固体介质中的冲击极化效应研究进展

常孟周; 范飞高; 唐恩凌; 贺丽萍; 郭凯; 崔化鹏; 陈闯; 韩雅菲; 王睿智; 曹洪祥

doi:10.11883/bzycj-2023-0473

固体介质中的冲击极化效应研究进展

doi: 10.11883/bzycj-2023-0473

辽宁省瞬态物理力学与能量转换材料重点实验室，沈阳理工大学，辽宁沈阳 110159

基金项目: 国家自然科学基金（12172232）

详细信息

作者简介:
常孟周（1990－　），男，博士，副教授，changmengzhou@163.com

通讯作者:
唐恩凌（1971－　），男，博士，教授，tangenling@126.com

中图分类号: O383
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- 文章访问数: 420
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- 被引次数: 10
出版历程
- 收稿日期: 2023-12-29
- 修回日期: 2024-05-20
- 网络出版日期: 2024-05-21
- 刊出日期: 2024-09-20

Research progress of shock induced polarization effect in solid mediums

Key Laboratory of Transient Physical Mechanics and Energy Conversion Materials of Liaoning Province, Shenyang Ligong University, Shenyang 110159, Liaoning, China

摘要

摘要: 冲击波在固体介质内传播时，内部电荷随冲击波作用向两极迁移形成电势差并对外输出电压/电流的极化效应称作冲击极化效应。针对晶体、金属、陶瓷以及聚合物等典型固体介质的冲击极化效应进行了系统梳理；总结了现阶段发展的冲击极化测试方法，分析了落锤/摆锤、SHPB、轻气炮以及炸药爆轰等加载方式诱发固体介质极化响应的差异；概述了有限元方法、分子动力学、近场动力学方法以及相场分析方法在固体介质冲击极化数值模拟领域的应用；围绕Allison理论、张裕恒理论、冲击挠曲电理论以及冲击波相关理论，总结了固体介质冲击极化的宏观唯象理论，并从固体介质微观结构、载流子输运模式、输运模型、迁移率以及态密度等方面说明了冲击极化的微观机理；分析了冲击极化效应在传感器、俘能器以及致动器等领域的应用前景，对固体介质冲击极化效应的发展趋势和需求进行了展望。
- 冲击极化 /
- 固体介质 /
- 载流子 /
- 冲击波
Abstract: When shock waves propagate in solid mediums, the internal charge carriers migrate to the electrodes under the action of shock waves to form electric potential and output voltage/current -shock induced polarization (SIP) effect. The development of SIP effect has challenged the traditional understanding of physical response of solid medium. In this paper, the SIP effects of typical solid mediums such as crystals, metals, ceramics and polymers are systematically reviewed. The SIP test methods developed at present are also summarized, and the characteristics of SIP induced by different loading methods such as drop hammer/pendulum, SHPB, light gas gun and explosive detonation are analyzed. The applications of finite element method, molecular dynamics, peridynamic and phase field analysis in the numerical simulation of SIP of solid mediums are summarized. The macroscopic phenomenology theories of SIP of solid mediums are summarized based on Allison theory, Zhang Yuheng theory, shock flexoelectric theory and shock wave theory, and the microscopic mechanism of SIP is explained considering the microstructure of solid medium, carrier transport mode, transport model, mobility and state density. Furthermore, the application prospect of SIP effect in sensors, energy harvesters and actuators are analyzed, also, the development trend and demand of SIP in solid media are also prospected.
- shock induced polarization /
- solid medium /
- charge carriers /
- shock wave

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高超声速飞行器热防护系统(thermal protection system, TPS)的结构材料必须具有轻质、耐高温、环境稳定等特点, 以满足长航程安全服役的要求。近年来, 陶瓷基复合材料在飞行器大面积热防护领域应用广泛, 这类材料可以比传统的金属TPS减重50%, 同时可以延长使用寿命, 降低制造成本^[1]。

基于化学气相沉积技术(chemical vapor infiltration, CVI)^[2]制造的碳纤维增韧碳化硅陶瓷基复合材料(C/SiC_S)以其低密度、抗氧化^[3]、耐高温^[4]等优异特性逐渐得到重视, 目前已经成功应用于飞行器的鼻锥和机翼前缘的防热结构中。然而, 与金属材料不同, 陶瓷基材料不具备产生塑性变形的能力, 材料的内部分层、基体裂纹、纤维断裂等破坏是其吸收冲击能量的主要方式, 这些损伤会降低材料的力学性能, 从而严重威胁材料的服役安全。因此, 为了飞行器内部结构及设备在服役过程中的安全性考虑, 这类结构材料在冲击载荷作用下的破坏行为及其数值预测一直是众多学者关注的热点。

目前, 针对平纹编织C/SiC(2D-C/SiC)复合材料的研究并不多见, 已有的相关文献多数仅限于对材料简单力学性能的实验研究^[5-6]。杨扬等^[7]基于电炮加载装置驱动Mylar飞片完成了冲击速度在3 400~9 300 m/s范围内2D-C/SiC材料的超高速撞击实验, 并对材料损伤演变和破坏特征进行了分析研究。然而, 关于材料在低速冲击载荷作用下的实验及其数值模拟目前尚未发现。

材料的各向异性特征使得在对其进行模拟计算时面临着巨大的挑战, R.A.Clegg等^[8]提出的正交各向异性模型可以较好地描述复合材料在冲击载荷作用下的力学响应, 已经在Kevlar-Epoxy复合材料的冲击模拟中得到了成功的应用^[9]。然而, 该模型重点关注了材料的非线性应变硬化行为, 在模拟C/SiC这种典型的脆性材料时无法得到理想的数值结果。本文中在已完成的球形弹丸低速冲击2D-C/SiC薄板的实验基础上, 引入材料的各向异性本构模型, 基于Autodyn商用软件对2D-C/SiC复合材料在钢球弹丸冲击作用下的数值模拟问题进行研究。

1. 2D-C/SiC材料低速冲击实验

基于空气炮加载装置对2D-C/SiC复合材料薄板在钢球撞击作用下的力学响应问题进行了实验研究。在实验过程中, 使用高速摄影技术记录了2D-C/SiC薄板受撞击后碎片云团的发展过程。

1.1 实验材料与计划

实验中选用的2D-C/SiC复合材料由西北工业大学超高温复合材料重点实验室研制。首先, T300碳纤维正交编织成碳布预制体, 在低压炉内基于CVI技术沉积6次SiC基体, 得到孔隙率约15%、体积分数约45%的2D-C/SiC材料, 而后基于CVI沉积2次SiC涂层。最终加工成型后, 切割为大小115 mm×115 mm、厚度3 mm、密度约2.11 g/cm³的正方形薄板作为目标靶板。实验中选用5、6 mm两种直径的普通钢球作为冲击弹丸。具体实验计划及主要结果列于表 1中, 表中D表示弹丸直径, v_p表示弹丸的冲击速度, E表示弹丸的动能。

表 1 低速冲击实验计划

Table 1. Low-speed impact experimental scheme

实验编号	D/mm	v_p/(m·s^-1)	E/J	实验结果
1	5	79.0	1.61	未穿透
2	5	92.7	2.22	未穿透
3	5	98.5	2.51	未穿透
4	5	130.0	4.37	穿透
5	6	144.0	9.26	穿透
6	6	147.0	9.65	穿透
7	6	211.0	19.89	穿透
8	6	218.0	21.23	穿透
9	6	219.0	21.43	穿透

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1.2 碎片云团结构

限于实验条件, 高速摄像机置于靶板斜后方位置, 图 1给出了试样8的高速摄影视频截图。由图 1可见, 2D-C/SiC碎片云团的发展过程大致分为2个阶段:(1)簇状飞散(图 1(a))。弹丸接触靶板时, 在作用点局部区域迅速发生纤维断裂、基体粉碎等多种形式破坏, 大量固体颗粒成簇状形式堆积飞出; (2)两区域形成(图 1(b)~(d))。簇状颗粒群运动发展, 逐渐在碎片云团中形成了界限清晰的两区域结构, 即柱状区和分散区。在柱状区内, 颗粒分布密度较高、颗粒尺寸较小、运动速度较快, 并沿轴线向前运动; 在分散区内, 颗粒分布较为稀疏、颗粒尺寸较大、运动速度较慢, 且沿一定的飞散角辐射飞出。2D-C/SiC碎片云结构与以往的金属材料及树脂基复合材料截然不同, 是其脆性特征的典型表现。在数值模拟中, 建立合理的本构模型对材料特有的力学响应特征进行模拟计算是本文研究的重点。

图 1 碎片云团发展过程

Figure 1. Expansion of the debris cloud

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2. 耦合的各向异性材料模型

2.1 正交各向异性弹性模型

复合材料在冲击载荷作用下的力学响应过程十分复杂, 通常是诸如体积响应、材料各向异性等多种因素的耦合。在数值计算中, 不仅球应变会影响偏应力的计算, 偏应变同样会在球应力的计算中起作用, 通常需要面临求解状态方程和本构关系这两个高度耦合的子模型^[10]。传统的计算方法是将这两个子模型独立计算, 然而这样的过程较为繁琐, C.E.Anderson等^[11]首先提出了一种适用于求解各向异性材料中多重响应耦合问题的计算方法。本文中也以此方法为基础, 建立如下的计算模型。

2D-C/SiC是典型的正交各向异性复合材料。基于塑性力学, 全应变可分解为与体积变化相关的球应变ε₀和与形状变化相关的偏应变之和。此时, 2D-C/SiC的本构关系可用如下全应力应变关系表述为:

$\left[\begin{array}{c} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{array}\right]=\left[\begin{array}{cccccc} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{array}\right]\left[\begin{array}{c} \varepsilon_{0}+\varepsilon_{11}^{\mathrm{d}}-1 \\ \varepsilon_{0}+\varepsilon_{22}^{\mathrm{d}} \\ \varepsilon_{0}+\varepsilon_{33}^{\mathrm{d}} \\ \varepsilon_{23} \\ \varepsilon_{31} \\ \varepsilon_{12} \end{array}\right]$

由于各向同性压力P是3个正应力的平均值的相反数, 因此由上式可以推导得出:

$P=-\frac{1}{3}\left(C_{11}+C_{22}+C_{33}+2 C_{12}+2 C_{13}+2 C_{23}\right) \varepsilon_{0}-\sum\limits_{i=1}^{3} \frac{1}{3}\left[C_{1 i}+C_{2 i}+C_{3 i}\right] \varepsilon_{i i}^{\mathrm{d}}$

(1)

从(1)式中可以清晰地看出, 与各向同性材料中球应力仅依赖于体积应变不同, 各向异性材料中偏应变也会在一定程度上影响球应力的计算。其中ε₀一项的系数可称为材料的等效体积模量, 记为A。

2.2 正交各向异性失效模型

失效模型主要包括失效初始化和后失效响应两个阶段。在失效初始化阶段, 采用最大应力判断准则, 即预先设置材料在3个拉伸方向和3个剪切方向的失效应力, 当任一方向的应力值达到失效应力时, 材料即在该方向失效, 进入后失效响应阶段。

对于材料的后失效响应, 需要在计算开始前按照如下准则进行相关设置:当材料在任一方向进入后失效阶段时, 材料在该方向的承载能力瞬间丧失, 但在其他方向的承载能力保持不变; 同时, 材料在3个剪切方向的承载能力下降^[8]。例如, 假设1方向为薄板厚度方向, 2、3方向为面内的2个正交方向。那么, 当1方向拉伸应力大于拉伸失效应力或12(或13)方向剪切应力大于剪切失效应力时, 材料会发生分层破坏。此时, 材料在1方向应力立即置为0, 不再具有承载能力, 但在面内的2个方向承载能力不变; 同时由于分层破坏的发生, 材料在剪切方向的承载能力也将有所下降, 需要对刚度矩阵中的剪切刚度系数进行退化, 可对材料的刚度矩阵作如下修正(α为刚度退化因子):

$\left[\begin{array}{c} 0 \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{array}\right]=\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & C_{22} & C_{23} & 0 & 0 & 0 \\ 0 & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \alpha C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \alpha C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & \alpha C_{66} \end{array}\right]\left[\begin{array}{l} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \varepsilon_{23} \\ \varepsilon_{31} \\ \varepsilon_{12} \end{array}\right]$

具体的材料参数及失效设置分别见表 2和表 3。

表 2 正交弹性模型材料参数

Table 2. Parameters of orthotropic elastic model

刚度系数									体积模量
C₁₁/GPa	C₂₂/GPa	C₃₃/GPa	C₁₂/GPa	C₂₃/GPa	C₃₁/GPa	G₁₂/GPa	G₂₃/GPa	G₃₁/GPa	A/GPa
19	120.5	120.5	7.8	20	7.8	8.8	23	8.8	36.8

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表 3 正交失效模型材料参数

Table 3. Parameters of orthotropic failure model

失效初始化
σ_fail-11/MPa	σ_fail-22/MPa	σ_fail-33/MPa	σ_fail-12/MPa		σ_fail-23/MPa	σ_fail-31/MPa
50	455	455	40		165	40

后失效响应
刚度退化因子	失效模式
0.2	11失效 11方向拉伸应力置0	22失效 22方向拉伸应力置0	33失效 33方向拉伸应力置0	12失效 11方向拉伸应力置0	23失效 11方向拉伸应力置0	31失效 11方向拉伸应力置0

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3. 数值模拟及分析

3.1 计算模型

由于正撞击的对称性, 基于Autodyn软件建立如图 2所示的二维半模型计算。2D-C/SiC靶板选用SPH求解器; 钢弹丸steel 4340基于FEM求解器计算; 弹丸与靶板之间设置接触, 接触间距0.01 mm; 在靶板边缘3 mm×4 mm范围内施加全局固定边界, 即限定此区域内的粒子在x和y方向速度恒为0。

图 2 Autodyn计算模型

Figure 2. Computational model in Autodyn

下载: 全尺寸图片幻灯片

下面通过对碎片云结构和典型破坏形式等实验结果的数值验证, 证明该模型对描述2D-C/SiC材料在冲击载荷作用下力学响应的合理性和可行性。

3.2 数值合理性的实验对比

3.2.1碎片云团结构的数值验证

图 3 (b)、(c)分别给出了试样8的数值模拟结果与对应的速度矢量图。通过对比可以看出, 实验结果和数值模拟结果吻合较好, 具有以下相似特征:(1)碎片云团均呈现明显的双区域结构, 即均存在沿冲击轴线方向的柱状区和轴线两侧的飞散区, 这是2D-C/SiC在冲击载荷作用下的特殊响应特征; (2)从速度矢量图可以看出, 柱状区粒子运动方向均为沿轴线方向; 而飞散区内的粒子速度方向与轴线有大小不同的夹角。这与实验结果是完全一致的, 表明该模型可以较好地表征2D-C/SiC在弹丸撞击下碎片云团的结构特征。

图 3 试样8计算结果

Figure 3. Simulation results of sample 8

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3.2.2 B扫描结果的数值验证

B扫描结果显示的实际上是试样内部损伤的二维截面图。图 4(a)给出试样8的B扫结果, 其中虚线区域的黄色部分表示试样; 图 4(b)~(c)给出工况8靶板扩展稳定后内部损伤的模拟计算结果。

图 4 B扫描结果与模拟结果对比图

Figure 4. Comparison between B-scan and simulation results

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从图 4看出:(1)数值模拟与B扫描显示的穿孔形状十分相似, 穿孔直径近似为弹丸直径(6 mm); (2)从B扫结果看, 靶材背面显示黄色较正面偏淡, 这是由于材料在靠近背面的区域损伤严重, 导致超声波信号无法完全被探头接收。相似的现象可以在数值模拟中显示得更加清晰, 图 4(c)给出了靶材穿孔区域的局部放大图。由图可见, 与正面相比, 靶材背面存在明显的材料分层与剥落损伤, 这与实验结果比较吻合。上述相似性也表明该模型可以较好地表征2D-C/SiC在弹丸撞击下的破坏特征。

3.2.3碎片云轴向速度的数值验证

碎片云的轴向速度是表征碎片云团发展状态的重要指标。实验中, 在靶板表面标记了刻度, 据此可以根据高速摄影记录计算出碎片云团柱状区前端的运动速度, 而将此速度视为碎片云轴向运动速度v_axis。同样, 在数值模拟中, 取靶板自由面与冲击轴线的交点为特征点近似柱状区端点。选取靶板穿透工况中冲击速度相差较大的4种情况进行验证, 图 5给出了不同冲击速度v_axis下随时间的变化历程。

图 5 不同冲击速度下特征点轴向速度历程

Figure 5. Axial velocity histories under different impact velocities

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从图 5可以看出, 在弹丸接触靶板后的短时间内, 特征点的轴向速度迅速增加并逐渐趋于稳定。将特征点最终的稳定速度视为对v_axis的近似, 图 6给出了实验数据与数值计算的对比结果。可以看出, 二者的吻合度较好, 并且随着冲击速度的增高, 相对误差逐渐减小, 其中冲击速度219 m/s时二者的相对误差仅为2.25%, 表明该模型可以较好地表征2D-C/SiC在冲击载荷作用下碎片云团的速度特性。

图 6 特征点轴向速度对比结果

Figure 6. Comparison of the axial velocities of simulation and experiment

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4. 极限侵彻深度预测模型

尽管碎片云团中柱状区内的粒子运动速度较高, 但由于颗粒质量较小, 对机体的损伤程度有限。碎片云团中对机体威胁较大的仍是穿透靶板的弹丸, 因此可以通过对弹丸剩余速度的分析研究, 建立合理有效的预测公式评估C/SiC靶板的抗冲击性能。以上通过3方面对比验证了文中使用的材料模型及相关参数的正确性和合理性, 下面将基于此模型对弹丸侵彻下C/SiC靶板的抗冲击特性进行预估。

考虑同一弹丸(直径6 mm)以不同冲击速度对不同厚度靶板进行撞击。设置靶板厚度δ为1、2、3、4、5 mm, 弹丸冲击速度v_i为50、100、150、200、250、300、350、400 m/s, 计算得到弹丸剩余速度变化规律如图 7所示。可以看出:(1)对同一厚度的靶板, 剩余速度随着冲击速度的减小不断减小; 而对相同冲击速度, 剩余速度会随着靶板厚度的减小而不断增加; (2)剩余速度在速度较低靶板较厚时出现了负值, 这是由于弹丸在这些工况下未能穿透靶板, 出现回弹现象造成的。

图 7 剩余速度随冲击速度与靶板厚度的变化规律

Figure 7. Variation of the residual velocity with impact velocity and plate thickness

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评估材料抗冲击性能比较关心的重要参数是靶板的极限穿透速度, 即上述曲面与剩余速度为0的平面的交线。然而, 如果固定靶板厚度分别观察剩余速度随冲击速度的变化趋势(图 8), 可以看出在临界穿透状态附近(图中虚线), 剩余速度变化曲线斜率陡然增高, 说明在临界穿透状态附近剩余速度变化较剧烈; 而当冲击速度高于极限穿透速度之后, 弹丸剩余速度变化基本稳定, 呈准线性增长趋势。因此, 如果直接对剩余速度曲面进行拟合来求解极限穿透速度, 会导致在临界穿透状态附近的计算误差较大。

图 8 不同厚度靶板剩余速度变化图

Figure 8. Variation of residual velocity with impact velocity for different plate thicknesses

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为了解决上述问题, 采用分步拟合方法:(1)假设靶板无限厚, 首先固定冲击速度, 分别拟合得到弹丸剩余速度随靶板厚度的关系, 进而求解在此冲击速度下弹丸的极限侵彻深度; (2)其次基于Poncelet型侵彻深度预测模型^[12], 利用得到的极限侵彻深度T与弹丸冲击速度v_i进行拟合, 得到如下关系:

$T=0.0005362 \ln \left(1+0.756 v_{\mathrm{i}}^{2}\right)$

此时, 便可以在已知靶板厚度的情况下, 通过反解上式预测弹丸的极限穿透速度。同时, 在工程设计中, 如果工程人员已经根据飞行器的服役环境预估出外来冲击物的撞击速度, 那么同样可以利用上式预估弹丸的极限侵彻深度, 也就可以为防护板厚度参数的确定提供一定的参考。

5. 结论

基于Autodyn软件选取正交各向异性材料模型, 推导了相关的材料参数, 并从碎片云团结构、B扫描内部损伤形貌和碎片云轴向速度三个方面对比了实验与模拟结果, 表明参数的合理性和正确性, 较好地描述了2D-C/SiC在钢球弹丸撞击下的主要损伤特征和力学特性。在此基础上, 基于数值模拟计算结果推导得到了钢球弹丸撞击2D-C/SiC材料的极限侵彻深度预测公式, 这可为TPS结构设计提供一定的工程参考价值。

图 1 不同厚度和电极形状6061Al的极化强度(P)与应变梯度^[13]

Figure 1. Polarization strength (P) and strain gradient of specimens with different thickness and electrodes^[13]

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图 2 6061Al薄板的冲击极化电压时程^[14]

Figure 2. Time history curves of SIP voltages of 6061Al thin plate^[14]

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图 3 冲击波前/后的电力特性^[17]

Figure 3. Electric conditions before and after the shock wave^[17]

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图 4 简正模式下矩形PZT铁电陶瓷的去极化示意图^[21]

Figure 4. Schematic of rectangular PZT depolarization under normalized mode^[21]

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图 5 极化BNT-BA-0.01NN陶瓷的介电特性和热感应电荷密度与温度的关系^[25]

Figure 5. Temperature-dependent dielectric properties and thermal induced charge density of poled BNT-BA-0.01NN ceramics^[25]

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图 6 未极化BNT-BA-0.01NN陶瓷在70 ℃时的极化特性^[25]

Figure 6. Polarization characteristics of non-polarized BNT-BA-0.01NN ceramics at 70 °C^[25]

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图 7 PMMA薄板的相对介电常数-应变曲线与极化电压时程^[14]

Figure 7. Relative dielectric constants and time history curves of SIP voltages of PMMA thin plate^[14]

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图 8 层内电压U_int对PDMS冲击力电响应的影响^[33]

Figure 8. Effect of U_int on the electromechanical characteristics of PDMS^[33]

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图 9 多脉冲下PDMS的极化电压^[33]

Figure 9. Polarization voltages of PDMS under multiple pulses^[33]

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图 10 试件结构与支撑^[39]

Figure 10. Specimen structure and support^[39]

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图 11 GFRP冲击极化实验示意图^[38]

Figure 11. Schematic of SIP of GFRP^[38]

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图 12 6061Al试件与实验系统^[13]

Figure 12. Schematic diagram of physical object and experimental system of 6061Al specimen^[13]

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图 13 PDMS动态力电响应测试系统

Figure 13. Dynamic electromechanical response testing system of PDMS

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图 14 不同冲击速度下BT块体的电压时程^[40]

Figure 14. Voltage time histories for BT bulk samples induced by the shock wave at different velocities^[40]

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图 15 基于一级轻气炮加载的固体介质极化特性测试系统^[14]

Figure 15. Polarization characteristic test system of solid medium based on one-stage light gas gun^[14]

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图 16 平面实验法原理示意图^[32]

Figure 16. Schematic diagram of the experimental principle of the planar test method^[32]

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图 17 爆炸冲击下铝的电输出特性测试装置^[41]

Figure 17. Test device for electrical output characteristics of aluminum under explosion impact^[41]

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图 18 层状PMMA的冲击极化效应测试装置^[42]

Figure 18. SIP test device of layered PMMA^[42]

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图 19 改进型冲击极化实验装置^[43]

Figure 19. Improved experimental device for SIP^[43]

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图 20 试件与测试电路^[44]

Figure 20. Specimen and test circuit^[44]

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图 21 基于泵浦-探针法的EFM测量装置原理图^[46]

Figure 21. Schematic diagram of the tip-synchronized pump-probe tr-EFM apparatus^[46]

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图 22 平均应变梯度的时空分布规律

Figure 22. Spatial and temporal distribution of average strain gradient

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图 23 棒体的力电耦合特性数值模拟^[47]

Figure 23. Numerical simulation of electromechanical coupling characteristics of rods^[47]

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图 24 PFM中压电材料BT产生的电弹性场^[45]

Figure 24. Electroelastic field generated by PFM acting on piezoelectric material BT^[45]

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图 25 不同电场下载流子迁移率^[48]

Figure 25. Carrier mobility under different electric fields^[48]

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图 26 计算中使用的超晶胞单元与应变剖面^[49]

Figure 26. Supercell and strain profile used in the calculations^[49]

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图 27 固定D_e条件下STO超晶胞中的电荷分布与力分布^[51]

Figure 27. Charge-density distribution and force distribution in STO supercell at fixed D_e^[51]

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图 28 金属Pt/Pd-甲烷分子结构及范德华能-距离曲线^[52]

Figure 28. Vander Waals energy-distance curves and structures of metal Pt/Pd-CH₄^[52]

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图 29 薄膜和厚膜挠曲电系数的温度依赖性^[53]

Figure 29. Temperature dependency of the flexoelectric coefficients in the thin and thick films^[53]

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图 30 基于核壳模型的分子动力学模拟结果^[57]

Figure 30. Molecular dynamics simulation results based on core-shell model^[57]

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图 31 不同应变下极化分布^[58]

Figure 31. Polarization distribution of BT under different strains^[58]

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图 32 考虑挠曲电效应时薄膜界面处不同取向的单位错对单畴结构的影响^[62]

Figure 32. Influence of single dislocations with different orientations at the film interface on the single domain structure when considering the flexoelectric effect^[62]

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图 33 考虑挠曲电效应时各构型畴中不同取向周期位错对90°畴结构影响示意图^[62]

Figure 33. Schematic of the influence of dislocations with different orientations in each configuration domain on the 90° domain structure considering the flexoelectric effect^[62]

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图 34 多种效应下PZT铁电薄膜电畴变化的阈值^[63]

Figure 34. Threshold of domain change in PZT ferroelectric thin films under multiple effects^[63]

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图 35 冲击诱发电位移原理图^[66]

Figure 35. Schematic of shock induced electric displacement^[66]

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图 36 基于张裕恒理论的冲击极化机理^[70]

Figure 36. Mechanism of SIP based on “Zhang Yuheng” theory^[70]

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图 37 冲击载荷诱发BT电极化机理示意图与实验结果^[74]

Figure 37. Mechanism and experimental results of SIP of BT^[74]

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图 38 冲击下偶极子极化示意图^[75]

Figure 38. Diagram of dipole polarization under impact^[75]

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图 39 铁电材料的冲击过程^[16]

Figure 39. Schematic diagram of the ferroelectric materials under impact^[16]

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图 40 孤子、极化子以及双极化子结构示意图^[83]

Figure 40. Schematic diagram of the structure of solitons, polarons and bipolaron^[83]

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图 41 线性离子链

Figure 41. Linear ionic chain

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图 42 包含黏性介质的有效偶极子（哑铃）模型^[76]

Figure 42. Effective dipole (dumbbell) model with viscous medium^[76]

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图 43 分支聚合物结构示意图^[86]

Figure 43. Structure of branching polymer^[86]

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图 44 压电/挠曲电环形传感器结构与动态响应^[102]

Figure 44. Structure and dynamic response of piezoelectric/flexoelectric ring sensor^[102]

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图 45 挠曲电效应传感器的典型结构^[103]

Figure 45. Typical structure of flexoelectric effect sensor^[103]

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图 46 俘能器结构示意图^[104]

Figure 46. Schematic diagram of the energy harvester structure^[104]

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图 47 线电极/梁/面电极致动器^[105]

Figure 47. Actuator with line electrode/beam/surface electrode structure^[105]

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表 1 典型动态加载技术的特点

Table 1. Characteristics of typical dynamic loading technology

实验方法	应变率/s⁻¹	响应特性	主要应用
落锤/摆锤	1～10²	弹塑性变形，伴随损伤破坏	测定抗冲击强度、变形与吸能
SHPB	10²～10⁴	弹塑相变，黏性与应变率效应明显	确定动态本构模型
轻气炮	10⁴～10⁶	出现流体性态，考虑密度与可压缩性	确定状态方程
爆轰	>10⁶	呈流体力学状态，伴有熔融与汽化	可计算爆轰波，确定状态方程

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参考文献(106)

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