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A review of dynamic analyses of single- and multi-layered graphene sheets/nanoplates using various nonlocal continuum mechanics-based plate theories

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Abstract

This article is intended to present an overview of dynamic analyses of single-/multi-layered graphene sheets (SLGSs/MLGSs) and nanoplates with combinations of simply supported, free, and clamped edge conditions embedded in an elastic medium using various two-dimensional (2D) nonlocal continuum mechanics-based plate theories. Based on Hamilton’s principle incorporating Eringen’s nonlocal constitutive relations, the authors derive strong formulations of assorted 2D nonlocal continuum mechanics-based plate theories for the free vibration analysis of embedded SLGSs/MLGSs, including the nonlocal classical plate theory, the nonlocal first-order shear deformation plate theory (SDPT), the nonlocal refined higher-order SDPT, the nonlocal sinusoidal SDPT, the nonlocal exponential SDPT, and the nonlocal hyperbolic SDPT. Navier-type solutions based on these 2D nonlocal plate theories for free vibration problems of simply supported, SLGSs/MLGSs and nanoplates embedded in an elastic medium are obtained. Articles examining various mechanical behaviors of SLGSs/MLGSs and nanoplates using different nonlocal plate theories available in the literature are tabulated into categories, including bending, free vibration, buckling, dynamic instability, wave propagation, geometrically nonlinear bending, geometrically nonlinear vibration, and forced vibration analyses.

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Acknowledgements

This work was supported by the Ministry of Science and Technology of the Republic of China through Grant MOST 109-2221-E-006-015-MY3.

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  1. Department of Civil Engineering, National Cheng Kung University, Tainan City, Taiwan

    Chih-Ping Wu & Hao-Xiang Hu

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Appendices

Appendix A: Expressions of the generalized moment and shear force resultants of the mth-layer GS based on the RSDPT

By using Eqs. (64) and (65), the generalized moment and shear force resultants of the mth-layer GS based on the RSDPT can be expressed in terms of displacement components as follows:

$$\left( {1 - \mu \,\nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {M_{xm} } \\ {M_{ym} } \\ {M_{xym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left[ {D_{11} - 4F_{11} /\left( {3h^{2} } \right)} \right]\phi_{xm,x} + \left[ {D_{12} - 4F_{11} /\left( {3h^{2} } \right)} \right]\phi_{ym,y} - \left[ {4F_{11} /\left( {3h^{2} } \right)} \right]w_{m,xx} - \left[ {4F_{12} /\left( {3h^{2} } \right)} \right]w_{m,yy} } \\ {\left[ {D_{12} - 4F_{12} /\left( {3h^{2} } \right)} \right]\phi_{xm,x} + \left[ {D_{22} - 4F_{22} /\left( {3h^{2} } \right)} \right]\phi_{ym,y} - \left[ {4F_{12} /\left( {3h^{2} } \right)} \right]w_{m,xx} - \left[ {4F_{22} /\left( {3h^{2} } \right)} \right]w_{m,yy} } \\ {\left[ {D_{66} - 4F_{66} /\left( {3h^{2} } \right)} \right]\phi_{xm,y} + \left[ {D_{66} - 4F_{66} /\left( {3h^{2} } \right)} \right]\phi_{ym,x} - \left[ {8F_{66} /\left( {3h^{2} } \right)} \right]w_{m,xy} } \\ \end{array} } \right\},$$
(92)
$$\left( {1 - \mu \nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {P_{xm} } \\ {P_{ym} } \\ {P_{xym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left[ {F_{11} - 4H_{11} /\left( {3h^{2} } \right)} \right]\phi_{xm,x} + \left[ {F_{12} - 4H_{12} /\left( {3h^{2} } \right)} \right]\phi_{ym,y} - \left[ {4H_{11} /\left( {3h^{2} } \right)} \right]w_{m,xx} - \left[ {4H_{12} /\left( {3h^{2} } \right)} \right]w_{m,yy} } \\ {\left[ {F_{12} - 4H_{12} /\left( {3h^{2} } \right)} \right]\phi_{xm,x} + \left[ {F_{22} - 4H_{22} /\left( {3h^{2} } \right)} \right]\phi_{ym,y} - \left[ {4H_{12} /\left( {3h^{2} } \right)} \right]w_{m,xx} - \left[ {4H_{22} /\left( {3h^{2} } \right)} \right]w_{m,yy} } \\ {\left[ {F_{66} - 4H_{66} /\left( {3h^{2} } \right)} \right]\phi_{xm,y} + \left[ {F_{66} - 4H_{66} /\left( {3h^{2} } \right)} \right]\phi_{ym,x} - \left[ {8H_{66} /\left( {3h^{2} } \right)} \right]w_{m,xy} } \\ \end{array} } \right\},$$
(93)
$$\left( {1 - \mu \nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {Q_{xm} } \\ {Q_{ym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left( {A_{55} - 4D_{55} /h^{2} } \right)w_{m} ,_{x} + \left( {A_{55} - 4D_{55} /h^{2} } \right)\phi_{xm} } \\ {\left( {A_{44} - 4D_{44} /h^{2} } \right)w_{m} ,_{y} + \left( {A_{44} - 4D_{44} /h^{2} } \right)\phi_{ym} } \\ \end{array} } \right\},$$
(94)
$$\left( {1 - \mu \nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {R_{xm} } \\ {R_{ym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left( {D_{55} - 4F_{55} /h^{2} } \right)w_{m} ,_{x} + \left( {D_{55} - 4F_{55} /h^{2} } \right)\phi_{xm} } \\ {\left( {D_{44} - 4F_{44} /h^{2} } \right)w_{m} ,_{y} + \left( {D_{44} - 4F_{44} /h^{2} } \right)\phi_{ym} } \\ \end{array} } \right\},$$
(95)

where \(F_{ij} = \int_{\, - h/2}^{\,h/2} {\,Q_{ij} } \,z^{4} \,dz\) and \(H_{ij} = \int_{\, - h/2}^{\,h/2} {\,Q_{ij} } \,z^{6} \,dz\;\left( {i{,}j = 1,2,{\text{and}}\;6} \right).\)

Appendix B: Euler–Lagrange equations of the mth-layer GS based on the RSDPT in terms of displacement components

Substituting Eqs. (92)–(95) in Eqs. (66)–(68), the Euler–Lagrange equations of the mth-layer GS based on the RSDPT can be expressed in terms of displacement components as follows:

$$\left[ {\begin{array}{*{20}c} {L_{11} } &\quad {L_{12} } &\quad {L_{13} } \\ {L_{21} } &\quad {L_{22} } &\quad {L_{23} } \\ {L_{31} } &\quad {L_{32} } &\quad {L_{33} } \\ \end{array} } \right]\,\left\{ {\begin{array}{*{20}c} {w_{m} } \\ {\phi_{xm} } \\ {\phi_{ym} } \\ \end{array} } \right\} = \left( {1 - \mu \nabla^{2} } \right)\left[ {\begin{array}{*{20}c} {\hat{L}_{11} } &\quad {\hat{L}_{12} } &\quad {\hat{L}_{13} } \\ {\hat{L}_{21} } &\quad {\hat{L}_{22} } &\quad 0 \\ {\hat{L}_{31} } &\quad 0 &\quad {\hat{L}_{33} } \\ \end{array} } \right]\,\left\{ {\begin{array}{*{20}c} {w_{m} } \\ {\phi_{xm} } \\ {\phi_{ym} } \\ \end{array} } \right\},$$
(96)

where

$$\begin{gathered} L_{11} = - \left[ {16H_{11} /\left( {9h^{4} } \right)} \right]\partial_{xxxx} - \left[ {16\left( {2H_{12} + 4H_{66} } \right)/\left( {9h^{4} } \right)} \right]\partial_{xxyy} - \left[ {16H_{22} /\left( {9h^{4} } \right)} \right]\partial_{yyyy} \hfill \\ \,\quad \quad \quad \quad \quad \quad \quad + \left( {A_{55} - 8D_{55} /h^{2} + 16F_{55} /h^{4} } \right)\partial_{xx} + \left( {A_{44} - 8D_{44} /h^{2} + 16F_{44} /h^{4} } \right)\partial_{yy} , \hfill \\ \end{gathered}$$
$$\begin{gathered} L_{12} = \left[ {4/\left( {3h^{2} } \right)} \right]\left[ {F_{11} - 4H_{11} /\left( {3h^{2} } \right)} \right]\partial_{xxx} + \left[ {4/\left( {3h^{2} } \right)} \right]\left[ {F_{12} + 2F_{66} - 4H_{12} /\left( {3h^{2} } \right) - 8H_{66} /\left( {3h^{2} } \right)} \right]\partial_{xyy} \hfill \\ \quad \quad \quad + \left( {A_{55} - 8D_{55} /h^{2} + 16F_{55} /h^{4} } \right)\partial_{x} , \hfill \\ \end{gathered}$$
$$\begin{aligned} L_{13} = & \left[ {4/\left( {3h^{2} } \right)} \right]\left[ {F_{22} - 4H_{22} /\left( {3h^{2} } \right)} \right]\partial_{yyy} + \left[ {4/\left( {3h^{2} } \right)} \right]\left[ {F_{12} + 2F_{66} - 4H_{12} /\left( {3h^{2} } \right) - 8H_{66} /\left( {3h^{2} } \right)} \right]\partial_{xxy} \\ & + \left( {A_{44} - 8D_{44} /h^{2} + 16F_{44} /h^{4} } \right)\partial_{y} , \\ L_{21} = & - L_{12} , \\ L_{22} = & \left[ {D_{11} - 8F_{11} /\left( {3h^{2} } \right) + 16H_{11} /\left( {9h^{4} } \right)} \right]\partial_{xx} + \left[ {D_{66} - 8F_{66} /\left( {3h^{2} } \right) + 16H_{66} /\left( {9h^{4} } \right)} \right]\partial_{yy} \\ & - \left( {A_{55} - 8D_{55} /h^{2} + 16F_{55} /h^{4} } \right), \\ L_{23} = & \left[ {D_{12} + D_{66} - 8F_{12} /\left( {3h^{2} } \right) - 8F_{66} /\left( {3h^{2} } \right) + 16H_{12} /\left( {9h^{4} } \right) + 16H_{66} /\left( {9h^{4} } \right)} \right]\partial_{xy} , \\ L_{31} = & - L_{13} ,L_{32} = L_{23} , \\ \end{aligned}$$
$$\begin{aligned} L_{33} = & \left[ {D_{66} - 8F_{66} /\left( {3h^{2} } \right) + 16H_{66} /\left( {9h^{4} } \right)} \right]\partial_{xx} + \left[ {D_{22} - 8F_{22} /\left( {3h^{2} } \right) + 16H_{22} /\left( {9h^{4} } \right)} \right]\partial_{yy} \\ & - \left( {A_{44} - 8D_{44} /h^{2} + 16F_{44} /h^{4} } \right), \\ \hat{L}_{11} = & I_{0} \partial_{tt} - \left[ {16I_{6} /\left( {9h^{4} } \right)} \right]\partial_{xxtt} - \left[ {16I_{6} /\left( {9h^{4} } \right)} \right]\partial_{yytt} ,\hat{L}_{12} = \left[ {4I_{4} /\left( {3h^{2} } \right) - 16I_{6} /\left( {9h^{4} } \right)} \right]\partial_{xtt} , \\ \hat{L}_{13} = & \left[ {4I_{4} /\left( {3h^{2} } \right) - 16I_{6} /\left( {9h^{4} } \right)} \right]\partial_{ytt} ,\hat{L}_{21} = - \hat{L}_{12} ,\hat{L}_{22} = \left[ {I_{2} - 8I_{4} /\left( {3h^{2} } \right) + 16I_{6} /\left( {9h^{4} } \right)} \right]\partial_{tt} , \\ \hat{L}_{31} = & - \hat{L}_{13} ,\hat{L}_{33} = \left[ {I_{2} - 8I_{4} /\left( {3h^{2} } \right) + 16I_{6} /\left( {9h^{4} } \right)} \right]\partial_{tt} . \\ \end{aligned}$$

Appendix C: Expressions of the generalized moment and shear force resultants of the mth-layer GS based on various nonlocal advanced shear deformation theories

By using Eqs. (64) and (65), the generalized moment and shear force resultants of the mth-layer GS based on various nonlocal advanced shear deformation theories can be expressed in terms of displacement components as follows:

$$\left( {1 - \mu \,\nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {M_{xm} } \\ {M_{ym} } \\ {M_{xym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {B_{11f} \,\phi_{xm} ,_{x} + B_{12f} \,\phi_{ym} ,_{y} - D_{11} \,w_{m} ,_{xx} - D_{12} \,w_{m} ,_{yy} } \\ {B_{12f} \,\phi_{xm} ,_{x} + B_{22f} \,\phi_{ym} ,_{y} - D_{12} \,w_{m} ,_{xx} - D_{22} \,w_{m} ,_{yy} } \\ {B_{66f} \,\phi_{xm} ,_{y} + B_{66f} \,\phi_{ym} ,_{x} - 2D_{66} w_{m} ,_{xy} } \\ \end{array} } \right\},$$
(97)
$$\left( {1 - \mu \nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {P_{xm} } \\ {P_{ym} } \\ {P_{xym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {F_{11f} \,\phi_{xm} ,_{x} + F_{12f} \,\phi_{ym} ,_{y} - B_{11f} \,w_{m} ,_{xx} - B_{12f} \,w_{m} ,_{yy} } \\ {F_{12f} \,\phi_{xm} ,_{x} + F_{22f} \,\phi_{ym} ,_{y} - B_{12f} \,w_{m} ,_{xx} - B_{22f} \,w_{m} ,_{yy} } \\ {F_{66f} \,\phi_{xm} ,_{y} + F_{66f} \,\phi_{ym} ,_{x} - 2B_{66f} \,w_{m} ,_{xy} } \\ \end{array} } \right\},$$
(98)
$$\left( {1 - \mu \nabla^{2} } \right)\,\left\{ {\begin{array}{*{20}c} {Q_{xm} } \\ {Q_{ym} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {H_{55f} \,\phi_{xm} } \\ {H_{44f} \,\phi_{ym} } \\ \end{array} } \right\},$$
(99)

where \(B_{ijf} = \int_{\, - h/2}^{\,h/2} {\,Q_{ij} } \,z\,f\,dz,F_{ijf} = \int_{\, - h/2}^{\,h/2} {\,Q_{ij} } f^{2} \,dz,\) and \(H_{ijf} = \int_{\, - h/2}^{\,h/2} {\,Q_{ij} } \left( {df/dz} \right)^{2} \,dz.\)

Appendix D: Euler–Lagrange equations of the mth-layer GS based on various advanced shear deformation plate theories in terms of displacement components

Substituting Eqs. (97)–(99) in Eqs. (85) and (87), the Euler–Lagrange equations of the mth-layer GS based on various advanced shear deformation plate theories can be expressed in terms of displacement components as follows:

$$\left[ {\begin{array}{*{20}c} {L_{11} } &\quad {L_{12} } &\quad {L_{13} } \\ {L_{21} } &\quad {L_{22} } &\quad {L_{23} } \\ {L_{31} } &\quad {L_{32} } &\quad {L_{33} } \\ \end{array} } \right]\,\left\{ {\begin{array}{*{20}c} {w_{m} } \\ {\phi_{xm} } \\ {\phi_{ym} } \\ \end{array} } \right\} = \left( {1 - \mu \nabla^{2} } \right)\left[ {\begin{array}{*{20}c} {\hat{L}_{11} } &\quad {\hat{L}_{12} } &\quad {\hat{L}_{13} } \\ {\hat{L}_{21} } &\quad {\hat{L}_{22} } &\quad 0 \\ {\hat{L}_{31} } &\quad 0 &\quad {\hat{L}_{33} } \\ \end{array} } \right]\,\left\{ {\begin{array}{*{20}c} {w_{m} } \\ {\phi_{xm} } \\ {\phi_{ym} } \\ \end{array} } \right\},$$
(100)

where

$$\begin{gathered} L_{11} = - D_{11} \partial_{xxxx} - \left( {2D_{12} + 4D_{66} } \right)\partial_{xxyy} - D_{22} \partial_{yyyy} ,L_{12} = B_{11f} \,\partial_{xxx} + \left( {B_{12f} + 2B_{66f} } \right)\partial_{xyy} , \hfill \\ L_{13} = \left( {B_{12f} + 2B_{66f} } \right)\,\partial_{xxy} + B_{22f} \,\partial_{yyy} ,L_{21} = - L_{12} , \hfill \\ L_{22} = F_{11f} \,\partial_{xx} + F_{66f} \,\partial_{yy} - H_{55f} ,L_{23} = \left( {F_{12f} + F_{66f} } \right)\partial_{xy} , \hfill \\ L_{31} = - L_{13} ,L_{32} = L_{23} ,L_{33} = F_{66} \,\partial_{xx} + F_{22f} \,\partial_{yy} - H_{44f} , \hfill \\ \hat{L}_{11} = I_{0} \partial_{tt} - I_{2} \,\partial_{xxtt} - I_{2} \,\partial_{yytt} ,\hat{L}_{12} = \hat{I}_{1} \,\partial_{xtt} , \hfill \\ \hat{L}_{13} = \hat{I}_{1} \,\partial_{ytt} ,\hat{L}_{21} = - \hat{L}_{12} ,\hat{L}_{22} = \overline{I}_{0} \,\partial_{tt} , \hfill \\ \hat{L}_{31} = - \hat{L}_{13} ,\hat{L}_{33} = \overline{I}_{0} \,\partial_{tt} . \hfill \\ \end{gathered}$$

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Wu, CP., Hu, HX. A review of dynamic analyses of single- and multi-layered graphene sheets/nanoplates using various nonlocal continuum mechanics-based plate theories. Acta Mech 232, 4497–4531 (2021). https://doi.org/10.1007/s00707-021-03068-4

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