Civil Engineering Reference
In-Depth Information
the clamping die made from A2 Steel,
A
2
is the element conduction area into the
dies,
T
die
is the present temperature of the clamping die,
L
die
is the die conduction
length,
e
gen,clamp
is the heat generation per unit volume for the sheet in the clamp-
ing region,
ρ
is the density of the sheet metal,
c
is the heat capacity of the sheet,
T
i
+
m
−
T
i
m
is the temperature change of the node being analyzed from the present
time to the future time, and
t
is the time step.
For the two nodes at each end of the specimen,
T
i
m
+
1
−
T
i
m
k
a
2
2
w
T
Tidie
−
T
i
m
kA
1
T
∞
−
T
i
m
+
2
+
htw
x
L
die
c
T
i
+
1
−
T
i
m
t
+
e
gen,clamp
A
1
x
2
= ρ
A
1
x
2
T
∞
−
T
i
m
m
+
htw
(6.4)
where
w
is the sheet width in the clamping region,
h
is the convection coefficient,
t
is the sheet thickness, and
T
∞
is the atmospheric temperature.
For a node in the testing region exposed to the environment,
kA
11
T
i
m
−
1
−
T
i
m
+
kA
11
T
i
m
+
1
−
T
i
m
T
∞
−
T
i
m
+
2
hA
22
x
x
(6.5)
+
e
gen,test
A
11
x
= ρ
A
1
xc
T
i
+
1
−
T
i
m
t
m
where
A
11
is the element conduction area of the sheet in the testing region,
A
22
is the element convection area in the test region, and
e
gen,test
is the heat genera-
tion per unit volume for the sheet in the testing region. And the material properties
were updated at each time step as a function of temperature,
T
i
m
k
,
k
a
2
,
ρ
,
c
=
f
(6.6)
in all the above equations [
2
].
Using an explicit solution approach, Eqs. (
6.3
)-(
6.5
) can be solved to deter-
mine the new nodal temperature after a given time step.
Thus, for an interior node in contact with the die interface,
kT
i
m
−
1
x
2
ρ
c
+
2
k
a
2
A
2
T
die
T
i
+
1
m
=
T
i
m
+
t
L
die
ρ
A
1
xc
(6.7)
T
i
m
+
kT
i
m
+
1
2
k
x
2
ρ
c
+
2
k
a
2
A
2
L
die
ρ
A
1
xc
x
2
ρ
c
+
e
gen,clamp
−
ρ
c
For the nodes at each end of the specimen,
