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Numerical and experimental investigation of seam welding with a pulsed laser
ARTICLE IN PRESS
Optics & Laser Technology 40 (2008) 289–296 www.elsevier.com/locate/optlastec
Numerical and experimental investigation of seam welding with a pulsed laser Jamshid Sabbaghzadeh, Maryam Azizi, M. Javad Torkamany Paya Partov Laser Research Center, P.O. Box 14665-576, Tehran, Iran Received 11 November 2006; received in revised form 24 May 2007; accepted 27 May 2007 Available online 16 July 2007
Abstract The present work develops two numerical models to compute thermal phenomena during pulsed laser welding. The first one which is based on finite difference model calculates the welding width and its penetration by utilizing heat transfer equations. Parametric design capabilities of the finite element code ANSYS were also employed for the simulation of the second model. The transient temperature profiles, the fusion dimensions and the heat affected zones (HAZ) have been calculated here. The thermo-physical properties are dependent on temperature and so a nonlinear solution is employed. It is found that the temperature profile and penetration depth are strongly dependent on the pulse parameters of laser beam. Finally, the results of the two models and the experimental outcomes are compared. r 2007 Elsevier Ltd. All rights reserved. Keywords: Nd:YAG pulsed laser; Finite difference model (FDM); Finite element model (FEM)
1. Introduction The theoretical and experimental study of laser welding began in 1962 [1]. However, laser welding using a CO2 laser source was first reported in 1971 [2]. Since then, the use of laser welding has grown swiftly, as the new manufacturing possibilities became better understood. Although it is possible to use CW lasers in welding process, the use of a pulsed laser offers the advantages of very low distortion and the ability to welding heat-sensitive components. During the pulsed laser welding process, the work piece is heated up to the melting point consecutively by shortduration pulses, producing a series of overlapping welding spots. The physical phenomena responsible for the welding spots in a low- or intermediate laser-power process may be simple, but due to vaporizing effects and keyhole formation, the welding process with a high power laser is a complicated phenomena and its analytical modeling is almost impossible. Without a comprehensive understanding of the physical phenomena associated with the welding Corresponding author. Tel.: +98 21 77630161.
E-mail address: [email protected] (M. Azizi). 0030-3992/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2007.05.005
process, the potential of lasers cannot be completely realized. Contrary to continuous laser welding processes where quasi-steady-state temperature distributions are established, the pulse process is characterized by a series of individual overlapping spot welds which result in very fast temperature variation in the welding area. The difficulties associated with analytical models of the laser keyhole welding may be partially overcome by employing numerical techniques. So numerical methods with very large number of small time steps (much smaller than those needed for continuous laser welding analysis) have to be used. There are few models that explain the physical mechanisms of welding processes using a pulsed laser [3–5]. The elementary models of laser welding were mostly based on Rosenthal’s solutions [6]. A numerical model for continuous laser welding based on heat transfer equations and the finite difference technique was developed by Solana and Ocana [7]. Since the complete data of temperaturedependent thermo-physical properties are difficult to obtain, most numerical modeling efforts have used constant or approximated values for these parameters [8]. In Ref. [4] a numerical model of laser spot welding for pulse durations greater than 150 ms has been developed.
ARTICLE IN PRESS 290
J. Sabbaghzadeh et al. / Optics & Laser Technology 40 (2008) 289–296
The analysis was based on conduction heat transfer. A two-dimensional asymmetric finite element analysis of heat flow during laser spot welding was employed, taking into account the temperature dependence of the physical properties and latent heat of transformations. In Ref. [5], a transient numerical model was developed to have a better understanding of heat transfer and fluid flow during single pulse laser welding of stainless steel. They considered surface tension and buoyancy forces to calculate transient convection of weld pool and temperature profiles. A more recent mathematical model for linear welding with a pulsed laser was introduced by DebRoy group in Penn State University (see Ref. [8]), which uses constant laser absorption coefficient and neglects the temperature dependency of thermo-physical properties during steel 304 welding. In our work, two numerical three-dimensional models based on thermal transient phenomenon are developed to describe seam welding with a pulsed laser. The laser light is absorbed by plasma inside the keyhole [9] by means of inverse bremsstrahlung phenomenon; however, some part of the light reflects many times between the walls of the keyhole and is absorbed by Fresnel absorption [7]. In the first numerical model we used finite difference model (FDM) and began with Rosenthal’s solution. Temperature dependency of thermo-physical properties was considered in finite element model (FEM) as the second numerical model which was coupled with parametric design potentials of the ANSYS code. In both models, we made some simplifications to compute the results in a reasonable time, and avoiding loss of physical insight with meaningless complexity of the equations. Finally, the results of the two models were compared with each other as well as with some experimental results which were carried out in our laboratory. 2. Model description After the absorption of the laser light by the material, the transfer and penetration of the heat into the bulk material is based on conduction, convection and radiation. Following the approach of P. Solana and J.L. Ocana, in the FDM model we considered only the conduction heat transfer, since it is the major phenomenon in our application and can be reasonably ignored. This is because the conductivity of the metal work piece is very high and the heat transfer through the surrounding area happens faster than convection. However, in the FEM model we considered both conduction and convection processes, since the formation of the plasma is needed. The welding process in the FDM model is basically assumed to be quasi-stationary. The justification for this assumption is that the shape of the keyhole and the temperature counters can be found from the timeindependent heat-conduction equation when the temperature of the initial condition is given. This is why we used the average power in our simulation. The origin of an x–y–z coordinate system was considered at the center of the laser
z
laser beam x velocity of workpiece, U
y spot weld
Fig. 1. Schematic model for laser welding. The laser beam and the coordinate system are fixed and the work piece moves at velocity U.
beam on the work piece surface. The depth of the work piece was aligned in z direction and increases with decreasing z. The work piece moves in the positive x direction with a constant velocity U as shown in Fig. 1. 2.1. Finite difference model (FDM) A numerical FDM is applied to solve the threedimensional steady-state heat-conduction problem with free keyhole boundary. The time-dependent heat-conduction equation for solid and liquid phase is q qT q qT q qT qT K K K , þ þ þ q_ ¼ rC qx qx qy qy qz qz qt (1) where K is the heat-conduction coefficient, r the density, C the specific heat, T the temperature, q_ the heat source per unit volume and x; y; z the coordinate system. In the same way, the following equation can be written for plasma phase: q qT q qT q qT Kg Kg Kg þ þ þ q_ qx qy qz qx qy qz qT , ð2Þ ¼ rg C pg qt where K g ; rg ; C pg ; T are relatively the heat-conduction coefficient, density, specific heat and temperature for plasma phase [7]. The boundary conditions are defined as follows: K g rT qabl ¼ KrTjG , T jG ¼ TjG ¼ T s , Lim T ¼ T 0 ,
ð3Þ
ðx;y!1Þ
where G; T 0 ; qabl are correspondingly keyhole boundary, ambient temperature and intensity ablation losses. The problem consists of solving the system of equations (1) and (2) with boundary conditions of Eq. (3). The independent unknown function of the problem is the temperature distribution both in the liquid/solid metal and in the plasma i.e. Tðx; y; zÞ and T ðx; y; zÞ. For evaluating ablation losses [7] in Knudsen layer near the plasma–liquid boundary conditions, we have to use the analysis of the pressure and energy balance.
ARTICLE IN PRESS J. Sabbaghzadeh et al. / Optics & Laser Technology 40 (2008) 289–296
291
x 10-4 4
x 10-3
3
z (m)
y (m)
2 1 0 -1 -2
0 -0.5 -1 -1.5 -2 -2.5 4 x 10-4
-3 -5
-4
-3
-2
-1 0 x (m)
1
2 3 x 10-4
2 0 -2 y( m) -4 -6
-4
-2 )
x (m
2 0 x 10-4
Fig. 2. (a) The cross-section of the keyhole for pulsed laser beam (11 J, 3 ms) and welding speed 1 cm/s. (b) Keyhole profile for the pulsed laser beam (11 J, 3 ms), focusing radius of 200 mm on steel 304, welding speed 1 cm/s.
Table 1 Chemical composition of st14 sheet metal vs. the weight percentage C
Si
Mn
P
Cr
Ni
S
Fe
0.04
0.01
0.21
0.007
0.006
0.03
0.007
Balance
where z is the average ionic charge in the plasma, h is Plank’s constant, c is the speed of light, me is the electron mass, o is the angular frequency, e0 is the permittivity of free space, ni is the ion density , ne is the electron density and g is the mechanical Gaunt factor. The Fresnel reflection factor is calculated against the incident angle y using the relation [11]
Table 2 Thermo-physical properties used in the finite difference model
Rz ¼ Temperature (1C)
Thermal conductivity (W/mK)
Specific heat (J/kg m3)
Density (kg/m3)
25
16.26
460
7818
(5) where e is the electric conductivity of metal that depends on the metal and laser specifications and is calculated by the following relation: 2 ¼
Fig. 3. The experimental result which shows the width of the welding area.
The heat is absorbed in the work piece by means of inverse bremsstrahlung and Fresnel absorption in the plasma within the keyhole. The inverse bremsstrahlung absorption coefficient is given by the following expression [10]: ne ni z2 e6 2p me 1=2 aðm1 Þ ¼ pffiffiffi 3 6 3m0 cho3 m2e 2pKT e _o 1 exp g, KT e
1 1 þ ð1 Cos yÞ2 2 2 Cos y þ 2 Cos2 y þ , 2 1 þ ð1 þ Cos yÞ2 2 þ 2 Cos y þ 2 Cos2 y
22 0 þ ½21 þ ½sst =ðo0 Þ2 1=2
.
(6)
Here, e0 is the electric susceptibility of vacuum and equals 8.85 102 C2/N m2, e1 and e2 are the real parts of the dielectric constants for the metal and the keyhole plasma, sst is the electric conductance per unit depth and o is the angular frequency of the laser light. Before we solve the equation, it is necessary to know the boundary limit of the keyhole. In this case, the keyhole geometry can be described by curves of practical ovoid shape as shown in Fig. 2. Here, the Rosenthal model is used [12], and the computer program is written by utilizing MATLAB software. The material used for welding was 1 mm thick st14 metal. The chemical composition of this metal is presented in Table 1.1 All thermo-physical properties which are listed in Table 2 are considered to be temperature independent. The laser source was Nd:YAG laser of 400 W average power. We used optimum mesh points which are available as an option in the software, and the following
ð4Þ 1
Refer to site http://www.engineersedge.com/properties_of_metals.htm
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Table 4 Input data for set number 1
Temperature (K)
5000 4000 3000 2000 1000
Energy (J)
Pulse duration (ms)
Frequency (Hz)
11 11 11 11 11 11
3 5 6 7 8 9
20 20 20 20 20 20
0 5 x10-4
5 0
Y(
m)
0 -5 -5
-10
x 10
-4
)
Table 3 Thermo-physical properties used in the finite element model Temperature (1C)
Thermal conductivity (W/mK)
Specific heat (J/kg m3)
Emissivity
Density (kg/m3)
0 75 100 175 200 225 275 300 325 375 400 475 500 575 600 675 700 730 750 775 800 1000 1500 1540 1690 1840 1890 2860
51.9 51.3 51.1 49.5 49 48.3 46.8 46.1 45.3 43.6 42.7 40.2 39.4 36.6 35.6 32.8 31.8 30.1 28.9 27.5 26 27.2 29.7 29.7 29.7 29.7 29.7 29.7
450 486 494 519 526 532 557 566 574 599 615 662 684 749 773 846 1139 1384 1191 950 931 779 400 400 847 847 400 400
0.2 0.35 0.4 0.44 0.45 0.46 0.47 0.48 0.48 0.5 0.51 0.53 0.54 0.55 0.56 0.57 0.57 0.58 0.58 0.58 0.58 0.59 0.6 0.6 0.6 0.6 0.6 0.62
7872 7852 7845 7824 7816 7809 7763 7740 7717 7727 7733 7720 7711 7680 7669 7636 7625 7612 7602 7590 7578 7552 7268 7218 7055 6757 6715 5902
laser energy: 11 J, pulse duration: 3 ms, welding speed: 1 cm/s, air convection coefficient: 50 W/m2 C,
ambient temperature: 25 1C, height of work piece: 0.7 mm.
X (m
Fig. 4. Temperature profile of keyhole for a pulsed laser beam (11 J, 3 ms), focusing radius of 200 mm on steel 304, welding speed 1 cm/s.
data were used:
The results of the numerical calculation of FDM are shown in Fig. 2. It is worth nothing to mention that the slight deviation of the center of the ovoid from the laser source arises from translational speed of the laser beam and increases with speed. The calculated width is about 1 mm, which was found at the same conditions. Fig. 2 shows that for the work piece with 0.7 mm height the penetration is complete. Fig. 3 shows the experimental result which is in good agreement with simulation (Fig. 4). 2.2. Finite element model (FEM) The FEM yields a set of simultaneous equations: ½Kfug ¼ fF a g,
(7)
where [K] is the coefficient matrix, {u} is the vector of unknown degree of freedom (DOF) values and {Fa} is the vector of applied loads. If the coefficient matrix [K] is a function of unknown DOF values (or their derivatives), then Eq. (7) is a nonlinear equation. The Newton–Raphson method is an iterative process of solving the nonlinear equations and can be written as follows: ½K Ti fDui g ¼ fF a g fF nr i g,
(8)
fuiþ1 g ¼ fui g þ fDui g,
(9)
T
where [Ki ] is the Jacobian matrix (tangent matrix), i is the subscript representing the current equilibrium iteration and fF nr i g is the vector of restoring loads corresponding to the element internal loads. Both [KiT] and fF nr i g are evaluated based on the values given by {ui}. The right-hand side of Eq. (8) is a residual or out-of-balance load vector; i.e. the amount of the system is out of equilibrium. In a thermal analysis, [KiT] is the conductivity matrix, {ui} is the temperature vector and fF nr i g is the resisting load vector calculated from the element heat flows [13]. FEM has the advantage that in meshing model we have many options for choosing elements. To reduce computation time, the simulation of this part has been done using ANSYS code, and the
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Fig. 5. (a) Temperature profile in three dimensions of stainless steel, energy ¼ 11 J, pulse duration ¼ 3 ms, welding speed ¼ 1 cm/s. (b) Penetration depth profile, energy ¼ 11 J, pulse duration ¼ 6 ms, welding speed ¼ 1 cm/s.
Fig. 6. The experimental result for the welding shape and its depth, energy ¼ 11 J, pulse duration ¼ 3 ms. As it can be seen, it confirms that at the above conditions the laser beam would have a destructive effect.
Fig. 8. The temperature variation of plasma plume with pulse duration.
Fig. 7. The experimental result for the welding shape and its depth, energy ¼ 11 J, pulse duration ¼ 6 ms. As it is clear, in the above conditions, the welding quality is better and satisfactory.
following assumptions were considered:
The laser beam enters to the work piece vertically. Both the laser beam and the coordinate system are fixed and the work piece moves in the positive x direction with a constant velocity U.
All thermo-physical properties are temperature dependent, and are shown in Table 3. Air convection coefficient is 50 W/m2 C. The ambient temperature is 25 1C.
The spatial and temporal temperature distribution T(x,y,z,t) satisfies the following differential equation for the threedimensional heat conduction: q qT q qT q qT Kx Ky Kz þ þ þQ qx qx qy qy qz qz qT qT U , ð10Þ ¼ rC qt qx where (x,y,z) is the coordinate system attached to the heat source, Q is the power generation per unit volume, K x ; K y ; K z are the thermal conductivities in the x; y; z directions, C is the specific heat capacity, r is the density, t is the time and U is the velocity of the work piece. The program ran for two sets of input data. In the first one the pulse duration was variable as in Table 4 and the results are shown in Fig. 5.
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Fig. 9. (a) The variations of the temperature for the lower layer of the work piece relative to the pulse duration. (b) The variations of the temperature for the upper layer of the work piece relative to the pulse duration. (c) The variations of the weld penetration of the work piece relative to the pulse duration.
Table 5 Input data for set number 2 Energy (J)
Pulse duration (ms)
Frequency (Hz)
12 11 10 9 8
6 6 6 6 6
20 20 20 20 20
Fig. 10. (a) Temperature profile in three dimensions of stainless steel, energy ¼ 12 J, pulse duration ¼ 6 ms, welding speed ¼ 1 cm/s. (b) Penetration depth profile, energy ¼ 12 J, pulse duration ¼ 6 ms, welding speed ¼ 1 cm/s.
Fig. 11. (a) The variation of the temperature for the lower layer of the work piece relative to the laser energy. (b) The variations of the temperature for the upper layer of the work piece relative to the laser energy. (c) The variations of the weld penetration of the work piece relative to the laser energy.
As it can be seen in Fig. 5, for the lowest pulse duration (3 ms) we have the maximum temperature in the top layer of the work piece. In this case the temperature is almost 5667 1C, which is higher than vapor temperature of stainless steel. Therefore, some of the metal infused out of the keyhole and the surface of the welding area will not
remain flat. In this case the overlapping of the pulses is so high. With increasing pulse duration the temperature and penetration depth are decreased as can be seen in Figs. 6–9. In other words, when the pulse duration reaches to 9 ms, the keyhole is not formed and the penetration is not completed. In the pulse duration equal to 5 or 6 ms, we
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Fig. 12. (a) Temperature profile in three dimensions for stainless steel, energy ¼ 11 J, pulse duration ¼ 3 ms, welding speed ¼ 1 cm/s, work piece depth ¼ 1.5 mm. (b) Penetration depth profile, energy ¼ 11 J, pulse duration ¼ 3 ms, welding speed ¼ 1 cm/s, work piece depth ¼ 1.5 mm.
1st Pulse
3rd Pulse
2nd Pulse
4th Pulse
3. Conclusions
5000 4500
Temperature (K)
4000 3500 3000 2500 2000 1500 1000 500 0 0
0.04
0.08
0.12
0.16
0.2
Time (s) Fig. 13. Evaluations of peak temperatures at four monitoring locations on the surface of the work piece. The four locations correspond to the middle of the laser beam spot on the work piece surface at the onset of the first, second, third and fourth pulses.
Three-dimensional numerical models of the heat flow during the pulsed laser welding have been developed using the FDM and the FEM. The results of the models have been compared with the experimental results. A very good agreement between theory and experiment has been achieved. The results clearly demonstrate the importance of the plasma formed above the weld area. The absorptivity of the Nd:YAG laser radiation by the work piece is clearly a complex function of a number of variables such as surface temperature, beam power density, focal position of the laser beam relative to the work piece surface. Full thermal histories of the weld have also been generated. A more direct check of the theoretical work, apart from the measurement of weld dimensions and the consequent estimations of temperature contours, was the measurement of temperature as a function of time during laser welding.
References have a good welding quality. In these cases the penetration is completed and the work piece surface is flat, and confirmed by the experimental results shown in Figs. 6–8. Fig. 8 explains that when the pulse duration increases, the temperature of the plasma plume is increased, which consequently causes the temperature of the work piece to go up [14]. In set number 2, the input energy was considered variable as in Table 5, and the results have been shown in Figs. 10 and 11. In the fixed pulse duration, the maximum temperature is linearly increased by the energy of the pulse, and when the energy reaches 10 J, the penetration will be completed. The simulation results show that we can weld two plates with 1.5 mm depth. Maximum and minimum temperatures for these conditions are 5667 and 2698 1C (Fig. 12a). We will also have completed penetration in this case (Fig. 12b). Fig. 13 shows the computed thermal cycles at four monitoring locations on the specimen surface in the middle of the laser spots for the first four pulses. The preheating of the monitoring locations from previous pulses is apparent.
[1] Dunlap GW, Williams DI. Proc Natl Electron Conf 1962;18:601. [2] Banas CM. UARL Report 125, June 1971. In: The 11th IEEE symposium on electron ion, laser beam technology. San Francisco: San Francisco Press. [3] Jouvard JM, Girard K, Perret O. Keyhole formation and power deposition in Nd:YAG laser spot welding. J Phys D 2001;34: 2894–901. [4] De A, Maiti SK, Walsh CA, Bhadeshia HKDH. Finite element simulation of laser spot welding. Sci Technol Weld Joining 2003;8(5): 377–84. [5] He X, Fuerschbach PW, DebRoy T. Heat transfer and fluid flow during laser spot welding of 304 stainless steel. J Phys D 2003;36: 1388–98. [6] Rosenthal D. The theory of moving sources of heat and its application to metal treatments. Trans ASME 1946;43(11):849–66. [7] Solana P, Ocana JL. A mathematical model for penetration laser welding as a free-boundary problem. J Phys D 1997;30:1300–13. [8] Zhang W, Roy GG, Elmer JW, DebRoy T. Modeling of heat transfer and fluid flow during gas tungsten arc spot welding of low carbon steel. J Appl Phys 2003;93(5):3022–33. [9] Torkamany MJ, Hamedi MJ, Malek F, Sabbaghzadeh J. The effect of process parameters on keyhole welding with a 400 W Nd:YAG pulsed laser. J Phys D Appl Phys 2006;39:4563–7. [10] Duley WW. Laser welding. New York: Wiley; 1999.
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[11] Ducharme R, Williams K, Kapadia P, Dowden J, Steen B, Glawacki M. A mathematical investigation of the penetration depth in keyhole welding with continuous CO2 laser. J Phys 1994;27:1619–27. [12] Steen WW. Laser material processing. 1998.
[13] ANSYS revision 9.0 user manual [chapter 15]. [14] Sabbaghzadeh J, Dadras S, Torkamany MJ. Comparison of pulsed Nd:YAG laser welding qualitative features with plasma plume thermal characteristics. J Phys D Appl Phys 2007;40:1047–51.
Optics & Laser Technology 40 (2008) 289–296 www.elsevier.com/locate/optlastec
Numerical and experimental investigation of seam welding with a pulsed laser Jamshid Sabbaghzadeh, Maryam Azizi, M. Javad Torkamany Paya Partov Laser Research Center, P.O. Box 14665-576, Tehran, Iran Received 11 November 2006; received in revised form 24 May 2007; accepted 27 May 2007 Available online 16 July 2007
Abstract The present work develops two numerical models to compute thermal phenomena during pulsed laser welding. The first one which is based on finite difference model calculates the welding width and its penetration by utilizing heat transfer equations. Parametric design capabilities of the finite element code ANSYS were also employed for the simulation of the second model. The transient temperature profiles, the fusion dimensions and the heat affected zones (HAZ) have been calculated here. The thermo-physical properties are dependent on temperature and so a nonlinear solution is employed. It is found that the temperature profile and penetration depth are strongly dependent on the pulse parameters of laser beam. Finally, the results of the two models and the experimental outcomes are compared. r 2007 Elsevier Ltd. All rights reserved. Keywords: Nd:YAG pulsed laser; Finite difference model (FDM); Finite element model (FEM)
1. Introduction The theoretical and experimental study of laser welding began in 1962 [1]. However, laser welding using a CO2 laser source was first reported in 1971 [2]. Since then, the use of laser welding has grown swiftly, as the new manufacturing possibilities became better understood. Although it is possible to use CW lasers in welding process, the use of a pulsed laser offers the advantages of very low distortion and the ability to welding heat-sensitive components. During the pulsed laser welding process, the work piece is heated up to the melting point consecutively by shortduration pulses, producing a series of overlapping welding spots. The physical phenomena responsible for the welding spots in a low- or intermediate laser-power process may be simple, but due to vaporizing effects and keyhole formation, the welding process with a high power laser is a complicated phenomena and its analytical modeling is almost impossible. Without a comprehensive understanding of the physical phenomena associated with the welding Corresponding author. Tel.: +98 21 77630161.
E-mail address: [email protected] (M. Azizi). 0030-3992/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2007.05.005
process, the potential of lasers cannot be completely realized. Contrary to continuous laser welding processes where quasi-steady-state temperature distributions are established, the pulse process is characterized by a series of individual overlapping spot welds which result in very fast temperature variation in the welding area. The difficulties associated with analytical models of the laser keyhole welding may be partially overcome by employing numerical techniques. So numerical methods with very large number of small time steps (much smaller than those needed for continuous laser welding analysis) have to be used. There are few models that explain the physical mechanisms of welding processes using a pulsed laser [3–5]. The elementary models of laser welding were mostly based on Rosenthal’s solutions [6]. A numerical model for continuous laser welding based on heat transfer equations and the finite difference technique was developed by Solana and Ocana [7]. Since the complete data of temperaturedependent thermo-physical properties are difficult to obtain, most numerical modeling efforts have used constant or approximated values for these parameters [8]. In Ref. [4] a numerical model of laser spot welding for pulse durations greater than 150 ms has been developed.
ARTICLE IN PRESS 290
J. Sabbaghzadeh et al. / Optics & Laser Technology 40 (2008) 289–296
The analysis was based on conduction heat transfer. A two-dimensional asymmetric finite element analysis of heat flow during laser spot welding was employed, taking into account the temperature dependence of the physical properties and latent heat of transformations. In Ref. [5], a transient numerical model was developed to have a better understanding of heat transfer and fluid flow during single pulse laser welding of stainless steel. They considered surface tension and buoyancy forces to calculate transient convection of weld pool and temperature profiles. A more recent mathematical model for linear welding with a pulsed laser was introduced by DebRoy group in Penn State University (see Ref. [8]), which uses constant laser absorption coefficient and neglects the temperature dependency of thermo-physical properties during steel 304 welding. In our work, two numerical three-dimensional models based on thermal transient phenomenon are developed to describe seam welding with a pulsed laser. The laser light is absorbed by plasma inside the keyhole [9] by means of inverse bremsstrahlung phenomenon; however, some part of the light reflects many times between the walls of the keyhole and is absorbed by Fresnel absorption [7]. In the first numerical model we used finite difference model (FDM) and began with Rosenthal’s solution. Temperature dependency of thermo-physical properties was considered in finite element model (FEM) as the second numerical model which was coupled with parametric design potentials of the ANSYS code. In both models, we made some simplifications to compute the results in a reasonable time, and avoiding loss of physical insight with meaningless complexity of the equations. Finally, the results of the two models were compared with each other as well as with some experimental results which were carried out in our laboratory. 2. Model description After the absorption of the laser light by the material, the transfer and penetration of the heat into the bulk material is based on conduction, convection and radiation. Following the approach of P. Solana and J.L. Ocana, in the FDM model we considered only the conduction heat transfer, since it is the major phenomenon in our application and can be reasonably ignored. This is because the conductivity of the metal work piece is very high and the heat transfer through the surrounding area happens faster than convection. However, in the FEM model we considered both conduction and convection processes, since the formation of the plasma is needed. The welding process in the FDM model is basically assumed to be quasi-stationary. The justification for this assumption is that the shape of the keyhole and the temperature counters can be found from the timeindependent heat-conduction equation when the temperature of the initial condition is given. This is why we used the average power in our simulation. The origin of an x–y–z coordinate system was considered at the center of the laser
z
laser beam x velocity of workpiece, U
y spot weld
Fig. 1. Schematic model for laser welding. The laser beam and the coordinate system are fixed and the work piece moves at velocity U.
beam on the work piece surface. The depth of the work piece was aligned in z direction and increases with decreasing z. The work piece moves in the positive x direction with a constant velocity U as shown in Fig. 1. 2.1. Finite difference model (FDM) A numerical FDM is applied to solve the threedimensional steady-state heat-conduction problem with free keyhole boundary. The time-dependent heat-conduction equation for solid and liquid phase is q qT q qT q qT qT K K K , þ þ þ q_ ¼ rC qx qx qy qy qz qz qt (1) where K is the heat-conduction coefficient, r the density, C the specific heat, T the temperature, q_ the heat source per unit volume and x; y; z the coordinate system. In the same way, the following equation can be written for plasma phase: q qT q qT q qT Kg Kg Kg þ þ þ q_ qx qy qz qx qy qz qT , ð2Þ ¼ rg C pg qt where K g ; rg ; C pg ; T are relatively the heat-conduction coefficient, density, specific heat and temperature for plasma phase [7]. The boundary conditions are defined as follows: K g rT qabl ¼ KrTjG , T jG ¼ TjG ¼ T s , Lim T ¼ T 0 ,
ð3Þ
ðx;y!1Þ
where G; T 0 ; qabl are correspondingly keyhole boundary, ambient temperature and intensity ablation losses. The problem consists of solving the system of equations (1) and (2) with boundary conditions of Eq. (3). The independent unknown function of the problem is the temperature distribution both in the liquid/solid metal and in the plasma i.e. Tðx; y; zÞ and T ðx; y; zÞ. For evaluating ablation losses [7] in Knudsen layer near the plasma–liquid boundary conditions, we have to use the analysis of the pressure and energy balance.
ARTICLE IN PRESS J. Sabbaghzadeh et al. / Optics & Laser Technology 40 (2008) 289–296
291
x 10-4 4
x 10-3
3
z (m)
y (m)
2 1 0 -1 -2
0 -0.5 -1 -1.5 -2 -2.5 4 x 10-4
-3 -5
-4
-3
-2
-1 0 x (m)
1
2 3 x 10-4
2 0 -2 y( m) -4 -6
-4
-2 )
x (m
2 0 x 10-4
Fig. 2. (a) The cross-section of the keyhole for pulsed laser beam (11 J, 3 ms) and welding speed 1 cm/s. (b) Keyhole profile for the pulsed laser beam (11 J, 3 ms), focusing radius of 200 mm on steel 304, welding speed 1 cm/s.
Table 1 Chemical composition of st14 sheet metal vs. the weight percentage C
Si
Mn
P
Cr
Ni
S
Fe
0.04
0.01
0.21
0.007
0.006
0.03
0.007
Balance
where z is the average ionic charge in the plasma, h is Plank’s constant, c is the speed of light, me is the electron mass, o is the angular frequency, e0 is the permittivity of free space, ni is the ion density , ne is the electron density and g is the mechanical Gaunt factor. The Fresnel reflection factor is calculated against the incident angle y using the relation [11]
Table 2 Thermo-physical properties used in the finite difference model
Rz ¼ Temperature (1C)
Thermal conductivity (W/mK)
Specific heat (J/kg m3)
Density (kg/m3)
25
16.26
460
7818
(5) where e is the electric conductivity of metal that depends on the metal and laser specifications and is calculated by the following relation: 2 ¼
Fig. 3. The experimental result which shows the width of the welding area.
The heat is absorbed in the work piece by means of inverse bremsstrahlung and Fresnel absorption in the plasma within the keyhole. The inverse bremsstrahlung absorption coefficient is given by the following expression [10]: ne ni z2 e6 2p me 1=2 aðm1 Þ ¼ pffiffiffi 3 6 3m0 cho3 m2e 2pKT e _o 1 exp g, KT e
1 1 þ ð1 Cos yÞ2 2 2 Cos y þ 2 Cos2 y þ , 2 1 þ ð1 þ Cos yÞ2 2 þ 2 Cos y þ 2 Cos2 y
22 0 þ ½21 þ ½sst =ðo0 Þ2 1=2
.
(6)
Here, e0 is the electric susceptibility of vacuum and equals 8.85 102 C2/N m2, e1 and e2 are the real parts of the dielectric constants for the metal and the keyhole plasma, sst is the electric conductance per unit depth and o is the angular frequency of the laser light. Before we solve the equation, it is necessary to know the boundary limit of the keyhole. In this case, the keyhole geometry can be described by curves of practical ovoid shape as shown in Fig. 2. Here, the Rosenthal model is used [12], and the computer program is written by utilizing MATLAB software. The material used for welding was 1 mm thick st14 metal. The chemical composition of this metal is presented in Table 1.1 All thermo-physical properties which are listed in Table 2 are considered to be temperature independent. The laser source was Nd:YAG laser of 400 W average power. We used optimum mesh points which are available as an option in the software, and the following
ð4Þ 1
Refer to site http://www.engineersedge.com/properties_of_metals.htm
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Table 4 Input data for set number 1
Temperature (K)
5000 4000 3000 2000 1000
Energy (J)
Pulse duration (ms)
Frequency (Hz)
11 11 11 11 11 11
3 5 6 7 8 9
20 20 20 20 20 20
0 5 x10-4
5 0
Y(
m)
0 -5 -5
-10
x 10
-4
)
Table 3 Thermo-physical properties used in the finite element model Temperature (1C)
Thermal conductivity (W/mK)
Specific heat (J/kg m3)
Emissivity
Density (kg/m3)
0 75 100 175 200 225 275 300 325 375 400 475 500 575 600 675 700 730 750 775 800 1000 1500 1540 1690 1840 1890 2860
51.9 51.3 51.1 49.5 49 48.3 46.8 46.1 45.3 43.6 42.7 40.2 39.4 36.6 35.6 32.8 31.8 30.1 28.9 27.5 26 27.2 29.7 29.7 29.7 29.7 29.7 29.7
450 486 494 519 526 532 557 566 574 599 615 662 684 749 773 846 1139 1384 1191 950 931 779 400 400 847 847 400 400
0.2 0.35 0.4 0.44 0.45 0.46 0.47 0.48 0.48 0.5 0.51 0.53 0.54 0.55 0.56 0.57 0.57 0.58 0.58 0.58 0.58 0.59 0.6 0.6 0.6 0.6 0.6 0.62
7872 7852 7845 7824 7816 7809 7763 7740 7717 7727 7733 7720 7711 7680 7669 7636 7625 7612 7602 7590 7578 7552 7268 7218 7055 6757 6715 5902
laser energy: 11 J, pulse duration: 3 ms, welding speed: 1 cm/s, air convection coefficient: 50 W/m2 C,
ambient temperature: 25 1C, height of work piece: 0.7 mm.
X (m
Fig. 4. Temperature profile of keyhole for a pulsed laser beam (11 J, 3 ms), focusing radius of 200 mm on steel 304, welding speed 1 cm/s.
data were used:
The results of the numerical calculation of FDM are shown in Fig. 2. It is worth nothing to mention that the slight deviation of the center of the ovoid from the laser source arises from translational speed of the laser beam and increases with speed. The calculated width is about 1 mm, which was found at the same conditions. Fig. 2 shows that for the work piece with 0.7 mm height the penetration is complete. Fig. 3 shows the experimental result which is in good agreement with simulation (Fig. 4). 2.2. Finite element model (FEM) The FEM yields a set of simultaneous equations: ½Kfug ¼ fF a g,
(7)
where [K] is the coefficient matrix, {u} is the vector of unknown degree of freedom (DOF) values and {Fa} is the vector of applied loads. If the coefficient matrix [K] is a function of unknown DOF values (or their derivatives), then Eq. (7) is a nonlinear equation. The Newton–Raphson method is an iterative process of solving the nonlinear equations and can be written as follows: ½K Ti fDui g ¼ fF a g fF nr i g,
(8)
fuiþ1 g ¼ fui g þ fDui g,
(9)
T
where [Ki ] is the Jacobian matrix (tangent matrix), i is the subscript representing the current equilibrium iteration and fF nr i g is the vector of restoring loads corresponding to the element internal loads. Both [KiT] and fF nr i g are evaluated based on the values given by {ui}. The right-hand side of Eq. (8) is a residual or out-of-balance load vector; i.e. the amount of the system is out of equilibrium. In a thermal analysis, [KiT] is the conductivity matrix, {ui} is the temperature vector and fF nr i g is the resisting load vector calculated from the element heat flows [13]. FEM has the advantage that in meshing model we have many options for choosing elements. To reduce computation time, the simulation of this part has been done using ANSYS code, and the
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Fig. 5. (a) Temperature profile in three dimensions of stainless steel, energy ¼ 11 J, pulse duration ¼ 3 ms, welding speed ¼ 1 cm/s. (b) Penetration depth profile, energy ¼ 11 J, pulse duration ¼ 6 ms, welding speed ¼ 1 cm/s.
Fig. 6. The experimental result for the welding shape and its depth, energy ¼ 11 J, pulse duration ¼ 3 ms. As it can be seen, it confirms that at the above conditions the laser beam would have a destructive effect.
Fig. 8. The temperature variation of plasma plume with pulse duration.
Fig. 7. The experimental result for the welding shape and its depth, energy ¼ 11 J, pulse duration ¼ 6 ms. As it is clear, in the above conditions, the welding quality is better and satisfactory.
following assumptions were considered:
The laser beam enters to the work piece vertically. Both the laser beam and the coordinate system are fixed and the work piece moves in the positive x direction with a constant velocity U.
All thermo-physical properties are temperature dependent, and are shown in Table 3. Air convection coefficient is 50 W/m2 C. The ambient temperature is 25 1C.
The spatial and temporal temperature distribution T(x,y,z,t) satisfies the following differential equation for the threedimensional heat conduction: q qT q qT q qT Kx Ky Kz þ þ þQ qx qx qy qy qz qz qT qT U , ð10Þ ¼ rC qt qx where (x,y,z) is the coordinate system attached to the heat source, Q is the power generation per unit volume, K x ; K y ; K z are the thermal conductivities in the x; y; z directions, C is the specific heat capacity, r is the density, t is the time and U is the velocity of the work piece. The program ran for two sets of input data. In the first one the pulse duration was variable as in Table 4 and the results are shown in Fig. 5.
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Fig. 9. (a) The variations of the temperature for the lower layer of the work piece relative to the pulse duration. (b) The variations of the temperature for the upper layer of the work piece relative to the pulse duration. (c) The variations of the weld penetration of the work piece relative to the pulse duration.
Table 5 Input data for set number 2 Energy (J)
Pulse duration (ms)
Frequency (Hz)
12 11 10 9 8
6 6 6 6 6
20 20 20 20 20
Fig. 10. (a) Temperature profile in three dimensions of stainless steel, energy ¼ 12 J, pulse duration ¼ 6 ms, welding speed ¼ 1 cm/s. (b) Penetration depth profile, energy ¼ 12 J, pulse duration ¼ 6 ms, welding speed ¼ 1 cm/s.
Fig. 11. (a) The variation of the temperature for the lower layer of the work piece relative to the laser energy. (b) The variations of the temperature for the upper layer of the work piece relative to the laser energy. (c) The variations of the weld penetration of the work piece relative to the laser energy.
As it can be seen in Fig. 5, for the lowest pulse duration (3 ms) we have the maximum temperature in the top layer of the work piece. In this case the temperature is almost 5667 1C, which is higher than vapor temperature of stainless steel. Therefore, some of the metal infused out of the keyhole and the surface of the welding area will not
remain flat. In this case the overlapping of the pulses is so high. With increasing pulse duration the temperature and penetration depth are decreased as can be seen in Figs. 6–9. In other words, when the pulse duration reaches to 9 ms, the keyhole is not formed and the penetration is not completed. In the pulse duration equal to 5 or 6 ms, we
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Fig. 12. (a) Temperature profile in three dimensions for stainless steel, energy ¼ 11 J, pulse duration ¼ 3 ms, welding speed ¼ 1 cm/s, work piece depth ¼ 1.5 mm. (b) Penetration depth profile, energy ¼ 11 J, pulse duration ¼ 3 ms, welding speed ¼ 1 cm/s, work piece depth ¼ 1.5 mm.
1st Pulse
3rd Pulse
2nd Pulse
4th Pulse
3. Conclusions
5000 4500
Temperature (K)
4000 3500 3000 2500 2000 1500 1000 500 0 0
0.04
0.08
0.12
0.16
0.2
Time (s) Fig. 13. Evaluations of peak temperatures at four monitoring locations on the surface of the work piece. The four locations correspond to the middle of the laser beam spot on the work piece surface at the onset of the first, second, third and fourth pulses.
Three-dimensional numerical models of the heat flow during the pulsed laser welding have been developed using the FDM and the FEM. The results of the models have been compared with the experimental results. A very good agreement between theory and experiment has been achieved. The results clearly demonstrate the importance of the plasma formed above the weld area. The absorptivity of the Nd:YAG laser radiation by the work piece is clearly a complex function of a number of variables such as surface temperature, beam power density, focal position of the laser beam relative to the work piece surface. Full thermal histories of the weld have also been generated. A more direct check of the theoretical work, apart from the measurement of weld dimensions and the consequent estimations of temperature contours, was the measurement of temperature as a function of time during laser welding.
References have a good welding quality. In these cases the penetration is completed and the work piece surface is flat, and confirmed by the experimental results shown in Figs. 6–8. Fig. 8 explains that when the pulse duration increases, the temperature of the plasma plume is increased, which consequently causes the temperature of the work piece to go up [14]. In set number 2, the input energy was considered variable as in Table 5, and the results have been shown in Figs. 10 and 11. In the fixed pulse duration, the maximum temperature is linearly increased by the energy of the pulse, and when the energy reaches 10 J, the penetration will be completed. The simulation results show that we can weld two plates with 1.5 mm depth. Maximum and minimum temperatures for these conditions are 5667 and 2698 1C (Fig. 12a). We will also have completed penetration in this case (Fig. 12b). Fig. 13 shows the computed thermal cycles at four monitoring locations on the specimen surface in the middle of the laser spots for the first four pulses. The preheating of the monitoring locations from previous pulses is apparent.
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