固支圆板在多次远场空爆载荷下位移响应的理论模型

郑晓波; 赵宏涛; 李腾; 姚伟光; 宋海生; 桂毓林; 王治

doi:10.11883/bzycj-2024-0488

固支圆板在多次远场空爆载荷下位移响应的理论模型

doi: 10.11883/bzycj-2024-0488

1.
中国工程物理研究院流体物理研究所，四川绵阳 621900
2.
哈尔滨工程大学船舶工程学院，黑龙江哈尔滨 150001

详细信息

作者简介:
郑晓波（1986－　），男，博士，高级工程师，zxb3710@163.com

通讯作者:
桂毓林（1973－　），男，博士，正高级工程师，guiyulin@21cn.com

中图分类号: O382; U674.7
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- 文章访问数: 268
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- 被引次数: 0
出版历程
- 收稿日期: 2024-12-17
- 修回日期: 2025-03-22
- 网络出版日期: 2025-03-26

Theoretical model of displacement response of clamped circular plate under multiple far-field blast loads

1.
Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, Sichuan, China
2.
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, Heilongjiang, China

摘要

摘要: 本文针对固支圆板在多次远场空爆载荷下的位移响应问题，基于膜理论能量方程，提出一种理论建模方法：通过将多次空爆载荷简化为线性衰减脉冲序列，首次建立了考虑应变率强化效应与累积硬化效应的固支圆板位移响应理论模型。首次加载阶段采用线性位移场近似，后续加载阶段引入二次函数位移场假设，推导出多次爆炸下中点位移的递推公式。通过LS-Dyna二次、三次空爆数值模拟验证表明，理论解与模拟结果的误差分别为20%~30%和20%以下。理论模型表明，圆板中点位移可表征为末次爆炸单独位移与前期累积位移的加权平方根函数，且后期爆炸的位移增量随前期累积位移增大而减小。该模型为多次空爆毁伤评估提供了理论依据，揭示了累积毁伤的非线性增长特性，对优化爆炸打击策略具有指导意义。
- 多次爆炸 /
- 圆板位移场 /
- 累积损伤
Abstract: Regarding the displacement response of clamped circular plates under multiple far-field blast loads, we proposes a novel theoretical modeling approach based on membrane theory energy equations, by simplifying multiple blast loads into linearly decaying pulse sequences, a theoretical displacement response model for clamped circular plates is established for the first time, considering both strain rate strengthening effects and cumulative hardening effects. The linear displacement field approximation is adopted for the initial loading phase, while a quadratic function displacement field assumption is introduced for subsequent loading phases, deriving recursive formulas for midpoint displacements under multiple blasts. Numerical validations were conducted using LS-DYNA for both double and triple blast scenarios. For double blast cases, theoretical predictions exhibited errors of 20%–30% compared to simulation results, while errors reduced to below 20% for triple blast conditions. The ASTM A415 steel circular plate model was used for the simulations, and the strain rate strengthening effect was described by the Cowper-Symonds model. Finite element models with quadrilateral shell elements demonstrated strong agreement with experimental data (errors <10%), confirming model reliability. The assumption of quadratic function displacement field for subsequent loading phases was verified by numerical displacement curves of the middle profiles of the plates. Further parametric analysis proved that the theoretical model is effective for different tangent modulus, which represents the strength of the strain strengthening effect. The model reveals that midpoint displacement can be characterized as a weighted square root function combining the final explosion’s individual displacement and prior cumulative displacement, with displacement increments from subsequent explosions decreasing as prior cumulative displacement increases. This study provides the first closed-form solution for multi-blast displacement prediction, addressing a critical gap in theoretical blast dynamics. This theoretical framework provides fundamental support for multiple blast damage assessments, unveils nonlinear cumulative damage growth characteristics, and offers guidance for optimizing explosive attack strategies.
- Multiple blast loads /
- displacement field of circular plate /
- cumulative damage

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图 1 多次载荷示意图

Figure 1. Schematic diagram of multiple loads

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图 2 线性强化刚塑性模型

Figure 2. Linear strengthening rigid plastic model

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图 3 圆板空爆仿真有限元模型

Figure 3. Finite element model for the circular plate under blast loads

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图 4 单次空爆载荷下圆板中点位移实验、数值模拟与理论结果的对比

Figure 4. Comparison of theoretical, test, and simulation result of midpoint displacement of the circular plate under single blast loads

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图 5 6 N·s+6 N·s冲量组合加载条件下圆板位移场

Figure 5. Displacement field of circular plate under 6 N·s+6 N·s impulse combination. (The displacement unit in the figure is cm)

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图 6 不同冲量组合加载条件下圆板中点位移时间曲线

Figure 6. Time curve of midpoint displacement of the circular plate under different impulse combinations

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图 7 典型二次空爆载荷下圆板中剖面位移曲线比较图

Figure 7. Comparison of displacement curves of middle profile of the circular plate under typical two blast loads

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图 8 典型三次空爆载荷下圆板中剖面位移曲线比较图（第3次加载后中剖面位移曲线）

Figure 8. Comparison of displacement curves of middle profile of the circular plate under typical three blast loads. (Displacement curve of the middle profile after the third loading)

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表 1 圆板尺寸与材料参数^[17]

Table 1. Size and material parameters of the circular plate

R/mm	H/mm	ρ/(kg·m⁻³)	$ {\dot{\varepsilon }}_{0} $/s⁻¹	n	σ₀/MPa	E_t/MPa
31.8	1.93	7800	40	5	223	10.8

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表 2 实验与数值模拟结果对比

Table 2. Comparison of numerical and experimental results

序号	实验数据^[16]		模拟数据
序号	I_t/(N·s)	W_f/H	p^m/MPa	τ/ms	W_f/mm	W_f/H	W_f/H误差
1	7.153	6.36	450.313	0.01	12.16	6.30	−0.94%
2	3.145	2.59	197.991	0.01	4.69	2.43	−6.18%
3	5.458	5.12	343.605	0.01	8.97	4.65	−9.18%
4	5.591	5.05	351.978	0.01	9.22	4.78	−5.35%
5	3.176	2.74	199.943	0.01	4.74	2.46	−10.22%
6	4.586	4.13	288.709	0.01	7.32	3.79	−8.23%
7	7.015	6.38	441.625	0.01	11.91	6.17	−3.29%

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表 3 二次空爆载荷下圆板中点位移理论与模拟结果的比较

Table 3. Comparison between theoretical and numerical results of midpoint displacement of the circular plate under two blast loads

冲量/(N·s)	$ W_{1}^{\mathrm{f}} $/mm		$ W_{1}^{\mathrm{f}} $误差/%	$ W_{2}^{\mathrm{f}} $/mm			$ W_{2}^{\mathrm{f}} $误差/%
冲量/(N·s)	模拟	理论	$ W_{1}^{\mathrm{f}} $误差/%	模拟	理论	修正理论	理论	修正理论
3+3	4.43	6.12	38.15	5.49	8.65	7.65	57.56	39.34
3+4	4.43	6.12	38.15	6.37	9.97	8.50	56.51	33.44
3+5	4.43	6.12	38.15	7.72	11.36	9.43	47.15	22.15
3+6	4.43	6.12	38.15	9.25	12.77	10.40	38.05	12.43
4+3	6.24	7.88	26.12	6.95	9.97	9.11	43.45	31.08
4+4	6.24	7.88	26.12	7.50	11.14	9.84	48.40	31.20
4+5	6.24	7.88	26.12	8.30	12.39	10.65	49.28	28.31
4+6	6.24	7.88	26.12	9.43	13.70	11.52	45.28	22.16
5+3	8.10	9.57	18.02	8.62	11.36	10.61	31.67	23.09
5+4	8.10	9.57	18.02	8.94	12.39	11.24	38.59	25.73
5+5	8.10	9.57	18.02	9.34	13.53	11.95	44.75	27.94
5+6	8.10	9.57	18.02	10.11	14.74	12.73	45.70	25.91
6+3	9.99	11.22	12.11	10.47	12.77	12.10	21.87	15.57
6+4	9.99	11.22	12.11	10.74	13.70	12.66	27.47	17.88
6+5	9.99	11.22	12.11	11.09	14.74	13.30	32.82	19.93
6+6	9.99	11.22	12.11	11.50	15.85	14.00	37.74	21.74

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表 4 三次空爆载荷下圆板中点位移理论与模拟结果的比较（E_t=10.8 MPa）

Table 4. Comparison between theoretical and numerical results of midpoint displacement of the circular plate under three blast loads (E_t=10.8 MPa)

冲量/(N·s)	$ W_{1}^{\mathrm{f}} $/mm		$ W_{1}^{\mathrm{f}} $误差/%	$ W_{2}^{\mathrm{f}} $/mm		$ W_{2}^{\mathrm{f}} $误差/%	$ W_{3}^{\mathrm{f}} $/mm		$ W_{3}^{\mathrm{f}} $误差/%
冲量/(N·s)	模拟	理论	$ W_{1}^{\mathrm{f}} $误差/%	模拟	理论	$ W_{2}^{\mathrm{f}} $误差/%	模拟	理论	$ W_{3}^{\mathrm{f}} $误差/%
3+3+3	4.43	6.12	38.15	5.49	7.65	39.34	6.76	8.92	31.95
4+4+4	6.24	7.87	26.12	7.50	9.84	31.20	9.42	11.48	21.87
5+5+5	8.10	9.56	18.02	9.34	11.95	27.94	11.79	13.94	18.24
6+6+6	9.99	11.20	12.11	11.50	14.00	21.74	15.17	16.33	7.65
4+5+6	6.24	7.87	26.12	8.30	10.65	28.31	11.84	13.57	14.61
4+6+5	6.24	7.87	26.12	9.43	11.52	22.16	12.37	13.57	9.70
5+4+6	8.10	9.56	18.02	8.94	11.24	25.73	12.00	14.03	16.92
5+6+4	8.10	9.56	18.02	10.11	12.73	25.91	11.81	14.03	18.80
6+4+5	9.99	11.20	12.11	10.74	12.66	17.88	12.07	14.55	20.55
6+5+4	9.99	11.20	12.11	11.09	13.30	19.93	12.20	14.55	19.26

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表 5 三次空爆载荷下圆板中点位移理论与模拟结果的比较（E_t=2 GPa）

Table 5. Comparison between theoretical and numerical results of midpoint displacement of the circular plate under three blast loads (E_t=2 GPa)

冲量/(N·s)	$ W_{1}^{\mathrm{f}} $/mm		$ W_{1}^{\mathrm{f}} $误差/%	$ W_{2}^{\mathrm{f}} $/mm		$ W_{2}^{\mathrm{f}} $误差/%	$ W_{3}^{\mathrm{f}} $/mm		$ W_{3}^{\mathrm{f}} $误差/%
冲量/(N·s)	模拟	理论	$ W_{1}^{\mathrm{f}} $误差/%	模拟	理论	$ W_{2}^{\mathrm{f}} $误差/%	模拟	理论	$ W_{3}^{\mathrm{f}} $误差/%
3+3+3	4.18	5.70	36.36	5.33	7.14	33.96	5.78	8.27	43.08
4+4+4	5.85	7.04	20.34	6.68	8.87	32.78	8.36	10.25	22.61
5+5+5	7.45	8.18	9.80	8.46	10.37	22.58	10.10	11.98	18.61
6+6+6	9.02	9.16	1.55	10.08	11.70	16.07	12.36	13.52	9.39
4+5+6	5.85	7.04	20.34	7.51	9.62	28.10	10.78	12.00	11.32
4+6+5	5.85	7.04	20.34	8.50	10.41	22.47	10.59	12.02	13.50
5+4+6	7.45	8.18	9.80	7.95	9.72	22.26	10.27	12.07	17.53
5+6+4	7.45	8.18	9.80	9.10	11.08	21.76	10.33	12.09	17.04
6+4+5	9.02	9.16	1.55	9.37	10.49	11.95	10.51	12.08	14.94
6+5+4	9.02	9.16	1.55	9.70	11.07	14.12	10.35	12.08	16.71

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