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Home / Technical Notes / Piezoelectric Analysis / Equation of Piezoelectricity
Equation of Piezoelectricity
The equation of piezoelectricity represents the relationship of stress T, strain S, electric field E, and electric flux density D.
Strain and stress are explained in detail at Strain Type and Stress Type respectively.
Here, strain, stress, electric field, and electric flux density are represented in the form of vector.
The equations of piezoelectricity are the elastic modulus (compliance matrix (s) or stiffness matrix (c)), the relative permittivity matrix ε (or inverse matrix of relative permittivity β), the equations of piezoelectric constants (e, g, h, d) as well as the above. There are four types of equations as below.
 |
e-type |
 |
g-type |
 |
h-type |
 |
d-type |
The suffix, t in et, gt, ht, dt indicates a transpose of the matrix.
d-type
For frequently used d-type, the components of the matrix correspond to the input items on the dialog box as below.
Input of the elastic compliance matrix (s)
The elasticity matrix is symmetric. In the input dialog box for the piezoelectricity, enter the lower left side only as below.
|
|
xx |
yy |
zz |
yz |
zx |
xy |
|
xx |
s11 |
-- |
-- |
-- |
-- |
-- |
|
yy |
s21 |
s22 |
-- |
-- |
-- |
-- |
|
zz |
s31 |
s32 |
s33 |
-- |
-- |
-- |
|
yz |
s41 |
s42 |
s43 |
s44 |
-- |
-- |
|
zx |
s51 |
s52 |
s53 |
s54 |
s55 |
-- |
|
xy |
s61 |
s62 |
s63 |
s64 |
s65 |
s66 |
Input of the piezoelectric constant d
|
|
x |
y |
z |
yz |
zx |
xy |
|
x |
d11 |
d12 |
d13 |
d14 |
d15 |
d16 |
|
y |
d21 |
d22 |
d23 |
d24 |
d25 |
d26 |
|
z |
d31 |
d32 |
d33 |
d34 |
d35 |
d36 |
Input of the relative permittivity matrix
The relative permittivity matrix is symmetric. In the input dialog box for the piezoelectricity, enter the lower left side only as below.
|
|
x |
y |
z |
|
x |
ε11 |
-- |
-- |
|
y |
ε21 |
ε22 |
-- |
|
z |
ε31 |
ε32 |
ε33 |
e-type
For frequently used e-type, the components of the matrix correspond to the input items on the dialog box as below.
Input of the elastic stiffness matrix
The elasticity matrix is symmetric. In the input dialog box for the piezoelectricity, enter the lower left side only as below.
|
|
xx |
yy |
zz |
yz |
zx |
xy |
|
xx |
c11 |
-- |
-- |
-- |
-- |
-- |
|
yy |
c21 |
c22 |
-- |
-- |
-- |
-- |
|
zz |
c31 |
c32 |
c33 |
-- |
-- |
-- |
|
yz |
c41 |
c42 |
c43 |
c44 |
-- |
-- |
|
zx |
c51 |
c52 |
c53 |
c54 |
c55 |
-- |
|
xy |
c61 |
c62 |
c63 |
c64 |
c65 |
c66 |
Input of the piezoelectric constant e
|
|
x |
y |
z |
yz |
zx |
xy |
|
x |
e11 |
e12 |
e13 |
e14 |
e15 |
e16 |
|
y |
e21 |
e22 |
e23 |
e24 |
e25 |
e26 |
|
z |
e31 |
e32 |
e33 |
e34 |
e35 |
e36 |
Input of the relative permittivity matrix
The relative permittivity matrix is symmetric. In the input dialog box for the piezoelectricity, enter the lower left side only as below.
|
|
x |
y |
z |
|
x |
ε11 |
-- |
-- |
|
y |
ε21 |
ε22 |
-- |
|
z |
ε31 |
ε32 |
ε33 |