3 Modeling of Friction Stir Welding Processes
107
where r is the radius at which a differential area dA exists with contact shear stress
of τ contact . The differential area is calculated for each portion by considering an
element of height dz, width dr in the radial direction subtending an angle θ at the
center and length rdθ in the circumferential direction Fig. 3.5b. Using Eq. (3.45),
dQ is integrated over the entire surfaces to arrive at the different heat contributions
expressed as
Q 1 =
2π
0
R sh
R pin
ωτ contact r
2
(1 + tan α)dr dθ =
2
3
πτ contact ω
R
3
sh − R
3
pin
(1 + tan α),
(3.46)
Q 2 =
2π
0
H pin
0
ωτ contact R
2
pin dzdθ = 2πτ contact ω R
2
pin H pin ,
(3.47)
Q 3 =
2π
0
R pin
0
ωτ contact r
2 dr dθ =
2
3
πτ contact ω R
3
pin .
(3.48)
From Eq. (3.44)
Q total =
2
3
πτ contact ω
R
3
sh − R
3
pin
(1 + tan α) + R
3
pin + 3R
2
pin H pin
.
(3.49)
For a flat shoulder tool (α = 0), which simplifies Eq. (3.49) to arrive at
Q total =
2
3
πτ contact ω
R
3
sh + 3R
2
pin H pin
.
(3.50)
However, while considering the force and torque on the inclined shoulder surface,
Schmidt et al. [72] split up the contributions from projected vertical and horizontal
areas of the inclined area. Perhaps a more accurate approach would have been to
consider the area of the inclined plane for force and torque calculations. This aspect
needs more investigation. Depending on the surface interaction condition, τ contact
varies as follows:
(a) Sticking condition
By using von Mises criterion,
τ contact = τ yield =
σ yield
√
3
.
(3.51)
From Eq. (3.49)