Optimal design of ballistic performance of fiber-metal laminates based on the response surface method
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摘要: 纤维金属层合板因其复合材料的各向异性和层合结构特征而具有较好的可设计性，开展金属纤维层合板的优化设计研究对其力学性能的增强和轻量化具有重要意义。为提高纤维金属层合板的抗弹性能，基于响应面分析法对纤维金属层合板的铺层方向和铺层厚度进行了优化设计。采用Box-Behnken方法进行方案设计，以纤维金属层合板各铺层相对厚度比为设计变量，以结构的比吸能为设计目标，根据设计的方案进行参数化建模获取样本点，在对设计样本进行方差分析和参数估计的基础上，建立了结构比吸能的响应面模型并验证了其精确度。采用遗传算法对响应面方程进行寻优分析，通过显式动力学计算程序ABAQUS/Explicit验证优化效果。最终，在最优的铺层方案下，层合板的质量减小了11.70%，能量吸收增加了19.40%，抗弹性能显著提升。Abstract: Fiber-metal laminates are highly designable due to the characteristics of their constituent materials and laminate structure. They have the characteristics of anisotropy, large interface differences, and flexible design. Optimizing the design of fiber-metal laminates is of great significance to the enhancement of its mechanical properties and weight reduction. In order to improve the ballistic performance of fiber-metal laminates, of which the layer direction and layer thickness are optimized based on the response surface analysis method. For layup direction optimization, several layup directions are designed based on the corresponding principles according to the composite material layup optimization design requirements, and the energy absorptions of the corresponding structures are calculated, respectively, then the design plan for the better layup direction is screened out. For the optimization of ply thickness, the relative thickness ratio of each ply of the fiber-metal laminate is used as the design variable, and the specific energy absorption of the structure is the design goal. The Box-Behnken method is used to design the experiment. According to the test plan, the explicit dynamic calculation program ABAQUS/Explicit is used for parametric modeling to obtain test sample points, and the design test samples are analyzed by using variance analysis and parameter estimation, and the response surface model of structural specific energy absorption (SEA) is established. The errors between the experimental values and the predicted values are compared, and the model can be used for prediction; the genetic algorithm is used to optimize the obtained response surface equation, and the optimization effect is verified by ABAQUS/Explicit. The optimization result shows that the accuracy of the obtained response surface model is high. Under the premise of not increasing the thickness and weight of the laminate, the best layup plan is finally obtained, which improves the energy absorption capacity of the laminate. Finally, the mass of laminates decreases by 11.70% and the energy absorption increases by 19.40% under the optimal lamination scheme.
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Key words:
- laminate /
- response surface method /
- genetic algorithm /
- composite material
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图 3 纤维金属层合板高速冲击实验布置
Figure 3. Layout diagram of high-speed impact experiment for fiber metal laminates
图 4 弹体剩余速度与入射速度的关系曲线
Figure 4. Change of projectile residual velocity with its incident velocity
图 5 实验与数值模拟的纤维金属层合板横截面破坏形貌
Figure 5. Experimental and numerical failure morphologies of fiber-metal laminates (a cross-sectioned view)
图 6 实验与数值模拟的纤维金属层合板背面破坏形貌
Figure 6. Experimental and numerical failure morphologies of fiber-metal laminates (a top view)
图 7 不同铺层方向下弹体动能-时间变化曲线
Figure 7. Kinetic energy-time curves of projectile under different layup directions
图 8 不同铺层方案下层合板的吸能效果
Figure 8. Energy absorption effects of the lower laminatesunder different layer schemes
图 10
${\alpha _1}$ 和${\alpha _4}$ 交互作用对比吸能影响的响应面和等高线Figure 10. Response surfaces and contour lines of interaction between
$ {\alpha _1} $ and$ {\alpha _4} $ on specific energy absorption
图 11
${\alpha _3}$ 和${\alpha _4}$ 交互作用对比吸能影响的响应面和等高线Figure 11. Response surfaces and contour lines of interaction between
$ {\alpha _3} $ and$ {\alpha _4} $ on specific energy absorption
图 17 优化前后层合板吸能效果
Figure 17. Energy absorption effect of original and optimized laminates
表 1 热塑性纤维增强材料的弹性参数
Table 1. Elastic parameters of thermoplastic fiber reinforced materials
ρ/（kg∙m−3） E1/GPa E2/GPa E3/GPa μ12 μ13 μ23 G12/GPa G13/GPa G23/GPa 1 800 13 13 4.8 0.1 0.3 0.3 1.72 1.69 1.69 
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表 2 热塑性纤维增强材料的强度参数
Table 2. Strength parameters of thermoplastic fiber reinforced materials
XT/MPa XC/MPa YT/MPa YC/MPa S12/MPa S13/MPa S23/MPa 300 200 300 200 120 120 120
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表 3 网格收敛性分析
Table 3. Analysis of grid convergence
网格尺寸/mm 网格数量 吸能/J 相对偏差/% 2.0 43 908 90.72 34.70 1.0 171 408 69.42 3.07 0.5 680 000 67.35 0
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ρc/(kg∙m−3) En/GPa Es/GPa Et/GPa $s_{\rm{n} }^{(0)}$/MPa $s_{\rm{s} }^{(0)}$/MPa $s_{\rm{t} }^{(0)}$/MPa $G_{\rm{n} }^{(0)}$/(J∙m−2) $G_{\rm{s} }^{(0)}$/(J∙m−2) $G_{\rm{t} }^{(0)}$/(J∙m−2) 900 2.05 0.72 0.72 70 100 100 300 700 700
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表 5 弹体剩余速度实验结果与数值模拟结果对比
Table 5. Comparison of projectile residual velocities between experimental and numerical results
实验工况 实验初速/(m·s−1) 实验剩余速度/(m·s−1) 模拟剩余速度/(m·s−1) 剩余速度相对误差/% 3Al(6-O)/2G-1 143 109 107.9 –1.0 3Al(6-O)/2G-2 172 125 127.6 2.1 3Al(6-O)/2G-3 195 150 151.1 0.7 3Al(6-O)/2G-4 214 181 172.1 –4.9 3Al(6-O)/2G-5 252 216 211.5 –2.1
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表 6 纤维金属层合板铺层方案
Table 6. Layer schemes of fiber metal laminates
铺层方案 铺层方向 铺层方案 铺层方向 1 Al/45°/90°/0°/Al/0°/90°/45°/Al 7 Al/90°/0°/–45°/Al/–45°/0°/90°/Al 2 Al/45°/90°/–45°/Al/–45°/90°/45°/Al 8 Al/90°/45°/0°/Al/0°/45°/90°/Al 3 Al/45°/0°/90°/Al/90°/0°/45°/Al 9 Al/90°/45°/–45°/Al/–45°/45°/90°/Al 4 Al/45°/0°/–45°/Al/–45°/0°/45°/Al 10 Al/90°/–45°/0°/Al/0°/–45°/90°/Al 5 Al/45°/–45°/0°/Al/0°/–45°/45°/Al 11 Al/0°/45°/–45°/Al/–45°/45°/0°/Al 6 Al/45°/–45°/90°/Al/90°/–45°/45°/Al 12 Al/0°/90°/–45°/Al/–45°/90°/0°/Al
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表 7 设计因子水平
Table 7. Design factor levels
设计因子 低水平 高水平 $ {\alpha _1} $ 0.029 0.200 $ {\alpha _2} $ 0.029 0.200 $ {\alpha _3} $ 0.029 0.200 $ {\alpha _4} $ 0.200 0.457
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表 8 BBD实验设计
Table 8. Experimental schemes designed by the BBD method
实验方案 $ {\alpha _1} $ $ {\alpha _2} $ $ {\alpha _3} $ $ {\alpha _4} $ $Q/({\rm{J}} \cdot {\rm{kg}^{ - 1} })$ ${Q_{\rm{{pre}}} }/({\rm{J} } \cdot {\rm{kg}^{ - 1} })$ 1 –1 0 0 –1 982.58 983.58 2 0 0 –1 1 820.35 815.75 3 1 0 0 –1 1 032.25 1 041.37 4 0 0 0 0 799.74 799.74 5 0 –1 0 –1 979.84 965.03 6 –1 0 0 1 675.37 671.43 7 –1 1 0 0 836.94 847.07 8 1 –1 0 0 846.44 859.65 9 0 0 –1 –1 1 135.93 1 143.95 10 –1 0 1 0 780.02 754.93 11 1 0 –1 0 1 015.36 1 002.85 12 0 –1 –1 0 914.80 912.54 13 –1 –1 0 0 730.39 746.48 14 0 1 0 1 808.02 808.85 15 1 1 0 0 960.88 960.23 16 1 0 1 0 881.44 868.10 17 0 0 0 0 799.74 799.74 18 0 0 1 –1 913.77 937.77 19 0 0 0 0 799.74 799.74 20 0 1 –1 0 976.15 985.71 21 0 0 1 1 741.05 752.42 22 1 0 0 1 835.80 839.97 23 –1 0 –1 0 887.89 889.69 24 0 1 0 –1 1 092.94 1 065.62 25 0 –1 0 1 716.09 708.26 26 0 –1 1 0 754.74 750.36 27 0 1 1 0 870.92 878.37
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表 9 方差分析和参数估计
Table 9. Analysis of variance and parameter estimation
方差来源 平方和 自由度 均方 F P 模型 366 000.00 11 33 247.56 156.55 < 0.000 1 $ {\alpha _1} $ 38 416.01 1 38 416.01 180.89 < 0.000 1 $ {\alpha _2} $ 30 354.44 1 30 354.44 142.93 < 0.000 1 $ {\alpha _3} $ 54 478.75 1 54 478.75 256.52 < 0.000 1 $ {\alpha _4} $ 198 000.00 1 198 000.00 931.36 < 0.000 1 $ {\alpha _1} \times {\alpha _4} $ 3 066.78 1 3 066.78 14.44 0.001 4 $ {\alpha _2} \times {\alpha _3} $ 751.72 1 751.72 3.54 0.077 2 $ {\alpha _3} \times {\alpha _4} $ 5 101.74 1 4 179.89 24.02 0.000 1 $ {\alpha _1} \times {\alpha _1} $ 4 179.89 1 5 170.59 19.68 0.000 4 $ {\alpha _2} \times {\alpha _2} $ 5 170.59 1 18 752.36 24.35 0.000 1 $ {\alpha _3} \times {\alpha _3} $ 18 752.36 1 22 551.08 88.30 < 0.000 1 $ {\alpha _4} \times {\alpha _4} $ 22 551.08 1 3 066.78 106.19 < 0.000 1 残差 3 610.39 17 212.38 总和 370 000.00 28 $ {R^2} $ 0.99 $R_{\rm{{adj}}}^2$ 0.98
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表 10 遗传算法优化结果
Table 10. Genetic algorithm optimization results
实验方案 $ {\alpha _1} $ $ {\alpha _2} $ $ {\alpha _3} $ $ {\alpha _4} $ $Q/({\rm{{J}}} \cdot {\rm{kg}^{ - 1} })$ 优化 0.183 0.198 0.041 0.217 1186.0
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表 11 优化前后层合板主要参数
Table 11. The main parameters of original and optimized laminates
层合板方案 $ {t_1}/{\text{mm}} $ $ {t_2}/{\text{mm}} $ $ {t_3}/{\text{mm}} $ $ {t_4}/{\text{mm}} $ $t_{0}/{\text{mm} }$ $m/{\text{g} }$ $Q/({\rm{{J}}} \cdot {\rm{kg}^{ - 1} })$ 原方案 0.50 0.50 0.50 1.00 3.50 78.70 882.12 优化后 0.64 0.69 0.14 0.76 3.00 69.50 1192.66
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