薄板在冲击载荷下线弹性理想塑性响应的相似性研究

李肖成; 徐绯; 杨磊峰; 王帅; 刘小川; 惠旭龙; 刘继军

doi:10.11883/bzycj-2020-0374

薄板在冲击载荷下线弹性理想塑性响应的相似性研究

doi: 10.11883/bzycj-2020-0374

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)；陕西省自然科学基础研究计划（2019JQ-625）；高等学校学科创新引智计划 (111计划) (BP0719007)；

详细信息

作者简介:
李肖成（1995－　），男，硕士研究生，lixiaocheng@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: O344; V214.4
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- 被引次数: 8
出版历程
- 收稿日期: 2020-10-09
- 修回日期: 2020-04-22
- 网络出版日期: 2021-09-30
- 刊出日期: 2021-11-23

Study on the similarity of elasticity and ideal plasticity response of thin plate under impact loading

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: 对于比例模型和原型使用不同弹塑性材料的冲击相似性问题，由于弹性和塑性阶段材料特性的差别及其在不同变形阶段的弹性塑性共存现象，将导致原有的结构冲击相似性理论失效。基于薄板冲击问题的理论模型，采用方程分析方法重新推导了材料线弹性以及理想刚塑性共存时的冲击响应的相似律。提出了一种能够同时考虑弹性变形和塑性变形的结构缩放响应相似性分析的厚度补偿方法，并利用数值分析验证了提出方法的适用性。分析结果表明，使用厚度补偿方法得到的比例模型结构响应能够准确地预测原型结构的冲击响应。
- 薄板结构 /
- 弹塑性耦合 /
- 不同材料 /
- 相似性分析
Abstract: For the impact similarity problem of the scaled model and the prototype usually have different materials with elastic and plastic properties, the differences of material properties and the coexistence of elastoplastic in different deformation stages will lead to the failure of the previous impact similarity theory. Based on the theory of the thin plate impact problem, the similarity law of impact response was derived by using the method of equation similarity analysis for the material with the linear elastic and ideal rigid-plastic properties. The basic equations of the thin plate structure, such as the energy conservation equation and the strain-displacement equation, were analyzed using equation similarity analysis methods, and the similarity scaling factor of the ideal elastic-plastic thin plate structure was derived. Based on the equation similarity analysis methods, a thickness compensation method that can simultaneously consider the similarity of elastic deformation and plastic deformation was proposed. For the impact similarity problem of the scaled model and the prototype using different ideal elastoplastic materials, this method can be used to calculate the geometric sizes and load conditions of the scaled model through the material properties when the response of the scaled model is similar to that of the prototype. Two finite element models of circular plate mass impact and circular plate velocity impact were established.The geometric sizes and load conditions of the scaled model were calculated through the thickness compensation method when the prototype uses aluminum alloy and the scaled model uses different materials such as steel, brass, etc. The applicability of the thickness compensation method was verified by the response of prototype and scaled model. The results show that the structural response of the scaled model obtained by the thickness compensation method can accurately predict the impact response of the prototype, even though the scale model and the scale model use different materials.
- thin-plate structure /
- elastic-plastic coupling /
- different materials /
- similarity analysis

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图 1 不同材料应力应变缩放结果

Figure 1. Stress-strain scaling results for different materials

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图 2 圆板冲击示意图

Figure 2. Schematic diagram of the circular plate under impact loading

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图 3 不同材料应力应变曲线

Figure 3. Stress-strain curves of different materials

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图 4 速度冲击比例模型与原型圆板中点处位移响应曲线对比

Figure 4. Comparison of displacement response between the velocity impact scale model and the prototype at the midpoint of the circular plate

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图 5 速度冲击比例模型与原型圆板中点处动能响应曲线对比

Figure 5. Comparison of kinetic energy response between the velocity impact scale model and the prototype at the midpoint of the circular plate

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图 6 速度冲击比例模型与原型圆板中点处应变响应曲线对比

Figure 6. Comparison of strain response between the velocity impact scale model and the prototype at the midpoint of the circular plate

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图 7 速度冲击比例模型与原型圆板中点处应力响应曲线对比

Figure 7. Comparison of stress response between the velocity impact scale model and the prototype at the midpoint of the circular plate

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图 8 质量冲击比例模型与原型圆板中点处位移响应曲线对比

Figure 8. Comparison of displacement response between the mass impact scale model and the prototype at the midpoint of the circular plate

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图 9 质量冲击比例模型与原型圆板中点处动能响应曲线对比

Figure 9. Comparison of kinetic energy response between the mass impact scale model and the prototype at the midpoint of the circular plate

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图 10 质量冲击比例模型与原型圆板中点处力响应曲线对比

Figure 10. Comparison of strain response between the mass impact scale model and the prototype at the midpoint of the circular plate

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图 11 质量冲击比例模型与原型圆板中点处应变响应曲线对比

Figure 11. Comparison of stress response between the mass impact scale model and the prototype at the midpoint of the circular plate

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图 12 不同冲击速度下圆板沿直径方向的位移

Figure 12. Displacement of circular plate along the diameter direction under different impact velocities

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图 13 不同冲击速度下圆板沿直径方向的Mises应力

Figure 13. Mises stress of circular plate along the diameter direction under different impact velocities

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图 14 不同冲击速度下圆板沿直径方向的等效应变

Figure 14. Equivalent strain of the circular plate along the diameter direction under different impact velocities

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表 1 刚塑性结构相似性缩放因子^{[6, 8]}

Table 1. Scaling factors of rigid-plastic structure ^{[6, 8]}

变量	缩放因子	变量	缩放因子
长度， $L$	$\lambda ={L}_{\mathrm{m}}/{L}_{\mathrm{p}}$	位移， $\delta$	${\lambda }_{\delta }=\lambda$
密度， $\rho$	${\lambda }_{\rho }={\rho }_{\mathrm{m}}/{\rho }_{\mathrm{p}}$	应力， $\sigma$	${\lambda }_{\sigma }={\lambda }_{\rho }{\lambda }_{v}^{2}$
速度， $v$	${\lambda }_{v}={v}_{\mathrm{m}}/{v}_{\mathrm{p}}$	应变， $\varepsilon$	${\lambda }_{\varepsilon }=1$
质量， $m$	${{\lambda }_{m}={\lambda }_{\rho }\lambda }^{3}$	应变率， $\dot{\varepsilon }$	${\lambda }_{\dot{\varepsilon }}={\lambda }_{v}/\lambda$
时间， $t$	${\lambda }_{t}=\lambda /{\lambda }_{v}$	力， $F$	${{\lambda }_{F}={\lambda }_{\rho }\lambda }^{2}{\lambda }_{v}^{2}$
加速度， $a$	${\lambda }_{a}={\lambda }_{v}^{2}/\lambda$	能量， ${E}_{{\rm{n}}}$	${ {\lambda }_{ {E}_{{\rm{n}}} }={\lambda }_{\rho }\lambda }^{3}{\lambda }_{v}^{2}$
注：在表中，比例模型与原型相关的物理量分别用下标m和p表示， ${\lambda }_{K}={K}_{\mathrm{m}}/{K}_{\mathrm{p}}$ 表示缩比模型和原型相关物理量的比值，例如 ${\lambda }_{v}={v}_{\mathrm{m} }/{v}_{\mathrm{p} }$ 表示比例模型和原型速度的比值，比例模型和原型的几何缩放系数 $\mathrm{\lambda }={L}_{\mathrm{m}}/{L}_{\mathrm{p}}$ 。

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表 2 理想弹塑性薄板结构冲击载荷作用下缩放因子

Table 2. Scaling factors of theideal elastic-plastic thin-plates under impact loading

变量	缩放因子	变量	缩放因子
x, y	${\lambda }_{x}={\lambda }_{y}={\left({L}_{x}\right)}_{\mathrm{m}}/{\left({L}_{x}\right)}_{\mathrm{p}}$	t	${\lambda }_{t}={\lambda }_{\zeta }/{\lambda }_{{v}_{\textit{z}}}$
z	${\lambda }_{\textit{z}}={\lambda }_{x}{\left({\lambda }_{{\sigma }_{\mathrm{Y}}}/{\lambda }_{E}\right)}^{1/2}$	χ, η	${\lambda }_{\chi }={\lambda }_{\eta }={\lambda }_{x}{\lambda }_{{\sigma }_{\mathrm{Y}}}/{\lambda }_{E}$
$\rho$	${\lambda }_{\rho }={\rho }_{\mathrm{m}}/{\rho }_{\mathrm{p}}$	ζ	${\lambda }_{\zeta }={\lambda }_{x}{\left({\lambda }_{{\sigma }_{\mathrm{Y}}}/{\lambda }_{E}\right)}^{1/2}$
m	${{\lambda }_{m}=\mathrm{\lambda }}_{\rho }{\lambda }_{x}{\lambda }_{y}{\lambda }_{\textit{z}}$	${v}_{\textit{z}}$	${\lambda }_{{v}_{\textit{z}}}={\lambda }_{{\sigma }_{\mathrm{Y}}}{\left({\lambda }_{\rho }{\lambda }_{E}\right)}^{-1/2}$
E	${\lambda }_{E}={E}_{\mathrm{m}}/{E}_{\mathrm{p}}$	${F}_{\textit{z}}$	${\lambda }_{{F}_{\textit{z}}}={\lambda }_{M}{\lambda }_{{A}_{\textit{z}}}$
${\sigma }_{\mathrm{Y}}$	${\lambda }_{ {\sigma }_{\mathrm{Y} } }={\left({\sigma }_{{\rm{Y}}}\right)}_{\mathrm{m} }/{\left({\sigma }_{\mathrm{Y} }\right)}_{\mathrm{p} }$	${a}_{\textit{z}}$	${\lambda }_{ {a}_{\textit{z} } }= {\lambda }_{ {v}_{\textit{z} } } ^{2}/{\lambda }_{\zeta }$
${\sigma }_{x},{\sigma }_{y}$	${\lambda }_{{\sigma }_{x}}={\lambda }_{{\sigma }_{y}}={\lambda }_{{\sigma }_{\mathrm{Y}}}$	${\varepsilon }_{x}{,\varepsilon }_{y}$	${\lambda }_{{\varepsilon }_{x}}{,\lambda }_{{\varepsilon }_{y}}={\lambda }_{{\sigma }_{\mathrm{Y}}}/{\lambda }_{E}$
${\tau }_{xy}$	${\lambda }_{{\tau }_{xy}}={\lambda }_{{\sigma }_{\mathrm{Y}}}$	${\gamma }_{xy}$	${\lambda }_{ {\gamma }_{xy} }={\lambda }_{ {\sigma }_{{\rm{Y}}} }/{\lambda }_{E}$

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表 3 圆板材料属性

Table 3. Material properties of circular plate

材料	密度/（kg·m⁻³）	弹性模量/GPa	屈服强度/MPa
原型铝	2.70	72.4	265
Ti-6Al-4V	4.43	114	1104
1006钢	7.89	200	350
黄铜	8.52	110	112
钨合金	17.0	400	1506

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表 4 模型缩放系数

Table 4. Scaling factors of themodels

${\lambda }_{x}$	模型材料	${\lambda }_{\rho }$	${\lambda }_{E}$	${\lambda }_{\sigma }$	${\lambda }_{\varepsilon }$	${\lambda }_{\textit{z}}$	${\lambda }_{{v}_{\textit{z}}}$	${\lambda }_{t}$
1	原型铝	1	1	1	1	1	1	1
1/50	Ti-6Al-4V	1.64	1.57	4.17	2.65	0.0325	2.59	0.0126
1/50	1006钢	2.92	2.76	1.32	0.48	0.0138	0.46	0.0298
1/50	黄铜	3.16	1.52	0.42	0.28	0.0106	0.19	0.0547
1/50	钨合金	6.30	5.52	5.68	1.03	0.0203	0.96	0.0211

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