Figure -2 Comparison between calculated weld dimensions and the
corresponding experimentally obtained weld macrographs.Laser power:
1.0 kW; Sheet thickness: 1.0 mm;On–times (a) 6 ms (b) 10 ms (c) 14
ms (d) 20 ms[Isotherm legends: A – 1773 K, B – 1473 K, C – 1203 K,
D –993 K, E – 623 K]
VI. REFERENCES
[1]D. Rosenthal: Mathematical theory of heat distribution during
welding and cutting, Welding Journal, 20(5), 1941, 220s-234s.
[2] Rykalin R R , Energy Source for Welding, Welding in the
World, Vol 12, No. 9/10, p 227-248, 1974.
[3] A. Paul and T. Debroy: Free surface flow and heat transfer
in conduction mode laser welding, Metallurgical Transactions B, 19,
1998, 851-857.
[4] S. Basu and T. Debroy: Liquid metal expulsion during laser
irradiation, Journal of Applied Physics, 72(8), 2003,
3317-3322.
[5] T. Zacharia, S. A. David, J. M. Vitek and T. Debroy: Heat
transfer during Nd: YAG pulsed laser welding and its effect on
solidification structure of austenitic stainless steels,
Metallurgical Transactions A, 20, 1989, 957-967.
[6] V. Pavelic, R. Tanabakuchi, O. Uyehara and P. Myers:
Experimental and computed temperature histories in gas tungsten arc
welding of thin plates, Welding Journal, 48(7), 1969,
295s-305s.
[7] J. A. Goldak, B. Chakravarti and M. J. Bibby: A new finite
element model for welding heat sources, Metallurgical Transactions
B, 15, 1984, 229-305.
[8] V. Kamla and J. A Goldak: Error due to two dimensional
approximations in heat transfer analysis of welds, Welding Journal,
72(9), 1993, 440s-446s.
[9] D. Grey, H. Long and P. Maropoulos: Effects of welding
speed, energy input and heat source distribution on temperature
variations in butt welding, Journal of Materials Processing
Technology, 167, 2005, 393-401.
[10] R. Akhter, M. Davis, J. Dowden, P. Kapadia, M. Ley and W.
M. Steen: A method for calculating the fused zone profile of laser
keyhole welds, Journal of Physics D: Applied Physics, 21, 1989,
23-28.
[11] N. Sonti and M. F. Amateau: Finite element modeling of heat
flow in deep penetration laser welds in aluminum alloys, Journal of
Numerical Heat Transfer A, 16, 1989, 351-370.
[12] R. Mueller: A study on heat source equations for prediction
of weld shape, Proc. ICALEO’94, Orlando, USA, October 1994,
509-518.[13] J. Song, J. Y. Shanghvi and P. Michaleris: Sensitivity
analysis and optimization of thermo-elasto-plastic processes with
application to welding side heater design, Computer methods in
applied mechanics and engineering, 193, 2004, 4541-4566.
13-14 May 2011 B.V.M. Engineering College,
V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering &
Technology
[14] R. G. Theissen, I. M. Richardson and J. Sietsma: Physically
based modeling of phase transformations during welding of low
carbon steel, Materials Science and Engineering A, 427, 2006,
223-231.
[15] D. Deng and H. Murakawa: Numerical simulation of
temperature field and residual stress in multi-pass weld in
stainless steel pipe and comparison with experimental measurements,
Computational Materials Science, 37, 2006, 269-277.
[16] M. A. Wahab and M. J. Painter: Numerical models of gas
metal arc welds using experimentally determined weld pool shapes as
the representation of the welding heat source, International
Journal of Pressure Vessels and Piping, 73, 1997, 153-159.
[17] W. S. Chang and S. J. Na: A study on the prediction of the
laser weld shape with varying heat source equations and thermal
distortion of a small structure in micro-joining, Journal of
Materials Processing Technology, 120, 2002, 208-214.
[18] E. Ranatowski and A. Pocwiardowski: An analytic-numerical
evaluation of the thermal cycle in the HAZ during welding, Proc. of
Fourth International Seminar on Numerical Analysis of Weldability
at Graz- Seggau, Austria- 4-5 September, 1997, 379-395.
[19] W. Sudnik, D. Radaj and E. Erofeew: Computerized simulation
of laser beam weld formation comprising joint gaps, Journal of
Physics D: Applied Physics, 31, 1998, 3475-3480.
[20] M. R. Frewin and D. A. Scott: Finite element model of
pulsed laser welding, Welding Research Supplement, 78(1), 1999,
15s-22s.
[21] Integrating Finite Element based Heat Transfer Analysis
with Multivariate Optimization for Efficient Weld Pool Modeling By
Amit Trivedi, Anand Suman, and Amitava De, ISIJ International, Vol.
46 (2006), No. 2, 267-275.
[22] H. Pantsar and V. Kujanappa: Diode laser beam absorption in
laser transformation hardening of low alloy steel, Journal of Laser
Applications, 16(3), 2004, 147-153.
[23] S. Katayama, S. Kohsaka, M. Mizutani, K. Nishizawa and A.
Matsunawa: Pulse shape optimization for defect prevention in pulsed
laser welding of stainless steels, ICALEO, Orlando, USA, 24-28
October 1993, 487-497.
13-14 May 2011 B.V.M. Engineering College,
V.V.Nagar,Gujarat,India
National Conference on Recent Trends in Engineering &
Technology
Modeling of Welding Heat Source for Laser Spot Welding
Process
Dr. Amit Trivedi
Department of Production Engineering,
B.V.M. Engineering College
Vallabh Vidhyanagar-388120, Gujarat, India
[email protected]
Prof. Purvi Chauhan
Department of Production Engineering,
B.V.M. Engineering College
Vallabh Vidhyanagar-388120, Gujarat, India
[email protected]
Prof. K. D. Bhatt
Department of Production Engineering,
B.V.M. Engineering College
Vallabh Vidhyanagar-388120, Gujarat, India
[email protected]
Prof. Hardik S. Berawala Prof. N.J.Manek
Department of Production Engineering, Department of Production
Engineering,
B.V.M. Engineering College B.V.M. Engineering College
Vallabh Vidhyanagar-388120, Gujarat, India Vallabh
Vidhyanagar-388120, Gujarat, India
Abstract: The laser spot welding results in a controlled energy
input, narrow heat affected zone and minimum thermal distortion.
The representation of laser beam as the heat source is critical in
numerical modeling as it governs the distribution of energy in and
around the weld. The models to represent laser as point, line,
surface or volume heat source has been reported. The volumetric
heat source models can widely be used if the weld pool dimensions
are known a priori. The present study aims at using an adaptive
heat source model which progressively changes its nature from
surface to volumetric heat source as the welding advances with
time. The dimensions of the adaptive heat source dynamically
changes as the weld pool starts growing and this s as the weld pool
starts growingically nates the wherein such dimensions are not to
be known a priori.
eliminates the need for knowing the weld dimensions a priori.
The present numerical model can thus be used for predicting thermal
stresses with great reliability.
Key Words: Laser Spot Welding, Heat Source Modeling, Finite
Element Method and Transient Heat Transfer
I. Introduction
The heating source, which melts the metal in fusion welding
processes, can be either arc, plasma, laser beam, electron beam or
the combination of them. The transient temperature in and around
the weld pool is greatly governed by the distribution of the energy
from this heat source. It is thereby necessary to properly
represent such a heat source. As the physics to represent arc,
laser beam, electron beam or plasma beam is often complex, thermal
modeling of welding heat sources has been developed.
Rosenthal [1] and Rykalin [2] represented the welding arc as
either a point, plane or line heat source depending on the
dimensionality of the problem. However, assumption of point heat
source resulted in a singularity of the temperature field.
Furthermore, the thermal energy supplied by a welding arc or
laser/plasma/electron beam is distributed over a finite area rather
than at a single point. Pavelic et al. [6] thus considered welding
arc as a distributed heat source with Gaussian distribution that
could avoid singularities in the solution of transient temperature
field. The heat flux distribution was represented in the form of a
Gaussian form as
)
cr
exp(
)
0
(
q
)
r
(
q
2
-
=
(1)
Where
)
0
(
q
is the maximum heat flux; c is the concentration co-efficient,
and r is termed as the radial distance of a point from the
symmetric heat source axis. The heat distribution by laser beam is
typical as its heat source has to account for the laser energy
reaching inside the weld pool volume [7]. The numerical modeling
needs sufficient trial–and–error exercises to obtain suitable
parameters to represent the parameters to represent the heat
source.
Goldak et al. [7] initiated the concept of volumetric heat
source in the context of moving arc welding that permitted the
energy to reach inside the weld pool volume and defining a pseudo
weld pool in the form of a double ellipsoid (Fig. 1). The double
ellipsoid heat source was defined by parameters a, b, c1 for
representing front of ellipsoid whereas the rear ellipsoid was
defined by parameter a, b and c2. The power density distribution in
the front and rear quadrant of weld pool was given as [7].
)
c
z
b
y
a
x
(
3
s
2
2
2
2
2
2
e
)
1
c
(
ab
Q
)
1
f
(
3
6
)
z
,
y
,
x
(
q
+
+
-
=
p
p
(2)
(For Frontal Ellipsoid)
)
c
z
b
y
a
x
(
3
s
2
2
2
2
2
2
e
)
2
c
(
ab
Q
)
2
f
(
3
6
)
z
,
y
,
x
(
q
+
+
-
=
p
p
(3)
(For Rear Ellipsoid)
Figure -1 Representation of double-ellipsoidal
heat source [8].
The heat input is represented as
s
q
whereas the fractions f1 and f2 take into consideration
asymmetry in magnitude of heat input at the front and the rear of
the volumetric heat source. As this heat source considers the
existence of pseudo weld pool from the beginning of the process,
only the equation of conservation of energy needs to be solved to
model the transient temperature field. The only limitation of this
approach is that it considers the existence of pseudo weld pool a
priori and hence the nature of fluid flow in a weld pool driven by
various convective forces cannot be predicted. The power density
distribution from the laser beam was observed to be conical.
Ranatowski and Pocwiardowski [18] had suggested a
cylindrical-involution-normal representation corresponding to a
laser beam as
(
)
))
s
z
(
u
1
(
e
)
s
K
exp(
1
Q
K
K
)
y
,
x
(
q
d
K
)
y
x
(
K
d
z
z
c
Z
2
2
C
-
-
-
-
=
-
+
-
p
(4)
where
Q
is the heat input,
z
K
involution factor of heat source,
c
K
the factor designating the heat source concentration,
d
s
the heat source penetration depth, and
)
s
z
(
u
d
-
the Heaveside’s function. The authors hypothesized that by
changing the values of
d
s
,
z
K
and
c
K
, equation (4) could account for point, line, plane, paraboloid,
conical, or Gaussian presentation of heat source. The value of the
depth of heat source
d
s
was taken as h at a point where the power at the said depth was
5% of the total input power. The values of
d
s
,
c
K
and
z
K
are rarely available and that has restricted the use of this
present model. The recourse is to either calibrate a heat source
expression against experimental data sets or evolve a route to
determine optimized value of such uncertain parameters for varying
weld conditions.
The numerical process models that had undertaken detailed
convection heat transfer analysis in weld pool preferred to use
surface heat flux input following a Gaussian distribution. The
numerical models that considered conduction heat transfer analysis
alone preferred to use a combination of Gaussian surface heat flux
distribution in association with a volumetric heat source
expression or only the later. In cases like laser spot welding
where weld pool size is small or the process involves rapid melting
and solidification offering very less time for convective flows to
develop fully, heat conduction based model with a volumetric heat
source representation can provide a fairly reliable estimation of
weld pool dimensions. In the present work, the energy input due to
the laser beam is considered as a surface heat flux with Gaussian
energy distribution and subsequently, an adaptive volumetric heat
source is introduced as the weld pool grows with elapse of time.
The experimental results of weld dimensions shows close agreement
with the model developed.
II. Heat transfer simulation
The axial symmetry of the laser beam permits us to apply the
two-dimensional axisymmetric heat conduction equation
t
T
C
Q
z
T
kr
z
r
1
r
T
kr
r
r
1
p
¶
¶
=
+
÷
ø
ö
ç
è
æ
¶
¶
¶
¶
+
÷
ø
ö
ç
è
æ
¶
¶
¶
¶
r
&
(5)
(1)
where r and z refer to radial and axial directions; k, ρ,
and
p
C
respectively refer to thermal conductivity, density, and
specific heat of material; T and t refer to temperature and time
variable respectively, and
Q
&
depicts internal heat generation per unit time and unit volume.
The associated boundary condition can be stated as
(
)
(
)
0
T
T
T
T
h
q
n
T
k
4
0
4
0
s
=
-
+
-
+
-
¶
¶
se
(6)
(2)
where k refers to thermal conductivity normal to surface; h
and
e
depict surface heat transfer co-efficient and emissivity
respectively;
s
is Stefan- Boltzmann constant, and T0 is the ambient
temperature. A lumped heat transfer co-efficient is used combining
the convective and radiative heat loss by considering h, as [2]
61
.
1
3
T
10
4
.
2
h
×
×
*
=
-
e
(7)
III. Adaptive Heat Source
In the present work, the energy input due to the laser beam is
considered as a surface heat flux with Gaussian energy distribution
and subsequently, as an adaptive volumetric heat source. The
surface heat flux is considered till the top surface under the
laser beam has reached melting point temperature. Once a melt pool
of finite dimensions is formed, an adaptively defined volumetric
heat source is considered that corresponds to the size and the
shape of the growing weld pool. The mathematical expression of
surface heat flux distribution considering Gaussian energy
distribution is given as
÷
÷
ø
ö
ç
ç
è
æ
-
=
2
eff
2
lb
2
eff
lb
gau
s
r
r
.
d
exp
r
d
)
(
P
q
p
h
(8)
Where P refers to the beam power,
gau
h
the absorption coefficient of the laser beam at the work-piece
surface,
eff
r
the effective radius of the focussed laser beam, and
lb
d
the beam distribution parameter of the laser beam profile. The
mathematical expression of adaptive volumetric heat source is given
as
ï
þ
ï
ý
ü
=
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
=
p
w
b
2
2
2
b
2
2
b
vol
w
p
2
w
l
where
p
z
3
l
r
3
exp
p
l
)
(
P
3
6
q
p
p
h
&
(9)
In equation (9), wp and ww represent the instantaneous values of
weld penetration (or depth) and weld width respectively that are
computed from the numerical model. The parameters,
vol
h
,
q
&
, lb, and p refer to absorption coefficient of laser beam within
weld pool, volumetric heat input, and the extent of heat source in
longitudinal and axial directions respectively. At time t = 0, the
numerical calculations for heat transfer analysis considers the
irradiated laser beam as a surface heat flux on the top surface.
Once a weld pool of finite size is formed in a subsequent
time-step, the volumetric heat source term is activated within the
melt volume. The calculated values of weld width (ww) and weld
penetration (wp) are assigned respectively to
b
l
and p as indicated in equation (9). The values of ww and wp
change as the weld pool grows in size in subsequent time-steps for
the total laser on-time. Thus, equation (9) does not need a-priori
knowledge of the final weld pool shape that is otherwise required
in similar expressions used earlier [7-8, 18-19].
The symmetry of a stationary laser beam, permits a symmetric
boundary condition to be applied along z-axis as
0
r
T
=
¶
¶
(10)
(6)
As, the process is transient in nature, an additional boundary
condition is defined at time t = 0 as
(
)
0
T
0
,
z
,
r
T
=
(11)
(7)
The details of finite element discretization followed in this
present study is as per the formulation details given by Amit et
al. [21].
IV. results & discussion
Figures 2(a) – (d) show the comparison of computed and the
corresponding measured weld dimensions for 1.0 mm low carbon steel
at a laser power of 1.0 kW and at on times 6, 10, 14, and 20 ms.
The isotherms are zoomed in a domain of 1.0 mm x 1.0 mm near the
laser beam axis, being an area of interest as regards to prediction
of fusion and heat affected zone. Figures 2(a) – (d) depict that
the computed dimensions of weld depth and width and they agree well
with the corresponding measured results at all the four
on-times.
The legends of the temperature isotherms plotted are given in
Figures 2(a)-(d) and hence, the intercepts of the 1773 K isotherm
(referred to A) in the radial and in the axial directions confirm
to half of the weld width and weld depth respectively. The fair
agreement between the calculated and the corresponding measured
shapes and sizes of the weld pool for all the on-times also
indicate the robustness of the model that has considered adaptive
heat source. However, such a procedure can be considered to be more
effective and practical when uncertain parameters like absorption
coefficient for the given laser power and laser on-time is
predicted. For example, Pantsar et al. [22] has showed that at a
constant value of power density, increase in laser on-time, causes
increase in absorption coefficient in laser welding process.
Although an increase in the value of the absorption coefficient is
a reality at higher laser on-time, the present model has considered
a time-averaged value of absorption coefficient and thermal
conductivity. The zone encompassed between the temperature
isotherms 1773 K (referred to A) and 993 K (referred to D) is
presumed extent of heat affected zone. The slight discrepancies
between the calculated and measured extent of heat affected zone
dimensions possibly indicate a greater value of lower critical
temperature than 993 K that is considered here. A greater value of
lower critical temperature indicates the need of superheating for
transformation from α-ferrite to γ-austenite. The non-equilibrium
cooling conditions prevailing due to rapid rate of heating and
cooling during laser spot welding process might have led to higher
value of lower critical temperature.
V. conclusion
The present study has attempted to bring out the prediction of
weld dimension by the numerical heat transfer model that has
considered adaptive heat source. The computed values of the weld
dimensions from the two-dimensional axisymmetric heat transfer
analysis are validated with the corresponding measured results from
experimental studies. It is also shown that the geometric
dimensions of the adaptive volumetric heat source, as adapted in
this work, closely follow the shape and size of the instantaneous
weld pool. It is thus conceived that the approach, presented in
this work, will alleviate the restricted use of volumetric heat
source in the context of modeling of laser welding, in particular,
and fusion welding, in general.
Figure -2 Comparison between calculated weld dimensions and the
corresponding experimentally obtained weld macrographs.
Laser power: 1.0 kW; Sheet thickness: 1.0 mm;
On–times (a) 6 ms (b) 10 ms (c) 14 ms (d) 20 ms
[Isotherm legends: A – 1773 K, B – 1473 K, C – 1203 K, D – 993
K, E – 623 K]
VI. references
[1]D. Rosenthal: Mathematical theory of heat distribution during
welding and cutting, Welding Journal, 20(5), 1941, 220s-234s.
[2] Rykalin R R , Energy Source for Welding, Welding in the
World, Vol 12, No. 9/10, p 227-248, 1974.
[3] A. Paul and T. Debroy: Free surface flow and heat transfer
in conduction mode laser welding, Metallurgical Transactions B, 19,
1998, 851-857.
[4] S. Basu and T. Debroy: Liquid metal expulsion during laser
irradiation, Journal of Applied Physics, 72(8), 2003,
3317-3322.
[5] T. Zacharia, S. A. David, J. M. Vitek and T. Debroy: Heat
transfer during Nd: YAG pulsed laser welding and its effect on
solidification structure of austenitic stainless steels,
Metallurgical Transactions A, 20, 1989, 957-967.
[6] V. Pavelic, R. Tanabakuchi, O. Uyehara and P. Myers:
Experimental and computed temperature histories in gas tungsten arc
welding of thin plates, Welding Journal, 48(7), 1969,
295s-305s.
[7] J. A. Goldak, B. Chakravarti and M. J. Bibby: A new finite
element model for welding heat sources, Metallurgical Transactions
B, 15, 1984, 229-305.
[8] V. Kamla and J. A Goldak: Error due to two dimensional
approximations in heat transfer analysis of welds, Welding Journal,
72(9), 1993, 440s-446s.
[9] D. Grey, H. Long and P. Maropoulos: Effects of welding
speed, energy input and heat source distribution on temperature
variations in butt welding, Journal of Materials Processing
Technology, 167, 2005, 393-401.
[10] R. Akhter, M. Davis, J. Dowden, P. Kapadia, M. Ley and W.
M. Steen: A method for calculating the fused zone profile of laser
keyhole welds, Journal of Physics D: Applied Physics, 21, 1989,
23-28.
[11] N. Sonti and M. F. Amateau: Finite element modeling of heat
flow in deep penetration laser welds in aluminum alloys, Journal of
Numerical Heat Transfer A, 16, 1989, 351-370.
[12] R. Mueller: A study on heat source equations for prediction
of weld shape, Proc. ICALEO’94, Orlando, USA, October 1994,
509-518
.
[13] J. Song, J. Y. Shanghvi and P. Michaleris: Sensitivity
analysis and optimization of thermo-elasto-plastic processes with
application to welding side heater design, Computer methods in
applied mechanics and engineering, 193, 2004, 4541-4566.
[14] R. G. Theissen, I. M. Richardson and J. Sietsma: Physically
based modeling of phase transformations during welding of low
carbon steel, Materials Science and Engineering A, 427, 2006,
223-231.
[15] D. Deng and H. Murakawa: Numerical simulation of
temperature field and residual stress in multi-pass weld in
stainless steel pipe and comparison with experimental measurements,
Computational Materials Science, 37, 2006, 269-277.
[16] M. A. Wahab and M. J. Painter: Numerical models of gas
metal arc welds using experimentally determined weld pool shapes as
the representation of the welding heat source, International
Journal of Pressure Vessels and Piping, 73, 1997, 153-159.
[17] W. S. Chang and S. J. Na: A study on the prediction of the
laser weld shape with varying heat source equations and thermal
distortion of a small structure in micro-joining, Journal of
Materials Processing Technology, 120, 2002, 208-214.
[18] E. Ranatowski and A. Pocwiardowski: An analytic-numerical
evaluation of the thermal cycle in the HAZ during welding, Proc. of
Fourth International Seminar on Numerical Analysis of Weldability
at Graz- Seggau, Austria- 4-5 September, 1997, 379-395.
[19] W. Sudnik, D. Radaj and E. Erofeew: Computerized simulation
of laser beam weld formation comprising joint gaps, Journal of
Physics D: Applied Physics, 31, 1998, 3475-3480.
[20] M. R. Frewin and D. A. Scott: Finite element model of
pulsed laser welding, Welding Research Supplement, 78(1), 1999,
15s-22s.
[21] Integrating Finite Element based Heat Transfer Analysis
with Multivariate Optimization for Efficient Weld Pool Modeling By
Amit Trivedi, Anand Suman, and Amitava De, ISIJ International, Vol.
46 (2006), No. 2, 267-275.
[22] H. Pantsar and V. Kujanappa: Diode laser beam absorption in
laser transformation hardening of low alloy steel, Journal of Laser
Applications, 16(3), 2004, 147-153.
[23] S. Katayama, S. Kohsaka, M. Mizutani, K. Nishizawa and A.
Matsunawa: Pulse shape optimization for defect prevention in pulsed
laser welding of stainless steels, ICALEO, Orlando, USA, 24-28
October 1993, 487-497.
_1365436713.unknown
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