高温、高压、高应变速率动态过程晶体塑性有限元理论模型及其应用

刘静楠; 叶常青; 刘桂森; 沈耀

doi:10.11858/gywlxb.20190874

高温、高压、高应变速率动态过程晶体塑性有限元理论模型及其应用

doi: 10.11858/gywlxb.20190874

上海交通大学材料科学与工程学院，上海　200240

基金项目: 科学挑战计划(TZ2018001)

详细信息

作者简介:
刘静楠(1993－)，女，硕士，主要从事动态晶体塑性有限元研究. E-mail：jingnanliu@sjtu.edu.cn

通讯作者:
沈　耀(1972－)，男，博士，教授，主要从事晶体缺陷行为、力学性能及塑性变形的微观机制研究. E-mail：yaoshen@sjtu.edu.cn

中图分类号: O344.1
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出版历程
- 收稿日期: 2019-12-31
- 修回日期: 2020-01-17

Crystal Plasticity Finite Element Theoretical Models and Applications for High Temperature, High Pressure and High Strain-Rate Dynamic Process

School of Material Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

摘要

摘要: 对于高温、高压、高应变速率加载条件下的材料冲击变形行为，动态晶体塑性模型能够直接反映晶体中塑性滑移的各向异性及其对温度、压力和微观组织结构的依赖性，因而广泛应用于材料的动态冲击力学响应、微观结构演化以及动态损伤破坏的模拟。本文综述了高压冲击下动态晶体塑性有限元的理论模型，主要包括变形运动学、包含状态方程的超弹性本构模型和晶体塑性本构模型，涉及位错滑移、相变、孪生等塑性变形机制，以及层裂、绝热剪切带等动态破坏方式。
- 动态晶体塑性有限元 /
- 动力学响应 /
- 微观结构演化
Abstract: For shock deformation behavior of materials under high temperature, high pressure and high strain-rate loading conditions, dynamic crystal plasticity models can directly reflect the anisotropy of plastic slip and its dependence of temperature, pressure and microstructure in crystals. In consequence, dynamic crystal plasticity models are widely used in simulations of material impact dynamic response, microstructure evolution and dynamic damage. Theoretical models of dynamic crystal plasticity under high pressure shock loading conditions were reviewed in this paper, mainly including: deformation kinetics, hyperelastic constitutive models incorporating equations of state, and crystal plasticity constitutive models. This paper also covers plastic deformation mechanisms, including dislocation slip, phase transition and twinning; as well as dynamic damage, including spalling and adiabatic shear band.
- dynamic crystal plasticity finite element method /
- dynamic response /
- microstructural evolution

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随着社会高速发展，特别是我国“双碳目标”的提出，对清洁能源的需求与日俱增，氢能作为一种环保、高效的新能源受到广泛关注。考虑到氢气管道建设成本高，利用现有的天然气管道，向甲烷中掺入一定量的氢气进而提升混合燃料热值，已成为一种经济可行的解决方案^[1]。值得关注的是，近年来天然气输运事故屡有发生，若向其掺入氢气，有可能进一步加剧爆炸的危险性。倪靖等^[2]讨论了不同掺氢比对甲烷-氧气爆轰特性的影响，发现掺氢后能够提高爆轰波的传播速度和爆轰敏感性。余明高等^[3]通过实验研究了障碍物对甲烷-氢气爆炸特性的影响，发现最大爆炸超压和火焰传播速度随着障碍物阻塞率以及氢气体积分数的增大而增大。Yu等^[4]研究了掺氢对甲烷-空气预混火焰传播特性的影响，发现随着氢气含量的增加，火焰前沿速度和爆炸超压显著升高。

鉴于可燃气体在运输过程中的危险性，大量学者对可燃气体的抑爆问题开展了研究。Li等^[5]发现CO₂对甲烷爆炸的抑制效果好于N₂。Luo等^[6]研究BC粉对氢气-甲烷-空气预混气的抑制后发现，BC粉对掺氢比低的预混气体有较好的抑制作用。陈鹏等^[7]发现当金属丝网层数大于3时，甲烷-空气预混气爆炸火焰经过金属丝网时会发生淬熄。徐海顺等^[8]通过研究发现铝镁合金泡沫对甲烷-空气预混气体爆炸有较好的抑制作用，材料对传播火焰的影响机制主要体现在湍流促进和冷却抑制两方面。袁必和等^[9] 研究了新型多孔聚丙烯复合材料对瓦斯爆炸的影响，发现多孔材料的抑爆性能受填充位置、材料内径以及填充长度的影响。An等^[10]通过对比球形非金属材料和铝合金材料对可燃气体的抑爆结果，发现由于球形非金属材料具有较高的结构强度，因而抑爆效果较好。邵继伟等^[11]研究多孔材料对可燃气体的抑爆效果后发现，组合型多孔材料在密闭容器管道系统内的抑爆效果更为突出。综合以上研究发现，非金属多孔材料对抑制甲烷爆炸具有良好的效果，但甲烷掺氢的爆炸特性较甲烷出现明显改变，并且前人对非金属多孔材料在甲烷掺氢抑爆方面的研究较少。传统的非金属材料存在易燃、爆炸后熔融物易黏附管道内壁等缺陷，严重限制了多孔非金属材料在可燃气体抑爆中的实际应用。球形抑爆材料中空多孔结构能够有效增大比表面积，扩大热损失，并且耐高温耐火焰，具有良好的阻火隔爆性质^[12-13]。

本研究通过实验探究掺氢比对甲烷掺氢爆炸特性的影响，比较单一球形多孔非金属材料和组合球形多孔非金属材料的抑爆效能，并通过改变材料填充长度，分析最佳阻火抑爆性能的搭配参数，为球形多孔非金属材料在混合燃料阻隔防爆领域的应用提供理论支撑与实验依据。

1. 实验装置与方案

1.1 实验装置与流程

为研究掺氢比对甲烷掺氢预混气体爆炸的影响并测试球形多孔材料的抑爆性能，自主搭建了气体管道爆炸实验平台，如图1所示。实验平台主要包括爆炸实验管道、点火系统、配气系统、同步控制、高速摄像系统（拍摄速度4 000帧每秒）以及数据采集系统。

图 1 实验装置示意图

Figure 1. Schematic diagram of the experiment setup

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实验管道由3节内径60 mm、管壁厚度10 mm、外径80 mm、长度2 m的水平圆形管道以及1节内径60 mm、长0.5 m、配备可视窗的方形管道组成，各个管道之间通过法兰-螺栓连接，实验管道为封闭状态。实验管道共安装2台压力传感器（CYG508微型高压传感器），分别距离管道左端1.2和6.3 m。点火系统（距离管道右端0.5 m处）由高压点火器以及2根钨丝组成，实验中点火能采用20 J。配气系统通过气瓶连接配气仪进行配气，预混气体中可燃气体的体积分数为10%。通过高速摄像系统采集火焰图像。实验过程如下：首先，清洁管道，连接管道和仪器；然后，检查管道气密性，采用抽真空法配气，向管道中通入预混气体；接着点火触发，数据采集系统对实验数据进行采集；最后，排出废气，重复实验，每组实验重复多次。

1.2 实验材料及实验方案

本实验的主要材料为球形多孔非金属材料和聚氨酯（polyurethane，PU）材料，如图2所示。球形多孔非金属材料的主要材质为聚偏氟乙烯，化学性质稳定，具有很强的抗冲击性和耐高温性，并且球形材料为多孔结构，孔隙率大，比表面积大。组合球形内部填充材料主要为聚氨酯，聚氨酯材料的可塑性较强，对爆燃火焰的传播具有一定的抑制作用^[14]。

图 2 多孔非金属抑爆材料

Figure 2. Porous non-metallic explosion suppression material

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最大爆炸压力（p_max）是评价爆炸强度以及材料抑爆性能的重要指标。为了判断掺氢比对可燃气体爆炸的影响以及单一球形多孔非金属材料和组合球形非金属材料的抑爆性能，设计如下实验方案：首先研究不同掺氢比x对p_max的影响，分别对掺氢比为0%、5%、10%的甲烷掺氢预混气体进行测试，记录管道内爆炸压力并进行对比分析；在此基础上，研究多孔抑爆材料对p_max的影响，选取p_max最大的预混气体，分别对单一球形材料和组合球形材料进行抑爆实验，填充于距管道左端1.5 m的位置，填充长度分别为20、30、40 cm，对不同填充长度的同一材料以及相同填充长度的不同材料进行对比分析，判断其抑爆性能。

掺氢比x表示为

$x=\frac{\mathrm{\varphi }({\rm H}_{2})}{\mathrm{\varphi }({\rm CH}_{4})+\mathrm{\varphi }({\rm H}_{2})}\times 100\text{%}$

(1)

式中： $\mathrm{\varphi }({\rm CH}_{4})$ 、 $\mathrm{\varphi }({\rm H}_{2})$ 分别为甲烷-氢气混合气体中甲烷和氢气的体积分数。

2. 实验结果及分析

2.1 掺氢比对爆炸压力的影响

管道中不同掺氢比对预混气体爆炸压力随时间的变化曲线如图3所示。从图3可以看出，1号压力传感器所采集的压力峰值随着氢气体积分数的增大而增大。不同掺氢比对火焰图像的影响如图4所示。对比发现，随着掺氢比的提高，火焰的亮度明显变强，甲烷-空气预混气体掺氢后火焰传播特性及爆炸特性发生显著改变，氢气对甲烷爆炸具有促进作用^[15]。

图 3 掺氢比对预混气体爆炸压力的影响

Figure 3. Effect of hydrogen doping ratio on explosion pressure of premixed gas

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图 4 不同掺氢比条件下甲烷掺氢爆炸火焰传播图像

Figure 4. Flame propagation images of methane hydrogen-doped syngas under different hydrogen doping ratios

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预混气体爆炸压力是反映爆炸破坏效应的重要指标，爆炸压力越大，爆炸造成的后果往往越严重。掺氢比对预混气体最大爆炸压力p_max和最大爆炸压力上升速率(dp/dt)_max的影响如图5所示。当掺氢比为0%、5%、10%时，1号压力传感器处的p_max分别为200、235、245 kPa，(dp/dt)_max分别为2 250、2 875、3 250 kPa/s。与未掺氢（掺氢比为0%）相比，掺氢比为10%时，1号压力传感器处的p_max提高22.50%，(dp/dt)_max提高44.44%，2号传感器处的(dp/dt)_max提高26.67%，说明掺氢比较低时，p_max和(dp/dt)_max均随着氢气含量的增加而增大，爆炸强度增大。这是由于氢气的活性高于甲烷，反应比甲烷剧烈，发生爆炸反应的时间更短，随着掺氢比的增加，单位体积内氢气的量增多，氢气比例的升高增强了能量释放的集中程度，气体燃烧速率加快，缩短了热量损失时间，导致爆炸压力上升速率提高^[16]。

图 5 不同掺氢比条件下最大爆炸压力和最大爆炸压力上升速率曲线

Figure 5. Maximum explosion pressure and maximum pressure rise rate for different hydrogen doping ratios

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2.2 单一球形材料对爆炸压力的影响

为研究球形非金属材料对甲烷掺氢后预混气体爆炸的抑制作用，采用爆炸压力较大的掺氢比为10%的预混气体进行对照。空管道与填充不同长度的抑爆材料对爆炸压力的影响如图6所示，其中0 cm代表无抑爆材料。由图6可知，填充抑爆材料后，各个工况下的p_max均小于无抑爆材料。1号、2号压力传感器测得无抑爆材料的p_max分别为245、200 kPa。当抑爆材料的填充长度分别为20、30、40 cm时，1号压力传感器处的p_max分别为205、185、175 kPa，对比无抑爆材料时的p_max分别降低了16.33%、24.49%、28.57%，爆炸压力得到有效抑制。这是由于球形抑爆材料是多孔中空塑料球形结构，有较大的比表面积，并且球形材料把管道空间切割成多个小空间，增大了反应面与球形材料表面的接触面积，促进了球形材料与火焰的热交换，增加了热损失，并且冲击波通过球形材料时，被切割的反应面与后面的球形材料发生碰撞导致能量损失，进而使爆炸压力降低^[12]。2号压力传感器处各个工况下的p_max相较于无抑爆材料分别降低5.00%、5.00%、12.50%，抑制效果较弱。这是由于随着火焰传播，火焰通过球形抑爆材料后再无障碍物，燃烧速率加快，火焰到达2号压力传感器时已充分反应，致使降压效果不明显。

图 6 不同填充长度条件下单一球形材料对预混气体爆炸压力的影响

Figure 6. Effect of single spherical material with different filling lengths on explosion pressure of premixed gas

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不同填充长度对(dp/dt)_max的影响如图7所示。在低填充长度下，(dp/dt)_max有所上升，当填充长度为20 cm时，1号压力传感器处的(dp/dt)_max提高7.69%，2号压力传感器处提高36.84%。重复实验均显示(dp/dt)_max提高，这是由于当球形材料的填充长度较低时，火焰在穿越过程中的湍流度增大，燃烧传质传热进程加快，燃烧强度增大，火焰传播速度进一步加快^[8]。随着填充长度的增加，火焰被分离成多个离散的湍流火焰，最大爆炸压力上升速率逐渐降低，填充长度为40 cm时，相较于无抑爆材料，1号压力传感器处的(dp/dt)_max降低33.85%。

图 7 不同填充长度条件下最大爆炸压力和最大爆炸压力上升速率曲线

Figure 7. Maximum explosion pressure and maximum pressure rise rate for different filling lengths

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2.3 组合球形材料对爆炸压力的影响

不同长度的组合球形材料对爆炸压力的影响曲线如图8所示。从图8可以看出：与无抑爆材料相比，填充组合球形材料后p_max明显降低，说明组合球形材料对管道内可燃气体爆炸有较好的抑制效果。当填充长度分别为20、30、40 cm时，1号压力传感器测得的p_max分别为160、155、120 kPa，与无抑爆材料相比，分别衰减了34.69%、36.73%、51.02%，相较于单一球形非金属材料，抑爆性能分别提升了112.43%、49.98%、78.58%；2号压力传感器测得的p_max为135、135、110 kPa，与无抑爆材料相比，分别衰减了32.50%、32.50%、45.00%。可以看到，当填充长度为20和30 cm时，抑爆性能差别不大，这是多孔材料冷却抑制与障碍物加压共同作用的结果：当火焰经过球形多孔抑爆材料时，材料会吸收热量从而对爆炸产生抑制效果，材料的填充长度越长，抑制效果越好；球形材料在管道中相当于障碍物，火焰经过障碍物时层流会转变成湍流，增加障碍物数量能够明显增大火焰传播过程中的湍流强度，导致压力上升^[3]。当填充长度较低时，材料冷却抑制效果与障碍物加压效果相互抵消，最终导致抑制作用较接近。

图 8 不同填充长度的组合球形材料对预混气体爆炸压力的影响

Figure 8. Effect of combined spherical material with different filling lengths on explosion pressure of premixed gas

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从图9中可以看出，(dp/dt)_max也得到了较大程度的抑制，与无抑爆材料相比，填充长度为20、30、40 cm时，1号压力传感器处的(dp/dt)_max分别衰减34.62%、46.15%、53.85%，2号压力传感器处的(dp/dt)_max最高衰减了47.37%，抑制效果明显。与单一球形材料相比，组合球形材料的抑爆性能得到大幅增强。这是由于爆炸穿越填充材料时，被球形材料多孔结构离散后的湍流火焰和冲击波能量与球形多孔材料中的聚氨酯材料接触，聚氨酯材料粗糙的壁面消耗了链式反应中的自由基，阻碍燃烧链式反应的进行，并使部分火焰热量转移到聚氨酯材料中，在火焰与聚氨酯材料发生碰撞的过程中，聚氨酯材料与冲击波产生摩擦，导致部分冲击波能量转化为热量并消散掉^[14]。

图 9 不同填充长度条件下最大爆炸压力和最大爆炸压力上升速率曲线

Figure 9. Maximum explosion pressure and maximum pressure rise rate for different filling lengths

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3. 结　论

基于自主搭建的气体爆炸管道平台，研究了不同掺氢比条件下甲烷掺氢的爆炸特性以及不同类型球形抑爆材料的抑爆性能，得到以下结论。

(1) 氢气对甲烷-空气爆炸具有一定的促进作用。在低掺氢比下，随着可燃气体中氢气体积分数的增加，火焰传播速率加快，最大爆炸压力和最大爆炸压力上升速率升高。

(2) 单一球形多孔非金属材料对可燃气体爆炸具有两方面作用：一方面为抑制作用，体现在填充材料后，球形材料会吸收能量，导致最大爆炸压力降低，并且填充长度越长，抑制效果越好；另一方面为促进作用，体现在填充长度较低时，会导致燃烧传质传热进程加快，火焰传播加速，导致最大爆炸压力上升速率上升。

(3) 组合球形多孔非金属材料与单一球形多孔非金属材料相比，抑爆性能大幅提升，降压效果明显，当填充长度为40 cm时，最大爆炸压力衰减51.02%。球形材料抑爆性能受填充长度影响，在冷却抑制与障碍物增压的共同作用下，填充20和30 cm的组合球形多孔非金属材料的抑爆效果相差不大。

图 1 经典的晶体运动学构型

Figure 1. Classical configurations of crystal kinematics

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图 2 引入热膨胀构型的变形梯度分解F = F^eF^θF^p^[19]

Figure 2. Decomposition of deformation gradient considered thermally-expanded configuration ${{F}}={{{F}}}^{\rm{e}}{{{{F}}}^{\theta }{{F}}}^{\rm{p}}$ \normalsize^[19]

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图 3 (a)热能协助位错克服势垒(T₀ < T₁ < T₂ < T₃)^[51]，(b)位错在运动过程中遇到的势垒^[52]

Figure 3. (a) Thermal energy assists dislocations to overcome barriers ( ${T}_{0}<{T}_{1}<{T}_{2}<{T}_{3}$ \normalsize)^[51], and (b) barriers encountered by a dislocation on its course^[52]

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图 4 热软化效应对多晶Ta在32 GPa冲击变形下累积塑性滑移量的影响^[29]

Figure 4. Influence of thermal softening on accumulated plastic slip of polycrystalline Ta during shock deformation under 32 GPa^[29]

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图 5 α-RDX单晶沿<210>晶向平板撞击变形过程中声子拖曳对(021)<100>滑移系上滑移阻力的影响^[26]

Figure 5. Influence of phonon drag on slip resistance of (021)<100> slip system, during α-RDX single crystal deformed in plate impact along <210> direction^[26]

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图 6 螺位错滑移的Kink-pair机制^[27]

Figure 6. Illustration of screw dislocation motion via a Kink-pair mechanism^[27]

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图 7 位错平均运动与热激活运动以及拖曳运动的对比^[27]

Figure 7. Comparison of the average dislocation velocity with the velocities of thermally-activated and drag-dominated dislocation motions^[27]

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图 8 不同压力加载下位错密度的演化机制^[83]

Figure 8. Dislocation density evolution mechanisms under different loading pressure^[83]

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图 9 RDX的α相与γ相Gibbs自由能之差与温度、压强的关系^[38]

Figure 9. Difference between Gibbs free energies of the α and γ RDX polymorphs as a function of pressure and temperature^[38]

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图 10 Fe冲击相变的单晶模拟与多晶实验结果^[111]

Figure 10. Single crystal Fe simulation data and polycrystal experimental data of shock-induced phase transformation^[111]

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图 11 冲击变形过程中波的传播及层裂现象(a)、3个时刻的应力波形(b)和3个位置的应力历史(c)^[52]

Figure 11. Wave propagation and spalling phenomenon (a), stress profiles at three different times (b), as well as stress histories at three different positions (c) during shock deformation^[52]

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图 12 铅合金动态晶体塑性有限元模拟结果：(a)层裂形核时的压力，(b)层裂形核时的弹性能密度，(c)经250 m/s冲击加载层裂面附近的等效应力；(d)经350 m/s冲击加载层裂面附近的等效应力^[24]

Figure 12. Dynamic crystal plasticity finite element simulation results of lead alloy: (a) pressure of spalling nucleation; (b) elastic energy density of spalling nucleation; (c) equivalent stress near the spalling surface under 250 m/s shock loading; (d) equivalent stress near the spalling surface under 350 m/s shock loading^[24]

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图 13 多孔晶体的变形梯度分解为弹性部分(F^e)、不可逆偏量部分(F^p)和不可逆体积变形部分(F^d)^[134]

Figure 13. Decompose deformation gradient of porous crystal into elastic part (F^e) and irreversible volumetric part (F^p) and irreversible volumetric part(F^d)^[134]

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图 14 晶粒取向和应力三轴度对孔洞合并的临界状态变量的影响^[133]

Figure 14. Influence of grain orientation and stress triaxiality on critical state variables for void coalescence^[133]

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图 15 经典应力-应变曲线上塑性变形的3个阶段(Stage 1：均匀变形；Stage 2：非均匀变形；Stage 3：宏观热塑性失稳)^[51]

Figure 15. Three stages of plastic deformation appeared on classical stress-strain curve (Stage1: homogeneous deformation; Stage2: inhomogeneous deformation; Stage3: macroscopic thermoplastic instability)^[51]

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图 16 不同累积滑移速率变形 $\bar \gamma$ =0.05时的温升云图^[143]

Figure 16. Distribution of temperature increase when $\bar \gamma = 0.05$ \normalsize for different accumulated slip rates^[143]

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图 17 hcp单晶和多晶样品在10⁵ s^–1应变率下的绝热剪切局域化^[147]

Figure 17. Adiabatic shear localization of hcp single crystal and polycrystalline samples under 10⁵ s^–1 strain rate^[147]

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图 18 动态冲击载荷下6种织构材料中形成绝热剪切带的临界应变(V_peak = 20 m/s)^[147]

Figure 18. Critical strain of adiabatic shear band nucleated in 6 different texture materials under dynamic shock loading ( ${V_{{\rm{peak}}}} = 20\;{\rm{m}}/{\rm{s}}$ \normalsize) ^[147]

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A1 运动学符号说明

A1. Symbol description of kinematics

Symbols	Description
F(F^e, F^p, F^θ )	Deformation gradient including elastic, plastic and thermal components
L(L^e, L^p, L^θ )	Velocity gradient including elastic, plastic and thermal components
R^e	Rotation tensor
U^e	Right stretch tensor
α	Thermal expansion coefficient tensor

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A2 热力学符号说明

A2. Symbol description of thermodynamics

Symbols	Description	Symbols	Description
D_int	Intrinsic dissipation of the system	K₀	Bulk modulus at zero pressure
ψ	Helmholtz free energy	K′	Pressure derivative of bulk modulus
s	Entropy of the system	T_D	Debye temperature
T	Temperature	R	Molar gas constant
K_T	Isothermal bulk modulus	M_mol	Molar mass of the material
c_V	Heat capacity at constant volume	k_B	Boltzmann constant
Γ	Grüneisen coefficient	X_TN	Variables related to the lattice thermal vibration
q_n	Internal variables for microscopic defects such as dislocations in materials	X_TE	Variables related to the electron activation

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A3 塑性本构符号说明

A3. Symbol description of plastic constitution

Symbols	Description	Symbols	Description
λ^α	Mean spacing between obstacles	${{\,\rho}_{\rm{for}}^{\alpha} }$	Forest dislocation density
τ^α	Resolved shear stress	${{t_{\rm r}^{\alpha}}}$	The drag-dominated mean transit time between obstacles
Q^α	Activation energy	B	Viscous drag coefficient
g^α	Slip resistance	${{\dot{\rho }}_{\rm{nuc}}^{\alpha }}$	The nucleation rate
${{g}_{\rm{ath}}^{\alpha} }$	Athermal slip resistance	${{\dot{\rho }}_{\rm{hom}}^{\alpha }}$	The homogeneous nucleation rate
h_αβ	Hardening coefficient	${ {\dot{\rho }}_{\rm{het}}^{\alpha }}$	The heterogeneous nucleation rate
ρ^α	Total dislocation density	${{\dot{\rho }}_{\rm{mult}}^{\alpha }}$	The multiplication rate
${{\rho}_{\rm{m}}^{\alpha} }$	Mobile dislocation density	${{\dot{\rho }}_{\rm{trap}}^{\alpha }}$	The trapping rate
${{\rho}_{\rm{i}}^{\alpha} }$	Immobile dislocation density	${{\dot{\rho }}_{\rm{ann}}^{\alpha }}$	The annihilation rate
b^α	Burgers vector	d_a	Capture distance of annihilation
v^α	Velocity of mobile dislocations	${{t}_{\rm{w}}^{\alpha}}$	The thermal activation-dominated waiting time at a barrier
${ {\dot{\gamma }}^{\alpha }}$	Slip rate on slip system α

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A4 超弹性本构符号说明

A4. Symbol description of hyper-elastic constitution

Symbols	Description	Symbols	Description
I	Second-order unit tensor	$\widehat{{{E}}^{\rm{e}}}$	Isochoric strain in expanded configuration
E^e	Elastic Green–Lagrange strain	$\widehat{\widehat{{{E}}^{\rm{e}}}}$	Isochoric strain in configuration I
C^e	Elastic right Cauchy-Green tensor	$\overline {{{{E}}^{\rm{e}}}}$	Volumetric strain in configuration I
$\widehat{{{{F}}}^{\rm{e}}}$	Isochoric part of elastic deformation	S	Second Piola–Kirchhoff stress
$\overline {{{{F}}^{\rm{e}}}}$	Volumetric expansion

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A5 相变、孪晶与动态破坏符号说明

A5. Symbol description of phase transformation, twining and damage

Symbols	Description	Symbols	Description
F^tr	Deformation gradient of phase transformation	$S_{\rm{tw}}^{\beta}$	Twin resistance of twin system
v_p	Volume fraction of the parent phase	ρ_deb	Dislocationdebris density
v_t	Volume fraction of the new phase t	d_mfp	Dislocation mean free path related to the volume fraction of twin
v_N	Volume fraction of all new phases	${\varphi}$	Void volume fraction
f_t	Driving force of phase transformation	F^d	Volumetricpartofplastic deformation gradient in porous crystal plastic model
f^β	Volume fraction of twin	Y_r	Resistance of damage evolution
γ_tw	Characteristic shear strain of twining

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1. 实验装置与方案
1.1 实验装置与流程
1.2 实验材料及实验方案
2. 实验结果及分析
2.1 掺氢比对爆炸压力的影响
2.2 单一球形材料对爆炸压力的影响
2.3 组合球形材料对爆炸压力的影响
3. 结　论

高温、高压、高应变速率动态过程晶体塑性有限元理论模型及其应用

doi: 10.11858/gywlxb.20190874

作者简介: 刘静楠(1993－)，女，硕士，主要从事动态晶体塑性有限元研究. E-mail：jingnanliu@sjtu.edu.cn

通讯作者: 沈 耀(1972－)，男，博士，教授，主要从事晶体缺陷行为、力学性能及塑性变形的微观机制研究. E-mail：yaoshen@sjtu.edu.cn

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