Journal of Applied Mathematics and Physics, 2013, 1, 48-53
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.16010
Open Access JAMP
Analysis of Fretting Fatigue Crack Initiation in a Riveted
Two Aluminum Specimen*
Li Huan1,2, Guo Ran2#, Cheng He Ming2, Wei Yeqing3
1College of Polytechnics, Yunnan Agriculture University, Kunming, China
2Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, China
3Yunnan Traffic Co., Ltd., Kunming, China
Received October 2013
ABSTRACT
Based on the existing experiment resu lts, the fretting fatigue contact geometry of a riveted two aluminum specimen was
studied using the finite element method. The contact stress fields of the inner and outer contact edges on the two speci-
men’s up and down surface under different contact friction coefficient and the fatigue loads were analyzed, the influ-
ences of the contact friction coefficient and remote stress on crack initiation were discussed. The results were in well
agreement with the observations of the existing experiments, that is, the crack initiated places of the first aluminum
specimen change from the area of 900 to 450, and the crack initiated places of the second aluminum specimen change
from the area of 900 to 1350 with the increase of the friction coefficient and the remote stress.
Keywords: A Riveted Two Aluminum Specimen; Crack Initiation; Fr ictio nal Coefficient; Fatigue Loads
1. Introduction
Fretting fatigue is one of the main reason s for the failure
of structure components, even the causes of many major
accidents in the fields of aviation, transportation and
mechanics etc. The fretting fatigue begins with the wear
and sometimes corrosion damage at the asperities of the
contact surfaces of the riveted components, which will
further induce the initiation and propagation of micro-
cracks. With the appearance of the micro-cracks, the fa-
tigue strength of the riveted components will significant-
ly reduce, leading to the decrease of the components’
service life [1]. Actually, the fre tting fatigue is a damage
process which is induced by the cyclic stress that works
on the asperities of the materials’ near -surface and causes
the locally permanent structure deformations on the sur-
face [2]. Under the action of the cyclic stress, the com-
ponents immediately enter into a fatigue development
process. The results of the damage accumulation during
this process are the initiation and propagation of the
cracks, following with the final fracture to end the
process. The researches of fretting fatigue are very sig-
nificant to ease fretting damage industry today. Due to
the complicate deformation process and the difficulties
on current experimental measurements, the researches of
the stress states in the contact area are mainly resorted to
the numerical simulation methods. For example, M.A.M
cCarthy and C.T.M cCarthy etc [3] have constructed the
FEM model of a 3-D bolt-alaminated plate to analyze the
plate’s contact stress. Mutoh, ect [4] analyzed the stress
distribution in the contact areas of a specimen contacting
with a rectangular fretting pad, based on which the fret-
ting fatigue life of the specimen was studied by using
fracture mechanics theories and was found to be well
consistent with the experimental resu lts.
Thus, following the methods and theories employed by
Mutoh [4], a riveted two aluminum plate specimen was
constructed and studied. The distributions of the normal
stress and shear stress in the contact area between the
two contacting aluminum plates were analyzed to find
out the posi tions of the ma ximum st ress a nd dis plac eme nt.
2. The Analysis of Influence Factors on
Fretting Fatigue
There are many factors influencing fretting fatigue, in-
cluding all kinds of fretting parameters, displacement
amplitude and environmental factors etc [5-8], the driv-
ing force of crack initiation is micro power, including
tangential force caused by surface friction, the tensile
stress and shear stress in the component body, and tensile
stress and shear stress generated by contact pressure
(namely the clamping force). The quantitative formula of
*Project supported by the National Natu ral Science Foundation of Chi-
na (Grant No. 11072092 and 11262007).
#Corresponding author.

H. LI ET AL.
Open Access JAMP
49
driving force of crack growth [9] is:
( )
fff 2
P01eS K σ σµ
−
= −−
(1)
Type: µ is the contact surface friction coefficient, P0 is
the contact pressure (Pa), S is the fretting slip amplitude
(M), K is formula constant having a length dimension
(m). Specimens of stress which fatigue crack initiation
required is
σ
f, load stress which lead to specimen’s fati-
gue crack is σ
ff.
σ
ff is the component's fatigue strength
fretting and
σ
f is the component’s fatigue strength no
fretting. Fatigue strength in a micro component is smaller
than which without fretting, which can be seen from this
formula, and the difference between the two depends on
P0,
µ and S, that is to say the fretting fatigue strength is
related to friction coefficient, contact pressure and slip
amplitude and so on, yet the slip amplitude is directly
related to fatigue loading. So, based on the above men-
tioned theory, the fretting fatigue contact geometry of a
riveted two aluminum specimen was studied using the
finite element method. The contact stress fields of the
inner and outer contact edges on the two specimen’s up
and down surface under different contact friction coeffi-
cient and the fatigue loads were analyzed, the influences
of the contact friction coefficient and remote stress on
crack initiation and propagation mechanism were dis-
cussed.
3. Modeling
3.1. Computational Model
The 3D finite element model of the rived aluminum spe-
cimen and its meshing result are showed in Figure 1. In
order to reduce the computational cost, only half of the
FEM model is constructed according to the symmetries
of the specimen. The model is composed of 8 parts, in-
cluding two aluminum plates, one screw bolt, one protec-
tive sleeve, two screw caps and two gaskets. In order to
further reduce the model size and computational cost, the
reducing of the meshing numbers and contact areas is →
are often adopted in the FEM simulation. Thus, we treat
the screw bolt, the protective sleeve, the screw caps and
the gaskets as an integrated section to neglect the con-
tacts between those parts. Three contacts regions are in-
vestigated in the simulations as shown in Figure 2: The
first region is the contact area between the upper protec-
tive sleeve’s lower surface and the aluminum plate I’s
upper surface; the second region is the contact area be-
tween the two aluminum plates; the third region is the
area between the lower protective sleeve’s upper surface
and the aluminum plate II’s lower surface. Among these
three regions, the stress distributions of upper and lower
surfaces of the two aluminum plates are emphatically
analyzed to evaluate the specimen’s fatigue life. The
eight-node hexahedral solid elements are employed in the
Alu mi n um plate I I
Alu mi n um plate I
Figure 1. FEM model of the rived aluminum specimen.
Figure 2. Contact regions.
simulation, where the total element number and node
number are 10336 and 13686. The contact area of the
two aluminum plates is 219.142 mm2 with a contact
width of 4.5 mm. The hole radius and thickness are set as
5.5 mm and 6 mm, respectively. The length and width
are respective 230 mm and 60 mm. The aluminum plates’
longitudinal axis is axis X, forward direction points to
longitudinal remote end. Transverse axis is axis Y, the
screw bolt’s axis is axis Z, origin of coordinates is lo-
cated in the centre of the hole. Normal chain bar con-
straints are exerted as boundary condition in fornter of
the model (y = 0).
3.2. Mechanics of Materia l Constants
In computing object this time, expect the two aluminum
plates whose Young’s modulus E and Poisson’s ratio
µ
are 40 GPa and 0.3, all the other parts of the specimen
are C45 steels with E = 210 GPa and
µ = 0.3. Due to the
elastic stress states of the screw bolt during its service
process, the screw bolt is regarded as an elastic material
in current simulation while the plasticity is taken into
consideration for all the other parts of the specimen. Be-
cause the analytical objects are connected components in
the fields of aviation and high speed train systems etc, so
the work temperature is really the same as environmental
temperature. Material temperatu re effect is not taken into
consideration. Material constants of every component are
showed in Table 1.
3.3. Computational Method
In order to simulate the fastening process of the screw
bolt, the FEM model with a clearance of d between the
spaces of the two gaskets and the aluminum plate’s

H. LI ET AL.
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50
Table 1. The mechanical parameters of the parts.
parts protective sleeve, screw
caps and gaskets screw bolt specimen
material C45 steels
(elastoplasticity) C45 steels
(elasticity) aluminum
Young’s modulus E 210 GPa 210 GPa 70 GPa
Poisson’s ratio µ 0.3 0.3 0.3
thickness is constructed firstly. Through the adjustment
of d, different fastening forces could be simulated. The
fastening process is realized through the following two
steps: on the first step, the tension load is applied on the
screw bolt to make the gaskets and the plates separate; on
the second step, the tension load is released gradually to
make the separation recover. During this recovery pro-
cess, the gaskets would gradually fasten the aluminum
plates when the contact relations between them are taken
into consideration. The fatigue load is applied on the
right end of the aluminum plate with the maximum stress
of s = 200 MPa and fatigue load str es s ratio of R = 0.1.
The contact stresses are studied using the direct con-
straint method, which would accurately track the move-
ment of the contact bodies and the occurrence of the
contact [10]. Once the contact occurs, the direct con-
straint method woul d directly adjust the displacements of
the contact nodes by modifying their boundary condi-
tions.
4. Results and Discussion
The contact surface is in compressive stress state when
the shear stress of nodes in contact areas is negative val-
ue, which makes against fatigue crack initiation and
propagation. The contact surface is in tensile stress state
when the shear stress of nodes in contact areas is positive
value, which provides the advantageous condition for
fatigue crack initiation and propagation. In consideration
of the larger relative slip distance of corresponding nodes
in contact edges, The variational condition of the cir-
cumferential normal stress fields of the inner contact
edges on the two specimen’s up and down surface (this
region are mainly subject to ordinary fatigue) and the
shear stress fields of the outer contact edges on the two
specimen’s up and down surface (fretting fatigue region)
are analyzed. Contact area is showed in Figure 3. The
distributions of stress amplitude in the inner and outer
circle contact edges of the aluminum plates under differ-
ent friction coefficient and fatigue load conditions are
discussed as follow.
4.1. Influence of Friction Coefficient on the
Initiation of Cracks
To investigate the influence of friction coefficient on the
Aluminum plate I
Fatigue load
Contact area
θ
x
y
①
②
(a)
Aluminum plate II
Fatigue load
Contact area
θ
x
y
①
②
(b)
Figure 3. Contact area. ① The outer circle contact edges;
② The inn er circle contac t edges.
initiation of cracks in the contact region, the fastening
force and fatigue stress are set constant as 5.5 KN and
200 MPa, while three different friction coefficients, 0.35 ,
0.5 and 0.7, are considered. The maximum and minimum
of the normal stress and shear stress on the aluminum
plate I’s lower surface and aluminum plate II’s upper
surface during one cycle of fatigue loading are analyzed
to obtain the stress amplitudes (
σ
max −
σ
min and
τ
max −
τ
min). The distributions of the stress amplitudes along the
inner and outer circle contact edges of the two plates
contact areas are calculated and shown in Figure 4.
Based on Figure 4, the influence of friction coefficient
on the stress amplitude can be obtained by analyzing the
extreme values in those curves in Figure 4.
It can be seen from Figures 4(b) and (d) that the maxi-
mum normal stress amplitudes ∆
σ
max of the inner contact
edges of the two aluminum plates loc ate at the about 90˚
from the fatigue loading direction. However, it is inter-
esting to find that the positions of ∆
σ
max on the plate I
tend to change slightly toward the fatigue loading direc-
tion, while those on the plate II are just opposite, which
tend to change slightly away from the fatigue loading
direction. For both two plates, ∆σmax decrease with the
increasing of friction coefficient. It indicates that the in-
creasing of friction coefficient, on the one hand, would
weaken the role of traditional fatigue damage on the
aluminum plates and delay the corresponding crack initi-
ation at the 90˚ region near the hole; and on the other
hand, would strengthen the role of fretting fatigue damage
and lead to the transition of the crack initiation positions
from 90˚ point at the inner edge to 45˚ point at the outer
edge for plate I and from 90˚ point at the inner edge to
135˚ point at the outer edge for plate II.

H. LI ET AL.
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51
020406080100 120 140 160 180 200
-150
-120
-90
-60
-30
0
30
60
90
120
150
stress amplitudes （ MPa）
angle （deg ree）
∆τ(ƒ=0.35)
∆τ(ƒ=0.5)
∆τ(ƒ=0.7)
020406080100 120 140 160 180 200
-200
-100
0
100
200
300
400
500
stress amplitudes （ MPa）
angle （degree）
∆σ(ƒ=0.35)
∆σ(ƒ=0.5)
∆σ(ƒ=0.7)
(a) (b)
020406080100120 140 160 180 200
-150
-120
-90
-60
-30
0
30
60
90
120
stress amplitudes （ MPa）
angle （degree）
∆τ (f=0.35)
∆τ (f=0.5)
∆τ (f=0.7)
020406080100120 140 160 180 200
-200
-100
0
100
200
300
400
stress amplitudes （ MPa）
angle （degree）
∆σ(ƒ=0.35)
∆σ(ƒ=0.5)
∆σ(ƒ=0.7)
(c) (d)
Figure 4. The distributions of stress amplitude in the inner and outer circle contact edges of the aluminum plates under dif-
ferent friction coefficient conditions. (a) The outer circle contact edges of the aluminum plate I’s upper surface; (b) The inner
circle contact edges of the aluminum plate I’s upper surface; (c) The outer circle contact edges of the aluminum plate II’s
lower su rface; (d) The inner circle contact edges of the aluminum plate II’s lower surface.
It can be observed from Figures 4(a) and (c) that he
maximum shear stress amplitudes ∆τmax of the outer con-
tact edges of plate I increase with the friction coefficient
at 0˚ - 90˚ region around the hole while decrease with the
friction coefficient at 90˚ - 180˚ region around the hole.
Thus, when the fastening force and fatigue stress are
fixed, increasing of the friction coefficient would in-
crease the friction force at 45˚ region of the plate I, im-
prove the driving force to initiate and propagate the
cracks and strengthen the role of fretting fatigue damage
at this region. The situation is just opposite for plate II,
∆
τ
max decrease with the friction coefficient at 0˚ - 90˚
region while increase with the friction coefficient at 90˚ -
1800˚ region. Thus , when the fastening force and fatigue
stress are fixed, increasing of the friction coefficient
would strengthen the role of fretting fatigue damage at
135˚ region. From the above discussion, we can reach the
conclusion that increasing the friction coefficient would
result in the initiation of cracks at 45˚ region of plate I’s
outer contact edge and the initiation of cracks at 135˚
region of plate II’s out er contact edge.
4.2. Influence of Fatigue Load on the Initiation
of Cracks
To investigate the influence of fatigue load on the initia-
tion of cracks in the contact region, the fastening force
and friction coefficient are set constant as 4 KN and 0.5,
while three different f atigue stresses, 150 MPa, 20 0 MPa
and 250 MP a, are considered. Following the same analy-
sis process in Section 4.1, the distributions of the stress
amplitudes and along the inner and outer circle contact
edges of the two plates contact areas are calculated and
shown in Figur e 5.
From Figures 5(b) and (d), it is easy to find that the
maximum normal stress amplitudes ∆ σ
max of the inner
contact edges of the two aluminum plates locate at about
90˚ region near the hole, and the position of ∆
σ
max grad-
ually shifts to the region whose angle is less than 90˚ and
more than 90˚ as fatigue stress increases, which demon-

H. LI ET AL.
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52
020406080100 120 140 160 180 200
50
20
90
60
30
0
30
60
90
20
50
（ MPa）
angle （deg ree）
∆τ (f=0.35)
∆τ (f=0.5)
∆τ (f=0.7)
020406080100 120 140 160 180 200
00
00
00
0
00
00
00
00
00
00
（ MPa）
angle （deg ree）
∆σ(f=0.35)
∆σ(f=0.5)
∆σ(f=0.7)
(a) (b)
020406080100 120 140 160 180 200
50
20
90
60
30
0
30
60
90
20
50
（ MPa）
angle （degree）
()
∆τ (f=0.5)
∆τ (f=0.7)
020406080100 120 140 160 180 200
00
00
00
0
00
00
00
00
00
00
（ MPa）
angle （degree）
∆σ(f 0.35)
∆σ(f=0.5)
∆σ(f=0.7)
(c) (d)
Figure 5. The distributions of stress amplitude in the inner and outer circle contact edges of the aluminum plates under dif-
ferent fatigue stress conditions; (a) The outer circle contact edges of the aluminum plate I’s upper surfac e; (b) The inner cir-
cle contact edges of the aluminum plate I’s upper surface; (c) The outer circle contact edges of the aluminum plate II’s upper
surface; (d) The inne r ci rcle contact edges of the aluminum pl ate II’s upper surface.
strates that when the traditional fatigue plays major role
for the failure of the specimen, increasing of fatigue
stress would accelerate the in itiation of the corresponding
fatigue cracks at the inner contact edge and shift the init-
iation position far away from its original 90˚ region.
For the maximum shear stress amplitude ∆
τ
max, we can
observe from Figures 5(a) and (c) that ∆
τ
max appear at
about 45˚ region and 135˚ region, respectively and they
both significantly increase with the fatigue stress.
Meanwhile, the position of ∆
τ
max can be seen to shift to
the region whose angle is less than 45˚ and the region
whose angle is greater than 135˚ with the fatigue stress. It
indicates that increasing of fatigue stress would streng-
then the role of the fretting fatigue damage at the outer
contact edges, causing the cracks tend to initiate at 45˚
region and 135˚ region at outer contact edges and in-
crease the tends to shift the crack initiation positions to-
ward the region whose angle is less than 45˚ or larger
than 135˚.
5. Discussions
The fretting fatigue contact geometry of a riveted two
aluminum specimen was studied. The distributions of
normal stress amplitude and shear stress amplitude in the
inner and outer circle contact edges of the aluminum
plates under different friction coefficient and fatigue load
conditions are elastically analyzed using the finite ele-
ment me thod and the following conclusions a re obtained.
1) In the light of traditional fatigue damage without
fretting damage, the main factor influencing fatigue is
hoop normal stress amplitude ∆
σ
max, the dangerous point
of traditional crack initiation is just the place w here hoop
normal stress amplitude ∆
σ
max reached the maximum.
2) The aluminum plate I’s fatigue crack initiation an-
gle shifts from 900 to 450 with the increasing of friction
coefficient and fatigue load, the aluminum plate II’s fa-
tigue crack initiation angle shifts from 900 to 1350 with
the increasing of friction coefficient and fatigue load.
The increasing of friction coefficient and fatigue load, on

H. LI ET AL.
Open Access JAMP
53
the one hand, would weaken the role of traditional fati-
gue damage on the aluminum plates and delay the cor-
responding crack initiation at the 90˚ region near the hole;
and on the other hand, would strengthen the role of fret-
ting fatigue damage and lead to the transition of the crack
initiation positions from 90˚ point at the inner edge to 45˚
point at the outer edge for plate I and from 90˚ point at
the inner edge to 135˚ point at the outer edge for plate II.
3) Either traditional fa tigue or fretting fatigue has been
accelerated with the increasing of fatigue loads, the in-
creasing of fatigue stress would increase the tends to shift
the initiation position far away from its original 90˚ re-
gion at the inner contact edge and shift the crack initia-
tion positions toward the region whose angle is less than
45˚ or larger than 135˚ at the outer contact edge.
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