高压物理学报

Group	Mass/g	Type	Cross section thickness/mm
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

Group	Mass/g	Type	Cross section thickness/mm
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

ρ/(g∙cm⁻³)	E/GPa	ν	σ_y/MPa	σ_u/MPa
2.7	68.0	0.33	71.0	130.7

ρ/(g∙cm⁻³)	E/GPa	ν	σ_y/MPa	σ_u/MPa
2.7	68.0	0.33	71.0	130.7

Tube	Thickness/mm	PCF/kN	MCF/kN	CFE	SEA/(J·g⁻¹)
MTTD^[25]	0.532	20.04	9.14	0.46	9.97
CS2^[22]	0.537	19.61	9.16	0.47	10.20
MSH^[26]	0.300	18.11	8.75	0.48	8.81
BHMC-R4-3^[21]	0.224	20.66	14.05	0.68	12.98
BFMC-E6-4	0.168	19.75	14.85	0.75	13.18

Tube	Thickness/mm	PCF/kN	MCF/kN	CFE	SEA/(J·g⁻¹)
MTTD^[25]	0.532	20.04	9.14	0.46	9.97
CS2^[22]	0.537	19.61	9.16	0.47	10.20
MSH^[26]	0.300	18.11	8.75	0.48	8.81
BHMC-R4-3^[21]	0.224	20.66	14.05	0.68	12.98
BFMC-E6-4	0.168	19.75	14.85	0.75	13.18

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

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

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} $

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} $

Group	Type	MCF
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

Group	Type	MCF
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

轴向荷载作用下仿生分形多胞圆管的耐撞性

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高建铭 , 张晓斌 , 刘志芳 , 雷建银 , 李世强

高压物理学报 | 轻质复合防护材料与结构的力学行为 2026,40(7): 070113-1-070113-22

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高压物理学报 |轻质复合防护材料与结构的力学行为 2026 , 40 (7) : 070113-1 -070113-22

轴向荷载作用下仿生分形多胞圆管的耐撞性

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高建铭（1999－），男，硕士研究生，主要从事结构耐撞性研究. E-mail：2590886959@qq.com

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高建铭, 张晓斌, 刘志芳, 雷建银, 李世强

作者信息

太原理工大学航空航天学院, 山西太原　030024

通讯作者:

刘志芳（1971－），女，博士，教授，主要从事冲击动力学研究. E-mail：liuzhifang@tyut.edu.cn

作者简介:

高建铭（1999－），男，硕士研究生，主要从事结构耐撞性研究. E-mail：2590886959@qq.com

Crashworthiness of Bionic Fractal Multi-Cell Circular Tubes under Axial Load

Jianming GAO, Xiaobin ZHANG, Zhifang LIU, Jianyin LEI, Shiqiang LI

Affiliations

College of Aeronautics and Astronautics, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China

出版时间: 2026-07-05 doi: 10.11858/gywlxb.20251250

文章导航

摘要

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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。

关键词

仿生分形多胞圆管 / 耐撞性能 / 超折叠单元理论 / 能量吸收

Abstract

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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.

Key words

bio-inspired fractal multi-cell circular tubes / crashworthiness / super folding element theory / energy absorption

引用本文

高建铭, 张晓斌, 刘志芳, 雷建银, 李世强. 轴向荷载作用下仿生分形多胞圆管的耐撞性. 高压物理学报, 2026 , 40 (7) : 070113-1 -070113-22 . DOI: 10.11858/gywlxb.20251250

Jianming GAO, Xiaobin ZHANG, Zhifang LIU, Jianyin LEI, Shiqiang LI. Crashworthiness of Bionic Fractal Multi-Cell Circular Tubes under Axial Load[J]. Chinese Journal of High Pressure Physics, 2026 , 40 (7) : 070113-1 -070113-22 . DOI: 10.11858/gywlxb.20251250

正文

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随着交通运输、军事、航天等行业的迅速发展，薄壁结构因其质量轻、力学性能优越及成本低等优势，被广泛应用于各类安全防护装置中。近年来，学者们对圆管^[1]、方管^[2]及多边形管^[3]等典型单胞薄壁结构的轴向压缩性能进行了系统研究。然而，随着对结构安全性能和吸能效率要求的不断提高，传统薄壁结构的能量吸收与轻量化之间的矛盾更加突出，亟需发展新型高比吸能的薄壁结构。

自然界中的许多动植物结构具有低密度、高吸能的特性，为新型吸能结构设计提供了重要启示。受竹材^[4]、甲虫鞘翅^[5]、马蹄壁^[6]等生物结构的启发，研究者提出了多种仿生结构，并验证了其在抗冲击与能量吸收方面优于传统管状结构。Yin等^[7]基于木贼类植物的茎部结构设计了6种仿生截面形式，揭示了几何参数对其耐撞性能的影响。Li等^[8]基于玻璃海绵的双对角线特征提出了仿生多胞管，其比吸能显著高于传统多胞管结构。上述研究表明，仿生几何特征有助于提升结构的吸能性能。

仿生分形结构因其自相似层级特征及优异的力学与吸能特性，已成为增强薄壁结构性能的重要设计路径。Wu等^[9]基于树状分形理念构建了三角形、正方形及五边形分形结构，发现三阶树状分形结构的承载力最高，比传统三角形管提升约180%。Nikkhah等^[10]受哈弗氏骨结构启发，设计了18种嵌套管，发现带加强壁的嵌套管（S-H-Cr）具有最优的耐撞性能。Zhang等^[11]提出了3种自相似分形结构，发现在等质量条件下，六边分形结构的比吸能分别比三角形和正方形分形结构提高48.8%和58.8%。Gao等^[12]结合梯度和层级构型提出了仿生分形层级多胞六边形管（BHMH），其比吸能比传统六边形管提高约146%。Zhang等^[13]采用小管单元对圆管进行完全填充（HCT-F）和部分填充（HCT-P）设计，结果表明，部分填充结构具有更优的力学性能。He等^[14]基于自相似分形理论构建了多种层级结构，揭示了不同子结构之间的协同作用可以进一步提升能量吸收能力。Gao等^[15]和Qin等^[16]提出了分形多胞六边形管及分形梯度多胞方管，并分别验证了分形层级特征在增强结构吸能与承载性能方面的有效性。

尽管上述研究为吸能结构的发展奠定了基础，但仍存在一些不足，例如：多数分形或仿生结构仍局限于六边形、正方形等单一截面形式，对于圆管内嵌不同几何构型的分形结构关注较少；现有研究主要聚焦于对层级或几何参数的优化，对分形尺度、分形维数等关键分形特征的系统分析仍然不足；缺乏在近似等质量条件下对吸能结构的评估，也缺少可用于预测分形多胞管平均压缩力的理论模型。基于上述不足，本研究提出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，以期实现生物启发结构与分形层级几何的融合。通过构建BFMC管的有限元模型，系统分析质量、分形维数以及内嵌多边形边数对其耐撞性能的影响，并将其耐撞性能与其他多胞管进行对比；基于超折叠单元理论构建BFMC管平均压缩力的预测模型，并通过数值模拟验证理论模型的准确性与适用性。

1 BFMC管的几何结构

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如图1所示，受草本植物及小麦等茎秆微观组织中梯度构型的启发，构建了3种内嵌正四边形、正五边形和正六边形的仿生分形多胞圆管。在外圆管内部分别嵌入正四边形、正五边形和正六边形，并通过连接内嵌正多边形各顶点与外圆管内壁的面板形成基本构型，定义为0阶的BFMC管，分别命名为BFMC-E4-0、BFMC-E5-0和BFMC-E6-0。图2给出了0阶BFMC管的几何构型，其周长可表示为

(1)

$ L_{j}^{0}=2{\text{π}} R+j\left(2r\sin \frac{{\text{π}} }{j}+R-r\right) $

式中：j为内嵌正多边形的边数，R为外圆管半径，r为正多边形外接圆半径。受分形结构的启发，在0阶BFMC管的基础上，通过在正多边形顶点与外圆弧中点之间添加肋骨，构建1阶BFMC管（BFMC-1）。进一步对1阶结构中生成的角形肋骨单元进行旋转分形迭代，依次获得2阶、3阶和4阶BFMC管，分别命名为BFMC-2、BFMC-3和BFMC-4。图2给出了所有BFMC管的构型及命名方式。每次分形迭代所增加的周长及其对应分形角单元的平分角θ可表示为

(2)

$ L_{{i}}^{}={2}^{\max (i-2,\,0)+1}j\frac{R\sin \dfrac{{\text{π}} }{{2}^{i-1}j}}{\sin \left(\theta +\dfrac{{\text{π}} }{{2}^{i-1}j}\right)}, \;\;\;\;\;\;\;\theta =\arctan \dfrac{r\sin \dfrac{{\text{π}} }{j}}{R-r\cos \dfrac{{\text{π}} }{j}} $

式中：i为分形阶次。则结构的总周长可表示为

(3)

$ {L}_{ij}=\begin{cases}L_{j}^{0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,i=0\\L_{j}^{0}+\displaystyle\sum \limits_{1}^{i}L_{i}^{}\;\;\;\;\;i=1,2,3,4\end{cases} $

为研究BFMC管的耐撞性，设置了4个实验组，分别记为组1、组2、组3和组4。所有BFMC管的高度均为150 mm，外径均为80 mm，内正多边形顶点至圆心的距离均为20 mm。每组均设置与BFMC管具有相同外径和高度的圆管作为对照组。为保证可比性，各组管件的质量保持一致，4组圆管的壁厚分别设为1.0、1.3、1.6和1.9 mm。如表1所示，BFMC管的壁厚随质量、分形维数和内嵌正多边形边数变化而调整，以确保在不同结构参数下实现质量一致。

2 耐撞性指标与数值模拟

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2.1 耐撞性评价指标

为定量评估BFMC管的耐撞性能，本研究选取以下4个典型指标进行分析。

总吸能（total energy absorption, EA）^[19–20]表示结构在压缩过程中吸收的能量，可由载荷-位移曲线下的面积计算得到

(4)

$ {E}_{\rm{a}}=\int \nolimits_{0}^{s}F(x){\rm{d}}x $

式中：E_a为总吸能；F(x)为载荷；s为有效压缩距离，即从结构开始压缩到致密化阶段的位移。

比吸能（specific energy absorption, SEA）表示单位质量结构吸收的能量，定义为总吸能与结构质量的比值

(5)

$ {E}_{\rm{sa}}={{E}_{\rm{a}}}/{M} $

式中：E_sa为比吸能，M为结构的质量。

平均压缩力（mean crushing force, MCF）反映结构在压缩过程中承受的平均载荷，定义为总吸能与压缩距离之比

(6)

$ {F}_{\rm{mc}}={{E}_{\rm{a}}}/{s} $

式中：F_mc为平均压缩力。

压缩力效率（crushing force efficiency, CFE）用于评估结构载荷分布的均匀性，定义为平均压缩力与峰值压缩力之比

(7)

$ \eta =\frac{{F}_{\rm{mc}}}{{F}_{\rm{pc}}}\times 100\text{%} $

式中：η为压缩力效率，F_pc为峰值压缩力（peak crushing force‌, PCF）。

2.2 有限元模型

采用ABAQUS软件对BFMC管的轴向压缩响应进行了数值模拟研究。BFMC管位于上、下2块刚性板之间，如图3所示。上刚性板以10 m/s的恒定速度垂直向下加载^[21]，压缩位移设为120 mm；下刚性板固定全部自由度。刚性板与BFMC管模型之间的接触以及BFMC管的自接触均采用通用接触模型，切向接触为罚函数形式，摩擦系数取0.2，法向接触为硬接触。模型材料均为AA6061-O铝合金^[22]，并采用理想弹塑性本构模型。该材料的应力-应变曲线如图4^[22]所示，材料参数如表2所示，其中，ρ为密度，E为杨氏模量，ν为泊松比，σ_y为初始屈服应力，σ_u为极限屈服应力。由于AA6061-O铝合金对应变率敏感性较低，本研究中不考虑应变率效应^[23]。

2.3 网格收敛性分析

采用线性四边形减缩积分壳单元（S4R）对BFMC管模型进行离散化。为了验证有限元模型的可靠性，开展了网格收敛性分析，选取的单元尺寸分别为0.5、1.0、1.5、2.0、2.5和3.0 mm。图5(a)给出了壁厚为1 mm的BFMC-E4-0管在不同单元尺寸下的力-位移曲线。当单元尺寸为1.0 mm时，力-位移曲线已基本收敛。图5(b)给出了不同单元尺寸下BFMC-E4-0管平均压缩力的变化趋势，当单元尺寸小于1.0 mm时，平均压缩力的变化趋于稳定。综合计算精度和计算效率，最终选用1.0 mm作为BFMC管模型的单元尺寸。

2.4 模型有效性验证

为验证有限元模型的有效性，将模拟结果与Chen等^[22]的实验数据进行对比。根据实验试件参数，建立了外径为80 mm、内部正方形对角线长度为40 mm、高度为120 mm的圆管有限元模型，材料性质采用表2所列的AA6061-O铝合金参数。加载速度设为0.5 m/s，以模拟准静态压缩过程。图6(a)给出了BFMC-E4-0管的实验测量和数值模拟的力-位移曲线。结果表明，2条曲线的整体变化趋势一致，均表现出典型的波动特征；微小的差异可能来自本构模型的简化、实验条件的变化或试件制造误差等因素。图6(b)给出了实验与模拟的变形模式对比，两者均呈现相似的渐进折叠特征，在压缩位移约为70 mm处折叠层数一致。峰值压缩力的实验值和模拟值分别为29.33和29.87 kN，相对误差为1.45%；平均压缩力的实验值和模拟值分别为15.56与15.33 kN，相对误差为1.48%。可以看出，建立的有限元模型能够较准确地反映BFMC管在压缩过程中的力学行为，验证了模型的可靠性和精度。

3 结果与讨论

收起

对BFMC管在轴向压缩载荷下的变形特征进行研究，并分析质量、分形维数以及内嵌多边形边数对其耐撞性能的影响。

3.1 变形模式与力-位移曲线

图7给出了组1中各类BFMC管在压缩至致密化阶段后的典型变形模式。可以看出，随着分形维数以及内嵌正多边形边数的增加，结构在压缩过程中形成的褶皱数量逐渐增多，变形模式也更加复杂。此类多褶皱、渐进式的折叠行为有利于提高结构的能量吸收能力。尽管不同类型的BFMC管在褶皱数量上存在差异，但其整体屈曲过程均表现出相似的渐进折叠特征。基于上述观察，后续分析将以BFMC-E4管为对象，对其压缩变形机理及力-位移响应特征进行深入讨论。

图8给出了BFMC-E4管在轴向压缩过程中的典型变形模式。从图8可以看出，分形维数对中部内嵌四边形区域各面板褶皱数量的影响较小，其中BFMC-E4-0、BFMC-E4-1和BFMC-E4-2管各面板褶皱数均为8，而BFMC-E4-3和BFMC-E4-4则增加至9。随着分形维数的增大，结构中更多面板参与渐进折叠过程。当分形维数较低时，各面板折叠几乎独立进行；而在较高分形维数下，外层面板之间出现明显的相互干涉与耦合。

以BFMC-E4管为例，BFMC-E4-0、BFMC-E4-1、BFMC-E4-2和BFMC-E4-3的外层面板褶皱数分别为8、9、10和14。当分形维数增至4时，折叠区域的面板数量显著增多。例如：BFMC-E4-1管截面右侧有2个面板发生折叠，褶皱数分别为7和9；而BFMC-E4-4管则表现出最显著的面板耦合行为，其截面外缘共有5个面板参与折叠，右侧面板的褶皱数由内向外依次达到16、18、18、26和21。

图9、图10、图11展示了4组不同BFMC管在动态轴向压缩载荷下的力-位移曲线。可以看出，BFMC-E4、BFMC-E5和BFMC-E6管的力-位移曲线具有相似的演变规律。所有曲线在压缩初期迅速达到峰值载荷，随后载荷明显下降，并进入相对平稳的平台阶段，该阶段对应结构的渐进压缩及能量吸收过程。当BFMC管发生致密化时，曲线斜率显著增大，承载力迅速提升。在相同质量组中，对照组圆管的力-位移曲线表现出明显的波动。随着分形维数的增加，载荷波动幅度逐渐减小；当分形维数达到4时，平台阶段的波动最小，表明其变形过程更加平稳。这主要归因于分形维数增加时，角单元数量增加，更有利于形成稳定的渐进折叠结构，从而提高能量吸收效率^[24]。另外，分形维数的变化会对BFMC管的有效压缩距离产生影响。模拟结果表明，随着分形维数的增加，对应的有效压缩距离呈逐渐减小的趋势。

3.2 耐撞性分析

3.2.1 质量的影响

图12、图13和图14分别给出了BFMC-E4、BFMC-E5和BFMC-E6管在质量递增条件下耐撞性指标的变化规律。结果表明，随着薄壁管结构质量的增加，PCF、MCF、EA和SEA均呈现单调递增的变化趋势。以分形维数为2的构型为例，在各质量组之间，SEA的提升情况如下：对于BFMC-E4-2管，组4相对于组3提升17.1%，组3相对于组2提升13.5%，组2相对于组1提升11.4%；对于BFMC-E5-2管，组4相对于组3提升15.3%，组3相对于组2提升12.8%，组2相对于组1提升8.3%；对于BFMC-E6-2管，组4相对于组3提升17.2%，组3相对于组2提升11.4%，组2相对于组1提升8.5%。

随着质量及对应壁厚的增加，BFMC管的SEA整体呈增加趋势。然而，壁厚增加所带来的失稳强度提高与质量增加之间存在竞争关系，在不同的壁厚区间，两者的主导作用不同，因而尽管SEA随壁厚增加而上升，但其增幅并未表现出递减规律。在组4中，BFMC-E6-4管表现出最佳的比吸能提升，其SEA达到组1圆管的221.2%，表现出显著的能量吸收优势。

压缩力效率通常随结构质量的增加而提高。然而，本研究中，组4的BFMC-E4-4、BFMC-E5-4和BFMC-E6-4管的CFE增幅相对有限。定量分析表明，在分形维数为4的构型下，组4中BFMC-E4-4、BFMC-E5-4和BFMC-E6-4管的压缩力效率分别比组1中相同构型管提高了9.7%、9.5%和9.3%。其原因在于，高维分形结构具有更大的截面周长。在质量相同的条件下，相较于低维分形或简单圆管，高维分形管的壁厚增量较小。而压缩力效率的提升通常与壁厚的增加密切相关，因此，在高维分形条件下，压缩力效率的增幅不如低维构型显著。尽管如此，在压缩力效率指标上表现最优的仍是组4中的BFMC-E6-4管，其压缩力效率达到了组1圆管的208.0%，表现出显著的压缩力效率优势。

3.2.2 分形维数的影响

图15、图16和图17展示了分形维数增加对BFMC管耐撞性指标的影响。对于组1和组2，BFMC管的PCF与同组圆管近似，且圆管的PCF在同组中最大，最大差异分别为10.6%和10.4%。随着质量增加，组3和组4中BFMC管的PCF随着分形维数呈现先降后升的非单调变化趋势，其中PCF在分形维数为2时最低，在分形维数为4时最高。以组4为例，BFMC-E4-2、BFMC-E5-2、BFMC-E6-2管的PCF分别比圆管降低了8.0%、7.6%、6.3%；而BFMC-E4-4、BFMC-E5-4、BFMC-E6-4管的PCF分别比圆管提高了6.9%、8.6%、8.0%。

BFMC管的其他耐撞性指标（MCF、CFE、EA、SEA）变化趋势类似，即在低分形维数时先降低，随分形维数增大而提升。当分形维数为1时，BFMC管的MCF、CFE、EA、SEA均处于最低水平；而当分形维数大于1时，这些指标随着分形维数增加而逐渐增大。以BFMC-E4管为例，BFMC-E4-4管的MCF显著提高，这是由于分形维数为4时，面板折叠过程中相互交叠较多，完成折叠所需载荷增加，从而提高了结构的MCF。4组BFMC-E4-4管的MCF分别比同组圆管高69.9%、68.9%、64.1%和67.3%。

在CFE和SEA方面，BFMC-E4-0管的指标均高于同组圆管，但BFMC-E4-1管的CFE和SEA低于圆管；而当分形维数增至4时，BFMC-E4-4的CFE和SEA达到最高。4组BFMC-E4-4管的CFE分别比同组圆管高81.1%、70.2%、58.3%和56.5%，SEA分别比同组圆管高45.6%、44.9%、40.6%和43.4%。当内嵌多边形边数为6时，CFE和SEA表现最佳，4组BFMC-E6-4管的CFE分别比同组圆管高88.7%、80.1%、65.7%和64.0%，SEA分别高55.6%、54.5%、51.2%和51.8%。主要原因是分形维数较低时新增角单元的数量有限。虽然这些角单元能够诱导局部屈曲的产生，但数量不足以与外层圆管共同形成稳定的轴对称折叠模式。例如，图8所示的BFMC-E4-1管在压缩过程中外侧褶皱分布不均，表现出不规则折叠特征。随着分形维数的增大，角单元数量显著增多，并逐渐在整体吸能机制中占主导地位，从而促进结构形成更加稳定的折叠模式，提高能量吸收效率^[24]，并提升结构的耐撞性能。然而，分形维数的增大也会增加结构制造难度和胞壁连接界面数量，可能引入制造缺陷并影响压溃过程中的性能稳定性。因此，在实际工程应用中，需要在结构性能与可制造性之间进行权衡。

3.2.3 内嵌正多边形边数的影响

通过比较同一质量组中相同分形维数的BFMC管的耐撞性指标，可以发现，MCF、CFE、EA及SEA均随内嵌多边形边数的增加而略有提升，但最大增幅不超过8%。以分形维数为3的构型为例，如图18所示。可以看出：4组BFMC-E5-3管的CFE分别比同组BFMC-E4-3管高2.8%、3.0%、4.5%、3.3%，4组BFMC-E6-3管的CFE分别比同组BFMC-E5-3管高2.8%、4.9%、4.2%、3.2%；4组BFMC-E5-3管的SEA分别比同组BFMC-E4-3管高5.7%、5.1%、5.5%、4.5%，4组BFMC-E6-3管的SEA分别比同组BFMC-E5-3管高2.0%、3.4%、4.8%、3.3%。

内嵌多边形边数的变化对PCF的影响较小。这主要是因为，在同一质量组中，结构质量保持不变，等效于横截面接触面积不变。根据碰撞动力学基本理论，PCF与材料屈服强度和接触面积的乘积成正比。因此，内嵌多边形边数的变化对PCF几乎无影响，表现出较强的鲁棒性（strong robustness），即内嵌多边形边数变化对峰值压缩载荷的影响可忽略不计。

3.3 多胞圆管耐撞性能的对比分析

根据3.2节的分析，BFMC-E6-4管表现出较好的耐撞性能。为进一步验证其优势，对组1中的BFMC-E6-4管与其他4种多胞圆管（MTTD^[25]、CS2^[22]、MSH^[26]和BHMC-R4-3^[21]）进行对比分析。5种管的外径（80 mm）、高度（150 mm）、质量（101.79 g）、材料（AA6061-O铝合金）及加载速度（10 m/s）均保持一致，仅根据保证等质量的要求对壁厚进行了调整。表3给出了5种多胞圆管的耐撞性能对比结果。可以看出，本研究提出的BFMC-E6-4管的MCF、CFE与SEA均优于其他多胞圆管。其中，SEA较MTTD、CS2、MSH和BHMC-R4-3分别提升了32.2%、29.2%、49.6%和15.4%，压缩力效率分别提升了63.0%、59.6%、56.3%和10.3%。

上述研究表明，将分形理论与仿生结构相结合用于分形多胞圆管的设计，可显著提升其耐撞性能。这一方法为开发新型高耐撞、轻量化结构提供了设计思路与技术途径。

4 理论预测

收起

为预测BFMC管在压缩过程的吸能特性，Chen等^[27]提出了超折叠单元理论（super folding element theory）。该理论假设在渐进折叠过程中，每一次折叠的波长相等。根据虚功原理，压缩过程中外功做的功由结构的弯曲变形和薄膜变形而耗散，可表示为

(8)

$ W=F_{ij}^{\rm{M}} s={E}_{\rm{b}}+{E}_{\rm{m}} $

式中：

$ F_{ij}^{\rm{M}} $

为压缩过程中的平均压缩力，E_b为弯曲变形能，E_m为薄膜耗散能。考虑到实际压缩过程中结构无法完全折叠，引入有效距离修正系数ξ_j，即有效压缩距离与管长之比，其取值受结构类型和角单元数量影响，一般在0.60～0.75范围内^{[25, 28–30]}，则有效压缩距离s可表示为

(9)

$ s=2{H}_{ij}{\xi }_{j} $

式中：H为折叠过程中的半波长，ξ_j为有效距离修正系数。前述分析表明，管的横截面形状及分形维数均会影响有效压缩距离，且分形维数越高，折叠越密集，其有效压缩距离越小。在本研究中，0阶、1阶和2阶BFMC管的修正系数均取0.68；对于角单元数量较多的3阶和4阶BFMC管，修正系数分别取0.63和0.60^{[15, 31]}。

4.1 弯曲变形能

如图19所示，弯曲变形能可视为3条塑性铰折叠弯曲的耗散能量之和，可表示为

(10)

$ {E}_{\rm{b}}=\sum \limits_{i=1}^{3}{\theta }_{i}{M}_{0}{L}_{ij}=2 {\text{π}}{M}_{0}{L}_{ij} $

式中：

$ {\theta }_{i}\left(i=1, 2, 3\right) $

为3条塑性铰线的转角，L_ij为结构横截面的周长，

$ {M}_{0} $

为塑性弯矩。

$ {M}_{0} $

可表示为

(11)

$ {M}_{0}=\frac{1}{4}{\sigma }_{0}{t}^{2}=\frac{1}{4}\sqrt{\frac{{\sigma }_{\rm{y}}{\sigma }_{\rm{u}}}{1+n}}{t}^{2} $

式中：t为薄壁管的壁厚；σ₀为材料的流变应力；n为应变硬化系数，本研究中n取0.2。

4.2 薄膜耗散能

当分形维数为零时，BFMC管由三面板T型和三面板Y型单元2种基本角单元组成；而当分形维数大于1时，BFMC管由三面板y型、四面板k型、五面板Ⅰ型和五面板Ⅱ型单元4种基本单元组成。如图20所示，BFMC-E6-0管由三面板T型和三面板Y型单元组成，而BFMC-E6-3管由三面板y型、四面板k型、五面板Ⅰ型和五面板Ⅱ型单元组成。表4列出了各个BFMC管中各角单元的数量分布。后续的分析基于上述角单元类型推导各单元的薄膜耗散能，为结构整体吸能分析提供理论依据。

4.2.1 三面板T型角单元

对称的三面板T型角单元可视为由三面板组成且相互夹角为直角的对称角单元^[32]，表示为

(12)

$ E_{\rm{m}}^{\text{T-shape}}=4\frac{{M}_{0}{H}^{2}}{t}\left(1+2\tan \frac{ {\text{π}}}{4}\right)=12\frac{{M}_{0}{H}^{2}}{t} $

4.2.2 三面板Y型角单元

对于对称的三面板Y型角单元，A面板仅发生弯曲而不发生膜变形^[33]。因此，Y型角单元的膜变形能主要由B板和C板的膜变形能以及厚度引起的耗散能组成，可表示为

(13)

$ \begin{aligned}E_{\rm{m}}^{\text{Y-shape}}&=E_{\rm{m}}^{\rm{A}}+E_{\rm{m}}^{\rm{B}}+E_{\rm{m}}^{\rm{C}}+E_{\rm{m}}^{\rm{thickening}}\\&=2\frac{{M}_{0}{H}^{2}}{t}\left(4\tan \frac{\theta }{4}+2\sin \frac{\theta }{2}+3\sin \theta \right)\end{aligned}$

4.2.3 三面板y型角单元

对于不对称的三面板y型角单元，其相邻面板之间的夹角满足α+β≥180°。由于A板仅发生弯曲变形，其膜变形能为零。因此，y型角单元的膜变形能由B板和C板的膜变形能以及厚度引起的耗散能组成^[33]，可表示为

(14)

$ \begin{aligned}E_{\rm{m}}^{\text{y-{shape}}}&=E_{\rm{m}}^{\rm{A}}+E_{\rm{m}}^{\rm{B}}+E_{\rm{m}}^{\rm{C}}+E_{\rm{m}}^{\rm{thickening}}\\&=8\frac{{M}_{0}{H}^{2}}{t}\tan \left(\frac{\alpha }{4}+\frac{\omega }{2}\right)+8\frac{{M}_{0}{H}^{2}}{t}\tan\left| \frac{\alpha }{4}-\frac{\omega }{2}\right|+ \\& 4\frac{{M}_{0}{H}^{2}}{t}\sin \left(\frac{\alpha }{2}+\omega \right)+4\frac{{M}_{0}{H}^{2}}{t}\sin \left| \frac{\alpha }{2}-\omega \right| \ +12\frac{{M}_{0}{H}^{2}}{t}\sin \alpha \end{aligned} $

式中：ω为B板与A板及C板对称轴的夹角，且ω=α/2。则三面板y型角单元的膜变形能可表示为

(15)

$ E_{\rm{m}}^{\text{y-shape}}=\frac{{M}_{0}{H}^{2}}{t}\left(8\tan \frac{\alpha }{4}+16\sin \alpha \right), \;\;\;\;\; \alpha =2\theta +{\text{π}} /({2}^{m-1}j) $

4.2.4 四面板k型角单元

对于对称的四面板k型角单元，可视为一个二面板角单元与2个附加面板组成。由于二面板角单元的夹角会影响其膜变形能^[34]，其膜变形能可表示为

(16)

$ E_{\rm{m}}^{\text{2-panel}}=4\frac{{M}_{0}{H}^{2}}{{t}} \frac{1.1\tan \varphi }{\tan \varphi +0.05/\tan \varphi } $

如图20所示，二面板角单元的夹角ϕ=2θ，附加面板与角单元面板之间的夹角为γ，且γ=π/2−θ。因此，四面板k型角单元在一个波长范围内的膜变形能可表示为

(17)

$ \begin{aligned}E_{\rm{m}}^{\text{k-shape}}&=E_{\rm{m}}^{2\text{-}\mathrm{panel}}\left(\varphi \right)+2E_{\rm{m}}^{\gamma \text{-}\mathrm{panel}}\\&=4\frac{{M}_{0}{H}^{2}}{{t}}\left[\frac{1.1\tan (2\theta )}{\tan (2\theta )+0.05/\tan (2\theta )}+2/\cos \left(\frac{{\text{π}} }{2}-\theta \right)\right]\end{aligned} $

4.2.5 五面板Ⅰ型结构

由于五面板Ⅰ型结构具有对称性，可由一个三面板对称的角单元与一个二面板角单元组合而成。在一个波长内，三面板对称角单元的膜变形能可表示为

(18)

$ E_{\rm{m}}^{3\text{-}{\rm{panel\text{-}sym}}}=4\frac{{M}_{0}{H}^{2}}{{t}}\left(\frac{1.1\tan \phi }{\tan \phi +0.05/\tan \phi }+2\tan \frac{\phi }{2}\right) $

如图20所示，三面板对称角单元中各板之间的夹角为

$ \phi $

，且

$ \phi = {\text{π}}/2+\theta -\left(j-2\right) {\text{π}}/2j $

，二面板角单元的夹角为φ=(j−2)π/j。因此，五面板Ⅰ型结构在一个波长范围内的膜变形能可表示为

(19)

$ \begin{aligned}E_{\rm{m}}^{{{Ⅰ}}\text{-}{\rm{shape}}}&=E_{\rm{m}}^{3\text{-}{\mathrm{panel\text{-}sym}}}+E_{\rm{m}}^{2\text{-}{\mathrm{panel}}}\\&=4\frac{{M}_{0}{H}^{2}}{t}\left(\frac{1.1\tan \phi }{\tan \phi +0.05/\tan \phi }+2\tan \frac{\phi }{2}+\frac{1.1\tan \varphi }{\tan \varphi +0.05/\tan \varphi }\right)\end{aligned} $

4.2.6 五面板Ⅱ型结构

五面板Ⅱ型结构具有对称性，可由一个三面板T型单元与一个二面板角单元组合而成^[35]。其中，T型结构可视为夹角为直角的对称三面板结构，二面板角单元的夹角为2θ。在一个波长的变形过程中，五面板Ⅱ型结构的膜变形能可表示为

(20)

$ \begin{aligned}E_{\rm{m}}^{{{Ⅱ}}\text{-}{\rm{shape}}}&=E_{\rm{m}}^{2\text{-}{\rm{panel}}}\left(2\theta \right)+E_{\rm{m}}^{{\rm{T\text{-}shape}}}\\&=4\frac{{M}_{0}{H}^{2}}{{t}}\left[\frac{1.1\tan (2\theta )}{\tan (2\theta )+0.05/\tan (2\theta )}+3\right]\end{aligned} $

4.3 平均压缩力

将式(9)和式(10)代入式(8)中，可得到

(21)

$ \begin{aligned}F_{ij}^{\rm{M}} 2{H}_{ij}{\xi }_{j}=&2{\text{π}} {M}_{0}{L}_{ij}+{N}_{1}{E}^{\rm{T\text{-}shape}}+{N}_{2}{E}^{\rm{Y\text{-}shape}}+\\& {N}_{3}{E}^{\rm{y\text{-}shape}}+{N}_{4}{E}^{\rm{k\text{-}shape}}+{N}_{5}{E}^{{{Ⅰ}}\text{-}{\rm{shape}}}+{N}_{6}{E}^{{{Ⅱ}}\text{-}{\rm{shape}}}\end{aligned} $

式中：N₁、N₂、N₃、N₄、N₅、N₆分别表示三面板T型、三面板Y型、三面板y型、四面板k型、五面板Ⅰ型和五面板Ⅱ型单元的数量。进一步化简，可得到

(22)

$ F_{ij}^{\rm{M}}=\frac{1}{2{\xi }_{i}}\left(\frac{2{\text{π}} {M}_{0}{L}_{ij}}{H}+{g}_{ij}\frac{{M}_{0}H}{{t}_{ij}}\right) $

其中，g_ij可表示为

(23)

$ g_{ij}^{\rm{T}}=\left( \begin{array}{ccc}\begin{array}{cc}\begin{array}{cc}\begin{array}{l}96.569\\125.093\\151.674\end{array} & \begin{array}{l}191.031\\227.501\\268.344\end{array}\end{array} & \begin{array}{l}363.962\\453.361\\539.000\end{array}\end{array} & \begin{array}{l}746.383\\917.685\\1077.978\end{array} & \begin{array}{l}1637.824\\2015.380\\2379.149\end{array}\end{array} \right) $

假设结构在压缩过程中发生理想变形，当平均压缩力对波长的偏导为零时，可求得最小折叠半波长

(24)

$ \frac{\partial F_{ij}^{\rm{M}}}{\partial {H}_{ij}}=0 $

由此，H_ij可表示为

(25)

$ {H}_{ij}=\sqrt{\frac{2{\text{π}} {L}_{ij}{t}_{ij}}{{g}_{ij}}} $

把式(25)代入式(22)中，可得到

$ F_{{\rm{mc}},{ij}}^{\rm{M}} $

的表达式为

(26)

$ F_{{\rm{mc}},{ij}}^{\rm{M}}=\frac{{M}_{0}}{{\xi }_{i}}\sqrt{\frac{2{\text{π}} {L}_{ij}{g}_{ij}}{{t}_{ij}}}=\frac{\sqrt{2{\text{π}} }}{4{\xi }_{i}}L_{ij}^{1/2}g_{ij}^{1/2}t_{ij}^{3/2}{\sigma }_{0} $

以上分析基于准静态加载条件，对平均压缩力进行了预测。表5总结了BFMC管的折叠半波长及平均压缩力的理论值。

在本研究的数值模拟中，采用了10 m/s的加载速度。已有研究表明，在准静态条件下，结构的平均压缩力通常低于动态加载下的平均压缩力。因此，引入动态修正系数K对平均压缩力进行校正。文献中K的取值范围一般为1.05～1.30^{[31, 36–37]}，本研究中取K=1.2。如表6所示，理论预测与数值模拟的平均压缩力的最大相对误差为8.93%，表明两者吻合较好，验证了理论公式能够较准确地预测BFMC管的平均压缩力。

5 结　论

收起

本研究提出了3种内嵌不同正多边形的仿生多胞分形圆（BFMC-E4/E5/E6）管，并通过数值模拟与理论分析研究了其耐撞性能，得到以下主要结论。

(1) 内嵌不同正多边形的仿生多胞分形圆管在压缩过程中均表现出相似的渐进折叠模式，且分形维数越大，形成的褶皱数量越多。力-位移曲线可分为3个阶段：初始峰值阶段、平台阶段及致密化阶段。随着分形维数增大，曲线的波动逐渐减小，表现出更为平稳的变形过程。

(2) 随着BFMC管质量的增加，其耐撞性指标显著提高。对于分形维数较大的构型，CFE的提升相对较小。在组4中，BFMC-E6-4管表现出最佳的SEA和CFE，分别达到组1圆管的221.2%和208.0%。随着分形维数的增加，其耐撞性指标呈现先下降后上升的非单调趋势，即在低分形维数时性能下降，而在高分形维数时性能显著提升。BFMC管的耐撞性指标随内嵌多边形边数的增加而略有提升，但PCF对内嵌正多边形边数的变化不敏感，表现出较强的鲁棒性。

(3) 基于超折叠单元理论和能量守恒原理，对轴向载荷下BFMC管的平均压缩力进行了理论推导，理论预测与数值模拟结果较为吻合，为阐明该类结构在轴向压缩载荷下的变形机制奠定了理论基础。

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2026年第40卷第7期

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文章信息

doi: 10.11858/gywlxb.20251250

接收时间：2025-11-05
首发时间：2026-11-05
出版时间：2026-07-05

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收稿日期：2025-11-05
修回日期：2025-12-22
录用日期：2026-06-08

基金

作者信息

太原理工大学航空航天学院, 山西太原　030024

通讯作者:

刘志芳（1971－），女，博士，教授，主要从事冲击动力学研究. E-mail：liuzhifang@tyut.edu.cn

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2种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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RefWorks
TxT

Group	Mass/g	Type	Cross section thickness/mm
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

Group

Mass/g

Type

Cross section thickness/mm

BFMC-0

BFMC-1

BFMC-2

BFMC-3

BFMC-4

101.79

Circular

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

132.32

Circular

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

162.86

Circular

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

193.40

Circular

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

ρ/(g∙cm⁻³)	E/GPa	ν	σ_y/MPa	σ_u/MPa
2.7	68.0	0.33	71.0	130.7

ρ/(g∙cm⁻³)

E/GPa

σ_y/MPa

σ_u/MPa

2.7

68.0

0.33

71.0

130.7

Tube	Thickness/mm	PCF/kN	MCF/kN	CFE	SEA/(J·g⁻¹)
MTTD^[25]	0.532	20.04	9.14	0.46	9.97
CS2^[22]	0.537	19.61	9.16	0.47	10.20
MSH^[26]	0.300	18.11	8.75	0.48	8.81
BHMC-R4-3^[21]	0.224	20.66	14.05	0.68	12.98
BFMC-E6-4	0.168	19.75	14.85	0.75	13.18

Tube

Cross-section

Thickness/mm

PCF/kN

MCF/kN

CFE

SEA/(J·g⁻¹)

MTTD^[25]

0.532

20.04

9.14

0.46

9.97

CS2^[22]

0.537

19.61

9.16

0.47

10.20

MSH^[26]

0.300

18.11

8.75

0.48

8.81

BHMC-R4-3^[21]

0.224

20.66

14.05

0.68

12.98

BFMC-E6-4

0.168

19.75

14.85

0.75

13.18

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

Type of angle element

Number

BFMC-0

BFMC-1

BFMC-2

BFMC-3

BFMC-4

3-panel-T-shape

N₁

3-panel-Y-shape

N₂

3-panel-y-shape

N₃

14j

4-panel-k-shape

N₄

5-panel-Ⅰ-shape

N₅

5-panel-Ⅱ-shape

N₆

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} $

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} $

Group	Type	MCF
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

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

−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

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