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Interlayer effects of Van der Waals interactions on transverse vibrational behavior of bilayer graphene sheets

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Abstract

This study focuses only on the interlayer effects of van der Waals (VdWs) interactions (including simultaneous effects of shear and tensile-compressive effects) on the free transverse vibrational behavior of bilayer graphene sheets by implementing the classical continuum mechanics theory. To this end, the classical sandwich plate theory and the Hamilton’s principle are involved to obtain the governing equations and the harmonic differential quadrature method is employed to calculate the natural frequencies and related mode shapes. The results show the shear effect of VdWs interactions has significant influences on primary natural frequencies and mode shapes. Therefore it is a main determinant and can safely assume the pure shear effect while designing sensors, actuators, accelerometers and resonators. Finally, the potential depth parameter is introduced to consider the simultaneous effects of shear and tensile-compressive forces.

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

    Kamran Kamali

  2. Department of Design, Fateh Sanat Kimia Company, Great Industrial Zone, Shiraz, Iran

    Kamran Kamali

  3. School of Engineering, Damghan University, Damghan, 36716-41167, Iran

    Reza Nazemnezhad

  4. Department of Mechanical and Manufacturing Engineering, Schulich School of Engineering, University of Calgary, 40 Research Place NW, Calgary, Alberta, T2L 1Y6, Canada

    Mojtaba Zare

Authors
  1. Kamran Kamali
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  2. Reza Nazemnezhad
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  3. Mojtaba Zare
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Corresponding author

Correspondence to Reza Nazemnezhad.

Appendix

Appendix

$$ - \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial^{2} u_{1} }}{{\partial x^{2} }} + \nu \frac{{\partial^{2} v_{1} }}{\partial x\partial y}} \right) - \frac{{E_{f} h_{f} }}{2(1 + \nu )}\left( {\frac{{\partial^{2} u_{1} }}{{\partial y^{2} }} + \frac{{\partial^{2} v_{1} }}{\partial x\partial y}} \right) + \frac{{G_{xz}^{c} }}{{h_{c} }}\left[ {u_{1} - u_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial x} + \frac{{\partial w_{2} }}{\partial x}} \right)} \right] + I_{0} \frac{{\partial^{2} u_{1} }}{{\partial t^{2} }} = 0 $$
(63)
$$ - \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial^{2} u_{2} }}{{\partial x^{2} }} + \nu \frac{{\partial^{2} v_{2} }}{\partial x\partial y}} \right) - \frac{{E_{f} h_{f} }}{2(1 + \nu )}\left( {\frac{{\partial^{2} u_{2} }}{{\partial y^{2} }} + \frac{{\partial^{2} v_{2} }}{\partial x\partial y}} \right) - \frac{{G_{xz}^{c} }}{{h_{c} }}\left[ {u_{1} - u_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial x} + \frac{{\partial w_{2} }}{\partial x}} \right)} \right] + I_{0} \frac{{\partial^{2} u_{2} }}{{\partial t^{2} }} = 0 $$
(64)
$$ - \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial^{2} v_{1} }}{{\partial y^{2} }} + \nu \frac{{\partial^{2} u_{1} }}{\partial x\partial y}} \right) - \frac{{E_{f} h_{f} }}{2(1 + \nu )}\left( {\frac{{\partial^{2} v_{1} }}{{\partial x^{2} }} + \frac{{\partial^{2} u_{1} }}{\partial x\partial y}} \right) + \frac{{G_{yz}^{c} }}{{h_{c} }}\left[ {v_{1} - v_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial y} + \frac{{\partial w_{2} }}{\partial y}} \right)} \right] + I_{0} \frac{{\partial^{2} v_{1} }}{{\partial t^{2} }} = 0 $$
(65)
$$ - \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial^{2} v_{2} }}{{\partial y^{2} }} + \nu \frac{{\partial^{2} u_{2} }}{\partial x\partial y}} \right) - \frac{{E_{f} h_{f} }}{2(1 + \nu )}\left( {\frac{{\partial^{2} v_{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} u_{2} }}{\partial x\partial y}} \right) - \frac{{G_{yz}^{c} }}{{h_{c} }}\left[ {v_{1} - v_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial y} + \frac{{\partial w_{2} }}{\partial y}} \right)} \right] + I_{0} \frac{{\partial^{2} v_{2} }}{{\partial t^{2} }} = 0 $$
(66)
$$ \begin{aligned} D\left( {\frac{{\partial^{4} w_{1} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{1} }}{{\partial y^{4} }} + 2\nu \frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }}} \right) + \frac{{E_{f} h_{f}^{3} }}{6(1 + \nu )}\frac{{\partial^{4} w_{1} }}{{\partial x^{2} \partial y^{2} }} + \frac{{E_{c} }}{{h_{c} }}(w_{1} - w_{2} ) \\ & \quad - \frac{{h_{c} }}{12}\left( {G_{xz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} - G_{xz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} + G_{yz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial y^{2} }} - G_{yz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial y^{2} }}} \right) \\ & \quad - \frac{1}{2}\left( {1 + \frac{{h_{f} }}{{h_{c} }}} \right)\left( \begin{aligned} G_{xz}^{c} \frac{{\partial u_{1} }}{\partial x} - G_{xz}^{c} \frac{{\partial u_{2} }}{\partial x} + G_{yz}^{c} \frac{{\partial v_{1} }}{\partial y} - G_{yz}^{c} \frac{{\partial v_{2} }}{\partial y} \hfill \\ + \frac{{h_{f} + h_{c} }}{2}\left( {G_{xz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} + G_{xz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} + G_{yz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial y^{2} }} + G_{yz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial y^{2} }}} \right) \hfill \\ \end{aligned} \right) + I_{0} \frac{{\partial^{2} w_{1} }}{{\partial t^{2} }} = 0 \\ \end{aligned} $$
(67)
$$ \begin{aligned} & D\left( {\frac{{\partial^{4} w_{2} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{2} }}{{\partial y^{4} }} + 2\nu \frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }}} \right) + \frac{{E_{f} h_{f}^{3} }}{6(1 + \nu )}\frac{{\partial^{4} w_{2} }}{{\partial x^{2} \partial y^{2} }} - \frac{{E_{c} }}{{h_{c} }}(w_{1} - w_{2} ) \\ & \quad + \frac{{h_{c} }}{12}\left( {G_{xz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} - G_{xz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} + G_{yz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial y^{2} }} - G_{yz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial y^{2} }}} \right) \\ & \quad - \frac{1}{2}\left( {1 + \frac{{h_{f} }}{{h_{c} }}} \right)\left( {G_{xz}^{c} \frac{{\partial u_{1} }}{\partial x} - G_{xz}^{c} \frac{{\partial u_{2} }}{\partial x} + G_{yz}^{c} \frac{{\partial v_{1} }}{\partial y} - G_{yz}^{c} \frac{{\partial v_{2} }}{\partial y}} \right. \\ & \quad \left. { + \frac{{h_{f} + h_{c} }}{2}\left( {G_{xz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} + G_{xz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} + G_{yz}^{c} \frac{{\partial^{2} w_{1} }}{{\partial y^{2} }} + G_{yz}^{c} \frac{{\partial^{2} w_{2} }}{{\partial y^{2} }}} \right)} \right) + I_{0} \frac{{\partial^{2} w_{2} }}{{\partial t^{2} }} = 0 \\ \end{aligned} $$
(68)
$$ \begin{aligned} y = 0(n = y,\,s = x); \\ \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial v_{1} }}{\partial y} + \nu \frac{{\partial u_{1} }}{\partial x}} \right) = \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial v_{2} }}{\partial y} + \nu \frac{{\partial u_{2} }}{\partial x}} \right) = G_{f} h_{f} \left( {\frac{{\partial u_{1} }}{\partial y} + \frac{{\partial v_{1} }}{\partial x}} \right) = G_{f} h_{f} \left( {\frac{{\partial u_{2} }}{\partial y} + \frac{{\partial v_{2} }}{\partial x}} \right) \\ & = - D\left( {\frac{{\partial^{2} w_{1} }}{{\partial y^{2} }} + \nu \frac{{\partial^{2} w_{1} }}{{\partial x^{2} }}} \right) = - D\left( {\frac{{\partial^{2} w_{2} }}{{\partial y^{2} }} + \nu \frac{{\partial^{2} w_{2} }}{{\partial x^{2} }}} \right) = \\ & \quad - D\left( {\frac{{\partial^{3} w_{1} }}{{\partial y^{3} }} + \nu \frac{{\partial^{3} w_{1} }}{{\partial y\partial x^{2} }}} \right) - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{1} }}{{\partial x^{2} \partial y}} \\ & \quad + \frac{{h_{f} + h_{c} }}{{2h_{c} }}G_{yz}^{c} \left[ {v_{1} - v_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial y} + \frac{{\partial w_{2} }}{\partial y}} \right)} \right] + \frac{1}{{h_{c} }}\frac{{G_{yz}^{c} h_{c}^{2} }}{12}\left( {\frac{{\partial w_{1} }}{\partial y} - \frac{{\partial w_{2} }}{\partial y}} \right) \\ & \quad - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{1} }}{{\partial x^{2} \partial y}} \\ & = - D\left( {\frac{{\partial^{3} w_{2} }}{{\partial y^{3} }} + \nu \frac{{\partial^{3} w_{2} }}{{\partial y\partial x^{2} }}} \right) - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{2} }}{{\partial x^{2} \partial y}} \\ & \quad + \frac{{h_{f} + h_{c} }}{{2h_{c} }}G_{yz}^{c} \left[ {v_{1} - v_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial y} + \frac{{\partial w_{2} }}{\partial y}} \right)} \right] - \frac{1}{{h_{c} }}\frac{{G_{yz}^{c} h_{c}^{2} }}{12}\left( {\frac{{\partial w_{1} }}{\partial y} - \frac{{\partial w_{2} }}{\partial y}} \right) \\ & \quad - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{2} }}{{\partial x^{2} \partial y}} = 0 \\ \end{aligned} $$
(69)
$$ \begin{aligned} x = 0(n = x,\,s = y); \\ \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial u_{1} }}{\partial x} + \nu \frac{{\partial v_{1} }}{\partial y}} \right) = \frac{{E_{f} h_{f} }}{{1 - \nu^{2} }}\left( {\frac{{\partial u_{2} }}{\partial x} + \nu \frac{{\partial v_{2} }}{\partial y}} \right) = G_{f} h_{f} \left( {\frac{{\partial u_{1} }}{\partial y} + \frac{{\partial v_{1} }}{\partial x}} \right) = G_{f} h_{f} \left( {\frac{{\partial u_{2} }}{\partial y} + \frac{{\partial v_{2} }}{\partial x}} \right) \\ & = - D\left( {\frac{{\partial^{2} w_{1} }}{{\partial x^{2} }} + \nu \frac{{\partial^{2} w_{1} }}{{\partial y^{2} }}} \right) = - D\left( {\frac{{\partial^{2} w_{2} }}{{\partial x^{2} }} + \nu \frac{{\partial^{2} w_{2} }}{{\partial y^{2} }}} \right) = \\ & \quad - D\left( {\frac{{\partial^{3} w_{1} }}{{\partial x^{3} }} + \nu \frac{{\partial^{3} w_{1} }}{{\partial x\partial y^{2} }}} \right) - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{1} }}{{\partial x\partial y^{2} }} \\ & \quad + \frac{{h_{f} + h_{c} }}{{2h_{c} }}G_{xz}^{c} \left[ {u_{1} - u_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial x} + \frac{{\partial w_{2} }}{\partial x}} \right)} \right] + \frac{1}{{h_{c} }}\frac{{G_{xz}^{c} h_{c}^{2} }}{12}\left( {\frac{{\partial w_{1} }}{\partial x} - \frac{{\partial w_{2} }}{\partial x}} \right) \\ & \quad - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{1} }}{{\partial x\partial y^{2} }} = \\ & \quad - D\left( {\frac{{\partial^{3} w_{2} }}{{\partial x^{3} }} + \nu \frac{{\partial^{3} w_{2} }}{{\partial x\partial y^{2} }}} \right) - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{2} }}{{\partial x\partial y^{2} }} \\ & \quad + \frac{{h_{f} + h_{c} }}{{2h_{c} }}G_{xz}^{c} \left[ {u_{1} - u_{2} + \frac{{h_{f} + h_{c} }}{2}\left( {\frac{{\partial w_{1} }}{\partial x} + \frac{{\partial w_{2} }}{\partial x}} \right)} \right] - \frac{1}{{h_{c} }}\frac{{G_{xz}^{c} h_{c}^{2} }}{12}\left( {\frac{{\partial w_{1} }}{\partial x} - \frac{{\partial w_{2} }}{\partial x}} \right) \\ & \quad - \frac{{E_{f} h_{f}^{3} }}{12(1 + \nu )}\frac{{\partial^{3} w_{2} }}{{\partial x\partial y^{2} }} = 0 \\ \end{aligned} $$
(70)
$$ \begin{aligned} y = b(n = y,\,s = x); \hfill \\ u_{1} = u_{2} = v_{1} = v_{2} = w_{1} = w_{2} = \frac{{\partial^{2} w_{1} }}{{\partial y^{2} }} = \frac{{\partial^{2} w_{2} }}{{\partial y^{2} }} = 0 \hfill \\ \end{aligned} $$
(71)
$$ \begin{aligned} x = a(n = x,\,s = y); \hfill \\ u_{1} = u_{2} = v_{1} = v_{2} = w_{1} = w_{2} = \frac{{\partial w_{1} }}{\partial x} = \frac{{\partial w_{2} }}{\partial x} = 0 \hfill \\ \end{aligned} $$
(72)
$$ \begin{aligned} x = y = 0; \hfill \\ \frac{{\partial^{2} w_{1} }}{\partial x\partial y} = \frac{{\partial^{2} w_{2} }}{\partial x\partial y} = 0 \hfill \\ \end{aligned} $$
(73)
$$ \begin{aligned} \left( {\frac{{h_{f}^{3} h}}{{12(1 - \nu^{2} )a^{4} }}} \right)\mathop \sum \limits_{n = 1}^{N} D_{in} \bar{W}_{1} (X_{n} ,Y_{j} ) + \left( {\frac{{h_{f}^{3} h}}{{12(1 - \nu^{2} )b^{4} }}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{D}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) \\ & \quad + \left( {\frac{{h_{f}^{3} h}}{{6(1 - \nu^{2} )a^{2} b^{2} }}} \right)\mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} B_{in} \bar{B}_{jm} \bar{W}_{1} (X_{n} ,Y_{m} ) + \left( {\frac{{E_{c} h}}{{E_{f} h_{c} }}} \right)\left( {\bar{W}_{1} (X_{i} ,Y_{j} ) - \bar{W}_{2} (X_{i} ,Y_{j} )} \right) \\ & \quad - \left( {\frac{{G_{xz}^{c} h_{c} h}}{{12E_{f} a^{2} }}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} B_{in} \bar{W}_{1} (X_{n} ,Y_{j} ) - \mathop \sum \limits_{n = 1}^{N} B_{in} \bar{W}_{2} (X_{n} ,Y_{j} )} \right) \\ & \quad - \left( {\frac{{G_{yz}^{c} h_{c} h}}{{12E_{f} b^{2} }}} \right)\left( {\mathop \sum \limits_{m = 1}^{M} \bar{B}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) - \mathop \sum \limits_{m = 1}^{M} \bar{B}_{jm} \bar{W}_{2} (X_{i} ,Y_{m} )} \right) \\ & \quad - \left( {\frac{{G_{xz}^{c} h(h_{f} + h_{c} )}}{{2E_{f} h_{c} a}}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} A_{in} \bar{U}_{1} (X_{n} ,Y_{j} ) - \mathop \sum \limits_{n = 1}^{N} A_{in} \bar{U}_{2} (X_{n} ,Y_{j} )} \right) \\ & \quad - \left( {\frac{{G_{yz}^{c} h(h_{f} + h_{c} )}}{{2E_{f} h_{c} b}}} \right)\left( {\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{V}_{1} (X_{i} ,Y_{m} ) - \mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{V}_{2} (X_{i} ,Y_{m} )} \right) \\ & \quad - \left( {\frac{{G_{xz}^{c} h(h_{f} + h_{c} )^{2} }}{{4E_{f} h_{c} a^{2} }}} \right)\left( {\mathop \sum \limits_{n = 1}^{N} B_{in} \bar{W}_{1} (X_{n} ,Y_{j} ) + \mathop \sum \limits_{n = 1}^{N} B_{in} \bar{W}_{2} (X_{n} ,Y_{j} )} \right) \\ & \quad - \left( {\frac{{G_{yz}^{c} h(h_{f} + h_{c} )^{2} }}{{4E_{f} h_{c} b^{2} }}} \right)\left( {\mathop \sum \limits_{m = 1}^{M} \bar{B}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) + \mathop \sum \limits_{m = 1}^{M} \bar{B}_{jm} \bar{W}_{2} (X_{i} ,Y_{m} )} \right) = \left( {\frac{{I_{0} \omega^{2} h}}{{E_{f} }}} \right)\bar{W}_{1} (X_{i} ,Y_{j} ) \\ \end{aligned} $$
(74)
$$ \begin{aligned} \left( {\frac{h}{b}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{V}_{1} (X_{i} ,Y_{m} ) + \left( {\frac{\nu h}{a}} \right)\mathop \sum \limits_{n = 1}^{N} A_{in} \bar{U}_{1} (X_{n} ,Y_{j} ) \\ & = \left( {\frac{h}{b}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{V}_{2} (X_{i} ,Y_{m} ) + \left( {\frac{\nu h}{a}} \right)\mathop \sum \limits_{n = 1}^{N} A_{in} \bar{U}_{2} (X_{n} ,Y_{j} ) \\ & = \left( {\frac{h}{b}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{U}_{1} (X_{i} ,Y_{m} ) + \left( {\frac{h}{a}} \right)\mathop \sum \limits_{n = 1}^{N} A_{in} \bar{V}_{1} (X_{n} ,Y_{j} ) \\ & = \left( {\frac{h}{b}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{U}_{2} (X_{i} ,Y_{m} ) + \left( {\frac{h}{a}} \right)\mathop \sum \limits_{n = 1}^{N} A_{in} \bar{V}_{2} (X_{n} ,Y_{j} ) \\ & = \left( {\frac{{h^{2} }}{{b^{2} }}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{B}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) + \left( {\frac{{\nu h^{2} }}{{a^{2} }}} \right)\mathop \sum \limits_{n = 1}^{N} B_{in} \bar{W}_{1} (X_{n} ,Y_{j} ) \\ & = \left( {\frac{{h^{2} }}{{b^{2} }}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{B}_{jm} \bar{W}_{2} (X_{i} ,Y_{m} ) + \left( {\frac{{\nu h^{2} }}{{a^{2} }}} \right)\mathop \sum \limits_{n = 1}^{N} B_{in} \bar{W}_{2} (X_{n} ,Y_{j} ) \\ & = - \left( {\frac{{h_{f}^{3} }}{{12(1 - \nu^{2} )b^{3} }}} \right)\mathop \sum \limits_{m = 1}^{M} \bar{C}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) - \left( {\frac{{h_{f}^{3} (2 - \nu )}}{{12(1 - \nu^{2} )a^{2} b}}} \right)\mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 1}^{M} B_{in} \bar{A}_{jm} \bar{W}_{1} (X_{n} ,Y_{m} ) \\ & \quad + \left( {\frac{{(h_{f} + h_{c} )G_{c} }}{{2E_{f} h_{c} }}} \right)\left( {\bar{V}_{1} (X_{i} ,Y_{j} ) - \bar{V}_{2} (X_{i} ,Y_{j} )} \right) \\ & \quad + \frac{{(h_{f} + h_{c} )^{2} G_{c} }}{{4E_{f} h_{c} b}}\left( {\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) + \mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} )} \right) \\ & \quad - \frac{{G_{c} h_{c} }}{{12E_{f} b}}\left( {\mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{W}_{1} (X_{i} ,Y_{m} ) - \mathop \sum \limits_{m = 1}^{M} \bar{A}_{jm} \bar{W}_{2} (X_{i} ,Y_{m} )} \right) = 0. \\ \end{aligned} $$
(75)

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Kamali, K., Nazemnezhad, R. & Zare, M. Interlayer effects of Van der Waals interactions on transverse vibrational behavior of bilayer graphene sheets. J Braz. Soc. Mech. Sci. Eng. 40, 54 (2018). https://doi.org/10.1007/s40430-018-0965-3

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