proc imeche part d: three-dimensional simulations of ...€¦ · 886 proc imeche part d: j...

Original Article

Proc IMechE Part D:J Automobile Engineering2015, Vol. 229(7) 885–897� IMechE 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0954407014548740pid.sagepub.com

Three-dimensional simulations ofresistance spot welding

Chris V Nielsen1, Wenqi Zhang2, William Perret3,Paulo AF Martins4 and Niels Bay1

AbstractThis paper draws from the fundamentals of electro-thermo-mechanical coupling to the main aspects of finite elementimplementation and three-dimensional modelling of resistance welding. A new simulation environment is proposed inorder to perform three-dimensional simulations and optimization of resistance welding together with the simulations ofconventional and special-purpose quasi-static mechanical tests. Three-dimensional simulations of resistance welding con-sider the electrical, thermal, mechanical and metallurgical characteristics of the material as well as the operating condi-tions of the welding machines. Simulations of the mechanical tests take into account material softening due to theaccumulation of ductile damage and cover conventional tests, such as tensile–shear tests, cross-tension test and peeltests, as well as the possibility of special-purpose tests designed by the users. The overall presentation is supported bynumerical simulations of electrode misalignment caused by the flexibility of the welding machine arms and electricalshunting due to consecutive welds in the resistance spot welding of two sheets.

KeywordsResistance welding, electrode misalignment, electrical shunting, mechanical testing, numerical simulations, finite elementmethod

Date received: 13 March 2013; accepted: 29 July 2014

Introduction

Resistance spot welding is the most preferred andwidely used technology for joining different types andcombinations of sheet metal parts in automotive body-in-white assemblies. An average midsized passenger carcontains several thousand spot welds which ensure therigidity and the strength of the overall assembly butalso introduce difficulties in the project, design and fab-rication methods.

The major difficulties in resistance spot welding arecaused by variations in the material properties andcombinations, the surface conditions and the electrodewear. The challenges to the welding engineer are todesign process operating parameters such as the cur-rent, the time, the force and the choice of electrode typein order to gain a weld schedule that meets the qualityrequirements of the weld, while being robust towardschanging conditions, e.g. in alignment or in surfaceconditions.

Numerical simulations can assist experimental workby providing a detailed understanding of the processand being a framework for trying different parameters,geometries and set-ups for guiding the experimental

work. In 1961, Greenwood1 introduced the first axisym-metric heat conduction model based on the finite differ-ence method and, in 1984, Nied2 introduced the firstnon-linear axisymmetric electro-thermo-mechanicalfinite element model for resistance spot welding, whichwas a large step forwards. However, because the modelwas based on small-strain formulations, like many oth-ers that were developed in the 1980s,3 it is nowadaysseen as inadequate for replicating the contact and mate-rial behaviour during the formation and growth of theweld nuggets.

The research work performed in the 1990s and 2000swas focused on the contact models, the formation and

1Department of Mechanical Engineering, Technical University of

Denmark, Kongens Lyngby, Denmark2SWANTEC Software and Engineering ApS, Kongens Lyngby, Denmark3Audi AG, Joining Technology Development, Ingolstadt, Germany4Instituto Superior Tecnico, Universidade de Lisboa, Lisboa, Portugal

Corresponding author:

Chris V Nielsen, Department of Mechanical Engineering, Technical

University of Denmark, Produktionstorvet 425, DK-2800 Kongens

Lyngby, Denmark.

Email: [email protected]

at DTU Library - Tech. inf. Center of Denmark on May 14, 2015pid.sagepub.comDownloaded from

http://pid.sagepub.com/

growth of nuggets and the occurrence of expulsion inresistance spot welding. Developments in large-strainformulations in conjunction with progress in computingtechnology allowed improvement and consolidation ofelectro-thermo-mechanical finite element models forresistance spot welding. The investigations performedin this period were mainly based on existing commercialsoftware. Tsai et al.,4 for example, utilized the commer-cial finite element program ANSYS to analyse the influ-ences of various operative parameters in resistance spotwelding, and Khan et al.5 employed a coupled electro-thermo-mechanical model based on the commercialfinite element program ABAQUS to predict the forma-tion and growth of nuggets. The developments in thisperiod also gave rise to the development of special-purpose computer programs whicht extended the rangeof potential users to professional welding engineers notworking at universities or research institutes. Zhang etal.6 presented the first electro-thermo-mechanical finiteelement computer program specifically developed forresistance welding that subsequently evolved into thecommercial program SORPAS, which is nowadays areference in the automotive industry worldwide.

The majority of the numerical developments in resis-tance spot welding that were performed in the 1990sand 2000s were carried out with two-dimensional axi-symmetric or simple three-dimensional finite elementmodels. Huh and Kang,7 for example, developed thefirst three-dimensional finite element computer pro-gram for resistance spot welding taking thetemperature-dependent electrical and thermal proper-ties of the materials into account but omitting the plas-tic deformation of the materials. Other researchers suchas Feulvarch et al.8 proposed general three-dimensionalfinite element formulations of the electro-thermal con-ditions at the contacting surfaces but restricted theirinvestigation to axisymmetric welding of sheets.

As a result of this, the current state of developmentin numerical simulations of resistance spot welding is inthe need for an environment for effectively handling thetechnical difficulties related to the electrode misalign-ment and shunt effects, which are known to have signif-icant influences on the overall quality of the weldedjoints as well as on the width of the acceptable processwindow. Moreover, this need is expected to grow in thecoming years because of the additional technical diffi-culties and challenges that are posed by the resistancespot welding of advanced high-strength steels and newultra-high-strength steels compared with conventionalmild steels.9, 10 Welding of aluminium alloys poses fur-ther challenges to welding engineers.

Evaluation and prediction of the weld strength areother challenges. Aslanlar et al.,11 for example, per-formed experimental investigations on the influence ofthe welding time on the peel strength and the tensile–shear strength of resistance-spot-welded joints. Morerecently, Xu et al.12 analysed the locations and onset offailure and necking in lap shear tests by analytical

stress functions and elasto-plastic finite element analy-sis under plane stress conditions; also, Radakovic andTumuluru13 performed finite element modelling of thecross-tension test and discussed its validity to representthe loading of the spot welds in a vehicle undergoing acrash event. Lin et al.14 proposed a different methodol-ogy to obtain the relationship between the spot-weldingparameters and the tensile–shear strength of test speci-mens by means of a Taguchi-based neural networkalgorithm. Finally, Ouisse and Cogan15 presented amethodology to design and compare spot-welded auto-mobile structures.

The aims and objectives of this paper is to presentthe theoretical and numerical fundamentals that pro-vide support to the first finite element computer envi-ronment (SORPAS 3D) which is specifically designedand developed to perform three-dimensional modelling,optimization and mechanical testing of resistance weld-ing. It is developed into an easy-to-use desktop envi-ronment to be utilized by professional engineers inindustry. The presentation includes results from thenumerical modelling of the electrode misalignment andshunt effects in resistance spot welding together withresults from the numerical modelling of tensile–sheartests. The example of the electrode misalignment is sub-stantiated by a corresponding experiment.

The simulations of strength testing are directly per-formed from the resistance-spot-welding models afterunloading the electrodes and cooling in air to roomtemperature. This is a novel approach, compared withthe traditional simulations available in the literaturewhich include a rigid nugget or an assumed initial stressfield to induce the weld properties.

The paper is organized in four sections. The follow-ing section provides an overview of the fundamentals ofthe electro-thermo-mechanical modelling of resistancewelding. Then follows a section addressing numericalmodelling topics related to the geometry and size of thetesting specimens, the discretization by hexahedralfinite elements, the temperature-dependent materialproperties and the welding conditions. The last section,before the conclusions, presents and discusses theresults obtained in the numerical modelling of the elec-trode misalignment, the shunt effects and the subse-quent tensile–shear tests.

Fundamentals of electro-thermo-mechanical modelling

The integrated resistance welding and mechanical test-ing simulation environment of SORPAS 3D was devel-oped by SWANTEC Software and Engineering ApS incollaboration with the Technical University ofDenmark and the University of Lisbon. The programperforms the coupled electro-thermo-mechanical finiteelement simulation of resistance welding and thedamage-coupled finite element simulations of quasi-static mechanical tests.

886 Proc IMechE Part D: J Automobile Engineering 229(7)



Mechanical module

The mechanical module allows the contact interfaces,the deformation of the welded parts and the distribu-tion of the major field variables (e.g. the stresses andthe strains) during the formation and growth of nuggetsto be modelled, taking into account the temperature-dependent mechanical properties of the materials. Themechanical module is built upon the finite element flowformulation, which is based on the weak variationalform

ðV

�s d_�e dV+K

ðV

_eii d _ejj dV�ðSt

ti dui dS=0 ð1Þ

where V is the domain volume limited by the surfacesSu and St, where the velocities ui and the tractions tiare prescribed, �s is the effective stress and _�e is the effec-tive plastic strain rate. The symbol K denotes a largepositive number related to the mean stress sm =skk=3through K _ekk =2sm, which is necessary to impose theincompressibility requirements _ekk =0 of the plasticdeformation of metallic materials.

Numerical implementation of the finite element flowformulation performed in SORPAS 3D allows plasticdeformation to follow either the isotropic von Misesyield criterion

�s2 =3 J2 ð2Þ

where J2 is the second invariant of the deviatoric stresstensor, or the anisotropic Hill quadratic yield criterion

2 �s2 =F sy � sz

� �2+ G sz � sxð Þ2 +H sx � sy

� �2+ 2L t2yz +2M t2zx +2N t2xy ð3Þ

where the constants F, G, H, L, M and N are to bedetermined from material testing.

In the case of quasi-static mechanical tests, it is pos-sible to account for the accumulation of damage bymeans of the constitutive equations of metallic materi-als with the initial porosity based on the yield criteriadue to Shima and Oyane16 or Doraivelu et al.17 andgiven by

�s2R=AJ2 +B I1 ð4Þ

In the above equation, I1 is the first invariant of thestress tensor sij and �s2

R=C �s2 is the effective stress

response of the material, taking into account thedecrease in the relative density R due to accumulationof damage (generation and coalescence of voids). A, Band C are material constants which depend on the rela-tive density.

The accountability of damage directly from the rela-tive density instead of the plastic strain13 is a step fur-ther for the numerical simulations of quasi-staticmechanical tests, which deals with the modern trends indamage mechanics.18 In fact, the main advantage ofusing the constitutive equations of metallic materials

with an initial porosity instead of the standard vonMises or Hill constitutive equation is the possibility ofdirectly influencing the predicted load–displacementcurves of the tests by the accumulation of ductile dam-age in the regions of the specimens undergoing largeplastic deformations. A measure of the damage D isdefined as

D=1� R ð5Þ

such that it is zero when the material is fully dense, anddecreasing relative density corresponds to increasingdamage.

Thermal module

The thermal module calculates the heat transfer andheat generation, including the temperature-dependentproperties of the materials. The module accounts forthe heat generated by the electrical Joule effect andplastic work and for the heat exchange with the electro-des and surrounding environment. The weak varia-tional form of the thermal module is built upon theclassical Galerkin treatment of the heat transferequation

ðV

kT, i dT, i dV+

ðV

rm cm _T dT dV�ðV

_qV dT dV

�ðS

_qS dT dS�ðS

kT, n dS=0 ð6Þ

In the above equation, the thermal conductivity k isrelated to heat conduction, the mass density rm and theheat capacity cm are related to the rate of stored energyper unit volume, giving rise to a temperature rate _T,and the rate of heat generation _qV in the material vol-ume includes the contributions due to dissipated energygiven by

_qplastic=b �s_�e

_qelectrical = r j2ð7Þ

where b is a constant taking values in the range between0.85 and 0.95, r is the electrical resistivity and j is thecurrent density. The rate _qS of heat generation alongsurfaces includes the contributions

_qconvection= � h Ts � Tf

� �_qradiation = � eemis sSB T4

s � T4f

� �_qfriction = tf vrj j

ð8Þ

where h is the heat transfer coefficient, Ts is the surfacetemperature, Tf is the temperature of the surroundings,eemis is the emissivity coefficient, sSB is the Stefan–Boltzmann coefficient, tf is the friction shear stress atthe contact interfaces between the parts (or sheets) tobe welded and vr is the relative sliding velocity at the

Nielsen et al. 887



contact interfaces between the parts (or sheets) to bewelded.

Electrical module

The electrical module calculates the distribution of theelectric potential F which, after differentiation andmultiplication by the conductivity, provides the currentdensity j at all locations in the electrodes and the partsto be welded. The electrical module takes into accountthe temperature-dependent electrical properties of thematerials, and its governing equation is built upon inte-gration of the Laplace equation for an arbitrary varia-tion dF in the electric potential, which by applying thedivergence theorem results in

ðV

F, i dF, i dV=

ðS

F, n dS ð9Þ

where F, n is the normal gradient of the electric poten-tial to the free surfaces (which is zero and therefore can-cels out the right-hand side).

Although this approach considers the distribution ofthe electric potential F to be solely determined by thegeometry under steady-state conditions ( _F=0), it isgenerally considered a good methodology because anelectric field has a much faster reaction rate than a tem-perature field does.19

The current density is calculated from ji =F, i=r,where the electrical resistivity r is temperature depen-dent. Its value is of special importance at the contactinterfaces where the majority of the heat is generated inthe early stages of welding. The contact resistivity rc ismodelled as20

rc =3ssoft

sn

r1 + r2

2+ rcontaminant

� �ð10Þ

In the above equation, the term placed outside theparentheses, including the flow stress of the softer mate-rial ssoft and the contact normal pressure sn, is the ratioof the real contact area to the apparent area obtainedby Bowden and Tabor.21 The bulk electrical resistivitiesof the materials in contact are r1 and r2, and rcontaminant

is the resistivity stemming from surface contaminantssuch as oxides, surface films and dirt. The increasedresistivity at the contact interfaces results in increasedheating by the Joule effect (see _qelectrical in equation (7)).

Numerical implementation

Figure 1 includes a schematic outline of the numericalimplementation of the couplings between the previ-ously described finite element modules. As shown inthe figure, the coupling between the electrical moduleand the thermal module is stronger than that with themechanical module, which is applied at the beginningof each step in order to obtain a velocity field and astress response of the material for subsequent

calculations. This is justified by the very small timesteps that are needed in order to represent all the physi-cal effects of the welding process. For example, whenusing AC with a frequency of 50 Hz as the energysource, each half-period has a duration of 10 ms, andproper numerical modelling therefore requires timesteps of 1 ms or preferably less. Consequently, insteadof having a strong coupling with the mechanical mod-ule, as for instance in the case of finite element com-puter programs for metal forming, the numericalimplementation relies on small time steps in a weakercoupling, where the new temperature-dependent mate-rial properties are affecting only the mechanicalresponse of the following time step. The correspondingchange in the heat generation due to plastic workwithin a particular step is neglected owing to its insig-nificant influence compared with that of the electricallygenerated heat.

The electrical module is applied after the mechanicalmodule in order to supply the thermal module with thecurrent density that is necessary to give rise to heat gen-eration. The electrical module is linear and thus inex-pensive to compute when compared with themechanical model. This is also the main reason whythe electrical module and the thermal module arestrongly coupled.

Further information on the overall numerical imple-mentation as well as the mechanical, thermal and elec-trical modules of SORPAS 3D can be foundelsewhere.22

Modelling conditions and effects underconsideration

All the test cases included in this paper make use of twoDC06 sheets of 0.8 mm thickness which are weldedbetween two type F1 electrodes with a tip diameter of 6mm. The sheets are zinc coated with layers 11 mm thick.

Tim

e st

eppi

ng

Itera�ons

Mechanical module

Electrical module

Thermal module

Input

Output

Itera

�ons

Itera

�ons

Figure 1. The proposed coupling between the mechanicalmodule, the thermal module and the electrical module of thefinite element computer program for resistance welding.




Electrode misalignment

The flexibility of the welding machine arms for posi-tioning the electrodes and reaching the locations ofspots in large sheet panels is a major source of electrodemisalignment in resistance spot welding. The problemis schematically illustrated in Figure 2(a), which showsthe rotation and off-centring of the electrodes whenapplying the electrode force through the weldingmachine arms. Depending on the type of machine andapplication of force, rotation may occur on one side oron both sides. Relative off-centring may be a result ofmore rotation on one side than on the other side.

In the case of the numerical simulations performedin this investigation, the discretization of the electrodes

was obtained by uploading their geometry directly from

a database of standard electrodes available in the com-

puter program and subsequently performing two clock-

wise rotations of 2� and 4� in order to replicate the

electrode misalignment against the ideal (aligned) per-

pendicular position between the electrodes and sheets

(Figure 2(b)). A relative off-centring of 0.5 mm is also

included in the misaligned test cases. The discretization

of the sheets was also performed by means of hexahe-

dral elements. The sheets of 40 mm width are clamped

30 mm away from the weld centre and are free on the

other end located 10 mm away from the weld centre.

The resulting meshes were refined in the region placed

in between the electrodes (where the welding nugget

Figure 2. Electrode misalignment and finite element modelling conditions: (a) schematic drawing of the electrode misalignment(left) and identification of the resulting rotation and off-centring of the electrodes (right); (b) discretization of the electrodes andsheets by means of hexahedral meshes, showing the full model with ideally aligned electrodes (left) and 0.5 m relative off-centringand rotations of the upper electrode of 2� and 4� respectively (right); (c) welding parameters in terms of the electrode force and thewelding current as functions of the process time.

Nielsen et al. 889



forms and the major deformation takes place). Layersof elements to model the coating and interface condi-tions were added subsequently by simple mesh extru-sion along the sheet surface normals. Only half of thegeometry is modelled because of the existence of a verti-cal symmetry plane. The model consists of a total num-ber of approximately 9000 hexahedral elements.

The choice of hexahedral elements instead of tetrahe-dral elements, which are the preferential and sometimesonly choice in commercial finite element computer pro-grams for metal forming, is justified by the fact thatmeshes based on tetrahedral elements result in muchlarger models than alternative meshes based on hexahe-dral elements for the same level of accuracy. This leadsto significant savings in computational requirementsand is of critical importance for the computationallyintensive electro-thermo-mechanical three-dimensionalalgorithm illustrated in Figure 1. Moreover, tetrahedralelements are known to cause critical errors when dis-torted, whereas hexahedral elements behave much bet-ter even when distorted.23 Furthermore, contact isnumerically more stable for first-order elements includ-ing the utilized hexahedral elements, as opposed tosecond-order elements that are sometimes applied intetrahedral meshes to overcome numerical difficulties.

Figure 2(c) shows the welding parameters that areutilized in all cases presented in this paper. The elec-trode force is constant at a level of 2.5 kN during thewelding and hold time. It is numerically raised from

zero force during 20 ms. The welding current is con-stant at 9 kA during the welding time of 275 ms. A holdtime of 100 ms, where the electrode force is maintained,was simulated after the end of the welding time.

Shunt effects

Electric shunting between two consecutive spot weldsdivert the welding current from the intended path andoften result in a current density that is not sufficient toproduce good-quality spot welds. The phenomenon isschematically illustrated in Figure 3(a). Figure 3(b)shows the hexahedral element discretization of the elec-trodes and sheets consisting of approximately 16,500 ele-ments. The mesh is again refined at the positions of thetwo spot welds, which are separated by a distance of 18mm between their centres. This value is close to, but stillabove, the recommended minimum distance betweenspots on 0.8 mm steel sheets.24 The overall sheet dimen-sions are 45 mm 3 83 mm. The welding conditions foreach weld are identical with those in Figure 2(c). Afterreleasing the force at the end of the hold time of the firstspot weld, there is an interval of 2 s before applying theforce that is necessary for the second spot weld. Coolingof the first spot weld is simulated during this period.

Mechanical testing

Tensile–shear testing of the sheet combination is alsosimulated as a continuation of the welding simulation

Figure 3. Electrical shunting between consecutive spot welds: (a) schematic drawing of the current being diverted from theintended path because the spot welds are placed close to each other; (b) discretization of the electrodes and sheets by means ofhexahedral meshes (the electrodes are positioned according to the first spot weld).




after cooling to room temperature, removing the elec-trodes and changing the tool connections (all prede-fined before starting the computer simulations).Tensile–shear testing is simulated under the same elec-trode alignment or misalignment as introduced in con-nection with Figure 2 (i.e. ideal conditions and relativeoff-centring of 0.5 mm with electrode rotations of 2�and 4� respectively). While the set-up in Figure 2 isassociated with corresponding experiments, the tensile–shear testing is only presented by numerical simula-tions. The sheet dimensions (except for the thickness)are changed and the clamping is removed in order tofollow the ISO 14273:2000(E) standard for tensile–shear testing. Each sheet has the dimensions 45 mm 3

65 mm (65 mm corresponds to the distance to theclamping tool), and they are placed with an overlap of35 mm. Figure 4(a) presents the mesh utilized, whichconsists of approximately 10,000 hexahedral elements,Figure 4(b) presents an example of a simulated weldnugget and initiated cooling, and Figure 4(c) shows thecontinuation of the tensile–shear testing.

The two sheets welded by two spots in Figure 3 arealready prepared as a lap joint which allows subsequenttensile–shear testing. This will be included only to show

the possibilities as it would be a similar procedure totest the damage, strength and/or deformation beha-viour of a welded structure.

Results and discussion

Electrode misalignment

The simulated weld nuggets for the different electrodealignments are shown in Figure 5, together with across-section of the experiment. In the simulations withideally aligned electrodes in Figure 5(a), the resultingweld nugget is symmetric and the amount of sheetseparation is also reasonable and close to symmetricdespite the fact that the left side is clamped (30 mmaway from the weld centre) while the right side is free(10 mm away from the weld centre). The nuggetbecomes asymmetric and the electrode indentations inthe sheets become severe when the electrodes are misa-ligned. Figure 5(b) shows the resulting weld when theupper electrode is rotated 2�, and Figure 5(c) shows theresulting weld when the rotation is 4�. The off-centringin both cases is equal to 0.5 mm. The initial contactbetween the upper electrode and the sheet is only a

Figure 4. Integrated resistance welding and mechanical testing simulation environment showing (a) the mesh utilized in thenumerical modelling, (b) the distribution of the process peak temperature (�C) in the simulated weld nugget after unloading andinitiated cooling in air and (c) the subsequent tensile–shear testing with simulated damage according to equation (5). The arrowsindicate the loading direction.

Nielsen et al. 891



partial ring contact when the electrode is misaligned byrotation. The electrode will indent the sheet in orderfor the contact area to develop during the process andto carry the applied electrode force. The indentationand separation of the sheets in combination with theasymmetric heat generation result in a high risk ofsplash, leading to uncontrolled joining conditions. Thesheet separation and lifting create problems to the over-all assembly of the sheets in production lines because itgives rise to relative movement of the sheets to a degreethat can move them off their positions for subsequentweld spots.

The cross-section in Figure 5(d) shows the experi-mental weld nugget and the final deformation of thesheets. The experiment was performed with a real weld-ing gun of the same type as in the production line. Inthis case the electrode misalignment is not fixed duringthe process, but rather a result of the process. This is infact the reason for carrying out the simulations becausethey provide further understanding. The electrodes startclose to ideally aligned in the real welding process but,on application of the electrode force, the misalignmentdevelops for the reasons described in connection withFigure 2(a).

The experimental weld in Figure 5(d) compares wellwith the finite element prediction in Figure 5(b),although it is somewhere in between Figure 5(b) andFigure 5(c). This suggests an effective electrode rotationof the upper electrode of around 2�.

The influence of the electrode misalignment on theweld strength is investigated numerically by performingfinite element simulations of the tensile–shear testing asoutlined in Figure 4. Following the ISO standard, weld-ing and tensile–shear testing were simulated under theabove conditions, including ideally aligned electrodesas well as misaligned electrodes. The three simulatedload–elongation curves are shown in Figure 6(a), andFigure 6(b) includes the evolution of the geometry of

the testing specimen at selected points on the curve forthe condition without electrode misalignment.

The differences in the load-bearing capacities aresurprisingly negligible in the three cases, leading to animmediate conclusion that the electrode misalignmenthas no influence on the weld strength. This, however, isnot confirmed by experiments, and the simulations didnot include the eventual splash and the related uncon-trolled effects. The results are therefore not presentedas a conclusion, but to show the possibilities of simula-tions when more detailed effects are included.

Figure 6(b) shows the tensile–shear testing procedureby the deformation and damage according to equation(5) at each 1 mm elongation. The load increases withincreasing elongation until a maximum weld strengthreached at 2.0 mm elongation with a correspondingload of 4.15 kN. This point is reached without severedamage, which accelerates after this point, reducing theload and finally resulting in facture. The region of highdamage reveals the location of the onset of fracture (seethe region indicated with a circle in Figure 6(b)), and itcan therefore be stated from the last figure in Figure6(b) that the failure type will be thinning and fractureof the sheet, resulting in plug failure. This is the casefor all three tensile–shear tests and is typical of soundweld nuggets in thin sheets.

Electrical shunting of welding current

The simulation results of the welding case including thetwo neighbouring spots introduced in Figure 3 are pre-sented in Figure 7. The figure includes an overview ofthe simulation procedure and the corresponding tem-perature development during the overall process. Thefirst welding time finishes after 315 ms, and the finite-element-predicted distribution of the temperature isshown in the upper left figure in Figure 7(a). The elec-trode force is maintained for an additional 100 ms and

Figure 5. Weld nugget cross-sections obtained from (a)–(c) the finite element predicted geometry and the distribution of theprocess peak temperature (�C) and (d) the experimental results. The finite element simulations include (a) the ideal electrodealignment and (b), (c) the electrode misalignment by 0.5 mm off-centring and clockwise rotations of the upper electrode of (b) 2�and (c) 4�.




Figure 6. Finite element analysis of tensile–shear testing for different electrode alignments showing (a) the predicted load–elongation curves and (b) the predicted evolution of specimen deformation and damage evolution (equation (5)) for the ideal case.

Figure 7. Distribution of the temperature (�C) during numerical simulations of the electric current shunting between twoconsecutive spot welds shown (a) at different instants of time and (b) in terms of the process peak temperature, revealing the nuggetsizes. Both figures share the same colour legend.

Nielsen et al. 893



then moved (instantaneously in the numerical simula-tions) to the location of the next spot weld. The upperright figure in Figure 7(a) shows the correspondingtemperature field, where the weld nugget has solidified.The electrodes are then closed after 2 s, such that thetotal simulated process time is 2415 ms when startingthe second spot weld. The lower left figure in Figure7(a) includes the corresponding temperature field afterthe additional cooling time of 2 s. Finally, the lowerright figure in Figure 7(a) shows the predicted distribu-tion of temperature after 2730 ms, which correspondsto the end of the second welding time. The second nug-get is now in its molten state. The simulations are thencompleted by an additional hold time of 100 ms andfurther cooling. At the end of the simulations, the pro-cess peak temperature distribution, shown in Figure7(b), reveals the weld nuggets. The first weld nuggethas a predicted diameter of 6.3 mm, while the secondweld nugget has a diameter of only 6.0 mm, which isalmost 5% smaller than that of the first spot.

The amount of shunting is presented in Figure 8 bythe current density distribution at an instant of timeand the current through each weld as a function oftime. The current density distribution in Figure 8(a)shows how part of the current flows through the sheetsand through the previous weld while welding the

second point. In the first spot, the current density ishigher around the periphery of the weld owing to thesharp geometrical change. Figure 8(b) shows the per-centage of the total current passing through each of thewelds as a function of time. The first weld will naturallyexperience the total welding current. The predicted cur-rent through the second weld, following the curve inFigure 8(b), is roughly 90%, while the remaining 10%passes through the first weld because of shunting. Theamount of shunting varies during the welding time as aresult of the changing contact properties, the materialproperties and the formation of contact area. From thecurve shown in Figure 8(b) it appears that the amountof shunt current initially increases and then decreasestowards the end of the welding time. After overcomingthe initially high contact resistance, the first increase inthe shunt current is explained by the temperatureincrease in the second weld and the associated increasein the electrical resistivity. The temperature in the firstweld and the sheet material in between remains moder-ate. The subsequent decrease in the amount of shuntcurrent is explained by the increasing contact area, andtherefore decreasing resistance, between the sheetswhere the second weld forms.

Finally, the total strength of the two neighbouringspots is simulated by tensile–shear testing in the length

Figure 8. Simulated current flow: (a) predicted current density distribution after 20 ms into the second weld time, where the scalecorresponds to 0–210 A/mm2 out of the maximum 1800 A/mm2 at this instant of time; (b) current through each of the weldingpoints as a percentage of the total applied welding current.




direction of the two spots. Figure 9 includes the load–elongation curve as well as the predicted deformationand accumulated damage D at elongations of 4.9 mmand 8.0 mm. The load–elongation curve reveals a maxi-mum load of 6.36 kN at an elongation of 4.9 mm. Thistesting does not follow any standards, but it illustratesthe potential of predicting the strength of a weldedstructure. The combination of two welds has a higherstrength than the single spot does (compare Figure 6(a))despite the fact that the loading direction sees only onespot. The presence of a neighbouring spot increases theresistance to bending of the sheets near the spots,thereby increasing the strength. Figure 9(b) shows thatthe first failure is predicted near the second weld (seethe region indicated with a circle) as may be expectedbecause of the smaller diameter of the nugget. It alsoshows that the failure mode is again plug failure.

The decrease in the diameter of the second welddepends on the distance between the spots and may

also be influenced by eventual other previouslywelded spots. In the simulated case, the current waskept constant to evaluate the effect on the nugget dia-meter, but it will also be possible to simulate the sec-ond weld with an increased welding current tocompensate for the shunt effect. An estimate for thenew current can thereby be predicted before carryingout further experiments and ultimately production.Simulations can also be utilized in the evaluation ofthe temperature rise in previous spots, which canreveal whether there is risk of influencing the micro-structure formation of previous spots during theircooling. This will depend on the distance and the timebetween the welds.

Conclusions

Three-dimensional numerical modelling is necessary forobtaining an insight into complex resistance welding

Figure 9. Finite element analysis of the tensile–shear testing of the specimen with two neighbouring spot welds: (a) predicted load–elongation curve; (b) predicted evolution of the specimen configuration during testing including the predicted damage according toequation (5).

Nielsen et al. 895



phenomena such as those included in the present paper.Moving from laboratory tests to the production lineis all about deviations from ideal conditions and theoccurrence of unexpected problems. Therefore,expanding laboratory tests to include the numericalmodelling of real production conditions is crucial forincreasing the accuracy and reliability of weldingoperating conditions. The cases presented in thispaper for resistance spot welding include some of themore typical challenges which are met in automotiveproduction lines and which may eventually lead topoor welds or ultimately to no joining even thoughthe applied welding schedules work out in ideallaboratory tests.

The modelling conditions that were utilized in thispaper replicate the operating conditions of the weldingmachines which are commonly employed in automotiveproduction lines. In specific terms, the overall flexibilityand the resulting electrode misalignment were modelledas closely as possible to real production conditions. Thethree-dimensional numerical simulations allow visuali-zation of physical effects that are not observed in two-dimensional numerical simulations nor in experimentaltesting, and it provides the possibility of obtaining ameasure of the robustness towards electrode rotation.It was also shown how continuation of the simulationsby tensile–shear testing can be successfully utilized toevaluate the strength and failure modes of the createdspot welds.

Another complication considered in the paper is theshunt effect between consecutively welded spots closeto each other. Again, three-dimensional numericalmodelling proves efficient to show how the currentflows through a previous spot weld, and to evaluate theamount of shunt current during the welding time. Theinfluence of the shunt effect on the resulting weld nug-get sizes is also evaluated. The analysis could be furtherextended to find the necessary increase in the currentduring the second weld to obtain a nugget size identicalwith that of the first nugget. Finally, the specimen con-taining the two spots was tested in order to demon-strate the overall testing capability of the new proposedsimulation environment.

Acknowledgements

Chris Nielsen and Niels Bay would like to acknowledgethe Danish Ministry of Science, Technology andInnovation, Research Council for Technology andProduction for financial support through the InnoJointproject. Paulo Martins would like to acknowledge thefinancial support provided by the Velux Foundationduring his sabbatical licence at the Technical Universityof Denmark.

Declaration of conflict of interest

The authors declare that there is no conflict ofinterest.

Funding

This work was supported by the Danish Ministry ofScience, Technology and Innovation Research Councilfor Technology and Production (grant number 274-05-0232).

References

1. Greenwood JA. Temperature in spot welding. Br Weld J

1961; 8: 316–322.2. Nied HA. The finite element modeling of the resistance

spot welding process. Weld J 1984; 63: 123s–132s.3. Nishiguchi K and Matsuyama K. Influence of current

wave form on nugget formation phenomena when spot

welding thin steel sheet. Weld World 1987; 25: 222–244.4. Tsai CL, Jammal OA, Papritan JC and Dickinson DW.

Analysis and development of a real time control metho-

dology in resistance spot welding. Weld J 1992; 70: 339s–

351s.5. Khan JA, Xu L and Chao Y. Numerical simulation of

resistance spot welding process. Numer Heat Transfer

Part A 2000; 37: 425–446.6. Zhang W, Hallberg H and Bay N. Finite element model-

ing of spot welding similar and dissimilar metals. In: 7th

international conference on computer technology in weld-

ing, San Francisco, California, USA, 8–11 July 1997,

NIST Special Publication 923, pp. 364–373. Gaithers-

burg, Maryland: NIST.7. Huh H and Kang WJ. Electrothermal analysis of

electric resistance spot welding processes by a 3-D finite

element method. J Mater Processing Technol 1997; 63:

672–677.8. Feulvarch E, Robin V and Bergheau JM. Resistance spot

welding simulation: a general finite element formulation

of electrothermal contact conditions. J Mater Processing

Technol 2006; 153–154: 436–441.9. Nielsen CV, Friis KS, Zhang W and Bay N. Three-sheet

spot welding of advanced high-strength steels. Weld J

2011; 90: 32s–40s.10. Larsson JK, Lundgren J, Asbjornsson E and Andersson

H. Extensive introduction of ultra high strength steels sets

new standards for welding in the body shop. Weld World

2009; 53(5–6): 4–14.11. Aslanlar S, Ogur A, Ozsarac U and Ilhan E. Welding

time effect on mechanical properties of automotive sheets

in electrical resistance spot welding. Mater Des 2008; 29:

1427–1431.12. Xu F, Sun G, Li G and Li Q. Failure analysis for resis-

tance spot welding in lap-shear specimens. Int J Mech Sci

2014; 78: 154–166.13. Radakovic DJ and Tumuluru M. An evaluation of the

cross-tension test of resistance spot welds in high-strength

dual-phase steels. Weld J 2012; 91: 8s–15s.14. Lin HL, Chou T and Chou CP. Modelling and optimiza-

tion of the resistance spot welding process via a Taguchi–

neural approach in the automobile industry. Proc

IMechE Part D: J Automobile Engineering 2008; 222(8):

1385–1393.15. Ouisse M and Cogan S. Robust design of spot welds in

automotive structures: a decision-making methodology.

Mech Systems Signal Processing 2010; 24: 1172–1190.16. Shima S and Oyane M. Plasticity theory for porous

metals. Int J Mech Sci 1976; 18: 285–291.




17. Doraivelu SM, Gegel HL, Gunasekera JS et al. A newyield function for compressible P/M materials. Int J Mech

Sci 1984; 26: 527–535.18. Nielsen KL. Modelling of damage development and ductile

failure in welded joints. PhD Thesis, Technical Universityof Denmark, Kongens Lyngby, Denmark, 2009.

19. Greenwood JA and Williamson JBP. Electrical conduc-tion in solids II. Theory of temperature-dependent con-ductors. Proc R Soc Lond, Ser A, Mathl Phys Sci 1958;246: 13–31.

20. Zhang W. Design and implementation of software forresistance welding process simulations. Trans ASME, J

Mater Mfg 2003; 112: 556–564.

21. Bowden FP and Tabor D. The fabrication and lubrication

of solids. Oxford: Oxford University Press, 1950.22. Nielsen CV, Zhang W, Alves LM et al. Modeling of

thermo-electro-mechanical manufacturing processes with

applications in metal forming and resistance welding.

Berlin: Springer, 2012.23. Kraft P. Automatic remeshing with hexahedral elements:

problems, solutions and applications. In: 8th international

meshing roundtable, South Lake Tahoe, California, USA,

10–13 October 1999, pp. 357–367.24. RWMA. Resistance welding manual. Miami, Florida:

American Welding Society, 2003.

Nielsen et al. 897



proc imeche part d: three-dimensional simulations of ...€¦ · 886 proc imeche part d: j...

Documents