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1 Hyde, Chrisopher J. and Hyde, T.H. and Sun, Wei and Beker, A.A. () Damage mehanis based prediions of reep rak growh in 36 sainless seel. Engineering Fraure Mehanis, 77 (). pp ISSN Aess from he Universiy of Noingham reposiory: hp://eprins.noingham.a.uk/465//hyde%-%eng.%fra.%meh. %Manusrip%-%ePrins.pdf Copyrigh and reuse: The Noingham eprins servie makes his work by researhers of he Universiy of Noingham available open aess under he following ondiions. This arile is made available under he Creaive Commons Aribuion Non-ommerial No Derivaives liene and may be reused aording o he ondiions of he liene. For more deails see: hp://reaiveommons.org/lienses/by-n-nd/.5/ A noe on versions: The version presened here may differ from he published version or from he version of reord. If you wish o ie his iem you are advised o onsul he publisher s version. Please see he reposiory url above for deails on aessing he published version and noe ha aess may ruire a subsripion. For more informaion, please ona eprins@noingham.a.uk

2 Damage mehanis based prediions of reep rak growh in 36 sainless seel C.J. Hyde, T.H. Hyde, W. Sun and A.A. Beker Deparmen of Mehanial, Maerials and Manufauring Engineering, Universiy of Noingham, Noingham, NG7 RD, UK Absra This paper desribes a novel modelling proess for reep rak growh prediion of a 36 sainless seel using oninuum damage mehanis, in onjunion wih finie elemen (FE) analysis. A damage maerial behaviour model, proposed by Liu and Murakami [], was used whih is believed o have advanages in modelling omponens wih raks. The mehods used o obain he maerial properies in he muliaxial form of he reep damage and reep srain uaions are desribed, based on uniaxial reep and reep rak growh es daa obained a 6 C. Mos of he maerial onsans were obained from uniaxial reep es daa. However, a novel proedure was developed o deermine he ri-axial sress sae parameer in he damage model by use of reep rak growh daa obained from esing of ompa ension (CT) speimens. The full se of maerial properies derived were hen used o model he reep rak growh for a se of humbnail rak speimen reep ess whih were also esed a 6 C. Exellen prediions have been ahieved when omparing he predied surfae profiles o hose obained from experimens. The resuls obained learly show he validiy and apabiliy of he oninuum damage modelling approah, whih has been esablished, in modelling he reep rak growh for omponens wih omplex iniial rak shapes. Keywords Creep rak growh, Damage, 36 sainless seel, Liu and Murakami model, Finie Elemen Mehod. Noaion a! rak growh rae A onsan in he Kahanov damage model (for e! ) B onsan in he Kahanov damage model (for w! ) C onsan in he Liu and Murakami damage model (for e! ) D onsan in he Liu and Murakami damage model (for w! ) C * reep onour inegral m onsan in he Kahanov damage model (for e! ) - -

3 M n n maerial onsan subsiuion (used wih he Kahanov damage model) onsan in he Kahanov damage model (for e! ) onsan in he Liu and Murakami damage model (for e! ) q onsan in he Liu and Murakami damage model (for w! ) p onsan in he Liu and Murakami damage model (for w! ) Q S ij f maerial onsan subsiuion (used wih he Liu and Murakami damage model) deviaori sress ime failure ime α d ij onsan of muliaxialiy Kroneker dela e uivalen reep srain rae s sress s uivalen sress s maximum prinipal sress s ij dire or shear sress definiion s h s r hydrosai sress rupure sress s 3 x hydrosai sress kk w W damage parameer inegral subsiuion onsan in he Kahanov damage model (for w! ) f onsan in he Kahanov damage model (for w! ). Inroduion Componens in power plan, hemial plan, manufauring proesses, aeroengines, e may operae a emperaures whih are high enough for reep o our []. Suh omponens may onain raks or mus be assumed o onain raks as par of design life or remaining life analyses whih are ruired [3]. In order o perform hese analyses a number of approahes have been used, based on, for example, a fraure mehanis approah using a! - C * relaionships [4], or a damage mehanis approah [5, 6, ]. This paper is relaed o he use of he damage mehanis approah. In pariular he mehods used o obain he maerial onsans in he muliaxial form of he reep - -

4 damage and reep srain uaions are desribed. Mos of he onsans are obained by fiing o uniaxial reep daa; his is a well esablished mehod [7]. However, in his paper, he deerminaion of he muliaxial sress sae parameer, α [8], is based on resuls from ompa ension (CT) ess; his approah is novel and resuls in properies whih are pariularly suied for prediing reep rak growh in omponens, where he rak growh is defined by a damage parameer, ω. When his damage parameer reahes a riial value (.99 hosen for he presened work) he maerial is regarded as ompleely damaged and hene a void or rak growh is assumed o be presen. A previously used ehnique for obaining he muliaxial sress sae parameer, based on he noh srenghening whih usually ours in Bridgman noh [9] reep rupure ess, relaive o orresponding uniaxial ess, does no losely represen he sress saes and onsrain whih our a rak ips. The validiy of he mehod proposed in his paper is esablished by omparing finie elemen prediions of reep rak growh in humbnail raked speimens wih experimenal daa [7] using he maerial onsans obained from uniaxial reep and CT reep es resuls. The maerial hosen for he invesigaion is 36 sainless seel beause of he ready availabiliy of uniaxial reep, uniaxial reep rupure, ompa ension reep rak growh and humbnail reep rak growh daa a a emperaure of 6ºC. The pariular form of damage uaion hosen for he invesigaion is ha proposed by Liu and Murakami []. By omparison wih he more ommonly used Kahanov damage uaions [5], i was found ha he Liu and Murakami uaions do no ause he ime seps in he finie elemen analyses o beome impraially small and unlike he Kahanov uaions, hey produe resuls whih are relaively insensiive o elemen size near he rak ip. These aspes are overed furher in he paper.. Experimenal Tesing Experimenal esing was arried ou using speimens of hree geomeries, namely, uniaxial, ompa ension (CT) and humbnail rak speimens. All ess presened were arried ou a 6 C using speimens made from 36 sainless seel ho rolled plae. The hemial omposiion for his maerial is given in Table. Table. Chemial omposiion of 36 sainless seel (% weigh). Cr Ni Mo Mn Si P S C Fe Balane.. Uniaxial reep esing Uniaxial es daa has been obained using ensile reep speimens, he geomery of whih is shown by Figure. The experimens were arried ou under onsan sress levels of 4MPa, 6MPa, 8MPa and 3MPa. The daa from hese ess is shown in Figure

5 Figure. Uniaxial reep speimen geomery Srain (%) 5 5 4MPa 4MPa 6MPa 6MPa 8MPa 3MPa Time (hours) Figure. Uniaxial reep daa for 36 sainless seel a 6 C. This daa (Figure ) has been used o obain he uniaxial maerial onsans in he reep damage model, as desribed in seion CT reep rak growh esing CT reep rak growh daa has been obained using speimens of he geomery shown in Figure

6 Figure 3. CT speimen geomery. Three CT reep rak growh ess were arried ou under onsan loads of 6.977kN (speimen ), 7.48kN (speimen 3) and 8.5kN (speimen ) as shown by Table. Figure 4 shows phoographs of he esed CT speimens where speimen has been annoaed in order o show he sages of rak growh for eah es. Region shows he sarer noh, region shows he iniial (faigue) rak, region 3 shows he reep rak, whih gives he valuable par of he es daa in he onex of his work, region 4 shows furher faigue raking in order o fraure he speimen for analysis and region 5 shows he end region where he speimen was orn open and fraured. The daa obained during region 3 is used o obain he muliaxial sress sae parameer, α, as desribed in seion 3.. and is also ompared o FE model prediions as shown in seion 4. Table. CT experimenal es ondiions. Tes no. Speimen no. Load (kn) Tes duraion (hours)

7 Figure 4. Phoographs of reep raked 36 sainless seel CT speimens..3. Thumbnail rak reep esing Thumbnail reep rak growh daa has been obained using speimens of he geomery shown in Figure 5. Figure 5. Thumbnail rak speimen (a) geomery, and (b) rak profile. Five humbnail reep rak growh ess were arried ou under onsan loads of 78.7kN (speimen 4), 9.7kN (speimen 6), 9.8kN (speimen ), 9.7kN (speimen 7) and.3kn (speimen 8) as shown by Table 3. Figure 6 shows phoographs of he esed humbnail rak speimens where speimen 5 has been annoaed in order o show he sages of rak growh for eah es. Region shows he sarer noh, region shows he iniial (faigue) rak, region 3 shows he reep rak, - 6 -

8 whih again gives he valuable par of he es daa in he onex of his work, region 4 shows furher faigue raking in order o fraure he speimen ino wo piees and region 5 shows he end region where he speimen was orn open and fraured. The daa obained during region 3 is ompared o FE model prediions as shown in seion 3.. and seion 4. Table 3. CT experimenal es ondiions. Tes no. Speimen no. Load (kn) Tes duraion (hours) Figure 6. Phoographs of reep raked 36 sainless seel humbnail rak speimens. 3. Liu & Murakami reep damage model 3.. Definiion of he maerial model The muliaxial form of he Liu and Murakami reep damage law is as follows: -7-

9 æ ö ç ( + ) æ ö = 3 S n ij n ç ç s 3 e! Cs exp.. ç + 3 w () s p è s ø è n ø where C and n are maerial onsans. e and s are he uivalen srain and uivalen sress, respeively, and s is he maximum priniple sress. S ij is he deviaori sress, i.e.: in whih d ij is he Kroneker dela and is defined as: and S ij d ij where s h is he hydrosai sress, defined as: h = s ij - d ijs kk () 3 ì = í î s = 3s kk s s = + s i = j i ¹ j i.e. s = s + s + s 33 (4) w is he damage variable, and is given as: kk 3 h + s 33 -q ( - e ) p q w! w = D s r e (5) q When his value reahes a riial value (.99 wihin he presen work), rak growh is assumed o have ourred ino he regions where his has happened. D, q and p are maerial onsans. s r is he rupure sress defined as: s r as + ( -a) s = (6) where a is a maerial onsan whih desribes he effe of muli-axial sress saes. (3) Under he uniaxial ondiion: and s an be subsiued for s. s = s = s (7) Therefore, subsiuing uaion (7) ino uaion (6) gives: and herefore, from uaion (7): s r = s s = s = s r = s (8) Hene, for he uniaxial ondiion, i is no possible o deermine he maerial onsan, α. Also, for he uniaxial ondiion (he -direion), uaion () an be simplified o: - 8 -

10 S = s - s = s (9) 3 3 as s and s are boh ual o zero and 33 s = s = s = s under he uniaxial ondiion. Therefore, subsiuing uaions (9) and (7) ino uaion () gives: æ ö ç ( ) n n + 3 e! = Cs expç. w () ç p + 3 è n ø Also, for he uniaxial ondiion, uaion (8) an be subsiued ino uaion (5) o give: -q ( - e ) p q w! w = D s e () q 3.. Deerminaion of he maerial onsans From uaions (), (5) and (6), i an be seen ha he onsans whih are ruired o be obained are C, n, D, q, p and α. Mehodologies for obaining hese onsans are desribed as follows: 3... Uniaxial maerial onsans (C, n, D, q and p) (a) C and n During he iniial sages of he reep of a maerial, w» and hene: æ ö ç ( n + ) 3 expç. w» exp ç p + 3 è n ø ( ) = Taking logs of boh sides of uaion () gives,. Therefore uaion () an be simplified o he following: n! () e = Cs ( e! ) = n log( s ) + log( C) log Therefore, using experimenal uniaxial reep daa o plo ( ) log e! vs. ( s ) log and fiing a sraigh line of bes fi hrough his daa allows he idenifiaion of n from he gradien and C from he y-axis inerep. An example of his plo is shown in Figure 7, for 36 sainless seel, a 6 C. The C and n values deermined for he 36 sainless seel, a 6 C, are inluded in Table

11 log(min reep srain rae [/h]) y =.63x log(σ [MPa]) Figure 7. Linear fi o reep srain rae vs. σ on a log-log sale. (b) D and p Equaion (5) an be wrien as follows: d = D d Separaing he variables for inegraion gives: -q ( - e ) p q w w q -q ( - e ) s e dw p = D s d qw (3) e q Sine he righ hand side of uaion (3) is made up of enirely onsans, uaion (3) an be rewrien as: where Q = D dw w e q = - ( - e ) q Qd q = (4) p s onsan (5) Inegraing uaion (4) beween he limis of and, for ω, and and f, for gives: ò e dw = Qò -qw f Equaion (6) an be furher simplified using he subsiuion: d (6) -q w = W (7) and herefore - q dw = dw (8) Therefore, uaion (6) beomes: ò e - q W -q f dw = Q ò Noe ha he damage inegraion limis have also hanged as a resul of he subsiuion shown by uaions (7) and (8). Equaion (9) an be solved o give: d (9) - -

12 q e Subsiuing uaion (5) ino uaion () gives he following: Taking logs of boh sides of uaion () gives: Therefore, ploing ( ) f ( s ) log( s ) log vs. ( s ) - - = Q f () q f -p = s () D ( ) ( ) æ ö = -p log + log ø log f s ç () è D log using daa obained from uniaxial (as for his ondiion, log r = ) experimens, allows he idenifiaion of boh p, from he gradien of he sraigh line of bes fi and D, from he y-inerep. Figure 8 shows an example of his plo for uniaxial, 36 sainless seel daa a 6 C. The D and p values obained for he 36 sainless seel, a 6 C, are inluded in Table 4. 4 log(f [h]) y = -.949x log(σ [MPa]) Figure 8. Linear fi o ( ) f log vs. ( s ) log. () q A his sage, all of he onsans ruired in he uniaxial version of he Liu and Murakami model are known, exep he q value. A urve fiing proess is used on he deermine he value of q whih is he opimum fi a all sress levels. In order o plo () shows, e vs. ime using he model, e vs. ime daa in order o e mus firs be found as a funion of. As uaion e! is a funion of ω as well as. ω is also a funion of, as shown by uaion (). Therefore, his expression for ω as a funion of mus firs be found, whih an hen be subsiued ino uaion () o give an expression for e as a funion of. Inegraing uaion (4), beween he limis of and ω, for ω, and and, for, leads o an expression for ω as a funion of, i.e., - -

13 w dw ò = Q q ò d w e Again, using he subsiuion shown by uaions (7) and (8) gives: ò e - q W -qw Equaion (4) an be solved and re-arranged for ω o give: (3) dw = Qò d (4) ( Qq ) ln - Subsiuing uaion (5) ino uaion (5) gives: w = - (5) q -q p ( - D( - e ) ) ln s w = - (6) q This is he expression for ω (as a funion of ) whih is needed in order o obain an expression for e as a funion of. Equaion (6) is subsiued ino uaion () o give: æ 3 ö ç -q ( ) æ p ( ( ) ) ö n n + ln - D - e s e! = Cs expç. ç - (7) ç p + 3 è q ø è n ø Equaion (7) anno be readily solved o produe a losed form soluion for ime. However, a ime marhing proedure an be used o obain he variaion of ime marhing proedure is arried ou by alulaing e as a funion of e wih ime. This e! for many small ime seps, up o he failure ime, and muliplying eah of hese values by he small ime inerval in order o give he reep srain inremen for ha ime inerval, as shown by he following uaion: where i denoes he urren ime sep. D e =! e D i These reep srain inremens are hen aumulaed in order o give he oal value of reep srain a he end of eah ime sep, i.e. e + De i = e i - Sress, σ, is assumed o be onsan for every ime inremen. Curves of i i e vs. an hen be ploed for eah sress value. q an hen be varied in order o opimise he general fi (for all σ values) of he model o he experimenal daa. An example of his plo using uniaxial reep daa for 36 sainless seel, a 6 C, is shown by Figure 9. The q value obained for he 36 sainless seel, a 6 C, is inluded in Table

14 Creep srain (abs) Time (hours) Exp - 4MPa - Exp - 4MPa - Model - 4MPa Exp - 6MPa - Exp - 6MPa - Model - 6MPa Exp - 8MPa Model - 8MPa Exp - 3MPa Model - 3MPa Figure 9. Comparison of he Liu and Murakami reep damage model o uniaxial, experimenal reep daa Muliaxialiy parameer, α Equaion (6) is used for he rupure sress, σ r, wihin he model o inlude he muliaxial sress effe. Wihin his uaion is he maerial onsan, α, whih is no ruired for he uniaxial ondiion. However, if a muliaxial sress ondiion exiss, he α value is ruired: Equaion (5) an be re-wrien as: d d -q ( - e ) p q w w = D s r e q Following he proess desribed by uaions (3) o (), wih s replaed by s r gives: Therefore, subsiuing uaion (6) ino uaion (8) gives: f f -p r = s (8) D = D (9) ( as + ( - a ) s ) p Two basi approahes have been used in order o deermine he α-value for a maerial. The mos sraighforward mehod involves performing ess on speimens wih speifi biaxial sress saes [] and o obain he α value whih fis uaion (9) o he experimenal daa. However, ess of his ype are ompliaed and ruire areful speimen design and ompliaed es failiies []. For his reason an alernaive approah based on he daa obained from nohed bar reep es speimens has been more widely used o obain he α-values. Nohed bars are esed under seady load ondiions and he failure imes obained. A series of finie elemen (FE) modelling of he experimenal ess are hen arried ou using he maerial properies - 3 -

15 (C, n, D, p, and q ) obained from he orresponding uniaxial es daa, ogeher wih a differen α- value for eah alulaion. The α-value whih resuls in he same failure ime as ha of he experimenal nohed bar es is aken o be he α-value in he expression for rupure sress (uaion (6)). The average α-value for a range of load levels applied in he experimens gives a reasonable esimae for he aual α-value. The proess is apable of giving α-values whih an be used wih onfidene when he riaxial sress sae in he noh region, where final fraure ours, is similar o ha in he omponens for whih damage zones and failure imes are o be deermined. However, rak ips have pariularly severe muliaxial sress saes and magniudes and hene he damage regions end o grow in a rak-like manner. Therefore, for hese siuaions, i would be advanageous if he α-value was obained from ess on raked omponens. This novel approah has been adoped in his paper. A series of FE alulaions, o predi he reep rak growh in he experimenal CT speimens, as shown in Figure 4, were arried ou for he experimenal es duraions, using he same load levels. Differen α-values were used for eah alulaion performed for eah es. A ypial hreedimensional FE mesh and.99 damage (rak) zone for his CT speimen geomery is shown in Figure, where due o wo axes of symmery in a CT speimen, only one quarer of he speimen has been modelled, wih he appropriae boundary ondiions applied. The α-value whih gave he bes overall fi o all of he experimenal reep rak growh es daa was found o be.48. I is worh noing ha he aurae deerminaion of he α-value is ruial in he auray of he damage prediions, herefore, hroughou his proess, are mus be aken in order o ensure ha he opimum α-value is deermined. The omparisons of he experimenal and FE reep rak growhs for he hree CT speimens are shown in Figure, from whih i an be seen ha he rak fron shapes, as well as he exens of reep rak growh were auraely predied wih α =.48. This α-value has been used o predi he rak growh obained for he humbnail rak speimens, hese prediions are ompared wih he experimenal daa in seion 4. Furher deails of he FE modelling of he CT speimens and alulaion of he opimum α-value are given in Appendix

16 Figure. 3D CT speimen FE mesh. Figure. Tesed speimen phoo o FE damage onour omparisons (a) Speimen (b) Speimen () Speimen 3. Table 4. Liu and Murakami reep damage model maerial onsans for 36 sainless seel a 6 C. C n D p q α.47e e

17 3.3. Advanage of he Lui & Murakami model The Lui & Murakami model has been favoured over he more widely used Kahanov model (he Kahanov model is briefly desribed in Appendix ) for he following reason. Figure shows uniaxial plos of reep srain versus ime for 36 sainless seel a 6 C under a onsan sress of 4MPa for boh models. I an be seen ha he urves from boh of he models orrespond very well..5. Creep srain (abs).5..5 Kah L+M Time (hours) Figure. Comparison of uniaxial reep urves from he Lui & Murakami and Kahanov models. Creep urves - 36SS - 4MPa However, omparing plos of damage versus ime, i an be seen ha he damage rae obained from he Kahanov model rapidly approahes infiniy a imes lose o he failure ime, as shown by he gradien of he dashed urve in Figure 3. This auses problems when running FE analyses, beause he ime sep is oninually redued in order o obain onverged soluions and he alulaion run ime beomes impraially large. In he Lui and Murakami model, however, alhough he damage rae neessarily beomes large, i mainains a manageably low gradien (a high imes) up o a damage value of, allowing analyses o be performed wih more praial ime seps and herefore relaively low alulaion imes

18 .8 Damage.6.4 Kah L+M Time (hours) Figure 3. Comparison of uniaxial damage urves from he Lui & Murakami and Kahanov models. Damage urves - 36SS - 4MPa 4. Prediive apabiliy of he model As he muliaxial onsan, α, was deermined using he CT rak growh daa, i is o some exen no surprising ha he FE rak growh prediions orrespond well o his experimenal daa, wih all of he oher maerial onsans having been deermined using daa from uniaxial reep daa. However, similar simulaions have been performed for humbnail rak geomeries based on he same onsans and an herefore be onsidered as pure prediion. Figure 4 shows an example of he 3- dimensional mesh (and.99 damage (rak) zone) used for he humbnail rak growh simulaions. As wih he CT speimens, due o wo axes of symmery in a humbnail rak speimen, only one quarer of he speimen has been modelled, wih he appropriae boundary ondiions applied

19 Figure 4. 3D humbnail rak speimen FE mesh. The omparisons of he experimenal and FE reep rak growhs for he five humbnail speimens are shown in Figure 5, from whih i an be seen ha similarly o he CT prediions, he rak fron shapes, as well as he exens of reep rak growh were auraely predied wih α =.48. Figure 5. Tesed speimen phoo o FE damage onour omparisons (a) Speimen 4 (b) Speimen 5 () Speimen 6 (d) Speimen 7 (e) Speimen

20 4... Mesh sensiiviy The sensiiviy of he humbnail rak growh simulaions o he mesh used has been invesigaed by onsideraion of he es ondiions used for speimen 6 (par () in Figure 5). Figure 6 shows he four meshes, (a), (b), () and (d) used in his invesigaion, where eah mesh is finer han he las, respeively. The number of elemens wihin eah mesh is shown in Table 5. The elemen ype used in all analyses presened wihin his paper are 8-noded linear briks. Figure 6. Tesed speimen phoo o FE damage onour omparisons, showing he mesh sensiiviy of he model prediions. (a) Mesh (b) Mesh () Mesh 3 (d) Mesh 4. Table 5. No of nodes/elemens in eah mesh shown in Figure 6. Mesh no. No. of nodes No. of elemens Figure 7 shows how he rak lengh a he axis beween he experimenal phoograph and he FE onour in Figure 6 varies wih elemen size

21 .6 Crak growh (mm) Elemen size (mm) Figure 7. Crak growh vs. elemen size for humbnail rak growh prediions (showing mesh sensiiviy). I an be seen from Figure 6 and Figure 7 ha as he mesh beomes finer, he predied soluion onverges owards he orre soluion. The differene beween he prediions from he wo fines meshes is 3.7%. Due o his small differene, he soluion an be onsidered o have onverged. Also, due o he small % differene, in order o balane he auray of soluion and ime of alulaion, a mesh using an elemen size of.mm (mesh ()) was hosen. 5. Disussion and fuure work A omprehensive proedure for he deerminaion of he maerial onsans for he Lui & Murakami reep damage model, based on experimenal daa has been desribed and implemened for 36 sainless seel a 6 C. These onsans have been applied o a user subrouine for he Lui & Murakami model whih has been used in onjunion wih Finie Elemen pakage ABAQUS, in order o provide heoreial prediions for reep rak growh in boh ompa ension speimen and humbnail speimen geomeries. Comparisons of he model prediions o orresponding experimenal daa show exremely enouraging resuls wih he rak frons mahing very losely for boh ompa ension and humbnail rak geomeries. Furher work inludes similar experimenaion and maerial modelling bu for maerials speifially used in high emperaure regions of aeroengines. Aknowledgemens The auhors would like o hank he EPSRC for he funding of his proje obained from he Dooral Training Programme. Thanks are also given o Dennis Cooper for his skilful ehnial suppor wihin he experimenaion. - -

22 Referenes. Lui, Y. and Murakami, S., "Damage Loalizaion of Convenional Creep Damage Models and Proposiion of a New Model for Creep Damage Analysis", JSME Inernaional Journal 4 (998), Penny, R. K. and Marrio, D. L., "Design for Creep, MGraw-Hill, Liverpool, Hyde, T. H., Sun, W. and Beker, A. A., "Creep rak growh in welds: A damage mehanis approah o prediing iniiaion and growh of irumferenial raks in CrMoV weldmens", In. J. Pres. Ves. & Piping 78 (), Dogan, B. and Provski, B., "Creep rak growh of high emperaure weldmen." Inernaional Journal of Pressure Vessel and Piping 78 (), Kahanov, L. M., "The ime o failure under reep ondiion", Izv. Akad. Nauk., SSSR. Tekh. Nauk 8 (958), Robonov, Y. N., "Creep Problems of Sruural Members, Norh-Holland, Hyde, T. H., "Creep rak growh in 36 sainless seel a 6 C", High Temperaure Tehnology 6 (988), no., R., H. D., R., D. P. and Morrison C. J., "Developmen of oninuum damage in he reep rupure of nohed bars", Phil. Trans. R. So. Lond. (A) 3 (984), Webser, G. A., Holdsworh, S. R., Loveday, M. S., Perrin, I. J., Nikbin, K., Purper, H., Skelon, R. P. and Spindler, M. W., "A ode of praie for onduing nohed bar reep rupure ess and for inerpreing he daa", J. Faigue and Faigue of Eng. Maerials and Sru. 4 (4), Hyde, T. H. and Sun, W., "Deermining high emperaure properies of weld maerials", JSME Inernaional Journal of Solid Mehanis & Maerial Engineering, Series A 43 (), no. 4, Hayhurs, D. R., "Creep rupure under muli-axial saes of sress", J. Meh. Phys. Solids. (97),

23 Appendix Kahanov reep damage model The muliaxial form of he model firs proposed by Kahanov and Robonov, whih has sine beome known as he Kahanov reep damage law is as follows: n æ ö 3 æ s ö S m ij e! = ç ç A è - ø (A.) w è s ø Where A, n and m are maerial onsans. Considering he uniaxial ondiion in he -direion, uaion (7) and uaion (9) an be subsiued ino uaion (A.) o give he uniaxial form of he model as: n æ s ö m e! = A ç (A.) ø è - w Deerminaion of he maerial onsans If m is onsidered o be zero, primary reep is negleed and uaion (A.) an be simplified o he following: where: n æ s e! = A (A.3) ö ç è - w ø s r! w = B (A.4) ( - w) f Where B, χ and φ are maerial onsans and s r is he rupure sress given as shown by uaion (6). As for he Lui and Murakami reep damage model, for he uniaxial ondiion, i is no neessary o deermine he maerial onsan, α. A and n During he iniial sages of reep of a maerial, w» and hene - w», herefore uaion (A.3) an be simplified o he following: n! (A.5) e = As This uaion an be seen o be analogous o uaion (), herefore he proess used o deermine C and n for he Liu and Murakami model an be used o deermine A and n for he Kahanov model, i.e., A = C n = n B, χ and φ For a uniaxial sress sae, uaion (A.4) an be wrien as: - -

24 dw = B d s ( - w) f (A.6) Hene, separaing he variables and inegraing beween he limis of and for ω and and failure ime, f, for gives: As f ( w ) dw = ò f ò - Bs d B s is a onsan, his uaion an be simplified o he following: f ( w ) dw = Bs ò f ò - d This uaion an be solved and re-arranged for f o give: f s - where M B( + ) = onsan = (A.7) M = f (A.8) Equaion (A.7) an be seen o be analogous o uaion (), herefore he proess used o deermine D and p for he Liu and Murakami model an be used o deermine M and χ for he Kahanov model, i.e., M = D p = A his sage, all of he onsans for he Kahanov damage model are known exep for he B and φ values, whih are inerrelaed. The aual values of B and φ are found by ploing reep srain urves ( e vs. ) using he model and varying eiher B or φ (if B hanges, so does φ, and vie versa, as indiaed by uaion (A.8)) in order o ahieve he bes fi o he experimenal reep daa. This proess is very similar o he proess desribed in seion 3.. in order o deermine q for he Liu and Murakami model. In order o plo shows, e vs. using he model, e mus firs be found as a funion of. As uaion (A.3) e! is also a funion of ω, whih is, in urn, a funion of, as shown by uaion (A.4). Therefore, his funion for ω in erms of, mus firs be found. By separaing he variables in uaion (A.6) and inegraing, beween he limis of and ω, for ω, and and, for, in order o ge a general expression for ω as a funion of gives: This uaion an be solved o give: f ( w ) dw = Bs d w ò ò - ( - B( f + ) s ) f+ - w = Subsiuing uaion (A.8) ino his uaion gives: - 3 -

25 Rearranging uaion (A.7) for M gives: Subsiuing his ino uaion (A.9) gives: and herefore ( - Ms ) f+ - = w (A.9) M = s r f f + æ ö - = ç - w (A.) ç è f ø f + æ ö = - ç - This is he expression for ω as a funion of whih is needed. Therefore, in order o ge (A.3) o give: w (A.) ç è f ø e as a funion of ime, uaion (A.) is subsiued ino uaion e! de = d æ ç ç A ç s = ç ç æ ö ç ç - ç è è f ø f + ö n ø By separaing he variables and inegraing beween he limis of and limis of and for in order o obain a general expression for This uaion an be solved o give: e ò de n = As ò æ ç - ç è f ö ø n f + e for e as a funion of gives: d e and beween he n n æ - ö ç æ ö f + As f ç - ç e = - n ç (A.) - ç è f ø f + è ø Alhough B and χ do no appear in uaion (A.) hey do appear in he expression for f (uaion (A.7)) whih mus be deermined before e is alulaed. Therefore, uaion (A.) an be used o plo e vs. for eah σ value and he value of φ is varied (ausing B o also vary aording o uaion (A.8)) unil he general fi of eah of he urves (for eah sress value) o he experimenal daa is opimum. An example of his plo using uniaxial reep daa for 36 sainless seel, a 6 C, is shown in Figure A

26 .35 Creep srain (abs) Exp - 4MPa - Exp - 4MPa - Kah - 4MPa Exp - 6MPa - Exp - 6MPa - Kah - 6MPa Exp - 8MPa Kah - 8MPa Exp - 3MPa Kah - 3MPa 3 4 Time (hours) Figure A.. Comparison of he Kahanov reep damage model o uniaxial, experimenal reep daa for 36 sainless seel. Appendix Prediion of α from CT FE alulaions One uaion (9) has been esablished, i.e. f = D ( as + ( - a ) s ) p he following proess an be used in order o obain an opimum value for α. Taking logs of boh sides of uaion (9) gives: log ( ) æ ö log - p log( as + ( - a ) s ) = (A.) è D ø f ç Equaion (A.) shows a relaionship beween he onsan a and log ( ) f. Therefore, running muliple FE analyses, for a given es omponen geomery (in his ase a CT speimen) using various α-values wihin he range of o allows a plo of α versus log ( f ) o be produed. As he es daa used here is rak growh daa (no failure daa), he f value used is srily he ime aken o reah a erain rak lengh raher han he failure ime of he speimen. Tradiionally a sraigh line fiing is applied o his daa, he uaion of whih an be used o obain he maerial a value by subsiuion of he experimenal f value. However, i has been found wihin his work ha for a daa se obained for a CT speimen, a logarihmi urve provides a more aurae fi o he daa. An example of his plo for a 36 sainless seel CT speimen geomery (for speimen 3 in Figure 4, i.e

27 using a load of 7.48kN), a 6 C, is shown by Figure A.. The appliaion of he experimenal f - value and reading of he maerial α value is indiaed by he dashed line. 4 log(f [h]) 3 y = -.963Ln(x) α Figure A.. Typial α deerminaion graph for 36 sainless seel from CT es daa, using a logarihmi fiing. In order o improve he auray of he deermined α-value, his proedure an be arried ou for muliple onsan load ondiions and he average α-value alulaed. Based on he hree experimens shown by Figure 4, he value of α for 36 sainless seel is alulaed as.48. This value for α has been used in all he prediions presened in seion 3.. and seion