# 压电材料：标准简介

2014年 10月 2日

### 两种方程形式：应变-电荷型和应力-电荷型

#### 应变-电荷型

\begin{array}{l}
\bf{S}=s_E \bf{T}+d^T \bf{E} \\[3mm]
\bf{D}=d \bf{T}+\epsilon_0 \epsilon_{rT} \bf{E}
\end{array}

\begin{array}{ll}
\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
s_{E11} & s_{E12} &s_{E13} &s_{E14} &s_{E15} &s_{E16}\\
s_{E21} & s_{E22} &s_{E23} &s_{E24} &s_{E25} &s_{E26}\\
s_{E31} & s_{E32} &s_{E33} &s_{E34} &s_{E35} &s_{E36}\\
s_{E41} & s_{E42} &s_{E43} &s_{E44} &s_{E45} &s_{E46}\\
s_{E51} & s_{E52} &s_{E53} &s_{E54} &s_{E55} &s_{E56}\\
s_{E61} & s_{E62} &s_{E63} &s_{E64} &s_{E65} &s_{E66}\\
\end{array}
\right)
\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
+
\left(
\begin{array}{lll}
d_{11} & d_{21} & d_{31} \\
d_{12} & d_{22} & d_{32} \\
d_{13} & d_{23} & d_{33} \\
d_{14} & d_{24} & d_{34} \\
d_{15} & d_{25} & d_{35} \\
d_{16} & d_{26} & d_{36} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\left(
\begin{array}{l}
D_{x} \\
D_{y} \\
D_{z} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
d_{11} & d_{12} &d_{13} & d_{14} & d_{15} & d_{16}\\
d_{21} & d_{22} &d_{23} & d_{24} & d_{25} & d_{26}\\
d_{31} & d_{32} &d_{33} & d_{34} & d_{35} & d_{36}\\
\end{array}
\right)\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
+
\epsilon_0 \left(
\begin{array}{lll}
\epsilon_{rT11} & \epsilon_{rT12} & \epsilon_{rT13} \\
\epsilon_{rT21} & \epsilon_{rT22} & \epsilon_{rT23} \\
\epsilon_{rT31} & \epsilon_{rT32} & \epsilon_{rT33} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\end{array}

#### 应力-电荷型

\begin{array}{l}
\bf{T}=c_E \bf{S}-e^T \bf{E} \\[3mm]
\bf{D}=e \bf{S}+\epsilon_0 \epsilon_{rS} \bf{E}
\end{array}

\begin{array}{ll}
\left(
\begin{array}{l}
T_{xx} \\
T_{yy} \\
T_{zz} \\
T_{yz} \\
T_{xz} \\
T_{xy} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
c_{E11} & c_{E12} &c_{E13} &c_{E14} &c_{E15} &c_{E16}\\
c_{E21} & c_{E22} &c_{E23} &c_{E24} &c_{E25} &c_{E26}\\
c_{E31} & c_{E32} &c_{E33} &c_{E34} &c_{E35} &c_{E36}\\
c_{E41} & c_{E42} &c_{E43} &c_{E44} &c_{E45} &c_{E46}\\
c_{E51} & c_{E52} &c_{E53} &c_{E54} &c_{E55} &c_{E56}\\
c_{E61} & c_{E62} &c_{E63} &c_{E64} &c_{E65} &c_{E66}\\
\end{array}
\right)
\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
+
\left(
\begin{array}{lll}
e_{11} & e_{21} & e_{31} \\
e_{12} & e_{22} & e_{32} \\
e_{13} & e_{23} & e_{33} \\
e_{14} & e_{24} & e_{34} \\
e_{15} & e_{25} & e_{35} \\
e_{16} & e_{26} & e_{36} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\left(
\begin{array}{l}
D_{x} \\
D_{y} \\
D_{z} \\
\end{array}
\right)
=
\left(
\begin{array}{llllll}
e_{11} & e_{12} &e_{13} & e_{14} & e_{15} & e_{16}\\
e_{21} & e_{22} &e_{23} & e_{24} & e_{25} & e_{26}\\
e_{31} & e_{32} &e_{33} & e_{34} & e_{35} & e_{36}\\
\end{array}
\right)
\left(
\begin{array}{l}
S_{xx} \\
S_{yy} \\
S_{zz} \\
S_{yz} \\
S_{xz} \\
S_{xy} \\
\end{array}
\right)
+
\epsilon_0 \left(
\begin{array}{lll}
\epsilon_{rS11} & \epsilon_{rS12} & \epsilon_{rS13} \\
\epsilon_{rS21} & \epsilon_{rS22} & \epsilon_{rS23} \\
\epsilon_{rS31} & \epsilon_{rS32} & \epsilon_{rS33} \\
\end{array}
\right)
\left(
\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z} \\
\end{array}
\right)
\\
\end{array}

\begin{array}{rl}
\left(
\begin{array}{cccccc}
c_{E11} & c_{E12} &c_{E13} & c_{E14} & 0 & 0\\
c_{E12} & c_{E11} &c_{E13} & -c_{E14} &0 & 0\\
c_{E13} & c_{E13} &c_{E33} & 0 & 0 & 0\\
c_{E14} & -c_{E14} & 0 & c_{E44} & 0 & 0 \\
0 & 0 & 0 & 0 & c_{E44} & c_{E14}\\
0 & 0 & 0 & 0 & c_{E14} & \frac{1}{2}\left(c_{E11}-c_{E12}\right)\\
\end{array}
\right)
&
\left(
\begin{array}{cccccc}
s_{E11} & s_{E12} &s_{E13} & s_{E14} & 0 & 0\\
s_{E12} & s_{E11} &s_{E13} & -s_{E14} &0 & 0\\
s_{E13} & s_{E13} &s_{E33} & 0 & 0 & 0\\
s_{E14} & -s_{E14} & 0 & s_{E44} & 0 & 0 \\
0 & 0 & 0 & 0 & s_{E44} & 2 s_{E14}\\
0 & 0 & 0 & 0 & 2 s_{E14} & 2\left(s_{E11}-s_{E12}\right)\\
\end{array}
\right)
\\
\left(
\begin{array}{cccccc}
e_{11} &-e_{11} & 0 & e_{14} & 0 & 0 \\
0 & 0 & 0 & 0 & -e_{14} & -e_{11}\\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)
&
\left( \begin{array}{cccccc}
d_{11} & -d_{11} & 0 & d_{14} & 0 & 0 \\
0 & 0 & 0 & 0 & -d_{14} & -2d_{11} \\
0 & 0 & 0 & 0 & 0 & 0\\
\end{array}
\right)
\\
\left(
\begin{array}{ccc}
\epsilon_{rS11} & 0 & 0 \\
0 & \epsilon_{rS11} & 0 \\
0 & 0 & \epsilon_{rS33} \\
\end{array}
\right)
&
\left(
\begin{array}{ccc} \epsilon_{rT11} & 0 & 0 \\
0 & \epsilon_{rT11} & 0 \\
0 & 0 & \epsilon_{rT33} \\
\end{array}
\right)
\\
\end{array}

### 两套标准：1949 IRE 和 1978 IEEE

• IEEE 1978标准
• 这是大多数文献中对除石英之外其他材料最常参考的标准。有时它也会被用来明确石英的材料属性，举个例子，B. A. Auld 所著的 Acoustic Fields and Waves in Solids 中用的就是这一标准。
• IRE 1949标准
• 各类文献通常会采用此标准中对石英材料属性的定义。

Quartz Page中介绍了许多常见的晶形，分别以 m、r、s、x、z 和 a 表示，并且还有专门的页面介绍对应晶形的 密勒指数，这非常有助于我们辨别各种晶形。各类标准通常会用晶形来进行坐标轴取向，下图即为 1978 和 1949 中所用的坐标轴。请注意图中同时有左对映和右对映石英的坐标轴图片。

IRE 1949 标准

IEEE 1978 标准

sE14

+

+

cE14

+

+

d11

+

+

d14

+

+

e11

+

+

e14

+

+

### 晶体切片的两个定义

Standard

AT Cut Definition

1949 IRE

(YXl) 35.25°

1978 IEEE

(YXl) -35.25°

### 下一步

#### 评论 (12)

2016-05-09

2016-05-16

2017-02-21

2017-02-24

##### 宇航 秦
2017-02-21

Email: support@comsol.com

2017-09-20

2018-01-18

##### 宇航 秦
2018-01-25

Email: support@comsol.com

2018-05-09

##### Tengyue Gao
2018-10-19

Email: support@comsol.com

##### Xuanwei Wei
2023-06-14

1.应力电荷型的方程错了吧，里面T=CS-eE而不是T=CS+eE？
2.第一张图片丢失左边D的矩阵表达式

##### Hao Li
2023-06-27 COMSOL 员工

Email: support@comsol.com