﻿ 空中爆炸冲击载荷下折叠式夹层板塑性动力响应研究
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 哈尔滨工程大学学报  2020, Vol. 41 Issue (6): 797-804  DOI: 10.11990/jheu.201901039 0

引用本文

KE Li, WANG Zili, WANG Zhe, et al. Plastic dynamic response of folded sandwich panels under air-blast loading[J]. Journal of Harbin Engineering University, 2020, 41(6): 797-804. DOI: 10.11990/jheu.201901039.

文章历史

KE Li , WANG Zili , WANG Zhe , LIU Kun , WANG Jiaxia
School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: Folded sandwich panels under impact resistance are mainly evaluated by experiments and numerical simulations, which are costly and time-consuming. To reduce cost, quickly evaluate the damage deformation of the structure, and effectively guide the design analysis of the ship structure in impact resistance, the structural response is divided into three stages according to the deformation characteristics of folded sandwich panels under air-blast loading. The analysis is performed based on the theorem of kinetic energy, the theory of energy conservation, and the theory of plastic mechanics, deriving the simplified theoretical prediction formula under air-blast loading. The applicability of the method is verified through a comparison with the calculations obtained from the finite element analysis. The findings provide a reference for the structure design and impact performance evaluation of folded sandwich panels.
Keywords: folded sandwich panel    air blast    impact load    equivalence theory    plastic dynamic response    plastic hinge    yield function    numerical simulation

1 冲击载荷响应分析

 Download: 图 1 折叠式夹层板动态响应过程 Fig. 1 Dynamic response process of folded sandwich panel
1.1 夹层板初始动能

 $I = 2{I_ + } = 2{A_i}\frac{{\sqrt[3]{{m_e^2}}}}{r}$ (1)

 ${E_i} = \frac{1}{2}S{\rho _f}{t_f}v_i^2 = \frac{{{I^2}S}}{{2{\rho _f}{t_f}}}$ (2)

1.2 夹层板芯层压缩

 ${E_k} = \frac{1}{2}S(2{\rho _f}{t_f} + {\rho _c}{h_c})v_k^2 = \frac{{{I^2}S}}{{2(2{\rho _f}{t_f} + {\rho _c}{h_c})}}$ (3)

 ${E_a} = {E_i} - {E_k}$ (4)

 $w(x,y) = {w_{mn}}{\rm{sin}}\left( {\frac{\pi }{2} + \frac{{m\pi x}}{L}} \right){\rm{sin}}\left( {\frac{\pi }{2} + \frac{{n\pi y}}{B}} \right)$ (5)

 $\begin{array}{*{20}{l}} {{E_a} = {\sigma _c}\int_{ - L/2}^{L/2} {\int_{ - B/2}^{B/2} {{w_{mn}}} } {\rm{sin}}\left( {\frac{\pi }{2} + \frac{{m\pi x}}{L}} \right) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{sin}}\left( {\frac{\pi }{2} + \frac{{n\pi y}}{B}} \right){\rm{d}}x{\rm{d}}y = {w_{mn}}\frac{{4LB}}{{{\pi ^2}}}{\sigma _c}} \end{array}$ (6)

 ${w_{mn}} = \frac{{{\pi ^2}}}{{4LB{\sigma _c}}}({E_i} - {E_k})$ (7)
1.3 夹层板整体变形 1.3.1 变形模式

 Download: 图 2 固支夹层板的塑性变形模式 Fig. 2 Plastic deformation mode of the clamed sandwich panel

 ${w_{i{\rm{I}}}} = \frac{{b{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi - 2{x^\prime }}}{{b{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}w,{w_{i{\rm{II}}}} = \frac{{b - 2y}}{b}w$ (8)

 ${\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi = \sqrt {3 + {\xi ^2}} - \xi ,\xi = b/a$ (9)

 ${E_D} = (M + N{w_i}){\theta _i}$ (10)

 ${E_p} = \sum\limits_{i = 1}^n {\int_{{l_i}} {(M + N{w_i})} } {\theta _i}{\rm{d}}{l_i}$ (11)

 $\begin{array}{*{20}{l}} {{E_p} = 2\int_{{l_{AB}}} {(M + N{w_i})} {\theta _{AB}}{\rm{d}}{l_{AB}} + 2\int_{{l_{AD}}} {(M + N{w_i})} {\theta _{AD}}{\rm{d}}{l_{AD}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4\int_{{l_{AE}}} {(M + N{w_i})} {\theta _{AE}}{\rm{d}}{l_{AE}} + \int_{{l_{EF}}} {(M + N{w_i})} {\theta _{EF}}{\rm{d}}{l_{EF}}} \end{array}$ (12)

 $\begin{array}{*{20}{l}} {{\theta _{AB}} = \frac{{2w}}{{b{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }},{\theta _{AD}} = \frac{{2w}}{b},{\theta _{EF}} = \frac{{4w}}{b},}\\ {{\theta _{AE}} = {\theta _{AB}}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi + {\theta _{AD}}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi = \frac{{2w}}{{b{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \end{array}$ (13)

 ${E_p} = \left( {\frac{{4a}}{b} + \frac{{2{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right)N{w^2} + \left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right)Mw$ (14)

 $\frac{{|M|}}{{{M_0}}} + \frac{{|N|}}{{{N_0}}} = 1$ (15)

 $\begin{array}{*{20}{l}} {{M_{0x}} = {\sigma _f}{t_f}[({h_c} - {w_{mn}}) + {t_f}] + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\sigma _c}{t_c}{{({h_c} - {w_{mn}})}^2}/4({d_p} - {d_f})} \end{array}$ (16)
 Download: 图 3 Ⅰ型折叠式夹层板及几何参数 Fig. 3 Ⅰ-type folded sandwich panel and geometric parameters

 ${N_{0x}} = 2{\sigma _f}{t_f} + {\sigma _c}{t_c}({h_c} - {w_{mn}})/({d_p} - {d_f})$ (17)

 $\begin{array}{*{20}{l}} {{M_{0x}} = {\sigma _f}{t_f}[({h_c} - {w_{mn}}) + {t_f}] + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\sigma _c}{{({h_c} - {w_{mn}})}^2}({d_p} - 2{d_f})/4{d_p}} \end{array}$ (18)
 Download: 图 4 U型折叠式夹层板及几何参数 Fig. 4 U-type folded sandwich panel and geometric parameters

 ${N_{0x}} = 2{\sigma _f}{t_f} + {\sigma _c}({h_c} - {w_{mn}})({d_p} - 2{d_f})/{d_p}$ (19)

 ${M_{0y}} = {\sigma _f}{t_f}[({h_c} - {w_{mn}}) + {t_f}] + {\sigma _c}{({h_c} - {w_{mn}})^2}/4$ (20)

 ${N_{0y}} = 2{\sigma _f}{t_f} + {\sigma _c}({h_c} - {w_{mn}})$ (21)

 ${M_0} = {\sigma _f}{t_f}[({h_c} - {w_{mn}}) + {t_f}] + {\sigma _c}{({h_c} - {w_{mn}})^2}/4$ (22)
 Download: 图 5 V型折叠式夹层板及几何参数 Fig. 5 V-type folded sandwich panel and geometric parameters
 Download: 图 6 Uc型折叠式夹层板及几何参数 Fig. 6 Uc-type folded sandwich panel and geometric parameters

 ${N_0} = 2{\sigma _f}{t_f} + {\sigma _c}({h_c} - {w_{mn}})$ (23)

 $\left\{ {\begin{array}{*{20}{l}} {{{\bar \rho }_I} = 2{t_c}/{d_p}}\\ {{{\bar \rho }_U} = {t_c}({d_p} - 2{d_f})/{h_c}{d_p}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }\\ {{{\bar \rho }_V} = {t_c}/({h_c} - {t_c}){\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta }\\ {{{\bar \rho }_{Uc}} = {t_c}(2{d_f}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta + {d_p} - 2{d_f})/{h_c}{d_p}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \theta } \end{array}} \right.$ (24)

 Download: 图 7 夹层板屈服曲线 Fig. 7 The yield curve of the sandwich panel

 $|N| = {N_0},|M| = {M_0}$ (25)

 $|N| = 0.5{N_0},|M| = 0.5{M_0}$ (26)

1.3.2 变形计算

 $\begin{array}{*{20}{l}} {{E_p} = \left( {\frac{{4a}}{b} + \frac{{2{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right){N_0}{w^2} + }\\ {\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_0}w} \end{array}$ (27)

 $\begin{array}{*{20}{l}} {{E_p} = \left( {\frac{{2a}}{b} + \frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{2}} \right){N_0}{w^2} + }\\ {\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\frac{{4a}}{b} + \frac{{4{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_0}w} \end{array}$ (28)

 ${E_p} = {E_k}$ (29)

 $\begin{array}{*{20}{c}} {\left( {\frac{{4a}}{b} + \frac{{2{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right){N_0}{w^2} + }\\ {\left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_0}w = {E_k}} \end{array}$ (30)
 $\begin{array}{*{20}{c}} {\left( {\frac{{2a}}{b} + \frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{2}} \right){N_0}{w^2} + }\\ {\left( {\frac{{4a}}{b} + \frac{{4{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_0}w = {E_k}} \end{array}$ (31)

M0xN0xM0yN0y分别代入式(30)和式(31)，舍去负值解，可得：

 ${w_{1x}} = \frac{{\sqrt {{{\left[ {\left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0x}}} \right]}^2} + 4{E_k}{N_{0x}}\left( {\frac{{4a}}{b} + \frac{{2{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right)} - \left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0x}}}}{{2{N_{0x}}\left( {\frac{{4a}}{b} + \frac{{2{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right)}}$ (32)
 ${w_{2x}} = \frac{{\sqrt {{{\left[ {\left( {\frac{{4a}}{b} + \frac{{4{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0x}}} \right]}^2} + 4{E_k}{N_{0x}}\left( {\frac{{2a}}{b} + \frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{2}} \right)} - \left( {\frac{{4a}}{b} + \frac{{4{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0x}}}}{{2{N_{0x}}\left( {\frac{{2a}}{b} + \frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{2}} \right)}}$ (33)
 ${w_{1y}} = \frac{{\sqrt {{{\left[ {\left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0y}}} \right]}^2} + 4{E_k}{N_{0y}}\left( {\frac{{4a}}{b} + \frac{{{\rm{2cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right)} - \left( {\frac{{8a}}{b} + \frac{{8{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0y}}}}{{2{N_{0y}}\left( {\frac{{4a}}{b} + \frac{{{\rm{2cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \right)}}$ (34)
 ${w_{2y}} = \frac{{\sqrt {{{\left[ {\left( {\frac{{4a}}{b} + \frac{{4{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0y}}} \right]}^2} + 4{E_k}{N_{0y}}\left( {\frac{{2a}}{b} + \frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{2}} \right)} - \left( {\frac{{4a}}{b} + \frac{{4{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}} \right){M_{0y}}}}{{2{N_{0y}}\left( {\frac{{2a}}{b} + \frac{{{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }} - \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{2}} \right)}}$ (35)

 $w = \frac{1}{4}({w_{1x}} + {w_{2x}} + {w_{1y}} + {w_{2y}})$ (36)
2 数值仿真结果验证 2.1 有限元模型

 $E = \frac{{|{w_T} - {w_s}|}}{{{w_T}}} \times 100\%$ (37)

 $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{I}=\frac{I}{M\sqrt{{{\sigma }_{f}}/{{\rho }_{f}}}}$ (38)

 Download: 图 8 四种典型折叠式夹层板有限元模型 Fig. 8 The finite element model of four typical folded sandwich panels

2.2 中心位移对比分析

 Download: 图 9 4种典型折叠式夹层板冲量-位移曲线 Fig. 9 Impulse-displacement curves of four typical folded sandwich panel

2.3 变形模式对比分析

 Download: 图 10 X轴方向中心线位移 Fig. 10 Displacement curves of the center line in X axis
 Download: 图 11 Y轴方向中心线位移 Fig. 11 Displacement curves of the center line in Y axis

3 结论

1) 将折叠式夹层板在爆炸冲击载荷作用下的动态响应分成3个阶段，结合每一阶段的响应特点，采用动能定理、能量守恒定律和刚塑性材料模型简化分析响应过程，研究夹层板在冲击载荷作用下的塑性响应求解方法，并推导出适用于多种折叠式夹层板的塑性变形简化计算公式。

2) 在小计算量内，计算不同冲量作用下多种形式折叠式夹层板的塑性变形，并与有限元仿真结果进行对比，发现2种结果吻合较好，证明了所得理论预报公式的可行性与实用性，并为夹层板的工程设计提供参考。

3) 在芯层数量相同的情况下Uc连续型夹层板结构具有更优的抗爆性能，并且利用内接屈服面函数能更好的反映夹层板局部区域的变形情况。