﻿ 基于应变梯度中厚板单元的石墨烯振动研究
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 力学学报  2015, Vol. 47 Issue (5): 751-761  DOI: 10.6052/0459-1879-15-074 0

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Xu Wei, Wang Lifeng, Jiang Jingnong. FINITE ELEMENT ANALYSIS OF STRAIN GRADIENT MIDDLE THICK PLATE MODEL ON THE VIBRATION OF GRAPHENE SHEETS[J]. Chinese Journal of Ship Research, 2015, 47(5): 751-761. DOI: 10.6052/0459-1879-15-074.
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### 文章历史

2015-03-05收稿
2015-06-29录用
2015-07-13网络版发表

1. 南京航空航天大学, 机械结构力学及控制国家重点实验室, 南京210016;
2. 中国航空动力机械研究所, 株洲412002

1 单层石墨烯等效明德林板模型} 1.1 基于应变梯度理论的明德林板的动力学方程

 ${\sigma _{ij}} = {C_{ijkl}}\left( {{\varepsilon _{kl}} + {g^2}{\nabla ^2}{{\tilde \varepsilon }_{kl}}} \right)$ (1)

 ${\varepsilon _{ij}} = \frac{1}{2}\left( {{u_{i,j}} + {u_{j,i}}} \right)$ (2a)
 ${\tilde \varepsilon _{kl}} = {\varepsilon _{kl}}$ (2b)

 $u = - z{\phi _x}\left( {x,y,t} \right),v = - z{\phi _y}\left( {x,y,t} \right),w = w\left( {x,y,t} \right)$ (3)
 图 1 明德林板变形示意图 Fig. 1 Configuration and coordinate system of the Mindlin plate

 $\left\{ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}}\\ {{\gamma _{yz}}}\\ {{\gamma _{zx}}} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} { - z\partial {\phi _x}/\partial x}\\ { - z\partial {\phi _y}/\partial y}\\ { - z\left( {\partial {\phi _x}/\partial y + \partial {\phi _y}/\partial x} \right)}\\ {\partial w/\partial y - {\phi _y}}\\ {\partial w/\partial x - {\phi _x}} \end{array}} \right.$ (4)

 $\begin{array}{l} {\sigma _x} = - \frac{{zE}}{{1 - {\nu ^2}}}\left( {\frac{{\partial {\phi _x}}}{{\partial x}} + \nu \frac{{\partial {\phi _y}}}{{\partial y}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\frac{{zE}}{{1 - {\nu ^2}}}\left( {\frac{{{\partial ^3}{\phi _x}}}{{\partial {x^3}}} + \frac{{{\partial ^3}{\phi _x}}}{{\partial x\partial {y^2}}} + \nu \frac{{{\partial ^3}{\phi _y}}}{{\partial {x^2}\partial y}} + \nu \frac{{{\partial ^3}{\phi _y}}}{{\partial {y^3}}}} \right) \end{array}$ (5a)
 $\begin{array}{l} {\sigma _y} = - \frac{{zE}}{{1 - {\nu ^2}}}\left( {\nu \frac{{\partial {\phi _x}}}{{\partial x}} + \frac{{\partial {\phi _y}}}{{\partial y}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\frac{{zE}}{{1 - {\nu ^2}}}\left( {\nu \frac{{{\partial ^3}{\phi _x}}}{{\partial {x^3}}} + \nu \frac{{{\partial ^3}{\phi _x}}}{{\partial x\partial {y^2}}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial {y^3}}}} \right) \end{array}$ (5b)
 $\begin{array}{l} {\tau _{xy}} = - \frac{{zE}}{{2\left( {1 + \nu } \right)}}\left( {\frac{{\partial {\phi _x}}}{{\partial y}} + \frac{{\partial {\phi _y}}}{{\partial x}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{zE}}{{2\left( {1 + \nu } \right)}}{g^2}\left( {\frac{{{\partial ^3}{\phi _x}}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}{\phi _x}}}{{\partial {y^3}}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial {x^3}}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial x\partial {y^2}}}} \right) \end{array}$ (5c)
 ${\tau _{yz}} = G\left( {\frac{{\partial w}}{{\partial y}} - {\phi _y}} \right) + G{g^2}\left( {\frac{{{\partial ^3}w}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}w}}{{\partial {y^3}}} - \frac{{{\partial ^2}{\phi _y}}}{{\partial {x^2}}} - \frac{{{\partial ^2}{\phi _y}}}{{\partial {y^2}}}} \right)$ (5d)
 ${\tau _{xz}} = G\left( {\frac{{\partial w}}{{\partial x}} - {\phi _x}} \right) + G{g^2}\left( {\frac{{{\partial ^3}w}}{{\partial {x^3}}} + \frac{{{\partial ^3}w}}{{\partial x\partial {y^2}}} - \frac{{{\partial ^2}{\phi _x}}}{{\partial {x^2}}} - \frac{{{\partial ^2}{\phi _x}}}{{\partial {y^2}}}} \right)$ (5e)

 $g = \frac{d}{{\sqrt {12} }}$ (6)

$d$为离散化微观模型内部质点间的距离.

 $\frac{{\partial {Q_x}}}{{\partial x}} + \frac{{\partial {Q_y}}}{{\partial y}} + q = \rho h\frac{{{\partial ^2}w}}{{\partial {t^2}}}$ (7a)
 $\frac{{\partial {M_x}}}{{\partial x}} + \frac{{\partial {M_{xy}}}}{{\partial y}} - {Q_x} = \rho J\frac{{{\partial ^2}{\phi _x}}}{{\partial {t^2}}}$ (7b)
 $\frac{{\partial {M_{xy}}}}{{\partial x}} + \frac{{\partial {M_y}}}{{\partial y}} - {Q_y} = \rho J\frac{{{\partial ^2}{\phi _y}}}{{\partial {t^2}}}$ (7c)

 $\begin{array}{l} {M_x} = \int_{ - h/2}^{h/2} {{\sigma _x}z} = - D\left( {\frac{{\partial {\phi _x}}}{{\partial x}} + \nu \frac{{\partial {\phi _y}}}{{\partial y}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{g^2}\left( {\frac{{{\partial ^3}{\phi _x}}}{{\partial {x^3}}} + \frac{{{\partial ^3}{\phi _x}}}{{\partial x\partial {y^2}}} + \nu \frac{{{\partial ^3}{\phi _y}}}{{\partial {x^2}\partial y}} + \nu \frac{{{\partial ^3}{\phi _y}}}{{\partial {y^3}}}} \right) \end{array}$ (8a)
 $\begin{array}{l} {M_y} = \int_{ - h/2}^{h/2} {{\sigma _y}z{\rm{d}}z} = - D\left( {\nu \frac{{\partial {\phi _x}}}{{\partial x}} + \frac{{\partial {\phi _y}}}{{\partial y}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{g^2}\left( {\nu \frac{{{\partial ^3}{\phi _x}}}{{\partial {x^3}}} + \nu \frac{{{\partial ^3}{\phi _x}}}{{\partial x\partial {y^2}}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial {y^3}}}} \right) \end{array}$ (8b)
 $\begin{array}{l} {M_{xy}} = \int_{ - h/2}^{h/2} {{\tau _{xy}}zdz} = - D\frac{{1 - \nu }}{2}\left( {\frac{{\partial {\phi _x}}}{{\partial y}} + \frac{{\partial {\phi _y}}}{{\partial x}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D\frac{{1 - \nu }}{2}{g^2}\left( {\frac{{{\partial ^3}{\phi _x}}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}{\phi _x}}}{{\partial {y^3}}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial {x^3}}} + \frac{{{\partial ^3}{\phi _y}}}{{\partial x\partial {y^2}}}} \right) \end{array}$ (8c)
 $\begin{array}{l} {Q_x} = \int_{ - h/2}^{h/2} {{\tau _{xz}}dz} = \kappa Gh\left( {\frac{{\partial w}}{{\partial x}} - {\phi _x}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\kappa Gh\left( {\frac{{{\partial ^3}w}}{{\partial {x^3}}} + \frac{{{\partial ^3}w}}{{\partial x\partial {y^2}}} - \frac{{{\partial ^2}{\phi _x}}}{{\partial {x^2}}} - \frac{{{\partial ^2}{\phi _x}}}{{\partial {y^2}}}} \right) \end{array}$ (8d)
 $\begin{array}{l} {Q_y} = \int_{ - h/2}^{h/2} {{\tau _{yz}}dz} = \kappa Gh\left( {\frac{{\partial w}}{{\partial y}} - {\phi _y}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\kappa Gh\left( {\frac{{{\partial ^3}w}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}w}}{{\partial {y^3}}} - \frac{{{\partial ^2}{\phi _y}}}{{\partial {x^2}}} - \frac{{{\partial ^2}{\phi _y}}}{{\partial {y^2}}}} \right) \end{array}$ (8e)

 $\begin{array}{l} \kappa Gh\left( {{\nabla ^2}w - \frac{{\partial {\phi _x}}}{{\partial x}} - \frac{{\partial {\phi _y}}}{{\partial y}}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\kappa Gh\left[{{\nabla ^4}w - {\nabla ^2}\left( {\frac{{\partial {\phi _x}}}{{\partial x}} + \frac{{\partial {\phi _y}}}{{\partial y}}} \right)} \right] - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \rho h\frac{{{\partial ^2}w}}{{\partial {t^2}}} + q = 0 \end{array}$ (9a)
 $\begin{array}{l} D\left( {\frac{{{\partial ^2}{\phi _x}}}{{\partial {x^2}}} + \frac{{1 - \nu }}{2}\frac{{{\partial ^2}{\phi _x}}}{{\partial {y^2}}} + \frac{{1 + \nu }}{2}\frac{{{\partial ^2}{\phi _y}}}{{\partial x\partial y}}} \right) + \\ D{g^2}\left( {\frac{{{\partial ^4}{\phi _x}}}{{\partial {x^4}}} + \frac{{3 - \nu }}{2}\frac{{{\partial ^4}{\phi _x}}}{{\partial {x^2}\partial {y^2}}} + \frac{{1 - \nu }}{2}\frac{{{\partial ^4}{\phi _x}}}{{\partial {y^4}}} + } \right.\left. {\frac{{1 + \nu }}{2}\frac{{{\partial ^4}{\phi _y}}}{{\partial {x^3}\partial y}} + \frac{{1 + \nu }}{2}\frac{{{\partial ^4}{\phi _y}}}{{\partial x\partial {y^3}}}} \right) + \\ \kappa Gh\left( {\frac{{\partial w}}{{\partial x}} - {\phi _x}} \right) + {g^2}\kappa Gh\left( {\frac{{{\partial ^3}w}}{{\partial {x^3}}} + \frac{{{\partial ^3}w}}{{\partial x\partial {y^2}}} - \frac{{{\partial ^2}{\phi _x}}}{{\partial {x^2}}} - \frac{{{\partial ^2}{\phi _x}}}{{\partial {y^2}}}} \right) + \\ \rho J\frac{{{\partial ^2}{\phi _x}}}{{\partial {t^2}}} = 0 \end{array}$ (9b)
 $\begin{array}{l} D\left( {\frac{{{\partial ^2}{\phi _y}}}{{\partial {y^2}}} + \frac{{1 - \nu }}{2}\frac{{{\partial ^2}{\phi _y}}}{{\partial {x^2}}} + \frac{{1 + \nu }}{2}\frac{{{\partial ^2}{\phi _x}}}{{\partial x\partial y}}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{\kern 1pt} {g^2}\left( {\frac{{{\partial ^4}{\phi _y}}}{{\partial {y^4}}} + \frac{{3 - \nu }}{2}\frac{{{\partial ^4}{\phi _y}}}{{\partial {x^2}\partial {y^2}}} + \frac{{1 - \nu }}{2}\frac{{{\partial ^4}{\phi _y}}}{{\partial {x^4}}} + } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{1 + \nu }}{2}\frac{{{\partial ^4}{\phi _x}}}{{\partial x\partial {y^3}}} + \frac{{1 + \nu }}{2}\frac{{{\partial ^4}{\phi _x}}}{{\partial {x^3}\partial y}}} \right) + \kappa Gh\left( {\frac{{\partial w}}{{\partial y}} - {\phi _y}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\kappa Gh\left( {\frac{{{\partial ^3}w}}{{\partial {x^2}\partial y}} + \frac{{{\partial ^3}w}}{{\partial {y^3}}} - \frac{{{\partial ^2}{\phi _y}}}{{\partial {x^2}}} - \frac{{{\partial ^2}{\phi _y}}}{{\partial {y^2}}}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \rho J\frac{{{\partial ^2}{\phi _y}}}{{\partial {t^2}}} = 0 \end{array}$ (9c)
1.2 应变梯度四边简支明德林板自由振动解析解

 $\begin{array}{*{20}{l}} \begin{array}{l} w\left( {0,y} \right) = 0{\mkern 1mu} ,\;\;{M_x}\left( {0,y} \right) = 0\\ w\left( {a,y} \right) = 0{\mkern 1mu} ,\;\;{M_x}\left( {a,y} \right) = 0\\ w\left( {x,0} \right) = 0{\mkern 1mu} ,\;\;{M_y}\left( {x,0} \right) = 0\\ w\left( {x,b} \right) = 0{\mkern 1mu} ,\;\;{M_y}\left( {x,b} \right) = 0 \end{array} \end{array}$

 $\begin{array}{l} w(0,y) = 0{\mkern 1mu} ,\;\;{\phi _x}(0,y) = 0\\ w(a,y) = 0{\mkern 1mu} ,\;\;{\phi _x}(a,y) = 0\\ w(x,0) = 0{\mkern 1mu} ,\;\;{\phi _y}(x,0) = 0\\ w(x,b) = 0{\mkern 1mu} ,\;\;{\phi _y}(x,b) = 0 \end{array}$

 $\begin{array}{*{20}{l}} \begin{array}{l} {Q_z}(0,y) = 0{\mkern 1mu} ,\;\;{M_x}(0,y) = 0{\mkern 1mu} ,\;\;{M_{xy}}(0,y) = 0\\ {Q_z}(a,y) = 0{\mkern 1mu} ,\;\;{M_x}(a,y) = 0{\mkern 1mu} ,\;\;{M_{xy}}(a,y) = 0\\ {Q_z}(x,0) = 0{\mkern 1mu} ,\;\;{M_y}(x,0) = 0{\mkern 1mu} ,\;\;{M_{yx}}(x,0) = 0\\ {Q_z}(x,b) = 0{\mkern 1mu} ,\;\;{M_y}(x,b) = 0{\mkern 1mu} ,\;\;{M_{yx}}(x,b) = 0 \end{array} \end{array}$

 $w\left( {x,y,t} \right) = {W_{mn}}\sin \frac{{m\pi x}}{a}\sin \frac{{n\pi y}}{b}{{\rm{e}}^{{\rm{i}}\omega t}}$ (10a)
 ${\phi _x}\left( {x,y,t} \right) = {\psi _{xmn}}\cos \frac{{m\pi x}}{a}\sin \frac{{n\pi y}}{b}{{\rm{e}}^{{\rm{i}}\omega t}}$ (10b)
 ${\phi _y}\left( {x,y,t} \right) = {\psi _{ymn}}\sin \frac{{m\pi x}}{a}\cos \frac{{n\pi y}}{b}{{\rm{e}}^{{\rm{i}}\omega t}}$ (10c)

 $\begin{array}{l} \kappa Gh\left[{ - {W_{mn}}\left( {{A^2} + {B^2}} \right) + {\psi _{xmn}}A + {\psi _{ymn}}B} \right] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {g^2}\kappa Gh\left[{{W_{mn}}{{\left( {{A^2} + {B^2}} \right)}^2} - } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {{A^2} + {B^2}} \right)\left( {{\psi _{xmn}}A + {\psi _{ymn}}B} \right)} \right] + \rho h{\omega ^2}{W_{mn}} = 0 \end{array}$ (11a)
 $\begin{array}{l} D\left( {{A^2}{\psi _{xmn}} + \frac{{1 - \nu }}{2}{B^2}{\psi _{xmn}} + \frac{{1 + \nu }}{2}AB{\psi _{ymn}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{g^2}\left( {{A^4}{\psi _{xmn}} + \frac{{3 - \nu }}{2}{A^2}{B^2}{\psi _{xmn}} + } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{1 - \nu }}{2}{B^4}{\psi _{xmn}} + \frac{{1 + \nu }}{2}{A^3}B{\psi _{ymn}} + \frac{{1 + \nu }}{2}A{B^3}{\psi _{ymn}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa Gh\left( {{W_{mn}}A - {\psi _{xmn}}} \right) + {g^2}\kappa Gh\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {{A^3}{W_{mn}} + A{B^2}{W_{mn}}} \right. - \left. {{A^2}{\psi _{xmn}} - {B^2}{\psi _{xmn}}} \right)\\ + \rho J{\omega ^2}{\psi _{xmn}} = 0 \end{array}$ (11b)
 $\begin{array}{l} D\left( {{B^2}{\psi _{ymn}} + \frac{{1 - \nu }}{2}{A^2}{\psi _{ymn}} + \frac{{1 + \nu }}{2}AB{\psi _{xmn}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{g^2}\left( {{B^4}{\psi _{ymn}} + \frac{{3 - \nu }}{2}{A^2}{B^2}{\psi _{ymn}} + } \right.\left. {\frac{{1 - \nu }}{2}{A^4}{\psi _{ymn}} + \frac{{1 + \nu }}{2}A{B^3}{\psi _{xmn}} + \frac{{1 + \nu }}{2}{A^3}B{\psi _{xmn}}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa Gh\left( {{W_{mn}}B - {\psi _{ymn}}} \right) + {g^2}\kappa Gh\left( {{A^2}B{W_{mn}} + {B^3}{W_{mn}}} \right. - \left. {{A^2}{\psi _{ymn}} - {B^2}{\psi _{ymn}}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \rho J{\omega ^2}{\psi _{ymn}} = 0 \end{array}$ (11c)

 $\left( {{Q_{\rm{s}}} - {\omega ^2}{M_{\rm{s}}}} \right){F_{\rm{s}}} = {\bf{0}}$ (12)

 ${Q_{\rm{s}}} = \left[{\begin{array}{*{20}{c}} {{e_1}}&{{e_2}}&{{e_3}}\\ {{e_2}}&{{e_4}}&{{e_5}}\\ {{e_3}}&{{e_5}}&{{e_6}} \end{array}} \right]$ (13)

${\pmb Q}_{\rm s}$中系数$e_{1}\sim e_{6}$的具体表达式如下

 ${e_1} = \kappa Gh\left( {{A^2} + {B^2}} \right) - {g^2}\kappa Gh{\left( {{A^2} + {B^2}} \right)^2}$ (14a)
 ${e_2} = - \kappa GhA + {g^2}\kappa Gh\left( {{A^2} + {B^2}} \right)A$ (14b)
 ${e_3} = - \kappa GhB + {g^2}\kappa Gh\left( {{A^2} + {B^2}} \right)B$ (14c)
 $\begin{array}{l} {e_4} = - D\left( {{A^2} + \frac{{1 - \nu }}{2}{B^2}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{g^2}\left( {{A^4} + \frac{{3 - \nu }}{2}{A^2}{B^2} + \frac{{1 - \nu }}{2}{B^4}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa Gh + {g^2}\kappa Gh\left( {{A^2} + {B^2}} \right) \end{array}$ (14d)
 ${e_5} = - \frac{{1 + \nu }}{2}DAB + D{g^2}\left( {\frac{{1 + \nu }}{2}{A^3}B + \frac{{1 + \nu }}{2}A{B^3}} \right)$ (14e)
 $\begin{array}{l} {e_6} = - D\left( {{B^2} + \frac{{1 - \nu }}{2}{A^2}} \right) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D{g^2}\left( {{B^4} + \frac{{3 - \nu }}{2}{A^2}{B^2} + \frac{{1 - \nu }}{2}{A^4}} \right) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa Gh + {g^2}\kappa Gh\left( {{A^2} + {B^2}} \right) \end{array}$ (14f)

${\pmb M}_{\rm s}$为广义质量矩阵

 ${M_{\rm{s}}}{\rm{ = }}\left[{\begin{array}{*{20}{c}} {\rho h}&0&0\\ 0&{\rho J}&0\\ 0&0&{\rho J} \end{array}} \right]$ (15)

${\pmb F}_{\rm s}$ 为振型函数的未知系数向量

 ${F_{\rm{s}}} = \left[{\begin{array}{*{20}{c}} \begin{array}{l} {W_{mn}}\\ {\psi _{xmn}}\\ {\psi _{ymn}} \end{array} \end{array}} \right]$ (16)

 $\begin{array}{l} {\omega _1} = \sqrt {\left[{1 - {g^2}\left( {{A^2} + {B^2}} \right)} \right]} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {\frac{{{\beta _t} - \sqrt {\beta _t^2 - 4\frac{\rho }{{\kappa G}}\frac{{\rho J}}{D}{{\left( {{A^2} + {B^2}} \right)}^2}} }}{{2\frac{\rho }{{\kappa G}}\frac{{\rho J}}{D}}}} \end{array}$ (17a)
 $\begin{array}{l} {\omega _2} = \sqrt {\left[{1 - {g^2}\left( {{A^2} + {B^2}} \right)} \right]} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {\frac{{{\beta _t} + \sqrt {\beta _t^2 - 4\frac{\rho }{{\kappa G}}\frac{{\rho J}}{D}{{\left( {{A^2} + {B^2}} \right)}^2}} }}{{2\frac{\rho }{{\kappa G}}\frac{{\rho J}}{D}}}} \end{array}$ (17b)
 $\begin{array}{l} {\omega _3} = \sqrt {\left[{1 - {g^2}\left( {{A^2} + {B^2}} \right)} \right]} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt {\frac{{\left( {1 - \nu } \right)D + 2\kappa Gh}}{{2\rho J}}} \end{array}$ (17c)

 ${\beta _t} = \left( {\frac{{\rho J}}{D} + \frac{\rho }{{\kappa G}}} \right)\left( {{A^2} + {B^2}} \right) + \frac{{\rho h}}{D}$ (18)

2 有限单元模型建立

 $w\left( {x,y,t} \right) = W\left( {x,y} \right){{\rm{e}}^{{\rm{i}}\omega t}}$ (19a)
 ${\phi _x}\left( {x,y,t} \right) = {\psi _x}\left( {x,y} \right){{\rm{e}}^{{\rm{i}}\omega t}}$ (19b)
 ${\phi _y}\left( {x,y,t} \right) = {\psi _y}\left( {x,y} \right){{\rm{e}}^{{\rm{i}}\omega t}}$ (19c)

 $\iiint_\Omega\left( {{\sigma _x}\delta {\varepsilon _x} + {\sigma _y}\delta {\varepsilon _y} + {\tau _{xy}}\delta {\gamma _{xy}} + {\tau _{yz}}\delta {\gamma _{yz}} + {\tau _{xz}}\delta {\gamma _{xz}}} \right){\rm{d}}V +$ $\iiint_\Omega\left( {\rho h\ddot w\delta w + \rho J{{\ddot \phi }_x}\delta {\phi _x} + \rho J{{\ddot \phi }_y}\delta {\phi _y}} \right){\rm{d}}V = 0$ (20)

 $\iint\left\{ {D\left[{2\frac{{\partial {\psi _x}}}{{\partial x}}\delta {\psi _{x,x}} + 2\nu \frac{{\partial {\psi _y}}}{{\partial y}}\delta {\psi _{x,x}}} \right.} \right. + 2\nu \frac{{\partial {\psi _x}}}{{\partial x}}\delta {\psi _{y,y}} +$ $2\frac{{\partial {\psi _y}}}{{\partial y}}\delta {\psi _{y,y}} + 4\left( {1 - \nu } \right)\left( {\frac{{\partial {\psi _x}}}{{\partial y}} + \frac{{\partial {\psi _y}}}{{\partial x}}} \right)\delta {\psi _{x,y}} +$ $\left. {4\left( {1 - \nu } \right)\left( {\frac{{\partial {\psi _x}}}{{\partial y}} + \frac{{\partial {\psi _y}}}{{\partial x}}} \right)\delta {\psi _{y,x}}} \right] - D{g^2}\left[{2\frac{{{\partial ^2}{\psi _x}}}{{\partial {x^2}}}\delta {\psi _{x,xx}}} \right. +$ $2\nu \frac{{{\partial ^2}{\psi _y}}}{{\partial x\partial y}}\delta {\psi _{x,xx}} + 4\left( {1 - \nu } \right)\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial y}} + \frac{{{\partial ^2}{\psi _y}}}{{\partial {x^2}}}} \right)\delta {\psi _{x,xy}} +$ $4\left( {1 - \nu } \right)\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial y}} + \frac{{{\partial ^2}{\psi _y}}}{{\partial {x^2}}}} \right)\delta {\psi _{y,xx}} + 2\nu \frac{{{\partial ^2}{\psi _x}}}{{\partial {x^2}}}\delta {\psi _{y,xy}} +$ $2\frac{{{\partial ^2}{\psi _y}}}{{\partial x\partial y}}\delta {\psi _{y,xy}} + 2\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial y}}\delta {\psi _{x,xy}} + 2\nu \frac{{{\partial ^2}{\psi _y}}}{{\partial {y^2}}}\delta {\psi _{x,xy}} +$ $2\nu \frac{{{\partial ^2}{\psi _y}}}{{\partial x\partial y}}\delta {\psi _{x,xx}} + 4\left( {1 - \nu } \right)\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial x\partial y}} + \frac{{{\partial ^2}{\psi _y}}}{{\partial {x^2}}}} \right)\delta {\psi _{x,xy}} +$ $4\left( {1 - \nu } \right)\left( {\frac{{{\partial ^2}{\psi _x}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{\psi _y}}}{{\partial x\partial y}}} \right)\delta {\psi _{y,xy}} + \left. {\left. {2\frac{{{\partial ^2}{\psi _y}}}{{\partial {y^2}}}\delta {\psi _{y,yy}}} \right]} \right\}{\rm{d}}x{\rm{d}}y +$ $\iint\left\{ {Gh\kappa \left[{\frac{{\partial W}}{{\partial x}}\delta {W_{,x}} - \frac{{\partial W}}{{\partial x}}\delta {\psi _x} - {\psi _x}\delta {W_{,x}} + {\psi _x}\delta {\psi _x}} \right.} \right. +$ $\begin{array}{l} \begin{array}{*{20}{l}} {\left. {\frac{{\partial W}}{{\partial y}}\delta {W_{,y}} - \frac{{\partial W}}{{\partial y}}\delta {\psi _y} - {\psi _y}\delta {W_{,y}} + {\psi _y}\delta {\psi _y}} \right] - }\\ {Gh\kappa {g^2}\left( {2\frac{{{\partial ^2}W}}{{\partial {x^2}}}\delta {W_{,xx}} + } \right.2\frac{{\partial {\psi _x}}}{{\partial x}}\delta {W_{,xx}} + 2\frac{{{\partial ^2}W}}{{\partial {x^2}}}\delta {\psi _{x,x}} + } \end{array}\\ \begin{array}{*{20}{l}} {2\frac{{\partial {\psi _x}}}{{\partial x}}\delta {\psi _{x,x}} + 2\frac{{{\partial ^2}W}}{{\partial x\partial y}}\delta {W_{,xy}} + 2\frac{{\partial {\psi _y}}}{{\partial x}}\delta {W_{,xy}} + }\\ {2\frac{{{\partial ^2}W}}{{\partial x\partial y}}\delta {\psi _{y,x}} + 2\frac{{\partial {\psi _y}}}{{\partial x}}\delta {\psi _{y,x}} + 2\frac{{{\partial ^2}W}}{{\partial x\partial y}}\delta {W_{,xy}} + }\\ {2\frac{{\partial {\psi _x}}}{{\partial y}}\delta {W_{,xy}} + 2\frac{{{\partial ^2}W}}{{\partial x\partial y}}\delta {\psi _{x,y}} + 2\frac{{\partial {\psi _x}}}{{\partial y}}\delta {\psi _{x,y}} + } \end{array}\\ \begin{array}{*{20}{l}} {2\frac{{{\partial ^2}W}}{{\partial {y^2}}}\delta {W_{,yy}} + 2\frac{{\partial {\psi _y}}}{{\partial y}}\delta {W_{,yy}} + 2\frac{{{\partial ^2}W}}{{\partial {y^2}}}\delta {\psi _{y,y}} + }\\ \begin{array}{l} 2\frac{{\partial {\psi _y}}}{{\partial y}}\delta {\psi _{y,y}})\} dxdy - \\ {\omega ^2}{\rm{{\mathop \iint \limits}}}(\rho hW\delta W + \rho J{\psi _x}\delta {\psi _x} + \rho J{\psi _y}\delta {\psi _y}){\rm{d}}x{\rm{d}}y = 0 \end{array} \end{array} \end{array}$ (21)

 图 2 4节点36自由度矩形明德林板单元示意图 Fig. 2 4-node 36-DOF rectangular Mindlin plate element

 $\xi = \left( {x - {x_0}} \right)/{a_e}{\mkern 1mu} ,\;\eta = \left( {y - {y_0}} \right)/{b_e}$ (22)

 $\begin{array}{l} W\left( {\xi ,\eta } \right) = {\alpha _1} + {\alpha _2}\xi + {\alpha _3}\eta + {\alpha _4}{\xi ^2} + {\alpha _5}\xi \eta + {\alpha _6}{\eta ^2} + \\ {\alpha _7}{\xi ^3} + {\alpha _8}{\xi ^2}\eta + {\alpha _9}\xi {\eta ^2} + {\alpha _{10}}{\eta ^3} + {\alpha _{11}}{\xi ^3}\eta + {\alpha _{12}}\xi {\eta ^3} \end{array}$ (23a)
 $\begin{array}{l} {\psi _x}\left( {\xi ,\eta } \right) = {\beta _1} + {\beta _2}\xi + {\beta _3}\eta + {\beta _4}{\xi ^2} + {\beta _5}\xi \eta + {\beta _6}{\eta ^2} + \\ {\beta _7}{\xi ^3} + {\beta _8}{\xi ^2}\eta + {\beta _9}\xi {\eta ^2} + {\beta _{10}}{\eta ^3} + {\beta _{11}}{\xi ^3}\eta + {\beta _{12}}\xi {\eta ^3} \end{array}$ (23b)
 $\begin{array}{l} {\psi _y}\left( {\xi ,\eta } \right) = {\gamma _1} + {\gamma _2}\xi + {\gamma _3}\eta + {\gamma _4}{\xi ^2} + {\gamma _5}\xi \eta + {\gamma _6}{\eta ^2} + \\ {\gamma _7}{\xi ^3} + {\gamma _8}{\xi ^2}\eta + {\gamma _9}\xi {\eta ^2} + {\gamma _{10}}{\eta ^3} + {\gamma _{11}}{\xi ^3}\eta + {\gamma _{12}}\xi {\eta ^3} \end{array}$ (23c)

 $\begin{array}{l} d_i^e = {\left\{ {{w_i},{\phi _{xi}},{\phi _{yi}},{w_{i,x}},{\phi _{xi,x}},{\phi _{yi,x}},{w_{i,y}},{\phi _{xi,y}},{\phi _{yi,y}}} \right\}^{\rm{T}}} = \\ {\left\{ {w,{\phi _{xi}},{\phi _{yi}},\frac{{\partial {w_i}}}{{\partial x}},\frac{{\partial {\phi _{xi}}}}{{\partial x}},\frac{{\partial {\phi _{yi}}}}{{\partial x}},\frac{{\partial {w_i}}}{{\partial y}},\frac{{\partial {\phi _{xi}}}}{{\partial y}},\frac{{\partial {\phi _{yi}}}}{{\partial y}}} \right\}^{\rm{T}}} = \\ {\left\{ {w,{\phi _{xi}},{\phi _{yi}},\frac{{\partial {w_i}}}{{{a_e}\partial \xi }},\frac{{\partial {\phi _{xi}}}}{{{a_e}\partial \xi }},\frac{{\partial {\phi _{yi}}}}{{{a_e}\partial \xi }},\frac{{\partial {w_i}}}{{{b_e}\partial \eta }},\frac{{\partial {\phi _{xi}}}}{{{b_e}\partial \eta }},\frac{{\partial {\phi _{yi}}}}{{{b_e}\partial \eta }}} \right\}^{\rm{T}}} \end{array}$ (24)

 $\left. \begin{array}{l} w = \sum\limits_{i = 1}^4 {\left( {{N_i}{w_i} + {N_{ix}}{\phi _{xi}} + {N_{iy}}{\phi _{yi}}} \right)} = \\ \sum\limits_{i = 1}^4 {{N_{{\phi _{xi}}}}d_i^e = {N_{{\phi _x}}}{d^e}} \\ {N_{wi}} = \left[{{N_i}\;\;0\;\;0\;\;{N_{ix}}\;\;0\;\;0\;\;{N_{iy}}\;\;0\;\;0} \right]\\ {N_w} = \left[{{N_{w1}},{N_{w2}},{N_{w3}},{N_{w4}}} \right] \end{array} \right\}$ (25a)
 $\left. \begin{array}{l} {\phi _x} = \sum\limits_{i = 1}^4 {\left( {{N_i}{\phi _{xi}} + {N_{ix}}{\phi _{xi,x}} + {N_{iy}}{\phi _{xi,y}}} \right)} = \\ \sum\limits_{i = 1}^4 {{N_{{\phi _{xi}}}}d_i^e} = {N_{{\phi _x}}}{d^e}\\ {N_{{\phi _{xi}}}} = \left[{0\;\;{N_i}\;\;0\;0\;\;{N_{ix}}\;\;0\;\;0\;\;{N_{iy}}\;\;0} \right]\\ {N_{{\phi _x}}} = \left[{{N_{{\phi _{x1}}}},{N_{{\phi _{x2}}}},{N_{{\phi _{x3}}}},{N_{{\phi _{x4}}}}} \right] \end{array} \right\}$ (25b)
 $\left. \begin{array}{l} {\phi _y} = \sum\limits_{i = 1}^4 {\left( {{N_i}{\phi _{yi}} + {N_{ix}}{\phi _{yi,x}} + {N_{iy}}{\phi _{yi,y}}} \right)} = \\ \sum\limits_{i = 1}^4 {{N_{{\phi _{yi}}}}d_i^e} = {N_{{\phi _y}}}{d^e}\\ {N_{{\phi _{yi}}}} = \left[{0\;\;0\;\;{N_i}\;\;0\;0\;\;{N_{ix}}\;\;0\;\;0\;\;{N_{iy}}} \right]\\ {N_{{\phi _y}}} = \left[{{N_{{\phi _{y1}}}},{N_{{\phi _{y2}}}},{N_{{\phi _{y3}}}},{N_{{\phi _{y4}}}}} \right] \end{array} \right\}$ (25c)

 ${N_i} = \left( {1 + {\xi _i}\xi } \right)\left( {1 + {\eta _i}\eta } \right)\left( {2 + {\xi _i}\xi + {\eta _i}\eta - {\xi ^2} - {\eta ^2}} \right)/8$ (26a)
 ${N_{ix}} = - {a_e}{\xi _i}\left( {1 + {\xi _i}\xi } \right)\left( {1 + {\eta _i}\eta } \right)\left( {1 - {\xi ^2}} \right)/8$ (26b)
 ${N_{iy}} = - {b_e}{\eta _i}\left( {1 + {\xi _i}\xi } \right)\left( {1 + {\eta _i}\eta } \right)\left( {1 - {\eta ^2}} \right)/8$ (26c)

 $\varepsilon = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\varepsilon _b}\\ {\varepsilon _s} \end{array} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} z{B_b}\\ {B_s} \end{array} \end{array}} \right\}{\delta ^e}$ (27)

 $\left. \begin{array}{l} {B_b} = \left[{{B_{b1}}\;{B_{b2}}\;{B_{b3}}\;\;{B_{b4}}} \right]\\ {B_{bi}} = \left[{\begin{array}{*{20}{c}} 0&{{N_{i,\xi }}/{a_e}}&0&0&{{N_{i,\xi }}/{a_e}}&0&0&{{N_{i,\xi }}/{a_e}}&0\\ 0&0&{{N_{i,\eta }}/{b_e}}&0&0&{{N_{i,\eta }}/{b_e}}&0&0&{{N_{i,\eta }}/{b_e}}\\ 0&{{N_{i,\eta }}/{b_e}}&{{N_{i,\xi }}/{a_e}}&0&{{N_{ix,\xi }}/{b_e}}&{{N_{ix,\xi }}/{a_e}}&0&{{N_{iy,\eta }}/{b_e}}&{{N_{iy,\xi }}/{a_e}} \end{array}} \right] \end{array} \right\}$ (28a)
 $\left. \begin{array}{l} {B_s} = \left[{{B_{s1}}\;\;{B_{s2}}\;\;{B_{s3}}\;\;{B_{s4}}} \right]\\ {B_{si}} = \left[{\begin{array}{*{20}{c}} {\partial {N_i}/{b_e}\partial \eta }&0&{{N_i}}&{\partial {N_{ix}}/{b_e}\partial \eta }&0&{{N_{ix}}}&{\partial {N_{iy}}/{b_e}\partial \eta }&0&{{N_{iy}}}\\ {\partial {N_i}/{a_e}\partial \xi }&{{N_i}}&0&{\partial {N_{ix}}/{a_e}\partial \xi }&{{N_{ix}}}&0&{\partial {N_{iy}}/{a_e}\partial \xi }&{{N_{iy}}}&0 \end{array}} \right] \end{array} \right\}$ (28b)

 $\sigma = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\sigma _b}\\ {\sigma _s} \end{array} \end{array}} \right\} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {D_b}{\varepsilon _b}\\ {D_s}{\varepsilon _s} \end{array} \end{array}} \right\}$ (29)

 ${D_b} = D\left[{\begin{array}{*{20}{c}} 1&\nu &0\\ \nu &1&0\\ 0&0&{\left( {1 - \nu } \right)/2} \end{array}} \right]$ (30a)
 ${D_s} = \left[{\begin{array}{*{20}{c}} {\kappa G}&0\\ 0&{\kappa G} \end{array}} \right]$ (30b)

 $\begin{array}{l} {\mathop \iint \limits}\frac{{{h^3}}}{{12}}\left( {B_b^{\rm{T}}{D_b}{B_b} - {g^2}B_{bx}^{\rm{T}}{D_b}{B_{bx}} - {g^2}B_{by}^{\rm{T}}{D_b}{B_{by}}} \right)dx{\rm{d}}y + \\ {\mathop \iint \limits}\left( {B_s^{\rm{T}}{D_s}{B_s} - {g^2}B_{sx}^{\rm{T}}{D_s}{B_{sx}} - {g^2}B_{sy}^{\rm{T}}{D_s}{B_{sy}}} \right)dxdy - \\ {\omega ^2}{\mathop \iint \limits}\left( {\rho hN_w^{\rm{T}}{N_w} + \rho JN_{{\phi _x}}^{\rm{T}}{N_{{\phi _x}}} + \rho JN_{{\phi _y}}^{\rm{T}}{N_{{\phi _y}}}} \right)dxdy = 0 \end{array}$ (31)

 $\begin{array}{l} {K^e} = \frac{{{h^3}}}{{12}}{\mathop \iint \limits}(B_b^{\rm{T}}{D_b}{B_b} - {g^2}B_{bx}^{\rm{T}}{D_b}{B_{bx}} - \\ {g^2}B_{by}^{\rm{T}}{D_b}{B_{by}})dxdy + \\ h{\mathop \iint \limits}(B_s^{\rm{T}}{D_s}{B_s} - {g^2}B_{sx}^{\rm{T}}{D_s}{B_{sx}} - \\ {g^2}B_{sy}^{\rm{T}}{D_s}{B_{sy}})dxdyn \end{array}$ (32)

 $\begin{array}{l} {M^e} = \rho h{\mathop \iint \limits}N_w^{\rm{T}}{N_w}{\rm{d}}x{\rm{d}}y + \rho J{\mathop \iint \limits}N_{{\phi _x}}^{\rm{T}}{N_{{\phi _x}}}dx{\rm{d}}y\\ + \rho J{\mathop \iint \limits}N_{{\phi _y}}^{\rm{T}}{N_{{\phi _y}}}dx{\rm{d}}y \end{array}$ (33)

 $\left( {K - {\omega ^2}M} \right)d = 0$

3 结果与讨论

 ${R_f} = \frac{\omega }{{\bar \omega }} = \frac{{\omega /2\pi }}{{\bar \omega /2\pi }} = \frac{f}{{\bar f\;{\mkern 1mu} }}$ (35)

 图 3 四边简支，长宽比$b/a=1$，石墨烯固有频率随边长的变化 Fig. 3 The relation between the natural frequencies and the size of graphene (SSSS, $b/a=1$)

 图 4 四边固支石墨烯的自由振动固有频率随边长的变化 Fig. 4 The relation between the natural frequencies and the size of graphene (CCCC)

 图 5 两对边简支两对边固支石墨烯的自由振动固有频率随长度的变化 Fig. 5 The relation between the natural frequencies and the size of graphene (SCSC)

 图 6 不同阶数石墨烯的固有频率比值随非局部参数的变化 Fig. 6 Effect of the small scale effect on the frequency ratios associated with various vibrational modes for graphene
4 结 论

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FINITE ELEMENT ANALYSIS OF STRAIN GRADIENT MIDDLE THICK PLATE MODEL ON THE VIBRATION OF GRAPHENE SHEETS
Xu Wei, Wang Lifeng, Jiang Jingnong
1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. China Aviation Powerplant Research Institute, Zhuzhou 412002, China
Fund: The project was supported by the Program for New Century Excellent Talents in University (NCET-11-0832), 333 Talents Program in Jiangsu Province and Fundamental Research Funds for the Central Universities of China.
Abstract: The dynamics equation of the Mindlin middle thick plate model based on strain gradient theory is formulated to study the vibration of single-layered graphene sheets (SLGSs). Analytical solution of the natural frequency for free vibration of Mindlin plate with all edges simply-supported is derived. A 4-node 36-degree-of-freedom (DOF) Mindlin plate element is proposed to build the nonlocal finite element (FE) plate model with second order gradient of strain taken into consideration. This FE method is used to study the influences of the size, vibration mode and nonlocal parameters on the scale effect of vibration behaviors of SLGSs, which validates the reliability of the FE model for predicting the scale effect on the vibrational SLGSs with complex boundary conditions. The natural frequencies obtained by the strain gradient Mindlin plate are lower than that obtained by classical Mindlin plate model. The natural frequencies of SLGSs obtained by Mindlin plate model with first-order shear deformation taken into account are lower than that obtained by Kirchhoff plate model for both strain gradient model and classical case. The small scale effect increases with the increase of the mode order and the decrease of the size of SLGSs.
Key words: strain gradient    finite element method    Mindlin plate    vibration    scale effect