1 - 20
Lase Beam Welding analy ical model when using wobble
s a egy
Au ho s: I. He nando, J.I. A izubie a, A. Lamikiz, E. Uka .
A ilia ion: Dep . o Mechanical Enginee ing, Uni e si y o he Basque Coun y (UPV/EHU),
To es Que edo 1, 48013 Bilbao, Spain.
Highligh s
An analy ical model o LBW ha conside s he wobble s a egy is de eloped.
Two-s ep model is p esen ed, combining Ca slaw-Jaege ´s and Rosen hal´s models.
An e o below 0.05 mm in wid hs and 0.3 mm in dep h is ob ained.
Mol en ma e ial mo emen is simula ed using monopole, dipole and quad upole
models.
The model p edic s he SDAS wi h a 1 μm e o .
Su ace cooling a e in he weld bead is calcula ed wi h an e o below 10%.
Abs ac
This a icle p esen s a model o es ima ing he he mal g adien , bead geome y and
mic os uc u e in he lase welding p ocess, when he Wobble s a egy is used. This me hod
combines he main eed mo ion wi h a seconda y high equency o bi al mo ion o he lase beam
in oduced by a gal anome e . The model is de eloped om an analy ical app oach and i is
pa icula ised o he case o he Wobble s a egy h ough he implemen a ion o wo co ec i e
ac o s. To his end, a wo-s ep analy ical model is p esen ed. Fi s , om Ca slaw-Jaege 's heo y,
he he mal ield o he uppe ace o he pla es is modelled, allowing he wid h o he gene a ed
weld bead o be de e mined. The de eloped model includes he e ec o he Wobble s a egy as
well as he ini ial ansien egime. In a second s ep, he in e nal mo emen o he mol en
ma e ial wi hin he mel -pool is modelled by means o he concep s o monopoles, dipoles and
quad upoles. Finally, he mic os uc u e calcula ion is also implemen ed based on he p e iously
es ima ed he mal g adien .
The model has been expe imen ally alida ed in Inconel 718 Nickel based alloy pla es welding,
using di e en p ocess pa ame e s and measu ing he esul ing bead sec ion and mic os uc u e.
E o s below 0.05 mm and 0.3 mm a e ob ained ega ding he bead wid h and dep h,
espec i ely, and di e ences below 10% a e ob ained be ween he es ima ed cooling a e by he
model and expe imen al measu emen s. Finally, he es ima ed alues o he Seconda y Dend i e
A m Spacing pa ame e s a e below 1 μm o e o in all es ed cases.
Keywo ds: Lase beam welding, wobble, analy ical, model, SDAS
1. In oduc ion
In a eas o high le el o excellence such as ae onau ics, i is necessa y o join componen s in a
eliable way wi h minimum addi ion o weigh . In uselage pa s, i e ing has been he mos
equen echnique o joining componen s om ini ial ai c a designs wi h aluminium skin
panels. Despi e he addi ion o he ex a-weigh o he i e o he s uc u e, his is s ill in use
because he e is no mic os uc u e change o he ma e ial. Besides, i o e s he damping
necessa y o a oid s ess concen a ions when he uselage is loaded and de o med du ing
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in In e na ional Jou nal o Hea and Mass T ans e 149 : (2020) 119248,
which has been published in inal o m a h ps://doi.o g/10.1016/j.ijhea mass ans e .2019.119248.
© 2020. This manusc ip e sion is made a ailable unde he CC-BY-NC-ND 4.0 license h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
2 - 20
ligh . In o he c i ical a eas, such as mo o s, high quali y welding is used o join di e en pa s
and educe he weigh o he en i e s uc u e. Mechanical p ope ies o he mel ed a ea and
hea -a ec ed zone (HAZ) mus be app op ia e and his is only possible wi h echnologies ha
p o ide homogeneous hea ing o he welding a ea. The elec on beam welding (EBW) has been
highly success ul since he 70s. In his p ocess, magne ic lenses guide accele a ed ee elec ons
owa ds he pa in a acuum chambe . When elec ons impac on he su ace, hey gene a e
non-elas ic collisions allowing a as and con olled beam pene a ion, and powe densi ies up
o 1012 W·cm-2 a e eached [1]. Despi e he high quali y o EBW, i s applica ions a e limi ed
because o he cos o he equipmen and he complex elec onic equi emen s o con ol he
wo kpiece manipula o . The e o e, lase beam welding (LBW) has a isen as an al e na i e o his
echnology.
LBW was i s de eloped in he 70s as one o he i s applica ions o he lase echnology o
high quali y welding. Nowadays powe densi ies o e 109 W·cm-2 a e possible esul ing in small
hea -a ec ed zones in combina ion wi h high hea ing and cooling a es. Wi h ypical spo size
be ween 0.2 and 13 mm and dep h pene a ion p opo ional o he amoun o powe supplied,
LBW is a e sa ile p ocess capable o joining a ious ma e ials in sec o s like ae onau ics [2].
Ne e heless, in c i ical applica ions, ac o s such as he ex ension o he HAZ and he
unce ain y ega ding he mel pool dynamics can a y he inal p ope ies o he joined egion,
so an accu a e model o he welding p ocess is con enien .
Indus ial lase sou ces p o ide a known powe densi y ha can be con olled accu a ely, hus,
he ene gy abso bed by he ma e ial, hea ans e by means o conduc ion and e ec s p oduced
du ing hea ing and la e cooling can be p edic ed using ma hema ics [3]. Howe e , he
simula ion o welding p ocesses is di icul due o he simul aneous appea ance o he mal,
mechanical and me allu gical phenomena [4].
The i s analy ical models we e de eloped in he 1940s o s udy he he mal ield in di e en
pa geome ies i adia ed by a hea sou ce [5], [6], and Rosen hal de eloped a ma hema ical
ool o unde s anding he hea low gene a ed by a mo ing hea sou ce in welding [7].
Ne e heless, hose models conside a s eady s a e si ua ion and conduc i i y and di usi i y
we e empe a u e independen . Simul aneously, Ca slaw and Jaege s udied he hea ans e
by conduc ion in me allic pa s [8]. Ne e heless, no phase change o empe a u e dependen
physical p ope ies we e conside ed.
In o de o make a p edic ion as close as possible o eali y, unde s anding phenomena such as
plasma gene a ion and mel -pool dynamics is essen ial [9]. Mazumde and S een de eloped he
i s h ee-dimensional model conside ing a Gaussian hea sou ce and sol ing hea ans e
equa ions by ini e di e ence nume ical echniques. In hei calcula ions, he keyhole o ma ion
pe iod was negligible, and once c ea ed, i was conside ed as a blackbody because o
abso p i i y inc ease o he mol en ma e ial. A same ime, an ene gy loss coe icien was
in oduced o conside he hea losses by con ec ion because o he shielding gas. The model
was p og ammed using a quasi-s a ic app oach in o de o simpli y he ma hema ics.
Ne e heless, hea ing cycles mus be accu a ely de ined o model he mechanical beha iou o
he welded a eas [10] and his is s ill challenging nowadays [11]. The physics behind keyhole
o ma ion and i s e olu ion in o mel -pool dynamics is s ill unde s udy [12]. Al hough
expe imen al alida ion es ing o LBW is di icul o pe o m, because o he simul aneous
exis ence o plasma, liquid and solid ma e ial, app op ia e nume ical simula ion can be help ul
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in hei unde s anding [13] specially in high-dep h LBW modelling whe e he e a e s ill many
aspec s o be analysed [14].
Tha is, 80 yea s a e he i s pionee ing esea ch, he modelling o hea sou ces used o
welding is imme sed in a con inuous imp o emen p ocess. In addi ion o exis ing models, i is
s ill necessa y o s udy he easibili y o complex weld-like echniques wi h new models. Fo his
eason, he aim o his wo k is o de elop an analy ical LBW model o nickel base alloys o
ae onau ic pa applica ion, ha conside s he mo emen o he mol en ma e ial and
empe a u e dependen p ope ies.
Table 1: Used symbols and hei physical desc ip ion.
Symbol
Uni s
Desc ip ion
(x,y,z)
[m]
Coo dina es o he poin whe e he empe a u e is
ob ained
(ξ,η,ζ)
[m]
Posi ion o he lase beam cen e
𝒬′′ζζ
[-]
Quad upole coe icien in Z di ec ion
𝒬′′ηη
[-]
Quad upole coe icien in Y di ec ion
𝒬′′ξξ
[-]
Quad upole coe icien in X di ec ion
𝑇0
[K]
Tempe a u e ield a he beginning o he ime s ep
𝑡′
[s]
Ini ial ime ins an
∆𝑥, ∆𝑦
[m]
Size o each elemen in he x and y axis, espec i ely
A
[-]
Abso p i i y
d
[m]
Thickness o he pla e
ds
[m]
Diame e o he lase beam
dw
[m]
Diame e o he wobble oscilla ion mo emen
I(ξ,η)
[W·m-2]
In ensi y unc ion
k
[W·m-2 ·K-1]
The mal conduc i i y
Lin
[m]
Sum o he pa ial pa hs
P eal
[W]
Real powe o he lase
Psim
[W]
Powe conside ed in he simula ion
R
[m]
Dis ance be ween lase beam and s udied poin
S ac o
[-]
A ea ac o
Sh
[m2]
A ea inside he oscilla ing mo emen desc ibed by he
lase beam
ac o
[-]
Time ac o
in
[s]
Time equi ed o he lase spo o ill he ci cle desc ibed
by he wobble mo emen
s ep
[s]
Time s ep
p
[m·s-1]
pe iphe al speed o he lase beam
W
[K·W-1]
Mo emen o he hea sou ce on a semi-in ini e pa
α
[m2·s]
The mal di usi i y
𝑇
[K]
Tempe a u e ield a he end o he ime s ep
𝑡
[s]
Time ins an whe e he he mal ield is calcula ed
𝑣
[m·s-1]
Feed a e
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2. Model basis
2.1. Summa y o he de eloped analy ical model
A wo-s ep model is p esen ed o he Lase Beam Welding (LBW). Fi s , based on Ca slaw-
Jaege ´s heo y he he mal ield on he uppe ace o he pla es is modelled, wha enables o
de e mine he wid h o he gene a ed weld bead. The de eloped model includes a new
app oach o he wobble s a egy, whe e he ansi o y egime is conside ed. Then, in a second
s ep, he in e nal mo emen o he mol en ma e ial is modelled based on Rosen hal´s model,
which enables o ob ain he pene a ion o he weld bead.
2.2. Weld bead uppe ace modelling
The modelling o he hea ans e om he lase beam o he wo kpiece is based on he
equa ions de eloped by Rosen hal [15]. Howe e , a ew modi ica ions a e in oduced. On he
one hand, he ac ha in LBW he hea is dis ibu ed in an a ea is conside ed by means o he
in ensi y unc ion 𝐼(𝜉,𝜂). On he o he hand, he model de eloped by Ca slaw and Jaege ha
ep esen he hea ans e in a semi-in ini e solid is conside ed [8]. Equa ion (1) de ines he
con olu ion mul iplica ion o he A·I·W p oduc , which de ines he empe a u e ise a any poin
(x,y,z) o he pa .
𝑇(𝑥,𝑦,𝑧)= ∫∫𝐴·𝐼(𝜉,𝜂)·𝑊(𝑥−𝜉,𝑦−𝜂,𝑧−𝜁,𝑣)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
(1)
𝑊=𝑒[−𝑣
2𝛼(𝑥−𝜉+𝑅)]
2𝜋𝑘𝑅
(2)
𝑅=√(𝑥−𝜉)2+(𝑦−𝜂)2+(𝑧−𝜁)2
(3)
The powe -densi y abso bed by he subs a e is de ined by mul iplying A and I, which enables
o de e mine he in ensi y dis ibu ion in each ime ins an . The unc ion W de ined in
equa ion (2) ep esen s he mo emen o he hea sou ce on a semi-in ini e pa . Equa ion (3)
ep esen s he dis ance om he lase beam cen e (ξ,η,ζ) o he poin whe e he empe a u e
is being calcula ed (x,y,z).
Ne e heless, be o e eaching a s able egime, in LBW he he mal ield mus go h ough a
ansi o y egime. The e o e, he in luence o ime in he welding p ocess mus be conside ed
in equa ion (1) in o de o model he ansien s age and equa ion (4) is achie ed.
𝑇(𝑥,𝑦,𝑧,𝑡)= ∫∫𝐴𝐼(𝜉,𝜂)𝑊(𝑥−𝜉,𝑦−𝜂,𝑧−𝜁)𝑈(𝑅,𝑡,𝑣)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
(4)
𝑈(𝑅,𝑡,𝑣)=𝑅
√𝜋∫ 𝑒−(𝑅𝜏2−𝑣𝛼)2
4𝜏2𝑑𝜏=
∞
1√𝛼𝑡
⁄12[1−e (𝑅−𝑣𝑡
2√𝛼𝑡)+𝑒𝑅𝑣/𝛼(1−e (𝑅+𝑣𝑡
2√𝛼𝑡))]
(5)
𝜏=[𝛼(𝑡−𝑡′)]−1/2
(6)
The e e o unc ion is de ined o a gene al a iable a acco ding o equa ion (7).
e (𝑎)=1
√𝜋∫𝑒−𝑡2𝑑𝑡=2
√𝜋∫𝑒−𝑡2𝑑𝑡
𝑎
0
𝑎
−𝑎
(7)
This way, he ime-dependen equa ion ha de e mines he empe a u e ield is ob ained.
Howe e , i empe a u e ise is o be calcula ed in he egions close o whe e he lase beam
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s ikes, he pa ame e R ends o ze o and in oduces a singula i y in he pa ame e W. In o de
o sol e his issue, in he cases whe e he R pa ame e alue is below he uni , ins ead o using
he pa ame e W, he ollowing W0= limz→0 W pa ame e is used.
𝑊0=𝑒−(𝑣𝑧)2
2𝛼
2𝜋𝑘∆𝑥∆𝑦[∆𝑥ln(𝑤1+∆𝑥
𝑤1−∆𝑥)+∆𝑦ln(𝑤2+∆𝑦
𝑤2−∆𝑦)−4𝑧(a c an(𝑤1
2𝑧)+a c an(𝑤2
2𝑧))+2𝑧𝜋]
(8)
𝑤1=√(∆𝑥2+4𝑧2cos2𝛽
sin2𝛽)
𝑤2=√(∆𝑦2+4𝑧2sin2𝛽
cos2𝛽)
𝛽=a c an∆𝑦
∆𝑥
(9)
2.3. Fou ie T ans o m
One o he main d awbacks o he analy ical me hods is he complexi y o sol e he ob ained
equa ions. Double in eg a es can be sol ed by means o a ious me hods, such as he mul iple
in eg a ion me hod. Howe e , hose me hods a e slow compa ed o he Fou ie T ans o m (FT).
Consequen ly, in o de o ensu e a low compu a ional cos and a high speed o he model, he
FT me hod is used. Fo his pu pose, he FFT2 module o Ma lab R2018b is employed.
Besides, hanks o he usage o he con olu ion mul iplica ion, he model sol es di ec ly he
double in eg al de ined o he empe a u e ield calcula ion o each z-le el su ace. The e o e,
he model ob ains he empe a u e ield o all nodes in each heigh cons an su ace, and each
su ace is calcula ed independen ly.
The calcula ed FT o he in ensi y, F(I), a ies as he hea sou ce mo es and he e o e, he model
includes he e ec o a mo ing hea sou ce. Simila o he in ensi y, he Fou ie ans o ma ion
o W and U mul iplica ion is calcula ed, F(W·U). A e wa d, he Fou ie ans o m o he
bi-dimensional empe a u e ield is ob ained by means o equa ion (10).
𝐹(𝑇)=𝐹(𝑊·𝑈)∗𝐹(𝐼)
(10)
In o de o calcula e he empe a u e inc ease in each poin o he hea ed pa he e e se FT is
employed, equa ion (11), whe e he abso p i i y, A, and a ea o he employed mesh a e
included, 𝑑𝑆=∆𝑥·∆𝑦 and he empe a u e ield a he beginning o he ime s ep, T0, is
conside ed.
𝑇(𝑥,𝑦,𝑧,𝑡)=𝐴·𝐹−1(𝐹(𝑇))·𝑑𝑆+𝑇0
(11)
2.4. Wobble s a egy
The coupling o as op ics in he lase welding head o e s he capabili y o ob ain di e en weld
sizes using a small lase spo , which inc eases he lexibili y o he employed lase equipmen .
Wobble s a egy combines wo mo emen s, as i is de ailed in Figu e 1, he main eed a e, a
linea mo emen , and a ci cula supe posed oscilla ion mo emen o he lase beam, which is
a ound 50 imes as e .
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Figu e 1: Wobble s a egy scheme, whe e linea and oscilla ion mo emen s a e combined.
Modelling he wobble s a egy is a complex ask and classical heo ies p o ide no possibili y o
include he oscilla ion mo emen o he lase beam when a wobble s a egy is used. Besides,
conside ing he hea ans e in e e y single loca ion o he lase beam as i mo es implies an
excessi e cos .
𝑃𝑠𝑖𝑚=𝑃𝑟𝑒𝑎𝑙·𝑆𝑓𝑎𝑐𝑡𝑜𝑟
𝑡𝑓𝑎𝑐𝑡𝑜𝑟
(12)
Because o he employed simpli ica ions, wo ac o s ha e been included in he model o ensu e
he co espondence be ween he model and he expe imen al esul s, see equa ion (12). These
pa ame e s a e applied o e he alue o he P eal in o de o ob ain he Psim.
2.4.1. A ea ac o
Based on he assump ion ha he oscilla ion eloci y o he spo is much highe han he eed
a e, i is conside ed ha he lase beam has an annula shape and he whole ing is hea ed
simul aneously. Ne e heless, he equa ions o he analy ical model can only sol e he he mal
ield in he case o a ci cula hea sou ce, see Figu e 2, and consequen ly, an a ea- ac o is
included, equa ion (13).
Figu e 2: App oxima ion o he hea sou ce.
𝑆𝑓𝑎𝑐𝑡𝑜𝑟=𝑆𝑠𝑖𝑚
𝑆𝑖𝑑𝑒𝑎𝑙
(13)
𝑆𝑠𝑖𝑚=𝜋(𝑑𝑤+𝑑𝑠)2
4
(14)
𝑆𝑖𝑑𝑒𝑎𝑙=𝜋(𝑑𝑤+𝑑𝑠)2
4−𝜋(𝑑𝑤−𝑑𝑠)2
4=𝜋𝑑𝑤𝑑𝑠
(15)
A ime s ep ha ensu es a minimum o e lap o he successi e loops swep by he lase is
conside ed in he analy ical model. The e o e, he alue o he employed ime s ep depends on
he p ocess pa ame e s.
𝑡𝑠𝑡𝑒𝑝=𝑑𝑠
𝑣
(16)
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2.4.2. Time ac o
In he eal si ua ion whe e he lase beam desc ibes an oscilla ing mo emen , he e is an a ea
in he cen e o he ing whe e no hea is in oduced. Ne e heless, in he p oposed assump ion
o using a ci cula hea sou ce, he lase powe ha i adia es he su ace o he subs a e needs
o be co ec ed acco ding o a ime ac o .
The ime ac o , ac o , is adjus ed o an empi ical equa ion, based on esul s o expe imen al
es . The aim is o de elop an equa ion ha is alid o di e en welding condi ions and
conside s he in luence o he mos ele an pa ame e s when a wobble s a egy is used. Fo
his pu pose, besides he lase eed a e and he wobble pa ame e s, he ime equi ed o he
lase spo o ill he ci cle desc ibed by he wobble mo emen ( in), he sum o he pa ial pa hs
(Lin) and he pe iphe al speed o he lase beam a e conside ed ( p). Simila ly, in o de o
conside he e ec o he hea ans e owa ds he cen e o he desc ibed ci cle in he wobble
s a egy, he a ea inside he ci cle is calcula ed, which is named as Sh.
𝐿𝑖𝑛=𝑑𝑤·∑𝑎𝑟𝑐𝑡𝑎𝑛(√(𝑑𝑤
𝑛·𝑑𝑠)2−1)
𝑑𝑤+𝑑𝑠
𝑑𝑠−1
𝑛=1
(17)
𝑡𝑖𝑛=𝐿𝑖𝑛/𝑣𝑝
(18)
𝑆ℎ=𝜋4·(𝑑𝑤−𝑑𝑠)2
(19)
2.5. Weld bead pene a ion modelling
Despi e he modi ica ions in oduced o Ca slaw-Jaege ´s heo y, i is no capable o modelling
he mo emen o he mol en ma e ial wi hin he mel pool. Fo his pu pose and based on he
mul ipole heo y, he equa ions published by Nunes a e conside ed, whe e he in e nal
mo emen o he mol en ma e ial is conside ed [16]. Fi s , he hea sou ce is ep esen ed as
epe i i e punc ual monopoles dis ibu ed among he pa , see he black do s in Figu e 3. In
o de o de e mine he in luence o he hea in a su ace si ua ed a z dep h, 2·P powe hea
sou ces sepa a ed by a 2·d dis ance mus be conside ed, whe e d is he hickness o he pla e.
Figu e 3: Monopole model ep esen a ion.
Being V one o he solu ions o he hea ans e equa ions and (ξ,η,ζ) he posi ion o he hea
sou ce, Rosen hal de ined he monopole model by means o equa ions (20-25).
𝜕2𝑇
𝜕𝑥2+𝜕2𝑇
𝜕𝑦2+𝜕2𝑇
𝜕𝑧2=−𝑣𝛼𝜕𝑇
𝜕𝑥
(20)
8 - 20
𝑉= 1
4𝜋𝑘𝑒−𝑣
2𝛼(𝑅+𝑥−ξ)
𝑅
(21)
𝑇−𝑇0=2𝑃[𝑉(𝑅0)+∑(𝑉(𝑅𝑛)+𝑉(𝑅𝑛´))
∞
𝑛=1 ]
(22)
𝑅0=√𝑥2+𝑦2+𝑧2
(23)
𝑅𝑛=√𝑥2+𝑦2+(𝑧−𝑛𝑑)2
(24)
𝑅𝑛´=√𝑥2+𝑦2+(𝑧+𝑛𝑑)2
(25)
A e wa d, Nunes ep esen ed geome ically he phase change in a mo ing mel pool by means
o dipoles. As he mel pool ad ances oge he wi h he lase beam, he ma e ial on he on is
mel ed, whe eas ma e ial on he backside is solidi ied. The e o e, he hea sou ce can be
ep esen ed by means o opposed dipoles. The dis ance be ween he poles is Δξ, see Figu e 4,
gene a ing an opposed se be ween he wo poles.
𝑇−𝑇0=∮𝜌𝐿𝑣𝑉(ξ,𝜂,𝜁)𝑑𝜂𝑑𝜁
𝑆´
(26)
𝑇−𝑇0≈2𝑃∆ξ[𝑉(∆ξ
2)−𝑉(∆ξ
2)
∆ξ ]
(27)
When Δξ ends o ze o:
𝑇−𝑇0≈𝒬′ξ(𝜕𝑉
𝜕ξ)0=𝒬′ξ·𝑉0·[(1+𝑥𝑅)𝑣
2𝛼+𝑥
𝑅2]
(28)
𝒬′ξ=lim
∆ξ→02𝑃∆ξ
(29)
𝑉0=1
4𝜋𝑘𝑒−𝑣
2𝛼(𝑅+𝑥)
𝑅
(30)
Figu e 4: Dipole ep esen a ion o phase change.
Once he in luence o monopoles and dipoles is included in he de eloped model, he in luence
o quad upoles needs o be conside ed in o de o simula e he mol en ma e ial mo emen , see
Figu e 5. The mo emen o he mol en ma e ial inside he mel pool is mainly due o he su ace
ension and is di ec ly in luenced by he sulphu concen a ion and he p ocess pa ame e s [17].
Mol en ma e ial mo es om highe -p essu e egions o lowe -p essu e egions. Consequen ly,
9 - 20
ins ead o accumula ing hea in he cen e o he mel pool, hea is ans e ed ou wa ds o
downwa ds, depending on he mo emen di ec ion.
Figu e 5: Quad upole ep esen a ion o he mol en ma e ial ci cula ion.
𝑇−𝑇0=2𝑃∆ξ2{[𝑉(ξ+∆ξ)−𝑉(ξ)
∆ξ ]−[𝑉(ξ)−𝑉(ξ−∆ξ)
∆ξ ]
∆ξ }
(31)
When Δξ ends o ze o:
𝒬′′ξξ=lim
∆ξ→02𝑃(∆ξ)2
(32)
𝒬′′ηη=lim
∆ξ→02𝑃(∆η)2
(33)
𝒬′′ζζ=lim
∆ξ→0𝑃(∆η)2
(34)
Unlike ξ and η di ec ions, in ζ di ec ion he e is no need o mi o ing he gene a ed ma e ial
mo emen . In equa ion (35), he in luence o he ma e ial mo emen in all di ec ions inside he
mel pool due o quad upoles is modelled.
𝑇−𝑇0=𝒬′′ξξ(𝜕2𝑉
𝜕ξ2)0+𝒬′′ηη(𝜕2𝑉
𝜕η2)0+𝒬′′ζζ(𝜕2𝑉
𝜕ζ2)0
(35)
(𝜕2𝑉
𝜕ξ2)0=𝑉0[(1+2𝑥𝑅+𝑥2
𝑅2)(𝑣
2𝛼)2+(−1+2𝑥𝑅+3𝑥2
𝑅2)(𝑣
2𝛼)(1𝑅)+(−1+3𝑥2
𝑅2)(1𝑅)2]
(36)
(𝜕2𝑉
𝜕η2)0=𝑉0[(𝑦2
𝑅2)(𝑣
2𝛼)2+(−1+3𝑦2
𝑅2)(𝑣
2𝛼)(1𝑅)+(−1+3𝑦2
𝑅2)(1𝑅)2]
(37)
(𝜕2𝑉
𝜕ζ2)0=𝑉0[(𝑧2
𝑅2)(𝑣
2𝛼)2+(−1+3𝑧2
𝑅2)(𝑣
2𝛼)(1𝑅)+(−1+3𝑦2
𝑅2)(1𝑅)2]
(38)
The posi i e o nega i e alue o he Q’’ξξ, Q’’ηη e a Q’’ζζ coe icien s has a di ec in luence on he
mol en ma e ial ci cula ion di ec ion. Fo ins ance, a posi i e alue o Q’’ξξ and Q’’ηη implies an
ou wa ds ma e ial mo emen in he x and y di ec ions. Mo eo e , i Q’’ζζ is nega i e he hea
will low owa ds he weld c own and he esul ing clad will ha e a wide and shallow geome y.
On he con a y, i Q’’ξξ and Q’’ηη p esen a nega i e alue and Q’’ζζ is posi i e, na ow and dep h
clads a e ob ained. The e o e, hei alues need o be es ablished acco ding o he welded
ma e ials and p ocess pa ame e s.
Consequen ly, conside ing he in luence o monopoles, dipoles and quad upoles, he
empe a u e inc ease gene a ed by a P powe hea ha mo es wi h a eed a e is de e mined
16 - 20
Table 7: Weld dep h esul s
Tes
Real dep h
[mm]
Sim dep h
[mm]
E o
[mm]
1
1.51
1.40
0.11
2
2.00
2.00
0.00
3
2.00
2.00
0.00
4
2.00
2.00
0.00
5
0.92
1.18
-0.27
6
1.21
1.36
-0.15
7
1.56
1.42
0.14
8
2.00
2.00
0.00
9
1.28
1.28
0.00
10
1.32
1.36
-0.04
11
1.71
1.44
0.27
12
2.00
2.00
0.00
13
0.72
1.00
-0.28
14
1.27
1.24
0.03
15
1.57
1.28
0.29
16
2.00
2.00
0.00
17
0.40
0.66
-0.26
18
0.56
0.78
-0.22
19
0.77
1.02
-0.25
20
0.88
1.16
-0.28
Ob ained e o is below 300 µm, which ensu es he accu acy o he model o di e en welding
condi ions. In Figu e 12 he longi udinal and c oss-sec ion o he weld bead co esponding o
es 1 a e shown, as well as a compa ison be ween he modelled and eal c oss sec ions.
Figu e 12: a) Longi udinal and b) oss sec ions o he weld bead co esponding o es 1, once he
welding p ocess is s abilized. c) Real and model c oss sec ion compa ison o es 1.
17 - 20
4.2. Tempe a u es and mic os uc u e compa ison
Besides he capabili y o he model o p edic he shape o he weld bead, he model is able o
p edic he he mal ield a ia ions on he welded pla es and he esul ing mic os uc u e.
Tempe a u e e olu ion on he su ace is measu ed di ec ly by using a wo-colou py ome e ,
whe eas he inne empe a u e a ia ions a e s udied based on he de eloped mic os uc u e
once he pa is cooled down.
4.2.1. Su ace cooling g adien measu emen
The empe a u e is measu ed in he cen e o he weld bead; egion named as R0. The signals
ob ained om he wo-colou py ome e a e il e ed in Ma lab in o de o ease hei analysis
and a oid oscilla ions. Resul ing da a is shown in Figu e 13.
Figu e 13: Tempe a u e measu emen s o di e en es s when a 1 mm wobble diame e is used.
A e wa d, he cooling a e is calcula ed by means o measu ing he amoun o ime equi ed
o lowe ing he empe a u e o he weld bead om sligh ly abo e he liquidus empe a u e
un il he solidus empe a u e. Ob ained alues, oge he wi h he p ocess pa ame e s in o de
o ease hei unde s anding, a e compa ed wi h hose p edic ed by he model and e o alues
a e shown in Table 8.
Table 8: Cooling a es a he su ace, in he cen e o he weld bead.
Tes
numbe
P [W]
[mm·s-1]
Cooling a e [K·s-1]
E o
[K·s-1]
E o
%
Expe imen al
Model
1
350
3
2459
159
159
6.5
2
400
3
1860
-96
-96
-5.2
3
450
3
1587
53
53
3.3
4
500
3
1025
82
82
8.0
5
350
5
6030
-220
-220
-3.6
6
400
5
4929
-154
-154
-3.1
7
450
5
4435
-273
-273
-6.2
8
500
5
1932
99
99
5.1
18 - 20
4.2.2. SDAS measu emen
Weld beads a e c oss sec ioned and e ched in o de o e eal he mic os uc u e in egions R1
and R2, Figu e 14. In each es pa 10 indi idual measu emen s a e pe o med in each egion
o ob aining he a e age SDAS.
Figu e 14: Mic os uc u e o es 1 in egions R1 (le ) and R2 ( igh ).
In pa allel, analysing he model esul s, he cooling g adien is calcula ed in bo h egions, R1 and
R2, by means o he amoun o ime equi ed o cool down om he liquidus (1609 K) o he
solidus empe a u e (1533 K). A e wa d, equa ion (41) is applied and he SDAS alue
co esponding o he calcula ed cooling a e is ob ained, see Table 9.
Table 9: A e age SDAS alues measu ed a di e en egions.
Tes
numbe
R1
R2
Cooling a e
[K·s-1]
SDAS
[µm]
Cooling a e
[K·s-1]
SDAS
[µm]
1
2400
4.046
2600
3.940
2
1933
4.349
1909
4.367
3
1627
4.606
1500
4.733
4
1111
5.231
1217
5.075
5
7000
2.832
6625
2.884
6
5333
3.101
5917
2.995
7
4375
3.312
4750
3.223
8
3708
3.500
3875
3.449
In o de o educe he numbe o analysed samples, he weld beads co esponding o he limi
welding pa ame e s a e s udied. The e o o he model when p edic ing he SDAS is de ailed in
Table 10, whe e esul s co esponding o bo h R1 and R2 egions a e shown. In all cases, an e o
below 1 μm is ob ained.
Table 10: Expe imen al and modelled SDAS alues compa ison
Tes
numbe
R1
R2
Expe imen al
Model
E o
[µm]
Expe imen al
Model
E o [µm]
A e age
SDAS [µm]
Cooling
a e [K·s-1]
SDAS
[µm]
A e age
SDAS [µm]
Cooling
a e [K·s-1]
SDAS
[µm]
1
3.701
2400
4.046
0.345
3.185
2600
3.940
0.755
4
4.249
1111
5.231
0.249
3.775
1217
5.075
-0.169
5
2.583
7000
2.832
0.982
3.053
6625
2.884
0.881
8
2.626
3708
3.500
0.874
3.537
3875
3.449
-0.088
19 - 20
5. Conclusions
In he p esen wo k, an analy ic model o LBW based on he classical hea ans e equa ions is
de eloped and alida ed. To ha end, an expe imen al se up is de eloped, and a ious es s a e
pe o med in o de o e alua e he model unde di e en condi ions. Acco ding o he ob ained
esul s, he ollowing conclusions can be d awn:
Thanks o a double analy ical model, bo h he weld bead wid h and dep h a e accu a ely
modelled. Besides, by in oducing he e ec o monopoles, dipoles and quad upoles,
he model can conside he e ec o mol en ma e ial mo emen inside he weld bead
a a low compu a ional cos .
I is e i ied ha he su ace and ime ac o s in oduced in he model in o de o
simula e he wobble s a egy enhance he accu acy o he model. E o s below 0.05 mm
and 0.3 mm a e ob ained ega ding he clad wid h and dep h, espec i ely.
The e o in he su ace cooling a e is below 300 K·s-1. On he one hand, his e o is
a ibu ed o he as na u e o he welding p ocess, which makes di icul o model his
pa ame e p ecisely. On he o he hand, an app oxima ion o he lase hea sou ce is
employed, which in oduces an inhe en e o in he model. Ne e heless, his accu acy
is adequa e conside ing he low compu a ional cos o an analy ical model.
The model esul s o p edic he SDAS wi h an e o below 1 μm. Besides, he model
shows he same beha iou as he expe imen al esul s, whe e an inc ease o he lase
powe esul s in a lowe cooling a e and highe SDAS, and an in e se si ua ion is
encoun e ed when he lase eed a e is aised.
Acknowledgemen s
Au ho s g a e ully acknowledge he Uni e si y o he Basque Coun y (UPV/EHU) o i s inancial
help. In addi ion, his wo k has been ca ied ou in he amewo k o he “En o no Vi ual de
Diseño y Fab icación de Tu binas Ae onáu icas” ENVIDIA p ojec (RTC-2017-6150-4) unded by
he Spanish Minis y o Indus y and Compe i i eness.
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