﻿ 热过屈曲功能梯度壁板的气动弹性颤振
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 力学学报  2016, Vol. 48 Issue (3): 609-614  DOI: 10.6052/0459-1879-15-361 0

### 引用本文 [复制中英文]

[复制中文]
Xia Wei, Feng Haocheng. AEROELASTIC FLUTTER OF POST-BUCKLED FUNCTIONALLY GRADED PANELS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 609-614. DOI: 10.6052/0459-1879-15-361.
[复制英文]

### 文章历史

2015-09-25 收稿
2015-11-23 录用
2015-12-01 网络版发表.

1 力学模型 1.1 材料等效力学性能

 ${{V}_{\text{c}}}={{\left( \frac{z}{h}+\frac{1}{2} \right)}^{n}},-h/2\le z\le h/2$ (1a)
 ${{V}_{\text{m}}}=1-{{V}_{\text{c}}}$ (1b)

 $P(z)={{P}_{\text{c}}}{{V}_{\text{c}}}+{{P}_{\text{m}}}{{V}_{\text{m}}}$ (2)

1.2 非线性壁板颤振方程

 $\left. \begin{matrix} u(x,y,z)={{u}_{0}}(x,y)+z{{\theta }_{y}}(x,y) \\ v(x,y,z)={{v}_{0}}(x,y)+z{{\theta }_{x}}(x,y) \\ w(x,y,z)={{w}_{0}}(x,y) \\ \end{matrix} \right\}$ (3)

 图1 功能梯度壁板示意图 Fig.1 Schematic of a functionally graded (FG) panel

 $\left. \begin{array}{*{35}{l}} {{\varepsilon }_{x}}=\frac{\partial {{u}_{0}}}{\partial x}+z\frac{\partial {{\theta }_{y}}}{\partial x}+\frac{1}{2}{{\left( \frac{\partial w}{\partial x} \right)}^{2}} \\ {{\varepsilon }_{y}}=\frac{\partial {{v}_{0}}}{\partial y}+z\frac{\partial {{\theta }_{x}}}{\partial y}+\frac{1}{2}{{\left( \frac{\partial w}{\partial y} \right)}^{2}} \\ {{\gamma }_{xy}}=\frac{\partial {{u}_{0}}}{\partial y}+\frac{\partial {{v}_{0}}}{\partial x}+z\left( \frac{\partial {{\theta }_{x}}}{\partial x}+\frac{\partial {{\theta }_{y}}}{\partial y} \right)+\frac{\partial w}{\partial x}\frac{\partial w}{\partial y} \\ \end{array} \right\}$ (4)

 $\left. \begin{array}{*{35}{l}} {{\gamma }_{yz}}={{\theta }_{x}}+\frac{\partial w}{\partial y} \\ {{\gamma }_{zx}}={{\theta }_{y}}+\frac{\partial w}{\partial x} \\ \end{array} \right\}$ (5)

 $\sigma =Q\left( \varepsilon -\alpha \Delta T \right)$ (6)

 $Q=\frac{E(z)}{1-\upsilon {{\left( z \right)}^{2}}}\left[ \begin{matrix} 1 & \upsilon (z) & 0 \\ \upsilon (z) & 1 & 0 \\ 0 & 0 & (1-\upsilon (z))/2 \\ \end{matrix} \right]$ (7)

 $\tau ={{Q}_{S}}\gamma$ (8)

 ${{Q}_{S}}=\frac{E(z)}{2\left( 1+\upsilon (z) \right)}\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$ (9)

 $\delta W_{\text{int}}^{e}=\int_{\Omega }{\left( \delta {{\varepsilon }^{\text{T}}}\sigma +\beta \delta {{\gamma }^{\text{T}}}\tau \right)}dV$ (10)

 $\Delta p=p-{{p}_{\infty }}=\frac{{{\rho }_{\text{a}}}{{v}^{2}}}{\sqrt{{{M}^{2}}-1}}\left( \frac{\partial w}{\partial x}+\frac{{{M}^{2}}-2}{{{M}^{2}}-1}\frac{1}{v}\frac{\partial w}{\partial t} \right)$ (11)

 $\lambda ={{\rho }_{\text{a}}}{{v}^{2}}{{l}^{3}}/({{D}_{\text{m}}}\sqrt{{{M}^{2}}-1})$ (12a)
 ${{R}_{M}}={{\left( \frac{{{M}^{2}}-2}{{{M}^{2}}-1} \right)}^{2}}\frac{{{\rho }_{\text{a}}}l}{{{\rho }_{\text{m}}}h\sqrt{{{M}^{2}}-1}}$ (12b)
(12b)

 $\delta W_{\text{ext}}^{\text{e}}=-\int_{\Omega }{\delta }w\rho \left( z \right)\ddot{w}dV-\int_{S}{\delta }w\Delta pdS$ (13)

 $m\ddot{w}+{{c}_{\text{a}}}\dot{w}+\left( {{k}_{0}}+{{k}_{\text{a}}}+{{k}_{\text{G}}}-{{k}_{\text{T}}}+\frac{1}{2}{{n}_{1}}+\frac{1}{3}{{n}_{2}} \right)w={{p}_{\text{T}}}$ (14)

2 热屈曲变形

 $\left( {{k}_{0}}+{{k}_{\text{a}}}+{{k}_{\text{G}}}-{{K}_{\text{T}}}+{{n}_{1}}/2+{{n}_{2}}/3 \right)w={{p}_{\text{T}}}$ (15)

 图2 功能梯度壁板有限元模型 Fig.2 Finite element mesh of a FG panel
 图3 功能梯度壁板的热屈曲变形 Fig.3 Thermal buckling deflection of a FG panel

 图4 热过屈曲变形形态 Fig.4 Thermal post-buckling deformation shape

3 壁板颤振

 $w={{w}_{\text{s}}}+\delta {{w}_{\text{d}}}$ (16)

 \begin{align} & m\delta {{{\ddot{w}}}_{\text{d}}}+{{c}_{\text{a}}}\delta {{{\dot{w}}}_{\text{d}}}+ \\ & \left( {{k}_{\text{L}}}+{{n}_{1}}\left( {{w}_{\text{s}}} \right)+{{n}_{2}}\left( {{w}_{\text{s}}} \right) \right)\delta {{w}_{\text{d}}}=0 \\ \end{align} (17)

 图5 功能梯度壁板颤振边界 Fig.5 Flutter boundary of a FG panel
4 结论

(1)沿板厚的材料梯度分布会破坏结构的对称性，导致陶瓷-金属功能梯度壁板在面内热应力作用下发生指向金属侧的热屈曲变形.

(2)超声速气流中壁板热屈曲变形最大的位置随气流速压增大向下游推移，并伴随有屈曲变形量的减小.无量纲气流速压足够大时，壁板热过屈曲变形的幅值会受到超声速气动力的压制，难以出现大挠度变形.

(3)热过屈曲壁板的几何非线性效应会提高壁板的颤振边界，这种影响在高温、低速压且壁板发生大挠度热屈曲变形时表现显著.较高无量纲气流速压下由于壁板的热屈曲变形被气动力限定在小挠度范围，几何非线性的影响不明显.