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. Booke , “Analysis o a sh ink- i ailu e on a gea
hub/sha assembly,” Eng. Fail. Anal., ol. 14, no. 4, pp. 557–572, 2007, doi:
10.1016/j.eng ailanal.2006.03.008.
[3] R. Wa e house, F e ing a igue. U.K: Applied science publishe s, 1981.
[4] R. E. Hobbs and M. Raoo , “Mechanism o e ing a igue in s eel cables,” In . J.
Fa igue, ol. 16, no. 4, pp. 273–280, 1994, doi: 10.1016/0142-1123(94)90341-7.
[5] Z. R. Zhou, A. Ca dou, M. Fise , and S. Goud eau, “F e ing a igue in elec ical
ansmission lines,” Wea , ol. 173, no. 1–2, pp. 179–188, 1994, doi:
10.1016/0043-1648(94)90271-2.
[6] C. R. F. Aze edo, A. M. D. Hen iques, A. R. Pulino Filho, J. L. A. Fe ei a, and
J. A. A aújo, “F e ing a igue in o e head conduc o s: Rig design and ailu e
analysis o a G osbeak aluminium cable s eel ein o ced conduc o ,” Eng. Fail.
22
Anal., ol. 16, no. 1, pp. 136–151, 2009, doi: 10.1016/j.eng ailanal.2008.01.003.
[7] J. Winkle , C. T. Geo gakis, and G. Fische , “F e ing a igue beha io o high-
s eng h s eel monos ands unde bending load,” In . J. Fa igue, ol. 70, pp. 13–
23, 2015, doi: 10.1016/j.ij a igue.2014.08.009.
[8] P. J. Golden, “De elopmen o a do e ail e ing a igue ix u e o u bine
engine ma e ials,” In . J. Fa igue, ol. 31, no. 4, pp. 620–628, 2009, doi:
10.1016/j.ij a igue.2008.03.017.
[9] B. Al edsson, “F e ing a igue o a sh ink- i pin subjec ed o o a ing bending:
Expe imen s and simula ions,” In . J. Fa igue, ol. 31, no. 10, pp. 1559–1570,
2009, doi: 10.1016/j.ij a igue.2009.04.019.
[10] Y. B. Zhang, L. T. Lu, L. Zou, D. F. Zeng, and J. W. Zhang, “Fini e elemen
simula ion o he in luence o e ing wea on e ing c ack ini ia ion in p ess-
i ed sha unde o a ing bending,” Wea , ol. 400, pp. 117–183, 2018, doi:
10.1016/j.wea .2018.01.008.
[11] L. Be ini and C. San us, “F e ing a igue es s on sh ink- i specimens and
in es iga ions in o he s eng h enhancemen induced by deep olling,” In . J.
Fa igue, ol. 81, pp. 179–190, 2015, doi: 10.1016/j.ij a igue.2015.08.007.
[12] C. Jiménez-Peña, R. H. Talemi, B. Rossi, and D. Deb uyne, “In es iga ions on
he e ing a igue ailu e mechanism o bol ed join s in high s eng h s eel
subjec ed o di e en le els o p e- ension,” T ibol. In ., ol. 108, pp. 128–140,
2017, doi: 10.1016/j. iboin .2016.11.014.
[13] Y. Zhang, L. Lu, Y. Gong, J. Zhang, and D. Zeng, “F e ing wea -induced
e olu ion o su ace damage in p ess- i ed sha ,” Wea , ol. 384, pp. 131–141,
2017, doi: 10.1016/j.wea .2017.05.014.
[14] N. L. Pede sen, “On op imiza ion o in e e ence i assembly,” S uc .
Mul idiscip. Op im., ol. 54, no. 2, pp. 349–359, 2016, doi: 10.1007/s00158-016-
1419-0.
[15] A. Neyman, “The in luence o oil p ope ies on he e ing wea o mild s eel,”
Wea , ol. 152, no. 1, pp. 171–181, 1992, doi: 10.1016/0043-1648(92)90212-Q.
[16] R. B. Wa e house and A. J. T owsdale, “Residual s ess and su ace oughness in
e ing a igue,” J. Phys. D. Appl. Phys., ol. 25, no. 1A, p. A236, 1992, doi:
10.1088/0022-3727/25/1A/036.
[17] C. S. Mon oss, T. Wei, L. Ye, G. Cla k, and Y. W. Mai, “Lase shock p ocessing
and i s e ec s on mic os uc u e and p ope ies o me al alloys: A e iew,” In . J.
23
Fa igue, ol. 24, no. 10, pp. 1021–1036, 2002, doi: 10.1016/S0142-
1123(02)00022-1.
[18] G. H. Majzoobi, K. Azadikhah, and J. Nema i, “The e ec s o deep olling and
sho peening on e ing a igue esis ance o Aluminum-7075-T6,” Ma e . Sci.
Eng. A, ol. 516, no. (1-2), pp. 235–247, 2009, doi: 10.1016/j.msea.2009.03.020.
[19] V. Ma ín, J. Vázquez, C. Na a o, and J. Domínguez, “E ec o sho peening
esidual s esses and su ace oughness on e ing a igue s eng h o Al 7075-
T651,” T ibol. In ., ol. 142, no. Oc obe 2019, 2020, doi:
10.1016/j. iboin .2019.106004.
[20] S. Be e a and S. Romano, “A compa ison o a igue s eng h sensi i i y o
de ec s o ma e ials manu ac u ed by AM o adi ional p ocesses,” In . J.
Fa igue, ol. 94, pp. 178–191, 2017, doi: 10.1016/j.ij a igue.2016.06.020.
[21] K. Solbe g, S. Guan, S. M. J. Raza i, T. Welo, K. C. Chan, and F. Be o,
“Fa igue o addi i ely manu ac u ed 316L s ainless s eel: The in luence o
po osi y and su ace oughness,” Fa igue F ac . Eng. Ma e . S uc ., ol. 42, no.
9, pp. 2043–2052, 2019, doi: 10.1111/ e.13077.
[22] D. E ena, J. Vázquez, C. Na a o, and J. Domínguez, “Nume ical analysis o
o oidal oids as s ess elie e s in sh ink- i ed sha s,” T ibol. In ., ol. 143,
2020, doi: 10.1016/j. iboin .2019.105996.
[23] D. E ena, J. Vázquez, C. Na a o, and J. Domínguez, “Voids as s ess elie e s
and a pallia i e in e ing,” Fa igue F ac . Eng. Ma e . S uc ., ol. 41, no. 12,
2018, doi: 10.1111/ e.12849.
[24] T. Bely schko and T. Black, “Elas ic c ack g ow h in ini e elemen s wi h
minimal emeshing,” In . J. Nume . Me hods Eng., ol. 45, no. 5, pp. 601–620,
1999, doi: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-
NME598>3.0.CO;2-S.
[25] R. Hojja i-Talemi, M. Abdel Wahab, J. De Pauw, and P. De Bae s, “P edic ion o
e ing a igue c ack ini ia ion and p opaga ion li e ime o cylind ical con ac
con igu a ion,” T ibol. In ., ol. 76, pp. 73–91, 2014, doi:
10.1016/j. iboin .2014.02.017.
[26] E. Gine , M. Sabsabi, J. J. Ródenas, and F. Ja ie Fuenmayo , “Di ec ion o
c ack p opaga ion in a comple e con ac e ing- a igue p oblem,” In . J. Fa igue,
ol. 58, pp. 172–180, 2014, doi: 10.1016/j.ij a igue.2013.03.001.
[27] L. Bohó quez, J. Vázquez, C. Na a o, and J. Domínguez, “On he p edic ion o
24
he c ack ini ia ion pa h in e ing a igue,” Theo . Appl. F ac . Mech., ol. 99,
no. Decembe 2018, pp. 140–146, 2019, doi: 10.1016/j. a mec.2018.11.014.
[28] J. Vázquez, C. Na a o, and J. Domínguez, “Analysis o e ing a igue ini ial
c ack pa h in Al7075-T651 using cylind ical con ac ,” T ibol. In ., ol. 108, pp.
87–94, 2017, doi: 10.1016/j. iboin .2016.09.023.
[29] K. N. Smi h, P. Wa son, and T. H. Toppe , “A s ess-s ain unc ion o he
a igue o me als,” J. Ma e ., ol. 5, no. 4, pp. 767–778, 1970.
[30] P. Pa is and F. E dogan, “A C i ical Analysis o C ack P opaga ion Laws,” J.
Basic Eng., ol. 85, no. 4, pp. 528–533, 1963, doi: 10.1115/1.3656900.
[31] G. B. Ma quis and D. F. Socie, Mul iaxial Fa igue. Socie y o Au omo i e
Enginee s, 2010.
[32] F. E dogan and G. C. Sih, “On he C ack Ex ension in Pla es Unde Plane
Loading and T ans e se Shea ,” J. Basic Eng., ol. 85, no. 4, pp. 519–525, 1963,
doi: 10.1115/1.3656897.
[33] K. Hayashi and S. Nema -Nasse , “Ene gy-Release Ra e and C ack Kinking
Unde Combined Loading,” J. Appl. Mech., ol. 48, no. 3, pp. 520–524, 1981,
doi: 10.1115/1.3157666.
[34] M. Hussain, S. Pu, and J. Unde wood, “S ain Ene gy Release Ra e o a C ack
Unde Combined Mode I and Mode II,” in Na ional Symposium on F ac u e
Mechanics, 2009.
[35] B. Co e ell and J. R. Rice, “Sligh ly cu ed o kinked c acks,” In . J. F ac ., ol.
16, no. 2, pp. 155–169, 1980, doi: 10.1007/BF00012619.
[36] M. C. Dubou g and V. Lamacq, “S age II c ack p opaga ion di ec ion
de e mina ion unde e ing a igue loading: A new app oach in acco dance wi h
expe imen al obse a ions,” in ASTM Special Technical Publica ion, 2000, doi:
10.1520/s p14746s.
[37] R. Ribeaucou , M. C. Baie o-Dubou g, and A. G a ouil, “A new a igue
ic ional con ac c ack p opaga ion model wi h he coupled X-FEM/LATIN
me hod,” Compu . Me hods Appl. Mech. Eng., ol. 196, no. 33–34, pp. 3230–
3247, 2007, doi: 10.1016/j.cma.2007.03.004.
[38] D. In an e-Ga cía, E. Gine , H. Miguélez, and M. Abdel Wahab, “Nume ical
analysis o he in luence o mic o- oids on e ing a igue c ack ini ia ion
li e ime,” T ibol. In ., ol. 135, pp. 121–129, 2019, doi:
10.1016/j. iboin .2019.02.032.