Numerical simulation of friction stir butt welding process for AA5083-H18 sheets

Numerical simulation of friction stir butt welding process for AA5083-H18sheetsDongun Kim a ,Harsha Badarinarayan b ,Ji Hoon Kim c ,Chongmin Kim d ,Kazutaka Okamoto e ,R.H.Wagoner c ,Kwansoo Chung a ,*aDepartment of Materials Science and Engineering,Research Institute of Advanced Materials,Seoul National University,599Gwanak-ro,Gwanak-gu,Seoul 151-742,Republic of Korea bResearch and Development Division,Hitachi America Ltd.,34500Grand River Avenue,Farmington Hills,MI 48335,USA cDepartment of Materials Science and Engineering,The Ohio State University,2041College Road,Columbus,OH 43210,USA dMaterials and Process Lab.,R&D Center &NAO Planning,General Motors Corporation,Warren,MI 48090-9055,USA eHitachi Research Laboratory,Hitachi Ltd.,Omika 7-1-1,Hitachi 319-1292,Japana r t i c l e i n f oArticle history:Received 11May 2009Accepted 20October 2009Available online 29October 2009Keywords:Friction stir weldingThermo-mechanical Simulation FVMAA5083-H18CFDSTAR-CCM þa b s t r a c tThermo-mechanical simulation of the friction stir butt welding (FSBW)process was performed for AA5083-H18sheets,utilizing a commercial finite volume method (FVM)code,STAR-CCM þ,which is based on the Eulerian formulation.Distributions of temperature and strain rate histories were calculated under the steady state condition and simulated temperature distributions (profiles and peak values)were compared with experiments.It was found that including proper thermal boundary condition for the backing plate (anvil)is critical for accurate simulation results.Based on the simulation,thermal and deformation histories of material elements were also calculated,useful to predict material characteristics of the weld such as hardness or grain size,and possibly for the susceptibility of weld to abnormal grain growth (AGG)after post-weld heat treatment.Ó2009Elsevier Masson SAS.All rights reserved.1.IntroductionDuring the past two decades,government legislation,environ-mental pressures,safety regulations and consumer demand have brought about the need for lighter automotive vehicles that are more fuel-efficient with lower emissions and improved handling and crashworthiness.In this regard,application of light-weight materials,such as aluminum and magnesium alloys has attracted great attention to partly replace conventional steels for structural parts.Also,the concept of combining various sheet metals into a single welded blank has been developed to enable products to have optimal properties.Friction stir welding (FSW)is a relatively new process to join metals such as aluminum,magnesium and steel alloys in the solid state.The process consists of a rotating tool which is plunged into two sheet or plates tightly abutted along a line,which line is then traversed by the rotating tool.(In fact,for most FSW processes the sheets are moved and the rotating tool is held stationary,but the distinction is insignificant.)During the process,heat is generated by plastic deformation as well as by the friction between the tool andthe work-pieces.The work-pieces are ultimately joined by the stirring action of the softened (but always solid)material.The advantages of FSW include low residual stress,low energy input,and fine grain size compared to the conventional liquid–solid welding method.One way to optimize the process is to utilize the thermo-mechanical modeling of FSW.A number of academic and industrial institutions have made efforts to develop numerical codes for FSW.Although FSW is simple in concept,the physics behind the process is complex,which includes mechanical heat generation,heat and mass transport.The large strains and strain rates make observing the details of the process difficult,which makes process modeling attractive or essential to understand it.The material database available in liter-ature does not typically include the constitutive data required to describe this phenomenon.It is not possible to directly observe the material mixing and flow either.Many attempts to optimize the process have been reported.Some of the published work include:Alcoa Technical Center (Ulysse,2002),Altair Engineering and Boeing (Reddy and Baumann,2004),CENAERO (De Vuyst et al.,2008),ExxonMobil Development Company (Kumar et al.,2008),Hitachi (Badarinarayan,2004;Badarinarayan et al.,2007;Hirasawa et al.,2004),Hydro Aluminum,REMAC (St-Georges et al.,2007)and The Welding Institute (Colegrove and Shercliff,2005;Russell and Shercliff,1999),*Corresponding author.Tel.:þ8228807189;fax:þ8228851748.E-mail address:kchung@snu.ac.kr (K.Chung).Contents lists available at ScienceDirectEuropean Journal of Mechanics A/Solidsjournal homepage:/locate/ejmsol0997-7538/$–see front matter Ó2009Elsevier Masson SAS.All rights reserved.doi:10.1016/j.euromechsol.2009.10.006European Journal of Mechanics A/Solids 29(2010)204–215as well as academic institutions,BYU (Fourment et al.,2004;Guerdoux et al.,2004,2007;Oliphant,2004),Cambridge (Cole-grove and Shercliff,2005;Russell and Shercliff,1999),Dalian University of Technology (Zhang et al.,2005,2007),Ecole des Mines de Paris (Fourment et al.,2004;Guerdoux et al.,2004,2007),EMSE (Heurtier et al.,2006),ENSAM (Lorrain et al.,2008),Kobe University (Badarinarayan et al.,2007;Hirasawa et al.,2004),Old Dominion University (Mandal et al.,2007),ORNL (Aramayo et al.,2002),OSU (Buffa et al.,2006a,b;Buffa et al.,2007;Fratini et al.,2006,2007),Seoul National University (Kim et al.,2009),SMU (Chen and Kovacevic,2006;Soundararajan et al.,2005),South Dakota school of Mines and Technology (Kalagara and Muci-Kuchler,2007),Technical University of Denmark (Schmidt and Hattel,2005,2008),The University of Adelaide (Colegrove et al.,2000),UMR,UNIPA (Buffa et al.,2006a,b,2007;Fratini et al.,2006,2007),USC (Khandkar et al.,2006;Yuh et al.,1999),University of Toronto (Bendzsak et al.,2000)and Queen’s University of Belfast (McCune et al.,2004).In the FSW process,many kind of commercial codes,such as ABAQUS (Lorrain et al.,2008;Kalagara and Muci-Kuchler,2007;Schmidt and Hattel,2005;Khandkar et al.,2006),ANSYS (Chen and Kovacevic,2006;Soundararajan et al.,2005),DEFORM (Buffa et al.,2006a,b;Buffa et al.,2007;Fratini et al.,2006,2007),FIDAP (Ulysse,2002),FLUENT (Colegrove and Shercliff,2005;Lorrain et al.,2008;Kim et al.,2009),LSDYNA (Aramayo et al.,2002),HYPERXTRUDE (Reddy and Baumann,2004),FORGE3(Fourment et al.,2004;Guerdoux et al.,2004;Oliphant,2004),THERCAST (Fourment et al.,2004;Guerdoux et al.,2004),as well as user-defined codes (Badarinarayan,2004,2007;Hirasawa et al.,2004;Heurtier et al.,2006)have been utilized to simulate the process.There are two basic approaches:1)a solid mechanics explicit finite element method (FEM)with an updated Lagrangian description,and 2)the computational fluid dynamics (CFD)finite volume method (FVM)with the Eulerian description.In the Lagrangian description model,the domain is divided into material nodal points and elements;these points and elements move and deform during the analysis.In the Eulerian description model,the domain is divided into spatial nodal points and elements;these points and elements remain stationary in space (in principle)during the analysis,with material flowing through and by them.At these points and elements,thermal and deformation histories including the temperature,velocity and strain rate histo-ries can be calculated (Oliphant,2004).Each of these types of codes has its own distinct advantages and disadvantages for FSW simulation.For the solid mechanics Lagrangian formulation (Zhang et al.,2005,2007;Buffa et al.,2006a,b,2007;Fratini et al.,2006,2007;Chen and Kovacevic,2006;Soundararajan et al.,2005;Khandkar et al.,2006),the material deformation as well as the stress,strain and temperature fields in terms of material points are calculated.This model maygiveTable 1Fig.1.(a)A CNC controlled 3D FSW system ‘‘Hitachi GR-3DM10T’’with 11KW spindle servomotor and (b)schematic view of the placement of thermocouples.Table 2Temperature dependent yield stress (OSU experiment)(partially measured at OSU_3Table 3Temperature dependent density,thermal conductivity and specific heat (calculated Experiment values are in parenthesis.D.Kim et al./European Journal of Mechanics A/Solids 29(2010)204–215205accurate results and be able to model a void formation and the slipping phenomenon using a proper friction boundary condition.However,since the FSW process involves severe deformation accompanied by large mesh distortion,the difficulty of obtaining proper converged solutions introduces dauntingly high computa-tional time.Re-meshing to improve accuracy and stability can be applied both for the Lagrangian and Eulerian formulations,often referred to as the Arbitrary Lagrangian–Eulerian (ALE)analysis (Reddy and Baumann,2004;Fourment et al.,2004;Guerdoux et al.,2004,2007;Mandal et al.,2007;Aramayo et al.,2002;Kalagara and Muci-Kuchler,2007;Schmidt and Hattel,2005),but high compu-tational time remains problematic.For the FVM codes based on the Eulerian formulation (De Vuyst et al.,2008;Kumar et al.,2008;St-Georges et al.,2007;Colegrove and Shercliff,2005;Lorrain et al.,2008;Kim et al.,in press;Schmidt and Hattel,2008),temperature variation as well as material flow patterns are spatially calculated.The histories of strain rate and temperature along the material flow path can be calculated in the post-process manner also.Such codes handle large deformations efficiently and accurately but cannot account for the hardening behavior of the material nor frictional slipping phenomenon at the interface.Some papers have appeared based on specially developed codes (Badarinarayan,2004;Badarinarayan et al.,2007;Hirasawa et al.,2004;Heurtier et al.,2006).Heurtier et al.(2006)set up the kinematic modeling in which three velocity components were implemented near the tool:circumventing,torsional and vortex motions generated by the screw driven material flow.This is similar to the CFD model.Badarinarayan (2004),Badarinarayan et al.(2007)and Hirasawa et al.(2004)incorporated a thermo-mechanical model utilizing particles connected by strings.Particle mixing as well as welding voids could thus be predicted.However,heat generation by plastic deformation and the friction boundary condition at the tool were not properly accounted for.As for material properties,some considered temperature and strain rate dependent mechanical properties either with plastichardening for the Lagrangian approach based on the elasto-visco plastic model (Fourment et al.,2004;Guerdoux et al.,2004,2007),the elasto-plastic Johnson-Cook model (Mandal et al.,2007;Schmidt and Hattel,2005)or the rigid-visco plastic model (Buffa et al.,2006a,b,2007;Fratini et al.,2006,2007),or without plastic hardening for the Lagrangian code based on the rigid-visco perfect plastic model (Ulysse,2002;Reddy and Baumann,2004)and without plastic hardening for the Eulerian code based on the rigid-visco perfect plastic model (Colegrove and Shercliff,2005)and the rigid-perfect plastic model (Kim et al.,2009).Some works consid-ered temperature dependent thermal properties such as for density,specific heat and thermal conductivity (Ulysse,2002;Fourment et al.,2004;Guerdoux et al.,2004,2007;Chen and Kovacevic,2006;Soundararajan et al.,2005).Peak temperatures and temperature histories measured with thermocouples are the main points of comparison with simulation results (Ulysse,2002;Reddy and Baumann,2004;De Vuyst et al.,2008;Kumar et al.,2008;Heurtier et al.,2006;Buffa et al.,2006b;Kim et al.,2008,2009;Soundararajan et al.,2005;Schmidt and Hattel,2008;Khandkar et al.,2006).Computed histories of effec-tive strains,strain rates,and stresses have been used to predict or evaluate the post-FSW hardness (Kumar et al.,2008;Heurtier et al.,2006;Kim et al.,in press;Kamp et al.,2007),residual stress (De Vuyst et al.,2008;Chen and Kovacevic,2006;Soundararajan et al.,2005;Khandkar et al.,2006)and grain size of welds (Buffa et al.,2006a,2007;Fratini et al.,2006,2009;Kim et al.,in press ).Recent work has shown abnormal grain growth (AGG)for heat-treated AA5083-H18following FSW (Gan et al.,2008).A critical part of interpreting possible mechanisms for such behavior is the establishment of the post-FSW grain size by dynamic recrystalli-zation throughout the weld,a phenomenon that depends inti-mately on the thermo-mechanical history of material elements during the FSW process.In order to provide such information,thermo-mechanical modeling for friction stir butt welding (FSBW)process is presented in this work for AA5083-H18using the CFD FVM code,STAR-CCM þ(CD-Adapco,2007).Temperature depen-dent material properties were implemented via a user subroutine.The process,consisting of plunging,short dwell,long linear weld-ing and retracting steps,was simplified as a steady state process wherein the process achieves an equilibrium operating tempera-ture in a relatively short time (Badarinarayan,2007).The continuity equation as well as the momentum and energy conservation equations for the process was solved with temperature dependent material properties along with the conduction and convection heat transfer boundary conditions.A full sticking condition at the interface between the tool and the material was adopted so that heat was generated only by plastic deformation withoutheatTable 4Material properties of AA5083-H18and AA5052-H32at room temperature (MetalsTable 5Simulation 1is the model with the proper thermal boundary condition at the interface between the work-piece and the backing plate (anvil)and simulation 2is the model without that (Kim et al.,2008).D.Kim et al./European Journal of Mechanics A/Solids 29(2010)204–215206generation by friction.This is a major assumption and a common drawback of Eulerian simulations.Experimental verification of temperature history was per-formed using six thermocouples under three process conditions chosen to change the peak temperature and temperature history in the practically achievable range.Since AGG is closely related to the amount of heat generation during FSW,three process conditions based on the rotation speed of the tool and the linear translation speed of work-pieces were considered in the previous AGG study (Gan et al.,2008);therefore,here also to account for different heat generation aspects (generally referred to jointly as ‘‘heat input’’in the FSW literature),even though this paper focuses on establish-ment and test of the model.Results obtained in the model will be utilized to investigate AGG in a separate paper (Chen et al.,submitted for publication ).2.Experimental proceduresFSBW was performed on AA5083-H18sheets,chemical composition appears in Table 1(Metals Handbook,1990),using a CNC controlled 3-D FSW system ‘‘Hitachi GR-3DM10T’’with a 11KW spindle servomotor,as shown in Fig.1(a)(Gan et al.,2008).The anvil below the work-piece is made of 1020plain carbon steel,while the tool is made of H13tool steel with the shoulder and pin diameters of 10mm and 4mm,respectively.The shoulder had a concave profile which acts as an escape volume for the material displaced by the pin,while preventing the material from extruding out of the sides of the shoulder and maintaining downward pres-sure.The pin is right-hand threaded.The pin length is slightly less than the thickness of work-pieces,with 1.34mm for AA5083-H18(sheet thickness of 1.64mm)and the plunging depth of the tool shoulder edge was 0.2mm.The work sheets were butted together and then firmly clamped in a clamping fixture.FSW was performed along the butt line which was parallel to the rolling direction of work sheets.The welding tool was tilted at an angle of 3degrees from the normal direction of the work sheets towards the trailing side of the tool.Three welding conditions were used for the current work:1000rpm–100mm/min (Case 1),1000rpm–300mm/min (Case 2)and 1500rpm–150mm/min (Case 3)which the first number refers to the rotation speed of the tool and the second refers to the linear translation speed of tool relative to the work-pieces.Note that FSW has been performed under various welding conditions before this simulation work (Gan et al.,2008)and the particular three conditions were selected for simulation to investigate the AGG phenomenon observed for AA5083-H18with different heat generation.Cases 1,2and 3account for the medium,low and high heat input systems,respectively.Fig.2.Dimensions of the thermo-mechanical model:(a)schematic view of work-piece and (b)section view of the tool (in mm).D.Kim et al./European Journal of Mechanics A/Solids 29(2010)204–215207Temperature measurements were conducted by inserting sheathed K-type thermocouples into three machined channels such that the tip of the thermocouple was located 0.6mm below the top surface of the sheet,Fig.1(b).The tip of the thermocouple was located 2mm,5mm,and 8mm away from the weld seam line,respectively.All the temperature measurements were made on the advancing side (AS),with some measurements also made for the retreating side (RS)of welds.The placement of thermocouples can affect material flow and therefore temperature variation during welding.Furthermore,since the thermocouples positioned 2mm away from the seam can touch or be very close to the edge of the tool pin,the thermocouple tip was frequently deformed or broken,thus decreasing the accuracy of those temperature measurements.Experimental temperature measurements for thermocouples are shown in Table 5and Fig.5(Gan et al.,2008).Fig.3.(a)Top views of 800,204meshes and (b)0.1mm average mesh size near the stirzone.Fig.4.Temperature dependent material properties forAA5052-H32.Fig.5.The calculated and experimental temperature history at locations away from the weld center at the advancing side under (a)the 1000rpm and 100mm/min,(b)the 1000rpm and 300mm/min and (c)1500rpm and 100mm/min welding conditions.D.Kim et al./European Journal of Mechanics A/Solids 29(2010)204–2152083.Thermo-mechanical modeling3.1.TheoryFor the thermo-mechanical analysis of FSW processes,the continuity equation as well as the momentum and energy conser-vation equations was solved with temperature dependent material properties.The mechanical property was assumed rate-insensitive and an incompressible rigid-perfect Mises plastic(Kim et al.,2009) constitutive model based on the normality rule was adopted;i.e.,S¼2m DD¼_3v sv S¼3_32sS¼S2mand s¼ffiffiffiffiffiffiffiffiffiffiffiffi32S:Sr;_3¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi23D:Dr(1)where D and S are the rate of deformation and deviatoric Cauchy stress tensors,respectively,and s,_3and m are the Mises effective yield stress,conjugate effective strain rate and viscosity,respectively.The yield stress sðTÞis temperature(T)dependent and the viscosity mðT;_3Þis dependent on temperature and strain rate as,from Eq.(1),m¼s3_3(2)The energy conservation equation(Spencer,2004)isr C p DTDt¼V$ðk V TÞþ_Q(3) where C p is the specific heat,k is the thermal conductivity and_Q is the heat generation rate by plastic dissipation.The density rðTÞ,the specific heat C pðTÞand the thermal conductivity kðTÞare temper-ature dependent,while the heat generation rate_Q becomes _Q¼a S:D(4) where a is the conversion factor(a is equal to1.0in this work).The convective heat boundary condition of Eq.(3)is given byV T¼hkðTÀT BÞ(5) where h is the convective heat transfer coefficient and T B is the surrounding temperature.As for the conductive heat boundary condition of Eq.(3)between solid materials(at the interface between the work-piece and the anvil in this work),a baffle heat transfer boundary condition was applied;i.e.,by assuming the equivalent thermal circuit for a series composite wall(Incropera et al.,2007),in which the relation between the heatflow rate,~q, across the contact boundary with an area A is given by~q¼ÀkAd TzÀD TR¼X niD n ii(6)where k i and D n i are the thermal conductivity and thickness, respectively,of the i th layer comprising the contact boundary.(The baffle contact boundary consists of two layers:one with AA5083-H18at the work-piece and the other with plain carbon steel at the anvil),while R is a representative thermal resistance for the area A (here,the n-direction is normal to the boundary surface).Eq.(6)is derived under the condition that the heatflow rate is maintained for each layer comprising the contact boundary without heat loss.3.2.Model descriptionThermo-mechanical simulation of the FSBW process for the AA5083-H18sheets was performed using the CFD FVM code,STAR-CCMþ(CD-Adapco,2007).Thecontinuity equation as well as theFig.6.Temperature profiles and iso-contours(every100 C starting from100 C)at the top surface for the welding condition of(a)1000rpm and100mm/min,(b)1000rpm and 300mm/min and(c)1500rpm and150mm/min(in C).D.Kim et al./European Journal of Mechanics A/Solids29(2010)204–215209momentum and energy conservation equations for the process were solved.The process was simplified as a steady state process (i.e.,dwelling only)wherein the process achieves an equilibrium operating temperature in a relatively short time (Badarinarayan,2007).The process consists of a rotating tool and moving work-pieces resting on an anvil,as shown in Fig.2.Heat is generated by plastic dissipation at the work-pieces,as stirred by the rotating tool.In the real process,the weld was then formed at the abutted work-piece interface by the stirring action of the rotating tool.Dimensions of the AA5083-H18sheet (work-piece)are 200mm Â600mm Â1.64mm and the work-piece lies on an anvil with similar dimensions but with the thickness of 10mm,as shown in Fig.2.Dimension of tool and welding conditions are same with experimental ones and a total of 800,204mesh elements were employed,as shown in Fig.3.Polyhedral meshes which typically have an average of 14cell faces were automatically generated by code.3.3.Material propertiesThe rate-insensitive and incompressible rigid-perfect Mises plastic property was applied for the work-piece as described in Section 3.1.Temperature dependent yield stresses for AA5083-H18were used in this work as shown in Table 2,which have beenpartially measured at OSU (with _3¼10À2:5).Since experimental yield stresses were available only up to 400 C,decreasing yield stresses were assumed above 400 C,considering that yield stresses vanish near the melting temperature of AA5083-H18(574 C)(Metals Handbook,1979).Based on the yield stress provided by Table 2,the viscosity was calculated from Eq.(2)in the user subroutine.As for the density and thermal properties,only the room temperature data were available for AA5083-H18(except partially for the thermal conductivity).Therefore,assumed temperature dependent properties were obtained from those of AA5052-H32(Zhu and Chao,2002)by proportionally modifying the values considering the ratios at room temperature values of the two materials,as shown in Table 3.Note that Fig.4shows the temper-ature dependent thermal properties of AA5052-H32and Table 4shows the room temperature properties of AA5052-H32and AA5083-H18(Metals Handbook,1961).Based on the difference of the room temperature values,the assumed temperature dependent thermal properties of AA5083-H18were obtained from Fig.4as shown in Table 3.Note that the thermal conductivity measured for AA5083-H18is very similar to the assumed properties,partially justifying the validity of the assumed values.The data in Table 3was implemented in the user subroutine.Material properties used for the type 1020steel anvil are shown in Table 4(Callister,2003).3.4.Boundary conditionsSteady state simulations were performed for the three welding conditions.As for the force equilibrium equation,the material at room temperature (20 C)flows in at the inlet boundary and flows out at the outlet boundary at the welding speed (the linear trans-lation speed of work-pieces in experiments)as shown in Fig.2.At all other boundaries (sides,top,bottom surfaces and the interface between the work-piece and the anvil),except the boundary con-tacting with the tool,a free traction boundary condition was applied.At the tool boundary,the material was assumed to be in perfect contact with the tool and rotate at the same speed as the tool,as shown in Fig.2.With this full sticking condition at the interface between the tool and the material,heat was generated only by plastic deformation without frictional heat wherein it might be the same with the total heat generation arising from the friction and the plastic dissipation (Schmidt and Hattel,2008).A convective heat transfer coefficient of 30W/m 2 C was used for the top and sides of the work-pieces,which is typical for natural convection heat transfer between aluminum and air (Chao et al.,2003).The baffle heat transfer boundary condition shown in Eq.(6)was applied with one layer for the work-piece and one layer for the anvil.A convective heat transfer coefficient of 35W/m 2 C was used for the sides of the anvil which is typical for natural convection heat transfer between steel and air (Totten,2007).At the bottom of the anvil,the coefficient was assumed 200W/m 2 C.Heat transfer at the interface between the tool and the work-piece was ignored by assigning a vanishing temperature gradient at the boundary (Schmidt and Hattel,2005),since heat transfer between the tool and the work-piece can be ignored wherein the tool reached high temperature in a short time during the welding process (Badar-inarayan,2007).4.ResultsIn order to verify the numerical simulation results,the simu-lated temperature profiles and peak temperatures were compared with experimental results (Gan et al.,2008)in Table 5and Fig.5,respectively.Calculated temperatures were collected at locations 2.0,5.0and 8.0mm away from the weld center on the advancing and retreating sides as shown in Fig.2.The simulation results matched reasonably well with experiments,even though the simulations slightly over predict temperature in most cases,resulting in that the peak temperature for 5mm data attheFig.7.Temperature profiles at the cross-section containing the middle of the tool for the welding condition of (a)1000rpm and 100mm/min,(b)1000rpm and 300mm/min and (c)1500rpm and 150mm/min (in C).D.Kim et al./European Journal of Mechanics A/Solids 29(2010)204–215210advancing side of Case 1is larger even than that of Case 3,unlike experiments.These three processes were simulated previously without the proper thermal boundary condition at the interface between the work-piece and the backing (anvil)plate (Kim et al.,2008)to simplify the boundary condition.As the results of both boundary conditions showed in Table 5,the simulations with the proper boundary condition improve the previous simulation results considerably,pointing to the importance of proper interface boundary condition between the work-piece and the anvil plate for accurate results.The temperature profiles at the top surface and at the cross-section containing the middle of the tool are shown in Figs.6and 7,respectively.In Fig.6,the tool rotation speed is related to the amount of heat generation near the pin,while the translation speed is related to the distribution speed of the heat into the trailing direction.With the lowest tool rotation speed (therefore,least heat generation rate)and highest translation speed (therefore,largest distribution rate),Fig.6(b)shows a thin and longtemperatureFig.8.Streamlines with effective strain rate (/s)and temperature ( C)for the welding condition of (a)1000rpm and 100mm/min,(b)1000rpm and 300mm/min and (c)1500rpm and 150mm/min.Fig.9.Model description and dimension for inverse trace of welds at the end side view of the work-piece and tool (in mm).D.Kim et al./European Journal of Mechanics A/Solids 29(2010)204–215211。