Modelling of the Resistance Spot Welding Process

Modelling of the Resistan e Spot Welding Pro ess

Master Thesis arried out atDivision of Solid Me hani s

Linköpings University

April 2009

Alexander Govik

Publi eringsdatum

Publi ation date

2008-04-23

Division of SolidMe hani s

Dept. of Management andEngineering

SE-581 83LINKÖPING Språk Language Svenska/Swedish Engelska/English X Rapporttyp Report ategory Li entiatavhandling Examensarbete C-uppsats D-uppsats Övrigrapport X ISBN: ISRN:LIU-IEI-TEK-A--09/00609--SE Serietitel: Titleofseries Serienummer/ISSN: Numberof series

URLför elektronisk version

URLforele troni version

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:di va-17835

Titel

Title

Modellingof the Resistan eSpot Welding Pro ess

Författare

Author

Alexander Govik

Sammanfattning

Abstra t

Aliterature survey onmodellingofthe resistan espot welding pro ess hasbeen

arriedout andsome of the more interesting modelson this subje thave been

reviewedin thiswork. The underlying physi s hasbeen studied and a brief

explanation ofHeat transfer,ele trokineti s andmetallurgy in a resistan espot

welding ontext have been presented.

Lastlya state ofthe artmodelanda simpliedmodel, with implementation in the

Abstra t

A 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 been reviewed in this work. The underlying physi s has been studied

and abrief explanation of Heattransfer, ele trokineti sand metallurgyina

resistan e spotwelding ontext have been presented.

Lastlya state ofthe art model and asimpliedmodel, with

Prefa e

Theworkpresented hereistheresultofamasterthesisthathasbeen arried

out atthe Division of Solid Me hani s atLinköpingUniversity.

I would like to express my gratitude to my supervisor Prof. Larsgunnar

Nilsson forthe support andguidan eduringmyworkwiththemasterthesis.

A spe ial thanks should also be given to all Ph.D. students on the division

for all theirhelp.

Lastly,Iwouldliketotaketheopportunitytothankmyfamilyandfriends

for all theirsupport during the years.

Linköping, April2009

Contents

1 Introdu tion 1

1.1 Ba kground . . . 1

1.2 Obje tive . . . 2

2 Theory 3 2.1 Resistan e spot welding pro ess . . . 3

2.1.1 Method des ription . . . 3 2.1.2 Current . . . 5 2.1.3 Heat generation . . . 5 2.2 Dynami resistan e . . . 9 2.3 Ele trokineti . . . 13 2.3.1 Basi s . . . 13

2.3.2 Ele tri eld analysis . . . 13

2.4 HeatTransfer . . . 16

2.4.1 Heat transfer me hanisms . . . 16

2.4.2 Heat transfer inthe resistan e spotwelding pro ess . . 17

2.5 Metallurgy . . . 21

2.5.1 Basi s . . . 21

2.5.2 Advan ed HighStrength Steel . . . 22

2.5.3 Welding metallurgy . . . 23

2.5.4 Modellingof mi rostru ture . . . 24

3 Previous modelling 25 4 Model 29 4.1 State of the art model . . . 29

4.1.1 Thermalmodel . . . 30 4.1.2 Ele tri almodel . . . 30 4.1.3 Metallurgi almodel . . . 31 4.1.4 Me hani al model . . . 31 4.2 Simplied model . . . 32 4.2.1 Thermalmodel . . . 33 4.2.2 Me hani al model . . . 33 4.2.3 Metallurgi almodel . . . 33 5 Dis ussion 35 5.1 Future work . . . 36

List of Figures

1 The welding y le . . . 3

2 (a) Weld growth urve, (b) 2-D weld lobe . . . 4

3 Dierent impulse arrangements . . . 5

4 S hemati representation of the resistan e spot weldingset-up 6 5 Dierent ele trode geometries . . . 8

6 Deformation of asperities . . . 10

7 S hemati urve of dynami resistan e . . . 11

8 Cylindri al element . . . 18

9 Phase diagram [14℄ . . . 21

10 CCT diagram with three dierent ooling s hemes [14℄. . . 22

11 Sket h of a ferrite-martensite DP mi rostru ture . . . 23

12 Summary of the presented models . . . 27

13 Couplings in the model . . . 29

1 Introdu tion

The automotive industry fa es great hallenges. The in reasing demand for

environmentallyfriendly vehi les requires the manufa turers toin rease the

use of advan ed high-strength materials to redu e weight. This ombined

with shortedprodu tlife y les andmore advan eddesignsin rease the ost

perprodu ed vehi le.

Byshortening the developmenttime of the produ t money an be saved.

The most ee tive way to a omplish this is to in rease the use of Virtual

Manufa turing Engineering (VME) te hniques. There are however still a

numberofissuesthathavetobeover omebeforeVMEisfullyrealized. The

predi tion a ura yof theforming simulationmustforexamplebeimproved

and reliablejoiningsimulationsmustbeestablishedsu hthat the properties

of an assembled part an bepredi ted.

Withthisinmind the proje tSimuPARTswasinitiatedby SSF(Swedish

FoundationforStrategi Resear h),SwedishAutomotiveIndustriesandsheet

metal working industries in ollaboration with Universities to over ome the

stated problems.

This master thesis has been arried out as a rst step in one of the

sub-proje tsof SimuPARTs.

1.1 Ba kground

Professor Elihu Thomson is redited to be the inventor of the idea to join

materials with ele tri resistivity heating. In the 1880's he rst applied this

te 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 ar

there are thousands of spotwelds.

RSW onsistsoftwoopposedele trodesthatarepressedagainstthesheets

of metal that are to be joined. When an ele tri urrent ows from one

ele trodethrough thesheetsof metaltothe otherele trode it ausesheating

due tothe resistan einthe joiningsurfa es andinthe sheets. Dependingon

the thi kness and type of materialthe urrent that auses aweld nugget to

form may be in the range of 1,000 to 100,000 amperes or even more, while

1.2 Obje tive

The aimofthis workistostudy howtheresistan e spotweldingpro ess has

been and is being modelled. The underlying physi s of the pro ess should

also be studied in order to present a suggestion on how a state of the art

model and a simplied model of RSW an be made in the general nite

2 Theory

Theresistan espotweldingpro esswillbeexplainedinthis hapter. Firstan

overviewofthepro ess,thensomeofthemoreimportanttopi sareexplained

more thorough.

2.1 Resistan e spot welding pro ess

2.1.1 Method des ription

The welding y le, Fig.1, an bedivided into fourdenite stages [2℄:

1. Squeeze time - the time between the appli ation of pressure by the

ele trodes tothe rst appli ationof the weld urrent.

2. Weldtime - the time during whi h welding urrent isapplied.

3. Hold time -the time duringwhi h thepressure atthe pointofwelding

is maintainedby the ele trodes after the weld urrent eases.

4. O time-thetimeduringwhi htheele trodesare separatedtopermit

movingofmaterial. The termisgenerallyused whenthewelding y le

is repetitive.

Figure1: The welding y le

The welding y le is required to develop su ient heat to raise a onned

volumeof metaltoitsmeltedstate. Thea hieved temperaturemust behigh

enough so that fusion or in ipient fusion is obtained, but not so high that

expulsiono urs. Expulsionhappenswhenthehydrostati pressurefromthe

melt ex eeds the onta t pressure and the liquid metal is radially dispersed

denes the weld lobe,see Fig. 2, i.e. the range in whi h aweld is formed. Weld diameter (a) Weld time (b) Welding urrent Welding urrent A eptable weld Expulsion In omplete weld

Figure2: (a) Weld growth urve, (b) 2-Dweld lobe

Themetalisthen ooled,underpressure,toatemperatureatwhi htheweld

has enoughstrength to hold the parts together.

Proper fun tioning of the ele trodes is vital for the pro ess. They must

have good ele tri al and thermal ondu tivity to lessen the heat generated

at the area of onta t between the ele trode and the work surfa e. Good

ele tri al ondu tivitygiveslowerresistan e, and goodthermal ondu tivity

gives better dissipation of heat from the weld zone. The ele trodes must

also have good me hani alproperties to withstand high stresses atelevated

temperatures withoutex essive deformation. The upholding of proper

ele -trode shapeisimportantsin ethe urrentneedstobe ondu tedtothework

within a xed area. The urrent an be appliedin either single or multiple

impulses as an beseen inFig. 3.

•

Singleimpulse -Onesingle ontinuous urrentisapplied,withor 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 and

is then alled pulsation welding. The impulses an also be of dierent

Singleimpulsewithupanddownslope upslope downslope

Multipleimpulsewith

preheatandtemper

Figure3: Dierent impulse arrangements

2.1.2 Current

Resistan e spotwelding anbedonewithbothalternating urrent(AC) and

dire t urrent(DC)powersupply. ACspotweldinghasbeenthemostwidely

used inthe automotive industry whereas DC spot welding has been used in

the aerospa eindustry duetothehigh powerneeded forweldingaluminium.

TheDCweldingpro essrequireslesswelding urrentthanthe

orrespond-ing AC welding pro ess, but despite this the automotive industry has been

relu tant touse the DCwelding pro ess sin e the energysaving isun ertain

due to the energy losses when onverting AC to DC as well as the extra

equipment ost and reliability.

In re ent years a new power supply for DC spot welding, mid-frequen y

DC (MFDC) inverter, has be ome more popular. The reliability has been

improved and more importantly the welding urrent an be ontrolled with

mu h higher resolution giving the possibility of higher quality welds. In a

re ent study [3℄it has been shown that MFDC ismore energy e ient than

AC, espe ially atlower welding urrents.

2.1.3 Heat generation

The transformationfromele tri urrent toheat an bederived fromOhm's

energy.

Q

= I

2

Rt

(1)

Where Qisthe generatedheatenergy,I isthe ele tri urrent,Ris thetotal

resistan e of the work area and t isthe weld time.

The physi alexplanation tothis heatingisthat whenthe urrentis applied,

ele trons will start to ow trough the metal. The kineti energy of these

ele trons istransferred tothe metalby ollisionswith theions, whi hare

vi-brating aroundtheirequilibriumpositioninthe rystallinestru ture. These

ollisions ausesahighervibrationalenergyoftheionsandthusanin reased

temperatureof the metal[4℄.

Inuen eof urrent. -As an beseenineq. (1),the urrenthasthe greatest

ee tonthegenerationofheat. On ein ipientfusiontemperatureisrea hed

the weld nugget size and strength in rease qui kly with further in rease of

the 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 weldingset-up 1 2 3 4 5 6 7 Faying surfa e Weld nugget Water ooling I I

Inuen e of resistan e. - The total resistan e is the sum of the series of

resistan es in the work area, as seen in Fig. 4. When welding together two

sheets ofmetalthere areseven majorpointsofresistan e, denedas follows.

1. Upper ele trode

2. Conta t between upperele trode and uppersheet

3. Body of uppersheet

4. Conta t between upperand lowersheets

5. Body of lower sheet

6. Conta t between lower sheet and lower ele trode

7. Lowerele trode

It is the resistan e atpoint4 between the two sheets that is of most

impor-tan e for the weld initiationand itis also this resistan e that is the largest,

thus generating the most heat. The se ond highest resistan e is at point

2 and 6 where the onta t between the sheet and the ele trode raises the

temperature rapidly,but due to water ooling in the ele trode and its good

ele tri alandthermal ondu tivity thetemperatureissu iently lownot to

have insipient fusion.

The resistan e inall onta ting surfa es dependens on the pressure. The

higher pressure the lowerresistan e. The internal resistan e in the sheets is

what makes the weld nugget grow after the melting of the onta t surfa es.

The resistan e inthe ele trodes is low.

Inuen eoftime. -Therateofheatgenerationmustbesu ientinthegiven

time intervalsothat the welding willbea hieved withproper ompensation

forheatlosses. Theheatlossesare ausedby ondu tionintothesurrounding

metal and into the ele trodes as well as by radiation into the air. Shorter

weld time de reases ex essive heat transfer to the surrounding metal and

thus de reases the heat ae tedzone.

Inuen e of themetal sheets. -The surfa e ondition of the sheets inuen e

theheatgenerationatthe onta tbetweenthesheetandtheele trode. Ifthe

sheets havedirtor anoxidelmonthemthe onta t resistan e willin rease

and ause more heating. It also leads to deterioration of the ele trodes due

to ontamination. All this leads to unpredi table results.

The omposition of the metal determines spe i heat, melting point,

latentheatoffusion, thermal ondu tivity anddensity. Italsoinuen esthe

rangeintemperaturebetweenthepointofsofteningandthepointofmelting,

Heat balan e. - If two sheets of the same materialand thi kness are welded

together with ele trodes of equal size and shape there willbe a orre t heat

balan e, and theresultingweldnuggetwillbesymmetri with respe t tothe

plane of the onta tsurfa e. If, however, one ofthe sheets isthi keror have

a dierent ele tri al ondu tivity orif one of the ele trodes have a dierent

size orshapethere willbe aheat unbalan e, resultingina weld nugget that

is more developed in one of the sheets than in the other.

Shunt ee t. - When pla ing a new spot weld next toan old one, a part of

the urrent is going to ow through the old spot weld and thus less heat is

generated. This phenomenon is logi alsin e the ow of urrent always tries

to take the easiest path, i.e. the path with the least resistan e. The loser

the two spotwelds are the stronger this ee t be omes.

Domed fa ed Flat fa ed

Figure5: Dierent ele trode geometries

Inuen e of ele trode geometry. - One of the important fun tions of the

ele trode is to deliver a uniform urrent to the metal sheet, sin e a

non-uniform urrent ause uneven heatinginthe weld. A high ele trode tosheet

angle, see Fig 5, gives more uniform urrent.

The size of the ele trode tip is determined by the thi kness of the metal

sheetsandthedesiredweldsize, a ommonrulefordeterminingtheele trode

tip 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 optimum weld size is the same asthe 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 the

middle, whi h leads to uneven heating. The domed fa ed tip gives a more

2.2 Dynami resistan e

To ompletelyunderstandthe resistan espotweldingpro ess itisne essary

to know the role of the dierent resistan es between the welding ele trodes

and how they ae t the heat generation. It is generally on luded that the

initial onta t resistan e has a marked inuen e on the required magnitude

of welding urrent. But whendeterminingwhat happens duringthe welding

pro ess the dynami resistan e ismore important[5℄.

Thedynami resistan e is onstitutedby thebulkresistivityof thesheets

and of the onta t resistan e in the ele trode/sheet and sheet/sheet

inter-fa es.

Thebulkresistivity ofametal in reaseswith the temperature. This

tem-perature dependen e an be explained with quantum me hani s in the

fol-lowing way. The thermal vibrationsin a solid an be said to be a swarm of

mi ros opi parti les alled phonons. The free ele trons of a metal swarm

around in the ele tron loud in random dire tions, but with a general drift

velo ity in the dire tion of the ele tri urrent. The phonons ontinuously

ollide withele trons andthis arbitrarydee tion ors atteringofele trodes

ausesadisturban einthegeneraldriftoftheele trons. Asthetemperature

rises, i.e. the vibrationalenergy in reases, the number of phonons in reases

and with itthe probability of a ollisionbetween 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 tion

resistan e. The lm resistan e originates from the surfa e ondition of the

sheets. The existen e of anoxide layeror some other oating oreven grease

ordirtonthesurfa ewillae tthelmresistan e. Inadditiontothesurfa e

ondition the lmresistan e alsodepends on urrentlevel,temperatureand

onta t pressure. The lmresistan e is more pronoun ed in the early stage

of the welding pro ess.

The onstri tion resistan e between the two onta ting surfa es originates

from the existen e of asperities on the surfa es. That is, the true onta t

area is that of the onta t between asperities asseen inFig. 6, this onta t

area is only a small fra tion of the apparent area. The true onta t area

will in rease with the onta t pressure be ause of lo al plasti deformation

of the asperities. Sin e the pressure atthe onta ting surfa es isnot evenly

distributed, the amount of onstri tion resistan e willvary in the radial

di-re tion.

The onta t resistan e is also temperature dependent, as the

Figure 6: Deformation of asperities

in reases. When the temperature rea hes the melting point, the onta t

re-sistan e eases toexist sin e the onta tingsurfa es have vanished.

Todevelop a orre tgeneralizedmodelofthe onta tresistan eisdi ult

sin e it is sensitive to variations and depends on several dierent physi al

phenomena. To experimentally measure the variation of resistan e in the

onta t interfa e is impossible. Only an average value over the surfa e an

be measured. Despite this a numberof models have been developed, see [6℄

where the three followinggeneralized models are mentioned.

The Wanheim and Bay model [6℄ al ulates the onta t resistivity

ρ

by taking into a ount the plasti deformation of asperities to determine the

real onta t area.

ρ

contact

=

1

3

 σ

f

σ

n

  ρ

1

+ ρ

2

2



+ ρ

contaminant

(3)

Where

σ

n

is the onta t pressure,

σ

f

is the ow stress,

ρ

1

and

ρ

2

are the temperaturedependent bulkresistivities and

ρ

contaminant

is the resistivity of surfa e agents.

depends on the surfa e asperities inthe following way

R

contact

= (ρ

1

− ρ

2

)



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

isthe averagein-planedistan e between asperities. Thesequantities shouldallbefun tionsof temperatureand

pres-sure.

A model developed by Kohlraus h [6℄ an des ribe the ele tri al onta t

ondu tan e asa gap ondu tan e as

σ

g

=

1

2πr

2

c

pL(T

2

s

− T

2

0

)

(5)

where

T

s

isthemaximumtemperatureattheinterfa e,

T

0

isthebulk temper-ature and

L

is the Lorentz onstant. In the above formulation itis assumed that the onta t area isunderintimatemetalli onta t,this meansthat the

radius

r

c

must be dependent on the onta t pressure.

Figure7: S hemati urve of dynami resistan e Asperity softening Temperaturerise Initialmelting Beta peak Nugget growth me hani al ollapse Expulsion

The shape of the dynami resistan e urve, as seen in Fig. 7, has been well

do umented in orrelation to the weld formation,more detailed work by [7℄

follows:

1. Deformation ofsurfa e asperities leadingtolarger onta t area givesa

rapid drop in resistan e.

2. Final ollapse of asperities oupled with in reasing resistivity of the

metal due to bulk heating. When the in reasing resistivity

predom-inates over the ee t of larger onta t area, an in rease of the total

resistan e starts.

3. In reasing resistan e due tobulk heating.

4. Melting at the faying surfa e o urs. The molten zone grows rapidly.

Whenthein reaseinresistivityofthemetalduetobulkheatingbalan e

with thede rease inresistan e duetothe in reaseinsize ofthemolten

zone, a peak value, alled Betapeak,of the resistan e isrea hed.

5. Ade rease ofthe resistan eowingtothe indentationofthe sheetsthat

givesa shorter urrent path.

Phases 3and 4are thepredominantphasesinthe developmentof a

2.3 Ele trokineti

2.3.1 Basi s

Lets start by dening somefundamentalentities.

•

Current

I

- isthe net harge owing through an area perunit time.

•

Current density

J

- isthe urrentper ross-se tional area.

•

Resistivity

ρ

-isameasure ofhowstronglyamaterialopposes theow of urrent, ondu tivity is itsre ipro al.

•

Resistan e

R

-is the resistan e of urrentow ina ondu tor, dened by.

R

=

L

A

ρ

(6)

Condu tan e isits re ipro al.

•

Ele tri eld

E

- is the for e per unit harge exerted on a harged parti le.

•

Ele tri Potential

V

- isthe potentialenergy per unit harge.

2.3.2 Ele tri eld analysis

Tobeableto al ulatetheheat generationinpartssubje tedtospotwelding

one must know the urrent density distribution.

The urrent density

J

is, a ording to Ohm's law, proportional to the ele tri eld

E

.

J

₌

1

ρ

E

(7)

The ele tri eld in turn an be expressed as the negative gradient of the

ele tri potential

V

[8℄.

E

_{= −∇V}

(8)

Thus the urrent density interms of the ele tri potential be omes.

J

_{= −}

1

The ontinuity equation des ribes the onservative transport of harge. It

states that the net ow outof a volumeshould bebalan ed by anet hange

of hargeheld insidethe volume.

Z

S

J · ndS +

Z

V

∂ρ

∂t

dV

= 0

(10)

Where

n

is the outward pointing normal to the surfa e

S

and

ρ

stands in this equationfor the harge density.

The spot welding pro ess an be looked upon as a losed ir uit, so the

net hange of harge is equal to zero, whi h means that also the net ow

must be equal to zero.

Z

S

J · ndS = 0

(11)

It an be rewritten with the help of the divergen e theorem and by stating

that it is true for every volume.

∇ · J = 0

(12)

With Eq. (9) inEq. (12) the ontinuity equation an beexpressed as

∇ ·

1

_ρ

∇

_V

_{= 0}

(13)

whi h in ylindri al oordinates with symmetry aroundthe z-axis be omes.

1

r

∂

∂r

 r

ρ

∂V

∂r



+

∂

∂z

 1

ρ

∂V

∂z



= 0

(14)

Thisisthegoverningequationfortheele tri potentialinthemodel. Tosolve

this se ond orderdierentialequation boundary onditions are needed. Two

types of boundary onditions an be used, either the Neumann boundary

ondition whi h spe ies the values that the derivative of a solution is to

take, in this ase the urrent density, or the Diri hlet boundary ondition

whi hspe ies thevaluesasolutionneeds totake,inthis asethe 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 is

zero. This means that the Neumann boundary ondition should beused.

1

ρ

∂V

On the top surfa e of the ele trode there is a pres ribed urrent density so

the Neumannboundary ondition is used here to.

1

ρ

∂V

2.4 Heat Transfer

Theexisten eofatemperaturedieren eisthedrivingfor eforheattransfer,

i.e. a heat transfer between two bodies of the same temperature an not

exist. 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 and

mole ules.

Therstlawofthermodynami statesthatenergy anneitherbe reatednor

destroyed, i.e. it an only hange forms. For a stationary body, this an be

formulatedas: The net hangeinthe internalenergy ofthe bodyis equal to

the dieren ebetween theinternalenergyenteringthebodyandtheinternal

energy leavingthe body.

∆U = Q − W

(17)

Where

∆

Uis the hange ininternalenergy,

Q

isthe heatadded tothe body and

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 of

plasti deformation for instan e, the internal energy will in rease whi h in

turn will lead toa highertemperature.

On a mi ros opi level the internal energy is the sum of the kineti and

potential energies in the body. The kineti energy originates in the

trans-lational, rotational and vibrationalmovements of the atoms and mole ules,

while the potential energy isthe sum of the hemi al and nu lear energies.

More informationon heat transfer an be found in[9℄ and [10℄.

2.4.1 Heat transfer me hanisms

Heat an be transferred in three dierent modes: ondu tion, onve tion

and radiation. As mentionedearlierthe heattransferrequiresatemperature

gradientand itwillalwaysbeinthe dire tionfromthe bodywith thehigher

temperatureto the bodywith the lower temperature.

Conduction.

In ondu tion the transfer of energy between the higher en-ergeti parti les tothe nearby lowerenergeti parti lesis due to intera tion

between the parti les.

Inametalthedominantfa toristhe ollisionsbetweenfreeele tronsthat

transferkineti energyfromthefastermovingtotheslowermovingele trons.

The vibrations of the atoms also ontributes to the ondu tion, the

lowerenergeti atoms. The thermal ondu tivity of amaterial is a measure

of thematerialsabilityto ondu theat. Itisatemperaturedependent

prop-erty andthereasonforthis isthatthehigherthe temperature, thefaster the

free ele tronsmoveswhi hinturn leadstomore ollisionsand thusthe heat

transfer in reases.

Convection.

In onve tiontheenergyistransferredbetweenasolidsurfa e and auid inmotion. Itinvolvesthe ombinedee t of ondu tionanduid

in motion, where the uid motion a ts as an enhan er of the ondu tion.

The faster the uid motion,the greaterthe heat transfer. It is alled for ed

onve tioniftheuidmotionisfor edtoowoverthesurfa eofthe solidby

some external work. If the uid motion is aused by density gradients due

to temperature dieren es it is alled natural onve tion.

Radiation.

A more orre t name is thermal radiationso that it is learly distinguishedfromothertypesofradiationlikegammaraysandx-rayswhi h

are not relatedto temperature.

Radiation is a volumetri phenomenon and all bodies with a

tempera-ture above absolute zero emits and absorbs thermal radiation. If a body

has highertemperaturethanitssurroundingitemitsmorethermalradiation

than itabsorbs, and thusitstemperaturegets lowerbe ause ofthe lessening

of the internalenergy.

The reason for this radiation is that an atom an only hange its energy

level in dis rete steps and su h a step in energy level involves the emitting

or absorbing of ele tromagneti waves (photons).

Unlike ondu tionand onve tion,radiationrequiresnomediumfor

trans-portingthe energy.

2.4.2 Heat transfer in the resistan e spot welding pro ess

In resistan espotweldingthe dominantmode ofheattransfer is ondu tion.

There also exists onve tion and radiationon the surfa es of the sheets and

ele trodes, and inthe weld poolthere isa mass transportation,due to

mag-neti for es and density gradients,whi h auses onve tion [11℄. To a ount

forthis onve tiona ontinuityequation,momentumequationandanenergy

equation is needed, see [11℄, [12℄ and [26℄. However these equationswill not

be 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 the

main topi of this se tion.

tem-perature gradient over the element and the area normal to the dire tion of

heat transfer, but inversely proportional tothe elementthi kness.

This isexpressed inFourier's law of heat ondu tion.

˙

Q

_{= −kA}

∆T

∆x

(18)

The dierentialform isgiven by letting

∆

x

→

0

˙

Q

_{= −kA}

dT

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 sheet

plane. Hen e the heat ondu tion equation in ylindri al oordinates with

an internal heat generation needs to be dedu ed from the Fourier's law of

heat ondu tion.

Figure 8: Cylindri al element

r z

˙

Q

z

˙

Q

z+∆z

˙

Q

r

Q

˙

r+∆r

∆r

∆z

Consideraninnitesimalshortandthin ylindri alelementasinFig. 8,with

density

̺

, spe i heat

C

and volume

V

=2

π

r

∆

r

∆

z.

The energy balan e of this element during a short time interval

∆

t be- omes

∆E

element

(First law of thermodynami s), where

Q

˙

is the heat transfer rate and

G

˙

is the heat generationrate.

˙

G

= I

2

R

(21)

The hangeinenergyoftheelement anbeexpressedintermsofthemass

m

, the spe i heat

C

(theenergyrequiredtoraisethetemperatureofaspe i quantityofasubstan ebyaspe i amount)andthe hangeintemperature.

∆E

element

= E

t+∆t

− E

t

= mC(T

t+∆t

− T

t

) = ̺C2πr∆r∆z(T

t+∆t

− T

t

)

(22)

The heat generation rate an be expressed in terms of heat generation rate

pervolume

˙g

.

˙

G

element

= V

element

˙g = 2πr∆r∆z ˙g

(23)

where

˙g

an be expressed as

˙g = J

2

ρ

(24)

Nowwith Eq. (22) and (23) inEq. (20) the energy balan e an be written

ρC2πr∆r∆z

T

t+∆t

− T

t

∆t

= ˙

Q

r

− ˙

Q

r+∆r

+ ˙

Q

z

− ˙

Q

z+∆z

+ 2πr∆r∆z ˙g

(25) Dividing by 2

π

r

∆

r

∆

z gives

ρC

T

t+∆t

− T

t

∆t

= −

1

2πr∆z

˙

Q

r+∆r

− ˙

Q

r

∆r

−

1

2πr∆r

˙

Q

z+∆z

− ˙

Q

z

∆z

+ ˙g

(26)

Now, by using the denition of derivatives and Fourier's law of heat

on-du tion, the terms in Eq. (26) an be expressed in the dierential form by

letting

∆

r,

∆

z,

∆

t

→

0

lim

∆r→0

1

2πr∆z

˙

Q

r+∆r

− ˙

Q

r

∆r

=

1

2πr∆z

∂ ˙

Q

r

∂r

=

1

2πr∆z

∂

∂r

(−2πr∆z

∂T

∂r

) =

= −

1

_r

_∂r

∂

(kr

∂T

∂r

)

(27)

lim

∆z→0

1

2πr∆r

˙

Q

z+∆z

− ˙

Q

z

∆z

=

1

2πr∆r

∂ ˙

Q

z

∂z

=

1

2πr∆r

∂

∂z

(−2πr∆r

∂T

∂z

) =

= −

_∂z

∂

(k

∂T

∂z

)

(28)

lim

∆t→0

ρC

T

t+∆t

− T

t

∆t

= ̺C

∂T

∂z

(29)

Eq. (26) then be omes

1

r

∂

∂r

(kr

∂T

∂r

) +

∂

∂z

(k

∂T

∂z

) + J

2

_ρ

_{= ̺C}

∂T

∂t

(30)

Thisse ondorderpartialdierentialequationisthegoverningequationofan

axisymmetri heattransferproblemwheretheheattransferinthe tangential

dire 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 be

expressed as

−k

∂T

∂n

= h(T − T

∞

)

(31)

where

h

is the onve tive heat transfer oe ient,

T

∞

is the temperature of the surrounding air and

n

is the normal dire tion to the surfa e. The boundary ondition at the interfa e between the ele trode and the sheet is

expressed as.

−k

contact

∂T

2.5 Metallurgy

2.5.1 Basi s

A modern steel often have several alloying materials other than iron and

arbon, e.g. manganese, hromium, ni kel and tungsten [13℄. The alloying

elements ontribute with dierent properties to the steel, su h as hardness,

du tilityor hardenability. The properties ofthe steel depend onhow the

al-loyingelementsae t themole ules and theway they arepositionedagainst

ea hother. A onvenientwaytodes ribethisisbythemi rostru ture,whi h

is a mi ros opi des ription of the individual phases of the steel.

A phase an be dened as any segment of a material, having the same

stru ture or atomi arrangement, roughly the same omposition and a

de-nite interfa e between the phase and itssurrounding.

Figure9: Phase diagram [14℄

There exists several dierent phases of steels e.g. ferrite, austenite, perlite

and martensite. They dierfromea hother in rystal stru ture or hemi al

omposition. The mi rostru ture depends on the temperature history and

the amount of alloyingelements. It an be derived with the help of a phase

diagram, see Fig. 9, whi h shows the phases and their omposition, and

a Continuous Cooling Transformation (CCT) diagram, see Fig. 10, whi h

Figure10: CCT diagram with three dierent ooling s hemes [14℄

2.5.2 Advan ed High Strength Steel

Advan edHighStrength Steel(AHSS)is arelativelynew lassof steelswith

several sub lasses e.g. transformation-indu ed plasti ity (TRIP), high hole

expansion (HHE)and dual-phase(DP). The DPsteels havedrawn the most

attention and the automotive industry has begun using them in some

om-ponents. The popularity and the fairly simple mi rostru ture makes thema

goodexample to explainmore thorough.

Theattentiongiven toDPsteels isdue totheir good ombinationofhigh

strength and du tility [15℄, i.e. they ombine the high strength of

onven-tional high strength steel (HSS) withthe goodelongation propertiesof mild

steels.

Dual-phaserefers tothat the mi rostru tureis onstitutedof twophases,

thebody- entred- ubi (BCC)

α

-ferriteandthebody- entred-tetragonal(BCT) martensite. The soft ferrite a t as the matrix with hard martensite

disper-sions init, see Fig. 11. This mi rostru ture isobtained by heating the steel

to the temperaturewhere austenite is formed, see Fig. 9. The steel is then

ooled and held ata ertaintemperatureuntil the rightamountof ferrite is

formed and Thereafter thesteel isquen hed. Ahigh degreeof under ooling

preventsanydiusiontoo urandthereforeindu estheremainingaustenite

Ferrite

Martensite

Figure11: Sket h of a ferrite-martensite DP mi rostru ture

of martensite, i.e. ahigh degree of martensitegivesa high strength but also

makesit less du tile.

Themore nelydispersedthe martensite isthebetteritsdynami energy

absorption 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, is

linearly proportionaltothe dynami absorbed energy [16℄. This is onlytrue

at high strain rates, sin e ata lowstrain rates this dependen e isnot seen.

AnunusualfeaturewithDPsteelsisthatthebakehardenabilityin reases

with in reasing work hardening.

2.5.3 Welding metallurgy

During a resistan e spot welding operation the steel is subje ted to an

ex-tensive heatingand then rapid ooling. This ausesa mi rostru tural

devel-opmentof the steel. The easiest way toexplain this is with anexample and

the DP steel fromthe previousse tion willbe used.

The DP steel onsistsof a ferrite matrix withmartensite dispersions. As

the temperaturerisesthe sheetsteelstartstomeltandformtheweldnugget.

Atthe sametimethe ferriteandmartensite intheimmediatesurroundingof

the melt are transforming intoaustenite.

Whenthe urrent eases arapid ooling ommen eswitharateof ooling

in the sheets rea hing as mu h as 10 000

◦

C/s. Thus all the melt and

sur-rounding austenite transformto martensite. Thestru ture of themartensite

in the nugget willbe oarse dendriti grains orientated in the solidi ation

dire tion, i.e. from the border to the enter, while the martensite in the

heat ae ted zone (HAZ) has a ne grain size and no orientation [17℄. The

amountof martensitede reaseswiththe distan efromthenuggetboundary.

Due totemperingthe amountofmartensiteinthe peripheryofthe HAZ an

2.5.4 Modelling of mi rostru ture

Mi rostru turaldevelopment anbemodelledusingthethermalhistory,

ma-terial ompositionandmi rostru tureasinput. Duringheatingmostmodels

des ribethe austenite formationand during ooling the austenite

de ompo-sition into ferrite, pearlite, bainiteand martensite [18℄.

The output of these models usually is the phase fra tions and the

hard-ness.

Leblond[19℄ suggeststhat transformationfromthe basestru ture to

austen-ite, when the ee t of austenite grain size is negle ted, an be expressed

as

dp

dt

=

p

eq

(T ) − p

τ(T )

(33)

where

p

is the proportion of austenite,

p

eq

(T )

is the equilibrium proportion of austenite at temperature

T

and

τ

(T )

is the time ne essary to rea h the proportion

p

at onstant temperature

T

. The latter two parameters an be determinedfromaphasediagramandfromknowingthestartaustenitization

and ompletedaustenitizationtemperatures.

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 e

spot welding is so high that no other stru tures are likely to be formed. A

model by Koistenen and Marburger [20℄ express the austenite tomartensite

transformation as

X

M

= 1 − e

−

α(M

s

−

T

)

(34)

where

X

M

is the volumefra tion of martensite attemperatureT,

M

s

is the martensiti start temperature and

α

is a onstant with a typi al value of 0.011

◦

C

−

1

for most typesof steel.

When the phase fra tions are known the hardness of the steel an be

esti-mated by using the rule of mixtures.

H

= H

M

X

M

+ H

B

X

B

+ H

F P

X

F P

(35) Where

H

is the total Vi kers hardness,

H

M

, H

B

and

H

F P

are the Vi kers hardness of martensite, bainite and ferrite-perlite mixture respe tively and

3 Previous modelling

In resistan e spot welding it is hard to monitor the a tual weld formation

sin e it takes pla e between the metal sheets. Therefore many resear hers

havetriedtosimulatethispro essandinthatwayenhan etheir

understand-ing of the pro ess. However, when simulating this pro ess the intera tion

between ele tri al, thermal,me hani al and metallurgi alphenomena needs

tobea ountedfor. Afully oupled model hasnot always been possibledue

to 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 is

presented.

Han et al. (1989) [21℄, ondu ted a heat transfer study of the resistan e

spot welding pro ess. An axisymmetri heat transfer model, with Eq. (30)

as governing equation, was developed and solved with the nite dieren e

method. Temperature dependent properties for the metal sheets were used.

The ele tri al onta t resistan e at the sheet/sheet interfa e was assumed

to 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 t

resistan e at the ele trode/sheet interfa e was assumed tobenegligible.

Temperaturemeasurementswereperformedexperimentallyandthe

agree-ment between these heat urvesand the simulatedheat urves is good.

Cho,Cho(1989)[22℄,developedathermoele tri axisymmetri model,solved

with the nite dieren e method, with governing equations in a ordan e

to the ones dedu ed in Se tion 2.3.2 and 2.4.2. Temperature dependent

properties for the metal sheets, but not for the ele trodes, were used. The

ele tri al and thermal onta t resistan e is taken to be a fun tion of the

temperature dependent hardness and an experimentally measured onta t

resistan e at roomtemperature.

The urrent density at the onta ting surfa es is evenly distributed and

thus the interfa eheat generation is uniform inthe radial dire tion.

The model is validatedby omparing the weld nugget diameters re eived

with experimentallymeasured nuggets. The error of the model was 15-20%.

Wei, Ho (1990) [23℄, developed an axisymmetri heat ondu tion model to

predi t the nugget growth. The governing equation for heat ondu tion in

the ele trodeisbasi allythesame asEq. (30),butthe urrentdensityinthe

heat generationtermhas beenadjustedtoa ountforthe onvergen e angle

of the ele trodes. Thegoverningequationfor themetal sheetsismodied so

that itisexpressed intermsofenthalpyandsothatthe mushyzonebetween

The ele tri al onta t resistan e is assumed to de line linearly with the

temperaturefromanexperimentallymeasuredstati resistan e tozero

resis-tan e when the melting temperature isrea hed.

Thematerialpropertiesdependonthephasebut notonthetemperature.

The re eived weld nuggetthi kness and shape are ompared with

experi-mentaldata produ ed by Gould [24℄. The agreement is very good,the error

is in generalless than 10 %.

Gupta,De(1998)[25℄,developedanaxisymmetri , oupled

thermal-ele tri al-me hani al model tosimulate the resistan e spotwelding pro ess. The

gov-erning equation for the heat transfer analysis is the same as Eq. (30) and

temperature dependent material properties are used. In the ele tri al eld

analysis the skin ee t isin orporated, i.e. the urrentdensity ina

ylindri- al ondu tor in reases fromthe 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 will

in reaseasthetemperaturerisesandthiswillae ttheheatgeneration. The

way of in orporating the onta t resistan e in the model is not dened.

The al ulatednuggetdiametersfordierentweldingparametersare

om-paredwithexperimentallymeasureddiameters. Theyagreeverywellandthe

error is less than 10 %.

Khanetal. (2000)[26℄,developedanaxisymmetri , oupled

thermal-ele tri al-me hani al model to predi tthe nugget development during resistan e spot

welding of aluminiumalloys.

The heat transfer analysis a ounts for onve tive transport in the weld

pooland thus the governing equationsused are the ontinuity equation, the

momentum equationand the energy equation.

The ele tri aleld analysis is done with Eq. (38).

Themodel forthe onta t resistan eisbased onexperimentalvaluesthat

depend on both temperatureand pressure.

Zhang (2003) [27℄, developed a ommer ial software, SORPAS,based on an

axisymmetri , oupled thermal-ele tri al-me hani al-metallurgi almodel.

The metallurgi al model is not des ribed in any detail but it al ulates

the phasetransformation,i.e. solidphasetoliquidphasenotmi rostru tural

phases. Thematerialpropertiesdependsontemperature. Thethermalmodel

negle ts the onve tion in the weld pool and use Eq. (30) as the governing

equationof theheat transfer. The ele tri almodel isbasedonthe governing

equation for the ele tri potential Eq. (38) and a ounts for the onta t

resistan e with the use of the Wanheimand Bay model Eq. (3).

materials, the stress and strain distribution and the onta t areas at the

interfa es.

The veri ation of the model is vague inthe arti le, but a ording tothe

author it has been extensively veried and gives almost identi al results as

experimentalobservations.

Feulvar h et al. (2006) [28℄, developed a fully oupled

thermal-ele tri al-me hani al-metallurgi almodel toinvestigate the weld nugget growth.

Thethermalandele tri alpart is al ulatedwithEq. (30),whi his

mod-ied to be expressed in terms of enthalpy instead of temperature, and Eq.

(38), respe tively. The metallurgi al ontribution to the thermal analysis

onsistsof phasedependentthermalproperties,thatare assembledtobethe

propertiesofthesteelbyamixturerule. Themi rostru turalevolutionis

al- ulatedwitha modelbasedonaContinuousCoolingTransformation (CCT)

diagram. Themetallurgi al ouplingalsoae tstheme hani al al ulations.

Plasti ity indu ed by metallurgi al transformations alters the plasti strain

rate.

The onta t resistan e used is experimentally measured as a fun tion of

temperature.

The re eived weld nuggets and HAZ are ompared with experimentally

obtained welds and a reallygood agreement, a deviation of less than 10 %,

is a hieved.

The models above are summarized in Fig. (12), where they are he ked

against the key features in modelling resistan e spot welding. As one an

expe tthe models get moreand moreadvan ed,but the resultsdonot seem

to improve atthe same rate as the in rease in omplexity.

4 Model

In this haptertwodierentmodelswillbepresented onhowtosimulatethe

resistan e spotweldingpro ess. The desiredoutputs fromthemodelare the

size of the weld nugget and the heat ae ted zone. The mi rostru ture of

the weld nugget and the heat ae ted zone is also of an interest, as well as

the residual stresses indu ed in the material. First a state of the art model

will bepresented, then a simpliedmodel is presented and motivated.

4.1 State of the art model

In this model asfewsimpli ationsaspossibleshouldbeused,whi hmeans

that a oupled thermal-ele tri al-me hani al-metallurgi almodel a ording

to Fig. 13must be used.

1 1 1 2 2 3 3 4 1 Material properties 2 Temperatureeld 3 Conta t ondition 4 urrent eld

Figure13: Couplings in the model

The me hani al model is used to evaluate the onta t area and the stress

and strain state in the materials. It needs the material properties from the

metallurgi al model and the temperature eld from the thermal model as

input in order todetermine the thermal strain.

The thermalmodel is used to evaluate the heat transfer and the

urrent density distribution from the ele tri al model and the deformed

ge-ometry and onta t onditions from the me hani almodel.

The ele tri al model is used to evaluate the urrent density distribution

and ele tri onta t resistivity. It needs the material properties from the

metallurgi almodelandthe deformedgeometryand onta t onditions from

the me hani al model.

The metallurgi al model is used to evaluate the mi rostru ture and the

material properties based onthe temperature eld fromthe thermal model.

4.1.1 Thermal model

A 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 the

ele trode/sheet interfa e. To in orporate onve tion in the weld poolin the

model, the governing equations for the sheets in the heat transfer analysis

must in lude ontinuity, momentum and energy equations.

The heat generation and heat transfer inthe ele trodes are governed by

1

r

∂

∂r

(kr

∂T

∂r

) +

∂

∂z

(k

∂T

∂z

) + J

2

_ρ

_{= ̺C}

∂T

∂t

(36)

whi h isidenti al toeq. (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 tion

2.2. Whi h implies that the thermal onta t ondu tivity also depends on

the onta t pressure and temperature. This an beexpressed by amodied

Wanheimand Bay's model [6℄ as

k

contact

=

1

3

(

σ

n

σ

Y

)(

k

1

+ k

2

2

)

(37)

Where

σ

n

is the onta t pressure,

σ

Y

is the temperature dependent yield strength and

k

1

and

k

2

are the temperature dependent bulk ondu tivities.

4.1.2 Ele tri al model

The ele tri al urrent density distribution is needed to evaluate the heat

solving the governing equation forthe ele tri potential, asspe iedin Se -tion 2.3.2.

1

r

∂

∂r

 r

ρ

∂V

∂r



+

∂

∂z

 1

ρ

∂V

∂z



= 0

(38)

Then the urrentdensity atall points an be evaluatedas

J

_{= −}

1

ρ

∇

V

(39)

It is of the utmostimportan eto have a orre t temperature dependen e of

the bulkresistivity andana urate model of the onta t resistan e, togeta

goodapproximationof the heat generationof the system.

Thereare twodierentapproa heswhenmodellingthe onta tresistan e.

Either the model an be based on a measured onta t resistan e at room

temperaturewhi h isthen made to be afun tion of temperature, ora more

generalized model likethe ones presented in Se tion 2.2, Eq. (3)-(5) an be

used. Sin ethe Wanheimand Baymodel[6℄isused to al ulate the thermal

onta t ondu tivity,itis onvenienttouseitto al ulatetheele tri onta t

resistivity as well. This model is also the one used in the ommer ial spot

welding simulation software SORPAS [27℄.

ρ

contact

=

1

3

 σ

f

σ

n

  ρ

1

+ ρ

2

2



+ ρ

contaminant

(40) 4.1.3 Metallurgi al model

The material properties, and espe ially its temperature dependen y, have a

high inuen eonthe a ura yofaresistan e spotweldingmodel. Therefore

it is appropriate to use phase dependent properties. The mi rostru tural

phases anbeevaluatedbysolvingtheequationsgoverningthede omposition

of austenite toferrite, pearlite, bainite and martensite as is outlined in [20℄

and then by using a mixture rule the ma ro properties of the metal an be

evaluated. At the end of the pro ess the nal mi rostru ture and hardness

of the weld nugget and the HAZ have been formed.

4.1.4 Me hani al model

The 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 onsiderationtoboth volume hanges duetophase transformationand

4.2 Simplied model

To develop a detailed model like the one outlined in the previous se tion

would require extensive work and the resulting model would be

omputa-tionally expensive. Therefore the aim for a simpliedmodel is to develop a

less ostly model that still delivers reasonable results.

Awayto utdownonthe omplexityofthemodelistoredu etheproblem

toa oupled thermal-me hani alproblem. However, a ording toFerro etal

[29℄, and the results from Ranjbar Nodeh et al [30℄ supports this, the

inu-en eofphasetransformationsonresidualstressesare onsiderable. Therefore

a metallurgi al model is needed to evaluate the phase transformations, see

Fig. (14). Su ient results should be a hieved with these simpli ations as

an be realizedwhen omparing tothe models presented inChapter 3.

2 3 1 2 1 1 Materialproperties 2 Temperatureeld 3 Conta t ondition

Figure 14: Couplings in the simplied model

The me hani al model is used to evaluate the onta t onditions and stress

and strain states in the materials. The temperature eld from the thermal

model and the material properties from the metallurgi al model is needed

for this purpose.

The thermal model is used toevaluate the heat transfer, the heat

gener-ation and the temperatureeld. The materialproperties fromthe

metallur-gi al model and thedeformed geometryand the onta t onditions fromthe

me hani al model isneeded for this purpose.

The metallurgi al model is used to evaluate the mi rostru ture and the

4.2.1 Thermal model

The heat transfer both in the ele trodes and in the metal sheets an be

approximated to be pure ondu tion. Thus the velo ity in the weld pool is

negle ted. Instead the onve tion an be taken intoa ount by an arti ial

in rease in the thermal ondu tivity. This approximation is very ommon

in the presented models and the ontributingerror fromitis not signi ant.

Then the governing equation for the heat transfer is as spe ied in Se tion

2.4.2.

1

r

∂

∂r

(kr

∂T

∂r

) +

∂

∂z

(k

∂T

∂z

) + J

2

_ρ

_{= ̺C}

∂T

∂t

(41)

Sin e no ele tri almodel exists, the urrent density distributionin the

ele -trodes and sheets needs to be predened 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 by

the Wanheim and Bay model [6℄.

ρ

contact

=

1

3

 σ

f

σ

n

  ρ

1

+ ρ

2

2



+ ρ

contaminant

(42) 4.2.2 Me hani al model

The me hani almodel doesnotneed tobesimpliedsin eitiseasily

imple-mented in existing ommer ial software. It al ulates the deformation and

geometry of the materials, the onta t area at the interfa e and the stress

and strain.

4.2.3 Metallurgi al model

The temperaturetogether with the material omposition and the initial

mi- rostru ture are inputto the metallurgi almodel. Itthen al ulates the

mi- rostru turea ordingtoSe tion2.5.4,i.e. theformationofaustenite during

heating is evaluated and then during ooling the formation of martensite is

evaluated. Thenalmi rostru tureandhardness ofthe weldnuggetand the

5 Dis ussion

One of the obje tives with this work has been to present a simpliedmodel

of the resistan e spot welding pro ess suitable for implementation in the

FEM software LS-DYNA. It isadi ultthing tosimplifythe RSWpro ess

sin eitinvolvesseveralinterrelatedphysi alphenomena,inuen ingdierent

aspe ts of the pro ess toa varying degree.

A lot of work onmodellingof the resistan e spotwelding pro ess have been

done and onlya small fra tion ofthe developed models is mentionedin this

work. Most of the work have been fo used on the weld formation, i.e. the

heat evolution in the sheets. There have not been as mu h work done on

residual stresses inthe RSW pro ess.

What isinteresting tosee isthat already twenty years ago the RSW

pro- ess ould besimulatedwith reasonable a ura y, interms of the size of the

weld 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 the

temperatureevolutionduring the weld with quitegooda ura y.

The onta t resistan e is the dominant ause of heat generation in the

earlystageoftheweld pro ess. It anbemodelleda ordingtotwodierent

methods, generalized orexperimentallybased. A third way an alsobe said

to exist among the very simple models where just a xed value is assigned

tothe onta t resistan e. The experimentallybased models an probablybe

the most a urate but abig disadvantage is that experiments must be done

for every new material whereas the generalized models only need to have

some material properties dened.

Thede isiontoleavethe ele tri modelout, i.e. not to al ulatethe

ele -tri potentialeld,inthe simpliedmodelismostly aquestionofsimplifying

the future implementationof the model. Touse apredened distribution of

the urrent density in the model should be su ient, as an be seen in the

work by Wei and Ho [23℄.

The residual stresses will be overestimated if the ee ts of phase

trans-formations are ignored. The reason for this is that Martensite has a higher

spe i volume than austenite and hen e the shrinkage during the ooling

stage will be less severe. Phase transformation plasti ity redu e the stress

levelseven further.

Oneofthekeyfa torswhenmodellingtheRSWpro ess istohave orre t

material properties sin e they have a big inuen e on the a ura y of the

model. It isanabsolutemust thatthey varywith the temperaturesin e the

variation an be onsiderable, e.g. The ele tri resistivity for a steel an be

A weakness with this work is that no implementation or veri ation of the

proposed model has been done. Sin e this master thesis was done as an

introdu tiontoaPhD-proje ttheplanwastoimplementthesimpliedmodel

intoLS-DYNA inthe latterproje t. Even though noveri ation is done on

the proposed model, the in orporated simpli ations are in a ordan e to

previous models and should not present any major errors.

5.1 Future work

The proposed model needs tobe implementedintoLS-DYNA andvalidated

to establishif itisa urate and e ient enough, orif further simpli ations

or any other alterations are needed.

It would beinteresting toimplementand ompare the resultsfrom dierent

onta t resistan emodels toexperimentalresults,both themoregeneralized

modelspresented inSe tion2.2andthemodelsbasedonexperimentalvalues

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