3 Modeling of Friction Stir Welding Processes
109
Q generated = η Q P,
(3.59)
where P is the mechanical power, and Q generated is the heat generated by the tool,
and the typical value of η Q varies from 0.6 to 1. A median value of 0.865 for the
complete weld cycle and 0.9 for the welding stage was obtained during the study.
Based on Schmidt et al.’s [72] work, Salimi et al. [68] developed a mathematical
relation to calculate the temperature distribution in the weld zone. Heat flux generated
beneath the shoulder q 1 , pin side q 2 and pin bottom surface q 3 were given as
q 1 = ωr τ c ,
(3.60)
q 2 = ω R pin τ c ,
(3.61)
q 3 = ωr τ c ,
(3.62)
where τ c represents equivalent shear stress, which depends on the contact condition.
The temperature field is described by heat conduction equation. Although Salimi
et al. [68] nicely described the heat conduction phenomenon, their expressions for
rate of heat generation per unit volume denoted by the function g(x, y, z, t) were
mathematically not correct. Here, the correct expressions are provided using Dirac
delta function. Dirac delta function converts heat flux into heat per unit volume.
Assuming that there are four heat flux sources, the function g is split into four parts,
i.e.,
g(x, y, z, t) = g 1 (x, y, z, t) + g 2 (x, y, z, t) + g 3 (x, y, z, t) + g 4 (x, y, z, t).
(3.63)
Thus, FSW tool is considered to be a three-dimensional non-uniform moving heat
source that supplies has three heat sources; the rate of heat generation by ith sources
is g i (x, y, z, t). Expressions for rate of heat supplied due to rubbing of the shoulder
on the top surface of the plates are given as (see Fig.3.6 for hint)
g 1 (x, y, z, t) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ωr τ c δ(z − h)
¯
x(t) −
R
2
sh − (y − ¯
y(t)) 2
< x
<
¯
x(t) +
R
2
sh − (y − ¯
y(t)) 2
,
( ¯
y(t) − R sh ) < y < ( ¯
y(t) + R sh );
0
otherwise,
(3.64)
where δ(·) is the delta function of Dirac, and centers of tool are denoted by x(t) and
y(t). However, expression given by Eq. (3.64) gives non-zero heat in the shoulder
area where there is pin and consequently no heat generation. To nullify it, negative
heat generation is taken at that portion. Thus,
109
Q generated = η Q P,
(3.59)
where P is the mechanical power, and Q generated is the heat generated by the tool,
and the typical value of η Q varies from 0.6 to 1. A median value of 0.865 for the
complete weld cycle and 0.9 for the welding stage was obtained during the study.
Based on Schmidt et al.’s [72] work, Salimi et al. [68] developed a mathematical
relation to calculate the temperature distribution in the weld zone. Heat flux generated
beneath the shoulder q 1 , pin side q 2 and pin bottom surface q 3 were given as
q 1 = ωr τ c ,
(3.60)
q 2 = ω R pin τ c ,
(3.61)
q 3 = ωr τ c ,
(3.62)
where τ c represents equivalent shear stress, which depends on the contact condition.
The temperature field is described by heat conduction equation. Although Salimi
et al. [68] nicely described the heat conduction phenomenon, their expressions for
rate of heat generation per unit volume denoted by the function g(x, y, z, t) were
mathematically not correct. Here, the correct expressions are provided using Dirac
delta function. Dirac delta function converts heat flux into heat per unit volume.
Assuming that there are four heat flux sources, the function g is split into four parts,
i.e.,
g(x, y, z, t) = g 1 (x, y, z, t) + g 2 (x, y, z, t) + g 3 (x, y, z, t) + g 4 (x, y, z, t).
(3.63)
Thus, FSW tool is considered to be a three-dimensional non-uniform moving heat
source that supplies has three heat sources; the rate of heat generation by ith sources
is g i (x, y, z, t). Expressions for rate of heat supplied due to rubbing of the shoulder
on the top surface of the plates are given as (see Fig.3.6 for hint)
g 1 (x, y, z, t) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ωr τ c δ(z − h)
¯
x(t) −
R
2
sh − (y − ¯
y(t)) 2
< x
<
¯
x(t) +
R
2
sh − (y − ¯
y(t)) 2
,
( ¯
y(t) − R sh ) < y < ( ¯
y(t) + R sh );
0
otherwise,
(3.64)
where δ(·) is the delta function of Dirac, and centers of tool are denoted by x(t) and
y(t). However, expression given by Eq. (3.64) gives non-zero heat in the shoulder
area where there is pin and consequently no heat generation. To nullify it, negative
heat generation is taken at that portion. Thus,