86
5 Finite Element Formulations
where δW
(1)
int and δW
(2)
int are the pure and piezoelectric coupled mechanical internal
virtual work, while δW
(3)
int and δW
(4)
int represent the coupled and pure electrical internal
virtual work, respectively.
By using the resultant strain and stress vectors, δW
(1)
int , δW
(2)
int , δW
(3)
int , δW
(4)
int , given
in (5.33), can be organized as
δW
(1)
int =
V
δε
T cε dV =
δ S
T H c S d ,
(5.34)
δW
(2)
int = −
V
δε
T e
T E dV =
δS
T H
T
e E d ,
(5.35)
δW
(3)
int = −
V
δ E
T eε dV =
δ E
T H e S d ,
(5.36)
δW
(4)
int = −
V
δ E
T
E dV =
δ E
T H g E d ,
(5.37)
with
H c =
h
H
T
s cH s μ d
3
,
(5.38)
H e = −
h
eH s μ d
3
,
(5.39)
H g = −
h
μ d
3
.
(5.40)
Furthermore, the external virtual work, δW ext , can be derived as [9, 10]
δW ext =
V
δu
T f b dV +
δu
T f s d + δu
T f c −
δφ
T
d − δφ
T Q c ,
(5.41)
where f b , f s and f c denote the body force, the surface distributed force and the
concentrated force vectors, associated with base vectors of curvilinear coordinate
axes. Additionally, is the surface charge vector and Q c the applied concentrated
electric charge vector.
5.5 Total Lagrangian Formulation
For linearization of nonlinear FE equations, total Lagrangian (TL) incremental formulation [4, 10–12] are adopted. Three configurations are defined and considered
for structures, listed in Table 5.2. The configurations are characterized by the left
superscripts 0, 1 or 2, the reference configurations are denoted by the left subscripts
0. Using the TL method, the stress vector, the strain vector, the displacement vector,
etc. in the virtual configuration can be expressed by those in the current configuration