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 哈尔滨工程大学学报  2018, Vol. 39 Issue (12): 1880-1886  DOI: 10.11990/jheu.201705103 0

### 引用本文

HAN Yueyang, ZHU Xiang, LI Tianyun, et al. Prediction of elastic critical load of functionally graded cylindrical shells[J]. Journal of Harbin Engineering University, 2018, 39(12), 1880-1886. DOI: 10.11990/jheu.201705103.

### 文章历史

1. 华中科技大学 船舶与海洋工程学院, 湖北 武汉 430074;
2. 船舶与海洋水动力湖北省重点实验室, 湖北 武汉 430074

HAN Yueyang 1, ZHU Xiang 1,2, LI Tianyun 1,2, YU Yunyan 1
1. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;
2. Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Wuhan 430074, China
Abstract: To solve the critical load problem of functionally graded cylindrical shells, a semi-analytic approach for predicting the elastic critical load is proposed in this paper. The coupled vibration equation was established using the thin shell theory, and natural frequencies of the fluid-filled functionally graded cylindrical shell were solved by the parabola method. The elastic critical load of the functionally graded cylindrical shell was obtained by an interpolation method through the linear relationship between natural frequency squared and the internal pressure. The accuracy of the proposed method was verified by comparison with published results. The elastic critical load of a silicon nitride-stainless steel functionally graded cylindrical shell was calculated, and the effects of different boundary conditions and structural parameters were analyzed. The proposed method for predicting the elastic critical loads of functionally graded cylindrical shells based on the natural frequencies under internal pressure conditions is a new nondestructive prediction method. Compared with the experimental and simulation methods under external pressure conditions, the method is simple and easier to implement.
Keywords: functionally graded material    cylindrical shell    internal pressure    coupled equation    natural frequency    critical load    interpolation prediction

1 结构模型及材料属性 1.1 功能梯度圆柱壳模型

1.2 功能梯度材料的特性

 $P = {P_0}\left( {{P_{ - 1}}{T^{ - 1}} + 1 + {P_1}T + {P_2}{T^2} + {P_3}{T^3}} \right)$ (1)

 $P = \sum\limits_{k = 1}^t {{P_k}{V_{fk}}}$ (2)

 $\sum\limits_{k = 1}^t {{V_{fk}}} = 1$ (3)

 ${V_f} = {\left( {\frac{z}{h} + 0.5} \right)^N}$ (4)

 $\left\{ \begin{array}{l} E = \left( {{E_o} - {E_i}} \right){\left( {\frac{z}{h} + 0.5} \right)^N} + {E_i}\\ \mu = \left( {{\mu _o} - {\mu _i}} \right){\left( {\frac{z}{h} + 0.5} \right)^N} + {\mu _i}\\ \rho = \left( {{\rho _o} - {\rho _i}} \right){\left( {\frac{z}{h} + 0.5} \right)^N} + {\rho _i} \end{array} \right.$ (5)

2 动力平衡方程

 $\begin{array}{*{20}{c}} {\left( {1 + {T_1}} \right){u_{xx}} + \left( {{T_2} + \frac{{1 - \mu }}{2}} \right){u_{\theta \theta }} + \frac{{1 + \mu }}{2}{v_{x\theta }} + }\\ {\left( {\mu - {T_2}} \right){w_x} + K\left( {\frac{{1 - \mu }}{2}{u_{\theta \theta }} - {w_{xxx}} + \frac{{1 - \mu }}{2}{w_{x\theta \theta }}} \right) = }\\ {\frac{{{\rho _s}{R^2}\left( {1 - {\mu ^2}} \right)}}{E}{u_{tt}}} \end{array}$ (6)
 $\begin{array}{*{20}{c}} {\frac{{1 + \mu }}{2}{u_{x\theta }} + \left( {{T_1} + \frac{{1 - \mu }}{2}} \right){v_{xx}} + \frac{{1 + \mu }}{2}{v_{\theta \theta }} + }\\ {\left( {1 + {T_2}} \right){w_\theta } + K\left( {\frac{{3\left( {1 - \mu } \right)}}{2}{v_{xx}} - \frac{{3 - \mu }}{2}{w_{xx\theta }}} \right) = }\\ {\frac{{{\rho _s}{R^2}\left( {1 - {\mu ^2}} \right)}}{E}{v_{tt}}} \end{array}$ (7)
 $\begin{array}{*{20}{c}} {\left( {\mu - {T_2}} \right){u_x} - K{u_{xxx}} + K\frac{{1 - \mu }}{2}{u_{x\theta \theta }} + }\\ {\left( {1 + {T_2}} \right){v_\theta } - K\frac{{3 - \mu }}{2}{v_{xx\theta }} + \left( {1 + K} \right)w + }\\ {K{w_{xxxx}} + 2K{w_{xx\theta \theta }} + K{w_{\theta \theta \theta \theta }} + \left( {2K - {T_2}} \right){w_{\theta \theta }} - }\\ {{T_1}{w_{xx}} + \frac{{{R^2}\left( {1 - {\mu ^2}} \right)}}{{Eh}}{P_f} = \frac{{{\rho _s}{R^2}\left( {1 - {\mu ^2}} \right)}}{E}{v_{tt}}} \end{array}$ (8)
 ${\left( {\;\;\;} \right)_x} = R\frac{{\partial \left( {\;\;\;} \right)}}{{\partial x}},{\left( {\;\;\;} \right)_\theta } = \frac{{\partial \left( {\;\;\;} \right)}}{{\partial \theta }},$
 ${\left( {\;\;\;} \right)_t} = \frac{{\partial \left( {\;\;\;} \right)}}{{\partial t}},K = \frac{{{h^2}}}{{12{R^2}}};{T_1} = \frac{{R\left( {1 - {\mu ^2}} \right)}}{{2Eh}}{p_0},$
 ${T_2} = \frac{{R\left( {1 - {\mu ^2}} \right)}}{{Eh}}{p_0},$

 $\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial {P_f}}}{{\partial r}}} \right) + \frac{1}{{{r^2}}}\frac{{{\partial ^2}{P_f}}}{{\partial {\theta ^2}}} - \frac{1}{{C_f^2}}\frac{{{\partial ^2}{P_f}}}{{\partial {t^2}}} = 0$ (9)

 $\left\{ \begin{array}{l} u\left( {x,\theta ,t} \right) = {A_m}\cos \left( {n\theta } \right)\exp \left( {{\rm{i}}\omega t - {\rm{i}}{k_n}x} \right)\\ v\left( {x,\theta ,t} \right) = {B_m}\sin \left( {n\theta } \right)\exp \left( {{\rm{i}}\omega t - {\rm{i}}{k_n}x} \right)\\ w\left( {x,\theta ,t} \right) = {C_m}\cos \left( {n\theta } \right)\exp \left( {{\rm{i}}\omega t - {\rm{i}}{k_n}x} \right) \end{array} \right.$ (10)

 ${P_f} = {P_{ns}}\cos \left( {n\theta } \right){\rm{H}}_n^{\left( 2 \right)}\left( {{k_r}r} \right)\exp \left( {{\rm{i}}\omega t - {\rm{i}}{k_n}x} \right)$ (11)

 $- \frac{1}{{{\rm{i}}\omega {\rho _f}}}\frac{{\partial {P_f}}}{{\partial r}}\left| {_{r = R}} \right. = \frac{{\partial w}}{{\partial t}}\left| {_{r = R}} \right.$ (12)

 ${P_{ns}} = \left[ {{\omega ^2}{\rho _f}/{k_r}H_n^{\left( 2 \right)}\left( {{k_r}r} \right)} \right]{C_m}$ (13)

 $\left[ {\begin{array}{*{20}{c}} {{c_{11}}}&{{c_{12}}}&{{c_{13}}}\\ {{c_{21}}}&{{c_{22}}}&{{c_{23}}}\\ {{c_{31}}}&{{c_{32}}}&{{c_{33}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{A_m}}\\ {{B_m}}\\ {{C_m}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 0\\ 0 \end{array}} \right]$ (14)

 ${c_{11}} = {\mathit{\Omega }^2} - \left( {1 + {T_1}} \right){\left( {{k_n}R} \right)^2} - \left[ {{T_2} + \left( {1 + \frac{{{h^2}}}{{12{R^2}}}} \right)\frac{{1 - \mu }}{2}} \right]{n^2}$
 ${c_{12}} = {c_{21}} = - {\rm{i}}{k_n}Rn\frac{{1 + \mu }}{2}$
 ${c_{13}} = {c_{31}} = - {\rm{i}}\left[ {\left( {\mu - {T_2}} \right){k_n}R + \frac{{{h^2}}}{{12}}{k_n}^3R - \frac{{{h^2}}}{{12R}}\frac{{1 - \mu }}{2}{k_n}{n^2}} \right]$
 ${c_{22}} = \left[ {{T_1} + \left( {1 + \frac{{{h^2}}}{{4{R^2}}}} \right)\frac{{1 - \mu }}{2}} \right]{k_n}^2{R^2} + \left( {1 + {T_2}} \right){n^2} - {\mathit{\Omega }^2}$
 ${c_{23}} = \left( {1 + {T_2}} \right)n + \frac{{{h^2}}}{{12}}\frac{{3 - \mu }}{2}{k_n}^2n$
 $\begin{array}{l} {c_{33}} = 1 + \frac{{{h^2}}}{{12{R^2}}} + \frac{{{h^2}}}{{12}}{k_n}^4{R^2} + \frac{{{h^2}}}{6}{n^2}{k_n}^2 + \frac{{{h^2}}}{{12{R^2}}}{n^4} - \\ \;\;\;\;\;\;\;\left( {\frac{{{h^2}}}{{4{R^2}}} - {T_2}} \right){n^2} + {T_1}{k_n}^2{R^2} - {\mathit{\Omega }^2} + {\rm{FL}} \end{array}$

 ${\rm{FL}} = - {\mathit{\Omega }^2}\frac{{{\rho _{\rm{f}}}}}{{{\rho _{\rm{s}}}}}\frac{R}{h}\frac{{H_n^{\left( 2 \right)}\left( {{k_r}R} \right)}}{{\left( {{k_r}R} \right)H{'}_n^{\left( 2 \right)}\left( {{k_r}R} \right)}}$ (15)

 $\left| \mathit{\boldsymbol{c}} \right| = 0$ (16)

 $f\left( \mathit{\Omega } \right) = 0$ (17)

3 功能梯度材料圆柱壳的临界载荷计算

3.1 计算方法正确性验证

3.1.1 钢制圆柱壳的临界载荷预测

 Download: 图 2 两端简支圆柱壳的固有频率的平方随内压的变化 Fig. 2 Variation of the square of natural frequencies of cylindrical shells with simply supported-simply supported boundary condition as variation of internal pressure

3.1.2 功能梯度圆柱壳固有频率计算

3.2 功能梯度材料圆柱壳的临界载荷计算

 Download: 图 3 功能梯度圆柱壳固有频率的平方与内压关系 Fig. 3 Relationship between square of natural frequency and internal pressure of functionally graded cylindrical shell

4 结论

1) 在相同模态阶数下，固有频率的平方与压力呈线性关系；

2) 通过固有频率的平方和内压的线性关系，插值求得临界载荷，并通过与文献对比，验证了本文方法的正确性；

3) 不同边界条件下，临界载荷在两端固支状态较其他两种状态大；

4) 功能梯度圆柱壳只改变厚度时，厚壳较稳定；

5) 功能梯度圆柱壳只改变长度时，短壳较稳定；

6) 在本文中梯度指数N越大，功能梯度圆柱壳的临界载荷值越小，这是由于N值越大，临界载荷值越接近材料i所构成圆柱壳的临界载荷值。

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