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1 Politecnico di Toino Poto Institutional Repositoy [Aticle] A geneal model fo cack gowth fom initial defect in Vey-High- Cycle Fatigue Oiginal Citation: Paolino, Davide S.; Tidello, Andea; Chiandussi, Giogio; Rossetto, Massimo (2017). A geneal model fo cack gowth fom initial defect in Vey-High-Cycle Fatigue. In: PROCEDIA STRUCTURAL INTEGRITY, vol. 3, pp ISSN Availability: This vesion is available at : since: May 2017 Publishe: Elsevie Published vesion: DOI: /j.post Tems of use: This aticle is made available unde tems and conditions applicable to Open Access Policy Aticle ("Ceative Commons: Attibution-Noncommecial-No Deivative Woks 3.0"), as descibed at http: //poto.polito.it/tems_and_conditions.html Poto, the institutional epositoy of the Politecnico di Toino, is povided by the Univesity Libay and the IT-Sevices. The aim is to enable open access to all the wold. Please shae with us how this access benefits you. You stoy mattes. (Aticle begins on next page)

2 Available online at ScienceDiect Pocedia Stuctual Integity 3 (2017) XXIV Italian Goup of Factue Confeence, 1-3 Mach 2017, Ubino, Italy A geneal model fo cack gowth fom initial defect in Vey-High- Cycle Fatigue Davide S. Paolino a *, Andea Tidello a, Giogio Chiandussi a, Massimo Rossetto a a Politecnico di Toino, Depatment of Mechanical and Aeospace Engineeing, Coso Duca degli Abuzzi 24, Tuin 10129, Italy Abstact It is well-known in the liteatue that intenal defects play a majo ole in the Vey-High-Cycle Fatigue (VHCF) esponse of metallic mateials. Geneally, VHCF failues nucleate fom intenal defects chaacteized by a limited size. Unexpectedly, it has been found that cacks can gow fom the initial defect even if the Stess Intensity Facto (SIF) is quite below the chaacteistic theshold fo cack gowth. Even though eseaches unanimously accept this singula expeimental evidence, they still dispute about its physical justification. Diffeent micomechanical explanations have been poposed in the liteatue: local gain efinement, cabide decohesion, matix fagmentation, hydogen embittlement, numeous cyclic pessue and fomation of pesistent slip bands ae the most famous poposals. Regadless of the specific micomechanical explanation, it is geneally acknowledged that a weakening mechanism occus aound the initial defect, thus pemitting cack gowth below the SIF theshold. The pesent pape poposes an innovative appoach fo the quantitative modeling of the weakening pocess aound the initial defect. The poposed model consides an additional SIF that educes the SIF theshold of the mateial. Stating fom a vey geneal fomulation fo the additional SIF, possible scenaios fo cack gowth fom the initial defect ae also identified and descibed. It is theoetically demonstated that, depending on the scenaio, a VHCF limit may also be pesent and its final fomulation ecalls the well-known expession peviously poposed by Muakami. Copyight 2017 The Authos. Published by Elsevie B.V. This is an open access aticle unde the CC BY-NC-ND license ( Pee-eview unde esponsibility of the Scientific Committee of IGF Ex-Co. Keywods: VHCF limit; SIF theshold; Cack gowth * Coesponding autho. Tel.: ; fax: addess: davide.paolino@polito.it Copyight 2017 The Authos. Published by Elsevie B.V. This is an open access aticle unde the CC BY-NC-ND license ( Pee-eview unde esponsibility of the Scientific Committee of IGF Ex-Co /j.post

3 412 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) Intoduction The inceasing demand fo high pefomance machiney able to sustain significant loads fo a vey lage numbe of cycles (lage than 10 8 cycles) is pesently diving the eseach on the Vey-High-Cycle Fatigue (VHCF) esponse of many metallic mateials. In the last decades, the extensive expeimental investigation on VHCF has shown that failues mainly oiginate fom intenal defects (inclusions, poes and inhomogeneities) with a typical fish-eye mophology. Within the fisheye, depending on the defect size and on the applied stess, factue sufaces may show the so-called Fine Ganula Aea (FGA) in the vicinity of the intenal defect. The FGA (also called Optically Dak Aea o ODA by Muakami, Ganula Bight Facet o GBF by Shiozawa and Rough Suface Aea o RSA by Ochi) is a esticted egion, dak at the optical micoscope, that plays a key ole in the initiation of the VHCF failue, since its fomation consumes moe than the 98% of the VHCF life. Reseaches still dispute about the actual mechanism behind the FGA fomation (Li et al., 2016), but they unanimously accept that, within the FGA, cack can gow even if the Stess Intensity Facto (SIF) is below the SIF theshold fo cack gowth. In the pesent pape, the eduction of the SIF theshold within the FGA is oiginally modeled in ageement with the diffeent weakening mechanisms poposed in the liteatue. Stating fom a vey geneal fomulation fo the SIF eduction, possible scenaios fo cack gowth fom the initial defect ae also identified and descibed. It is theoetically demonstated that, depending on the scenaio, a VHCF limit may also be pesent and its final fomulation ecalls the well-known expession peviously poposed by Muakami (Muakami, 2002). An illustative numeical example, based on expeimental data, is finally epoted in the pape in ode to show the applicability of the poposed model and its potentialities. Nomenclatue FGA HV SIF VHCF a c, a d, a d,0, a FGA, max c th, g, th, g, c th,, th, c I, Fine Ganula Aea Vickes Hadness Stess Intensity Facto Vey-High-Cycle Fatigue a pojected aea of defects, FiE m I, c II, m II, c III, m III, c s, k d k th, g, th, l paametes involved in SIF thesholds Pais constants in the thee stages of cack gowth SIF of defect m s, k, k th, SIF thesholds s stess amplitude s l fatigue limit N f numbe of cycles to failue N I, I, min N, N I, max, N II, N III numbe of cycles in the thee stages of cack gowth

4 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) Methods A geneal expession fo modeling the SIF theshold within the FGA is pesented in Section 2.1. Stating fom the poposed model, an expession fo the fatigue limit is defined in Section 2.2 and a model fo the cack gowth ate is intoduced in Section SIF theshold within the FGA It is geneally acknowledged that cack can gow within the FGA even if the SIF is below the SIF theshold of the mateial. Regadless of the physical justification fo this unexpected expeimental evidence, cack gowth can occu in the FGA only if a local eduction of the global SIF theshold of the mateial is accepted. The following assumptions pemit to define a geneal model fo the local eduction of the SIF theshold within the FGA: 1. the global SIF theshold, efeed to as k th, g, can be expessed as (Shiozawa et al. 2001; Muakami, 2002; Tanaka and Akiniwa, 2002; Chapetti et al., 2003; Liu et al., 2006; Li et al., 2010; Hong et al., 2014; Matsunaga et al., 2015): th, g kth, g cth, g HV 120 a d, (1) whee a d is the pojected aea of the defect, HV is the Vickes hadness of the mateial in the vicinity of the defect and c th, g 0 and 0th, g 1/2 (being th, g 0 in case of global SIF theshold fo long cacks) ae two mateial coefficients. 2. the SIF fo an intenal defect, efeed to as k d, is given by (Muakami, 2002): 1/2 d k 0.5s a, (2) d whee s is the local stess amplitude at the defect location. 3. within the FGA, the local SIF theshold, efeed to as k th, l, is defined as: k k k, (3) th, l th, g th, whee k th, accounts fo the eduction of the SIF theshold induced by the diffeent weakening mechanisms poposed in the liteatue: local gain efinement (Sakai, 2009; Nakamua et al., 2010; Gad et al., 2012; Sakai et al, 2015; Hong et al., 2016), hydogen embittlement (Muakami, 2002; Liu et al., 2010), cabide decohesion (Shiozawa et al., 2001), matix fagmentation (Shanyavskiy, 2013) o fomation of pesistent slip bands (Huang et al., 2010). 4. the SIF theshold eduction has the most geneal fomulation fulfilling the following thee basic conditions: a. the pinciple of dimensional homogeneity, fo which k th, must be popotional to the stess amplitude and to the squae-oot of the defect size. b. the initial condition, fo which k th, must be popotional to the squae-oot of the initial defect size when cack stats gowing. c. the defect size dependency, fo which k th, may depend on the defect size.

5 414 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) Accoding to the conditions a)-c), the easiest and most geneal fomulation fo k th, is: 1/2 th, th, d,0 d / d,0 th, k c s a a a, (4) whee a d,0 is the pojected aea of the initial defect and c th, 0 and th, 0 (being, 0 constant value) ae two mateial coefficients that ensue 0 kth, l k th, g fo any a d value. 5. FGA foms until kth, l kd k th, g. th if th, k is a Fom Eqs. (1), (3) and (4), the geneal expession fo the local SIF theshold is finally given by: th, l th, g d th, d,0 d d,0 th, g 1/2 120 / k c HV a c s a a a. (5) 2.2. Fatigue limit expession Accoding to assumption 5) in Section 2.1, the following fou distinct cases may occu: k a k a : fatigue life is finite and the FGA does not fom. Cack gows until failue 1. d d,0 th, g d,0 without the assistance of any weakening mechanism. k a k a k a : fatigue life is finite, the FGA foms and it eaches its maximum 2. th, l d,0 d d,0 th, g d,0 extension, a FGA, max. Within the FGA, cack can gow thanks to the weakening mechanisms poposed in the liteatue (Shiozawa et al., 2001; Muakami, 2002; Sakai, 2009; Huang et al., 2010; Liu et al., 2010; Nakamua et al., 2010; Gad et al., 2012; Shanyavskiy, 2013; Sakai et al, 2015; Hong et al., 2016). When k d eaches k th, g, the FGA attains its maximum extension. Then cack gows until failue without the assistance of any weakening mechanism. k a k a k a : fatigue life is infinite, the FGA foms but it does not each the 3. th, l d,0 d d,0 th, g d,0 maximum extension. Within the FGA, cack can gow thanks to any of the weakening mechanisms poposed in the liteatue (Shiozawa et al., 2001; Muakami, 2002; Sakai, 2009; Huang et al., 2010; Liu et al., 2010; Nakamua et al., 2010; Gad et al., 2012; Shanyavskiy, 2013; Sakai et al, 2015; Hong et al., 2016). When k th, l eaches k d, the FGA stops foming at afga a FGA, max and cack aests. 4. d d,0 th, l d,0 k a k a : fatigue life is infinite and the FGA does not fom. Cack cannot gow fom the initial defect. Fig. 1 qualitatively depicts the fou cases in a double logaithmic plot of the SIF with espect to the defect size. The tansition between case 2) and case 3) disciminates between finite and infinite fatigue life and it occus when, fo a given initial defect size, the stess amplitude equals the mateial fatigue limit. It can be demonstated (Paolino et al., 2016) that the mateial fatigue limit, efeed to as s l, can be expessed as: th, s l c th, g cs l 1/2th, g d,0 HV 120, (6) a

6 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) /2 th, g 1/ 2 1/2,, 0.5 th th g th, g th, th, g th, cth, 0.5 1/ 2 th, whee c s. Eq. (6) ecalls the well-known expession poposed l by Muakami (Muakami, 2002) and it can be obtained by imposing the condition of tangency (Fig. 2) between the kd a d cuve (Eq. (1)) and the th, l d k a cuve (Eq. (5)). a) 76 b) c) d) Fig. 1. Vaiation of elevant SIFs with defect size in VHCF: a) Finite life without FGA fomation; b) Finite life with FGA fomation; c) Infinite life with FGA fomation; d) Infinite life without FGA fomation Cack gowth ate within the FGA Fig. 2. Vaiation of elevant SIFs with defect size in fatigue limit condition. In the VHCF liteatue (Tanaka and Akiniwa, 2002; Maines-Gacia et al., 2008; Su et al., in pess), the cack gowth ate within the FGA is usually modeled with the Pais law. Thee stages can be pesent in sigmoidal cack gowth ate diagams elated to VHCF failues fom intenal defects (see Fig. 3): the below-theshold egion within

7 416 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) the FGA (up to k th, g ), the steady cack popagation egion fom the bode of the FGA (with SIF equal to k, the bode of the fish-eye (with SIF equal to k FiE bode (with SIF lage than k FiE ). th g ) to ), the unsteady cack popagation egion beyond the fish-eye Fig. 3. The thee stages of cack popagation in a cack gowth ate diagam fo VHCF failues fom intenal defects. In ode to model the below-theshold egion, the modified Pais law poposed by Donahue et al. (1972) (see also Sun et al., 2014) is hee adopted (a stess atio equal to -1 is assumed in the following, yielding the effective stess equal to the stess amplitude): da dn c k k I d th, l m I, (7) whee c I and m I ae the two Pais constants elated to the fist popagation stage, fom d,0 Fom the bode of the FGA to the bode of the fish-eye (with size a FiE a to a FGA, max. ), the cack gowth ate is modeled with the conventional Pais law, in ageement with the liteatue (Tanaka and Akiniwa, 2002; Maines-Gacia et al., 2008; Su et al., in pess): da dn mii II d c k, (8) whee c II and m II ae the two Pais constants elated to the second popagation stage, fom a FGA, max to a FiE. Final factue may occu when the cack size eaches the bode of the fish-eye. In these cases, the thid stage of cack popagation is not visible on factue sufaces and it can be neglected. In some othe cases, cack can popagate beyond the fish-eye bode until it eaches the bode of the final factue, with size a c.. In these cases, a thid stage of cack popagation is visible on factue sufaces and it can be modeled again with the conventional Pais law (Su et al., in pess):

8 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) da dn c m III kd III, (9) whee c III and m III ae the two Pais constants elated to the thid popagation stage, fom a FiE to a c. By taking into account the thee stages of popagation, the numbe of cycles to failue, N f, can be expessed as: whee N f NI NII N III, (10) N I, N II. and N III ae the numbe of cycles consumed within stages I, II and III, espectively. Following the pocedue usually adopted in the VHCF liteatue (e.g., Su et al., in pess), subtacting, fom the expeimental N I can be estimated by N f, the numbe of cycles N II and N III obtained though integation of Eqs. (8) and (9). The Pais constants in Eq. (8) ae those typical fo suface cacks in the steady phase of cack gowth; wheeas, the Pais constants in Eq. (9) ae fo suface cacks in the unsteady phase of cack gowth, nea the final factue. If the Pais constants in Eq. (9) ae assumed equal to those of Eq. (8), the cack gowth ate is undeestimated and, consequently, N is oveestimated. Theefoe, it can be concluded that: III N N N N N N N, (11) I, min f II III I f II I, max whee N N 1 ms/2 1 ms/2 ac afga, max II III ms II a 1 ms / 2cs0.5s 1 ms/2 1 ms/2 FiE afga, max ms 1 ms / 2cs0.5s, (12) being c s and m s the two Pais constants fo suface cacks in the steady phase of cack gowth. The diffeence between N I, min and N I, max is geneally negligible if f aveage value between N I, min and N I, max is a good appoximation fo. N I.. The appoximated expeimental N is lage than 8 10 cycles. Thus, the N I values can be used fo the estimation of the fou paametes c I, m I, c th, and th, : accoding to nonlinea least squaes method, the paamete estimates ae obtained by minimizing the sum of squaed pecent eos between the expeimental log N values and the log 10 I 10 N values computed though integation of Eq. (7). The othe two paametes involved in Eq. (7) (i.e., c th, g and th, g ) ae estimated though application of the odinay least squaes method to the expeimental data elated to the a FGA, max values measued on the factue sufaces. In paticula, c th, g and, the a FGA, max values, in a log-log plot (Paolino et al., 2016). th g ae obtained with a linea fit of the d FGA, max I k a values vs.

9 418 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) Application to an expeimental dataset In ode to show the applicability of the poposed appoach, model paametes ae fitted to an expeimental dataset. VHCF tests ae caied out on Gaussian specimens (Paolino et al., 2014) made of an AISI H13 steel with Vickes hadness 560 kg f /mm 2. Details on the testing setup and on the tested mateial ae epoted in Tidello et al. (2015) and Tidello et al. (2016) and will not be ecalled hee fo the sake of bevity. Twelve specimens ae loaded at a constant stess amplitude up to failue. The numbe of cycles to failue anges fom to cycles. Factue sufaces ae seen though a Scanning-Electon-Micoscope (SEM) in ode to measue the initial defect size (i.e., a d 0 ) in each specimen; wheeas, the FGA sizes (i.e., a, FGA max ) ae measued fom pictues taken at the optical micoscope. Fom the SEM analysis, all the fatigue factues nucleated fom non-metallic inclusions (oxidetype inclusions). The local stess amplitude in the vicinity of the initial defect is consideed as the stess amplitude applied duing the test. As shown in the S-N plot of the expeimental dataset (Fig. 4), the local stess amplitudes ae in the ange MPa. Fig. 4. S-N plot of the expeimental dataset. As a fist step, the paametes and ae estimated fom the FGA sizes. As shown in Fig. 5, the linea model of Eq. (1) is in good ageement with the expeimental data. Fig. 5. Global SIF theshold vaiation as a function of the FGA size.

10 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) In paticula, accoding to Eq. (1), the least squaes estimates of c th, g and th, g ae given by: cth, g th, g , (13) whee denotes the paamete estimate. It is woth noting that the estimates c th, g and th, g ae in ageement with the values poposed in the liteatue fo c th, g (Li et al., 2010; Liu et al., 2008) and fo th, g (Muakami, 2002; Liu et al., 2008; Li et al., 2010; Matsunaga et al., 2015). In ode to estimate c I, m I, c th, and th,, the expeimental numbe of cycles consumed in stage I must be computed fom Eqs. (11) and (12). The two Pais constants in Eq. (12) (i.e., c s = and m s 3.21 ) ae taken fom the available liteatue (Schuchta and Plumtee, 1988) fo a vey simila steel type. Fig. 6 shows the vaiation of the atios of NI, min / N f and NI, max / N f with N f. Fig. 6. Vaiation of the atios and with the numbe of cycles to failue. As shown in Fig. 6, the diffeence between N I, min and N I, max is negligible. Theefoe, the aveage value between N I, min and N I, max can be consideed as a good appoximation of the actual N I. In ageement with the liteatue (Tanaka and Akiniwa, 2002; Hong et al., 2014; Su et al., in pess), the atio N / N inceases apidly with N f and, fo the expeimental dataset, is lage than 99.5%. Fom the expeimental N I values and fom the measued a d,0 and a FGA, max values, it is also possible to compute the aveage cack gowth ate within stage I: I f v ai, a FGA, max d,0 N I a, (14) whee v ai, denotes the aveage da / dn in stage I. Fig. 7 shows the vaiation of v ai, with N f.

11 420 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) Fig. 7. Vaiation of the aveage cack gowth ate within stage I with the numbe of cycles to failue. As shown in Fig. 7, the cack gowth ate deceases with the numbe of cycles to failue. It is woth noting that, fo N f lage than 10 9, the cack gowth ate is smalle than the physical theshold of m/cycle suggested by Pippan et al. (2002) and is fa below one Buges vecto (Tanaka and Akiniwa, 2002; Zhao et al., 2011; Sun et al., 2014). Theefoe, it can be agued that cack gowth within the FGA is not unifom: cack altenatively aests and gows with an aveage ate that can be smalle than m/cycle. Application of the nonlinea least squaes method yields the following estimates fo paametes c I, m I, c th, and th, : 15 ci m I 4.249, (15) cth, th, Fig. 8 shows the good ageement obtained between the expeimental and the estimated values of afte the fitting pocedue. N I and ai, v, a) b) Fig. 8. Compaison between estimated and expeimental data: a) Numbe of cycles consumed in stage I; b) Aveage cack gowth ate in stage I. Fom the estimates in Eqs. (13) and (15), it is also possible to estimate an aveage fatigue limit, accoding to the expession in Eq. (6). The fatigue limit depends on the initial defect size. Fig. 9 shows the vaiation of the fatigue limit with the initial defect size, fo the expeimental dataset.

12 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) Fig. 9. Vaiation of the fatigue limit with the initial defect size. As shown in Fig. 9 and epoted in the liteatue (Muakami, 2002; Fuuya, 2011), the fatigue limit deceases with the initial defect size. As expected fom the definition of fatigue limit, the values in Fig. 9 ae fa below the stess amplitudes that induce failue fo the investigated mateial. Depending on the stess amplitude and on the initial defect size, the diffeent scenaios in Figs. (1) and (2) may occu. Fig. (10) shows the Pais diagam fo an initial defect size equal to 40 m and fo diffeent elevant values of the stess amplitude. Fig. 10. Vaiation of the fatigue limit with the initial defect size. As shown in Fig. (10), fo stess amplitudes below the fatigue limit of 384 MPa the cack aests and the fatigue life is infinite. In paticula, fo stess amplitudes below 350 MPa, the FGA does not fom; wheeas, in the ange MPa, the FGA foms but it does not each its maximum size. Fo stess amplitudes above the fatigue limit, the cack does not aest and the fatigue life is finite. In paticula, fo stess amplitudes lage than 704 MPa, the FGA does not fom; wheeas, in the ange MPa the FGA foms and it eaches its maximum size. 4. Conclusions A simple and geneal fomulation fo the eduction of the SIF theshold in the FGA was poposed in the pape. It was shown that, with the poposed fomulation, the diffeent weakening mechanisms involved in the FGA fomation

13 422 Davide S. Paolino et al. / Pocedia Stuctual Integity 3 (2017) can be quantitatively modeled. Fom the poposed fomulation, a geneal expession fo the fatigue limit and a cack gowth ate model fo cack popagation fom intenal defect up to failue wee defined. The pocedue fo the estimation of the six paametes in the cack gowth ate model was also shown. The model was successfully applied to an expeimental dataset. The estimated and the expeimental aveage cack gowth ates within the FGA wee found to be fa below the physical theshold fo cack gowth. This expeimental evidence suggested that cack does not gow unifomly within the FGA and that it athe altenates aest and popagation phases. Refeences Chapetti, M. D., Tagawa, T., Miyata, T., Ulta-long cycle fatigue of high-stength cabon steels pat II: estimation of fatigue limit fo failue fom intenal inclusions. Mateials Science and Engineeing: A 356, Donahue, R. J., Clak, H. M., Atanmo, P., Kumble, R., McEvily, A. 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