{ }
2
2
y
x
z
T
T
T
u
DIV T
u
x
y
z
t
ρ
ρ
∂
∂
∂
∂
=
+
+
=
=
∂
∂
∂
∂
ɺɺ
(6)
Where
DIV
is divergence of a dyadic.
ρ
is density of the piezoelectric medium.
Whereas the electric behavior is described by Maxwell’s
Equation considering that piezoelectric media are insulating
(no free volume charge).
{
}
0
DIV
D
=
(7)
Equations (1)-(7) constitute a complete set of differential
equations which can be solved with appropriate mechanical
(displacements and forces) and electrical (potential and
charge) boundary conditions. [1]
The dynamic equations of a piezoelectric continuum can be
derived from the Hamilton principle, in which the Lagrangian
and the virtual work are properly adapted to include the
electrical contributions, as well as the mechanical ones. The
potential energy density of a piezoelectric material includes
contributions from the strain energy and from the electrostatic
energy. [5], [6]
{ } { }
{
} {
}
1
2
T
T
H
S
T
E
D
=
−
(8)
Similarly, the virtual work density
{
} {
}
t
W
u
F
δ
δ
δφσ
=
−
(9)
Where
{
}
F
is the external force and
σ
is the electric
charge. From (8), (9), into the Hamilton principle. [7]
{
} { }
{
}
{ }
{
}
[ ]
{
}
{
}
[ ]
{ }
{
}
{
}
{
} {
}
{
} {
}
{
} {
}
1
2
0
t
t
E
t
t
t
V
t
t
S
b
t
S
S
t
c
S
u
u
S
c
S
S
e
E
E
e
S
dV
E
E
u
P
u
P
dS
u
P
dS
Q
ρ δ
δ
δ
δ
δ
ε
δ
δ
δ
δφσ
δφ
−
= −
+
+
+
+
+
+
−
−
∫
∫
∫
ɺɺ
(10)
In the finite element formulation, the displacement field
{ }
u
and the electric potential
φ
over an element are related
to the corresponding node values
{
}
i
u
and
{
}
i
φ
by the mean
of the shape functions
{
}
u
N
,
{
}
N
φ
.
{ }
[
]
{
}
u
i
u
N
u
=
(11)
{
}
i
N
φ
φ
φ
=
(12)
And therefore, the strain field
{ }
S
and the electric field
{
}
E
are related to the nodal displacements and potential by
the shape functions derivatives
[
]
u
B
and
B
φ
defined by:
{ }
[
]
{ }
[
][
]
{
}
[
]
{
}
u
i
u
i
S
D
u
D
N
u
B
u
=
=
=
(13)
{
}
{
}
{
}
i
i
E
N
B
φ
φ
φ
φ
φ
= −∇
= −∇
= −
(14)
Substituting expressions (11) – (14), into the variation
principle (10), yields:
{
}
[
] [
]
{ }
{
}
[
]
[
]
{
}
{
}
[
]
[ ]
{
}
{
}
[ ] [
]
{
}
{
}
{
}
{
}
[
]
{
}
{
}
[
]
{
}
{
}
[
]
{
}
{
}
{
}
1
2
0
t
t
i
u
u
V
t
t
E
i
u
u
i
V
t
t
i
u
i
V
t
t
t
i
u
i
V
t
t
S
i
i
V
t
t
i
u
b
V
t
t
t
t
i
u
S
i
u
c
S
t
t
t
t
i
i
S
u
N
N
dV u
u
B
c
B
dV u
u
B
e
B
dV
B
e
B
dV u
B
B
dV
u
N
P dV
u
N
P
dS
u
N
P
N
dS
N
Q
φ
φ
φ
φ
φ
φ
δ
ρ
δ
δ
φ
δφ
δφ
ε
φ
δ
δ
δ
δφ
σ
δφ
= −
−
−
−
+
+
+
+
−
−
∫
∫
∫
∫
∫
∫
∫
∫
ɺɺ
(15)
Which must be verified for any arbitrary variation of the
displacements
{
}
i
u
δ
and
electrical
potential
{
}
i
δφ
compatible with the essential boundary conditions. For an
element, (15), can be written under the form:
[
]
{
}
[
]
{
}
{
}
{
}
i
uu
i
u
i
i
M
u
K
u
K
f
φ
φ
+
+
=
ɺɺ
(16)
{
}
{
}
{
}
u
i
i
i
K
u
K
g
φ
φφ
φ
+
=
(17)
With
[
]
[
] [
]
t
u
u
V
M
N
N
dV
ρ
=
∫
(18)
[
]
[
]
[
]
t
E
uu
u
u
V
K
B
c
B
dV
=
∫
(19)
[
] [ ]
t
t
u
u
V
K
B
e
B
dV
φ
φ
=
∫
(20)
t
S
V
K
B
B
dV
φφ
φ
φ
ε
= −
∫
(21)

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- Fall '19
- Finite Element Method, Piezoelectricity, International Journal of Electrical and Computer Engineering, Electrical and Computer Engineering Vol