NUMERICAL SIMULATION OF THIN METAL PLATES
WELDING
1
1
A.N. Cherepanov, 2V.P. Shapeev
Khristianovich Institute of Theoretical and Applied Mechanics SB RAS,
2
Novosibirsk State University,
Novosibirsk, 630090, Russia
INTRODUCTION
In the last years increasing attention has been paid to development of
technology for laser welding of metal products. Laser welding has some advantages
over the other technologies for materials joining, but its widespread expansion is
restrained by the low stability of the weld joint properties. Experimental investigation
and determination of the optimal technological parameters is accompanied by serious
methodological difficulties and considerable expenses due to particularities of the
welding process. In this connection, development of appropriate mathematical models
and numerical algorithms for their implementation is a pressing problem.
Mathematical models are developed in this paper for description of
thermophysical processes at laser welding of metal plates: 1) without use of
nanopowdered modifying agents, and 2) for the case when modifying nanoparticles of
refractory compounds (nitrides, oxides, etc.) are introduced into the weldpool.
Specially prepared nanoparticles serve here as crystallization centers, i.e. in fact they
are seeding agents on which surface individual clusters group. Such a combination of
a nucleus-inoculant and the cluster shell surrounding it should be thermodynamically
stable not only at the crystallization temperature, but at its higher values as well.
The first model is based on quasi-equilibrium description of melting and
crystallization processes in multicomponent alloys with creation of steam channel
taken into account [1]. The second one is based on non-equilibrium emergence and
growth of crystal phase of inoculants which are nanoparticles, with use of
Kolmogorov’s theory [2]. At that, as shown in [3], homogeneous nucleation can be
neglected. The melting process is considered to be quasi-equilibrium.
As an example, simulation results of butt welding of two AL2 alloy plates are
presented. Depth of the steam channel, size of the weldpool, width of the two-phase
zone are calculated.
Fig. 1. Layout of the weld zone: 1 – laser beam, 2 – liquid phase pool, 3 – two-phase
zone, 4 – welding joint.
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Let us introduce Cartesian coordinate system, where the laser beam incident
upon the tight joint of the welded plates is immobile, while the plates move with the
welding speed v. Axis z is directed downward along the beam axis, x axis is oriented
along the joint in the direction of the plates movement, and y axis is perpendicular to
the joint. The coordinate origin is located on the beam axis at the plates upper surface
(Fig. 1).
1. PHYSICOMATHEMATICAL MODEL OF THE PROCESS
The welded plates are blown by inert gas (argon) to protect the alloy from
oxidation. We assume the thermophysical parameters constant and equal to their
average values in the temperature range under consideration. With these assumptions,
the three-dimensional equation of heat transfer in the weldpool and solid metal takes
the form:
  2T  2T  2T 
T

cei v
 i 


 x 2 y 2 z 2  ,
x


T  Te ,
 c11 ,




f

l 
c 2  2 1 
, Te  T  Tl 0 ,

c 2 T 
cei  

 c3  3 ,
Tl 0  T ,


(1)
where ci, i, I are specific heat, heat conductivity, and density of the i-th phase,
respectively (indices i = 1, 2, 3 denote parameters of solid, two-phase, and liquid
states of the metal); Tl0, Te are temperatures of the beginning and ending of
solidification; fl is a share of liquid phase in the two-phase zone;  is latent melting
heat. On the assumption of quasi-equilibrium in the two-phase zone [3], expression
for f l takes the form:
T
f
1
l 
 T 1  k   C
0 0
  0 C0 


 T T 
 A

2k
1 k
(2)
At simulation of the welding process with use of nanopowdered inoculators we
consider all nanoparticles to have spherical form and be the crystallization centers.
Then, for f l we will have [4,5]
f l  e  x, y , z  ,
3


Kv x
4



r
,
z

N
r

 Td   is volume of crystal phase emerged in the
where

p
p
3
v
xl 0


overcooled melt. Here, xl0 is coordinate of point on isotherm with liquidus
temperature Tl0 ( f l Tl 0   1), N p is the number of nanoparticles in the volume unit,
rp is the nanoparticles radius.
According to (1), in the crystallization zone ( Te  T  Tl 0 ) a heat source
emerges in the original equation, related to heat generation at melt crystallization.
Because of nonlinear dependence fl (T ) , in order to take contribution of this heat
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generation into account, the heat conduction equation can be solved iteratively,
adjusting at each iteration f l and, therefore, T in the crystallization zone.
We assume that welding is performed by CO2-laser with wavelength 0=10.6
μm. The radiation intensity is described by the Gaussian normal distribution:
I  x, y, z   I 0 exp 2( r / rF )2 ,
(3)


where I 0  2W /   r 02  ; W is the radiation density; r  x 2  y 2 . Let us write down


expression for the radiation power density absorbed by the work surface in the form:
2 W
p
2

qx, y, Z x, y  
exp   2( r / r )  ,
(4)
F
2


r
where
 p is the absorption coefficient,
F
rF is radius of the focal spot.
Interaction of the welding zone with the environment is described by
corresponding conditions of heat balance on the calculation zone boundaries [1, 2].
They are formulated as boundary conditions for the heat conduction equation. The
main contribution and the critical role in temperature distribution picture in the
welding area belong to heat flux from the laser radiation (4).
2. MODEL OF STEAM CHANNEL FORMATION
On lower and upper surfaces of the plates, outside the steam channel, boundary
conditions take into account heat losses caused by heat irradiation and convective heat
exchange with the environment [5]. In formulation of the model of steam and gas
channel, we make the following simplifying assumptions.
1. We will assume that the steam channel is a surface of revolution relative to an
axis parallel to Z and located in the plane of symmetry, which is monotonously
converging with the depth. This corresponds to experimental observations.
2. The laser beam and steam channel are positioned with respect to each other as
shown in Fig. 3. Namely, the axis of revolution of the steam channel surface is
located one laser beam radius from the laser beam , while at Z=0 (on the plates
surface) the steam channel radius is twice larger than the laser beam radius.
This assumption is also confirmed experimentally with good accuracy in the
cases when the steam channel depth is sufficiently large (e.g., exceeds the
laser beam radius more than thrice for plates made of aluminum alloy AL2).
3. The steam channel bottom has the form of spherical surface with radius,
according to theoretical estimate, defined by expression
2
R 
(5)
c Pmax  3 gh
where σ is coefficient of surface tension for liquid metal (alloy), Pmax is excessive
pressure of the metal vapor, g is the gravitational acceleration, h is depth of the steam
channel..
All made assumptions to some extent are confirmed experimentally in the cases when
the steam channel depth is sufficiently large.
Temperature on the steam channel surface can not exceed that of the alloy
boiling-point (however, it can be lower than the boiling-point temperature on a part of
the channel surface). At that, the maximal heat flux takes place in zone of direct laser
radiation action (laser beam spot). Among all possible generatrices of the steam
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channel surface (which is a surface of revolution according to assumption 1), the line
situated in plane y = 0 on its front wall will be the most “warmed-up” (i.e., according
to the model adopted, of all surface elements this line will have the longest section
with the temperature reaching the boiling-point). Therefore, we will take it as
generatrix of the steam channel surface. When constructing this line, we will use the
following assumption: on its largest possible part near the laser beam axis, the
temperature must be close to the boiling-point temperature Tsat for the alloy.
Fig. 2. Schematic layout of the steam channel and laser beam.
Generatrix AC of vapor channel surface, lying in plane y = 0, is sought in the
form of spline – line consisting of two parts AB and BC (AC = AB U BC). AB
represents a cubic polynomial, tangentially conjugating with BC which is a part of a
circle (see Fig. 3). At that, it is expedient to consider AB belonging to two-parametric
family


3
 r  ch 
(c, h)   x  z  : x  z    F 3   z  h   c  z  h  , 0  c  rF / h, h  0 
 h 


with independent parameters c and h. Lines of this family, under limitations indicated,
pass through point A (see Fig. 3) and possess symmetry property relative to the point
of their intersection with line x=0, y=0 (laser beam axis). At that, parameter h is Zcoordinate of AB symmetry point, while c is tangent of inclination angle θ of line x(z)
to axis Z in the symmetry point
(this is the minimal inclination angle θ of line AC to axis Z). In point B line x(z) is
conjugated with a part of circle which radius is given by formula (5). Surface of the
steam channel with generatrix AC constructed by the described method satisfies
requirements 1) – 3).
Let us assume that we can solve a problem of finding the temperature field in
the plate at a known and fixed shape of the steam channel surface. In this case,
problem of finding this surface can be reduced to the following problem: varying the
channel surface shape it is necessary to find such a solution (i.e. the temperature
distribution) which satisfies the principle: on the largest possible section AC near the
laser beam axis temperature is close to the boiling-point temperature Tboil for the
alloy. In other words, by governing parameters c and h we try to construct line AC
which would be the best from the above-mentioned principle perspective. This is
achieved by the following method.
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Governing parameter h. Let us calculate average temperature Тср on a part of
line AB, symmetrical relative to point of its intersection with Z axis and situated close
to the laser beam axis. If Тср> Tsat, then the melt is “overheated”, and the calculated
depth of the steam channel needs to be increased (which is achieved through increase
of parameter h). At that, the area of the channel surface absorbing direct laser
radiation increases, which in turn leads to decrease of temperature on it. Similarly, if
Тср< Tsat, then h needs to be decreased. The channel depth is adjusted in this way.
Fig. 3. Scheme of vapor channel representation.
Governing parameter c. After finding the proper h, it may turn out that at
intersection of line AC and the beam laser axis the temperature exceeds (or,
alternatively, does not reach) the boiling-point temperature Tboil, while at a short
distance from AC on both sides of the beam axis the opposite picture is observed. This
testifies to a wrong choice of parameter c characterizing the angle of inclination of the
channel walls to axis Z. Through decreasing (or increasing) inclination angle θ, we
achieve that the nearest neighborhood of line AC point lying on the laser beam axis
will receive less (more) heat compared to the periphery. (This owes to the fact that the
power density of the laser radiation has Gaussian distribution (4), while the heat
absorbed by the channel walls from direct laser radiation is proportional to sinθ).
Thus, varying c and h, we find optimal form of the steam channel surface.
Further adjustment of the steam channel wall form is connected with
accounting for heat balance condition and condition of dynamic equilibrium
P( z )  K c  g 3 z, z  Z c on its surface [1, 2]. Here, P ( z ) is pressure on the
wall, K c is its curvature, Zc is coordinate of z point on the surface. Due to smallness
of the steam channel cross-sections sizes in directions perpendicular to Z axis and
significance of  for small K c , the channel surface still can be considered to be a
surface of revolution accurate to values of higher order of smallness. Therefore,
adjustment of its form can be reduced to adjustment of its generatrix. It is shown in [1,
2] that P ( z ) represents a sum of static pressure at surface evaporation Ps(z) and
pressure of return (reaction) Pr . Using simplifying hypothesis, values Ps(z) and Pr
can be expressed through the temperature. As a result, two values T and x(z) remain
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in conditions of heat balance and dynamic equilibrium, which can be adjusted
iteratively.
For numerical solution of the problem, iterative finite-difference scheme of
steadying [5] is used:
 
 
 

n 1 
n 1 
n 1
  I  Lx uijk  I  L y uijk  I  Lz uijk



n
u n 1uijk

n L un L un 
  ijk
  Lx uijk

y ijk
z ijk 




     
(6)
Here, σ is weight parameter ( 0    1 ). For approximation of differential operators
  
x 

 
T   T  
T
T

,  ,

x y  y  z z
x
taken as
the following difference operators were
Lx , Ly , Lz :
Lx T   Lnx T  
Tijk  Ti 1 jk
1  n Ti 1 jk  Tijk
 n 1
  1
i  jk
hx  i  2 jk
hx
hx
2

Ti 1 jk  Ti 1 jk
n
,
   ijk
2hx

(7)
Ly T   Lny T  
1
hy
 n Tij 1k  Tijk
Tijk  Tij 1k
 n 1
  1
ij  k
ij  k
hy
hy
2
 2

 ,

(8)
Lz T   Lnz T  
1
hz
 n Tijk 1  Tijk
Tijk  Tijk 1 
 n 1
  1
 ,
ijk 
ijk 
hz
hz
2
2


(9)
where
  cei v ;

1
i
2
 1 , T  Tl
.
2 , T  Tl
 T T 
   i 1 i  ,
 2 
 T   
(10)
Applying the method of approximate factorization to (6) we obtain difference scheme
 Tijkn 1  Tijkn 
n
n
n
(11)
 I   Lx  I   Ly  I   Lz  
  Lx Tijk  Ly Tijk  Lz Tijk



or equivalent scheme in fractional steps
 n  Lx u n  Ly u n  Lz u n ,


 n 1 
 I   Lx    3    n ,




2
1
n
 n 3 

 I   Ly      3 ,
(12)



2

n
 I   Lz   n 1   3 ,

u n 1  u n   n 1.


With its help, temperature distribution in the plates is determined, and location of
internal boundary between phases is found. On the other hand, the steam channel
surface is constructed also iteratively by the method described above. In the computer


  


  

 



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 
 
 
program realized these iterative processes are combined. Namely, after each iteration
of the scheme, the steam channel surface form is corrected. At that, h can take values
divisible by the grid step along Z axis (i.e. the generatrix of steam and gas channel in
cross-section y=0 passes through a grid node lying on Z axis), and it is changed in the
case if
T  Tboil 
1
T
 hz
2
z
x  0,
y  0,
z h
,
where T is the average temperature on the steam and gas channel surface in plain y=0
near the laser beam axis, μ is parameter controlling “passage” boundaries, within
which value h is not changed. Such a mechanism of discrete (“stepped”) variation of
parameter h is realized in order to avoid small oscillations of the channel shape, which
interfere with establishing of the temperature field. Variation of parameter c is also
performed discretely in a similar way.
The above-described method for constructing the channel surface was coded
and allowed us to automate process of its formation. Owing to it, the generatrix of its
surface is constructed in one simulation with the temperature field.
3. RESULTS OF NUMERICAL SIMULATION
The results include temperature fields in the item, location of internal
boundaries between phases of the item material, shape and depth of the steam
channel. The calculations results allow one to predict sizes of zones occupied by
various phases during welding and to determine which speed of moving laser beam
along the welding seam at a given laser power can provide a sufficiently large size of
liquid phase zone without a through steam channel (without breakdown of the item by
the laser beam). Numerical simulations on a sequence of grids with decreasing steps
were carried out, first of all, in a calculation domain with fixed length. Calculation
parameters of the problem were observed (temperature distribution, isotherms
location, boundaries of the steam channel and those between phases of the material).
First order of their convergence was obtained in calculations on a sequence of grids
with decreasing of grid steps.
Results of calculation of temperature fields in liquid pool and adjacent layers
of solid alloy are shown in figures below. The role of length scale along all axes is
played by the beam radius rF in focal plane. A picture in a small calculation domain
is presented for the purposes of obviousness. Numerical calculations are carried out
for alloy Аl + 10% Si (% of mass) at the same thermophysical parameters as in [1, 2]
in the problem with creation of steam channel: 1=155,7 W/(mK); 2=127,85
W/(mK); 3=100 W/(mK); c1=1000 J/(kgK); c2=1050 J/(kgK); c3=1100 J/(kgK);
1=2,6103 kg/m3; 2=2,45103 kg/m3; 3=2,3103 kg/m3; ТА=933 K; Tl0=862 K;
Tl=850 K; =5,37105 J/kg; L=1,11107 J/kg; C1=10 % (of mass); 1=7,1 K/%;
k1=0,14; A=0,65; xF=10-4 m; zF=0; Tg=293 K; g=0,024 W/(mK); g=1,3710-5 m2
/s; T1v=2720 K; T2v=2628 K; A1=33,294; B1=37723,14 K; A2=43,584;
B2=63590,782 K; P10=P20=1 N/m2; =0,57 N/m;  1=0,176; 2=3 = 0,18; vg = 0,5
m/s; l =0,1 m; b = 0,55, KV = 0,05; ED =0,78 10-20 Дж; D0 =5,8 10-5 m 2/с, kB = 1,38
10-23 Дж; mp= 0,05; rp= 10-7 m; ρp = 3500 кг/ m 3, la= a0 =4,5 10-10 m,
3
N p  m p 3 /( rp  p ) ,1/ m 3.
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Example: Laser power W=3.18kW, welding rate v=4.7 m/min, plate thickness
h=1.5mm, 350 × 200 × 150 – number of steps along x, y, and z.
Fig. 4. Temperature field and isotherms in calculation domain (cross-section y=0)
1 – steam channel, 2 – 2629.1K (boiling-point temperature), ), 3 – T = 2155 K, 4 – T
= 1724 K, 5 – Tl0 = 862 K (liquidus temperature, 6 – T = 420 K.
Fig. 5. Temperature field and isotherms in calculation domain (view from above,
plane z=0)). 1 – steam channel, 3. - T = K, 4 – T = 1293 K, 5 – Tl0 = 862 K (liquidus
temperature), 6 – T = 560.3 K.
Fig. 6. Temperature field and isotherms in calculation domain
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(cross-section x=0.6 mm, corresponding to the maximal width of the liquid pool):
4 – 1293 K, 5 – Tl0 = 862 K (liquidus temperature), 6 – T = 560.3 K.
One can see from the figures that the most warmed-up areas as well as the
largest temperature gradients in calculation domain are located near the steam channel
surface. It also follows from calculations that area of the largest temperature gradients
is located in neighborhood of the channel front wall (in the melting area). Isothermal
surfaces on the left and right from the channel (in the areas of cooling and
solidification of the melt) are convex and their curvature decreases with their distance
from the channel, i.e. the temperature field in the plates levels out. One can also
notice that the temperature on a sufficiently extensive part of the front surface of the
steam channel in neighborhood of the laser axis is close to the boiling-point, which is
in agreement with the above-formulated assumptions made at description of the
channel construction algorithm.
Selecting from the calculations a cross-section of the liquid pool by plane
x=const corresponding to its maximal width (on the plates upper surface), one can
compare the calculated size and shape of the region occupied by liquid metal with
experimentally observed sizes
of the weld seam obtained at
the same welding parameters as in the simulation.
Example of such a comparison is shown in Fig. 7.
a
b
Fig. 7. a: Photo of microsection of the weld seam in the plane orthogonal to the weld
seam (the boundary of solidified pool is seen). b: section of the computational domain
with the maximal pool width, orthogonal to X axis (pool region is marked with a
brighter color).
Comparison of cross-section sizes of weld seam (Fig. 7a) and simulated liquid pool
(Fig. 7b) shows their satisfactory agreement. Difference of a somewhat concave shape
of the experimental joint from the convex shape of the welding pool, obviously, is
caused by the shrink processes which are not taken into account in the model.
Therefore, considering the simplifying assumptions made in the problem statement,
the results of comparison can be considered satisfactory.
Particularity of alloy crystallization with a modifying nanopowder is
overcooling taking place in the region of emergence and growth of crystal phase, that
causes non-monotony of the temperature variation along the x coordinate (Fig. 8).
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Fig. 8. Non-dimensional temperature profile in the crystallization zone. The x axis
18
19
begins at xl 0 . Curve 1 corresponds to N p = 10 m-3, curve 2 – to N p = 10 m-3.
Maximal value of overcooling is ~5 K and it depends on nanoparticles
quantity N p in the melt volume unit. Due to large specific surface of nanoparticles a
fine-grained structure is formed in the melt. The crystals form is essentially changed –
it becomes equiaxed instead of acicular dendritic one. Therefore, the joint elasticity
and breaking strength are increased.
a
b
Fig. 9. Structure of weld seam of carbon steel
without modifying powder (a) and with powder (b).
Figure 9 shows photos of test samples of welded joints obtained by laser
welding of steel samples without modifying agents (a) and with application of mixture
of titanium nitride and yttrium oxide nanopowders (b).
The work was supported by RFBR grants 06-01-00080-а and 08-08-00249.
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