An approximate method for calculating elastic-plastic stress and strain on notched specimens

  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 Publishe ’s No e: MDPI s ays neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a il- ia ions. Copy igh : © 2022 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). 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 Re e ences 1. 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