单晶与纳米多晶锡层裂的分子动力学研究

杨鑫; 赵晗; 高学军; 陈臻林; 王放; 曾祥国

doi:10.11883/bzycj-2022-0203

单晶与纳米多晶锡层裂的分子动力学研究

doi: 10.11883/bzycj-2022-0203

杨鑫^1,,
赵晗²,
高学军¹,
陈臻林¹,
王放³,
曾祥国^2, ,

1.
成都理工大学环境与土木工程学院，四川成都 610059
2.
四川大学建筑与环境学院深地科学与工程教育部重点实验室，四川成都 610065
3.
西南大学材料与能源学院，重庆 400715

基金项目: 国家自然科学基金（11972095，12202081）；四川省自然科学基金（2022NSFSC0443）；四川省科技厅项目（2021YJ0525）；工程材料与结构冲击振动四川省重点实验室资助项目（20kfgk02）

详细信息

作者简介:
杨　鑫（1988－　），男，博士，讲师，scsnyangxin@sina.com

通讯作者:
曾祥国（1960－　），男，博士，教授，xiangguozeng@scu.edu.cn

中图分类号: O383; O347.4
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- 被引次数: 2
出版历程
- 收稿日期: 2022-05-12
- 修回日期: 2022-10-07
- 网络出版日期: 2022-10-13
- 刊出日期: 2023-02-25

Molecular dynamics study on spallation in single-crystal and nanocrystalline tin

1.
College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu 610059, Sichuan, China
2.
Loboratory of Deep Underground Science and Engineering of Minitry of Education, College of Architecture and Environment, Sichuan University, Chengdu 610065, Sichuan, China
3.
School of Materials and Energy, Southwest University, Chongqing 400715, China

摘要

摘要: 低熔点金属的层裂是目前延性金属动态断裂的基础科学问题之一。采用非平衡态分子动力学方法模拟了冲击压力在13.5～61.0 GPa下单晶和纳米多晶锡的经典层裂和微层裂过程。研究结果表明：在加载阶段，冲击速度不影响单晶模型中的波形演化规律，但影响纳米多晶模型中的波形演化规律，其中经典层裂中晶界滑移是影响应力波前沿宽度的重要因素；在单晶模型中，经典层裂和微层裂中孔洞成核位置位于高势能处；在纳米多晶模型中，经典层裂中的孔洞多在晶界（含三晶界交界处）处成核，并沿晶定向长大，产生沿晶断裂，而微层裂中孔洞在晶界和晶粒内部成核，导致沿晶断裂、晶内断裂和穿晶断裂；孔洞体积分数呈现指数增长，相同冲击速度下单晶和纳米多晶Sn孔洞体积分数变化规律一致；经典层裂中孔洞体积分数曲线的两个转折点分别表示孔洞成核与长大的过渡和材料从损伤到断裂的灾变性转变。
- 非平衡态分子动力学 /
- 单晶与纳米多晶锡 /
- 应力波演化 /
- 断裂方式 /
- 孔洞体积分数
Abstract: One of the fundamental scientific problems of dynamic fracture of ductile metals is spallation of low melting point metals. The classical spallation and micro-spallation of single-crystal (SC) and nanocrystal (NC) tin were carried out using the non-equilibrium molecular dynamics (NEMD) at shock pressures of 13.5−61.0 GPa. In order to achieve the spallation in the SC and NC models, the piston-target method was utilized. Specifically, the rigid piston was assigned an initial velocity, then the piston impacted the target to generate stress wave, and the stress waveform was controlled by adjusting the loading time after the length of the model along the shock direction was determined. The simulation results show that: during the loading stage, the shock wave velocity has no influence on the waveform evolution of the SC Sn model, but it does have an effect on the waveform evolution of the NC Sn model, in which the front width of the stress wave in classical spallation of the NC Sn model is mainly affected by grain boundary sliding. The void nucleation sites in classical spallation and micro-spallation are found at high potential energies in the SC model. In the NC model, for the classic spallation, voids mostly nucleate at grain boundaries (including the triple junctions of the grain boundaries) and grow along grain boundaries, resulting in intergranular fractures; for the micro-spallation, voids nucleate at the grain boundary and inside the grain, resulting in intergranular fracture, intragranular fracture, and transgranular fracture. The void volume fraction increases exponentially, and the variation law of void volume fraction of SC and NC Sn is the same under the same impact velocity. The two turning points of the void volume fraction curve in classical spallation represent the transition from nucleation to growth and the catastrophic transition from damage to fracture.
- NEMD /
- single-crystal and nanocrystal tin /
- Stress wave evolution /
- fracture mode /
- void volume fraction

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可膨胀石墨受热膨胀后成为中空状粒子, 形似蠕虫, 漂浮在空中能够有效干扰毫米波; 不同规格的可膨胀石墨膨胀后可以得到1~8 mm甚至更长的粒子, 因此有望使用一种材料遮蔽干扰不同的波长, 它在发烟剂的应用中具有很大的潜力。实验表明, 爆炸法能够快速并有效地在指定空域形成膨胀石墨型气溶胶云团, 但爆炸实验成本高、风险大^[1-3]。

若能计算出膨胀石墨燃爆剂的JWL^[4-5]状态方程参数, 就可以用LS-DYNA数值模拟研究代替部分实验研究, 从而达到降低实验成本、提高研究效率的目的。一般来说, JWL状态方程参数需要通过圆筒实验及二维流体动力学程序确定。本文中提出一种JWL状态方程的参数近似解法, 即通过凝聚炸药等熵线的物态方程推导出目标方程, 通过差分进化法拟合得出膨胀石墨燃爆剂JWL状态方程的参数。

1. JWL状态方程及应用

JWL状态方程是由J.W.Kury等^[6-7]提出的, 该方程的未知参数需要通过圆筒实验及二维流体动力学程序来确定, 它不显含化学反应, 能够比较精确地描述爆轰产物的膨胀驱动做功过程。JWL状态方程的具体形式如下:

$p=A\left(1-\frac{\omega}{R_{1} V}\right) \mathrm{e}^{-R_{1} V}+B\left(1-\frac{\omega}{R_{2} V}\right) \mathrm{e}^{-R_{2} V}+\frac{\omega E}{V}$

(1)

式中:p为爆轰产物的压力, V为爆轰产物的相对比容, E为爆轰产物的比内能。A、B、R₁、R₂、ω为待拟合参数, 也称经验(调节)常数。

在LS-DYNA中, 对于炸药材料的定义需要输入JWL状态方程的A、B、R₁、R₂、ω等参数值。

2. 膨胀石墨燃爆剂JWL状态方程关键参数的拟合

要拟合得到膨胀石墨燃爆剂JWL状态方程的参数, 需要找到能够描述膨胀石墨燃爆剂爆炸时p -V关系的方程作为目标方程。那么对于凝聚炸药, 忽略初始压力p₀和初始内能E₀, 凝聚炸药的质量守恒、动量守恒、能量守恒与C-J条件(相切条件)方程, 即凝聚炸药爆轰参数方程^[8]为:

$\left\{\begin{array}{l} D^{2}=v_{0}^{2} \frac{p_{\mathrm{H}}-p_{0}}{v_{0}-v_{\mathrm{H}}} \\ u_{\mathrm{H}}^{2}=\left(p_{\mathrm{H}}-p_{0}\right)\left(v_{0}-v_{\mathrm{H}}\right) \\ e_{\mathrm{H}}-e_{0}=(1 / 2)\left(p_{\mathrm{H}}+p_{0}\right)\left(v_{0}-v_{\mathrm{H}}\right)+Q_{\mathrm{v}} \\ \left(\frac{\partial p}{\partial v}\right)_{S}=-\frac{p_{\mathrm{H}}-p_{0}}{v_{0}-v_{\mathrm{H}}} \end{array}\right.$

(2)

式中:u₀、u_H分别表示原始爆炸物的质点速度和爆轰波反应末端介质的质点速度; p₀、T₀、ρ₀、v₀分别表示原始爆炸物的压力、温度、密度和比容; p_H、T_H、ρ_H、v_H分别表示爆轰波反应末端断面处的压力、温度、密度和比容; e₀、e_H分别表示原始爆炸物和爆轰波反应末端的内能; Q_v表示爆轰反应释放出的化学能。

凝聚炸药爆轰产物状态方程为:

$p=A \rho^{k}+\frac{B}{v} T$

(3)

式中:A、B、k都是与炸药性质有关的常数。Aρ^k表示冷压强, $\frac{B}{v} T$ 表示热压强, 热压强对压力的作用比冷压强小得多^[9], 因此可将(3)式简化为:

$p=A \rho^{k}=\frac{A}{v^{k}}$

(4)

方程(4)为凝聚炸药爆轰产物的近似状态方程, 由于其中没有温度项, 该方程可近似为等熵方程, 本文中称为凝聚炸药等熵线物态方程。其中, k是等熵常数, 在CJ点的k值一般为3左右, 本文中取为3。

那么方程组(2)可以简化成如下形式:

$\left\{\begin{array}{l} \rho_{\mathrm{H}}=\frac{k+1}{k} \rho_{0} \\ p_{\mathrm{H}}=\frac{1}{k+1} \rho_{0} D^{2} \\ u_{\mathrm{H}}=\frac{1}{k+1} D \\ D=\sqrt{2\left(k^{2}-1\right) Q_{\mathrm{v}}} \end{array}\right.$

(5)

方程组(5)即凝聚态炸药爆轰波参数近似计算方程。

将中的v用JWL状态方程中的相对比容V来表示。令v=v₀V, 则

$p=\frac{A}{\left(v_{0} V\right)^{k}}=\frac{A \rho_{0}^{k}}{V^{k}}=\frac{M}{V^{k}}$

(6)

式中: 为常数。将方程(5)代入方程(6), 得:

$A=p v^{k}=p_{\mathrm{H}} v_{\mathrm{H}}^{k}=\frac{\rho_{0} D^{2}}{k+1}\left(\frac{k}{k+1} v_{0}\right)^{k}$

(7)

$M=A \rho_{0}^{k}=\frac{\rho_{0} D^{2}}{k+1}\left(\frac{k}{k+1} v_{0}\right)^{k} \rho_{0}^{k}=\frac{\rho_{0} D^{2}}{k+1}\left(\frac{k}{k+1}\right)^{k}$

(8)

那么只要已知炸药的密度ρ₀、爆速D, 就可以根据方程(5)和方程(7)求解得到目标方程:

$p=\frac{M}{V^{k}}=\frac{\frac{\rho_{0} D^{2}}{k+1}\left(\frac{k}{k+1}\right)^{k}}{V^{k}}$

(9)

式中:V取0~7, 因为在=7之前圆筒实验中圆柱壳体运动规律与实验运动规律相符合, 本文中取0.5~3。然后利用LstOpt或MATLAB软件中的差分进化法拟合出A、B、R₁、R₂、ω。

实验时采用的膨胀石墨燃爆剂配方为黑火药和可膨胀石墨的混合物, 黑火药/可膨胀石墨质量之比为3:2, 其装药密度ρ=1.2 g/cm³; 根据经验公式^[9]可计算出膨胀石墨燃爆剂的爆速D≈850 m/s。当k=3时, 根据公式(5)和(8), 计算得到p_H=216.8 MPa, M=0.091 5, 那么拟合目标函数方程可表示为:

$y=\frac{0.0915}{x^{3}}$

(10)

利用LstOpt或MATLAB数据处理软件, 采用差分进化法和遗传算法相结合的方法, 对方程(1)参数进行拟合。参数拟合结果如表 1所示, 表中E_max表示最大误差。与其对应的拟合曲线如图 1所示。

表 1 几组拟合参数及其误差

Table 1. Several groups of fitting parameters and max error

数据组	A	B	R₁	R₂	ω	E_max/%
Value(a)	0.413	8.000	1.977	5.570	0.103	3.02
Value(b)	0.058	1.729	1.046	3.107	0.043	8.02
Value(c)	0.105	0.869	1.813	2.401	0.109	20.05
Value(d)	0.412	3.841	2.057	4.597	0.166	2.30

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图 1 几组拟合曲线与目标曲线对比

Figure 1. Comparison between fitting and objective curves

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从表 1中可以看出, Value(d)组数据最大误差为2.30%, 精度比较高, 选取其作为最终拟合结果。当然这5个参数值不是确定不变的, 在满足一定精度条件下, 只要选取其中精度较高的一组参数即可。

3. 数值模拟与分析

圆筒为轴对称结构, 可在柱坐标系中建立轴对称计算模型, 如图 2所示。图 2中OCDE区为圆筒内混合炸药部分, ABCD区为铜管部分, 在E点直接起爆。OE边的长度为圆筒长度, ED和EA的长度为铜管的内径和外径。JWL参数采用Value(d)组数据带入计算。

图 2 计算模型简图

Figure 2. Diagram of model

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图 3为圆筒初分网格图, 图 4为45 μs时圆筒网格变形图, 此刻圆筒上端部分已经充分膨胀, 炸药爆轰波继续向下传播。

图 3 网格划分图

Figure 3. Diagram of meshing

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图 4 圆筒45 μs时网格变形图

Figure 4. Gridding distortion at 45 μs

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利用后处理软件LS-PREPOST, 得出圆筒A点的Δr -t(半径变化-时间)曲线如图 5所示。

图 5 数值模拟得到的Δr-t曲线

Figure 5. Simulation curve of Δr-t

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4. 圆筒实验验证

圆筒实验^[10-11]是专门用于确定炸药爆轰产物JWL状态方程参数和评定炸药做功的标准化实验, 其实验原理图如图 6所示, 圆筒平行放置于支架上, 高速扫描相机通过金属板狭缝记录燃爆剂稳定爆轰段圆筒膨胀距离。

图 6 圆筒实验原理示意图

Figure 6. Diagram of cylinder test

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圆筒实验数据一般被拟合成如下形式:

$\left\{\begin{array}{l} t=a+b\left(R-R_{0}\right)+r \mathrm{e}^{d\left(R-R_{0}\right)} \\ v=\left[{ }^{b}+r d{\mathrm{e}}^{d\left(R-R_{0}\right)}\right]-1 \\ E_{\mathrm{kc}}=v_{\mathrm{c}}^{2} / 2 \end{array}\right.$

(11)

式中:t为圆筒壁膨胀的时间; (R-R₀)为圆筒壁膨胀的距离, 用Δr表示; a、b、c、d为根据实验数据得到的拟合系数; v_c为不同膨胀距离(R-R₀)相对应的圆筒壁的速度; E_kc为不同膨胀距离相对应的比动能。

实验时, 圆筒半径R₀=25 mm, 膨胀石墨燃爆剂装药密度ρ₀=1.2 g/cm³, 爆速D≈850 m/s。膨胀石墨燃爆剂圆筒实验结果如表 2所示。

表 2 圆筒实验结果

Table 2. Data of cylinder test

t/μs	(R-R₀)/mm
25	0.0
30	0.0
35	0.5
40	1.0
45	2.0
50	2.5
55	3.0
60	3.5
65	4.0
70	4.5
75	5.0
80	5.5
85	6.0
90	6.6
95	7.2
100	8.5

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实验所得圆筒Δr -t曲线如图 7所示。

图 7 实验得到的Δr-t曲线

Figure 7. Test curve of Δr-t

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分析图 5、7中数据, 可以得出模拟曲线与实验曲线符合非常好, 经过MATLAB对2组数据进行分析后, 得出其误差最大为3.3%。

5. 结束语

把凝聚炸药等熵线物态方程作为目标方程拟合出膨胀石墨燃爆剂的JWL状态方程的关键参数, 然后利用拟合的JWL参数对圆筒实验模型进行数值模拟得到Δr -t曲线, 最后用圆筒实验得出圆筒的Δr -t曲线。通过MATLAB对两组数据进行分析, 得出其误差最大为3.3%。实验结果表明:基于凝聚炸药等熵线物态方程拟合膨胀石墨燃爆剂JWL状态方程参数的方法可行, 满足实际应用需求。

图 1 分子动力学模拟模型

Figure 1. Simulation models of molecular dynamics

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图 2 Hugoniot压力p_H与冲击速度u_p的关系

Figure 2. Relation of Hugoniot pressure p_H and shock velocity u_p

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图 3 不同冲击速度下应力波（p_zz）波形演化过程

Figure 3. Evolutionary processes of stress wave (p_zz) profiles at different shock velocities

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图 4 应力波剖面与原子构型的关系

Figure 4. The relation of stress wave profile and atomic structure

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图 5 u_p = 0.5 km/s时单晶Sn的孔洞成核、长大与贯穿过程

Figure 5. Process of void nucleation, growth and coalescence in SC Sn at u_p = 0.5 km/s

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图 6 u_p = 0.5 km/s时纳米多晶Sn的孔洞成核、长大与贯穿过程

Figure 6. Process of void nucleation, growth and coalescence in NC Sn at u_p = 0.5 km/s

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图 7 u_p = 0.5 km/s时单晶Sn层裂区域孔洞演化过程的截面

Figure 7. Section of spallation zone in SC Sn at u_p = 0.5 km/s

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图 8 u_p = 0.5 km/s时纳米多晶Sn层裂区域孔洞演化过程的截面

Figure 8. Section of spallation zone in NC Sn at u_p = 0.5 km/s

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图 9 u_p = 1.5 km/s时单晶Sn的孔洞成核、长大与贯穿过程

Figure 9. Process of void nucleation, growth and coalescence in SC Sn at u_p = 1.5 km/s

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图 10 u_p = 1.5 km/s时纳米多晶Sn的孔洞成核、长大与贯穿过程

Figure 10. Process of void nucleation, growth and coalescence in NC Sn at u_p = 1.5 km/s

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图 11 u_p = 1.5 km/s时单晶Sn层裂区域孔洞演化过程的截面

Figure 11. Section of spallation zone in SC Sn at u_p = 1.5 km/s

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图 12 u_p = 1.5 km/s时纳米多晶Sn层裂区域孔洞演化过程的截

Figure 12. Section of spallation zone in NC Sn at u_p = 1.5 km/s

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图 13 微层裂后期过程（u_p = 0.5 km/s）

Figure 13. Later process of micro-spallation for u_p = 0.5 km/s

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图 14 微层裂后期过程（u_p = 1.0 km/s）

Figure 14. Later process of micro-spallation for u_p = 1.0 km/s

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图 15 微层裂后期过程（u_p = 1.5 km/s）

Figure 15. Later process of micro-spallation for u_p = 1.5 km/s

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图 16 温度表征的纳米多晶Sn微层裂演化

Figure 16. Micro-spallation evolution characterized by temperature in the NC Sn

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图 17 压力表征的纳米多晶Sn微层裂演化

Figure 17. Micro-spallation evolution characterized by pressure in the NC Sn

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图 18 孔洞体积分数V_f与体积分数差值ΔV_f演化过程

Figure 18. Evolutionary processes of void volume fraction V_f and its difference ΔV_f

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表 1 材料物态变化与层裂类型

Table 1. Matter state variation and spallation classification

Sn材料	u_p/(km·s⁻¹)	p_H/GPa			T_H/K	物态	冲击熔化	分类
Sn材料	u_p/(km·s⁻¹)	本文	文献[35]	文献[42]	T_H/K	物态	冲击熔化	分类
单晶	0.5	14.0	13.9	15.58	485.0	固态	未熔化	经典层裂
	1.0	34.6	32.1	36.94	1086.0	固液混合态	卸载熔化	微层裂
	1.5	60.5	55.3	60.1	2311.0	液态	加载熔化	微层裂
纳米多晶	0.5	13.5	−	−	484.0	固态	未熔化	经典层裂
	1.0	33.5	−	−	1062.0	固液混合态	卸载熔化	微层裂
	1.5	61.0	−	−	2391.0	液态	加载熔化	微层裂

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参考文献(61)

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