Civil Engineering Reference
In-Depth Information
assumed that the thermal gradient in the thickness and width direction is rela-
tively small as compared to the gradient along the length of the specimen).
2. The material is homogeneous and isotropic.
3. The geometric model does not incorporate the fillets which link the testing
region to the clamping region of the specimen.
4. The clamping dies are analyzed as a lumped mass to conserve the 1D nature of
the model.
5. The radiation heat transfer is incorporated into the convection coefficient (i.e.,
h
combined
and it is just denoted as
h
in the below equations for simplicity).
6. The electrical resistivity and specific heat of the clamping dies are not tempera-
ture dependent (there is not a large temperature change of the clamping die so
this is an accurate assumption).
The general expression to characterize the balance of energy for the system given
in terms of power is as follows:
Q
+
E
generation
=
E
System
t
(6.1)
All Sides
where
Q
is the respective rate of heat transfer into all of the system sides depend-
ing on the boundary conditions (i.e., conduction or convection),
E
generation
is the
heat generated within the specimen from resistive heating,
E
System
is the change
of internal energy associated with the system, and
Δ
t
is the change in time.
Considering only one element,
Q
+
E
gen,element
=
E
Element
t
(6.2)
Element Sides
In constructing the model, there are three separate areas which had varying bound-
ary conditions. These include the interior nodes in contact with the clamping die
interface, the two nodes at each end of the specimen, and the exterior nodes in the
testing region exposed to the environment.
For an interior node in contact with the clamping die interface, the power bal-
ance can be written as follows:
T
i
m
−
1
−
T
i
m
T
i
m
+
1
−
T
i
m
T
Tidie
−
T
i
m
kA
1
+
kA
1
k
a
2
A
2
+
2
x
x
L
die
(6.3)
+
e
gen,clamp
A
1
x
= ρ
A
1
xc
T
i
+
1
−
T
i
m
t
m
where
k
is the thermal conductivity of the sheet,
A
1
is the element conduction
area of the sheet in the clamping die region,
x
is the nodal spacing,
T
i
m
−
1
is the
present temperature at the node to the left of the node being analyzed,
T
i
m
is the
temperature of the node being analyzed,
T
i
m
+
1
is the present temperature at the
node to the right of the node being analyzed,
k
a
2
is the thermal conductivity of
