A New Approach to the Fatigue Life Prediction for Notched Components Under Multiaxial Cyclic Loading. Zhi-qiang TAO and De-guang SHANG *

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1 2017 2nd Intrnational Conrnc on Applid Mchanics, Elctronics and Mchatronics Enginring (AMEME 2017) ISBN: A Nw Approach to th Fatigu Li Prdiction or Notchd Componnts Undr Multiaxial Cyclic Loading Zhi-qiang TAO and D-guang SHANG * Collg o Mchanical Enginring and Applid Elctronics Tchnology, Bijing Univrsity o Tchnology, Bijing, , P.R. China *Corrsponding author Kywords: Multiaxial cyclic loading, Notchd componnts, Fatigu li prdiction, Non-proportional hardning. Abstract. A nw atigu li prdiction approach or notchd componnts is proposd undr multiaxial cycl loading. Firstly, th psudo quivalnt strain-ral quivalnt strss rlationship is dtrmind by utilizing th Nubr s rul and th matrial strss-strain rlation undr proportional loading. Scondly, th psudo lastic strain historis at th notch nd to b solvd or th invstigatd notchd componnts undr multiaxial cyclic loading. Thirdly, or th psudo lastic strain historis, it is proposd that th Shang Wang multiaxial atigu damag paramtr can b usd to substitut th quivalnt strain undr proportional loading to tak account o th additional hardning du to th non-proportionality o xtrnal loadings. Manwhil, th ral quivalnt strss amplitud at th notch can b solvd. Fourthly, basd on th ral quivalnt strss amplitud at th notch intgratd with th Nubr s rul, th ral quivalnt strain amplitud can b dtrmind. Finally, th ral quivalnt strain amplitud is utilizd to prdict th atigu crack initiation litim o notchd componnts. Th proposd mthod is vriid by th xprimntal data o SAE 1045 notchd shat spcimns undr proportional and non-proportional cyclic loading. Th rsults showd th prdiction rrors ar mostly within a actor o 2. Introduction Many mchanical componnts invitably contain som notchs and gomtrical irrgularitis. Morovr, ths componnts ar subjctd to complx multiaxial loadings, which can rsult in th chang o th principal strss dirctions and th ratio o th principal strsss at th notch aras [1, 2]. Howvr, atigu ailur usually occurs at th notch ara du to signiicant strss concntrations. Thus, atigu li prdiction o notchd componnts undr multiaxial atigu loading is o grat importanc in practic. Various kinds o mthods hav bn proposd or prdicting th atigu li or notchd componnts subjctd to multiaxial loading [3-5], ths mthods ar composd o two principl stps, namly, multiaxial lastic-plastic strss and strain analysis at th notch ara [6-8] and multiaxial atigu li analysis. Although xprimntal masurmnt mthod and init lmnt analysis mthod can b usd to obtain rliabl strss and strain rsponss at th notch ara, such mthods ar not asibl in practic bcaus o cost and tim considrations. Thror, an icint approach to th atigu li prdiction o notchd componnt is an indispnsabl tool in th notchd componnt dsign. Th main objctiv o th prsnt papr is to dvlop a atigu li assssmnt approach or th notchd componnt, which is convnint or nginring dsign and application. In th proposd approach, th psudo lastic strain historis ar utilizd as basic input data. For th psudo lastic strain history, it is proposd that th quivalnt strain amplitud can b rplacd by th Shang Wang multiaxial atigu damag paramtr to tak account o th additional hardning du to th non-proportionality o xtrnal loadings. Th ral quivalnt strain amplitud can b solvd by using th Nubr s rul and th matrial strss-strain rlationship. Th validity o th proposd approach is 19

2 assssd by th xprimntal data o SAE 1045 notchd shat spcimns undr proportional and non-proportional cyclic loading. Proposd Approach or Prdicting th Fatigu Livs o Notchd Componnts In this sction, a nw approach is proposd to prdict th atigu li prdiction o notchd componnts undr multiaxial cyclic loading. Th dtaild procdur o th proposd approach is outlind as ollows: 1. Dtrmin th psudo quivalnt strain-ral quivalnt strss rlationship by utilizing th Nubr s rul and th matrial strss-strain rlation undr proportional loading. Nubr s rul: N N σ ε σ ε = (1) whr σ = E ε, E is th lastic modulus, and th lt-hand suprscript and N dnot th psudo and ral quantity at th notch. Th matrial strss-strain rlationship undr proportional loading: 1 n N N N ε σ σ = + (2) 2 2E 2K whr n and K ar th cyclic strain hardning xponnt and strngth coicint, rspctivly. From Eqs. (1) and (2), th psudo quivalnt strain-ral quivalnt strss rlationship can b dtrmind as ollow: N ( ) 2 1 σ N N n ε σ σ = + (3) 2 4E 2E 2K 2. Calculat th psudo lastic strain history at th notch rom th nominal multiaxial cyclic ε t ) can b obtaind rom nominal strsss and loading. Th psudo lastic strain history ( ( ) ij corrsponding strss concntration actors by th linar lastic thory or linar lastic init lmnt analysis. 3. Comput psudo quivalnt multiaxial atigu damag paramtr utilizing th psudo lastic cr ε t ). Th Shang Wang multiaxial atigu damag paramtr ( ε 2 ) [9], which strain history ( ( ) ij can b rducd to quivalnt strain approach orm ( ε 2 ) in th cas o th proportional loading, is usd in th prsntd invstigation. Th paramtr is xprssd as ollow: cr ε * 2 1 γ max = ( ε n ) whr ( * γ max 2 ) is th largst shar strain amplitud on th maximum shar plan, ε n is th largst normal strain xcursion btwn adjacnt turning points o th largst shar strain on th critical plan. N cr 4. Calculat th ral quivalnt strss amplitud ( σ 2 ) th psudo quivalnt strain-ral quivalnt strss rlationship (Eq.(3)). For th psudo lastic strain history, it is proposd that th cr Shang Wang multiaxial atigu damag paramtr ( ε 2 ) can b usd to substitut th quivalnt strain ( ε 2) undr proportional loading to tak account o th additional hardning du to th (4) 20

3 non-proportionality o xtrnal loadings. Manwhil, th ral quivalnt strss amplitud ( N cr σ 2 ) at th notch can b solvd. 5. Dtrmind th ral quivalnt strain amplitud ( N cr ε 2 ) by using th ral quivalnt strss amplitud ( N cr σ 2 ) obtaind in th prvious stp and th Nubr s rul (Eq.(1)). For non-proportional cyclic loading path, th non-proportional cyclic strngth coicint ( K NP ) can b obtaind by th matrial strss-strain rlationship undr non-proportional loading, i.. 1 n N cr N cr N cr σ σ ε = + (5) E K NP 6. Comput th atigu livs o notchd componnts utilizing th Manson-Coin quation, i.. N cr ε σ b c = ( 2N ) + ε ( 2N ) (6) 2 E whr σ, ε, b, and c ar th atigu strngth coicint, atigu ductility coicint, atigu strngth xponnt and atigu ductility xponnt, rspctivly. N is th atigu litim. In ordr to accomplish th proposd approach or prdicting th atigu livs o notchd componnts in this study, a computr program writtn in Fortran 95 is prormd. Exprimntal Vriication In ordr to chck th accuracy and rliability o th proposd approach or prdicting th atigu livs o notchd componnts, a numbr o atigu tst data on SAE 1045 stl undr proportional and non-proportional bnding and torsional loading wr slctd rom th tchnical litraturs [10]. Th gomtry and dimnsions o th SAE 1045 stl notchd shat ar dpictd in Figur 1. Th bnding-torsional atigu tst rsults o SAE 1045 stl notchd shat spcimns ar listd in Tabl 1. Th atigu proprtis o th SAE 1045 stl ar shown in Tabl 2. Figur 1. Gnral gomtry and dimnsions or SAE 1045 stl notchd shat spcimn (all dimnsions in mm). Tabl 1. Bnding-torsional atigu tst rsults o SAE 1045 stl notchd shat spcimns [10]. Spcimn ID Phas angl θ (dg.) Bnding momnt (N m) M b Torsional momnt M (N m) t N i (cycls) JD-XR JD-ZR JD-XR JD-YR JD-YR

4 JD-ZR JD-ZR JD-XR JD-XR JD-YR JD-ZR JD-XR JD-XR Tabl 2.Th mchanical proprtis o SAE 1045 stl and constants in Manson-Coin strain li quation [10]. Matrial E ( GPa ) K ( MPa) n σ ( MPa) ε b c SAE 1045 stl Figur 2. Comparison o xprimntal and prdictd atigu livs basd on proposd approach or SAE 1045 stl notchd shat spcimn undr proportional and non-proportional cyclic loading. A linar lastic FE analysis has bn prormd to dtrmin psudo lastic strain rsponss or th SAE 1045 stl notchd shat spcimn. Atr th psudo lastic strain historis ar dtrmind, th atigu livs o th invstigatd SAE 1045 stl notchd shat spcimn can b calculatd by th proposd approach. Comparisons o prdictd rsults with xprimntal data undr proportional and non-proportional cyclic loading ar shown in Fig. 2. It is ound that th prdictiv atigu livs agr wll with th xprimntal data, in which prdictiv litim rrors ar mostly within a actor o 2. Th rsults indicat that th proposd approach can provid accptabl li prdiction rsults or th SAE 1045 stl notchd shat spcimns. Summary In th currnt study, a nw approach is proposd to prdict atigu li o notchd componnts subjctd to multiaxial cyclic loading. Th proposd approach can tak account o th additional hardning du to th non-proportionality o xtrnal loadings. Th xprimntal data o SAE 1045 stl notchd shat componnts hav bn utilizd to vriy th proposd approach undr proportional and non-proportional cyclic loading. Th rsults show th proposd approach can wll prdict atigu livs or multiaxial cyclic loading, and most prdictiv rsults ar alling within an rror actor o 2. 22

5 Acknowldgmnt Th authors gratully acknowldg th inancial support o th National Natural Scinc Foundation o China (Nos , ). Rrncs [1] S. Babai, A. Ghasmi-Ghalbahman. Damag-basd modiication or atigu li prdiction undr non-proportional loadings. Int. J. Fatigu 77 (2015) [2] S. Babai, A. Ghasmi-Ghalbahman, R. Hajighorbani. A atigu modl or snsitiv matrials to non-proportional loadings. Int. J. Fatigu 80 (2015) [3] A. Inc, G. Glinka. Innovativ computational modling o multiaxial atigu analysis or notchd componnts. Int. J. Fatigu 82 (2016) [4] M. Firat. A numrical analysis o combind bnding-torsion atigu o SAE notchd shat. Finit Elm. Anal. Ds. 54 (2012) [5] N.R. Gats, A. Fatmi. Multiaxial variabl amplitud atigu li analysis including notch cts. Int. J. Fatigu 91 (2016) [6] V.B. Köttgn, M.E. Barky, D.F. Soci. Psudo strss and psudo strain basd approachs to multiaxial notch analysis. Fatigu Fract. Eng. Matr. Struct. 18 (1995) [7] A. Inc, G. Glinka, A. Buczynski. Computational modling o multiaxial lasto-plastic strss strain rspons or notchd componnts undr non-proportional loading. Int. J. Fatigu 62 (2014) [8] D.Y. Y, O. Hrtl, M. Vormwald. A uniid xprssion o lastic plastic notch strss strain calculation in bodis subjctd to multiaxial cyclic loading. Int. J. Solids Struct. 45 (2008) [9] D.G. Shang, D.J. Wang. A nw multiaxial atigu damag modl basd on th critical plan approach. Int. J. Fatigu 20 (1998) [10] J.C. Simo, T.J.R. Hughs, Computational Inlasticity, Springr-Vrlag Nw York Inc., Nw York,