基于Galerkin法研究应力波作用下复合材料板的动力学失稳

王志鹏; 韩志军; 王龙飞

doi:10.11858/gywlxb.20210705

基于Galerkin法研究应力波作用下复合材料板的动力学失稳

doi: 10.11858/gywlxb.20210705

王志鹏^1,,
韩志军^2,*, ,,
王龙飞²

1.
太原理工大学机械与运载工程学院，山西太原　030024
2.
太原理工大学生物医学工程学院，山西太原　030024

基金项目: 国家自然科学基金（11802195）

详细信息

作者简介:
王志鹏（1994－），男，硕士，主要从事非线性动力屈曲研究. E-mail：694596123@qq.com

通讯作者:
韩志军（1964－），男，博士，教授，主要从事非线性动力屈曲研究. E-mail：13073578705@126.com

中图分类号: O343.5
计量
- 文章访问数: 2810
- HTML全文浏览量: 987
- PDF下载量: 24
出版历程
- 收稿日期: 2021-01-10
- 修回日期: 2021-02-05

Dynamic Instability of Composite Plate under Stress Wave Based on Galerkin Method

1.
College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China
2.
College of Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China

摘要

摘要: 基于Kirchhoff薄板理论和Hamilton原理，考虑应力波效应，对含初始几何缺陷的三边简支、一边固支的复合材料板，建立了振动控制方程，得到了其屈曲临界荷载表达式。采用MATLAB编程进行数值计算，讨论了初始几何缺陷、初相位、铺层角度、屈曲模态阶数及铺层层数对板屈曲临界荷载的影响。结果表明：复合材料板的屈曲临界载荷随临界长度增大、铺设厚度减小、初始几何缺陷系数增大、振型函数初相位减小而减小。此外，复合材料板的各层铺层角度与荷载作用方向的夹角越小，屈曲临界载荷越大，当板的对称铺设层数达到7层时，临界荷载趋于稳定。
- 复合材料板 /
- 应力波效应 /
- 初始几何缺陷 /
- 屈曲 /
- 临界荷载
Abstract: Based on the Kirchhoff thin plate theory and Hamilton principle, the vibration control equation of composite plate is established. The equation is simply supported on three sides and fixed on one side with initial geometric imperfections. The expression of buckling critical load is obtained. The numerical calculation is carried out by MATLAB programming. The effects of initial geometric imperfections, initial phase, ply angle, buckling mode order and layer number on the critical buckling load of the plate are discussed. Results show that the critical buckling load is decrease with the increasing of the critical length, the decreasing of the laying thickness, the increasing of the initial geometric defect coefficient, and the decreasing of the initial phase of the mode function. In addition, the smaller the angle between the laying angle of each layer and the load direction is, the greater the buckling critical load is. And the buckling critical load tends to be stable when the layer number of symmetrical laminate reaches seven.
- composite plate /
- effect of stress wave /
- initial geometric defect /
- buckling /
- critical load

HTML全文

复合材料板具有较高的强度质量比、良好的耐腐蚀性和优异的可设计性，被广泛应用于航空航天和工业制造等领域^[1]。在实际使用中复合材料板经常受到不同形式的冲击荷载，从而产生振动和屈曲问题，因此冲击载荷作用下复合材料板的动力稳定性问题备受关注。

近年来关于复合材料板的研究越来越多，尤其是冲击荷载作用下复合材料板的动力稳定性问题研究^[2]，对工程部件结构设计和使用具有重要的意义。Sun等^[3]研究了在加热环境中应力波对功能梯度圆柱壳轴向冲击屈曲的影响；毛柳伟等^[4]对弹性直杆在应力波作用下的动力分叉屈曲进行了分析与探讨，提出了求解应力波作用下直杆动力屈曲的数值方法；Lepik^[5]讨论了在应力波影响下轴向压缩的弹塑性梁的屈曲；Zhang等^[6]分析了不确定初始几何缺陷对薄板屈曲的影响；Abdelaziz等^[7]利用双曲线剪切变形理论，分析了在各种边界条件下复合材料板的弯曲变形和屈曲；Kouchakzadeh等^[8]采用线性和旋转弹簧的均匀分布来模拟边界条件，对矩形层压复合板的屈曲进行了分析；Czapski等^[9]通过数值和实验方法，研究了残余应力对压缩至破坏期间薄壁层压板屈曲性能的影响。

在实际工程中，复合材料板多应用于振动环境，其在动力响应下的动态特性和振动分析是必不可少的，因而对该类材料的振动屈曲研究至关重要。Kuo^[10]研究了两种非均匀分布纤维复合材料板的振动屈曲问题，Villarreal等^[11]对典型正交异性板的本征频率和振动屈曲进行了理论分析，Eftekhari等^[12]提出了通过组合应用有限元方法和微分正交方法求解矩形板的振动屈曲问题，Rehman等^[13]探讨了壳体结构的缺陷和损坏对结构振动屈曲的影响，Sayyad等^[14]将三角剪切变形理论应用于复合板的变形和振动屈曲研究。

关于复合材料板的振动屈曲问题已开展了较多的研究，但大多未考虑应力波效应对振动屈曲的影响，而动力屈曲一般与应力波相联系且具有局部发生的特点，研究含初始缺陷的复合材料板能更好地揭示实际工程中复合材料板在不同工况下发生动力屈曲的机理。基于此，本研究利用Kirchhoff薄板理论和Hamilton原理，考虑应力波效应，建立含初始几何缺陷的四边简支复合材料板的振动控制方程，得到板的屈曲临界荷载表达式，在此基础上通过数值计算讨论初始几何缺陷、振型函数初相位、铺层角度、屈曲模态阶数和铺层层数对复合材料板振动屈曲临界荷载的影响，为工程实际提供理论依据。

1. 复合材料板的控制方程

复合材料板在x = L_a处为固定边界条件，其余3边为简支边界条件，在 $z$ = 0的中性面上受x方向的面内阶跃荷载N作用，如图1所示。板在 $z$ 方向上含初始几何缺陷w₁，且x、y、 $z$ 方向的位移分别为u、v、w。复合材料板的长度、宽度和厚度分别为L_a、L_b、h，由n层单层板组成， $\theta$ 为单层板的铺层角度，即纤维材料铺设方向与x方向的夹角（见图1）。

图 1 复合材料板结构示意图

Figure 1. Schematic diagram of composite plate structure

下载: 全尺寸图片幻灯片

根据Kirchhoff薄板理论及经典弹性理论，复合材料板的位移与应变、弹性曲面的曲率和扭率的表达式为

${\begin{cases} {w = {w_1} + {w_0}} \\ {u = {u_0} - z\displaystyle\frac{{\partial \left( {w - {w_1}} \right)}}{{\partial x}}} \\ {v = {v_0} - z\displaystyle\frac{{\partial \left( {w - {w_1}} \right)}}{{\partial y}}} \end{cases}}$

(1)

${\begin{cases} {\varepsilon _x^0 = \displaystyle\frac{{\partial {u_0}}}{{\partial x}}} \\ {\varepsilon _y^0 = \displaystyle\frac{{\partial {v_0}}}{{\partial y}}} \\ {\gamma _{xy}^0 = \displaystyle\frac{{\partial {u_0}}}{{\partial y}} + \displaystyle\frac{{\partial {v_0}}}{{\partial x}}} \end{cases}}$

(2)

${\begin{cases} {{\kappa _x} = - \displaystyle\frac{{{\partial ^2}\left( {w - {w_1}} \right)}}{{\partial {x^2}}}} \\ {{\kappa _y} = - \displaystyle\frac{{{\partial ^2}\left( {w - {w_1}} \right)}}{{\partial {y^2}}}} \\ {{\kappa _{xy}} = - \displaystyle\frac{{2{\partial ^2}\left( {w - {w_1}} \right)}}{{\partial x\partial y}}} \end{cases}}$

(3)

${\begin{cases} {{\varepsilon _x} = \varepsilon _x^0 + z{\kappa _x}} \\ {{\varepsilon _y} = \varepsilon _y^0 + z{\kappa _y}} \\ {{\gamma _{xy}} = \gamma _{_{xy}}^0 + z{\kappa _{xy}}} \end{cases}}$

(4)

式中：u₀、v₀、w₀分别为复合材料板在x、y、 $z$ 方向上的中面位移， $\varepsilon _x^0$ 、 $\varepsilon _y^0$ 、 $\gamma _{xy}^0$ 分别为复合材料板中面应变分量， ${\kappa _x}$ 、 ${\kappa _y}$ 、 ${\kappa _{xy}}$ 为中面的曲率和扭率， ${\varepsilon _x}$ 、 ${\varepsilon _y}$ 、 ${\gamma _{xy}}$ 为复合材料板任意一点的应变。

板的内力（N_x、N_y、N_xy）与内力矩（M_x、M_y、M_xy）之间的关系为

$\left[ \begin{array}{l} {N_x} \\ {N_y} \\ {N_{xy}} \\ {M_x} \\ {M_y} \\ {M_{xy}} \end{array} \right] = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}&{{A_{16}}}&{{B_{11}}}&{{B_{12}}}&{{B_{16}}} \\ {{A_{12}}}&{{A_{22}}}&{{A_{26}}}&{{B_{12}}}&{{B_{22}}}&{{B_{26}}} \\ {{A_{16}}}&{{A_{26}}}&{{A_{66}}}&{{B_{16}}}&{{B_{26}}}&{{B_{66}}} \\ {{B_{11}}}&{{B_{12}}}&{{B_{16}}}&{{D_{11}}}&{{D_{12}}}&{{D_{16}}} \\ {{B_{12}}}&{{B_{22}}}&{{B_{26}}}&{{D_{12}}}&{{D_{22}}}&{{D_{26}}} \\ {{B_{16}}}&{{B_{26}}}&{{B_{66}}}&{{D_{16}}}&{{D_{26}}}&{{D_{66}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\varepsilon _x^0} \\ {\varepsilon _y^0} \\ {\gamma _{xy}^0} \\ {{\kappa _x}} \\ {{\kappa _y}} \\ {{\kappa _{xy}}} \end{array}} \right]$

(5)

式中： ${A}_{ij}$ 、 ${B}_{ij}$ 、 ${D}_{ij} \left(i,j=1,2,6\right)$ 分别表示板的拉伸刚度、耦合刚度和弯曲刚度系数^[15]，表达式如下

${\begin{cases} {{A_{ij}} = \displaystyle\sum\limits_{k = 1}^n {{{\overline Q_{ij}^k }}\left( {{h_k} - {h_{k - 1}}} \right)} } \\ {{B_{ij}} = \displaystyle\frac{1}{2}\displaystyle\sum\limits_{k = 1}^n {{{\overline Q_{ij}^k }}( {{h^2_k} - h_{k - 1}^2} )} } \\ {{D_{ij}} = \displaystyle\frac{1}{3}\displaystyle\sum\limits_{k = 1}^n {{{\overline Q_{ij}^k }}( {{h^3_k} - h_{k - 1}^3} )} } \end{cases}}$

(6)

式中： ${\overline Q_{ij}^k }$ 为复合材料板第 $k$ 层的偏轴刚度系数

$\overline {\boldsymbol{Q}} = {{\boldsymbol{P}}^{ - 1}}{\boldsymbol{Q}}{({{\boldsymbol{P}}^{ - 1}})^{\rm{T}}}$

(7)

式中：P为坐标转换矩阵，Q为刚度矩阵。

${\boldsymbol{P}} = \left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\theta }&{{{\sin }^2}\theta }&{2\sin \theta \cos \theta } \\ {{{\sin }^2}\theta }&{{{\cos }^2}\theta }&{ - 2\sin \theta \cos \theta } \\ { - \sin \theta \cos \theta }&{\sin \theta \cos \theta }&{{{\cos }^2}\theta - {{\sin }^2}\theta } \end{array}} \right]$

(8)

${\boldsymbol{Q}}{\rm{ = }}\left[ {\begin{array}{*{20}{c}} {{Q_{11}}}&{{Q_{12}}}&0 \\ {{Q_{12}}}&{{Q_{22}}}&0 \\ 0&0&{{Q_{66}}} \end{array}} \right]$

(9)

考虑材料为正交各向异性材料，设E₁、E₂、G₁₂、μ₁₂、μ₂₁分别为板材料x和y方向的拉压弹性模量、剪切弹性模量、主泊松比和副泊松比，则有

${Q_{11}}{\rm{ = }}\frac{{{E_1}}}{{1 - {\mu _{12}}{\mu _{21}}}},\;\;\;{Q_{22}}{\rm{ = }}\frac{{{E_2}}}{{1 - {\mu _{12}}{\mu _{21}}}},\;\;\;{Q_{12}}{\rm{ = }}\frac{{{\mu _{12}}{E_2}}}{{1 - {\mu _{12}}{\mu _{21}}}}{\rm{ = }}\frac{{{\mu _{21}}{E_1}}}{{1 - {\mu _{12}}{\mu _{21}}}},\;\;\;{Q_{66}}{\rm{ = }}{G_{12}}$

(10)

复合材料板在左端受面内冲击荷载N作用（见图1）时，应力波沿x方向在板内传播，其应力变化如图2所示。

图 2 应力波传播示意图

Figure 2. Schematic diagram of stress wave propagation

下载: 全尺寸图片幻灯片

当应力波传播至波阵面位置L_cr（临界长度）时，板发生振动屈曲，板的内力N_t和应力波波速c分别表示为

${N_{\rm{t}}} = {\begin{cases} N&{0 \leqslant x \leqslant {L_{{\rm{cr}}}}} \\ 0&{x > {L_{{\rm{cr}}}}} \end{cases}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} c{\rm{ = }}\sqrt {\frac{{{A_{11}}}}{{\rho h}}}$

(11)

板发生振动屈曲时的变形能可以表示为

$\begin{split} U = &\displaystyle\frac{1}{2}\displaystyle\int_0^{{L_{\rm{cr}}}} {\displaystyle\int_0^{{L_{\rm{b}}}} {\left( {{N_x}\varepsilon _x^0 + {N_y}\varepsilon _y^0 + {N_{xy}}\gamma _{xy}^0 + {M_x}{\kappa _x} + {M_y}{\kappa _y} + {M_{xy}}{\kappa _{xy}}} \right)} } {\rm{d}}x{\rm{d}}y {\rm{ = }}\\ &\displaystyle\frac{1}{2}\displaystyle\int_0^{{L_{\rm{cr}}}} {\displaystyle\int_0^{{L_{\rm{b}}}} {\left[ {{N_x}\frac{{\partial {u_0}}}{{\partial x}} + {N_y}\frac{{\partial {v_0}}}{{\partial y}} + {N_{xy}}\left( {\frac{{\partial {u_0}}}{{\partial x}} + \frac{{\partial {v_0}}}{{\partial y}}} \right) + {M_x}\left( { - \frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}}} \right) + {M_y}\left( { - \frac{{{\partial ^2}{w_0}}}{{\partial {y^2}}}} \right) + {M_{xy}}\left( { - \frac{{2{\partial ^2}{w_0}}}{{\partial x\partial y}}} \right)} \right]} } {\rm{d}}x{\rm{d}}y \end{split}$

(12)

发生振动屈曲时的动能（考虑转动惯量）可以表示为

$\begin{split} T =& \displaystyle\frac{1}{2}\displaystyle\int_{ - {h}/{2}}^{{h}/{2}} {\int_0^{{L_{\rm{cr}}}} {\int_0^{{L_{\rm{b}}}} {{\rho _{\left( k \right)}}\left[ {{{\left( {\displaystyle\frac{{\partial u}}{{\partial t}}} \right)}^2} + {{\left( {\displaystyle\frac{{\partial v}}{{\partial t}}} \right)}^2} + {{\left( {\displaystyle\frac{{\partial {w_0}}}{{\partial t}}} \right)}^2}} \right]} } } {\rm{d}}x{\rm{d}}y{\rm{d}}z {\rm{ = }}\\ &\displaystyle\frac{1}{2}\int_0^{{L_{\rm{cr}}}} {\int_0^{{L_{\rm{b}}}} {\left[ {{I_0}{{\left( {\displaystyle\frac{{\partial {u_0}}}{{\partial t}}} \right)}^2} + {I_0}{{\left( {\displaystyle\frac{{\partial {v_0}}}{{\partial t}}} \right)}^2} + {I_0}{{\left( {\displaystyle\frac{{\partial {w_0}}}{{\partial t}}} \right)}^2} - 2{I_1}\displaystyle\frac{{\partial {u_0}}}{{\partial t}}\displaystyle\frac{{{\partial ^2}{w_0}}}{{\partial x\partial t}} - 2{I_1}\displaystyle\frac{{\partial {v_0}}}{{\partial t}}\displaystyle\frac{{{\partial ^2}{w_0}}}{{\partial y\partial t}} + {I_2}{{\left( {\displaystyle\frac{{{\partial ^2}{w_0}}}{{\partial x\partial t}}} \right)}^2} + {I_2}{{\left( {\displaystyle\frac{{{\partial ^2}{w_0}}}{{\partial y\partial t}}} \right)}^2}} \right]} } {\rm{d}}x{\rm{d}}y{\rm{ }} \end{split}$

(13)

式中： $\left( {{I_0},\; {I_1},\; {I_2}} \right) = \displaystyle\sum\limits_{k = 1}^{{N_k}} {\int_{{h_{k - 1}}}^{{h_k}} {{\rho _{\left( k \right)}}} } \left( {1, \;z, \;{z^2}} \right){\rm{d}}z$ ， ${{\rho _{\left( k \right)}}}$ 为第k层材料的密度。

板发生振动屈曲时的外力功可以表示为

$W = \frac{1}{2}\int_0^{{L_{\rm{cr}}}} {\int_0^{{L_{\rm{b}}}} {{N_{\rm{t}}}{{\left( {\frac{{\partial w}}{{\partial x}}} \right)}^2}} } {\rm{d}}x{\rm{d}}y$

(14)

考虑Hamilton变分原理，即

${\rm{\delta}} \int_{{{{t}}_1}}^{{{{t}}_2}} {\left( {T - U + W} \right)} {\rm{ d}}t = 0$

(15)

将式(12)～式(14)代入式(15)并进行变分计算，可得

$\frac{{\partial {N_x}}}{{\partial x}} + \frac{{\partial {N_{xy}}}}{{\partial x}} - {I_0}\frac{{{\partial ^2}{u_0}}}{{\partial {t^2}}} + {I_1}\frac{{{\partial ^3}{w_0}}}{{\partial x\partial {t^2}}}{\rm{ = }}0$

(16)

$\frac{{\partial {N_y}}}{{\partial y}} + \frac{{\partial {N_{xy}}}}{{\partial y}} - {I_0}\frac{{{\partial ^2}{v_0}}}{{\partial {t^2}}} + {I_1}\frac{{{\partial ^3}{w_0}}}{{\partial y\partial {t^2}}}{\rm{ = }}0$

(17)

$\frac{{{\partial ^2}{M_x}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{M_y}}}{{\partial {y^2}}} - 2\frac{{{\partial ^2}{M_{xy}}}}{{\partial x\partial y}} - {N_{\rm t}}\left( {\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{w_1}}}{{\partial {x^2}}}} \right) = {I_0}\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}}{\kern 1pt} + {I_1}\left( {\frac{{{\partial ^3}{u_0}}}{{\partial x\partial {t^2}}} + \frac{{{\partial ^3}{v_0}}}{{\partial x\partial {t^2}}}} \right) - {I_2}\left( {\frac{{{\partial ^4}{w_0}}}{{\partial {x^2}\partial {t^2}}} + \frac{{{\partial ^4}{w_0}}}{{\partial {y^2}\partial {t^2}}}} \right)$

(18)

对于正交各向异性正规对称正交铺设的复合材料板，其刚度矩阵满足^[16]

${D}_{16}={D}_{26}=0,\;\;{A}_{16}={A}_{26}=0,\;\;{B}_{ij}=0$

(19)

根据 Kirchhoff 薄板理论及经典弹性理论，薄板中面在变形过程中没有伸长变形，将板的本构关系代入式(16)～式(18)中，略去含 ${u_0}$ 和 ${v_0}$ 的项，得到复合材料板在面向阶跃荷载激励下的控制方程

${D_{11}}\frac{{{\partial ^4}{w_0}}}{{\partial {x^4}}} + \left( {2{D_{12}} + 4{D_{66}}} \right)\frac{{{\partial ^4}{w_0}}}{{\partial {x^2}\partial {y^2}}} + {D_{22}}\frac{{{\partial ^4}{w_0}}}{{\partial {y^4}}} + {N_{\rm t}}\left( {\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{w_1}}}{{\partial {x^2}}}} \right) - {I_0}\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}} + {I_2}\left( {\frac{{{\partial ^4}{w_0}}}{{\partial {x^2}\partial {t^2}}} + \frac{{{\partial ^4}{w_0}}}{{\partial {y^2}\partial {t^2}}}} \right) = 0$

(20)

设三边简支和应力波传播到L_cr（ ${w_0}\left( {{L_{{\rm{cr}}}},y,t} \right) = {w'_0}\left( {{L_{{\rm{cr}}}},y,t} \right) = 0$ ）时的振型函数^[17]为

${w_0}\left( {x,y,t} \right) = {R_{ij}}\left[ {\sin \frac{{i{\text{π}} x}}{{{L_{{\rm{cr}}}}}} + \frac{i}{{i + 1}}\sin \frac{{\left( {i + 1} \right){\text{π}}x }}{{{L_{{\rm{cr}}}}}}} \right]\sin \frac{{j{\text{π}} {{y}}}}{{{L_{\rm{b}}}}}\sin \left( {\omega {\kern 1pt} t + \varphi } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}$

(21)

由屈曲模态确定的缺陷分布形式是板结构最有可能发生的屈曲形式，能够很好地确定结构的缺陷敏感性^[18]。对于复合材料板在制造过程中出现的初始几何缺陷，引入屈曲模态的 $\varepsilon$ 倍变形作为初始几何缺陷^[19]，可以表示为

${w_1}\left( {x,y} \right) = \varepsilon {R_{ij}}\left[ {\sin \frac{{i{\text{π}}x }}{{{L_{{\rm{cr}}}}}} + \frac{i}{{i + 1}}\sin \frac{{\left( {i + 1} \right){\text{π}} x}}{{{L_{{\rm{cr}}}}}}} \right]\sin \frac{{j{\text{π}} {{y}}}}{{{L_{\rm{b}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}$

(22)

式中：i、j为屈曲模态阶数， $i,j = 1,2,3, \cdots$ ；R_ij为板的第(i, j)阶模态幅值； $\varepsilon$ 表示初始几何缺陷系数。

根据式(21)和式(22)，利用棣莫弗公式对控制方程式(20)中的各项进行求导并化简，得到

$\begin{split} &{D_{11}}{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)^4}\left[ {{i^4} + {{\left( {i + 1} \right)}^4}} \right]\sin \left( {\omega t + \varphi } \right) + \left( {2{D_{12}} + 4{D_{66}}} \right){\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)^2}\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right]{\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)^2}\sin \left( {\omega t + \varphi } \right) + \\ &2{D_{22}}{\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)^4}\sin \left( {\omega t + \varphi } \right) - {N_{\rm{t}}}{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)^2}\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right]\left[ {\sin \left( {\omega t + \varphi } \right) + \varepsilon } \right] + 2{I_0}{\omega ^2}\sin \left( {\omega t + \varphi } \right) + \\ &{I_2}{\omega ^2}\left\{ {{{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)}^2}\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right] + 2{{\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)}^2}} \right\}\sin \left( {\omega t + \varphi } \right) = 0 \end{split}$

(23)

根据式(23)可以得到N的表达式

$\begin{split} N{\rm{ = }}&\left\{ {{D_{11}}{{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)}^4}\left[ {{i^4} + {{\left( {i + 1} \right)}^4}} \right] + \left( {2{D_{12}} + 4{D_{66}}} \right){{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)}^2}\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right]{{\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)}^2}} \right. + 2{D_{22}}{\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)^4} + 2{I_0}{\omega ^2} + \\ &{\left.{I_2}{\omega ^2}\left\{ {{{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)}^2}\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right] + 2{{\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)}^2}} \right\}\right\}} \displaystyle\frac{{\sin \left( {\omega t + \varphi } \right)}}{{{{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)}^2}\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right]\left[ {\sin \left( {\omega t + \varphi } \right) + \varepsilon } \right]}} \end{split}$

(24)

板发生屈曲时，临界条件为 $\omega = 0$ ^[20]，代入式(24)可得振动屈曲临界荷载为

${N_{\rm{cr}}}{\rm{ = }}{D_{11}}{\left( {\displaystyle\frac{{\text{π}} }{{{L_{\rm{cr}}}}}} \right)^2}\displaystyle\frac{{\left[ {{i^4} + {{\left( {i + 1} \right)}^4}} \right]\sin \varphi }}{{\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right]\left( {\sin \varphi + \varepsilon } \right)}} + \left( {2{D_{12}} + 4{D_{66}}} \right){\left( {\displaystyle\frac{{j{\text{π}} }}{{{L_{\rm{b}}}}}} \right)^2}\displaystyle\frac{{\sin \varphi}}{{\left( {\sin \varphi + \varepsilon } \right)}} + 2{D_{22}}{\left( {\displaystyle\frac{{{j^2}{\text{π}} }}{{L_{\rm{b}}^2}}} \right)^2}L_{{\rm{cr}}}^2\displaystyle\frac{{\sin \varphi}}{{\left[ {{i^2} + {{\left( {i + 1} \right)}^2}} \right]\left( {\sin \varphi + \varepsilon } \right)}}$

(25)

2. 算例分析

利用MATLAB数值分析应力波未反射时初始几何缺陷、初相位、铺层角度、屈曲模态阶数、铺设厚度以及铺层层数对复合材料板振动屈曲临界荷载的影响，使用的材料参数见表1^[21]。

表 1 复合材料板参数^[21]

Table 1. Material parameters of composite plate^[21]

E₁/GPa	E₂/GPa	G₁₂/GPa	μ₁₂	L_a/m	L_b/m
140.0	8.6	5.0	0.35	0.60	0.50

下载: 导出CSV

| 显示表格

图3显示了复合材料板的模态阶数i为1、2、3，j为1时的屈曲模态。当板的x方向模态增大时，x方向的屈曲模态第一峰值增大且波数增加，而y方向的屈曲模态呈正对称分布。模态阶数的增加使板振动的屈曲模态变得更复杂。

图 3 x方向模态取值增大时板的屈曲模态

Figure 3. Buckling mode of composite plate with increasing mode value in x direction

下载: 全尺寸图片幻灯片

设置7组算例，分别以初始几何缺陷、初相位、铺层角度、x和y两个方向屈曲模态阶数、铺层层数及铺设厚度为变量进行算例分析，研究以上因素对板振动屈曲临界荷载的影响，算例参数见表2。

表 2 算例分析参数表

Table 2. Example analysis parameter table

Group	Initial defect coefficient	Order of mode		Laying angle/(°)	Initial phase	Number of layers laid	Thickness of the plate/m
Group	Initial defect coefficient	x direction	y direction	Laying angle/(°)	Initial phase	Number of layers laid	Thickness of the plate/m
A	Variable	i = 1	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	0.01
B	0.1	Variable	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	0.01
C	0.1	i =1	Variable	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	0.01
D	0.1	i =1	j = 1	Variable	${\text{π}}$ /2	5	0.01
E	0.1	i =1	j = 1	[0, 0, 0, 0, 0]	Variable	5	0.01
F	0.1	i =1	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	Variable	0.01
G	0.1	i =1	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	Variable

下载: 导出CSV

| 显示表格

将A组数据代入式(25)中，可以得到不同初始缺陷系数对复合材料板振动屈曲的影响，如图4所示。由L_cr-N_cr曲线可知：在应力波传播过程中，N_cr呈指数下降，分为敏感区和非敏感区。应力波在L_cr < 0.4 m区域传播时，N_cr的变化较陡峭，该区域为敏感区；应力波在L_cr > 0.4 m区域传播时，N_cr的变化趋于平缓，该区域为非敏感区，因此敏感分界点为0.4 m。当选取的初始缺陷系数增大时，临界荷载N_cr也随之增大。在敏感区，初始缺陷系数对临界荷载N_cr的影响较大，且随应力波传播呈减小趋势。此外，初始缺陷系数对非敏感区的影响较小。图4表明，初始几何缺陷系数越大，板越容易发生屈曲。

图 4 不同初始缺陷系数条件下N_cr与L_cr的关系曲线

Figure 4. Relationship between N_cr and L_cr under different initial defect coefficients

下载: 全尺寸图片幻灯片

将B组数据代入式(25)中，可以得到不同x方向模态阶数对复合材料板振动屈曲的影响，如图5所示。由L_cr-N_cr曲线可知：当选取的x方向模态阶数增大时，临界荷载N_cr随之明显增大。在敏感区，x方向模态阶数对临界荷载N_cr的影响很大，并随应力波的传播不断减小，到达非敏感区之后影响较小并趋于稳定。图5表明，x方向模态阶数的增加会显著增大板的屈曲临界荷载。

图 5 x方向模态阶数不同时N_cr与L_cr的关系曲线

Figure 5. Relationship between N_cr and L_cr with different order of modes in x direction

下载: 全尺寸图片幻灯片

将C组数据代入式(25)中，可以得到不同的y方向模态阶数对复合材料板振动屈曲的影响，如图6所示。由L_cr-N_cr曲线可知：当选取的y方向模态阶数增大时，临界荷载N_cr也随之变大。在应力波传播过程中，在敏感区y方向模态阶数对临界载荷基本没有影响，而在非敏感区有极小的影响。图6表明，y方向模态阶数的变化对板屈曲临界荷载基本没有影响。

图 6 y方向模态阶数不同时N_cr与L_cr的关系曲线

Figure 6. Relationship between N_cr and L_cr with different order of modes in y direction

下载: 全尺寸图片幻灯片

将D组数据代入式(25)中，得到不同铺层角度对复合材料板振动屈曲的影响，如图7所示。由L_cr-N_cr曲线可知：在敏感区，不同的铺设角度对临界荷载N_cr的影响较大，且随应力波的传播不断减小，到达非敏感区后趋于稳定。图7表明，铺层角度小的单层板的层数越多，板的临界屈曲荷载越大，说明复合材料板的铺设角度直接影响板的屈曲临界荷载。

图 7 不同铺层角度条件下N_cr与L_cr的关系曲线

Figure 7. Relationship between N_cr and L_cr under different laying angles

下载: 全尺寸图片幻灯片

将E组数据代入式(25)中，得到不同初相位对复合材料板振动屈曲的影响，如图8所示。由L_cr-N_cr曲线可知：振型函数的初相位越大，对应的临界荷载越大。在敏感区，振型函数的初相位对临界荷载N_cr的影响较小，且随应力波的传播不断减小；到达非敏感区之后，影响趋于平缓。图8表明，振型函数的初相位越大，板的屈曲临界荷载越大。

图 8 不同初相位条件下N_cr与L_cr的关系曲线

Figure 8. Relationship between N_cr and L_cr under the condition of initial phase of different mode functions

下载: 全尺寸图片幻灯片

将F组数据代入式(25)中，得到不同铺层层数对复合材料板振动屈曲的影响，如图9所示。由L_cr-N_cr曲线可知：当按照不同层数铺设时，敏感区的临界荷载N_cr的变化较大，且随应力波的传播不断减小；到达非敏感区之后变化较小并趋于平缓。图9表明，对于厚度固定且对称铺设的板，当铺设层数达到7时，其屈曲荷载随层数增加趋于稳定。

图 9 不同铺层层数下N_cr与L_cr的关系曲线

Figure 9. Relationship between N_cr and L_cr under different laying modes

下载: 全尺寸图片幻灯片

将G组数据代入式(25)中，得到不同铺设厚度对复合材料板振动屈曲的影响，如图10所示。由L_cr-N_cr曲线可知：板的铺设厚度越大，对应的临界荷载越大。在敏感区，不同的板厚对临界荷载N_cr的影响很大，且随应力波的传播不断减小，到达非敏感区后趋于稳定。图10表明，复合材料板的铺设厚度将直接决定板的屈曲临界荷载。

图 10 不同铺设厚度下N_cr与L_cr的关系曲线

Figure 10. Relationship between N_cr and L_cr under different thicknesses

下载: 全尺寸图片幻灯片

3. 结　论

基于Kirchhoff薄板理论和Hamilton变分原理，建立了具有初始几何缺陷的四边简支复合材料板的振动控制方程。采用伽辽金法，选取符合边界条件的振型函数求解控制方程，得到屈曲临界载荷表达式。数值计算结果表明：应力波在未发生反射前的传播过程中，复合材料板的振动屈曲临界载荷随着临界长度的增大、铺设厚度的减小、初始几何缺陷系数的增大、振型函数初相位的减小而减小；复合材料板的各层铺层角度与荷载作用方向的夹角越小，屈曲临界载荷越大，当对称铺设层数达7层时，临界荷载趋于稳定。研究结果可为工程中复合材料板的结构设计与应用提供一定的参考。

图 1 复合材料板结构示意图

Figure 1. Schematic diagram of composite plate structure

下载: 全尺寸图片幻灯片

图 2 应力波传播示意图

Figure 2. Schematic diagram of stress wave propagation

下载: 全尺寸图片幻灯片

图 3 x方向模态取值增大时板的屈曲模态

Figure 3. Buckling mode of composite plate with increasing mode value in x direction

下载: 全尺寸图片幻灯片

图 4 不同初始缺陷系数条件下N_cr与L_cr的关系曲线

Figure 4. Relationship between N_cr and L_cr under different initial defect coefficients

下载: 全尺寸图片幻灯片

图 5 x方向模态阶数不同时N_cr与L_cr的关系曲线

Figure 5. Relationship between N_cr and L_cr with different order of modes in x direction

下载: 全尺寸图片幻灯片

图 6 y方向模态阶数不同时N_cr与L_cr的关系曲线

Figure 6. Relationship between N_cr and L_cr with different order of modes in y direction

下载: 全尺寸图片幻灯片

图 7 不同铺层角度条件下N_cr与L_cr的关系曲线

Figure 7. Relationship between N_cr and L_cr under different laying angles

下载: 全尺寸图片幻灯片

图 8 不同初相位条件下N_cr与L_cr的关系曲线

Figure 8. Relationship between N_cr and L_cr under the condition of initial phase of different mode functions

下载: 全尺寸图片幻灯片

图 9 不同铺层层数下N_cr与L_cr的关系曲线

Figure 9. Relationship between N_cr and L_cr under different laying modes

下载: 全尺寸图片幻灯片

图 10 不同铺设厚度下N_cr与L_cr的关系曲线

Figure 10. Relationship between N_cr and L_cr under different thicknesses

下载: 全尺寸图片幻灯片

表 1 复合材料板参数^[21]

Table 1. Material parameters of composite plate^[21]

E₁/GPa	E₂/GPa	G₁₂/GPa	μ₁₂	L_a/m	L_b/m
140.0	8.6	5.0	0.35	0.60	0.50

下载: 导出CSV

表 2 算例分析参数表

Table 2. Example analysis parameter table

Group	Initial defect coefficient	Order of mode		Laying angle/(°)	Initial phase	Number of layers laid	Thickness of the plate/m
Group	Initial defect coefficient	x direction	y direction	Laying angle/(°)	Initial phase	Number of layers laid	Thickness of the plate/m
A	Variable	i = 1	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	0.01
B	0.1	Variable	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	0.01
C	0.1	i =1	Variable	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	0.01
D	0.1	i =1	j = 1	Variable	${\text{π}}$ /2	5	0.01
E	0.1	i =1	j = 1	[0, 0, 0, 0, 0]	Variable	5	0.01
F	0.1	i =1	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	Variable	0.01
G	0.1	i =1	j = 1	[0, 0, 0, 0, 0]	${\text{π}}$ /2	5	Variable

下载: 导出CSV

参考文献(21)

[1]	GALOS J. Thin-ply composite laminates: a review [J]. Composite Structures, 2020, 236: 11192. doi: 10.1016/j.compstruct.2020.111920
[2]	KAYRAN A, VINSON J R. Free vibration analysis of laminated composite truncated circular conical shells [J]. AIAA Journal, 1990, 28(7): 1259–1269. doi: 10.2514/3.25203
[3]	SUN J B, XU X S, LIM C W. Stress waves and dynamic buckling of functionally graded cylindrical shells under combined axial impact and thermal load [J]. Acta Mechanica, 2015, 226(5): 1323–1339. doi: 10.1007/s00707-014-1244-8
[4]	毛柳伟, 王安稳, 邓磊, 等. 应力波作用下弹性直杆动力分叉屈曲研究 [J]. 振动与冲击, 2014, 33(6): 174–178. doi: 10.13465/j.cnki.jvs.2014.06.032 MAO L W, WANG A W, DENG L, et al. Dynamic bifurcation buckling of elastic rods under stress wave [J]. Journal of Vibration and Shock, 2014, 33(6): 174–178. doi: 10.13465/j.cnki.jvs.2014.06.032
[5]	LEPIK Ü. Dynamic buckling of elastic-plastic beams including effects of axial stress waves [J]. International Journal of Impact Engineering, 2001, 25(6): 537–552. doi: 10.1016/S0734-743X(00)00070-1
[6]	ZHANG S, XIAO H, QIANG F W, et al. Analytical model and numerical simulation for the effect of uncertain initial geometrical imperfection on the buckling of thin plate [C]//Proceedings of the 5th International Conference on Frontiers of Manufacturing Science and Measuring Technology. Taiyuan: Computer Science and Electronic Technology International Society, 2017: 8.
[7]	ABDELAZIZ H H, MEZIANE M A A, BOUSAHLA A A, et al. An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions [J]. Steel and Composite Structures, 2017, 25(6): 693–704. doi: 10.12989/SCS.2017.25.6.693
[8]	KOUCHAKZADEH M A, RAHGOZAR M, BOHLOOLY M. Buckling of laminated composite plates with elastically restrained boundary conditions [J]. Structural Engineering and Mechanics, 2020, 74(5): 577–588. doi: 10.12989/SEM.2020.74.5.577
[9]	CZAPSKI P, KUBIAK T. Influence of residual stresses on the buckling behaviour of thin-walled, composite tubes with closed cross-section: numerical and experimental investigations [J]. Composite Structures, 2019, 229: 111407. doi: 10.1016/j.compstruct.2019.111407
[10]	KUO S Y. Thermal buckling, vibration and flutter of composite laminates containing two non-uniformly distributed fibers [J]. Journal of Aeronautics, Astronautics and Aviation, 2016, 48(3): 173–182. doi: 10.6125/16-0505-888
[11]	VILLARREAL E, ABAJO D. Buckling and modal analysis of rotationally restrained orthotropic plates [J]. Progress in Aerospace Sciences, 2015, 78: 116–130. doi: 10.1016/j.paerosci.2015.06.005
[12]	EFTEKHARI S A, JAFARI A A. Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates [J]. Applied Mathematics and Mechanics, 2012, 33(1): 81–98. doi: 10.1007/s10483-012-1535-6
[13]	REHMAN S M, RAO C S. Vibration buckling and fracture analysis of a cracked cylindrical shell [J]. International Journal of Design Engineering, 2017, 7(1): 33–53. doi: 10.1504/IJDE.2017.10006427
[14]	SAYYAD A S, SHINDE B M, GHUGAL Y M. Bending, vibration and buckling of laminated composite plates using a simple four variable plate theory [J]. Latin American Journal of Solids and Structures, 2019, 13(3): 516–535. doi: 10.1590/1679-78252241
[15]	王震鸣. 复合材料力学和复合材料结构力学 [M]. 北京: 机械工业出版社, 1991. WANG Z M. Composite materials and structural mechanics [M]. Beijing: China Machine Press, 1991.
[16]	沈观林, 胡更开. 复合材料力学 [M]. 北京: 清华大学出版社, 2006. SHEN G L, HU G K. Mechanics of composite materials [M]. Beijing: Tsinghua University Press, 2006.
[17]	王鑫. 功能梯度材料板动力屈曲及混沌行为的研究 [D]. 太原: 太原理工大学, 2018. WANG X. Research on dynamic buckling and chaos behavior of functionally graded materials plate [D]. Taiyuan: Taiyuan University of Technology, 2018.
[18]	AMABILI M. Nonlinear vibrations and stability of shells and plates [M]. Cambridge: Cambridge University Press, 2008.
[19]	顾明剑, 张其林. 引入一阶屈曲模态作为初始缺陷的研究[C]//第五届全国现代结构工程学术研讨会论文集. 广州: 全国现代结构工程学术研讨会学术委员会, 2005.
[20]	KHDEIR A A, REDDY J N, FREDERICK D. A study of bending, vibration and buckling of cross-ply circular cylindrical shells with various shell theories [J]. International Journal of Engineering Science, 1989, 27(11): 1337–1351. doi: 10.1016/0020-7225(89)90058-X
[21]	矫桂琼, 贾普荣. 复合材料力学 [M]. 西安: 西北工业大学出版社, 2008. JIAO G Q, JIA P R. Mechanics of composite materials [M]. Xi’an: Northwestern Polytechnical University Press, 2008.