The aim of this work is to propose a mechanical behavior law dedicated to the mushy zone located between the solid phase and the weld pool in welding.  The objective is to take into account of the influence of the mushy zone in the simulation of welding in order to improve the computation of induced effects such as residual stresses

Amin-el-Sayed, H.

Bergheau, J. M.

Feulvarch, E.

Leblond, J. B.

English

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A mechanical behavior law for the numerical simulation of the mushy zone in welding

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XII International Conference on Computational Plasticity. Fundamentals and ApplicationsCOMPLAS XIIE. On˜ate, D.R.J. Owen, D. Peric and B. Sua´rez (Eds)A MECHANICAL BEHAVIOR LAW FOR THE NUMERICALSIMULATION OF THE MUSHY ZONE IN WELDINGH. AMIN-EL-SAYED∗†, E. FEULVARCH∗, F. BOITOUT†, J.-B LEBLOND††AND J.-M BERGHEAU∗∗ Laboratoire de Tribologie et Dynamique des Syste`mes (LTDS)Universite´ de Lyon, Ecole Nationale d’Inge´nieur de Saint-Etienne, UMR 5513 CNRS58 rue Jean Parot, 42100 Saint-Etienne, Francee-mail: hussein.amin-el-sayed@enise.fr, eric.feulvarch@enise.fr, jean-michel.bergheau@enise.fr -web page: http://www.enise.fr, http://ltds.ec-lyon.fr/spip/†ESI-Group70, rue Robert, 69006 Lyon, Francee-mail: frederic.boitout@esi-group.com - web page: http://www.semni.org††Institut Jean le Rond d’Alembert (IJLRDA)Universite´ Pierre et Marie Curie4, place Jussieu, 75252 Paris cedex 05, Francee-mail: jbl@lmm.jussieu.fr - web page: http://www.dalembert.upmc.frKey words: Behavior law, welding, mushy zone, numerical simulationAbstract. The aim of this work is to propose a mechanical behavior law dedicatedto the mushy zone located between the solid phase and the weld pool in welding. Theobjective is to take into account of the influence of the mushy zone in the simulation ofwelding in order to improve the computation of induced effects such as residual stresses.1 INTRODUCTIONDuring welding, the material can be liquid in the weld pool or solid in the rest of theassembly. These two states can co-exist in the mushy zone located between the weldpool and the structure as shown in figure 1. This zone is caracterized by a temperaturebetween the liquidis temperature and the solidus temperature.From the simulation point of view, the mushy zone is not always considered for thecomputation of residual effects such as residual stresses. For example, Heuze´ suggeststo use two independent behavior laws without considering a mushy zone. His approachconsists in coupling a Lagrange solid and a Euler fluid using a melting temperature cri-terion. This criterion defines the temperature at which, the base material is switched1A Mechanical behaviour law for the simulation of the mushy zone in welding  164H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAU“sharply”between solid and liquid phases, as for an isothermal transformation. This tech-nique has been implemented in the software SYSWELDR© [4].Unfortunately, the computation of residual stresses requires to take accurately intoaccount the behavior of the solid state during all the cooling stage including the solid-ification in the mushy zone. The finite element simulation of such phenomenon is notalways obvious. To take account of the mixture mechanical behavior in the mushy zone,Decultieux [1] and Vicente [3] propose a specific law depending on the solidification state.They consider a half fluid visco plastic behavior for a fluid fraction fL between 50% and100% and a half solid elasto visco plastic behavior for 0% ≺ fL ≺ 50%. Jaouen [2] hasused this approach to model the solidification of alloys in casting. He has defined a refer-ence temperature called coherence temperature corresponding to fL = 50%, at which themixture behavior changes.In this paper, we propose a new approach based on a continuous transition betweenthe solid and the liquid phases through the mushy zone. It is a phenomenological andpragmatic approach to compute the mixture behavior in a solid context. An example ofa spot weld is proposed to show the interest of such an approach.Weld poolMushy zoneBase materialABFigure 1: Schematic description of welding2 Behaviour law2.1 FormulationThe new behavior law proposed in this work is based on a combination of parallel andserie behaviors. The use of a pure parallel behaviour is unrealistic because the deforma-tion is therefore the same in the solid and the liquid which do not behave in the sameway. Indeed, the liquid is assumed to be incompressible unlike solid which is not. Withsuch an approach, as soon as a fluid drop appears, the behavior of the mixture becomes2165H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAUincompressible. To overcome this difficulty, we propose a more complex combination ofparallel and serie behaviors. The parallel behavior is modeling the deviatoric part of thestress tensor whereas the serie behavior is modeling the spherical part as shown in thefigure 2.LiquidSolidLiquid Solidp pε′s←− s−→sL←− sL−→sS←− sS−→σ←− σ−→Figure 2: Rheological scheme of the combined modelThe pressure and the deviatoric part of the deformation are supposed to be the samein the solid and the liquid. The stress tensor is obtained by:=s= fL=sL +fS=sS (1)pS = pL = p (2)=σ==s +p=I (3)Where T , p,=σ,=s represent respectively the temperature, the pressure, the stress tensorand it’s deviatoric part. The indexes L and S are related to the solid and the liquidphases. fL and fS denote the liquid and solid fraction depending on the temperature onlywith fS = 1− fL. Using a linear mixture rule, we getfL = 1 si T  TLfL =T−TSTL−TS si TS  T  TLfL = 0 si T  TS(4)where TL and TS denote the liquidis temperature and the solidus temperature.2.2 ImplementationConsidering that the pressure is identical in the solid and the liquid, it’s computed bymeans of the compressibility coefficient of the solid phase. As shown above, the stressesstate is an overlay of both the solid and the fluid behaviour for the deviatoric part. From3166H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAUthe numerical point of view, the deviatoric part is computed in each phase separately. Atfirst,=sS is calculated using the solid mechanical behavior as follows:=sS= 2µs=ε′e(5)where=ε′edenotes the deviatoric part of the elastic strain rate tensor=εe. µs is respectivelyone of the Lame´ parameters. In the fluid part,=sL is then obtained as follows=sL= 2µf=.ε′ (6)where=.ε′ denotes the deviatoric part of the strain rate tensor and µf is the viscosity of theliquid. The stress tensor of the mixture is therefore given by:=σ= 2µs (1− fL)=ε′e+2µffL=.ε′ +p=I (7)The interest of such a model remains in the fact that we can use any kind of solidbehaviour law. For the computation of the solid mechanical behavior, the history of thematerial in terms of plastic strain needs to be known at each time step. Because thishistory is deleted during the liquefaction in heating, it is necessary to distinguish severalcases during the solidification:• the existing solid fraction which has all its history,• the existing fluid fraction which has not any history,• the new formed solid fraction which has a virgin state.In this way, the plastic strain and the cumulative plastic strain [6] need to take intoaccount of the new formed solid fraction. The new values are those of a new solid fractionand they are given by:〈Xi〉 = fSfSXi +12f S + fSfS(Xi −Xi ) (8)where Xi denotes the hardening strain or the cumulative plastic deformation, 〈Xi〉 is themean value of Xi. The exponent “star”denotes the value of the variable at the previoustime step.3 Application3.1 Description of the problemWe are interested in the 3D simulation of a SPOT welding. We consider a square plateof 40 mm side and 3 mm thick. This plate admits two plane of symmetry. Thus, onlythe quarter of the plate is modeled.The mesh is represented on the figure 3. It contains 3,389 node and 12,261 linear tetra-hedron element.4167H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAUFigure 3: The mesh related to the quarter of the plate. The symmetry conditions areapplied on the red nodes.3.2 Boundary conditionsThe thermal load is represented by an equivalent heat source applied on the top centerof the plate. The heating is active during the first two seconds. A temperature of 293 Kis imposed on the lateral side of the plate. The maximal temperature reached is 2368 Kon the centre of the top side. After the first two heating seconds, the plate is cooled byconduction to reach the ambient temperature. The figure 4 shows the temperature of thecenter of the plate top side during the welding.0 20 40 60 80 100 12005001,0001,5002,000TimeTemperatureFigure 4: Temperature on the center of the plate top sideWe consider an elasto-plastic solid behaviour with an isotropic hardening strain. TheYoung modulus, the yield stress, the thermal strain and the hardening strain depend onthe temperature. They are represented on the figure 5. The Poisson’s ratio is equal to 0, 3.The solidification and liquefaction temperature range is situated between TS = 1673[K]and TL = 1773[K].5168H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAU0 500 1,000 1,50000.511.52×105(a) Young modulus [MPa]0 200 400 600 800 1,0001,2001,400050100150200250(b) Yield stress [MPa]0 200 400 600 800 1,000 1,20000.511.52×10−2(c) Thermal strain0 0.2 0.4 0.6 0.8 1050100150 20 C˚100 C˚200 C˚300 C˚400 C˚500 C˚600 C˚700 C˚800 C˚(d) Hardening strain [MPa]Figure 5: Material data3.3 Numerical resultsThe temperature fields obtained at different times are presented on the figure 6.Concerning the pressure or the spherical part of the stress tensor, the figure 7 shows thatxyz0208.6991906.99397.397586.096774.794963.4931152.191340.891529.591718.29   Max = 2095.68    Min = 20 Comput.Ref GlobalTime 2TempCONTOURSSYSWELD - 12.1  -  allsts  -  Tue Feb 19 16:56:16 2013(a) 2 secxyz020.157221.572420.314520.471720.62920.786220.943421.100721.257921.4151   Max = 21.7296    Min = 20 Comput.Ref GlobalTime 100TempCONTOURSSYSWELD - 12.1  -  allsts  -  Tue Feb 19 16:56:31 2013(b) 110 secFigure 6: The temperature fieldsthe pressure is uniform in the fluid and the mushy zone. We also mention that the new6169H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAUsolid formed is under a tension state. Finally, the interest of the numerical simulation ofxyz0-86.03116.5788-74.6299-63.2288-51.8277-40.4266-29.0255-17.6244-6.223355.17774   Max = 27.9799    Min = -97.4321 Comput.Ref GlobalTime 2mean stressCONTOURSSYSWELD - 12.1  -  allsts  -  Wed Feb 20 01:41:27 2013(a) 2 secxyz0-37.8481197.856-11.658714.530640.7266.909393.0987119.288145.477171.667   Max = 224.045    Min = -64.0375 Comput.Ref GlobalTime 110mean stressCONTOURSSYSWELD - 12.1  -  allsts  -  Wed Feb 20 01:41:53 2013(b) 110 secFigure 7: The pressure fieldswelding remain in the calculation of the residual stresses. The figure 8(a) shows that thestresses become weak in the mushy zone and approach zero in the molten pool. However,the figure 8(b) shows that, on the ambient temperature, we obtained a large Von misesstresses.xyz00303.333336.666671013.333316.66672023.333326.6667   Max = 176.553    Min = 0 Comput.Ref GlobalTime 2Von mises stressSECTIONSSYSWELD - 12.1  -  allsts  -  Wed Feb 20 17:20:56 2013(a) Von mises at the time 2 secxyz031.7438311.59562.838493.933125.028156.122187.217218.311249.406280.501   Max = 342.69    Min = 0.649224 Comput.Ref GlobalTime 110Von mises stressCONTOURSSYSWELD - 12.1  -  allsts  -  Wed Feb 20 01:18:56 2013(b) Von mises at the ambient temperatureFigure 8: Von mises stresses field7170H. AMIN-EL-SAYED, E. FEULVARCH, F. BOITOUT, J.-B LEBLOND AND J.-M BERGHEAU3.4 ConclusionIn this paper, we propose a new approach of a behaviour law able to simulate the mushyzone during the material transition in the welding process. This pragmatic approach isalready implemented in the finite element software SYSWELD. The first results convergesproperly and leads to a satisfactory results.REFERENCES[1] F. Decultieux, Caracte´risation du comportement thermome´canique d’alliages defonderie pendant la solidification: Application la pre´diction des de´fauts de ruptureet a` la mode´lisation des transferts de chaleur entre pie`ce et outils, The`se de doctorat,Ecole des mines de Paris, 1996.[2] O. Jaouen, mode´lisation tridimensionnelle par e´le´ments finis pour l’analyse ther-mome´canique du refroidissement des pie`ces coule´es, The`se de doctorat, Ecole desmines de Paris, 1998.[3] P. Vicente-Hernandez, Vers une approche thermome´canique du proce´de´ de fonderie:caracte´risation rhe´ologique et microstructurale d’alliage me´tallique a`l’e´tat semi-solide, The`se de doctorat, Ecole des mines de Paris, 1994.[4] SYSWELDR©, User’s manual, ESI-group, 2012.[5] T. Heuze, Mode´lisation des proce´de´s de soudage par friction et malaxage, The`se dedoctorat, Universite´ Pierre et Marie Curie, 2011.[6] J.-B Leblond, Mathmatical modelling of transformation plasticity in steels - II cou-pling with strain hardening phenomena, International journal of plasticity, vol 5,573-591, 1989.8171

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A mechanical behavior law for the numerical simulation of the mushy zone in welding

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