelectromechanical resonators. Engineering with Computers 37, 1835–1850 (2021). https://doi.org/10.1007/s00366-019-00916-9

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