5.5 Total Lagrangian Formulation
89
1 F uφ =
B
T
u H
T
e
1
0 E d ,
(5.52)
1 K uφ =
B
T
u H
T
e Bφ d .
(5.53)
From Eq. (5.36), the coupled electrical internal virtual work in the virtual configuration,
2
0 δW
(3)
int , can be calculated as
2
0 δW
(3)
int =
2
0 δ E
T H e
2
0 S d
= δφ
T
B
T
φ H e
1
0 S d +
0
B
T
φ H e B u d q
= δφ
T
1 F φu +
1 K φu q
,
(5.54)
Here,
1 F φu and
1 K φu denote the mechanically induced in-balance charge vector and
the piezoelectric coupled capacity matrix, which are respectively given by
1 F φu =
B
T
φ H e
1
0 S d ,
(5.55)
1 K φu =
B
T
φ H e B u d .
(5.56)
From Eq. (5.37), the pure electric internal virtual work in the virtual configuration,
2
0 δW
(4)
int , can be expressed as
2
0 δW
(4)
int =
2
0 δ E
T H g
2
0 E d
= δφ
T
B
T
φ H g
1
0 E d +
B
T
φ H g B φ d φ
= δφ
T
1 F φφ +
1 K φφ φ
,
(5.57)
in which the electrically induced in-balance charge vector
1 F φφ and the piezoelectric
capacity matrix
1 K φφ are calculated as
1 F φφ =
B
T
φ H g
1
0 E d ,
(5.58)
1 K φφ =
B
T
φ H g B φ d .
(5.59)
The variation of the external work in the virtual configuration,
2
0 δW ext , including
the mechanical force and electric charge loads, are expressed as