﻿ 热电材料断裂力学的若干进展 Some Advances in Fracture Mechanics of Thermoelectric Material

International Journal of Mechanics Research
Vol.07 No.02(2018), Article ID:25507,12 pages
10.12677/IJM.2018.72007

Some Advances in Fracture Mechanics of Thermoelectric Material

Qingnan Liu, Shenghu Ding*

School of Mathematical Statistics, Ningxia University, Yinchuan Ningxia

Received: Jun. 1st, 2018; accepted: Jun. 14th, 2018; published: Jun. 21st, 2018

ABSTRACT

Thermoelectric material is a kind of functional material which converts heat energy and electric energy to each other. Since the constitutive equations of thermoelectric materials are nonlinear and the corresponding mechanical analysis is very challenging, it is of great significance to sort out the research progress of thermoelectric materials in fracture mechanics. In this paper, the development status and application prospect of thermoelectric materials are introduced. The latest research progress of fracture mechanics of thermoelectric materials is reviewed, including the theoretical research and numerical study of thermoelectric materials fracture mechanics, and experimental research in fracture mechanics of thermoelectric materials. Finally, the future research on the fracture mechanics of thermoelectric materials is prospected.

Keywords:Thermoelectric Materials, Crack, Fracture Mechanics, Numeral Calculation

1. 引言

2.热电材料

2.1. 热电材料的发展与背景

2.2. 影响热电材料能量转换的因素

$ZT={\alpha }^{2}\sigma T/\lambda .$ (1)

Hicks等人提出热电材料的低维化有利于提高热电材料的热电优值，而电子和孔洞的存在会降低热电材料的优值 [9] 。优值Z受材料的种类、组分、掺杂比例和具体结构影响。优值与材料的塞贝克系数以及电导率、热导率有关，同时这三个参数都与温度相关。热电材料是一种非均匀功能梯度材料，现阶段提升热电材料转换能力的措施有以下几种 [10] ：首先，最直接的措施是提高热电材料的塞贝克系数。通过改变热电材料内部的构造以及化学合成成分来提高热电材料的塞贝克系数，具体方法有参杂、杂质替代等。通过理论分析及大量实验来寻找塞贝克系数较大的热电材料。其次，可以通过提高载流子的密度、提高载流子迁移率等方式来提高电导率 [11] ，但这些方法同时会使塞贝克系数降低，整体效果不太明显。最后，降低材料的热导率成为提高材料热电性能的主要方法。Bi2Te3基热电材料是最常见具有低的热导率和较高的热导率的热电材料，因而广泛应用于热电制冷和利用温差发电技术等领域。常见的热电材料还有硫属化铅基热电材料，用高能球磨法制备的BiSbTe纳米晶块体材料和p型MgAgSb基热电材料，三种热电材料的晶体结构图见图1 [12] 。因其晶体结构的不同而被应用于热电能量转换器和热电制冷等不同领域。例如市面上的一种移动型冰箱，具有无噪音体积小便于携带等优点，天气冷时还可变为保温器，这种冰箱应用的就是热电材料。还有利用温差产生电力来驱动手表等技术都是热电材料的发展及应用。

Zhang和Wang用解析法研究了对矩形热电板施加电和温度载荷，计算了裂纹尖端的电流密度和能量通量强度因子，结果显示可以通过增加裂纹尺寸来降低热导率 [13] 。化学家们将钠原子添加到一种具有有序晶格结构的热电材料中以增加材料的导电性，允许电子穿过材料的裂缝以减缓热量的传播 [14] 。在用保角映射函数和Mori-Tanaka方法研究椭圆体热电复合材料的性能时，发现通过增加填充率可以降低电导率和热导率 [15] 。其次高温高压也可以降低材料的电阻率得到极低的热导率，通过电学测试发现高压对材料的热电性能影响不大 [16] 。

3. 热电材料断裂力学的理论分析

${\sigma }_{ij}=\frac{K}{\sqrt{2\text{π}r}}{f}_{ij}\left(\theta \right).$ (2)

Figure 1. Crystal structure diagram of three different thermoelectric materials

${j}_{e}=-\gamma \Delta V-\gamma \epsilon \Delta T,$ (3)

$q=-\gamma \epsilon T\Delta V-\left(k-\gamma {\epsilon }^{2}T\right)\Delta V.$ (4)

${j}_{u}=q+{j}_{e}V$ (5)

Zhang和Wang用复变函数方法解决了无限介质中的裂纹问题 [20] ，得出了热通量，电通量和裂纹尖端附近的应力场表达式

${\sigma }_{xx}+{\sigma }_{yy}=4\mathrm{Re}\left[{\phi }^{\prime }\left(z\right)\right]+\frac{\mu {\alpha }^{\prime }\gamma }{2k}{f}_{1}\left(z\right)\stackrel{¯}{{f}_{1}\left(z\right)},$ (6)

${\sigma }_{yy}-{\sigma }_{xx}+2i{\sigma }_{xy}=2\left[z{\phi }^{″}\left(z\right)+{\varphi }^{\prime }\left(z\right)\right]+\frac{\mu {\alpha }^{\prime }\gamma }{2k}{f}_{2}\left(z\right)\stackrel{¯}{{{f}^{\prime }}_{1}\left(z\right)},$ (7)

$2\mu \left[{u}_{x}+i{u}_{y}\right]=\kappa \phi \left(z\right)-z\stackrel{¯}{{\phi }^{\prime }\left(z\right)}-\stackrel{¯}{\varphi \left(z\right)}+2\mu {\alpha }^{\ast }\int g\left(z\right)\text{d}z-\frac{\mu {\alpha }^{\prime }\gamma }{4k}{f}_{2}\left(z\right)\stackrel{¯}{{f}_{1}\left(z\right)}.$ (8)

$\varphi \left(z\right)+z\stackrel{¯}{{\varphi }^{\prime }\left(z\right)}+\stackrel{¯}{\phi \left(z\right)}=i\int \left({P}_{x}+i{P}_{y}\right)\text{d}s-\frac{\mu {\alpha }^{\prime }\gamma }{4k}{f}_{2}\left(z\right)\stackrel{¯}{{f}_{1}\left(z\right)}+\text{constant}.$ (9)

Huang对热电薄膜制冷器的热应力进行了研究，假定汤姆逊效应的冷却功率，对一维和二维温度场和应力场进行了分析，发现了热应力的分层结构和温度差引起的非耦合热弹性理论可以作为判定薄膜结构是否受热应力破坏，并对相邻层间的剪切应力提供了初步认识 [31] 。Hikage等人研究了热电发电器的热膨胀，得到了热电材料的热膨胀系数受材料内部晶体大小的影响的结论 [32] 。在此之后Ravi等人对高温热电材料的热膨胀性能进行了研究，证明了热电材料的热膨胀系数对材料的内部应力大小具有重要影响 [33] 。Chuan-Bin研究了无限远热电介质中圆弧裂纹的平面问题。结果表明，电通量强度因子和能量通量强度因子分别取决于远场电通量负荷和施加的总能量通量负荷 [34] 。Song等在研究热电材料裂纹问题中用复变函数方法，讨论裂纹对热电材料的转换效率的影响，分析了当电流流动和热通量分离时，热电转换效率可以高于在相同温度条件下一维热电偶的最大转换效率 [35] 。文中给出电流密度和热通量的表达式为

${J}_{y}=-\frac{{\sigma }_{i}}{2}\left[{f}^{\prime }\left(z\right)-\stackrel{¯}{{f}^{\prime }\left(z\right)}\right]={J}_{y}^{\infty }+\frac{a{J}_{y}^{\infty }}{\sqrt{2a\left(z-a\right)}},$ (10)

$\begin{array}{c}{J}_{Qx}-i{J}_{Qy}=\frac{{\sigma }^{2}\epsilon }{\kappa }f\left(a\right)\stackrel{¯}{f\left(a\right)}{f}^{\prime }\left(a\right)-\sigma \epsilon {C}_{2}{f}^{\prime }\left(a\right)+2\sigma {f}^{\prime }\left(a\right)\stackrel{¯}{f\left(a\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\sigma \epsilon {f}^{\prime }\left(a\right)\mathrm{Re}\left[{f}_{2}\left(a\right)\right]-2\kappa {{f}^{\prime }}_{2}\left(a\right).\end{array}$ (11)

${J}_{y}=-\frac{{\sigma }_{i}}{2}\left[{f}^{\prime }\left(z\right)-\stackrel{¯}{{f}^{\prime }\left(z\right)}\right]={J}_{y}^{\infty },$ (12)

$\begin{array}{c}{J}_{Qx}-i{J}_{Qy}=\frac{{\sigma }^{2}\epsilon }{\kappa }f\left(a\right)\stackrel{¯}{f\left(a\right)}{f}^{\prime }\left(a\right)-\sigma \epsilon {C}_{2}{f}^{\prime }\left(a\right)+2\sigma {f}^{\prime }\left(a\right)\stackrel{¯}{f\left(a\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-2\sigma \epsilon {f}^{\prime }\left(a\right)\mathrm{Re}\left[{f}_{2}\left(a\right)\right]-2\kappa {{f}^{\prime }}_{2}\left(a\right).\end{array}$ (13)

Wang采用有限厚度切口方法讨论了裂纹表面热电边界条件对材料的影响 [36] 。之后用复变函数法研究了填充在裂纹中的空气的导热率以及裂纹厚度对裂纹尖端场的影响 [37] 。如图3图4为热电材料中裂纹为a时裂纹表面的温度场和电势场，发现裂纹内的空气的导热性不能忽略，并且应该考虑裂纹的有限厚度的影响。

Ding等人研究了梯度非均匀材料的热弹性断裂，探究材料非均匀性参数和热阻对热应力强度因子的影响，热阻的变化会显著影响到裂纹表面的温度 [38] 。为更好的研究梯度层的热行为提供了依据。之后Ding和Li使用傅立叶变换、叠加原理和奇异积分方程等方法研究受热与机械应力载荷下部分绝缘的两种不同功能梯度材料间的界面裂纹，推导出温度和位移场的表达式。同时对材料的非均匀性参数，热应力强度因子和最小应变能密度进行了研究 [39] 。

4. 热电材料断裂的数值研究

Figure 2. Thermal shock function of different thickness plates

Figure 3. The temperature field on the crack surface and the lamellar thermoelectric materials of the crack extension line which the length is a

Figure 4. The electric potential field on the crack surface and the lamellar thermoelectric materials of the crack extension line which the length is a

5. 热电材料断裂力学的实验研究

Figure 5. Experiment data on cooling performance of monocrystalline silicon

(a) (b)

Figure 6. The inherent crack tip toughness data of two materials

6. 结束语

Some Advances in Fracture Mechanics of Thermoelectric Material[J]. 力学研究, 2018, 07(02): 54-65. https://doi.org/10.12677/IJM.2018.72007

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