Civil Engineering Reference
In-Depth Information
σ =
C
(
T
)ε
n
(11.13)
The temperature rise is obtained from the energy balance in Eq. (
11.14
).
∂ρ
U
∂
t
+
∂
∂
x
j
Q
conv
+
Q
cond
=
Q
rad
+
Q
e
(11.14)
where
∂ρ
U
∂
t
are the convective and conduction components of the heat flux,
Q
rad
is the radiative
heat, and
Q
e
is heat generated in the part from the electric energy dissipated. Using
constitutive equations for each component, the heat equation to be solved to deter-
mine the temperature rise for particular electric parameters is shown in Eq. (
11.15
):
is the rate of change of the internal energy of the part,
∂
x
j
Q
conv
+
Q
cond
ρ
V
v
C
p
∂
T
∂
2
T
∂
x
j
T
4
−
T
4
∞
∂
t
=−
A
s
[
h
(
T
−
T
∞
)
]
−
k
2
A
d
+
A
p
−
A
s
εσ
SB
+
VI
,
(11.15)
where
ρ
is the density of the material,
V
v
is the volume of the part,
C
p
is the spe-
cific heat of the material,
T
is the temperature,
t
is time,
A
s
is the lateral surface
of the part,
h
is the convection heat transfer coefficient,
T
∞
is the surrounding
temperature,
k
is thermal conductivity for the die and punch material,
A
d
and
A
p
are the cross-sectional area of the dies and punch,
x
j
are coordinates,
ε
is radia-
tive emissivity for the part,
σ
SB
is the Stefan-Boltzmann constant,
V
is the electric
voltage, and
I
is the intensity of the current, given by the product of the current
density and cross-sectional area.
After each pulse, when the current is not applied any longer, the temperature
decreases and can be calculated from Eq. (
11.16
):
ρ
V
v
C
p
∂
T
∂
t
=−
A
s
[
h
(
T
−
T
∞
)
]
−
k
(
2
A
d
+
A
p
)
∂
2
T
T
4
−
T
4
∞
−
A
s
εσ
SB
,
∂
x
j
(11.16)
which is obtained from Eq. (
11.15
) after taking out the heat source.
11.1.4.3 Summary of Electrical Effects
The analytical model for EAB uses the energy-based approach to incorporate
the effects of electricity on the bending force model from the classical bending
process. The same punch displacement,
s
, and the same die velocity,
u
, are con-
sidered for both classical and EAB processes. It is assumed that the same total
energy is required for deformation in both cases. Thus, the force needed in the
EAB process can be determined and compared with the force in the classical
