4.2 Fundamental Theory of Piezoelectricity
61
˘
c
1111
= Y 1
1 − ν 23 ν 32
Δ
,
˘
c
2222
= Y 2
1 − ν 31 ν 13
Δ
,
˘
c
3333
= Y 3
1 − ν 12 ν 21
Δ
,
˘
c
1122
= Y 1
ν 21 − ν 31 ν 23
Δ
,
˘
c
1133
= Y 3
ν 13 − ν 12 ν 23
Δ
, ˘
c
2233
= Y 2
ν 32 − ν 12 ν 31
Δ
,
˘
c
2323
= G 23 ,
˘
c
1313
= G 13 ,
˘
c
1212
= G 12 ,
(4.18)
where Δ = 1 − ν 12 ν 21 − ν 23 ν 32 − ν 31 ν 13 − ν 21 ν 32 ν 13 , Y i denotes the Young’s moduli, G i j the shear moduli, and ν i j the Poisson’s ratios.
The components of the third-order piezoelectric constant tensor and the secondorder dielectric constant tensor in (4.15) and (4.16) can be obtained by
e
mkl
= d
m
·i j c
i jkl
,
(4.19)
χ
mn
=
mn
− d
m
·i j e
i jn
.
(4.20)
Similarly, they can be arranged by matrix form in
[e
mkl
] =
⎡
⎣
0 0 0 0 e
113 0
0 0 0 e
223 0 0
e
311 e
322 e
333 0 0 0
⎤
⎦ , [χ
mn
] =
⎡
⎣
χ
11 0 0
0 χ
22 0
0 0 χ
33
⎤
⎦ .
(4.21)
4.3 Coordinate Transformation in Plates and Shells
In the simulation of piezo-laminated plates and shells, isotropic and orthotropic
materials are mostly used in the analysis. We define two coordinate systems, one is
material coordinate system, denoted by ˘
Θ
i ; the other one is curvilinear coordinate
system representing structural geometries, denoted by Θ
i . For isotropic material,
the material coordinate axes can be set the same as the curvilinear coordinate axes.
However, in case that the fiber reinforcement direction of orthotropic material is
not parallel to the curvilinear coordinate axes, like in the case shown in Fig. 4.3, a
transformation matrix is necessary for converting the constitutive equations from the
material coordinate axes to the curvilinear coordinate axes.
The components of the elasticity constant tensor used in (4.15) are associated with
the unit covariant base vectors ˘
i a in the material coordinate system. They must be
transformed to the base vectors g i , since the formulations of strain field are developed
in the curvilinear coordinate system. The transformation matrix is determined by
means of the following equations
61
˘
c
1111
= Y 1
1 − ν 23 ν 32
Δ
,
˘
c
2222
= Y 2
1 − ν 31 ν 13
Δ
,
˘
c
3333
= Y 3
1 − ν 12 ν 21
Δ
,
˘
c
1122
= Y 1
ν 21 − ν 31 ν 23
Δ
,
˘
c
1133
= Y 3
ν 13 − ν 12 ν 23
Δ
, ˘
c
2233
= Y 2
ν 32 − ν 12 ν 31
Δ
,
˘
c
2323
= G 23 ,
˘
c
1313
= G 13 ,
˘
c
1212
= G 12 ,
(4.18)
where Δ = 1 − ν 12 ν 21 − ν 23 ν 32 − ν 31 ν 13 − ν 21 ν 32 ν 13 , Y i denotes the Young’s moduli, G i j the shear moduli, and ν i j the Poisson’s ratios.
The components of the third-order piezoelectric constant tensor and the secondorder dielectric constant tensor in (4.15) and (4.16) can be obtained by
e
mkl
= d
m
·i j c
i jkl
,
(4.19)
χ
mn
=
mn
− d
m
·i j e
i jn
.
(4.20)
Similarly, they can be arranged by matrix form in
[e
mkl
] =
⎡
⎣
0 0 0 0 e
113 0
0 0 0 e
223 0 0
e
311 e
322 e
333 0 0 0
⎤
⎦ , [χ
mn
] =
⎡
⎣
χ
11 0 0
0 χ
22 0
0 0 χ
33
⎤
⎦ .
(4.21)
4.3 Coordinate Transformation in Plates and Shells
In the simulation of piezo-laminated plates and shells, isotropic and orthotropic
materials are mostly used in the analysis. We define two coordinate systems, one is
material coordinate system, denoted by ˘
Θ
i ; the other one is curvilinear coordinate
system representing structural geometries, denoted by Θ
i . For isotropic material,
the material coordinate axes can be set the same as the curvilinear coordinate axes.
However, in case that the fiber reinforcement direction of orthotropic material is
not parallel to the curvilinear coordinate axes, like in the case shown in Fig. 4.3, a
transformation matrix is necessary for converting the constitutive equations from the
material coordinate axes to the curvilinear coordinate axes.
The components of the elasticity constant tensor used in (4.15) are associated with
the unit covariant base vectors ˘
i a in the material coordinate system. They must be
transformed to the base vectors g i , since the formulations of strain field are developed
in the curvilinear coordinate system. The transformation matrix is determined by
means of the following equations