

Ci a ion: Lu o ino , M.; Halama, R.;
Papuga, J.; Ba ošák, M.; Kuželka, J.;
R˚užiˇcka, M. An App oxima e
Me hod o Calcula ing
Elas ic–Plas ic S ess and S ain on
No ched Specimens. Ma e ials 2022,
15, 1432. h ps://doi.o g/10.3390/
ma15041432
Academic Edi o : Wojciech Bo ek
Recei ed: 30 Decembe 2021
Accep ed: 10 Feb ua y 2022
Published: 15 Feb ua y 2022
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ma e ials
A icle
An App oxima e Me hod o Calcula ing Elas ic–Plas ic S ess
and S ain on No ched Specimens
Maxim Lu o ino 1, Radim Halama 2,* , Jan Papuga 1, Michal Ba ošák1, Jiˇ íKuželka 1and Milan R˚užiˇcka 1
1Facul y o Mechanical Enginee ing, Czech Technical Uni e si y in P ague, Technická 4,
16607 P ague, Czech Republic; [email p o ec ed] (M.L.); [email p o ec ed] (J.P.);
[email p o ec ed] (M.B.); [email p o ec ed] (J.K.); [email p o ec ed] (M.R.)
2Facul y o Mechanical Enginee ing, VSB-Technical Uni e si y o Os a a, 17.lis opadu 2172/15,
708 00 Os a a, Czech Republic
*Co espondence: [email p o ec ed]
Abs ac :
The pape deals wi h an app oxima e me hod o calcula ing elas ic–plas ic s esses and
s ains on he su ace o no ched samples. The me hod is based on he Abdel–Ka im–Ohno cyclic
plas ici y model. The plane s ess condi ion is conside ed wi hin he e alua ion. The ou pu o he
app oxima ion on se e al mul iaxial axial– o sion load pa hs is compa ed o ou own expe imen-
al esul s. Expe imen s we e ca ied ou on samples o wo no ch ypes manu ac u ed om he
2124-T851
aluminum alloy. S ain dis ibu ion in he no ch a ea was measu ed by digi al image
co ela ion. The compa ison be ween compu a ional solu ion and measu ed esponse shows ha he
new me hod allows o ob aining easonably good app oxima ion, e en o ela i ely complica ed
mul iaxial load cases.
Keywo ds: plas ici y; mul iaxial loading; pseudos ess; s ess–s ain es ima ion; 2124-T851
1. In oduc ion
Mos ini ially iso opic enginee ing ma e ials exhibi elas ic–plas ic beha io . Howe e ,
he ini e-elemen calcula ion o plas ici y is ime-consuming and equi es mo e inpu
da a. Enginee s dealing wi h a igue li e es ima ion, he e o e, o en use app oaches ha
conside only elas ic beha io o assess he du abili y o s uc u es. Such a simpli ica ion
is possible in he domain o high-cycle a igue, bu i is unaccep able i he low-cycle
egime is e alua ed, whe e he scale o plas ici y is much mo e subs an ial. In o de o
assess he elas ic–plas ic s ess s a e, app oxima e me hods ha ake elas ic–plas ic ma e ial
beha io in o conside a ion migh be a good al e na i e o ime–consuming ini e-elemen
elas ic–plas ic analyses whene e a longe load his o y should be analyzed.
Many me hods ha e al eady been sugges ed o he es ima ion o elas ic–plas ic
s esses and s ains. The i s g oup o me hods con ains hose in ended o mono onic
loading only [
1
–
5
]. The me hods do no ake cyclic ha dening o cyclic so ening in o
accoun , and hey do no desc ibe he mo emen o he yield su ace. The e o e, his g oup
o me hods is no sui able o cyclic loading. The in es iga ion o hese me hods can be
ound in [6].
The second g oup o me hods [
7
–
16
] deals wi h cycling loading and inco po a es
plas ici y models o desc ibe cyclic ha dening o so ening and yield su ace mo emen .
Unlike ini e-elemen analyses (FEA), app oxima e me hods do no deal wi h elas ic–plas ic
s i ness ma ices o ob ain a solu ion. Ins ead, hey use an elas ic solu ion ha hey con e
in o an elas ic–plas ic solu ion by using a ela ion ei he be ween pseudoma e ial and eal
ma e ial o be ween linea –elas ic and elas ic–plas ic s ain ene gies.
Ba key in oduced one o he i s elas ic–plas ic s ess–s ain app oxima ion me hods
o cyclic loading [
7
]. Nominal s esses and plas ic s ains we e ela ed in his wo k in o de
o e ie e he app oxima ion. The kinema ic wo k-ha dening model o M óz [
17
] was used
Ma e ials 2022,15, 1432. h ps://doi.o g/10.3390/ma15041432 h ps://www.mdpi.com/jou nal/ma e ials
Ma e ials 2022,15, 1432 2 o 22
o desc ibe elas ic–plas ic beha io . Ba key epo ed good co ela ion be ween es ima es
wi h he expe imen s and FEA.
Two app oaches, pseudono ch s ess and pseudono ch s ain, we e p esen ed in [
8
].
Bo h app oaches inco po a ed he M óz model o plas ici y. The essence o he me hods
is in ela ing en i ies om a pu ely elas ic solu ion o eal elas ic–plas ic ones, e.g., he
equi alen pseudos ess o eal plas ic equi alen s ain, c ea ing a pseudoma e ial wi h i .
The beha io o he pseudoma e ial is simul aneously cha ac e ized by s esses om he
elas ic solu ion and by plas ic s ains om he eal elas ic–plas ic ma e ial esponse.
In conjunc ion wi h Koe gen’s model, Langlais [
9
] used he in ini e su ace ha dening
ule, which is a modi ica ion o M óz’s model p esen ed by Chu [
18
]. Ins ead o he low
ule, Langlais used D ucke ’s equa ion, which ela ed plas ic s ain a e and gene alized
plas ic modulus. Langlais epo ed ha he same le el o p ecision was achie ed as in [
7
,
8
].
Fi a [
10
] used a pseudos ess me hod simila o he me hod p esen ed by Koe gen.
The app oach was combined wi h he a e-independen plas ici y model by Chaboche [
19
].
The au ho epo ed a high accu acy o he p edic ed alues.
Ince e al. [
11
] combined he P and l–Reuss low ule [
20
,
21
], an assump ion abou he
equi alence o inc emen s o he o al dis o ional s ain ene gy densi y, and he Ga ud
mul isu ace plas ici y model [
22
]. The au ho s epo ed nonconse a i e es ima es, as
s ain anges we e p edic ed o be 4–15% smalle han he expe imen al esul s ha had
been used o compa ison. Rega ding ene gy app oaches, an o e iew o he equi alen
s ain ene gy densi y (ESED) app oach de elopmen by Glinka and cowo ke s is desc ibed
in [23].
Ye e al. [
12
] p oposed a new uni ied exp ession based on he he modynamic analysis
o cyclic plas ic de o ma ion. The au ho s used his solu ion wi h he ma e ial cons i u i e
model p oposed by Jiang and Sehi oglu [
24
] o es ima e no ch s esses and s ains. The mean
ela i e e o s we e epo ed o be
−
3.5% o he axial s ain componen , and
−
3.9% o he
shea s ain componen . The au ho s indica ed ha he uni ied exp ession de eloped in he
pape had he ange o applicabili y limi ed o chosen geome ies and loading condi ions,
and ha u he e i ica ions o he p oposed app oxima e me hod we e needed.
One o he ecen wo ks on he opic o pseudocu e app oaches was conduc ed by
Li e al. [13]
. They combined he pseudos ain me hod wi h he Jiang–Sehi oglu plas ici y
model. The au ho s epo ed easonable esul s unde mul iaxial cyclic
axial– o sion loading
.
The app oach in [
14
] used angen moduli o pseudo and eal cu es o calcula e
he eal s ess his o y. The Ga ud plas ici y model was used o desc ibe he beha io o
he ma e ial. Expe imen al no ch s ain da a we e p esen ed o samples om he TC21
i anium alloy. To he au ho s’ knowledge, i is he i s wo k in which no ch s ess–s ain
co ec ion me hod es ima es we e alida ed on o he ma e ial han s eel.
Simila ly o [
14
], Li e al. [
15
] used angen moduli o s ess es ima ion, bu hei
me hod also ook in o accoun he in luence o empe a u e. This was achie ed by inco -
po a ing he Rambe g–Osgood equa ion in high- empe a u e o m [
25
]. Es ima es we e
alida ed agains FEA.
K a and Vo mwald [
16
] combined he uni ied exp ession o [
12
] wi h he Ohno–
Wang plas ici y model [
26
]. The in eg a ion algo i hm used o calcula e he elas ic–plas ic
a iables was desc ibed in dep h.
One d awback o some o he me hods men ioned abo e, also no ed by o he au-
ho s [
13
], is ha he me hods a e no su icien ly desc ibed o eplica ion by o he esea ch
eams. The second d awback o he cu en s a e o such me hods is ha expe imen al
e i ica ion was only ca ied ou on a limi ed numbe o ma e ials and es pa hs (Table 1).
Table 1shows ha mos o he ma e ials used o he alida ion o he me hods
we e s eel ypes, and only ou sou ces o expe imen al da a we e used. This p o es ha
a wide expe imen al in es iga ion is necessa y o explo e he ange o applicabili y o
such me hods.
Ma e ials 2022,15, 1432 3 o 22
Table 1.
Ma e ials used in o he pape s o he alida ion o he me hods o calcula ing he elas ic–
plas ic s ess and s ain.
Au ho Ma e ial Used Da a
Ba key [7] 1070 s eel [7]
Koe gen e al. [8] s eel only FEA
Langlais [9] 1070 s eel [7]
Fi a [10] 1070 s eel [7]
Ince e al. [11] 1070 s eel [7]
Ye e al. [12] S460N s eel [12]
Li e al. [13] 1070 s eel, S460N s eel [7,12]
Tao e al. [14]
TC21 i anium alloy, 1070 s eel
[7,14]
Li e al. [15] GH4169 supe alloy only FEA
K a [16] s eel only FEA
The p esen pape in oduces a new combina ion o he pseudos ess me hod and he
Abdel–Ka im–Ohno (AKO) plas ici y model [
27
]. The a icle desc ibes he pseudos ess–
plas ic s ain app oach. Valida ion was pe o med on he own new expe imen al da a
measu ed on no ched samples manu ac u ed om he 2124-T851 aluminum alloy and
on a ailable expe imen al da a o 1070 s eel [
7
] and TC21 i anium alloy [
14
]. The AKO
model has i s ad an ages in an addi ional pa ame e ha in luences he a che ing esponse.
I can be enhanced by a memo y su ace in oduc ion [
28
] in a u u e applica ion o he
pseudos ess–plas ic s ain app oach.
2. Ma e ials and Me hods
2.1. App oxima e Me hod
Pseudoma e ial app oaches a e based on ma e ial beha io ha couples ei he elas ic
s ess wi h elas ic–plas ic s ain o elas ic s ain wi h elas ic–plas ic s ess. The beha io
o he ma e ial can be ep esen ed by a pseudocu e ha is analogous o he s a ic/cyclic
s ess–s ain cu e (Figu e 1). In he case o he pseudos ess– eal plas ic s ain cu e,
which is used in his wo k, he pseudoma e ial expe iences s esses ha co espond o he
s esses ela ed o a pu ely elas ic solu ion, while i plas icizes acco ding o i s eal ma e ial
esponse a he same ime.
Ma e ials 2022, 15, x FOR PEER REVIEW 3 o 24
Table 1. Ma e ials used in o he pape s o he alida ion o he me hods o calcula ing he elas ic–
plas ic s ess and s ain.
Au ho Ma e ial Used Da a
Ba key [7] 1070 s eel [7]
Koe gen e al. [8] s eel only FEA
Langlais [9] 1070 s eel [7]
Fi a [10] 1070 s eel [7]
Ince e al. [11] 1070 s eel [7]
Ye e al. [12] S460N s eel [12]
Li e al. [13] 1070 s eel, S460N s eel [7,12]
Tao e al. [14] TC21 i anium alloy, 1070 s eel [7,14]
Li e al. [15] GH4169 supe alloy only FEA
K a [16] s eel only FEA
Table 1 shows ha mos o he ma e ials used o he alida ion o he me hods we e
s eel ypes, and only ou sou ces o expe imen al da a we e used. This p o es ha a wide
expe imen al in es iga ion is necessa y o explo e he ange o applicabili y o such me h-
ods.
The p esen pape in oduces a new combina ion o he pseudos ess me hod and he
Abdel–Ka im–Ohno (AKO) plas ici y model [27]. The a icle desc ibes he pseudos ess–
plas ic s ain app oach. Valida ion was pe o med on he own new expe imen al da a
measu ed on no ched samples manu ac u ed om he 2124-T851 aluminum alloy and on
a ailable expe imen al da a o 1070 s eel [7] and TC21 i anium alloy [14]. The AKO model
has i s ad an ages in an addi ional pa ame e ha in luences he a che ing esponse. I
can be enhanced by a memo y su ace in oduc ion [28] in a u u e applica ion o he pseu-
dos ess–plas ic s ain app oach.
2. Ma e ials and Me hods
2.1. App oxima e Me hod
Pseudoma e ial app oaches a e based on ma e ial beha io ha couples ei he elas ic
s ess wi h elas ic–plas ic s ain o elas ic s ain wi h elas ic–plas ic s ess. The beha io
o he ma e ial can be ep esen ed by a pseudocu e ha is analogous o he s a ic/cyclic
s ess–s ain cu e (Figu e 1). In he case o he pseudos ess– eal plas ic s ain cu e,
which is used in his wo k, he pseudoma e ial expe iences s esses ha co espond o he
s esses ela ed o a pu ely elas ic solu ion, while i plas icizes acco ding o i s eal ma e-
ial esponse a he same ime.
Figu e 1. Pseudos ess– eal plas ic s ain cu e and cyclic s ess–s ain cu e; poin s ep esen dis-
c e e e sions o cu es based on he Hollomon pa ame e s.
Figu e 1.
Pseudos ess– eal plas ic s ain cu e and cyclic s ess–s ain cu e; poin s ep esen
disc e e e sions o cu es based on he Hollomon pa ame e s.
The e a e wo ypes o pseudos ess app oaches. In he i s app oach, pseudos ess is
pai ed wi h he o al s ain [
13
]; in he second ype, pseudos ess is pai ed wi h he plas ic
s ain [
8
,
10
]. The main di e ence be ween he wo app oaches is in he s ain componen
ha is ob ained when a plas ici y model is applied o pseudos ess his o y, namely, i i is a
o al s ain enso o a plas ic s ain enso . In he p esen pape , he second ype o solu ion
is used.
Ma e ials 2022,15, 1432 4 o 22
2.1.1. Es ablishing Pseudos ess Ma e ial Cu e
A pseudocu e was es ablished by combining elas ic s ess wi h plas ic s ain. The
plas ic s ain alues we e he same as he plas ic s ain alues o he eal cyclic s ess–s ain
(CSS) cu e. The CSS cu e could be ob ained by he Rambe g–Osgood exp ession using
Hollomon pa ame e s (Sec ion 2.3). Because he numbe o selec ed plas ic s ain alues
was ini e, bo h cu es we e disc e e.
Elas ic s ess was calcula ed on he basis o he modi ica ion o he Neube ule:
σe= σεp+σ
EE, (1)
whe e
σ
and
εp
a e eal s ess and plas ic s ain desc ibed by he cyclic s ess–s ain cu e,
espec i ely, and Eis Young’s modulus.
When he cu e was es ablished, he pa ame e s o plas ici y model C
i
and
γi
we e
calcula ed. The numbe o in e als be ween disc e e poin s io pseudo o eal cu es
a ec ed he numbe o backs esses used in he app oxima ion because pai s o C
i
and
γi
pa ame e s we e calcula ed he e o each disc e e in e al o ei he a pseudo o eal cu e,
and he numbe o pai s o Ciand γiga e he numbe o backs esses.
Du ing he app oxima ion p ocess, cu es we e ep esen ed solely by he C
i
and
γi
pa ame e s. They a e no e e enced in any o he way.
2.1.2. Ge ing Real S ain and Real S ess
When pa ame e s C
i
and
γi
ep esen ing pseudo and eal cu es had been de ined,
he plas ici y model was applied o he elas ic s ess his o y. Elas ic s ess his o y can
be ob ained, e.g., o a chosen loading pa h om an elas ic FEA. Because o he way he
pseudocu e was buil , his s ep p o ided a eal plas ic s ess enso and accumula ed
plas ic s ain as i s ou pu s. De ailed analyses o his p ope y o pseudoma e ial a e
p esen ed in Sec ion 2.1.3.
Once he plas ic s ain enso had been ob ained, he plas ici y model was applied
again, and he eal s ess and eal o al s ain we e es ima ed. Howe e , his ime, he
inc emen al algo i hm o acqui e he accumula ed plas ic s ain was no in ol ed because
i had been calcula ed in he p e ious s ep.
2.1.3. Equi alence o Pseudo and Real Plas ic S ain Tenso s
The key pa in calcula ing he eal esponse om he pseudo a iables is he equi a-
lence o he accumula ed plas ic s ain dp o he pseudocu e and o he eal s ess–s ain
cu e. This was ensu ed by he way he pseudocu e had been es ablished (Sec ion 2.1.1).
Unlike he case o accumula ed plas ic s ain, he equi alence o he pseudoplas ic
s ain enso and o he eal plas ic s ain enso was no explici ly s a ed in [
8
] o in [
10
],
whe e simila app oaches we e used. Howe e , i was s a ed in [
8
] ha applying he
plas ici y model o he pseudos ess his o y esul s in a eal plas ic s ain enso . This
s a emen suppo s he claim o plas ic s ain enso equi alence.
Jus i ica ion can be ound when analyzing he widely used ela ionship be ween
accumula ed plas ic s ain inc emen dp and plas ic s ain enso inc emen dεp
dp = 2
3dεp:dεp, (2)
and he low ule:
dεp=3
2dps−a
σy(3)
whe e
s
is he de ia o ic pa o he s ess enso ,
a
is he de ia o ic pa o he backs ess,
and σyis yield s eng h. dp and σywe e iden ical o he eal cu e and he pseudocu e.
Due o he in insic di e ence be ween he elas ic and elas ic–plas ic ma e ial beha io
o iso opic ma e ials, he eal s ess was smalle han he pseudos ess unde he same
Ma e ials 2022,15, 1432 5 o 22
load. Backs ess
α
ollowed he s ess while main aining he adius o he yield sphe e
du ing he kinema ic ha dening. Because o his, i was sa e o assume ha enso s
s−a
o he eal and he pseudoma e ial simila ly changed; mo e speci ically, he co esponding
componen s o
s−a
changed in a simila manne o bo h ma e ials. I one componen
inc eased o he pseudoma e ial, he co esponding componen o he eal ma e ial also
inc eased. In o he wo ds, he inc emen s should ha e a leas he same sign.
The inc emen o plas ic s ain was ob ained by mul iplying
s−a
. The e o e, he same
assump ions as o
s−a
enso we e alid o he inc emen o plas ic s ain. Howe e ,
hen, i all he co esponding componen s o he eal/pseudoplas ic s ain changed in a
simila manne by ei he inc easing o dec easing, hey could no p o ide he same dp in
Equa ion (2) unless hey we e equal. Hence, he componen s o pseudo and eal plas ic
s ains had o be he same.
2.1.4. App oxima ion Me hod S ep by S ep
This sec ion summa izes he app oxima ion me hod desc ibed in Sec ions 2.1.1 and 2.1.2:
1. Pseudoma e ial cu e is es ablished.
2.
Pseudos ess his o y is ob ained ei he by elas ic FEA o using s ess concen a ion
ac o s [10].
3.
The plas ici y model is applied o he pseudos ess his o y. In his s ep, plas ici y
pa ame e s Ci and
γ
iob ained o he pseudoma e ial a e used. The plas ic s ain
enso and he accumula ed s ain a e calcula ed.
4.
The plas ici y model is applied o he plas ic s ain enso ob ained and o he accu-
mula ed s ain. In his s ep, he plas ici y pa ame e s Ci and
γ
i o he eal ma e ial
a e used. Real s ess and eal backs ess a e calcula ed.
2.2. Abdel–Ka im–Ohno Plas ici y Model
The ime-independen heo y o plas ici y was conside ed because he aluminum alloy
is no sensi i e o he s ain a e. I was assumed ha he o al s ain enso was composed
o elas ic and plas ic s ain enso s εe,εpacco ding o he addi i e ule:
ε=εe+εp. (4)
S esses a e compu ed using Hooke’s law:
σ=D:εe, (5)
whe e
D
is he elas ic s i ness enso o he ou h o de . The yield su ace is in oduced in
he de ia o ic space based on he on Mises condi ion:
(σ)= 3
2(s−a):(s−a)–σY=0, (6)
whe e
σY
ep esen s he adius o he yield su ace (and yield s eng h). The low ule and he
accumula ed plas ic s ain we e de ined in Equa ions (2) and (3). Acco ding o Chaboche [
29
],
he backs ess is cons uc ed by supe posing ele an pa s o he backs esses:
a=∑M
i=1a(i). (7)
The kinema ic ha dening ule is in oduced in acco dance wi h Abdel–Ka im and Ohno
heo y ([27], AKO), i.e., he e olu ion o backs ess is de ined by he di e en ial equa ion:
da(i)=2
3Cidεp−µiγia(i)dp −γiH( i)hdλiia(i), (8)

Ma e ials 2022,15, 1432 6 o 22
whe e
i=3
2a(i):a(i)−Ci
γi2
, (9)
dλi=dεp:a(i)
Ci/(γi)–µidp. (10)
In Equa ions (8)–(10),
Ci
,
γi
a e basic ma e ial pa ame e s,
µi
is he a che ing pa am-
e e , he symbol
hxi
ep esen s Macaulay b acke s (
hxi=(x+|x|)/
2) and
H( i)
is he
Hea iside s ep unc ion. Unde uniaxial loading, he model gi es plas ic shakedown o
µi=
0 o all
i
. Then, mo e p ecisely, he mul ilinea model o Ohno and Wang [
26
] is
ob ained. This op ion is called OWI he ea e (Ohno-Wang model I, see [
26
]). The maximal
a che ing a e is gi en by
µi=
1 o all
i
(A ms ong and he F ede ick ule [
30
]), and he
co esponding op ion is ma ked as CHAB (Chaboche model [19]).
The same alue o he a che ing pa ame e is usually se o all backs ess pa s,
µi=µ
o all
i
, and i s alue is usually small. The e o e, he AKO model is calib a ed in
he same way as he OWI model [
26
]. The selec ed poin s o he cyclic s ess–s ain cu e
di ec ly de ine he alues o he basic ma e ial pa ame e s by ela ions
Ci=σa(i)−σa(i−1)
εap(i)−εap(i−1)−σa(i+1)−σa(i)
εap(i+1)−εap(i)
o i6=M, (11)
CM=σa(M)−σa(M−1)
εap(M)−εap(M−1)
, (12)
γi=1
εap(i)
o all i, (13)
whe e
σa(i)
,
εap(i)
a e he s ess ampli ude and he plas ic s ain ampli ude o he gi en
i
- h
poin , espec i ely. The esul ing basic ma e ial pa ame e s o he 2124-T851 aluminum
alloy a e lis ed in Table 2.
Table 2. Ma e ial pa ame e s o plas ici y model ob ained o 2124-T851 aluminum alloy.
Real Pseudo
Pa ame e Value Pa ame e Value
C1(MPa) 16,401 C1(MPa) 51,760
γ1(-) 500 γ1(-) 500
C2(MPa) 4561 C2(MPa) 7205
γ2(-) 250 γ2(-) 250
C3(MPa) 1948 C3(MPa) 3908
γ3(-) 166.7 γ3(-) 166.7
C4(MPa) 1097 C4(MPa) 2637
γ4(-) 125 γ4(-) 125
C5(MPa) 4216 C5(MPa) 27,589
γ5(-) 100 γ5(-) 100
2.3. Expe imen al Da a
To e i y he app oxima ion me hod, a igue expe imen s we e ca ied ou on wo
ypes o no ched samples (Figu es 2and 3) manu ac u ed om aluminum alloy
2124-T851
.
The Young’s modulus o he ma e ial was 73,100 MPa [31], Poisson numbe was 0.33, and
Hollomon pa ame e s o he Rambe g–Osgood cu e we e K= 646 MPa and
n= 0.089
.
Cyclic yield s eng h
σy
o 330 MPa was used o he FEA simula ion and o he app oxi-
ma ion me hod.
Ma e ials 2022,15, 1432 7 o 22
Ma e ials 2022, 15, x FOR PEER REVIEW 7 o 24
Cyclic yield s eng h 𝜎 o 330 MPa was used o he FEA simula ion and o he app ox-
ima ion me hod.
Figu e 2. Specimen wi h U-no ch. A and B in ames indica e su aces based on which a da um
axis o geome ical un-ou ole ance is de ined.
Figu e 3. Specimen wi h ille . A and B in ames indica e su aces based on which a da um axis
o geome ical un-ou ole ance is de ined.
The aim o he expe imen s was o measu e he no ch ip s ains, speci ically he axial
and shea s ain componen s. These componen s a e commonly used o compa e es i-
ma es wi h expe imen al esul s [7–16].
Expe imen s we e ca ied ou unde o ce and momen con ol. The i s eason o
he load-con olled expe imen s is he possibili y o ecalcula ing he loading o ces and
momen s in o he local elas ic no ch s ess his o y using he s ess concen a ion ac o s as
desc ibed in [10]. Howe e , in his wo k, he elas ic FEA was used o ob ain he local s ess
his o y. The second eason o using o ce con ol is ha he s ain con ol o no ched
specimens would equi e complex eal- ime no ch s ain measu emen and p ocessing.
The s ess a io o nominal axial s ess o shea s ess was 1 o pa hs Squa e, NV
shape, X, and pa h 7 (see Figu e 4). In he case o pa h Ci cle, esul s co esponding o
s ess a ios 1 and 1.73 a e p esen ed. Fo s ess a io 1, maximal o ce was 65.8 kN, and
maximal momen was 329 Nm. Fo he 1.73 a io, maximal o ce was 100.5 kN, and max-
imal momen was 290 Nm. Expe imen s we e ca ied ou a oom empe a u e.
Figu e 2.
Specimen wi h U-no ch. A and B in ames indica e su aces based on which a da um axis
o geome ical un-ou ole ance is de ined.
Ma e ials 2022, 15, x FOR PEER REVIEW 7 o 24
Cyclic yield s eng h 𝜎 o 330 MPa was used o he FEA simula ion and o he app ox-
ima ion me hod.
Figu e 2. Specimen wi h U-no ch. A and B in ames indica e su aces based on which a da um
axis o geome ical un-ou ole ance is de ined.
Figu e 3. Specimen wi h ille . A and B in ames indica e su aces based on which a da um axis
o geome ical un-ou ole ance is de ined.
The aim o he expe imen s was o measu e he no ch ip s ains, speci ically he axial
and shea s ain componen s. These componen s a e commonly used o compa e es i-
ma es wi h expe imen al esul s [7–16].
Expe imen s we e ca ied ou unde o ce and momen con ol. The i s eason o
he load-con olled expe imen s is he possibili y o ecalcula ing he loading o ces and
momen s in o he local elas ic no ch s ess his o y using he s ess concen a ion ac o s as
desc ibed in [10]. Howe e , in his wo k, he elas ic FEA was used o ob ain he local s ess
his o y. The second eason o using o ce con ol is ha he s ain con ol o no ched
specimens would equi e complex eal- ime no ch s ain measu emen and p ocessing.
The s ess a io o nominal axial s ess o shea s ess was 1 o pa hs Squa e, NV
shape, X, and pa h 7 (see Figu e 4). In he case o pa h Ci cle, esul s co esponding o
s ess a ios 1 and 1.73 a e p esen ed. Fo s ess a io 1, maximal o ce was 65.8 kN, and
maximal momen was 329 Nm. Fo he 1.73 a io, maximal o ce was 100.5 kN, and max-
imal momen was 290 Nm. Expe imen s we e ca ied ou a oom empe a u e.
Figu e 3. Specimen wi h ille . A and B in ames indica e su aces based on which a da um axis o
geome ical un-ou ole ance is de ined.
The aim o he expe imen s was o measu e he no ch ip s ains, speci ically he axial
and shea s ain componen s. These componen s a e commonly used o compa e es ima es
wi h expe imen al esul s [7–16].
Expe imen s we e ca ied ou unde o ce and momen con ol. The i s eason o
he load-con olled expe imen s is he possibili y o ecalcula ing he loading o ces and
momen s in o he local elas ic no ch s ess his o y using he s ess concen a ion ac o s
as desc ibed in [
10
]. Howe e , in his wo k, he elas ic FEA was used o ob ain he local
s ess his o y. The second eason o using o ce con ol is ha he s ain con ol o no ched
specimens would equi e complex eal- ime no ch s ain measu emen and p ocessing.
The s ess a io o nominal axial s ess o shea s ess was 1 o pa hs Squa e, NV
shape, X, and pa h 7 (see Figu e 4). In he case o pa h Ci cle, esul s co esponding o s ess
a ios 1 and 1.73 a e p esen ed. Fo s ess a io 1, maximal o ce was 65.8 kN, and maximal
momen was 329 Nm. Fo he 1.73 a io, maximal o ce was 100.5 kN, and maximal momen
was 290 Nm. Expe imen s we e ca ied ou a oom empe a u e.
Pa h Squa e was achie ed by mul iaxial loading by apezoidal wa e o ms o o ce and
momen signals wi h a mu ual phase shi o 90
◦
. The common load equency o he en i e
es was 0.0417 Hz. O he au ho s used a simila pa h o alida e hei es ima es [7,9–16].
Fo he NV pa h, bo h channels had sinusoidal wa e o ms. The loading equency
o he o sion channel was i e imes as e han ha o he ension comp ession channel.
Fo he digi al image co ela ion (DIC) measu emen , which was ca ied ou o he i s
Ma e ials 2022,15, 1432 8 o 22
49 cycles
, equencies we e 0.004 Hz o he o que channel, and 0.02 Hz o he o ce chan-
nel. O he au ho s used his pa h o alida e o he app oxima ion me hods [7–9,11,14,15].
Ma e ials 2022, 15, x FOR PEER REVIEW 8 o 24
Figu e 4. Loading pa hs: ( op le ) Squa e; ( op middle) NV; ( op igh ) X; (bo om le and bo om
middle) Ci cle; (bo om igh ) 7.
Pa h Squa e was achie ed by mul iaxial loading by apezoidal wa e o ms o o ce
and momen signals wi h a mu ual phase shi o 90°. The common load equency o he
en i e es was 0.0417 Hz. O he au ho s used a simila pa h o alida e hei es ima es
[7,9–16].
Fo he NV pa h, bo h channels had sinusoidal wa e o ms. The loading equency o
he o sion channel was i e imes as e han ha o he ension comp ession channel. Fo
he digi al image co ela ion (DIC) measu emen , which was ca ied ou o he i s 49
cycles, equencies we e 0.004 Hz o he o que channel, and 0.02 Hz o he o ce channel.
O he au ho s used his pa h o alida e o he app oxima ion me hods [7–9,11,14,15].
Pa h Ci cle consis ed o wo sinusoidal wa e o ms o axial o ce and o que wi h a
phase shi o 90°. The loading equency was 0.1 Hz. In con as o pa hs Squa e and NV,
which appea ed in se e al publica ions by o he au ho s, pa h Ci cle has so a only been
p esen ed in [7,12,16].
Pa h 7 ep esen s loading by a cons an o que in one channel and a sinusoidal wa e-
o m o ension–comp ession in he o he . The loading equency when measu ing he
s ains was 0.1 Hz. Pa h 7 was used o alida ion in [7,8] bu only in he o m o FEA
simula ions.
Loading equencies we e se acco ding o he abili y o he es ing machine used o
main ain he loading pa hs on each channel wi hou dis o ions. The sampling equency
o he DIC came as was also aken in o accoun , as he loading equencies had o be 20
imes smalle han he sampling equency o he used came as o a oid aliasing.
The es ing machine used o he expe imen s was INOVA FU 250 (dis ibu ed by
Ino a P aha s. .o.), mul iaxial ension–comp ession and o sion load ame wi h hyd aulic
ac ua o o dynamic loading. The maximal alue o he axial o ce channel o he machine
is 250 kN, and he maximal momen is 2000 Nm.
The Dan ec Dynamics 3D Q-450 high-speed image co ela ion sys em was used o
he DIC measu emen . The sys em consis s o MKII-NanoSense came as wi h a CCD sen-
so wi h esolu ion o 1024 × 1280 pixels and Is a 4-D so wa e ( e sion 4.4.3.414). The
so wa e was used o calib a ion, measu emen , displacemen e alua ion, and displace-
men expo .
Fo calcula ing axial and shea s ains on he basis o expo ed displacemen s om
Is a 4D , a p og am was w i en in MATLAB language. The eason o p ocessing he
Figu e 4.
Loading pa hs: (
op le
) Squa e; (
op middle
) NV; (
op igh
) X; (
bo om le
and
bo om
middle) Ci cle; (bo om igh ) 7.
Pa h Ci cle consis ed o wo sinusoidal wa e o ms o axial o ce and o que wi h a
phase shi o 90
◦
. The loading equency was 0.1 Hz. In con as o pa hs Squa e and NV,
which appea ed in se e al publica ions by o he au ho s, pa h Ci cle has so a only been
p esen ed in [7,12,16].
Pa h 7 ep esen s loading by a cons an o que in one channel and a sinusoidal
wa e o m o ension–comp ession in he o he . The loading equency when measu -
ing he s ains was 0.1 Hz. Pa h 7 was used o alida ion in [
7
,
8
] bu only in he o m o
FEA simula ions.
Loading equencies we e se acco ding o he abili y o he es ing machine used o
main ain he loading pa hs on each channel wi hou dis o ions. The sampling equency o
he DIC came as was also aken in o accoun , as he loading equencies had o be
20 imes
smalle han he sampling equency o he used came as o a oid aliasing.
The es ing machine used o he expe imen s was INOVA FU 250 (dis ibu ed by
Ino a P aha s. .o.), mul iaxial ension–comp ession and o sion load ame wi h hyd aulic
ac ua o o dynamic loading. The maximal alue o he axial o ce channel o he machine
is 250 kN, and he maximal momen is 2000 Nm.
The Dan ec Dynamics 3D Q-450 high-speed image co ela ion sys em was used o he
DIC measu emen . The sys em consis s o MKII-NanoSense came as wi h a CCD senso
wi h esolu ion o 1024
×
1280 pixels and Is a 4-D so wa e ( e sion 4.4.3.414). The so wa e
was used o calib a ion, measu emen , displacemen e alua ion, and displacemen expo .
Fo calcula ing axial and shea s ains on he basis o expo ed displacemen s om
Is a 4D, a p og am was w i en in MATLAB language. The eason o p ocessing he da a
ou side he DIC sys em was he possibili y o applying a highe le el o au oma ion and
mo e con ol o e displacemen smoo hing.
3. Fini e Elemen Analyses
FE analyses wi h pu ely elas ic and elas ic–plas ic ma e ial da a we e ca ied ou in
Abaqus 6.14-5. The elas ic ma e ial model aimed o achie e no ch ip s ess his o ies o
use hem as inpu s o he app oxima ions. Analyses wi h elas ic–plas ic ma e ial da a we e
pe o med o e i y he co espondence o he ma e ial da a and he expe imen al esul s.
Ma e ials 2022,15, 1432 9 o 22
Specimens we e modeled as axisymme ic. The same mesh was used o bo h elas ic
and elas ic–plas ic analyses. The inal mesh size o he quad a ic axisymme ic s ess ele-
men s CGAX8R (8-node biquad a ic, educed in eg a ion) was 0.1 mm (Figu e 5). A emp s
o u he dec ease he elemen size did no a ec he esul s by mo e han by 0.007%. All
elemen s passed he mesh quali y check wi hou e o s and wa nings.
Ma e ials 2022, 15, x FOR PEER REVIEW 9 o 24
da a ou side he DIC sys em was he possibili y o applying a highe le el o au oma ion
and mo e con ol o e displacemen smoo hing.
3. Fini e Elemen Analyses
FE analyses wi h pu ely elas ic and elas ic–plas ic ma e ial da a we e ca ied ou in
Abaqus 6.14-5. The elas ic ma e ial model aimed o achie e no ch ip s ess his o ies o
use hem as inpu s o he app oxima ions. Analyses wi h elas ic–plas ic ma e ial da a
we e pe o med o e i y he co espondence o he ma e ial da a and he expe imen al
esul s.
Specimens we e modeled as axisymme ic. The same mesh was used o bo h elas ic
and elas ic–plas ic analyses. The inal mesh size o he quad a ic axisymme ic s ess ele-
men s CGAX8R (8-node biquad a ic, educed in eg a ion) was 0.1 mm (Figu e 5). A -
emp s o u he dec ease he elemen size did no a ec he esul s by mo e han by
0.007%. All elemen s passed he mesh quali y check wi hou e o s and wa nings.
Figu e 5. Axisymme ic model o U-no ched specimen in Abaqus.
Fo elas ic–plas ic analyses, combined ha dening beha io was chosen wi h he s a-
bilized da a ype. The numbe o backs esses was se o 5. The cyclic s ess–s ain cu e
used in he model was calcula ed on he basis o he Hollomon pa ame e s p esen ed in
Sec ion 2.3. I s alues a e shown in Table 3.
The coincidence o he esponses om he expe imen s and om FEA (Figu e 6) a
he ini ial loading and a he beginning o cyclic loading sugges s ha he elas ic da a we e
alid. The e we e small di e ences in cyclic egions ha could ha e been caused by he
absence o a a che ing pa ame e in he combined ha dening model o plas ici y in
Abaqus.
Figu e 5. Axisymme ic model o U-no ched specimen in Abaqus.
Fo elas ic–plas ic analyses, combined ha dening beha io was chosen wi h he s a-
bilized da a ype. The numbe o backs esses was se o 5. The cyclic s ess–s ain cu e
used in he model was calcula ed on he basis o he Hollomon pa ame e s p esen ed in
Sec ion 2.3. I s alues a e shown in Table 3.
Table 3. Cyclic s ess–s ain cu e used o elas ic–plas ic simula ion.
S ess (MPa) Plas ic S ain (-)
330 0.000
371.56 0.002
395.19 0.004
409.72 0.006
420.35 0.008
428.78 0.010
435.79 0.012
441.81 0.014
447.09 0.016
451.81 0.018
The coincidence o he esponses om he expe imen s and om FEA (Figu e 6) a
he ini ial loading and a he beginning o cyclic loading sugges s ha he elas ic da a we e
Ma e ials 2022,15, 1432 16 o 22
%% %%%%%%%%%%%%%%%%%%%%%%%%% THE ESTIMATION PROCEDURE
ii=1;% index o es ima ion poin s, all ma ices a 1 consis o ze os.
o i = 2:1:(leng h(sigYyFI))
% i . . . index h ough loading s eps
SPsS a = SPs; % de ia o ic pseudos ess a he beginning o he s ep
al aPsI = al aPs; % I as ini ial
ii = ii+1;
%% ind he s a o he plas iciza ion
sigPs = [0 0 0;
0 sigYyFI(i sigYzFI(i);
0 sigYzFI(i) sigZzFI(i);];
SPs = sigPs - ace(sigPs)/3*eye(3); % pseudos ess de ia o
% ind s a o he plas iciza ion
ol = 1; %(MPa), ole ance
beginA = 0.01;
o i2 = beginA :beginA /10:1
SIn = SPsS a +(SPs-SPsS a ) * i2;
sEq = calc_equi _s ess(SIn -al aPsI);
i (sEq - yieldS eng h) < ol && (sEq - yieldS eng h) > 0
elas ic_pa _ins ep = 1; b eak;
elsei (sEq - yieldS eng h) > 0
% no pu ely elas ic loading du ing he s ep
elas ic_pa _ins ep = 0; b eak;
end
end
% check i elas ic loading/unloading has occu ed
i i2 ~= beginA
%% PURELY ELASTIC egion solu ion
dSigYPs =(sigYyFI(i)-sigYyFI(i-1))*i2;dSigZPs =(sigZzFI(i)-sigZzFI(i-1))*i2;
dSigYzPs = (sigYzFI(i)-sigYzFI(i-1))*i2;
elas ic_ egion_solu ion; % >> unc ion
epsXxPl(ii) = epsXxPl(ii-1);epsYyPl(ii) = epsYyPl(ii-1);
epsZzPl(ii) = epsZzPl(ii-1);epsYzPl(ii) = epsYzPl(ii-1);
else
% no pu ely elas ic loading du ing he s ep
sigYPs( ii)=sigYyFI(i-1);sigZPs( ii)=sigZzFI(i-1); sigYzPs(ii)=sigYzFI(i-1);
ii= ii-1; % in o de o no skip index when elas ic a iables we en’ calcula ed
end
%% plas ic egion solu ion (elas ic+plas ic s esses and s ains)
i i2 < 1
% plas iciza ion has occu ed
% pseudono ch s esses inc emen s ill he end o cu en s ep
dSigYPs =(sigYyFI(i) - sigYPs( ii))/incPe PlR;
dSigZPs =(sigZzFI(i) - sigZPs( ii))/incPe PlR;
dSigYzPs = (sigYzFI(i) - sigYzPs(ii))/incPe PlR;
o j = 1:1:incPe PlR
%% cycle ough inc emen s o plas ic ange
ii=ii+1;% index o calcula ed a iables – s esses and s ains
sigYPs(ii) = sigYPs(ii-1)+dSigYPs;sigZPs(ii) = sigZPs(ii-1)+dSigZPs;
sigYzPs(ii) = sigYzPs(ii-1)+dSigYzPs;% pseudos ess
sigPs = [0 0 0;
0 sigYPs(ii) sigYzPs(ii);
0 sigYzPs(ii) sigZPs(ii)];
SPs = sigPs- ace(sigPs)/3*eye(3);% de ia o o pseudos ess

Ma e ials 2022,15, 1432 17 o 22
[dp, he a] = calc_dp_AKO(gPC,cPC,SPs,al a_pa Ps);
[al aPs, al a_pa Ps, dEpsPlPs] = ...
calc_al a_and_dEpsPl(dp,gPC,cPC,al a_pa Ps,SPs, he a);
dEpsPl = dEpsPlPs;
[al a, al a_pa , SReal] = ...
calc_al a_and_Snp1(dp,gRC,cRC,al a_pa ,dEpsPl);
%% non-de ia o ic eal s ess componen s
sigRealH = -SReal(1,1);%% plain s ess condi ion
sigReal = SReal + sigRealH*eye(3);
sigY(ii) = sigReal(2,2);sigZ(ii) = sigReal(3,3);sigYz(ii) = sigReal(2,3);
dSigY = sigY(ii) - sigY(ii-1);dSigZ = sigZ(ii) - sigZ(ii-1);
dSigYz = sigYz(ii) - sigYz(ii-1);
%% s o ing calcula ed s ains
% plas ic s ain inc emen s - enso ial shea s ain o m!
x(1) = dEpsPl(1,1); x(2) = dEpsPl(2,2); x(3) = dEpsPl(3,3); x(4) = dEpsPl(2,3);
% elas ic s ain
dEpsXxE = 1/E * (-ny) * (dSigY + dSigZ);
dEpsYyE = 1/E * (dSigY + (-ny) * dSigZ);
dEpsZzE = 1/E * (dSigZ + (-ny) * dSigY);dGammaYzE = 1/G* dSigYz;
dEpsYzE = dGammaYzE/ 2;
epsXxE(ii) = epsXxE(ii-1) + dEpsXxE; epsYyE(ii) = epsYyE(ii-1) + dEpsYyE;
epsZzE(ii) = epsZzE(ii-1) + dEpsZzE;epsYzE(ii) = epsYzE(ii-1) + dEpsYzE;
% plas ic s ain
epsXxPl(ii) = epsXxPl(ii-1) + x(1); epsYyPl(ii) = epsYyPl(ii-1) + x(2);
epsZzPl(ii) = epsZzPl(ii-1) + x(3);epsYzPl(ii) = epsYzPl(ii-1) + x(4);
% o al s ain (elas ic o al new + plas ic o al)
epsXx(ii) = epsXxE(ii) + epsXxPl(ii); epsYy(ii) = epsYyE(ii) + epsYyPl(ii);
epsZz(ii) = epsZzE(ii) + epsZzPl(ii);epsYz(ii) = epsYzE(ii) + epsYzPl(ii);
end %j = 1:1:(incPe PlR)
end
end
g aphs; % unc ion o plo ing esul s
%% %%%%%%%%%%%%%%%%%% FUNCTIONS: %% %%%%%%%%%%%%%%%
unc ion [ePlRC, sRC, ePlPC, sPC] = ma e ial_cu es(E, K, nRO)
% RC . . . eal cu e (cyclic s ess s ain cu e); Pl . . . plas ic
% PC . . . pseudocu e;e . . . epsilon, s . . . sigma
%% REAL CURVE
% 4 in e als o gamma and C and 1 co esponding o he yield
% s eng h - added la e -> 5 in e als == 5 backs ess pa s
ePlRC = [0.002 0.004 0.006 0.008 0.01]0;
n = size(ePlRC,1); %numbe o disc e e poin s on he cu e
o i = 1:n
sRC(i,1) = K*ePlRC(i)ˆnRO;
end
%% PSEUDOCURVE
ePlPC= ePlRC;
o i = 1:n
%sig_e = sig *eps_ o *E
sPC(i,1) = sq ( sRC(i)* (ePlRC(i)+sRC(i)/E) *E );
end
end%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
unc ion [K, n] = ambe g_osgood_coe icien s(eps, sig)
= i (eps,sig,‘powe 1’);
egCs = coe alues( );
Ma e ials 2022,15, 1432 18 o 22
K = egCs(1); %RO wi h Hollomon pa ame e s: eps = sig/E + (sig/K)ˆ(1/n)
n = egCs(2);
end%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
unc ion [gamma, cMa ix] = calc_C_gamma(epsPl,s ess,K,n)
global yieldS eng h
slope = ze os(size(epsPl,1),1);gamma = ze os(size(epsPl,1)-1,1);
cMa ix = ze os(size(epsPl,1)-1,1);epsPlA YS = (yieldS eng h/K)ˆ(1/n); nn = size(epsPl,1);
o i = 1:nn
i i == 1
slope(1) = (s ess(1)-yieldS eng h)/(epsPl(i)-epsPlA YS);
else
slope(i) = (s ess(i)-s ess(i-1))/(epsPl(i)-epsPl(i-1));
end
end
slope(nn+1) = 0;
o i = 1:nn
gamma(i) = 1/epsPl(i);
cMa ix(i) = slope(i) - slope(i+1);
end
end%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
%% %%%%%%%%%%%%% PROCEDURE de ine_no ch_s ess_impu s.m
i s cmp(pa h_name,‘squa e’)
% Pa h Squa e,F = 66 kN, M = 329.133 Nm, D 26 mm
es _spcm = ‘u3’;
num_cycles_ ex = ‘1s 100 cycles’;
sigYyFI = [0112 109 109 -3-116-120-120-7 112 ];
sigZzFI = [0427 428 428 0 -429-427-4271427];
sigYzFI = [0-291 1 292 292 292 1 -291-291 -291];
elsei s cmp(pa h_name,‘NV’)
% Pa h NV;F = 66 kN, M = 329.133 Nm, D 26 mm
es _spcm = ‘u4’;
num_cycles_ ex = ‘1s 49 cycles’;
% coa se inpu - mul ilinea
sigYyFI = [0366591105 112 104 906333-4-37 -71 -95 ...
-113 -117-114-96 -73 -41 -8];
sigZzFI = [0132 252 346 407 428 408 346 253 133 1 -131-250 . . .
-345 -405-427-405-345-249-1303];
sigYzFI = [0-2920 292 0 -2920 292 0 -2920 ...
2910 -2910 291 0 -2910 291 0];
elsei s cmp(pa h_name,‘ci cle’)
% Pa h Ci cle;F = 66 kN, M = 329.133 Nm, D 26 mm
es _spcm = ‘u73’;
num_cycles_ ex = ‘1s 100 cycles’;
sigYyFI = [0112 106 9167361 -34 -66 -91 -108-114 ...
-108 -92 -67 -35 0 356590105 110];
sigZzFI = [0427 406 345 251 131 -1-133-253-347-408 ...
-429 -408-347-253-133-1131 251 345 406 427];
sigYzFI = [00 90172 236 277 292 277 236 171 900 -90 -171 . . .
-236 -277-291-277-236-171-90 0];
elsei s cmp(pa h_name,‘ci cle_1p73’)
% F = 101 kN, M = 290 Nm, D 26 mm
sigYyFI = [0170 170 162 138 101 531 -53 -101-140-165-174-165-140-102-53 0 53100 137
161 169];
Ma e ials 2022,15, 1432 19 o 22
sigZzFI = [0651 651 619 527 382 200 -2-204-386-531-624-656-624-531-386-204-2200 382
526 619 651];
sigYzFI = [00 0 80151 208 245 257 244 208 151 790 -79 -151-208-244-257-245-208-151-80
0];
elsei s cmp(pa h_name,‘7’)
% Pa h “7”; F = 66 kN, M = 329.133 Nm, D 26 mm
es _spcm = ‘ 9’;
num_cycles_ ex = ‘1s 100 cycles’;
sigYyFI = [080.8 -78.280.7];% angen ial di ec ion
sigZzFI = [0373.3-371.7 372.9]; % axial di ec ion,
sigYzFI = [0 -271.4-270.5-271.5]; % FI as o inc emen a ion
elsei s cmp(pa h_name,‘X’)
% F = 66 kN, M = 329.133 Nm, D 26 mm
sigYyFI = [0112-4109-7-120-10-123-14];
sigZzFI = [0427 04281-428 1 -428 1];
sigYzFI = [0 -292 12920 292 1 -291 0];
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF PROCEDURE
%% %%%%%%%%%%%%% PROCEDURE p ealoca e_ a iables.m
n = size(gPC,1);% numbe o backs ess pa s
al a = ze os(3,3);% o al backs ess
al a_pa = ze os(3,3,n); % backs ess pa s
al aPs = ze os(3,3);% o al pseudobacks ess
al a_pa Ps = ze os(3,3,n); % pa s o he pseudobacks ess
al a_pa PsI = ze os(3,3,n); % pa s o he pseudobacks ess a he beginning o he s ep
he a = ones(1,n);
SPs = ze os(3,3);SPsS a = ze os(3,3);SReal = ze os(3,3);
dEpsPl = ze os(3,3);dEpsPlPs = ze os(3,3); dp = 0;
i sigYyFI(leng h(sigYyFI)) == 0
% m = numbe o poin s in elas ic ange + numbe o poin s in plas ic
% ange + one ini ial ze o s a e
m = leng h(sigYyFI)-1 + incPe PlR*(leng h(sigYyFI)-2) + 1;
else
m = leng h(sigYyFI)-1 + incPe PlR*(leng h(sigYyFI)-1) + 1;
end
epsXx(1) = 0; epsYy(1) = 0; epsZz(1) = 0; epsYz(1) = 0;
epsXxE(1) = 0;epsYyE(1) = 0;epsZzE(1) = 0;epsYzE(1) = 0;
epsXxPl(1) = 0; epsYyPl(1) = 0; epsZzPl(1) = 0; epsYzPl(1) = 0;
sigY = ze os(m, 1); sigZ = ze os(m, 1); sigYz = ze os(m, 1);
sigYPs(1) = 0; sigYPs(2) = 0; sigZPs = ze os(m, 1); sigYzPs = ze os(m, 1);
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF PROCEDURE
unc ion equi alen _s ess = calc_equi _s ess(A)
aMa ix = A.*A;
equi alen _s ess = sq (3/2* sum(aMa ix(:)) );
end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
%% %%%%%%%%%%%%% PROCEDURE elas ic_ egion_solu ion.m
sigYPs(ii)= sigYPs(ii-1) + dSigYPs;sigZPs(ii)= sigZPs(ii-1) + dSigZPs;
sigYzPs(ii) = sigYzPs(ii-1)+ dSigYzPs;sigY(ii)= sigY(ii-1) + dSigYPs;
% in elas ic egime inc emen s o eal and pseudos esses a e he same
sigZ(ii)= sigZ(ii-1) + dSigZPs;sigYz(ii) = sigYz(ii-1)+ dSigYzPs;
sigReal = [0 0 0;
0 sigY(ii)sigYz(ii);
0 sigYz(ii) sigZ(ii)];
SReal = sigReal - ace(sigReal)/3*eye(3);
Ma e ials 2022,15, 1432 20 o 22
dEpsXxE = 1/E * (-ny) * (dSigYPs + dSigZPs);dEpsYyE = 1/E * (dSigYPs + (-ny) * dSigZPs);
dEpsZzE = 1/E * (dSigZPs + (-ny) * dSigYPs);dGammaYzE = 1/G *dSigYzPs;
dEpsYzE = dGammaYzE/ 2;
epsXxE(ii) = epsXxE(ii-1) + dEpsXxE;epsYyE(ii) = epsYyE(ii-1) + dEpsYyE;
epsZzE(ii) = epsZzE(ii-1) + dEpsZzE;epsYzE(ii) = epsYzE(ii-1) + dEpsYzE;
epsXx(ii) = epsXx(ii-1) + dEpsXxE;epsYy(ii) = epsYy(ii-1) + dEpsYyE;
epsZz(ii) = epsZz(ii-1) + dEpsZzE;epsYz(ii) = epsYz(ii-1) + dEpsYzE;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF PROCEDURE
unc ion [dp, he a] = calc_dp_AKO(gamma,cMa ix,S,al a_pa )
global yieldS eng h
n = size(gamma,1); he a = ones(1,n);% ec o s p ealoca ion
mu1 = 0; % a che ing pa ame e ; 0 . . . OWI; 0.1 . . . AKO; 1 . . . CHAB
dpkm1 = 0;dpkm2 = 0;% a iables o check con e gence
o k =1:100
aMa ix = ze os(3,3); % a suppo i e a iable o he calcula ion
aNum = 0;
o i2 = 1:n
aMa ix = aMa ix + he a(i2)*al a_pa (:,:,i2);
aNum = aNum + cMa ix(i2)* he a(i2);
end
aNum2 = calc_equi _s ess(S-aMa ix);
dp = (aNum2 - yieldS eng h)/aNum; % dp om he i s i e a ion
% The Ai ken’s del aˆ2 p ocess o sho en he con e gence —–
i mod(k,3) == 0
con = dp-(dp-dpkm1)*(dp-dpkm1)/(dp-2*dpkm1+dpkm2);
i con > 0
dp = con;
end
end % ———————————————————
SminusA = yieldS eng h/(yieldS eng h+aNum*dp)*(S-aMa ix);
dEpsPl = 3/2*dp*SminusA/yieldS eng h;
i abs(1-dpkm1/dp) < 10ˆ(-4)
b eak;% solu ion ound
end
o i2 = 1:n
aMa ix = ze os(3,3);alnp1_s (:,:,i2) = al a_pa (:,:,i2)+2/3*cMa ix(i2)*dEpsPl;
alnp1_dash(i2) = calc_equi _s ess(alnp1_s (:,:,i2));
mu(i2) = mu1;c(i2) = 1/(1+mu(i2)*gamma(i2)*dp);
alnp1_hash(:,:,i2) = c(i2)*(alnp1_s (:,:,i2));
np1_hash = calc_equi _s ess(alnp1_hash(:,:,i2))ˆ2-(cMa ix(i2)/gamma(i2))ˆ2;
he a(i2)=c(i2)+hea iside( np1_hash)* ...
(cMa ix(i2)/gamma(i2)/alnp1_dash(i2)-c(i2));
end
dpkm2 = dpkm1;dpkm1 = dp;
i k == 100
e o (‘E o : numbe o i e a ions has exceeded he allowed alue’);
k
end
end
end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
unc ion [al a, al a_pa , dEpsPl] = ...
calc_al a_and_dEpsPl(dp,gamma,cMa ix,al a_pa ,S, he a)
global yieldS eng h
m2 = size(gamma,1); % numbe o backs ess pa s
Ma e ials 2022,15, 1432 21 o 22
aMa ix = ze os(3,3);al a = ze os(3,3);aNum = 0;
o i = 1:m2
aMa ix =aMa ix + he a(i)*al a_pa (:,:,i);aNum = aNum + cMa ix(i)* he a(i);
end
% plas ic s ain enso
SminusA = yieldS eng h/(yieldS eng h+aNum*dp)*(S-aMa ix);
dEpsPl = 3/2*dp*SminusA/yieldS eng h;% calcula e backs ess pa s ( o he nex i e a-
ion)
o i = 1:m2
al a_pa (:,:,i) = (al a_pa (:,:,i)+2/3*cMa ix(i)*dEpsPl)* he a(i);
al a = al a + al a_pa (:,:,i);
end
end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
unc ion [al a, al a_pa , Snp1] = ...
calc_al a_and_Snp1(dp,gamma,cMa ix,al a_pa ,dEpsPl)
global yieldS eng h
m2 = size(gamma,1); % numbe o backs ess pa s
al a = ze os(3,3);
% calcula e backs ess pa s
o i = 1:m2
al a_pa (:,:,i) = (al a_pa (:,:,i)+2/3*cMa ix(i)*dEpsPl)/(1+gamma(i)*dp);
al a = al a + al a_pa (:,:,i);
end
Snp1 = dEpsPl*yieldS eng h*2/3/dp + al a;
end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION
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