Acoustic and Elastic Properties of Polycrystalline Potassium Sodium Niobate under High Pressures

XIAO Likang; FENG Qiu; FANG Leiming; ZHOU Zhangyang; XIONG Zhengwei; LAN Jianghe; YANG Jia; LIU Yi; GAO Zhipeng

doi:10.11858/gywlxb.20230660

Volume 37 Issue 6

Dec 2023

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References

Chinese Journal of High Pressure Physics > 2023 > 37(6): 061101.

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

Citation:

XIAO Likang, FENG Qiu, FANG Leiming, ZHOU Zhangyang, XIONG Zhengwei, LAN Jianghe, YANG Jia, LIU Yi, GAO Zhipeng. Acoustic and Elastic Properties of Polycrystalline Potassium Sodium Niobate under High Pressures[J]. Chinese Journal of High Pressure Physics, 2023, 37(6): 061101. doi: 10.11858/gywlxb.20230660

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Acoustic and Elastic Properties of Polycrystalline Potassium Sodium Niobate under High Pressures

doi: 10.11858/gywlxb.20230660

1.
School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, China
2.
School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang 621010, Sichuan, China
3.
Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621999, Sichuan, China
4.
Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621999, Sichuan, China

Received Date: 15 May 2023
Rev Recd Date: 14 Jun 2023

Available Online: 22 Oct 2023

Issue Publish Date: 15 Dec 2023

Abstract

Abstract

Polycrystalline Na_0.5K_0.5NbO₃ ceramics were prepared under the condition of 10 GPa and 1050 ℃. The compression and shear wave velocities of the samples under various hydrostatic pressures were measured using ultrasonic interferometry. By fitting the third-order finite strain state equation, the bulk and shear moduli of polycrystalline Na_0.5K_0.5NbO₃ and their pressure dependency were determined as 172.6 GPa, 54.6 GPa, 0.3 and 2.1, respectively. The sample shows a positive pressure dependence of Young’s modulus. Based on the elastic modulus data, the Poisson’s ratio was obtained as 0.342, indicating that the material was ductile and shows brittleness under high pressure. The Vickers hardness and fracture toughness also show positive pressure dependence. Using the empirical models, the Vickers hardness and fracture toughness of the ceramics were obtained as 2.40 GPa and 2.33 MPa·m^1/2. Meanwhile, based on the elastic wave velocity and density data, the Debye temperature (513.1 K) and the Grüneisen constant (2.113) of Na_0.5K_0.5NbO₃ ceramics were derived. Our results provide an experimental reference for evaluating the acoustic and elastic-related comprehensive properties of Na_0.5K_0.5NbO₃ under high pressure, and a foundation for its engineering applications under extreme high-pressure conditions as well.
- Na_0.5K_0.5NbO₃,
- high pressure,
- sound velocity,
- elasticity,
- hardness,
- fracture toughness

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功能梯度材料(Functionally Graded Materials, FGM)的概念是1984年在航空飞机计划中首次提出的^[1]，FGM的特性在于它的组成和结构随着体积的变化而变化，从而导致材料相应性质发生改变。因其材料特性呈幂律分布^[2-3]，FGM被广泛应用于工程领域，如航空航天、机械工程、生物医学等。圆柱壳在联合荷载作用下的屈曲分析备受学术界关注^[4-5]。目前，对FGM板壳的研究较为深入^[6-8]。Beni等^[9]利用改进的偶应力理论，对FGM圆柱壳在不同边界条件下的动力屈曲进行了分析；Kargarnovin等^[10]研究了轴向荷载作用下FGM圆柱壳的动力屈曲；Sofiyev等^[11]研究了横向压力下功能梯度正交各向异性圆柱壳的动力屈曲，推导出基于一阶剪切变形理论的功能梯度正交各向异性圆柱壳的稳定性和相容性方程；Khazaeinejad等^[12]研究了弹性模量在厚度方向上连续变化的FGM圆柱壳在复合外压和轴向压缩载荷作用下的动力屈曲；Khalili等^[13]研究了横向冲击载荷作用下FGM圆柱壳的动力屈曲；Alashti等^[14]对变厚度FGM圆柱壳外压和轴向压缩的动力屈曲问题进行了分析。

基于以上研究，本研究讨论了FGM圆柱壳在轴向荷载作用下的动力屈曲。根据Donnell壳体理论和经典板壳理论，利用Hamilton变分原理得到FGM圆柱壳的动力屈曲控制方程；采用分离变量法求得动力屈曲临界荷载表达式；通过MATLAB软件计算动力屈曲临界荷载，讨论由不同材料(陶瓷和钛、陶瓷和铁、陶瓷和铜)组成的FGM圆柱壳的径厚比(R/h)、梯度指数(k)、环向模态数(m)、轴向模态数(n)等对临界荷载的影响。

1. FGM圆柱壳的屈曲控制方程

如图 1所示，FGM圆柱壳长度为l，半径为R，总厚度为h，选取柱坐标系(x, θ, z)，其相应位移为(u, v, w)。FGM的材料属性(弹性模量E、密度ρ、泊松比μ等)呈幂律分布^[2-3]，表示为

$P\left( z \right) = \left( {{P_1} - {P_2}} \right){\left( {\frac{{2z + h}}{{2h}}} \right)^k} + {P_2}$

(1)

图 1 圆柱壳坐标系统

Figure 1. Cylindrical shell coordinates

下载: 全尺寸图片幻灯片

式中：P为物性参数，下标“1”和“2”分别代表组分1和组分2；k为梯度指数，k∈(0, ∞)。圆柱壳内任意点的物性参数为

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

(2)

根据Donnell壳体理论，圆柱壳的小挠度几何方程为

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

(3)

式中：ε为正应变，γ为切应变，上、下标“0”表示壳体中面，K为壳体曲率。

根据经典板壳理论，FGM圆柱壳的内力N与内力矩M可表示为

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

(4)

式中：A_ij、B_ij、D_ij(i, j=1, 2, 6)分别为FGM圆柱壳的拉伸刚度、耦合刚度和弯曲刚度系数矩阵分量。 ${A_{11}} = {A_{22}} = \int_{ - h/2}^{h/2} {\frac{{E\left( z \right)}}{{1 - {\mu ^2}\left( z \right)}}} {\rm{d}}z$ , ${A_{12}} = {A_{21}} = \int_{ - h/2}^{h/2} {\frac{{\mu \left( z \right)E\left( z \right)}}{{1 - {\mu ^2}\left( z \right)}}} {\rm{d}}z$ , ${A_{66}} = \int_{ - h/2}^{h/2} {\frac{{E\left( z \right)}}{{2\left[ {1 + \mu \left( z \right)} \right]}}} {\rm{d}}z$ , ${B_{11}} = {B_{22}} = \int_{ - h/2}^{h/2} {\frac{{E\left( z \right)}}{{1 - {\mu ^2}\left( z \right)}}z\;}{\rm{d}}z$ , ${B_{12}} = {B_{21}} = \int_{ - h/2}^{h/2} {\frac{{\mu \left( z \right)E\left( z \right)}}{{1 - {\mu ^2}\left( z \right)}}z\;} {\rm{d}}z$ , ${B_{66}} = \int_{ - h/2}^{h/2} {\frac{{E\left( z \right)}}{{2\left[ {1 + \mu \left( z \right)} \right]}}z\;} {\rm{d}}z$ , ${D_{11}} = {D_{22}} = \int_{ - h/2}^{h/2} {\frac{{E\left( z \right)}}{{1 - {\mu ^2}\left( z \right)}}{z^2}\;} {\rm{d}}z$ , ${D_{12}} = {D_{21}} = \int_{ - h/2}^{h/2} {\frac{{\mu \left( z \right)E\left( z \right)}}{{1 - {\mu ^2}\left( z \right)}}{z^2}\;} {\rm{d}}z$ , ${D_{66}} = \int_{ - h/2}^{h/2} {\frac{{E\left( z \right)}}{{2\left[ {1 + \mu \left( z \right)} \right]}}{z^2}\;} {\rm{d}}z$ 。FGM圆柱壳的力学性能为各向同性^[15]，那么：A₁₆=A₂₆=B₁₆=B₂₆=D₁₆=D₂₆=0。

对于圆柱壳，系统的应变能(不考虑剪力)为

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

(5)

动能为

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

(6)

外力功为

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

(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)

将(3)式~(7)式代入(8)式中，由Donnell壳体理论可知，圆柱壳内力沿环向均匀分布，忽略中面位移^[16]，由u₀、v₀、w₀的变分系数为零，整理得到FGM圆柱壳的动力屈曲控制方程为

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

(9)

2. 控制方程的求解

设径向位移表示为^[17]

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

(10)

将(10)式代入(9)式中，分离变量得

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

(11)

其中

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

(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)

当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)

式中：C₁~C₄为系数， ${k_1} = \sqrt {\frac{{{\alpha ^2} - \sqrt {{\alpha ^4} - 4{\beta ^2}} }}{2}} , {k_2} = \sqrt {\frac{{{\alpha ^2} + \sqrt {{\alpha ^4} - 4{\beta ^2}} }}{2}}$ 。

(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)

将(14)式代入(15)式中，整理得到如下齐次线性方程组

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

(17)

若要(17)式有非平凡解，其系数行列式必为零，于是

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

(18)

由k₁和k₂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)

将(19)式代入(12)式和(13)式中，得一端夹支另一端固支时FGM圆柱壳动力屈曲临界荷载N_cr，即

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

(20)

式中：n₁=n=1, 2, 3, …；m=1, 2, 3, …；n₂=n+2。

同理可得一端简支另一端固支时FGM圆柱壳动力屈曲临界荷载

(21)

此时，n₁=n=1, 2, 3, …；m=1, 2, 3, …；n₂=n+1。

将FGM退化成金属材料，得到金属材料圆柱壳动力屈曲临界荷载

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

(22)

(22)式与文献[16]中的表达式相同。

根据(10)式，取一端夹支另一端固支时圆柱壳动力屈曲解的表达式^[16]

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

(23)

将(23)式代入控制方程(9)式中，计算并化简整理得到临界荷载表达式

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

(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

采用MATLAB软件编程，对FGM圆柱壳动力屈曲临界荷载进行计算。讨论由不同材料(陶瓷-钛、陶瓷-铁、陶瓷-铜)组成的FGM圆柱壳(见图 2)的径厚比(R/h)、梯度指数(k)、环向模态数(m)、轴向模态数(n)对临界荷载N_cr的影响。基本材料参数如表 1所示。

图 2 材料沿壁厚分布

Figure 2. Distribution of material along wall thickness

下载: 全尺寸图片幻灯片

表 1 材料参数

Table 1. Material parameters

Material	E/GPa	ρ/(g·cm^-3)	μ
Ceramic	385	3.96	0.230
Ti	109	4.54	0.410
Fe	155	7.86	0.291
Cu	119	8.96	0.326

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| 显示表格

图 3表示n=1、m=2、k=1、R/h=20时，不同材料组成下N_cr与临界长度l(本研究中临界长度即为圆柱壳长度)的关系曲线。从图 3可以看出：N_cr随l的增加而减小；当l＜0.5 m时，N_cr随l的增加而迅速减小；当l＞0.5 m时，N_cr随l的增大缓慢减小，且逐渐趋于常数；同一l下，陶瓷-铜的N_cr最大，陶瓷-铁次之，陶瓷-钛的N_cr最小。以下均以陶瓷-钛为例进行讨论。

图 3 不同材料组成下临界荷载与临界长度的关系

Figure 3. Critical load vs.critical length under different material composition conditions

下载: 全尺寸图片幻灯片

图 4和图 5分别表示冲击端为夹支和简支时n=1、m=2、k=1时不同R/h下N_cr与l的关系曲线。可以看出：当l增加时，N_cr减小，且逐渐趋于常数；在同一l下圆柱壳的N_cr随着R/h的增大而减小；当R/h和l一定时，冲击端为夹支时的N_cr明显比冲击端为简支时的N_cr大，说明约束条件对N_cr有较大影响。

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

Figure 4. Critical load vs.critical length under clamped edge and different diameter-thickness ratios conditions

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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

Figure 5. Critical load vs.critical length under simple support and different diameter-thickness ratios conditions

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图 6和图 7分别表示冲击端为夹支和简支条件下n=1、m=2、R/h=20时不同k下N_cr与l的关系曲线。可见：FGM圆柱壳的N_cr随着l的增加而减小；在同一l下，FGM圆柱壳的N_cr随着k的增加而增加；当l＜0.5 m时，N_cr随l的增加迅速减小，当l＞0.5 m时，N_cr随l的增加缓慢减小并逐渐趋于常数；当k=1且l一定时，冲击端为夹支时的N_cr明显比冲击端为简支时的N_cr大，再次说明约束条件对N_cr的影响较大。

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

Figure 6. Critical load vs.critical length under clamped edge and different gradient indexes conditions

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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

Figure 7. Critical load vs.critical length under simple support and different gradient indexes conditions

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图 8和图 9分别表示冲击端为夹支和简支，n=1、k=1、R/h=20时不同m下N_cr与l的关系曲线。可以看出：当l在一定范围内时，N_cr随l的增加而迅速减小，超出这一范围后N_cr随l的增加而缓慢减小且逐渐趋于常数；同一l下，随着m的增大，FGM圆柱壳的N_cr增大，表明N_cr越大，高阶模态越易被激发。当m=6且l一定时，冲击端为夹支时的N_cr明显比冲击端为简支时的N_cr大，表明约束条件对N_cr有较大影响。

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

Figure 8. Critical load vs.critical length under clamped edge and different circumferential modal number conditions

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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

Figure 9. Critical load vs.critical length under simple support and different circumferential modal number conditions

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图 10和图 11分别表示冲击端为夹支和简支，m=1、k=1、R/h=20时不同n下N_cr与l的关系曲线。图 10和图 11显示：N_cr随着l的增加而减小；不同n条件下，N_cr随l的增加逐渐趋于同一值；当l＜1 m时，在同一l下N_cr随n的增加而增加，说明N_cr越大，高阶模态越容易被激发。

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

Figure 10. Critical load vs.critical length under clamped edge and different axial modal number conditions

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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

Figure 11. Critical load vs.critical length under simple support and different axial modal number conditions

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图 12为不同环向模态数m下FGM圆柱壳的动力屈曲模态。可以看出：随着m的增大，圆柱壳的模态变得越来越复杂，俯视图由单一形变为多瓣形；当m=6时，俯视图为12瓣形。

图 12 不同环向屈曲模态图(n=1;m=1, 2, 3, 4, 5, 6)

Figure 12. Different circumferential buckling modes (n=1;m=1, 2, 3, 4, 5, 6)

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图 13为不同轴向模态数n下FGM圆柱壳的动力屈曲模态图。由图 13可知：随着n的增加，模态图变得越来越复杂。由FGM圆柱壳的俯视图可知，各阶模态数下动力屈曲模态图为轴对称。

图 13 不同轴向屈曲模态图(m=2;n=1, 2, 3, 4, 5, 6)

Figure 13. Different axial buckling modes (m=2;n=1, 2, 3, 4, 5, 6)

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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n、m、k、l一定的情况下，临界荷载随着径厚比的增大而减小；在n、m、R/h、l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。

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