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 哈尔滨工程大学学报  2019, Vol. 40 Issue (4): 676-682  DOI: 10.11990/jheu.201805001 0

### 引用本文

DING Hu. Free vibration analysis of a functionally graded rectangular plate with central circular cutout[J]. Journal of Harbin Engineering University, 2019, 40(4), 676-682. DOI: 10.11990/jheu.201805001.

### 文章历史

Free vibration analysis of a functionally graded rectangular plate with central circular cutout
DING Hu
No. 704 Research Institute, China Shipbuilding Industry Corporation, Shanghai 200031, China
Abstract: Based on first-order shear deformation theory and isogeometric analysis, a model of functionally graded rectangular plate with a circular cutout was built to study its free vibration characteristics.The non-uniform rational B-spline (NURBS) basis functions used to precisely model the geometry also described the solution field, which achieved seamless integration of computer-aided design and finite element analysis.The accuracy of the present method was validated through comparison with other solutions.The influence of boundary conditions, gradient index, aspect ratio, plate thickness, and radius of circular cutout on the natural frequency of the plate was investigated in detail.Numerical examples show that the isogeometric analysis has relatively high accuracy, and the method can effectively solve the free vibration of a functionally graded rectangular plate with a central circular cutout.
Keywords: isogeometric analysis    functionally graded    first-order shear deformation theory    rectangular plate    circular cutout    NURBS basis functions    free vibration

1 功能梯度Mindlin板的基本方程 1.1 带圆开孔的功能梯度矩形板

 Download: 图 1 带圆开孔的功能梯度矩形板 Fig. 1 Geometry of FG rectangle plate with a circular cutout
 ${P_e} = {P_c}{V_c} + {P_m}\left( {1 - {V_c}} \right)$ (1)
 ${V_c} = {\left( {1/2 + z/h} \right)^n}$ (2)

 Download: 图 2 体积分数随厚度的变化 Fig. 2 Variation of volume fraction Vc with the plate thickness
1.2 Mindlin板理论

 $\left\{ \begin{array}{l} U\left( {x,y,z} \right) = {u_0}\left( {x,y} \right) + z{\theta _x}\left( {x,y} \right)\\ V\left( {x,y,z} \right) = {v_0}\left( {x,y} \right) + z{\theta _y}\left( {x,y} \right)\\ W\left( {x,y,z} \right) = {w_0}\left( {x,y} \right) \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\varepsilon }} = \left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}} \end{array}} \right] = {\mathit{\boldsymbol{\varepsilon }}_m} + z{\mathit{\boldsymbol{\varepsilon }}_b} = \left[ {\begin{array}{*{20}{c}} {{u_{0,x}}}\\ {{v_{0,y}}}\\ {{u_{0,y}} + {v_{0,x}}} \end{array}} \right] + z\left[ {\begin{array}{*{20}{c}} {{\theta _{x,x}}}\\ {{\theta _{y,y}}}\\ {{\theta _{x,y}} + {\theta _{y,x}}} \end{array}} \right]\\ \mathit{\boldsymbol{\gamma }} = \left[ {\begin{array}{*{20}{c}} {{\gamma _{xz}}}\\ {{\gamma _{yz}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\theta _x} + {w_{0,x}}}\\ {{\theta _y} + {w_{0,y}}} \end{array}} \right] \end{array} \right.$ (5)

 $\mathit{\boldsymbol{\sigma }} = \left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\tau _{xy}}} \end{array}} \right] = \frac{{{E_e}}}{{1 - \nu _e^2}}\left[ {\begin{array}{*{20}{c}} 1&{{\nu _e}}&0\\ {{\nu _e}}&1&0\\ 0&0&{\frac{{1 - {\nu _e}}}{2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}} \end{array}} \right] = \mathit{\boldsymbol{D\varepsilon }}$ (6)
 $\mathit{\boldsymbol{\tau }} = \left[ {\begin{array}{*{20}{c}} {{\tau _{xz}}}\\ {{\tau _{yz}}} \end{array}} \right] = \frac{{k{E_e}}}{{2\left( {1 + {\nu _e}} \right)}}\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\gamma _{xz}}}\\ {{\gamma _{yz}}} \end{array}} \right] = {\mathit{\boldsymbol{D}}_s}\mathit{\boldsymbol{\gamma }}$ (7)

2 Mindlin板自由振动的等几何分析 2.1 B样条基函数

p=0时，

 ${N_{i,0}}\left( \xi \right) = \left\{ \begin{array}{l} 1,\;\;\;\;{\xi _i} \le \xi \le {\xi _{i + 1}}\\ 0,\;\;\;\;其他 \end{array} \right.$ (8)

p>0时，

 ${N_{i,p}}\left( \xi \right) = \frac{{\xi - {\xi _i}}}{{{\xi _{i + p}} - {\xi _i}}}{N_{i,p - 1}}\left( \xi \right) + \frac{{{\xi _{i + p + 1}} - \xi }}{{{\xi _{i + p + 1}} - {\xi _{i + 1}}}}{N_{i + 1,p - 1}}\left( \xi \right)$ (9)
2.2 NURBS基本概念

NURBS基函数是B样条基函数的有理形式，其定义为[19]

 $R_i^p\left( {\xi ,\eta } \right) = \frac{{{w_i}{N_{i,p}}\left( \xi \right)}}{{\sum\limits_{i = 1}^n {{w_i}{N_{i,p}}\left( \xi \right)} }}$ (10)

 $R_{i,j}^{p,q}\left( {\xi ,\eta } \right) = \frac{{{w_{i,j}}{N_{i,p}}\left( \xi \right){M_{j,q}}\left( \eta \right)}}{{\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{w_{i,j}}{N_{i,p}}\left( \xi \right){M_{j,q}}\left( \eta \right)} } }}$ (11)

NURBS曲面可由控制点Bi, j与NURBS基函数得[19]

 $S\left( {\xi ,\eta } \right) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {R_{i,j}^{p,q}\left( {\xi ,\eta } \right){B_{i,j}}} }$ (12)
2.3 基于NURBS的等几何分析

 ${\mathit{\boldsymbol{x}}^h}\left( {\xi ,\eta } \right) = \sum\limits_{A = 1}^N {{R_A}\left( {\xi ,\eta } \right){\mathit{\boldsymbol{x}}_A}}$ (13)
 ${\mathit{\boldsymbol{u}}^h}\left( {\xi ,\eta } \right) = \sum\limits_{A = 1}^N {{R_A}\left( {\xi ,\eta } \right){\mathit{\boldsymbol{u}}_A}}$ (14)

 ${\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\varepsilon }}_m^{\rm{T}}}&{\mathit{\boldsymbol{\varepsilon }}_b^{\rm{T}}}&{{\mathit{\boldsymbol{\gamma }}^{\rm{T}}}} \end{array}} \right]^{\rm{T}}} = \sum\limits_{A = 1}^N {\left[ {\begin{array}{*{20}{c}} {{{\left( {\mathit{\boldsymbol{B}}_A^m} \right)}^{\rm{T}}}}&{{{\left( {\mathit{\boldsymbol{B}}_A^b} \right)}^{\rm{T}}}}&{{{\left( {\mathit{\boldsymbol{B}}_A^s} \right)}^{\rm{T}}}} \end{array}} \right]{\mathit{\boldsymbol{u}}_A}}$ (15)

 $\mathit{\boldsymbol{B}}_A^m = \left[ {\begin{array}{*{20}{c}} {{R_{A,x}}}&0&0&0&0\\ 0&{{R_{A,y}}}&0&0&0\\ {{R_{A,y}}}&{{R_{A,x}}}&0&0&0 \end{array}} \right]$ (16)
 $\mathit{\boldsymbol{B}}_A^b = \left[ {\begin{array}{*{20}{c}} 0&0&0&{{R_{A,x}}}&0\\ 0&0&0&0&{{R_{A,y}}}\\ 0&0&0&{{R_{A,y}}}&{{R_{A,x}}} \end{array}} \right]$ (17)
 $\mathit{\boldsymbol{B}}_A^s = \left[ {\begin{array}{*{20}{c}} 0&0&{{R_{A,x}}}&{{R_A}}&0\\ 0&0&{{R_{A,y}}}&0&{{R_A}} \end{array}} \right]$ (18)

 $\left( {\mathit{\boldsymbol{K}} - {\omega ^2}\mathit{\boldsymbol{M}}} \right)\mathit{\boldsymbol{X}} = 0$ (19)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{K}} = \int\limits_\Omega {\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}^m}}&{{\mathit{\boldsymbol{B}}^b}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{D}}&{z\mathit{\boldsymbol{D}}}\\ {z\mathit{\boldsymbol{D}}}&{{z^2}\mathit{\boldsymbol{D}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}^m}}\\ {{\mathit{\boldsymbol{B}}^b}} \end{array}} \right]{\rm{d}}\mathit{\Omega }} + }\\ {\int\limits_\mathit{\Omega } {{{\left( {{\mathit{\boldsymbol{B}}^s}} \right)}^{\rm{T}}}{\mathit{\boldsymbol{D}}_s}{\mathit{\boldsymbol{B}}^s}{\rm{d}}\mathit{\Omega }} } \end{array}$ (20)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{M}} = \int\limits_\mathit{\Omega } {{\mathit{\boldsymbol{N}}^{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{\rho _e}}&0&0&{z{\rho _e}}&0\\ 0&{{\rho _e}}&0&0&{z{\rho _e}}\\ 0&0&{{\rho _e}}&0&0\\ {z{\rho _e}}&0&0&{{z^2}{\rho _e}}&0\\ 0&{z{\rho _e}}&0&0&{{z^2}{\rho _e}} \end{array}} \right]\mathit{\boldsymbol{N}}{\rm{d}}\mathit{\Omega }} }\\ {\mathit{\boldsymbol{N}} = \left[ {\begin{array}{*{20}{c}} {{R_{\rm{A}}}}&0&0&0&0\\ 0&{{R_{\rm{A}}}}&0&0&0\\ 0&0&{{R_{\rm{A}}}}&0&0\\ 0&0&0&{{R_{\rm{A}}}}&0\\ 0&0&0&0&{{R_{\rm{A}}}} \end{array}} \right]} \end{array}$ (22)
3 数值验证

 $\begin{array}{l} {\rm{SSSS}}:\\ \;\;\;\;\;\;\;\;x = 0\;与\;x = a:{v_0} = {w_0} = {\theta _y} = 0 \end{array}$ (23)
 $y = 0\;与\;y = b:{u_0} = {w_0} = {\theta _x} = 0$ (24)
 $\begin{array}{l} {\rm{CSCS}}:\\ \;\;\;\;\;\;\;\;x = 0\;与\;x = a:{v_0} = {w_0} = {\theta _y} = 0 \end{array}$ (25)
 $y = 0\;与\;y = b:{u_0} = {v_0} = {w_0} = {\theta _x} = {\theta _y} = 0$ (26)
 $\begin{array}{l} {\rm{CCCC:}}\\ \;\;\;\;\;\;\;\;四边:{u_0} = {v_0} = {w_0} = {\theta _x} = {\theta _y} = 0 \end{array}$ (27)
3.1 各向同性的带圆开孔矩形板

3.2 边界条件、梯度指数及长宽比对固有频率的影响

 Download: 图 3 CCCC矩形板的前6阶模态振型 Fig. 3 The first six mode shapes of CCCC rectangle plate
3.3 板厚度及圆孔半径对固有频率的影响

 Download: 图 4 固支矩形板前四阶无量纲频率与圆孔半径r的关系 Fig. 4 The variation of the first four non-dimensional frequencies for clamped rectangle plate with radius r
4 结论

1) 边界的约束越大，对应的频率越高；随着梯度指数的增加，前6阶固有频率逐渐降低；随着长宽比的增大，前6阶固有频率逐渐增大。

2) 前6阶固有频率随板厚度的升高而减小；开孔半径对前四阶频率的影响较为复杂。

3) 借助NURBS强大的几何建模能力，本方法适用于更加复杂的工程实际问题；此外，可以进一步研究力、热、电等多场耦合作用下的功能梯度板的动力学行为。

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