﻿ 基于分形理论的雷电冲击土体等离子体通道发展研究
 上海理工大学学报  2023, Vol. 45 Issue (6): 610-619 PDF

1. 上海理工大学 环境与建筑学院，上海 200093;
2. 同济大学 土木工程学院，上海 200092

Development of the plasma channels in lightning-impacted soil based on the fractal theory
RAO Pingping1, WANG Qiqing1, WU Jian2
1. School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. College of Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: The plasma channel has an internal dynamic effect on the breaking process of the geotechnical body. For the development of plasma channel of the geotechnical body after the lightning strikes, the stochastic probability model of the geotechnical body breakdown channel was constructed from the viewpoint of the electric breakdown. The electric field was calculated by the Laplace equation, and the development probability was calculated by introducing the fractal theory. Then the simulation diagram of the plasma channel trajectory was obtained. The fractal dimension was calculated by the box dimension method to reflect the development process of the plasma channel. The influence of soil resistivity, lightning current amplitude and internal voltage drop on the fractal dimension was analyzed. The results showed that the fractal dimension decreased as the soil resistivity increased. With the increase of lightning current amplitude, the fractal dimension gradually increased, and the change of fractal dimension was more significant when the resistivity of soil was smaller. As the voltage drop inside the soil increased, the fractal dimension gradually increased. In this paper, the development of plasma channels within the geotechnical body was visualized by means of the two-dimensional images, and the development of plasma channels within the geotechnical body was quantitatively described by using fractal geometric relations, which can help predict the development law of the damage process of the geotechnical body and facilitates the physical nature of the electric breakdown of the geotechnical body.
Key words: plasma channel     electric breakdown     Laplace equation     fractal theory     box dimension method

 图 1 雷电冲击土体实例 Fig. 1 Examples of the lightning striking soil

Andres等[7]指出电脉冲在放电过程中击穿矿体形成的等离子体通道会发生爆炸，产生上升的冲击波，使得矿体内部产生“内伤”导致矿物破碎。Chen等[8]通过野外观测、实验和理论分析，对花岗岩遭受雷击后形成的闪电熔岩及熔化区域进行研究。Song等[9]采用有限单元法模拟雷击对埋地管道的损害影响。Charalambous等[10]基于实际案例进行试验，模拟雷电冲击电流通过埋地管道涂层缺陷进入管道内壁时，土体击穿电离对埋地管道的破坏影响。雷击后等离子体通道膨胀产生的电、热及冲击波效应对周围介质会产生破坏。对于击穿过程，根据等离子体通道的形成特点，固体击穿模型可以分为电击穿、热击穿、电–机械击穿和局部放电击穿[11]。已有充分的证据表明，电介质的击穿表现出分形特征[12-14]。其中，Yan等[14]在高压电脉冲处理无烟煤后利用电子显微镜等仪器分析了煤中微观裂纹的分形特征及等离子体通道膨胀现象。分形是指在局部和整体上具有自相似性的物质结构和规律。

1 基本原理和计算模型

1.1 模拟雷击点的描述

 图 2 等离子体通道示意图 Fig. 2 Schematic diagram of plasma channels

 图 3 电场计算的几何模型 Fig. 3 Geometric models for the electric field calculation
1.2 电场模型的数学描述

 $J = \sigma E$ (1)

 ${\boldsymbol{E}} = - \nabla \varphi$ (2)

 $\nabla \cdot J = \nabla \cdot \left( {\frac{1}{{{\rho _s}}}} \right)\nabla V = 0$ (3)

 $\left\{\begin{array}{l}-n\dfrac{1}{{\rho }_{s}}\nabla V=0,\text{ }\;\;\;边界\text{1}，边界\text{3}\\ V={V}_{{\rm{g}}},\text{ }边界\text{2}\\ V=0,\text{ }边界\text{4}\end{array}\text{ } \right.$

1.3 基于分形理论的土体击穿通道的预测

 图 4 等离子体通道发展的随机模型 Fig. 4 Stochastic model for the development of plasma channels

 图 5 模拟流程图 Fig. 5 Flow diagram of the simulation

a. 计算冲击点的电势Vg

 ${V_\text{g}} = I R$ (4)
 $R = \frac{{{\rho _s}}}{{2\text{π} r}}$ (5)
 $r = \sqrt {\frac{{I{\rho _s}}}{{2{\text{π}} {E_{\text{c}}}}}}$ (6)

b. 电场计算。

c. 选择新的击穿点。

 $p\left(P, P^{\prime}\right)=\left\{\begin{array}{ll} \dfrac{\left(\left|E\left(P, P^{\prime}\right)\right|-E_{\rm{c}}\right)^\eta {A_{\rm{rand}} }}{\displaystyle\sum_{P^{\prime}}\left(\left|E\left(P, P^{\prime}\right)\right|-E_{\rm{c}}\right)^\eta}, & E> E_{\rm{c}} \\ 0, & E \leqslant E_{\rm{c}} \end{array}\right.$ (7)

d. 修改边界条件。

 $U(l) = {U_0} - {E_{\text{i}}l}$ (8)

e. 计算等离子体通道的分形维数。

 $N(a) \propto {a^{ - D}}$ (9)

 $D = \mathop {\lim }\limits_{a = 0} \frac{{\ln\, N(a)}}{{\ln\, (1/a)}}$ (10)
2 对比验证

 图 6 分形维数与临界击穿场强的关系曲线 Fig. 6 Relationship between the fractal dimension and critical breakdown field strength

 图 7 不同生长概率指数下等离子体通道模拟图 Fig. 7 Simulation diagrams of plasma channels under different growth probability indexes
3 结果与分析 3.1 分形维数与土体电阻率的关系

 图 8 分形维数与土体电阻率的关系 Fig. 8 Relationship between the fractal dimension and soil resistivity

 图 9 不同土体电阻率下的等离子体通道模拟图 Fig. 9 Simulation diagram of plasma channels under different soil resistivities
3.2 分形维数与电流幅值的关系

 图 10 分形维数与电流幅值的关系 Fig. 10 Relationship between the fractal dimension and current amplitude

 图 11 不同电流幅值下的等离子体通道模拟图（ ρs=50 Ω·m） Fig. 11 Simulation diagrams of plasma channels under different current amplitudes（ρs=50 Ω·m）
3.3 分形维数与内部电压降的关系

 图 12 分形维数与内部电压降的关系 Fig. 12 Relationship between the fractal dimension and internal voltage drop

 图 13 等离子体通道模拟图 Fig. 13 Simulation diagram of plasma channels
4 结　论

a. 本文建立的随机概率模型能够有效模拟雷击土体内等离子体通道的发展过程，得到了直观的等离子通道图像，通过理论分析与数值模拟可分析土体及雷电各项参数对击穿通道的影响。

b. 在冲击电流幅值一定时，分形维数随着土体电阻率的增加而减少。而在土体电阻率一定时，分形维数随着冲击电流幅值增大而缓慢增大。等离子体通道曲折变化并伴有分支呈树枝状的情况，电阻率对击穿路径分支主干长度及分支发展情况的影响，与冲击电流幅值对其的影响相比较弱。

c. 在高电压作用下，分形维数随着内部电压降的增大而增大。在内部电压降增大过程中，等离子体通道分支发展情况越来越丰富，分形维数显著增大。

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