爆炸地震动下矩形储液结构动力响应理论研究

张浩天; 宋春明; 王明洋; 赵雪川; 吴红晓; 郑际镜

doi:10.11883/bzycj-2023-0099

爆炸地震动下矩形储液结构动力响应理论研究

doi: 10.11883/bzycj-2023-0099

张浩天^1,,
宋春明^1, ,,
王明洋^{1, 2},
赵雪川³,
吴红晓³,
郑际镜³

1.
陆军工程大学爆炸冲击防灾减灾国家重点实验室，江苏南京 210007
2.
南京理工大学机械工程学院，江苏南京 210094
3.
中国人民解放军96911部队，北京 100010

基金项目: 国家自然科学基金（51808553）

详细信息

作者简介:
张浩天（1996－　），男，博士研究生，335733770@qq.com

通讯作者:
宋春明（1979－　），男，博士，副教授，ming1979@126.com

中图分类号: O342; TV31
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- 文章访问数: 377
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- 被引次数: 0
出版历程
- 收稿日期: 2023-03-17
- 修回日期: 2023-05-10
- 网络出版日期: 2023-07-04
- 刊出日期: 2023-08-31

Theoretical study on the dynamic response of rectangular liquid storage structure under explosion-induced ground shock

1.
Disaster Prevention and Mitigation of Explosion and Impact, Army Engineering University of PLA, Nanjing 210007, iangsu, China
2.
School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China
3.
Unit 96911 of PLA, Beijing 100010, China

摘要

摘要: 为完善防护工程储液结构设计与评估体系，开展了爆炸冲击地震动作用下储液结构动力响应理论研究。将矩形储液结构简化为具有分布弹性的广义单自由度体系，采用虚功原理建立水平地震动下结构运动方程，通过双向梁函数组合法、Rayleigh法和Duhamel积分法分别得到储液结构壁板振型、振动频率和动力响应，进而构建地震响应谱。利用爆炸冲击震动模拟平台开展模型试验，结构测点应变、动水压力计算值与试验数据基本一致，验证了理论方法。通过算例分析储液率、地震动要素对模型结构动力响应的影响，构建爆炸地震动下储液结构挠度响应谱，结果表明：随储液率增加，结构基频降低，地震动激励特征因子先提高后降低，后者反映流固耦合对地震作用的强化效应先增强后减弱；弹性范围内，随地震动加速度峰值提高，结构挠度响应线性提高；地震动加速度持时和波形改变影响频谱特性，使挠度响应发生非线性变化；典型波形爆炸地震动的作用效果均可划分为相对于等效静力作用的缓和区、增强区和等效区；以响应谱峰值作为最不利响应进行防护设计偏于保守，考虑场地爆炸参数范围进行计算可提高工程设计的经济性。
- 储液结构 /
- 广义单自由度体系 /
- 虚功原理 /
- 流固耦合 /
- 爆炸地震动 /
- 响应谱
Abstract: To improve the design and evaluation system of liquid storage structure (LSS) in protection engineering, theoretical research on the dynamic response of LSS subjected to explosion-induced ground shock has been carried out. The rectangular LSS was simplified into a generalised single-degree-of-freedom system with distributed elasticity. The motion equation under horizontal ground shock was established based on the virtual work principle. The vibration mode function, vibration frequency, and dynamic response of the rectangular plate were obtained using the two-way beam function combination, the Rayleigh method, and the Duhamel’s integration method, respectively. The influences of liquid filling ratio, and ground-shock essentials (i.e. the peak, duration, waveform of ground acceleration) on the dynamic response of the model LSS were analysed by calculation examples. The maximum deflection was used as an index to build the dynamic response spectrum of the LSS subjected to explosion-induced ground shocks. The results showed that, with the increase of liquid filling ratio, the fundamental frequency of the structure decreases, and the characteristic factor of ground motion excitation first increase and then decrease. The latter reflects that the strengthening effect of fluid-structure interaction on seismic action is first enhanced and then weakened. Within the elastic range, as the peak value of ground acceleration increases, the deflection response of the LSS increases linearly. The varations in the duration and waveform of ground acceleration affect the spectrum characteristics, causing the nonlinear changes of deflection response. The effects of explosion-induced ground shocks featured by various typical waveforms can be divided into the mitigation, enhancement, and equality regions relative to the equivalent static action. It is conservative to take the peak of response spectrum as the most adverse response for protection design, whereas the calculation considering the range of site explosion parameters would improve the economy of engineering design. The proposed simplified theoretical method meets the requirement of preliminary rapid calculation and provides a reference for the protection design of LSS.
- liquid storage structure (LSS) /
- generalised single-degree-of-freedom system /
- virtual work principle /
- fluid-structure interaction /
- explosion-induced ground shock /
- response spectrum

HTML全文

压电材料可以制造成执行器或传感器等智能元件，广泛应用于国防工业与实际生活中。由于压电材料中力学与电学性质相互耦合，在SH波作用下压电材料中夹杂或圆孔等缺陷处的动应力集中及电场强度集中问题也比一般材料更复杂。近年来，许多学者对缺陷问题进行了研究，并取得了丰富的成果^[1-12]。X.F.Li等^[1]基于电磁材料弹性理论研究了径向非均匀性的压电压磁球壳的静态响应问题；时朋朋等^[2]利用分离变量法和Hilbert核奇异积分方程理论研究了功能梯度压电压磁双材料的周期界面裂纹问题；靳静等^[3]利用积分变换法和奇异积分方程技术研究了压电压磁双材料界面裂纹的二维断裂问；舒小平^[4-5]基于等效单层理论的位移场和电势场求解了正交压电复合材料层板在各类边界条件下的解析解；宋天舒等^[6-7]研究了双相压电介质中圆孔与界面裂纹相互作用的动力学问题。但是，以上工作中大部分是关于径向非均匀介质的静态响应问题的求解，对含圆孔的压电介质在SH波作用下的动态响应问题，目前仍未见报道。

1. 控制方程

含圆孔的全空间非均匀压电介质如图 1所示，已知其密度ρ(r)=ρ₁β²r^2(β-1)，其中ρ₁为常数，β为幂次。弹性常数、压电常数、介电常数分别为c₄₄、e₁₅、κ₁₁；圆孔内部可以形成电场，其压电常数为e₁₅^c，介电常数为κ₁₁^c。在直角坐标系中：r²=x²+y²，ρ(x, y)=ρ₁β²(x²+y²)^(β-1)。满足控制方程：

$\left\{ \begin{array}{l} {c_{44}}{\nabla ^2}w + {e_{15}}{\nabla ^2}\varphi + \rho \left( {x, y} \right){\omega ^2}w = 0\\ {e_{15}}{\nabla ^2}w - {\kappa _{11}}{\nabla ^2}\varphi = 0 \end{array} \right.$

(1)

图 1 含圆孔径向非均匀压电介质模型

Figure 1. Model of the radial inhomogeneous piezoelectric medium with a circular cavity

下载: 全尺寸图片幻灯片

式中：w和φ分别为压电材料的位移和电势，ω为SH波的圆频率。令φ=e₁₅(w+f)/κ₁₁，对式(1)化简得：

$\left\{ \begin{array}{l} {\nabla ^2}w + k_0^2{\beta ^2}{\left( {{x^2} + {y^2}} \right)^{\left( {\beta - 1} \right)}}w = 0\\ {\nabla ^2}f = 0 \end{array} \right.$

(2)

波数满足：

${k^2} = \rho {w^2}/{c^ * } = k_0^2{\beta ^2}{\left( {{x^2} + {y^2}} \right)^{\left( {\beta - 1} \right)}}$

(3)

式中：k为波数；k₀²=ρ₁ω²/c^*，c^*为压电介质的剪切波速，且c^*=c₄₄+e₁₅²/κ₁₁。

利用复变函数法，令z=x+iy, z=x-iy，在复平面(η, η)中控制方程可化为：

$\left\{ \begin{array}{l} \frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + \frac{1}{4}{\beta ^2}{\left( {z\bar z} \right)^{\beta - 1}}k_0^2w = 0\\ \frac{{{\partial ^2}f}}{{\partial z\partial \bar z}} = 0 \end{array} \right.$

(4)

引入变量ζ=z^β，ζ=z^β，控制方程可进一步转化为：

$\frac{{{\partial ^2}w}}{{\partial \zeta \partial \bar \zeta }} + \frac{1}{4}k_0^2w = 0$

(5)

本构方程为：

$\left\{ \begin{array}{l} {\tau _{rz}} = \left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\left( {\frac{{\partial w}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} + \frac{{\partial w}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right) + \frac{{e_{15}^2}}{{{\kappa _{11}}}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} + \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)\\ {\tau _{\theta z}} = {\rm{i}}\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\left( {\frac{{\partial w}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial w}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right) + {\rm{i}}\frac{{e_{15}^2}}{{{\kappa _{11}}}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)\\ {D_r} = - {e_{15}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} + \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)\\ {D_\theta } = - {\rm{i}}{e_{15}}\left( {\frac{{\partial f}}{{\partial z}}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial f}}{{\partial \bar z}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right) \end{array} \right.$

(6)

式中：τ_rz和τ_θz分别为非均匀压电介质的径向应力和切向应力，D_r和D_θ分别为圆孔中电场的径向电位移和切向电位移。

2. 介质中的位移场

SH波散射过程中，入射波引起的压电材料位移wⁱⁿ表达式为：

${w^{{\rm{in}}}} = {w_0}\exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]$

(7)

散射波引起的压电材料位移w^s表达式为：

${w^{\rm{s}}} = \frac{{\rm{i}}}{{2{c_{44}}\left( {1 + \lambda } \right)}}\sum\limits_{n = - \infty }^{ + \infty } {{A_n}H_n^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^n}}$

(8)

式中：H_n⁽¹⁾( $k\left| \zeta \right|$ )为n阶第一类Hankel函数，λ=e₁₅²/(c₄₄κ₁₁)，A_n为系数。

$\varphi = \frac{{{e_{15}}}}{{{\kappa _{11}}}}\left( {{w^{{\rm{in}}}} + {w^{\rm{s}}} + {f^{\rm{s}}}} \right)$

(9)

散射波引起的电场附加函数f^s表达式为：

${f^{\rm{s}}} = \sum\limits_{n = 1}^{ + \infty } {{B_n}{z^{ - n}}} + {C_n}{{\bar z}^{ - n}}$

(10)

式中：B_n和C_n为系数。由此得到：

$\left\{ \begin{array}{l} \tau _{rz}^{{\rm{in}}} = \frac{{{\rm{i}}k}}{2}\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\beta {w_0}\left( {{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\left( {\theta - {\alpha _0}} \right)}} + {{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\left( {\theta - {\alpha _0}} \right)}}} \right)\\ \exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]\\ \tau _{rz}^{\rm{s}} = \frac{{{\rm{i}}\beta k}}{4}\sum\limits_{n = - \infty }^{ + \infty } \\ {{A_n}\left[ {H_{n - 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n - 1}}{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\theta }} - H_{n + 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n + 1}}{{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right]} \\ \; - \frac{{e_{15}^2}}{{{\kappa _{11}}}}n\left( {\sum\limits_{n = 1}^{ + \infty } {{B_n}{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} } \right)\\ D_r^{\rm{s}} = {e_{15}}n\left( {\sum\limits_{n = 1}^{ + \infty } {{B_n}{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} } \right) \end{array} \right.$

(11)

式中：上标“in”、“s”分别表示物理量与入射波、反射波相关。圆孔内部存在电场，满足方程：

$\frac{{{\partial ^2}{f^{\rm{c}}}}}{{\partial z\partial \bar z}} = 0$

(12)

式中：f^c为圆孔内部的电场附加函数。求解式(12)可得：

${f^{\rm{c}}} = \sum\limits_{n = 0}^{ + \infty } {{D_n}{z^n}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}{{\bar z}^n}}$

(13)

式中：D_n和E_n为系数。由此可得：

$\left\{ \begin{array}{l} \tau _{rz}^{\rm{c}} = 0\\ {\varphi ^{\rm{c}}} = \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{f^{\rm{c}}}\\ D_r^{\rm{c}} = - e_{15}^{\rm{c}}n\left( {\sum\limits_{n = 0}^{ + \infty } {{D_n}{z^{n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}{{\bar z}^{n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} } \right) \end{array} \right.$

(14)

式中：上标“c”表示物理量与圆孔中空气形成的电场相关。

3. 边界条件与定解方程

圆孔处的边界条件为：

$\begin{array}{*{20}{c}} {{\tau _{rz}} = \tau _{rz}^{{\rm{in}}} + \tau _{rz}^{\rm{s}} = \tau _{rz}^{\rm{c}} = 0}&{\varphi = {\varphi ^{\rm{c}}}}&{D_r^{\rm{s}} = D_r^{\rm{c}}} \end{array}$

(15)

利用以上边界条件式(15)建立关于A_n、B_n、C_n、D_n、E_n的方程组：

$\begin{array}{l} {\xi ^{\left( 1 \right)}} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}\xi _n^{\left( {11} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{B_n}\xi _n^{\left( {12} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}\xi _n^{\left( {13} \right)}} \\ {\xi ^{\left( 2 \right)}} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}\xi _n^{\left( {21} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{B_n}\xi _n^{\left( {22} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}\xi _n^{\left( {23} \right)}} + \sum\limits_{n = 0}^{ + \infty } {{D_n}\xi _n^{\left( {24} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}\xi _n^{\left( {25} \right)}} \\ {\xi ^{\left( 3 \right)}} = \sum\limits_{n = 1}^{ + \infty } {{B_n}\xi _n^{\left( {32} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{C_n}\xi _n^{\left( {33} \right)}} + \sum\limits_{n = 0}^{ + \infty } {{D_n}\xi _n^{\left( {34} \right)}} + \sum\limits_{n = 1}^{ + \infty } {{E_n}\xi _n^{\left( {35} \right)}} \end{array}$

(16)

式中：

$\left\{ \begin{array}{l} \xi _n^{\left( {11} \right)} =\\ \frac{{{\rm{i}}\beta k}}{4}\left[ {H_{n - 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n - 1}}{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\theta }} - H_{n + 1}^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){{\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)}^{n + 1}}{{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right]\\ \xi _n^{\left( {12} \right)} = - \frac{{e_{15}^2}}{{{\kappa _{11}}}}n{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}\;\;\;\;\;\xi _n^{\left( {13} \right)} = - \frac{{e_{15}^2}}{{{\kappa _{11}}}}n{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}\;\;\;\;\\ \xi _n^{\left( {21} \right)} = \frac{{{\rm{i}}{e_{15}}}}{{2{c_{44}}{\kappa _{11}}\left( {1 + \lambda } \right)}}H_n^{\left( 1 \right)}\left( {k\left| \zeta \right|} \right){\left( {\frac{\zeta }{{\left| \zeta \right|}}} \right)^n}\\ \xi _n^{\left( {22} \right)} = \frac{{{e_{15}}}}{{{\kappa _{11}}}}{z^{ - n}}\;\;\;\;\;\;\xi _n^{\left( {23} \right)} = \frac{{{e_{15}}}}{{{\kappa _{11}}}}{{\bar z}^{ - n}}\;\;\;\;\;\;\;\xi _n^{\left( {24} \right)} = - \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{z^n}\;\;\;\;\;\;\\\ xi _n^{\left( {25} \right)} = - \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{{\bar z}^n}\\ \xi _n^{\left( {32} \right)} = {e_{15}}n{z^{ - n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}\;\;\;\;\;\;\xi _n^{\left( {33} \right)} = {e_{15}}n{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}\\ \xi _n^{\left( {34} \right)} = e_{15}^{\rm{c}}n{z^{n - 1}}{{\rm{e}}^{{\rm{i}}\theta }}\;\;\;\;\;\;\xi _n^{\left( {35} \right)} = e_{15}^{\rm{c}}n{{\bar z}^{ - n - 1}}{{\rm{e}}^{ - {\rm{i}}\theta }}\\ {\xi ^{\left( 1 \right)}} = - \frac{{{\rm{i}}k}}{2}\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right)\beta {w_0}\left( {{z^{\beta - 1}}{{\rm{e}}^{{\rm{i}}\left( {\theta - {\alpha _0}} \right)}} + {{\bar z}^{\beta - 1}}{{\rm{e}}^{ - {\rm{i}}\left( {\theta - {\alpha _0}} \right)}}} \right)\\ \exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]\\ {\xi ^{\left( 2 \right)}} = - \frac{{{e_{15}}}}{{{\kappa _{11}}}}{w_0}\exp \left[ {\frac{{{\rm{i}}k}}{2}\left( {\zeta {{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + \bar \zeta {{\rm{e}}^{{\rm{i}}{\alpha _0}}}} \right)} \right]\;\;\;\;\;\;\;{\xi ^{\left( 3 \right)}} = 0 \end{array} \right.$

(17)

将式(16)取有限截断项，等式两边同时乘以e^-imθ(m=0, ±1, ±2, ±3, …)，从(-π, π)进行积分得到多元一次方程组，从而求解出未知系数A_n、B_n、C_n、D_n、E_n。

4. 动应力集中系数与电场强度系数

根据文献[11]可知，动应力集中系数τ_θz^*(dynamic stress concentration factor, DSCF)和电场强度集中系数E_θ^*(electric field intensity concentration factor, EFICF)表达式分别为：

$\begin{array}{*{20}{c}} {\tau _{\theta z}^ * = \left| {{\tau _{\theta z}}/{\tau _0}} \right|}&{E_\theta ^ * = \left| {{E_\theta }/{E_0}} \right|} \end{array}$

(18)

式中：

$\begin{array}{*{20}{c}} {{\tau _0} = {\rm{i}}k\left( {{c_{44}} + \frac{{e_{15}^2}}{{{\kappa _{11}}}}} \right){w_0}}&{{E_0} = \frac{{k{e_{15}}{w_0}}}{{{\kappa _{11}}}}}&{{E_\theta } = - {\rm{i}}\left( {\frac{{\partial \varphi }}{{\partial \eta }}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial \varphi }}{{\partial \bar \eta }}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)} \end{array}$

5. 算例分析

当β=1时，本文模型退化为均匀压电介质模型。为对本文方法进行验证，采用与文献[7]中相同的参数，求解得到动应力系数τ_θz^*沿圆孔周边的分布情况，如图 2所示。可以看出，计算结果与文献[7]中结果吻合较好，说明本文方法精确可行。以下取κ₁₁/κ₁₁^c=1000进行建模，分析各参数对动应力集中系数及电场强度系数的影响。

图 2 方法验证(与文献[7]比较)

Figure 2. Verification of the present method (compared with reference [7])

下载: 全尺寸图片幻灯片

图 3给出了SH波以不同角度(α₀)入射时圆孔周围动应力系数的变化情况。由图 3可知：SH波垂直入射时，τ_θz^*达到最大值3.8；SH波水平入射时，τ_θz^*最大值约为均匀压电介质的2~3倍。由此可见，入射角度α₀对非均匀介质具有一定的影响。

图 3 SH波入射角度不同时动应力集中系数的变化

Figure 3. Varition of DSCF around the circular cavity edge by SH-wave with different incident angles

下载: 全尺寸图片幻灯片

图 4给出了SH波水平入射时圆孔周边动应力集中系数随波数ka的变化情况。图 4显示：τ_θz^*随波数ka的增大而减小，SH波低频入射时，τ_θz^*的最大值约为高频入射时的2倍。

图 4 SH波水平入射时圆孔周边动应力集中系数随波数ka的变化情况

Figure 4. DSCF around circular cavity edge vs.ka by horizontal SH-wave

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图 5给出了SH波垂直入射时圆孔周边动应力集中系数随波数ka的变化情况。图 5显示：τ_θz^*随波数ka增大而减小，与图 4中规律相同，但图 5中τ_θz^*的最大值比图 4中约大18%。由图 3~5可知，SH波低频垂直入射对径向非均匀压电介质破坏较大，在工程中应该对这种情况引起注意。

图 5 SH波垂直入射时圆孔周边动应力集中系数随波数ka的变化情况

Figure 5. DSCF around circular cavity edgevs.ka by vertical SH-wave

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图 6给出了SH波水平入射时圆孔周边动应力集中系数随λ的变化情况。图 6显示：压电参数λ对τ_θz^*几乎没有影响。图 7给出了SH波水平入射时圆孔周边动应力集中系数随β的变化情况。由图 7可知，τ_θz^*随幂次β的增大而增大，当β=4时，τ_θz^*达到最大值3.2，约为均匀压电材料τ_θz^*最大值的2倍，因此工程中应该合理调整参数，避免幂次β过大。

图 6 SH波水平入射时圆孔周边动应力集中系数随λ变化情况

Figure 6. DSCF around circular cavity edge vs. λ by horizontal SH-wave

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图 7 SH波水平入射时圆孔周边动应力集中系数随幂次β的变化情况

Figure 7. DSCF around circular cavity edge vs. β by horizontal SH-wave

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图 8给出了SH波水平入射时圆孔θ=π/2处动应力集中系数随波数ka的变化情况。图 8显示：τ_θz^*随ka值的增大而减小，下降率约为1.1%。不同压电参数λ条件下得到的τ_θz^*曲线几乎完全重合，说明λ对τ_θz^*几乎没有影响，与图 6中的结论一致。

图 8 SH波垂直入射时圆孔θ=π/2处动应力集中系数随ka的变化

Figure 8. DSCF around circular cavity edge vs. ka by vertical SH-wave

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图 9给出了SH波高频入射时圆孔周边电场强度系数随SH波入射角度的变化情况。由图 9可知：入射角度对E_θ^*最大值的影响不大；斜入射时，E_θ^*达到最大值3.1。

图 9 SH波以不同角度入射时圆孔周边电场强度系数的变化情况

Figure 9. Variation of EFICF around circular cavity edge by SH-wave with different incident angles

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图 10给出了SH波水平入射时圆孔周边电场强度系数随λ的变化情况。由图 10可知，E_θ^*随压电参数λ的增大而减小，当λ=0.2时，E_θ^*达到最大值6.2，因此工程中需要注意λ取值较小的情况。

图 10 SH波水平入射时圆孔周边电场强度系数随λ的变化情况

Figure 10. EFICF around circular cavity edge vs. λ by horizontal SH-wave

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图 11给出了SH波水平入射时圆孔周边电场强度系数随β的变化情况。由图 11可知，E_θ^*随幂次β增大而增大，与图 7中τ_θz^*变化规律一致。当β=4时，E_θ^*达到最大值3.1。

图 11 SH波水平入射时圆孔周边电场强度系数随β变化情况

Figure 11. EFICF around circular cavity edge vs. β by horizontal SH-wave

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图 12给出了SH波水平入射时圆孔θ=π/2处电场强度系数随波数ka的变化情况。由图 12可知：当ka < 0.2时，E_θ^*无明确的变化规律；当ka>0.2时，E_θ^*随ka的增大而减小。

图 12 SH波水平入射时圆孔θ=π/2处电场强度系数随波数ka变化情况

Figure 12. EFICF at the circular cavity edge vs. ka by horizontal SH-wave

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6. 结论

利用复变函数理论，对径向非均匀压电介质中圆孔对SH波的散射问题进行了研究。结果表明，SH波低频垂直入射对径向非均匀压电介质破坏较大；高频入射时，压电参数λ对τ_θz^*几乎没影响，但E_θ^*随λ的减小而增大，τ_θz^*与E_θ^*均随幂次β的增大而增大。另外，SH波水平入射时，τ_θz^*随ka的减小而增大，当ka>0.2时，E_θ^*也随ka的减小而增大。在实际工程中应该对这些规律引起注意，以避免非均匀压电介质发生破坏。

图 1 作为主体结构内部设备的储液结构^[8]

Figure 1. Liquid storage structure (LSS) as an internal device in the main structure

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图 2 储液结构计算模型

Figure 2. Calculation model of the LSS

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图 3 试验平台、结构模型与传感器布置

Figure 3. Test platform, structural model, and sensor layout

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图 4 不同摆锤高度下振动台输入加速度时程曲线

Figure 4. Time-history curves of input acceleration of shaking table at different pendulum heights

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图 5 结构应变理论计算结果与试验数据对比

Figure 5. Comparison between the calculation results and test data for structural strain

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图 6 动水压力理论计算结果与试验数据对比

Figure 6. Comparison between the calculation results and test data for hydrodynamic pressure

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图 7 典型爆炸地震动波形的归一化加速度时程曲线

Figure 7. Normalised acceleration time-history curves of typical explosion-induced ground shock waveforms

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图 8 广义质量随储液率的变化

Figure 8. Variation in generalised mass with liquid filling ratio

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图 9 激励因子随储液率的变化

Figure 9. Variation in excitation factor with liquid filling ratio

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图 10 广义刚度、基频、特征因子随储液率的变化

Figure 10. Variation in generalised stiffness, fundamental frequency, and characteristic factors with liquid filling ratio

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图 11 不同储液率下储液结构挠度时程曲线

Figure 11. Deflection time-history curves of the LSS with different liquid filling ratios

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图 12 不同地震动加速度峰值下储液结构挠度时程曲线

Figure 12. Deflection time-history curves of the LSS under ground shocks with different peak accelerations

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图 13 不同地震动加速度持时下储液结构挠度时程曲线

Figure 13. Deflection time-history curves of the LSS under ground shocks with different acceleration durations

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图 14 典型爆炸地震动下储液结构标准化挠度响应谱

Figure 14. Standardised deflection response spectra of the LSS under typical explosion-induced ground shock

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表 1 储液结构模型计算参数

Table 1. Calculation parameters of LSS model

2L_x/m 2L_y/m H_s/m d_s/m ρ_s/(kg·m⁻³) E/GPa ν H_l/m ρ_l/(kg·m⁻³)

1.18 0.88 0.74 0.01 7930 203 0.3 0.30 1000

　注：储液结构长、宽、高计算参数由外壁尺寸减去结构厚度，取内壁尺寸。

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表 2 弹性嵌固梁一阶频率系数

Table 2. First order frequency coefficient value of elastic embedded beam

K 0 0.5 1 3 5 7 10 20 30 100 ∞

β₁ 3.142 3.284 3.399 3.710 3.897 4.025 4.156 4.374 4.471 4.641 4.730

下载: 导出CSV

表 3 储液结构动力特性参数

Table 3. Dynamic characteristic parameters of the LSS

m_s^*/kg m_l^*/kg m^*/kg F_s^*/kg F_l^*/kg F^*/kg $\tilde F$ k^*/(N·m⁻¹) ω/Hz

10.79 1.19 11.98 18.08 5.87 23.95 2.00 1.75×10⁶ 381.82

下载: 导出CSV

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1. 控制方程
2. 介质中的位移场
3. 边界条件与定解方程
4. 动应力集中系数与电场强度系数
5. 算例分析
6. 结论

2L_x/m	2L_y/m	H_s/m	d_s/m	ρ_s/(kg·m⁻³)	E/GPa	ν	H_l/m	ρ_l/(kg·m⁻³)
1.18	0.88	0.74	0.01	7930	203	0.3	0.30	1000
注：储液结构长、宽、高计算参数由外壁尺寸减去结构厚度，取内壁尺寸。

K	0	0.5	1	3	5	7	10	20	30	100	∞
β₁	3.142	3.284	3.399	3.710	3.897	4.025	4.156	4.374	4.471	4.641	4.730

m_s^*/kg	m_l^*/kg	m^*/kg	F_s^*/kg	F_l^*/kg	F^*/kg	$\tilde F$	k^*/(N·m⁻¹)	ω/Hz
10.79	1.19	11.98	18.08	5.87	23.95	2.00	1.75×10⁶	381.82

爆炸地震动下矩形储液结构动力响应理论研究

doi: 10.11883/bzycj-2023-0099

作者简介: 张浩天（1996－ ），男，博士研究生，335733770@qq.com

通讯作者: 宋春明（1979－ ），男，博士，副教授，ming1979@126.com

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