Temperature field has an important influence on the phase changes and, finally, on the microstructure and mechanical properties, residual stresses and strains development in the welded joints. During the welding, temperature field shape and its profile depend on many factors but, assuming some assumptions regarding homogeneity and materials isotropy, the mathematical relations for different practical cases can be obtained. Besides, welded bodies can be infinite or semi-infinite as the plates, bars and massive bodies are considered. The instantaneous or permanent thermal sources, fixed or mobile have an important influence on the temperatures distribution. Therefore, the paper presents some analytical solutions for the temperatures prediction in the welded joints in case of the mobile sources with 2D and 3D Gauss distribution

Georgescu, Bogdan

Iordachescu, Mihaela

Scutelnicu, Elena

Archivo Digital UPM

Analytical models for temperature prediction on welding

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

313 
ANALYTICAL MODELS FOR TEMPERATURE PREDICTION ON WELDING 
 
Elena SCUTELNICU1, Mihaela IORDACHESCU2, Bogdan GEORGESCU1 
 1Robotics and Welding Department, Dunarea de Jos University of Galati - Romania 
 2Dep. Ciencia de Materiales, Universidad Politécnica de Madrid – España 
E-mail: elena.scutelnicu@ugal.ro  
 
 
Abstract 
 
Temperature field has an important influence on the phase changes and, finally, on the 
microstructure and mechanical properties, residual stresses and strains development in the welded 
joints. During the welding, temperature field shape and its profile depend on many factors but, 
assuming some assumptions regarding homogeneity and materials isotropy, the mathematical 
relations for different practical cases can be obtained. Besides, welded bodies can be infinite or 
semi-infinite as the plates, bars and massive bodies are considered. The instantaneous or 
permanent thermal sources, fixed or mobile have an important influence on the temperatures 
distribution. Therefore, the paper presents some analytical solutions for the temperatures 
prediction in the welded joints in case of the mobile sources with 2D and 3D Gauss distribution. 
 
1. Equations of the temperature field 
 
The general equation of the energy which is the start point for the temperature field analysis has 
the following expression:  
vzyx Qz
Tv
y
Tv
x
Tvc
z
T
zy
T
yx
T
xt
Tc +⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂+∂
∂⋅−⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
∂
∂+⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂=∂
∂⋅ ρλλλρ   (1) 
where vx, vy, vz are the speed components on the three directions. 
 
Considering the thermal source moving on the direction x (vx=v, vy=vz=0) and the coordinates 
system mobile solidary with the thermal source, the energy balance equation can be written as:  
0=+∂
∂⋅⋅−⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
∂
∂+⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂
vQx
Tvc
z
T
zy
T
yx
T
x
ρλλλ    (2) 
Due to the existence of the solid (s) and liquid (l) phases in the weld pool, the energy balance 
equations, in the quasi-stationary period, can be expressed as:  
- in case of solid phase: 
0=+∂
∂⋅⋅−⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
∂
∂+⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂
vssss Qx
Tvc
z
T
zy
T
yx
T
x
ρλλλ ,  (3) 
- in case of liquid phase: 
0=+∂
∂⋅⋅−⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
∂
∂+⎟⎠
⎞⎜⎝
⎛
∂
∂
∂
∂
vllll Qx
Tvc
z
T
zy
T
yx
T
x
ρλλλ ,   (4) 
at solid-liquid interface: 
( ) top
l
l
s
s TTn
T
n
THvn =⎟⎠
⎞⎜⎝
⎛
∂
∂−⎟⎠
⎞⎜⎝
⎛
∂
∂=∆⋅⋅⋅ ;λλρ .   (5) 
The upper surface of the piece is under the influence of the thermal source and also of the heat 
lost in the environment. Therefore, the contour condition can be written as:  
q
z
T
z
=∂
∂−
=0
λ       (6) 
 
314 
2. Mobile thermal source of 2D Gauss distribution 
 
2.1. Temperature field analysis in the butt welded joints 
 
Considering the mobile thermal source of Gauss distribution, Eagar and Tsai changed the 
Rosenthal’s theory [2]. Their solution represents an important step for the temperatures’ 
approximation in the vicinity of the thermal source. Assuming an exponential repartition of the heat 
flow, its mathematical expression can be written as:  
2
max
rkeqq ⋅−= ,     (7) 
where k is the concentration factor which depends on the thermal source type. 
 
 
 
Figure 1. Gauss distribution of the heat flow.  
 
Taking account of Gauss distribution of the heat flow, Pavelic et al. [1], developed the calculus 
expressions of the heat flow for the contact area and outside of the contact area:  
3
exp3 2
22
2 r
yx
r
Qq −
+
⋅= π  for: ( ) ryx ≤+ 2122 ,   (8) 
where Q=η·U·I and r=r0 is the effective radius of the thermal source when the heat flow decreases 
at 5% of maximum value  3Q/πr2.  
Outside of the area, which is under the influence of the thermal source, the heat flow’s expression 
is:  
( ) ( )[ ]4040 TTCTTq −⋅⋅+−⋅−= εα  
for: ( ) ryx f2122 + .           (9) 
 
 
Figure 2. 2D Gauss distribution of heat flow.  
 
2.2. Temperature field analysis in the fillet welds  
 
Fillet welds represent 80% of the total naval production. Therefore, taking account of two-
dimensional distribution of heat flow, Jeong and Cho [3] analysed temperature field in fillet welds 
and found the analytical solutions for the temperatures calculus. (Figure 2). 
315 
 
Since the geometrical shape and the electric arc distribution are complex, the authors replaced the 
real system with an equivalent one (Figure 3). 
 
 
 
Figure 3. The real and equivalent systems in case of fillet welds [3]. 
 
Finally, the mathematical model of the heat transfer in the changed coordinates system is given by 
the following equations (10)...(16). 
¾ Conduction equation in case of solids: 
u
u T
wvu
a
t
T
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂+∂
∂=∂
∂
2
2
2
2
2
2
    (10) 
¾ Contour condition on the upper surface: ( ) ( )( ) 00,,0,, 01 =−+∂
∂− TVUT
W
VUT
u
u αλ    (11) 
¾ Contour condition on the lower surface: ( ) ( )( ) 0,,,, 02 =−+∂
∂
TdVUT
W
dVUT
u
u αλ    (12) 
¾ Boundary condition at the infinite distance from the thermal source: 
( ) 0,,,lim TtWVUTur =∞→ ,  ( ) ( ) 222 '' WVVUUr +−+−= , (13) 
Where r is the distance from the thermal source located in the coordinates point (U’, V’, 0). 
¾ Temperatures distribution at time t=t0: ( ) 00,,, TWVUTu =      (14) 
¾ Heat flow two-dimensional distribution is given by the equation: 
( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛ +−= 2
2
2
2
22
exp
2
,,
vuvu
vutqtvuQ σσσπσ ,   (15) 
if σ2= σu x σv, the equation (15) can be written as: 
( ) ( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛ +−= 4
2222
2 2
exp
2
,, σ
σσ
πσ
vutqtvuQ uv .    (16) 
316 
In the equation (16) σu, σv are the distribution parameters and q(t)=ηUI(t) is the  useful power of the 
electric arc. Solving the equations 10)...(16), the analytical solutions of the temperatures in the 
changed coordinates system can be obtained for three analyzed cases: 
 
1. Temperature at t moment, in the point P of (U,V,W) coordinates, in the infinite plate which is 
under the influence of the thermal source qi, located in the point of (U ’, V ’, 0) coordinates at t1 
moment 
( ) ( )
( ) ( )
( ) ( )( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅
−+−−⋅−⋅⋅⋅= ∑
∞
=
W
a
aW
a
ttA
tta
VVUU
ttd
qtWVUT
n
n
n
n
nn
i
u
µ
µ
βµ
µλπ
sincos
exp
4
''exp
2
,,,
1
0
1
2
1
22
1
(17) 
where: ρλλ
αβλ
αβββµ
µ ⋅⋅===
++
= ca
d
aa
A
n
n
n ,,,2
2
2
1
1
12
1
2
2
 
 and µn is a positive value which satisfies the equality: 
( )
a
a
a
d
tg
n
nn
⋅⋅−
+=⎟⎟⎠
⎞⎜⎜⎝
⎛
21
2
21
ββµ
ββµµ
. 
 
2. Temperature in the infinite plate which is under the influence of the two-dimensional Gauss 
thermal source located in a fixed point at t1 moment and obtained  by overlapped instantaneous 
thermal sources  
( ) ( ) ( ) [ ] [ ]
( )( ) ( )( )
( )( ) ( )( )
( )( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−−⋅
⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
−⋅+⋅−⋅+
⋅−⋅++⋅−⋅+−⋅
⋅
−⋅+⋅−⋅+
⋅⋅⋅⋅
⋅=
∑∞
=
W
a
aW
a
ttA
ttatta
VttaUtta
ttattadc
tdtqtdT
n
n
n
n
n
n
vu
uv
uv
u
µ
µ
βµµ
σσ
σσ
σσσσ
σ
ρπ
sincosexp
222
22
exp
22
1
1
2
0
1
2
1
2
2
1
22
1
2
2
1
42
1
4
2
11
1
  (18) 
3. Temperature in the infinite plate which is under the influence of the two-dimensional Gauss 
thermal source, located in the point of (Ua, Va, 0) coordinates at t1 moment and obtained by 
overlapped instantaneous thermal sources  
( ) ( ) ( ) [ ] [ ]
( ) ( )( ) ( ) ( )( )
( )( ) ( )( )
( )( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−−⋅
⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+⋅−+
−⋅−++−⋅−⋅+−⋅
⋅
−⋅+⋅−⋅+
⋅⋅⋅⋅
⋅=
∑∞
=
W
a
aW
a
ttA
ttatta
VVttaUUtta
ttattadc
tdtqtdT
n
n
n
n
n
n
vu
auav
uv
u
µ
µ
βµµ
σσ
σσ
σσσσ
σ
ρπ
sincosexp
222
22
exp
22
1
1
2
0
1
2
1
2
2
1
22
1
2
2
1
42
1
4
2
11
1
 (19) 
In case of mobile system, solidary with the electrode, the temperature Tu(u,v,w), at t moment, can 
be computed with the following mathematical relation:  
317 
( ) ( ) [ ] [ ]
( ) ( ) ( )( )( ) ( ) ( ) ( )( )( )
( )( ) ( )( )
( )( ) 1112
0
1
2
1
2
2
11
22
11
2
2
1
42
1
4
2
0
1
sincosexp
222
22
exp
22
,,,
dtW
a
a
W
a
ttA
ttatta
tVtVvttatUtUutta
ttattadc
tq
twvuT
n
n
n
n
n
n
vu
aauaav
uv
t
u
⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛+⎟⎟⎠
⎞
⎜⎜⎝
⎛⋅−−⋅
⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+⋅−+
−+⋅−++−+⋅−⋅+−⋅
⋅
⋅−+⋅⋅−+
⋅⋅⋅⋅=
∑
∫
∞
=
µ
µ
βµµ
σσ
σσ
σσσσ
σ
ρπ
  (20) 
According to Jeong and Cho [3], the analytical solution for the temperatures calculus in the fillet 
joints is: 
( ) ( ) ( ) ( )( ) ( ) d
wB
d
vA
BeBe
BeaBe
H
dtwvuTtzyxT
AA
AA
u
⋅=⋅=
⋅+−⋅
⋅++⋅⋅= ππ ,,
sin1cos
sincos
,,,,,, 2
2
22
 (21) 
 
3. Mobile thermal source of 3D Gauss distribution 
 
Due to the important penetration in case of the semi-infinite welded bodies, Goldak, Chakravarti 
and Bibby proposed the three-dimensional mobile source to predict the temperatures. 
 
3.1. Semi-ellipsoidal thermal source 
 
Firstly, Goldak et al. proposed a semi-ellipsoidal thermal source of the heat flow in any point of (x, 
y, z) which can be computed with the equation [4]: 
⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−−⋅⋅⋅⋅
⋅⋅= 2
2
2
2
2
2 333exp36),,,(
hhhhhh b
z
a
y
c
x
cba
IUzyxQ ππ
η
,   (22) 
where ah, bh, ch are the parameters of the semi-ellipsoidal thermal source (Fig. 4) and x, y, z are 
the coordinates of the thermal source.  
 
 
Figure 4. 3D Gauss distribution of the heat flow (chf=chb=ch). 
3.2. Temperature field developed by the semi-ellipsoidal thermal source  
 
Taking account of the solution obtained by Carslaw and Jaeger in case of the instantaneous 
source, the researchers [2] established the temperature field produced by the semi-ellipsoidal 
thermal source:  
( )[ ]
( ) ( ) ( )
( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅
−+−+−−⋅−⋅⋅⋅⋅
⋅=
'4
'''exp
'4
' 222
23' tta
zzyyxx
ttac
dtQdTt πρ
δ
.  (23) 
318 
Overlapping more instantaneous thermal sources and taking account of semi-ellipsoidal thermal 
source distribution, it can be written the equation (24) [2]: 
( )[ ]
( ) ( ) ( )
( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅
−+−+−−⋅
⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−−⋅⋅⋅⋅
⋅⋅⋅−⋅⋅⋅⋅= ∫ ∫ ∫
∞
∞−
∞
∞−
∞
∞−
'4
'''exp
'3'3'3exp36
'4
''''
2
1
222
2
2
2
2
2
2
23'
tta
zzyyxx
b
z
a
y
c
x
cba
IU
ttac
dtdzdydxdT
hhhhhh
t ππ
η
πρ
(24) 
and rewritten as: 
( ) ( ) ( )
( ) ( ) ( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅−+−⋅−+−⋅−⋅
⋅
+−⋅
⋅
+−⋅
⋅
+−⋅
⋅⋅⋅
⋅⋅⋅=
2
2
2
2
2
2
222'
'12
3
'12
3
'12
3exp
'12
1
'12
1
'12
1'33
hhh
hhh
t
btta
z
atta
y
ctta
x
cttabttaattac
dtIUdT ππρ
η
  (25) 
Considering that the thermal source moves with constant speed from initial moment t’=0 to the 
moment t’=t, the increase of temperature in this range time can be computed as [2]: 
( ) ( ) ( )
( )
( ) ( ) ( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅−+−⋅−+−⋅
−−⋅
⋅
+−⋅⋅+−⋅+−⋅
⋅⋅⋅
⋅⋅=− ∫
2
2
2
2
2
2
0
2220
'12
3
'12
3
'12
'3exp
'12'12'12
'33
hhh
t
hhh
btta
z
atta
y
ctta
vtx
cttabttaatta
dt
c
IUTT ππρ
η
 (26) 
Appling the equation from above, the analytical solutions can be obtained for three specific cases 
as [2]: 
 
1. If ah=bh=ch=rh, the thermal source becomes a semi-sphere with radius of rh, and the relation (26) 
has the expression: 
( )[ ] ( )( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅
++−−⋅
+−⋅
⋅⋅⋅
⋅⋅=− ∫ 2 22
2
0
2320 '12
33'3exp
'12
'33
h
t
h
rtta
zyvtx
rtta
dt
c
IUTT ππρ
η
. (27) 
When σ= 3rh  the equation (2.40) becomes: 
( )[ ] ( )( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅
++−−⋅
+−⋅
⋅⋅⋅
⋅⋅=− ∫ 2 22
2
0
2320 '4
'exp
'4
'
σσππρ
η
tta
zyvtx
tta
dt
c
IUTT
t
.  (28) 
2. If the distribution parameter is σ=0, the thermal source becomes a point one, the analytical 
solution confirming the solution of Carslaw and Jaeger: 
 
( )[ ]
( )
( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅
++−−⋅−⋅⋅⋅
⋅⋅=− ∫ '4 'exp'4
' 222
0
230 tta
zyvtx
tta
dt
c
IUTT
t
πρ
η
.             (29) 
 
3. If bh=0, then the semi-ellipsoidal thermal source becomes a surface semi-elliptical thermal 
source and the equation (26) is the following: 
( ) ( ) ( )
( )
( ) ( ) ( )⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅−+−⋅−+−⋅
−−⋅
⋅
+−⋅+−⋅−⋅⋅
⋅⋅⋅
⋅⋅=− ∫
'4'12
3
'12
'3exp
'12'12'4
'3
2
2
2
2
2
0
220
tta
z
atta
y
ctta
vtx
cttaattatta
dt
c
IUTT
hh
t
hhπρπ
η
  (30) 
319 
When σ⋅== 6hh ca , the thermal source changes in a circular one. In this case, the equation 
(28) is similar to the solution of Eagar and Tsai: 
( ) ( )[ ] ( )( ) ( )⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅−+−⋅
+−−⋅+−⋅⋅−⋅⋅⋅⋅⋅
⋅⋅=− ∫ '42'4 'exp2'4'4
' 2
2
22
0
20 tta
z
tta
yvtx
ttatta
dt
c
IUTT
t
σσπρπ
η
(31) 
 
3.3. Double ellipsoidal thermal source 
 
Combining two semi-ellipses, Goldak, Chakravarti and Bibby proposed the double ellipsoidal 
thermal source (Figure 4) [4]. For the semi-ellipse located in front of the welding arc, the heat flow 
can be computed in each point of (x, y, z) coordinates using the following mathematical relation:  
⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−−⋅⋅⋅⋅
⋅⋅⋅⋅= 2
2
2
2
2
2 333exp
36
),,,(
hhhfhfhh
f
b
z
a
y
c
x
cba
IUr
zyxQ ππ
η
.   (32) 
For the second semi-ellipse located behind of the welding arc, the heat flow can be computed in 
each point of (x, y, z) coordinates using the following mathematical relation:  
⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−−⋅⋅⋅⋅
⋅⋅⋅⋅= 2
2
2
2
2
2 333exp
36
),,,(
hhhbhbhh
b
b
z
a
y
c
x
cba
IUrzyxQ ππ
η
.   (33) 
In the equations (32) and (33) rf and rb represent the repartition coefficients of the heat in front and 
behind of the thermal source and their mathematical relations are the following [4]: ( ) ( )hbhfhbbhbhfhff cccrcccr +=+= 2,2  ,  (34) 
where the thermal source dimensions are similar to the weld pool geometrical characteristics [4]. 
 
3.4. Temperature field developed by the double ellipsoidal thermal source  
 
In case of the double ellipsoidal distribution of the thermal source, for the volume corresponding to 
the areas located in front and behind of the thermal source, the following equation can be written: 
 
( )[ ]
( ) ( ) ( )
( )
''''3'3'3exp'3'3'3exp
'4
'''exp
'4
'36
4
1
2
2
2
2
2
2
2
2
2
2
2
2
222
23'
dzdydx
b
z
a
y
c
x
c
r
b
z
a
y
c
x
c
r
tta
zzyyxx
ttabac
dtIUdT
hhhbhb
b
hhhfhf
f
hh
t
⋅⋅⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−−⋅+⎟⎟⎠
⎞
⎜⎜⎝
⎛ −−−⋅⋅
⋅⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅
−+−+−−−⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅= ∫ ∫ ∫
∞
∞−
∞
∞−
∞
∞−πππρ
η
(35) 
 
The equation (35) can be adapted as: 
( )( ) ( )( ) ( ) ( ) ⎟⎟⎟⎠
⎞
⎜⎜
⎜
⎝
⎛
+−⋅
+
+−⋅
⋅
+−⋅⋅+−⋅⋅⋅⋅⋅
⋅⋅⋅⋅=
2222' '12'12'12'122
'33
hbhfhh
t
ctta
B
ctta
A
bttaattac
dtIUdT
πρπ
η
  (36) 
( ) ( ) ( )
( ) ( ) ( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅−+−⋅−+−⋅−⋅=
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅−+−⋅−+−⋅−⋅=
2
2
2
2
2
2
2
2
2
2
2
2
'12
3
'12
3
'12
3exp
'12
3
'12
3
'12
3expw
hhhb
b
hhhf
f
btta
z
atta
y
ctta
xrB
btta
z
atta
y
ctta
xrAhere
. 
Similarly, considering that the thermal source moves with constant speed from initial moment t’=0 
to the moment t’=t, the increase of temperature in this range time can be computed as [2]: 
320 
( )( ) ( )( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( ) ⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅−+−⋅−+−⋅
−−⋅=
⎟⎟⎠
⎞
⎜⎜⎝
⎛
+−⋅−+−⋅−+−⋅
−−⋅=
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+−⋅
+
+−⋅+−⋅⋅+−⋅
⋅⋅⋅⋅
⋅⋅⋅=− ∫
2
2
2
2
2
2
2
2
2
2
2
2
22
0
220
'12
3
'12
3
'12
'3exp
'12
3
'12
3
'12
'3exp
'12'12'12'12
'
2
33
hhhb
b
hhhf
f
hbhf
t
hh
btta
z
atta
y
ctta
vtxrB
btta
z
atta
y
ctta
vtxrAwhere
ctta
B
ctta
A
bttaatta
dt
c
IUTT πρπ
η
 (37) 
The equation from above represents the analytical solution of temperature calculus in the transitory 
regime in case of the semi-infinite body which is under the influence of the double ellipsoidal 
thermal source. If chf=chb=ch, the equations (36) and (37) can be written as the equations (25) and 
(26) which describe the temperature variation of the semi-infinite body under the influence of the 
semi-ellipsoidal thermal source.  
 
4. Conclusion 
 Investigations concerning the analytical solutions of the thermal transfer in the welded joints 
in case of 2D and 3D thermal sources are presented in this paper. The analytical models 
don’t take account neither of thermo-physical properties dependent of temperature nor of 
heat lost by convection and radiation.  
 Goldak [4] proposes the different distribution of heat flow in front and behind the thermal 
source and its real dimensions in the analytical modelling of the thermal transfer in the 
welded joints. 
 
Notations  
a – thermal diffusivity; 
ah, bh, chf, chb – ellipsoidal thermal source; 
c – specific heat at ambient temperature; 
cS – specific heat for the solid phase; 
cL – specific heat for the liquid phase; 
∆H – fusion heat per mass unit; 
λ - material thermal conductivity; 
λx, λy, λz – material thermal conductivities on x, y, z directions; 
rf, rb – repartition coefficients of the heat in front and behind of the thermal source; 
σ – distribution parameter specific for the thermal source;  
Vx, Vy, Vz – speed components on x, y, z directions. 
 
References 
 
[1] Kou, S.: Welding Metallurgy, University of Wisconsin, John Wiley & Sons, Inc., 1987. 
[2] Nguyen, N., T., Ohta, A., Matsuoka, K., Suzuki, N., Maeda, Y. – Analytical solutions for 
transient temperature of semi-infinite body subjected to 3-D moving heat sources, Welding Journal, 
August, 1999, pag. 265s-273s. 
[3] Jeong, S., K., Cho, H., S. – An analytical solution to predict the transient temperature 
distribution in fillet arc welds, Welding Journal 76(6), 1997, pag. 223s-232s. 
[4] Goldak,J., Chakravarti, A., Bibby, M. – A double ellipsoid finite element model for welding heat 
sources, IIW Doc. 212-603-85, 1985. 
 


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Analytical models for temperature prediction on welding

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