Civil Engineering Reference
In-Depth Information
where
ξ
is defined as the EEC, and
P
e
is the power of the electrical current pass-
ing through the workpiece. It is known from previous experimental research that
the EEC is a new material property coefficient introduced here, and it reflects the
ratio of the electrical power that contributes toward plastic deformation to the total
input electrical power [
1
]. The EEC,
ξ
, depends on the material, applied current
density and time, and can be determined through the tests defined below. Electric
current will also increase the part temperature, thereby lowering the flow stress of
the material and contributing to decreased required work.
Through resistive heating, the electric current results in bulk thermal soften-
ing of the material, thereby decreasing flow stress as represented by the strength
coefficient,
C
in Eq. (
5.5
). In our analysis, a stepwise approach is used, whereby
the material is strained by
d
ε
,
the contribution of electrical energy to the deforma-
tion work calculated, the remaining electrical energy converted to heat, then the
material workpiece temperature rise and subsequent effect on strength coefficient
derived. The process is repeated to the final desired strain.
When the electric current passes through the metallic workpiece, heat is gen-
erated and the temperature of the part rises. Heat transfer and thermodynamics
knowledge was gathered from Cengel [
8
] and Cengel and Boles [
9
] to derive the
thermal equations for this section. The temperature rise can be determined from an
energy balance conforming to Eq. (
5.9
)
∂ρ
U
∂
t
+
∂
∂
x
j
Q
conv
+
Q
cond
=
Q
rad
+ (
1
− ξ)
P
e
(5.9)
where
∂ρ
U
∂
t
is the rate of change of the internal energy of the part,
∂
x
j
Q
conv
+
Q
cond
are the convective and conduction components of the heat flux,
Q
rad
is the radiation
heat, and
(
1
−
ξ)
P
e
is heat generated in the part from the electric energy dissipated.
Using constitutive equations for each component, the heat equation to be solved for
that determines the temperature rise for particular electric parameters is Eq. (
5.10
):
ρ
V
v
C
p
∂
T
∂
t
=−
A
s
[
h
(
T
−
T
∞
)
]
−
2
kA
c
∂
2
T
T
4
−
T
4
∞
−
A
s
εσ
SB
+ (
1
− ξ)
VI
∂
x
j
(5.10)
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 tem-
perature,
k
is thermal conductivity of the die material,
A
c
is the cross-sectional
area,
x
j
are coordinates,
ε
is radiative 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.
The local heating of the workpiece influences the flow of the material. At room
temperature, the flow is given by the power law presented earlier, but as the tempera-
ture rises, the flow curve depends strongly on the temperature. If the temperature is
higher than the temperature at which recovery and recrystallization take place, then
the flow depends also on the strain rate of the process, conforming to Eq. (
5.11
)
