60
4 Nonlinear Constitutive Relations
ν i j Y j = ν ji Y i .
(4.10)
Isotropic material can be considered as specific simplification of the orthotropic case,
which yields
Y = Y 1 = Y 2 = Y 3 ,
(4.11)
ν = ν 12 = ν 13 = ν 23 ,
(4.12)
G = G 23 = G 13 = G 12 =
Y
2(1 + ν)
.
(4.13)
The third-order tensor of piezoelectric constant tensor d
m
·kl and the second-order
tensor of dielectric constant tensor
mn are arranged as
d
m
·kl =
⎡
⎢
⎢
⎢
⎣
0 0 0 0 d
1
·13 0
0 0 0 d
2
·23 0 0
d
3
·11 d
3
·22 d
3
·33 0 0 0
⎤
⎥
⎥
⎥
⎦
, [
mn
] =
⎡
⎢
⎢
⎣
11 0 0
0
22 0
0 0
33
⎤
⎥
⎥
⎦ .
(4.14)
In d
m
·kl , the superscript m represent the direction of electric field applied on piezoelectric material, while the subscript pair kl is the stress direction due to the driving
electric field.
Alternatively, the constitutive relations of piezoelectric materials can be expressed
by stiffness way
σ
i j
= c
i jkl
ε kl − e
i jm E m ,
(4.15)
D
m
= e
mkl
ε kl + χ
mn E n ,
(4.16)
where (4.15) is the actuator equation and (4.16) is the sensor equation.
Using the Voigt notations, the components of the fourth-order elasticity constant
tensor in (4.15) can be arranged in matrix form as
[ ˘
c
i jkl
] =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
˘
c
1111
˘
c
1122
˘
c
1133
0
0
0
˘
c
1122
˘
c
2222
˘
c
2233
0
0
0
˘
c
1133
˘
c
2233
˘
c
3333
0
0
0
0
0
0 ˘
c
2323
0
0
0
0
0
0 ˘
c
1313
0
0
0
0
0
0 ˘
c
1212
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.17)
with