Crashworthiness of Bionic Fractal Multi-Cell Circular Tubes under Axial Load
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摘要: 针对传统薄壁圆管与高吸能需求之间的矛盾，提出了一种内嵌不同正多边形的仿生分形多胞圆（bio-inspired fractal multi-cell circular, BFMC）管。基于生物结构启发与分形层级理论，构建了内嵌正四边形、正五边形及正六边形的BFMC管几何模型，通过数值模拟系统研究了质量、分形维数及内嵌正多边形边数等关键参数对结构耐撞性能的影响，并与典型多胞管进行了对比分析。结果表明：在近似等质量条件下，BFMC管凭借分形层级与仿生构型，能够显著提升比吸能和承载能力；其耐撞性能随质量增加而增强，随分形维数的增大呈先降后升的趋势，并随内嵌多边形边数的增加而增强，而峰值力受边数的影响较小。基于超折叠单元理论，建立了用于预测BFMC管平均压缩力的理论模型，并通过数值模拟验证了其准确性与适用范围。研究结果可为高比吸能薄壁结构的设计提供理论依据与结构构型途径。Abstract: A bio-inspired fractal multi-cell circular (BFMC) tube with embedded regular polygons is proposed to address the gap between the need for high absorption and the limited performance of traditional thin-walled circular tubes. Inspired by biological structures and fractal hierarchy theory, geometric models of BFMC tubes embedded with square, pentagonal, and hexagonal cells are constructed. Numerical simulations are carried out to systematically investigate the effects of mass, fractal dimension, and the number of sides of the embedded polygons on the axial crushing performance, and the results are compared with those of typical multi-tubes. The results indicate that, under approximately equal mass conditions, the BFMC tube can significantly enhance the specific energy absorption and the load-bearing capacity owing to its fractal hierarchical and bio-inspired configurations. Its crashworthiness increases with mass, first decreases and then rises as the fractal dimension increases, and improves further as the number of polygon sides increases, while the peak force is only weakly affected. A theoretical model for predicting the mean crushing force of BFMC tubes is developed based on the super folding element theory and is validated through numerical simulations. This study provides theoretical support and structural design guidelines for developing high performance thin-walled energy absorption structures.
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图 6 BFMC-E4-0管的实验和模拟结果的对比
Figure 6. Comparison of experimental and simulated results for BFMC-E4-0 tube
图 9 BFMC-E4管和圆管的力-位移曲线
Figure 9. Force-displacement curves of BFMC-E4 and circular tubes
图 10 BFMC-E5管和圆管的力-位移曲线
Figure 10. Force-displacement curves of BFMC-E5 and circular tubes
图 11 BFMC-E6管和圆管的力-位移曲线
Figure 11. Force-displacement curves of BFMC-E6 and circular tubes
图 15 分形维数对BFMC-E4管耐撞性的影响
Figure 15. Effect of fractal dimension on crashworthiness of BFMC-E4 tube
图 16 分形维数对BFMC-E5管耐撞性的影响
Figure 16. Effect of fractal dimension on crashworthiness of BFMC-E5 tube
图 17 分形维数对BFMC-E6管耐撞性的影响
Figure 17. Effect of fractal dimension on crashworthiness of BFMC-E6 tube
图 18 内嵌正多边形边数对BFMC-3管耐撞性的影响
Figure 18. Effect of the number of embedded regular polygon edges on the crashworthiness of BFMC-3 tubes
表 1 圆管和BFMC管的结构参数
Table 1. Structural parameters of circular and BFMC tubes
Group Mass/g Type Cross section thickness/mm BFMC-0 BFMC-1 BFMC-2 BFMC-3 BFMC-4 1 101.79 Circular 1 1 1 1 1 BFMC-E4 0.565 0.369 0.300 0.243 0.200 BFMC-E5 0.536 0.342 0.275 0.223 0.183 BFMC-E6 0.512 0.319 0.255 0.205 0.168 2 132.32 Circular 1.3 1.3 1.3 1.3 1.3 BFMC-E4 0.735 0.480 0.390 0.317 0.260 BFMC-E5 0.697 0.445 0.358 0.290 0.238 BFMC-E6 0.665 0.414 0.331 0.266 0.218 3 162.86 Circular 1.6 1.6 1.6 1.6 1.6 BFMC-E4 0.905 0.591 0.480 0.390 0.321 BFMC-E5 0.858 0.547 0.441 0.356 0.293 BFMC-E6 0.818 0.510 0.407 0.327 0.268 4 193.40 Circular 1.9 1.9 1.9 1.9 1.9 BFMC-E4 1.074 0.702 0.570 0.463 0.381 BFMC-E5 1.018 0.650 0.524 0.423 0.348 BFMC-E6 0.972 0.605 0.484 0.389 0.318 
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表 2 AA6061-O铝合金的材料参数
Table 2. Material parameters of AA6061-O aluminum alloy
ρ/(g∙cm−3) E/GPa ν σy/MPa σu/MPa 2.7 68.0 0.33 71.0 130.7
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表 3 5种多胞圆管的耐撞性对比
Table 3. Comparison of crashworthiness for five multi-cell circular tubes
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表 4 BFMC管的角单元数量
Table 4. Number of angle elements in BFMC tubes
Type of angle element Number BFMC-0 BFMC-1 BFMC-2 BFMC-3 BFMC-4 3-panel-T-shape N1 j j 0 0 0 3-panel-Y-shape N2 j 0 0 0 0 3-panel-y-shape N3 0 0 2j 6j 14j 4-panel-k-shape N4 0 j j 2j 6j 5-panel-Ⅰ-shape N5 0 j j j j 5-panel-Ⅱ-shape N6 0 0 j j j
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表 5 BFMC管半波长和MCF理论值
Table 5. Theoretical values of half-wavelength and MCF for BFMC tubes
Type Half wavelength MCF BFMC-E4-0 $ {H}_{0,4}=\sqrt{\dfrac{{\text{π}} {L}_{0,4}{t}_{0,4}}{48.28}} $ $ F_{{\rm{mc}},{0,4}}^{\rm{M}}=\dfrac{\sqrt{48.28{\text{π}} }}{{\xi }_{0}}L_{0,4}^{1/2}t_{0,4}^{3/2}{\sigma }_{0} $ BFMC-E4-1 $ {H}_{1,4}=\sqrt{\dfrac{\text{π} {L}_{1,4}{t}_{1,4}}{95.52}} $ $ F_{{\rm{mc}},{1,4}}^{\rm{M}}=\dfrac{\sqrt{95.52\text{π} }}{{\xi }_{1}}L_{1,4}^{1/2}t_{1,4}^{3/2}{\sigma }_{0} $ BFMC-E4-2 $ {H}_{2,4}=\sqrt{\dfrac{\text{π} {L}_{2,4}{t}_{2,4}}{181.98}} $ $F_{{\rm{mc}},{2,4}}^{\rm{M}}=\dfrac{\sqrt{181.98\text{π} }}{{\xi }_{2}}L_{2,4}^{1/2}t_{2,4}^{3/2}{\sigma }_{0} $ BFMC-E4-3 $ {H}_{3,4}=\sqrt{\dfrac{\text{π} {L}_{3,4}{t}_{3,4}}{373.19}} $ $ F_{{\rm{mc}},{3,4}}^{\rm{M}}=\dfrac{\sqrt{373.19\text{π} }}{{\xi }_{3}}L_{3,4}^{1/2}t_{3,4}^{3/2}{\sigma }_{0} $ BFMC-E4-4 $ {H}_{4,4}=\sqrt{\dfrac{\text{π} {L}_{4,4}{t}_{4,4}}{818.91}} $ $F_{{\rm{mc}},{4,4}}^{\rm{M}}=\dfrac{\sqrt{818.91\text{π} }}{{\xi }_{4}}L_{4,4}^{1/2}t_{4,4}^{3/2}{\sigma }_{0} $ BFMC-E5-0 $ {H}_{0,5}=\sqrt{\dfrac{\text{π} {L}_{0,5}{t}_{0,5}}{62.55}} $ $ F_{{\rm{mc}},{0,5}}^{\rm{M}}=\dfrac{\sqrt{62.55\text{π} }}{{\xi }_{0}}L_{0,5}^{1/2}t_{0,5}^{3/2}{\sigma }_{0} $ BFMC-E5-1 $ {H}_{1,5}=\sqrt{\dfrac{\text{π} {L}_{1,5}{t}_{1,5}}{113.75}} $ $ F_{{\rm{mc}},{1,5}}^{\rm{M}}=\dfrac{\sqrt{113.75\text{π} }}{{\xi }_{1}}L_{1,5}^{1/2}t_{1,5}^{3/2}{\sigma }_{0} $ BFMC-E5-2 $ {H}_{2,5}=\sqrt{\dfrac{\text{π} {L}_{2,5}{t}_{2,5}}{226.68}} $ $ F_{{\rm{mc}},{2,5}}^{\rm{M}}=\dfrac{\sqrt{226.68\text{π} }}{{\xi }_{2}}L_{2,5}^{1/2}t_{2,5}^{3/2}{\sigma }_{0} $ BFMC-E5-3 $ {H}_{3,5}=\sqrt{\dfrac{\text{π} {L}_{3,5}{t}_{3,5}}{458.84}} $ $ F_{{\rm{mc}},{3,5}}^{\rm{M}}=\dfrac{\sqrt{458.84\text{π} }}{{\xi }_{3}}L_{3,5}^{1/2}t_{3,5}^{3/2}{\sigma }_{0} $ BFMC-E5-4 $ {H}_{4,5}=\sqrt{\dfrac{\text{π} {L}_{4,5}{t}_{4,5}}{1\;007.69}} $ $ F_{{\rm{mc}},{4,5}}^{\rm{M}}=\dfrac{\sqrt{1\;007.69\text{π} }}{{\xi }_{4}}L_{4,5}^{1/2}t_{4,5}^{3/2}{\sigma }_{0} $ BFMC-E6-0 $ {H}_{0,6}=\sqrt{\dfrac{\text{π} {L}_{0,6}{t}_{0,6}}{75.84}} $ $ F_{{\rm{mc}},{0,6}}^{\rm{M}}=\dfrac{\sqrt{75.84\text{π} }}{{\xi }_{0}}L_{0,6}^{1/2}t_{0,6}^{3/2}{\sigma }_{0} $ BFMC-E6-1 $ {H}_{1,6}=\sqrt{\dfrac{\text{π} {L}_{1,6}{t}_{1,6}}{134.17}} $ $ F_{{\rm{mc}},{1,6}}^{\rm{M}}=\dfrac{\sqrt{134.17\text{π} }}{{\xi }_{1}}L_{1,6}^{1/2}t_{1,6}^{3/2}{\sigma }_{0} $ BFMC-E6-2 $ {H}_{2,6}=\sqrt{\dfrac{\text{π} {L}_{2,6}{t}_{2,6}}{269.50}} $ $F_{{\rm{mc}},{2,6}}^{\rm{M}}=\dfrac{\sqrt{269.50\text{π} }}{{\xi }_{2}}L_{2,6}^{1/2}t_{2,6}^{3/2}{\sigma }_{0} $ BFMC-E6-3 $ {H}_{3,6}=\sqrt{\dfrac{\text{π} {L}_{3,6}{t}_{3,6}}{538.99}} $ $ F_{{\rm{mc}},{3,6}}^{\rm{M}}=\dfrac{\sqrt{538.99\text{π} }}{{\xi }_{3}}L_{3,6}^{1/2}t_{3,6}^{3/2}{\sigma }_{0} $ BFMC-E6-4 $ {H}_{4,6}=\sqrt{\dfrac{\text{π} {L}_{4,6}{t}_{4,6}}{1\,189.57}} $ $ F_{{\rm{mc}},{4,6}}^{\rm{M}}=\dfrac{\sqrt{1\,189.57\text{π} }}{{\xi }_{4}}L_{4,6}^{1/2}t_{4,6}^{3/2}{\sigma }_{0} $
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表 6 BFMC管的理论与模拟MCF的对比
Table 6. Comparison of theoretical and simulated MCF of the BFMC tubes
Group Type MCF BFMC-E4 BFMC-E5 BFMC-E6 Theory/kN Sim./kN Error/% Theory/kN Sim./kN Error/% Theory/kN Sim./kN Error/% Group1 BFMC-0 8.57 8.69 −1.40 9.24 9.08 1.77 9.71 9.57 1.46 BFMC-1 7.87 7.99 −1.42 7.96 8.24 −3.38 8.05 8.37 −3.80 BFMC-2 8.83 8.12 8.64 9.05 8.63 4.89 9.11 8.76 4.08 BFMC-3 11.07 10.16 8.93 11.23 10.80 4.00 11.18 11.00 1.60 BFMC-4 14.17 13.81 2.54 14.35 14.45 −0.69 14.28 14.75 −3.15 Group2 BFMC-0 12.70 13.11 −3.18 13.70 13.70 −0.01 14.40 14.64 −1.69 BFMC-1 11.67 12.20 −4.38 11.79 12.37 −4.69 11.93 12.66 −5.78 BFMC-2 13.08 12.28 6.56 13.41 12.95 3.60 13.51 13.35 1.22 BFMC-3 16.41 15.33 7.09 16.64 16.12 3.28 16.57 16.66 −0.55 BFMC-4 21.00 20.83 0.79 21.27 21.84 −2.61 21.17 22.21 −4.69 Group3 BFMC-0 17.34 18.28 −5.13 18.71 19.49 −4.02 19.66 20.13 −2.34 BFMC-1 15.93 16.59 −3.96 16.10 17.25 −6.63 16.29 17.51 −6.97 BFMC-2 17.86 17.14 4.24 18.32 17.96 1.96 18.44 18.31 0.72 BFMC-3 22.41 21.28 5.28 22.72 22.45 1.23 22.63 23.37 −3.20 BFMC-4 28.67 28.67 0.01 29.04 30.24 −3.94 28.90 30.81 −6.20 Group4 BFMC-0 22.44 24.23 −7.39 24.21 25.17 −3.82 25.44 26.31 −3.32 BFMC-1 20.62 21.68 −4.89 20.84 22.04 −5.44 21.08 22.34 −5.68 BFMC-2 23.12 22.70 1.85 23.70 23.12 2.51 23.87 23.59 1.16 BFMC-3 29.00 28.26 2.62 29.41 29.52 −0.39 29.28 30.52 −4.05 BFMC-4 37.10 37.65 −1.48 37.58 39.41 −4.64 37.40 39.87 −6.20
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