改进的Whipple防护结构与相关数值模拟方法研究进展

陈莹; 陈小伟

doi:10.11883/bzycj-2020-0289

改进的Whipple防护结构与相关数值模拟方法研究进展

doi: 10.11883/bzycj-2020-0289

陈莹^1,,
陈小伟^{2, 3, ,}

1.
北京理工大学机电学院，北京 100081
2.
北京理工大学爆炸科学与技术国家重点实验室，北京 100081
3.
北京理工大学前沿交叉科学研究院，北京 100081

基金项目: 国家自然科学基金（11627901，11872118）

详细信息

作者简介:
陈　莹（1996－　），女，博士研究生，604544512@qq.com

通讯作者:
陈小伟（1967－　），男，博士，教授，chenxiaoweintu@bit.edu.cn

中图分类号: O385
计量
- 文章访问数: 1208
- HTML全文浏览量: 600
- PDF下载量: 224
- 被引次数: 2
出版历程
- 收稿日期: 2020-08-24
- 修回日期: 2020-09-03
- 网络出版日期: 2021-02-02
- 刊出日期: 2021-02-05

A review on the improved Whipple shield and related numerical simulations

CHEN Ying^1
,,
CHEN Xiaowei^{2, 3
, ,}

1.
School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China
2.
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China
3.
Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China

摘要

摘要: 基于弹丸在超高速撞击薄板时破碎形成碎片云的机理，Whipple防护结构能够对航天器所面临的空间碎片及微流星体等威胁形成有效防护。通过回顾Whipple防护结构的研究和发展历程，对多层板结构、填充式防护结构、夹芯板结构等进行对比，分析其力学效应和防护性能；总结可应用于含泡沫、蜂窝、梯度和编织等材料的防护结构超高速撞击的数值模拟方法及其改进方法；结合相关材料的超高速撞击试验及数值模拟结果，为防护结构未来的研究方向提出建议。
- 超高速碰撞 /
- Whipple防护结构 /
- 碎片云 /
- 多层板结构 /
- 填充/夹芯板结构 /
- FEM-SPH自适应耦合算法
Abstract: Based on the formation mechanism of the debris cloud caused by the projectile hypervelocity impacting onto a thin plate, the Whipple shield can effectively protect the spacecraft from space debris and micrometeoroid. By reviewing the research and development of the Whipple shield, and compares the mechanical effects and protective performance of multilayer, stuffed and sandwich shield. The paper also summarizes the application of numerical simulation methods and their improvement for the hypervelocity impact of protective structures containing materials such as foam and honeycomb, etc. By addressing the results of hypervelocity impact tests and numerical simulations of relevant materials, suggestions are made for the future research of the Whipple shield.
- hypervelocity impact /
- Whipple shield /
- debris cloud /
- multilayer shield /
- stuffed/sandwich shield /
- FEM-SPH adaptive method

HTML全文

随着重要军事目标的坚固化和地下化，世界各国加速发展钻地武器，使其打击更加精确、侵彻更深、破坏力更大，对地下防护工程带来极大威胁。与空中爆炸或触地爆炸相比，钻入地下的武器再爆炸能使绝大多数爆炸能量耦合至岩土中，使地冲击威力大大增强。同时，钻地爆炸对地下工程产生的破坏效应与其地冲击能量特征密切相关。大当量地下爆炸现场实测数据均表明^[1-5]，地下爆炸耦合至岩土中的地冲击能量随装药埋深增加而迅速增大，在某一临界深度时增速减缓，而后随埋深增大逐渐趋近于地下封闭爆炸。深埋封闭爆炸的耦合地冲击能量可达同当量触地爆炸的10倍以上，因此对于防护工程设计来讲，必须要建立爆炸深度和耦合地冲击能量间的定量关系。

近几十年来，对于岩土介质中的封闭爆炸和触地爆炸的研究已经较为完善，可获得较丰富试验数据和较为成熟的计算方法。梁霍夫等^[6]在土壤(包括饱和土和非饱和土)、砂(包括饱和砂和非饱和砂)、岩石中进行了平面波、球面波和柱面波的地冲击效应研究，综合考虑现场的地质特征和实验所用介质的物理力学参数建立了较为简练的地冲击效应计算方法。在不同岩土介质中地冲击衰减规律方面，学者们也做了大量的研究工作。Yankelevsky等^[7]通过对已有实验数据的分析和数值模拟，得到冲击波峰值压力衰减的特点。穆朝民等^[8-9]在黄土和砂土中进行了一系列爆炸成坑试验，并结合爆炸宏观特征，确定了黄土及饱和砂土中发生封闭爆炸的临界比例埋深，得到了变埋深条件下应力波在土中传播规律。施鹏等^[10]通过模拟手段计算了不同装药比例埋深下土中爆炸能量耦合问题，通过实验得到了耦合系数数据，并给出了公式使用范围和对象。叶亚齐等^[11]在砂质黏土中进行了不同装药比例埋深爆炸自由场试验，给出了砂质黏土中不同深度爆炸自由场地冲击参数的衰减规律，建立了砂质黏土中不同深度爆炸自由场地冲击参数的预计公式。赵红玲等^[12]研究了石灰岩中常规装药不同埋深爆炸自由场地冲击参数的传播规律，得到了石灰岩介质中变埋深爆炸地冲击参数随比例距离的预计公式。何翔等^[13]基于试验手段研究了常规装药爆炸不同深度对自由场直接地冲击参数的影响，建立了石灰岩中爆炸成坑经验公式和地冲击传播特性。但由于土中浅埋爆炸是复杂的耦合效应问题，其理论和实验研究均存在诸多困难，目前尚无可靠的计算方法。目前对于浅埋爆炸地冲击效应的计算，通常采用由美国陆军工程兵水道试验站给出的地冲击耦合系数，将浅埋爆炸转变成等效的封闭或者触地爆炸^[14-15]。然而，为验证此方法的有效性，在饱和砂土中共进行了58次爆炸试验^[16]，发现该计算与试验结果存在较大偏差，特别是在爆炸远区，预测公式不能准确求得地冲击参数，只能定性分析。关于地冲击耦合系数的计算，目前尚无准确的计算方法，给出的实验数据误差较大，难以适应钻地爆炸等效当量及耦合地冲击参数的准确计算。同时，现有文献^[17-18]中存在多种“地冲击耦合系数”，如能量耦合系数、等效当量系数、地冲击应力耦合系数等，相互间未建立准确的换算方法，若不仔细区分其物理本质而加以混淆使用，容易造成计算错误。

为了研究黏土中爆炸成坑体积与耦合地冲击能量的关系，本文在已有研究成果的基础上，系统开展不同埋深下成坑地冲击耦合效应实验研究，探索弹坑体积以及地冲击压力随装药埋深增加的变化规律，寻找不同耦合系数间的换算关系以及等效封闭当量计算方法，以期为地下工程的抗爆防护提供设计依据。

1. 实验介绍

1.1 黏土试样

本实验所用的黏土取自南京孟墓地区，首先清除地表覆土，然后将黏土平摊放置地表捡去里面大块杂质，之后倒入爆炸试验容器中，分层夯实。该黏土中主要化学组成及质量分数分别为：SiO₂ (61.16 %)、Al₂O₃ (23.10 %)、Fe₂O₃ (8.94 %)、K₂O (2.65 %)、MgO (1.56 %)，等。黏土试样的密度是2.242×10³ kg/m³，含水率12.8 %，纵波速度1832 m/s，其波阻抗近似为0.4×10⁷ N∙s/m³。根据国家标准^[19]采用筛析法计算绘出黏土的粒径级配曲线如图1所示。

图 1 黏土的粒径级配曲线

Figure 1. Particle size gradation curve of clay

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1.2 爆炸容器及球形爆源

采用一种分层可拼装的筒体结构作为爆炸容器，如图2所示。该装置高度是1 490 mm，容器内径1 500 mm，钢板厚度为15 mm，由7个子单元自下而上垂直拼装而成。子单元由Q345钢板经弯、卷、焊等工序加工而成，包括上下两片宽75 mm、厚16 mm的法兰盘，起到提高容器半径法向刚度和连接拼装单元的作用。法兰盘上均匀加工24个内径16 mm的通孔，为增强法兰和圆环之间的强度，在圆环外侧铺设12个肋板。子单元之间通过强度等级为8.8级的高强度M14螺栓连接。按照薄壁圆筒公式计算，容器可承受内壁上强度为5.2 MPa的均匀荷载。

图 2 分层式爆炸容器

Figure 2. Layered explosion vessel

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在黏土介质爆炸实验研究中，大多使用块状TNT、雷管和乳化炸药，当测点距离爆心较远时，可以把爆源近似当作点源，但是测点距离爆心较近时，炸药的形状、密度和种类均会给试验结果产生显著的影响，Krauthammer^[20]指出在爆炸试验中采用球形装药所采集的试验数据更加科学。为获取爆心距较近范围内黏土中的爆炸应力波衰减规律，本实验选用球形装药作为爆源，如图3所示。药球由三硝基甲苯（TNT）采用一体成型技术压装而成，药球质量为10.5 g，直径为24.4 mm，装药密度1.5 g/cm³。药球顶部预留直径8 mm、深度12 mm的雷管安装孔，其尺寸和标准雷管中猛炸药尺寸相近，安装孔的体积占比约为7.93 %。采用电雷管起爆，实验前将雷管插入装药安装孔中，并使用绝缘胶带固定。

图 3 球形爆源

Figure 3. Spherical explosion source

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1.3 测点布置

为获取装药埋深对地冲击压力传播衰减的影响规律，共设计7组不同埋深的爆炸实验，每组实验中布置5组测点（记为1#、2#、3#、4#、5#），各实验装药埋深（h）和地冲击压力测点比例爆心距（R）见表1。

表 1 装药埋深（

$h$ ）及爆心距（

$R$ ）设计

Table 1. Design of burial depth of charge (

$h$ ) and burst core distance (

$R$ )

工况	h/(m·kg^−1/3)	R/(m·kg^−1/3)
工况	h/(m·kg^−1/3)	1#	2#	3#	4#	5#
1	−0.056	0.799	1.427	2.295	3.274	4.000
2	0	0.799	1.427	2.295	3.274	4.000
3	0.14	0.799	1.427	2.295	3.274	4.000
4	0.37	0.799	1.427	2.295	3.274	4.000
5	0.55	0.799	1.427	2.295	3.274	4.000
6	1.19	0.799	1.427	2.295	3.274	4.000
7	1.46	0.479	1.155	2.068	3.046	3.772

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| 显示表格

药球位于筒体容器的中轴线上，在试样制备过程中预留装药孔，然后将药球放入装药孔后回填。爆心下方共铺设5层土压力传感器，每层铺设2个，于中轴线两侧对称布置，距离容器轴线50 mm，具体如图4(a)所示。采用DNS123型土压力传感器获取不同测点处爆炸波法向应力的时程曲线，其尺寸为 $\varnothing$ 50 mm×10 mm，如图4(b)所示。为了减少传感器本身对地冲击传播的干扰，采用上下层交叉铺设的方式使传感器位于不同方位，交叉角度约36°，如图4(c)所示（图中编号表示传感器铺设所在的层数）。采用东华DH8302高性能动态信号测试系统进行地冲击压力数据采集，采样频率为100 kHz，如图5所示。

图 4 药球位置及传感器布置

Figure 4. Drug package location and sensor arrangement

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图 5 测量采集记录图

Figure 5. Measurement acquisition record diagram

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为方便对比不同装药埋深对地冲击传播的影响，7次实验中保持各组传感器距离爆心的比例距离不变。每做完一次实验后首先利用3D扫描仪对弹坑进行扫描，然后沿弹坑中轴面将弹坑剖开，观察弹坑周围介质的压缩破坏情况，分析完成后将破坏的黏土铲除而后重新铺设新的黏土层并进行重新夯实。

2. 实验结果分析

2.1 成坑特征

2.1.1 可视弹坑分析

黏土中爆炸时，爆炸冲击波和爆生气体压缩爆炸中心周围黏土介质，并形成冲击波向四周传播，随着传播距离增加，冲击波逐渐衰减为塑性波、弹性波，同时在黏土介质中形成爆炸空腔区、破坏区等区域。当冲击波遇到自由面时，在自由面反射作用下形成反向传播的拉伸波，对黏土介质产生层裂或者剥离。如果爆炸埋深较浅，爆轰产物和爆炸应力波激发近地表土层土颗粒发生飞散形成抛掷弹坑；而随着埋深进一步增大，弹坑体积也进一步增大，直至在某一最佳临界深度处体积达到最大；而后随着埋深进一步增加，地表可视弹坑逐渐消失，地下逐渐形成完整的爆炸空腔，但由于空腔膨胀作用，在地面处形成鼓包和破裂；最后，当埋深超出封闭爆炸临界深度时，地下爆炸破坏效应完全被封闭在地下，地表无反应，一般将地表面上无明显可见变化的深度称为封闭爆炸临界深度。图6和图7给出了不同埋深情况下爆炸弹坑的宏观破坏情况，相关弹坑尺寸数据见表2。从结果可知：可视弹坑，在装药埋深达到0.55 m/kg^1/3时，弹坑体积达到最大，在装药比例埋深到1.19 m/kg^1/3时，地表弹坑几乎将近消失，同时地表发生明显隆起鼓包(图6(c)和图7(c) )，当比例埋深达到1.46 m/kg^1/3时，地表鼓包几近消失，但在土层表面形成不规则裂纹(图6(d)和图7(d) )。对于封闭爆炸临界深度，目前公开文献的外观点基本一致，穆朝民等^[8]所做的土中爆炸试验指出在比例埋深到达1.96 m/kg^1/3时，地表面接近没有明显可见变化，梁霍夫^[6]在归纳整理的资料中，也得出土中集团装药封闭爆炸临界深度为2.0 m/kg^1/3。

图 6 成坑的俯视与剖面图

Figure 6. Overhead view and profile of the crater

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图 7 3D扫描成坑形貌

Figure 7. 3D scanning into crater morphology

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表 2 不同埋深条件下弹坑尺寸数据

Table 2. Size data of craters under different burial depths

工况	h/ (m·kg^−1/3)	r_v / (m·kg^−1/3)	d_v / (m·kg^−1/3)	V_v / (m³·kg⁻¹)	r_a / (m·kg^−1/3)	d/ (m·kg^−1/3)	V/ (m³·kg⁻¹)
1	−0.056	0.261	0.247	0.035	0.179	0.251	0.017
2	0	0.280	0.292	0.048	0.184	0.260	0.018
3	0.14	0.580	0.539	0.380	0.289	0.379	0.066
4	0.37	0.682	0.685	0.667	0.340	0.416	0.101
5	0.55	0.896	0.776	1.304	0.453	0.459	0.197
6	1.19	0.615	0.502	0.397	0.478	0.478	0.228
7	1.46	0	0	0.000	0.479	0.479	0.230
注：r_v、d_v、V_v分别为可视弹坑的半径、深度和体积，r_a、d、V为有效弹坑的半径、深度和体积。

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| 显示表格

图8为ConWep爆炸荷载计算程序^[21]预测的质量为10.5 g TNT药球在黏土中可视爆坑的深度和直径随装药比例埋深变化曲线，及其与本文实测结果的对比。在装药比例埋深h≤0.55 m/kg^1/3时，可视爆坑深度实测值和ConWep计算程序预测的值偏差最高可达20.6 %，而实测的可视爆坑直径与ConWep预测值具有很好的一致性，可视爆坑深度产生的偏差可能由于夯实的黏土起炸后，飞散的黏土粘连并带出爆坑里面黏土，回填变少，现场实测的可视爆坑比ConWep中预测的可视爆坑要大。

图 8 可视弹坑实测值与ConWep计算值的对比

Figure 8. Comparison of the measured values of the visible burst crater with those computed by ConWep

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2.1.2 有效弹坑分析

装药爆炸过程中，装药爆生产物通过压缩周围介质形成地冲击波，装药中心下方的压缩弹坑对于应力波的形成具有重要作用，将装药中心下方的压缩弹坑称为有效弹坑。理论分析表明^[22]：耦合至岩土中的地冲击能量与有效弹坑体积呈正比例关系，在以往历史试验数据中，往往只关注可视弹坑，而忽略了有效弹坑体积的统计。为了便于描述有效弹坑的演化过程，不妨引入球形度和有效弹坑体积比两个量；其中，球形度是有效弹坑半径r_a与弹坑深度d的比值。有效弹坑体积比是不同装药比例埋深下的有效弹坑体积V与工况7有效弹坑体积V₇的比值，其比值记为V/V₇。可通过表2数据计算得到有效弹坑的球形度和有效弹坑体积比。图9给出了随装药埋深增加有效弹坑球形度和有效弹坑体积比值的变化规律。当装药比例埋深为h=0时，有效弹坑轮廓呈抛物线型（图6(a)），装药中心所在的球形度约为0.708；随装药埋深增大，有效弹坑的球形度和体积比均在增加（图9），同时爆心下方弹坑轮廓线逐渐由抛物线型演化为半球形（图6(b)～(d)）；当装药比例埋深h≥0.55 m/kg^1/3时，装药中心下方有效弹坑体积达到趋近于极限，其形状也接近于完全封闭爆炸。

图 9 随装药埋深增加有效弹坑的变化规律

Figure 9. The change rule of effective crater with the increase of charge depth

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2.2 地冲击传播规律特征

图10分别给出了装药比例埋深为0.55 m/kg^1/3 （工况5）和1.19 m/kg^1/3 （工况6）时，不同测点处的地冲击应力实测波形，从图10中可以看出，随距离装药比例距离的增加，地冲击应力峰值呈指数衰减特征。

图 10 不同装药比例埋深时实测地冲击应力波形

Figure 10. Measured stress waveform of clay at different scaled buried depths

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表3给出了不同装药埋深情况下地冲击应力峰值统计数据，图11给出了不同比例爆心距离处地冲击应力峰值比值随装药比例埋深增加的变化情况，地冲击应力峰值比值是不同装药比例埋深下爆炸应力峰值σ_pk与工况7下爆炸应力峰值σ_pk7的比值。从表3和图11中可以看出，在相同比例爆心距离处，当装药比例埋深h≤0.55 m/kg^1/3时，随装药埋深增加，耦合至黏土中的地冲击压力峰值也急剧增大，而当装药比例埋深h≥0.55 m/kg^1/3时，耦合至黏土中的地冲击压力峰值增势趋近平缓，接近于完全封闭爆炸，这一变化趋势与有效弹坑体积随装药埋深的变化趋势(图9(b) )基本一致。

表 3 黏土中各比例埋深下地冲击应力峰值数据

Table 3. Subsurface impact stress peak data of each proportion buried depth in clay

工况	h/ (m·kg^−1/3)	σ_pk/MPa					工况	h/ (m·kg^−1/3)	σ_pk/MPa
工况	h/ (m·kg^−1/3)	1#	2#	3#	4#	5#	工况	h/ (m·kg^−1/3)	1#	2#	3#	4#	5#
1	−0.056	0.047	0.015	0.019	0.011	0.002	5	0.55	0.358	0.155	0.115	0.086	0.022
2	0	0.050	0.038	0.021	0.015	0.006	6	1.19	0.334	0.156	0.126	0.115	0.032
3	0.14	0.141	0.092	0.074	0.054	0.015	7	1.46	1.080	0.196	0.124	0.100	0.024
4	0.37	0.184	0.100	0.070	0.052	0.014	7	1.46	(0.389)	(0.166)	(0.047)	(0.023)	(0.015)
注：(1)1～6炮次1#、2#、3#、4#、5#测点比例距离分别为0.799、1.427、2.295、3.274、4.000；(2)第7炮次1#、2#、3#、4#、5#测点比例距离分别为0.479、1.155、2.068、3.046、3.772；(3)第7炮括号内数据为将第7炮次数据在比例距离0.799、1.427、2.295、3.274、4.000处换算数据。

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图 11 不同比例爆心距离处地冲击应力峰值比值随装药比例埋深增加的变化情况

Figure 11. Peak of the ground impact stress varied with scaled butied depth charge at different scaled blast center distances

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将爆炸地冲击应力峰值接近于封闭爆炸时的爆炸比例埋深称为等效封闭爆炸临界埋深，可知等效封闭爆炸临界埋深与封闭爆炸临界埋深为两个不同概念：对于土中化学爆炸，当装药埋深与封闭爆炸空腔半径相当时，漏斗坑位于装药中心以下的部分呈半球型，半径与封闭爆炸半径基本相同，此时地下爆炸辐射至地下的地冲击参数基本不受来自地表的影响，介质中爆炸压缩波的幅值与完全封闭爆炸产生的幅值几乎接近，因此可以将封闭爆炸空腔半径作为等效封闭爆炸临界埋深。

对于岩土体介质中爆炸，目前一致认为可以利用公式 $\sigma = A{\left( {r/{Q^{1/3}}} \right)^ - }^n$ 描述地冲击应力传播的衰减规律；其中：A和n分别为应力经验表达式的衰减系数和衰减指数；r为爆心距，单位m；Q为炸药质量，单位kg。图12给出了不同装药比例埋深条件下，地冲击压力传播衰减曲线，表4给出了对于的A、n值。衰减指数n反映了随爆心比例距离的增加地冲击传播的衰减规律，只与岩土体介质的物理力学性质相关，与装药埋深无关。为了上述公式简洁性，对不同装药比例埋深条件下衰减指数n求平均值，其平均值记为 $\bar n$ ，可得实验所用黏土的地冲击传播平均衰减指数 $\bar n$ 为1.14。衰减系数A则反映了随装药埋深增加耦合至岩土中的地冲击应力的变化规律；将平均衰减指数为1.14条件下的不同比例埋深的衰减系数记为A'，从表4可以看出，当装药比例埋深h≥0.55 m/kg^1/3时，随装药埋深增加，A和A'值变化均趋近平缓，进一步说明对于黏土，可将h≈0.55 m/kg^1/3作为等效封闭爆炸临界埋深。

图 12 不同装药比例埋深时地冲击应力峰值衰减曲线

Figure 12. Peak groud impact stress attenuation with scaled blast center distance at different scaled buried depths of charge

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表 4 不同装药比例埋深条件下拟合参数

Table 4. Fitting parameters with different scaled buried depths of charge

h/(m·kg^−1/3)	n	A	$\bar n$	A'
−0.056	1.16	0.035	1.14	0.031
0	1.05	0.038		0.041
0.14	1.09	0.120		0.123
0.37	1.07	0.124		0.148
0.55	1.23	0.261		0.258
1.19	1.16	0.267		0.258
1.46	1.19	0.269		0.260

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3. 成坑地冲击耦合规律分析与等效当量计算

3.1 成坑与地冲击耦合规律理论分析

岩土介质中发生爆炸时，爆炸空腔中的爆轰产物挤压四周的岩土介质扩张，并形成冲击波向外传播，耦合进介质中的能量可以通过压缩边界（非弹性变形区边界）径向力沿边界位移所做的功进行计算，有约一半的能量变为动能^[23]：

$W = \frac{1}{2}{S_r}\int_0^{{u_r}(\infty )} {{\sigma _r}{\rm{d}}{u_r}}$

(1)

式中： ${S_r}$ 为半径为r的球形表面积， ${u_r}$ 为r处岩土的径向位移， ${u_r}(\infty )$ 时间趋于无穷的最终位移大小， ${\sigma _r}$ 为r处的径向应力。

文献[24-25]给出了封闭爆炸和触地爆炸时从弹性边界传播出的能量，推导发现对于浅埋爆炸而言，弹性区边界处传播出的能量W_failure依然由弹性边界所包围的破坏区岩体体积V^*所控制，即W_failure与V^*呈线性关系：

${W_{{\rm{failure}}}} = \frac{B}{{12}}\frac{{{\tau _{\rm{s}}}^2}}{G}\frac{{c_{\rm{p}}^2}}{{c_{\rm{s}}^2}}{V^*}$

(2)

式中： ${c_{\rm{p}}}$ 为介质中纵波速度， ${c_{\rm{s}}}$ 为介质中剪切波速， ${\tau _s}$ 为剪切强度， $G$ 为剪切模量， $B = \left[ {5 + 3{{\left( {1 + 24\upsilon } \right)}^2}} \right]/64$ 取决于爆源周围岩石介质的性质， $\upsilon$ 为岩土介质的泊松比。

图13给出了依据爆炸成坑最终形态岩土介质中爆坑常见的三种形式。对于地下爆炸，从装药中心传播出的地冲击能量向四周传播，但对防护工程而言一般均处于装药中心下方。对于有效的地冲击能量，通常只考虑装药中心以下的半空间范围，因此在本文中所述的地冲击能量均指有效地冲击能量。

图 13 地下爆炸的三种形式

Figure 13. Three types of underground explosion

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耦合入岩土介质中的地冲击能量源自于爆炸空腔扩张压缩做功，当爆炸空腔扩张停止后，爆炸产生的能量耦合入介质的过程基本结束。根据已有的数据发现，可以用Boltzmann函数来表达爆炸地冲击能量耦合系数η_e(h)变化规律，结合函数的特点，只要分别求出触地爆炸和封闭爆炸的能量耦合系数（η_e(0)和η_e(∞)），即可描绘能量耦合系数的发展规律：

${\eta _{\text{e}}}(h) = 1 - \frac{{2(1 - {D_1})}}{{{{\rm{e}}^{h/{D_2}}} + 1}}$

(3)

式中：D₁为触地爆炸时的能量耦合系数，即 ${\eta _{\rm{e}}}(0) = {D_1}$ ；理论上，埋深为无限大时， $h \to \infty$ ， ${\eta _{\rm{e}}} \to 1$ ，通过本实验发现，存在着最小封闭爆炸的最小比例埋深，该值与D₂相关联，决定着曲线的发展轨迹，可取 ${\eta _{\rm{e}}} = 0.99$ 时，当作封闭爆炸能量完全耦合。当装药比例埋深大于等效封闭爆炸临界埋深时，传入岩土介质中下方的有效地冲击能量已达到饱和，本文中选取h = 1.19 m/kg^1/3作为参考点，代入式(3)中可得：

${D_2} = \frac{{1.19}}{{\ln (199 - 200{D_1})}}$

(4)

根据表2数据得：

$\left\{ \begin{gathered} {D_1} = {\eta _{\rm{e}}}(0)= \frac{{\left( {2/3} \right){\text{π}} \times {{0.184}^2} \times 0.260}}{{\left( {2/3} \right){\text{π}} \times {{0.478}^3}}} = 0.081 \\ {D_2} = \frac{{1.19}}{{\ln (199 - 200 \times 0.081)}} = 0.228 \\ \end{gathered} \right.$

(5)

代入式(3)，得到地冲击能量耦合系数 ${\eta _e}(h)$ 曲线如图14所示。

图 14 地冲击能量耦合系数随比例埋深的变化

Figure 14. Variation of ground impact energy coupling coefficient with scaled burial depth

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对于相等的爆炸当量地下爆炸，在装药比例埋深 $h=1.19\;{{\rm{m}}} /{\rm{k}}{{\rm{g}}^{{1 / 3}}}$ ，爆炸地冲击能量W超出触地爆炸地冲击能量W₀的倍数为：

$\frac{W}{{{W_0}}} = \frac{{\left( {2/3} \right){\text{π}} \times {{0.478}^3}}}{{\left( {2/3} \right){\text{π}} \times {{0.184}^2} \times 0.260}} \approx 12.4$

(6)

3.2 浅埋爆炸等效当量耦合系数与地冲击参数应力/粒子速度耦合系数

对于封闭爆炸，质点峰值速度计算公式为：

${v}_{\rm{pk}}(r)={K}_{v}{\left(\frac{r}{{Q}^{1/3}}\right)}^{-n}\text{，}\qquad {\sigma }_{{\rm{pk}}}(r)=\rho {c}_{{\rm{p}}}{v}_{\rm{pk}}(r)$

(7)

式中： ${K_v}$ 为地冲击质点峰值速度经验表达式的衰减系数， ${K_v}$ 与 $n$ 均可由实验得到； ${v_{{\rm{pk}}}}$ 为质点峰值速度； ${\sigma _{{\rm{pk}}}}$ 为峰值应力；ρ为介质密度； $Q$ 为药球当量； ${c_{\rm{p}}}$ 为纵波波速。

对于浅埋爆炸，实际计算时采用当量耦合系数或地冲击应力耦合系数将浅埋爆炸变为等效的封闭爆炸或者触地爆炸，即利用式(8)和(9)进行求解。

当量耦合系数 ${\eta _Q}$ 为在同种介质中，比例爆心距相等时产生相同地冲击参数(应力、速度、加速度等)大小的封闭爆炸与浅埋爆炸的当量比值，即为：

${\eta _Q} = \frac{{{Q_{{\text{eff}}}}}}{Q}$

(8)

式中： $Q$ 为装药比例埋深为 $h$ 的爆炸当量； ${Q_{{\rm{eff}}}}$ 为等效的封闭爆炸当量。

地冲击参数应力耦合系数 ${\eta _\sigma }$ 为在同种介质中，爆炸当量和比例爆心距均相等时浅埋爆炸地冲击参数与封闭爆炸地冲击参数比值，即为：

${\eta _\sigma } = \frac{\sigma }{{{\sigma _{{\text{close}}}}}} = \frac{v}{{{v_{{\text{close}}}}}} = \frac{a}{{{a_{{\text{close}}}}}} = \frac{u}{{{u_{{\text{close}}}}}}$

(9)

式中：σ、v、a、u分别为地冲击应力、粒子速度、加速度以以及介质位移，下标close表示封闭爆炸。

由此，对于浅埋爆炸，地冲击质点峰值速度可表达为

${v_{{\text{pk}}}}(r) = {K_v}{\left( {\frac{r}{{Q_{{\rm{eff}}}^{1/3}}}} \right)^{ - n}} = {\eta _\sigma }{K_v}{\left( {\frac{r}{{{Q^{1/3}}}}} \right)^{ - n}}$

(10)

于是得到 ${\eta _\sigma }$ 与 ${\eta _Q}$ 的对应关系：

${\eta _\sigma } = {\eta _Q}^{n/3}$

(11)

为准确计算浅埋爆炸等效封闭当量，还需要建立当量耦合系数与能量耦合系数间的关系。对于地下爆炸，距离装药中心为 $r$ 处介质最大质点径向速度的表达式为：

${v_{r,\max }} = {v^{*}}{\left( {\frac{r}{{{r^{{\text{*}}}}}}} \right)^{ - n}}$

(12)

式中： ${v^{*}}$ 为破坏边界处粒子速度， ${r^{*}}$ 为破坏边界与装药中心距离。

对于封闭爆炸，装药中心下方破坏区呈半球形，且有 ${r^{_*}} = k{Q^{{1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}}}$ ，式(12)可改写为：

${v_{r,\max }} = {v^{*}}{\left( {\frac{r}{{{r^{{{*}}}}}}} \right)^{ - n}} = {K_v}{\left( {\frac{r}{{{Q^{1/3}}}}} \right)^{ - n}}$

(13)

式中： ${K_v} = {v^{*}}{k^n}$ ，k为破坏区半径比例系数，和岩土介质的力学性质有关。

就岩土介质中的浅埋爆炸，装药中心下方破坏区截面边界为抛物线型，如图13(a)和图13(b)所示，可定义装药中心下方破坏区竖直高度为破坏区水平半径的 $\lambda \left( h \right)$ 倍，随着h变大， $\lambda \left( h \right)$ 接近于1，即随着装药比例埋深的变大，装药中心下方破坏区的轮廓由抛物线形渐变为半球形，如图13(c)。如果按破坏区体积进行等效，即将浅埋爆炸破坏区体积等效为封闭爆炸破坏区体积： $\dfrac{2}{3}{\text{π }}{r^{*}}{\left[ {{r^{*}}/\lambda (h)} \right]^2} \approx \dfrac{2}{3}{\text{π }}r_{{\rm{eff}}}^{*3}$ ，利用式(13)，进行地冲击粒子速度计算，则对于装药中心正下方：

${v_{r,\max }} = {v^*}{\left( {\frac{r}{{{r^*}}}} \right)^{ - n}} = {v^*}{\left( {\frac{r}{{{\lambda ^{{2 \mathord{\left/ {\vphantom {2 3}} \right. } 3}}}r_{{\text{eff}}}^{\text{*}}}}} \right)^{ - n}} = {K_v}{\left( {\frac{r}{{Q_{{\text{eff}}}^{{1 / 3}}}}} \right)^{ - n}} = {v^*}{\left( {\frac{r}{{kQ_{{\text{eff}}}^{{1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}}}}} \right)^{ - n}}$

(14)

通过式(14)得， ${\lambda ^{{2 \mathord{\left/ {\vphantom {2 3}} \right. } 3}}}r_{{\text{eff}}}^* = kQ_{{\rm{eff}}}^{1/3}$ ，依据式(2)爆炸冲击波能量与破坏区体积间关系，可以建立三种耦合系数间的关系：

${\eta _{\rm{e}}} = \frac{{{\eta _Q}}}{{{\lambda ^2}}} = \frac{{\eta _\sigma ^{3/n}}}{{{\lambda ^2}}}$

(15)

3.3 黏土中化学爆炸地冲击应力参数耦合系数和当量耦合系数

由计算而来的 ${\eta _{\text{e}}}$ ，依据式(15)计算地冲击参数应力耦合系数 ${\eta _\sigma }$ ，结果如图15(a)所示，其中n取1.14。计算得到 ${\eta _\sigma }$ 并与实测值对比，如图15(a)所示。从图15(a)中可以看出，地冲击参数应力耦合系数 ${\eta _\sigma }$ 计算值与地冲击粒子速度实测值整体上吻合度较好。与TM5-855-1^[2] 地冲击参数应力耦合曲线和施鹏^[10]根据数值模拟得到的曲线进行比较，可以看出，本实验黏土的地冲击参数应力耦合系数 ${\eta _\sigma }$ 起点较施鹏^[10]数值计算的数据较相近，当装药比例埋深增加到0.40～0.60 m/kg^1/3之间时，地冲击参数应力耦合系数 ${\eta _\sigma }$ 和施鹏^[10]数值计算曲线符合度高，之后随着装药比例埋深的增加，地冲击参数应力耦合系数 ${\eta _\sigma }$ 逐渐趋近1，从而也验证了利用有效弹坑体积计算地冲击耦合系数的可行性与可信性，在工程方面偏向安全。

图 15 黏土中耦合系数随装药比例埋深变化关系

Figure 15. Relationship between coupling coefficient and scaled burial depth in clay

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图15(b)中当量耦合系数 ${\eta _Q}$ 同样是由式(15)换算而得，随着装药比例埋深的增加，当量耦合系数 ${\eta _Q}$ 先快速增加，装药比例埋深增加到0.55 m/kg^1/3后，当量耦合系数 ${\eta _Q}$ 增加变得缓慢，最后逐渐趋近于1。为便于实际工程应用，将图15(b)中黏土当量耦合系数表达成如下关系：

${\eta _Q} = 1 - 0.91\;{{\rm{e}}^{ - h/0.33}}$

(16)

4. 结　论

通过在 $\varnothing$ 1500 mm×1490 mm分层式爆炸装置开展了变埋深黏土中爆炸实验，利用3D扫描仪和预埋土压力传感器分别测得不同埋深的弹坑尺寸和爆炸冲击应力，从现场实验和理论分析给出了弹坑压缩体积与耦合地冲击能量之间关系，得到如下结论：

(1) 地下爆炸发生后，随着埋深增加，有效弹坑轮廓逐渐由抛物线型发展为半球型，其形状演化过程与地冲击耦合过程同步；黏土的等效封闭爆炸临界埋深约为0.55 m/kg^1/3，数值上略大于地下封闭爆炸空腔半径，与Conwep计算程序预测的值基本一致；通过黏土中可视弹坑演化过程可知，黏土的封闭爆炸临界埋深约为1.46 m/kg^1/3；

(2) 对于本实验中的黏土，当−0.056 m/kg^1/3≤h≤0.37 m/kg^1/3，埋深增加对爆炸效应有加强的作用；在h≥0.55 m/kg^1/3时，埋深增加，衰减系数处于稳定状态，爆炸地冲击基本完全耦合，埋深再增加爆炸耦合地冲击能量此时可忽略；

(3) 爆炸耦合进介质中能量正比于有效弹坑的体积，此结论适用于浅埋爆炸和封闭爆炸；对于浅埋爆炸，可以把不规则轮廓通过等效面积转化为规则的球体进行计算；黏土爆炸实验证实了通过有效弹坑体积方式计算地冲击耦合系数的可行性与可信性；

(4) 建立了三种地冲击耦合系数的关系，最终都可与有效弹坑体积建立联系，引入Boltzmann函数给出了黏土耦合系数与比例埋深的关系；为了便于实际工程应用，文中直接给出了黏土的当量耦合系数与比例埋深的函数关系，此公式具有较好的预估精度。

图 1 Whipple防护结构^[5]

Figure 1. Whipple shield^[5]

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图 2 碎片云结构（球形弹丸）^[6]

Figure 2. Debris cloud structure (spherical projectile)^[6]

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图 3 Multi-shock (MS)防护结构^[10]

Figure 3. Multi-shock shield^[10]

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图 4 Mesh double-bumper（MDB）防护结构^[10]

Figure 4. Mesh double-bumper shield^[10]

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图 5 填充式防护结构^[14]

Figure 5. Stuffed Whipple shield^[14]

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图 6 填充式泡沫铝防护结构^[30]

Figure 6. Al-form stuffed Whipple shield^[30]

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图 7 填充式泡沫铝防护结构与传统Whipple防护结构的弹道极限曲线^[33]

Figure 7. Ballistic limit curves of Al-form stuffed Whipple shield and Whipple shield^[33]

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图 8 弹丸高速撞击蜂窝铝夹芯板的破碎过程^[41]

Figure 8. Simulation of hypervelocity impacts on honeycomb sandwich structure^[41]

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图 9 不同入射角度蜂窝铝夹芯板防护结构的弹道极限曲线^[39]

Figure 9. Projectile critical perforation diameter as a function of impact velocity against the same honeycomb panel^[39]

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图 10 泡沫铝夹芯板防护结构示意图^[32]

Figure 10. Schematic configuration of Al-foam sandwiched shield^[32]

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图 11 开孔泡沫夹芯板和蜂窝夹芯板损伤对比^[50]

Figure 11. Comparison of damages in open-cell foam core and honeycomb core sandwiches^[50]

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图 12 等质量的不同Whipple防护结构弹道极限曲线对比^[48]

Figure 12. Comparison of ballistic limit curves for comparable weight/standoff Whipple shield types^[48]

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图 13 开孔泡沫孔隙与泡沫孔大小和不同孔隙率的孔隙形状示意图^[48]

Figure 13. Open cell foam pore and cell size and ligament cross section variation with relative density^[48]

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图 14 双层蜂窝结构(DL-H) 和双层泡沫结构(DL-F) ^[16]

Figure 14. Schematic of the double-layer honeycomb target and the double-layer foam target ^[16]

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图 15 350 μs时SPH和FER方法碎片的比较^[95]

Figure 15. Comparison of debris cloud results between SPH and FER at 350 μs^[95]

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图 16 5种不同冲击速度下球形弹体的损伤对比^[111]

Figure 16. Damage in aluminum spheres due to impact with aluminum bumpers at five different velocities^[111]

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图 17 自适应耦合算法所得碎片云形态^[119]

Figure 17. The debris cloud of FEM-SPH adaptive method^[119]

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图 18 泡沫材料试件与Voronoi泡沫模型对比^[135]

Figure 18. Comparison of the Al-foam with the Voronoi tessellation model^[135]

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图 19 泡沫结构生成算法示意图^[138]

Figure 19. Schematic illustration of the algorithm for foam structure generation^[138]

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图 20 相邻两个孔间胞壁厚度示意图^[139]

Figure 20. Schematic diagram of the cell-wall thickness between the two adjacent pores^[139]

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图 21 闭孔金属泡沫的有限元单元^[139]

Figure 21. Finite element grid of closed-cell metallic foams^[139]

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图 22 有限元模型不同截面和CT扫描得到的灰度图像对比^[152]

Figure 22. Comparison of different section and gray images of finite element model^[152]

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