22
H. Li et al.
Substituting Eq. (2.9) into Eqs. (2.1) and (2.4) yields Eqs. (2.10) and (2.11):
ρc
∂ T (x
i
, t)
∂t
=
H x i
K [δ]ω(ξ
i j
)ϕ
x
i
, x
j
, t
T
x
j
, t
− T
x
i
, t
ξ
i j
2
dV x i + Q x i ,
(2.10)
ρ ¨
u
x
i
, t
=
H x i
ϕ
x
i
, x
j
, t
f(ξ, η, T, t)dV x j + b
i
.
(2.11)
It is noted that Eq. (2.11) is a dynamic equation. However, the mechanical distortion caused by the changes of temperature should be under a quasi-static process.
Therefore, in order to weaken the dynamic effects of the structure, an artificial
damping method is utilized for the peridynamic equation of motion, which yields
Eq. (2.12):
ρ ¨
u
x
i
, t
+ λ ˙
u =
H x i
ϕ
x
i
, x
j
, t
f(ξ, η, T, t)dV x j + b
i
,
(2.12)
where λ is a damping coefficient obtained by an adaptive dynamic relaxation
technique [28].
On the other hand, it is assumed that 60% of the total heat input of MIG welding
is transferred to the work piece via the molten metal droplets [13], and the total heat
input is calculated by Eq. (2.13):
Q x i = ηV I/V x i ,
(2.13)
where η is the arc efficiency, V is the arc voltage and I is the arc current. In addition,
the heat flux of the molten metal droplets in the Gaussian distribution [9] is adapted
in this paper.
So far, Eqs. (2.10) and (2.12) show the decoupled peridynamic formulations for
the numerical simulation of MIG welding.
2.4 Results and Discussions
To verify the present method for MIG welding process, a classical model of two plates
being joined together is considered in the following situations. The geometry and
welding configuration of the test are shown in Fig. 2.2. The plate is measured 500 mm
long, 250 mm wide and 4 mm thick, respectively. The plate is made of aluminium
alloy 6061-T6 and the temperature dependent physical properties of the material are
shown in Tables 2.1 and 2.2. The solidification temperature of aluminium 6061-T6
H. Li et al.
Substituting Eq. (2.9) into Eqs. (2.1) and (2.4) yields Eqs. (2.10) and (2.11):
ρc
∂ T (x
i
, t)
∂t
=
H x i
K [δ]ω(ξ
i j
)ϕ
x
i
, x
j
, t
T
x
j
, t
− T
x
i
, t
ξ
i j
2
dV x i + Q x i ,
(2.10)
ρ ¨
u
x
i
, t
=
H x i
ϕ
x
i
, x
j
, t
f(ξ, η, T, t)dV x j + b
i
.
(2.11)
It is noted that Eq. (2.11) is a dynamic equation. However, the mechanical distortion caused by the changes of temperature should be under a quasi-static process.
Therefore, in order to weaken the dynamic effects of the structure, an artificial
damping method is utilized for the peridynamic equation of motion, which yields
Eq. (2.12):
ρ ¨
u
x
i
, t
+ λ ˙
u =
H x i
ϕ
x
i
, x
j
, t
f(ξ, η, T, t)dV x j + b
i
,
(2.12)
where λ is a damping coefficient obtained by an adaptive dynamic relaxation
technique [28].
On the other hand, it is assumed that 60% of the total heat input of MIG welding
is transferred to the work piece via the molten metal droplets [13], and the total heat
input is calculated by Eq. (2.13):
Q x i = ηV I/V x i ,
(2.13)
where η is the arc efficiency, V is the arc voltage and I is the arc current. In addition,
the heat flux of the molten metal droplets in the Gaussian distribution [9] is adapted
in this paper.
So far, Eqs. (2.10) and (2.12) show the decoupled peridynamic formulations for
the numerical simulation of MIG welding.
2.4 Results and Discussions
To verify the present method for MIG welding process, a classical model of two plates
being joined together is considered in the following situations. The geometry and
welding configuration of the test are shown in Fig. 2.2. The plate is measured 500 mm
long, 250 mm wide and 4 mm thick, respectively. The plate is made of aluminium
alloy 6061-T6 and the temperature dependent physical properties of the material are
shown in Tables 2.1 and 2.2. The solidification temperature of aluminium 6061-T6