受轴向冲击薄壁圆管的几何畸变相似律研究

杨磊峰; 常新哲; 徐绯; 王帅; 刘小川; 惠旭龙; 李肖成

doi:10.11883/bzycj-2021-0452

受轴向冲击薄壁圆管的几何畸变相似律研究

doi: 10.11883/bzycj-2021-0452

1.
西北工业大学航空学院计算力学与工程应用研究所，陕西西安 710072
2.
中国飞机强度研究所结构冲击动力学航空科技重点实验室，陕西西安 710065

基金项目: 国家自然科学基金（11972309）；中央高校基本科研业务费专项资金（310201901A012）；高等学校学科创新引智计划（111计划）（BP0719007）

详细信息

作者简介:
杨磊峰（1997－　），男，硕士研究生，409332963@mail.nwpu.edu.cn

通讯作者:
徐　绯（1970－　），女，博士，教授，xufei@nwpu.edu.cn

中图分类号: O347.3；V214.4
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- 被引次数: 0
出版历程
- 收稿日期: 2021-11-02
- 修回日期: 2022-01-04
- 网络出版日期: 2022-04-24
- 刊出日期: 2022-05-27

Study on the scaling law of geometrically-distorted thin-walled cylindrical shells subjected to axial impact

1.
Institute for Computational Mechanics and Its Applications, School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China
2.
Aviation Key Laboratory of Science and Technology on Structures Impact Dynamics, Aircraft Strength Research Institute of China, Xi’an 710065, Shaanxi, China

摘要

摘要: 对于受轴向冲击载荷作用的薄壁圆管动态响应的相似律问题，由于圆管的薄壁特性导致厚度无法与高度和半径按相同的比例进行结构缩放，从而产生模型的几何畸变，此时传统的相似律已无法描述原型与畸变模型之间的动态响应规律。基于薄壁圆管轴向冲击问题的控制方程，通过能量守恒和量纲分析，推导了考虑几何畸变条件下轴向冲击载荷作用的理想弹塑性薄壁圆管动态响应的相似律。通过在给定应变与应变率区间上建立比例模型预测的流动屈服应力与原型流动屈服应力的最佳逼近关系，将几何畸变相似律进一步推广至包含应变率和应变硬化的材料。通过数值方法验证了提出的几何畸变模型相似律的适用性。分析结果表明，提出的考虑厚度畸变的受轴向冲击薄壁圆管的相似律可用于预测原型结构的冲击动态响应，并显著降低比例模型与原型结构平均载荷和能量的偏差。
- 薄壁圆管 /
- 应变率效应 /
- 应变硬化 /
- 几何畸变 /
- 相似律
Abstract: In scaling the dynamic responses of thin-walled cylindrical shells subjected to axial impact loading, the thickness cannot be adjusted according to the same scale as the radius and height due to the thin wall characteristics. Hence, geometrically-distorted models would be used, and the traditional scaling law cannot describe the relationship between the dynamic responses of the prototype and the geometrically-distorted model. In this paper, the scaling law for this case was derived for elastic-ideal plastic thin-walled cylindrical shells under axial impact loading. For strain hardening and strain-rate hardening material, based on the average load, deformation energy, and displacement of the shell in the axisymmetric deformation mode, the dimensionless numbers of three key design parameters, namely the stress, mass, and displacement, were obtained through the law of energy conservation. Then, the optimal approximation of the flow stress predicted by the distorted scaled model to the flow stress of the prototype was established on a given strain and strain rate interval. In this way, the derived scaling law can be applied to the case considering the coupling effects of geometric distortion, strain-rate sensitivity, and strain hardening. Finally, several finite element models of thin-walled cylindrical shell models subject to axial mass impact were established. These models use the elastic-ideal plastic material model and the general material model with strain-rate hardening and strain hardening effects. The modified impact velocity and impact mass were obtained by the present method using the geometrically-distorted model, which verified the effectiveness and correctness of the proposed scaling law. The results show that the geometrically distorted model corrected by the method proposed in this article can quite accurately predict the dynamic responses of the prototype, and significantly reduce the errors in the dynamic responses of the thin-walled cylindrical shell subjected to axial impact loading, especially the average load and deformation energy.
- thin-walled cylindrical shell /
- strain-rate sensitivity /
- strain hardening /
- geometric distortion /
- scaling law

HTML全文

图 1 轴向受压薄壁圆管的轴对称压溃模式

Figure 1. Axisymmetric crushing mode of the thin-walled cylindrical shell under axial compression

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图 2 金属材料应力-应变曲线

Figure 2. Stress-strain curves of metal materials

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图 3 受轴向冲击的薄壁圆管示意图

Figure 3. Schematic diagram of a thin-walled cylindrical shell under axial impact

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图 4 原型与比例模型的动态响应-时间曲线

Figure 4. Dynamic response-time curves of the scale models and the prototype

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图 5 修正后的比例模型与原型的动态响应-时间曲线

Figure 5. Dynamic response-time curves of the modified scaled models and the prototype

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表 1 纯几何相似律比例因子^[10-11]

Table 1. Scaling factors by pure geometric similarity^[10-11]

变量	比例因子	变量	比例因子
长度L^[10]	$\ \beta = {L_{\rm{m}}}/{L_{\rm{p}}}$	位移δ^[10]	$\ {\beta _\delta } = \beta $
密度ρ^[10]	$\ {\beta _\rho } = {\rho _{\rm{m}}}/{\rho _{\rm{p}}}$	应力${\sigma _{\rm{d}}}$^[10]	$\ {\beta _{{\sigma _{\rm{d}}}}} = {\beta _\rho }\beta _v^2$
速度v^[10]	$\ {\beta _v} = {v_{\rm{m}}}/{v_{\rm{p}}}$	应变ε^[10]	$\ {\beta _\varepsilon } = 1$
质量m^[10]	$\ {\beta _{{m} } } = {\beta _\rho }{\beta ^3}$	应变率$\dot \varepsilon $^[10]	$\ {\beta _{\dot \varepsilon }} = {\beta _v}/\beta $
时间t^[10]	$\ {\beta _{{t} } } = \beta /{\beta _v}$	载荷P^[11]	$\ {\beta _{{P} } } = {\beta _\rho }{\beta ^2}\beta _v^2$
加速度a^[10]	$\ {\beta _a} = \beta _v^2/\beta $	动能E_k^[11]	$\ {\beta _{ {E_{\rm{k}}} } } = {\beta _\rho }{\beta ^3}\beta _v^2$

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表 2 受轴向冲击的理想弹塑性薄壁圆管比例因子

Table 2. Scaling factors of the elastic-ideal plastic thin-walled cylindrical shell under axial impact loading

变量	比例因子	变量	比例因子
长度L	$\ \beta = {L_{\rm{m}}}/{L_{\rm{p}}}$	位移δ	$\ {\beta _\delta } = \beta $
密度ρ	$\ {\beta _\rho } = {\rho _{\rm{m}}}/{\rho _{\rm{p}}}$	应力${\sigma _{\rm{d}}}$	$\ {\beta _{{\sigma _{\rm{d}}}}} = {\beta _\rho }\beta _v^2\sqrt {\beta /{\beta _{\rm{h}}}} $
速度v	$\ {\beta _v} = {v_{\rm{m}}}/{v_{\rm{p}}}$	应变ε	$\ {\beta _\varepsilon } = \sqrt {{\beta _h}/\beta } $
质量m	$\ {\beta _{{m} } } = {\beta _\rho }{\beta ^2}{\beta _h} $	应变率$\dot \varepsilon $	$\ {\beta _{\dot \varepsilon }} = \left( {{\beta _v}/\beta } \right)\sqrt {{\beta _L}/{\beta _h}} $
时间t	$\ {\beta _{{t} } } = \beta /{\beta _v}$	载荷P	$\ {\beta _{{P} } } = {\beta _\rho }\beta {\beta _h}\beta _v^2$
加速度 a	$\ {\beta _a} = \beta _v^2/\beta $	动能E_k	$\ {\beta _{E_{\rm{k} } } } = {\beta _\rho }{\beta _h}{\beta ^2}\beta _v^2$

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表 3 1006 钢的材料参数^[24]

Table 3. Material parameters of 1006 steel^[24]

ρ/(g∙cm⁻³)	E/GPa	μ	A/MPa	B/MPa	C	n	${\dot \varepsilon _0}/\text{s}^{-1}$
7.89	207	0.3	350	275	0.022	0.36	1

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表 4 理想弹塑性模型几何畸变比例因子

Table 4. Scaling factors of geometrically-distorted models of elastic-ideal plastic material

$\ \beta $	η	${\ \beta _h}$	${\ \beta _M}$	${\ \beta _v}$	${\ \beta _t}$	${\ \beta _P}$	${\ \beta _\delta }$	${\ \beta _\varepsilon }$
0.1	1.2	0.12	0.12	1.0466	0.0955	0.0131	0.1	1.0954
0.1	1.5	0.15	0.15	1.1067	0.0904	0.0184	0.1	1.2247
0.1	1.8	0.18	0.18	1.1583	0.0863	0.0241	0.1	1.3416
0.1	2.0	0.20	0.20	1.1892	0.0841	0.0283	0.1	1.4142

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表 5 考虑应变率效应和应变硬化效应几何畸变模型比例因子

Table 5. Scaling factors of geometrically-distorted models considering strain-rate sensitivity and strain hardening

$\ \beta $	η	${\ \beta _h}$	${\ \beta _M}$	${\ \beta _v}$	${\ \beta _t}$	${\ \beta _P}$	${\ \beta _\delta }$	${\ \beta _\varepsilon }$
0.1	1.2	0.12	0.12	1.0756	0.0930	0.0139	0.1	1.0954
0.1	1.5	0.15	0.15	1.1442	0.0874	0.0196	0.1	1.2247
0.1	1.8	0.18	0.18	1.2038	0.0831	0.0261	0.1	1.3416
0.1	2.0	0.20	0.20	1.2396	0.0807	0.0306	0.1	1.4142

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表 6 比例模型与原型的峰值位移和平均载荷相对误差

Table 6. Relative errors in the peak displacement and average force between the scale models and prototype

模型	（δ/β_δ）/mm	相对误差/%	（P/β_P）/kN	相对误差/%
原型	124.258	−	71.95	−
β=1/10, η=1.2	121.669	2.084	73.43	2.057
β=1/10, η=1.5	120.651	2.903	69.48	3.433
β=1/10, η=1.8	118.642	4.520	75.85	5.420
β=1/10, η=2.0	117.678	5.295	72.25	0.417

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表 7 考虑应变率与应变硬化效应的比例模型位移与平均载荷相对误差

Table 7. Relative errors in the peak displacement and average force of the scaled models considering strain-rate sensitivity and strain hardening

模型	(δ/β_δ) /mm	相对误差/%	(P/β_P)/kN	相对误差/%
原型	90.740	−	97.79	−
β=1/10, η=1.2	89.619	1.235	98.42	0.644
β=1/10, η=1.5	89.222	1.673	99.47	1.718
β=1/10, η=1.8	86.201	5.002	101.06	3.344
β=1/10, η=2.0	86.232	4.968	105.14	7.516

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