on

Ger

Carreau uid

theFWredeldinstnted

Lagrange bulk viscosity term, a purely displacement based form of the numerical solution procedure is

eldingng a wg theeachedconve

tion of stable and suitable process parameters and effective feed-

necessity therefor is the revelation of the physical phenomena dur-ing the process. Hence, the demand of predictive tools in modelingthe transactions during RFW being able to understand and opti-mize the process is of crucial interest for industrial applications.

The main process parameters are illustrated in Fig. 1 being theaxial force FA, the torque T, the rotary speed n and the axial feed sf .Therein the shaft A is the static part and shaft B is the rotating one

this process have been identied being the heat generation, heatfriction surfacesce [18]. Incts rst at

In 1995, Stokes modeled the molten lm of frictionpolymers by introducing a temperature and shear dependecosity [20]. Bendzsak tried to capture the ow regime of frictionwelds by solving the NavierStokes equation directly [2]. Thereinthe focus has been set on the ow traces within aluminium frictionwelds. Moal et al. proposed a fully coupled viscoplastic model [14]for the simulation of Inertia Friction Welding (IFW). While a Nor-ton power law was used for the deviator stress calculation, a purelyvelocity dependent friction law was implemented for contact forceand heat generation prediction. In order to map the isochoric,

Corresponding author. Tel.: +49 391 67 12679; fax: +49 391 67 11379.

Comput. Methods Appl. Mech. Engrg. 265 (2013) 186194

Contents lists available at

A

.eE-mail address: david.schmicker@ovgu.de (D. Schmicker).back control algorithms. A perseverative problem, for instance, isthe identication of an optimal control of a position based frictionwelding process, where for reasons of nish worked parent parts anal axial shortening during welding has to be achieved. Basic

conduction, plastic deformation, abrasion of theand the diffusion of the weld partners, for instanoping simulation models to cover all these effecan be dated back to the mid 90s.0045-7825/$ - see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2013.06.007devel-tempts

weldednt vis-Moreover, neither ller material nor any inert gas is needed forjoining providing a very efcient and uncontaminated fuse of theweld. Benets are the high quality and symmetry of the joint, thegood potential for process automation and the short cycle times.In contrasts to all these virtues the main problem is the identica-

mentioned nal shortening of the shafts lengths. In Fig. 2 qualita-tive runs of the process parameters are displayed [8]. The threestages therein are the rubbing stage (I), the breaking stage (II)and the forging stage (III).

Already in the 1970s the main driving physical phenomena of1. Introduction

The process of Rotary Friction Wfusing process allowing for combinimaterials. Due to the fact that durinmaterials melt temperature is not rzone of the weld is smaller than atapplicable. A penalty contact approach together with a regularized friction law allow for an efcient deter-mination of the interface pressure and friction forces. Themodels performance is demonstrated bymeansof a test problem highlighting the potential of adopting it to industrial applications later on.

2013 Elsevier B.V. All rights reserved.

(RFW) is a solid-stateide range of dissimilarwelding procedure the[18], the heat affectedntional fusion welding.

inducing the frictional heat on the contact surface. Basically, twokinds of drives are used in RFWmachines, either a y wheel servesas an energy storage being referred to as Inertia Friction Welding(IFW) or a continuously driven engine generates the power ofwelding being known as Direct Drive Friction Welding (DDFW).Concerning DDFW usually the rotary speed time line is given whilethe axial force is feedback controlled in order to achieve the above-Rotary Friction WeldingNon-Newtonian uid modeling

robust simulation of the process and the physical palpability of the utilized material parameters. Using aspecial element formulation on the general axisymmetric frame in conjunction with an AugmentedA robust simulation of Direct Drive FrictiCarreau uid constitutive model

D. Schmicker , K. Naumenko, J. StrackeljanInstitute for Mechanics, Otto-von-Guericke University of Magdeburg, 39106 Magdeburg,

a r t i c l e i n f o

Article history:Received 14 April 2013Accepted 19 June 2013Available online 13 July 2013

Keywords:Welding simulation

a b s t r a c t

The subject of this paper isDrive FrictionWelding (DDthe consistent and stable padopting Rotary FrictionWThe Carreau type uid co

ature dependency is accou

Comput. Methods

journal homepage: wwwWelding with a modied

many

presentation of a holistic, fully-temperature-coupled simulation of Direct) based on themodied Carreau uidmodel. Themainmotivation therein isiction of suitable process parameters, which is still the major problem inng (RFW) processes to new pairs of weld partners at industrial applications.itutive equation is derived from a NortonBailey law wherein the temper-for by a JohnsonCook power approach. Its main benet is the stable and

SciVerse ScienceDirect

ppl. Mech. Engrg.

l sevier .com/locate /cma

s of

D. Schmicker et al. / Comput. Metho AppFig. 1. Parameterplastic deformations, an augmented variational formulation wasused and the elements were formulated in the r2-z space. Extensiveremeshing of the Lagrangian mesh allowed for a complete simula-tion of the process yielding plausible results of the ash evolutionand the process parameters. DAlvise et al. presented simulationresults of dissimilar weld parts [7]. The implemented friction lawhas been adopted by Moal with the slight change, that a certaintemperature is introduced at which the friction law turns from aclassical Coulomb law into a thermo-dependent viscous one. In2002 Ulysse suggested a Non-Newtonian uid law linked with aSheppardWright viscosity approach1 for the simulation of FrictionStir Welding (FSW) [21]. While the simulated temperature eldstend to be over predicted, parametric studies like the inuence ofthe tools speed on its forces have been successfully conducted withthis model. Wang made further suggestions for estimating the heatinput at IFW, one of which is known as Energy-Input-Based method[22]. Here, the modeling of a frictional law is circumvented byneglecting the twist deformation and implementing a thermody-namically consistent heat-source being ruled by the energy decayof the y-wheel. The disadvantage of this approach is that the heatsource is rather an input than an output quantity of the modelwhich means that it is not predictive. Another member of this classof model has been pursuit by Grant [11]. Also here the process

Fig. 2. Parameter runs of the Direct Drive Friction welding process.

1 It shall be mentioned that the SheppardWright viscosity is technically speakinga Garofalo uid law [9], with the difference that the material parameters therein aredened in a slight different way.ds,parameters and properties cannot be predicted, but residual stressesare able to be calculated which has been shown to be in good agree-ment with measurement results. In 2010 Hamilton published aframework for mapping the FSW process [12] wherein a classicplasticity law incorporating elastic deformations has been imple-mented as constitutive equation. In this context the temperaturedependency has been taken into account by a JohnsonCook powerapproach [13]. More recently, Chiumenti proposed a model for thesimulation of FSW incorporating an Arbitrary-Lagrangian-Formulation(ALE) in order to resolve the large plastic distortions [5]. A mixedvelocitypressure formulation being stabilized by the OrthogonalSubgrid Scale (OSS) technique accounts for non-volumetric lockingwhile using lower order nite elements. A classical NortonHoff law and a SheppardWright model have been suggested asthe constitutive relation.

Basic ambition of our approach in modeling DDFW is to increasethe robustness and efciency of the simulation, and to express thematerial parameters in a more palpable way. We believe that forthe sake of a robustness increase the material law needs to incor-porate a steady transition from the Non-Newtonian solid to theNewtonian liquid phase, which is not accounted for in prior mod-els. Moreover, due to the lack of experimental data for the inelasticmaterial behavior in wide strain-rate and temperature ranges, thesimulation should predict the welding process parameters withcoarse values of material properties. Therefore, the constitutiveequation, the response functions of stress and temperature as wellas material parameters have to be formulated in an easy under-standable and accessible way.

Our constitutive model is based upon a Carreau uid law featur-ing the solidliquid phase transition by means of a constant, New-tonian viscosity of the materials cast. It is derived from Nortonspower law [5,1416] and formulated in terms of the materialsyield limit. The temperature dependency of this measure is linkedto a JohnsonCook power law [12,13], enabling the pursued possi-bility of assessment of the plastic ow behavior even if hardly anyexperimental data is at hand. A distinct time integration schemewith adaptive step size regulation, in conjunction with remeshingprocedures of the Lagrangian formulated nite elements ensure avery robust and smooth simulation of the DDFW process.

the RFW process.

l. Mech. Engrg. 265 (2013) 186194 187The structure of the paper is as follows: We start with the for-mulation of the constitutive model that reects the non-linear uidbehavior for a wide stress and temperature range. Then we presentthe local and the weak forms of governing equations. Spatial dis-cretization techniques including the nite element formulationand the remeshing procedure are discussed. After outlining the uti-lized contact approach, the time marching scheme is depicted. Abenchmark example is introduced, demonstrating the performanceand capacities of the model. Finally the conclusions are drawn withregard to efciency and accuracy of the proposed approaches.

1.1. A remark on notation

Scalars are expressed using simple non-boldfaced letters liker; J2D and n, for instance. Tensorial expressions of higher order than

3 _e0l0 _e00 1 1

Appzero are stated as boldfaced symbols like in x; er ;r and D. Theircomponents according to a vectorial basis as well as nodal valuesand scalars being arranged to matrices are denoted by blackboardbold symbols as in M;v;O and Acap. Domains are marked by calli-graphic letters like Bt or B0, for instance.

2. Constitutive modeling

2.1. Viscoplastic material model

The constitutive law is based on a NortonBailey potential ofthe form

W r0 _e0n 1

rvMr0

n11

where r0 is the materials yield limit, _e0 the plastic strain rate corre-sponding to the yield limit r0 and n the Norton-exponent [15,16].The law is a classic J2-material law since the potential only dependson the second invariant of the deviatoric part of the stress tensor,that is to say on the von-Mises equivalent stresses

rvM 32s s

r

3J2D

p2

Consequently, the inelastic strain rate is

D @W@r

32

_e0rvMr0

n srvM

3

where r is Cauchys stress tensor, s its deviatoric part

s devr r 13trr1 4

and 1 the second rank unit tensor. Inversion of (3) gives the morecommon stressstrain relation

s 23r0_e0

_evM_e0

1nn

D 5

with the von-Mises equivalent strain rate

_evM 23devD devD

r6

The full stress tensor is then given by

r lB trD1 2ldevD: 7involving the abbreviation for the shear viscosity

l _evM ;H r0H3 _e0_evM_e0

1nn

8

The spherical part of r determines the pressure state of the materialwhich in our model is assessed by an Augmented Lagrange methodinvolving a bulk viscosity lB. The choice of this measure is basicallya trade-off between the strictness of keeping the volume constraintand the condition corruption in solving the system of equations la-ter on. The temperature dependency in Eq. (8) is now incorporatedby linking the yield limit r0 to a JohnsonCook power ansatz of theform

r0H r0;R 1HHRHMHR

mh i; H < HM

0; HP HM

8 e 28

acts as a trigger for remeshing corresponding to a certain thresholdvalue e. In our simulation the threshold value was chosen to bee 0:5, yielding reasonable results on the remeshing time in-stances. In transferring the state variables from one mesh to theother the element shape functions are used for interpolation.

5. Contact modeling

The realistic simulation of the DDFW process inevitably dependson the implementation of the contact and friction model. For thesake of reasonable computation times a simple penalty contactapproach has been utilized in our model, being illustrated inFig. 5. Therein the two weld partners as well as the self-contactof the ash with the workpiece are allowed to penetrate eachother. Hence, the pressure p at the surface Gauss-points isgoverned by the intersection depth s according to the function

p 0; s < 0aps2; sP 0

29Fig. 5. Penalty contact approach.

The penalty term ap can be chosen in regard to the dimension ofthe workpiece and the to be expected contact pressures. Forinstance when saying that a penetration of 0:1 mm is acceptableand typical contact pressures are about 60 MPa, then a good choicefor ap is 6000 MPa=mm2. The tangential stresses szu obey anCoulomb friction law of the form

the state variables at the proceeding time step both equations can behandled separately according to their individual best suiting timemarching scheme, without the necessity of implementing a loopbetween the two subproblems (cf. [5]). Let us assume that at acertain time t0 the conguration is given by the state variables v0and O0. Then rstly, the mechanic subproblem is integrated explic-itly. Therefore the nodal force balance evaluated at t0

0 fextt0;v0; _v0 fficv0; _v0 fint _v0;O0 35is solved in terms of _v0 and the conguration at the end of the timestep t t1 is calculated by a forward Euler differencev1 _v0 Dt v0 36involving the time step size Dt t1 t0. Secondly, the thermody-namic subproblem is solved implicitly by a backward Euler strategy.Thus, with putting

_v v1 v0Dt

; _O O1 O0Dt

; v v1 and O O1 37

into (34)2 and solving the equation in terms of O1 the temperaturesat the end of the time step are obtained. The time step size Dt of thisprocedure is adaptively controlled by a standard step size regula-tion algorithm being described in [19], resulting in a very robustand stable temporal discretization scheme for the simulation ofthe process.

7. Simulation results

In this section the model will be demonstrated in simulating thewelding process of a forge part being illustrated in Figs. 6 and 7.Therein, the focus lies on the qualitative demonstration of the

D. Schmicker et al. / Comput. Metho Appl. Mech. Engrg. 265 (2013) 186194 191szu gp Dvu > 0< gp Dvu 0

30

where Dvu is the relative velocity difference of the contact inter-faces of the two workpieces and g is a temperature dependent fric-tion coefcient. Since, Eq. (30) involves a strong singularity at thetransition from slipping to sticking condition the regularizedapproach

szu 2gpp arctanDvuareg

31

involving a regularization factor areg is used instead. In fact, the sim-ulation results reveal that during the friction phase the sticking con-dition is basically never met, hence the regularization is more offormal nature, not really entering the model. The heat source atthe contact interface of workpiece A and B is then given by

rq;A aAaA aB szuDvu aABHB HA and

rq;B aBaA aB szuDvu aABHA HB32

Here, the thermal diffusity parameters

aA jc;AqAcAand aB jc;BqBcB

33

are used for portioning the frictional heat onto the two weld part-ners following the suggestions of Chiumenti [5]. The heat ux be-tween the two surfaces is incorporated by the heat transmissioncoefcient aAB.

6. Temporal discretization

Since elastic wave propagation phenomena are of subordinatenature at the process of RFW, inertia effects can be neglected.Hence, the problem will be treated in a quasi static way. Fictitiousforces, following from the formulation of the governing equationusing cylindrical coordinates, are able to be accounted for never-theless. The coupled to be solved differential equation takes theform

f t; z; _z 0 fextt;v; _v fficv; _v fint _v;OAcap _OAconO qint _v;O qextv; _v;O

" #

34involving the state vector z v;OT , consisting of the nodal loca-tion and temperature values. The system is solved by dividing prob-lem (34) into its mechanic and thermodynamic subproblem. Since,the sensitivity of the temperatures on the nodal force balance iscomparatively low, a decoupled treatment of the two subproblemson the time step level is conductible.4 That is to say, that for gaining

4 The temperatures enter the linear momentum balance only due to the temper-ature-dependent viscosity of the viscoplastic material law. Since for usual materialsthis dependency is of very steady nature exhibiting no singularities (see Eq. (9) forinstance), the impact of the extent of temperature change during one time step on thesolution of the mechanic subproblem is rather marginal. Hence, an isothermaltreatment of the nodal force balance is employable. When the velocities anddisplacements at the end of the time step can be determined only by means of the

rst sub problem at constant temperature, they can be regarded as given whensolving the thermodynamic subproblem. Hence, at the time step stage bothsubproblems can be treated separately.dsFig. 6. Friction weld sample serving as a simulation example.Fig. 7. Observed friction welding problem.

tentials and the performance of the model without claimingantitative exactness of the material properties and boundarynditions. The simulation is performed on basis of a self-writtenite element code within MATLAB. The employed material con-nts are comprehended in Table 1. The considered process con-ts of only one rubbing stage at p 40N/mm2 and800 min1 for a friction time tr 7 s. The weld partners areilar, offering the possibility of cutting the computational effortthe half by modeling only one side. However, one has to keep inind that this step is only admissible if inertia forces including thetitious ones are neglected and the heat transport coefcient asurfassumed to be independent of the rotary speed. The presentmputation example conforms these requirements, hence thehavior of the rotary side is mapped by a symmetry boundary

condition. The meshing of the stationary side is performed with anaverage element size of de 0.4 mm at the contact surface. Theaugmented Lagrange bulk viscosity is assessed to 1:175 106 Ns/mm2 being a suited compromise between satisfying the volumeconstraint and keeping the system of equations well conditioned.The penalty parameter for the calculation of the contact forces isap 1000 N/mm4. For regularization of the friction shear stressesa factor of areg 200 mm/s is taken into account. During the simu-lation 103 remeshing steps have been performed. Therein, theaverage amount of elements was about 600 resulting in approxi-mately 4000 displacement- and 1300 temperature degrees of free-dom. In Fig. 8 the deformation shape at the end of the process att 8 s is compared with the actual ash formation. A qualitativeagreement between the ashes can be identied. In Fig. 9 the tem-perature eld at the end of the rubbing stage at t 7 s is displayed.It is found out that the materials melt temperature is not reachedat no time during the simulation. In fact, it can be easily reasonedthat during the process of RFW the melt temperature is very unli-kely to be reached or exceeded neither in simulation nor in prac-tice. The explanation therefor is that in the vicinity of the meltpoint the material is owing and not able to transmit shear stres-ses anymore.5 Hence, once the temperature gets as high as the meltpoint the rubbing surfaces will stick and frictional heating will sus-pend avoiding the temperature to exceed the melt point. Lookingcloser at the contact pressure condition in Fig. 10 it can be seen thatat the beginning the axial force is distributed evenly across the cross-

Table 1Utilized parameters of the simulation example.

Parameter Value Parameter Value

r0;R 235 N/mm2 HM 1420 C_e0 1 s1 q 7.85 g/cm3

m 1.5 c 460 J/kgKn 5 k 50W/mKl0 108 MPas asurf 100 W/m

2K

l1 102 MPas g 0.3HR 20 C

Fig. 8. Comparison of the ash formatio

2

192 D. Schmicker et al. / Comput. Methods Appl. Mech. Engrg. 265 (2013) 1861942530350

2

4

6

8

10

12

14

16

z [m

m]

500 C

600 C

700 C

950 C

1000 C

1100 C 1200 C 1300 C poquconstasisn simtomciscobeFig. 9. Temperatures immediateln between model and experiment.

0510150

Temperature [C]r [mm]

y after the rubbing stage at t 7 s.

section. However, as soon as the material starts to ow a distinctpressure prole develops reaching its maximum in the middle ofthe cross section at r 17 mm. Comparison of this prole with thetemperatures over the contact surface in Fig. 9 reveals that the max-ima of temperature and pressure do not coincide. Whereas the pres-sure is highest near the inner diameter at r 17 mm, the maximumtemperature is reached at r 22 mm. The unsteadiness of the con-tour lines in Fig. 10 is due to the remeshing time instances duringthe simulation which slightly alters the contact force distribution.Fig. 11 displays the nal twist of the material. It is clearly seen thatthe largest plastic deformations during the process do not lie within

the r zplane but in tangential direction. Hence, they should notbe neglected in modeling RFW. In Fig. 12 the power input and outputduring the process is contrasted with each other. When the weldstarts twisting an increasing part of heat power is generated by plas-tic dissipation. The loss of energy due to thermal emission and con-vection reaches its maximum at the end of the rubbing stage. InFig. 13 the process parameter runs are shown. Since at this examplethe process is force controlled, the left two diagrams namely the ax-ial force and the rotary speed are input quantities, whereas the righttwo diagrams are simulation results. The nal upset of the welded

Fig. 10. Contact interface pressure evolution during the process.

Fig. 11. Twist of the weld at t 8 s.

25

30Plastic dissipation work

work

Con

D. Schmicker et al. / Comput. Methods Appl. Mech. Engrg. 265 (2013) 186194 1930 1 2 3-10

-5

0

5

10

15

20

P [k

W]

Friction t

Fig. 12. Energy prol4 5 6 7

vection and radiation heat [s]

e of the process.

0 2 4 6 80

200400600800

n [m

in-1

]0 2 4 6 8

0

200

400

T [N

m]

ter

194 D. Schmicker et al. / Comput. Methods Appl. Mech. Engrg. 265 (2013) 186194part is 12.8 mm which is about the shortening of the referencesample.

8. Conclusions

In the current paper a holistic model approach based on a Car-reau type uid constitutive model for the simulation of DirectDrive Friction Welding (DDFW) is presented. The advantage ofthe proposed material law is the formulation in terms of the yieldlimit such that its material parameters can be easily identied.Having in mind that in practical applications experimental dataare usually available for narrow stress and temperature ranges,the current model is able to give qualitative predictions by meansof quite fundamental parameters like the melt temperature andthe yield limit at room temperature, for instance. Moreover, thepaper summarizes a framework for a stable, efcient and robustsimulation of the friction welding process, featuring a steady tran-sition from solid to liquid material behavior when passing the melttemperature. With the implemented time integration and niteelement procedure, the whole process can be simulated in lessthan one hour on a usual personal computer as the presented testproblem reveals. As a next step it is planned to extend the model todissimilar welds, which broadens the effort in contact mapping.Moreover, a coupling of the model with the diffusion differentialequation is pursued. Parallel to these model extensions we workon a quantitative validation of the model and determination guide-

t [s]

0 2 4 6 80

20

40

60

t [s]

F A [k

N]

Fig. 13. Process paramelines for the involved material constants in the friction weldingprocess itself. Their results will be communicated in a forthcomingpaper.

References

[1] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice Hall,1996.

[2] G.J. Bendzsak, T.H. North, Z. Li, Numerical model for steady-state ow infriction welding, Acta Mater. 45 (4) (1997) 17351745.

[3] A. Bertram, Elasticity and Plasticity of Large Deformations An Introduction,Springer-Verlag, 2005.

[4] P.J. Carreau, Rheological equations from molecular network theories, Trans.Soc. Rheol. 16 (1) (1972) 99127.[5] M. Chiumenti, M. Cervera, C.A. Saracibar, N. Dialami, Numerical modeling offriction stir welding processes, Comput. Methods Appl. Mech. Engrg. 1 (2012)145.

[6] G.R. Cowper, Gaussian quadrature formulas for triangles, Int. J. Numer.Methods Engrg. 7 (3) (1973) 405408.

[7] L. DAlvise, E. Massoni, S.J. Walloe, Finite element modelling of the inertiafriction welding process between dissimilar materials, J. Mater. Process.Technol. 125126 (2002) 387391.

[8] DIN EN ISO 15620, Reibschweien von metallischen Werkstoffen, DeutschesInstitut fr Normung, 2000.

[9] F. Garofalo, Fundamentals of Creep and Creep-rupture, Macmillan, New York,1965.

[10] H. Giesekus, Phnomenologische Rheologie Eine Einfhrung, Springer-Verlag, 1994.

[11] B. Grant, M. Preuss, P.J. Withers, G. Baxter, M. Rowlson, Finite element processmodelling of inertia friction welding advanced nickel-based superalloy, Mater.Sci. Engrg.: A 513514 (2009) 366375.

[12] R. Hamilton, D. MacKenzie, H. Li, Multi-physics simulation of friction stirwelding process, Engrg. Comput. 27 (2010) 967985.

[13] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected tolarge strains, high strain rates and high temperatures, in: Proceedings of the7th International Symposium on Ballistics, 1983, pp. 541547.

[14] A. Moal, E. Massoni, Finite element simulation of the inertia welding of twosimilar parts, Engrg. Comput. 12 (1995) 497512.

[15] K. Naumenko, Modeling of Creep for Structural Analysis, Springer, Berlin,Heidelberg, 2007.

[16] F.H. Norton, The Creep of Steel at High Temperature, McGraw-Hill BookCompany Inc., 1924.

[17] M. Rappaz, M. Bellet, M. Deville, Continuous media, in: Numerical Modeling inMaterials Science and Engineering, Springer Series in ComputationalMathematics, vol. 32, Springer, Berlin, Heidelberg, 2003, pp. 145.

[18] G. Schfer, A. Dietzel, D. Schober, S. Buchholz, Handbuch Reibschweiszen, VVBGetriebe und Kupplungen, 1976.

[19] J. Stoer, R. Bulirsch, Numerische Mathematik II, Springer Verlag, 2005.[20] V.K. Stokes, A.J. Poslinski, Effects of variable viscosity on the steady melting of

t [s]

0 2 4 6 80

5

10

15

t [s]

s [m

m]

runs of the simulation.thermoplastics during spin welding, Polym. Engrg. Sci. 35 (5) (1995) 441459.[21] P. Ulysse, Three-dimensional modeling of the friction stir-welding process, Int.

J. Mach. Tools Manuf. 42 (2002) 15491557.[22] L. Wang, M. Preuss, P.J. Withers, G. Baxter, P. Wilson, Energy-input-based

nite-element process modeling of inertia welding, Metall. Mater. Trans. B 36(2005) 513523.

[23] P. Wriggers, Nonlinear Finite Element Methods, Springer-Verlag, 2008.[24] H.S. Yu, G.T. Houlsby, H.J. Burd, A novel isoparametric nite element

displacement formulation for axisymmetric analysis of nearlyincompressible materials, Int. J. Numer. Methods Engrg. 36 (14) (1993)24532472.

[25] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method The Basis,Butterworth-Heinemann, 2000.

A robust simulation of Direct Drive Friction Welding with a modified Carreau fluid constitutive model1 Introduction1.1 A remark on notation

2 Constitutive modeling2.1 Viscoplastic material model

3 Weak formulation of the governing equations3.1 The principle of virtual power3.2 The principle of virtual temperatures

4 Spatial discretization4.1 Formulation in cylindrical coordinates4.2 Finite element formulation4.3 Remeshing procedure

5 Contact modeling6 Temporal discretization7 Simulation results8 ConclusionsReferences