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Residual stress and distortion calculation of laser beam welding for aluminum lap joints
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 260–269
journal homepage: www.elsevier.com/locate/jmatprotec
Residual stress and distortion calculation of laser beam welding for aluminum lap joints G.A. Moraitis, G.N. Labeas ∗ Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Patras 26 500, Greece
a r t i c l e
i n f o
a b s t r a c t
Article history:
A numerical simulation model of the laser beam welding (LBW) process is developed, aiming
Received 2 August 2006
to a reliable prediction of the residual stress and distortion fields. As LBW is a thermo-
Received in revised form
mechanical process, a thermal analysis is conducted to analyze the spatial temperature
26 June 2007
distribution history, coupled to a mechanical analysis to calculate the residual stresses and
Accepted 5 July 2007
distortions. An innovative and efficient keyhole model, independent of any empirical parameter, is introduced for the prediction of the keyhole size and shape required for the thermal analysis. All the major physical phenomena associated to the LBW process, such as, heat
Keywords:
radiation, thermal conduction and convection heat losses are taken into account in the
Laser beam welding
model development. The thermal and mechanical material properties are introduced as
Keyhole modelling
temperature dependent functions, due to the high temperature variations and the material
Lap joint
phase changes occurring during the welding. The simulation algorithm is programmed as
Thermo-mechanical analysis
a macro routine within the ANSYS finite element code. The model is validated for the case
Welding simulation
of butt joint welding DH-36 steel plates and its efficiency is demonstrated for the lap-joint welding of two aluminum 6061-T6 plates. The main advantage of the developed model is its generality and flexibility, as it is independent of any empirical parameter, enabling its application in parametric studies of a wide range of LBW problems of different geometrical, material and joint type, requiring only the basic mechanical and thermal material properties. © 2007 Elsevier B.V. All rights reserved.
1.
Introduction
Laser beam welding (LBW) is based on high power density welding technologies, which have the possibility of focusing the beam power to a very small spot diameter. As a result, the LBW process is advantageous compared to the conventional welding processes, e.g. arc welding, leading to smaller Heat Affected Zones (HAZ), lower distortions, residual stresses and strains. Most welding processes operate in conduction-limited mode, implying that the heat supplied to the surface of the
∗
Corresponding author. Tel.: +30 2610 991027; fax: +30 2610 997190. E-mail address: [email protected] (G.N. Labeas). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.07.013
irradiated material via laser light absorption is basically transferred through conduction to the surrounding material. When the laser power density is high (around 1 MW/cm2 ), a keyhole is formatted, accompanied by phase changing phenomena, i.e. melting and evaporation. Due to the complicated physical phenomena taking place [1], the optimization of laser welding process is usually based on welding experiments and on trial and error approaches, which have many operative difficulties, increasing the total cost of the process. Simulation models of the welding process, which have been validated through experimental results, are of major
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 260–269
importance for a high number of reasons: the deep understanding of the LBW physics, the reliable extension of the process applicability to modern demanding industrial applications and the efficient definition of the LBW process parameters, without enormous cost penalty. A number of analytical and numerical models of welding processes have been developed to evaluate temperature, residual stress and distortion distributions during the welding process of structural components. These include analytical models [2–5], two-dimensional finite element models [6–7] and three-dimensional finite element models [8–11]. However, most of the investigations are limited to local problems of laser beam butt joint, while simulations of full-scale LB lap-joint components are not available so far. Furthermore, most of the investigations refer to steel components and only a handful to aluminum alloys. The most critical issue of the thermal analysis of LBW is the proper representation of the heat input to the material. The keyhole phenomenon has the principal contribution to the non-homogeneous local heating of the material volume. It arises from experimental measurements that in deep penetration welding, the keyhole shape is nearly conic and its vertex angle decreases as keyhole depth increases [12–15]. Several approaches to the mathematical modelling of keyhole formation in laser welding of steel material can be found in the literature [16,17], however, for aluminum material no reference may be found. A number of researchers have developed mathematical models for the shape and location of the weld pool and the keyhole of steel LBW [18–24], by considering appropriately the energy and pressure balance; however, these models are very complicated and their practical applicability to different geometrical, material and joint type becomes difficult. In refs. [25–28] simple moving point and line heat source models are presented. In refs. [29–33] three-dimensional models with Gaussian heat sources have been used in the thermal analysis of steel materials treated by various laser beam applications. In the present paper, a thermo-mechanical FE model based on the keyhole theory is developed in order to predict the stress, strain and distortion fields for simulating the lap-joint welding of aluminum components. Major innovations of the model are an innovative keyhole prediction approach, which has the capability to handle any steel or aluminum alloy, as well as, an advanced numerical approach enabling efficient prediction of residual stress and distortions of LBW components. Furthermore, the proposed simulation methodology is especially efficient for lap- and T-joint LBW, which are not covered sufficiently in the literature, although, they are widely used in many industrial applications. The entire simulation process comprises two numerical models of different scale level. The first model aims to a reliable prediction of the keyhole formation process, while the second one calculates the thermal stresses and strains. The keyhole prediction model is not based on experimentally defined keyhole parameters, but approaches the real thermal phenomena, through an efficient route, applicable to different material types, laser conditions and welded joint geometry. The keyhole shape and size are calculated using local finite element analysis, which simulates all the basic phenomena taking place locally at the heat introduction area. The keyhole model and its advantages are presented in Section 2.
261
Consequently, the calculated keyhole is approximated by a conical volume having a three-dimensional Gaussian heat flux distribution, moving along the weld line. A global non-linear thermal analysis step of the entire welded configuration is carried out in Section 3, to calculate the spatial temperature distribution; based on it, the corresponding mechanical analysis step is implemented, to calculate deformation, stresses and strains of the welded components. In Section 4, various numerical results from the simulation of LBW aluminum components are presented.
2.
Keyhole prediction model
The laser heat absorbed by the welded surfaces, leads to temperature values far above the material’s melting point at specific locations; this results in local material melting creating a molten pool, which increases, as more heat is input into the material volume. As shown in Fig. 1, when the power density is high, in the range of 1 MW/cm2 , the laser “drills” through the components, creating a conical or cylindrical column of material vapours, forming the keyhole volume. It should be mentioned that the vapour pressure within the keyhole maintains the equilibrium and stops the keyhole from collapsing. As the laser heat source is moving, the keyhole is dragged through the welding line, by melting in front and solidifying at behind, resulting to weld creation. The above-described physical phenomena are simulated through a detailed numerical FE model of the local volume affected by the laser beam. The FE mesh comprises threedimensional eight-node thermal elements (type ‘Solid70’), with temperature as the single degree of freedom at each node. The area nearby and along the weld line is discretized by a very dense mesh, comprising about 80,000 elements as resulted from a mesh convergence parametric study, which is illustrated in Fig. 2. It is usual assumption that the laser energy is applied on the surface of the welded component as a surface load having the Gaussian distribution shown in Fig. 3. The surface heat flux distribution q0 (x,y) may be calculated according to the formula [36]: q0 (x, y) =
c · n · P −c(x/r)2 +(y/r)2 ·e · r2
Fig. 1 – Keyhole formation during the welding process (modified from ref. [37]).
(1)
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Fig. 4 – Thermal properties for Al alloy 6061-T6 (after ref. [34]). Fig. 2 – Temperature plot for mesh convergence parametric study at the first step of the analysis.
Fig. 3 – Heat flux Gaussian distribution of laser beam.
where n is the laser absorption coefficient of the irradiated surface, P the laser beam power, r the laser beam radius, c a shape parameter of the heat-flux distribution and x, y, z are the Cartesian coordinates. The laser absorption coefficient n of an irradiated surface depends mainly on material type, surface treatment, color and roughness. It has been calculated after measurements that for pure aluminum the absorption coefficient is about 0.18. The thermal material properties of Al 6061-T6 are presented in Fig. 4, after ref. [34]. By solving the FE model for a selected solution time interval, which depends on the welding velocity, element size and beam radius, the heat transfer through conduction
and the heat losses through convection are calculated. The temperature increases locally creating a molten pool, while simultaneously, more heat is applied to the top surface. When the temperature of an element exceeds the vaporization point, the corresponding material volume is converted to vapours; the developed algorithm considers this element as ‘invisible’ and ‘transparent’, therefore, successive surface loads are introduced to the element(s) below the vaporized one. This procedure is based on an element selection routine, which is repeated as the heat source moves along the weld line with the welding velocity. When the heat source has moved enough, i.e. more than one laser beam diameter, the temperature distribution behind the heat source becomes almost steady, therefore, the final keyhole shape and size can be defined. In Fig. 5, the flow chart of the keyhole prediction procedure using FE numerical simulation is presented. The numerically predicted keyhole is verified through Eqs. (2)–(4), which result from a validated semi-experimental procedure [12,16]. Due to the lack of data for Al, the verification is performed for steel and yields that the numerically predicted keyhole is in good agreement with the semi-experimental one, as shown in Fig. 6. d=
Qin h(Tv − T0 )(c1 + (c2 /2) Pe + (c3 /3) Pe2 + (c4 /4) Pe3 )
(2)
Pe =
uw 2˛(ˇ1 e−1 Pe + ˇ2 e−2 Pe + ˇ3 e−3 Pe )
(3)
r0 =
2˛Pe u
(4)
Fig. 5 – Flow chart of the keyhole simulation approach.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 260–269
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where d is the keyhole depth; T the ambient temperature; w the weld pool width; Tv the vaporization temperature; u the welding velocity; h the thermal conductivity; Qin the absorbed laser power; a the thermal diffusivity; ci the constants (independent of material); i is the constants (dependent of material).
3.
Fig. 6 – Comparison between FE and semi-experimental keyhole for steel DH-36.
Thermo-mechanical analysis
The global coupled thermo-mechanical model for the LBW simulation assumes a three-dimensional Gaussian heat flux distribution in a conical volume, which moves along the weld line. This conical volume arises as an approximation of the predicted keyhole, as described in Section 2. By applying the thermal and mechanical boundary conditions a coupled thermo-mechanical analysis may be executed and temperature distributions, residual stresses and distortions can be calculated. An overview of this numerical procedure
Fig. 7 – Flow chart of the thermo-mechanical global LBW model.
Fig. 8 – (a) Lap-joint geometry and (b) detail of the mesh of the weld area.
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Fig. 10 – Predicted keyhole shape. Fig. 9 – Mechanical properties for Al alloy 6061-T6.
is described in the flow chart of Fig. 7. In this model the thermal and mechanical material properties dependency on temperature is introduced. The coupled thermo-mechanical analysis is demonstrated in the simulation of LBW lap joint of two thin aluminum 6061-T6 plates of dimensions 150 mm × 150 mm × 2 mm. The welding line has 120 mm length, the welding velocity is 50 mm/s and the lap width is 50 mm (Fig. 8a). A dense mesh is used in the area along and nearby the weld line (Fig. 8b), in order to simulate the complex physical phenomena that take place in the area, while the remaining volume has a coarser mesh, leading to a total of about 50,000 elements; the element type is ‘Solid 70’ for the thermal analysis and ‘Solid 45’ for
the mechanical one. For each time interval, the solution is executed in two successive steps; first, a transient heat transfer analysis is performed and the resulting temperature field is used as input to the mechanical analysis. The same procedure is repeated for every time interval.
3.1.
Global thermal analysis
According to the keyhole theory, the laser beam is modelled as a three-dimensional moving heat source, simulating the heat distribution and flow in the welding direction. The keyhole is assumed to have a conical shape the radius r0 and the depth d, which are predicted by the methodology described in Section 2; the heat flux in the keyhole volume follows a Gaussian distribution, which can be computed according to
Fig. 11 – Temperature distribution during the welding process, at the (a) run-in, (b) mid and (c) run-out section of the workpiece.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 260–269
265
Fig. 12 – Run-in and run-out temperature during (a) the welding process, (b) the cooling stage.
the formula [36]:
Q=
2P r02 d
e1−(r/r0 )
2
1−
z d
(5)
where r0 is the initial radius at the top of the keyhole, d the depth of the keyhole, r the current radius (the distance from the cone axis), z the vertical axis and P is the absorbed laser power. For aluminum lap and butt joints, where the laser beam
Fig. 13 – Distributions of equivalent stains at the time of run-in (a), mid-pass (b), run-out (c) and after cooling (d).
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Fig. 14 – Distributions of equivalent stresses at cross-sections of the workpiece; (a) at the run-in area, (b) around the weld line middle, (c) at the run-out area and (d) at weld line middle after cooling and clamp released.
is normal to the component surfaces, the reflected energy is approximately 80–90% of the nominal power of the source. The absorbed energy is transferred to the metal by conduction and convection, while is partly lost afterwards from the metal to the environment by convection (cooling). A film or convection coefficient, h, is introduced in the numerical model as a thermal material property, in addition to conductivity, specific heat and enthalpy (Fig. 4). The heat transfer problem is described by the transient Eqs. (6a) and (6b): [C(T)]{T (t)} + [K(T)]{T(t)} + {v} = {Q(t)}
(6a)
qc = A · h · (T − Tair )
(6b)
where [K] is the conductivity matrix, [C] the specific heat matrix, {T} the vector of nodal temperatures, {T } the vector of time derivative of {T}, {v} is the velocity vector, which is equal to zero as no mass transport is assumed in the current problem, {Q} the nodal heat flow vector, A the plate surface area, h
the convection (or film) coefficient and T and Tair are the plate and air temperatures, respectively. In each time interval, time integration of the heat conduction and heat convection equations (Eqs. (6a) and (6b)) lead to the calculation of the required transient temperature distributions.
3.2.
Global mechanical analysis
A non-linear mechanical analysis is executed just after each thermal analysis step, in order to calculate stress and strain distributions of the structure. The results of the thermal analysis step (transient temperature distribution) serve as loading to the corresponding mechanical analysis step. The introduced temperature dependent material mechanical properties Young’s Modulus, Poisson’s Ration, density and thermal expansion coefficient are presented in Fig. 9. The boundary conditions assumed in the mechanical analysis, is clamping of one, parallel to the weld line, plate edge.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 8 ( 2 0 0 8 ) 260–269
267
Fig. 15 – Residual stress distributions (a) longitudinal–transverse to the weld line, (b) transversal–along the weld line, (c) longitudinal and transversal through-the-thickness and (d) definition of lines for residual stress presentation.
The non-linear mechanical analysis problem is described by the following general FE equation: [K(T)]{u(t)} + {F(t)} + {Fth (t)} = 0
(7)
where K(T) is the temperature-dependent stiffness matrix; F(t) the external load vector; Fth (t) the temperature load vector; {u(t)} is the displacement vector. More details about Eqs. (6) and (7) may be found in ref. [35]. Material plasticity is represented using the von Mises criterion and a kinematic strain hardening law, requiring activation of the material nonlinear option in the FE analysis. The geometrical non-linear option of the analysis is not activated, as large displacement or strains were not expected to be developed in the structure. The Newton–Raphson algorithm is utilized for solving the non-linear equation system and the Newmark integration scheme is applied for the numerical integration in the time domain.
4.
Results and discussions
The lap joint is welded using a CO2 laser of 4 kW power with 2 mm focused radius and 50 mm/s welding velocity. From the local analysis model, it arises that the keyhole has an elliptical cone shape with upper radius 1.98 mm and depth 2.18 mm. The elliptical shape has been approximated as a perfect cone (Fig. 10), which reduces the modelling difficulties without affecting the analysis results significantly. By using Eqs. (5)–(7),
a global coupled FE thermo-mechanical analysis is carried out. The simulation of the welding process is followed by a number of loading steps simulating the cooling stage, which lasts for about 10 min. The temperature distributions developed during the laser beam pass are presented in Fig. 11, from which, the very high temperature gradients in the vicinity of the weld line close to the upper surfaces of the welded components may be clearly observed. In Fig. 12a and b, the temperature of run-in and runout points during the welding process and during the cooling stage, respectively, are presented. It can be observed that the heating rate is very steep, leading to almost immediate local heating, which is followed by a very fast (less than half second) cooling period. In Fig. 13, the strains and distortions of the workpiece during the welding process are presented. It may be observed that at the beginning of the welding (run-in) the strains are nonuniform and the specimen is distorted more at the side of the laser entry. At the end of the welding process and especially after cooling, the strains become more uniform. In Fig. 14a–c, the equivalent stress (von Mises) distribution at the run-in, mid-plate and run-out cross-section during the welding of the plates is plotted. In Fig. 14d, the von Mises stress is presented at the mid-plate cross-section after cooling and clamp release of the workpiece. Finally, in Fig. 15 are presented distributions of longitudinal residual stress x transverse to the weld line, transversal y – along the weld line (at middle plate thickness), as well as, x and y – across the plate thickness, after releasing and 600 s of cooling time. The longitudinal residual stress x
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transverse to the weld line, presented in Fig. 15a, follows a typical residual stress distribution, which has a maximum value of about 160 MPa at the weld line and decreases quickly to almost zero values at very short distance away from the weld pass. From Fig. 15b, it may be observed that if the run-in and run-out regions are excluded, the transversal residual stress y distribution along the weld line does not change significantly and has values ranging between 60 and 70 MPa (compression). However, in the run-in and run-out region a considerable y increases, up to 250 MPa, may be observed. The through-thethickness residual stress distributions x and y , presented in Fig. 15c, indicate that high stresses are developing close to the plate surface, with their maximum values occurring at about 1 mm depth from the upper surface, while these stresses decrease rapidly to almost zero values at the lower surface.
5.
Conclusions
A three-dimensional numerical model has been developed to simulate the entire LBW process, by considering all the important physical mechanisms of the laser welding process during the heating and cooling stages. The keyhole model is independent of any expensive experimental tests and measurements required for the keyhole characterization and can be widely applied in a very flexible way. The global coupled thermo-mechanical LBW model predicts reliably the transient temperature, stress, strain and distortion fields for various weld types, different material and varying process parameters, i.e. laser power, welding velocity and angle. Main contribution of the current work comprises the development of a validated local model for keyhole prediction, as well as, simulation results of full-scale lap-joint LBW of aluminum components.
references
Solana, P., Negro, G., 1997. A study of the effect of multiple reflections on the shape of the keyhole in the laser processing of materials. J. Phys. D: Appl. Phys. 30, 3216–3222. Okerblow, N., 1958. The Calculations of Deformations of Welded Metal Structures. Her Majesty’s Stationery Office, London, UK. Masubuchi, K., 1980. Analysis of Welded Structures. Pergamon Press, Oxford, UK. Yang, L.J., Xiao, Z.M., 1995. Elastic–plastic modeling of the residual stress caused by welding. J. Mater. Process. Technol. 48, 589–601. Tanigawa, Y., Akai, T., Kawamura, R., Oka, N., 1996. Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material properties. J. Therm. Stresses 19, 77–102. Chakravarti, L., Malik, M., Goldak, J., 1986. Prediction of distortion and residual stresses in panel welds. In: Computer Modeling of Fabrication Processes and Constitutive Behavior of Metals. Canadian Government Publishing Centre, Ottawa, Ontario, pp. 547–561. Fujii, S., Takahashi, N., Sakai, S., Nakabayashi, T., Muro, M., 2000. Development of 2D simulation model for laser welding. In: Proceedings of SPIE, vol. 3888.
Frewin, M.R., Scott, D.A., 1999. Finite element model of pulsed laser welding. Weld. J. Weld. Res. Suppl. (January), 15s–22s. Reinhart, G., Lenz, B., Rick, F., 1999. Finite element simulation for the planning of laser welding applications. In: Proceedings of the 18th International Congress on Applications of Lasers and Electro-Optics (ICALEO’99), San Diego, CA. Carmignani, C., Mares, R., Toselli, G., 1999. Transient finite element analysis of deep penetration laser welding process in a single-pass butt-welded thick steel plate. Comput. Methods Appl. Mech. Eng. 179, 197–214. Kang, D.-H., Son, K.-J., Yang, Y.-S., 2001. Analysis of laser weldment distortion in the EDFA LD pump packaging. Finite Elem. Anal. Des. 37, 749–760. Lankalapalli, K.N., Tu, J.F., Gartner, M., 1996. A model for estimating penetration depth of laser welding processes. Phys. D: Appl. Phys. 29, 1831–1841. Lampa, C., Kaplan, A.F.H., Powell, J., Magnusson, C., 1997. An analytical thermodynamic model of laser welding. Phys. D: Appl. Phys. 30, 1293–1299. Williams, K., 1999. Development of laser welding theory with correlation to experimental welding data. Laser Eng. 8, 197–214. Duley, W.W., 1999. Laser Welding. John Wiley & Sons, New York. Solana, P., Ocana, J.L., 1997. A mathematical model for penetration laser welding as a free-boundary problem. J. Phys. D: Appl. Phys. 30, 1300–1313. Kar, A., Mazumder, J., 1995. Mathematical-modeling of key-hole laser-welding. J. Appl. Phys. 78, 6353–6360. Dowden, J., Wu, S.C., Kapadia, P., Strange, C., 1991. Dynamics of the vapor flow in the keyhole in penetration welding with a laser at medium welding speeds. J. Phys. D 24, 519–532. Kar, A., Rockstroh, T., Mazumder, J., 1992. 2-dimensional model for laser-induced materials damage-effects of assist gas and multiple reflections inside the cavity. J. Appl. Phys. 71, 2560–2569. Metzbower, E.A., 1993. Keyhole formation. Met. Trans. B Process Met. 24, 875–880. Klein, T., Vicanek, M., Kroos, J., Decker, I., Simon, G., 1994. Oscillations of the keyhole in penetration laser-beam welding. J. Phys. D 27, 2023–2030. Matsunawa, A., Semak, V., 1997. The simulation of front keyhole wall dynamics during laser welding. J. Phys. D 30, 798–809. Kapadia, P., Solana, P., Dowden, J., 1998. Stochastic model of the deep penetration laser welding of metals. J. Laser Appl. 10, 170–173. Fabbro, R., Chouf, K., 2000. Keyhole modeling during laser welding. J. Appl. Phys. 87, 4075–4083. Rosenthal, D., 1946. The theory of moving sources of heat and its application to metal treatments. Trans. Am. Soc. Mech. Eng. 68, 849–866. Swift-Hook, D.T., Gick, A.E.F., 1973. Penetration welding with lasers. Weld. Res. Suppl., 492s–498s. Bunting, K.A., CornDeld, G., 1975. Toward a general theory of cutting: a relationship between the incident power density and the cut speed. Trans. ASME J. Heat Transfer, 116–122. O’Neill, W., Steen, W., 1994. Review of mathematical models of laser cutting of steel. Lasers Eng. 3, 281–299. Lax, M., 1977. Temperature rise induced by a laser beam. J. Appl. Phys. 48, 3919–3924. Nissim, Y.I., Lietoila, A., Gold, R.B., Gibbons, J.F., 1980. Temperature distributions produced in semiconductors by a scanning elliptical or circular CW laser beam. J. Appl. Phys. 51, 274–279. Moody, J.E., Hendel, R.H., 1982. Temperature profiles induced by a scanning cw laser-beam. J. Appl. Phys. 53, 4364–4371. Miyazaki, T., Giedt, W.H., 1982. Heat-transfer from an elliptical cylinder moving through an infinite-plate applied to electron-beam welding. Int. J. Heat Mass Transfer 25, 807–814.
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ANSYS User’s Manual, Release 10, 2005. Tsirkas, S.A., Papanikos, P., Kermanidis, Th., 2003. Numerical simulation of the laser welding process in butt-joint specimens. Mater. Process. Technol. 134, 59–69. www.twi.co.uk TWI World Center for Materials Joining Technology.
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journal homepage: www.elsevier.com/locate/jmatprotec
Residual stress and distortion calculation of laser beam welding for aluminum lap joints G.A. Moraitis, G.N. Labeas ∗ Laboratory of Technology and Strength of Materials, Department of Mechanical Engineering and Aeronautics, University of Patras, Patras 26 500, Greece
a r t i c l e
i n f o
a b s t r a c t
Article history:
A numerical simulation model of the laser beam welding (LBW) process is developed, aiming
Received 2 August 2006
to a reliable prediction of the residual stress and distortion fields. As LBW is a thermo-
Received in revised form
mechanical process, a thermal analysis is conducted to analyze the spatial temperature
26 June 2007
distribution history, coupled to a mechanical analysis to calculate the residual stresses and
Accepted 5 July 2007
distortions. An innovative and efficient keyhole model, independent of any empirical parameter, is introduced for the prediction of the keyhole size and shape required for the thermal analysis. All the major physical phenomena associated to the LBW process, such as, heat
Keywords:
radiation, thermal conduction and convection heat losses are taken into account in the
Laser beam welding
model development. The thermal and mechanical material properties are introduced as
Keyhole modelling
temperature dependent functions, due to the high temperature variations and the material
Lap joint
phase changes occurring during the welding. The simulation algorithm is programmed as
Thermo-mechanical analysis
a macro routine within the ANSYS finite element code. The model is validated for the case
Welding simulation
of butt joint welding DH-36 steel plates and its efficiency is demonstrated for the lap-joint welding of two aluminum 6061-T6 plates. The main advantage of the developed model is its generality and flexibility, as it is independent of any empirical parameter, enabling its application in parametric studies of a wide range of LBW problems of different geometrical, material and joint type, requiring only the basic mechanical and thermal material properties. © 2007 Elsevier B.V. All rights reserved.
1.
Introduction
Laser beam welding (LBW) is based on high power density welding technologies, which have the possibility of focusing the beam power to a very small spot diameter. As a result, the LBW process is advantageous compared to the conventional welding processes, e.g. arc welding, leading to smaller Heat Affected Zones (HAZ), lower distortions, residual stresses and strains. Most welding processes operate in conduction-limited mode, implying that the heat supplied to the surface of the
∗
Corresponding author. Tel.: +30 2610 991027; fax: +30 2610 997190. E-mail address: [email protected] (G.N. Labeas). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.07.013
irradiated material via laser light absorption is basically transferred through conduction to the surrounding material. When the laser power density is high (around 1 MW/cm2 ), a keyhole is formatted, accompanied by phase changing phenomena, i.e. melting and evaporation. Due to the complicated physical phenomena taking place [1], the optimization of laser welding process is usually based on welding experiments and on trial and error approaches, which have many operative difficulties, increasing the total cost of the process. Simulation models of the welding process, which have been validated through experimental results, are of major
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importance for a high number of reasons: the deep understanding of the LBW physics, the reliable extension of the process applicability to modern demanding industrial applications and the efficient definition of the LBW process parameters, without enormous cost penalty. A number of analytical and numerical models of welding processes have been developed to evaluate temperature, residual stress and distortion distributions during the welding process of structural components. These include analytical models [2–5], two-dimensional finite element models [6–7] and three-dimensional finite element models [8–11]. However, most of the investigations are limited to local problems of laser beam butt joint, while simulations of full-scale LB lap-joint components are not available so far. Furthermore, most of the investigations refer to steel components and only a handful to aluminum alloys. The most critical issue of the thermal analysis of LBW is the proper representation of the heat input to the material. The keyhole phenomenon has the principal contribution to the non-homogeneous local heating of the material volume. It arises from experimental measurements that in deep penetration welding, the keyhole shape is nearly conic and its vertex angle decreases as keyhole depth increases [12–15]. Several approaches to the mathematical modelling of keyhole formation in laser welding of steel material can be found in the literature [16,17], however, for aluminum material no reference may be found. A number of researchers have developed mathematical models for the shape and location of the weld pool and the keyhole of steel LBW [18–24], by considering appropriately the energy and pressure balance; however, these models are very complicated and their practical applicability to different geometrical, material and joint type becomes difficult. In refs. [25–28] simple moving point and line heat source models are presented. In refs. [29–33] three-dimensional models with Gaussian heat sources have been used in the thermal analysis of steel materials treated by various laser beam applications. In the present paper, a thermo-mechanical FE model based on the keyhole theory is developed in order to predict the stress, strain and distortion fields for simulating the lap-joint welding of aluminum components. Major innovations of the model are an innovative keyhole prediction approach, which has the capability to handle any steel or aluminum alloy, as well as, an advanced numerical approach enabling efficient prediction of residual stress and distortions of LBW components. Furthermore, the proposed simulation methodology is especially efficient for lap- and T-joint LBW, which are not covered sufficiently in the literature, although, they are widely used in many industrial applications. The entire simulation process comprises two numerical models of different scale level. The first model aims to a reliable prediction of the keyhole formation process, while the second one calculates the thermal stresses and strains. The keyhole prediction model is not based on experimentally defined keyhole parameters, but approaches the real thermal phenomena, through an efficient route, applicable to different material types, laser conditions and welded joint geometry. The keyhole shape and size are calculated using local finite element analysis, which simulates all the basic phenomena taking place locally at the heat introduction area. The keyhole model and its advantages are presented in Section 2.
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Consequently, the calculated keyhole is approximated by a conical volume having a three-dimensional Gaussian heat flux distribution, moving along the weld line. A global non-linear thermal analysis step of the entire welded configuration is carried out in Section 3, to calculate the spatial temperature distribution; based on it, the corresponding mechanical analysis step is implemented, to calculate deformation, stresses and strains of the welded components. In Section 4, various numerical results from the simulation of LBW aluminum components are presented.
2.
Keyhole prediction model
The laser heat absorbed by the welded surfaces, leads to temperature values far above the material’s melting point at specific locations; this results in local material melting creating a molten pool, which increases, as more heat is input into the material volume. As shown in Fig. 1, when the power density is high, in the range of 1 MW/cm2 , the laser “drills” through the components, creating a conical or cylindrical column of material vapours, forming the keyhole volume. It should be mentioned that the vapour pressure within the keyhole maintains the equilibrium and stops the keyhole from collapsing. As the laser heat source is moving, the keyhole is dragged through the welding line, by melting in front and solidifying at behind, resulting to weld creation. The above-described physical phenomena are simulated through a detailed numerical FE model of the local volume affected by the laser beam. The FE mesh comprises threedimensional eight-node thermal elements (type ‘Solid70’), with temperature as the single degree of freedom at each node. The area nearby and along the weld line is discretized by a very dense mesh, comprising about 80,000 elements as resulted from a mesh convergence parametric study, which is illustrated in Fig. 2. It is usual assumption that the laser energy is applied on the surface of the welded component as a surface load having the Gaussian distribution shown in Fig. 3. The surface heat flux distribution q0 (x,y) may be calculated according to the formula [36]: q0 (x, y) =
c · n · P −c(x/r)2 +(y/r)2 ·e · r2
Fig. 1 – Keyhole formation during the welding process (modified from ref. [37]).
(1)
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Fig. 4 – Thermal properties for Al alloy 6061-T6 (after ref. [34]). Fig. 2 – Temperature plot for mesh convergence parametric study at the first step of the analysis.
Fig. 3 – Heat flux Gaussian distribution of laser beam.
where n is the laser absorption coefficient of the irradiated surface, P the laser beam power, r the laser beam radius, c a shape parameter of the heat-flux distribution and x, y, z are the Cartesian coordinates. The laser absorption coefficient n of an irradiated surface depends mainly on material type, surface treatment, color and roughness. It has been calculated after measurements that for pure aluminum the absorption coefficient is about 0.18. The thermal material properties of Al 6061-T6 are presented in Fig. 4, after ref. [34]. By solving the FE model for a selected solution time interval, which depends on the welding velocity, element size and beam radius, the heat transfer through conduction
and the heat losses through convection are calculated. The temperature increases locally creating a molten pool, while simultaneously, more heat is applied to the top surface. When the temperature of an element exceeds the vaporization point, the corresponding material volume is converted to vapours; the developed algorithm considers this element as ‘invisible’ and ‘transparent’, therefore, successive surface loads are introduced to the element(s) below the vaporized one. This procedure is based on an element selection routine, which is repeated as the heat source moves along the weld line with the welding velocity. When the heat source has moved enough, i.e. more than one laser beam diameter, the temperature distribution behind the heat source becomes almost steady, therefore, the final keyhole shape and size can be defined. In Fig. 5, the flow chart of the keyhole prediction procedure using FE numerical simulation is presented. The numerically predicted keyhole is verified through Eqs. (2)–(4), which result from a validated semi-experimental procedure [12,16]. Due to the lack of data for Al, the verification is performed for steel and yields that the numerically predicted keyhole is in good agreement with the semi-experimental one, as shown in Fig. 6. d=
Qin h(Tv − T0 )(c1 + (c2 /2) Pe + (c3 /3) Pe2 + (c4 /4) Pe3 )
(2)
Pe =
uw 2˛(ˇ1 e−1 Pe + ˇ2 e−2 Pe + ˇ3 e−3 Pe )
(3)
r0 =
2˛Pe u
(4)
Fig. 5 – Flow chart of the keyhole simulation approach.
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where d is the keyhole depth; T the ambient temperature; w the weld pool width; Tv the vaporization temperature; u the welding velocity; h the thermal conductivity; Qin the absorbed laser power; a the thermal diffusivity; ci the constants (independent of material); i is the constants (dependent of material).
3.
Fig. 6 – Comparison between FE and semi-experimental keyhole for steel DH-36.
Thermo-mechanical analysis
The global coupled thermo-mechanical model for the LBW simulation assumes a three-dimensional Gaussian heat flux distribution in a conical volume, which moves along the weld line. This conical volume arises as an approximation of the predicted keyhole, as described in Section 2. By applying the thermal and mechanical boundary conditions a coupled thermo-mechanical analysis may be executed and temperature distributions, residual stresses and distortions can be calculated. An overview of this numerical procedure
Fig. 7 – Flow chart of the thermo-mechanical global LBW model.
Fig. 8 – (a) Lap-joint geometry and (b) detail of the mesh of the weld area.
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Fig. 10 – Predicted keyhole shape. Fig. 9 – Mechanical properties for Al alloy 6061-T6.
is described in the flow chart of Fig. 7. In this model the thermal and mechanical material properties dependency on temperature is introduced. The coupled thermo-mechanical analysis is demonstrated in the simulation of LBW lap joint of two thin aluminum 6061-T6 plates of dimensions 150 mm × 150 mm × 2 mm. The welding line has 120 mm length, the welding velocity is 50 mm/s and the lap width is 50 mm (Fig. 8a). A dense mesh is used in the area along and nearby the weld line (Fig. 8b), in order to simulate the complex physical phenomena that take place in the area, while the remaining volume has a coarser mesh, leading to a total of about 50,000 elements; the element type is ‘Solid 70’ for the thermal analysis and ‘Solid 45’ for
the mechanical one. For each time interval, the solution is executed in two successive steps; first, a transient heat transfer analysis is performed and the resulting temperature field is used as input to the mechanical analysis. The same procedure is repeated for every time interval.
3.1.
Global thermal analysis
According to the keyhole theory, the laser beam is modelled as a three-dimensional moving heat source, simulating the heat distribution and flow in the welding direction. The keyhole is assumed to have a conical shape the radius r0 and the depth d, which are predicted by the methodology described in Section 2; the heat flux in the keyhole volume follows a Gaussian distribution, which can be computed according to
Fig. 11 – Temperature distribution during the welding process, at the (a) run-in, (b) mid and (c) run-out section of the workpiece.
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Fig. 12 – Run-in and run-out temperature during (a) the welding process, (b) the cooling stage.
the formula [36]:
Q=
2P r02 d
e1−(r/r0 )
2
1−
z d
(5)
where r0 is the initial radius at the top of the keyhole, d the depth of the keyhole, r the current radius (the distance from the cone axis), z the vertical axis and P is the absorbed laser power. For aluminum lap and butt joints, where the laser beam
Fig. 13 – Distributions of equivalent stains at the time of run-in (a), mid-pass (b), run-out (c) and after cooling (d).
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Fig. 14 – Distributions of equivalent stresses at cross-sections of the workpiece; (a) at the run-in area, (b) around the weld line middle, (c) at the run-out area and (d) at weld line middle after cooling and clamp released.
is normal to the component surfaces, the reflected energy is approximately 80–90% of the nominal power of the source. The absorbed energy is transferred to the metal by conduction and convection, while is partly lost afterwards from the metal to the environment by convection (cooling). A film or convection coefficient, h, is introduced in the numerical model as a thermal material property, in addition to conductivity, specific heat and enthalpy (Fig. 4). The heat transfer problem is described by the transient Eqs. (6a) and (6b): [C(T)]{T (t)} + [K(T)]{T(t)} + {v} = {Q(t)}
(6a)
qc = A · h · (T − Tair )
(6b)
where [K] is the conductivity matrix, [C] the specific heat matrix, {T} the vector of nodal temperatures, {T } the vector of time derivative of {T}, {v} is the velocity vector, which is equal to zero as no mass transport is assumed in the current problem, {Q} the nodal heat flow vector, A the plate surface area, h
the convection (or film) coefficient and T and Tair are the plate and air temperatures, respectively. In each time interval, time integration of the heat conduction and heat convection equations (Eqs. (6a) and (6b)) lead to the calculation of the required transient temperature distributions.
3.2.
Global mechanical analysis
A non-linear mechanical analysis is executed just after each thermal analysis step, in order to calculate stress and strain distributions of the structure. The results of the thermal analysis step (transient temperature distribution) serve as loading to the corresponding mechanical analysis step. The introduced temperature dependent material mechanical properties Young’s Modulus, Poisson’s Ration, density and thermal expansion coefficient are presented in Fig. 9. The boundary conditions assumed in the mechanical analysis, is clamping of one, parallel to the weld line, plate edge.
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Fig. 15 – Residual stress distributions (a) longitudinal–transverse to the weld line, (b) transversal–along the weld line, (c) longitudinal and transversal through-the-thickness and (d) definition of lines for residual stress presentation.
The non-linear mechanical analysis problem is described by the following general FE equation: [K(T)]{u(t)} + {F(t)} + {Fth (t)} = 0
(7)
where K(T) is the temperature-dependent stiffness matrix; F(t) the external load vector; Fth (t) the temperature load vector; {u(t)} is the displacement vector. More details about Eqs. (6) and (7) may be found in ref. [35]. Material plasticity is represented using the von Mises criterion and a kinematic strain hardening law, requiring activation of the material nonlinear option in the FE analysis. The geometrical non-linear option of the analysis is not activated, as large displacement or strains were not expected to be developed in the structure. The Newton–Raphson algorithm is utilized for solving the non-linear equation system and the Newmark integration scheme is applied for the numerical integration in the time domain.
4.
Results and discussions
The lap joint is welded using a CO2 laser of 4 kW power with 2 mm focused radius and 50 mm/s welding velocity. From the local analysis model, it arises that the keyhole has an elliptical cone shape with upper radius 1.98 mm and depth 2.18 mm. The elliptical shape has been approximated as a perfect cone (Fig. 10), which reduces the modelling difficulties without affecting the analysis results significantly. By using Eqs. (5)–(7),
a global coupled FE thermo-mechanical analysis is carried out. The simulation of the welding process is followed by a number of loading steps simulating the cooling stage, which lasts for about 10 min. The temperature distributions developed during the laser beam pass are presented in Fig. 11, from which, the very high temperature gradients in the vicinity of the weld line close to the upper surfaces of the welded components may be clearly observed. In Fig. 12a and b, the temperature of run-in and runout points during the welding process and during the cooling stage, respectively, are presented. It can be observed that the heating rate is very steep, leading to almost immediate local heating, which is followed by a very fast (less than half second) cooling period. In Fig. 13, the strains and distortions of the workpiece during the welding process are presented. It may be observed that at the beginning of the welding (run-in) the strains are nonuniform and the specimen is distorted more at the side of the laser entry. At the end of the welding process and especially after cooling, the strains become more uniform. In Fig. 14a–c, the equivalent stress (von Mises) distribution at the run-in, mid-plate and run-out cross-section during the welding of the plates is plotted. In Fig. 14d, the von Mises stress is presented at the mid-plate cross-section after cooling and clamp release of the workpiece. Finally, in Fig. 15 are presented distributions of longitudinal residual stress x transverse to the weld line, transversal y – along the weld line (at middle plate thickness), as well as, x and y – across the plate thickness, after releasing and 600 s of cooling time. The longitudinal residual stress x
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transverse to the weld line, presented in Fig. 15a, follows a typical residual stress distribution, which has a maximum value of about 160 MPa at the weld line and decreases quickly to almost zero values at very short distance away from the weld pass. From Fig. 15b, it may be observed that if the run-in and run-out regions are excluded, the transversal residual stress y distribution along the weld line does not change significantly and has values ranging between 60 and 70 MPa (compression). However, in the run-in and run-out region a considerable y increases, up to 250 MPa, may be observed. The through-thethickness residual stress distributions x and y , presented in Fig. 15c, indicate that high stresses are developing close to the plate surface, with their maximum values occurring at about 1 mm depth from the upper surface, while these stresses decrease rapidly to almost zero values at the lower surface.
5.
Conclusions
A three-dimensional numerical model has been developed to simulate the entire LBW process, by considering all the important physical mechanisms of the laser welding process during the heating and cooling stages. The keyhole model is independent of any expensive experimental tests and measurements required for the keyhole characterization and can be widely applied in a very flexible way. The global coupled thermo-mechanical LBW model predicts reliably the transient temperature, stress, strain and distortion fields for various weld types, different material and varying process parameters, i.e. laser power, welding velocity and angle. Main contribution of the current work comprises the development of a validated local model for keyhole prediction, as well as, simulation results of full-scale lap-joint LBW of aluminum components.
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