Lamellar Spacing Modelling for LPBF Aluminum Parts

Ci a ion: Anglada, E.; Ga cía, J.C.;
A ue, M.; Cea solo, X.; Ga mendia, I.
Lamella Spacing Modelling o LPBF
Aluminum Pa s. J. Manu . Ma e .
P ocess. 2022,6, 164. h ps://doi.o g/
10.3390/jmmp6060164
Academic Edi o : S e en Y. Liang
Recei ed: 18 No embe 2022
Accep ed: 15 Decembe 2022
Published: 17 Decembe 2022
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Manu ac u ing and
Ma e ials P ocessing
Jou nal o
A icle
Lamella Spacing Modelling o LPBF Aluminum Pa s
E a Anglada 1, JoséCa los Ga cía1, Ma io A ue 2, Xabie Cea solo 2and Iñaki Ga mendia 3,*
1TECNALIA, Basque Resea ch and Technology Alliance (BRTA), Mikele egi Pasealekua, 2,
E-20009 Donos ia-San Sebas ián, Spain
2IMH, Azkue Auzoa 1, E-20870 Elgoiba , Spain
3Mechanical Enginee ing Depa men , Enginee ing School o Gipuzkoa, Uni e si y o he Basque Coun y
UPV/EHU, Plaza de Eu opa, 1, E-20018 Donos ia-San Sebas ián, Spain
*Co espondence: [email p o ec ed]
Abs ac :
The high cooling a es eached du ing me al addi i e manu ac u ing (MAM) gene a e
mic os uc u es e y di e en om hose ob ained by o he con en ional manu ac u ing me hods.
The e o e, esea ch abou he modeling o his ype o mic os uc u e is o g ea in e es o he MAM
communi y. In his wo k, he p edic ion o he lamella spacing o an AlSi10Mg sample manu ac u ed
by lase powde bed usion (LPBF), is p esen ed. A mul iscale app oach is used, combining a
CALPHAD (Compu e Coupling o Phase Diag ams and The mochemis y) model o p edic he
ma e ial p ope ies, wi h a mac oscale model o he sample manu ac u ing and wi h a mic oscale
model o p edic he mic os uc u e. The manu ac u ing and me allog aphic cha ac e iza ion o
he sample is also included. The esul s p o e ha he mul iscale s a egy ollowed is a alid
app oxima ion o simula e his ype o manu ac u ing p ocess. In addi ion, i is shown ha he use
o a gene ic simula ion so wa e ocused on me al cas ing p ocesses can be use ul in p edic ing he
lamella spacing o he mic os uc u e manu ac u ed by LPBF. Finally, he ela ionship be ween he
cooling a e and he esul ing lamella spacing has been es ablished o his AlSi10Mg unde he
speci ic manu ac u ing condi ions conside ed.
Keywo ds: LPBF; mic os uc u e; simula ion
1. In oduc ion
Addi i e manu ac u ing (AM) is a dis up i e echnology ha makes i possible o
manu ac u e geome ies ha a e almos impossible o achie e by o he manu ac u ing
me hods. This ac , oge he wi h i s good adap abili y o cus omized p oduc ion and
wi h he possibili y o being in eg a ed in o in elligen p oduc ion sys ems, makes AM an
essen ial ing edien o Indus y 4.0, whe e i may become a key echnology [
1
]. Al hough
he e a e s ill some doub s abou i s applicabili y in mass p oduc ion, he u iliza ion o AM
in he indus y is on he ise in di e en sec o s such as ae ospace, biomedical, au omo i e,
e c. In ac , he o e all AM ma ke o me al and polyme sys ems was alued a 8.33 billion
EUR in 2021 wi h a p ojec ed g ow h o compound annual g ow h a e (CAGR) o 18.2 %
un il 2026. Rela ed o he me al AM (MAM) ma ke alone, i was alued a 2.50 billion EUR
in 2021 wi h an expec ed CAGR o 25.5% un il 2026 [2].
The main ypes o p ocesses used in MAM a e powde bed usion (PBF), di ec ene gy
deposi ion, ma e ial je ing, and binde je ing. Conside ing powe bed usion, he wo
main echnologies o his g oup a e lase powde bed usion (LPBF), also known as selec i e
lase mel ing (SLM), and elec on beam mel ing (EBM). The LPBF, one o he mos ex ended
in his indus y, basically consis s o he deposi ion o a me allic powde laye on a pla o m
and he selec i e mel ing o he powde by he lase . Once he laye has been selec i ely
mel ed, he pla o m descends, and a new laye o me allic powde is deposi ed o be
selec i ely mel ed; his p ocess is epea ed un il he comple e pa is manu ac u ed laye
by laye [3].
J. Manu . Ma e . P ocess. 2022,6, 164. h ps://doi.o g/10.3390/jmmp6060164 h ps://www.mdpi.com/jou nal/jmmp
J. Manu . Ma e . P ocess. 2022,6, 164 2 o 14
An impo an cons ain o MAM is ela ed o he esul an mic os uc u e o he
manu ac u ed pa , as i is di ec ly co ela ed wi h he ma e ial
'
s mechanical pe o mance.
The high empe a u es eached du ing he p ocess, oge he wi h he ex emely as hea ing
and cooling a es, may p omo e he appea ance o ce ain mic os uc u es, which a e
de imen al o he mechanical p ope ies. Fo example, he ine mic os uc u e ha esul s
om high cooling a es can be de imen al o c ack p opaga ion esis ance in a igue
beha io [4].
In ac , cu en ly, he e a e only a limi ed numbe o alloys ha can be p ocessed
by MAM since he es p esen a poo pe o mance ela ed o ho c acking and/o low
mechanical p ope ies [5–9]. The e o e, mic os uc u e p edic ion is a e y use ul ool no
only o suppo he p ocess pa ame e ’s op imiza ion bu also o help in he de elopmen
o new alloys mo e sui able o be used in his manu ac u ing p ocess.
Se e al au ho s ha e s udied di e en app oaches o he modeling o he LPBF. Fo
example, Bida e e al. s udy he in e ac ion be ween he lase beam and he powde bed
by means o FEM analysis, including he dynamics o he gas and plasma phases o
s ainless s eel 316 L [
10
]. Acha ya e al. de eloped a model combining a CFD model
wi h a phase- ield model o p edic he mic os uc u e o he Inconel 718 [
11
]. Zhang e al.
use a 3-D hea ans e ini e elemen model o p edic ing mel pool dimensions in he
LPBF o s ainless s eel 17-4PH [
12
]. Denlinge e al. model he he mal beha io o he
Inconel 718 du ing he LPBF manu ac u ing by means o a 3D Lag angian ansien he mal
analysis [
13
]. The e a e also wo ks de o ed o he pa icula case o aluminum alloys,
al hough he numbe is lowe . Fo example, Azizi e al. s udied he LPBF o AlSi by means
o a phase- ield o mula ion [
14
], and Rehman e al. s udied he spa e o ma ion and
splashing-induced de ec s in AlSi10Mg LPBF [
15
]. Many o he s ha e also made in e es ing
con ibu ions [16–22].
The mo i a ion o he p esen wo k is o con ibu e o he LPBF modeling by p esen ing
an al e na i e app oach o ackle his ype o simula ion. In addi ion, he expe imen al
esul s by hemsel es a e also an in e es ing con ibu ion o o he esea che s in e es ed in
his ype o manu ac u ing me hod.
The esea ch hypo hesis is he possibili y o p edic ing he mic os uc u e o an LPBF
pa h ough a mul iscale and mul i-so wa e app oach, being one o he so wa e speci i-
cally de eloped o o he ypes o me allic manu ac u ing p ocesses, whe e cooling a es
a e e y di e en om hose expe imen ed in LPBF. To alida e his hypo hesis, he man-
u ac u ing p ocess simula ion was ca ied ou , combining h ee di e en models. One
he modynamic model, one model a he mac oscale le el ep esen a i e o LPBF manu ac-
u ing, and one model a he mic oscale le el ocused on he mic os uc u e e olu ion. The
models ha e been adjus ed, aking as a e e ence he me allog aphic s udy o expe imen-
ally manu ac u ed samples.
The main no el y o he app oach is he mul iscale modeling: he so wa e used o
p edic he mic os uc u e was no de eloped o he LPBF mic os uc u es. This app oach
may be use ul o hose esea che s ha ha e no a ailable speci ic so wa e ools o his
manu ac u ing p ocess bu a e in e es ed in hei modeling. The wo k con ibu ion is
in ended o p o ide an al e na i e app oach o his ype o modeling a he han closing a
speci ic esea ch gap in his opic.
2. Ma e ials and Me hods
In o de o be able o alida e he simula ion models de eloped in his wo k, sample pa s
ha e been manu ac u ed and cha ac e ized o e alua e hei mic os uc u e expe imen ally.
The alloy s udied is he AlSi10Mg. I s chemical composi ion has been analyzed by
op ical emission spec ome y, ob aining he composi ion shown in Table 1.
J. Manu . Ma e . P ocess. 2022,6, 164 3 o 14
Table 1. Alloy chemical composi ion.
Al Si Fe Zn Mg Cu Ti Mn Ni
Base 10.7 0.21 <0.01 0.26 <0.01 0.02 <0.01 <0.01
2.1. Samples Manu ac u ing
The sample pa s we e buil using a Samylabs ALBA300 machine (Figu e 1), which
has a ibe lase wi h a maximum powe ou pu o 250 W and 1080 nm wa eleng h. The
lase spo ocalized in he subs a e co esponds o a diame e o 71 µm (1/e2 me hod).
J. Manu . Ma e . P ocess. 2022, 6, 164 3 o 15
Table 1. Alloy chemical composi ion.
Al
Si
Fe
Zn
Mg
Cu
Ti
Mn
Ni
Base
10.7
0.21
<0.01
0.26
<0.01
0.02
<0.01
<0.01
2.1. Samples Manu ac u ing
The sample pa s we e buil using a Samylabs ALBA300 machine (Figu e 1), which
has a ibe lase wi h a maximum powe ou pu o 250 W and 1080 nm wa eleng h. The
lase spo ocalized in he subs a e co esponds o a diame e o 71 μm (1/e2 me hod).
Figu e 1. Samylabs ALBA300 machine a IMH Campus.
The build chambe was illed wi h a gon (pu i y le el 99.999%) be o e ope a ion and
a cons an shielding gas low was used du ing he build o spa e emo al ollowing
[23]. The oxygen con en in he build chambe was main ained below 0.43 ol% (4300
ppm) du ing he p ocess ollowing [24,25].
The samples we e buil di ec ly on EN AW-5083 aluminum subs a e wi hou p e-
hea ing because he equipmen in which he expe imen was ca ied ou does no ha e a
subs a e hea e . The 20 mm hick subs a e pla e unc ioned as a hea sink o allow hea
o be conduc ed away om he samples.
The manu ac u ing pa h was gene a ed by SamyS udio 5.1 so wa e using a powde
laye hickness o 25 µm. Di e en pa ame e s we e se o he co e and he edge o he
pa o imp o e pa densi y and su ace quali y (Figu e 2). The co e was buil by a bidi-
ec ional meande ha ching pa e n wi h 67 deg ees clockwise u n be ween laye s. The
edge o he pa was buil by a single con ou pa h.
Figu e 2. Manu ac u ing pa h s a egy: laye pa h and pa pa h.
Figu e 1. Samylabs ALBA300 machine a IMH Campus.
The build chambe was illed wi h a gon (pu i y le el 99.999%) be o e ope a ion and a
cons an shielding gas low was used du ing he build o spa e emo al ollowing [
23
].
The oxygen con en in he build chambe was main ained below 0.43 ol% (4300 ppm)
du ing he p ocess ollowing [24,25].
The samples we e buil di ec ly on EN AW-5083 aluminum subs a e wi hou p e-
hea ing because he equipmen in which he expe imen was ca ied ou does no ha e a
subs a e hea e . The 20 mm hick subs a e pla e unc ioned as a hea sink o allow hea o
be conduc ed away om he samples.
The manu ac u ing pa h was gene a ed by SamyS udio 5.1 so wa e using a powde
laye hickness o 25
µ
m. Di e en pa ame e s we e se o he co e and he edge o he pa
o imp o e pa densi y and su ace quali y (Figu e 2). The co e was buil by a bidi ec ional
meande ha ching pa e n wi h 67 deg ees clockwise u n be ween laye s. The edge o he
pa was buil by a single con ou pa h.
J. Manu . Ma e . P ocess. 2022, 6, 164 3 o 15
Table 1. Alloy chemical composi ion.
Al
Si
Fe
Zn
Mg
Cu
Ti
Mn
Ni
Base
10.7
0.21
<0.01
0.26
<0.01
0.02
<0.01
<0.01
2.1. Samples Manu ac u ing
The sample pa s we e buil using a Samylabs ALBA300 machine (Figu e 1), which
has a ibe lase wi h a maximum powe ou pu o 250 W and 1080 nm wa eleng h. The
lase spo ocalized in he subs a e co esponds o a diame e o 71 μm (1/e2 me hod).
Figu e 1. Samylabs ALBA300 machine a IMH Campus.
The build chambe was illed wi h a gon (pu i y le el 99.999%) be o e ope a ion and
a cons an shielding gas low was used du ing he build o spa e emo al ollowing
[23]. The oxygen con en in he build chambe was main ained below 0.43 ol% (4300
ppm) du ing he p ocess ollowing [24,25].
The samples we e buil di ec ly on EN AW-5083 aluminum subs a e wi hou p e-
hea ing because he equipmen in which he expe imen was ca ied ou does no ha e a
subs a e hea e . The 20 mm hick subs a e pla e unc ioned as a hea sink o allow hea
o be conduc ed away om he samples.
The manu ac u ing pa h was gene a ed by SamyS udio 5.1 so wa e using a powde
laye hickness o 25 µm. Di e en pa ame e s we e se o he co e and he edge o he
pa o imp o e pa densi y and su ace quali y (Figu e 2). The co e was buil by a bidi-
ec ional meande ha ching pa e n wi h 67 deg ees clockwise u n be ween laye s. The
edge o he pa was buil by a single con ou pa h.
Figu e 2. Manu ac u ing pa h s a egy: laye pa h and pa pa h.
Figu e 2. Manu ac u ing pa h s a egy: laye pa h and pa pa h.
J. Manu . Ma e . P ocess. 2022,6, 164 4 o 14
The co e pa ame e s used o he manu ac u ing o he samples we e 800 mm/s
scanning speed, con inuous 250 W lase powe and 0.15 mm ha ch wid h. These p ocess
pa ame e alues we e se by he p e ious Box–Behnken Design o Expe imen s analysis
op imized o minimizing po osi y. Mo e in o ma ion abou Box-Behnken Design o Ex-
pe imen s can be ound a [
26
]. The con ou pa h was buil by 800 mm/s scanning speed,
con inuous 200 W lase powe and 0.07 mm con ou dis ance, which we e de e mined
based on p e ious wo ks.
The sample geome y was designed in o de o a oid addi ional suppo s uc u es,
o ule ou any e ec hey migh ha e on he inished pa , and o acili a e he manual
emo al o pa s om he subs a e (Figu e 3).
J. Manu . Ma e . P ocess. 2022, 6, 164 4 o 15
The co e pa ame e s used o he manu ac u ing o he samples we e 800 mm/s scan-
ning speed, con inuous 250 W lase powe and 0.15 mm ha ch wid h. These p ocess pa-
ame e alues we e se by he p e ious Box–Behnken Design o Expe imen s analysis
op imized o minimizing po osi y. Mo e in o ma ion abou Box-Behnken Design o Ex-
pe imen s can be ound a [26]. The con ou pa h was buil by 800 mm/s scanning speed,
con inuous 200 W lase powe and 0.07 mm con ou dis ance, which we e de e mined
based on p e ious wo ks.
The sample geome y was designed in o de o a oid addi ional suppo s uc u es,
o ule ou any e ec hey migh ha e on he inished pa , and o acili a e he manual
emo al o pa s om he subs a e (Figu e 3).
Figu e 3. Manu ac u ed samples o me allog aphic cha ac e iza ion.
2.2. Me allog aphic Cha ac e iza ion
Me allog aphic samples we e p epa ed in h ee di e en planes (Figu e 4) o check
he ela ionship o mic os uc u e–di ec ion, assuming ha he componen s manu ac-
u ed by LPBF show a high aniso opic empe a u e dependence.
Figu e 4. Cu ing planes o ca y ou he mic os uc u e inspec ion.
Samples we e p epa ed acco ding o he classical me hod [27]: esin embedded, pol-
ished, and e ched (2 mL o HF, 3 mL o HCl, 5 mL o HNO3, and 190 mL o wa e ). The
obse a ion o mic os uc u es was pe o med by means o ligh (b igh and da k ields)
and elec onic mic oscopy.
2.3. Modeling
A mul iscale app oach has been used o model he mic os uc u e e olu ion. Fi s ,
he he mo-physical ma e ial p ope ies ha e been calcula ed based on he alloy's chemi-
cal composi ion. Nex , he sample manu ac u ing p ocess has been simula ed by means
o a mac oscale simula ion, aking in o accoun he p e iously calcula ed ma e ial p op-
Figu e 3. Manu ac u ed samples o me allog aphic cha ac e iza ion.
2.2. Me allog aphic Cha ac e iza ion
Me allog aphic samples we e p epa ed in h ee di e en planes (Figu e 4) o check he
ela ionship o mic os uc u e–di ec ion, assuming ha he componen s manu ac u ed by
LPBF show a high aniso opic empe a u e dependence.
J. Manu . Ma e . P ocess. 2022, 6, 164 4 o 15
The co e pa ame e s used o he manu ac u ing o he samples we e 800 mm/s scan-
ning speed, con inuous 250 W lase powe and 0.15 mm ha ch wid h. These p ocess pa-
ame e alues we e se by he p e ious Box–Behnken Design o Expe imen s analysis
op imized o minimizing po osi y. Mo e in o ma ion abou Box-Behnken Design o Ex-
pe imen s can be ound a [26]. The con ou pa h was buil by 800 mm/s scanning speed,
con inuous 200 W lase powe and 0.07 mm con ou dis ance, which we e de e mined
based on p e ious wo ks.
The sample geome y was designed in o de o a oid addi ional suppo s uc u es,
o ule ou any e ec hey migh ha e on he inished pa , and o acili a e he manual
emo al o pa s om he subs a e (Figu e 3).
Figu e 3. Manu ac u ed samples o me allog aphic cha ac e iza ion.
2.2. Me allog aphic Cha ac e iza ion
Me allog aphic samples we e p epa ed in h ee di e en planes (Figu e 4) o check
he ela ionship o mic os uc u e–di ec ion, assuming ha he componen s manu ac-
u ed by LPBF show a high aniso opic empe a u e dependence.
Figu e 4. Cu ing planes o ca y ou he mic os uc u e inspec ion.
Samples we e p epa ed acco ding o he classical me hod [27]: esin embedded, pol-
ished, and e ched (2 mL o HF, 3 mL o HCl, 5 mL o HNO3, and 190 mL o wa e ). The
obse a ion o mic os uc u es was pe o med by means o ligh (b igh and da k ields)
and elec onic mic oscopy.
2.3. Modeling
A mul iscale app oach has been used o model he mic os uc u e e olu ion. Fi s ,
he he mo-physical ma e ial p ope ies ha e been calcula ed based on he alloy's chemi-
cal composi ion. Nex , he sample manu ac u ing p ocess has been simula ed by means
o a mac oscale simula ion, aking in o accoun he p e iously calcula ed ma e ial p op-
Figu e 4. Cu ing planes o ca y ou he mic os uc u e inspec ion.
Samples we e p epa ed acco ding o he classical me hod [
27
]: esin embedded, pol-
ished, and e ched (2 mL o HF, 3 mL o HCl, 5 mL o HNO3, and 190 mL o wa e ). The
obse a ion o mic os uc u es was pe o med by means o ligh (b igh and da k ields)
and elec onic mic oscopy.
2.3. Modeling
A mul iscale app oach has been used o model he mic os uc u e e olu ion. Fi s , he
he mo-physical ma e ial p ope ies ha e been calcula ed based on he alloy
'
s chemical
J. Manu . Ma e . P ocess. 2022,6, 164 5 o 14
composi ion. Nex , he sample manu ac u ing p ocess has been simula ed by means o a
mac oscale simula ion, aking in o accoun he p e iously calcula ed ma e ial p ope ies.
A e wa ds, he empe a u e p o iles ob ained as a esul o he mac oscale modeling ha e
been used o de ine bounda y condi ions ha , applied o he mic os uc u e e olu ion
model, p oduce equi alen cooling a es in he ma e ial (see Figu e 5).
J. Manu . Ma e . P ocess. 2022, 6, 164 5 o 15
e ies. A e wa ds, he empe a u e p o iles ob ained as a esul o he mac oscale model-
ing ha e been used o de ine bounda y condi ions ha , applied o he mic os uc u e e o-
lu ion model, p oduce equi alen cooling a es in he ma e ial (see Figu e 5).
Figu e 5. Flow cha o he mul iscale models' connec ion.
2.3.1. The mo–Physical Ma e ial P ope ies Calcula ion
Ma e ial p ope ies ha e been calcula ed by a CALPHAD (Compu e Coupling o
Phase Diag ams and The mochemis y) based modeling.
The CALPHAD me hodology is based on compu a ional he modynamics models,
which use expe imen al in o ma ion collec ed o bina y and e na y subsys ems o ex-
apola e he p ope ies o highe o de alloys (mul icomponen sys ems), p edic ing as-
pec s like he phases o some he mo-physical p ope ies, such as, o example, he densi y
o he en halpy among o he s [28].
In his case, comme cial CALPHAD so wa e [29] has been used o calcula e he ma-
e ial p ope ies o he alloy. A mic oseg ega ion model known as back di usion has been
used o p edic he ma e ial p ope ies, assuming a cooling a e acco ding o alues ound
in he bibliog aphy o he LPBF p ocess (750 ×·103 K/s) [30] and he chemical composi ion
p e iously measu ed (Table 1). The back di usion model [31] can be conside ed a balance
be ween he well-known le e ule model (which conside s a comple e mixing o he so-
lu e in he solid, ha is a e y good di usion) [32] and he Scheil model (which assumes
no di usion a all in he solid phase) [33]. The solid ac ion cu e has been ob ained o-
ge he wi h he densi y, he he mal conduc i i y, and he en halpy, all o hem empe a-
u e-dependen (Figu e 6).
Figu e 5. Flow cha o he mul iscale models' connec ion.
2.3.1. The mo–Physical Ma e ial P ope ies Calcula ion
Ma e ial p ope ies ha e been calcula ed by a CALPHAD (Compu e Coupling o
Phase Diag ams and The mochemis y) based modeling.
The CALPHAD me hodology is based on compu a ional he modynamics models,
which use expe imen al in o ma ion collec ed o bina y and e na y subsys ems o ex ap-
ola e he p ope ies o highe o de alloys (mul icomponen sys ems), p edic ing aspec s
like he phases o some he mo-physical p ope ies, such as, o example, he densi y o he
en halpy among o he s [28].
In his case, comme cial CALPHAD so wa e [
29
] has been used o calcula e he ma e-
ial p ope ies o he alloy. A mic oseg ega ion model known as back di usion has been
used o p edic he ma e ial p ope ies, assuming a cooling a e acco ding o alues ound
in he bibliog aphy o he LPBF p ocess (750
×·
10
3
K/s) [
30
] and he chemical compo-
si ion p e iously measu ed (Table 1). The back di usion model [
31
] can be conside ed a
balance be ween he well-known le e ule model (which conside s a comple e mixing
o he solu e in he solid, ha is a e y good di usion) [
32
] and he Scheil model (which
assumes no di usion a all in he solid phase) [
33
]. The solid ac ion cu e has been
ob ained oge he wi h he densi y, he he mal conduc i i y, and he en halpy, all o hem
empe a u e-dependen (Figu e 6).

J. Manu . Ma e . P ocess. 2022,6, 164 6 o 14
J. Manu . Ma e . P ocess. 2022, 6, 164 6 o 15
Figu e 6. The mo-physical p ope ies ob ained.
The so wa e used o model he LPBF p ocess is no able o wo k wi h empe a u e-
dependen p ope ies, he e o e, conside ing he as solidi ica ion a es and he punc ual
hea sou ce, he a e age alue a solid s a e has been used o he densi y and he he mal
conduc i i y. The empe a u e co esponding o 50% o he solid ac ion has been as-
sumed as he mel ing poin . Ano he es ain o he so wa e is he equi emen o use
he speci ic hea (𝐶𝑝) alue ins ead o he en halpy alue (𝐻). The e o e, he speci ic hea
cu e has been calcula ed om he en halpy cu e a solid-s a e, ollowing Equa ion (1),
whe e (T) ep esen s he empe a u e, and i s a e age alue has been used in he simu-
la ion.
𝐶𝑝(T)=𝜕H
𝜕T
(1)
2.3.2. Sample Manu ac u ing Modeling
The sample manu ac u ing has been modeled by means o comme cial so wa e spe-
ci ically de eloped o he LPBF simula ion [34]. The sample geome y is a small cube wi h
a py amidal suppo whose dimensions can be seen in Figu e 7 (le ).
Figu e 7. Sample geome y (le ), mesh ( igh ).
Figu e 6. The mo-physical p ope ies ob ained.
The so wa e used o model he LPBF p ocess is no able o wo k wi h empe a u e-
dependen p ope ies, he e o e, conside ing he as solidi ica ion a es and he punc ual
hea sou ce, he a e age alue a solid s a e has been used o he densi y and he he mal
conduc i i y. The empe a u e co esponding o 50% o he solid ac ion has been assumed
as he mel ing poin . Ano he es ain o he so wa e is he equi emen o use he speci ic
hea
(Cp)
alue ins ead o he en halpy alue
(H)
. The e o e, he speci ic hea cu e has
been calcula ed om he en halpy cu e a solid-s a e, ollowing Equa ion (1), whe e
(T)
ep esen s he empe a u e, and i s a e age alue has been used in he simula ion.
Cp(T)=∂H
∂T(1)
2.3.2. Sample Manu ac u ing Modeling
The sample manu ac u ing has been modeled by means o comme cial so wa e
speci ically de eloped o he LPBF simula ion [
34
]. The sample geome y is a small cube
wi h a py amidal suppo whose dimensions can be seen in Figu e 7(le ).
J. Manu . Ma e . P ocess. 2022, 6, 164 6 o 15
Figu e 6. The mo-physical p ope ies ob ained.
The so wa e used o model he LPBF p ocess is no able o wo k wi h empe a u e-
dependen p ope ies, he e o e, conside ing he as solidi ica ion a es and he punc ual
hea sou ce, he a e age alue a solid s a e has been used o he densi y and he he mal
conduc i i y. The empe a u e co esponding o 50% o he solid ac ion has been as-
sumed as he mel ing poin . Ano he es ain o he so wa e is he equi emen o use
he speci ic hea (𝐶𝑝) alue ins ead o he en halpy alue (𝐻). The e o e, he speci ic hea
cu e has been calcula ed om he en halpy cu e a solid-s a e, ollowing Equa ion (1),
whe e (T) ep esen s he empe a u e, and i s a e age alue has been used in he simu-
la ion.
𝐶𝑝(T)=𝜕H
𝜕T
(1)
2.3.2. Sample Manu ac u ing Modeling
The sample manu ac u ing has been modeled by means o comme cial so wa e spe-
ci ically de eloped o he LPBF simula ion [34]. The sample geome y is a small cube wi h
a py amidal suppo whose dimensions can be seen in Figu e 7 (le ).
Figu e 7. Sample geome y (le ), mesh ( igh ).
Figu e 7. Sample geome y (le ), mesh ( igh ).
J. Manu . Ma e . P ocess. 2022,6, 164 7 o 14
The manu ac u ing p ocess has been modeled conside ing he expe imen al condi ions
p e iously de ailed in Sec ion 2.1. The model includes he powde laye s deposi ion and
he hea ans e phenomena, including he powde usion by he lase and he cooling.
Re e ing o he bounda y condi ions, he con ec ion coe icien has been ixed o a
alue equi alen o na u al con ec ion (12.7 W/m
2
K). The powe abso p ion is included
as a pe cen age, and i has been ixed a 11%. The p ocess condi ions co espond o hose
used du ing he pa manu ac u ing (Table 2).
Table 2. P ocess pa ame e s.
Ha ch Space
(mm)
Lase Diame e
(mm) Veloci y (mm/s) Lase Powe
(W)
Powde Laye
Thickness (mm)
0.15 0.071 800 200 0.025
The main phenomenon conside ed in he simula ion is hea ans e , he e o e, he
main equa ion sol ed is he hea ans e equa ion shown in (2).
Q=ρ·Cp∂T
∂ − ∇·(k·∇T)(2)
The model is sol ed by he ini e elemen me hod. Al hough ou in e es is ocused on
he he mal beha io du ing he LPBF manu ac u ing, he he mo-mechanical sol e has
been used due o i s be e s abili y when compa ed wi h he s abili y o he he mal sol e
alone, included in he so wa e. In ac , using he he mal so wa e alone, con e gence
p oblems we e expe ienced ha p e en ed i s use in his case.
Re e ing o he ma e ial p ope ies, he mel ing poin , he densi y, he he mal con-
duc i i y, and he speci ic hea alues p e iously calcula ed ha e been used (Table 3). The
emissi i y alue co esponds o he de aul alue included in he so wa e da abase o
his alloy. Fo he mechanical p ope ies (Young’s modulus, Poisson a io, e c.), he de aul
alues included in he da abase o his alloy ha e been used.
Table 3. Ma e ial p ope ies.
Densi y (kg/m3)Speci ic Hea
(J/kgK)
Conduc i i y
(W/mK)
Mel ing Poin
(K) Emissi i y
2608.79 1005.11 151.55 846.82 0.18
The geome y has been disc e ized by a egula mesh o med by oxel elemen s wi h
0.25 mm side (Figu e 7 igh ); his size is 10 imes bigge han he powde laye hickness,
bu i has no been possible o employ smalle elemen s due o so wa e limi a ions.
The empe a u e– ime cu e o e e y node has been ex ac ed and p ocessed o ob ain
he solidi ica ion cooling a e (SCR), which has been calcula ed ollowing Equa ion (3),
whe e
T
ep esen s empe a u es exp essed in K and
ep esen imes exp essed in seconds.
The SCR is exp essed in K/s.
SCR (K/s)=Tliquidus(K)−Tsolidus(K)
liquidus(s)− solidus (s)(3)
The SCR dis ibu ion in he manu ac u ed pa has been analyzed by applying clus e -
ing echniques. Mo e p ecisely, a KMeans algo i hm has been used o classi y he cooling
a es in 5 di e en clus e s. KMeans is a classi ica ion algo i hm ha clus e s he da a by
ying o sepa a e he samples in
n
g oups o equal a iance, minimizing a c i e ion known
as he ine ia o wi hin-clus e sum-o -squa es [35].
J. Manu . Ma e . P ocess. 2022,6, 164 8 o 14
2.3.3. Mic os uc u e P edic ion
The inal objec i e is o p edic he lamella spacing o he mic os uc u e. Conside ing
ha he magni ude o de expec ed o he lamella spacing is mic ons, a educed model
based on a ep esen a i e olume elemen (RVE) has been used o equilib a e he use o a
mesh smoo h enough wi h a easonable calcula ion ime.
The RVE used is a cube 0.25 mm side, whose heigh is equi alen o 10 powde laye s
(0.025 mm/laye ). I has been d awn and disc e ized by a ini e elemen mesh o med by
e ahed al elemen s 0.025 mm side, gi ing a mesh o med by 1479 nodes and 6786 elemen s
(Figu e 8le ).
J. Manu . Ma e . P ocess. 2022, 6, 164 8 o 15
2.3.3. Mic os uc u e P edic ion
The inal objec i e is o p edic he lamella spacing o he mic os uc u e. Conside -
ing ha he magni ude o de expec ed o he lamella spacing is mic ons, a educed
model based on a ep esen a i e olume elemen (RVE) has been used o equilib a e he
use o a mesh smoo h enough wi h a easonable calcula ion ime.
The RVE used is a cube 0.25 mm side, whose heigh is equi alen o 10 powde laye s
(0.025 mm/laye ). I has been d awn and disc e ized by a ini e elemen mesh o med by
e ahed al elemen s 0.025 mm side, gi ing a mesh o med by 1479 nodes and 6786 ele-
men s (Figu e 8 le ).
Figu e 8. RVE model (le ). Hea ex ac ion condi ion ( igh ).
Comme cial so wa e speci ically de eloped o me al cas ing simula ions [36] has
been used o p edic he mic os uc u e. As a consequence, some assump ions mus be
made in o de o se up he case o be ep esen a i e o he LPBF p ocess. The idea is o
ep oduce only he solidi ica ion p ocess. Fo his eason, he ini ial empe a u e o he
alloy has been ixed, equal o liquidus empe a u e (585 °C), and a hea ex ac ion bound-
a y condi ion has been applied o he op su ace o he RVE (Figu e 8 igh ).
The hea ex ac ion bounda y condi ion has been adjus ed o p oduce an SCR in he
RVE simila o he alues p edic ed in he LPBF p ocess simula ion, ollowing equa ion
(4). Conside ing a olume equal o 1.5625 ·10−11 m3, an a e age alue in he solidi ica-
ion ange o he densi y and he speci ic hea (𝜌 = 2507 kg/m3,𝐶𝑝 =1132.8 J/kgK) and
a SCR equal o 698,062 K/s (see Sec ion 3), a alue equal o 30.98 W is ob ained.
𝑄 (W) = 𝑉(m3)· 𝜌(kg m3
⁄ ) ·𝐶𝑝(J/kgK) · 𝑆𝐶𝑅(K/s)
(4)
The main go e ning equa ions o he physics in ol ed in alloy cooling and solidi i-
ca ion, o his case, a e he hea ans e Equa ion (2) and he nuclea ion and g ow h o
he g ains. The so wa e sol es he nuclea ion and g ow h o he g ains ollowing Equa-
ions (5) and (6) p oposed by Old ield [37], whe e 𝑁𝑛𝑢𝑐𝑙𝑒𝑖 is he numbe o nuclei, 𝐴𝑒 and
𝑛 a e he nuclea ion cons an s, T is he empe a u e, (𝑑𝑅_𝑛𝑢𝑐𝑙𝑒𝑖)/𝑑 is he g ow h, and
𝜇𝑔 is he g ow h coe icien .
N𝑛𝑢𝑐𝑙𝑒𝑖 = 𝐴𝑒·(∆T)𝑛
(5)
𝑑𝑅𝑛𝑢𝑐𝑙𝑒𝑖
𝑑 = 𝜇𝑔·(∆T)2
(6)
Ma e ial p ope ies we e calcula ed by he CALPHAD me hod p e iously men-
ioned, bu he alues o he nuclea ion and g ow h cons an s (𝐴𝑒, 𝑛, 𝜇𝑒) mus be adjus ed
depending on he alloy, he cooling a e, he me al ea men , e c. In his case, an op imi-
za ion algo i hm has been used o adjus hem.
The model adjus men can be conside ed an op imiza ion p oblem, whe e he objec-
i e is o minimize he di e ences be ween he alues p edic ed by he simula ion and he
Figu e 8. RVE model (le ). Hea ex ac ion condi ion ( igh ).
Comme cial so wa e speci ically de eloped o me al cas ing simula ions [
36
] has
been used o p edic he mic os uc u e. As a consequence, some assump ions mus be
made in o de o se up he case o be ep esen a i e o he LPBF p ocess. The idea is o
ep oduce only he solidi ica ion p ocess. Fo his eason, he ini ial empe a u e o he
alloy has been ixed, equal o liquidus empe a u e (585
◦
C), and a hea ex ac ion bounda y
condi ion has been applied o he op su ace o he RVE (Figu e 8 igh ).
The hea ex ac ion bounda y condi ion has been adjus ed o p oduce an SCR in he
RVE simila o he alues p edic ed in he LPBF p ocess simula ion, ollowing equa ion
(4). Conside ing a olume equal o 1.5625
·
10
−11 m3
, an a e age alue in he solidi ica ion
ange o he densi y and he speci ic hea
ρ=2507 kg/m3,Cp =1132.8 J/kgK
and a
SCR equal o 698, 062 K/s (see Sec ion 3), a alue equal o 30.98 W is ob ained.
Q(W)=Vm3·ρkg/m3·Cp(J/kgK)·SCR(K/s)(4)
The main go e ning equa ions o he physics in ol ed in alloy cooling and solidi ica-
ion, o his case, a e he hea ans e Equa ion (2) and he nuclea ion and g ow h o he
g ains. The so wa e sol es he nuclea ion and g ow h o he g ains ollowing Equa ions
(5) and (6) p oposed by Old ield [
37
], whe e
Nnuclei
is he numbe o nuclei,
Ae
and
n
a e
he nuclea ion cons an s,
T
is he empe a u e,
(dR_nuclei)/d
is he g ow h, and
µg
is he
g ow h coe icien .
Nnuclei =Ae·(∆T)n(5)
dRnuclei
d =µg·(∆T)2(6)
Ma e ial p ope ies we e calcula ed by he CALPHAD me hod p e iously men ioned,
bu he alues o he nuclea ion and g ow h cons an s
(Ae,n,µe)
mus be adjus ed de-
pending on he alloy, he cooling a e, he me al ea men , e c. In his case, an op imiza ion
algo i hm has been used o adjus hem.
The model adjus men can be conside ed an op imiza ion p oblem, whe e he objec i e
is o minimize he di e ences be ween he alues p edic ed by he simula ion and he
e e ence alues measu ed expe imen ally. This is achie ed modi ying he alues assigned
J. Manu . Ma e . P ocess. 2022,6, 164 9 o 14
o he a iables o be co ela ed (in his case, he nuclea ion and g ow h cons an s) un il
he alue p edic ed (in his case, he lamella spacing) ma ches he alues measu ed ex-
pe imen ally. This minimiza ion can be made by manually modi ying he alues assigned
o he a iables [
38
–
41
] o by means o op imiza ion algo i hms. Se e al au ho s ha e
explo ed he use o di e en ypes o op imiza ion algo i hms o model adjus men s [
42
–
48
]
in di e en ields. In his case, he NEWUOA algo i hm included in he Py hon Pa allel
Global Mul iobjec i e Op imize [
49
] was used o pe o m he adjus men . NEWUOA is a
de e minis ic local op imiza ion algo i hm de eloped by M.F.D. Powell is well known o
his wo k in nume ical analysis, especially in op imiza ion [50].
3. Resul s and Discussion
3.1. Me alog aphycal Cha ac e iza ion
The mic os uc u e in planes YZ and XZ, as de ined in Figu e 4, shows a s uc u e
e y simila o hose ob ained in a ypical welding p ocess. The di e en weld seams can
be obse ed in Figu e 9.
J. Manu . Ma e . P ocess. 2022, 6, 164 9 o 15
e e ence alues measu ed expe imen ally. This is achie ed modi ying he alues as-
signed o he a iables o be co ela ed (in his case, he nuclea ion and g ow h cons an s)
un il he alue p edic ed (in his case, he lamella spacing) ma ches he alues measu ed
expe imen ally. This minimiza ion can be made by manually modi ying he alues as-
signed o he a iables [38–41] o by means o op imiza ion algo i hms. Se e al au ho s
ha e explo ed he use o di e en ypes o op imiza ion algo i hms o model adjus men s
[42–48] in di e en ields. In his case, he NEWUOA algo i hm included in he Py hon
Pa allel Global Mul iobjec i e Op imize [49] was used o pe o m he adjus men .
NEWUOA is a de e minis ic local op imiza ion algo i hm de eloped by M.F.D. Powell is
well known o his wo k in nume ical analysis, especially in op imiza ion [50].
3. Resul s and Discussion
3.1. Me alog aphycal Cha ac e iza ion
The mic os uc u e in planes YZ and XZ, as de ined in Figu e 4, shows a s uc u e
e y simila o hose ob ained in a ypical welding p ocess. The di e en weld seams can
be obse ed in Figu e 9.
Figu e 9. Mic os uc u e on he XZ (le ) and YZ ( igh ) planes (×100).
In he plane XY, a di e en pa e n can be obse ed (Figu e 10). This is mo e ela ed
o he welding pa h along he lase beam mo emen on he uppe su ace o he sample
(diagonally in e laced).
Figu e 10. Mic os uc u e on he XY plane ×50 (le ), ×100 ( igh ).
To ha e a close iew o he mic os uc u e, samples we e obse ed in a scanning
elec on mic oscope. The images (Figu e 11) showed a needle-like s uc u e (eu ec ic ype)
qui e ypical in as -cooled me als. These needles appea ine in he co e o he weld seam
and sligh ly coa se in he bo de o adjacen seams.
Figu e 9. Mic os uc u e on he XZ (le ) and YZ ( igh ) planes (×100).
In he plane XY, a di e en pa e n can be obse ed (Figu e 10). This is mo e ela ed
o he welding pa h along he lase beam mo emen on he uppe su ace o he sample
(diagonally in e laced).
J. Manu . Ma e . P ocess. 2022, 6, 164 9 o 15
e e ence alues measu ed expe imen ally. This is achie ed modi ying he alues as-
signed o he a iables o be co ela ed (in his case, he nuclea ion and g ow h cons an s)
un il he alue p edic ed (in his case, he lamella spacing) ma ches he alues measu ed
expe imen ally. This minimiza ion can be made by manually modi ying he alues as-
signed o he a iables [38–41] o by means o op imiza ion algo i hms. Se e al au ho s
ha e explo ed he use o di e en ypes o op imiza ion algo i hms o model adjus men s
[42–48] in di e en ields. In his case, he NEWUOA algo i hm included in he Py hon
Pa allel Global Mul iobjec i e Op imize [49] was used o pe o m he adjus men .
NEWUOA is a de e minis ic local op imiza ion algo i hm de eloped by M.F.D. Powell is
well known o his wo k in nume ical analysis, especially in op imiza ion [50].
3. Resul s and Discussion
3.1. Me alog aphycal Cha ac e iza ion
The mic os uc u e in planes YZ and XZ, as de ined in Figu e 4, shows a s uc u e
e y simila o hose ob ained in a ypical welding p ocess. The di e en weld seams can
be obse ed in Figu e 9.
Figu e 9. Mic os uc u e on he XZ (le ) and YZ ( igh ) planes (×100).
In he plane XY, a di e en pa e n can be obse ed (Figu e 10). This is mo e ela ed
o he welding pa h along he lase beam mo emen on he uppe su ace o he sample
(diagonally in e laced).
Figu e 10. Mic os uc u e on he XY plane ×50 (le ), ×100 ( igh ).
To ha e a close iew o he mic os uc u e, samples we e obse ed in a scanning
elec on mic oscope. The images (Figu e 11) showed a needle-like s uc u e (eu ec ic ype)
qui e ypical in as -cooled me als. These needles appea ine in he co e o he weld seam
and sligh ly coa se in he bo de o adjacen seams.
Figu e 10. Mic os uc u e on he XY plane ×50 (le ), ×100 ( igh ).
To ha e a close iew o he mic os uc u e, samples we e obse ed in a scanning
elec on mic oscope. The images (Figu e 11) showed a needle-like s uc u e (eu ec ic ype)
qui e ypical in as -cooled me als. These needles appea ine in he co e o he weld seam
and sligh ly coa se in he bo de o adjacen seams.