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1 Avalable onlne at Proceda Engneerng (010) Proceda Engneerng 4 (010) 5 3 ISAB-010 Proceda Engneerng A smulaton on mcrostructure senstvty to very-hgh-cycle fatgue behavor of metallc materals Zhengqang Le, Jja Xe, Aguo Zhao, Yoush Hong* State Key Laboratory of Nonlnear Mechancs (LNM), Insttute of Mechancs, Chnese Academy of Scences, Bejng , Chna Receved July 010; revsed 7 July 010; accepted 30 July 010 Abstract Ths paper studed the mcrostructure senstvty to very-hgh-cycle fatgue of metallc materals based on Tanaka and Mura model by usng Monte-Carlo method. In ths smulaton, local stress, frctonal stress and ncluson sze are assumed to be random varables, and two senstvty ndcators are proposed: senstvty of mean value and senstvty of standard devaton. Results ndcate that fatgue falure s most senstve to frctonal stress, and the dspersty of varables also nfluences the fatgue lfe. 010 Publshed by Elsever Ltd. Open access under CC BY-NC-ND lcense. Keywords: very-hgh-cycle fatgue; Tanaka and Mura model; mcrostructure senstvty; Monte-Carlo smulaton; metallc materals 1. Introducton Under hydrodynamc actons, such as water currents, waves and tdes, supportng system of Submerged Floatng Tunnels (SFT) endurances a fluctuated tensle stress n water [1], and the desgn lfe s over ffty years, so fatgue falure under hydraton medum envronment can not be neglected for the desgn of anchor rope. Very-hgh-cycle fatgue (VHCF) s the study of fatgue falure behavors of materals and structures beyond and equal to hundreds mllons of cycles []. A number of expermental data has been accumulated and expermental phenomenon has been reported durng the research of VHCF of metallc materals n recent years. Especally for hgh strength steels, t s often shown a step-wse S-N curve [3, 4], the plateau present to the regon of 10 5 to 10 7 cycles. The plateau of the S-N curve,.e. the transton stress at whch the crack ntaton changes from surface to subsurface s the tradtonal fatgue lmt. When the nomnal stress s bellow the tradtonal fatgue lmt, fatgue falure occurs n the VHCF (N>10 7 ) regon. Ths s a new challenge to the desgn of long lfe components. Qan et al. [5] showed that the fatgue strength of the steel n water was remarkably degraded compared wth that n ar and the fatgue strength n 3.5% NaCl soluton was even lower than that n water. Therefore, t s necessary to further understand the fatgue behavor of very hgh strength steels, whch may be used n the SFT supportng system. * Correspondng author. Tel.: ; fax: E-mal address: hongys@mech.ac.cn c 010 Elsever Ltd. Open access under CC BY-NC-ND lcense. do: /j.proeng

2 6 Z. Le et al. / Proceda Engneerng 4 (010) 5 3 Z. Le et al. / Proceda Engneerng (010) Accordng to the present research stuaton, the fatgue falure s dvded nto two stages: crack ntaton stage and crack propagaton, and the crack ntaton control the fatgue lfe of VHCF [6]. Fatgue lfe s nfluenced by many factors at dfferent stages. Surface roughness [7], materal strength, loadng frequency, loadng envronment [8], ncluson sze [6] and dstrbuton play an mportant part n determnng the VHCF lfe. Tanaka and Mura [9] proposed a mcromechancal model for estmatng the fatgue lfe of metallc materals. In ther model, local shear stress, frctonal stress and characterstc sze of mcrostructure were mportant parameters n fatgue crack ntaton. The prncpal objectve of ths work s to show the scatter of fatgue lfe and mcrostructures senstvty of VHCF of metallc materals based on Tanaka and Mura model by Monte-Carlo analyss.. Fatgue crack ntaton model.1. Crack ntaton model Tanaka and Mura [9] model has been wdely used to fatgue crack ntaton lfe predcton [10 1]. It can be smplfed as AWs N ( k) (1) where N s the cycles to crack ntaton, W s s the specfc fracture energy, s the range of local shear stress, k s frctonal stress. A s a functon dependng on the materals propertes and the type of ntal cracks. Three types of ntal cracks have been proposed, they are crack ntates along slp bands, crack ntates along gran boundary, and cracks ntates along the nterface of ncluson. In ths paper, the thrd type s proposed, and A has the followng form: 4 GG ( G ) h A G ( h l) a () where G s the bulk shear modulus of matrx, G s the shear modulus of ncluson, l s the sem-length of slp band, h s the sem-mnor length of the ellptcal slp band area, a s the ncluson sze. A physcal nterpretaton of the Von Msses yeld crtera s that yeldng occurs when the resolved shear stress on the octahedral plane exceeds the octahedral shear strength, where 0 0 (3) 3 where 0 s the yeld strength from unaxal tensle test. The relatonshp between the shear stress ampltude and the appled stress ampltude can be defned as [6]: (4) 3 Usng Grffth crack theory we can calculate the W s as follows [13]: Kth Ws (5) E where s the fatgue crack growth threshold, ts value s determned by the three ponts notch test done by Qu Kth et al. [14]. In ths paper, the followng assumptons are made:

3 Z. Le et al. / Proceda Engneerng 4 (010) Z. Le et al. / Proceda Engneerng (010) (1) The lfe of fatgue crack propagaton s gnored, so the fatgue crack ntaton lfe s consdered to be the whole fatgue lfe. () The sem-length of slp band s assumed to be equal to the sem-mnor length of the ellptcal slp band area. (3) The shear modulus of ncluson s assumed to be equal to the bulk shear modulus... Prmtve random varables..1. Local stress In rotaton bendng fatgue test, local stress s changed along radus drecton, so ncluson at dfferent locaton has dfferent stress. Experment results show that the maxmum depth of ncluson where crack ntates s about 50 mcrons [15] away from surface of the materals, so the local stress around ncluson where fatgue cracks ntate s close to the maxmum stress wth a small fluctuaton. Here local stress s assumed to follow a normal dstrbuton wth a Coeffcent of Varaton (COV) of Incluson sze In order to research the nfluence of the ncluson sze on the fatgue lmt, Murakam et al. [16] used the parameter area to remark the ncluson sze and got satsfyng predcton result. Now t s wdely accepted n VHCF study. Incluson sze s manly nfluenced by productve technology, Murakam used Gumbel dstrbuton to manfest the maxmum sze of the ncluson, but few data can be found about the dstrbuton of ncluson sze. Here we assume the normal dstrbuton wth a COV of Frctonal stress The varaton n frctonal stress depends on the nteracton between moble dslocatons and sub-gran obstacles. Only a few papers can be found about the determnaton of the value of frctonal strength. Wang et al. [4] deduced a new expresson that the frctonal strength of hgh strength steel was about a quarter of fatgue lmt at stress rato R=-1. Tanaka [17] ndcated that a two-parameter Webull dstrbuton wth a COV between 0.3 and 0.7 can be used to descrbe the frctonal stress. In the present paper, the mean frctonal strength s derved from fatgue lmt and a two-parameter Webull dstrbuton wth a COV of 0.3 wll be used. 3. Computatonal method Here a structure relablty method [18] wll be used for fatgue falure possblty analyss. The relablty of the crack ntaton lfe R s defned as the probablty that the crack ntaton lfe N(X) wll exceed a specfed number of cycles N s. R s determned usng the performance functon defned as: g X NX Ns (6) n drect Monte-Carlo smulaton method. N(X) s determned by Tanaka and Mura model. Monte-Carlo method begns wth samplng of all random varables, and then the value of the performance functon s solved for further probablty analyss. To the present problem, a vector X wll be generated frstly, where X s a vector of three random varables, ka,. Then the value of g(x) can be obtaned. Repeat ths process N tmes, f N s large enough, the probablty of g(x) <0 wll be equal to the frequences of g(x) <0 accordng the law of large numbers, thus we obtan the fatgue falure probablty P f=1-r. Cumulatve Dstrbuton Functon (CDF) P f of a response g can be represented as f 0... X d (7) P P g X f x x where f X s the probablty densty functon of X, s the doman n X whch satsfy the performance functon constrant g(x)<0. Based on ntegral n Eq. (7), samplng based CDF senstvtes of P f wth respect to a dstrbuton parameter (mean or standard devaton of X) can be defned as

4 8 Z. Le et al. / Proceda Engneerng 4 (010) Z. Le et al. / Proceda Engneerng (010) f Pf f X dx (8) P presents the change n CDF due to the change n a statstcal meanng that characterzes the dstrbuton functon of the nput varable X. Eq. (8) can be presented by an expectaton functon: Pf Pf f X f P X f fx... dx E fx P f fx (9) n whch P f s the probablty n the doman and f X /P f s the samplng densty functon n.whch s now the samplng doman. Based on Eq. (9) the mean senstvty S and the standard-devaton senstvty S are defned as [19]: Pf / Pf S / Pf / Pf S / (10) (11) where and are respectvely the mean and the standard devaton of the random varable X. For ndependent varables, the jont probablty densty functon (PDF) of X s a product of the PDF of X, Eq. (4) can be smplfed to Pf / Pf f X f X f f X X E E E / fx fx fx f X (1) Eq. (1) s a functon of only the varable X and ts probablty dstrbuton parameters. It can be smplfed further by transform the non-normal varables X to standard normal varables U usng the followng transformaton: 1 X U F X (13) 1 s the nverse standard normal CDF and U s a normal varable wth zero mean and unt standard where devaton usng U varables. Thus we can get S S Pf / Pf EU / U Pf / Pf EU 1 / U (14) (15) where the expectatons are only over the reduced subspace defned by, U s the mean of U wth a nomnal value of zero and U s the standard devaton of U wth a nomnal value of one. The algorthm mentoned above s compled and performed by MATLAB program.

5 Z. Le et al. / Proceda Engneerng 4 (010) Z. Le et al. / Proceda Engneerng (010) Computatonal result 4.1. Fatgue falure probablty The predcted dstrbuton of the fatgue falure probablty s shown n Fg. 1 usng Eq. (1) by drect Monte Carlo smulaton. The bulk shear modulus G and the specfc fracture energy W s are assumed to be determnstc varables wth values 79GPa and 14kN/m respectvely where the value of W s s obtaned from Eq. (5). The local stress s, frctonal strength k and ncluson sze a are assumed to be random varables. The local stress s s normally dstrbuted a COV of 0.1 at three stress level (mean stress): 1600MPa, 1400MPa and 100MPa. The frctonal strength s a Webull dstrbuton wth a mean value of 500MPa and a COV of 0.3. The ncluson sze s normally dstrbuton wth a mean value of 15 mcrons and a COV s 0.3. Probablty of falure, P f predcted n ths paper 1600MPa 1400MPa MPa 0.1 experment data[0] 1400MPa Number of cycles to falure, N f Fg. 1. Fatgue lfe dstrbuton predcton at dfferent stress level Fg. 1 shows the change of probablty of falure wth the loadng cycles under dfferent local stress level. In ths fgure, wth the ncrease of the stress level, the probablty of falure ncreased at the same fatgue lfe, and falure probablty converges to a certan value at each stress level. The convergence value also ncreased wth the ncrease of the stress. For the falure probablty at 100MPa converges to 0.69 as N f ncreases, ths s because fatgue lfe can reach a nfnte value when the local stress s close to the frctonal stress n Tanaka and Mura model, and t s nevtable that a porton of sample values occur n ths doman n the statstcal analyses. The predcted result s compared to expermental data. The shape of the falure probablty curve s smlar to the statstcal result whch s obtaned by fttng the rotatng bendng fatgue test data through a three parameters Webull dstrbuton provded by Saka research group [0], although the falure probablty s some dfferent from experment data. 4.. S-N scatter property predcton Fg. s the predcted S N curve wth the probablstc scatter bars at four stress levels. Takng no account of small probablty event, the left pont at each stress level s the.5% fatgue falure probablty pont, the mddle pont s the mean fatgue lfe pont and the rght pont s the 97.5% fatgue falure probablty pont or the begnnng convergence pont. Fg. ndcates that both the mean fatgue lfe and the scatter of S-N curve ncrease wth the stress decrease. Fatgue lfe of metallc materals ranges from 10 4 to nearly 10 8 at a certan stress level, and the mean value of fatgue lfe reaches 10 9 at 1000Mpa, ths means that specmen at ths stress level s lkely to reach an nfnte fatgue lfe, ths s smlar to SUJ steel experment data [0].

6 30 Z. Le et al. / Proceda Engneerng 4 (010) Z. Le et al. / Proceda Engneerng (010) Stress ampltude,mpa Number of cycles to falure, N f Fg.. Probablstc stress versus lfe plot 4.3. Mcrostructure varable senstvty to fatgue falure Ths relablty and senstvty analyss reflects the quanttatve nfluence of mean value and standard devatons of varables on the fatgue falure probablty. Besdes the varaton trend of fatgue falure probablty can be obtaned by checkng the sgn of S and S [1]. (a) Senstvty of mean value, S ncluson sze stress frctonal stress Probablty of falure, P f (b) Senstvty of standard devaton, S ncluson sze stress frctonal stress Probablty of falure, P f Fg. 3. (a) Senstvty of mean value ( S ); (b) Senstvty of standard devaton ( S ) Senstvty analyss results n Fg. 3 are calculated wth local stress mean value of 1400MPa, and other condtons are as same as these n fatgue falure probablty analyss. Fg. 3 shows that mean value senstvtes of stress and frctonal stress are at same order of magntude throughout the regme, and fatgue falure beng most senstve to frctonal stress, then the local stress and ncluson sze. It s noteworthy that ncluson sze almost has no effect to the senstvty to fatgue falure n Fg. 3, n another words nfluence of ncluson sze s much less than nfluence of frctonal stress and local stress. From the sgn of S, t seems that the senstvty to fatgue falure ncreases wth ncreasng frctonal strength, and on the contrary, t decreases wth ncreasng local stress. From the sgn of S, the senstvty wll be enhanced by decreasng dspersty of local stress or frctonal strength, or by ncreasng dspersty of ncluson sze.

7 Z. Le et al. / Proceda Engneerng 4 (010) Z. Le et al. / Proceda Engneerng (010) Conclusons Drect Monte-Carlo smulaton n relablty analyss s used to fatgue falure probablty evaluaton n ths paper, and local stress, frctonal stress, ncluson sze are assumed to be random varables. For the problem of estmatng fatgue falure possbltes and fatgue lfe scatter, Tanaka and Mura model s well performed n Monte-Carlo smulaton. The man results are as follows: (1) Wth the ncrease of the stress level, the probablty of falure ncreased at the same fatgue lfe, and falure probablty converges to a certan value at each stress level. Both the mean fatgue lfe and the scatter of S-N curve ncreases wth the decrease of stress. () Fatgue falure s senstve to mcrostructures. Accordng to senstvty the rankng order of the three varables are frctonal stress, local stress, and ncluson sze. The senstvty ncreases wth ncreasng frctonal strength, and on the contrary, t decreases wth ncreasng local stress. Also the senstvty wll be enhanced by decreasng dspersty of local stress or frctonal strength, or by ncreasng dspersty of ncluson sze. (3) Anchor rope of SFT wll endurance a fluctuated stress up to 10 7 cycles, so metallc materals wth hgh frctonal stress or fewer nclusons wll sutable for the anchor rope consderng very hgh cycle fatgue property of the materals. Acknowledgements Ths paper s supported by Natonal Natural Scence Foundaton of Chna (Grant nos , ) and Knowledge Innovaton Program of Chnese Academy of Scences (Grant no. KJCX-YW-L07). References [1] Mazzolan FM, Landolfo R, Faggano B, Esposto M, Perott F, Barbella G. Structural analyses of the submerged floatng tunnel prototype n Qandao Lake (P.R. of Chna). Advances n Structural Engneerng 009; 11: [] Mayer H, Haydn W, Schuller R, Issler S, Furtner B, Bacher-Höchst M.Very hgh cycle fatgue propertes of bantc hgh carbon chromum steel. Internatonal Journal of Fatgue 009; 31:4 9. [3] Nato T, Ueda H, Kkuch M. Observaton of fatgue fracture surface of carburzed steel. Journal of Socety Materals Scence 1983; 3: [4] Saka T, Sato Y, Oguma N. Characterstc S N propertes of hgh-carbon chromum-bearng steel under axal loadng n long-lfe fatgue. Fatgue & Fracture of Engneerng Materals & Strucutures 00; 5: [5] Qan GA, Hong YS. Effect of envronmental meda on hgh cycle and very-hgh-cycle fatgue behavors of structural steel 40Cr. Acta Metallurgca Snca 009; 45: [6] Wang QY, Bathas C, Kawagosh N, Chen Q. Effect of ncluson on subsurface crack ntaton and ggacycle fatgue strength. Internatonal Journal of Fatgue 00; 4: [7] Itoga H, Tokaj K, Nakajma M, Ko HN. Effect of surface roughness on step-wse S-N characterstcs n hgh strength steel. Internatonal Journal of Fatgue 003; 5: [8] Hong Y, Zhao A, Qan G. Essental characterstcs and nfluental factors for very-hgh-cycle fatgue behavor of metallc materals. Acta Metallurgca Snca 009; 45: [9] Tanaka K, Mura T. A dslocaton model for fatgue crack ntaton. ASME J App Mech 1981; 48: [10] Murakam Y, Nomoto T, Ueda T, Murakam Y, Ohor M. Analyss of the mechansm of superlong fatgue falure by optcal mcroscope and SEM/AFM observatons. Journal of the Socety of Materals Scence 1999; 48: [11] Tryon RG, Cruse TA. A relablty-based model to predct scatter n fatgue crack nucleaton lfe. Fatgue & Fracture of Engneerng Materals & Strucutures 1998; 1: [1] Tryon RG, Anmesh D, Ganapath K, Chandran KS. Identfyng senstve parameters at fatgue crack nucleaton stes usng mcrostructual smulaton models. The Mnerals, Metals & Materals Socety 006, p [13] Anderson TL. Fracture mechancs, fundamental and applcaton. Boca Raton, FL: CRC Press; [14] Qu S. Study on the fatgue strength factor and the threshold of bearng steel. Materal for Mechancal Engneerng 1998; : 16 7.

8 3 Z. Le et al. / Proceda Engneerng 4 (010) Z. Le et al. / Proceda Engneerng (010) [15] Suh CM, Km JH. Fatgue characterstcs of bearng steel n very hgh cycle fatgue. Journal of Mechancal Scence and Technology 009; 3: [16] Murakam Y, Yokoyama NN, Nagata J. Mechansm of fatgue falure n ultralong lfe regme. Fatgue & Fracture of Engneerng Materals & Strucutures 00; 8: [17] Tanaka K, Knefuch M, Yokomaku T. Modelng of statstcal characterstcs of the propagaton ofsmallfatgue cracks. In: Mller KJ, Ros ER, edtors. Short Fatgue Cracks ESIS 13, London: Mechancal Engneerng; 199, p [18] Wu YT, Mllwater HR, Cruse TA. An advanced probablstc structural analyss method for mplct performance functons. AIAA Journal 1990; 8: [19] Wu YT, Mohanty S. Varable screenng and rankng usng samplng-based senstvty measures. Relablty Engneerng and System Safety 006; 91: [0] Saka T et al. Statstcal duplex S N characterstcs of hgh carbon chromum bearng steel n rotatng bendng n very hgh cycle regme. Internatonal Journal of Fatgue 010; 3: [1] Song SF, Lv ZZ. Structural relablty senstvty analyss method based on Markov Chan Monte Carlo subset smulaton. Journal of Mechancal Engneerng 009; 45: 33 8.