20
H. Li et al.
In addition, on the basis of the method to impose the thermal boundary conditions listed in literature [25], both the temperature and heat-flux boundary conditions are considered in this paper by adding the fictitious material points outside the
corresponding boundaries.
2.2.2 Mechanical Distortion Mechanism with Thermal Effect
On the basis of the bond-based peridynamic theory and the thermal expansion, the
peridynamic equation of motion for the material point x
i can be expressed as Eq. (2.4)
[18]:
ρ ¨
u
x
i
, t
=
H x i
f(ξ, η, T, t)dV x j + b
i
,
(2.4)
in which, ¨
u is the accelerated velocity, f is the pairwise force and b
i is the density
force. Investigating the MIG welding process, the high temperature will cause the
regions near the welding seam to plastically deform, and thereby develop a significant residual tensive stresses in the welded structures. Consequently, the plastic
deformation is considered in the simulation of welding process and the coresspoding
expression of f based on the microplastic model can be defined by Eq. (2.5) [18]:
f(ξ, η, T, t) = f
η − ξ
η − ξ
,
(2.5)
in which, ˙
f can be defined as Eq. (2.6):
˙
f =
⎧
⎨
⎩
c (ξ)
˙
s − α ˙
T
_
, if | f | < f y or f
˙
s − α ˙
T
_
< 0,
0,
otherwise
(2.6)
where c is the micro-modulus, s =
η+ξ−−ξ
η+ξ
is the bond strain, α is the coefficient of
thermal expansion, T
_
is the average temperature of the related two material points,
and f y is a prescribed density force at yield that can be calibrated according to
Eq. (2.7):
f y =
⎧
⎪ ⎨
⎪ ⎩
2σ y
Aδ 2 , 1D,
12σ y
π hδ 2 , 2D,
.
(2.7)
H. Li et al.
In addition, on the basis of the method to impose the thermal boundary conditions listed in literature [25], both the temperature and heat-flux boundary conditions are considered in this paper by adding the fictitious material points outside the
corresponding boundaries.
2.2.2 Mechanical Distortion Mechanism with Thermal Effect
On the basis of the bond-based peridynamic theory and the thermal expansion, the
peridynamic equation of motion for the material point x
i can be expressed as Eq. (2.4)
[18]:
ρ ¨
u
x
i
, t
=
H x i
f(ξ, η, T, t)dV x j + b
i
,
(2.4)
in which, ¨
u is the accelerated velocity, f is the pairwise force and b
i is the density
force. Investigating the MIG welding process, the high temperature will cause the
regions near the welding seam to plastically deform, and thereby develop a significant residual tensive stresses in the welded structures. Consequently, the plastic
deformation is considered in the simulation of welding process and the coresspoding
expression of f based on the microplastic model can be defined by Eq. (2.5) [18]:
f(ξ, η, T, t) = f
η − ξ
η − ξ
,
(2.5)
in which, ˙
f can be defined as Eq. (2.6):
˙
f =
⎧
⎨
⎩
c (ξ)
˙
s − α ˙
T
_
, if | f | < f y or f
˙
s − α ˙
T
_
< 0,
0,
otherwise
(2.6)
where c is the micro-modulus, s =
η+ξ−−ξ
η+ξ
is the bond strain, α is the coefficient of
thermal expansion, T
_
is the average temperature of the related two material points,
and f y is a prescribed density force at yield that can be calibrated according to
Eq. (2.7):
f y =
⎧
⎪ ⎨
⎪ ⎩
2σ y
Aδ 2 , 1D,
12σ y
π hδ 2 , 2D,
.
(2.7)