Available online at ScienceDirect. Procedia Engineering 74 (2014 )

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1 Avilble online t ScienceDirect Procei Engineering 74 (2014 ) XVII Interntionl Colloquium on Mechnicl Ftigue of Metls (ICMFM17) On the sher stress prmeter of thin-wlle tubulr specimens uner torsionl loing Tin-Xing Xi Wei-Xing Yo b* Jin-Guo Lu Key Lbortory of Funmentl Science for Ntionl Defense-Avnce Design Technology of Flight Vehicle, Nnjing University of Aeronutics n Astronutics, Nnjing , Chin b Stte Key Lbortory of Mechnics n Control of Mechnicl StructuresNnjing University of Aeronutics n Astronutics, Nnjing , Chin Abstrct In the most of existing ftigue life preiction moels, the imum sher stress on tubulr specimen s outer surfce is chosen s the sher stress prmeter to preict ftigue lifetime for torsionl loing. It hs been experimentlly foun tht the ftigue lifetime observe uner fully reverse torsion is lwys higher thn tht uner tension-compression for the sme uivlent stress level. In this pper, n improve sher stress ftigue mge prmeter bse on Buch s two-prmeter moel ws propose. In this prmeter, the influences of both sher stress grient n cyclic strin hrening were tken into ccount. A seril of tensile-compressive n torsionl loing tests were crrie out on 2024-T4 luminum lloy to vlite the new prmeter. Four -N f curves were plotte using ifferent sher stress prmeters. Compre with other sher stress prmeters, the -N f curve trnslte from the new prmeter me best coincie with the tension-compression S-N curve The Publishe Authors. by Publishe Elsevier Lt. by Elsevier This is Lt. n open ccess rticle uner the CC BY-NC-ND license Selection ( n peer-review uner responsibility of the Politecnico i Milno, Diprtimento i Meccnic. Selection n peer-review uner responsibility of the Politecnico i Milno, Diprtimento i Meccnic Keywors: Multixil ftigue; Sher stress grient; Supporting effect; Cyclic hrening Nomenclture F Axil force mplitue M Torsionl moment mplitue D Outer imeter Inner imeter * Corresponing uthor. Tel.: ; fx: E-mil ress: wxyo@nu.eu.cn Publishe by Elsevier Lt. This is n open ccess rticle uner the CC BY-NC-ND license ( Selection n peer-review uner responsibility of the Politecnico i Milno, Diprtimento i Meccnic oi: /j.proeng

2 192 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) R A h von Mises uivlent stress Outsie rius Sher stress on the outer surfce Mteril constnt Mteril constnt Resiul strin 1. Introuction Engineering components n structures often unergo multixil loing. These proucts esigne by irect ppliction of unixil ftigue theory cnnot meet ftigue strength esign ruirements generlly. For this reson, the problem of multixil ftigue ssessment hs long been investigte n continues to be investigte by mny reserchers. One propose of the multixil ftigue stuy is to fin moel which cn be use to estimte the multixil ftigue life by using only unixil ftigue t. Most existing moels nee unixil ftigue t obtine from both fully reverse tension-compression n torsion tests [1-4]. In the fully reverse torsion tests, thin-wlle tubulr specimens re commonly use. The specimen s wll thickness shoul be lrge enough to voi instbilities uring cyclic loing without violting the thin-wlle tube criterion, i.e. the rtio of verge imeter to wll thickness shoul be 10:1 or lrger [5]. At present, the imum sher stress on the outer surfce is the most commonly use prmeter to represent the ftigue mge [6-10]. In ition, the sher stress on the surfce of men imeter hs lso been use [11, 12]. There is no resonble finl conclusion on the selection of sher stress prmeter until now. It hs lrey been experimentlly inicte tht, for certin mteril, uivlent (von Mises) stress-lifetime curves, which re respectively trnsforme from S-N curve n -N curve, o not coincie with ech other. In orer to eliminte this iscrepncy, n improve sher stress ftigue mge prmeter bse on Buch s twoprmeter moel ws propose. Menwhile, series of multixil ftigue tests were conucte to vlite the new prmeter. Owing to its physicl bsis, this prmeter is prticulrly suitble for the representtion of specimen s ftigue mge conition n the lifetime preiction for torsionl loing. 2. Test progrms 2.1. Mteril n Specimen 2024-T4 ws use in this stuy. The geometry of the specimen is shown in Figure 1. The specimen ws mnufcture from br 30mm in imeter of luminium lloy, 2024-T4. The mteril properties t the room temperture re shown in Tble 1. Eleven specimens wre use for tensile-compressive ftigue tests n 13 specimens for torsionl ftigue tests on MTS809 electro-hyrulic servo ftigue testing mchine. Figure 1 the geometry of the specimen (mm)

3 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) Tble 1 Mechnicl properties of 2024-T4 E/GP y u f f K n K n The results for the 24 specimens re given in Tble 2. The tensile-compressive stress n sher stress were respectively clculte by 4F (1) 2 2 D - 16MD 4 4 ( D ) where F n M re xil force mplitue n torsionl moment mplitue respectively, D n re outer imeter n inner imeter of the guge section. The test results re liste in Tble 2. In Tble 2, stns for von Mises uivlent stresses. For tensioncompression, n for torsion, 3. (2) Tble 2 Ftigue life uner tensile-compressive n torsion loing Lo cse Specimen No. N f N 50 AT AT AT AT Tension 250 AT AT Compression 300 AT AT AT AT AT AT AT-88 * Torsion AT AT-78 * AT AT

4 194 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) Lo cse Specimen No. N f N 50 AT-70 * (* Ftigue crck in t occur in the test section) AT AT AT AT AT AT Novel sher stress prmeter It hs been experimentlly foun tht the ftigue lifetime observe uner fully reverse bening re lwys higher thn tht uner tension-compression for the sme stress level [13]. This beneficil influence of the stress grient is clle supporting effect. At present, mny pproches hve lrey been estblishe to moelling the supporting effect uner unixil norml cyclic stress. Among these pproches, Buch s two-prmeter moel [14] hs both simplifiction n effectivity Buch s two-prmeter moel In Buch s two-prmeter moel, the supporting effect of stress grient is tken into ccount. It is conclue tht ftigue crck initites only if the stress ( ), which is cquire on the mteril lyer of criticl epth h, excees K criticl vlue A. is the ftigue strength t the sme ftigue lifetime corresponing to the lo with no stress grient, e.g. reserve tension-compression. Mteril constnts h n A re relte to the types of mterils n loings, n etermine experimentlly. For tubulr specimen subjecte to torsion loing, the theoreticl sher stress grient is, if the mteril is in liner elstic stte, (1) r R where R is the outsie rius, is the sher stress on the outer surfce which cn be clculte s follow = M (2) W where M is the mplitue of torsionl moment, W is section moulus in torsion. As previously stte, the stress on mteril lyer of criticl epth h is enote s K, so the sher stress grient cn be lso expresse s K r r h (3) Therefore,

5 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) h = - h - h 1 r R R K (4) Thus, the sher stress prmeter is 1 h 1 A A R k (5) where h n A re mteril constnts, is the sher stress on the outer surfce, R is the outsie rius The effect of cyclic strin hrening or softening Most of mterils show cyclic strin hrening or softening uner cyclic loing. The behvior of cyclic hrening or softening les to the iscrepncy between monotonous n cyclic stress-strin curve. Although mteril t bout cyclic stress-strin curve uner torsionl loing is limite, t uner tensile-compressive loing is vilble. Since the shpes of the hysteresis loops re quite similr for these two kins of loing cses, it cn be ssume tht cyclic stress-strin curves re the sme for torsionl n tensile-compressive loing. The cyclic stress-strin curve n monotonic stress-strin curve for 2024-T4 is shown in Figure 2. Figure 2 cyclic stress-strin curve n monotonic stress-strin curve for 2024-T4 In Figure 2, it cn be observe tht the iscrepncy between these two kins of stress-strin curves vrious with the mgnitue of lo. This mens tht the egree of cyclic hrness vrious with the mgnitue of lo. To tke this phenomenon into ccount, vritionl prmeter A, which is moifie by the mgnitue of loing, is propose s follow, A A M C 0 (6) where A 0 is the reference vlue which is set to be 1.05 for 2024-T4 luminum lloy[15], is the stress corresponent to the resiul strin in the cyclic stress-strin curve, n is the stress corresponent to the resiul strin in the monotonic stress-strin curve. In etil, the fctor C / M, which is relte to the lo mgnitue, cn be solve by following utions:

6 196 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) E C M C- M- (7) E 1 C n C C E K (8) 1 M n M M E K (9) where is the resiul strin, M n C re von Mises uivlent stresses for monotonic n cyclic stressstrin curve, M n C re the strins which re relevnt to M n C on the stress-strin curves. For certin torsionl loing, C is set to be 3. Then, lining Eq.(9), Eq. (10) n Eq.(11), M cn be clculte. The iscussion in Section 3.1 is restricte in liner elstic conition. With the increse of loing, the specimen s section enters plstic eformtion grully from outsie to insie. However, owing to the newly propose prmeter A, the cyclic plsticity cn be tken into ccount. So, in elstic-plstic stte, the sher stress prmeter cn still be obtine by Eq. (7) Determintion of prmeter h In the originl Buch s two-prmeter moel, the prmeter h is etermine by t-fitting. This metho is teious n lck of operbility. Accoring to its essentil conception, it cn be foun tht there re some similrities between Buch s two-prmeter moel n Stress Fiel Intensity Approch (SFI pproch). They both tke chrcteristic stress of certin re s criterion to estimte the filure of the mteril. The sizes of the re re both mteril constnts. In SFI pproch, this is clle fiel-imension n hs been efine clerly. It is regre s the inherent ttribute of mteril n oes not vry with the loing. The fiel-imension is etermine by fitting the experimentl t. Consiering of the escription of Buch s two-prmeter moel, the criticl epth h oesn t chnge with the mplitue of loing either. This is similr to the fiel-imension of SFI pproch. So, mteril s fiel-imension is use s the criticl epth h in this pper. For 2024-T4, the fiel-imension is 0.185mm [16]. 4. Vlition of the propose prmeter Before the vlition, relevnt fctors, such s C / M n A, shoul be etermine first. Detile vlues of these fctors re liste in Tble 3. It cn be seen from the tble tht, the prmeter A, which is tking the cyclic strin hrening into ccount, inee vries with the loing mplitue. Then, ccoring to Eq.(7), the newly propose sher stress prmeter is reclculte. In Tble 3, is the von Mises uivlent stress clculte by. Tble 3 Detile vlues for the sher stress prmeter clcultion A C / M

7 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) C / M A Then, four -N curves re plotte in Figure 3. Among these curves: Tensile-Compression stns for the tensile-compressive S-N curve. Torsion ( Out ) stns for the uivlent stress-lifetime curve, of which the von Mises stress is clculte by the sher stress prmeter which is obtine on the outer surfce. The sher stress prmeter is common use in the reserches to represent the ftigue mge conition. Torsion ( Misection ) stns for the uivlent stress-lifetime curve of which the von Mises stress is clculte by the sher stress prmeter mi which is obtine on the misection surfce. Accoring to Ref. [12], mi cn be given by (10) mi = M ra m where M is the mplitue of torsionl moment, A is specimen cross-section re, n r m is misection rius. Torsion ( Propose ) stns for the uivlent stress-lifetime curve of which the von Mises stress is clculte by the newly propose sher stress prmeter. In Figure 3, it cn be observe tht the N f curve trnslte from common use is lrgely ifferent from the tensile-compressive S-N curve t the sme ftigue lifetime. The Nf curve trnslte from mi is closer to the S-N curve. To some extent, this lso inictes tht it is inpproprite to regr the sher stress on the outer surfce s the mge prmeter for ftigue lifetime preiction, without consiering the influence of sher stress grient. By contrst, the Nf curve trnslte from mkes best coincie with the S-N curve in tension-compression, owing to its consiertion of both sher stress grient n cyclic strin hrening. Figure 3 -N f curves of ifferent sher stress prmeters

8 198 Tin-Xing Xi et l. / Procei Engineering 74 ( 2014 ) Conclusion For thin-wlle tubulr specimen uner torsionl loing, there is stress grient on the cross section. This stress grient les to the fct tht ftigue lifetime observe uner fully reverse torsion is lwys higher thn tht uner tension-compression for the sme uivlent stress level. The newly propose sher stress ftigue mge prmeter tkes the influences of both sher stress grient n cyclic strin hrening into ccount. Compre with other two sher stress prmeters, the N f curve trnslte from mkes best coincie with the S-N curve in tension-compression, lmost hving eliminte the iscrepncy. Acknowlegements The uthors grtefully cknowlege the finncil support of the Ntionl Nturl Science Fountion of Chin(No ), the Reserch Fun of Stte Key Lbortory of Mechnics n Control of Mechnicl Structures (Nnjing University of Aeronutics n stronutics)(no. 0213G02), the Funing for Outstning Doctorl Disserttion in NUAA(No. BCXJ13-02), the Funing of Jingsu Innovtion Progrm for Grute Euction(No. CXZZ13_0175), n the Funmentl Reserch Funs for the Centrl Universities. References [1] Wng YY, Yo WX. Evlution n comprison of severl multixil ftigue criteri. Int J Ftigue 2004;26(1): [2] Ppopoulos IV, Dvoli P, Gorl C, Filippini M, Bernsconi A. A comprtive stuy of multixil high-cycle ftigue criteri for metls. Int J Ftigue 1997;19(3): [3] You B, Lee S. A criticl review on multixil ftigue ssessments of metls. Int J Ftigue 1996;18(4): [4] Ppug J. A survey on evluting the ftigue limit uner multixil loing. Int J Ftigue 2011;33(2): [5] ASTM Stnr E Stnr prctice for strin-controlle xil-torsionl testing with thin wlle tube specimens. Annul Book of ASTM Stnrs 2007(03.01): [6] McDirmi DL. A generl criterion for high cycle multixil ftigue filure. Ftigue Frct Eng Mt 1991;14(4): [7] -66. [8] Mtke T. An explntion on ftigue limit uner combine stress. Bulletin of the JSME 1977(20): [9] Zenner H, Simburger A, Liu J. On the ftigue limits of uctile metls uner complex multixil loing. Int J Ftigue 2000(22): [10] Liu Y, Mhevn S. Multixil high-cycle ftigue criterion n life preiction for metls. Int J Ftigue 2005(27): [11] Wng YY, Yo WX. Experimentl investigtion on the ftigue crcks propgtion uner tension-torsion loing. Trnsctions of Nnjing University of Aeronutics &Astronutics 2004;4(21): [12] Shmsei N, Glskyi M, Pnsovskyi K, Shukev S, Ftemi A. Multixil ftigue of titnium incluing step loing n lo pth ltertion n suence effects. Int J Ftigue 2010;32(11): [13] Ppopoulosi IV, Pnoskltsis VP. Invrint formultion of grient epenent multixil high-cycle ftigue criterion. Eng Frct Mech 1996;55(4): [14] Buch A. Anlyticl Approch to Size n Notch-size Effects in Ftigue of Aircrft Mteril Specimens. Mterils Science n Engineering 1974(15): [15] Buch A. Ftigue Strength clcultion. USA: Trns Tech Publictions, [16] Yo WX. Ftigue life preiction of structures. Beijing: Ntionl Defence Inustry Press, 2003.