﻿ 轴向荷载下功能梯度材料圆柱壳的动力屈曲
 高压物理学报   2018, Vol. 32 Issue (5): 054102.  DOI: 10.11858/gywlxb.20180502.

## 引用本文 [复制中英文]

[复制中文]
ZHOU Jiahua, YANG Qiang, HAN Zhijun, LU Guoyun. Dynamic Buckling of Functionally Graded Cylindrical Shells under Axial Loading[J]. Chinese Journal of High Pressure Physics, 2018, 32(5): 054102. DOI: 10.11858/gywlxb.20180502.
[复制英文]

### 文章历史

( 1. 太原理工大学力学学院, 山西 太原 030024
2. 太原理工大学建筑与土木工程学院, 山西 太原 030024 )

1 FGM圆柱壳的屈曲控制方程

 $P\left( z \right) = \left( {{P_1} - {P_2}} \right){\left( {\frac{{2z + h}}{{2h}}} \right)^k} + {P_2}$ (1)
 图 1 圆柱壳坐标系统 Fig.1 Cylindrical shell coordinates

 $\left\{ \begin{array}{l} E\left( z \right) = \left( {{E_1} - {E_2}} \right){\left( {\frac{{2z + h}}{{2h}}} \right)^k} + {E_2}\\ \rho \left( z \right) = \left( {{\rho _1} - {\rho _2}} \right){\left( {\frac{{2z + h}}{{2h}}} \right)^k} + {\rho _2}\\ \mu \left( z \right) = \left( {{\mu _1} - {\mu _2}} \right){\left( {\frac{{2z + h}}{{2h}}} \right)^k} + {\mu _2} \end{array} \right.$ (2)

 $\begin{array}{*{20}{c}} {\left\{ \begin{array}{l} {\varepsilon _x} = \varepsilon _x^0 + z{K_x}\\ {\varepsilon _\theta } = \varepsilon _\theta ^0 + z{K_\theta }\\ {\gamma _{x\theta }} = \gamma _{x\theta }^0 + z{K_{x\theta }} \end{array} \right.,}&{\left\{ \begin{array}{l} u = {u_0} - z\frac{{\partial w}}{{\partial x}}\\ v = {v_0} - z\frac{{\partial w}}{{R\partial \theta }}\\ w = {w_0} \end{array} \right.,}&{\left\{ \begin{array}{l} \varepsilon _x^0 = \frac{{\partial {u_0}}}{{\partial x}}\\ \varepsilon _\theta ^0 = \frac{{\partial {v_0}}}{{R\partial \theta }} - \frac{{{w_0}}}{R}\\ \gamma _{x\theta }^0 = \frac{{\partial {u_0}}}{{R\partial \theta }} - \frac{{\partial {v_0}}}{{\partial x}} \end{array} \right.,}&{\left\{ \begin{array}{l} {K_x} = - \frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}}\\ {K_\theta }{\rm{ = }} - \frac{{{\partial ^2}{w_0}}}{{{R^2}\partial {\theta ^2}}}\\ {K_{x\theta }}{\rm{ = }} - \frac{{2{\partial ^2}{w_0}}}{{R\partial x\partial \theta }} \end{array} \right.} \end{array}$ (3)

 $\left( {\begin{array}{*{20}{c}} {{N_x}}\\ {{N_\theta }}\\ {{N_{x\theta }}}\\ {{M_x}}\\ {{M_\theta }}\\ {{M_{x\theta }}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}&{{A_{16}}}&{{B_{11}}}&{{B_{12}}}&{{B_{16}}}\\ {{A_{21}}}&{{A_{22}}}&{{A_{26}}}&{{B_{21}}}&{{B_{22}}}&{{B_{26}}}\\ {{A_{16}}}&{{A_{26}}}&{{A_{66}}}&{{B_{16}}}&{{B_{26}}}&{{B_{66}}}\\ {{B_{11}}}&{{B_{12}}}&{{B_{16}}}&{{D_{11}}}&{{D_{12}}}&{{D_{16}}}\\ {{B_{21}}}&{{B_{22}}}&{{B_{26}}}&{{D_{21}}}&{{D_{22}}}&{{D_{26}}}\\ {{B_{16}}}&{{B_{26}}}&{{B_{66}}}&{{D_{16}}}&{{D_{26}}}&{{D_{66}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\varepsilon _x^0}\\ {\varepsilon _\theta ^0}\\ {\gamma _{x\theta }^0}\\ {{K_x}}\\ {{K_\theta }}\\ {{K_{x\theta }}} \end{array}} \right)$ (4)

 $U = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {\int_0^l {\left( {{N_x}\varepsilon _x^0 + {N_\theta }\varepsilon _\theta ^0 + {N_{x\theta }}\gamma _{x\theta }^0 + {M_x}{K_x} + {M_\theta }{K_\theta } + {M_{x\theta }}{K_{x\theta }}} \right)} R{\rm{d}}x{\rm{d}}\theta }$ (5)

 $T = \frac{1}{2}\int_{ - h/2}^{h/2} {\int_0^{2{\rm{ \mathsf{ π} }}} {\int_0^l {\rho \left( z \right)\left[ {{{\left( {\frac{{\partial u}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial v}}{{\partial t}}} \right)}^2} + {{\left( {\frac{{\partial w}}{{\partial t}}} \right)}^2}} \right]R{\rm{d}}x{\rm{d}}\theta {\rm{d}}z} } }$ (6)

 $W = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {\int_0^l {N\left( t \right){{\left( {\frac{{\partial {w_0}}}{{\partial x}}} \right)}^2}} R{\rm{d}}x{\rm{d}}\theta }$ (7)

Hamilton变分原理为

 ${\rm{ \mathsf{ δ} }}\int_{{t_2}}^{{t_1}} {\left( {T - U + W} \right){\rm{d}}t} = 0$ (8)

 $\begin{array}{l} 4{I_0}\frac{{{\partial ^2}{w_0}}}{{\partial t}} - {I_2}\left( {\frac{{{\partial ^4}{w_0}}}{{{R^2}\partial {\theta ^2}\partial {t^2}}} + \frac{{{\partial ^4}{w_0}}}{{\partial {x^2}\partial {t^2}}}} \right) = - 4{A_{22}}\frac{{{w_0}}}{{{R^2}}} - 4\frac{{{B_{12}}}}{{{R^2}}}\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} - 4\frac{{{B_{22}}}}{R}\frac{{{\partial ^2}{w_0}}}{{{R^2}\partial {\theta ^2}}} - {D_{11}}\frac{{{\partial ^4}{w_0}}}{{\partial {x^4}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\left( {{D_{12}} + 2{D_{66}}} \right)\frac{{{\partial ^4}{w_0}}}{{{R^2}\partial {x^2}\partial {\theta ^2}}} - {D_{22}}\frac{{{\partial ^4}{w_0}}}{{{R^4}\partial {\theta ^4}}} + N\left( t \right)\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} \end{array}$ (9)
2 控制方程的求解

 $w = Y\left( x \right)T\left( t \right){{\rm{e}}^{{\rm{i}}m\theta }}$ (10)

 $\left\{ \begin{array}{l} {Y^{\left( 4 \right)}} = {\alpha ^2}Y'' + {\beta ^2}Y = 0\\ \ddot T - \lambda T = 0 \end{array} \right.$ (11)

 ${\alpha ^2} = \frac{1}{{{D_{11}}}}\left[ {4\frac{{{B_{12}}}}{{{R^2}}} + \frac{2}{{{R^2}}}\left( {{D_{12}} + 2{D_{66}}} \right){m^2} - N\left( t \right) - {I_2}\lambda } \right]$ (12)
 ${\beta ^2} = \frac{1}{{{D_{11}}}}\left[ {4\frac{{{A_{22}}}}{{{R^2}}} + 4\frac{{{B_{22}}}}{{{R^3}}} + \frac{{{D_{22}}}}{{{R^4}}}{m^4} + 4{I_0}\lambda - \frac{{{I_2}}}{{{R^2}}}{m^2}\lambda } \right]$ (13)

α4＞4β2＞0且λ＞0时，圆柱壳屈曲[18-20]，其动力屈曲解为

 $Y\left( x \right) = {C_1}\sin \left( {{k_1}x} \right) + {C_2}\cos \left( {{k_1}x} \right) + {C_3}\sin \left( {{k_2}x} \right) + {C_4}\cos \left( {{k_2}x} \right)$ (14)

(14)式满足下列两种边界条件：

(1) 对于一端夹支另一端固支的圆柱壳，其边界条件为

 $\left\{ \begin{array}{l} Y\left( 0 \right) = Y'\left( 0 \right) = 0\\ Y\left( l \right) = Y'\left( l \right) = 0 \end{array} \right.$ (15)

(2) 对于一端简支另一端固支的圆柱壳，其边界条件为

 $\left\{ \begin{array}{l} Y\left( 0 \right) = Y''\left( 0 \right) = 0\\ Y\left( l \right) = Y'\left( l \right) = 0 \end{array} \right.$ (16)

 $\left( {\begin{array}{*{20}{c}} 0&1&0&1\\ {{k_1}}&0&{{k_2}}&0\\ {\sin \left( {{k_1}l} \right)}&{\cos \left( {{k_1}l} \right)}&{\sin \left( {{k_2}l} \right)}&{\cos \left( {{k_2}l} \right)}\\ {{k_1}\cos \left( {{k_1}l} \right)}&{ - {k_1}\sin \left( {{k_1}l} \right)}&{{k_2}\cos \left( {{k_2}l} \right)}&{ - {k_2}\sin \left( {{k_2}l} \right)} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{C_1}}\\ {{C_2}}\\ {{C_3}}\\ {{C_4}} \end{array}} \right) = 0$ (17)

 $2{k_1}{k_2} - 2{k_1}{k_2}\cos \left( {{k_1}l} \right)\cos \left( {{k_2}l} \right) - \left( {k_1^2 + k_2^2} \right)\sin \left( {{k_1}l} \right)\sin \left( {{k_2}l} \right) = 0$ (18)

k1k2可得

 $\left\{ \begin{array}{l} k_1^2 + k_2^2 = \left( {n_1^2 + n_2^2} \right){{\rm{ \mathsf{ π} }}^2}/{l^2} = {\alpha ^2}\\ k_1^2k_2^2 = n_1^2n_2^2{{\rm{ \mathsf{ π} }}^4}/{l^4} = {\beta ^2} \end{array} \right.$ (19)

 ${N_{{\rm{cr}}}} = \frac{{{D_{11}}{{\rm{ \mathsf{ π} }}^2}\left( {n_1^2 + n_2^2} \right)}}{{{l^2}}} - \frac{{{B_{12}}}}{{{R^2}}} + \frac{{2\left( {{D_{12}} + 2{D_{66}}} \right){m^2}}}{{{R^2}}} + \frac{{{h^2}\left( {{D_{11}}n_1^2n_2^2{{\rm{ \mathsf{ π} }}^4}{R^4} - {A_{22}}{l^4}{R^2} + 4{B_{22}}{m^2}{R^2}{l^4} - {D_{22}}{l^4}{m^4}} \right)}}{{{l^4}{R^2}\left( {{h^2}{m^2} - 12{R^2}} \right)}}$ (20)

 ${N_{{\rm{cr}}}} = \frac{{{D_{11}}{{\rm{ \mathsf{ π} }}^2}\left( {n_1^2 + n_2^2} \right)}}{{{l^2}}} - \frac{{{B_{12}}}}{{{R^2}}} + \frac{{2\left( {{D_{12}} + 2{D_{66}}} \right){m^2}}}{{{R^2}}} + \frac{{{h^2}\left( {{D_{11}}n_1^2n_2^2{{\rm{ \mathsf{ π} }}^4}{R^4} - {A_{22}}{l^4}{R^2} + 4{B_{22}}{m^2}{R^2}{l^4} - {D_{22}}{l^4}{m^4}} \right)}}{{{l^4}{R^2}\left( {{h^2}{m^2} - 12{R^2}} \right)}}$ (21)

 ${N_{{\rm{cr}}}} = \frac{{{D_{11}}{{\rm{ \mathsf{ π} }}^2}\left( {n_1^2 + n_2^2} \right)}}{{{l^2}}} + \frac{{2{D_{12}}{m^2}}}{{{R^2}}} + \frac{{{h^2}\left( {{D_{11}}n_1^2n_2^2{{\rm{ \mathsf{ π} }}^4}{R^4} - {A_{22}}{l^4}{R^2} - {D_{22}}{l^4}{m^4}} \right)}}{{{l^4}{R^2}\left( {{h^2}{m^2} - 12{R^2}} \right)}}$ (22)

(22)式与文献[16]中的表达式相同。

 $w = T\left( t \right)\left[ {\sin \left( {\frac{{{n_1}{\rm{ \mathsf{ π} }}x}}{l}} \right) - \frac{{{n_1}}}{{{n_2}}}\sin \left( {\frac{{{n_2}{\rm{ \mathsf{ π} }}x}}{l}} \right)} \right]\sin \left( {m\theta } \right)$ (23)

 ${N_{{\rm{cr}}}} = \frac{{\left( {n_1^2 + n_2^2} \right){D_{11}}{{\rm{ \mathsf{ π} }}^2}}}{{{l^2}}} + \frac{{2{m^2}\left( {{D_{12}} + 2{D_{66}}} \right)}}{{{R^2}}}$ (24)

(24)式与不考虑转动惯量时用分离变量得到的结果相同，此时n1=n=1, 2, 3, …; m=1, 2, 3, …; n2=n+2。同理可得当边界条件为一端简支另一端固支时的临界荷载表达式，与(24)式相同，此时n1=n=1, 2, 3, …; m=1, 2, 3, …; n2=n+1。

3 算例分析

 图 2 材料沿壁厚分布 Fig.2 Distribution of material along wall thickness

 图 3 不同材料组成下临界荷载与临界长度的关系 Fig.3 Critical load vs.critical length under different material composition conditions

 图 4 冲击端为夹支时不同径厚比下临界荷载与临界长度的关系 Fig.4 Critical load vs.critical length under clamped edge and different diameter-thickness ratios conditions
 图 5 冲击端为简支时不同径厚比下临界荷载与临界长度的关系 Fig.5 Critical load vs.critical length under simple support and different diameter-thickness ratios conditions

 图 6 冲击端为夹支时不同梯度指数下临界荷载与临界长度的关系 Fig.6 Critical load vs.critical length under clamped edge and different gradient indexes conditions
 图 7 冲击端为简支时不同梯度指数下临界荷载与临界长度的关系 Fig.7 Critical load vs.critical length under simple support and different gradient indexes conditions

 图 8 冲击端为夹支时不同环向模态数下临界荷载与临界长度的关系 Fig.8 Critical load vs.critical length under clamped edge and different circumferential modal number conditions
 图 9 冲击端为简支时不同环向模态数下临界荷载与临界长度的关系 Fig.9 Critical load vs.critical length under simple support and different circumferential modal number conditions

 图 10 冲击端为夹支时不同轴向模态数下临界荷载与临界长度的关系 Fig.10 Critical load vs.critical length under clamped edge and different axial modal number conditions
 图 11 冲击端为简支时不同轴向模态数下临界荷载与临界长度的关系 Fig.11 Critical load vs.critical length under simple support and different axial modal number conditions

 图 12 不同环向屈曲模态图(n=1;m=1, 2, 3, 4, 5, 6) Fig.12 Different circumferential buckling modes (n=1;m=1, 2, 3, 4, 5, 6)

 图 13 不同轴向屈曲模态图(m=2;n=1, 2, 3, 4, 5, 6) Fig.13 Different axial buckling modes (m=2;n=1, 2, 3, 4, 5, 6)
4 结论

(1) 根据Donnell壳体理论和经典板壳理论，由Hamilton变分原理得到轴向荷载作用下FGM圆柱壳的动力屈曲控制方程。

(2) 由圆柱壳周向连续性设出径向位移的周向形式，并用分离变量法得到不同约束条件下FGM圆柱壳动力屈曲临界荷载的表达式和屈曲解式。

(3) 利用MATLAB对临界荷载进行计算，得到：在轴向模态数(n)、环向模态数(m)、梯度指数(k)、径厚比(R/h)一定的情况下，同种材料组成的圆柱壳的临界荷载随着临界长度的增加而减小；在nmkl一定的情况下，临界荷载随着径厚比的增大而减小；在nmR/hl一定的情况下，临界荷载随着梯度指数k的增加而增加；不同约束条件下，冲击端为夹支的临界荷载大于冲击端为简支的临界荷载，表明约束条件对临界荷载有较大影响；圆柱壳的临界荷载随模态数的增加而增大，表明临界荷载越大，越容易激发高阶模态；圆柱壳的动力屈曲模态随模态数的增加变得更为复杂。