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基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析

陈卫, 方耀楚, 孙冰, 彭林欣

陈卫, 方耀楚, 孙冰, 彭林欣. 基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析. 力学学报, 2023, 55(6): 1355-1370. DOI: 10.6052/0459-1879-23-040
引用本文: 陈卫, 方耀楚, 孙冰, 彭林欣. 基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析. 力学学报, 2023, 55(6): 1355-1370. DOI: 10.6052/0459-1879-23-040
shu
Chen Wei, Fang Yaochu, Sun Bing, Peng Linxin. Meshless analysis of linear bending and free vibration of functionally graded carbon nanotube-reinforced composite plate on elastic foundation based on improved Reddy type third-order shear deformation theory. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1355-1370. DOI: 10.6052/0459-1879-23-040
Citation: Chen Wei, Fang Yaochu, Sun Bing, Peng Linxin. Meshless analysis of linear bending and free vibration of functionally graded carbon nanotube-reinforced composite plate on elastic foundation based on improved Reddy type third-order shear deformation theory. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1355-1370. DOI: 10.6052/0459-1879-23-040
shu
陈卫, 方耀楚, 孙冰, 彭林欣. 基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析. 力学学报, 2023, 55(6): 1355-1370. CSTR: 32045.14.0459-1879-23-040
引用本文: 陈卫, 方耀楚, 孙冰, 彭林欣. 基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析. 力学学报, 2023, 55(6): 1355-1370. CSTR: 32045.14.0459-1879-23-040
shu
Chen Wei, Fang Yaochu, Sun Bing, Peng Linxin. Meshless analysis of linear bending and free vibration of functionally graded carbon nanotube-reinforced composite plate on elastic foundation based on improved Reddy type third-order shear deformation theory. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1355-1370. CSTR: 32045.14.0459-1879-23-040
Citation: Chen Wei, Fang Yaochu, Sun Bing, Peng Linxin. Meshless analysis of linear bending and free vibration of functionally graded carbon nanotube-reinforced composite plate on elastic foundation based on improved Reddy type third-order shear deformation theory. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1355-1370. CSTR: 32045.14.0459-1879-23-040
shu

基于改进Reddy型TSDT的弹性地基上FG-CNTRC板线性弯曲与自由振动无网格分析

  • *.

    南华大学土木工程学院, 湖南衡阳 421001

  • †.

    广西大学土木建筑工程学院, 南宁 530004

  • **.

    广西大学广西防灾减灾与工程安全重点实验室, 工程防灾与结构安全教育部重点实验室, 南宁 530004

基金项目: 国家自然科学基金(11562001, 12162004)和南华大学博士科研启动基金(Y00043-13)资助项目
详细信息
    通讯作者:

    彭林欣, 教授, 主要研究方向为计算复合板壳力学中的无网格法. E-mail: penglx@gxu.edu.cn

  • 中图分类号: TU339

MESHLESS ANALYSIS OF LINEAR BENDING AND FREE VIBRATION OF FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE PLATE ON ELASTIC FOUNDATION BASED ON IMPROVED REDDY TYPE THIRD-ORDER SHEAR DEFORMATION THEORY

  • *.

    School of Civil Engineering, University of South China, Hengyang 421001, Hunan, China

  • †.

    School of Civil Engineering and Architecture, Guangxi University, Nanning 530004, China

  • **.

    Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning 530004, China

摘要

    摘要
  • 摘要: 基于改进Reddy型3阶剪切变形理论(third-order shear deformation theory, TSDT)假设, 考虑碳纳米管(carbon nanotubes, CNTs)转向及功能梯度材料的不均匀性, 建立弹性地基上功能梯度碳纳米管增强复合材料(functionally graded carbon nanotube-reinforced composite, FG-CNTRC)板的线性弯曲和自由振动无网格分析模型. 利用改进Reddy型TSDT推导FG-CNTRC板的势能和动能, 给出弹性地基势能的表达式, 再将其分别进行叠加, 通过最小势能原理及Hamilton原理推导出弹性地基上FG-CNTRC板的线性弯曲和自由振动控制方程. 采用稳定移动克里金插值(stabilized moving Kriging interpolation, SMKI)对问题域内的节点进行离散, 该近似形函数的构造方法满足克罗内克条件, 可以直接施加边界条件. 文中首先给出基于三阶剪切变形理论的弹性地基FG-CNTRC板线性弯曲与自由振动无网格离散模型. 随后通过基准算例, 研究本文方法的有效性及精度问题. 最后数值分析了CNTs的分布形式、转向角、体积分数、地基系数、宽厚比和边界条件等对FG-CNTRC板的线性弯曲及自振频率的影响. 研究表明: 采用本文方法计算FG-CNTRC薄板、中厚板、甚至厚板的线性弯曲和自振频率均具有较高的计算精度; 随着CNTs体积分数和地基系数的增加, FG-CNTRC板结构刚度逐渐增大; FG-CTRC板结构刚度与宽厚比成正相关, 厚度增加的剪切效应会让CNTs转向角对结构刚度的影响逐渐降低.
    Abstract: Based on the improved Reddy type third-order shear deformation theory (TSDT), and considering the orientation of carbon nanotubes (CNTs) and the inhomogeneity of functionally gradient materials, a meshless analysis model for the linear bending and free vibration of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates on elastic foundation is established. The potential energy and kinetic energy of FG-CNTRC plate are derived by the improved Reddy type TSDT, and then the expression of the potential energy of elastic foundation is given, and then they are respectively superposed. The linear bending and free vibration control equations of FG-CNTRC plate on elastic foundation are derived by the principle of minimum potential energy and Hamilton principle. Stable moving Kriging interpolation (SMKI) is used to discretize the nodes in the problem domain. The construction method of the approximate shape function satisfies the Kronecker condition and can directly apply the boundary conditions. In this paper, a meshless discrete model of linear bending and free vibration of FG-CNTRC plate on elastic foundation based on the third-order shear deformation theory is presented. Then, the effectiveness and accuracy of the proposed method are studied by a benchmark example. Finally, the effects of CNTs distribution, orientation angle, volume fraction, foundation coefficient, width thickness ratio and boundary conditions on the linear bending and natural frequency of FG-CNTRC plate are numerically analyzed. The results show that the proposed method has a good accuracy in calculating the linear bending and natural frequencies of FG-CNTRC thin, medium-thick, and even thick plates. As the volume fraction of CNTs and the foundation coefficient increase, the stiffness of the FG-CNTRC plate structure gradually increases. The stiffness of the FG-CTRC plate structure is positively correlated with the width-thickness ratio, and the shear effect of increasing thickness gradually reduces the influence of the CNTs orientation angle on the plate stiffness.
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  • 引言

    自1991年日本电镜学家Iijima[1]发现碳纳米管(carbon nanotubes, CNTs)以来, 因其超高强度、低密度等特性而引起了研究人员的广泛关注. 传统的纤维增强复合材料通常是将具有微尺度的高强纤维嵌入不同的基体制造. CNTs的发现可将增强体改为纳米级纤维来增强复合材料的性能. 因此, CNTs被认为是一种非常有潜力的增强材料和多功能元件, 并用于制造了新一代聚合物复合材料, 即碳纳米管增强复合材料(carbon nanotube reinforced composites, CNTRCs)[2-3]. 研究表明, 通过添加少量CNTs可以有效增强复合材料的电−热及力学性能[4-5].

    功能梯度材料(functionally graded materials, FGMs)组分的空间变化使其性能连续变化, 可有效缓解应力集中等问题. FGM凭借其优越的力学性能, 广泛应用于海洋结构、汽车和航空航天工业等领域. 受FGM概念的启发, Shen[6]于2009年考虑CNTs的尺寸效应及温度依赖性, 并引入CNTs的效能参数提出了功能梯度碳纳米管增强复合材料(functionally graded carbon nanotube-reinforced composite, FG-CNTRC). 此后, 众多学者进行了大量的理论和数值计算研究, 探索FG-CNTRC板的静态弯曲[7-10]、自由振动[11-15]和动态瞬态[16-17]、屈曲及后屈曲[18-21]等力学行为.

    从文献调查来看, FG-CNTRC板力学分析目前集中在3个板理论上: 经典板理论(classical plate theory, CPT)、一阶剪切变形理论(first-order shear deformation theory, FSDT)和三阶剪切变形理论(third-order shear deformation theory, TSDT). CPT忽略了横向剪切应力, 仅对薄板分析有效且需满足C1连续性. 基于Reissner-Mindlin建立的FSDT可用于分析中厚板, 该理论考虑了横向剪切效应, 实现简单, 只需满足C0连续性即可. 然而FSDT用于分析薄板时, 容易出现剪切锁死现象, 导致计算结果不准确. 通常需要采取特殊处理来克服这一现象, 但需额外的计算成本. 此外, FSDT不能准确描述板的横向剪切应变和应力, 并违反了板的底部和顶部横向剪应力为零的原则. 因此, 需要引进人工剪切修正因子(shear correction factor, SCF)来纠正. 为了克服这一缺点, TSDT便应运而生. TSDT可用于分析薄板、中厚板, 甚至厚板, 但TSDT的位移场列式中通常包含纵向位移的一阶导数, 使其同CPT一样需要满足C1连续性.

    无网格伽辽金法作为近20年来迅速发展的数值算法, 摆脱了网格类数值算法对高质量网格的依赖性, 只需对问题域进行节点离散. 此类算法易实现自适应分析, 且在求解超大变形、裂纹扩展及高速冲击等问题具有一定的优势[22]. 采用FSDT时, 只需提高基函数阶次, 则可避免剪切锁死现象, 但当采用CPT和TSDT时, 处理第二类边界条件存在固有缺陷, 无法像有限元那样通过简单的Hermite插值来实现. 2006年, 王东东等[23]通过构造类似的Hermite型单元来进行施加, 研究表明该方法有损计算精度. 2013年, Nguyen等[24]提出流函数列式(stream function formulation)技术, 需要在边界节点处内置对应的临近节点来施加第二类边界条件. 该方法使得问题域内的节点数迅速增加, 增大了计算量. 为避免基于TSDT的无网格法中第二类边界条件的施加, Selim等[17, 25]于2016年对TSDT做了改进, 假设3阶项为一个独立变量, 即ψx = φx + ∂w0/∂x, ψy = φy + ∂w0/∂y而提出的TSDT, 并结合无网格将其成功应用于求解FG-CNTRC板的自由振动及受迫振动问题中.

    对弹性地基FG-CNTRC板的研究有: 2015年, Zhang等[26-27]基于FSDT理论框架下, 采用改进的移动最小二乘−瑞兹法(improved moving least squares-ritz, IMLS-RITZ)研究了Winkler地基上FG-CNTRC的自由振动与屈曲问题. 同年, Wattanasakuplong等[28]基于TSDT和正弦剪切变形理论(sinusoidal shear deformation theory, SSDT), 采用Navier解答给出了Pasternak地基上四边简支FG-CNTRC板线性弯曲、自由振动及屈曲的解析解. 2016年, Shams 等[29]利用FSDT和再生核粒子法(reproducing kernel particle method, RKPM)分析了Pasternak地基FG-CNTRC板的屈曲问题. 2017年Nguyen等[30]采用Navier解答给出了FSDT下Pasternak地基上四边简支FG-CNTRC板线性弯曲和自由振动的解析解. 2019年, Keleshteri等[31]运用广义微分求积法(generalized differential quadrature method, GDQM)和TSDT, 研究了Pasternak地基上变厚度FG-CNTRC环形板的非线性弯曲问题. 同年, Adhikari 等[32]采用高阶准三维理论(high-order quasi-3D theory)和有限元法(finite element method, FEM)分析了Pasternak地基上FG-CNTRC板的振动响应问题.

    综上可知, 目前研究FG-CNTRC力学性能的文献大部分是基于FSDT, 采用TSDT的较少, 考虑CNTs转向问题较为罕见. 基于此, 本文采用稳定移动克里金插值, 结合Selim等[25]改进Redddy型的TSDT推导了含碳纳米管转向的弹性地基上FG-CNTRC板的线性弯曲与自由振动无网格控制方程, 研究成果未见相关报道. 此外, 文末还讨论了CNTs体积分数、转向角、分布形式和地基系数等对FG-CNTRC板线性弯曲和自由振动的影响.

    1.   稳定移动克里金插值

    本节简要介绍SMKI形函数插值技术, 详细描述SMKI方法及数学特性可参考文献[33-34].

    假设点x的子域Ωx中的MK插值uh(x)是线性回归模型和偏差的组合, 可定义为

    $$ {u^h}({\boldsymbol{x}}) = \sum\limits_{i = 1}^m {{p_i}} ({\boldsymbol{x}}){a_i} + z({\boldsymbol{x}}) = {{\boldsymbol{p}}^{\text{T}}}({\boldsymbol{x}}){\boldsymbol{a}} + z({\boldsymbol{x}}) $$ (1)

    式中, pi(x)是单项基函数, ai(x)为相应的系数, m为基函数的个数. 常采用的多项式基函数在移动克里金无网格离散节点间距较小时, 会使得插值变得不稳定, 为了消除由于节点间距的影响, 归一化多项式基函数P(x)被Tu等[34]于2019年提出

    $$\begin{split} & {{\boldsymbol{P}}^{\text{T}}}({\boldsymbol{x}}) = \left[ 1\quad {\frac{{x - {x_e}}}{{{d_m}}}}\quad {\frac{{y - {y_e}}}{{{d_m}}}}\quad {\frac{{{{\left( {x - {x_e}} \right)}^2}}}{{{d_m}^2}}} \right. \\ &\qquad \left. {\frac{{\left( {x - {x_e}} \right)\left( {y - {y_e}} \right)}}{{{d_m}^2}}}\quad {\frac{{{{\left( {y - {y_e}} \right)}^2}}}{{{d_m}^2}}} \right] \end{split} $$ (2)

    式中, xe或(xe, ye)是在x的支撑域Ωx内任意点xe的坐标, dm是影响域Ωx的半径

    $$ {d_m} = \alpha {d_{{\rm{ave}}}} $$ (3)

    式中, dave表示平均节点间距, α表示比例因子.

    式(1)中z(x)表示点x的随机偏差, 均值为0, 方差σ2, z(xi)和z(xj)之间的协方差表示为

    $$ {\text{Cov}}\left\{ {z\left( {{{\boldsymbol{x}}_i}} \right),z\left( {{{\boldsymbol{x}}_j}} \right)} \right\} = {\sigma ^2}{\boldsymbol{R}}\left[ {R\left( {{{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}} \right)} \right] $$ (4)

    式中, R[R(xi, xj)] 是相关矩阵, R(xi, xj) 是位于Ωx的任意两个节点(xi 和 xj)之间的相关函数. 本文采用常用高斯函数

    $$ R\left( {{{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}} \right) = \exp \left( { - {\theta ^2}\frac{{r_{ij}^2}}{{a_0^2}}} \right) $$ (5)

    式中, a0表示支持域Ωx中一对节点之间的最大距离, θ = 2.0.

    基于最佳线性无偏预测(best linear unbiased prediction, BLUP), 式(1)可改写为

    $$ {u^h}({\boldsymbol{x}}) = {\boldsymbol{N}}\left( {\boldsymbol{x}} \right){\boldsymbol{u}} = \left( {{{\boldsymbol{P}}^{\text{T}}}\left( {\boldsymbol{x}} \right){\boldsymbol{A}} + {{\boldsymbol{R}}^{\text{T}}}\left( {\boldsymbol{x}} \right){\boldsymbol{B}}} \right){\boldsymbol{u}} $$ (6)

    式中, N = [N(x, x1), N(x, x2), ···, N(x, xn)]是形函数矩阵, uT = [u1, u2, ···, un] 表示节点xi处的节点值. A和B 表示如下

    $$ {\boldsymbol{A}} = {\left( {{\boldsymbol{P}}_m^{\text{T}}{\boldsymbol{R}}_m^{ - 1}{{\boldsymbol{P}}_m}} \right)^{ - 1}}{\boldsymbol{P}}_m^{\text{T}}{\boldsymbol{R}}_m^{ - 1},{\boldsymbol{B}} = {\boldsymbol{R}}_m^{ - 1}\left( {{\boldsymbol{I}} - {{\boldsymbol{P}}_m}{\boldsymbol{A}}} \right) $$ (7)

    其中

    $$ {{\boldsymbol{R}}_m} = \left[ {\begin{array}{*{20}{c}} {R\left( {{{\boldsymbol{x}}_1},{{\boldsymbol{x}}_1}} \right)}&{R\left( {{{\boldsymbol{x}}_1},{{\boldsymbol{x}}_2}} \right)}& \cdots &{R\left( {{{\boldsymbol{x}}_1},{{\boldsymbol{x}}_n}} \right)} \\ {R\left( {{{\boldsymbol{x}}_2},{{\boldsymbol{x}}_1}} \right)}&{R\left( {{{\boldsymbol{x}}_2},{{\boldsymbol{x}}_2}} \right)}& \cdots &{R\left( {{{\boldsymbol{x}}_2},{{\boldsymbol{x}}_n}} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {R\left( {{{\boldsymbol{x}}_n},{{\boldsymbol{x}}_1}} \right)}&{R\left( {{{\boldsymbol{x}}_n},{{\boldsymbol{x}}_2}} \right)}& \cdots &{R\left( {{{\boldsymbol{x}}_n},{{\boldsymbol{x}}_n}} \right)} \end{array}} \right] $$
    $$ {{\boldsymbol{R}}^{\text{T}}} = \left[ {\begin{array}{*{20}{c}} {R\left( {{{\boldsymbol{x}}_1},{{\boldsymbol{x}}_0}} \right)}&{R\left( {{{\boldsymbol{x}}_2},{{\boldsymbol{x}}_0}} \right)}& \cdots &{R\left( {{{\boldsymbol{x}}_n},{{\boldsymbol{x}}_0}} \right)} \end{array}} \right] $$
    $$ {\boldsymbol{P}}_m^{\text{T}} = \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{p}}\left( {{{\boldsymbol{x}}_1}} \right)}&{{\boldsymbol{p}}\left( {{{\boldsymbol{x}}_2}} \right)}& \cdots &{{\boldsymbol{p}}\left( {{{\boldsymbol{x}}_n}} \right)} \end{array}} \right] $$
    $$ {\boldsymbol{p}} = {\boldsymbol{p}}\left( {{{\boldsymbol{x}}_0}} \right) $$

    稳定移动克里金形函数具有Kronecker delta特性, 表达式如下

    $$ \begin{split} &N({{\boldsymbol{x}}_i},{{\boldsymbol{x}}_j}) = {\delta _{ij}} = \left\{ \begin{array}{*{20}{l}} {1,\quad i = j} \\ {0,\quad i \ne j} \end{array}{\text{ }}\right.\\ &\qquad (i,j = 1,2, \cdots ,n)\end{split} $$ (8)

    2.   数学模型

    2.1   弹性地基FG-CNTRC 板

    如图1所示, FG-CNTRC板的长、宽和高分别为a, b和h. 以板中面建立x-y-z坐标系, 其中z为沿着厚度方向的坐标. 假设板由单壁碳纳米管(single-walled carbon nanotubes, SWCNTs)和基体混合而成, 碳纳米管在厚度方向的分布形式具有UD, FG-V, FG-O和FG-X 四种, 相应的表达式如下[8]

    图  1  弹性地基上FG-CNTRC板的等效模型
    Figure  1.  Equivalent model of FG-CNTRC plate on elastic foundation
    下载: 全尺寸图片 幻灯片
    $$ {V_{{\text{CNT}}}} = \left\{ \begin{split} & {V_{{\text{CNT}}}^*}\quad {({\text{UD}})} \\ & {\left( {1 + \frac{{2z}}{h}} \right)V_{{\text{CNT}}}^*}\quad{({\text{FG}} - {\text{V}})} \\ & {2\left( {1 - \frac{{2|z|}}{h}} \right)V_{{\text{CNT}}}^*}\quad{({\text{FG}} - {\text{O}})} \\ & {2\left( {\frac{{2|z|}}{h}} \right)V_{{\text{CNT}}}^*}\quad{({\text{FG}} - {\text{X}})} \end{split} \right. $$ (9)

    其中

    $$ V_{{\text{CNT}}}^* = \frac{{{w_{{\text{CNT}}}}}}{{{w_{{\text{CNT}}}} + \left( {{\rho ^{{\text{CNT}}}}/{\rho ^{\text{m}}}} \right)\left( {1 - {w_{{\text{CNT}}}}} \right)}} $$ (10)

    式中, wCNT为碳纳米管的质量分数, ρCNT和ρm分别为碳纳米管和基体的密度, 4种分布形式的CNTRC具有相同的质量分数和体积分数.

    为了在复合材料物理模型参数中能够计入尺寸、界面及应变梯度效应等影响, Shen等[6]于2009年考虑了碳纳米管的尺寸及温度依赖性, 引入碳纳米管的效能参数而提出了广义混合律模型. 根据广义混合律模型, CNTRC板的等效杨氏模量和剪切模量可表示为

    $$ {E_{11}} = {\eta _1}{V_{{\text{CNT}}}}E_{11}^{{\text{CNT}}} + {V_{\text{m}}}{E^{\text{m}}} $$ (11)
    $$ \frac{{{\eta _2}}}{{{E_{22}}}} = \frac{{{V_{{\text{CNT}}}}}}{{E_{22}^{{\text{CNT}}}}} + \frac{{{V_{\text{m}}}}}{{{E^{\text{m}}}}} $$ (12)
    $$ \frac{{{\eta _3}}}{{{G_{12}}}} = \frac{{{V_{{\text{CNT}}}}}}{{G_{12}^{{\text{CNT}}}}} + \frac{{{V_{\text{m}}}}}{{{G^{\text{m}}}}} $$ (13)

    式中, ${{E}}_{\text{11}}^{\text{CNT}}$, ${{E}}_{\text{22}}^{\text{CNT}}$和${{G}}_{\text{11}}^{\text{CNT}}$分别为碳纳米管的杨氏模量和剪切模量; Em和Gm分别为基体的杨氏模量和剪切模量; VCNT和Vm分别为碳纳米管和基体的体积分数, 且VCNT + Vm = 1; ηj (j = 1, 2, 3)为碳纳米管的效能参数; FG-CNTRC泊松比和质量密度定义为

    $$ {v_{12}} = V_{{\text{CNT}}}^*v_{12}^{{\text{CNT}}} + {V_{\text{m}}}{v^{\text{m}}} $$ (14)
    $$ \rho = {V_{{\text{CNT}}}}{\rho ^{{\text{CNT}}}} + {V_{\text{m}}}{\rho ^{\text{m}}} $$ (15)

    式中, ν12和ρ分别为FG-CNTRC的等效泊松比和等效密度; $ {\nu }_{\text{12}}^{\text{CNT}} $和νm分别为碳纳米管和基体的泊松比; ρCNT, ρm分别为碳纳米管和基体的质量密度.

    Pasternak地基模型[35]为

    $$ {q_e}(x,y) = {K_w}w(x,y) - {K_s}{\nabla ^2}w(x,y) $$ (16)

    式中, Kw为弹性地基系数; Ks为地基剪切模量; $\nabla $2为拉普拉斯算子; 当Ks = 0时, Pasternak地基模型则简化为Winkler模型.

    2.2   总能量泛函

    根据改进Reddy型的3阶剪切变形理论[17, 25], 板的位移场U = (u, v, w)T可表示为

    $$ \left. \begin{split} & {u(x,y,z) = {u_0}(x,y) + z{\varphi _x}(x,y) + c{z^3}{\psi _x}(x,y)} \\ & {v(x,y,z) = {v_0}(x,y) + z{\varphi _y}(x,y) + c{z^3}{\psi _y}(x,y)} \\ & {w(x,y,z) = {w_0}(x,y)} \end{split} \right\} $$ (17)

    式中, (u0, v0, w0)T为板中面任意一点在x, y, z方向的位移, φx和φy分别为绕y轴和x轴的转动; ψx和ψy分别为φx + ∂w0/∂x, φy + ∂w0/∂y; c = −4h2/3.

    根据几何方程, 板的面内及剪切应变分别为

    $$ {\boldsymbol{\varepsilon }} = {\left\{ {\begin{array}{*{20}{c}} {{\varepsilon _x}}&{{\varepsilon _y}}&{{\gamma _{xy}}} \end{array}} \right\}^{\text{T}}} = {{\boldsymbol{\varepsilon }}_0} + z{{\boldsymbol{\kappa }}_1} + {z^3}{{\boldsymbol{\kappa }}_2} $$ (18)
    $$ {\boldsymbol{\gamma }} = {\left\{ {\begin{array}{*{20}{c}} {{\gamma _{yz}}}&{{\gamma _{xz}}} \end{array}} \right\}^{\text{T}}} = {{\boldsymbol{\varepsilon }}_s} + {z^2}{{\boldsymbol{\kappa }}_s} $$ (19)

    其中

    $$ {{\boldsymbol{\varepsilon }}_0} = {\left\{ {\begin{array}{*{20}{c}} {{u_{0,x}}}&{{v_{0,y}}}&{{u_{0,y}} + {v_{0,x}}} \end{array}} \right\}^{\text{T}}} $$
    $$ {{\boldsymbol{\kappa }}_1} = {\left\{ {\begin{array}{*{20}{c}} {{\varphi _{x,x}}}&{{\varphi _{y,y}}}&{{\varphi _{x,y}} + {\varphi _{y,x}}} \end{array}} \right\}^{\text{T}}} $$
    $$ {{\boldsymbol{\kappa }}_2} = c{\left\{ {\begin{array}{*{20}{c}} {{\psi _{x,x}}}&{{\psi _{y,y}}}&{{\psi _{x,y}} + {\psi _{y,x}}} \end{array}} \right\}^{\text{T}}} $$
    $$ {{\boldsymbol{\varepsilon }}_s} = {\left\{ {\begin{array}{*{20}{c}} {{\varphi _x} + {w_{0,x}}}&{{\varphi _y} + {w_{0,y}}} \end{array}} \right\}^{\text{T}}} $$
    $$ {{\boldsymbol{\kappa }}_s} = 3c{\left\{ {\begin{array}{*{20}{c}} {{\psi _x}}&{{\psi _y}} \end{array}} \right\}^{\text{T}}} $$

    正交各向异性材料的本构关系为[36]

    $$ \left\{ {\begin{array}{*{20}{c}} {{\sigma _x}} \\ {{\sigma _y}} \\ {{\tau _{xy}}} \\ {{\tau _{xz}}} \\ {{\tau _{yz}}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {{{\bar Q}_{11}}}&{{{\bar Q}_{12}}}&{{{\bar Q}_{16}}}&0&0 \\ {{{\bar Q}_{12}}}&{{{\bar Q}_{22}}}&{{{\bar Q}_{26}}}&0&0 \\ {{{\bar Q}_{16}}}&{{{\bar Q}_{26}}}&{{{\bar Q}_{66}}}&0&0 \\ 0&0&0&{{{\bar Q }_{55}}}&{{{\bar Q }_{45}}} \\ 0&0&0&{{{\bar Q }_{45}}}&{{{\bar Q }_{44}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\varepsilon _x}} \\ {{\varepsilon _y}} \\ {{\gamma _{xy}}} \\ {{\gamma _{xz}}} \\ {{\gamma _{yz}}} \end{array}} \right\} $$ (20)

    式中, ${\bar{{Q}}}_{{ij}}$是全局x, y坐标系中弹性矩阵的分量, 可通过以下坐标转换方程变换得到[36]

    $$ \bar {\boldsymbol{Q}} = {\boldsymbol{T}}{\boldsymbol{Q}}{{\boldsymbol{T}}^{\text{T}}} $$ (21)

    其中

    $$ {\boldsymbol{T}} = \left[ {\begin{array}{*{20}{c}} {{m^2}}&{{n^2}}&{ - 2mn}&0&0 \\ {{n^2}}&{{m^2}}&{2mn}&0&0 \\ {mn}&{ - mn}&{{m^2} - {n^2}}&0&0 \\ 0&0&0&m&{ - n} \\ 0&0&0&n&m \end{array}} \right] $$
    $$ {\boldsymbol{Q}} = \left[ {\begin{array}{*{20}{c}} {{Q_{11}}(z)}&{{Q_{12}}(z)}&0&0&0 \\ {{Q_{12}}(z)}&{{Q_{22}}(z)}&0&0&0 \\ 0&0&{{Q_{66}}(z)}&0&0 \\ 0&0&0&{{Q_{55}}(z)}&0 \\ 0&0&0&0&{{Q_{44}}(z)} \end{array}} \right] $$

    式中, m和n分别为cosθ和sinθ; θ为材料主方向和全局坐标系的夹角, 也可称为碳纳米管的旋转角度; Qij为碳纳米管横向分布时FG-CNTRC物理关系矩阵, 定义如下[8]

    $$\left. \begin{split} &{Q}_{11} = \frac{{E}_{11}}{1-{\nu }_{12}{\nu }_{21}}\text{, }{Q}_{22} = \frac{{E}_{22}}{1-{\nu }_{12}{\nu }_{21}}\text{, }{Q}_{12} = \frac{{\nu }_{21}{E}_{11}}{1-{\nu }_{12}{\nu }_{21}}\\ &{Q}_{66} = {G}_{12},{Q}_{44} = {G}_{23},{Q}_{55} = {G}_{13}\text{, }{\nu }_{21} = \frac{{E}_{22}}{{E}_{11}}{\nu }_{12}\end{split}\right\} $$ (22)

    式中, E11和E22分别为1, 2方向上的杨氏模量; G12, G31和G23分别为1-2, 1-3和2-3平面的剪切模量; νij为应力在i方向作用时j方向的横向应变的泊松比.

    板的内力和应变关系为

    $$ \left\{ {\begin{array}{*{20}{c}} {\boldsymbol{N}} \\ {\boldsymbol{M}} \\ {\boldsymbol{P}} \\ {{{\boldsymbol{Q}}^{{s}}}} \\ {{{\boldsymbol{R}}^{{s}}}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {\boldsymbol{A}}&{\boldsymbol{B}}&{\boldsymbol{E}}&{\boldsymbol{0}}&{\boldsymbol{0}} \\ {\boldsymbol{B}}&{\boldsymbol{D}}&{\boldsymbol{F}}&{\boldsymbol{0}}&{\boldsymbol{0}} \\ {\boldsymbol{E}}&{\boldsymbol{F}}&{\boldsymbol{H}}&{\boldsymbol{0}}&{\boldsymbol{0}} \\ {\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{{{\boldsymbol{A}}^{{s}}}}&{{{\boldsymbol{D}}^{{s}}}} \\ {\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{{{\boldsymbol{D}}^{{s}}}}&{{{\boldsymbol{F}}^{{s}}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{l}} {{{\boldsymbol{\varepsilon }}_0}} \\ {{{\boldsymbol{\kappa }}_1}} \\ {{{\boldsymbol{\kappa }}_2}} \\ {{{\boldsymbol{\varepsilon }}_s}} \\ {{{\boldsymbol{\kappa }}_s}} \end{array}} \right\} = {\boldsymbol{S\varepsilon }} $$ (23)

    其中

    $$ {\boldsymbol{N}} = {\left\{ {\begin{array}{*{20}{c}} {{N_x}} & {{N_y}} & {{N_{xy}}} \end{array}} \right\}^{\text{T}}} = {\int_{ - h/2}^{h/2} {\left\{ {\begin{array}{*{20}{c}} {{\sigma _x}} & {{\sigma _y}} & {{\tau _{xy}}} \end{array}} \right\}} ^{\text{T}}}{\text{d}}z $$
    $$ {\boldsymbol{M}} = {\left\{ {\begin{array}{*{20}{c}} {{M_x}} & {{M_y}} & {{M_{xy}}} \end{array}} \right\}^{\text{T}}} = \int_{ - h/2}^{h/2} {{{\left\{ {\begin{array}{*{20}{c}} {{\sigma _x}} & {{\sigma _y}} & {{\tau _{xy}}} \end{array}} \right\}}^{\text{T}}}} z{\text{d}}z $$
    $$ {\boldsymbol{P}} = {\left\{ {\begin{array}{*{20}{c}} {{P_x}} & {{P_y}} & {{P_x}} \end{array}} \right\}^{\text{T}}} = {\int_{ - h/2}^{h/2} {\left\{ {\begin{array}{*{20}{c}} {{\sigma _x}}&{{\sigma _y}}&{{\tau _{xy}}} \end{array}} \right\}} ^{\text{T}}}{z^2}{\text{d}}z $$
    $$ {{\boldsymbol{Q}}^s} = {\left\{ {\begin{array}{*{20}{c}} {Q_x^s}&{Q_y^s} \end{array}} \right\}^{\text{T}}} = {\int_{ - h/2}^{h/2} {\left\{ {\begin{array}{*{20}{c}} {{\tau _{xz}}}&{{\tau _{yz}}} \end{array}} \right\}} ^{\text{T}}}{\text{d}}z $$
    $$ {{{\boldsymbol R}}^{{s}}} = {\left\{ {\begin{array}{*{20}{c}} {R_x^s}&{R_y^s} \end{array}} \right\}^{\text{T}}} = \int_{ - h/2}^{h/2} {{{\left\{ {\begin{array}{*{20}{c}} {{\tau _{xz}}}&{{\tau _{yz}}} \end{array}} \right\}}^{\text{T}}}} {z^2}{\text{d}}z $$

    将式(20)代入上式中, 则有

    $$ \begin{split} & \left( {{A_{ij}},{B_{ij}},{D_{ij}},{E_{ij}},{F_{ij}},{H_{ij}}} \right)= \\ &\qquad \int_{ - h/2}^{h/2} {{{\bar Q}_{ij}}} \left( {1,z,{z^2},{z^3},{z^4},{z^6}} \right){\text{d}}z,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i,j = 1,2,6 \end{split} $$ (24)
    $$ \left( {A_{ij}^s,D_{ij}^s,F_{ij}^s} \right) = \int_{ - h/2}^{h/2} {{{\bar Q}_{ij}}} \left( {1,{z^2},{z^4}} \right){\text{d}}z,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i,j = 4,5 $$ (25)

    FG-CNTRC板的应变能为

    $$ {U_p} = \frac{1}{2}\int_\varOmega {{{\boldsymbol{\varepsilon}} ^{\rm{T}}}} {\boldsymbol{S\varepsilon }}{\text{d}}\varOmega $$ (26)

    由式(16), 可得Pasternak地基势能的一阶变分为

    $$ \begin{split} & \delta {U_e} = \int_\varOmega {{q_e}\delta {w_p}{\text{d}}\varOmega } = \\ & \qquad \int_\varOmega {\left[ {{K_w}{w_p} - {K_s}\left( {\frac{{{\partial ^2}{w_p}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{w_p}}}{{\partial {y^2}}}} \right)} \right]\delta {w_p}} {\text{d}}\varOmega \end{split} $$ (27)

    Pasternak地基上FG-CNTRC板受横向载荷q(x, y)的总势能泛函为

    $$ {\varPi _u} = {U_p} + {U_e} - \int_\varOmega{q(x,y){w_p}{\text{d}}} \varOmega$$ (28)

    FG-CNTRC板的动能为

    $$ T = \frac{1}{2}\int_\varOmega {\int_{ - h/2}^{h/2} \rho } (z)\left( {{{\dot u}^2} + {{\dot v}^2} + {{\dot w}^2}} \right){\text{d}}z{\text{d}}\varOmega $$ (29)

    将式(26)、式(27)和式(29)进行叠加, 则得Pasternak地基上FG-CNTRC板自由振动时的总势能泛函为

    $$ {\varPi _s} = {U_p} + {U_e} - T $$ (30)

    2.3   控制方程

    对FG-CNTRC板的节点位移及转角进行离散, 利用式(6)近似得到

    $$ {\boldsymbol{u}}_0^h = \left( {\begin{array}{*{20}{c}} {u_0^h} \\ {v_0^h} \\ {w_0^h} \\ {\phi _x^h} \\ {\phi _y^h} \\ {\psi _x^h} \\ {\psi _y^h} \end{array}} \right) = \sum\limits_{I = 1}^n {{N_I}} (x){u_I} = \sum\limits_{I = 1}^n {{N_I}} (x)\left( {\begin{array}{*{20}{c}} {{u_l}} \\ {{v_I}} \\ {{w_I}} \\ {{\phi _{xI}}} \\ {{\phi _{yI}}} \\ {{\psi _{xI}}} \\ {{\psi _{yl}}} \end{array}} \right) $$ (31)

    将式(31)代入式(28), 根据最小势能原理, 由泛函的变分为0, 则可得到Pasternak地基上FG-CNTRC板线性弯曲的控制方程为

    $$ {\boldsymbol{KU}} = {\boldsymbol{F}} $$ (32)

    将式(31)代入式(30), 根据Hamilton原理: $\delta \displaystyle\int_{{t_1}}^{{t_2}} {\left( {{U_p} + {U_e} - T} \right)} {\rm{d}}t = 0$可得到Pasternak地基上FG-CNTRC板自由振动的控制方程为

    $$ {\boldsymbol{K}}{\boldsymbol{U}} + {\boldsymbol{M}}\ddot {\boldsymbol{U}} = {\boldsymbol{0}} $$ (33)

    相应的特征方程为

    $$ \left( {{\boldsymbol{K}} - {\lambda ^2}{\boldsymbol{M}}} \right){\boldsymbol{U}} = {{{\boldsymbol{0}}}} $$ (34)

    其中

    $$ {\boldsymbol{{ K}}} = \int_\varOmega {\left( {\left\{ {\begin{array}{*{20}{l}} {{\boldsymbol{B}}_m^{\rm{T}}} \\ {{\boldsymbol{B}}_{b1}^{\rm{T}}} \\ {{\boldsymbol{B}}_{b2}^{\rm{T}}} \\ {{\boldsymbol{B}}_{s1}^{\rm{T}}} \\ {{\boldsymbol{B}}_{s2}^{\rm{T}}} \end{array}} \right\}\left[ {\begin{array}{*{20}{c}} {\boldsymbol{A}}&{\boldsymbol{B}}&{\boldsymbol{E}}&{\boldsymbol{0}}&{\boldsymbol{0}} \\ {\boldsymbol{B}}&{\boldsymbol{D}}&{\boldsymbol{F}}&{\boldsymbol{0}}&{\boldsymbol{0}} \\ {\boldsymbol{E}}&{\boldsymbol{F}}&{\boldsymbol{H}}&{\boldsymbol{0}}&{\boldsymbol{0}} \\ {\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{{{\boldsymbol{A}}_s}}&{{{\boldsymbol{D}}_s}} \\ {\boldsymbol{0}}&{\boldsymbol{0}}&{\boldsymbol{0}}&{{{\boldsymbol{D}}_s}}&{{{\boldsymbol{F}}_s}} \end{array}} \right]\left\{ {\begin{array}{*{20}{l}} {{{\boldsymbol{B}}_m}} \\ {{{\boldsymbol{B}}_{b1}}} \\ {{{\boldsymbol{B}}_{b2}}} \\ {{{\boldsymbol{B}}_{s1}}} \\ {{{\boldsymbol{B}}_{s2}}} \end{array}} \right\}} \right)} {\text{d}}\varOmega $$
    $$ {\boldsymbol{M}} = \int_\varOmega {{\boldsymbol{G}}_I^{\rm{T}}{\boldsymbol{m}}{{\boldsymbol{G}}_J}} {\text{d}}\varOmega $$
    $$ {\boldsymbol{U}} = {\left[ {\begin{array}{*{20}{l}} {{u_I}}&{{v_I}}&{{w_I}}&{{\phi _{xI}}}&{{\phi _{yI}}}&{{\psi _{xI}}}&{{\psi _{yI}}} \end{array}} \right]^{\text{T}}} $$
    $$ {\boldsymbol{F}} = {\int_\varOmega {q(x,y)[0,0,{N_I},0,0,0,0]} ^{\text{T}}}{\text{d}}\varOmega $$

    式中

    $$ {{\boldsymbol{B}}_m} = \left[ {\begin{array}{*{20}{c}} {{N_{I,x}}}&0&0&0&0&0&0 \\ 0&{{N_{I,y}}}&0&0&0&0&0 \\ {{N_{I,y}}}&{{N_{I,x}}}&0&0&0&0&0 \end{array}} \right] $$
    $$ {{\boldsymbol{B}}_{b1}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&{{N_{I,x}}}&0&0&0 \\ 0&0&0&0&{{N_{I,y}}}&0&0 \\ 0&0&0&{{N_{I,y}}}&{{N_{I,x}}}&0&0 \end{array}} \right] $$
    $$ {{\boldsymbol{B}}_{b2}} = c\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&{{N_{I,x}}}&0 \\ 0&0&0&0&0&0&{{N_{I,y}}} \\ 0&0&0&0&0&{{N_{I,y}}}&{{N_{I,x}}} \end{array}} \right] $$
    $$ {{\boldsymbol{B}}_{s1}} = \left[ {\begin{array}{*{20}{c}} 0&0&{{N_{I,x}}}&{{N_I}}&0&0&0 \\ 0&0&{{N_{I,y}}}&0&{{N_I}}&0&0 \end{array}} \right] $$
    $$ {{\boldsymbol{B}}_{s2}} = 3c\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&{{N_I}}&0 \\ 0&0&0&0&0&0&{{N_I}} \end{array}} \right] $$
    $$ {{\boldsymbol{G}}_I} = \left[ {\begin{array}{*{20}{c}} {{N_I}}&0&0&0&0&0&0 \\ 0&{{N_I}}&0&0&0&0&0 \\ 0&0&{{N_I}}&0&0&0&0 \\ 0&0&0&{{N_I}}&0&0&0 \\ 0&0&0&0&{{N_I}}&0&0 \\ 0&0&0&0&0&{{N_I}}&0 \\ 0&0&0&0&0&0&{{N_I}} \end{array}} \right] $$
    $$ {\boldsymbol{m}} = \left[ {\begin{array}{*{20}{c}} {{I_0}}&0&0&{{I_1}}&0&{c{I_3}}&0 \\ 0&{{I_0}}&0&0&{{I_1}}&0&{c{I_3}} \\ 0&0&{{I_0}}&0&0&0&0 \\ {{I_1}}&0&0&{{I_2}}&0&{c{I_4}}&0 \\ 0&{{I_1}}&0&0&{{I_2}}&0&{c{I_4}} \\ {c{I_3}}&0&0&{c{I_4}}&0&{{c^2}{I_6}}&0 \\ 0&{c{I_3}}&0&0&{c{I_4}}&0&{{c^2}{I_6}} \end{array}} \right] $$
    $$ {I_i} = \int_{ - h/2}^{h/2} \rho \times {z^i}{\text{d}}z,\quad i = 0,1,2,3,4,6$$

    2.4   边界条件

    本文考虑板结构常见的2种边界条件, 以x = ± a/2边界为例, 则有:

    (1)固支边界(clamped, C): u = v = w = φx = φy = ψx = ψy = 0;

    (2)简支边界(simply, S): v = w = φy = ψy = 0.

    3.   数值结果与讨论

    本节首先以FG-CNTRC板的线性弯曲及自由振动为例分析SMKI的收敛性及有效性, 接着运用SMKI对含碳纳米管转向的弹性地基上FG-CNTRC板线性弯曲和自由振动进行数值分析和讨论.

    文中所有算例采用均布节点, 方形影响域dm = 4.0, 高斯点为4 × 4. SWCNTs为增强体材料, 基体材料为聚酰亚胺薄膜[37](poly-cp-vinylene, PmPV), 根据分子动力学模拟[38], 材料参数见表1. $ {{V}}_{\text{CNT}}^{\text{*}} $分别考虑0.11, 0.14和0.17这3种体积分数, 与其对应的效能参数ηi (i = 1, 2, 3)见表2. 如无特别说明, q = −0.1 MPa, a/b = 1, 采用如下无量纲参量

    表  1  材料参数
    Table  1.  The properties of material
    ParametersMatrixCNTs
    Poisson’s ratioνm = 0.34$ {\nu }_{\text{12}}^{\text{CNT}} $ = 0.175
    density/(kg·m−3)ρm = 1150ρCNT = 1400
    Young’s modulus/GPaEm = 2.1${{E} }_{\text{11} }^{\text{CNT} }$ = 5646.6,
    ${{E} }_{\text{22} }^{\text{CNT} }$ = 7080
    shear modulus/GPa—${{G} }_{\text{12} }^{\text{CNT} }$ = 1944.5
    下载: 导出CSV 
    | 显示表格
    表  2  CNTs的效能参数
    Table  2.  The efficiency parameters of CNTs
    ${{V} }_{\text{CNT} }^{\text{*} }$η1η2η3
    0.110.1490.9340.934
    0.140.1500.9410.941
    0.170.1491.3811.381
    下载: 导出CSV 
    | 显示表格
    $$ \bar w = \frac{{{w_0}}}{h}, {\bar \sigma _{xx}} = \frac{{{\sigma _{xx}}{h^2}}}{{|q|{a^2}}} , \bar z = \frac{z}{h},\bar \omega = \omega \frac{{{b^2}}}{h}\sqrt {\frac{{{\rho ^{\text{m}}}}}{{{E^{\text{m}}}}}} $$
    $$ {k_w} = \frac{{{K_w}{b^4}}}{{{D_{\text{m}}}}},{k_s} = \frac{{{K_{\text{s}}}{b^{\text{2}}}}}{{{D_{\text{m}}}}},{D_{\text{m}}} = \frac{{{E^{\text{m}}}{h^3}}}{{12[1 - {{({v^{\text{m}}})}^2}]}} $$

    3.1   收敛性及有效性分析

    图2计算了宽厚比b/h = 10, 体积分数为0.11, 四边简支FG-CNTRC板下中点挠度随离散点数的收敛情况. 由图2可知: 随着离散点数的增加, 本文计算结果与Phung等[16]采用IGA-HSDT的结果迅速逼近, 说明了本文方法具有较好的收敛性; 在节点数达到300左右时计算结果近乎完全收敛, 故本文后续算例采用19 × 19节点数进行离散.

    图  2  四边简支FG-CNTRC板中点挠度收敛性分析(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)
    Figure  2.  Convergence analysis of central deflection of simply supported FG-CNTRC plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)
    下载: 全尺寸图片 幻灯片

    表3计算了均布载荷作用下, 体积分数为0.11, 不同CNTs分布型下四边简支FG-CNTRC板中点无量纲挠度, 并将结果与Zhu等[8]采用FEM-FSDT的结果和Phung等[16]采用IGA-HSDT的结果进行对比. 可以看出本文方法与上述两者数值结果非常接近, 说明了基于TSDT的SMKI求解FG-CNTRC板线性弯曲的有效性及正确性.

    表  3  四边简支FG-CNTRC板中点无量纲挠度(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)
    Table  3.  Dimensionless central deflection of simply supported FG-CNTRC plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)
    b/hCNT typeFEM-FSDT[8]IGA-TSDT[16]Present
    10UD3.739 × 10−33.717 × 10−33.716 × 10−3
    FG-V4.466 × 10−34.427 × 10−34.447 × 10−3
    FG-O5.230 × 10−35.438 × 10−35.430 × 10−3
    FG-X3.177 × 10−33.102 × 10−33.140 × 10−3
    20UD3.628 × 10−33.624 × 10−33.629 × 10−3
    FG-V4.879 × 10−34.877 × 10−34.880 × 10−3
    FG-O6.155 × 10−36.248 × 10−36.243 × 10−3
    FG-X2.701 × 10−32.685 × 10−32.697 × 10−3
    50UD1.1551.1561.156
    FG-V1.6531.6541.652
    FG-O2.1572.1632.158
    FG-X0.7900.7910.792
    下载: 导出CSV 
    | 显示表格

    图3给出了宽厚比b/h = 50, 体积分数为0.17的FG-CNTRC板中点应力随厚度坐标变化的情况, 与Zhu等[8]文献中的图2 ~ 图4基本一致, 这里不再列出. 从图3中可以看出UD, FG-O和FG-X分布型的CNTRC板应力与z = 0成中心对称, 原因是此3种分布型CNTs呈对称性分布. 此外, 由于固支边界条件下的弯曲挠度较小导致四边固支FG-CNTRC板的应力比简支的小.

    图  3  不同边界条件下FG-CNTRC板中点无量纲轴向应力${\bar \sigma }_{xx}$(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.17, b/h = 50)
    Figure  3.  Normalized axial stress ${{\bar \sigma }}_{xx}$ of central point of FG-CNTRC plate under different boundary condition (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.17, b/h = 50)
    下载: 全尺寸图片 幻灯片

    表4展示了体积分数为0.11四边简支FG-CNTRC板的无量纲基础频率, 并与已有文献Zhu等[8]和Adhikari等[32]分别基于不同理论得到的结果进行参考比较, 其中SMKI-FSDT只需去掉式(17)的高阶项, 剪切修正因子取5/6, 其余类推, 最终可得FG-CNTRC板线性弯曲和自由振动控制表达式. 表4括号中的数据为相对误差, 定义为: |文献解[32]−本文解或SMKI-FSDT|/文献解[32] × 100%. 从表4可知: 当b/h = 10, 20, 50 (中厚/薄板)时, 采用本文方法及SMKI-FSDT求解FG-CNTRC板自由振动均具有计算精度高的特点; 当b/h = 5 (厚板)时, 基于SMKI-FSDT得出的相对误差明显高于本文方法, 而本文所提方法相对误差最高为1.154%.

    表  4  四边简支FG-CNTRC板无量纲基础频率(${\boldsymbol{V}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)
    Table  4.  Dimensionless fundamental frequencies of simply supported FG-CNTRC plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)
    b/hCNT
    type    		FEM-
    FSDT[8]    		FEM-
    TSDT[32]    		SMKI-
    FSDT (error)    		Present (error)
    5UD—8.8328.627 (2.321)8.753 (0.894)
    FG-V—8.4078.551 (1.713)8.504 (1.154)
    FG-O—8.0298.133 (1.295)7.946 (1.034)
    FG-X—9.1228.877 (2.686)9.040 (0.899)
    10UD13.53213.60113.519 (0.603)13.553 (0.353)
    FG-V12.45212.35212.423 (0.575)12.458 (0.858)
    FG-O11.55011.37111.544 (1.521)11.322 (0.431)
    FG-X14.61614.72714.601 (0.856)14.677 (0.340)
    20UD17.35517.35417.333 (0.121)17.320 (0.196)
    FG-V15.11015.03815.050 (0.080)15.081 (0.286)
    FG-O13.53213.43213.526 (0.700)13.405 (0.201)
    FG-X19.93919.94519.905 (0.201)19.914 (0.155)
    50UD19.22319.18119.268 (0.454)19.199 (0.094)
    FG-V16.25216.21816.261 (0.265)16.252 (0.210)
    FG-O14.30214.27514.410 (0.946)14.302 (0.189)
    FG-X22.98422.93022.993 (0.275)22.935 (0.022)
    下载: 导出CSV 
    | 显示表格

    3.2   计算效率分析

    通过与SMKI-FSDT的计算耗时对比来开展效率研究, 程序均在相同的软件和硬件(Intel Core i7-8700 CPU @ 3.20 GHz和RAM 16 GB)环境中运行. 图4记录了所提方法和SMKI-FSDT求解四边固支UD型板线性弯曲和自由振动时CPU计算时间随离散点数的变化曲线. 从图4可知, 随着离散点数的增加, 本文方法与SMKI-FSDT计算UD型板线性弯曲和自由振动所需CPU时间差增大, 且求解自由振动时更为明显. 这是因为本文方法是基于含有7个自由度变量的改进Reddy型TSDT, 比FSDT多2个未知量, 这对自由振动时的特征值求解更为耗时.

    图  4  本文方法与SMKI-FSDT之间的计算效率比较(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)
    Figure  4.  Comparison of computational efficiency between the present method and SMKI-FSDT (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)
    下载: 全尺寸图片 幻灯片

    3.3   弯曲分析

    表5计算了宽厚比为b/h = 10, 分别承受均布载荷与正弦载荷(q0(x, y) = qsin(πx/a)sin(πy/b)), 在不同体积分数及地基系数下四边简支FG-CNTRC板中点无量纲挠度${\tilde w}$ = 103Dmw/(a2q), 并与Wattanasakuplong等[28]基于TSDT和SSDT, 采用Navier解答获得四边简支FG-CNTRC板的解析解进行对比, 发现本文结果与解析解吻合良好, 证明了本文方法计算弹性地基上FG-CNTRC线性弯曲的准确性. 从表5可知: 均布载荷作用下FG-CNTRC板的挠度比正弦载荷作用下FG-CNTRC板的挠度大; 地基系数因子对FG-CNTRC板的挠度影响较大, 对整个地基FG-CNTRC板结构刚度具有增强作用, 随着地基系数因子的增加挠度减少; FG-O分布型板的挠度最大, FG-X分布型板的挠度最小, UD及FG-V分布型的结果介于两者之间.

    表  5  不同体积分数及地基系数下四边简支FG-CNTRC板中点无量纲挠度${{\tilde w}}$
    Table  5.  Dimensionless central deflections ${\tilde w}$ of simply supported FG-CNTRC plates with different volume fraction and foundation coefficient
    (kw, ks)Theory$\text{}{{V} }_{\text{CNT} }^{\text{*} }$ = 0.11$\text{}{{V} }_{\text{CNT} }^{\text{*} }$ = 0.17
    UDFG-VFG-OFG-XUDFG-VFG-OFG-X
    uniform load
    (0, 0)TSDT0.73560.88061.07050.62060.47120.56630.68440.4012
    SSDT0.73400.87921.07430.61790.47020.56550.68620.4001
    present0.73520.87971.07410.62140.47100.56560.68540.4013
    (100, 0)TSDT0.69830.82860.99550.93500.45560.54400.65300.3897
    SSDT0.69690.82740.99880.59100.45480.54370.65460.3887
    present0.69800.82780.99870.59420.45540.54370.65400.3898
    (100, 50)TSDT0.47730.53460.59910.42620.35000.40000.45570.3098
    SSTD0.47670.53410.60030.42500.34950.39970.45650.3092
    present0.47740.53460.60080.42660.35000.39990.45650.3100
    sinusoidal load
    (100, 0)TSDT0.49640.58690.70810.42270.31770.37690.45260.2723
    SSDT0.49530.58590.71040.42080.31700.37630.45370.2715
    present0.49630.58650.71020.42350.31760.37650.45320.2726
    (100, 0)TSDT0.47290.55440.66120.40560.30790.36320.43300.2651
    SSDT0.47190.55340.66330.40380.30720.36260.43400.2644
    present0.47280.55400.66310.40630.30780.36280.43350.2653
    (100, 50)TSDT0.32400.35830.40010.28960.23610.26740.30340.2101
    SSDT0.32190.35790.40090.28880.23570.26710.30390.2097
    present0.32260.35840.40110.29020.23620.26730.30380.2104
    Notes: The data TSDT and SSDT in the table are from the analytical solution of Ref. [28]
    下载: 导出CSV 
    | 显示表格

    为研究宽厚比b/h对不同地基系数下FG-CNTRC板的影响, 表6绘制了体积分数为0.11, 不同宽厚比下弹性地基上四边固支FG-CNTRC板中点无量纲挠度. 从表6可知, 不同地基系数下的4种分布型CNTRC板中点挠度均随着宽厚比的增大而增大, 且宽厚比越大, 地基上系数的影响越加明显. 这是因为随着板宽厚比的增加, 即板厚的逐渐减小, 地基刚度在地基板整个结构刚度中的占比越大导致.

    图5给出了体积分数为0.14, 不同宽厚比下四边固支UD板中点归一化挠度随转向角θ的变化, 可以看出UD板中点挠度随转向角呈现先增后减的规律, 在θ = 45°达到最大值, 曲线关于θ = 45°对称. 此外, 随着宽厚比的增加, 厚度减少的剪切效应使得挠度受CNTs转向角的影响越大. 图6计算了不同地基系数下四边固支FG-CNTRC板的无量纲中点挠度随转向角θ的变化, 板的宽厚比为50, 体积分数为0.14. 从图6可知, 不同地基系数及分布型下FG-CNTRC板中点挠度随着CNTs转向角呈现周期性变化; 不同CNTs分布型下的数据曲线均关于θ = 0°对称, 在θ = 45°和θ = −45°时, 其抗弯刚度最小, 中点挠度达到最大值, 在θ = −90°, θ = 0°和θ = 90°挠度为最小值; 随着地基系数的增加, 整个结构的抗弯刚度增加, 板中点挠度明显减小.

    图  5  不同宽厚比下四边固支UD板中点归一化挠度随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)
    Figure  5.  Normalized central deflection versus CNT orientation angle θ for the clamped UD plate with different width-thickness ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)
    下载: 全尺寸图片 幻灯片
    图  6  弹性地基上四边固支FG-CNTRC板的无量纲中点挠度随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    Figure  6.  Dimensionless central deflection versus CNT orientation angle θ for the clamped FG-CNTRC plate on elastic foundation (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    下载: 全尺寸图片 幻灯片

    3.4   自振频率分析

    下面采用SMKI对含CNTs转向问题的弹性地基FG-CNTRC板自由振动进行分析和讨论. 表7给出了不同体积分数及地基系数下四边简支FG-CNTRC板的无量纲基础频率, 并与Wattanasakuplong等[28]基于Navier解答得到的四边简支结果进行对比, 说明了本文方法计算弹性地基上FG-CNTRC板自由振动的正确性. 从表7可知, 随着体积分数的增大, 不同分布型的CNTRC板自振频率均增大; CNTs分布形式对自振频率的影响较大, 自振频率大小依次为FG-X > UD > FG-V > FG-O; 随着地基系数的增大, 自振频率增大.

    表  6  不同宽厚比及地基系数下四边固支FG-CNTRC板中点无量纲挠度(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)
    Table  6.  Dimensionless central deflections of clamped FG-CNTRC plates with width-thick ratio and foundation coefficient (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)
    (kw, ks)CNT typeb/h
    5102050100
    (0, 0)UD3.833 × 10−42.119 × 10−31.315 × 10−22.627 × 10−13.632
    FG-V3.875 × 10−42.252 × 10−31.572 × 10−23.663 × 10−15.293
    FG-O4.251 × 10−42.604 × 10−31.934 × 10−24.797 × 10−17.040
    FG-X3.722 × 10−41.985 × 10−31.120 × 10−21.896 × 10−12.470
    (100, 0)UD3.792 × 10−42.056 × 10−31.195 × 10−21.434 × 10−10.521
    FG-V3.832 × 10−42.181 × 10−31.406 × 10−21.708 × 10−10.538
    FG-O4.201 × 10−42.511 × 10−31.693 × 10−21.926 × 10−10.544
    FG-X3.683 × 10−41.929 × 10−31.030 × 10−21.181 × 10−10.492
    (100, 50)UD3.428 × 10−41.592 × 10−36.580 × 10−32.900 × 10−20.064
    FG-V3.462 × 10−41.665 × 10−37.110 × 10−32.980 × 10−20.065
    FG-O3.759 × 10−41.847 × 10−37.730 × 10−33.030 × 10−20.065
    FG-X3.340 × 10−41.516 × 10−36.080 × 10−32.810 × 10−20.064
    下载: 导出CSV 
    | 显示表格
    表  7  不同体积分数及地基系数下四边简支FG-CNTRC板的无量纲基础频率
    Table  7.  Dimensionless fundamental frequencies of simply supported FG-CNTRC plates with different volume fraction and foundation coefficient
    $\text{}{{V} }_{\text{CNT} }^{\text{*} }$CNT type(kw, ks) = (0, 0)(kw, ks) = (100, 0)(kw, ks) = (100, 50)
    TSDTSSDTPresentTSDTSSDTPresentTSDTSSDTPresent
    0.11UD13.5513.5713.5513.8813.9013.8916.8216.8316.81
    FG-V12.4512.4612.4612.8112.8212.8215.9415.9515.93
    FG-O11.3411.2011.3211.7311.7211.7215.0915.0715.06
    FG-X14.6914.7214.6815.0015.0314.9817.7517.7717.73
    0.14UD14.3614.3814.3614.6714.6914.6717.4617.4717.44
    FG-V13.2813.2913.2813.6213.6313.6216.5716.5916.57
    FG-O12.1321.1012.1212.5012.4812.4815.6715.6615.65
    FG-X15.4115.4515.4015.7015.7415.6918.3318.3618.31
    0.17UD16.8316.8516.8417.1017.1217.1019.5319.5419.52
    FG-V15.4415.4615.4515.7315.7415.7418.3418.3518.33
    FG-O14.0914.0814.0814.4114.3914.4017.2117.2017.20
    FG-X18.1918.2118.1818.4318.4618.4320.7020.7320.69
    Notes: The data TSDT and SSDT in the table are from the analytical solution of Ref. [28]
    下载: 导出CSV 
    | 显示表格
    表  8  不同宽厚比及地基系数下四边固支FG-CNTRC板的无量纲基础频率(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)
    Table  8.  Dimensionless fundamental frequencies of clamped FG-CNTRC plates with different width-thickness ratio and foundation coefficient (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)
    b/h(kw, ks) = (0, 0)(kw, ks) = (100, 0)(kw, ks) = (100, 50)
    UDFG-VFG-OFG-XUDFG-VFG-OFG-XUDFG-VFG-OFG-X
    510.6110.5710.1210.7610.6610.6210.1710.8111.2011.1610.7411.34
    1018.0417.5616.4018.5918.2917.8216.6818.8420.7720.3719.3821.25
    2028.5726.3923.9530.7429.8227.7425.4331.9140.8239.3937.8342.32
    5039.5333.9829.9546.0152.0848.0045.2457.15121.31119.69118.51123.44
    10042.5635.8131.3350.98104.94102.39100.91108.62323.58322.49321.73325.06
    下载: 导出CSV 
    | 显示表格

    表8计算了体积分数为0.11, 不同宽厚比及地基系数下四边固支FG-CNTRC板的无量纲基础频率, 可以看出FG-CNTRC板自振频率与宽厚比成正相关, 厚度减少的剪切效应使得地基系数对自振频率的影响越加明显.

    接下来FG-CNTRC板体积分数取为0.14, 讨论边界条件、长宽比、宽厚比、转向角对其自振频率的影响.

    图7展示了宽厚比为50, 不同分布型下CNTRC板的无量纲基础频率随边界条件的变化, 可以看出随着边界条件的加强, 自振频率增大. 图8计算了不同长宽比下四边固支UD板的无量纲基础频率随转向角的变化, 发现当CNTs在竖向放置θ = 90°时, 自振频率大小基本相等, 在横向放置θ = 0°时, 自振频率随不同的长宽比a/b产生不一样的自振效果; 当长宽比a/b < 1.8时, 随着CNTs转向角θ的增大, 自振频率先减少后增大, 而当a/b = 1.8, 2.0时, 随着CNTs转向角θ的增大, 自振频率逐渐增大.

    图  7  FG-CNTRC板的无量纲基础频率随边界条件的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    Figure  7.  Dimensional fundamental frequency versus boundary condition for the FG-CNTRC square plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    下载: 全尺寸图片 幻灯片
    图  8  不同长宽比下四边固支UD板的无量纲基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)
    Figure  8.  Dimensional fundamental frequency versus CNT orientation angle θ for the clamped UD plate with different length-width ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)
    下载: 全尺寸图片 幻灯片

    图9讨论了不同CNTs转向角下宽厚比b/h对四边固支UD板的归一化基础频率影响, 可以看出UD板的基础频率与宽厚比成正相关, 且厚度增加的剪切效应使得CNTs转向角对基础频率的影响逐渐降低. 图10计算了不同地基系数、不同分布型及CNTs转向角θ下四边固支FG-CNTRC板的无量纲基础频率. 从图10可知, 不同地基系数及分布型下FG-CNTRC板基础频率随CNTs转向角呈现周期性变化; FG-CNTRC板的基础频率在CNTs转向角θ = 0°, 90°和−90°时, 其结构获得相同的自振效果, CNTs转向角θ = 45°和−45°时, 其结果抗弯刚度最小, 基础频率最小; 随着地基系数的增大, 基础频率逐渐增大. 图11给出了CNTs转向角为0°, 45°和90°时四边固支UD板前5阶自振模态.

    图  9  不同宽厚比下四边固支UD板的归一化基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)
    Figure  9.  Normalized fundamental frequency versus CNT orientation angle θ for the clamped UD plate with different width-thickness ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)
    下载: 全尺寸图片 幻灯片
    图  10  弹性地基上四边固支FG-CNTRC板的无量纲基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    Figure  10.  Dimensionless fundamental frequencies versus CNT orientation angle θ for the clamped FG-CNTRC plate on elastic foundation (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    下载: 全尺寸图片 幻灯片
    图  11  四边固支UD板前5阶自振模态(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    Figure  11.  The first five natural vibration modes of clamped UD plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)
    下载: 全尺寸图片 幻灯片

    4.   结论

    本文采用SMKI求解了基于改进Reddy型TSDT含CNTs转向的弹性地基FG-CNTRC板的线性弯曲及自由振动问题. 首先通过基准算例验证了本文方法的有效性及准确性. 接着探讨了CNTs的体积分数、分布模式、转向角、板的宽厚比及长宽比、地基系数等对FG-CNTRC板线性弯曲和自振频率的影响. 本文得出结论如下.

    (1)采用本文方法求解FG-CNTRC薄板、中厚板, 甚至厚板的线性弯曲和自振频率均具有较好的计算精度. 与SMKI-FSDT相比, 基于含7个自由度变量的改进Reddy型TSDT计算效率有所降低.

    (2) CNTs分布形式对板结构刚度的影响具有较大差异, 不同分布型下FG-CNTRC板的刚度大小依次为FG-X > UD > FG-V > FG-O型.

    (3)随着CNTs体积分数和地基系数的增加, FG-CNTRC板结构刚度逐渐增大.

    (4) CNTs在竖向放置θ = 90°时, 不同长宽比FG-CNTRC板结构刚度基本一致, 在横向放置θ = 0°时, 不同长宽比FG-CNTRC板结构刚度存在较大差异, 且随着长宽比的增大而减少; 当长宽比a/b < 1.8时, 随着CNTs转向角θ的增大, 结构刚度先增大后减小, 当a/b = 1.8和2.0时, 随着CNTs转向角θ的增大, 结构刚度逐渐增大.

    (5) FG-CTRC板结构刚度与宽厚比成正相关, 厚度增加的剪切效应会让CNTs转向角对结构刚度的影响逐渐降低.

    (6) FG-CNTRC板的挠度和自振频率随着CNTs转向角呈现周期性变化, 其结构的刚度在CNTs转向角θ = 0°, 90°和−90°时最大, 在CNTs转向角θ = 45°和−45°时最小.

  • 图  1   弹性地基上FG-CNTRC板的等效模型

    Figure  1.   Equivalent model of FG-CNTRC plate on elastic foundation

    下载: 全尺寸图片 幻灯片

    图  2   四边简支FG-CNTRC板中点挠度收敛性分析(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    Figure  2.   Convergence analysis of central deflection of simply supported FG-CNTRC plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    下载: 全尺寸图片 幻灯片

    图  3   不同边界条件下FG-CNTRC板中点无量纲轴向应力${\bar \sigma }_{xx}$(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.17, b/h = 50)

    Figure  3.   Normalized axial stress ${{\bar \sigma }}_{xx}$ of central point of FG-CNTRC plate under different boundary condition (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.17, b/h = 50)

    下载: 全尺寸图片 幻灯片

    图  4   本文方法与SMKI-FSDT之间的计算效率比较(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    Figure  4.   Comparison of computational efficiency between the present method and SMKI-FSDT (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11, b/h = 10)

    下载: 全尺寸图片 幻灯片

    图  5   不同宽厚比下四边固支UD板中点归一化挠度随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    Figure  5.   Normalized central deflection versus CNT orientation angle θ for the clamped UD plate with different width-thickness ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    下载: 全尺寸图片 幻灯片

    图  6   弹性地基上四边固支FG-CNTRC板的无量纲中点挠度随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  6.   Dimensionless central deflection versus CNT orientation angle θ for the clamped FG-CNTRC plate on elastic foundation (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    下载: 全尺寸图片 幻灯片

    图  7   FG-CNTRC板的无量纲基础频率随边界条件的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  7.   Dimensional fundamental frequency versus boundary condition for the FG-CNTRC square plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    下载: 全尺寸图片 幻灯片

    图  8   不同长宽比下四边固支UD板的无量纲基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    Figure  8.   Dimensional fundamental frequency versus CNT orientation angle θ for the clamped UD plate with different length-width ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    下载: 全尺寸图片 幻灯片

    图  9   不同宽厚比下四边固支UD板的归一化基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    Figure  9.   Normalized fundamental frequency versus CNT orientation angle θ for the clamped UD plate with different width-thickness ratio (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14)

    下载: 全尺寸图片 幻灯片

    图  10   弹性地基上四边固支FG-CNTRC板的无量纲基础频率随转向角θ的变化(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  10.   Dimensionless fundamental frequencies versus CNT orientation angle θ for the clamped FG-CNTRC plate on elastic foundation (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    下载: 全尺寸图片 幻灯片

    图  11   四边固支UD板前5阶自振模态(${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    Figure  11.   The first five natural vibration modes of clamped UD plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.14, b/h = 50)

    下载: 全尺寸图片 幻灯片

    表  1   材料参数

    Table  1   The properties of material

    ParametersMatrixCNTs
    Poisson’s ratioνm = 0.34$ {\nu }_{\text{12}}^{\text{CNT}} $ = 0.175
    density/(kg·m−3)ρm = 1150ρCNT = 1400
    Young’s modulus/GPaEm = 2.1${{E} }_{\text{11} }^{\text{CNT} }$ = 5646.6,
    ${{E} }_{\text{22} }^{\text{CNT} }$ = 7080
    shear modulus/GPa—${{G} }_{\text{12} }^{\text{CNT} }$ = 1944.5
    下载: 导出CSV

    表  2   CNTs的效能参数

    Table  2   The efficiency parameters of CNTs

    ${{V} }_{\text{CNT} }^{\text{*} }$η1η2η3
    0.110.1490.9340.934
    0.140.1500.9410.941
    0.170.1491.3811.381
    下载: 导出CSV

    表  3   四边简支FG-CNTRC板中点无量纲挠度(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  3   Dimensionless central deflection of simply supported FG-CNTRC plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    b/hCNT typeFEM-FSDT[8]IGA-TSDT[16]Present
    10UD3.739 × 10−33.717 × 10−33.716 × 10−3
    FG-V4.466 × 10−34.427 × 10−34.447 × 10−3
    FG-O5.230 × 10−35.438 × 10−35.430 × 10−3
    FG-X3.177 × 10−33.102 × 10−33.140 × 10−3
    20UD3.628 × 10−33.624 × 10−33.629 × 10−3
    FG-V4.879 × 10−34.877 × 10−34.880 × 10−3
    FG-O6.155 × 10−36.248 × 10−36.243 × 10−3
    FG-X2.701 × 10−32.685 × 10−32.697 × 10−3
    50UD1.1551.1561.156
    FG-V1.6531.6541.652
    FG-O2.1572.1632.158
    FG-X0.7900.7910.792
    下载: 导出CSV

    表  4   四边简支FG-CNTRC板无量纲基础频率(${\boldsymbol{V}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  4   Dimensionless fundamental frequencies of simply supported FG-CNTRC plate (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    b/hCNT
    type    		FEM-
    FSDT[8]    		FEM-
    TSDT[32]    		SMKI-
    FSDT (error)    		Present (error)
    5UD—8.8328.627 (2.321)8.753 (0.894)
    FG-V—8.4078.551 (1.713)8.504 (1.154)
    FG-O—8.0298.133 (1.295)7.946 (1.034)
    FG-X—9.1228.877 (2.686)9.040 (0.899)
    10UD13.53213.60113.519 (0.603)13.553 (0.353)
    FG-V12.45212.35212.423 (0.575)12.458 (0.858)
    FG-O11.55011.37111.544 (1.521)11.322 (0.431)
    FG-X14.61614.72714.601 (0.856)14.677 (0.340)
    20UD17.35517.35417.333 (0.121)17.320 (0.196)
    FG-V15.11015.03815.050 (0.080)15.081 (0.286)
    FG-O13.53213.43213.526 (0.700)13.405 (0.201)
    FG-X19.93919.94519.905 (0.201)19.914 (0.155)
    50UD19.22319.18119.268 (0.454)19.199 (0.094)
    FG-V16.25216.21816.261 (0.265)16.252 (0.210)
    FG-O14.30214.27514.410 (0.946)14.302 (0.189)
    FG-X22.98422.93022.993 (0.275)22.935 (0.022)
    下载: 导出CSV

    表  5   不同体积分数及地基系数下四边简支FG-CNTRC板中点无量纲挠度${{\tilde w}}$

    Table  5   Dimensionless central deflections ${\tilde w}$ of simply supported FG-CNTRC plates with different volume fraction and foundation coefficient

    (kw, ks)Theory$\text{}{{V} }_{\text{CNT} }^{\text{*} }$ = 0.11$\text{}{{V} }_{\text{CNT} }^{\text{*} }$ = 0.17
    UDFG-VFG-OFG-XUDFG-VFG-OFG-X
    uniform load
    (0, 0)TSDT0.73560.88061.07050.62060.47120.56630.68440.4012
    SSDT0.73400.87921.07430.61790.47020.56550.68620.4001
    present0.73520.87971.07410.62140.47100.56560.68540.4013
    (100, 0)TSDT0.69830.82860.99550.93500.45560.54400.65300.3897
    SSDT0.69690.82740.99880.59100.45480.54370.65460.3887
    present0.69800.82780.99870.59420.45540.54370.65400.3898
    (100, 50)TSDT0.47730.53460.59910.42620.35000.40000.45570.3098
    SSTD0.47670.53410.60030.42500.34950.39970.45650.3092
    present0.47740.53460.60080.42660.35000.39990.45650.3100
    sinusoidal load
    (100, 0)TSDT0.49640.58690.70810.42270.31770.37690.45260.2723
    SSDT0.49530.58590.71040.42080.31700.37630.45370.2715
    present0.49630.58650.71020.42350.31760.37650.45320.2726
    (100, 0)TSDT0.47290.55440.66120.40560.30790.36320.43300.2651
    SSDT0.47190.55340.66330.40380.30720.36260.43400.2644
    present0.47280.55400.66310.40630.30780.36280.43350.2653
    (100, 50)TSDT0.32400.35830.40010.28960.23610.26740.30340.2101
    SSDT0.32190.35790.40090.28880.23570.26710.30390.2097
    present0.32260.35840.40110.29020.23620.26730.30380.2104
    Notes: The data TSDT and SSDT in the table are from the analytical solution of Ref. [28]
    下载: 导出CSV

    表  6   不同宽厚比及地基系数下四边固支FG-CNTRC板中点无量纲挠度(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  6   Dimensionless central deflections of clamped FG-CNTRC plates with width-thick ratio and foundation coefficient (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    (kw, ks)CNT typeb/h
    5102050100
    (0, 0)UD3.833 × 10−42.119 × 10−31.315 × 10−22.627 × 10−13.632
    FG-V3.875 × 10−42.252 × 10−31.572 × 10−23.663 × 10−15.293
    FG-O4.251 × 10−42.604 × 10−31.934 × 10−24.797 × 10−17.040
    FG-X3.722 × 10−41.985 × 10−31.120 × 10−21.896 × 10−12.470
    (100, 0)UD3.792 × 10−42.056 × 10−31.195 × 10−21.434 × 10−10.521
    FG-V3.832 × 10−42.181 × 10−31.406 × 10−21.708 × 10−10.538
    FG-O4.201 × 10−42.511 × 10−31.693 × 10−21.926 × 10−10.544
    FG-X3.683 × 10−41.929 × 10−31.030 × 10−21.181 × 10−10.492
    (100, 50)UD3.428 × 10−41.592 × 10−36.580 × 10−32.900 × 10−20.064
    FG-V3.462 × 10−41.665 × 10−37.110 × 10−32.980 × 10−20.065
    FG-O3.759 × 10−41.847 × 10−37.730 × 10−33.030 × 10−20.065
    FG-X3.340 × 10−41.516 × 10−36.080 × 10−32.810 × 10−20.064
    下载: 导出CSV

    表  7   不同体积分数及地基系数下四边简支FG-CNTRC板的无量纲基础频率

    Table  7   Dimensionless fundamental frequencies of simply supported FG-CNTRC plates with different volume fraction and foundation coefficient

    $\text{}{{V} }_{\text{CNT} }^{\text{*} }$CNT type(kw, ks) = (0, 0)(kw, ks) = (100, 0)(kw, ks) = (100, 50)
    TSDTSSDTPresentTSDTSSDTPresentTSDTSSDTPresent
    0.11UD13.5513.5713.5513.8813.9013.8916.8216.8316.81
    FG-V12.4512.4612.4612.8112.8212.8215.9415.9515.93
    FG-O11.3411.2011.3211.7311.7211.7215.0915.0715.06
    FG-X14.6914.7214.6815.0015.0314.9817.7517.7717.73
    0.14UD14.3614.3814.3614.6714.6914.6717.4617.4717.44
    FG-V13.2813.2913.2813.6213.6313.6216.5716.5916.57
    FG-O12.1321.1012.1212.5012.4812.4815.6715.6615.65
    FG-X15.4115.4515.4015.7015.7415.6918.3318.3618.31
    0.17UD16.8316.8516.8417.1017.1217.1019.5319.5419.52
    FG-V15.4415.4615.4515.7315.7415.7418.3418.3518.33
    FG-O14.0914.0814.0814.4114.3914.4017.2117.2017.20
    FG-X18.1918.2118.1818.4318.4618.4320.7020.7320.69
    Notes: The data TSDT and SSDT in the table are from the analytical solution of Ref. [28]
    下载: 导出CSV

    表  8   不同宽厚比及地基系数下四边固支FG-CNTRC板的无量纲基础频率(${{{\boldsymbol{V}}}}_{\bf{CNT}}^{{{\boldsymbol{*}}}}$ = 0.11)

    Table  8   Dimensionless fundamental frequencies of clamped FG-CNTRC plates with different width-thickness ratio and foundation coefficient (${{V}}_{\text{CNT}}^{\text{*}}$ = 0.11)

    b/h(kw, ks) = (0, 0)(kw, ks) = (100, 0)(kw, ks) = (100, 50)
    UDFG-VFG-OFG-XUDFG-VFG-OFG-XUDFG-VFG-OFG-X
    510.6110.5710.1210.7610.6610.6210.1710.8111.2011.1610.7411.34
    1018.0417.5616.4018.5918.2917.8216.6818.8420.7720.3719.3821.25
    2028.5726.3923.9530.7429.8227.7425.4331.9140.8239.3937.8342.32
    5039.5333.9829.9546.0152.0848.0045.2457.15121.31119.69118.51123.44
    10042.5635.8131.3350.98104.94102.39100.91108.62323.58322.49321.73325.06
    下载: 导出CSV
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    1. 郝玉琪，李凤莲，吕梅. 压电FG-CNTRC夹层板自由振动与隔声量分析. 科学技术与工程. 2024(30): 12901-12908 . 百度学术

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  • 文章访问数:  768
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  • 被引次数: 2
出版历程
  • 收稿日期:  2023-02-12
  • 录用日期:  2023-05-09
  • 网络出版日期:  2023-05-10
  • 刊出日期:  2023-06-17

目录

摘要
HTML全文

  • 引言

  • 1.   稳定移动克里金插值

  • 2.   数学模型

  • 2.1   弹性地基FG-CNTRC 板

  • 2.2   总能量泛函

  • 2.3   控制方程

  • 2.4   边界条件

  • 3.   数值结果与讨论

  • 3.1   收敛性及有效性分析

  • 3.2   计算效率分析

  • 3.3   弯曲分析

  • 3.4   自振频率分析

  • 4.   结论

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