铝弹丸超高速撞击防护结构的研究进展

林健宇; 罗斌强; 徐名扬; 宋卫东; 柏劲松; 裴晓阳; 于继东; 李平

doi:10.11858/gywlxb.20190774

铝弹丸超高速撞击防护结构的研究进展

doi: 10.11858/gywlxb.20190774

1.
中国工程物理研究院流体物理研究所，四川绵阳　621999
2.
北京理工大学爆炸科学与技术国家重点实验室，北京　100081

基金项目: 国家自然科学基金青年科学基金（11702272）; 国家自然科学基金（11532012）

详细信息

作者简介:
林健宇（1987－），男，博士，助理研究员，主要从事多介质力学计算研究. E-mail：linjiany@mail.ustc.edu.cn

通讯作者:
宋卫东（1975－），男，博士，教授，主要从事材料与结构冲击动力学研究. E-mail：swdgh@bit.edu.cn

中图分类号: O347; V423
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出版历程
- 收稿日期: 2019-05-13
- 修回日期: 2019-05-21

Progress of Aluminum Projectile Impacting on Plate with Hypervelocity

1.
Institute of Fluid Physics, CAEP, Mianyang 621999, China
2.
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China

摘要

摘要: 以空间碎片防护为背景，回顾了超高速铝弹丸正撞击单层和双层铝合金防护结构的研究进展，讨论了目前针对超高速撞击的弹丸发射技术和数值模拟方法（如Euler方法、无网格方法等）的优缺点。数值模拟不仅建立在离散方法上，还需要提供准确的材料本构模型和状态方程。介绍了常用材料模型（包括Johnson-Cook、Steinberg-Guinan模型）和状态方程（包括Tillotson、ANEOS、SESAME、GRAY三相状态方程）。基于实验和数值模拟，目前对7 km/s以下的超高速撞击物理过程已经认识得比较清楚。对单层板，重点讨论了板的穿孔特征和孔径模型；对双层板，除了前板的穿孔外，还讨论了碎片云的分布特征、材料相变、碎片云的相态分布、弹丸形状的影响、碎片云的散布模型以及碎片云对后板造成的破坏特征。最后介绍了工程防护中较为重要的防护结构的弹道极限方程。单层板和双层板的弹道极限方程研究近年来取得了较大进展。本文回顾了国内外常用的弹道极限方程以及近年来新提出的理论模型和基于人工神经网络的模型等。
- 碎片云 /
- 高速撞击 /
- 弹道极限方程
Abstract: This paper focuses on the protection of debris clouds in space and the progress of the studies of Al projectile impacting on single shield and Whipple shields are discussed. The advantages and drawbacks of the widely used experimental method for launching hypervelocity projectile and numerical methods for hypervelocity impact such as Euler methods and meshfree methods are introduced. The numerical simulation is usually based on the constitutive laws and the equation of states. In this paper we have reviewed the constitutive law including Johnson-Cook and Steinberg-Gruinan, while for the equation of states we include the Tillotson, ANEOS, SESAME, GRAY. The mechanics and physics for the hypervelocity impact below 7 km/s are now well understood based on the progresses made by the experiments and numerical simulations. Then, for the single shield we mainly focus on the perforation by the projectile and the models predicting the hole size. For the Whipple shield we have discussed the characteristics of the debris clouds evolution, the phase states of the materials, the models predicting the evolution of the debris clouds and damage features of the second wall of the Whipple structure induced by the debris clouds. Finally we discussed the ballistic limit equations which are very important to the protection in the engineering. Great progresses have been achieved for the ballistic equations for the single and double shields structures based on the experiments and numerical simulations. We have discussed the commonly used ones and the models which are newly developed recently including theoretical models and the models from artificial intelligence.
- debris clouds /
- hypervelocity impact /
- ballistic limit equation

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复合材料是一种性能优良、可设计性强的新型材料，具有比强度、比模量高等优点，被广泛应用于航空航天、汽车制造、风力发电叶片以及压力容器等领域^[1-3]。其中玻璃纤维增强环氧树脂基壳体的应用率较高，也是近年来的研究重点^[4-5]。用于管道和压力容器等的复合材料壳体因承受较大的力学载荷，壳体内部会发生基体开裂、纤维与基体脱粘、层合板分层和纤维基体同时断裂等损伤，损伤部位产生应力集中，当应力大于材料的许用应力后，会对整个系统的安全构成潜在的威胁^[6-8]。因此，通过研究应力、应变等重要参数从而理解复合材料结构壳体的力学响应行为，对提高含有断裂缺陷的复合材料的安全性能具有重要意义^[9-11]。

目前，科研人员对复合材料壳体的损伤已进行了大量的有限元模拟研究，通过建立渐进损伤有限元模型预测了复合材料壳体的损伤失效过程。Leh等^[12]建立了连续和渐进损伤两种有限元模型，对纤维缠绕压力容器的爆破压力进行预测，与试验结果相比，渐进损伤模型能够更好地模拟复合材料储氢容器的爆破过程。Wang等^[13]将材料递减规则和基于表面行为单元结合建立渐进损伤模型，预测了铝内胆碳纤维缠绕复合材料储氢气瓶的极限承载能力和复杂的失效行为。Liu等^[14]对3种不同尺寸的碳纤维缠绕压力容器进行失效分析，采用Hashin失效准则并基于能量损伤演变规律，得到与实验结果相符的失效性质参数和爆破强度。杨斌等^[15]为了分析混杂复合材料层合板在冲击载荷下的损伤演变过程，利用有限元分析软件Abaqus建立了预测复合材料层合板在低速冲击作用下损伤的三维有限元模型。

还有很多学者采用应变片或光纤埋入复合材料结构以实时监测的实验方法，研究了复合材料层合板和壳体在外压作用下的应变响应过程。Okabe等^[16]采用光纤布拉格光栅传感器识别碳纤维增强塑料层合板的裂纹位置，将光纤光栅传感器嵌入层合板交叉层中，并在拉伸载荷作用下测量反射光谱，从测量的光谱判断裂纹位置，证明了小直径光纤光栅传感器也可用于裂纹位置的识别。肖飚等^[17]在玻璃纤维缠绕金属内胆复合材料压力容器的制备过程中，将应变传感器埋在金属内胆与玻璃纤维/环氧树脂复合材料层之间，得到了具有原位监测功能的纤维缠绕压力容器。Kanerva等^[18]原位测试了碳纤维增强复合材料的层间变化，结果表明埋入应变片可以精准监测复合材料的层间状态。应变参数可以有效反映复合材料层合板及壳体的受力状态，而将有限元模拟和应力和应变响应结合考虑复合材料壳体的损伤行为从而判断整个压力容器健康情况的相关研究还很少。开展含有断裂缺陷的复合材料壳体的力学行为分析，能够为复合材料壳体检测及容器的安全检测提供一定的数据基础。

本研究将在玻璃纤维缠绕环氧树脂基压力容器的圆柱体上截取试样，通过拉伸、双悬臂梁（DCB）和三点端部开口弯曲（3ENF）试验对复合材料层合板进行基本力学性能评估，编写进用于判断材料失效的VUMAT用户子程序中，并采用有限元软件Abaqus分析不同深度断裂缺陷对复合材料压力容器应力、应变的影响。

1. 试样制备

将圆柱形模具固定在旋转操作箱上，使旋转操作箱不停转动，将含有环氧树脂预浸料的玻璃纤维缠绕在模具表面，缠绕到一定厚度后脱模，形成完整的压力容器，如图1（a）所示。该容器的内径、高度和厚度分别为1600、1620和18 mm，容积为3 m³，设计压力为0.375 MPa。玻璃纤维部分包括玻璃纤维缠绕线层（Y）、布层（C）和毡层（M），3种纤维层的铺层顺序为M/Y/C/Y/C/Y/C/M/Y/C/Y/C/Y/C/M。按照ASTM D3039/D3039M-08标准，采用水切割方法在压力容器筒体段的0°、90°和45°方向截取试样，试样尺寸分别为220 mm × 25 mm × 18 mm、220 mm × 25 mm × 18 mm和160 mm × 25 mm × 18 mm，每种试样各取5份。

图 1 试样制备及其力学性能测试：（a）复合材料压力容器，（b）双悬臂梁试验，（c）三点端部开口弯曲试验

Figure 1. Sample preparation and mechanical properties tests: (a) composite pressure vessel，(b) double cantilever beam test，(c) three-point end opening bending test

下载: 全尺寸图片幻灯片

根据裂纹产生原因不同，复合材料可以划分为3种分层模式：Ⅰ型断裂又称张开型裂纹，由与层合板面垂直的力产生，分层扩展方向与外载荷方向垂直，Ⅱ型和Ⅲ型分别称滑移型断裂和撕开型断裂，与面内剪应力有关，分别由平行和垂直于裂纹扩展方向的面内力产生。考虑到复合材料压力容器是由内压而产生的力学行为特点，主要考虑Ⅰ型和Ⅱ型两种基本断裂模式，测量这两种类型分层的断裂韧性，从而确定该材料的层间性能。根据ISO 15024标准、ASTM D5528-13标准和航空工业HB7402-1996标准，采用DCB试验方法测量I型复合材料的层间断裂韧性，如图1（b）所示。试样采用单向层板，厚度（h）为15 mm，长度（l）为220 mm，预制分层长度（a₀）为25 mm，试样宽度（b）为25 mm。根据ASTM D7905标准、日本JSAK7086标准和中国航空工业HB7403-1996标准，采用3ENF试验方法测定Ⅱ型复合材料的分层断裂韧性，如图1（c）所示。试样采用单向层板，厚度（h）为15 mm，长度（l）为100 mm，预制分层长度（a₀）为25 mm，试样宽度（b）为10 mm。

2. 基本力学性能测试

力学性能测试在INSTRON-8032电液伺服动态万能材料疲劳试验机上进行，试验机的最大载荷为100 kN，测试速度为2 mm/min。通过拉伸试件测试获得试样的刚度参数：横向拉伸模量E_x、纵向拉伸模量E_y、泊松比v_xy和剪切模量G_xy。由于所取试样的纤维方向与其主方向的夹角θ为12°，通过偏轴公式式（1）～式（4）得到最终的刚度参数：纤维主方向拉伸模量E₁、与纤维垂直方向拉伸模量E₂、泊松比v₁₂和剪切模量G₁₂。

$\frac{1}{{{E_x}}} = \frac{1}{{{E_1}}}{\cos ^4}\theta + \left( {\frac{1}{{{G_{12}}}} - \frac{{2{v_{12}}}}{{{E_1}}}} \right){\sin ^2}\theta {\cos ^2}\theta + \frac{1}{{{E_2}}}{\sin ^4}\theta$

(1)

$\frac{1}{{{E_y}}} = \frac{1}{{{E_1}}}{\sin ^4}\theta + \left( {\frac{1}{{{G_{12}}}} - \frac{{2{v_{12}}}}{{{E_1}}}} \right){\sin ^2}\theta {\cos ^2}\theta + \frac{1}{{{E_2}}}{\cos ^4}\theta$

(2)

$- \frac{{{v_{xy}}}}{{{E_x}}} = - \frac{{{v_{12}}}}{{{E_1}}}({\sin^4}\theta + {\cos ^4}\theta ) + \left( {\frac{1}{{{E_1}}} + \frac{1}{{{E_2}}} - \frac{1}{{{G_{12}}}}} \right){\sin ^2}\theta {\cos ^2}\theta$

(3)

$\frac{1}{{{G_{xy}}}} = \frac{1}{{{G_{12}}}}({\sin^4}\theta + {\cos ^4}\theta ) + 4\left( {\frac{{1 + 2{v_{21}}}}{{{E_1}}} + \frac{1}{{{E_2}}} - \frac{1}{{2{G_{12}}}}} \right){\sin ^2}\theta {\cos ^2}\theta$

(4)

根据修正梁理论，DCB试验中Ⅰ型分层试样的层间应变能释放率

${R_{\!\!\rm{\text{Ⅰ\!\!}c}}} = \frac{{3F{p_{\rm{c}}}d}}{{2b(a + \varDelta )}}$

(5)

其中

$F = 1 - \frac{3}{{10}}{\left( {\frac{d}{a}} \right)^2} - \frac{2}{3}\left( {\frac{{d{l_1}}}{{{a^2}}}} \right)$

(6)

式中：p_c为试样端部施加的载荷，d为端部位移即分层开口位移，a为分层长度，F为考虑大位移和加载块影响时的修正因子，Δ为分层长度的修正量。对于Ⅱ型分层，应变能释放率

${R_{\!\!\rm{\text{Ⅱ\!}c}}} = \frac{{9{a^2_0}{p_{\rm{c}}}d}}{{2b\left( {\displaystyle\frac{3}{8}{{l^3_0}} + 3{a^3_0}} \right)}}$

(7)

式中：l₀为弯曲加载跨距的一半，d为端部位移即弯曲试样中心挠度。试样采用预制分层的单向层板，通过加载平台施加载荷。

3. 有限元模拟方法

如图2所示，利用ABAQUS建立含有不同深度断裂缺陷的复合材料壳体的有限元分析模型，预测其在内压载荷作用下的应力和应变响应。同时，编写用于定义材料参数及判断材料失效的VUMAT用户子程序，采用Hashin 3D失效准则，并考虑材料的刚度递减规律，参数的选取参考文献[19]。复合材料壳体的厚度为18 mm，根据纤维的缠绕工艺特点，将复合材料壳体分为6层。断裂缺陷的长度、宽度和厚度分别为50、2和3 mm。第1层断裂缺陷深度为18 mm，第2层为15 mm，第3层为12 mm，第4层为9 mm，第5层为6 mm，第6层为3 mm。内部压力分别设定为0.1、0.2和0.3 MPa。

图 2 含有断裂缺陷的复合材料壳体的模拟：（a）有限元模型及缺陷位置，（b）边界条件及加载位置

Figure 2. Simulation of composite shell with fracture defect: (a) finite element model and defect position，(b) boundary condition and loading position

下载: 全尺寸图片幻灯片

4. 结果与讨论

玻璃纤维增强环氧树脂基复合材料的刚度和强度参数列于表1和表2中，表中数据为5个试样的平均值和方差。表中：E₁、E₂和E₃分别为纤维方向、垂直纤维方向和试样厚度方向的拉伸模量，G₁₂、G₁₃和G₂₃为3个面内的剪切模量，v₁₂、v₁₃和v₂₃为3个面内的泊松比，X_T和X_C分别为纤维方向的拉伸和压缩强度，Y_T和Y_C分别为垂直纤维方向的拉伸和压缩强度，Z_T和Z_C分别为试样厚度方向的拉伸和压缩强度，S₁₂、S₁₃和S₂₃表示3个面内的剪切强度，其中厚度方向的相关参数取自文献[20-21]。可以看到，复合材料主向的拉伸强度为（222.7 ± 18）MPa，弹性模量为39.39 GPa，横向的拉伸强度为（136 ± 22）MPa，弹性模量为18.1 GPa，层间强度Ⅰ型和Ⅱ型断裂韧性分别为（4.67 ± 0.24）kJ/m²和（4.98 ± 0.26）kJ/m²。

表 1 玻璃纤维/环氧树脂复合材料板的刚度参数^[20-21]

Table 1. Stiffness parameters of glass fiber/epoxy resin composite plate^[20-21]

E₁/GPa	E₂/GPa	E₃/GPa	G₁₂/GPa	G₁₃/GPa	G₂₃/GPa	v₁₂	v₁₃	v₂₃
39.39 ± 0.36	18.10 ± 0.24	18.10 ± 0.24	6.31 ± 0.38	6.00 ± 0.27	6.00 ± 0.27	0.270 ± 0.012	0.350 ± 0.015	0.350 ± 0.015

下载: 导出CSV

| 显示表格

表 2 玻璃纤维/环氧树脂复合材料板的强度参数^[20-21]

Table 2. Strength parameters of glass fiber/epoxy resin composite plate^[20-21]

MPa　

X_T	X_C	Y_T	Y_C	Z_T	Z_C	S₁₂	S₁₃	S₂₃
222.7 ± 18.0	200 ± 22	136 ± 22	100 ± 4	50 ± 2	100 ± 4	90.5 ± 4.2	50.00 ± 3.12	50.00 ± 3.12

下载: 导出CSV

| 显示表格

图3为不同深度断裂损伤的应力分布结果，H为断裂深度，p_H为内压。如图3(a)所示，当断裂损伤深度为18 mm，内压载荷为0.1 MPa时，应力范围为3.621～13.140 MPa；内压载荷为0.2 MPa时，应力范围为5.002～21.840 MPa；内压载荷为0.3 MPa时，应力范围为5.116～28.880 MPa。最大应力分布在筒体的底端，主要是由于此处不仅受到内压载荷的作用，还受到位移的限制，并承受着筒体的整体重力。含有不同断裂深度壳体的应力结果均有此特点。如图3(b)所示，当断裂损伤深度为15 mm，内压载荷为0.1、0.2和0.3 MPa时，筒体底端的应力高达8.614、13.710和17.020 MPa，而断裂位置处的应力相对较小，主要是由于断裂缺陷位置与其相邻的壳体无接触，内部受到的载荷并没有传递到最外层壳体上，导致最外层壳体断裂缺陷位置处的应力较小。如图3(c)所示，当断裂损伤深度为12 mm，内压载荷为0.1、0.2和0.3 MPa时，缺陷位置处的应力分别为6.664、11.190和12.860 MPa，均小于最大应力。如图3(d)和图3(e)所示，当断裂损伤的深度分别为9和6 mm时，应力的分布较为相似，依然具有以上特点，不同的是随着深度减小，最大应力基本保持不变，内压载荷为0.1 MPa时，最大应力分别为8.066和8.130 MPa。这主要是因为断裂缺陷距离内压载荷面越来越远，对整个壳体应力分布的影响也越来越小，直到断裂缺陷出现在最外层壳体上，如图3(f)所示，当内压分别为0.1、0.2和0.3 MPa时，最大应力分别为8.122、11.410和15.150 MPa，断裂缺陷对整个壳体的应力分布影响最小。

下载: 全尺寸图片幻灯片

图 3 含有不同断裂缺陷深度的壳体的应力分布

Figure 3. Stress distribution of the shell with different fracture defect depths

下载: 全尺寸图片幻灯片

由图3可以看出，随着内压载荷增大，最大应力也不断增大，且最大应力出现在壳体底端附近，缺陷位置处的应力相对较小。统计上述不同内压载荷下的最大应力值，如图4所示。可以明显看出，随着内压增加，应力也不断增大。当断裂缺陷发生在第1层时，复合材料压力容器的应力最大。内压为0.1 MPa时，最大Mises应力高达13.14 MPa；内压为0.2 MPa时，最大Mises应力高达21.84 MPa；内压为0.3 MPa时，最大Mises应力高达28.8 MPa。当断裂缺陷逐渐靠近外壁，即与内压作用面较远时，最大应力趋于不变，保持平稳状态，断裂缺陷在第1层产生的最大应力是在第6层时的2倍。

图 4 含不同深度断裂缺陷的复合材料壳体的应力分布

Figure 4. Stress distribution of composite shell with different depth fracture defects

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为了探究壳体外壁应变的变化情况，选择穿过断裂缺陷中心位置处的圆周路径，得到含有不同深度断裂缺陷的压力容器壳体的应变曲线，如图5所示。在图5中， $\varepsilon$ _CS为周向应变， $\varepsilon$ _LS为纵向应变，横坐标零处为缺陷中心位置，横轴为与缺陷中心的周向距离。如图5(a)所示，当内压为0.1 MPa时，周向应变在3.0 × 10⁻⁴～1.50 × 10⁻³之间波动，曲线的形状与反正弦曲线相似，因此可进一步拟合，得到周向应变与距断裂缺陷距离的关系函数，而纵向应变在6.3 × 10⁻⁴～8.3 × 10⁻⁴之间波动。随着距离增加，纵向应变比较稳定，尤其是当断裂缺陷位于第1层和第6层时，几乎稳定在6.5 × 10⁻⁴～8.0 × 10⁻⁴范围内，这是由研究路径决定的，因研究路径位于壳体的最外层，当断裂缺陷位于第1层时，缺陷对研究路径上的应变影响较小，而缺陷位于第6层时，则是内压对最外层的影响小而导致应变分布较为均匀。如图5(b)和图5(c)所示，当内压为0.2和0.3 MPa时，周向和纵向应变的范围增大，且最大值也相应增加。当内压为0.2 MPa时，周向应变在5.0 × 10⁻⁴～2.50 × 10⁻³之间波动，纵向应变在1.35 × 10⁻³～1.70 × 10⁻³之间波动；当内压为0.3 MPa时，周向应变在5.0 × 10⁻⁴～3.50 × 10⁻³之间波动，纵向应变在2.20 × 10⁻³～2.60 × 10⁻³之间波动。

图 5 含有不同深度断裂缺陷的复合材料壳体的应变分布

Figure 5. Strain distribution of composite shell with different depth fracture defects

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综上所述，当内压为0.3 MPa时，周向应变的变化范围最大，可根据应变值确定断裂缺陷的位置和深度。因此，对该内压下的周向应变与距离曲线进行拟合，拟合采用waveform的sine公式，得到不同深度断裂缺陷的周向应变（ $\varepsilon$ _CS）与缺陷周向位置（x）的关系式（即式（8）～式（13）），可以较准确地确定不同深度断裂缺陷的周向位置和周向应变的关系。在实际工程实践中，采用应变检测方法确定壳体的应变分布，再通过关系式即可确定缺陷的周向位置和深度，为复合材料压力容器和管道检测提供一定的数据基础。

$18\;{\rm{mm}}\;\left( {1{\rm{st}}} \right):\;{\varepsilon _{{\rm{CS}}}} = 1\;989.3 + 1\;272.4\sin \left( {{\text{π}} \frac{{x - 2\;414.0}}{{2\;546.6}}} \right)$

(8)

$15\;{\rm{mm}}\;\left( {2{\rm{nd}}} \right):\;{\varepsilon _{{\rm{CS}}}} = 2\;086.3 + 1\;390.2\sin \left( {{\text{π}} \frac{{x - 2\;421.2}}{{2\;546.5}}} \right)$

(9)

$12\;{\rm{mm}}\;\left( {3{\rm{rd}}} \right):\;{\varepsilon _{{\rm{CS}}}} = 2\;026.8 + 1\;304.3\sin \left( {{\text{π}} \frac{{x - 2\;548.0}}{{2\;543.0}}} \right)$

(10)

$9\;{\rm{mm}}\;\left( {4{\rm{th}}} \right):\;{\varepsilon _{{\rm{CS}}}} = 2\;027.8 + 1\;345.4\sin \left( {{\text{π}} \frac{{x - 2\;421.3}}{{2\;552.7}}} \right)$

(11)

$6\;{\rm{mm}}\;\left( {5{\rm{th}}} \right):\;{\varepsilon _{{\rm{CS}}}} = 2\;035.4 + 1\;340.3\sin \left( {{\text{π}} \frac{{x - 2\;509.8}}{{2\;564.6}}} \right)$

(12)

$3\;{\rm{mm}}\;\left( {6{\rm{th}}} \right):\;{\varepsilon _{{\rm{CS}}}} = 2\;045.2 + 1\;333.6\sin \left( {{\text{π}} \frac{{x - 2\;509.8}}{{2\;564.6}}} \right)$

(13)

5. 结　论

对玻璃纤维增强环氧树脂基复合材料层合板进行了基本力学性能试验，并利用Abaqus有限元软件建立了含有不同深度断裂缺陷的复合材料壳体的三维有限元模型，将所得的基本力学性能参数用于有限元模拟中，通过对模拟结果与实验结果进行对比分析，得到以下结论：

（1）复合材料主向的拉伸强度和弹性模量均大于横向，层间强度Ⅰ型和Ⅱ型断裂韧性相差不大；

（2）应力和应变参数可以反映出复合材料壳体断裂缺陷的位置，通过有限元模拟，能够很方便地得到含有断裂缺陷壳体的应力及应变分布情况，复合材料壳体的最大应力在筒体端部位置，缺陷位置处的应力相对较小；

（3）随着内压增加，复合材料壳体的Mises应力也不断增加，当断裂缺陷在第1层时，Mises应力最大，应变随着与缺陷位置距离的增加，呈现规律性变化，通过拟合公式，可以初步判断断裂损伤的周向位置和深度，为复合材料壳体的损伤检测提供一定的数据基础。

图 1 横截面图和孔洞形状^[154]（左侧横截面图的上方为撞击侧，箭头为横截面图放大的位置）

Figure 1. Cross-sections and the shape of the hole^[154]（Cross sections are shown with the impacted side toward the top of the page. Arrow in hole indicates where material in micrograph was obtained.）

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图 2 Rosenberg公式^[161]与实验^[154]的对比（虚线斜率分别为0.95和1.05）

Figure 2. Comparison between Rosenberg’s model^[161] and experiments^[154] (The slopes of dashed lines are separately 0.95 and 1.05.)

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图 3 数值模拟^[76]和实验^[62]对比（v=6.15 km/s，撞击时间分别是8.1 ${\text{μ}}{\rm{s}}$ 、23.2 ${\text{μ}}{\rm{s}}$ ，初始弹丸形状叠加在图中）

Figure 3. Comparison of debris from calculation （top）^[76] and experiment （bottom）^[62] （v=6.15 km/s, time at 8.1 ${\text{μ}}{\rm{s}}$ and 23.2 ${\text{μ}}{\rm{s}}$ , the initial shape of the projectile is also shown in the left picture.）

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图 4 二阶Euler数值模拟和实验^[62]对比（v=6.15 km/s，撞击时间是8.1 ${\text{μ}}{\rm{s}}$ ）

Figure 4. Comparison of 2^nd Eulerian simulation and experiment^[62]（v=6.15 km/s, time at 8.1 ${\text{μ}}{\rm{s}}$ ）

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图 5 不同厚度下碎片云的变化^[166]

Figure 5. Debris clouds of different thicknesses of the plate^[166]

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图 6 碎片云随弹丸速度变化^[166]（t/D=0.049）

Figure 6. Debris clouds for different velocities of the projectile^[166] （t/D=0.049）

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图 7 （a）（b） Mo和Pb弹丸高速撞击碎片云形态对比^{[169, 171]}；（c） Pb弹丸数值模拟密度云图^[136]；（d） MPM数值模拟结果^[110]；（e）铝弹丸的数值模拟相态分布^[153]（红色为气态，绿色为液态，淡蓝色为固态）

Figure 7. （a）（b） Comparison of the debris clouds of Mo and Pb projectiles^{[169, 171]}; （c） density clouds from the simulation of Pb projectile^[136]; （d） results from MPM simulation; （e） phase clouds from the simulation of Al projectile^[153]（ red for gases, green for liquids and cyan for solids）

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图 8 弹丸质量相同但形状不同的碎片云分布

Figure 8. The distribution of debris cloud generated by hypervelocity impact of projectiles with the same mass and different shapes

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图 9 三维SPH模拟柱状弹丸撞击刚性壁的温度云图

Figure 9. The temperature cloud diagram of the cylindrical projectile impacting rigid wall by 3D SPH simulation

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图 10 六种不同速度下三维模拟得到的电荷数随时间变化

Figure 10. The charges change with time at six different impact velocities by 3D simulation

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图 11 相同速度、不同角度碰撞产生的等离子体电量

Figure 11. Plasma charges generated along different impacting angles under the same impact velocity

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图 12 相同碰撞速度、不同撞击角度下产生等离子体引发的磁感应强度随时间变化曲线

Figure 12. The time-depedent magnetic induction intensity generated by hypervelocity impacts at different impact angles under the same impact velocity

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图 13 直径为6.35 mm的铝弹丸以5 km/s撞击1 mm厚铝板碎片云模型^[184]与SPH模拟对比

Figure 13. Comparison between the model and the simulation of SPH^[184] （The diameter of the Al projectile is 6.35 mm, the velocity is 5 km/s and the thickness of the Al plate is 1 mm.）

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图 14 中低速后撞击板损伤破坏特征：（a）（b）取自文献[185]；（c）取自文献[184]

Figure 14. The damage patterns impacted by low and medium velocity: （a）（b） are from Ref.[185] and （c） is from Ref.[184]

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图 15 高速撞击后板损伤破坏特征^[186]

Figure 15. The damage patterns for high speed projectile^[186]

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图 16 数值模拟结果与二级轻气炮实验结果的比较

Figure 16. Comparions between simulation and two-stage light gas gun experiment

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图 17 ANN的弹道极限曲线与JSC模型（BLE）的比较

Figure 17. Comparison between ANN model and JSC （BLE） ballistic model

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表 1 实验加载方式和典型克/亚克级发射参数

Table 1. Experimental methods and typical parameters for projectile with mass in gram or sub-gram

Methods	Year	Material	Velocity/（km·s^–1）	Shape	Mass/g	Comments and sources
Three-stage light gas gun	1993	Al	9.52	Flyer plate	0.78	Sandia National Laboratories^[48]
Three-stage light gas gun	2017	Al	10.1	Flyer plate	0.22	Institute of Fluid Physics^[49]
Magnetically driven device	2011	Al	45	Flyer plate	0.79	Z accelerator, Sandia National Laboratories^[50]
	2014	Al	8.7	Flyer plate	0.12	CQ-4, Institute of Fluid Physics^[51]
	2014	Al	11.5	Flyer plate	0.15	PTS, Institute of Fluid Physics^[52]
Electric gun	2019	Mylar	10	Flyer plate	0.30	Institute of Fluid Physics
ISCL （Inhibited Shaped Charge Launcher）	1995	Al	11.16	Cylinder	1.02	Southwest Research Institute^[53]

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表 2 铝平面对称碰撞时相变对应的速度和压力

Table 2. The velocity of the Al projectile and the pressure from impacting

Source/Phase change	Incipient melting due to release	Complete melting due to release	Incipient vaporization due to release	Complete vaporization due to release
Hopkins et al.^[169]	2.7 km/s, 65 GPa	3.38 km/s, 89 GPa
Anderson et al.^[170]	2.85 km/s, 71 GPa	3.45 km/s, 94 GPa	5.2 km/s, 174 GPa
Bjork^[171]			6.2 km/s, 225 GPa	2700 GPa
Shockey et al.^[172]	2.6–3.6 km/s	3.3–4.6 km/s	5.5–7.5 km/s	12.5–16.5 km/s
Pierazzo et al.^[173]	73 GPa	106 GPa	315 GPa

Source/Phase change	Incipient melting due to shock	Complete melting due to shock
Tang^[153]	125 GPa	160 GPa

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表 3 弹丸质量相同、形状不同所得到碎片云的参数

Table 3. The parameters of debris cloud generated by hypervelocity impact of projectiles with the same mass and different shapes

Shape of projectile	Dimensions	Mass/g	Axial length/mm	Radical length/mm
Sphere	$\varnothing$ 5.02 mm	0.180 89	44.5	40.5
Cylinder	$\varnothing$ 5.02 mm×4.6 mm	0.182 61	46.5	44.0
Disk	$\varnothing$ 5.02 mm×1.0 mm	0.181 06	45.5	32.2

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