Ci a ion: Paska, Z.; Halama, R.;
Dymacek, P.; Go inda aj, B.; Rojicek,
J. Compa ison o Tensile and C eep
P ope ies o SAC305 and SACX0807
a Room Tempe a u e wi h DIC
Applica ion. Appl. Sci. 2024,14, 604.
h ps://doi.o g/10.3390/
app14020604
Academic Edi o : Ma in Haga a
Recei ed: 26 Decembe 2023
Re ised: 5 Janua y 2024
Accep ed: 8 Janua y 2024
Published: 10 Janua y 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
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applied
sciences
A icle
Compa ison o Tensile and C eep P ope ies o SAC305 and
SACX0807 a Room Tempe a u e wi h DIC Applica ion
Zbynek Paska 1, Radim Halama 1,* , Pe Dymacek 2, Bhu anesh Go inda aj 1and Ja osla Rojicek 1
1
Depa men o Applied Mechanics, Facul y o Mechanical Enginee ing, VŠB—Technical Uni e si y o Os a a,
17. lis opadu 2172/15, 708 00 Os a a, Czech Republic; [email p o ec ed] (Z.P.);
[email p o ec ed] (B.G.); ja osla . [email p o ec ed] (J.R.)
2Ins i u e o Physics o Ma e ials, Czech Academy o Sciences, Zizko a 22, 616 00 B no, Czech Republic;
[email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac :
The con ibu ion p esen s he e i ica ion o he me hodology o accele a ed c eep es s om
he poin o iew o ob aining mo e in o ma ion abou he s ess–s ain beha iou o he in es iga ed
ma e ials using he Digi al Image Co ela ion me hod. C eep es s a e pe o med on SAC305 and
SACX0807 lead- ee solde s and a e supplemen ed by nume ical modelling using he ini e elemen
me hod, conside ing he iscoplas ic model based on he heo y o Pe zyna, Chaboche, and No on.
The s ess–s ain beha iou o bo h solde s appea s o be e y simila a applied s ain a es o 0.0002–
0.0026%/s and applied c eep s esses o 15–28 MPa. Ini ially, he iscoplas ic model is calib a ed
using an analy ical app oach. Then, he ini e elemen model upda ing app oach is used o op imise
he ma e ial pa ame e s based on he simul aneous simula ions o c eep and ensile es s. As a esul ,
he o al objec i e unc ion alue is educed almos i e imes due o op imisa ion. The p oposed ype
o accele a ed es wi h an hou glass specimen p o es o be sui able o calib a ing he conside ed
class o iscoplas ic models. The main bene i is ha a single specimen is equi ed o ob ain c eep
cu es on a ious s ess le els.
Keywo ds:
c eep; SAC305; SACX0807; DIC; iscoplas ici y; FEM; No on model; Pe zyna model;
Chaboche model
1. In oduc ion
The e olu ion o lead- ee solde alloys, pa icula ly in he con ex o elec onic manu-
ac u ing, ep esen s a signi ican shi in ma e ial science and enginee ing. While he use
o solde ing o mechanical and elec ical connec ions has a long his o y, ecen ad ance-
men s ha e been d i en by he need o en i onmen ally iendly and high-pe o mance
al e na i es o adi ional in–lead (Sn-Pb) solde s. Solde ing elec onic componen s o-
ge he using ‘so solde ing’, which in ol es much lowe empe a u es, is a much mo e
ecen de elopmen .
Ea ly so solde ing employed pu e in. Alloys we e c ea ed o e ime o sol e
challenges such as he mal cycling, shock ole ance, elec on mig a ion, and whiske
o ma ion in in-based alloys [
1
]. Al hough lead could play his ole in mos solde ing
applica ions, he elimina ion o lead om de ices and he implemen a ion o new s anda ds
o mo e ine-pi ched pa s necessi a ed he de elopmen o new solde alloys. To be
conside ed a iable al e na i e o Sn-Pb solde s, lead- ee candida es mus o e quali ies
and unc ionali y compa able o o g ea e han hose o eu ec ic o close eu ec ic Sn-Pb
solde s. The p esence o Pb in solde s imp o es he o e all e iciency o he Sn-Pb solde .
In addi ion o ca ying elec ici y, solde join s p o ide mechanical suppo o elec i-
cal de ices. As a esul , he mechanical quali ies o solde alloys a e c i ical in he p oduc ion
o long-las ing p oduc s. Tensile cha ac e is ics and c eep a e pa icula ly impo an o
Appl. Sci. 2024,14, 604. h ps://doi.o g/10.3390/app14020604 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2024,14, 604 2 o 21
e alua ing he mechanical p ope ies o solde s and ha e consequen ly been widely e-
sea ched [
2
]. SAC is an abb e ia ion o he alloy SnAgCu ( in, sil e , and coppe ). SAC
alloys a e he mos widely u ilised alloys in he elec onics indus y. SAC305, a lead- ee
g ade alloy, is made up o 96.5% in, 3% sil e , and 0.5% coppe . I s 3% sil e componen
ensu es op imum we ing quali ies as well as balanced he mal a igue, solde link powe ,
and mechanical s ess esis ance [
3
]. Solde pas es a e made up o a me al alloy powde
(abou 90% by weigh ) and a c eam-like o ganic chemical componen (app oxima ely 10%
by weigh ). ‘Flux’, he c eam o o ganic componen s, is usually a ade sec e o pa en -
p o ec ed [
4
]. This s udy aims o del e deepe in o he mechanical beha iou o Pb- ee
solde alloys, speci ically hei esponse o di e en s ain a es and cons an s esses. A
model conside ing bo h elas ic and plas ic de o ma ion along wi h c eep beha iou is being
de eloped o accu a ely ep esen he s ess–s ain cha ac e is ics o hese alloys [5,6].
The mechanical cha ac e is ics o lead- ee solde s, pa icula ly SAC305, a e highly
dependen on pa ame e s such as empe a u e and s ain a e. This eliance is caused by he
high homologous empe a u e o hese solde alloys, which makes compa ing esul s ac oss
esea ch di icul . Such dispa i ies a e caused by a lack o consis en es ing ci cums ances,
which includes di e ences in specimen p epa a ion p ocesses and s ain a es. Despi e
hese di icul ies, a b oad spec um o mechanical cha ac e is ics has been de e mined. The
elas ic modulus o SAC alloys, including SAC305, no mally anges om 30 o 54 GPa, wi h
he majo i y o alues alling be ween 40 and 50 GPa. In e ms o ensile pa ame e s, he
Sn-3.9Ag-0.5Cu a ia ion has an elas ic modulus o 50.3 GPa, an ul ima e ensile s eng h
(UTS) o 54 MPa, and a yield s eng h (YS) o 36.2 MPa. Va ia ions in hese alues a e
obse ed be ween esea ch, which can be a ibu ed o he a ious es ing p ocedu es used.
As a esul , hese s a ed da a mus be e alua ed wi h cau ion, aking in o accoun he high
he e ogenei y due o he a ious p oduc ion echnologies and condi ions [7].
SACX0807 (Alpha SACX Plus 0807) is a low-sil e lead- ee alloy in ended o gi e
solde ing and eliabili y cha ac e is ics equal o hose o highe -sil e SAC305. Tensile
p ope ies and mic os uc u e a e compa ed wi h SAC305 in [8]. The co osion beha iou
o SAC305 and SACX0807 unde applica ion condi ions was co ela ed wi h he mic os uc-
u e o he alloy and he elec ochemical beha iou in [
9
,
10
]. The c eep beha iou o
SACX0807 has no ye been epo ed. This is an impo an gap in he ield ha should be
co e ed by his wo k (a oom empe a u e).
Wi h a ocus on SAC alloys such as SAC305 o SACX0807, i is necessa y o in es i-
ga e he S ain Ra e Sensi i i y (SRS) o he ma e ial. The peculia mic os uc u e o his
alloy, which includes Sn, Ag3Sn, and Cu6Sn5, has a subs an ial impac on i s mechanical
beha iou a a ied s ain a es. In pa icula , SAC305 exhibi s a dec ease in SRS as he
empe a u e and s ain a e inc ease, which is an impo an elemen in de e mining he
mechanical s abili y o solde connec ions in elec onic de ices [
11
]. This disco e y is
especially impo an in ope a ional se ings in ol ing dynamic mechanical loads. Unde -
s anding hese co ela ions is c i ical o o ecas ing solde pe o mance and gua an eeing
elec onic assembly du abili y [12].
The a igue li e o SAC305 unde di e en wa e loading si ua ions, as in es iga ed
in publica ions [
13
–
15
], gi es impo an insigh s in o he solde ’s beha iou . By including
bo h plas ic and c eep s ain componen s, hey conside ably imp o e ou knowledge o
SAC305 a igue beha iou . Ano he c i ical conside a ion is he e ec o ageing on he
a igue li e o SAC305 [
16
]. The lowe mechanical a igue li e due o age, no ably in SAC305,
shows he impo ance o a ho ough examina ion o s ess elaxa ion and low cycle a igue
ea u es. A ecen s udy [
5
] emphasises he impo ance o he cons i u i e iscoplas ic
model o SAC305 and p o ides an in-dep h look a he a che ing beha iou o SAC305
unde cyclic loading, highligh ing i s lowe ime-dependen de o ma ion compa ed o Sn-
37Pb. This di e ence is due o he dis inc mic os uc u al and mechanical cha ac e is ics
o SAC305, which emphasises he excep ional capaci y o he ma e ial o ca y cyclic loads.
This is c i ical o he dependabili y o mechanically s essed elec onic sys ems.
Appl. Sci. 2024,14, 604 3 o 21
Some au ho s o he pape de eloped a special expe imen al echnique based on he
Digi al Image Co ela ion (DIC) me hod, which enables he accele a ion o he es ing e-
ga ding he necessi y o ma e ial model calib a ion, as well as he ob aining o de o ma ion
cha ac e is ics unde di e en modes o loading [
17
–
21
]. I is based on measu ing he his-
o y o s ain on he cu ed pa o he s anda d specimen o mechanical es ing. The e o e,
i b ings ex a da a o he measu emen by he ex ensome e , leading o a mo e e icien
way o es ing s ess–s ain beha iou . Using he newly de eloped DIC echnique, i is
possible o e icien ly ob ain he cyclic s ess–s ain cu e [
18
] o se e al a che ing cu es
a lowe loading le els [
17
]. This ime, o he knowledge o he au ho s, he de eloped
DIC echnique is i s applied o he c eep o solde alloys. The i s applica ions o he
expe imen al analysis o c eep o addi i ely p epa ed composi es and me als a e epo ed
in [19–21].
Fo nume ical simula ions o he SAC305 ma e ial, he Anand ma e ial model is o en
used, see [
22
]. The Anand ma e ial model was designed o ho wo king me als [
23
] and
can cap u e he e ec o empe a u e and s ain a e. Anand model is desc ibed by using
nine pa ame e s. Tensile es s and c eep es s a e equen ly sol ed sepa a ely [
24
]. In
he li e a u e, di e en pa ame e alues can be ound o simila o e en he same ma-
e ials, see [
25
]. The pa ame e alue is usually de e mined by he leas squa es me hod
as a non-linea eg ession om comp ession [
23
], ensile o c eep es s [
24
]. A icle [
25
]
p esen s he pa ame e alues o a ma e ial labelled (Sn95.5Ag3.8Cu0.7) om h ee di -
e en pape s. The pa ame e alues di e e y signi ican ly, o example, pa ame e
A=15.773 ÷1.63 ×106
. The p oblem is also sol ed in [
26
]. In addi ion, he model is
no easible o p edic ing he beha iou o complex loading condi ions, such as mul iax-
ial loading o cyclic loading, see [
27
,
28
]. The Anand model does no include backs ess;
he e o e, he model is no able o desc ibe kinema ic ha dening du ing cyclic loading.
This ype o beha iou can also be desc ibed by a combina ion o ma e ial models.
The e is a ma e ial model o iso opic o kinema ic ha dening, a model o cap u ing
iscoplas ic beha iou ( ime-dependen , iscosi y), and a model o solu ions o c eep.
The Chaboche model wi h an op ional numbe o pa ame e s can be used o desc ibe
kinema ic ha dening [
29
]. The Pe zyna ma e ial model wi h wo pa ame e s can be used
o desc ibe ime-dependen beha iou [
30
]. The beha iou du ing seconda y c eep can be
desc ibed using he No on ela ion [
31
]. Comme cial so wa e ( o example, Ansys 2022
R2, MSC.Ma c 2023.1) can combine hese ma e ial models and can he e o e be used o
simula e he complex beha iou o he ma e ial.
The implici s ess in eg a ion algo i hm is c i ical in he con ex o building cons i-
u i e models o solde s, pa icula ly in he cons uc ion o algo i hms o empe a u e-
cons an de o ma ion. In his special case, he combina ion o Chaboche–Pe zyna–No on
models can be e icien ly implemen ed in o an FE code by using he algo i hm o Kobayashi
e al. [
32
], which is based on successi e subs i u ion wi h an applica ion o a damped
New on–Raphson me hod. Tempe a u e in luence is usually in oduced using exponen ial
and polynomial unc ions o some pa ame e s. These unc ions a e c i ical in p ecisely
p edic ing he beha iou o solde ma e ials a a ious empe a u es [
33
]. Fu he mo e,
o accu a e cap u ing o he elaxa ion beha iou , an algo i hm sui ed o a iscoplas ic
model wi h he inclusion o a s a ic eco e y e m mus be implemen ed [
34
]. The e u n
mapping me hod is used in his app oach, which inco po a es common ac ics o he elas ic
p edic o and he plas ic co ec o wi hin he in eg a ion algo i hm.
The p ima y pu pose o his s udy is o compa e he solde alloys o SAC305 and
SACX0807 om ensile p ope ies and c eep beha iou poin s o iew. C eep and ensile
es s we e pe o med a oom empe a u e, whe eas axial s ain was measu ed by using
he DIC me hod, which allowed he cap u e o s ain con ou s. The bene i o DIC was
ully used in c eep es s, which we e conduc ed on an hou glass- ype specimen. Cap u -
ing s ains on pa icula c oss-sec ions o he specimen b ings mo e c eep cu es om a
single c eep es . This is bene icial o calib a ing iscoplas ic models, as p esen ed in he
nume ical s udy o SAC305 based on he esul s o he classic mechanical es s [35,36].
Appl. Sci. 2024,14, 604 4 o 21
The me hod o de e mining he alues o ma e ial pa ame e s o ma e ial models
is usually desc ibed in he sou ce a icles o he gi en ma e ial model. Howe e , hese
p ocedu es usually do no in ol e combina ions o ma e ial models. Ano he me hod is
o use he Fini e Elemen Model Upda ing (FEMU) app oach. He e, an expe imen (o
se o expe imen s) is simula ed using he ini e elemen me hod wi h a selec ed ma e ial
model. The esul o he simula ion is compa ed wi h he esul o he expe imen , and by
g adually changing he alues o he ma e ial pa ame e s, he di e ence be ween hem is
minimised. Nowadays, he e is comme cial so wa e ( o example, Op iSLang 2022 R2)
ha allows his me hod o be used, including a ew minimisa ion algo i hms (g adien ,
e olu iona y, simplex, e c.).
Howe e , he main no el y o he a icle is he p esen a ion o a new accele a ed
echnique using he DIC me hod o he e alua ion o he c eep p ope ies o me allic
ma e ials. The s ains cap u ed ia he op ical me hod on he cu ed su ace o an hou glass
specimen in ime a e e alua ed in he o m o c eep cu es o selec ed c oss-sec ions
co esponding o di e en nominal s esses. The new app oach can help expe imen e s
es ima e he No on exponen and plan classic c eep es s. The e o e, he accele a ed
app oach can se e as a ool o compa ing he c eep p ope ies o a ious ma e ials (in
his a icle, he compa ison o SAC305 and SACX0807).
2. Ma e ials and Me hods
The chemical composi ion o he SAC305 and SACX0807 solde s acco ding o he
manu ac u e is shown in Table 1. The specimens we e machined om ingo s wi h 21 mm
diame e and 290 mm o al leng h. All specimens we e es ed a e wo yea s o ageing o
na u al ma e ial. The geome y o he specimens and he speci ic es ing condi ions a e
desc ibed in he ollowing sec ions.
Table 1. Chemical composi ion o SAC305 and SACX0807 (chosen elemen s).
Solde Alloy Ag [%] Cu [%] Pb [%] Sb [%] Bi [%]
SAC305 [3] 3 0.5 0.07 max. 0.1 max. 0.1 max.
SACX0807 [
8
]
1.0 1.0 0.1 0.2 0.2
2.1. Con en ional Mechanical Tes ing
2.1.1. Tensile Tes s
The basic mechanical p ope ies o solde s we e de e mined using a uni e sal es ing
machine wi h ensile pneuma ic jaws a he Depa men o Applied Mechanics o he
Facul y o Mechanical Enginee ing o VSB—Technical Uni e si y o Os a a. Tin solde s
a e gene ally sensi i e o s ain a e e en a oom empe a u e due o hei low mel ing
poin (216–218
◦
C). The e o e, he ensile es s we e pe o med a h ee posi ion a es o
1 mm/min, 5 mm/min and 10 mm/min. The geome y o he specimen o he ensile es s
is shown in Figu e 1. Tes s ealized a 1 and 10 mm/min posi ion a es we e pe o med on
an elec omechanical es ing machine Tes ome ic 500-50CT (50 kN) wi h a semi-au oma ic
ex ensome e . Un o una ely, he es ing machine does no o e au oma ic ze o o ce con ol
be o e s a ing he es s. The o ce was manually ze oed by changing posi ion. The hi d
posi ion a e (5 mm/min) was applied on he LabCon ol 20 kN es ing machine, which
con ols he o ce un il he es is s a ed.
2.1.2. C eep Tes s
The same cylind ical specimens (Figu e 1) we e used o s anda d c eep es s pe -
o med in he cons an o ce egime a he Ins i u e o Physics o Ma e ials o he Czech
Academy o Sciences (IPM CAS, . .i.). In he i s es , he o ce co esponding o 15 MPa
o axial s ess was applied o 1000 hou s (up o s abilisa ion o he ma e ial esponse). The
load was inc eased o gene a e an axial s ess o abou 20 MPa, which was main ained un il
up u e. The ime o up u e is speci ied o all c eep es s in Table 2.
Appl. Sci. 2024,14, 604 5 o 21
Appl. Sci. 2024, 14, x FOR PEER REVIEW 5 o 23
Figu e 1. A scheme o he geome y o he ensile es specimen.
2.1.2. C eep Tes s
The same cylind ical specimens (Figu e 1) we e used o s anda d c eep es s
pe o med in he cons an o ce egime a he Ins i u e o Physics o Ma e ials o he Czech
Academy o Sciences (IPM CAS, . .i.). In he i s es , he o ce co esponding o 15 MPa
o axial s ess was applied o 1000 hou s (up o s abilisa ion o he ma e ial esponse).
The load was inc eased o gene a e an axial s ess o abou 20 MPa, which was main ained
un il up u e. The ime o up u e is speci ied o all c eep es s in Table 2.
Table 2. Time o up u e in he c eep es s pe o med on SAC305 and SACX0807.
Tes Numbe T
[h] s [MPa]
C1—SAC305 31.6 20 (second block)
C2—SAC305 2.55 25
C1—SACX0807 77.8 20 (second block)
C2—SACX0807 2.51 25
2.2. Accele a ed C eep Tes ing
The accele a ion o c eep es ing consis s o measu ing he de o ma ion on he cu ed
su ace o he specimen using he DIC me hod. The accele a ion o c eep es ing leads o
he de e mina ion o mul iple c eep cu es om a single expe imen , no o he educ ion
in es ime.
2.2.1. DIC Equipmen Desc ip ion
Fo de o ma ion measu emen s, he DIC op ical sys em wi h Alpha so wa e om
X-Sigh , s. .o. e sion 2.1.52 was used. The sys em con ains wo MCR100 came as wi h a
esolu ion o 2.3 Mpx and a maximum possible ame a e o 40 ps (a ull esolu ion).
The came as a e equipped wi h lenses wi h a ocal leng h o 12 mm and a minimum
possible objec dis ance o 0.1 m.
2.2.2. C eep Tes s on Hou glass Specimen
Fo c eep es s, he shape o he specimen was designed acco ding o Figu e 2a. The
designed specimen wi h a con inuously a ying c oss-sec ion o he es sec ion is sui able
o op ical DIC measu emen s. C eep es s o SAC305 and SACX0807 we e also pe o med
on a LabCon ol 20 kN single-axis machine equipped wi h a empe a u e chambe .
(a) (b)
Figu e 2. (a) geome y o he hou glass es specimen, (b) me hodology o ma king he posi ion o
c oss-sec ions on hou glass samples.
Figu e 1. A scheme o he geome y o he ensile es specimen.
Table 2. Time o up u e in he c eep es s pe o med on SAC305 and SACX0807.
Tes Numbe T [h] s [MPa]
C1—SAC305 31.6 20 (second block)
C2—SAC305 2.55 25
C1—SACX0807 77.8 20 (second block)
C2—SACX0807 2.51 25
2.2. Accele a ed C eep Tes ing
The accele a ion o c eep es ing consis s o measu ing he de o ma ion on he cu ed
su ace o he specimen using he DIC me hod. The accele a ion o c eep es ing leads o
he de e mina ion o mul iple c eep cu es om a single expe imen , no o he educ ion
in es ime.
2.2.1. DIC Equipmen Desc ip ion
Fo de o ma ion measu emen s, he DIC op ical sys em wi h Alpha so wa e om
X-Sigh , s. .o. e sion 2.1.52 was used. The sys em con ains wo MCR100 came as wi h a
esolu ion o 2.3 Mpx and a maximum possible ame a e o 40 ps (a ull esolu ion). The
came as a e equipped wi h lenses wi h a ocal leng h o 12 mm and a minimum possible
objec dis ance o 0.1 m.
2.2.2. C eep Tes s on Hou glass Specimen
Fo c eep es s, he shape o he specimen was designed acco ding o Figu e 2a. The
designed specimen wi h a con inuously a ying c oss-sec ion o he es sec ion is sui able
o op ical DIC measu emen s. C eep es s o SAC305 and SACX0807 we e also pe o med
on a LabCon ol 20 kN single-axis machine equipped wi h a empe a u e chambe .
Appl. Sci. 2024, 14, x FOR PEER REVIEW 5 o 23
Figu e 1. A scheme o he geome y o he ensile es specimen.
2.1.2. C eep Tes s
The same cylind ical specimens (Figu e 1) we e used o s anda d c eep es s
pe o med in he cons an o ce egime a he Ins i u e o Physics o Ma e ials o he Czech
Academy o Sciences (IPM CAS, . .i.). In he i s es , he o ce co esponding o 15 MPa
o axial s ess was applied o 1000 hou s (up o s abilisa ion o he ma e ial esponse).
The load was inc eased o gene a e an axial s ess o abou 20 MPa, which was main ained
un il up u e. The ime o up u e is speci ied o all c eep es s in Table 2.
Table 2. Time o up u e in he c eep es s pe o med on SAC305 and SACX0807.
Tes Numbe T
[h] s [MPa]
C1—SAC305 31.6 20 (second block)
C2—SAC305 2.55 25
C1—SACX0807 77.8 20 (second block)
C2—SACX0807 2.51 25
2.2. Accele a ed C eep Tes ing
The accele a ion o c eep es ing consis s o measu ing he de o ma ion on he cu ed
su ace o he specimen using he DIC me hod. The accele a ion o c eep es ing leads o
he de e mina ion o mul iple c eep cu es om a single expe imen , no o he educ ion
in es ime.
2.2.1. DIC Equipmen Desc ip ion
Fo de o ma ion measu emen s, he DIC op ical sys em wi h Alpha so wa e om
X-Sigh , s. .o. e sion 2.1.52 was used. The sys em con ains wo MCR100 came as wi h a
esolu ion o 2.3 Mpx and a maximum possible ame a e o 40 ps (a ull esolu ion).
The came as a e equipped wi h lenses wi h a ocal leng h o 12 mm and a minimum
possible objec dis ance o 0.1 m.
2.2.2. C eep Tes s on Hou glass Specimen
Fo c eep es s, he shape o he specimen was designed acco ding o Figu e 2a. The
designed specimen wi h a con inuously a ying c oss-sec ion o he es sec ion is sui able
o op ical DIC measu emen s. C eep es s o SAC305 and SACX0807 we e also pe o med
on a LabCon ol 20 kN single-axis machine equipped wi h a empe a u e chambe .
(a) (b)
Figu e 2. (a) geome y o he hou glass es specimen, (b) me hodology o ma king he posi ion o
c oss-sec ions on hou glass samples.
Figu e 2.
(
a
) geome y o he hou glass es specimen, (
b
) me hodology o ma king he posi ion o
c oss-sec ions on hou glass samples.
The DIC can be used o measu e de o ma ion on a cu ed su ace whe e he c oss-
sec ion o he specimen con inuously a ies wi h he alue o he co esponding nominal
s ess. A g aphical ep esen a ion o he accele a ed c eep es p ocedu e is shown in
Figu e 3.

Appl. Sci. 2024,14, 604 6 o 21
Appl. Sci. 2024, 14, x FOR PEER REVIEW 6 o 23
The DIC can be used o measu e de o ma ion on a cu ed su ace whe e he c oss-
sec ion o he specimen con inuously a ies wi h he alue o he co esponding nominal
s ess. A g aphical ep esen a ion o he accele a ed c eep es p ocedu e is shown in
Figu e 3.
Figu e 3. (a) he specimen placed in he he mal u nace is imaged using 2 came as, (b) he DIC
me hod can be used o de e mine he c eep cu e a diffe en nominal s ess le els (c).
C eep es s on he hou glass specimen we e pe o med a oom empe a u e 22 °C
and wi h a cons an ensile o ce o 1430 N o bo h SAC305 and SACX0807.
3. Expe imen al Resul s
The main esul s e alua ed om mechanical es s pe o med on bo h solde alloys
a e p esen ed in he ollowing sec ions.
3.1. Tensile Tes s Resul s
The yield and ul ima e s eng hs e alua ed om he ensile es s a e a ailable o
SAC305 and SACX0807 in Table 3 and Table 4, espec i ely. The dependency o bo h
quan i ies on he s ain a e is shown in Figu e 4. The s ain a e was de e mined as he
slope o he angen c ea ed in he ime his o y o axial s ain a he ul ima e s eng hs. As
isible om he compa ison o eg ession line pa ame e s o bo h solde alloys (Figu e
4a,b), he SACX0807 shows sligh ly highe sensi i i y o he s ain a e.
Table 3. Tensile p ope ies o SAC305.
Posi ion Ra e [mm/min] Rp
0.2
[MPa] UTS [MPa]
1 23.3 31.2
5 24.1 34.5
10 25.6 35.6
Table 4. Tensile p ope ies o SACX0807.
Posi ion Ra e [mm/min] Rp
0.2
[MPa] UTS [MPa]
1 19.8 31.4
5 21.6 33.5
10 26.1 37.6
Figu e 3.
(
a
) he specimen placed in he he mal u nace is imaged using 2 came as, (
b
) he DIC
me hod can be used o de e mine he c eep cu e a di e en nominal s ess le els (c).
C eep es s on he hou glass specimen we e pe o med a oom empe a u e 22
◦
C
and wi h a cons an ensile o ce o 1430 N o bo h SAC305 and SACX0807.
3. Expe imen al Resul s
The main esul s e alua ed om mechanical es s pe o med on bo h solde alloys a e
p esen ed in he ollowing sec ions.
3.1. Tensile Tes s Resul s
The yield and ul ima e s eng hs e alua ed om he ensile es s a e a ailable o
SAC305 and SACX0807 in Tables 3and 4, espec i ely. The dependency o bo h quan i ies
on he s ain a e is shown in Figu e 4. The s ain a e was de e mined as he slope o
he angen c ea ed in he ime his o y o axial s ain a he ul ima e s eng hs. As isible
om he compa ison o eg ession line pa ame e s o bo h solde alloys (Figu e 4a,b), he
SACX0807 shows sligh ly highe sensi i i y o he s ain a e.
Table 3. Tensile p ope ies o SAC305.
Posi ion Ra e [mm/min] Rp0.2 [MPa] UTS [MPa]
1 23.3 31.2
5 24.1 34.5
10 25.6 35.6
Table 4. Tensile p ope ies o SACX0807.
Posi ion Ra e [mm/min] Rp0.2 [MPa] UTS [MPa]
1 19.8 31.4
5 21.6 33.5
10 26.1 37.6
3.2. C eep Tes s Resul s
3.2.1. C eep Tes s on Cylind ical Specimens
A oom empe a u e, c eep es s on cylind ical specimens (Figu e 1) we e pe o med
a he IPM CAS, . .i. and a e shown in Figu e 5.
Appl. Sci. 2024,14, 604 7 o 21
Appl. Sci. 2024, 14, x FOR PEER REVIEW 7 o 23
(a) (b)
Figu e 4. In luence o s ain a e on yield and ul ima e s eng h: (a) SAC305; (b) SACX0807.
3.2. C eep Tes s Resul s
3.2.1. C eep Tes s on Cylind ical Specimens
A oom empe a u e, c eep es s on cylind ical specimens (Figu e 1) we e pe o med
a he IPM CAS, . .i. and a e shown in Figu e 5.
Figu e 5. C eep cu es o SACX0807 and SAC305 cylind ical solde samples a 22 °C.
3.2.2. C eep Tes s on Hou glass Specimen
Calcula ing he de o ma ion alues is unde aken in he DIC so wa e (Alpha
so wa e om X-Sigh , s. .o. e sion 2.1.52). The en i e s ain enso is a ailable o each
calcula ion poin o he DIC ne wo k. Figu e 2b schema ically shows he posi ions o he
c oss-sec ions in which he c eep s ain e alua ion was pe o med ( o solde SAC305,
hou glass sample).
The c eep cu es o he SAC305 and SACX0807 alloys ob ained by using DIC
measu emen s on he hou glass specimens a e shown in Figu e 6a,b. The i s numbe o
he legend e e s o he dis ance o he e alua ed poin om he loca ion o he smalles
c oss-sec ion, while he second numbe indica es he co esponding nominal s ess alue.
The HMH index means he equi alen alue calcula ed acco ding o on Mises heo y.
Figu e 4. In luence o s ain a e on yield and ul ima e s eng h: (a) SAC305; (b) SACX0807.
Appl. Sci. 2024, 14, x FOR PEER REVIEW 7 o 23
(a) (b)
Figu e 4. In luence o s ain a e on yield and ul ima e s eng h: (a) SAC305; (b) SACX0807.
3.2. C eep Tes s Resul s
3.2.1. C eep Tes s on Cylind ical Specimens
A oom empe a u e, c eep es s on cylind ical specimens (Figu e 1) we e pe o med
a he IPM CAS, . .i. and a e shown in Figu e 5.
Figu e 5. C eep cu es o SACX0807 and SAC305 cylind ical solde samples a 22 °C.
3.2.2. C eep Tes s on Hou glass Specimen
Calcula ing he de o ma ion alues is unde aken in he DIC so wa e (Alpha
so wa e om X-Sigh , s. .o. e sion 2.1.52). The en i e s ain enso is a ailable o each
calcula ion poin o he DIC ne wo k. Figu e 2b schema ically shows he posi ions o he
c oss-sec ions in which he c eep s ain e alua ion was pe o med ( o solde SAC305,
hou glass sample).
The c eep cu es o he SAC305 and SACX0807 alloys ob ained by using DIC
measu emen s on he hou glass specimens a e shown in Figu e 6a,b. The i s numbe o
he legend e e s o he dis ance o he e alua ed poin om he loca ion o he smalles
c oss-sec ion, while he second numbe indica es he co esponding nominal s ess alue.
The HMH index means he equi alen alue calcula ed acco ding o on Mises heo y.
Figu e 5. C eep cu es o SACX0807 and SAC305 cylind ical solde samples a 22 ◦C.
3.2.2. C eep Tes s on Hou glass Specimen
Calcula ing he de o ma ion alues is unde aken in he DIC so wa e (Alpha so wa e
om X-Sigh , s. .o. e sion 2.1.52). The en i e s ain enso is a ailable o each calcula ion
poin o he DIC ne wo k. Figu e 2b schema ically shows he posi ions o he c oss-sec ions
in which he c eep s ain e alua ion was pe o med ( o solde SAC305, hou glass sample).
The c eep cu es o he SAC305 and SACX0807 alloys ob ained by using DIC mea-
su emen s on he hou glass specimens a e shown in Figu e 6a,b. The i s numbe o
he legend e e s o he dis ance o he e alua ed poin om he loca ion o he smalles
c oss-sec ion, while he second numbe indica es he co esponding nominal s ess alue.
The HMH index means he equi alen alue calcula ed acco ding o on Mises heo y.
3.2.3. C eep Compa ison o Di e en Specimen Shape
F om he poin o iew o modelling and desc ibing he c eep beha iou , he a ea o
seconda y c eep is pa icula ly impo an . Seconda y c eep is cha ac e ised by a linea
dependence o he s ain on ime and is desc ibed by a minimum s ain a e. A e inding
he alues o he minimum c eep s ain a es om Figu es 5and 6, he esul s can be
ob ained in he o m o Figu e 7a,b in a loga i hmic scale o SAC305 and SACX0807
solde , espec i ely.
As he s ess alue o he hou glass specimen dec eases, he e is a de ia ion om he
esul s ob ained on he cylind ical specimens owa d highe s ain a es o bo h solde
alloys. This de ia ion can be explained by a mul iaxial s ess s a e, which can be p esen ed
by using he esul s o a nume ical simula ion.
Appl. Sci. 2024,14, 604 8 o 21
Appl. Sci. 2024, 14, x FOR PEER REVIEW 8 o 23
(a) (b)
Figu e 6. C eep es o alloy (a) SAC305 and (b) SACX0807, hou glass specimen, empe a u e 22 °C,
axial o ce 1430 N.
3.2.3. C eep Compa ison o Diffe en Specimen Shape
F om he poin o iew o modelling and desc ibing he c eep beha iou , he a ea o
seconda y c eep is pa icula ly impo an . Seconda y c eep is cha ac e ised by a linea
dependence o he s ain on ime and is desc ibed by a minimum s ain a e. A e inding
he alues o he minimum c eep s ain a es om Figu es 5 and 6, he esul s can be
ob ained in he o m o Figu e 7a,b in a loga i hmic scale o SAC305 and SACX0807
solde , espec i ely.
(a) (b)
Figu e 7. Room empe a u e c eep s ain a es o SAC305 (a) and SACX0807 (b) solde , loga i hmic
scale (compa ison o diffe en samples o ew selec ed poin s).
As he s ess alue o he hou glass specimen dec eases, he e is a de ia ion om he
esul s ob ained on he cylind ical specimens owa d highe s ain a es o bo h solde
alloys. This de ia ion can be explained by a mul iaxial s ess s a e, which can be p esen ed
by using he esul s o a nume ical simula ion.
4. Nume ical Simula ions
Based on he esul s o he ensile es s, i can be s a ed ha o a nume ical simula ion
o he co ec mechanical beha iou in mono onic loading, i is necessa y o selec a
iscoplas ic ma e ial model in combina ion wi h a sui able ha dening model. The Pe zyna
ma e ial model was chosen o cap u e he iscoplas ic beha iou , and he Chaboche
model was chosen o cap u e he ha dening beha iou o he ma e ial. In c eep es s,
a en ion is only paid o he simula ion o seconda y c eep. This beha iou is cap u ed by
using he No on model. The e o e, he non-uni ied Pe zyna–Chaboche–No on ma e ial
model was chosen o he nume ical simula ion o he pe o med es s.
Figu e 6.
C eep es o alloy (
a
) SAC305 and (
b
) SACX0807, hou glass specimen, empe a u e 22
◦
C,
axial o ce 1430 N.
Appl. Sci. 2024, 14, x FOR PEER REVIEW 8 o 23
(a) (b)
Figu e 6. C eep es o alloy (a) SAC305 and (b) SACX0807, hou glass specimen, empe a u e 22 °C,
axial o ce 1430 N.
3.2.3. C eep Compa ison o Diffe en Specimen Shape
F om he poin o iew o modelling and desc ibing he c eep beha iou , he a ea o
seconda y c eep is pa icula ly impo an . Seconda y c eep is cha ac e ised by a linea
dependence o he s ain on ime and is desc ibed by a minimum s ain a e. A e inding
he alues o he minimum c eep s ain a es om Figu es 5 and 6, he esul s can be
ob ained in he o m o Figu e 7a,b in a loga i hmic scale o SAC305 and SACX0807
solde , espec i ely.
(a) (b)
Figu e 7. Room empe a u e c eep s ain a es o SAC305 (a) and SACX0807 (b) solde , loga i hmic
scale (compa ison o diffe en samples o ew selec ed poin s).
As he s ess alue o he hou glass specimen dec eases, he e is a de ia ion om he
esul s ob ained on he cylind ical specimens owa d highe s ain a es o bo h solde
alloys. This de ia ion can be explained by a mul iaxial s ess s a e, which can be p esen ed
by using he esul s o a nume ical simula ion.
4. Nume ical Simula ions
Based on he esul s o he ensile es s, i can be s a ed ha o a nume ical simula ion
o he co ec mechanical beha iou in mono onic loading, i is necessa y o selec a
iscoplas ic ma e ial model in combina ion wi h a sui able ha dening model. The Pe zyna
ma e ial model was chosen o cap u e he iscoplas ic beha iou , and he Chaboche
model was chosen o cap u e he ha dening beha iou o he ma e ial. In c eep es s,
a en ion is only paid o he simula ion o seconda y c eep. This beha iou is cap u ed by
using he No on model. The e o e, he non-uni ied Pe zyna–Chaboche–No on ma e ial
model was chosen o he nume ical simula ion o he pe o med es s.
Figu e 7.
Room empe a u e c eep s ain a es o SAC305 (
a
) and SACX0807 (
b
) solde , loga i hmic
scale (compa ison o di e en samples o ew selec ed poin s).
4. Nume ical Simula ions
Based on he esul s o he ensile es s, i can be s a ed ha o a nume ical simula ion
o he co ec mechanical beha iou in mono onic loading, i is necessa y o selec a is-
coplas ic ma e ial model in combina ion wi h a sui able ha dening model. The Pe zyna
ma e ial model was chosen o cap u e he iscoplas ic beha iou , and he Chaboche model
was chosen o cap u e he ha dening beha iou o he ma e ial. In c eep es s, a en ion is
only paid o he simula ion o seconda y c eep. This beha iou is cap u ed by using he
No on model. The e o e, he non-uni ied Pe zyna–Chaboche–No on ma e ial model was
chosen o he nume ical simula ion o he pe o med es s.
An es ima ion o he ini ial alues o he pa ame e s is p esen ed in de ail o he
SAC305 ma e ial. Fo he SACX0807 ma e ial, only he di e ences compa ed o he SAC305
ma e ial a e lis ed in he las chap e o his sec ion.
4.1. Cons i u i e Model
The Pe zyna iscoplas ic ma e ial model, when combined wi h he kinema ic ha den-
ing ule, gi es he a e o accumula ed plas ic s ain ( .
p). All expe imen s we e ca ied ou
in he ensile mode, hus he cons i u i e model is desc ibed o he uniaxial loading case
(pu e ension). The alue o he accumula ed plas ic s ain a e is calcula ed as ollows [
30
]:
.
p=νσ−α
σy
−11/m
, (1)
Appl. Sci. 2024,14, 604 9 o 21
whe e
α
is a kinema ic s ess (“backs ess”),
ν
is he iscosi y o he ma e ial,
m
is he
ha dening exponen ,
σy
is he ini ial yield s eng h o he ma e ial and
σ
is he alue o he
axial s ess. I is clea om Equa ion (1) ha he model is no dependen on empe a u e
and i i is used, hen i mus be calib a ed o each empe a u e sepa a ely.
The backs ess is gi en by a supe posi ion o
M
pa s (Chaboche kinema ic ha dening
model [
37
]) and he analy ical solu ion o mono onic ensile loading is exp essed as ollows:
α=CMεp+∑M−1
i=1
Ci
γi1−e−γiεp, (2)
whe e
εp
is he plas ic s ain and
Ci
,
γi
a e ma e ial pa ame e s. I he cyclic c eep o he
ma e ial does no occu , he las supe posi ion pa ame e
γM=
0 is chosen; he e o e,
he linea e m
CMεp
appea s in Equa ion (2). In his s udy, h ee backs ess pa s a e
conside ed (M=3).
In combina ion wi h a nonlinea kinema ic ha dening ule, he Pe zyna model can
only cap u e he p ima y c eep. The e o e, i is necessa y o add a sui able c eep model o
desc ibe he seconda y c eep. The No on model [
31
] appea s o be sui able, and o he
case o a cons an es empe a u e, he a e o c eep s ain can be exp essed as a powe
unc ion o s ess, as ollows: .
εc =C4σC5, (3)
whe e
C4
and
C5
a e he ma e ial pa ame e s o he No on model. We mus iden i y
Young’s modulus, yield s eng h, wo pa ame e s o Pe zyna model, 2
M−
1 o Chaboche
model and wo pa ame e s o he No on model.
4.2. FE Models
The ini e elemen model was c ea ed using Ansys Wo kbench 2022 R2 so wa e.
Tensile and c eep es s a e simula ed as an axisymme ic ask. In he simula ion o he
ensile and c eep es s, quad a ic FEM elemen s PLANE183 we e used. The bounda y
condi ions o he simula ion o he ensile es on he cylind ical sample a e shown in
Figu e 8a. A displacemen in he y-axis di ec ion is applied on he igh side o he model;
he alues a e p esen ed in he o m o a ime displacemen g aph (see Figu e 9). The c eep
es simula ion model is shown in Figu e 8b. In he gi en igu e, he es poin s and hei
coo dina es on he y-axis a e highligh ed. Tes poin s a e places whe e s ain alues we e
compa ed in he expe imen and i s simula ion.
Appl. Sci. 2024, 14, x FOR PEER REVIEW 10 o 23
(a) (b)
Figu e 8. Bounda y condi ions and loading me hod o cylind ical specimen in ensile es
simula ion (a) and hou glass specimen in c eep es (b).
(a) (b)
Figu e 9. Displacemen bounda y condi ion o (a) ensile es a a e 1 mm/min and (b) ensile
es s a a e 5 and 10 mm/min (SAC305 solde alloy).
The c eep es was pe o med unde cons an o ce on he igh side o he sample
model (Figu e 8b). The o ce is inc eased a a speed o 1500 N/s in he FEM simula ion,
and he loading is applied in wo s eps by a able desc ibing he ime his o y o he o ce.
The e o e, he maximal o ce o 1430 N is eached a 0.95 s, and hen he o ce is kep
cons an . The c eep es was simula ed un il he inal ime o 960 s, when he a ea o
e ia y c eep occu ed.
4.3. Ini ial Values o Ma e ial Pa ame e s o Th ee Ma e ial Models (SAC305)
The ini ial pa ame e alues we e de e mined sepa a ely o he Pe zyna, Chaboche
and No on models. The ela ionships usually used o he gi en ma e ial models we e
used.
The ini ial alues o he pa ame e s o he No on model we e de e mined acco ding
o Figu e 10, whe e he a es o he seconda y pa o he c eep a e gi en as a unc ion o
he nominal s ess. The pa ame e alues o he No on model 𝐶 and 𝐶 can be
de e mined di ec ly by compa ing Equa ion (3) wi h he powe app oxima ion unc ion
o he equa ion shown in Figu e 10 o he hou glass specimen (i is necessa y o conside
he s ess uni s used in he Ansys 2022 R2 so wa e). Then, he ini ial alues o he
pa ame e s o he No on model a e 𝐶 = 9.62728 and 𝐶 = 7.97016 × 10
−19
s
−1
.
Figu e 8.
Bounda y condi ions and loading me hod o cylind ical specimen in ensile es simula ion
(a) and hou glass specimen in c eep es (b).
The c eep es was pe o med unde cons an o ce on he igh side o he sample
model (Figu e 8b). The o ce is inc eased a a speed o 1500 N/s in he FEM simula ion,
and he loading is applied in wo s eps by a able desc ibing he ime his o y o he o ce.
The e o e, he maximal o ce o 1430 N is eached a 0.95 s, and hen he o ce is kep
cons an . The c eep es was simula ed un il he inal ime o 960 s, when he a ea o e ia y
c eep occu ed.
Appl. Sci. 2024,14, 604 16 o 21
Table 6. Pa ame e alues o he Chaboche–Pe zyna–No on model o SAC305 alloy, T = 22 ◦C.
Ma . Model Pa ame e Ini ial Final
elas ic iso opic E[MPa]42,045.29 75,945.1
σy[MPa]21.51 10.4411
Chaboche
C1[MPa]1782.9 3148.46
C2[MPa]396.88 56.4921
C3[MPa]35.117 16.1326
γ1[−]678.79 251.575
γ2[−]57.675 121.283
γ3[−]0 0
Pe zyna m[−]0.969242 0.50316
νs−15.30916 ×10−39.84942 ×10−4
No on ( o [MPa]) C4s−17.97016 ×10−19 4.02 ×10−19
C5[−]9.62728 6.8239
o al objec i e
unc ion ERRA[−]1226.4 61.2
Appl. Sci. 2024, 14, x FOR PEER REVIEW 17 o 23
(a) (b)
(c) (d)
Figu e 16. Tensile es s and hei FEM p edic ions o alloy SAC305: (a) 1 mm/min; (b) 5 mm/min;
(c) 10 mm/min; (d) c eep es s and hei FEM p edic ions.
The la ges diffe ences in he ensile es p edic ions a e in he i s pa o he cu e,
a ound 20–25 MPa, whe e he simula ion shows a much sha pe ansi ion han he
expe imen . The bes ag eemen was achie ed be ween he FEM simula ion and he
expe imen da a o he ensile es o 5 mm/min and o he c eep a he lowes s ess o
25.05 MPa.
6.2. Resul s o Calib a ion o he SACX0807 Ma e ial
The ma e ial pa ame e s o SACX0807 a e shown in Table 7. The able con ains he
same in o ma ion as o ma e ial SAC305 in he p e ious chap e .
Table 7. Pa ame e alues o he Chaboche–Pe zyna–No on model o he SAC0807 alloy, T = 22
°C.
Ma . Model Pa ame e Ini ial Final
elas ic iso opic 𝐸 󰇟MPa󰇠 23,817 41,127.7
𝜎 󰇟MPa󰇠 13.43 10.873
Chaboche
𝐶 󰇟MPa󰇠 6165.4 16,210.5
𝐶 󰇟MPa󰇠 594.44 20.0247
𝐶 󰇟MPa󰇠 43.033 288.226
𝛾 󰇟−󰇠 924.01 1344.29
𝛾 󰇟−󰇠 57.265 250.142
𝛾 󰇟−󰇠 0 0
Pe zyna 𝑚 󰇟−󰇠 0.969242 0.677308
𝜈 󰇟s󰇠 5.30916 × 10
−3
7.14427 × 10
−3
No on ( o [MPa]) 𝐶 󰇟s󰇠 1.12583 × 10
−19
7.01215 × 10
−20
𝐶 󰇟−󰇠 10.13748 10.3626
o al objec i e unc ion 𝐸𝑅𝑅 󰇟−󰇠 230.1 50.6
Figu e 16.
Tensile es s and hei FEM p edic ions o alloy SAC305: (
a
) 1 mm/min; (
b
) 5 mm/min;
(c) 10 mm/min; (d) c eep es s and hei FEM p edic ions.
The la ges di e ences in he ensile es p edic ions a e in he i s pa o he cu e,
a ound 20–25 MPa, whe e he simula ion shows a much sha pe ansi ion han he expe i-
men . The bes ag eemen was achie ed be ween he FEM simula ion and he expe imen
da a o he ensile es o 5 mm/min and o he c eep a he lowes s ess o 25.05 MPa.
6.2. Resul s o Calib a ion o he SACX0807 Ma e ial
The ma e ial pa ame e s o SACX0807 a e shown in Table 7. The able con ains he
same in o ma ion as o ma e ial SAC305 in he p e ious chap e .

Appl. Sci. 2024,14, 604 17 o 21
Table 7.
Pa ame e alues o he Chaboche–Pe zyna–No on model o he SAC0807 alloy, T = 22
◦
C.
Ma . Model Pa ame e Ini ial Final
elas ic iso opic E[MPa]23,817 41,127.7
σy[MPa]13.43 10.873
Chaboche
C1[MPa]6165.4 16,210.5
C2[MPa]594.44 20.0247
C3[MPa]43.033 288.226
γ1[−]924.01 1344.29
γ2[−]57.265 250.142
γ3[−]0 0
Pe zyna m[−]0.969242 0.677308
νs−15.30916 ×10−37.14427 ×10−3
No on ( o [MPa]) C4s−11.12583 ×10−19 7.01215 ×10−20
C5[−]10.13748 10.3626
o al objec i e
unc ion ERRA[−]230.1 50.6
F om Table 7, i is possible o each he same conclusions as in he p e ious chap e o
SAC305. Thus, he p oposed p ocedu e gi es signi ican ly be e esul s han when only
analy ical es ima ion is used.
The p edic ion o ensile es s o he SACX0807 alloy using FEA can be seen in
Figu e 17a–c. The p edic ion o he SACX0807 c eep es s is shown in Figu e 17d.
Appl. Sci. 2024, 14, x FOR PEER REVIEW 18 o 23
F om Table 7, i is possible o each he same conclusions as in he p e ious chap e
o SAC305. Thus, he p oposed p ocedu e gi es signi ican ly be e esul s han when
only analy ical es ima ion is used.
The p edic ion o ensile es s o he SACX0807 alloy using FEA can be seen in Figu e
17a–c. The p edic ion o he SACX0807 c eep es s is shown in Figu e 17d.
(a) (b)
(c) (d)
Figu e 17. Tensile es s and hei FEM p edic ions o alloy SACX0807: (a) 1 mm/min; (b) 5 mm/min;
(c) 10 mm/min; (d) c eep es s and hei FEM p edic ions.
The bes ag eemen be ween he FEM simula ion and he expe imen da a was
achie ed again o he ensile es o 5 mm/min. In he c eep es s, he cu es almos
coincide om he ime o 200 s.
7. Discussion
In his wo k, an accele a ion es wi h an hou glass specimen and DIC measu emen
is p esen ed o calib a e he ma e ial model. The ma e ial model consis s o h ee pa s:
Chaboche model, Pe zyna model, and he No on model. Each o he h ee pa s cap u es
a diffe en componen o he ma e ial beha iou , and he pa ame e s o hese models can
be de e mined sepa a ely based on analy ical ela ionships and selec ed expe imen s. This
me hodology is p esen ed in Sec ion 4.3, and he pa ame e s ound a e used as an ini ial
o u he calib a ion. In p ac ice, such a ma e ial model is hen used o FEM simula ions
in a ious comme cial so wa e. In he case o FE models, he so-called FEMU app oach
is hen used o iden i ica ion, which was used in his a icle o “ une” he pa ame e
alues and is b ie ly desc ibed in Sec ion 5. The solu ion esul s, see Sec ion 6, show ha
he second calib a ion using he FEMU app oach signi ican ly imp o es he quali y o he
solu ion, which is exp essed by he alue o he objec i e unc ion (o g aphs). Se e al
p oblem a eas we e also iden i ied du ing he solu ion. These we e no analysed in mo e
de ail in he a icle, as he goal was o es he basic p ocedu e.
Hou glass samples seem o gene a e sligh ly diffe en esul s compa ed o classic
samples; he p oblem is shown in Sec ion 3.2.3. Diffe en s ess s a es (mul i-axial s ess
Figu e 17.
Tensile es s and hei FEM p edic ions o alloy SACX0807: (
a
) 1 mm/min; (
b
) 5 mm/min;
(c) 10 mm/min; (d) c eep es s and hei FEM p edic ions.
The bes ag eemen be ween he FEM simula ion and he expe imen da a was
achie ed again o he ensile es o 5 mm/min. In he c eep es s, he cu es almos
coincide om he ime o 200 s.
Appl. Sci. 2024,14, 604 18 o 21
7. Discussion
In his wo k, an accele a ion es wi h an hou glass specimen and DIC measu emen
is p esen ed o calib a e he ma e ial model. The ma e ial model consis s o h ee pa s:
Chaboche model, Pe zyna model, and he No on model. Each o he h ee pa s cap u es a
di e en componen o he ma e ial beha iou , and he pa ame e s o hese models can be
de e mined sepa a ely based on analy ical ela ionships and selec ed expe imen s. This
me hodology is p esen ed in Sec ion 4.3, and he pa ame e s ound a e used as an ini ial
o u he calib a ion. In p ac ice, such a ma e ial model is hen used o FEM simula ions
in a ious comme cial so wa e. In he case o FE models, he so-called FEMU app oach is
hen used o iden i ica ion, which was used in his a icle o “ une” he pa ame e alues
and is b ie ly desc ibed in Sec ion 5. The solu ion esul s, see Sec ion 6, show ha he
second calib a ion using he FEMU app oach signi ican ly imp o es he quali y o he
solu ion, which is exp essed by he alue o he objec i e unc ion (o g aphs). Se e al
p oblem a eas we e also iden i ied du ing he solu ion. These we e no analysed in mo e
de ail in he a icle, as he goal was o es he basic p ocedu e.
Hou glass samples seem o gene a e sligh ly di e en esul s compa ed o classic
samples; he p oblem is shown in Sec ion 3.2.3. Di e en s ess s a es (mul i-axial s ess
s a es in he hou glass sample) o sligh ly di e en manu ac u ing echnologies can be a
p oblem he e. Howe e , simila p oblems can also a ise in p ac ice.
The FEMU app oach does no cap u e he s anda d physical meaning o he pa ame e s
( o example, Young’s modulus
E
, he alue o he yield s eng h
σy
) bu a he he o e all
e ec o he pa ame e on he gi en cu e and app oaches he ma e ial model as a black
box. When using he FEMU app oach, he alue co esponding o he physical meaning
o he pa ame e may de ia e om his in e p e a ion. The p oblem is shown in Sec ion 6.
We can emo e such pa ame e s om FEMU iden i ica ion o signi ican ly s eng hen he
in luence o he pa ame e on he esul ing objec i e unc ion alue. Howe e , bo h s eps
can lead o a dec ease in he quali y o he esul ing es ima e.
Ano he p oblem is de e mining he weigh s o indi idual expe imen s when calcu-
la ing he alue o he o al objec i e unc ion. Tensile es s and c eep es s we e used o
iden i ica ion. In c eep es s, a la ge pa o he e o alue is ela ed o only wo pa ame e s
in he No on model. The pa ame e alues o he Pe zyna model will a ec he beginning
o he cu e in c eep es s (p ima y c eep) and also he ensile cu e a di e en speeds. Fo
he Chaboche model, he e will be a p edominan in luence in ensile es s. In his pape ,
c eep and ensile es s we e gi en equal weigh . Th ough a isual compa ison o he esul s,
see Figu es 16 and 17, we came o he opinion ha he c eep es s show a be e ag eemen
wi h he expe imen . The e is a ques ion as o whe he he numbe o pa ame e s ied o i
should no be aken in o accoun when designing weigh s o indi idual expe imen s.
Wi h epe i i e FEM calcula ions, he ques ion a ises how o educe he equi ed
compu a ion ime? The o al compu a ional ime equi ed o une he ma e ial model used
has no been accu a ely measu ed bu is es ima ed o ake se en days. When he numbe
o elemen s is educed, signi ican ime sa ings can be achie ed, bu he accu acy may be
a ec ed. This e ec has been in es iga ed and can be exp essed using Figu e 18.
F om Figu e 18a, i can be concluded ha he size o he elemen does no ha e a
signi ican e ec on he p ecision o he ensile es esul s. All h ee cu es o e lap. F om
Figu e 18b, i can be seen ha he e ec o he size o he elemen on he accu acy o he
p edic ion o he c eep es is e y small. Thus, in bo h cases, an elemen size o abou 3 mm
was used o minimise compu a ional ime. In Figu e 18b, he in luence o ime s ep size is
in es iga ed o 3 mm elemen size (1 s e sus 5 s), wi h he conclusion ha i is negligible
oo.
The beha iou o he es ed ma e ials is signi ican ly a ec ed by empe a u es. The
Pe zyna ma e ial model was designed o one empe a u e and was he e o e also used in
his pape . Modi ica ion o Pe zyna model is possible in se e al ways. Fo example, in [
42
],
a modi ica ion was es ed ha addi ionally adds a pa ame e as a unc ion o empe a u e
o he Anand model.
Appl. Sci. 2024,14, 604 19 o 21
Appl. Sci. 2024, 14, x FOR PEER REVIEW 19 o 23
s a es in he hou glass sample) o sligh ly diffe en manu ac u ing echnologies can be a
p oblem he e. Howe e , simila p oblems can also a ise in p ac ice.
The FEMU app oach does no cap u e he s anda d physical meaning o he
pa ame e s ( o example, Young’s modulus 𝐸 , he alue o he yield s eng h 𝜎 ) bu
a he he o e all effec o he pa ame e on he gi en cu e and app oaches he ma e ial
model as a black box. When using he FEMU app oach, he alue co esponding o he
physical meaning o he pa ame e may de ia e om his in e p e a ion. The p oblem is
shown in Sec ion 6. We can emo e such pa ame e s om FEMU iden i ica ion o
signi ican ly s eng hen he in luence o he pa ame e on he esul ing objec i e unc ion
alue. Howe e , bo h s eps can lead o a dec ease in he quali y o he esul ing es ima e.
Ano he p oblem is de e mining he weigh s o indi idual expe imen s when
calcula ing he alue o he o al objec i e unc ion. Tensile es s and c eep es s we e used
o iden i ica ion. In c eep es s, a la ge pa o he e o alue is ela ed o only wo
pa ame e s in he No on model. The pa ame e alues o he Pe zyna model will affec
he beginning o he cu e in c eep es s (p ima y c eep) and also he ensile cu e a
diffe en speeds. Fo he Chaboche model, he e will be a p edominan in luence in ensile
es s. In his pape , c eep and ensile es s we e gi en equal weigh . Th ough a isual
compa ison o he esul s, see Figu es 16 and 17, we came o he opinion ha he c eep
es s show a be e ag eemen wi h he expe imen . The e is a ques ion as o whe he he
numbe o pa ame e s ied o i should no be aken in o accoun when designing weigh s
o indi idual expe imen s.
Wi h epe i i e FEM calcula ions, he ques ion a ises how o educe he equi ed
compu a ion ime? The o al compu a ional ime equi ed o une he ma e ial model used
has no been accu a ely measu ed bu is es ima ed o ake se en days. When he numbe
o elemen s is educed, signi ican ime sa ings can be achie ed, bu he accu acy may be
affec ed. This effec has been in es iga ed and can be exp essed using Figu e 18.
(a) (b)
Figu e 18. Effec o elemen size on he p edic ion o ensile es (a) and c eep es (b).
F om Figu e 18a, i can be concluded ha he size o he elemen does no ha e a
signi ican effec on he p ecision o he ensile es esul s. All h ee cu es o e lap. F om
Figu e 18b, i can be seen ha he effec o he size o he elemen on he accu acy o he
p edic ion o he c eep es is e y small. Thus, in bo h cases, an elemen size o abou 3
mm was used o minimise compu a ional ime. In Figu e 18b, he in luence o ime s ep
size is in es iga ed o 3 mm elemen size (1 s e sus 5 s), wi h he conclusion ha i is
negligible oo.
The beha iou o he es ed ma e ials is signi ican ly affec ed by empe a u es. The
Pe zyna ma e ial model was designed o one empe a u e and was he e o e also used in
his pape . Modi ica ion o Pe zyna model is possible in se e al ways. Fo example, in [42],
a modi ica ion was es ed ha addi ionally adds a pa ame e as a unc ion o empe a u e
o he Anand model.
Figu e 18. E ec o elemen size on he p edic ion o ensile es (a) and c eep es (b).
I is no possible o di ec ly compa e he ob ained ma e ial model pa ame e s wi h
o he sou ces as hey a e no ye widely a ailable in he li e a u e (SACX0807). In addi-
ion, he pa ame e s depend on he empe a u e and he way he mechanical es s a e
pe o med. The li e a u e [
27
,
43
], which is mo e simila o he p oblem p esen ed he e, can
be ecommended.
The inclusion o addi ional expe imen s o pa ame e calib a ion o alida ion o hei
esul ing design was also conside ed. Fo he selec ed ma e ial model, o example, his
in ol ed he p oblem o a che ing. Fo h ee ma e ial models and wo ypes o expe imen s,
he iden i ica ion p ocedu e consis ed o 10 op imisa ion s eps (see Sec ion 5.3). Tha is
why we decided o es he basic p ocedu e and no es empe a u es o mo e complex
s a es ( a che ing).
8. Conclusions
As shown in his a icle, he ensile p ope ies o he SAC305 and SACX0807 solde s
a oom empe a u e a e compa able (wi h s ain a es s udied). A simila conclusion can
also be s a ed o he uniaxial c eep beha iou . In he accele a ed c eep es , he s ain
con ou s we e measu ed using he digi al image co ela ion (DIC) me hod. Due o he
hou glass- ype specimen, he DIC cap u es c eep s ains on pa icula c oss-sec ions o
he specimen (ha ing di e en ac ing nominal s ess), b inging mo e c eep cu es om
a single c eep es . Subsequen ly, hese c eep cu es a e used o calib a e a iscoplas ic
model. Thus, he accele a ion o he c eep es is in ob aining mo e s ain da a om a single
es . The p ocedu e is desc ibed in de ail and can help esea che s wi h i s applica ion o
o he enginee ing ma e ials.
The iscoplas ic model used in his s udy o bo h solde alloys consis s o h ee pa s:
he Chaboche model, he Pe zyna model and he No on model. The pa ame e s ob ained
ia op imisa ion in Op iSLang signi ican ly imp o ed he accu acy o he p edic ions
compa ed o he adi ional analy ical app oach (almos i e imes less objec i e unc ion
alue). The ad an age is mainly in he possibili y o mul iaxial s ess s a e conside a ion
and he usage o mo e expe imen s in he op imisa ion p ocedu e simul aneously.
The in luence o empe a u e on he mechanical p ope ies o bo h solde alloys is
unde in es iga ion. The capabili ies o he ma e ial model o desc ibe he empe a u e
e ec will be p esen ed in a u u e s udy. The accele a ed echnique wi h DIC can be
applied o he case o ele a ed empe a u e analogously. Especially o SAC alloys, whe e
he in es iga ed empe a u es a e below 130
◦
C, s anda d sp ays can be used o c ea e
a pa e n.
Au ho Con ibu ions:
Concep ualisa ion, R.H., P.D. and Z.P.; w i ing—o iginal d a p epa a ion,
Z.P., R.H., J.R. and B.G.; me hodology, R.H., P.D. and Z.P.; da a cu a ion, Z.P., R.H., B.G. and P.D.;
in es iga ion, Z.P., R.H., B.G. and P.D.; o mal analysis, J.R. and R.H.; so wa e, Z.P.; w i ing—
e iew and edi ing, Z.P., R.H., J.R. and P.D.; isualisa ion, Z.P.; supe ision, R.H. and P.D.; p ojec
Appl. Sci. 2024,14, 604 20 o 21
adminis a ion, R.H.; esou ces, R.H. All au ho s ha e ead and ag eed o he published e sion o
he manusc ip .
Funding:
This esea ch was unded by he Minis y o Educa ion, You h and Spo s o Czech Republic,
g an numbe SP2023/027 and by The Technology Agency o he Czech Republic in he ame o he
p ojec TN01000024 Na ional Compe ence Cen e -Cybe ne ics and A i icial In elligence.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
The o iginal con ibu ions p esen ed in he s udy a e included in he
a icle, u he inqui ies can be di ec ed o he co esponding au ho .
Acknowledgmen s:
The au ho s a e hank ul o he suppo o Vi esco Technologies Czech Republic
s. .o. company.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
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