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Effect of Magnetic Field on Vibration of Electrorheological Fluid Nanoplates with FG-CNTRC Layers

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Abstract

Background

Functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) have attracted significant attention in the field of structural engineering due to their unique mechanical properties.

Objective

In this study, the free vibration of a functionally graded carbon nanotube-reinforced composite sandwich nanoplate with an electrorheological fluid (ERF) layer as its core under a longitudinal magnetic field is examined using the nonlocal elasticity theory.

Results

A comparison with some other item of existing literature is used to evaluate the established solution's accuracy and correctness. The obtained results demonstrate the dependency of the vibration behavior on the aforementioned factors. Specifically, the influences of electric field strength, boundary conditions, magnetic field intensity, volume fraction of carbon nanotubes (CNTs), CNTs distribution, and nonlocal parameter are comprehensively analyzed through a parametric study.

Methods

The governing equations are derived based on Hamilton's principle and solved using the Galerkin technique. While the continuity of physical quantities is required between all layers, the rule of mixing allows us to analyze the distribution of characteristics in this system's thickness direction. Changing the electric field also alters the ERF parameters in the pre-yield region. The developed mathematical model incorporates the nonlocal elasticity theory and the third-order shear deformation theory (TSDT) for different boundary conditions.

Conclusion

The findings contribute to a better understanding of the dynamic response of such composite structures and can aid in their optimal design for various engineering applications.

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Data Availability

Data sharing not applicable.

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Acknowledgements

This study was supported by Thammasat Postdoctoral Fellowship, Thammasat University research Division, Thammasat University. In addition, this research was supported by Thammasat University research Unit in Structural and Foundation Engineering, Thammasat University.

Funding

This research received no external funding.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Thammasat School of Engineering, Faculty of Engineering, Thammasat University, Pathumthani, 12121, Thailand

    Peyman Roodgar Saffari, Chanachai Thongchom & Sayan Sirimontree

  2. Department of Engineering, School of Physics, Engineering and Computer Science, University of Hertfordshire, Hatfield, AL10 9AB, England, UK

    Sikiru Oluwarotimi Ismail

  3. Department of Mechanical Engineering, Faculty of Engineering, Thammasat School of Engineering, Thammasat University, Pathumthani, 12121, Thailand

    Thira Jearsiripongkul

Authors
  1. Peyman Roodgar Saffari
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  2. Sikiru Oluwarotimi Ismail
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  3. Chanachai Thongchom
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  4. Sayan Sirimontree
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  5. Thira Jearsiripongkul
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Contributions

PRS: conceptualization, data curation, formal analysis, project administration, software, investigation, validation, supervision, and writing—review and editing. SOI: conceptualization, formal analysis, project administration, software, investigation, validation, supervision, writing—review and editing, and writing—original draft. CT: conceptualization, data curation, formal analysis, project administration, visualization, investigation, validation, supervision, and writing—review and editing. SS: investigation, data curation, validation, and writing—original draft. TJ: investigation, software, validation, and formal analysis. All the authors have read and agreed to the published version of the manuscript.

Corresponding authors

Correspondence to Peyman Roodgar Saffari, Sikiru Oluwarotimi Ismail or Chanachai Thongchom.

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Appendices

Appendix A

The governing equations of the sandwich nanoplate are provided as

$$\delta u_{0t} \to \frac{{\partial N_{xx}^{t} }}{\partial x} + \frac{{\partial N_{xy}^{t} }}{\partial y} - \frac{{3Q_{xz}^{c} }}{{4h_{2} }} + \frac{{9C_{1} R_{xz}^{c} }}{{4h_{2} }} = I_{0}^{t} \ddot{u}_{0t} + \overline{I}_{1}^{t} - I_{3}^{t} C_{1} \frac{{\partial \ddot{W}_{0} }}{\partial x},$$
(A1)
$$\delta u_{0b} \to \frac{{\partial N_{xx}^{b} }}{\partial x} + \frac{{\partial N_{xy}^{b} }}{\partial y} + \frac{{3Q_{xz}^{c} }}{{4h_{2} }} - \frac{{9C_{1} R_{xz}^{c} }}{{4h_{2} }} = I_{0}^{b} \ddot{u}_{0b} + \overline{I}_{1}^{b} \ddot{\theta }_{xb} - I_{3}^{b} C_{1} \frac{{\partial \ddot{W}_{0} }}{\partial x},$$
(A2)
$$\delta v_{0t} \to \frac{{\partial N_{yy}^{t} }}{\partial y} + \frac{{\partial N_{xy}^{t} }}{\partial x} - \frac{{3Q_{yz}^{c} }}{{4h_{2} }} + \frac{{9C_{1} R_{yz}^{c} }}{{4h_{2} }} = I_{0}^{t} \ddot{v}_{0t} + \overline{I}_{1}^{t} \ddot{\theta }_{yt} - I_{3}^{t} C_{1} \frac{{\partial \ddot{W}_{0} }}{\partial y},$$
(A3)
$$\delta v_{0b} \to \frac{{\partial N_{yy}^{b} }}{\partial y} + \frac{{\partial N_{xy}^{b} }}{\partial x} + \frac{{3Q_{yz}^{c} }}{{4h_{2} }} - \frac{{9C_{1} R_{yz}^{c} }}{{4h_{2} }} = I_{0}^{b} \ddot{v}_{0b} + \overline{I}_{1}^{b} \ddot{\theta }_{yb} - I_{3}^{b} C_{1} \frac{{\partial \ddot{W}_{0} }}{\partial y},$$
(A4)
$$\delta {W}_{0}\to \frac{\partial {Q}_{xz}^{t}}{\partial x}+\frac{\partial {Q}_{xz}^{b}}{\partial x}+\frac{\partial {Q}_{yz}^{t}}{\partial y}+\frac{\partial {Q}_{yz}^{b}}{\partial y}-3{C}_{1}\left(\frac{\partial {R}_{xz}^{t}}{\partial x}+\frac{\partial {R}_{yz}^{t}}{\partial y}\right)-3{C}_{1}\left(\frac{\partial {R}_{xz}^{b}}{\partial x}+\frac{\partial {R}_{yz}^{b}}{\partial y}\right)+{C}_{1} \left(\frac{{\partial }^{2}{P}_{xx}^{t}}{\partial {x}^{2}}+2\frac{{\partial }^{2}{P}_{xy}^{t}}{\partial x\partial y}+\frac{{\partial }^{2}{P}_{yy}^{t}}{\partial {y}^{2}}\right)+{C}_{1}\left(\frac{{\partial }^{2}{P}_{xx}^{b}}{\partial {x}^{2}}+2\frac{{\partial }^{2}{P}_{xy}^{b}}{\partial x\partial y}+\frac{{\partial }^{2}{P}_{yy}^{b}}{\partial {y}^{2}}\right) +\left(\frac{3}{4}-\frac{{h}_{1}}{8{h}_{2}}-\frac{{h}_{3}}{8{h}_{2}}\right) \left(\frac{\partial {Q}_{xz}^{c}}{\partial x}+\frac{\partial {Q}_{yz}^{c}}{\partial y}\right)-\left(\frac{9}{4}-\frac{{3h}_{1}}{8{h}_{2}}-\frac{{3h}_{3}}{8{h}_{2}}\right) {C}_{1}\left(\frac{\partial {R}_{xz}^{c}}{\partial x}+\frac{\partial {R}_{yz}^{c}}{\partial y}\right)=q+\left({I}_{0}^{t}+{I}_{0}^{c}+{I}_{0}^{b}\right){\ddot{W}}_{0}+{I}_{3}^{t}{C}_{1}\left(\frac{\partial {\ddot{u}}_{0t}}{\partial x}+\frac{\partial {\ddot{v}}_{0t}}{\partial y}\right)+{I}_{3}^{b}{C}_{1}\left(\frac{\partial {\ddot{u}}_{0b}}{\partial x}+\frac{\partial {\ddot{v}}_{0b}}{\partial y}\right)-\left({I}_{6}^{t}+{I}_{3}^{b}\right) {C}_{1}^{2}\left(\frac{{\partial }^{2}{\ddot{W}}_{0}}{\partial {x}^{2}}+\frac{{\partial }^{2}{\ddot{W}}_{0}}{\partial {y}^{2}}\right)+{\bar{I}}_{4}^{t}{C}_{1}\left(\frac{\partial {\ddot{\theta }}_{xt}}{\partial x}+\frac{\partial {\ddot{\theta }}_{yt}}{\partial y}\right)+{\bar{I}}_{4}^{b}{C}_{1}\left(\frac{\partial {\ddot{\theta }}_{xb}}{\partial x}+\frac{\partial {\ddot{\theta }}_{yb}}{\partial y}\right),$$
(A5)
$$\delta \theta_{xt} \to \frac{{\partial M_{xx}^{t} }}{\partial x} + \frac{{\partial M_{xy}^{t} }}{\partial y} - C_{1} \left( {\frac{{\partial P_{xx}^{t} }}{\partial x} + \frac{{\partial P_{xy}^{t} }}{\partial y}} \right) - Q_{xz}^{t} + 3C_{1} R_{xz}^{t} + \frac{{h_{1} Q_{xz}^{c} }}{{2h_{2} }} - \frac{{3C_{1} h_{1} R_{xz}^{c} }}{{2h_{2} }} = \overline{I}_{1}^{t} \ddot{u}_{0t} + \overline{I}_{2}^{t} \ddot{\theta }_{xt} - C_{1} \overline{I}_{4}^{t} \frac{{\partial \ddot{W}_{0} }}{\partial x},$$
(A6)
$$\delta \theta_{xb} \to \frac{{\partial M_{xx}^{b} }}{\partial x} + \frac{{\partial M_{xy}^{b} }}{\partial y} - C_{1} \left( {\frac{{\partial P_{xx}^{b} }}{\partial x} + \frac{{\partial P_{xy}^{b} }}{\partial y}} \right) - Q_{xz}^{b} + 3C_{1} R_{xz}^{b} + \frac{{h_{3} Q_{xz}^{c} }}{{2h_{2} }} - \frac{{3C_{1} h_{3} R_{xz}^{c} }}{{2h_{2} }} = \overline{I}_{1}^{b} \ddot{u}_{0b} + \overline{I}_{2}^{b} \ddot{\theta }_{xb} - C_{1} \overline{I}_{4}^{b} \frac{{\partial \ddot{W}_{0} }}{\partial x},$$
(A7)
$$\delta \theta_{yt} \to \frac{{\partial M_{yy}^{t} }}{\partial y} + \frac{{\partial M_{xy}^{t} }}{\partial x} - C_{1} \left( {\frac{{\partial P_{yy}^{t} }}{\partial y} + \frac{{\partial P_{xy}^{t} }}{\partial x}} \right) - Q_{yz}^{t} + 3C_{1} R_{yz}^{t} + \frac{{h_{1} Q_{yz}^{c} }}{{2h_{2} }} - \frac{{3C_{1} h_{1} R_{yz}^{c} }}{{2h_{2} }} = \overline{I}_{1}^{t} \ddot{v}_{0t} + \overline{I}_{2}^{t} \ddot{\theta }_{yt} - C_{1} \overline{I}_{4}^{t} \frac{{\partial \ddot{W}_{0} }}{\partial y},$$
(A8)
$$\delta \theta_{yb} \to \frac{{\partial M_{yy}^{b } }}{\partial y} + \frac{{\partial M_{xy}^{b} }}{\partial x} - C_{1} \left( {\frac{{\partial P_{yy}^{b} }}{\partial y} + \frac{{\partial P_{xy}^{b} }}{\partial x}} \right) - Q_{yz}^{t} + 3C_{1} R_{yz}^{b} + \frac{{h_{3} Q_{yz}^{c} }}{{2h_{2} }} - \frac{{3C_{1} h_{3} R_{yz}^{c} }}{{2h_{2} }} = \overline{I}_{1}^{b} \ddot{v}_{0b} + \overline{I}_{2}^{b} \ddot{\theta }_{yb} - C_{1} \overline{I}_{4}^{b} \frac{{\partial \ddot{W}_{0} }}{\partial y},$$
(A9)

where

$$\begin{gathered} \left( {I_{0}^{t} ,I_{1}^{t} ,I_{2}^{t} ,I_{3}^{t} ,I_{4}^{t} ,I_{6}^{t} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} \rho \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)dz, \hfill \\ \left( {I_{0}^{b} ,I_{1}^{b} ,I_{2}^{b} ,I_{3}^{b} ,I_{4}^{b} ,I_{6}^{b} } \right) = \mathop \int \limits_{{ - h_{3} /2}}^{{h_{3} /2}} \rho \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)dz, I_{0}^{c} = \mathop \int \limits_{{ - h_{2} /2}}^{{h_{2} /2}} \rho_{f} dz , \hfill \\ \end{gathered}$$
(A10)
$$\overline{I}_{1}^{t} = I_{1}^{t} - I_{3}^{t} C_{1} , \overline{I}_{2}^{t} = I_{2}^{t} - 2C_{1} I_{4}^{t} + C_{1}^{2} I_{6}^{t} , \overline{I}_{4}^{t} = I_{4}^{t} - C_{1} I_{6}^{t} ,$$
$$\overline{I}_{1}^{b} = I_{1}^{b} - I_{3}^{b} C_{1} , \overline{I}_{2}^{b} = I_{2}^{b} - 2C_{1} I_{4}^{b} + C_{1}^{2} I_{6}^{b} , \overline{I}_{4}^{b} = I_{4}^{b} - C_{1} I_{6}^{b} ,$$
$$\left( {N_{xx}^{t} ,M_{xx}^{t} ,P_{xx}^{t} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} \sigma_{xxt} \left( {1,z,z^{3} } \right)dz,$$
$$\left( {N_{xy}^{t} ,M_{xy}^{t} ,P_{xy}^{t} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} \tau_{xyt} \left( {1,z,z^{3} } \right)dz,$$
$$\left( {N_{yy}^{t} ,M_{yy}^{t} ,P_{yy}^{t} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} \sigma_{yyt} \left( {1,z,z^{3} } \right)dz,$$
$$\left( {Q_{xz}^{t} ,R_{xz}^{t} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} \tau_{xzt} \left( {1,z^{2} } \right)dz,\left( {Q_{xz}^{t} ,R_{xz}^{t} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} \tau_{xzt} \left( {1,z^{2} } \right)dz,$$
$$\left( {N_{xx}^{b} ,M_{xx}^{b} ,P_{xx}^{b} } \right) = \mathop \int \limits_{{ - h_{3} /2}}^{{h_{3} /2}} \sigma_{xxb} \left( {1,z,z^{3} } \right)dz,$$
$$\left( {N_{xy}^{b} ,M_{xy}^{b} ,P_{xy}^{b} } \right) = \mathop \int \limits_{{ - h_{3} /2}}^{{h_{3} /2}} \tau_{xyb} \left( {1,z,z^{3} } \right)dz,$$
$$\left( {N_{yy}^{b} ,M_{yy}^{b} ,P_{yy}^{b} } \right) = \mathop \int \limits_{{ - h_{3} /2}}^{{h_{3} /2}} \sigma_{yyb} \left( {1,z,z^{3} } \right)dz,$$
$$\left( {Q_{xz}^{b} ,R_{xz}^{b} } \right) = \mathop \int \limits_{{ - h_{3} /2}}^{{h_{3} /2}} \tau_{xzb} \left( {1,z^{2} } \right)dz,\left( {Q_{xz}^{b} ,R_{xz}^{b} } \right) = \mathop \int \limits_{{ - h_{3} /2}}^{{h_{3} /2}} \tau_{xzb} \left( {1,z^{2} } \right)dz$$
$$\left( {Q_{xz}^{c} ,R_{xz}^{c} } \right) = \mathop \int \limits_{{ - h_{2} /2}}^{{h_{2} /2}} \tau_{xzc} \left( {1,z^{2} } \right)dz, \left( {Q_{yz}^{c} ,R_{yz}^{c} } \right) = \mathop \int \limits_{{ - h_{2} /2}}^{{h_{2} /2}} \tau_{yzc} \left( {1,z^{2} } \right)dz.$$

Appendix B

The size-dependent governing partial differential equations are provided as

$$\delta {u}_{0t}\to {A}_{11}\left(\frac{{\partial }^{2}{u}_{0t}}{\partial {x}^{2}}\right)+{A}_{66}\left(\frac{{\partial }^{2}{u}_{0t}}{\partial {y}^{2}}\right)+\left[{A}_{12}+{A}_{66}\right]\left(\frac{{\partial }^{2}{v}_{0t}}{\partial x\partial y}\right)+\left[{B}_{11}-{C}_{1}{E}_{11}\right]\left(\frac{{\partial }^{2}{\theta }_{xt}}{\partial {x}^{2}}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xt}}{\partial {y}^{2}}\right)+\left[{B}_{12}+{B}_{66}-{C}_{1}{E}_{12}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yt}}{\partial x\partial y}\right)-\left[{C}_{1}{E}_{12}+2{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial x\partial {y}^{2}}\right)-{C}_{1}{E}_{11}\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{3}}\right)-\left(\frac{3}{4{h}_{2}}{f}_{1}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{4}\right)\left({u}_{0t}-{u}_{0b}\right)+\left(\frac{3}{4{h}_{2}}{f}_{2}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{5}\right)\left({h}_{1}{\theta }_{xt}+{h}_{3}{\theta }_{xb}\right)+\left(\frac{3}{4{h}_{2}}{f}_{3}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{6}\right)\frac{\partial {W}_{0}}{\partial x}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({I}_{0}^{t}{\ddot{u}}_{0t}+{\bar{I}}_{1}^{t}{\ddot{\theta }}_{xt}-{I}_{3}^{t}{C}_{1}\frac{\partial {\ddot{W}}_{0}}{\partial x}\right),$$
(B1)
$$\delta {u}_{0b}\to {A}_{11}\left(\frac{{\partial }^{2}{u}_{0b}}{\partial {x}^{2}}\right)+{A}_{66}\left(\frac{{\partial }^{2}{u}_{0b}}{\partial {y}^{2}}\right)+\left[{A}_{12}+{A}_{66}\right]\left(\frac{{\partial }^{2}{v}_{0b}}{\partial x\partial y}\right)+\left[{B}_{11}-{C}_{1}{E}_{11}\right]\left(\frac{{\partial }^{2}{\theta }_{xb}}{\partial {x}^{2}}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xb}}{\partial {y}^{2}}\right)+\left[{B}_{12}+{B}_{66}-{C}_{1}{E}_{12}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yb}}{\partial x\partial y}\right)-\left[{C}_{1}{E}_{12}+2{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial x\partial {y}^{2}}\right)-{C}_{1}{E}_{11}\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{3}}\right)+\left(\frac{3}{4{h}_{2}}{f}_{1}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{4}\right)\left({u}_{0t}-{u}_{0b}\right)-\left(\frac{3}{4{h}_{2}}{f}_{2}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{5}\right)\left({h}_{1}{\theta }_{xt}+{h}_{3}{\theta }_{xb}\right)-\left(\frac{3}{4{h}_{2}}{f}_{3}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{6}\right)\frac{\partial {W}_{0}}{\partial x}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({I}_{0}^{b}{\ddot{u}}_{0b}+{\bar{I}}_{1}^{b}{\ddot{\theta }}_{xb}-{I}_{3}^{b}{C}_{1}\frac{\partial {\ddot{W}}_{0}}{\partial x}\right),$$
(B2)
$$\delta {v}_{0t}\to {A}_{22}\left(\frac{{\partial }^{2}{v}_{0t}}{\partial {y}^{2}}\right)+{A}_{66}\left(\frac{{\partial }^{2}{v}_{0t}}{\partial {x}^{2}}\right)+\left[{A}_{21}+{A}_{66}\right]\left(\frac{{\partial }^{2}{u}_{0t}}{\partial x\partial y}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yt}}{\partial {x}^{2}}\right)+\left[{B}_{22}-{C}_{1}{E}_{22}\right]\left(\frac{{\partial }^{2}{\theta }_{yt}}{\partial {y}^{2}}\right)+\left[{B}_{21}+{B}_{66}-{C}_{1}{E}_{21}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xt}}{\partial x\partial y}\right)-\left[{C}_{1}{E}_{21}+2{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{2}\partial y}\right)-{C}_{1}{E}_{22}\left(\frac{{\partial }^{3}{W}_{0}}{\partial {y}^{3}}\right)-\left(\frac{3}{4{h}_{2}}{f}_{1}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{4}\right)\left({v}_{0t}-{v}_{0b}\right)+\left(\frac{3}{4{h}_{2}}{f}_{2}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{5}\right)\left({h}_{1}{\theta }_{yt}+{h}_{3}{\theta }_{yb}\right)+\left(\frac{3}{4{h}_{2}}{f}_{3}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{6}\right)\frac{\partial {W}_{0}}{\partial y}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({I}_{0}^{t}{\ddot{v}}_{0t}+{\bar{I}}_{1}^{t}{\ddot{\theta }}_{yt}-{I}_{3}^{t}{C}_{1}\frac{\partial {\ddot{W}}_{0}}{\partial y}\right),$$
(B3)
$${v}_{0b}\to {A}_{22}\left(\frac{{\partial }^{2}{v}_{0b}}{\partial {y}^{2}}\right)+{A}_{66}\left(\frac{{\partial }^{2}{v}_{0b}}{\partial {x}^{2}}\right)+\left[{A}_{21}+{A}_{66}\right]\left(\frac{{\partial }^{2}{u}_{0b}}{\partial x\partial y}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yb}}{\partial {x}^{2}}\right)+\left[{B}_{22}-{C}_{1}{E}_{22}\right]\left(\frac{{\partial }^{2}{\theta }_{yb}}{\partial {y}^{2}}\right)+\left[{B}_{21}+{B}_{66}-{C}_{1}{E}_{21}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xb}}{\partial x\partial y}\right)-\left[{C}_{1}{E}_{21}+2{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{2}\partial y}\right)-{C}_{1}{E}_{22}\left(\frac{{\partial }^{3}{W}_{0}}{\partial {y}^{3}}\right)+\left(\frac{3}{4{h}_{2}}{f}_{1}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{4}\right)\left({v}_{0t}-{v}_{0b}\right)-\left(\frac{3}{4{h}_{2}}{f}_{2}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{5}\right)\left({h}_{1}{\theta }_{yt}+{h}_{3}{\theta }_{yb}\right)-\left(\frac{3}{4{h}_{2}}{f}_{3}-\frac{9}{4{h}_{2}}{{C}_{1}f}_{6}\right)\frac{\partial {W}_{0}}{\partial y}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({I}_{0}^{t}{\ddot{v}}_{0t}+{\bar{I}}_{1}^{t}{\ddot{\theta }}_{yt}-{I}_{3}^{t}{C}_{1}\frac{\partial {\ddot{W}}_{0}}{\partial y}\right),$$
(B4)
$$\delta {W}_{0}\to 2\left[{A}_{44}-6{C}_{1}{H}_{44}+9{C}_{1}^{2}{T}_{44}\right]\left(\frac{{\partial }^{2}{W}_{0}}{\partial {x}^{2}}+\frac{{\partial }^{2}{W}_{0}}{\partial {y}^{2}}\right)+\left[{A}_{44}-6{C}_{1}{H}_{44}+9{C}_{1}^{2}{T}_{44}\right]\left(\frac{\partial {\theta }_{xt}}{\partial x}+\frac{\partial {\theta }_{yt}}{\partial y}+\frac{\partial {\theta }_{xb}}{\partial x}+\frac{\partial {\theta }_{yb}}{\partial y}\right)+{C}_{1}{E}_{11}\left(\frac{{\partial }^{3}{u}_{0t}}{\partial {x}^{3}}+\frac{{\partial }^{3}{u}_{0b}}{\partial {x}^{3}}\right)+{C}_{1}{E}_{22}\left(\frac{{\partial }^{3}{v}_{0t}}{\partial {y}^{3}}+\frac{{\partial }^{3}{v}_{0b}}{\partial {y}^{3}}\right)+\left[{C}_{1}{E}_{12}+2{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{3}{u}_{0t}}{\partial x\partial {y}^{2}}+\frac{{\partial }^{3}{u}_{0b}}{\partial x\partial {y}^{2}}+\frac{{\partial }^{3}{v}_{0t}}{\partial {x}^{2}\partial y}+\frac{{\partial }^{3}{v}_{0b}}{\partial {x}^{2}\partial y}\right)+\left[{C}_{1}{T}_{11}-{C}_{1}^{2}{D}_{11}\right]\left(\frac{{\partial }^{3}{\theta }_{xt}}{\partial {x}^{3}}+\frac{{\partial }^{3}{\theta }_{xt}}{\partial {x}^{3}}\right)+\left[{C}_{1}{T}_{22}-{C}_{1}^{2}{D}_{22}\right]\left(\frac{{\partial }^{3}{\theta }_{yt}}{\partial {y}^{3}}+\frac{{\partial }^{3}{\theta }_{yb}}{\partial {y}^{3}}\right)+\left[{C}_{1}{T}_{12}-{C}_{1}^{2}{D}_{12}+2\left({C}_{1}{T}_{66}-{C}_{1}^{2}{D}_{66}\right)\right]\left(\frac{{\partial }^{3}{\theta }_{xt}}{\partial x\partial {y}^{2}}+\frac{{\partial }^{3}{\theta }_{yt}}{\partial {x}^{2}\partial y}+\frac{{\partial }^{3}{\theta }_{xb}}{\partial x\partial {y}^{2}}+\frac{{\partial }^{3}{\theta }_{yb}}{\partial {x}^{2}\partial y}\right)-2{C}_{1}^{2}{D}_{11}\left(\frac{{\partial }^{4}{W}_{0}}{\partial {x}^{4}}\right)-2{C}_{1}^{2}{D}_{22}\left(\frac{{\partial }^{4}{W}_{0}}{\partial {y}^{4}}\right)-2\left[2{C}_{1}^{2}{D}_{12}+4{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{4}{W}_{0}}{\partial {x}^{2}\partial {y}^{2}}\right)-\left(r{f}_{1}-r{C}_{1}{f}_{4}\right)\left(\frac{\partial {u}_{0t}}{\partial x}-\frac{\partial {u}_{0b}}{\partial x}\right)+\left(r{f}_{2}-r{C}_{1}{f}_{5}\right)\left({h}_{1}\frac{\partial {\theta }_{xt}}{\partial x}+{h}_{2}\frac{\partial {\theta }_{xb}}{\partial x}\right)+\left(r{f}_{3}-r{C}_{1}{f}_{6}\right)\frac{{\partial }^{2}{W}_{0}}{\partial {x}^{2}}-\left(r{f}_{1}-r{C}_{1}{f}_{4}\right)\left(\frac{\partial {v}_{0t}}{\partial y}-\frac{\partial {v}_{0b}}{\partial x}\right)+\left(r{f}_{2}-r{C}_{1}{f}_{5}\right)\left({h}_{1}\frac{\partial {\theta }_{yt}}{\partial y}+{h}_{2}\frac{\partial {\theta }_{yb}}{\partial y}\right)+\left(r{f}_{3}-r{C}_{1}{f}_{6}\right)\frac{{\partial }^{2}{W}_{0}}{\partial {y}^{2}}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left\{q+\left({I}_{0}^{t}+{I}_{0}^{c}+{I}_{0}^{b}\right){\ddot{W}}_{0}+{I}_{3}^{t}{C}_{1}\left(\frac{\partial {\ddot{u}}_{0t}}{\partial x}+\frac{\partial {\ddot{v}}_{0t}}{\partial y}\right)+{I}_{3}^{b}{C}_{1}\left(\frac{\partial {\ddot{u}}_{0b}}{\partial x}+\frac{\partial {\ddot{v}}_{0b}}{\partial y}\right)-\left({I}_{6}^{t}+{I}_{3}^{b}\right){C}_{1}^{2}\left(\frac{{\partial }^{2}{\ddot{W}}_{0}}{\partial {x}^{2}}+\frac{{\partial }^{2}{\ddot{W}}_{0}}{\partial {y}^{2}}\right)+{\bar{I}}_{4}^{t}{C}_{1}\left(\frac{\partial {\ddot{\theta }}_{xt}}{\partial x}+\frac{\partial {\ddot{\theta }}_{yt}}{\partial y}\right)+{\bar{I}}_{4}^{b}{C}_{1}\left(\frac{\partial {\ddot{\theta }}_{xb}}{\partial x}+\frac{\partial {\ddot{\theta }}_{yb}}{\partial y}\right)\right\},$$
(B5)
$$\delta {\theta }_{xt}\to \left[{B}_{11}-{C}_{1}{E}_{11}\right]\left(\frac{{\partial }^{2}{u}_{0t}}{\partial {x}^{2}}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{u}_{0t}}{\partial {y}^{2}}\right)+\left[{B}_{12}-{C}_{1}{E}_{12}+{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{v}_{0t}}{\partial x\partial y}\right)+\left[{H}_{11}-2{C}_{1}{T}_{11}+{C}_{1}^{2}{D}_{11}\right]\left(\frac{{\partial }^{2}{\theta }_{xt}}{\partial {x}^{2}}\right)+\left[{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xt}}{\partial {y}^{2}}\right)+\left[{H}_{12}-2{C}_{1}{T}_{12}\right.\left.+{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{12}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yt}}{\partial x\partial y}\right)+\left[6{C}_{1}{H}_{44}-{A}_{44}-9{C}_{1}^{2}{T}_{44}\right]\left({\theta }_{xt}+\frac{\partial {W}_{0}}{\partial x}\right)+\left({C}_{1}^{2}{D}_{11}-{C}_{1}{T}_{11}\right)\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{3}}\right)+\left[{C}_{1}^{2}{D}_{12}-{C}_{1}{T}_{12}+2\left({C}_{1}^{2}{D}_{66}-{C}_{1}{T}_{66}\right)\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial x\partial {y}^{2}}\right)+\left(\frac{{h}_{1}}{2{h}_{2}}{f}_{1}-\frac{{3{C}_{1}h}_{1}}{2{h}_{2}}{f}_{4}\right)\left({u}_{0t}-{u}_{0b}\right)-\left(\frac{{h}_{1}}{2{h}_{2}}{f}_{2}-\frac{{3{C}_{1}h}_{1}}{2{h}_{2}}{f}_{5}\right)\left({h}_{1}{\theta }_{xt}+{h}_{3}{\theta }_{xb}\right)-\left(\frac{{h}_{1}}{2{h}_{2}}{f}_{3}-\frac{{3{C}_{1}h}_{1}}{2{h}_{2}}{f}_{6}\right)\frac{\partial {W}_{0}}{\partial x}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({\bar{I}}_{1}^{t}{\ddot{u}}_{0t}+{\bar{I}}_{2}^{t}{\ddot{\theta }}_{xt}-{C}_{1}{\bar{I}}_{4}^{t}\frac{\partial {\ddot{W}}_{0}}{\partial x}\right),$$
(B6)
$$\delta {\theta }_{xb}\to \left[{B}_{11}-{C}_{1}{E}_{11}\right]\left(\frac{{\partial }^{2}{u}_{0b}}{\partial {x}^{2}}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{u}_{0b}}{\partial {y}^{2}}\right)+\left[{B}_{12}-{C}_{1}{E}_{12}+{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{v}_{0b}}{\partial x\partial y}\right)+\left[{H}_{11}-2{C}_{1}{T}_{11}+{C}_{1}^{2}{D}_{11}\right]\left(\frac{{\partial }^{2}{\theta }_{xb}}{\partial {x}^{2}}\right)+\left[{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xb}}{\partial {y}^{2}}\right)+\left[{H}_{12}-2{C}_{1}{T}_{12}\right.\left.+{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{12}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yb}}{\partial x\partial y}\right)+\left[6{C}_{1}{H}_{44}-{A}_{44}-9{C}_{1}^{2}{T}_{44}\right]\left({\theta }_{xb}+\frac{\partial {W}_{0}}{\partial x}\right)+\left({C}_{1}^{2}{D}_{11}-{C}_{1}{T}_{11}\right)\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{3}}\right)+\left[{C}_{1}^{2}{D}_{12}-{C}_{1}{T}_{12}+2\left({C}_{1}^{2}{D}_{66}-{C}_{1}{T}_{66}\right)\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial x\partial {y}^{2}}\right)+\left(\frac{{h}_{3}}{2{h}_{2}}{f}_{1}-\frac{{3{C}_{1}h}_{3}}{2{h}_{2}}{f}_{4}\right)\left({u}_{0t}-{u}_{0b}\right)-\left(\frac{{h}_{3}}{2{h}_{2}}{f}_{2}-\frac{{3{C}_{1}h}_{3}}{2{h}_{2}}{f}_{5}\right)\left({h}_{1}{\theta }_{xt}+{h}_{3}{\theta }_{xb}\right)-\left(\frac{{h}_{3}}{2{h}_{2}}{f}_{3}-\frac{{3{C}_{1}h}_{3}}{2{h}_{2}}{f}_{6}\right)\frac{\partial {W}_{0}}{\partial x}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({\bar{I}}_{1}^{b}{\ddot{u}}_{0b}+{\bar{I}}_{2}^{b}{\ddot{\theta }}_{xb}-{C}_{1}{\bar{I}}_{4}^{b}\frac{\partial {\ddot{W}}_{0}}{\partial x}\right),$$
(B7)
$$\delta {\theta }_{yt}\to \left[{B}_{22}-{C}_{1}{E}_{22}\right]\left(\frac{{\partial }^{2}{v}_{0t}}{\partial {y}^{2}}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{v}_{0t}}{\partial {x}^{2}}\right)+\left[{B}_{12}-{C}_{1}{E}_{12}+{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{u}_{0t}}{\partial x\partial y}\right)+\left[{H}_{22}-2{C}_{1}{T}_{22}+{C}_{1}^{2}{D}_{22}\right]\left(\frac{{\partial }^{2}{\theta }_{yt}}{\partial {y}^{2}}\right)+\left[{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yt}}{\partial {x}^{2}}\right)+\left[{H}_{12}-2{C}_{1}{T}_{12}\right.\left.+{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{12}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xt}}{\partial x\partial y}\right)+\left[6{C}_{1}{H}_{44}-{A}_{44}-9{C}_{1}^{2}{T}_{44}\right]\left({\theta }_{yt}+\frac{\partial {W}_{0}}{\partial y}\right)+\left({C}_{1}^{2}{D}_{22}-{C}_{1}{T}_{22}\right)\left(\frac{{\partial }^{3}{W}_{0}}{\partial {y}^{3}}\right)+\left[{C}_{1}^{2}{D}_{12}-{C}_{1}{T}_{12}+2\left({C}_{1}^{2}{D}_{66}-{C}_{1}{T}_{66}\right)\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{2}\partial y}\right)+\left(\frac{{h}_{1}}{2{h}_{2}}{f}_{1}-\frac{{3{C}_{1}h}_{1}}{2{h}_{2}}{f}_{4}\right)\left({v}_{0t}-{v}_{0b}\right)-\left(\frac{{h}_{1}}{2{h}_{2}}{f}_{2}-\frac{{3{C}_{1}h}_{1}}{2{h}_{2}}{f}_{5}\right)\left({h}_{1}{\theta }_{yt}+{h}_{3}{\theta }_{yb}\right)-\left(\frac{{h}_{1}}{2{h}_{2}}{f}_{3}-\frac{{3{C}_{1}h}_{1}}{2{h}_{2}}{f}_{6}\right)\frac{\partial {W}_{0}}{\partial y}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({\bar{I}}_{1}^{t}{\ddot{v}}_{0t}+{\bar{I}}_{2}^{t}{\ddot{\theta }}_{yt}-{C}_{1}{\bar{I}}_{4}^{t}\frac{\partial {\ddot{W}}_{0}}{\partial y}\right),$$
(B8)
$$\delta {\theta }_{yb}\to \left[{B}_{22}-{C}_{1}{E}_{22}\right]\left(\frac{{\partial }^{2}{v}_{0b}}{\partial {y}^{2}}\right)+\left[{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{v}_{0b}}{\partial {x}^{2}}\right)+\left[{B}_{12}-{C}_{1}{E}_{12}+{B}_{66}-{C}_{1}{E}_{66}\right]\left(\frac{{\partial }^{2}{u}_{0b}}{\partial x\partial y}\right)+\left[{H}_{22}-2{C}_{1}{T}_{22}+{C}_{1}^{2}{D}_{22}\right]\left(\frac{{\partial }^{2}{\theta }_{yb}}{\partial {y}^{2}}\right)+\left[{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{yb}}{\partial {x}^{2}}\right)+\left[{H}_{12}-2{C}_{1}{T}_{12}\right.\left.+{H}_{66}-2{C}_{1}{T}_{66}+{C}_{1}^{2}{D}_{12}+{C}_{1}^{2}{D}_{66}\right]\left(\frac{{\partial }^{2}{\theta }_{xb}}{\partial x\partial y}\right)+\left[6{C}_{1}{H}_{44}-{A}_{44}-9{C}_{1}^{2}{T}_{44}\right]\left({\theta }_{yb}+\frac{\partial {W}_{0}}{\partial y}\right)+\left({C}_{1}^{2}{D}_{22}-{C}_{1}{T}_{22}\right)\left(\frac{{\partial }^{3}{W}_{0}}{\partial {y}^{3}}\right)+\left[{C}_{1}^{2}{D}_{12}-{C}_{1}{T}_{12}+2\left({C}_{1}^{2}{D}_{66}-{C}_{1}{T}_{66}\right)\right]\left(\frac{{\partial }^{3}{W}_{0}}{\partial {x}^{2}\partial y}\right)+\left(\frac{{h}_{3}}{2{h}_{2}}{f}_{1}-\frac{{3{C}_{1}h}_{3}}{2{h}_{2}}{f}_{4}\right)\left({v}_{0t}-{v}_{0b}\right)-\left(\frac{{h}_{3}}{2{h}_{2}}{f}_{2}-\frac{{3{C}_{1}h}_{3}}{2{h}_{2}}{f}_{5}\right)\left({h}_{1}{\theta }_{yt}+{h}_{3}{\theta }_{yb}\right)-\left(\frac{{h}_{3}}{2{h}_{2}}{f}_{3}-\frac{{3{C}_{1}h}_{3}}{2{h}_{2}}{f}_{6}\right)\frac{\partial {W}_{0}}{\partial y}=\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]\left({\bar{I}}_{1}^{b}{\ddot{v}}_{0b}+{\bar{I}}_{2}^{b}{\ddot{\theta }}_{yb}-{C}_{1}{\bar{I}}_{4}^{b}\frac{\partial {\ddot{W}}_{0}}{\partial y}\right),$$
(B9)

where

$$\left( {A_{11} ,B_{11} ,H_{11} ,E_{11} ,T_{11} ,D_{11} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} Q_{11} \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)dz,$$
$$\left( {A_{12} ,B_{12} ,H_{12} ,E_{12} ,T_{12} ,D_{12} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} Q_{12} \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)dz,$$
$$\left( {A_{22} ,B_{22} ,H_{22} ,E_{22} ,T_{22} ,D_{22} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} Q_{22} \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)dz,$$
$$\left( {A_{44} ,B_{44} ,H_{44} ,E_{44} ,T_{44} ,D_{44} } \right) = \mathop \int \limits_{{ - h_{1} /2}}^{{h_{1} /2}} Q_{44} \left( {1,z,z^{2} ,z^{3} ,z^{4} ,z^{6} } \right)dz,$$
$$r = \left( {\frac{{h_{1} }}{{8h_{2} }} + \frac{{h_{3} }}{{8h_{2} }} - \frac{3}{4}} \right),f_{1} = \frac{3}{{4h_{2} }}G^{\left( c \right)} \overline{a} - \frac{{9C_{1} }}{{4h_{2} }}G^{\left( c \right)} \overline{b},f_{2} = \frac{1}{{2h_{2} }}G^{\left( c \right)} \overline{a} - \frac{{3C_{1} }}{{2h_{2} }}G^{\left( c \right)} \overline{b}$$
$$f_{3} = G^{\left( c \right)} \overline{a}r - 3C_{1} G^{\left( c \right)} \overline{b}r,f_{4} = \frac{3}{{4h_{2} }}G^{\left( c \right)} \overline{b} - \frac{{9C_{1} }}{{4h_{2} }}G^{\left( c \right)} \overline{d},f_{5} = \frac{1}{{2h_{2} }}G^{\left( c \right)} \overline{b} - \frac{{3C_{1} }}{{2h_{2} }}G^{\left( c \right)} \overline{d}$$
$$f_{6} = G^{\left( c \right)} \overline{b}r - 3C_{1} G^{\left( c \right)} \overline{d}r,\left( {\overline{a},\overline{b},\overline{d}} \right) = \mathop \int \limits_{{ - h_{2} /2}}^{{h_{2} /2}} \left( {1,z^{2} ,z^{4} } \right)dz$$

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Saffari, P.R., Ismail, S.O., Thongchom, C. et al. Effect of Magnetic Field on Vibration of Electrorheological Fluid Nanoplates with FG-CNTRC Layers. J. Vib. Eng. Technol. 12, 3335–3354 (2024). https://doi.org/10.1007/s42417-023-01048-7

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