88
5 Finite Element Formulations
where
1 F ut and
1 M uu represent the inertial in-balance force and mass matrix, which
are respectively calculated by
1 F ut =
N
T
u T
T
v H u
1
0 ¨
v d ,
(5.46)
1 M uu =
N
T
u T
T
v H u T v N u d .
(5.47)
From Eq. (5.34), the pure mechanical induced virtual work in the virtual configuration,
2
0 δW
(1)
int , can be expressed as
2
0 δW
(1)
int =
2
0 δ S
T H c
2
0 S d
= δq
T
B
T
u H c
1
0 S d +
B
T
u H c B u d q
= δq
T
1 F uu +
1 K uu q
,
(5.48)
where
1 F uu and
1 K uu denote the mechanically induced in-balance force vector and
the linearized stiffness matrix, respectively. The linearized and geometrically nonlinear stiffness matrices will be updated after every iteration. The mechanically induced
in-balance force vector
1 F uu and the linearized stiffness matrix
1 K uu can be respectively obtained as
1 F uu =
B
T
u H c
1
0 S d ,
(5.49)
1 K uu =
B
T
u H c B u d .
(5.50)
From Eq. (5.35), the coupled mechanical internal virtual work in the virtual configuration,
2
0 δW
(2)
int , can be expressed as
2
0 δW
(2)
int =
2
0 δ S
T H
T
e
2
0 E d
= δq
T
B
T
u H
T
e
1
0 E d +
B
T
u H
T
e B φ d φ
= δq
T
1 F uφ +
1 K uφ φ
,
(5.51)
where
1 F uφ ,
1 K uφ are the electrically induced in-balance force vector, the coupled
stiffness matrix. They can be calculated by