Numerical study on the influence of artificial internal stress relief groove on fretting fatigue in a shrink-fitted assembly

Depósi o de In es igación de la Uni e sidad de Se illa h ps://idus.us.es/ This is an Accep ed Manusc ip o an a icle published by Else ie in T ibology In e na ional, Vol. 151, on No embe 2020, a ailable a : h ps://doi.o g/10.1016/j. iboin .2020.106443 © 2020 Else ie . En idUS Licencia C ea i e Commons CC BY-NC-ND 1 NUMERICAL STUDY ON THE INFLUENCE OF ARTIFICIAL INTERNAL STRESS RELIEF GROOVE ON FRETTING FATIGUE IN A SHRINK- FITTED ASSEMBLY Diego E ena1, Jesús Vázquez1, Ca los Na a o1, Reza Talemi2 1 Depa men o Mechanical Enginee ing, Uni e si y o Se ille, Spain 2 Depa men o Ma e ials Enginee ing, KU Leu en, Belgium E-mail add ess: [email p o ec ed] KEYWORDS: F e ing Fa igue, Addi i e Manu ac u ing, C ack Ini ia ion, C ack P opaga ion, XFEM ABSTRACT F e ing a igue ailu e occu s in sh ink- i ed assemblies due o he combina ion o high s esses and ela i e displacemen s nea he con ac edge. Due o hese high s esses, a igue c ack ini ia es ollowed by c ack p opaga ion un il inal up u e. Addi i e manu ac u ing (AM) is a game changing echnology, which enables new componen capabili ies ha canno be manu ac u ed wi h con en ional echniques. This esea ch wo k analyses nume ically he in luence o an a i icial in e nal s ess elie o oidal g oo e inside a sh ink- i ed sha , which could be manu ac u ed using AM echnology. Due o he o oidal oid, he s ess/s ain ields a e edis ibu ed imp o ing he e ing a igue c ack ini ia ion and p opaga ion li e imes. To do so, 2D ini e elemen models a e c ea ed in Abaqus so wa e wi h and wi hou he in e nal g oo e. To es ima e he e ing a igue ini ia ion and p opaga ion li e ime and c ack p opaga ion di ec ion, c i ical plane me hods a e used. In e ms o he c ack p opaga ion, eX ended Fini e Elemen Me hod (XFEM) is used o simula e mixed mode c ack ad ancing in a single mesh s uc u e. Finally, he ob ained esul s wi h and wi hou oid we e compa ed concluding wi h signi ican imp o emen s in e ms o o al a igue li e ime. 1. INTRODUCTION The ailu e o sh ink- i assemblies is commonly p oduced by e ing a igue phenomena [1-2]. The con ac p essu e p oduced by he in e e ence i ing and ela i e displacemen s ound a he sha /hub edge, p oduced by ex e nal loads, lead o he ailu e o he sha due o e ing a igue. In gene al, he combina ion o con ac s esses and ela i e displacemen s be ween bodies lead o h ee well-di e en ia ed damage ypes: wea , oxida ion and c ack nuclea ion [3]. F e ing phenomena is obse ed in se e al mechanical 2 componen s, such as me allic cables [4-7], u bine blade do e ails [8], sh ink i ed sha s assemblies [9–11] and bol ed join s [12] among o he s. Sh ink- i ed assemblies a e commonly used a gea sha s assemblies and ailway wheelse s among o he s. The e o e, hey a e commonly subjec ed o o a y bending and o sional cyclic loads. In he p esen wo k only o a y bending bounda y condi ions a e analysed. Se e al pape s s udied he ype o ailu e o hese assemblies, no icing ha he ailu e is obse ed in he icini y o he con ac edge a he sha side, in which he e ec o e ing wea is signi ican , go e ning he c ack ini ia ion poin [9-10,13]. To delay he ailu e, di e en pallia i es a e ound in p ac ice, such as he applica ion o deep- olling o he con ac su aces [11], and he use o sha o e hangs and slo s in he hub side [14]. Se e al au ho s ha e analysed di e en pallia i es in o de o educe he nega i e e ec o e ing a igue applied o di e en e ing p oblems. These pallia i es can be di ided in h ee well di e en ia ed g oups. Those ha modi y some geome ical aspec [14], hose ha modi y he ma e ial p ope ies [15-16] and hose ha induce comp essi e esidual s esses below he su aces in con ac [17-19]. All hese pallia i es, in one way o ano he , achie e signi ican li e enhancemen s mi iga ing he in luence o e ing. The p oposed pallia i e could be conside ed in he i s g oup. The pallia i e analysed in he cu en wo k is di ec ly linked wi h AM echnology, which is he only way o ep oduce he sugges ed in e nal geome ical ea u e in an ac ual case. This ield is s ill unde de elopmen so he e a e s ill many unknowns. F om he mechanical poin o iew, he po osi y o he new ma e ial, he su aces oughness and he appa en aniso opy o he ma e ial due o he laye -by-laye deposi ion play an impo an ole in he mechanical beha iou o componen s manu ac u ed by AM [20-21]. All hese a iables a e no conside ed in he p esen wo k, assuming ha he beha iou o he new ma e ial is he same as he adi ional one. Al hough AM is s ill unde de elopmen , he e is no doub ha i is gaining a s ong oo hold in he manu ac u ing ield so i is in e es ing o in es iga e abou he bene i s ha could o e in he nea u u e. The main objec i e o cu en esea ch wo k is mo ing o wa d in he analysis o oids as e ing a igue s ess elie e s o he speci ic case o sh ink- i ed sha assemblies as p oposed in o me wo ks o he au ho s [22-23]. In [22] i is s udied he in luence o a 3 o oidal a i icial oid, wi h he shape o a g oo e, inside he sha o a sh ink- i ed assembly subjec ed o o a y bending. Concluding ha , wi h he adequa e oid posi ion and size, a dec ease in e ms o a mul iaxial a igue pa ame e was achie able i compa ed wi h a adi ional case wi hou oids. In basis o he a o emen ioned idea, he objec i e o he p esen wo k is applying his pallia i e o an ac ual assembly and nume ically s udy i s in luence a he ini ia ion and p opaga ion c ack s ages. Besides li e enhancemen s p oduced by he oid, c ack o ien a ion is also analysed a bo h s ages. In his sense, a SWT based c ack p opaga ion o ien a ion me hod is applied by means o XFEM. XFEM app oach is an ex ension o he con en ional Fini e Elemen Me hod (FEM) and is based on he concep o pa i ion o uni y [24]. Based on a ini e elemen model, local en ichmen unc ions a e in oduced in he model, and he e o e new deg ees o eedom (DOFs). These new DOFs a e linked wi h he nodes o he elemen s ha a e pene a ed by he c ack. This ea u e allows modelling he discon inui y p oduced by a c ack and i s p opaga ion wi hou modi ying he disc e iza ion. The e o e, i is possible o p opaga e a c ack in a single mesh. Fo he pu pose o ac u e analysis, hese unc ions consis o he nea - ip asymp o ic unc ions ha cap u e he singula i y a ound he c ack ip. The app oxima ion o a displacemen ec o unc ion u wi h he pa i ion o uni y en ichmen is shown in Eq. (1). 𝑢=∑𝑁𝐼(𝑥)[𝑢𝐼+𝐻(𝑥)𝑎𝐼+∑𝐹𝛼(𝑥)𝑏𝐼𝛼 4 𝛼=1 ] (1) 𝑁 𝐼=1 whe e NI(x) a e he adi ional shape unc ions and uI is he nodal displacemen ec o o he ini e elemen solu ion. The second addend is he p oduc o he nodal en iched deg ee o eedom ec o , aI, and he associa ed discon inuous jump unc ion H(x) ac oss he c ack su aces. The jump unc ion can only ake he alues H(x)=±1, depending on he ela i e posi ion o he en iched node wi h espec o he c ack ace. The hi d e m is he p oduc o he nodal en iched deg ee o eedom ec o , bIα, and he associa ed elas ic asymp o ic c ack- ip unc ions. The i s e m on he igh -hand side is applicable o all he nodes in he model; he second e m is alid o nodes whose shape unc ion suppo is cu by he c ack in e io ; and he hi d e m is used only o nodes whose shape unc ion suppo is cu by he c ack ip. The asymp o ic c ack ip unc ions in an iso opic elas ic ma e ial, Fα(x), a e shown in Eq. (2) and ob ained om [24]. 4 𝐹𝛼(𝑥)=[√𝑟 𝑠𝑖𝑛𝜃 2,√𝑟 𝑐𝑜𝑠𝜃 2,√𝑟sin𝜃𝑠𝑖𝑛𝜃 2,√𝑟sin𝜃𝑐𝑜𝑠𝜃 2] (2) whe e and 𝜃 a e pola coo dina es wi h i s o igin a he c ack ip and 𝜃 = 0 is angen o he c ack a he ip. The e o e, wi h he XFEM o mula ions i is possible o p opaga e a p ede ined ini ial c ack in a single mesh. In he li e a u e he e a e se e al c i e ia o es ima e he c ack p opaga ion di ec ion speci ically designed o p opo ional loading condi ions. Some o hem a e b ie ly de eloped in he ollowing sec ion. Some au ho s ha e obse ed ha when applying hese p opaga ion me hods o e ing a igue condi ions, e oneous c ack pa hs a e p edic ed [25-26]. The main eason o he ailu e o hese app oaches is he non- p opo ional loading condi ions ha appea in e ing a igue condi ions. To ackle his p oblem, a c ack p opaga ion di ec ion me hod based on he SWT pa ame e is applied wi h sa is ac o y esul s. The me hod was p e iously de eloped by he au ho ´s and applied o a FEM model [27]. Besides, he adi ional p opo ional me hods o es ima e he c ack p opaga ion di ec ion and some non-p opo ional ones a e s udied in he nex sec ion and compa ed la e in he esul s sec ion. 2. Backg ound 2.1. C ack ini ia ion The alue o he SWT mul iaxial a igue pa ame e is used as a c ack ini ia ion c i e ion using he p ocedu e de eloped by Vázquez e al. [28] and applied o e ing a igue p oblems in subsequen wo ks [23]. The equa ion de ining he adi ional SWT pa ame e o a non-p opo ional loading s a e is shown in Eq. (3) [29]. SWT= (σn∆ε 2)max. (3) Whe e σn is he no mal s ess o he ma e ial plane and Δε is he ange o he no mal s ain along he loading cycle, bo h o a speci ic o ien a ion. The alue o SWT is he p oduc o hese pa ame e s a he o ien a ion whe e i is maximum. Fig. 1 shows a schema ic ep esen a ion o he me hod p oposed based on he adi ional SWT pa ame e wi h some modi ica ions. A i s , i is necessa y o seek o he mos un a ou able con ac su ace poin , conside ing i as he one wi h he maximum alue o 5 he SWT pa ame e acco ding o Eq. (3). This c i ical poin is conside ed o be he o igin o di e en ma e ial lines dis ibu ed homogeneously be ween θ = 0 ° and θ = 180 ° wi h adius R, de ining he size o he in luence a ea (see Fig. 1). The SWT pa ame e is calcula ed along each o hese lines (de ined by θ) a di e en poin s, bu wi h he no el y ha , o each o hese poin s, he o ien a ion o he ma e ial plane conside ed o e alua e he pa ame e is no he c i ical one, i.e., acco ding o Eq. (3), bu he one ha coincides wi h he o ien a ion imposed by θ. Fo example, o he 90 deg ees’ plane ( e ical) he s esses and s ains used o ob ain he SWT pa ame e a e σxx and εxx (pe pendicula o he ma e ial line), which could p oduce a c ack in e ical di ec ion. Wi h his p ocedu e a SWT dis ibu ion is ob ained along each ma e ial line. Nex , he mean alue o he SWT dis ibu ion along each line is calcula ed. Finally, he o ien a ion ha ing he highes mean alue o he pa ame e is conside ed o be he mos likely o ini ia e a c ack as shown in Fig. 1. Figu e 1. SWT c i ical plane ini ia ion p ocedu e. The mean SWT alues depends on he R pa ame e . To es ima e he ini ia ion li e imes, he R pa ame e should be calib a ed o each p oblem by means o he a igue cu e o he ma e ial. The p ocedu e is de eloped a he esul s sec ion. I is impo an o emphasise ha his me hod is only applicable o compa ison pu poses. In he p esen wo k his me hodology is used o compa e he esul s ob ained o wo case s udies o he same ma e ial and subjec ed o he same bounda y condi ions bu wi h di e en geome ies. F om now on e e ences o SWT pa ame e a e e e ed o he mean alue, a he c i ical o ien a ion de ined by he angle 𝜃0 ob ained wi h he de ined me hod and no acco ding o Eq. (3). 2.2. C ack p opaga ion 6 The c ack p opaga ion phase is s udied ia XFEM implemen ed in Abaqus so wa e. This nume ical echnique allows he simula ion o c acks, which a e in oduced a e he meshing p ocess. Due o his ea u e, i is no necessa y o de elop di e en models wi h di e en c ack sizes and ine meshes a he c ack ip [24]. To es ima e he c ack p opaga ion li e ime, s ess in ensi y ac o s KI and KII a e compu ed ia he in e ac ion in eg al me hod implemen ed in Abaqus o each loading s ep. The in e ac ion in eg al is a modi ica ion o he J-in eg al in which he line in eg al is con e ed o an a ea in eg al. In he case o Abaqus so wa e i is possible o de ine he c ack ip and se e al con ou s o e alua e he in eg al and ob ain he S ess In ensi y Fac o (SIFs) and c ack p opaga ion di ec ion a each con ou . Each con ou is de ined by one ing o elemen s su ounding he c ack ip as shown in Fig.2. Figu e 2. Con ou s o e alua e SIFs. Once he SIFs a e known i is possible o es ima e he c ack p opaga ion li e ime by in eg a ion o he well know Pa is’ law shown in Eq. (4) [30]. 𝑑𝑎 𝑑𝑁=𝐶(∆𝐾𝑒𝑞)𝑚 (4) Whe e da/dN is he c ack g ow h a e, C and m a e ma e ial cons an and ΔKeq is ob ained om Eq. (5) [31]. ∆𝐾𝑒𝑞 =√∆𝐾𝐼2+∆𝐾𝐼𝐼 2 (5) Whe e ΔKI and ΔKII is he ange o he SIF along he loading cycle o he modes I and II espec i ely. I will be conside ed ha he c ack is closed o nega i e alues o KI, he e o e any nega i e alue o KI will be conside ed as ze o. C ack C ack ip closes node Con ou s 7 C ack p opaga ion di ec ion could be es ima ed i s ess and s ain ields a e known a he c ack ip. Th ee di e en me hods a e commonly used o p opo ional loading s a es: Maximum Tangen ial S ess c i e ion (MTS) [32], Maximum Ene gy Release a e c i e ion (MERR) [33] and KII = 0 c i e ion (KII0) [35]. MTS c i e ion conside he nex c ack p opaga ion di ec ion o be o hogonal o he o ien a ion wi h maximum angen ial s ess a c ack ip. In plane s ain beha iou he s ess ield close o he c ack ip is de ined by Eq. (6) and Eq. (7) o an iso opic linea elas ic ma e ial. 𝜎𝜃𝜃 =1 √2𝜋𝑟 𝑐𝑜𝑠𝜃 2[𝐾𝐼cos2𝜃 2−3 2 𝐾𝐼𝐼 𝑠𝑖𝑛 𝜃] (6) 𝜏𝑟𝜃 =1 2√2𝜋𝑟 𝑐𝑜𝑠𝜃 2[𝐾𝐼 𝑠𝑖𝑛 𝜃−𝐾𝐼𝐼 (3cos𝜃−1)] (7) Whe e and θ a e pola coo dina es and he o igin is a he c ack ip in a plane pe pendicula o he c ack ace. The c ack p opaga ion di ec ion is ob ained wi h he condi ion 𝜏𝑟𝜃 =0. The e o e, he c ack p opaga ion o ien a ion θp , measu ed wi h espec o he o me c ack di ec ion is ob ained wi h Eq. (8). 𝜃𝑝=𝑎𝑐𝑜𝑠[3𝐾𝐼𝐼 2+√𝐾𝐼4+8𝐾𝐼2𝐾𝐼𝐼 2 𝐾𝐼2+9𝐾𝐼𝐼 2] (8) A nega i e alue o θp should be conside ed i KII ≥ 0 and a posi i e alue o KII ≤ 0. MERR c i e ion conside s he c ack p opaga ion in he di ec ion in which he ene gy elease a e Gk, de ined by Eq. (9), is maximum. 𝐺𝑘=1 𝐸[(𝐾𝐼𝑘(𝜃))2+(𝐾𝐼𝐼 𝑘(𝜃))2] (9) Whe e KIk and KIIk a e de ined by: 𝐾𝐼𝑘(𝜃)=𝐶11(𝜃)𝐾𝐼+𝐶12(𝜃)𝐾𝐼𝐼 (10) 𝐾𝐼𝐼 𝑘(𝜃)=𝐶21(𝜃)𝐾𝐼+𝐶22(𝜃)𝐾𝐼𝐼 (11) And he cons an s Cij a e gi en in [33]. 8 Finally, he KII0 c i e ion pos ula es ha he p opaga ion is de ined by he di ec ion in which KII is ze o o minimum in some loading condi ions whe e a ze o alue is no possible. Howe e , i is well known ha unde e ing loading condi ions hese c i e ia do no p edic he co ec di ec ion due o high s ess g adien zones and he exis ence o mixed mode non-p opo ional loading [25-26]. Besides, due o he o mula ion o hese me hods, hey do no ake in o conside a ion he comple e ange o he loading cycle. In he p esen wo k, he c ack o ien a ion di ec ions ob ained acco ding o he a o emen ioned c i e ia a e ob ained o he loading s ep in which he c ack is open. Wi h he aim o sol ing hese di icul ies, assuming ha he s ess ange in a igue ield is i al, di e en c ack p opaga ion c i e ia a e ound in p ac ice o he cases o non- p opo ional loading condi ions. Dubou g e al.[36] p oposed he c ack p opaga ion in he di ec ion o which he ange o he ci cum e en ial s ess is maximum wi hou aking in o accoun comp essi e s esses, max(Δσe (θ)). Ribeauco e al. [37] compa ed he esul s ob ained using wo di e en app oaches. The i s one conside s ha he c ack p opaga ion di ec ion is de ined by he maximum alue o he c ack g ow h a e, max(𝑑𝑎 𝑑𝑁 (θ)). The second one is based on he combina ion o mode I and II s ess in ensi y ac o s, his c i e ion looks o he di ec ion wi h he maximum alue o he ange o Eq. (10), max(ΔKIk(θ)). Gine e al.[26] de eloped a non-p opo ional c i e ion de ining he c ack o ien a ion in he di ec ion wi h he minimum alue o he shea s ess ange, min(Δτ(θ)). Finally, he me hod applied in he cu en wo k is based in he p ocedu e de eloped a he c ack ini ia ion s age and was p oposed and alida ed by Bohó quez e al [27].The e o e a mo e de ailed explana ion o he me hod is de eloped nex . A schema ic iew o he p oposed me hodology is depic ed in Fig.3. 15 Besides he ini ia ion li e es ima ion ob ained, he s esses and he sliding ampli ude along he sha con ac su ace (Pa h in Fig. 8) o he i s loading s ep (i.e. upwa ds bending load) a e shown in Fig. 8. I can be no iced ha he axial s ess σxx, shea s ess σxy and no mal s ess σyy dec ease subs an ially in he case wi h he oid wi h espec o he e e ence case close o he c i ical zone. In special, i is impo an o no e he dec ease in he alue o σxx (~25%). Tha is he main eason o he nume ically p edic ed c ack ini ia ion li e ime imp o emen . A ema kable ea u ed p oduced by he oid in he assembly he e analysed, is he inc ease in he slip. FEM simula ions show ha he sliding dis ance, δ, o he case wi h he oid is la ge han he co esponding o he e e ence case. Howe e , his inc emen in he slip is no aduced in an inc emen in he di ec s ess a he con ac edge, which could be he main agen o he c ack ini ia ion. Acco ding o A cha d´s law, e ing wea is p opo ional o he p oduc o he con ac no mal p essu e, σyy, and he sliding dis ance [43]. This p oduc is analysed in o de o gi e a quali a i ely idea o he e ing wea p ocess in he con ac zone. The esul s a e shown in Fig. 8b, which shows ha due o he g oo e, wea is expec ed o be mo e p onounced a om he con ac edge. 16 Figu e 8. a) Con ac s ess and sliding dis ibu ion along pa h, b) δσyy pa ame e . 4.2. F e ing a igue c ack p opaga ion The e ing a igue c ack p opaga ion s age is pe o med acco ding o he SWT me hod p oposed in sec ion 2.2 and compa ed wi h he min(Δτ(θ)) me hod. Fi s , an ini ial c ack is de ined wi h a leng h o 0.1 mm and an ini ial o ien a ion o 108º a 50 µm inside he con ac om he con ac edge. These ini ial c ack pa ame e s a e ob ained acco ding o he expe imen al esul s measu emen s shown in e e ence [13] and he mesh size used in he icini y o he c ack. Once he s ess/s ain ields a e known o each c ack leng h, a he pos p ocessing s age, he SIFs a e ob ained by means o he in e ac ion in eg al me hod and he nex c ack p opaga ion di ec ion a e ob ained acco ding o he p opaga ion me hods. C ack inc emen s, Δa, o 0.1 mm a e conside ed. The p opaga ion p ocess is ca ied ou au oma ically by means o a Py hon sc ip . The compa ison o he esul s ob ained o he me hods al eady implemen ed in Abaqus, he p oposed one as desc ibed in sec ion 2.2 and he non-p opo ional me hod min(Δτ(θ)), a e shown oge he wi h wo ac ual c acks, in Fig. 9. The e alua ion o min(Δτ(θ)) is ca ied ou in he same Re e ence case Void case -1.5 -1 -0.5 0 x –Pa h (mm) 0 50 100 150 200 250 300 350 0.5 1.0 1.5 2.0 2.5 3.0 3.5 δ δ(µm) σ(MPa) σyy σxy σxx -1.5 -1 -0.5 0 x –Pa h (mm) δσyy a) b) 17 manne as shown in Fig. 3 o calcula e bo h me hods wi h he same s ess/s ain ields. The c ack ob ained wi h adi ional cases (MTS, MERR and KII0), quickly ends o he e ical di ec ion and con e gence p oblems a ose. Ne e heless, he p edic ed c ack p opaga ion pa h using he SWT p oposed me hod and he min(Δτ(θ)) a e in good ag eemen wi h he obse ed expe imen al esul s in Fig. 9. Figu e 9. C ack p opaga ion di ec ion [13]. The e o e, he non-p opo ional c ack p opaga ion me hods a e used o in es iga e he e ing a igue c ack p opaga ion esponse o he sh ink- i assembly. The p edic ed c acks o he e e ence case, he case wi h he g oo e and he ac ual c acks o Fig. 9 a e plo ed in Fig. 10. The maximum c ack leng h analysed is 2.6 mm. The limi is imposed by he heigh o he e ined mesh. The esul s ob ained o bo h p opaga ion me hods a e e y cong uen . The slope o he min(Δτ(θ)) me hod is sligh ly la ge han he slope o he SWT me hod, al hough he di e ence is almos negligible. The c ack pa h in he cases wi h he oid, independen ly o he me hod, end mo e quickly o he e ical di ec ion han he e e ence case c ack. This di e ence is mo e isible wi h he min(Δτ(θ)) me hod. HUB SHAFT SWT MERR MTS KII0 Ini ial c ack 200 μm P opaga ed c acks Tes 1 c ack & Tes 2 c ack min( )Δτ 18 Figu e 10. P edic ed and expe imen al c ack pa hs wi h SWT p opaga ion me hod. The p opaga ion cycles o each c ack inc emen , Δa, a e compu ed using he Pa is’ law (see Eq. (4)). The SIFs alues a e ob ained a each c ack inc emen wi h he con ou in eg al me hod by de ining 9 con ou s and a e aging he esul s o he las i e (see Fig. 1). The ma e ial´s cons an s C and m de ining Eq. (4) depends in a g ea ex en o he speci ic ype o ma e ial. Due do he lack o da a wo di e en cons an o wo ypes o s eel a e analysed wi h compa a i e pu poses [44]. The cons an used and he p edic ed p opaga ion li e imes (Np) a e shown in Table 3. Table 3. P opaga ion li e ime compa ison. Me hod Ma e ial C m Np (Re e ence case) Np (Void case) Imp o emen SWT S eel (Tempe ed ma ensi e) 1.35e- 11 2.8 264500 377700 1.4 S eel (Pea li e) 7.15e- 13 3.4 1436400 2426500 1.7 min(Δτ(θ)) S eel (Tempe ed ma ensi e) 1.35e- 11 2.8 301200 412400 1.4 S eel (Pea li e) 7.15e- 13 3.4 1663400 2674300 1.6 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 -0.6 -0.4 -0.2 0 x (mm) Tes 2 Tes 1-Ini ial c ack SWT Re e ence case SWT Void case min(Δτ) Re e ence case min(Δτ) Void case y (mm) 19 The p edic ed li e imes alues depend on he Pa is’ law cons an s. Howe e , li e imp o emen s a ound 1.5 imes a e obse ed in he case wi h he oid wi h espec o he e e ence case independen ly o he p opaga ion me hod used. This inc emen is subs an ially smalle han he enhancemen s achie ed a he ini ia ion s age. Fo a be e unde s anding o he oid in luence along he c ack p opaga ion p ocess, he numbe o cycles o p opaga e an ini ial c ack, wi h a leng h a=a0 = 0.1 mm, up o a c ack leng h o a = a0+Δa a e shown in Fig. 11. Analysing Fig. 11 i can be obse ed ha he p opaga ion li e ime enhancemen s shown in Table 3 a e mainly due o he imp o emen s ob ained a he e y beginning o he c ack, independen ly o he p opaga ion c i e ia. F om his igu e i can be obse ed ha he di e ence in he c ack p opaga ion cycles a e p oduced du ing he i s s eps. Then bo h c ack p opaga ion cu es end o be almos pa allel due o he small di e ence be ween he p opaga ion cycles; his beha iou is obse ed app oxima ely om a c ack inc emen a ound Δa=0.9 mm. The e o e, he in luence o he oid is ema kable o he ini ia ion s age and o c ack leng hs below 1 mm app oxima ely. Figu e 11. Accumula i e cycles o p opaga e he ini ial c ack a0 up o a0+Δa. The cycles equi ed o p opaga e he c ack om 2.5 mm up o 2.6 mm a e lowe han he 0.8% o he o al p opaga ion li e ime. The e o e, and al hough he c ack could p opaga e mo e han he imposed limi o 2.6 mm (up o 3 mm in he oid case), his ex a numbe o cycles a e negligible. This e ec can be obse ed a Fig.11, whe e he slope o he cu es ends o in ini y. 0.10 0.30 0.50 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30 2.50 0.5 2.0 3.0 4.0 5.0 Δa (mm) Np(accumula i e cycles)x105 SWT Re e ence case SWT Void case min(Δτ) Re e ence case min(Δτ) Void case Tempe ed ma ensi e 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 a (mm) 20 Fig. 12 compa es he i s p incipal s ess a loading s ep 1 (i.e. load Q upwa ds) be ween he e e ence case and he case wi h he a i icial oid o a c ack leng h o 1 mm and he SWT p opaga ion me hod. The zone wi h alues highe han he yield s ess is ep esen ed wi h g ey colou . I can be no iced ha , hanks o he oid in oduced, lowe maximum p incipal s ess is obse ed in he case wi h he oid al hough he di e ence is negligible. In gene al, he s ess ields a e e y simila , he e o e o his c ack leng h he oid in luence is null as was obse ed om Fig.11. Figu e 12. σI o 1mm c ack, a) Assembly wi h oid, b) Re e ence case assembly. 5. CONCLUSIONS The conclusions could be di ided in o wo well di e en ia ed poin s. On he one hand a ecen me hod o es ima e he c ack p opaga ion o ien a ion has been analysed o he case o mixed mode non-p opo ional loading. The me hod is based in a mul iaxial a igue pa ame e . The esul s show ha he p oposed me hod es ima es co ec ly he c ack p opaga ion di ec ion while adi ional me hods, al eady implemen ed in Abaqus, p edic w ong c ack p opaga ion di ec ions when hey a e applied o a e ing a igue p oblem. Besides, he SWT p opaga ion me hod has been compa ed wi h he min(Δτ(θ)) me hod. In gene al, al hough some di e ences a e obse ed disc epancies a e minimum. On he o he hand, a new e ing a igue pallia i e has been analysed o he case o an ac ual sh ink- i ed sha assembly subjec ed o o a y bending loading condi ions. The p oposed pallia i e consis s on he in oduc ion o a o oidal oid wi h a p e iously de ined op imum posi ion and geome y, benea h he con ac edge. To analyse he oid a = 1 mm Void case σI (MPa) Re e ence case σI (MPa) a = 1 mm Q Q 21 in luence, wo s a egies ha e been de eloped by means o a 2D simpli ied model o an ac ual es . The i s s a egy was analysing he c ack ini ia ion li e ime o he cases wi h and wi hou oid, no icing li e imp o emen s o mo e han i e imes he ac ual case li e ime. Besides, he ini ial c ack o ien a ion es ima ed ag ees e y well wi h he one measu ed om he es s. The second s a egy was analysing he c ack p opaga ion o ien a ion and li e ime. The esul s sugges ha imp o emen s in e ms o c ack p opaga ion li e ime a e also obse ed due o he pallia i e p oposed. These enhancemen s o li e ime a e o he o de o one and a hal imes he one es ima ed wi h he e e ence case. The e o e, he imp o emen s a he ini ia ion s age a e mo e signi ican han hose obse ed a he p opaga ion s age. In gene al, i can be d awn ha wi h he adequa e oid posi ion and size, impo an imp o emen s in e ms o c ack ini ia ion and p opaga ion li e imes could be achie ed. Acknowledgemen s The au ho s wish o exp ess hei g a i ude o he Minis y o Economy and Compe i i eness o unding he esea ch o he DPI2014-59160-P p ojec . Re e ences [1] M. Madia, S. Be e a, and U. Ze bs , “An in es iga ion on he in luence o o a y bending and p ess i ing on s ess in ensi y ac o s and a igue c ack g ow h in ailway axles,” Eng. F ac . Mech., ol. 75, no. 8, pp. 1906–1920, 2008, doi: 10.1016/j.eng acmech.2007.08.015. [2] C. E. T uman and J. D. 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