Towards a probabilistic concept of the Kitagawa-Takahashi diagram

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1 Tords probbilistic concept of the Kitg-Tkhshi digr Alfonso Fernández-Cnteli 1, Roberto Brighenti, Enrique Cstillo 3 1 Dept. of Construction nd Mnufcturing Engineering, University of Oviedo, Cpus de Viesques, 333 Gijón, Spin fc@uniovi.es Dept. of Civil-Environentl Engineering nd Architecture, University of Pr, Vile G.P. Usberti 181/A, 431 Pr, Itly brigh@unipr.it 3 Dept. of Applied Mthetics nd Coputtionl Sciences, University of Cntbri, Avd de los Cstros s/n, 395 Sntnder, Spin cstie@unicn.es ABSTRACT. The Kitg-Tkhshi (K-T digr, ipleented by the El Hddd eqution, reltes the conventionl ftigue liit to the crck size, enbling boundry for the cyclic stress rnge to be estblished, belo hich n infinite life of the structurl coponent y be, theoreticlly, ensured for ny crck size due to the non-propgtion of icro- nd crocrcks. In order to ccount for the inherent rndo chrcter of the ftigue phenoenon in rel terils nd the need of extending the K-T pplicbility to ny prefixed nuber of cycles, dvnced probbilistic S- odels should be considered to define the ftigue liit. In this y, ne bsis tords probbilistic Kitg-Tkhshi-El Hddd pproch is provided in greeent ith the syptotic tching proposed by Civrell-Monno. ITRODUCTIO AD MOTIVATIO The Kitg-Tkhshi (KT digr [1] represents boundry in ters of crck size nd stress rnge for hich infinite ftigue lifetie of structurl or echnicl coponents cn be sfely ensured due to non-propgting icro- nd crocrcks. Such ftigue life ssessent cn be relted to both the clssicl ftigue liit concept, resulting fro the experientl-bsed S- pproch, nd the threshold stress intensity fctor rnge, s defined by the crck propgtion l. Even fter the trnscendent iproveent provided by the intrinsic crck concept of El Hddd (EH [], to issues need to be delt ith: the extension of the KT-EH digr to ftigue liit for finite nuber of cycles, hich is not necessrily identified ith the endurnce liit for, nd b stochstic definition of the KT-EH digr s consequence of the vribility of the bsic ftigue functions being considered (S- nd crck groth rte curves. Both represent prcticl requireents relted to structurl integrity design. In this ork, ne pproch to the proble is supplied by considering the probbilistic S- developed by Cstillo nd Fernández-Cnteli [3], hich provides sound bsis in the definition of the KT-EH line, peritting lso the odel to be 141

2 extended fro infinite to ny finite life, s lredy proposed by Civrell nd Monno [4], hich provide n interesting lterntive to solve the first issue, heres the liittions of the siplistic S- field used, bsed on truncted Bsquin pproch ith bsence of probbilistic considertions, evidence the need of further enhnceent. Thus, the boundry beteen frcture nd non-frcture cn be optionlly referred to finite nuber of cycles for certin probbility of filure, becuse in the engineering prctice high, but finite, nuber of cycles, rther thn infinity, re usully ssued for ftigue life. PROBABIISTIC COCEPTS APPIED TO THE K-T APPROACH In this ork, only vribility of the S- field is considered, hile deterinistic concept of the crck groth rte curve is for the present ssued [5] s first ttept of introducing probbilistic considertions in the definition of the KT-EH digr. Figure 1. S- field ith percentile curves ccording to the regression odel[3]. The proposed probbilistic odel. According to the probbilistic regression odel proposed by Cstillo nd Fernández- Cnteli [3] the ftigue life given stress rnge, nd the stress rnge given lifetie re rndo vribles folloing Weibull distribution (Fig. 1: β (log B(log C λ p 1 exp δ, (1 here p is the probbility of filure, the lifetie in cycles, Δ σ the pplied stress rnge, B represents the threshold vlue of the lifetie nd C the endurnce liit, or ftigue liit for, nd β, δ nd λ re, respectively, the shpe, scle nd loction Weibull preters. As soon s the five preters re estited, the odel provides coplete nlyticl description of the probbilistic S- field being delt ith. The percentile curves re hyperbols shring the syptotes log B nd log C (see Fig. 1, the zero percentile curve represents the iniu possible required nuber of 14

3 cycles to chieve filure for different vlues of log. The percentile curves cn be interpreted s representing different initil fl sizes. Since the odel extends the Wöhler field up to n unliited nuber of cycles, n extrpoltion of the lifetie is possible outside the rnge of nuber of cycles tested in the experientl progr. The in liittion of the odel, consisting in the bsence of n upper bound in the CF region, y be overcoe by considering ne ftigue vrible, Δ σ ε x, s the reference preter controlling the ftigue process in the previous odel. Once the odel preters in ters of Δ σ ε x hve been estited, the S- field y be reconverted to the conventionl Δ σ vrible using the stress-strin cyclic digr of the teril, for instnce s Rberg-Osgood (R-O curve. K-T digr for finite nuber of cycles to filure Unless otherise specified, the S- curves re generlly relted to unique probbility of filure, p.5 or p.5, though the sctter of experientl dt requires the definition of the hole S- field s percentile curves bsed on sttisticl principles. The ftigue behviour of structurl coponent is inly governed by its surfce (or volue stte qulity, iplying roughness nd iperfections tht re preferentil sites for crck nucletion nd subsequent propgtion. According to [3], the percentile curves cn be ssued to be ssocited ith the probbility of the existence of crck being less thn certin initil crck size, i, initilly unknon. Consequently, the ftigue filure is governed by the xiu crck size present in the specien being tested so tht percentile curves ith incresing probbilities of filure re relted to diinishing crck sizes: the percentile curve p, corresponding to the gretest, or orst, of the xiu crck sizes of the popultion, i.e., i, x( x, hich is denoted xx crck size (Fig.. Siilrly, the upper percentile curve, p1, corresponds to the iniu, or best, of the xiu crck sizes of the popultion, i.e., i, b in( x, hich is denoted in-x crck size. For prcticl purposes, the definition of the ltter cn be relxed identifying, s n initil crck size relted to high probbility of i b filure, for instnce, p b.9 or.95. The to curves ssocited ith i, nd i,b represent the to liiting sizes of the initil xiu defect corresponding to the prticulr surfce finishing of the tested teril. According to [3], i,, relted to the percentile curve p, is deterined s here ΔKth 1 ΔK is the threshold SIF rnge, Y the geoetric fctor of the crck nd ( the endurnce liit. For given nuber of cycles to filure Δ σ b nd Δ σ ( Δ σ b > respectively. Consequently, for th i, π Y (, (, see Fig., to different stress rnges, re identified ith the best nd orst surfce defects s reference lifetie, n engineering SIF 143

4 threshold Δ K th, eng cn be found, prticulrly for the defect sizes i, nd i,b, but lso for ny crck size i, ( i, > i, > i,b, to be given by Δ Kth, eng, Y π i,, (3 here the stress rnge Δ σ results fro the percentile curve corresponding to (see Fig.. For, the Δ K th, eng becoes the true threshold vlue Δ Kth of the crck groth rte curve. ( ( ( Figure. Probbilistic concept pplied to n experientl S- curve for given teril. ote tht, ccording to El Hddd [], the intrinsic crck size is given s: 1 ΔK th, (4 π Y here Δ σ represents in this cse the ftigue liit for nuber of cycles, pointing out tht only for the intrinsic initil crck size coincides ith the orst crck size i,. Applying the El Hddd eqution to the evlution of testing sples ith vrying surfce sttes, i.e. intrinsic crck sizes, ould provide different ftigue liits Δ σ, i.e., unlike KT digrs, evidencing dependency ith respect to the prticulr surfce stte tested, thus, proving tht such KT digrs re not teril chrcteristic. On the contrry, unique KT digr is expected fro the proposed pproch despite the different endurnce liits, ssocited ith the orst crck sizes, obtined for the respective surfce sttes. The ssuption of deterinistic crck groth rte curve iplies deterinistic reltion beteen crck size nd nuber of cycles to filure ening tht the KT-EH digr is deterinistic too [5]. Accordingly, the sctter of the experientl dt re only relted to the initil surfce stte of the teril, iplying siply uncertinties relted to esureent precision or to teril heterogeneity. For certin initil crck size i,, the filure condition, K f K Ic, resulting for given nuber of cycles to filure,, is equivlent to 144

5 Δ K( Y π ( ΔK lloing us to express the finl crck size rtio, f f,, / s: f Ic f, > 1. (5 f, Assuing constnt geoetry fctor Y during the crck propgtion, the current crck size rtio r ( ( r (6 reins constnt long the hole propgtion process (see Fig. 3(b, in prticulr, fro the beginning of the ftigue process, i.e., for the initil crck sizes i, nd i,, up to the finl filure stte, proving tht on logrithic scle, the verticl distnce clog r beteen the curves relted to i, nd i, reins constnt ll long the propgtion process (see Fig. 3(b. ( (b Figure 3. Crck groth curves for the best nd orst initil defect cses for given teril ( plotted in nturl scle, nd (b plotted in logrithic scle. While the initil orst crck size i,, relted to p, is deterined fro Eq. (, nother crck size, i,, relted to given probbility p (prticulrly the best one i, b relted to p1, reining, in principle, unknon cn be obtined by integrtion of the crck groth rte l, d / d G( ΔK, due to its uniqueness, for generic nuber of cycles to filure, i.e.: f, d d f, f, ( i,,, G( ΔK(. (7 i, Since Eq. (6 pplies irrespective of the nuber of cycles considered, the bove quntities cn be used to rite syste of to equtions: 145

6 ( ( i,,, i,,, r nd f, f, f, f, ( ( i, i,,, r,, ith r ( /, the solution of hich provides the quntities i, nd i, tht cn be finlly ssocited ith the orst nd generic surfce stte of the teril, respectively. (8 APPICATIOS OF THE EXTEDED K-T DIAGRAM TO STEE 145 In order to deonstrte its utility for ftigue life ssessent of echnicl nd structurl coponents, the bove proposed pproch is pplied to the stndrd crbon steel 145 tking into ccount the echnicl nd ftigue properties s reported in Tble 1 [6]. Steel 145 Cheicl coposition Crbon Iron Mngnese Phosphorus Sulphur Blnce x.5 x Mechnicl properties E (GP σ y (MP σ u (MP K n E Ftigue properties Δ σ (MP ( K fc K th C (/cycle E-4 8 MP 7.1 MP 8.E Tble 1. Min cheicl, echnicl nd ftigue properties of the crbon steel 145 (fro [6]. In the present cse, rther thn using the probbilistic odel proposed by [3], the S- field is defined by deterining the percentile curves corresponding, respectively, to the probbilities of filure p nd p.9, fro the results of previous testing progr crried out for three different ftigue stress rnges ( 3, 4, 5 MP, s illustrted in Fig. 4. The quntities orst x( x nd best in( x re deterined ccording to 5 Eq. (6 by considering filure conditions relted to 1 cycles, heres, r ( b / (455/ (Fig. 4 or, equivlently, c ln r By dopting the crck groth rte l by Donhue et l. [7], expressed by: d G( ΔK C ( Y π ΔK th (9 d F ( the crck groth l ( e is ssessed nd used to get the expressions to be inserted into the syste given by Eqs (8. Since this syste of equtions is not esily solvble by nlyticl or nuericl stndrd techniques, it hs been tckled by using genetic lgorith (GA [8-9]. 146

7 Stress plitude, (MP E+1 1E+ 1E+3 p % p9% lo-cycle ft. energy pproch bsed on R-O odel 1E+4 1E+5 1E+6 Steel 145 1E+7 1E+8 1E+9 uber of cycles, Figure 4. S- curves corresponding to the p nd p.9 for the steel 145. b σy38 MP The ethod opertes by selecting, long the itertion process, the best solution ong different possibilities, trying to fulfil s better s possible the required vlue of the objective function, here identified s the knon rtio r ( / Δ b σ 1.79 for both the initil nd the finl crck sizes. After short nuber of itertions, the sought vlues for i, b nd i, re obtined. The GA process produces stble result for the to bove quntities, -4 nely, nd i b -4 i, , irrespectively of the stress rnge nd the finl nuber of cycles Stress plitude, (MP 1/ 1 1 dopted (see Eq.(8. σ 75 MP 1.E-4 1K 5K M 1M i,b i, ( 1.E-3 i ( Initil crck size, i ( Figure 5. Kitg-Tkhshi digr for the steel 145 for different vlues of cycles to filure ( ith probbility distribution relted to the finishing surfce stte (b. Finlly, the generlised Kitg-Tkhshi digr hs been obtined by integrting the Donhue eqution strting fro the initil crck size up to the fulfilent of the criticl condition ( Δ K I ( i, ΔK Ic for given nuber of cycles to filure; the initil crck size i hs been ssued to be siply uniforly distributed in the intervl i, b - i,. As cn be observed in Fig. 5, the initil crck size to be p( i 1 i,b (b i, 147

8 considered in the KT digr corresponds to the probbility of filure ssued, this being dependent on the finishing surfce stte. COCUSIOS The in conclusions to be drn fro the present study re the folloing: - A probbilistic concept is proposed for being ipleented in the Kitg- Tkhshi-El Hddd digr iing t providing higher relibility in prcticl design cses. - The pproch llos us to estblish connection beteen the probbilistic S- field nd the crck groth rte curve, the ltter being ssued for the present to be deterinistic. - The pproch cn be pplied indistinctly either for n infinite or finite liit nuber of cycles. - Further study is needed to llo n extension of the proposed pproch to define the KT digr for sll crcks. i.e., in the CF region, s ell s the considertion of stochstic crck groth rte curves, prticulrly in the threshold regie. ACKOWEGEDMETS The uthors cknoledge prtil support supplied by the Spnish Ministry of Science nd Innovtion MICI (Ref: BIA1-199 nd the Itlin Ministry for University nd Technologicl nd Scientific Reserch (MIUR. REFERECES [1] Kitg, H., Tkhshi, S. (1976 Proc. of the nd Int. Conf on Mech. Behviour of Mterils., ASM, [] El Hddd, M., Topper, T., Sith, K. (1979 Eng. Frct. Mech. 11, [3] Cstillo, E., Fernández-Cnteli, A. (1 Int. J. Frct. 17, [4] Civrell, M., Monno, F. (6 Int. J. Ft., 8, [5] Fernández-Cnteli A., Cstillo, E., Siegele, D. (1 18th Europen Conference on Frcture, ECF18, Dresden. [6] Mteril Reference. A Copendiu of Ftigue Thresholds nd Groth Rtes, EMAS, [7] Donhue, R.J., Clrk H.M., Atno P., et l. (197 Int J Frct Mech. 8, [8] Goldberg, D.E. (1989, Genetic lgoriths in serch, optiiztion, nd chine lerning. MA, Addison-Wesley Publishing Copny Inc. [9] Brighenti, R., Crpinteri, A., Vntdori, S. (6 Cop. Meth. App. Mech. nd Engng. 196,