1. Constitutive relations of piezoelectric material
The constitutive relations of linear anisotropic piezoelectric material are [1]
T i j = c i j k l E S k l − e k i j E k , D i = e i k l S k l + ε i j S E j , (1) T_{ij} = c_{ijkl}^E S_{kl} - e_{kij}E_k, \\
D_i = e_{ikl}S_{kl} + \varepsilon_{ij}^S E_j,
\tag{1} T ij = c ij k l E S k l − e k ij E k , D i = e ik l S k l + ε ij S E j , ( 1 )
where $\textbf{T}$ is stress tensor, $\textbf{S}$ is strain tensor, $\textbf{E}$ is electric field, $\textbf{D}$ is electric displacement, $c_{ijkl}^E$ are elastic constants with independent variables $\textbf{E}$, $e_{kij}$ are piezoelectric constants and $\varepsilon_{ij}^S$ are dielectric constants with independent variables $\textbf{S}$.
The linear constitutive relations can also be expressed as
S i j = s i j k l E T k l + d k i j E k , D i = d i k l T k l + ε i j T E j , (2) S_{ij} = s_{ijkl}^E T_{kl} + d_{kij}E_k,\\
D_i = d_{ikl}T_{kl} + \varepsilon_{ij}^T E_j,
\tag{2} S ij = s ij k l E T k l + d k ij E k , D i = d ik l T k l + ε ij T E j , ( 2 )
and
S i j = s i j k l D T k l − g k i j D k , E i = − g i k l D k l + β i j T D j . (3) S_{ij} = s_{ijkl}^D T_{kl} - g_{kij}D_k,\\
E_i = -g_{ikl}D_{kl} + \beta_{ij}^T D_j.
\tag{3} S ij = s ij k l D T k l − g k ij D k , E i = − g ik l D k l + β ij T D j . ( 3 )
2. Matrix notation
We now introduce a compact matrix notation which is more convenient to calculation by the following transformation
i j ij ij k l kl k l 11 22 33 23 or 32 13 or 31 12 or 21 p p p q q q 1 2 3 4 5 6
Thus we have the mapping
c i j k l → c p q , e i k l → e i p , T i j → T p , (4) c_{ijkl} \rightarrow c_{pq}, e_{ikl} \rightarrow e_{ip}, T_{ij}\rightarrow T_p,
\tag{4} c ij k l → c pq , e ik l → e i p , T ij → T p , ( 4 )
and the relation between S p S_p S p S p q S_{pq} S pq
S 1 = S 11 , S 2 = S 22 , S 3 = S 33 , S 4 = S 23 , S 5 = S 13 , S 6 = S 12 . (5) S_1=S_{11}, S_2=S_{22}, S_{3}=S_{33}, \\
S_4=S_{23}, S_5=S_{13}, S_{6}=S_{12}. \tag{5} S 1 = S 11 , S 2 = S 22 , S 3 = S 33 , S 4 = S 23 , S 5 = S 13 , S 6 = S 12 . ( 5 )
Using Eq. (4) and (5), Eq. (1-3) are transformed to
T p = c p q E S q − e k p E k , D i = e i q S q + ε i j S E j , (6) T_{p} = c_{pq}^E S_{q} - e_{kp}E_k, \\
D_i = e_{iq}S_{q} + \varepsilon_{ij}^S E_j,
\tag{6} T p = c pq E S q − e k p E k , D i = e i q S q + ε ij S E j , ( 6 )
S p = s p q E T q + d k p E k , D i = d i q T q + ε i j T E j , (7) S_{p} = s_{pq}^E T_{q} + d_{kp}E_k,\\
D_i = d_{iq}T_{q} + \varepsilon_{ij}^T E_j,
\tag{7} S p = s pq E T q + d k p E k , D i = d i q T q + ε ij T E j , ( 7 )
and
S p = s p q D T q − g k p D k , E i = − g i q T q + β i j T D j . (8) S_{p} = s_{pq}^D T_{q} - g_{kp}D_k,\\
E_i = -g_{iq}T_{q} + \beta_{ij}^T D_j.
\tag{8} S p = s pq D T q − g k p D k , E i = − g i q T q + β ij T D j . ( 8 )
The matrices of material constants have the following relations [2]
c p r E s q r E = δ p q , c p r D s q r D = δ p q , β i k S ε j k S = δ i j , β i k T ε j k T = δ i j , c p q D = c p q E + e k p h k q , s p q D = s p q E − d k p g k q , ε i j T = ε i j S + D i q e j q , β i j T = β i j S − g i q h j q , e i p = d i q c q p E , d i p = ε i k T g k p , g i p = β i k T d k p , h i p = g i q c q p D . (9) c^E_{pr}s^E_{qr}=\delta_{pq}, \quad \quad \quad \quad c^D_{pr}s^D_{qr}=\delta_{pq},\\
\beta_{ik}^S\varepsilon^S_{jk}=\delta_{ij},\quad \quad \quad \quad \beta^T_{ik}\varepsilon^T_{jk}=\delta_{ij},\\
c^D_{pq} = c^E_{pq}+e_{kp}h_{kq},\quad\quad s^D_{pq}=s^E_{pq}-d_{kp}g_{kq},\\
\varepsilon^T_{ij}=\varepsilon^S_{ij}+D_{iq}e_{jq},\quad\quad
\beta^T_{ij}=\beta^S_{ij}-g_{iq}h_{jq},\\
e_{ip}=d_{iq}c^E_{qp},\quad\quad\quad\quad d_{ip}=\varepsilon^T_{ik}g_{kp},\\
g_{ip}=\beta^T_{ik}d_{kp},\quad\quad\quad\quad h_{ip}=g_{iq}c^D_{qp}.\\
\tag{9} c p r E s q r E = δ pq , c p r D s q r D = δ pq , β ik S ε j k S = δ ij , β ik T ε j k T = δ ij , c pq D = c pq E + e k p h k q , s pq D = s pq E − d k p g k q , ε ij T = ε ij S + D i q e j q , β ij T = β ij S − g i q h j q , e i p = d i q c q p E , d i p = ε ik T g k p , g i p = β ik T d k p , h i p = g i q c q p D . ( 9 )
The values of c p q E , e k p and ε i j S c^E_{pq}, e_{kp} \; \text{and} \; \varepsilon^S_{ij} c pq E , e k p and ε ij S c p q E , e k p and ε i j S , c^E_{pq}, e_{kp} \; \text{and} \; \varepsilon^S_{ij}, c pq E , e k p and ε ij S ,
Many piezoelectric materials such as quartz, piezoelectric ceramics et al. are anisotropic material. The structures we often used are plates which are cut from matrices. Due to the anisotropicity, the material constants are different if the structures are cut from the matrices with different rotational angles. To obtain the material constants with different rorational angles from the original coordinate system, we need coordinate transformation.
For dielectric constants matrix [ ε S ] [\varepsilon^S] [ ε S ] [ ∙ ] [\bullet] [ ∙ ]
[ ε ′ S ] = [ a ] [ ε S ] [ a ] − 1 , (10) [\varepsilon'^{S}]=[a][\varepsilon^S][a]^{-1},
\tag{10} [ ε ′ S ] = [ a ] [ ε S ] [ a ] − 1 , ( 10 )
where [ a ] [a] [ a ]
For elasctic constants matrix [ c E ] [c^E] [ c E ] 6 × 6 6\times6 6 × 6
[ c ′ E ] = [ M ] [ c E ] [ M ] − 1 , (11) [c'^{E}]=[M][c^E][M]^{-1},
\tag{11} [ c ′ E ] = [ M ] [ c E ] [ M ] − 1 , ( 11 )
where [ M ] [M] [ M ] [ a ] [a] [ a ]
[ a x x 2 a x y 2 a x z 2 2 a x y a x z 2 a x z a x x 2 a x x a x y a y x 2 a y y 2 a y z 2 2 a y y a y z 2 a y z a y x 2 a y x a y y a z x 2 a z y 2 a z z 2 2 a z y a z z 2 a z z a z x 2 a z x a z y a y x a z x a y y a z y a y z a z z a y y a z z + a y z a z y a y x a z z + a y z a z x a y y a z x + a y x a z y a z x a x x a z y a x y a z z a x z a x y a z z + a x z a z y a x z a z x + a x x a z z a x x a z y + a x y a z x a x x a y x a x y a y y a x z a y z a x y a y z + a x z a y y a x z a y x + a x x a y z a x x a y y + a x y a y z ] . (12) \begin{bmatrix}
a_{xx}^2 & a_{xy}^2 & a_{xz}^2 & 2a_{xy}a_{xz} & 2a_{xz}a_{xx} & 2a_{xx}a_{xy}\\
a_{yx}^2 & a_{yy}^2 & a_{yz}^2 & 2a_{yy}a_{yz} & 2a_{yz}a_{yx} & 2a_{yx}a_{yy}\\
a_{zx}^2 & a_{zy}^2 & a_{zz}^2 & 2a_{zy}a_{zz} & 2a_{zz}a_{zx} & 2a_{zx}a_{zy}\\
a_{yx}a_{zx} & a_{yy}a_{zy} & a_{yz}a_{zz} & a_{yy}a_{zz}+a_{yz}a_{zy} & a_{yx}a_{zz}+a_{yz}a_{zx} & a_{yy}a_{zx}+a_{yx}a_{zy} \\
a_{zx}a_{xx} & a_{zy}a_{xy} & a_{zz}a_{xz} & a_{xy}a_{zz}+a_{xz}a_{zy} & a_{xz}a_{zx}+a_{xx}a_{zz} & a_{xx}a_{zy}+a_{xy}a_{zx} \\
a_{xx}a_{yx} & a_{xy}a_{yy} & a_{xz}a_{yz} & a_{xy}a_{yz}+a_{xz}a_{yy} & a_{xz}a_{yx}+a_{xx}a_{yz} & a_{xx}a_{yy}+a_{xy}a_{yz} \\
\end{bmatrix}.
\tag{12} a xx 2 a y x 2 a z x 2 a y x a z x a z x a xx a xx a y x a x y 2 a y y 2 a z y 2 a y y a z y a z y a x y a x y a y y a x z 2 a y z 2 a z z 2 a y z a z z a z z a x z a x z a y z 2 a x y a x z 2 a y y a y z 2 a z y a z z a y y a z z + a y z a z y a x y a z z + a x z a z y a x y a y z + a x z a y y 2 a x z a xx 2 a y z a y x 2 a z z a z x a y x a z z + a y z a z x a x z a z x + a xx a z z a x z a y x + a xx a y z 2 a xx a x y 2 a y x a y y 2 a z x a z y a y y a z x + a y x a z y a xx a z y + a x y a z x a xx a y y + a x y a y z . ( 12 )
For piezoelectric constants, the transformation is
[ e ′ ] = [ M ] [ e ] [ a ] − 1 , (13) [e']=[M][e][a]^{-1},
\tag{13} [ e ′ ] = [ M ] [ e ] [ a ] − 1 , ( 13 )
where [ M ] and [ a ] [M] \; \text{and} \; [a] [ M ] and [ a ]
4. Reference
[1] J. S. Yang. An Introduction to the Theory of Piezoelectricity, 2nd (Switzerland: Springer Nature, 2018).
[2] A. H. Meitzler H. F. Tiersten, A. W. Warner, D. Berlincourt, G. A. Couqin, F. S. Welsh, IEEE Standard on Piezoelectricity (New York: IEEE , 1988).
[3] B. A. Auld. Acoustic Fields and Wave in Solids, Vol. 1 (New York: John Wiley & Sons, 1973).