58
4 Nonlinear Constitutive Relations
strains given as
ε 1 = d 13 E 3 ,
ε 2 = d 23 E 3 ,
ε 3 = d 33 E 3 .
(4.2)
In the same way, applying an electric field along x 1 or x 2 direction yields additional
shear strains as
ε 4 = d 42 E 2 ,
ε 5 = d 51 E 1 .
(4.3)
Here, d 13 = d 31 , d 23 = d 32 , d 42 = d 24 and d 51 = d 15 for isotropic piezoelectric material. More detailed information can be found e.g. in [7, 8].
4.2 Fundamental Theory of Piezoelectricity
Most piezoelectric materials are composed by either single crystals or polycrystalline. Piezoceramics are the most widely used piezoelectric materials, also known
as ferroelectric ceramics, which have much larger piezoelectric coefficients than natural crystals. In the original unprocessed form, these materials have no piezoelectric
properties. However, the materials can be polarized by applying a strong electric field
Piezoceramics can be considered as isotropic material. Using the assumptions of
small strains and weak electric field for piezoelectric patches or layers, the constitutive relations can be expressed as [9]
ε i j = s i jkl σ
kl
+ d
m
i j· E m ,
(4.4)
D
m
= d
m
·kl σ
kl
+
mn E n .
(4.5)
Here ε i j is the strain tensor, σ
kl is the stress tensor, s i jkl is the compliance tensor,
d
m
·kl is the mixed piezoelectric constants tensor (d
m
i j· is the transposed tensor),
mn
is the dielectric constant tensor, E m is the electric field tensor, and D
m is the electric displacement tensor. Furthermore, the second-order strain and stress tensors are
organized as
[σ
i j
] =
⎡
⎣
σ
11
σ
12
σ
13
σ
21
σ
22
σ
23
σ
31
σ
32
σ
33
⎤
⎦ , [ε i j ] =
⎡
⎣
ε 11 ε 12 ε 13
ε 21 ε 22 ε 23
ε 31 ε 32 ε 33
⎤
⎦ .
(4.6)
Due to the symmetry of the stress and strain tensors, σ
i j
= σ
ji and ε i j = ε ji , the
Voigt notations are introduced to describe the second-order strain and stress tensors
in vector form, which are defined as listed in Table 4.1. In such a way, the strains
and stresses can be arranged in vector form as
4 Nonlinear Constitutive Relations
strains given as
ε 1 = d 13 E 3 ,
ε 2 = d 23 E 3 ,
ε 3 = d 33 E 3 .
(4.2)
In the same way, applying an electric field along x 1 or x 2 direction yields additional
shear strains as
ε 4 = d 42 E 2 ,
ε 5 = d 51 E 1 .
(4.3)
Here, d 13 = d 31 , d 23 = d 32 , d 42 = d 24 and d 51 = d 15 for isotropic piezoelectric material. More detailed information can be found e.g. in [7, 8].
4.2 Fundamental Theory of Piezoelectricity
Most piezoelectric materials are composed by either single crystals or polycrystalline. Piezoceramics are the most widely used piezoelectric materials, also known
as ferroelectric ceramics, which have much larger piezoelectric coefficients than natural crystals. In the original unprocessed form, these materials have no piezoelectric
properties. However, the materials can be polarized by applying a strong electric field
Piezoceramics can be considered as isotropic material. Using the assumptions of
small strains and weak electric field for piezoelectric patches or layers, the constitutive relations can be expressed as [9]
ε i j = s i jkl σ
kl
+ d
m
i j· E m ,
(4.4)
D
m
= d
m
·kl σ
kl
+
mn E n .
(4.5)
Here ε i j is the strain tensor, σ
kl is the stress tensor, s i jkl is the compliance tensor,
d
m
·kl is the mixed piezoelectric constants tensor (d
m
i j· is the transposed tensor),
mn
is the dielectric constant tensor, E m is the electric field tensor, and D
m is the electric displacement tensor. Furthermore, the second-order strain and stress tensors are
organized as
[σ
i j
] =
⎡
⎣
σ
11
σ
12
σ
13
σ
21
σ
22
σ
23
σ
31
σ
32
σ
33
⎤
⎦ , [ε i j ] =
⎡
⎣
ε 11 ε 12 ε 13
ε 21 ε 22 ε 23
ε 31 ε 32 ε 33
⎤
⎦ .
(4.6)
Due to the symmetry of the stress and strain tensors, σ
i j
= σ
ji and ε i j = ε ji , the
Voigt notations are introduced to describe the second-order strain and stress tensors
in vector form, which are defined as listed in Table 4.1. In such a way, the strains
and stresses can be arranged in vector form as