Forschung und Technik, Daimler-Benz AG, D-70546 Stuttgart, Germany. Received 27 March ... For roughly twenty years, researchers have been working towards a ... and energy loss, thus allowing the coupling efficiency to be calculated. 2.
J. Phys. D: Appl. Phys. 29 (1996) 2811–2817. Printed in the UK
Computerized simulation of laser beam welding, modelling and verification W Sudnik†, D Radaj‡ and W Erofeew† † Department of Welding, University of Tula, Pr Lenin 92, 300600 Tula, Russia ‡ Forschung und Technik, Daimler-Benz AG, D-70546 Stuttgart, Germany Received 27 March 1996, in final form 19 June 1996 Abstract. The theoretical basis for the numerical simulation of stationary penetration welding with a laser beam is presented. The characteristics of the self-consistent model are the following. Vapour channel, weld pool and solid are considered as a nonlinear thermodynamic continuum. The laser-induced channel formation, multiple reflection of the laser beam in the channel, and plasma generation are taken into account. The absorption coefficient of the plasma is assumed as dependent on the degree of ionization. The geometry of the channel is determined on the basis of the pressure equilibrium at the channel surface taking the enthalpy of the molten mass into account. The input parameters for the simulation are the geometry of the workpiece, the material characteristic values dependent on temperature and the technological parameters such as laser beam power, beam diameter at focus, focus position, divergence angle of the beam, type of working gas and welding velocity. The results comprise the distribution of enthalpy and temperature, the shape and dimensions of the vapour channel of the weld pool and of the completed weld, and the energy losses by reflection, vaporization, thermal radiation and plasma shielding. The model was verified with welding experiments on steel and aluminium alloy. Data taken from the literature were also used for verification.
1. Introduction The computerized simulation of laser beam welding is of great practical importance (Sudnik et al 1996, Radaj et al 1996). For roughly twenty years, researchers have been working towards a model for the laser beam welding process. Andrews and Atthey (1976) were the first to develop a model of the vapour channel. This was expanded by Dowden et al (1987), incorporating the absorption of radiation in the plasma. The version of Beck et al (1992) of this expanded model considered the radiation reflection from the channel wall. Rapp et al (1994) incorporated the thermodynamic dependency of the vaporization temperature on the chemical composition of the alloy and on the vapour pressure in the channel. This could be used to explain the increase of the penetration depth observed in certain alloys, in spite of raised heat conductivity, based on the lowered vaporization temperature. The three-dimensional models published to date cover the subareas of the laser welding process without considering their interdependencies, i.e. the existence of the channel, the distribution of power in the beam cross section, the radiation reflection and the effect of the plasma working gas (Mazumder and Steen 1980), the flow in the molten pool, the plasma formation and its absorption of radiation (Grigorjanc 1989), and additionally c 1996 IOP Publishing Ltd 0022-3727/96/112811+07$19.50
the hydrostatics of the channel (Matsuhiro et al 1994). Ducharme et al (1994) described an integrated model of laser beam welding of thin metal sheets, by which the molten pool dimensions are calculated based on the interaction of the laser radiation with the plasma in the channel and on the heat transfer from the plasma into the weld pool. The first ‘self-consistent’ model by Kroos et al (1993), which considers the interdependencies of the subareas of the laser welding process, makes it possible to determine the shape, position and size of the channel, as well as the pressure and temperature in the channel, using the process parameters and the material properties. It should be noted, however, that in the models mentioned, the coupling efficiency of the welding process is a specified parameter. The self-consistent model described below considers the fundamental processes of radiation absorption and energy loss, thus allowing the coupling efficiency to be calculated. 2. Physical-mathematical model 2.1. Overall survey The numerical simulation of penetration welding with lasers with the changes of state of fusion and vaporization requires the enthalpy to be incorporated as the basic state quantity 2811
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of the material in the weld formation zone. State and temperature are dependent on enthalpy. The overall model consists of submodels for the laser beam, plasma formation, vapour channel, radiation absorption, fusion zone and the solid. The boundary between the weld pool and the solid is given by the enthalpy values at the start and the completion of solidification or at the start and the completion of fusion. The channel surface is assigned an enthalpy value at which the vaporization of the alloy constituents produces an equilibrium between vapour pressure, surface tension and hydrostatic pressure of the molten pool. The plasma has an effect between cross sections of the laser beam within which the spontaneous ionization of the metal vapour in the laser beam is possible due to local thermodynamic equilibrium. The equations are formulated with a coordinate system which is bound to the laser beam while the workpiece is moved with the welding velocity vW in the x-direction. The stationary thermodynamic state of the medium is expressed by the equation for the conservation of energy in the form of enthalpy H (x, y, z) in connection with the type III boundary condition for radiation losses at the workpiece surface: ∂H −ρvW + ρ(u, gradH ) = div[λ(T )gradT ] ∂x (1) +Q[x, y, ZK (x, y)] ∂T = εσ0 (T 4 − T04 ). (2) −λ ∂z ρ is the density of the medium, u the flow velocity vector, λ(T ) the temperature-dependent heat conductivity, Q(x, y, ZK ) the intensity distribution of the absorbed laser and plasma radiation in the channel with surface ZK (x, y), ε the degree of emission and σ0 the Stefan–Boltzmann constant. The weld follows the x-coordinate and the channel follows the z-coordinate. The temperature dependency of the enthalpy is expressed by the Kirchhoff equation in which the temperature dependency of the specific heat capacity cS and cL (indices S and L for solid and liquid) and the thermal energy of fusion (HL ) and vaporization (HV ) are considered: Z TL H (T ) = cS (T ) dT + ψL (T )HL Z TTV0 + cL (T ) dT + ψV (T )HV . (3) TL
The parameters ψL and ψV are the volume proportions of molten metal and vapour respectively in the ‘solid– liquid’ and ‘liquid–vapour’ transitional zones. Their size is dependent on the temperature and the kinetics of the processes.
where n is the normal to the channel surface, εL and εP are the absorption coefficients of radiation from the laser beam and plasma respectively, I and IP the intensities of radiation from the laser beam and plasma respectively, IR and IRL are the intensities of laser beam radiation after the first and second reflection respectively, IRP the intensity of reflected plasma radiation and IV the intensity of loss from vaporization. The description of the power density in the laser beam considers the focus radius rF (equated to the value of 86.5% in the Gauss distribution), the focus position ZF , the divergence angle and the inclination of the laser beam to the surface of the workpiece. The Gauss distribution of the intensity in the laser beam is: l(r, z) =
π {rF2
2PL exp(−2r 2 /rF2 ) (5) + [(z − ZF )DS /FS ]2 }
where PL is the laser power, DS the beam diameter on the focusing lens and FS the focal length. 2.3. Plasma formation submodel The absorption of the laser beam depends on photoelectric absorption by excited atoms and the process of inverse Bremsstrahlung of electrons in the field of ions and neutral atoms. The energy of the inverse Bremsstrahlung of electrons is proportional to their concentration which, in turn, depends on the degree of ionization. The formation of plasma from the optical discharge in the vapour jet as it emerges from the channel and its interaction with the laser beam is calculated considering: (i) the equation for the energy balance resulting from the absorption of infrared laser radiation and the heat exchange by radiation and convection; (ii) the dependency of the absorption coefficients of radiation in the plasma on its degree of ionization; and (iii) the dependency of the degree of ionization on the enthalpy and chemical composition of the plasma. The laser radiation is completely shielded in the case of severe ionization of the plasma. This process is not used for welding in practice. Another process is considered in the model. The plasma is assumed to be diluted by the working gas so that only a minor portion of the radiation is absorbed. This allows one to neglect both self-absorption of radiation and heat conduction in the plasma. The state of the plasma is described on the basis of an energy balance. This balance is given by the following equation in which the term on the left-hand side designates the intensity of laser energy absorption in the plasma and the terms on the right-hand side refer to the loss of energy by thermal radiation and heat conduction in the radial direction: αP IL = εσ0 (TP4 − T04 ) + λg (TP − T0 )/rF
2.2. Laser beam submodel The distribution of the intensity of heat input to the channel surface is described as follows: ∂ [εL (I + IR ) + εP IP + εRL IRL Q[x, y, ZK (x, y)] = ∂n +εRP IRP − IV ] (4) 2812
(6)
where αP is the absorption coefficient of radiation in the plasma (this depends on the degree of ionization of the gas), TP the maximum temperature of the plasma and λg the thermal conductivity of the working gas. The degree of ionization of the gas which rises in a cumulative manner with increased intensity of plasma
Computerized simulation of laser beam welding
The plasma is a source of radiation, the wavelength of which is dependent on the plasma temperature. As a simplification, it is assumed here that the plasma radiation acts at equal strength in all directions from a point source in the centre C (with the coordinates x = 0, y = 0 and z = ZC ) of the beam cross section with the maximum plasma temperature. The intensity IP of the plasma radiation at a point with the coordinates x, y, z is thus: IP = PP /{4π [x 2 + y 2 + (z − ZC )2 ]}
(7)
where ZC is the coordinate for the height of the section with maximum plasma temperature. 2.4. Radiation absorption submodel The description of the absorption in the metal considers: (i) the dependency of the absorption coefficient on the wavelength of the radiation, on the temperature-dependent electrical conductivity of the alloy and on the beam angle of incidence, as well as (ii) the multiple reflection of radiation in the channel. Figure 1. Schematic representation of the zones of optical discharge (i.e. plasma formation) and of the distribution of the intensity of the laser beam with the focus positioned at the surface of the workpiece (a), with increased power of the laser (b) and with the focus positioned below the surface of the workpiece (c). Beginning and end of plasma formation in cross section A and B, centre of the plasma in C. Intensities of the laser beam without plasma (IL ) and with plasma (I ).
radiation is determined with the Saha equation, considering the temperature of the plasma and the chemical composition of the vapour. The combination of equation (6) and the Saha equation describes the process of plasma formation. The solution of the equation system with increasing plasma radiation intensity IP allows its threshold IP S to be calculated. The threshold is defined as that intensity at which the ionization snowballs in the sense of an optical discharge. If I ≥ IP S , plasma is formed and determines the section positions in a laser beam where the continuous optical discharge begins, (figure 1, cross section A). With the sequence of solutions of the equation system for the laser beam cross sections considering radiation losses (according to the Beer–Lambert law) in the previous sections, it is possible to determine the plasma temperature, degree of ionization and coefficient of radiation absorption in the respective cross sectional layer. Using this procedure, it is possible to find the section in which the intensity has dropped to the threshold IP S at which extinction of the plasma occurs (cross section B). The maximum plasma temperature occurs in the centre point C of a beam cross section between sections A and B. The change in the intensity of the primary flow of radiation through the plasma is described considering R r the power losses in the plasma according to PP = 0 [I (r, ZF ) − IP S ] dr for I (r, ZF ) > IP S . The plasma may have an effect only above the surface or it may reach into the vapour channel depending on its extent and the focus position.
The absorption coefficients for the laser radiation (εL ) and plasma radiation (εP ) are established using the approximate equation according to Hagen and Rubens considering the beam angle of incidence and the Fresnel equations for circular polarization. The incidence zones of the laser beam on the channel surface are determined for the first and second reflections. The power of the reflected radiation after the first and second reflections is PR1 = (1 − εL )(PL − PP ) and PR2 = (1 − εL )2 (PL − PP ) respectively. The power of the (simple) reflected plasma radiation is PR3 = (1 − εP )PP . It is impossible to track the course of the laser beam and plasma radiation in the channel precisely. This can only be done for the first reflection, whereas the further multiple reflections are assumed as evenly distributed over definite parts of the channel surface inclusive of entry and exit opening. The first reflection of the laser beam with its intensity following a Gauss distribution is calculated based on geometrical optics. The power remaining after the first reflection is evenly distributed over the channel surface portion below the minimum depth Z0 . The latter is defined as the depth below which the reflected beam hits the channel surface. It is approximated from the diameter DK of the entry opening and the channel depth ZK : Z0 = 2 2 − DK )/(2ZK ). The power remaining after the second (ZK reflection is evenly distributed as scattered radiation over the complete channel surface SS , the surface ST of the entry opening and the surface SB of the exit opening. The radiation intensities from the plasma point source at the first reflection are calculated according to equation (7). The power remaining after the first reflection is once more evenly distributed over the total surface mentioned above. The intensities of the scattered radiation of laser and plasma result as follows: IRL =
(1 − εL )2 (PL − PP ) SS + ST + SB
(8) 2813
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(1 − εP )PP . (9) SS + ST + SB The proportion of scattered radiation falling on the channel openings is lost as far as the welding process is concerned. For the absorption of the scattered radiation from the laser beam and the plasma in the rest of the channel surface, the equivalent absorption coefficient εR is incorporated: ∞ X (1 − εL )SS n εR = εL . (10) SS + ST + SB n=1 IRP =
2.5. Vapour channel submodel The initial approximation to the channel surface is performed according to the boiling point of the basic metal of the alloy. The precise conditions are determined assuming a pressure equilibrium at the channel surface. Since the sum of the vapour pressure, capillary pressure and hydrostatic pressure must be equal to zero at every point on the surface, the surface curvature is not constant and the pressure and temperature vary from point to point. The model described by Rapp et al (1994) for surface vaporization which combines the vapour pressure, the temperature and the quantity of volatile alloy constituents was expanded by the effect of the rise in vapour pressure caused by the transport of such elements to the vaporization front at welding velocity. The dependency of the vaporization pressure on the temperature describes the non-equilibrium vaporization of the real alloy, considering its chemical composition, the welding velocity and the volume-related vaporization of the volatile elements which have a boiling temperature lower than the temperature of the vaporization surface of the channel (Sudnik et al 1995). As an example, the dependency of the excess vapour pressure p(T , vW ) for the aluminium alloys AA6013, AlSi12 and its mixture is presented in figure 2 for a welding speed vW = 4.8 m min−1 (p0 is the ambient atmospheric pressure). The curves have a step at the boiling point of the more volatile magnesium. The excess pressure p is the sum of the static vapour pressure of the surface (pi,S ), volume-related (pi,V OL ) vaporization and the recoil or ablation pressure pi,ABL of the ith alloy element on the channel surface: X (pi,ABL + pi,V OL + pi,S ). (11) p= i
In order to determine the coordinates of the points of the channel surface, the thermodynamic and hydrostatic equilibrium conditions are examined. Figure 3 illustrates the interactions between pressure and temperature. The pressure and temperature vary over the channel depth. The maximum excess vapour pressure pmax occurs at the base of the channel if there is no penetration: pmax =
σ (T ) + ρL gZK . rx + ry
(12)
σ (T ) is the temperature-dependent surface tension, rx and ry are the main curvature radii of the semi-ellipsoid base which is determined by the approximation of the surface ZK (x, y) at the deepest point of the channel and ρL is the density of the molten pool. 2814
Figure 2. Dependency used in the calculation of excess vapour pressure increase on temperature for the aluminium alloys AA6013, AlSi12 and their mixture in the fusion zone with welding speed v W = 4.8 m min−1 .
Figure 3. Determination of the coordinates of the vapour channel surface ZK (x , y ) proceeding from the isotherms (T1 , T2 , T3 ), the pressure distribution along the channel axis, p (z ), and the dependency of vapour pressure from temperature p (T ), schematic graph.
The excess pressure in the other channel sections and in the base with full penetration is calculated according to: p(z) =
σ (T ) + ρL gz rz
(13)
where rz is the average curvature radius of the surface ZK (x, y) at a depth z in the channel. Using the dependency of the pressure on the temperature p(T , vW ), the temperature of the respective section at the surface of the channel is corrected. This results in the new position of the surface ZK (x, y). For the creation of the equilibrated excess pressure in the sections √ of the channel surface, the vaporization speed ˙ mass flow per surface unit and m(z) ˙ = p(z)ρg (with m(z) ρg vapour density) is required. This is used to establish the ˙ heat flow of vaporization IV (z) = m(z)H V (vaporization cooling). This heat flow is required as a minimum quantity to maintain the pressure equilibrium at the channel wall. It forms the basis of further calculations.
Computerized simulation of laser beam welding
2.6. Weld pool submodel
2.7. Solid submodel
The three-dimensional flow in the weld pool is approximated using a two-dimensional flow both in the horizontal and vertical section planes. In the horizontal section, a two-dimensional laminar flow around the channel opening with the formation of a double eddy behind the channel is assumed. It is fed from the fusion process at the front face of the molten pool and is determined according to the equations for fusion power and continuity of the molten pool. The diameter of the eddy corresponds to the radius of the channel and the maximum flow velocity in the eddy is equal to the average flow velocity beside the channel. A two-dimensional laminar double eddy produced by the Marangoni effect is assumed in the cross-section, the velocities of which result from an analytical solution according to Wwedenow and Gladusch (1985). The enthalpy values in the molten pool are corrected in accordance with the calculated components of the flow velocity in this zone. This is done introducing an effective turbulent heat conductivity of the molten pool in which the molecular heat flow is increased by a corresponding factor and the convective heat flow is completely ignored. The fusion zone boundary surfaces between the solid and the channel are connected by an upper and a lower free surface (ZST (x, y) and ZSB (x, y) respectively) the equilibrium of which is described by the following differential equation:
The distribution of the enthalpy in the solid based on the effect of the heat source considers:
∇ZS pABL p = ∓σ (T )∇ p − ρL gZS + 0. 1 + (∇ZS )2 1 + (∇ZS )2 (14) The minus or plus sign is chosen according to the position of the molten pool either above or below the surface (i.e. the minus sign for the upper and the plus sign for the lower surface). The constant 0 has the dimensions of pressure and is selected in accordance with the mass balance: Z
+b/2 −b/2
(ZST + ZSB ) dy = s1q
(15)
where b is the weld width, s the sheet thickness and 1q = AαηPL /(svW cρ) the transverse shrinkage after welding according to a semi-empirical formula given by Vinokurov (1977); α is the coefficient of heat expansion, η the process efficiency (here coupling efficiency), ρ the metal density and A an adjustment factor. The boundary conditions in the fusion zone (a), the vaporization zone (b) and the crystallization zone (c) consider the profiles ZF = ZF (x, y) of the solid and ZK of the channel. These are as follows for the lower and upper sides respectively: ( 0 for ZST ≥ ZF (b) ZST = ZK (a)ZST = (16) ST ZF for ZST < ZF (c) ∂Z ∂x = 0 (a)ZSB = s
(b)ZSB = ZK
(c)
∂ZSB = 0. ∂x
(17)
(i) conductive heat transfer; (ii) the rate of feed or welding velocity; (iii) losses from heat radiation at the entry section and, if present, the exit section of the channel.
2.8. Thermodynamic characteristic values of the material The dependencies of the enthalpy, heat conductivity and electrical conductivity as well as the surface tension and viscosity on the temperature are represented for the alloys under investigation by linear functions in length sections and stored in a special database. The atomic quantum coefficients and the ionization energies are provided to calculate the degree of ionization of the metal vapour. 3. Numerical approximation The equations for the energy balance in the medium and for the pressure equilibrium at the surfaces are solved by means of the method of finite differences in a homogeneous three-dimensional rectangular grid. The solution of the equations for stationary behaviour results as a limiting value of an iterative calculation for non-stationary behaviour with a sufficiently large number of time steps. The scheme of differences has a high degree of nonlinearity, as the characteristic values for the material are introduced as temperature dependent. In addition, couplings occur between and within the submodels which necessitate an iterative procedure. For instance, the channel size depends on the vapour pressure, the vapour pressure on the wall temperature and the latter on the channel size. The weld pool flow influences the channel size via convective heat transfer, but is in turn dependent on the channel size. The complexity of the connection between the coefficients of the nonlinear three-dimensional problem in an inhomogeneous medium makes the application of traditional solution schemes impossible. Instead, it is necessary to go over to an economically viable additive scheme which combines the advantages of explicit and implicit schemes. The scheme is based on the method of summary approximation according to Samarskij (1984): the approximation solution for the complex three-dimensional problem is divided into a series of steps. Each step comprises the solution of a simpler (usually one-dimensional) problem, e.g. the solution for the boundary value problem of the differences using a three-point scheme with reduced Gauss elimination. The individual intermediate schemes cannot approximate the initial problem. The approximation is performed by summing the individual defects. The initial equation (1) is expanded by the nonstationary additional element ρ∂H /∂t and converted into a difference equation. The problem is solved iteratively by splitting it up into physical subprocesses: radiation 2815
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absorption, heat conduction, flow and rate of travel of the workpiece. The process of three-dimensional heat conduction is replaced by a series of one-dimensional heat conduction processes. The nonlinearity which is essential to consider in nonstationary problems need not be considered for calculations up to the stationary limiting value, because the preceding and the following values of the function are practically the same when approximating the stationary solution. The overall algorithm consists of individual procedures which are performed iteratively. The latter calculate: (i) the radiation absorption in the plasma; (ii) the coordinates of the channel surface in accordance with the current distribution of enthalpy and vaporization temperature; (iii) the absorption of laser and plasma radiation at the channel surface; (iv) the heat conduction in the metal; (v) the vapour pressure in the channel and the vaporization temperature in accordance with this pressure; (vi) the correction of the enthalpy in the area of the rear channel surface in accordance with the vaporization energy. The geometrical parameters of the weld are calculated with an error of less than (0.1–0.15)dF . For the typical value of dF ≈ 0.4 mm, a calculation error of about 0.05 mm is to be expected. Using a PC 586 with a frequency of 100 MHz, a calculation with a grid of 20 × 20 × 100 nodes is carried out within 2–3 min. The implementation of the model and the numerical procedure in a computer program designed for engineers is described by Radaj et al (1996). The input and output parameters are summarized in the abstract of the article in hand. 4. Verification of the model The verification of the model and the associated calculation programme was performed with welding experiments on steel 16MnCr5 (square butt joint, s = 5 mm, vW = 2.4 and 0.7 m min−1 , PL = 4.1 and 2.4 kW) and on the aluminium alloy AlMgSiCu with cladding material AlSi12 (tee butt joint, s = 1.6 mm, vW = 10 m min−1 , PL = 4.1 kW), (see figures 4 and 5). The correspondence between the calculation and the experiment with regard to the geometry, fusion zone and the temperature cycles in the heat-affected zone (not shown here) is very satisfactory. A comparison of the geometry of the weld pool and channel established experimentally by Beyer (1995) with the corresponding values calculated according to the above model is presented in figure 6. The heat-affected zone is also shown in the right-hand section of the figure. The agreement here is also quite good. The coupling efficiency for laser welding of 16MnCr5 steel with sheet thickness s = 5 mm with insufficient and complete penetration as determined by calculation to be ηE = 80–85% is in agreement with the data given by Dausinger (1995) and Fuerschbach (1996). Considering the aluminium alloy AA6013, the threshold intensity for 2816
Figure 4. Comparison of weld seam formation in laser beam welding determined experimentally and by calculation (cross sectional halves on left-hand and right-hand side respectively); steel 16MnCr5, butt joint, s = 5 mm, v W = 2.5 m min−1 , PL = 4.1 kW, dF = 0.56 mm (a) and v W = 0.7 m min−1 , PL = 2.4 kW, dF = 0.55 mm (b); drill holes for thermocouples visible at the left-hand sides.
Figure 5. Comparison of weld seam formation in laser beam welding determined experimentally (a) and by calculation (b); aluminium alloy AA6013 clad with a layer of G-AlSi12, T-joint, s = 1.6 mm, v W = 10 m min−1 , PL = 4.1 kW, dF = 0.54 mm.
penetration welding is calculated to be 2.5 × 106 W cm−2 , which corresponds to the values observed in practice. The temperature at the channel surface is an extremely important parameter for verification. However, there are only theoretical assessments and no measurements available for laser beam welding. On the other hand, some experimental data are available for electron beam welding from Schauer et al (1978). Therefore, a calculation was performed with the above model for electron beam welding in vacuum and with a relatively low welding speed, vW = 0.8 m min−1 . When comparing the vapour pressure under ambient and vacuum conditions respectively, only the term pi s in equation (11) must be changed. This term was assumed to be increased by a factor of six in the calculation for the technical vacuum. The appertaining maximum temperature at the channel surface for the aluminium alloy AA6013 (0.95% Mg) was 1760 ◦ C. For the condition of ambient atmospheric pressure, the relevant temperature was 1850 ◦ C. The calculated values correspond well to the measured data 1800–1890 ◦ C from Schauer et al (1978) in electron beam welding (vW = 0.8 m min−1 , PL = 2.7 kW) of the comparable alloy AA6061 (0.95% Mg, 0.063% Zn). The calculated maximum temperature in welding
Computerized simulation of laser beam welding
vapour pressure at a given temperature caused by volatile alloy constituents. Therefore, the capillary is produced at a lowered temperature. References
Figure 6. Comparison of the geometry of the vapour channel and fusion zone in laser beam welding of low-alloy steel determined experimentally (cross sectional half on left-hand side according to Beyer (1995), his figure 6.1.2) and by calculation (cross sectional half on right-hand side); PL = 5.3 kW, rF = 0.115 mm, v W = 3.6 m min−1 , ZF = +1 mm.
of 16MnCr5 steel is approximately 2500–2750 ◦ C for the typical focal diameter dF = 0.4 mm. The calculated and measured temperatures at the channel surface of the aluminium alloy AA6103 are about 670 ◦ C below the boiling point of pure aluminium. This temperature reduction underlines the important role which the alloy constituents play in laser beam penetration welding. The appertaining maximum pressure increase within the channel works out at 4% of the ambient pressure. 5. Conclusion The self-consistent model described and the associated calculation program make it possible to simulate laser beam penetration welding on computer. Experimental data on the geometry of the channel and fusion zone for a steel and an aluminium alloy show good correspondence with the calculated results. In addition, the coupling efficiency calculated is close to the real value. The model allows not only the determination of the geometric characteristics of the weld which are needed for quality assessments, based solely on the process and workpiece parameters, but also of the coupling efficiency which may be used for the energetic optimization of the welding process. An essential element of the model is the increase of
Andrews J G and Atthey D R 1976 Hydrodynamic limit to penetration of a material by a high-power beam J. Phys. D: Appl. Phys. 9 2181–94 Beck M, Berger P and H¨ugel H 1992 Modelling of keyhole/melt interaction in laser deep penetration welding Laser Treatment of Materials, ECLAT ’92 ed B L Modrike (Oberursel: DGM Informationsgesellschaft) pp 963–98 Beyer E 1995 Schweißen mit Laser, Grundlagen (Berlin: Springer) Dausinger F 1995 Strahlwerkzeug Laser, Energieeinkopplung und Prozeßeffektivit¨at (Stuttgart: Teubner) Dowden J, Postacioglu N, Davis M and Kapadia P D 1987 A keyhole model in penetration welding with a laser J. Phys. D: Appl. Phys. 20 36–44 Ducharme R, Williams K, Kapadia P, Dowden J, Steen B and Golawcki M 1994 The laser welding of thin metal sheets: an integrated keyhole and weld pool model with supporting experiments J. Phys. D: Appl. Phys. 27 1619–27 Fuerschbach P W 1996 Measurements and prediction of energy transfer efficiency in laser beam welding. Welding J. 75 24s–34s Grigorjanc A G 1989 Fundamentals of Laser Treatment of Materials (Moscow: Mashinostrojenie) in Russian Kroos J, Gratzke U and Simon G 1993 Towards a self-consistent model of the keyhole in penetration laser beam welding J. Phys. D: Appl. Phys. 26 474–80 Matsuhiro Y, Inaba Y and Ohji T 1994 Mathematical modelling of laser welding with keyhole. Part I Quart. J. Japan. Weld. Soc. 11 479–83 Mazumder J and Steen W M 1980 Heat transfer model for cw laser material processing J. Appl. Phys. 51 941–7 Radaj D, Sudnik W and Erofeev W 1996 Computerized modelling of laser beam welding, general approach and implementation Welding J. submitted for publication Rapp J, Beck M, Dausinger F and H¨ugel H 1994 Fundamental approach to the laser weldability of aluminium and copper alloys Laser Treatment of Materials, ECLAT ’94 (D¨usseldorf: DVS-Bericht) 163 Samarskij A A 1984 Theorie der Differenzenverfahren (Leipzig: Akad. Verlagsges.) Schauer D A, Giedt W H and Shintaku S M 1978 Electron beam welding cavity temperature distributions in pure metals and alloys Welding J. 57 127s–133s Sudnik W, Radaj D and Erofeev W 1996 Computerized simulation of laser beam welding, an introductory survey Computer Technology in Welding ed W Lucas (Cambridge, UK: Welding Institute) Sudnik V, Zaytsev O I and Protopopov A A 1995 Mathematical model of metal evaporation in fusion welding CAD and Expert Systems in Welding ed V Sudnik (Staatl. Univ. Tula) pp 92–9 in Russian Vinokurov V A 1977 Welding Stresses and Distortion (Boston Spa, The British Library) Wwedenow A A and Gladusch G G 1985 Physical Processes in Laser Treatment of Metals (Moscow: Energoatomizdat) in Russian
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