DEPARTMENT OF MANAGEMENT AND ENGINEERINGModelling of the Resistan e Spot Welding Pro essMaster Thesis arried out at Division of Solid Me hani sLinköpings UniversityApril 2009Alexander GovikLIU-IEI-TEK-A--09/00609--SE

Institute of Te hnology, Dept. of Management and Engineering,SE-581 83 Linköping, Sweden

FramläggningsdatumPresentation date2009-04-17Publi eringsdatumPubli ation date2008-04-23Avdelning, institutionDivision, departmentDivision of Solid Me hani sDept. of Management and EngineeringSE-581 83 LINKÖPINGSpråkLanguageSvenska/SwedishEngelska/EnglishX RapporttypReport ategoryLi entiatavhandlingExamensarbeteC-uppsatsD-uppsatsÖvrig rapportX ISBN:ISRN: LIU-IEI-TEK-A--09/00609--SESerietitel:Title of seriesSerienummer/ISSN:Number of seriesURL för elektronisk versionURL for ele troni versionhttp://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-17835TitelTitle Modelling of the Resistan e Spot Welding Pro essFörfattareAuthor Alexander GovikSammanfattningAbstra tA literature survey on modelling of the resistan e spot welding pro ess has been arried out and some of the more interesting models on this subje t have beenreviewed in this work. The underlying physi s has been studied and a briefexplanation of Heat transfer, ele trokineti s and metallurgy in a resistan e spotwelding ontext have been presented.Lastly a state of the art model and a simpli�ed model, with implementation in theFEM software LS-DYNA in mind, have been presented.

Ny kelord:Keyword Resistan e spot welding, modelling, heat transfer, ele trokineti s, metallurgy, DP

IV

� Modelling of the Resistan e Spot Welding Pro ess �

VAbstra tA literature survey on modelling of the resistan e spot welding pro ess hasbeen arried out and some of the more interesting models on this subje thave been reviewed in this work. The underlying physi s has been studiedand a brief explanation of Heat transfer, ele trokineti s and metallurgy in aresistan e spot welding ontext have been presented.Lastly a state of the art model and a simpli�ed model, with implementa-tion in the FEM software LS-DYNA in mind, have been presented.

� Govik �

VI

� Modelling of the Resistan e Spot Welding Pro ess �

VIIPrefa eThe work presented here is the result of a master thesis that has been arriedout at the Division of Solid Me hani s at Linköping University.I would like to express my gratitude to my supervisor Prof. LarsgunnarNilsson for the support and guidan e during my work with the master thesis.A spe ial thanks should also be given to all Ph.D. students on the divisionfor all their help.Lastly, I would like to take the opportunity to thank my family and friendsfor all their support during the years.Linköping, April 2009Alexander Govik

� Govik �

VIII

� Modelling of the Resistan e Spot Welding Pro ess �

CONTENTS IXContents1 Introdu tion 11.1 Ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Obje tive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Theory 32.1 Resistan e spot welding pro ess . . . . . . . . . . . . . . . . . 32.1.1 Method des ription . . . . . . . . . . . . . . . . . . . . 32.1.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Heat generation . . . . . . . . . . . . . . . . . . . . . . 52.2 Dynami resistan e . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Ele trokineti . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Basi s . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Ele tri �eld analysis . . . . . . . . . . . . . . . . . . . 132.4 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Heat transfer me hanisms . . . . . . . . . . . . . . . . 162.4.2 Heat transfer in the resistan e spot welding pro ess . . 172.5 Metallurgy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.1 Basi s . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.2 Advan ed High Strength Steel . . . . . . . . . . . . . . 222.5.3 Welding metallurgy . . . . . . . . . . . . . . . . . . . . 232.5.4 Modelling of mi rostru ture . . . . . . . . . . . . . . . 243 Previous modelling 254 Model 294.1 State of the art model . . . . . . . . . . . . . . . . . . . . . . 294.1.1 Thermal model . . . . . . . . . . . . . . . . . . . . . . 304.1.2 Ele tri al model . . . . . . . . . . . . . . . . . . . . . . 304.1.3 Metallurgi al model . . . . . . . . . . . . . . . . . . . . 314.1.4 Me hani al model . . . . . . . . . . . . . . . . . . . . . 314.2 Simpli�ed model . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.1 Thermal model . . . . . . . . . . . . . . . . . . . . . . 334.2.2 Me hani al model . . . . . . . . . . . . . . . . . . . . . 334.2.3 Metallurgi al model . . . . . . . . . . . . . . . . . . . . 335 Dis ussion 355.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36� Govik �

X LIST OF FIGURESList of Figures1 The welding y le . . . . . . . . . . . . . . . . . . . . . . . . . 32 (a) Weld growth urve, (b) 2-D weld lobe . . . . . . . . . . . . 43 Di�erent impulse arrangements . . . . . . . . . . . . . . . . . 54 S hemati representation of the resistan e spot welding set-up 65 Di�erent ele trode geometries . . . . . . . . . . . . . . . . . . 86 Deformation of asperities . . . . . . . . . . . . . . . . . . . . . 107 S hemati urve of dynami resistan e . . . . . . . . . . . . . 118 Cylindri al element . . . . . . . . . . . . . . . . . . . . . . . . 189 Phase diagram [14℄ . . . . . . . . . . . . . . . . . . . . . . . . 2110 CCT diagram with three di�erent ooling s hemes [14℄ . . . . . 2211 Sket h of a ferrite-martensite DP mi rostru ture . . . . . . . . 2312 Summary of the presented models . . . . . . . . . . . . . . . . 2713 Couplings in the model . . . . . . . . . . . . . . . . . . . . . . 2914 Couplings in the simpli�ed model . . . . . . . . . . . . . . . . 32

� Modelling of the Resistan e Spot Welding Pro ess �

11 Introdu tionThe automotive industry fa es great hallenges. The in reasing demand forenvironmentally friendly vehi les requires the manufa turers to in rease theuse of advan ed high-strength materials to redu e weight. This ombinedwith shorted produ t life y les and more advan ed designs in rease the ostper produ ed vehi le.By shortening the development time of the produ t money an be saved.The most e�e tive way to a omplish this is to in rease the use of VirtualManufa turing Engineering (VME) te hniques. There are however still anumber of issues that have to be over ome before VME is fully realized. Thepredi tion a ura y of the forming simulation must for example be improvedand reliable joining simulations must be established su h that the propertiesof an assembled part an be predi ted.With this in mind the proje t SimuPARTs was initiated by SSF (SwedishFoundation for Strategi Resear h), Swedish Automotive Industries and sheetmetal working industries in ollaboration with Universities to over ome thestated problems.This master thesis has been arried out as a �rst step in one of the sub-proje ts of SimuPARTs.1.1 Ba kgroundProfessor Elihu Thomson is redited to be the inventor of the idea to joinmaterials with ele tri resistivity heating. In the 1880's he �rst applied thiste hnique to join lengths of opper wire. Today Resistan e Spot Welding(RSW) is used extensively in the automotive industry, i.e. in a typi al arthere are thousands of spot welds.RSW onsists of two opposed ele trodes that are pressed against the sheetsof metal that are to be joined. When an ele tri urrent �ows from oneele trode through the sheets of metal to the other ele trode it auses heatingdue to the resistan e in the joining surfa es and in the sheets. Depending onthe thi kness and type of material the urrent that auses a weld nugget toform may be in the range of 1,000 to 100,000 amperes or even more, whilethe voltage typi ally is between 1 and 30 volts [1℄.� Govik �

2 1 INTRODUCTION1.2 Obje tiveThe aim of this work is to study how the resistan e spot welding pro ess hasbeen and is being modelled. The underlying physi s of the pro ess shouldalso be studied in order to present a suggestion on how a state of the artmodel and a simpli�ed model of RSW an be made in the general �niteelement framework.

� Modelling of the Resistan e Spot Welding Pro ess �

32 TheoryThe resistan e spot welding pro ess will be explained in this hapter. First anoverview of the pro ess, then some of the more important topi s are explainedmore thorough.2.1 Resistan e spot welding pro ess2.1.1 Method des riptionThe welding y le, Fig.1, an be divided into four de�nite stages [2℄:1. Squeeze time - the time between the appli ation of pressure by theele trodes to the �rst appli ation of the weld urrent.2. Weld time - the time during whi h welding urrent is applied.3. Hold time - the time during whi h the pressure at the point of weldingis maintained by the ele trodes after the weld urrent eases.4. O� time - the time during whi h the ele trodes are separated to permitmoving of material. The term is generally used when the welding y leis repetitive.

Figure 1: The welding y leThe welding y le is required to develop su� ient heat to raise a on�nedvolume of metal to its melted state. The a hieved temperature must be highenough so that fusion or in ipient fusion is obtained, but not so high thatexpulsion o urs. Expulsion happens when the hydrostati pressure from themelt ex eeds the onta t pressure and the liquid metal is radially dispersedbetween the sheet-to-sheet interfa e. This upper and lower limit of heat input� Govik �

4 2 THEORYde�nes the weld lobe, see Fig. 2, i.e. the range in whi h a weld is formed.Welddiameter(a)Weldtime(b) Welding urrent

Welding urrentA eptableweld ExpulsionIn ompleteweldFigure 2: (a) Weld growth urve, (b) 2-D weld lobeThe metal is then ooled, under pressure, to a temperature at whi h the weldhas enough strength to hold the parts together.Proper fun tioning of the ele trodes is vital for the pro ess. They musthave good ele tri al and thermal ondu tivity to lessen the heat generatedat the area of onta t between the ele trode and the work surfa e. Goodele tri al ondu tivity gives lower resistan e, and good thermal ondu tivitygives better dissipation of heat from the weld zone. The ele trodes mustalso have good me hani al properties to withstand high stresses at elevatedtemperatures without ex essive deformation. The upholding of proper ele -trode shape is important sin e the urrent needs to be ondu ted to the workwithin a �xed area. The urrent an be applied in either single or multipleimpulses as an be seen in Fig. 3.

• Single impulse - One single ontinuous urrent is applied, with or with-out up/down slope• Multiple impulse - Two or more appli ations of urrent with an o�time in between. The impulses may have the same magnitude andis then alled pulsation welding. The impulses an also be of di�erentmagnitude, e.g. �rst a preheat then the weld and last a temper urrent.� Modelling of the Resistan e Spot Welding Pro ess �

2.1 Resistan e spot welding pro ess 5Single impulse with up and down slopeup slope down slope

Multiple impulse withpreheat and temperFigure 3: Di�erent impulse arrangements2.1.2 CurrentResistan e spot welding an be done with both alternating urrent (AC) anddire t urrent (DC) power supply. AC spot welding has been the most widelyused in the automotive industry whereas DC spot welding has been used inthe aerospa e industry due to the high power needed for welding aluminium.The DC welding pro ess requires less welding urrent than the orrespond-ing AC welding pro ess, but despite this the automotive industry has beenrelu tant to use the DC welding pro ess sin e the energy saving is un ertaindue to the energy losses when onverting AC to DC as well as the extraequipment ost and reliability.In re ent years a new power supply for DC spot welding, mid-frequen yDC (MFDC) inverter, has be ome more popular. The reliability has beenimproved and more importantly the welding urrent an be ontrolled withmu h higher resolution giving the possibility of higher quality welds. In are ent study [3℄ it has been shown that MFDC is more energy e� ient thanAC, espe ially at lower welding urrents.2.1.3 Heat generationThe transformation from ele tri urrent to heat an be derived from Ohm'slaw ombined with Joule's law, when heat is onsidered as synonymous with� Govik �

6 2 THEORYenergy.Q = I2Rt (1)Where Q is the generated heat energy, I is the ele tri urrent, R is the totalresistan e of the work area and t is the weld time.The physi al explanation to this heating is that when the urrent is applied,ele trons will start to �ow trough the metal. The kineti energy of theseele trons is transferred to the metal by ollisions with the ions, whi h are vi-brating around their equilibrium position in the rystalline stru ture. These ollisions auses a higher vibrational energy of the ions and thus an in reasedtemperature of the metal [4℄.In�uen e of urrent. - As an be seen in eq. (1), the urrent has the greateste�e t on the generation of heat. On e in ipient fusion temperature is rea hedthe weld nugget size and strength in rease qui kly with further in rease ofthe urrent. If to mu h urrent is applied it will result in weld expulsion,weld ra king and redu ed me hani al properties.

Figure 4: S hemati representation of the resistan e spot welding set-up

1234567Faying surfa eWeld nuggetWater oolingII

� Modelling of the Resistan e Spot Welding Pro ess �

2.1 Resistan e spot welding pro ess 7In�uen e of resistan e. - The total resistan e is the sum of the series ofresistan es in the work area, as seen in Fig. 4. When welding together twosheets of metal there are seven major points of resistan e, de�ned as follows.1. Upper ele trode2. Conta t between upper ele trode and upper sheet3. Body of upper sheet4. Conta t between upper and lower sheets5. Body of lower sheet6. Conta t between lower sheet and lower ele trode7. Lower ele trodeIt is the resistan e at point 4 between the two sheets that is of most impor-tan e for the weld initiation and it is also this resistan e that is the largest,thus generating the most heat. The se ond highest resistan e is at point2 and 6 where the onta t between the sheet and the ele trode raises thetemperature rapidly, but due to water ooling in the ele trode and its goodele tri al and thermal ondu tivity the temperature is su� iently low not tohave insipient fusion.The resistan e in all onta ting surfa es dependens on the pressure. Thehigher pressure the lower resistan e. The internal resistan e in the sheets iswhat makes the weld nugget grow after the melting of the onta t surfa es.The resistan e in the ele trodes is low.In�uen e of time. - The rate of heat generation must be su� ient in the giventime interval so that the welding will be a hieved with proper ompensationfor heat losses. The heat losses are aused by ondu tion into the surroundingmetal and into the ele trodes as well as by radiation into the air. Shorterweld time de reases ex essive heat transfer to the surrounding metal andthus de reases the heat a�e ted zone.In�uen e of the metal sheets. - The surfa e ondition of the sheets in�uen ethe heat generation at the onta t between the sheet and the ele trode. If thesheets have dirt or an oxide �lm on them the onta t resistan e will in reaseand ause more heating. It also leads to deterioration of the ele trodes dueto ontamination. All this leads to unpredi table results.The omposition of the metal determines spe i� heat, melting point,latent heat of fusion, thermal ondu tivity and density. It also in�uen es therange in temperature between the point of softening and the point of melting,the so alled readily molded state.� Govik �

8 2 THEORYHeat balan e. - If two sheets of the same material and thi kness are weldedtogether with ele trodes of equal size and shape there will be a orre t heatbalan e, and the resulting weld nugget will be symmetri with respe t to theplane of the onta t surfa e. If, however, one of the sheets is thi ker or havea di�erent ele tri al ondu tivity or if one of the ele trodes have a di�erentsize or shape there will be a heat unbalan e, resulting in a weld nugget thatis more developed in one of the sheets than in the other.Shunt e�e t. - When pla ing a new spot weld next to an old one, a part ofthe urrent is going to �ow through the old spot weld and thus less heat isgenerated. This phenomenon is logi al sin e the �ow of urrent always triesto take the easiest path, i.e. the path with the least resistan e. The loserthe two spot welds are the stronger this e�e t be omes.Domed fa edFlat fa edFigure 5: Di�erent ele trode geometriesIn�uen e of ele trode geometry. - One of the important fun tions of theele trode is to deliver a uniform urrent to the metal sheet, sin e a non-uniform urrent ause uneven heating in the weld. A high ele trode to sheetangle, see Fig 5, gives more uniform urrent.The size of the ele trode tip is determined by the thi kness of the metalsheets and the desired weld size, a ommon rule for determining the ele trodetip size and weld size is the equation

d = 5√

t (2)where d is the diameter of the ele trode, it is assumed that the optimumweld size is the same as the ele trode diameter, and t is the sheet thi kness.The ele trode tip an be either domed of �at fa ed, Fig. ??angle). The�at fa ed ele trode tip gives a higher pressure at the periphery than in themiddle, whi h leads to uneven heating. The domed fa ed tip gives a moreuniform heating but the indention be omes larger.� Modelling of the Resistan e Spot Welding Pro ess �

2.2 Dynami resistan e 92.2 Dynami resistan eTo ompletely understand the resistan e spot welding pro ess it is ne essaryto know the role of the di�erent resistan es between the welding ele trodesand how they a�e t the heat generation. It is generally on luded that theinitial onta t resistan e has a marked in�uen e on the required magnitudeof welding urrent. But when determining what happens during the weldingpro ess the dynami resistan e is more important [5℄.The dynami resistan e is onstituted by the bulk resistivity of the sheetsand of the onta t resistan e in the ele trode/sheet and sheet/sheet inter-fa es.The bulk resistivity of a metal in reases with the temperature. This tem-perature dependen e an be explained with quantum me hani s in the fol-lowing way. The thermal vibrations in a solid an be said to be a swarm ofmi ros opi parti les alled phonons. The free ele trons of a metal swarmaround in the ele tron loud in random dire tions, but with a general driftvelo ity in the dire tion of the ele tri urrent. The phonons ontinuously ollide with ele trons and this arbitrary de�e tion or s attering of ele trodes auses a disturban e in the general drift of the ele trons. As the temperaturerises, i.e. the vibrational energy in reases, the number of phonons in reasesand with it the probability of a ollision between an ele tron and a phonon.Thus, when temperature rises the resistivity in reases.The onta t resistan e an be divided into �lm resistan e and onstri tionresistan e. The �lm resistan e originates from the surfa e ondition of thesheets. The existen e of an oxide layer or some other oating or even greaseor dirt on the surfa e will a�e t the �lm resistan e. In addition to the surfa e ondition the �lm resistan e also depends on urrent level, temperature and onta t pressure. The �lm resistan e is more pronoun ed in the early stageof the welding pro ess.The onstri tion resistan e between the two onta ting surfa es originatesfrom the existen e of asperities on the surfa es. That is, the true onta tarea is that of the onta t between asperities as seen in Fig. 6, this onta tarea is only a small fra tion of the apparent area. The true onta t areawill in rease with the onta t pressure be ause of lo al plasti deformationof the asperities. Sin e the pressure at the onta ting surfa es is not evenlydistributed, the amount of onstri tion resistan e will vary in the radial di-re tion.The onta t resistan e is also temperature dependent, as the tempera-ture rises the yield strength and sti�ness lessen, so the true onta t area� Govik �

10 2 THEORY

Figure 6: Deformation of asperitiesin reases. When the temperature rea hes the melting point, the onta t re-sistan e eases to exist sin e the onta ting surfa es have vanished.To develop a orre t generalized model of the onta t resistan e is di� ultsin e it is sensitive to variations and depends on several di�erent physi alphenomena. To experimentally measure the variation of resistan e in the onta t interfa e is impossible. Only an average value over the surfa e anbe measured. Despite this a number of models have been developed, see [6℄where the three following generalized models are mentioned.The Wanheim and Bay model [6℄ al ulates the onta t resistivity ρ bytaking into a ount the plasti deformation of asperities to determine thereal onta t area.ρcontact =

1

3

(

σf

σn

) (

ρ1 + ρ2

2

)

+ ρcontaminant (3)Where σn is the onta t pressure, σf is the �ow stress, ρ1 and ρ2 are thetemperature dependent bulk resistivities and ρcontaminant is the resistivity ofsurfa e agents.The Vogler and Sheppard model [6℄ suggests that the onta t resistan e� Modelling of the Resistan e Spot Welding Pro ess �

2.2 Dynami resistan e 11depends on the surfa e asperities in the following wayRcontact = (ρ1 − ρ2)

(

1

4ηa+

3π

32ηl

) (4)where η is the number density of asperities in onta t, a is the average ra-dius of onta ting asperities, and 2l is the average in-plane distan e betweenasperities. These quantities should all be fun tions of temperature and pres-sure.A model developed by Kohlraus h [6℄ an des ribe the ele tri al onta t ondu tan e as a gap ondu tan e asσg =

1

2πr2c

√

L(T 2s − T 2

0)(5)where Ts is the maximum temperature at the interfa e, T0 is the bulk temper-ature and L is the Lorentz onstant. In the above formulation it is assumedthat the onta t area is under intimate metalli onta t, this means that theradius rc must be dependent on the onta t pressure.

Figure 7: S hemati urve of dynami resistan e

Asperity softeningTemperature riseInitial meltingBeta peakNugget growthme hani al ollapse Expulsion

The shape of the dynami resistan e urve, as seen in Fig. 7, has been welldo umented in orrelation to the weld formation, more detailed work by [7℄suggests that the dynami resistan e urve an be divided into �ve phases as� Govik �

12 2 THEORYfollows:1. Deformation of surfa e asperities leading to larger onta t area gives arapid drop in resistan e.2. Final ollapse of asperities oupled with in reasing resistivity of themetal due to bulk heating. When the in reasing resistivity predom-inates over the e�e t of larger onta t area, an in rease of the totalresistan e starts.3. In reasing resistan e due to bulk heating.4. Melting at the faying surfa e o urs. The molten zone grows rapidly.When the in rease in resistivity of the metal due to bulk heating balan ewith the de rease in resistan e due to the in rease in size of the moltenzone, a peak value, alled Beta peak, of the resistan e is rea hed.5. A de rease of the resistan e owing to the indentation of the sheets thatgives a shorter urrent path.Phases 3 and 4 are the predominant phases in the development of a satisfa -tory weld.

� Modelling of the Resistan e Spot Welding Pro ess �

2.3 Ele trokineti 132.3 Ele trokineti 2.3.1 Basi sLets start by de�ning some fundamental entities.• Current I - is the net harge �owing through an area per unit time.• Current density J - is the urrent per ross-se tional area.• Resistivity ρ - is a measure of how strongly a material opposes the �owof urrent, ondu tivity is its re ipro al.• Resistan e R - is the resistan e of urrent �ow in a ondu tor, de�nedby.

R =L

Aρ (6)Condu tan e is its re ipro al.

• Ele tri �eld E - is the for e per unit harge exerted on a hargedparti le.• Ele tri Potential V - is the potential energy per unit harge.2.3.2 Ele tri �eld analysisTo be able to al ulate the heat generation in parts subje ted to spotweldingone must know the urrent density distribution.The urrent density J is, a ording to Ohm's law, proportional to theele tri �eld E.

J =1

ρE (7)The ele tri �eld in turn an be expressed as the negative gradient of theele tri potential V [8℄.

E = −∇V (8)Thus the urrent density in terms of the ele tri potential be omes.J = −

1

ρ∇V (9)� Govik �

14 2 THEORYThe ontinuity equation des ribes the onservative transport of harge. Itstates that the net �ow out of a volume should be balan ed by a net hangeof harge held inside the volume.∫

S

J · ndS +

∫

V

∂ρ

∂tdV = 0 (10)Where n is the outward pointing normal to the surfa e S and ρ stands inthis equation for the harge density.The spot welding pro ess an be looked upon as a losed ir uit, so thenet hange of harge is equal to zero, whi h means that also the net �owmust be equal to zero.

∫

S

J · ndS = 0 (11)It an be rewritten with the help of the divergen e theorem and by statingthat it is true for every volume.∇ · J = 0 (12)With Eq. (9) in Eq. (12) the ontinuity equation an be expressed as∇ ·

1

ρ∇V = 0 (13)whi h in ylindri al oordinates with symmetry around the z-axis be omes.

1

r

∂

∂r

(

r

ρ

∂V

∂r

)

+∂

∂z

(

1

ρ

∂V

∂z

)

= 0 (14)This is the governing equation for the ele tri potential in the model. To solvethis se ond order di�erential equation boundary onditions are needed. Twotypes of boundary onditions an be used, either the Neumann boundary ondition whi h spe i�es the values that the derivative of a solution is totake, in this ase the urrent density, or the Diri hlet boundary onditionwhi h spe i�es the values a solution needs to take, in this ase the potential.All surfa es with boundaries to the surrounding air is taken to be isolated,i.e. the �ow of harge in the normal dire tion of the surfa e into the air iszero. This means that the Neumann boundary ondition should be used.1

ρ

∂V

∂n= 0 (15)� Modelling of the Resistan e Spot Welding Pro ess �

2.3 Ele trokineti 15On the top surfa e of the ele trode there is a pres ribed urrent density sothe Neumann boundary ondition is used here to.1

ρ

∂V

∂z= j0 (16)

� Govik �

16 2 THEORY2.4 Heat TransferThe existen e of a temperature di�eren e is the driving for e for heat transfer,i.e. a heat transfer between two bodies of the same temperature an notexist. The temperature of a body an be des ribed as the average kineti energy asso iated with the disordered mi ros opi motion of the atoms andmole ules.The �rst law of thermodynami states that energy an neither be reated nordestroyed, i.e. it an only hange forms. For a stationary body, this an beformulated as: The net hange in the internal energy of the body is equal tothe di�eren e between the internal energy entering the body and the internalenergy leaving the body.∆U = Q − W (17)Where ∆U is the hange in internal energy, Q is the heat added to the bodyand W is the work done by the body.So if the body is heated or if work is done upon the body in the form ofplasti deformation for instan e, the internal energy will in rease whi h inturn will lead to a higher temperature.On a mi ros opi level the internal energy is the sum of the kineti andpotential energies in the body. The kineti energy originates in the trans-lational, rotational and vibrational movements of the atoms and mole ules,while the potential energy is the sum of the hemi al and nu lear energies.More information on heat transfer an be found in [9℄ and [10℄.2.4.1 Heat transfer me hanismsHeat an be transferred in three di�erent modes: ondu tion, onve tionand radiation. As mentioned earlier the heat transfer requires a temperaturegradient and it will always be in the dire tion from the body with the highertemperature to the body with the lower temperature.

Conduction. In ondu tion the transfer of energy between the higher en-ergeti parti les to the nearby lower energeti parti les is due to intera tionbetween the parti les.In a metal the dominant fa tor is the ollisions between free ele trons thattransfer kineti energy from the faster moving to the slower moving ele trons.The vibrations of the atoms also ontributes to the ondu tion, the vibra-tions auses waves in the latti e that transfer kineti energy from higher to� Modelling of the Resistan e Spot Welding Pro ess �

2.4 Heat Transfer 17lower energeti atoms. The thermal ondu tivity of a material is a measureof the materials ability to ondu t heat. It is a temperature dependent prop-erty and the reason for this is that the higher the temperature, the faster thefree ele trons moves whi h in turn leads to more ollisions and thus the heattransfer in reases.Convection. In onve tion the energy is transferred between a solid surfa eand a �uid in motion. It involves the ombined e�e t of ondu tion and �uidin motion, where the �uid motion a ts as an enhan er of the ondu tion.The faster the �uid motion, the greater the heat transfer. It is alled for ed onve tion if the �uid motion is for ed to �ow over the surfa e of the solid bysome external work. If the �uid motion is aused by density gradients dueto temperature di�eren es it is alled natural onve tion.Radiation. A more orre t name is thermal radiation so that it is learlydistinguished from other types of radiation like gamma rays and x-rays whi hare not related to temperature.Radiation is a volumetri phenomenon and all bodies with a tempera-ture above absolute zero emits and absorbs thermal radiation. If a bodyhas higher temperature than its surrounding it emits more thermal radiationthan it absorbs, and thus its temperature gets lower be ause of the lesseningof the internal energy.The reason for this radiation is that an atom an only hange its energylevel in dis rete steps and su h a step in energy level involves the emittingor absorbing of ele tromagneti waves (photons).Unlike ondu tion and onve tion, radiation requires no medium for trans-porting the energy.2.4.2 Heat transfer in the resistan e spot welding pro essIn resistan e spot welding the dominant mode of heat transfer is ondu tion.There also exists onve tion and radiation on the surfa es of the sheets andele trodes, and in the weld pool there is a mass transportation, due to mag-neti for es and density gradients, whi h auses onve tion [11℄. To a ountfor this onve tion a ontinuity equation, momentum equation and an energyequation is needed, see [11℄, [12℄ and [26℄. However these equations will notbe explained further in this work. Instead the heat ondu tion equation,whi h is used by most models to approximate the heat transfer, will be themain topi of this se tion.The rate of heat ondu tion through an element is proportional to the tem-� Govik �

18 2 THEORYperature gradient over the element and the area normal to the dire tion ofheat transfer, but inversely proportional to the element thi kness.This is expressed in Fourier's law of heat ondu tion.Q = −kA

∆T

∆x(18)The di�erential form is given by letting ∆x → 0

Q = −kAdT

dx(19)Where k is the thermal ondu tivity of the material.The resistan e spot welding pro ess is often modelled in ylindri al oordi-nates with symmetry around the z-axis, whi h is dire ted out of the sheetplane. Hen e the heat ondu tion equation in ylindri al oordinates withan internal heat generation needs to be dedu ed from the Fourier's law ofheat ondu tion.

Figure 8: Cylindri al elementr

z

Qz

Qz+∆z

Qr Qr+∆r∆r

∆z

Consider an in�nitesimal short and thin ylindri al element as in Fig. 8, withdensity , spe i� heat C and volume V=2πr∆r∆z.The energy balan e of this element during a short time interval ∆t be- omes∆Eelement

∆t= Qr − Qr+∆r + Qz − Qz+∆z + Gelement (20)� Modelling of the Resistan e Spot Welding Pro ess �

2.4 Heat Transfer 19(First law of thermodynami s), where Q is the heat transfer rate and G isthe heat generation rate.G = I2R (21)The hange in energy of the element an be expressed in terms of the mass m,the spe i� heat C (the energy required to raise the temperature of a spe i� quantity of a substan e by a spe i� amount) and the hange in temperature.

∆Eelement = Et+∆t−Et = mC(Tt+∆t−Tt) = C2πr∆r∆z(Tt+∆t−Tt) (22)The heat generation rate an be expressed in terms of heat generation rateper volume g.Gelement = Velementg = 2πr∆r∆zg (23)where g an be expressed asg = J2ρ (24)Now with Eq. (22) and (23) in Eq. (20) the energy balan e an be written

ρC2πr∆r∆zTt+∆t − Tt

∆t= Qr − Qr+∆r + Qz − Qz+∆z + 2πr∆r∆zg (25)Dividing by 2πr∆r∆z gives

ρCTt+∆t − Tt

∆t= −

1

2πr∆z

Qr+∆r − Qr

∆r−

1

2πr∆r

Qz+∆z − Qz

∆z+ g (26)Now, by using the de�nition of derivatives and Fourier's law of heat on-du tion, the terms in Eq. (26) an be expressed in the di�erential form byletting ∆r, ∆z, ∆t → 0

lim∆r→0

1

2πr∆z

Qr+∆r − Qr

∆r=

1

2πr∆z

∂Qr

∂r=

1

2πr∆z

∂

∂r(−2πr∆z

∂T

∂r) =

= −1

r

∂

∂r(kr

∂T

∂r)

(27)lim

∆z→0

1

2πr∆r

Qz+∆z − Qz

∆z=

1

2πr∆r

∂Qz

∂z=

1

2πr∆r

∂

∂z(−2πr∆r

∂T

∂z) =

= −∂

∂z(k

∂T

∂z)

(28)� Govik �

20 2 THEORYlim

∆t→0ρC

Tt+∆t − Tt

∆t= C

∂T

∂z(29)Eq. (26) then be omes

1

r

∂

∂r(kr

∂T

∂r) +

∂

∂z(k

∂T

∂z) + J2ρ = C

∂T

∂t(30)This se ond order partial di�erential equation is the governing equation of anaxisymmetri heat transfer problem where the heat transfer in the tangentialdire tion has been negle ted.Boundary onditions are needed to solve this problem. The boundary ondition between a body and the surrounding air is onve tion, it an beexpressed as

−k∂T

∂n= h(T − T∞) (31)where h is the onve tive heat transfer oe� ient, T∞ is the temperatureof the surrounding air and n is the normal dire tion to the surfa e. Theboundary ondition at the interfa e between the ele trode and the sheet isexpressed as.

−kcontact

∂T

∂z= helectrode(T − Telectrode) (32)

� Modelling of the Resistan e Spot Welding Pro ess �

2.5 Metallurgy 212.5 Metallurgy2.5.1 Basi sA modern steel often have several alloying materials other than iron and arbon, e.g. manganese, hromium, ni kel and tungsten [13℄. The alloyingelements ontribute with di�erent properties to the steel, su h as hardness,du tility or hardenability. The properties of the steel depend on how the al-loying elements a�e t the mole ules and the way they are positioned againstea h other. A onvenient way to des ribe this is by the mi rostru ture, whi his a mi ros opi des ription of the individual phases of the steel.A phase an be de�ned as any segment of a material, having the samestru ture or atomi arrangement, roughly the same omposition and a de�-nite interfa e between the phase and its surrounding.

Figure 9: Phase diagram [14℄There exists several di�erent phases of steels e.g. ferrite, austenite, perliteand martensite. They di�er from ea h other in rystal stru ture or hemi al omposition. The mi rostru ture depends on the temperature history andthe amount of alloying elements. It an be derived with the help of a phasediagram, see Fig. 9, whi h shows the phases and their omposition, anda Continuous Cooling Transformation (CCT) diagram, see Fig. 10, whi hrepresents phase hanges due to the rate of ooling.� Govik �

22 2 THEORY

Figure 10: CCT diagram with three di�erent ooling s hemes [14℄2.5.2 Advan ed High Strength SteelAdvan ed High Strength Steel (AHSS) is a relatively new lass of steels withseveral sub lasses e.g. transformation-indu ed plasti ity (TRIP), high holeexpansion (HHE) and dual-phase (DP). The DP steels have drawn the mostattention and the automotive industry has begun using them in some om-ponents. The popularity and the fairly simple mi rostru ture makes them agood example to explain more thorough.The attention given to DP steels is due to their good ombination of highstrength and du tility [15℄, i.e. they ombine the high strength of onven-tional high strength steel (HSS) with the good elongation properties of mildsteels.Dual-phase refers to that the mi rostru ture is onstituted of two phases,the body- entred- ubi (BCC) α-ferrite and the body- entred-tetragonal (BCT)martensite. The soft ferrite a t as the matrix with hard martensite disper-sions in it, see Fig. 11. This mi rostru ture is obtained by heating the steelto the temperature where austenite is formed, see Fig. 9. The steel is then ooled and held at a ertain temperature until the right amount of ferrite isformed and Thereafter the steel is quen hed. A high degree of under oolingprevents any di�usion to o ur and therefore indu es the remaining austeniteto transform to martensite. The strength of the steel depends on the amount� Modelling of the Resistan e Spot Welding Pro ess �

2.5 Metallurgy 23FerriteMartensiteFigure 11: Sket h of a ferrite-martensite DP mi rostru tureof martensite, i.e. a high degree of martensite gives a high strength but alsomakes it less du tile.The more �nely dispersed the martensite is the better its dynami energyabsorption be omes. This is be ause the ferrite-martensite perimeter, i.e.the length of the interfa e between ferrite and martensite in a unit area, islinearly proportional to the dynami absorbed energy [16℄. This is only trueat high strain rates, sin e at a low strain rates this dependen e is not seen.An unusual feature with DP steels is that the bake hardenability in reaseswith in reasing work hardening.2.5.3 Welding metallurgyDuring a resistan e spot welding operation the steel is subje ted to an ex-tensive heating and then rapid ooling. This auses a mi rostru tural devel-opment of the steel. The easiest way to explain this is with an example andthe DP steel from the previous se tion will be used.The DP steel onsists of a ferrite matrix with martensite dispersions. Asthe temperature rises the sheet steel starts to melt and form the weld nugget.At the same time the ferrite and martensite in the immediate surrounding ofthe melt are transforming into austenite.When the urrent eases a rapid ooling ommen es with a rate of oolingin the sheets rea hing as mu h as 10 000 ◦C/s. Thus all the melt and sur-rounding austenite transform to martensite. The stru ture of the martensitein the nugget will be oarse dendriti grains orientated in the solidi� ationdire tion, i.e. from the border to the enter, while the martensite in theheat a�e ted zone (HAZ) has a �ne grain size and no orientation [17℄. Theamount of martensite de reases with the distan e from the nugget boundary.Due to tempering the amount of martensite in the periphery of the HAZ anbe lower than in the base material.� Govik �

24 2 THEORY2.5.4 Modelling of mi rostru tureMi rostru tural development an be modelled using the thermal history, ma-terial omposition and mi rostru ture as input. During heating most modelsdes ribe the austenite formation and during ooling the austenite de ompo-sition into ferrite, pearlite, bainite and martensite [18℄.The output of these models usually is the phase fra tions and the hard-ness.Leblond [19℄ suggests that transformation from the base stru ture to austen-ite, when the e�e t of austenite grain size is negle ted, an be expressedasdp

dt=

peq(T ) − p

τ(T )(33)where p is the proportion of austenite, peq(T ) is the equilibrium proportionof austenite at temperature T and τ(T ) is the time ne essary to rea h theproportion p at onstant temperature T . The latter two parameters an bedetermined from a phase diagram and from knowing the start austenitizationand ompleted austenitization temperatures.When evaluating the austenite de omposition a suitable simpli� ation,in the resistan e spot welding ase, is that only the formation of marten-site needs to be al ulated. This is be ause the ooling rate in resistan espot welding is so high that no other stru tures are likely to be formed. Amodel by Koistenen and Marburger [20℄ express the austenite to martensitetransformation as

XM = 1 − e−α(Ms−T ) (34)where XM is the volume fra tion of martensite at temperature T, Ms is themartensiti start temperature and α is a onstant with a typi al value of0.011◦C−1 for most types of steel.When the phase fra tions are known the hardness of the steel an be esti-mated by using the rule of mixtures.H = HMXM + HBXB + HFPXFP (35)Where H is the total Vi kers hardness, HM , HB andHFP are the Vi kershardness of martensite, bainite and ferrite-perlite mixture respe tively and

XM , XB andXFP are the orresponding volume fra tions.� Modelling of the Resistan e Spot Welding Pro ess �

253 Previous modellingIn resistan e spot welding it is hard to monitor the a tual weld formationsin e it takes pla e between the metal sheets. Therefore many resear hershave tried to simulate this pro ess and in that way enhan e their understand-ing of the pro ess. However, when simulating this pro ess the intera tionbetween ele tri al, thermal, me hani al and metallurgi al phenomena needsto be a ounted for. A fully oupled model has not always been possible dueto the la k of software that an handle the four ways oupling.In the rest of this hapter a review of some of the developed models ispresented.Han et al. (1989) [21℄, ondu ted a heat transfer study of the resistan espot welding pro ess. An axisymmetri heat transfer model, with Eq. (30)as governing equation, was developed and solved with the �nite di�eren emethod. Temperature dependent properties for the metal sheets were used.The ele tri al onta t resistan e at the sheet/sheet interfa e was assumedto vary with the applied ele trode for e, whi h means that the onta t re-sistan e is onstant and evenly distributed during the pro ess. The onta tresistan e at the ele trode/sheet interfa e was assumed to be negligible.Temperature measurements were performed experimentally and the agree-ment between these heat urves and the simulated heat urves is good.Cho, Cho (1989) [22℄, developed a thermoele tri axisymmetri model, solvedwith the �nite di�eren e method, with governing equations in a ordan eto the ones dedu ed in Se tion 2.3.2 and 2.4.2. Temperature dependentproperties for the metal sheets, but not for the ele trodes, were used. Theele tri al and thermal onta t resistan e is taken to be a fun tion of thetemperature dependent hardness and an experimentally measured onta tresistan e at room temperature.The urrent density at the onta ting surfa es is evenly distributed andthus the interfa e heat generation is uniform in the radial dire tion.The model is validated by omparing the weld nugget diameters re eivedwith experimentally measured nuggets. The error of the model was 15-20%.Wei, Ho (1990) [23℄, developed an axisymmetri heat ondu tion model topredi t the nugget growth. The governing equation for heat ondu tion inthe ele trode is basi ally the same as Eq. (30), but the urrent density in theheat generation term has been adjusted to a ount for the onvergen e angleof the ele trodes. The governing equation for the metal sheets is modi�ed sothat it is expressed in terms of enthalpy and so that the mushy zone betweenthe pure solid and liquid is taken into a ount.� Govik �

26 3 PREVIOUS MODELLINGThe ele tri al onta t resistan e is assumed to de line linearly with thetemperature from an experimentally measured stati resistan e to zero resis-tan e when the melting temperature is rea hed.The material properties depend on the phase but not on the temperature.The re eived weld nugget thi kness and shape are ompared with experi-mental data produ ed by Gould [24℄. The agreement is very good, the erroris in general less than 10 %.Gupta, De (1998) [25℄, developed an axisymmetri , oupled thermal-ele tri al-me hani al model to simulate the resistan e spot welding pro ess. The gov-erning equation for the heat transfer analysis is the same as Eq. (30) andtemperature dependent material properties are used. In the ele tri al �eldanalysis the skin e�e t is in orporated, i.e. the urrent density in a ylindri- al ondu tor in reases from the interior towards the surfa e.Unlike the earlier mentioned models, the ele trode/sheet onta t area is al ulated and not pre-determined. This means that the onta t area willin rease as the temperature rises and this will a�e t the heat generation. Theway of in orporating the onta t resistan e in the model is not de�ned.The al ulated nugget diameters for di�erent welding parameters are om-pared with experimentally measured diameters. They agree very well and theerror is less than 10 %.Khan et al. (2000) [26℄, developed an axisymmetri , oupled thermal-ele tri al-me hani al model to predi t the nugget development during resistan e spotwelding of aluminium alloys.The heat transfer analysis a ounts for onve tive transport in the weldpool and thus the governing equations used are the ontinuity equation, themomentum equation and the energy equation.The ele tri al �eld analysis is done with Eq. (38).The model for the onta t resistan e is based on experimental values thatdepend on both temperature and pressure.Zhang (2003) [27℄, developed a ommer ial software, SORPAS, based on anaxisymmetri , oupled thermal-ele tri al-me hani al-metallurgi al model.The metallurgi al model is not des ribed in any detail but it al ulatesthe phase transformation, i.e. solid phase to liquid phase not mi rostru turalphases. The material properties depends on temperature. The thermal modelnegle ts the onve tion in the weld pool and use Eq. (30) as the governingequation of the heat transfer. The ele tri al model is based on the governingequation for the ele tri potential Eq. (38) and a ounts for the onta tresistan e with the use of the Wanheim and Bay model Eq. (3).The me hani al model al ulates the deformation and geometry of the� Modelling of the Resistan e Spot Welding Pro ess �

27materials, the stress and strain distribution and the onta t areas at theinterfa es.The veri� ation of the model is vague in the arti le, but a ording to theauthor it has been extensively veri�ed and gives almost identi al results asexperimental observations.Feulvar h et al. (2006) [28℄, developed a fully oupled thermal-ele tri al-me hani al-metallurgi al model to investigate the weld nugget growth.The thermal and ele tri al part is al ulated with Eq. (30), whi h is mod-i�ed to be expressed in terms of enthalpy instead of temperature, and Eq.(38), respe tively. The metallurgi al ontribution to the thermal analysis onsists of phase dependent thermal properties, that are assembled to be theproperties of the steel by a mixture rule. The mi rostru tural evolution is al- ulated with a model based on a Continuous Cooling Transformation (CCT)diagram. The metallurgi al oupling also a�e ts the me hani al al ulations.Plasti ity indu ed by metallurgi al transformations alters the plasti strainrate.The onta t resistan e used is experimentally measured as a fun tion oftemperature.The re eived weld nuggets and HAZ are ompared with experimentallyobtained welds and a really good agreement, a deviation of less than 10 %,is a hieved.The models above are summarized in Fig. (12), where they are he kedagainst the key features in modelling resistan e spot welding. As one anexpe t the models get more and more advan ed, but the results do not seemto improve at the same rate as the in rease in omplexity.

Figure 12: Summary of the presented models� Govik �

28 3 PREVIOUS MODELLING

� Modelling of the Resistan e Spot Welding Pro ess �

294 ModelIn this hapter two di�erent models will be presented on how to simulate theresistan e spot welding pro ess. The desired outputs from the model are thesize of the weld nugget and the heat a�e ted zone. The mi rostru ture ofthe weld nugget and the heat a�e ted zone is also of an interest, as well asthe residual stresses indu ed in the material. First a state of the art modelwill be presented, then a simpli�ed model is presented and motivated.4.1 State of the art modelIn this model as few simpli� ations as possible should be used, whi h meansthat a oupled thermal-ele tri al-me hani al-metallurgi al model a ordingto Fig. 13 must be used.11 12

23341 Material properties2 Temperature �eld3 Conta t ondition4 urrent �eld

Figure 13: Couplings in the modelThe me hani al model is used to evaluate the onta t area and the stressand strain state in the materials. It needs the material properties from themetallurgi al model and the temperature �eld from the thermal model asinput in order to determine the thermal strain.The thermal model is used to evaluate the heat transfer and the temper-ature �eld. It needs the material properties from the metallurgi al model,� Govik �

30 4 MODEL urrent density distribution from the ele tri al model and the deformed ge-ometry and onta t onditions from the me hani al model.The ele tri al model is used to evaluate the urrent density distributionand ele tri onta t resistivity. It needs the material properties from themetallurgi al model and the deformed geometry and onta t onditions fromthe me hani al model.The metallurgi al model is used to evaluate the mi rostru ture and thematerial properties based on the temperature �eld from the thermal model.4.1.1 Thermal modelA realisti thermal model of the resistan e spot welding pro ess must ne -essarily in lude a thorough heat transfer analysis that onsiders the on-ve tion in the weld pool as well as the thermal onta t onditions in theele trode/sheet interfa e. To in orporate onve tion in the weld pool in themodel, the governing equations for the sheets in the heat transfer analysismust in lude ontinuity, momentum and energy equations.The heat generation and heat transfer in the ele trodes are governed by1

r

∂

∂r(kr

∂T

∂r) +

∂

∂z(k

∂T

∂z) + J2ρ = C

∂T

∂t(36)whi h is identi al to eq. (30).The heat transfer in the ele trode/sheet interfa e depends on the ther-mal onta t ondu tivity. There is a lear analogy between the thermal onta t ondu tivity and the ele tri onta t resistivity des ribed in Se tion2.2. Whi h implies that the thermal onta t ondu tivity also depends onthe onta t pressure and temperature. This an be expressed by a modi�edWanheim and Bay's model [6℄ as

kcontact =1

3(σn

σY

)(k1 + k2

2) (37)Where σn is the onta t pressure, σY is the temperature dependent yieldstrength and k1 and k2 are the temperature dependent bulk ondu tivities.4.1.2 Ele tri al modelThe ele tri al urrent density distribution is needed to evaluate the heatgeneration in the resistan e spot welding pro ess. It an be formed by �rst� Modelling of the Resistan e Spot Welding Pro ess �

4.1 State of the art model 31solving the governing equation for the ele tri potential, as spe i�ed in Se -tion 2.3.2.1

r

∂

∂r

(

r

ρ

∂V

∂r

)

+∂

∂z

(

1

ρ

∂V

∂z

)

= 0 (38)Then the urrent density at all points an be evaluated asJ = −

1

ρ∇V (39)It is of the utmost importan e to have a orre t temperature dependen e ofthe bulk resistivity and an a urate model of the onta t resistan e, to get agood approximation of the heat generation of the system.There are two di�erent approa hes when modelling the onta t resistan e.Either the model an be based on a measured onta t resistan e at roomtemperature whi h is then made to be a fun tion of temperature, or a moregeneralized model like the ones presented in Se tion 2.2, Eq. (3)-(5) an beused. Sin e the Wanheim and Bay model [6℄ is used to al ulate the thermal onta t ondu tivity, it is onvenient to use it to al ulate the ele tri onta tresistivity as well. This model is also the one used in the ommer ial spotwelding simulation software SORPAS [27℄.

ρcontact =1

3

(

σf

σn

) (

ρ1 + ρ2

2

)

+ ρcontaminant (40)4.1.3 Metallurgi al modelThe material properties, and espe ially its temperature dependen y, have ahigh in�uen e on the a ura y of a resistan e spot welding model. Thereforeit is appropriate to use phase dependent properties. The mi rostru turalphases an be evaluated by solving the equations governing the de ompositionof austenite to ferrite, pearlite, bainite and martensite as is outlined in [20℄and then by using a mixture rule the ma ro properties of the metal an beevaluated. At the end of the pro ess the �nal mi rostru ture and hardnessof the weld nugget and the HAZ have been formed.4.1.4 Me hani al modelThe me hani al model al ulates the deformation and geometry of the ma-terials, the onta t area at the interfa e and the stress and strain �elds,with onsideration to both volume hanges due to phase transformation andtransformation plasti ity. � Govik �

32 4 MODEL4.2 Simpli�ed modelTo develop a detailed model like the one outlined in the previous se tionwould require extensive work and the resulting model would be omputa-tionally expensive. Therefore the aim for a simpli�ed model is to develop aless ostly model that still delivers reasonable results.A way to ut down on the omplexity of the model is to redu e the problemto a oupled thermal-me hani al problem. However, a ording to Ferro et al[29℄, and the results from Ranjbar Nodeh et al [30℄ supports this, the in�u-en e of phase transformations on residual stresses are onsiderable. Thereforea metallurgi al model is needed to evaluate the phase transformations, seeFig. (14). Su� ient results should be a hieved with these simpli� ations as an be realized when omparing to the models presented in Chapter 3.23 121 1 Material properties2 Temperature �eld3 Conta t onditionFigure 14: Couplings in the simpli�ed modelThe me hani al model is used to evaluate the onta t onditions and stressand strain states in the materials. The temperature �eld from the thermalmodel and the material properties from the metallurgi al model is neededfor this purpose.The thermal model is used to evaluate the heat transfer, the heat gener-ation and the temperature �eld. The material properties from the metallur-gi al model and the deformed geometry and the onta t onditions from theme hani al model is needed for this purpose.The metallurgi al model is used to evaluate the mi rostru ture and thematerial properties based on the temperature �eld from the thermal model.� Modelling of the Resistan e Spot Welding Pro ess �

4.2 Simpli�ed model 334.2.1 Thermal modelThe heat transfer both in the ele trodes and in the metal sheets an beapproximated to be pure ondu tion. Thus the velo ity in the weld pool isnegle ted. Instead the onve tion an be taken into a ount by an arti� ialin rease in the thermal ondu tivity. This approximation is very ommonin the presented models and the ontributing error from it is not signi� ant.Then the governing equation for the heat transfer is as spe i�ed in Se tion2.4.2.1

r

∂

∂r(kr

∂T

∂r) +

∂

∂z(k

∂T

∂z) + J2ρ = C

∂T

∂t(41)Sin e no ele tri al model exists, the urrent density distribution in the ele -trodes and sheets needs to be prede�ned so that the heat generation term an be evaluated. In the sheet/sheet and ele trode/sheet interfa es the ad-ditional heating due to the onta t resistan e an be taken into a ount bythe Wanheim and Bay model [6℄.

ρcontact =1

3

(

σf

σn

) (

ρ1 + ρ2

2

)

+ ρcontaminant (42)4.2.2 Me hani al modelThe me hani al model does not need to be simpli�ed sin e it is easily imple-mented in existing ommer ial software. It al ulates the deformation andgeometry of the materials, the onta t area at the interfa e and the stressand strain.4.2.3 Metallurgi al modelThe temperature together with the material omposition and the initial mi- rostru ture are input to the metallurgi al model. It then al ulates the mi- rostru ture a ording to Se tion 2.5.4, i.e. the formation of austenite duringheating is evaluated and then during ooling the formation of martensite isevaluated. The �nal mi rostru ture and hardness of the weld nugget and theHAZ is the result.� Govik �

34 4 MODEL

� Modelling of the Resistan e Spot Welding Pro ess �

355 Dis ussionOne of the obje tives with this work has been to present a simpli�ed modelof the resistan e spot welding pro ess suitable for implementation in theFEM software LS-DYNA. It is a di� ult thing to simplify the RSW pro esssin e it involves several interrelated physi al phenomena, in�uen ing di�erentaspe ts of the pro ess to a varying degree.A lot of work on modelling of the resistan e spot welding pro ess have beendone and only a small fra tion of the developed models is mentioned in thiswork. Most of the work have been fo used on the weld formation, i.e. theheat evolution in the sheets. There have not been as mu h work done onresidual stresses in the RSW pro ess.What is interesting to see is that already twenty years ago the RSW pro- ess ould be simulated with reasonable a ura y, in terms of the size of theweld nugget and the HAZ, even though only a heat transfer study was ar-ried out. This shows that even with a fairly simple model one an get thetemperature evolution during the weld with quite good a ura y.The onta t resistan e is the dominant ause of heat generation in theearly stage of the weld pro ess. It an be modelled a ording to two di�erentmethods, generalized or experimentally based. A third way an also be saidto exist among the very simple models where just a �xed value is assignedto the onta t resistan e. The experimentally based models an probably bethe most a urate but a big disadvantage is that experiments must be donefor every new material whereas the generalized models only need to havesome material properties de�ned.The de ision to leave the ele tri model out, i.e. not to al ulate the ele -tri potential �eld, in the simpli�ed model is mostly a question of simplifyingthe future implementation of the model. To use a prede�ned distribution ofthe urrent density in the model should be su� ient, as an be seen in thework by Wei and Ho [23℄.The residual stresses will be overestimated if the e�e ts of phase trans-formations are ignored. The reason for this is that Martensite has a higherspe i� volume than austenite and hen e the shrinkage during the oolingstage will be less severe. Phase transformation plasti ity redu e the stresslevels even further.One of the key fa tors when modelling the RSW pro ess is to have orre tmaterial properties sin e they have a big in�uen e on the a ura y of themodel. It is an absolute must that they vary with the temperature sin e thevariation an be onsiderable, e.g. The ele tri resistivity for a steel an befour times higher at the melting point than at room temperature.� Govik �

36 5 DISCUSSIONA weakness with this work is that no implementation or veri� ation of theproposed model has been done. Sin e this master thesis was done as anintrodu tion to a PhD-proje t the plan was to implement the simpli�edmodelinto LS-DYNA in the latter proje t. Even though no veri� ation is done onthe proposed model, the in orporated simpli� ations are in a ordan e toprevious models and should not present any major errors.5.1 Future workThe proposed model needs to be implemented into LS-DYNA and validatedto establish if it is a urate and e� ient enough, or if further simpli� ationsor any other alterations are needed.It would be interesting to implement and ompare the results from di�erent onta t resistan e models to experimental results, both the more generalizedmodels presented in Se tion 2.2 and the models based on experimental valuesthat many of the models reviewed in Se tion 3 use.

� Modelling of the Resistan e Spot Welding Pro ess �

REFERENCES 37Referen es[1℄ Andersson B., Swedenborg H., 1948, Svetsteknisk handbok. Bd 2, Sto k-holm[2℄ Phillips A. L., 1969, Welding handbook. 2, Welding pro esses: gas, ar and resistan e, New York : Ameri an welding so iety[3℄ Wei L., Cerjane D., Grzadzinski G. A., 2005, A Comparative Study ofSingle-Phase AC and Multiphase DC Resistan e Spot Welding, Journal ofManufa turing S ien e and Enginering 127, pp. 583-589[4℄ Young H. D., Freedman R. A., 2004, Sears and Zemansky's UniversityPhysi s: with Moder Physi s 11th Edition, San Fran is o: Pearson Edu- ation[5℄ Williams N. T., Parker J. D., 2004, Review of resistan e spot weldingof steel sheets Part 1: Modelling and ontrol of weld nugget formation,Internantional Materials Review, 49(2), pp. 45-75[6℄ Zhang H., Senkara J.. 2006, Resistan e Welding, Fundamentals and Ap-pli ations, Bo a Raton: Taylor & Fran is Group LLC[7℄ Di kinson D. W., Franklin J. E., Stanya A., 1980, Chara terization of SpotWelding Behavior by Dynami Ele tri al Parameter monitoring, WeldingJournal, 59(6), pp. 170-176[8℄ Serway R. A., Jewett J. W., 2005,Prin iples of Physi s: A al ulus-basedtext 4th edition, Cengage Learning[9℄ Çengel Y. A., 2003, Heat Transfer: A Pra ti al Approa h, Prin eton:M Graw-Hill Professional[10℄ Rohsenow M.,Hartnett J. P.,Cho Y. I., 1998, Handbook of Heat Transfer,3rd edition, New York: M Graw-Hill Professional[11℄ Wang, S. C., Wei, P. S., 2001, Modeling Dynami Ele tri al Resistan eDuring Resistan e Spot Welding, ASME J. Heat Transfer, 123, pp. 576-585[12℄ Wei P. S., Wang S. C., and Lin, M. S., 1996, Transport PhenomenaDuring Resistan e Spot Welding, ASME J. Heat Transfer, 118, pp. 762-773[13℄ Askeland D.R., 2001, The S ien e and Engineering of Material, Chel-tenham: Nelson Thornes Ltd� Govik �

38 REFERENCES[14℄ Sjöström S., 1982, The Cal ulation of Quen h Stresses in Steel,Linköping University, Linköping Studies in S ien e and Te hnology Dis-sertation No. 84[15℄ Marya M., Gayden X. Q., 2005, Development of Requirements for Re-sistan e Spot Welding Dual-Phase steels Part 1 - The Causes of Interfa ialFra ture, Welding Journal, 84(11), pp. 172-182[16℄ Senuma T., 2001, Physi al Metallurgy of Modern High Strength SteelSheets, SIJ International 41(6), pp. 520-532[17℄ Mimer M., 2002, Fra ture Me hanisms of Resistan e Spot Welded HighStrength Steel, Linköping University, Linköping, Mater thesis LiTH-IKP-Ex-1970.[18℄ Babu S. S., Riemer B. W., Santella M. L., Feng Z., 1998, Integratedthermal-mi rostru ture model to predi t the property gradients in resistan espot steel welds, Pro . Sheet Metal Welding Conferen e VIII, AWS DetroitSe tion.[19℄ Leblond J. B., Devaux J., 1984, A new kineti model for anisothermalmetallurgi al transformations in steels in luding e�e t of austenite grainsize, A ta metall, 32(1), pp 137-146[20℄ Goldak J. A., Akhlaghi, M., 2005, Computational Welding Me hani hs,New York: Springer[21℄ Han Z., Oroz o J., Inda o hea J. E., Chen C. H., 1989, Resistan e SpotWelding: A Heat Transfer Study, Welding Journal, 67(9), pp. 363-371[22℄ Cho H. S., Cho Y. J., 1989, A Study of the Thermal Behavior in Resis-tan e Spot Welding, Welding Journal, 67(6), pp. 236-244[23℄ Wei P. S., Ho C. Y., 1990, Axisymmetri Nugget Growth During Resis-tan e Spot Welding, ASME J. Heat Transfer, 112, pp. 309-316[24℄ Gould J. E., 1987, An Examination of Nugget Development During SpotWelding, Using Both Experimental and Analyti al Te hniques, WeldingJournal 66, pp 1-10[25℄ Gupta O. P., De A., 1998, An Improved Numeri al Modeling for Re-sistan e Spot Welding Pro ess and Its Experimental Veri� ation, J. Man-ufa turing S ien e and Engineering, Transa tion of the ASME, 120, pp.246-251 � Modelling of the Resistan e Spot Welding Pro ess �

REFERENCES 39[26℄ Khan J. A., Xu L., Chao Y. J., Broa h K. , 2000, Numeri al Simulationof Resistan e Spot Welding Pro ess, Numeri al Heat Transfer, Part A, 37,pp. 425-446[27℄ Zhang W, 2003, Design and Implementation of Software for Resistan eWelding Pro ess Simulations, SAE 2003 Transa tions: Journal of Materialsand Manufa turing, 112(5), pp. 556-564[28℄ Feulvar h E., Rogeon P., Carré P., Robin V., Sibilia G., Bergheau J.M., 2006, Resistan e Spot Welding Pro ess: Experimental and Numeri alModeling of the Weld Growth Me hanisms with Consideration of Conta tConditions, Numeri al Heat Transfer, Part A, 49, pp. 345-367[29℄ Ferro P., Porzner H., Tiziani A., Bonollo F., 2006, The in�uen e of phasetransformations on residual stresses indu ed by the welding pro ess - 3Dand 2D numeri al models, Modelling Simul. S i. Eng. 14, pp 117-136[30℄ Ranjbar Nodeh I., Serajzadeh S., Kokabi A. H., 2008, Simulation ofwelding residual stresses in resistan e spot welding, FE modeling and X-ray veri� ation, J. of materials pro essing te hnology 205, pp 60-69

� Govik �