Environmental Engineering Reference
In-Depth Information
=
ʴμ
f
P
(ˉ
−
),
q
f
U
(6)
where
μ
f
is the friction coefficient,
P
is the pressure plunge,
ˉ
and
U
are the
angular and weld speed, respectively. Here
ʴ
is the contact state variable or extent of
slip which defines the influence of welding tool slipping in the heat generation (Deng
et al.
2001
):
exp
ˉ
ˉ
0
R
p
R
s
1
ʴ
0
ʴ
=
−
−
.
1
(7)
In the above expression
ˉ
is the welding tool normalized angular frequency,
R
p
and
R
s
are the pin and shoulder radii, respectively;
ʴ
0
is a constant; and
ˉ
0
is the angular
velocity of a reference welding condition.
3 Mathematical Formulation
For this problema steady-state process is assumed. AISI 1018 steel is incompressible,
thermal conductivity is constant in the mathematical domain. In the velocity vector,
only the
e
i
component prevails, which is in
x
direction. Velocity and temperature
are functions that depend only the
y
direction. Dimensionless variables are defined
in Table
1
. The weld speed
U
, the difference between shoulder and pin radii
H
,
reference temperatures
T
0
and
T
1
are macroscopic variables that are established in
the process.
Figure
1
depicts the mathematical domain and boundary conditions for the 1-D
case. The continuity (
8
), momentum (
9
) and energy (
10
) equations are solved using
a series expansion.
dž
∂ˆ
∂ʱ
=
0
,
(8)
Ta b l e 1
Dimensionless
variables
V
U
y
H
x
H
ˆ
=
ʱ
=
ʳ
=
pH
n
m
0
U
n
T
0
T
1
−
T
−
ʸ
=
ˀ
=
b
=−
a
(
T
1
−
T
0
)
T
0
Fig. 1
Scheme and
boundary conditions
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