Out-of-plane dynamic instability of nonlocal shear deformable nanoplates made of polyvinylidene fluoride materials subjected to electromechanical forces

Abstract

Nowadays, piezoelectric materials are used as smart nanostructures in many engineering applications. Polyvinylidene fluoride (PVDF) is one of the piezoelectric polymeric materials which has fascinated the attention of the scientific community due to its remarkable properties. In the present study, dynamic stability and the parametric resonance of a Mindlin PVDF nanoplate lying on an elastic medium under electromechanical loadings and a moving nanoparticle are investigated. First, Hamilton's principle is used to obtain the partial equations governing the transverse oscillations of the PVDF nanoplate. By utilizing the theory of nonlocal piezoelasticity, the small-scale effects are applied to the equations. Navier’s approach is employed to procure the solution for the simply supported nanoplate. Then, the boundaries of instability in the plane of parameters are obtained by applying the energy-rate method to the governing time-varying ODEs. The results demonstrate that the increase in the nonlocal parameter and reduction of the nanoparticle movement path decrease the dynamic stability of the PVDF nanoplate. Also, the nanoplate made of PVDF materials is dynamically more stable than the nanoplate made of PZT-4 piezoceramic materials. Moreover, the piezoelectric voltage can be used to control the parametric resonance conditions of nanostructures. The results of this study have the potential to be utilized in the accurate design of nanoscale piezoelectric mass sensors in nanoelectromechanical systems.

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Correspondence to Davood Toghraie.

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Appendices

Appendix A

Dimensionless parameters are defined as follows:

$$ \begin{aligned} &\alpha = \frac{m}{{\rho hab}},\,\,\,\,\,\,\mu ^{*} = \frac{\mu }{{ab}},\,\,\,\,\,\,\,a_{r} = \frac{a}{b},\,\,\,\,\,\,\tau = \omega t,\,\,\,\,\,\,W = \frac{w}{{\sqrt {ab} }},\,\,\,\,\,\,\,\Psi _{x} = \psi _{x} ,\,\,\,\,\,\,\,\Psi _{y} = \psi _{y} \hfill \\ &\zeta = \frac{{a_{0} }}{a},\,\,\,\,\,\,\,\eta = \frac{{b_{0} }}{b},\,\,\,\,\,\,\,r_{a} = \frac{h}{a},\,\,\,\,\,\,\,r_{b} = \frac{h}{b},\,\,\,\,\,\,\,\,\Omega = \frac{\omega }{{\pi \sqrt {\frac{\kappa }{\rho }\left( {\frac{{C_{{55}}^{{{\text{PVDF}}}} }}{{a^{2} }} + \frac{{C_{{44}}^{{{\text{PVDF}}}} }}{{b^{2} }}} \right)} }}, \hfill \\ &\bar{\Phi } = \frac{\Phi }{{\sqrt {\frac{{C_{{11}}^{{{\text{PVDF}}}} }}{{ \in _{{11}} }}ab} }},\,\,\,\,\,k_{w}^{*} = \frac{{k_{w} a^{2} b^{2} }}{{\pi ^{2} \kappa h\left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,k_{g}^{*} = \frac{{k_{g} \left( {a^{2} + b^{2} } \right)}}{{\kappa h\left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\ &N_{{xm}}^{*} = \frac{{N_{{xm}} b^{2} }}{{\kappa h\left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,N_{{ym}}^{*} = \frac{{N_{{ym}} a^{2} }}{{\kappa h\left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\ &Q_{1} = \frac{{e_{{31}} V_{0} b^{2} }}{{\kappa h\left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,\,\,\,\,Q_{2} = \frac{{e_{{32}} V_{0} a^{2} }}{{\kappa h\left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\&T_{1} = \frac{{C_{{55}}^{{{\text{PVDF}}}} ab^{2} }}{{\pi \sqrt {ab} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,T_{2} = \frac{{C_{{44}}^{{{\text{PVDF}}}} a^{2} b}}{{\pi \sqrt {ab} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\ &T_{3} = \frac{{C_{{11}}^{{{\text{PVDF}}}} b^{2} }}{{\kappa \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{4} = \frac{{C_{{66}}^{{{\text{PVDF}}}} a^{2} }}{{\kappa \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\&T_{5} = \frac{{C_{{55}}^{{{\text{PVDF}}}} a^{2} b^{2} }}{{\pi h^{2} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,\,\,\,\,T_{6} = \frac{{C_{{12}}^{{{\text{PVDF}}}} ab}}{{\kappa \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\&T_{7} = \frac{{C_{{66}}^{{{\text{PVDF}}}} ab}}{{\kappa \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{8} = \frac{{C_{{55}}^{{{\text{PVDF}}}} ab^{2} \sqrt {ab} }}{{\pi h^{2} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\&T_{9} = \frac{{C_{{11}}^{{{\text{PVDF}}}} a^{2} }}{{\kappa \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T_{{10}} = \frac{{C_{{66}}^{{{\text{PVDF}}}} b^{2} }}{{\pi h^{2} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\ &T_{{11}} = \frac{{C_{{44}}^{{{\text{PVDF}}}} a^{2} b^{2} }}{{\pi h^{2} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}},\,\,\,\,\,\,\,\,\,\,\,\,\,T_{{12}} = \frac{{C_{{44}}^{{{\text{PVDF}}}} a^{2} b\sqrt {ab} }}{{\pi h^{2} \left( {a^{2} C_{{44}}^{{{\text{PVDF}}}} + b^{2} C_{{55}}^{{{\text{PVDF}}}} } \right)}}, \hfill \\ &A_{1} = \sqrt {\frac{{C_{{44}}^{{{\text{PVDF}}}} }}{{C_{{11}}^{{{\text{PVDF}}}} }}} ,\,\,\,\,\,\,\,A_{2} = \sqrt {\frac{{C_{{55}}^{{{\text{PVDF}}}} }}{{C_{{11}}^{{{\text{PVDF}}}} }}} ,\,\,\,\,\,\,\,E_{1} = \frac{{\sqrt {C_{{44}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{15}} }},\,\,\,\,\,\,\,\,E_{2} = \frac{{\sqrt {C_{{55}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{15}} }},\,\,\,\,\,\, \hfill \\& E_{3} = \frac{{\sqrt {C_{{44}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{24}} }},\,\,\,\,\,\,\,E_{4} = \frac{{\sqrt {C_{{55}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{24}} }},\,\,\,\,\,\,\,E_{5} = \frac{{\sqrt {C_{{44}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{31}} }},\,\,\,\,\,\,\,E_{6} = \frac{{\sqrt {C_{{55}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{31}} }}, \hfill \\& E_{7} = \frac{{\sqrt {C_{{44}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{32}} }},\,\,\,\,\,\,\,E_{8} = \frac{{\sqrt {C_{{55}}^{{{\text{PVDF}}}} \in _{{11}} } }}{{e_{{32}} }},\,\,\,\,\,\,\,E_{9} = \frac{{\sqrt {C_{{11}}^{{{\text{PVDF}}}} \in _{{33}} } }}{{e_{{24}} }},\,\,\,\,\,\,\,E_{{10}} = \frac{{\sqrt {C_{{11}}^{{{\text{PVDF}}}} \in _{{33}} } }}{{e_{{15}} }}, \hfill \\& E_{{11}} = \frac{{\sqrt {C_{{11}}^{{{\text{PVDF}}}} \in _{{33}} } }}{{e_{{31}} }},\,\,\,\,\,\,\,E_{{12}} = \frac{{\sqrt {C_{{11}}^{{{\text{PVDF}}}} \in _{{33}} } }}{{e_{{32}} }},\,\,\,\,\,\,\,\Xi _{1} = \sqrt {\frac{{ \in _{{33}} }}{{ \in _{{11}} }}} ,\,\,\,\,\,\,\,\Xi _{2} = \sqrt {\frac{{ \in _{{22}} }}{{ \in _{{33}} }}} ,\,\,\,\,\,\,\,\Xi _{3} = \sqrt {\frac{{ \in _{{11}} }}{{ \in _{{33}} }}} , \hfill \\& g^{*} = \frac{g}{{\frac{{\pi ^{2} \kappa \sqrt {ab} }}{\rho }\left( {\frac{{C_{{55}}^{{{\text{PVDF}}}} }}{{a^{2} }} + \frac{{C_{{44}}^{{{\text{PVDF}}}} }}{{b^{2} }}} \right)}}. \hfill \\ \end{aligned}$$
(A.1)

Appendix B

The components of vectors and matrices appearing in Eq. (48) are as follows:

$$ \begin{aligned} &{\textbf{M}}_{1} = \left[ {\begin{array}{*{20}c} {1 + \pi ^{2} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right)} & 0 & 0 \\ 0 & {1 + \pi ^{2} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right)} & 0 \\ 0 & 0 & {1 + \pi ^{2} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right)} \\ \end{array} } \right], \hfill \\ &{\textbf{M}}_{2} \left( \tau \right) = \left[ {\begin{array}{*{20}c} {P_{1} (\tau ) + \pi ^{2} \mu ^{*} P_{1} (\tau )\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right)} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right], \hfill \\ &{\textbf{C}}_{1} (\tau ) = \left[ {\begin{array}{*{20}c} {c_{{11}} } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right], \hfill \\ &{\textbf{K}}_{1} = \left[ {\begin{array}{*{20}c} {k_{{11}} } & {k_{{12}} } & {k_{{13}} } \\ {k_{{21}} } & {k_{{22}} } & {k_{{23}} } \\ {k_{{31}} } & {k_{{32}} } & {k_{{33}} } \\ \end{array} } \right],\,{\textbf{K}}_{2} (\tau ) = \left[ {\begin{array}{*{20}c} {\bar{k}_{{11}} } & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right],\,{\textbf{Q}} = \left[ {W,\Psi _{x} ,\Psi _{y} } \right]^{T} , \hfill \\ \end{aligned} $$
(B.1)

in which

$$ \begin{aligned} c_{{11}} &= \pi \zeta \,\sin \left( \tau \right)P_{2} \left( \tau \right) + \pi ^{3} \mu ^{*} \zeta \,\sin \left( \tau \right)P_{2} \left( \tau \right)\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) - \pi \eta \cos \left( \tau \right)P_{3} \left( \tau \right) \hfill \\ &\quad - \pi ^{3} \mu ^{*} \eta \cos \left( \tau \right)P_{3} \left( \tau \right)\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right), \hfill \\ k_{{11}}& = 1 + k_{w}^{*} + \pi ^{2} k_{w}^{*} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) + k_{g}^{*} + \pi ^{2} G^{*} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) + N_{{xm}}^{*} + \pi ^{2} N_{{xm}}^{*} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) \hfill \\&\quad + N_{{ym}}^{*} + \pi ^{2} N_{{ym}}^{*} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) + 2Q_{1} + 2\pi ^{2} Q_{1} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) + 2Q_{2} + 2\pi ^{2} Q_{2} \mu ^{*} \left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) \hfill \\ &\quad + \frac{{\left( {\frac{2}{{\pi \left( {a_{r}^{2} A_{1} E_{1} + A_{2} E_{2} } \right)}} + \frac{2}{{\pi \left( {A_{1} E_{3} + \frac{1}{{a_{r}^{2} }}A_{2} E_{4} } \right)}}} \right)\left( {\frac{{4r_{b}^{2} }}{{\pi \,\Xi _{1} E_{9} }} + \frac{{4r_{a}^{2} }}{{\pi \,\Xi _{1} E_{{10}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }}, \hfill \\ k_{{12}} &= T_{1} + \frac{{\left( {\frac{2}{{\pi \left( {a_{r}^{2} A_{1} E_{1} + A_{2} E_{2} } \right)}} + \frac{2}{{\pi \left( {A_{1} E_{3} + \frac{1}{{a_{r}^{2} }}A_{2} E_{4} } \right)}}} \right)\left( {\frac{{4\sqrt {r_{a}^{3} r_{b} } }}{{\pi ^{2} \,\Xi _{1} E_{{10}} }} + \frac{{4\sqrt {r_{a}^{3} r_{b} } }}{{\pi ^{2} \,\Xi _{1} E_{{11}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }},\,\,\,\,\,\,\,\,\, \hfill \\ \,k_{{13}} &= T_{2} + \frac{{\left( {\frac{2}{{\pi \left( {a_{r}^{2} A_{1} E_{1} + A_{2} E_{2} } \right)}} + \frac{2}{{\pi \left( {A_{1} E_{3} + \frac{1}{{a_{r}^{2} }}A_{2} E_{4} } \right)}}} \right)\left( {\frac{{4\sqrt {r_{a} r_{b}^{3} } }}{{\pi ^{2} \,\Xi _{1} E_{9} }} + \frac{{4\sqrt {r_{a} r_{b}^{3} } }}{{\pi ^{2} \,\Xi _{1} E_{{12}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }},\,\,\,\, \hfill \\ k_{{21}} &= 12T_{8} + \frac{{\left( {\frac{{24}}{{\pi ^{2} \kappa \left( {a_{r}^{2} \sqrt {r_{a}^{3} r_{b} } A_{1} E_{5} + \sqrt {r_{a}^{3} r_{b} } A_{2} E_{6} } \right)}} + \frac{{24}}{{\pi ^{2} \left( {a_{r}^{2} \sqrt {r_{a}^{3} r_{b} } A_{1} E_{1} + \sqrt {r_{a}^{3} r_{b} } A_{2} E_{2} } \right)}}} \right)\left( {\frac{{4r_{b}^{2} }}{{\pi \,\Xi _{1} E_{9} }} + \frac{{4r_{a}^{2} }}{{\pi \,\Xi _{1} E_{{10}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }},\, \hfill \\ k_{{22}}& = T_{3} + T_{4} + 12T_{5} + \frac{{\left( {\frac{{24}}{{\pi ^{2} \kappa \left( {a_{r}^{2} \sqrt {r_{a}^{3} r_{b} } A_{1} E_{5} + \sqrt {r_{a}^{3} r_{b} } A_{2} E_{6} } \right)}} + \frac{{24}}{{\pi ^{2} \left( {a_{r}^{2} \sqrt {r_{a}^{3} r_{b} } A_{1} E_{1} + \sqrt {r_{a}^{3} r_{b} } A_{2} E_{2} } \right)}}} \right)\left( {\frac{{4\sqrt {r_{a}^{3} r_{b} } }}{{\pi ^{2} \,\Xi _{1} E_{{10}} }} + \frac{{4\sqrt {r_{a}^{3} r_{b} } }}{{\pi ^{2} \,\Xi _{1} E_{{11}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }}, \hfill \\ k_{{23}}& = T_{6} + T_{7} + \frac{{\left( {\frac{{24}}{{\pi ^{2} \kappa \left( {a_{r}^{2} \sqrt {r_{a}^{3} r_{b} } A_{1} E_{5} + \sqrt {r_{a}^{3} r_{b} } A_{2} E_{6} } \right)}} + \frac{{24}}{{\pi ^{2} \left( {a_{r}^{2} \sqrt {r_{a}^{3} r_{b} } A_{1} E_{1} + \sqrt {r_{a}^{3} r_{b} } A_{2} E_{2} } \right)}}} \right)\left( {\frac{{4\sqrt {r_{a} r_{b}^{3} } }}{{\pi ^{2} \,\Xi _{1} E_{9} }} + \frac{{4\sqrt {r_{a} r_{b}^{3} } }}{{\pi ^{2} \,\Xi _{1} E_{{12}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }}, \hfill \\ k_{{31}} &= 12T_{{12}} + \frac{{\left( {\frac{{24}}{{\pi ^{2} \kappa \left( {\sqrt {r_{a} r_{b}^{3} } A_{1} E_{7} + \frac{{\sqrt {r_{a} r_{b}^{3} } }}{{a_{r} }}A_{2} E_{8} } \right)}} + \frac{{24}}{{\pi ^{2} \left( {\sqrt {r_{a} r_{b}^{3} } A_{1} E_{3} + \frac{{\sqrt {r_{a} r_{b}^{3} } }}{{a_{r} }}A_{2} E_{4} } \right)}}} \right)\left( {\frac{{4r_{b}^{2} }}{{\pi \,\Xi _{1} E_{9} }} + \frac{{4r_{a}^{2} }}{{\pi \,\Xi _{1} E_{{10}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }}, \hfill \\ k_{{32}} &= T_{6} + T_{7} + \frac{{\left( {\frac{{24}}{{\pi ^{2} \kappa \left( {\sqrt {r_{a} r_{b}^{3} } A_{1} E_{7} + \frac{{\sqrt {r_{a} r_{b}^{3} } }}{{a_{r} }}A_{2} E_{8} } \right)}} + \frac{{24}}{{\pi ^{2} \left( {\sqrt {r_{a} r_{b}^{3} } A_{1} E_{3} + \frac{{\sqrt {r_{a} r_{b}^{3} } }}{{a_{r} }}A_{2} E_{4} } \right)}}} \right)\left( {\frac{{4\sqrt {r_{a}^{3} r_{b} } }}{{\pi ^{2} \,\Xi _{1} E_{{10}} }} + \frac{{4\sqrt {r_{a}^{3} r_{b} } }}{{\pi ^{2} \,\Xi _{1} E_{{11}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }},\, \hfill \\ k_{{33}} &= T_{9} + T_{{10}} + 12T_{{11}} + \frac{{\left( {\frac{{24}}{{\pi ^{2} \kappa \left( {\sqrt {r_{a} r_{b}^{3} } A_{1} E_{7} + \frac{{\sqrt {r_{a} r_{b}^{3} } }}{{a_{r} }}A_{2} E_{8} } \right)}} + \frac{{24}}{{\pi ^{2} \left( {\sqrt {r_{a} r_{b}^{3} } A_{1} E_{3} + \frac{{\sqrt {r_{a} r_{b}^{3} } }}{{a_{r} }}A_{2} E_{4} } \right)}}} \right)\left( {\frac{{4\sqrt {r_{a} r_{b}^{3} } }}{{\pi ^{2} \,\Xi _{1} E_{9} }} + \frac{{4\sqrt {r_{a} r_{b}^{3} } }}{{\pi ^{2} \,\Xi _{1} E_{{12}} }}} \right)}}{{1 + r_{b}^{2} \,\Xi _{2} + r_{b}^{2} \,\Xi _{3} }},\,\,\,\,\,\,\,\, \hfill \\ \bar{k}_{{11}} &= - 4\alpha \pi ^{2} \zeta ^{2} \,\Omega ^{2} \,\sin ^{2} \left( \tau \right)P_{1} \left( \tau \right) - 4\alpha \pi ^{4} \mu ^{*} \zeta ^{2} \,\Omega ^{2} \,\sin ^{2} \left( \tau \right)P_{1} \left( \tau \right)\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) \hfill \\ &\quad - 4\alpha \pi ^{2} \eta ^{2} \,\Omega ^{2} \,\cos ^{2} \left( \tau \right)P_{1} \left( \tau \right) - 4\alpha \pi ^{4} \mu ^{*} \eta ^{2} \,\Omega ^{2} \,\cos ^{2} \left( \tau \right)P_{1} \left( \tau \right)\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) \hfill \\&\quad - 8\alpha \pi ^{2} \zeta \eta \,\Omega ^{2} \sin \left( \tau \right)\,\cos \left( \tau \right)P_{4} \left( \tau \right)P_{5} \left( \tau \right) \hfill \\ &\quad - 8\alpha \pi ^{4} \mu ^{*} \zeta \eta \,\Omega ^{2} \sin \left( \tau \right)\,\cos \left( \tau \right)P_{4} \left( \tau \right)P_{5} \left( \tau \right)\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) \hfill \\ &\quad + 4\alpha \pi \zeta \,\Omega ^{2} \,\cos \left( \tau \right)P_{2} \left( \tau \right) + 4\alpha \pi ^{3} \mu ^{*} \zeta \,\Omega ^{2} \,\cos \left( \tau \right)P_{2} \left( \tau \right)\left( {\frac{{1 + a_{r}^{2} }}{{a_{r} }}} \right) \hfill \\ &\quad + 4\alpha \pi \eta \,\Omega ^{2} \,\sin \left( \tau \right)P_{3} \left( \tau \right) + 4\alpha \pi ^{3} \mu ^{*} \eta \,\Omega ^{2} \,\sin \left( \tau \right)P_{3} \left( \tau \right). \hfill \\ \end{aligned} $$
(B.2)

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Pirmoradian, M., Torkan, E., Hashemian, M. et al. Out-of-plane dynamic instability of nonlocal shear deformable nanoplates made of polyvinylidene fluoride materials subjected to electromechanical forces. J Braz. Soc. Mech. Sci. Eng. 43, 145 (2021). https://doi.org/10.1007/s40430-021-02846-4

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Keywords

  • Dynamic stability
  • Parametric resonance
  • Polyvinylidene fluoride material
  • Nonlocal piezoelasticity theory
  • Mindlin plate theory