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