Nonlinear thermo-electromechanical vibration analysis of size-dependent functionally graded flexoelectric nano-plate exposed magnetic field

Abstract

In the present study, a continuous-based thermo-electromechanic model has been developed by the Kirchhoff plate’s theory and the modified flexoelectric theory in order to study the size-dependent nonlinear free vibration of functionally graded flexoelectric nano-plate under the magnetic field. Using the Hamilton’s principle and variation method, the nonlinear governing differential equations of the nano-plate and their associated boundary conditions have been extracted and the governing equations solved by using Galerkin’s and perturbation methods. The electromechanical coupling (electromechanical stress) in the internal energy function causes nonlinearity in the governing equations. The applied magnetic field is a type of external static field along with the nano-plate thickness. The natural frequencies and related mode shapes have been determined in two modes of direct and inverse flexoelectric effects. Also, the effects of such factors as length scale parameters, geometric parameters, thermal, magnetic and electrical loadings were investigated. In the presence of flexoelectric effect, the results showed that the dependence of electromechanical behavior of the structure on size is found to be significant in nanoscales. Regarding the application of this type of nano-plate in the oscillators and considering the flexoelectric effect, the applied potential difference can play an important role in adjusting and controlling the frequency.

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Appendices

Appendix A

$$\begin{aligned} g_{1} ({{\hat{n}}})= & {} \rho _{2} \left( {h-2{\bar{z}}_{c} } \right) +h\left( {\rho _{2} -\rho _{1} } \right) f_{1} ({{\hat{n}}}), \nonumber \\ g_{2} ({{\hat{n}}})= & {} -2\left( {\frac{\rho _{2} h}{2}\left( {h-2{\bar{z}}_{c} } \right) +h^{2}\left( {\rho _{2} -\rho _{1} } \right) f_{2} ({{\hat{n}}})} \right) , \nonumber \\ g_{3} ({{\hat{n}}})= & {} \frac{\rho _{2} h}{3}\left( {h^{2}-3h{\bar{z}}_{c} +3{\bar{z}}_{c}^{2} } \right) +h^{3}\left( {\rho _{2} -\rho _{1} } \right) f_{3} ({{\hat{n}}}), \end{aligned}$$
(A.1)
$$\begin{aligned} f_{1} ({{\hat{n}}})= & {} \frac{({1+{\bar{h}}})^{{\hat{n}}+1} -({{\bar{h}}})^{{\hat{n}}+1}}{n+1}, \nonumber \\ f_{2} ({{\hat{n}}})= & {} \frac{({1+{\bar{h}}})^{{\hat{n}}+2}- ({{\bar{h}}})^{{\hat{n}}+2}}{n+2}-\frac{({1+{\bar{h}}})^{{\hat{n}}+1}- ({{\bar{h}}})^{{\hat{n}}+1}}{2({n+1})},\nonumber \\ f_{3} ({{\hat{n}}})= & {} \frac{({1+{\bar{h}}})^{{\hat{n}}+3}- ({{\bar{h}}})^{{\hat{n}}+3}}{n+3}-\frac{({1+{\bar{h}}})^{{\hat{n}}+2} -({{\bar{h}}})^{{\hat{n}}+2}}{n+2}-\frac{({1+{\bar{h}}})^{{\hat{n}}+1} -({{\bar{h}}})^{{\hat{n}}+1}}{4({n+1})}, \end{aligned}$$
(A.2)

where:

$$\begin{aligned} {\bar{h}}=\frac{1}{2}-\frac{{\bar{z}}_{c} }{h}. \end{aligned}$$

Appendix B

$$\begin{aligned} N_{xx}= & {} {\hat{a}}_{1} \left( {\frac{\partial ^{2}w}{\partial x^{2}}} \right) ^{2}+{\hat{a}}_{2} \left( {\frac{\partial ^{2}w}{\partial y^{2}}} \right) ^{2}+{\hat{a}}_{3} \left( {\frac{\partial u_{\circ } }{\partial x}} \right) ^{2}+{\hat{a}}_{4} \left( {\frac{\partial v_{\circ } }{\partial y}} \right) ^{2}+{\hat{a}}_{5} +\,{\hat{a}}_{6} \left( {\frac{\partial ^{2}w}{\partial x^{2}}} \right) \nonumber \\&+{\hat{a}}_{7} \left( {\frac{\partial u_{\circ } }{\partial x}\frac{\partial ^{2}w}{\partial x^{2}}} \right) +{\hat{a}}_{8} \left( {\frac{\partial v_{\circ } }{\partial y}\frac{\partial ^{2}w}{\partial x^{2}}} \right) +{\hat{a}}_{9} \left( {\frac{\partial ^{2}w}{\partial y^{2}}\frac{\partial ^{2}w}{\partial x^{2}}} \right) \,+\,{\hat{a}}_{10} \left( {\frac{\partial ^{2}w}{\partial y^{2}}} \right) \nonumber \\&+{\hat{a}}_{11} \left( {\frac{\partial u_{\circ } }{\partial x}\frac{\partial ^{2}w}{\partial y^{2}}} \right) +{\hat{a}}_{12} \left( {\frac{\partial v_{\circ } }{\partial y}\frac{\partial ^{2}w}{\partial y^{2}}} \right) \,+\,{\hat{a}}_{13} \left( {\frac{\partial u_{\circ } }{\partial x}} \right) +{\hat{a}}_{14} \left( {\frac{\partial v_{\circ } }{\partial y}\frac{\partial u_{\circ } }{\partial x}} \right) \,\,+\,{\hat{a}}_{15} \left( {\frac{\partial v_{\circ } }{\partial y}} \right) , \nonumber \\ N_{xx}^{hx}= & {} {\hat{b}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{b}}_{2} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{b}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) +{\hat{b}}_{4} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) +{\hat{b}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) , \nonumber \\ N_{yy}^{hx}= & {} {\hat{c}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{c}}_{2} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{c}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) +{\hat{c}}_{4} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) +{\hat{c}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) , \nonumber \\ N_{xy}^{hx}= & {} {\hat{d}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{2}\partial y}} \right) +{\hat{d}}_{2} \left( {\frac{\partial ^{3}w}{\partial y^{3}}} \right) +{\hat{d}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x\partial y}} \right) +{\hat{d}}_{4} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x^{2}}} \right) +{\hat{d}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial y^{2}}} \right) , \nonumber \\ Q= & {} {\hat{e}}_{1} \left( {\frac{\partial ^{2}w}{\partial x\partial y}} \right) +{\hat{e}}_{2} \left( {\frac{\partial u_{\circ } }{\partial y}} \right) +{\hat{e}}_{3} \left( {\frac{\partial v_{\circ } }{\partial x}} \right) , \end{aligned}$$
(B.1)
$$\begin{aligned} N_{yy}= & {} {\hat{f}}_{1} \left( {\frac{\partial ^{2}w}{\partial x^{2}}} \right) +{\hat{f}}_{2} \left( {\frac{\partial ^{2}w}{\partial y^{2}}} \right) +{\hat{f}}_{3} \left( {\frac{\partial u_{\circ } }{\partial x}} \right) +{\hat{f}}_{4} \left( {\frac{\partial v_{\circ } }{\partial y}} \right) +{\hat{f}}_{5}, \nonumber \\ N_{xx}^{hy}= & {} {\hat{g}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{g}}_{2} \left( {\frac{\partial ^{3}w}{\partial y^{3}}} \right) +{\hat{g}}_{3} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{g}}_{4} \left( {\frac{\partial ^{3}w_{\circ } }{\partial y\partial x^{2}}} \right) +{\hat{g}}_{5} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) \nonumber \\&+{\hat{g}}_{6} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) +{\hat{g}}_{7} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x\partial y}} \right) +{\hat{g}}_{8} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x^{2}}} \right) +{\hat{g}}_{9} \left( {\frac{\partial ^{2}v_{\circ } }{\partial y^{2}}} \right) +{\hat{g}}_{10} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) , \nonumber \\ N_{yy}^{hy}= & {} {\hat{h}}_{1} \left( {\frac{\partial ^{3}w}{\partial y^{3}}} \right) +{\hat{h}}_{2} \left( {\frac{\partial ^{3}w}{\partial y\partial x^{2}}} \right) +{\hat{h}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x\partial y}} \right) +{\hat{h}}_{4} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x^{2}}} \right) +{\hat{h}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial y^{2}}} \right) , \nonumber \\ N_{xy}^{hy}= & {} {\hat{i}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{i}}_{2} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{i}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) +{\hat{i}}_{4} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) +{\hat{i}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) , \, \hbox {uik} \end{aligned}$$
(B.2)
$$\begin{aligned} M_{xx}= & {} {\hat{j}}_{1} \left( {\frac{\partial ^{2}w}{\partial x^{2}}} \right) ^{2}+{\hat{j}}_{2} \left( {\frac{\partial ^{2}w}{\partial y^{2}}} \right) ^{2}+{\hat{j}}_{3} \left( {\frac{\partial u_{\circ } }{\partial x}} \right) ^{2}+{\hat{j}}_{4} \left( {\frac{\partial v_{\circ } }{\partial y}} \right) ^{2}+{\hat{j}}_{5} +{\hat{j}}_{6} \left( {\frac{\partial ^{2}w}{\partial x^{2}}} \right) \nonumber \\&+{\hat{j}}_{7} \left( {\frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial ^{2}w}{\partial y^{2}}} \right) +{\hat{j}}_{8} \left( {\frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial u_{\circ } }{\partial x}} \right) +{\hat{j}}_{9} \left( {\frac{\partial ^{2}w}{\partial x^{2}}\frac{\partial v_{\circ } }{\partial y}} \right) +{\hat{j}}_{10} \left( {\frac{\partial ^{2}w}{\partial y^{2}}} \right) +{\hat{j}}_{11} \left( {\frac{\partial ^{2}w}{\partial y^{2}}\frac{\partial u_{\circ } }{\partial x}} \right) \nonumber \\&+{\hat{j}}_{12} \left( {\frac{\partial ^{2}w}{\partial y^{2}}\frac{\partial v_{\circ } }{\partial y}} \right) +{\hat{j}}_{13} \left( {\frac{\partial u_{\circ } }{\partial x}} \right) +{\hat{j}}_{14} \left( {\frac{\partial v_{\circ } }{\partial y}\frac{\partial u_{\circ } }{\partial x}} \right) +{\hat{j}}_{15} \left( {\frac{\partial v_{\circ } }{\partial y}} \right) +{\hat{j}}_{16} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) , \nonumber \\ M_{yy}= & {} {\hat{k}}_{1} \left( {\frac{\partial ^{2}w}{\partial x^{2}}} \right) +{\hat{k}}_{2} \left( {\frac{\partial ^{2}w}{\partial y^{2}}} \right) +{\hat{k}}_{3} \left( {\frac{\partial ^{3}u_{\circ } }{\partial x\partial y^{2}}} \right) +{\hat{k}}_{4} \left( {\frac{\partial u_{\circ } }{\partial x}} \right) +{\hat{k}}_{5} \left( {\frac{\partial v_{\circ } }{\partial y}} \right) +{\hat{k}}_{6}, \nonumber \\ M_{xy}= & {} {\hat{l}}_{1} \left( {\frac{\partial ^{2}w}{\partial x\partial y}} \right) +{\hat{l}}_{2} \left( {\frac{\partial u_{\circ } }{\partial y}} \right) +{\hat{l}}_{3} \left( {\frac{\partial v_{\circ } }{\partial x}} \right) , \nonumber \\ M_{xx}^{h}= & {} {\hat{m}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{m}}_{2} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{m}}_{3} \left( {\frac{\partial ^{3}w}{\partial y\partial x^{2}}} \right) +{\hat{m}}_{4} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) +{\hat{m}}_{5} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) \nonumber \\&+{\hat{m}}_{6} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x\partial y}} \right) +{\hat{m}}_{7} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x^{2}}} \right) +{\hat{m}}_{8} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) , \nonumber \\ M_{yy}^{h}= & {} {\hat{n}}_{1} \left( {\frac{\partial ^{3}w}{\partial y^{3}}} \right) +{\hat{n}}_{2} \left( {\frac{\partial ^{3}w}{\partial y\partial x^{2}}} \right) +{\hat{n}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x\partial y}} \right) +{\hat{n}}_{4} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x^{2}}} \right) +{\hat{n}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial y^{2}}} \right) , \nonumber \\ M_{xxy}^{h}= & {} {\hat{o}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{o}}_{2} \left( {\frac{\partial ^{3}w}{\partial y^{3}}} \right) +{\hat{o}}_{3} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{o}}_{4} \left( {\frac{\partial ^{3}w}{\partial y\partial x^{2}}} \right) +{\hat{o}}_{5} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) \nonumber \\&+{\hat{o}}_{6} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) +{\hat{o}}_{7} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x\partial y}} \right) +{\hat{o}}_{8} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x^{2}}} \right) +{\hat{o}}_{9} \left( {\frac{\partial ^{2}v_{\circ } }{\partial y^{2}}} \right) +{\hat{o}}_{10} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) , \nonumber \\ M_{xyy}^{h}= & {} {\hat{p}}_{1} \left( {\frac{\partial ^{3}w}{\partial x^{3}}} \right) +{\hat{p}}_{2} \left( {\frac{\partial ^{3}w}{\partial x\partial y^{2}}} \right) +{\hat{p}}_{3} \left( {\frac{\partial ^{2}u_{\circ } }{\partial x^{2}}} \right) +{\hat{p}}_{4} \left( {\frac{\partial ^{2}u_{\circ } }{\partial y^{2}}} \right) +{\hat{p}}_{5} \left( {\frac{\partial ^{2}v_{\circ } }{\partial x\partial y}} \right) . \end{aligned}$$
(B.3)

The coefficients \({\hat{a}}_{i}\), \({\hat{b}}_{i}\), \({\hat{c}}_{i}\), \({\hat{d}}_{i}\), \({\hat{e}}_{i}\), \({\hat{f}}_{i}\), \({\hat{g}}_{i}\), \({\hat{h}}_{i}\), \({\hat{i}}_{i}\), \({\hat{j}}_{i}\), \({\hat{k}}_{i}\), \({\hat{l}}_{i}\), \({\hat{m}}_{i}\), \({\hat{n}}_{i} ,{\hat{o}}_{i}\), \({\hat{p}}_{i}\) are the material derivatives determined by integrating along the z-axis. In this section, the process of determining one of these coefficients such as \({\hbox {a}}_{\mathrm {i}}\) is described. For example, \({\hbox {N}}_{\mathrm {xx}}\) is obtained using equation (33) as follows:

$$\begin{aligned} N_{xx} =N_{\sigma _{11} } +N_{\sigma _{11}^{ES} } +N_{\sigma _{11}^{MS} } =\int \limits _{-{\bar{z}}_{c} }^{h-{\bar{z}}_{c} } {\left( {\sigma _{11} +\sigma _{11}^{ES} +\sigma _{11}^{MS} } \right) } \mathrm{d}z. \end{aligned}$$
(B.4)

Using relations (2), (12) and (13), the parameters \(\sigma _{11}\), \(\sigma _{11}^{ES}\), \(\sigma _{11}^{MS}\) are simplified as follows:

$$\begin{aligned} \sigma _{11}= & {} \frac{\partial U}{\partial \varepsilon _{11} }=k\varepsilon _{nn} +2\mu {\varepsilon }'_{11} -f_{1}^{E} Q_{kk} -f_{1}^{M} S_{kk} -\beta _{11} \Delta T, \nonumber \\ \sigma _{11}^{ES}= & {} \frac{1}{2}E_{3} P_{3} +\frac{1}{2}V_{33} Q_{33} -\frac{1}{2}\varepsilon _{0} \varphi _{,3} \varphi _{,3} +\varphi _{,3} P_{3} \nonumber \\ \sigma _{11}^{MS}= & {} \frac{1}{2}H_{3} B_{3} +\frac{1}{2}Z_{33} S_{33} -\frac{1}{2}\mu \psi _{,3} \psi _{,3}. \end{aligned}$$
(B.5)

Using strain–displacement relations (Eq. 9), the above quantities are obtained in terms of displacement components. Therefore, substituting the obtained parameters in Appendix (B.4) and integrating along the z-axis, the constants \({\hat{a}}_{i}\) are determined. A similar process will be used to determine the other coefficients.

Appendix C

$$\begin{aligned} F_{1}= & {} \frac{A_{11} B_{12}^{2} -A_{12} B_{11} B_{12} +A_{15} B_{11}^{2} }{B_{11}^{2}}, \quad F_{2} =\frac{A_{16} B_{11}^{2} -\left( {A_{12} B_{13} -A_{13} B_{12} } \right) B_{11} +2B_{12} B_{13} A_{11} }{B_{11}^{2}}, \nonumber \\ F_{3}= & {} \frac{({-2A_{11} B_{12} +A_{12} B_{11}}) B_{14}}{B_{11}^{2}}, \quad F_{4} =\frac{-A_{14} B_{12} +A_{17} B_{11} }{B_{11}}, \nonumber \\ F_{5}= & {} \frac{A_{11} B_{13}^{2} -A_{13} B_{11} B_{13} +A_{18} B_{11}^{2} }{B_{11}^{2}}, \quad F_{6} =\frac{({2A_{11} B_{13} -A_{13} B_{11}}) B_{14}}{B_{11}^{2}}, \nonumber \\ F_{7}= & {} \frac{-A_{14} B_{13} +A_{19} B_{11} }{B_{11}}, F_{8} =\frac{-A_{11} B_{14}^{2} }{B_{11}^{2}}, \nonumber \\ F_{9}= & {} \frac{({-A_{20} B_{11} +A_{14} B_{14}})}{B_{11}}, \end{aligned}$$
(C.1)
$$\begin{aligned} G_{1}= & {} -4F_{1} F_{8} +F_{3}^{2},\quad G_{2} =-4F_{1} F_{6} +2F_{2} F_{3},\quad G_{3} =-4F_{9} F_{1} +2F_{3} F_{4}, \nonumber \\ G_{4}= & {} -4F_{1} F_{5} +F_{2}^{2}, \quad G_{5} =-4F_{1} F_{7} +2F_{2} F_{4} ,\quad G_{6} =F_{4}^{2}. \end{aligned}$$
(C.2)

Appendix D

$$\begin{aligned} \zeta _{1}= & {} \frac{1}{2F_{1}^{2} B_{11}^{2} }\left[ {2\zeta _{11} F_{1}^{2} +\zeta _{12} F_{1} +\zeta _{13} F_{2}^{2} } \right] ,\quad \zeta _{2} =\frac{1}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{21} F_{1}^{2} +\zeta _{22} F_{1} +2\zeta _{23} F_{2} F_{3} } \right] , \nonumber \\ \zeta _{3}= & {} \frac{{\tilde{g}}}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{31} F_{1} +\zeta _{32} F_{2} } \right] , \quad \zeta _{4} =\frac{1}{2F_{1}^{2} B_{11} }\left[ {\zeta _{41} F_{1}^{2} +\zeta _{42} F_{1} +2\zeta _{43} F_{2} F_{4} } \right] , \nonumber \\ \zeta _{5}= & {} \frac{1}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{51} F_{1}^{2} +\zeta _{52} F_{1} +\zeta _{53} F_{3}^{2} } \right] , \quad \zeta _{6} =\frac{{\tilde{g}}}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{61} F_{1} +\zeta _{62} F_{3} } \right] , \nonumber \\ \zeta _{7}= & {} \frac{1}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{71} F_{1}^{2} +\zeta _{72} F_{1} +2\zeta _{73} F_{3} F_{4} } \right] ,\quad \zeta _{8} =\frac{{\tilde{g}}}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{81} F_{1} +\zeta _{82} F_{4} } \right] , \nonumber \\ \zeta _{9}= & {} \frac{1}{2F_{1}^{2} B_{11}^{2} }\left[ {\zeta _{91} F_{1}^{2} +\zeta _{92} F_{1} F_{4} +\zeta _{93} F_{4}^{2} } \right] , \end{aligned}$$
(D.1)

where:

$$\begin{aligned} \zeta _{11}= & {} B_{11}^{2} C_{8} -B_{11} B_{13} C_{13} +B_{13}^{2} C_{11}, \nonumber \\ \zeta _{12}= & {} \left( {F_{2} C_{16} -2F_{5} C_{15} } \right) B_{11}^{2} +\left( {\left( {F_{2} C_{13} +2F_{5} C_{12} } \right) B_{12} +B_{13} F_{2} C_{12} } \right) B_{11} \nonumber \\&-2B_{12} C_{11} \left( {F_{2} B_{13} +F_{5} B_{12} } \right) , \nonumber \\ \zeta _{13}= & {} B_{11}^{2} C_{5} -B_{11} B_{12} C_{12} +B_{12}^{2} C_{11},\nonumber \\ \zeta _{21}= & {} -2B_{14} \left( {B_{11} C_{13} -2B_{13} C_{11} } \right) , \nonumber \\ \zeta _{22}= & {} \left( {F_{3} C_{16} -2F_{6} C_{15} } \right) B_{11}^{2} +\left( {\left( {F_{3} C_{13} +2F_{6} C_{12} } \right) B_{12} +C_{12} \left( {F_{2} B_{14} +F_{3} B_{13} } \right) } \right) B_{11} \nonumber \\&-2B_{12} C_{11} \left( {F_{2} B_{14} +F_{3} B_{13} +F_{6} B_{12} } \right) , \nonumber \\ \zeta _{23}= & {} B_{11}^{2} C_{15} -B_{11} B_{12} C_{12} +B_{12}^{2} C_{11}, \nonumber \\ \zeta _{31}= & {} B_{11}^{2} C_{16} -\left( {B_{12} C_{13} +B_{13} C_{12} } \right) B_{11} +2B_{12} B_{23} C_{11}, \nonumber \\ \zeta _{32}= & {} -\left( {B_{11}^{2} C_{15} -B_{11} B_{12} C_{12} +B_{12}^{2} C_{11} } \right) , \nonumber \\ \zeta _{41}= & {} \left( {2B_{11}^{2} C_{19} -2B_{11} B_{13} C_{14} } \right) , \nonumber \\ \zeta _{42}= & {} -\left( {F_{2} C_{17} +2F_{4} C_{16} +2F_{7} C_{15} } \right) B_{11}^{2} +\left[ {\left( {F_{2} C_{14} +F_{4} C_{13} +2F_{7} C_{12} } \right) B_{12} +B_{13} F_{4} C_{12} } \right] B_{11} \nonumber \\&-2B_{12} C_{11} \left( {F_{4} B_{13} +F_{7} B_{12} } \right) , \nonumber \\ \zeta _{43}= & {} 2\left( {B_{11}^{2} C_{15} -B_{11} B_{12} C_{12} +B_{12}^{2} C_{11} } \right) ,\nonumber \\ \zeta _{51}= & {} 2\left( {B_{14}^{2} C_{11} } \right) ,\nonumber \\ \zeta _{52}= & {} \left( {-2F_{8} C_{15} } \right) B_{11}^{2} +\left( {B_{14} F_{3} +2B_{12} F_{8} } \right) C_{12} B_{11} -2\left( {B_{14} F_{3} +B_{12} F_{8} } \right) C_{11} B_{12}, \nonumber \\ \zeta _{53}= & {} B_{11}^{2} C_{15} -B_{11} B_{12} C_{12} +C_{11} B_{12}^{2}, \nonumber \\ \zeta _{61}= & {} -\left( {B_{11} C_{12} -2B_{12} C_{11} } \right) B_{14}, \nonumber \\ \zeta _{62}= & {} -\left( {C_{15} B_{11}^{2} -B_{11} B_{12} C_{12} +C_{11} B_{12}^{2} } \right) , \nonumber \\ \zeta _{71}= & {} 2\left( {C_{20} B_{11}^{2} -B_{11} B_{14} C_{14} } \right) , \nonumber \\ \zeta _{72}= & {} -\left( {F_{3} C_{17} +F_{9} C_{15} +2F_{7} C_{15} } \right) B_{11}^{2} +\left[ {\left( {F_{3} C_{14} +2F_{9} C_{12} } \right) B_{12} +B_{14} F_{4} C_{12} } \right] B_{11} \nonumber \\&-2B_{12} C_{11} \left( {F_{4} B_{14} +F_{9} B_{12} } \right) , \nonumber \\ \zeta _{73}= & {} \left( {C_{15} B_{11}^{2} -B_{11} B_{12} C_{12} +C_{11} B_{12}^{2} } \right) ,\nonumber \\ \zeta _{81}= & {} \left( {B_{11} C_{17} -B_{12} C_{14} } \right) B_{11}, \nonumber \\ \zeta _{82}= & {} -\left( {C_{15} B_{11}^{2} -B_{11} B_{12} C_{12} +C_{11} B_{12}^{2} } \right) , \nonumber \\ \zeta _{91}= & {} 2\left( {B_{11}^{2} C_{11} } \right) , \nonumber \\ \zeta _{92}= & {} -\left( {B_{11} C_{17} -B_{12} C_{14} } \right) B_{11}, \nonumber \\ \zeta _{93}= & {} \left( {C_{15} B_{11}^{2} -B_{11} B_{12} C_{12} +C_{11} B_{12}^{2} } \right) . \end{aligned}$$
(D.2)

Appendix E

$$\begin{aligned} \lambda _{1}= & {} \frac{G_{5} }{2G_{6}},\quad \lambda _{2} =\frac{G_{3} }{2G_{6}},\quad \lambda _{3} =\frac{G_{2} }{2G_{6}}-\frac{G_{3} G_{5} }{4G_{6}^{2}}, \nonumber \\ \lambda _{4}= & {} \frac{G_{4} }{2G_{6} }-\frac{1}{8}\left( {\frac{G_{5} }{G_{6} }} \right) ^{2}, \quad \lambda _{5} =\frac{G_{1} }{2G_{6} }-\frac{1}{8}\left( {\frac{G_{3} }{G_{6} }} \right) ^{2}, \quad \lambda _{6} =-\frac{G_{2} G_{5} +G_{3} G_{4} }{4G_{6}^{2} }+\frac{3G_{5}^{2} G_{3} }{16G_{6}^{3}}, \nonumber \\ \lambda _{7}= & {} -\frac{G_{1} G_{5} +G_{2} G_{3} }{4G_{6}^{2} }+\frac{3G_{5} G_{3}^{2} }{16G_{6}^{3} },\quad \lambda _{8} =\frac{G_{4} G_{5} }{4G_{6}^{2} }+\frac{1}{16}\left( {\frac{G_{5} }{G_{6} }} \right) ^{3}, \nonumber \\ \lambda _{9}= & {} -\frac{G_{1} G_{3} }{4G_{6}^{2} }+\frac{1}{16}\left( {\frac{G_{3} }{G_{6} }} \right) ^{3}. \end{aligned}$$
(E.1)

Appendix F

$$\begin{aligned} \eta _{1}= & {} \zeta _{9} +\zeta _{8} G_{6}^{1/2} ,\quad \eta _{2} =\zeta _{3} G_{6}^{1/2} +\zeta _{4} +\zeta _{8} G_{6}^{1/2} \lambda _{1}, \nonumber \\ \eta _{3}= & {} \zeta _{6} G_{6}^{1/2} +\zeta _{7} +\zeta _{8} G_{6}^{1/2} \lambda _{2} ,\quad \eta _{4} =\zeta _{2} +G_{6}^{1/2} \left( {\zeta _{3} \lambda _{2} +\zeta _{6} \lambda _{1} +\zeta _{8} \lambda _{3}} \right) , \nonumber \\ \eta _{5}= & {} \zeta _{1} +G_{6}^{1/2} \left( {\zeta _{3} \lambda _{1} +\zeta _{8} \lambda _{4} } \right) ,\quad \eta _{6} =\zeta _{5} +G_{6}^{1/2} \left( {\zeta _{6} \lambda _{2} +\zeta _{8} \lambda _{5} } \right) , \nonumber \\ \eta _{7}= & {} G_{6}^{1/2} \left( {\zeta _{3} \lambda _{3} +\zeta _{6} \lambda _{4} +\zeta _{8} \lambda _{6} } \right) , \quad \eta _{8} =G_{6}^{1/2} \left( {\zeta _{3} \lambda _{5} +\zeta _{6} \lambda _{3} +\zeta _{8} \lambda _{7} } \right) , \nonumber \\ \eta _{9}= & {} G_{6}^{1/2} \left( {\zeta _{3} \lambda _{4} +\zeta _{8} \lambda _{8} } \right) ,\quad \eta _{10} =G_{6}^{1/2} \left( {\zeta _{6} \lambda _{5} +\zeta _{8} \lambda _{9} } \right) . \end{aligned}$$
(F.1)

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Ghobadi, A., Beni, Y.T. & Golestanian, H. Nonlinear thermo-electromechanical vibration analysis of size-dependent functionally graded flexoelectric nano-plate exposed magnetic field. Arch Appl Mech (2020). https://doi.org/10.1007/s00419-020-01708-0

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Keywords

  • Flexoelectric theory
  • Nonlinear free vibration
  • Functionally graded behavior
  • Size effect
  • Static magnetic field