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metals Article

Numerical Modeling of the Effect of Randomly Distributed Inclusions on Fretting Fatigue-Induced Stress in Metals Qingming Deng 1,2 , Nadeem Bhatti 2 , Xiaochun Yin 1 and Magd Abdel Wahab 3,4, * 1 2 3 4

*

Department of Mechanics and Engineering Science, Nanjing University of Science and Technology, Nanjing 210094, China; [email protected] (Q.D.); [email protected] (X.Y.) Department of Electrical energy, Metals, Mechanical Constructions, and Systems, Faculty of Engineering and Architecture, Ghent University, 9052 Zwijnaarde, Belgium; [email protected] Institute of Research and Development, Duy Tan University, 03 Quang Trung, Da Nang, Vietnam Soete Laboratory, Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde 903, B-9052 Zwijnaarde, Belgium Correspondence: [email protected]; Tel.: +32-9-331-04-81

Received: 10 September 2018; Accepted: 15 October 2018; Published: 17 October 2018







Abstract: The analysis of fretting fatigue plays an important role in many engineering fields. The presence of heterogeneity may affect the performance of a machine or a structure, including its lifetime and stability. In this paper, the effect of randomly distributed micro inclusions on the fretting fatigue behaviour of heterogeneous materials is analysed using the finite element method (FEM) for different sizes, shape and properties of inclusions. The effect of micro inclusions on macroscopic material properties is also considered by representative volume element (RVE). It is shown that the influence of micro inclusions on macroscopic material properties cannot be ignored, and the shape and size of the inclusions have less effect on the macroscopic material properties as compared to the material properties of inclusion and volume ratio. In addition, various parameters of inclusions have little effect on the peak tensile stress, which remains almost the same as homogeneous material. Peak shear stress occurs at many places inside the specimen, which can result in multiple cracking points inside the specimen, as well as at the contact surface. Moreover, the stress band formed by the stress coupling between adjacent inclusions may have an important influence on the direction of crack growth. Keywords: fretting fatigue; heterogeneous material; finite element method; inclusions

1. Introduction Fretting fatigue can significantly affect structural performance in many engineering applications [1]. The fatigue life of the structure may be reduced by the fretting phenomenon, which occurs due to small oscillatory motion between two contact surfaces [2]. The reduction in fatigue life can reach up to 50% [3]. The failure process is generally characterized by two phases, crack nucleation [4,5] and crack propagation [6,7]. Researchers are concerned with contact stresses because it directly affects the initiation of cracks [8]. The contact stresses are affected by various factors, such as loading magnitude, contact geometry, surface imperfections and slip amplitude, which consequently affect the failure process. On the other hand, material properties also have significant influence on contact stresses. By common consensus, heterogeneity of material has a great influence on the life and stability of the structure [9]. For many engineering components, the required design lifetime can be affected significantly by heterogeneity. Therefore, material heterogeneity has been widely studied in recent years [10–14]. Generally, materials contain fibers [15–19], particles [20,21], Metals 2018, 8, 836; doi:10.3390/met8100836

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precipitates, porosities, or voids/cracks [14,22] at the micro level, which are common causes of the heterogeneity. Generally, scanning electron microscopy (SEM) is used to observe the microstructure of material [23]. There are several kinds of typical inclusions in alloys; Al2 O3 , MgO, Al2 MgO4 , CaSO4, TiB2 , Al3 Ti and refractory brick (Al, Si, O). Typical geometric form of inclusions are particles, films or group of films and rods [24]. As a result, stress concentration will appear at the interface between different materials or voids, which may give rise to shorter fatigue life. Thus, in addition to contact area, the subsurface area can also be affected by the heterogeneity, and in some cases it is strongly affected [25]. There are many ways to analyze the behavior of heterogeneous materials. Previous researchers have proposed several models to predict the effect of inclusions and defects on metal fatigue strength. Murakami and Endo [25] proposed an engineering guide to predict the fatigue strength of components with heterogeneity. Some experimental results about high-strength steels showed that fatigue failures were mostly caused by the inclusions inside the matrix [26–31]. In short, both fretting fatigue and heterogeneity may significantly influence the lifetime and stability of mechanical components. Numerous factors can affect micro stress field, thus affecting the fretting fatigue behavior [8]. The macroscopic fatigue failure occurs due to the distribution of micro-stress. In the literature, homogeneous materials are widely assumed to study fretting fatigue problems [14]. Generally, the micro cracks are observed inside the slip zone and at the contact edge. For experimental methods, it is difficult to get the details of stress field and initiation of crack. Therefore, numerical modeling is an efficient way to solve the problem of fretting fatigue [8,32–38] and fracture [39–43]. However, only few studies have taken into account the heterogeneity of material under fretting fatigue conditions [14,44,45]. Kumar et al. used numerical analysis to study the influence of heterogeneity on stress distribution in fretting fatigue problems [14]. They studied heterogeneity by considering micro-voids in the material, which is found in metal alloys [46]. They found that the effect of heterogeneity on shear stress was greater than on normal stress. Normally, for homogeneous material, the peak shear stress appears at the contact interface. However, in the case of heterogeneous materials, the peak may shift to the micro-voids. In the practical situation, the materials always have some heterogeneity due to heat treatment process. The inclusions can be regarded as foreign particles, which exist in metals [47]. According to the relative position of the inclusions and the material surface, there can be surface inclusions and internal inclusions [48]. In this study, we focus on two kinds of common inclusions in aluminum alloy 2024-T3. Considering the effect of micro inclusions on macroscopic material properties by means of representative volume element (RVE), a numerical analysis is presented to study the influence of local randomly distributed inclusions on the stress distribution of fretting fatigue specimens. The influence of different variables, namely material properties, volume ratio and shape of the inclusions is investigated. This study presents the variation of different stresses at the contact interface, as well as under the surface, and thus allows predicting probable crack initiation sites. From the result of stress analysis, conclusions are drawn. 2. Line Contact under Partial Slip Before we study the influence of inclusions on fretting fatigue behavior in heterogeneous material, a brief review of Hertzian contact is presented here, which also can be used to verify the finite element (FE) model for homogenous material. The contact is said to be line contact if it is between a cylinder and a flat body (or two parallel cylinders). We have an analytical solution [49] to get the normal stress of the contact surface, if there is only a normal force F to compress the two bodies: r p( x ) = − pmax

x 2 1−( ) a

(1)

where p( x ) represents the normal stress distribution, a is the semi contact width and pmax the peak value of normal contact stress at the centre of the contact zone, and is given by [49] :

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pmax

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FE*  t R* r

(2)

FE∗ = flat specimen where t is the thickness of both cylinderpand and R * is the equivalent radius(2) and max the tπR∗ * E is the equivalent modulus of elasticity, i.e.,: where t is the thickness of both cylinder and the flat specimen and R∗ is the equivalent radius and E∗ 1 1 1 is the equivalent modulus of elasticity, i.e.,:   (3) * R1 R2 R 1 1 1 = + (3) ∗ 1  v12 R11 v22 R2 1 R   (4) E *1 E11 − v2E2 1 − v2 1 2 = + (4) E∗ 𝑅 isEthe E2 of the specimen, which is flat for our 1 𝑅 is the radius of the pad and radius 1

2

the𝐸pad andYoung’s R2 is themoduli, radius ofand the specimen, is flat for our configuration. R1 is the radius configuration. 𝐸1 of and 𝑣1 and 𝑣2 which are Poisson’s ratios for the two 2 are E1bodies, and E2respectively. are Young’sThe moduli, and v1𝑎and v2 are Poisson’s ratios for the two bodies, respectively. parameter is given by: The parameter a is given by: r FR* ∗ a a2= 2 * FR (5)(5) t E tπE∗ Figure 1 shows thethe normal stress distribution ofof thethe contact interface of of thethe specimen. On the Figure 1 shows normal stress distribution contact interface specimen. On the whole contact surface (between -1 ≤ x/a ≤ 1), it is a parabolic distribution. whole contact surface (between ‒1 ≤ x/a ≤ 1), it is a parabolic distribution. 30

Normal stress(MPa)

0 -30 -60 -90 -120 -150 -180 -210 -2

-1

0 x/a

1

2

Figure 1. 1. AnAn example of of normal stress distribution of of thethe contact surface. Figure example normal stress distribution contact surface.

ToTo illustrate thethe effects of tangential loading Q a modified formulation is required [50,51]. Near the illustrate effects of tangential loading 𝑄 a modified formulation is required [50,51]. Near contact edge, c ≤ x < a, this is the slip zone and there will be relative sliding between the contact | | the contact edge, 𝑐 ≤ |𝑥| < 𝑎, this is the slip zone and there will be relative sliding between the interfaces. Therefore,Therefore, we can express shear the traction µp( x ). as Whereas, near the contact contact interfaces. we canthe express shearas traction 𝜇𝑝(𝑥). Whereas, near thecentre, contact is a centrally symmetrical stick zone (indicated as stick zone 1 in Figure 2), means2), | xcentre, | < c, there |𝑥| < 𝑐, there is a centrally symmetrical stick zone (indicated as stick zone which 1 in Figure that the contact surfaces will move together. thattogether. in this zone the shear q(shear x ) does not reach which means that the contact surfaces willSo move So that in thistraction zone the traction 𝑞(𝑥) the critical value µp x . ( ) does not reach the critical value 𝜇𝑝(𝑥). The contact shear traction can bebe modeled byby thethe Coulomb friction law, which is given by:by: The contact shear traction can modeled Coulomb friction law, which is given q ( x ) = q ( x ) + q2 ( x )

q( x)  q1 ( x)  q12 ( x)

(6) (6)

where q1 (qx )( x= µp( x ). q represents shear traction due to slip. q is added as a perturbation in the where 1 )   p( x) . 𝑞1 1 represents shear traction due to slip. 2𝑞2 is added as a perturbation in the solution of slip condition, to represent shear traction in the stick zone. solution of slip condition, to represent shear traction in the stick zone. r  x 2 q1 ( x ) = −µpmax 21 − (7) a x (7) q1 ( x)    pmax 1    a  Moreover, in the slip zones, the perturbation q2 ( x ) is zero.

Moreover, in the slip zones, the perturbation q2 ( x) is zero. q2 ( x ) = 0, c ≤ | x | ≤ a

q2 ( x)  0, c  x  a

(8)

(8)

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F Q

e -a

Stick zone 2

-c

Stick zone 1

c

a

-Q F Figure2.2.Slip Slipand andstick stickregions regionsfor forthe thecontact contactof oftwo twoparallel parallelcylinders. cylinders. Figure

However, for the stick zones, it is given by: However, for the stick zones, it is given by: r  x 2 2 c c  q2 ( x ) = µpmax x 1 − , |x| ≤ c (9) (9) q2 ( x)   pmax 1 a  , x c c a c where: s where: c Q = 1− (10) a µF c Q  F1 and tangential loads Q simultaneously, the shear stress (10) Finally, when there are normal loads a F distribution on the contact surface is expressed as: Finally, when there are  normal loads 𝐹 q and tangential loads 𝑄 simultaneously, the shear
 x 2  1 − , c < x| < a − µp | stress distribution on the contact surface is expressed as:a max  q q (11) q( x ) =

x 2 c x 2  − µp 1 − − 1 − , |x| ≤ c 2  max  a a c  x    pmax 1    , c  x  a  a Furthermore, there q( xalso )  may be bulk stresses σaxial acting on the body, thus the perturbation shear (11) 2 2   traction q2 ( x ) is given by [52]:   x c  x     pmax  1   a s a 1   c   , x  c      2   c x+e q2 ( x ) = µpmax 1− , | x + e| ≤ c (12) a c acting on the body, thus the perturbation Furthermore, there also may be bulk stresses 𝜎𝑎𝑥𝑖𝑎𝑙 shear traction 𝑞2 (𝑥)is given by [52]: aσaxial In the above formula, there is an eccentric displacement e = 4µp in the stick zones (the stick max 2 zone changes into stick zone 2) as shown c in Figure  x  e 2. For the slip zone, the shear traction still follows (12) q2 ( x)   pmax 1   , xe  c the Coulomb friction model q( x ) = µp(ax ). Therefore,  c  the shear stress is given by:  q
 a axial  −µpmax 1 − xa 2 , − a ≤ x < −ce− −e < x ≤ a  e&c In the above formula, there is an eccentric displacement  p in the stick zones (the stick q q 4  (13) q( x ) =

2 e 2 max  1 − xa − ac 1 − x+ , | x + e| ≤ c  −µpmax c zone changes into stick zone 2) as shown in Figure 2. For the slip zone, the shear traction still follows the Coulomb friction model q( x)   p( x) . Therefore, the shear stress is given by: An example of the above analytical solutions is shown in Figure 3. 2    pmax 1   x  ,  a  x  c  e & c  e  x  a  a q( x)   (13) 2 2     x c  xe      pmax  1   a   a 1   c   , x  e  c       

An example of the above analytical solutions is shown in Figure 3.

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Shear stress(MPa) Shear stress(MPa) Shear stress(MPa)

90 90 60 60 90 30 30 60 0 0 30 -30 -30 0 -60 -60 -30 -90 -90 -60 -120 -120 -90 -150 -150-120 -180 -180-150 -2 -2 -180 -2

q1(x) q1(x) q(x) q (x) q(x) q2(x)1 q2(x) q(x) q2(x)

-(c+e)/a -(c+e)/a -(c+e)/a

(c-e)/a (c-e)/a 1 (c-e)/a2

-1 -1

0 0 1 2 x/a -1 x/a 0 1 2 x/a Figure 3. Shear stress distribution at the contact interface.

Figure 3. Shear stress distribution at the contact interface. Figure 3. 3. Shear Shear stress stress distribution distribution at at the the contact contact interface. interface. Figure

3. Finite Element Model and Validation 3. Finite Element Model andand Validation 3. Finite Element Model Validation 3. Finite Element Model and From Figure 4, we can see Validation the experimental setup by a schematic view [53] for contact of two From Figure 4, we cancan seesee thethe experimental setup by by a schematic view [53][53] for for contact of two From Figure 4, awe experimental setup a schematic view contact of two cylindrical pads and flat specimen. Under the action of anormal loadview F, the two fretting pads From Figure 4, we can see the experimental setup by schematic [53] for contact of two cylindrical pads and a aflat specimen. Underthe theaction actionof of normal load F, two the fretting two fretting pads cylindrical pads and flat specimen. Under normal load F, the pads maintain maintain the contact. In aaddition, the coefficient friction between theload contact surfaces is 0.65 [53]. cylindrical pads and flat specimen. Underofof the actionbetween of normal F, the two isfretting pads maintain the contact. In addition, the coefficient friction the contact 0.65 [53]. the contact. In addition, the coefficient of friction between the contact surfacessurfaces is 0.65 [53]. The cyclic The cyclic axial load  axial is acting on right sideofoffriction the specimen. attached to the maintain the contact. In addition, thethe coefficient betweenTwo the springs contact are surfaces is 0.65 [53]. Theaxial cyclic axial load on the side of the specimen. Twoare springs are to attached to thepad, load σaxial is acting onacting the right sideright of the specimen. Two springs attached the fretting axial is fretting pad,axial which willgenerate the tangential the combined effect of these loads,toso The cyclic load is acting on the rightload sideQ, ofunder the specimen. Two springs are attached the axial which willwhich generate tangential load Q, under combined effect of these so that fretting fretting pad, willthe generate the tangential load the Q, under the combined effectloads, of these loads, so that fretting fatigue will occur around the contact load area. Q, under the combined effect of these loads, so pad, which will generate thethe tangential fatigue will occur around the contact area. thatfretting fretting fatigue will occur around contact area. that fretting fatigue will occur around the contact area.

F F

F

Q Q Q

Pad Pad Pad

axial axial  axial

Specimen Specimen Specimen y y

Q Q

yxx x

Q F F F

Figure 4. Schematic view of a fretting fatigue experiment. Figure 4. Schematic view of aof fretting fatigue experiment. Figure 4. Schematic view a fretting fatigue experiment. Figure 4. Schematic view of a fretting fatigue experiment.

As geometries and loads are symmetric, a simplified model of of thethe structure can bebe constructed, geometries loads symmetric, a simplified model structure constructed, As As geometries andand loads are are symmetric, a simplified model of the structure cancan be constructed, which is one pad and half of the specimen. In the same way as previous research [8,14,35,38], we can As geometries and loads are symmetric, a simplified model of the structure can be constructed, which is one pad and half of the specimen. In the same way as previous research [8,14,35,38], which is one pad and half of the specimen. In the same way as previous research [8,14,35,38], we we cancan model the load and boundary conditions as shown in Figure 5. which is one pad and half of the specimen. In the same way as previous research [8,14,35,38], we can model load boundary conditions as shown in Figure model thethe load andand boundary conditions as shown in Figure 5. 5. model the load and boundary conditions as shown in Figure 5. 10 mm 10 mm 10Fmm

F

10 mm 1010mm mm

 axial  axial  axial

5mm 5mm 5mm

 reaction  reaction  reaction

F

40 mm 40 mm Figure 5. Configuration of 40 themm fretting fatigue numerical model.

Figure 5. Configuration of the fretting fatigue numerical model. Figure 5. Configuration of the fretting fatigue numerical model. Figure 5. Configuration of the fretting fatigue numerical model.

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The radius of the pad and width of the specimen are taken from [53]. As shown in Figure 5, the The radius of the pad and width of the specimen are taken from [53]. As shown in Figure 5, length of the specimen is 40 mm and the width is equal to 5 mm. In addition, the thickness of two the length of the specimen is 40 mm and the width is equal to 5 mm. In addition, the thickness of two bodies and the radius of the cylindrical pad are 4 mm and 50 mm, respectively. At the top surface of bodies and the radius of the cylindrical pad are 4 mm and 50 mm, respectively. At the top surface of the cylindrical pad, a normal load F is applied. The pad is restrained from both sides in the the cylindrical pad, a normal load F is applied. The pad is restrained from both sides in the x-direction. x-direction. On the right hand-side and left hand-side of the specimen, an axial stress  axial and a On the right hand-side and left hand-side of the specimen, an axial stress σaxial and a reaction stress   stress are applied, respectively. The value of by is given by [54]: σreaction applied, respectively. The value of σreaction is given [54]: reaction reaction reaction are

Q Q  reaction   axial σreaction = -σaxial − As

(14) (14)

As

whereQQisisthe thetangential tangentialforce forcebetween between the contact surfaces. addition, cross sectional of where the contact surfaces. In In addition, thethe cross sectional areaarea of the specimen is expressed as A . The bottom side of the specimen is fixed in the y-direction. In order to the specimen is expressed sas As . The bottom side of the specimen is fixed in the y-direction. In verify ourverify FE model and studyand the study effect the of inclusions on frettingon fatigue, thefatigue, experimental data used order to our FE model effect of inclusions fretting the experimental in this paper is taken from the work of Talemi and Wahab [53]. In this study, both in validation models data used in this paper is taken from the work of Talemi and Wahab [53]. In this study, both in and in parametric the FF2 [53]studies, load case been = 543N, σaxial = 115 FMPa and validation modelsstudies, and in parametric thehas FF2 [53]used, load with case Fhas been used, with = 543N, Q = 186.25 N. max axial = 115 MPa and Qmax = 186.25 N. Both pad and specimen are made of aluminum alloy 2024-T3, which are widely used in the Both pad and specimen are made of aluminum alloy 2024-T3, which are widely used in the aviation field. Here, we choose two kinds of inclusions Al2 CuMg and Al2 O3 that are very commonly aviation field. Here, we choose two kinds of inclusions Al2CuMg and Al2O3 that are very commonly embedded in aluminum alloy 2024-T3 [55]. Their SEM observations showed that inclusions in embedded in aluminum alloy 2024-T3 [55]. Their SEM observations showed that inclusions in material are highly discrete and randomly distributed. So we model the heterogeneity of materials by material are highly discrete and randomly distributed. So we model the heterogeneity of materials representative volume element method using DIGIMAT-FE which is a tool that considers the effects of by representative volume element method using DIGIMAT-FE which is a tool that considers the microstructure on macroscopic material properties as shown in Figure 6. It should be noted that this effects of microstructure on macroscopic material properties as shown in Figure 6. It should be noted study does not consider the heterogeneity of cylindrical pad material. that this study does not consider the heterogeneity of cylindrical pad material. Inclusions

RVE

Equivalent homogeneous material

Figure Figure 6. 6. Schematic of representative volume element element (RVE) (RVE) method. method.

In this way, we can get the macro material properties (equivalent elastic modulus E E∗* and In this way, we can get the macro material properties (equivalent elastic modulus and equivalent Poisson’s ratio u∗ )* that consider the effect of microscopic inclusions [56–59]. Here, we just  ) that consider the effect of microscopic inclusions [56–59]. Here, we equivalent ratio response, consider thePoisson’s elastic material because for all specimens with inclusions, von-Mises stress just consider thethe elastic response, becauseconditions. for all specimens with inclusions, von-Mises is always below yieldmaterial limit under such loading This is common in fretting fatigue stress is always below the yield limit under such loading conditions. This is common in fretting problems. According to the previous literature, the original material properties of aluminum alloy fatigue problems. According to Al the2 O previous literature, the original material properties of aluminum 2024-T3 [53], Al2 CuMg [60] and 3 [61] in this paper are given in Table 1. This article assumes that alloy 2024-T3 pad [53],isAl CuMg [60] and Al2O3 [61] in 2024-T3. this paper are given in Table 1. This article the cylindrical a 2homogeneous aluminum alloy assumes that the cylindrical pad is a homogeneous aluminum alloy 2024-T3. Table 1. The original material properties involved in this paper. Table 1.The original material properties involved in this paper. Material Modulus (GPa) Poisson’s Ratio

Material

Modulus (GPa)

Poisson’s Ratio

Αl 2 CuMg

120.5

0.2

Αl 2 Ο3

380

Aluminum alloy 2024-T3 Aluminum alloyAl2024-T3 2 CuMg Al2 O3

72.1 120.5 72.1 380

0.33 0.2 0.33 0.2

0.2

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According to the SEM study by Merati [55], here 2%, 4%, and 6% volume ratio υ between 7 of 20 inclusions and matrix material was chosen. From the experimental observation by Hashimoto et al. [62] and other FEM research about the inclusion [62,63], we consider the inclusions as idealized According to the SEM study by65Merati [55], here 2%, 4%, and 6% volume υ between spherical and ellipsoid with 23 μm to μm diameter, perfectly bonded with matrixratio material. Due to inclusions and matrix material was chosen. From the experimental observation by Hashimoto et [62] the randomness and uniformity of the inclusions’ distribution, the RVE is constructed as aal.cube, and other FEM research about boundary the inclusion [62,63], we consider the inclusions spherical which is subjected to periodic conditions. In order to study the effectasofidealized inclusion, various and with 23 are µmconsidered. to 65 µm diameter, perfectly bonded matrixAlmaterial. Due to sizesellipsoid and distribution As an example, an RVE withwith spherical 2O3 inclusions, 65 the μm randomness and uniformity of the inclusions’ distribution, the RVE is constructed as a cube, which is diameter, and 6% volume ratio has been studied first. subjected to periodic boundary conditions. In order to study the effect of inclusion, various sizes and distribution are considered. As an example, an RVE with spherical Al2 O3 inclusions, 65 µm diameter, and 6% volume ratio has been studied first. Metals 2018,case 8, x FOR 7 of for 20 it. This hasPEER theREVIEW strongest inclusions, and the convergence of RVE size is studied Four different kinds and sizes of RVEs and corresponding mesh models are shown in Figure 7. According to the SEM study by Merati [55], here 2%, 4%, and 6% volume ratio υ between For the convergence study nine different sizes have been calculated and the corresponding macro inclusions and matrix material was chosen. From the experimental observation by Hashimoto et al. material properties are shown in Figure 8. When the RVE size reaches 325 µm, the macro elastic [62] and other FEM research about the inclusion [62,63], we consider the inclusions as idealized modulus remains around 78.84 GPa, Poisson’s ratio will be around 0.32 and the relative difference spherical and ellipsoid with 23 μm to 65 μm diameter, perfectly bonded with matrix material. Due to from adjacent results is less than 1%. Therefore, the converged RVE size of this case is 325 µm and it the randomness and uniformity of the inclusions’ distribution, the RVE is constructed as a cube, can be applied to all other cases. In order to be safe, each case has been calculated three times, and then which is subjected to periodic boundary conditions. In order to study the effect of inclusion, various the average of the results is taken. InAs this we can get the macroscopic properties of a sizes and distribution are considered. anway, example, an RVE with spherical Almaterial 2O3 inclusions, 65 μm specific heterogeneous aluminum alloy. diameter, and 6% volume ratio has been studied first. Metals 2018, 8, 836

RVE size = 100 μm

RVE size = 325 μm

RVE size = 200 μm

RVE size = 325 μm

Figure 7. Four difference size RVEs and corresponding mesh models.

This case has the strongest inclusions, and the convergence of RVE size is studied for it. Four different kinds and sizes of RVEs and corresponding mesh models are shown in Figure 7. For the convergence study nine different sizes have been calculated and the corresponding macro material properties are shown in Figure 8. When the RVE size reaches 325 μm, the macro elastic modulus remains around 78.84 GPa, Poisson’s ratio will be around 0.32 and the relative difference from adjacent results is less than 1%. Therefore, the converged RVE size of this case is 325 μm and it can be applied to all other cases. In order to be safe, each case has been calculated three times, and then the average of the results is taken. In this way, we can get the macroscopic material properties of a RVE size = 325 μm RVE size = 325 μm RVE size = 100 μm RVE size = 200 μm specific heterogeneous aluminum alloy. Figure 7. Four difference size RVEs and corresponding mesh models. Figure 7. Four difference size RVEs and corresponding mesh models. 92

0.326

Macro Poisson's Ratio

Macro Elastic Modulus(GPa)

This 90 case has the strongest inclusions, and the convergence of RVE size is studied for it. Four 0.324 88 Macro Elastic different kinds and sizes of RVEs andModulus corresponding 0.322 mesh models are shown in Figure 7. For the 86 study nine different sizes have been calculated and the corresponding macro material convergence 0.320 84 properties are shown in Figure 8. When the RVE size0.318 reaches 325 μm, the macro elastic modulus 82 Macro Poisson's ratio remains around 78.84 GPa, Poisson’s ratio will be around 0.316 0.32 and the relative difference from 80 0.314RVE size of this case is 325 μm and it can adjacent results is less than 1%. Therefore, the converged 78 0.312has been calculated three times, and then be applied76to all other cases. In order to be safe, each case (a) (b) 74 the average of the results is taken. In this way, we can0.310 get the macroscopic material properties of a 100 200 300 400 500 100 200 300 400 500 specific heterogeneousRVE aluminum alloy. RVE Size(m) Size(m) Figure 8. Convergence of (a) macro elastic modulus and and (b) (b) Poisson’s Poisson’s ratio. ratio. Macro Poisson's Ratio

Macro Elastic Modulus(GPa)

92 0.326 90 0.324 The fretting fatigue FE model of heterogeneous The fretting fatigue FE model of heterogeneous material material can can be be built builtin intwo twoways: ways:(a) a) use use the the 88 Macro Elastic Modulus 0.322 equivalent material in the whole specimen, or (b) model a small area near the contact equivalent homogenized homogenized material in the whole specimen, or b) model a small area near the contact 86 0.320 region and use equivalent homogenized material 84the region using using the heterogeneous heterogeneous material material with with inclusions inclusions 0.318 and use equivalent homogenized material 82the specimen. Since inclusion will cause stress concentration Macro[25,30,48,55], Poisson's ratio and fretting in the rest of in the rest of the specimen. Since inclusion will cause stress concentration [25,30,48,55], and fretting 0.316 80 fatigue has78 maximum stress near the contact area 0.314 [8], the second way is chosen in order to study 0.312 76 (a) (b) 74 0.310 100 200 300 400 500 100 200 300 400 500 RVE Size(m) RVE Size(m)

Figure 8. Convergence of (a) macro elastic modulus and (b) Poisson’s ratio.

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fatigue has maximum stress near the contact area [8], the second way is chosen in order to study the fatigue has maximum stress near the contact area secondarea. way is chosen in order to study effect of inclusion on the stress distribution near [8], thethe contact The partition diagram ofthe the theeffect effectof of inclusion inclusionon onthe the stress stressdistribution distributionnear nearthe thecontact contactarea. area.The Thepartition partitiondiagram diagram the ofofthe specimen is shown in Figure 9. specimen is shown in Figure 9. specimen is shown in Figure 9. Area 1: Heterogeneous material with inclusions Area 1: Heterogeneous material with inclusions

Area 2: Equivalent homogeneous material Area 2: Equivalent homogeneous material

Figure Figure9.9.AApartition partitiondiagram diagramof ofthe thespecimen specimenwith withtwo twoarea. area. Figure 9. A partition diagram of the specimen with two area.

Shear stress(MPa) Shear stress(MPa)

Afterthat, that,aaparametric parametric2D 2Dfinite finiteelement element modelisiscreated created inABAQUS ABAQUS[64] [64]using usingPython Pythonscript. script. After After that, a parametric 2D finite elementmodel model is createdin in ABAQUS [64] using Python script. A higher-order higher-order element always causes an instability in the stressvalue valueon onthe thecontact contactsurfaces surfaces [65]. A element always causes an the contact surfaces [65]. A higher-order element always causes aninstability instabilityin inthe thestress stress value on [65]. Therefore, we chose the CPE4R element (plane strain element, 2D, four nodes) instead of eight-node Therefore, we chose the CPE4R element (plane strain element, 2D, four nodes) instead of eight-node Therefore, we chose the CPE4R element (plane strain element, 2D, four nodes) instead of eight-node elements to mesh mesh both parts. As shown in Figure the stress distributionin inthe thecontact contactregion regionisisis elements to both parts. As shown in the contact region elements to mesh both parts. As shownin inFigure Figure3,3, 3,the thestress stress distribution distribution very complicated and stress amplitude large. In particular, near theborder borderbetween betweenslip slipand andstick stick very complicated and stress amplitude isisislarge. between slip and stick very complicated and stress amplitude large.In Inparticular, particular,near near the the zone, it changes very rapidly. In order to get precisely the stress distribution and contact stresses, zone, it changes very rapidly. In order to get precisely the stress distribution and contact stresses, the zone, it changes very rapidly. In order to get precisely the stress distribution and contact stresses, the the model is refined near the region of contact and inclusions. The boundary and loading conditions model is refined near region contactand andinclusions. inclusions.The Theboundary boundary and loading model is refined near thethe region of of contact loadingconditions conditionsare are aredescribed as described described atthe the beginning ofthis this section. The contact behavior describedby bythe themaster-slave master-slave as at beginning section. The contact behavior isisdescribed described as at the beginning of of this section. The contact behavior is by the master-slave algorithm. A Lagrange multiplier used establishthe contactbetween between specimen. algorithm. Lagrange multiplier used totoestablish establish thecontact contact betweenthe thepad padand andthe the specimen. algorithm. AALagrange multiplier isisis used to pad and the specimen. The other contact methods can also be applied as shown by other researchers [66]. The slave surface The other other contact contact methods can also The also be be applied appliedas asshown shownby byother otherresearchers researchers[66]. [66].The Theslave slavesurface surfaceis is defined ontop the top surface of the specimen and the master surface is defined the bottom surface onon the surface ofof the specimen and the master surface is is defined onon the bottom surface of isdefined defined the top surface the specimen and the master surface defined on the bottom surface of the pad. For homogeneous material, the stress distribution at the contact interface can be obtained the pad. For homogeneous material, the stress distribution at the contact interface can be obtained of the pad. For homogeneous material, the stress distribution at the can be obtained analytically, if the assumptions of the Hertz solution are met. The most important two assumptions analytically, if the assumptions of Hertz solution are met. The most important two assumptions are: analytically, the assumptions ofthe the Hertz solution are met. The most important two assumptions are: (a) pure elasticity and (b) the size of the contact area is small enough relative to both contact (a) pure elasticity and (b) theofcontact area isarea small relative to bothtocontact bodies. are: (a) pure elasticity andthe (b)size the of size the contact is enough small enough relative both contact The first assumption is met as only lineariselasticity is considered in this The second Thebodies. firstThe assumption is met asisonly elasticity considered in this study. Thestudy. second one is also bodies. first assumption metlinear as only linear elasticity is considered in this study. The second one is also called the half-space assumption [52]. The contact width for all load cases in the called The contact all load cases the load experiment [53] one is the alsohalf-space called theassumption half-space [52]. assumption [52].width The for contact width forin all cases in theis experiment [53] is 0.47 mm, which is less than one tenth of the height (5 mm) of the sample. A 0.47 mm, which is less than one tenth of the height (5 mm) of the sample. A comparison between experiment [53] is 0.47 mm, which is less than one tenth of the height (5 mm) of the sample. A comparison between the analytical solution of Equation (13) and simulation results with different the analytical solution Equationsolution (13) andofsimulation results different meshwith sizes, for the comparison between theofanalytical Equation (13) andwith simulation results different mesh sizes, for the case of homogeneous material, is shown in Figure 10a. Element sizes of 5 μm, 3 case of homogeneous material, is shown in Figure 10a. Element sizes ofElement 5 µm, 3sizes µm of and 2 µm mesh sizes, for the case of homogeneous material, is shown in Figure 10a. 5 μm, 3 μm and 2 μm around the contact zone have been chosen for the convergence study. The mesh around the contact zone have been chosen for the convergence study. The mesh refinement showed μm refinement and 2 μmshowed aroundconvergence the contacttowards zone have beenanalytical chosen for the convergence mesh the peak solution for the shear study. traction.The Finally, convergence towards the peak analytical solution for the shear traction. Finally, according to the refinement showed convergence towards the peak analytical solution for the shear traction. Finally, according to the results, a 2 µ m × 2 µ m element size is used around the contact zone, which is results, a 2 µm × 2 µm element size is used around the contact zone, which is smaller than in most of according to the results, a 2 µ m × 2 µ m element size is used around the contact zone, which smaller than in most of the previous numerical studies [8,32–38]. The simulation results and theis the previous numerical studies [8,32–38]. The simulation results and the theoretical results may not smaller than in mostmay of the [8,32–38].errors The simulation results and the theoretical results notprevious be exactlynumerical the same, studies due to numerical and geometric constraints be exactly the same, due to be numerical errors anddue geometric constraints [67]. However, the difference theoretical results may not exactly the same, totheoretical numerical errors and geometric constraints [67]. However, the difference between simulation and solution is less than 2% (green line between simulation and theoretical solution is less than 2% (green line and red line in the Figure 10a), [67].and However, difference between and theoretical is less thanour 2%FE (green line red linethe in the Figure 10 a), whichsimulation can be considered as goodsolution enough to validate contact which enough our FE contact model.to validate our FE contact model. and redcan linebeinconsidered the Figureas 10good a), which canto bevalidate considered as good enough model. 0

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(b) Figure Meshsensitivity sensitivity diagram diagram (effect (effect of distribution on on Figure 10.10.(a)(a)Mesh of mesh mesh refinement refinementon onshear shearstress stress distribution contact interface);(b) (b)shear shearstress stressdistribution distribution near case with 2 μm contact interface); near contact contactinterface interfacefor forhomogenous homogenous case with 2 µm contact element size. contact element size.

For comparativeanalysis, analysis, all all the the cases cases involved The aspect For comparative involvedin inthis thisstudy studyare aregiven givenininTable Table2. 2. The aspect ratio describes the evolution of inclusion from sphere to ellipsoid. The size refers to the diameter of of ratio describes the evolution of inclusion from sphere to ellipsoid. The size refers to the diameter ball thelength lengthofofthe thelong longaxis axis of of the the ellipsoid. ellipsoid. In thethe ellipsoid thethe ball oror the Incase case77and andcase case8,8,we weconsider consider ellipsoid inclusions and in order to control them, they have the same cross-sectional area as the spherical inclusions and in order to control them, they have the same cross-sectional area as the spherical inclusion witha adiameter diameterof of44 44µm µ m and and the the long long axis to to 53.889 µ mµm inclusion with axis length lengthof ofthe theellipsoid ellipsoidconverted converted 53.889 and 62.225 µ m, respectively. and 62.225 µm, respectively. From the results of case 3 and case 5 to 9, it can be seen that the macroscopic material properties From the results of case 3 and case 5 to 9, it can be seen that the macroscopic material properties have little to do with the size and shape of inclusions. However, by comparing case 1 with case 3, it is have little to do with the size and shape of inclusions. However, by comparing case 1 with case 3, it is shown that the material properties of inclusions have a significant impact on macro elastic modulus. shown that the material properties of inclusions have a significant impact on macro elastic modulus. Similarly, by comparing case 3, 4 and 5, it is shown that the volume fraction ratio of inclusion also Similarly, by comparing case 3, 4 and 5, it is shown that the volume fraction ratio of inclusion also obviously affects the macroscopic material properties. The macroscopic Poisson's ratio has hardly obviously affects the macroscopic material properties. The macroscopic Poisson’s ratio has hardly changed for the different cases. changed for the different cases. Table 2. All the cases involved in this study. Table 2. All the cases involved in this study. Number

Volume Ratio

Type

Size (µm)

Number Case 1

Volume 4% Ratio

Al2Type CuMg

Size44(µm)

Aspect 1 Ratio

Case 12 Case Case 23 Case 34 Case 45 Case Case 56 Case 67 Case Case 78 Case 89 Case Case 10 9

4% 2% 4% 2% 4% 6%

44

1

4% 4% 4%

Al2O3 Void Al2 O3

23 65 23 65 53.889 62.225 53.889 62.225 23 to 65 44 23 to 65

Case 10

4%

Void

44

6%

Al2 CuMg

Al2O3

44

Al2 O3

44

4%

Al2O3

4%

Al2 O3

4%

Al2O3

4%

Al2 O3

Aspect Ratio

1 1

1 1

1.5 21.5 12 11 1

E * (GPa) E∗ (GPa) 73.7005 73.7005 74.269 76.615 74.269 76.615 78.2946 78.2946 76.5524 76.5046 76.5524 76.5046 76.318 76.163 76.318 76.163 76.2028 69.1 76.2028 69.1

* ∗

µ 0.32584 0.32584 0.32791 0.32579 0.32791 0.32579 0.3243 0.3243 0.32503 0.32537 0.32503 0.32537 0.3256 0.32255 0.3256 0.32255 0.32532 0.327 0.32532 0.327

As the contact width for all load cases in the experiment [53] is 0.47 mm. A 2 mm × 2 mm rectangle is chosen in Area 1 in Figure 9. The numerical study in this paper is divided into two parts. thepart contact for allofload casesinclusion-structures in the experiment in [53] is 0.47 mm. have A 2 mm × 2 mm TheAsfirst is thewidth simulation the nine Table 2, which completely rectangle is chosen in Area 1 in Figure 9. The numerical study this paper is divided intocase two are parts. randomly distributed inclusions. For the second part, since theininclusion locations of each The first part is the simulation of the nine inclusion-structures in Table 2, which have completely randomly generated, each case cannot have the same inclusion order. This makes it impossible to randomly distributed inclusions. the second part, since the inclusion locations of each case are use the control variable method toFor explore the effect of the shape, size and position of the inclusions on the contact stresseach distribution. Therefore, wesame artificially placed fourThis inclusions the sample randomly generated, case cannot have the inclusion order. makes below it impossible to use to compare different theand surface stress These thecontact controlarea variable methodthe to effects exploreofthe effect ofinclusions the shape,onsize position ofdistribution. the inclusions on the inclusions in size, shape, and material corresponding to different cases. Figure 11 shows a contact stressvary distribution. Therefore, we artificially placed four inclusions below the sample contact completed finite element corresponding case in Table stress 2. area to compare the effectsmodel, of different inclusionstoon the3 surface distribution. These inclusions vary in size, shape, and material corresponding to different cases. Figure 11 shows a completed finite element model, corresponding to case 3 in Table 2.

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Figure 11.11. Finite element model of case 3 with artificially placed four inclusions. Figure Finite element model of case 3 with artificially placed four inclusions.

4. Numerical Results and Discussion 4. Numerical Results and Discussion Completely Randomly Distributed Inclusions 4.1.4.1. Completely Randomly Distributed Inclusions 4.1.1. Stress Peak and Its Location 4.1.1. Stress Peak and Its Location There are many experimental studies [48,55,68,69] and numerical studies [63] on the fatigue There are many experimental studies [48,55,68,69] and numerical studies [63] on the fatigue problem of heterogeneous materials. However, limited numerical research on fretting fatigue of problem of heterogeneous materials. However, limited numerical research on fretting fatigue of heterogeneous materials is available [14]. Therefore, the first part of our research is to study the cases heterogeneous materials is available [14]. Therefore, the first part of our research is to study the cases with random distribution inclusions. In real metal materials, inclusions and defects are common with random distribution inclusions. In real metal materials, inclusions and defects are common and and randomly distributed. In the fretting fatigue problem of homogeneous materials, the peak of randomly distributed. In the fretting fatigue problem of homogeneous materials, the peak of shear shear stress appears between the stick zone and the slip zone, as shown in Figure 10b. The peak stress appears between the stick zone and the slip zone, as shown in Figure 10b. The peak tensile tensile stress and peak von-Mises stress in the whole specimen appears near the edge of contact [8]. stress and peak von-Mises stress in the whole specimen appears near the edge of contact [8]. From From experimental materials and loading conditions, the point of occurrence of von-Mises stress and experimental materials and loading conditions, the point of occurrence of von-Mises stress and tensile stress peaks was observed at x = 0.47 mm, the shear stress peak occurred at x = 0.21 mm on the tensile stress peaks was observed at x = 0.47 mm, the shear stress peak occurred at x = 0.21 mm on the contact surface. But for the heterogeneous materials with randomly distributed inclusions (Table 2), contact surface. But for the heterogeneous materials with randomly distributed inclusions (Table 2), significant stress concentration inside the specimen is observed, as shown in Figure 12. It is the stress significant stress concentration inside the specimen is observed, as shown in Figure 12. It is the stress distribution below the contact surface of the specimen. It can be seen, from comparison between distribution below the contact surface of the specimen. It can be seen, from comparison between Figures 10b and 12c, that the shear stress peak is transferred from the contact surface to the inside of Figures 10b and 12c, that the shear stress peak is transferred from the contact surface to the inside of the specimen in the heterogeneous situation. It also showed that there may be multiple high stress the specimen in the heterogeneous situation. It also showed that there may be multiple high stress concentration points inside the structure, hereby forming an influencing group of intrusions, eventually concentration points inside the structure, hereby forming an influencing group of intrusions, causing multiple fatigue cracking points inside the material as observed in the experiment [55]. eventually causing multiple fatigue cracking points inside the material as observed in the experiment [55].

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(b)

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Figure 12. Stress distributionofofCase Case33 below below the (a)(a) Mises stress, (b) tensile stress,stress, Figure 12. Stress distribution thecontact contactsurface, surface, Mises stress, (b) tensile and (c) shear stress. and (c) shear stress.

Due to the complete randomness of the inclusion distribution, it is difficult to investigate the Due to the complete randomness of the inclusion distribution, it is difficult to investigate the effect of particle size, shape and other factors on the stress distribution by the control variable effect of particle size, shape and other factors on the stress distribution by the control variable method; method; such as maintaining the same inclusion material, shape, size but different volume ratio such(case as maintaining the the same inclusion material, size but different volume (case 2, 3, 4) to 2, 3, 4) to study effect of volume ratio onshape, the surface stress distribution of theratio sample. There studyis the effect of volume ratio on the stress of the sample. no regularity no regularity in the results, that is,surface the stress peakdistribution is almost determined by the There criticalisdefects in in the results, that is, the stress peak is almost determined by the critical defects in each case. each case. It is similar as the experimental observation, that the size of the particles is not necessarily It is related to the fatigue life [55]. similar as the experimental observation, that the size of the particles is not necessarily related to the Based fatigue life [55].on the performed simulations, peak stress value and locations are determined, as shown in Figure Based on13. the performed simulations, peak stress value and locations are determined, as shown

in Figure 13. Although various parameters (type, volume ratio, size, shape) of inclusions are considered in the numerical model, the data from Figure 13a indicates that tensile stress and shear stress are similar to homogeneous materials except for the von-Mises stress, which is significantly higher than the case of homogeneous materials. As shown in Figure 13b, the stress peak location is the same as the homogeneous specimen on the contact surface as mentioned before, the ordinate value is equal to 1, otherwise (below the contact surface), is equal to 2. These figures show that, for a fretting fatigue problem, heterogeneous materials containing randomly distributed inclusions have almost the same tensile and shear stress peaks and locations as homogeneous materials on the contact surface. Due to the presence of hard inclusion, the equivalent elastic modulus of the material becomes larger, resulting in a situation where the stress peak in heterogeneous material is sometimes even lower than that of the homogeneous material. But at the same time, the shear stress inside the specimen is also relatively large, comparable to peak value on the surface. Therefore, for materials with shear stress as the main fatigue index, it is likely that cracks will occur simultaneously on the surface as well as inside (Figure 12c). The von-Mises stress will be higher and appear below the contact surface with a high probability. Moreover, we can notice that for Case 10, which uses voids instead of inclusions, all of the three kinds of stress peak are significantly increased and appear inside the specimen near the edge of voids. This confirms experimental observations, that porosity is the main cause of fatigue damage, followed by oxide [68] and provides more useful results data.

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Figure 13. (a)13. Comparison of stressof peaks between materials andmaterials homogenous Figure (a) Comparison stress peaksheterogeneous between heterogeneous and materials, homogenous and (b) their location of all cases. materials, and (b) their location of all cases.

4.1.2. Stress Peak Location Characteristic Although various parameters (type, volume ratio, size, shape) of inclusions are considered in From Figure 13b, we can the shear randomly appearing insidestress or on and the surface of the are the numerical model, thesee data from stress Figureis13a indicates that tensile shear stress material. In Figure 12c, it is also indicated that there will be multiple cracking points appearing onthan similar to homogeneous materials except for the von-Mises stress, which is significantly higher the surface the material becauseAs both have in roughly same For homogeneous the caseorofinside homogeneous materials. shown Figurethe 13b, the shear stress stress. peak location is the same as materials, the peak value of shear stress appears on the contact surface. Also, in the left side ofisthe the homogeneous specimen on the contact surface as mentioned before, the ordinate value equal sample the surface, stress is relatively higher in Figure 10b. that, So when to 1,below otherwise (belowthe theshear contact surface), is equal to as 2. shown These figures show for athere fretting are inclusions in the material, the peakmaterials value of shear stress randomly is likely todistributed transfer to this area (left sidealmost of fatigue problem, heterogeneous containing inclusions have sample inside). The local inclusion alignment of the stress and peak location of case 3 and case 4 are the same tensile and shear stress peaks and locations as homogeneous materials on the contact shown in Figure this shows that the inclusion in the high elastic stress region is more likely to surface. Due14. to Therefore, the presence of hard inclusion, the equivalent modulus of the material cause the transfer of stress peaks from surface to inside the specimen. There is also an effect of mutual becomes larger, resulting in a situation where the stress peak in heterogeneous material is coupling between thelower inclusions. Forming stress band between theBut inclusions may directly sometimes even than that of theahomogeneous material. at the same time, theaffect shearthe stress expansion of cracks at relatively a later stage. inside direction the specimen is also large, comparable to peak value on the surface. Therefore, for

materials with shear stress as the main fatigue index, it is likely that cracks will occur simultaneously on the surface as well as inside (Figure 12c). The von-Mises stress will be higher and appear below the contact surface with a high probability. Moreover, we can notice that for Case 10, which uses voids instead of inclusions, all of the three kinds of stress peak are significantly increased and appear

left side of the below the material, surface, the higher as shown in Figure 10b. So when there aresample inclusions in the theshear peakstress valueisofrelatively shear stress is likely to transfer to this So when there are inclusions in the material, the peak value of shear stress is likely to transfer to this area (left side of sample inside). The local inclusion alignment of the stress and peak location of case area (left side of sample inside). The local inclusion alignment of the stress and peak location of case 3 and case 4 are shown in Figure 14. Therefore, this shows that the inclusion in the high stress region 3 and case 4 are shown in Figure 14. Therefore, this shows that the inclusion in the high stress region is more likely to cause the transfer of stress peaks from surface to inside the specimen. There is also is more likely to cause the transfer of stress peaks from surface to inside the specimen. There is also an effect mutual coupling between the inclusions. Forming a stress band between the inclusions Metals 2018,of 8, 836 13 of 20 an effect of mutual coupling between the inclusions. Forming a stress band between the inclusions may directly affect the expansion direction of cracks at a later stage. may directly affect the expansion direction of cracks at a later stage.

(a) (a)

(b) (b)

Figure 14. The The local local inclusion alignment of of thethe shear stress peak for (a) case 3 and (b) case 4. Figure 14. inclusion alignment of the shear stress peak for Figure 14. The local inclusion alignment shear stress peak for(a)(a)case case3 3and and(b) (b)case case4.4.

In all cases, except for ofof void (case 10),10), case 5 has the highest von-Mises stress. The local In all except forthe thecase case void (case case In cases, all cases, except for the case of void (case 10), case5 5has hasthe thehighest highestvon-Mises von-Misesstress. stress. The The inclusion alignment of the of stress peak location of case of 5 iscase shown in Figurein15a. Stress coupling between local inclusion alignment the stress peak location 5 is local inclusion alignment of the stress peak location of case 5 isshown shown inFigure Figure15a. 15a.Stress Stresscoupling coupling the between inclusions also appears here; smaller and denser inclusions a more pronounced stress band between the the inclusions also appears here; smaller and inclusions form aamore inclusions also appears here; smaller anddenser denserform inclusions form morepronounced pronounced as shown inas Figure 15.in Figure stress band as shown in Figure stress band shown 15.15.

(a)(a)

(b) (b)

(c) (c)

Figure 15. 15．Stress Stress distribution of Case 5 below thethe contact surface, (a)(a)Mises Figure distribution of Case 5 below contact surface, Misesstress, stress,(b) (b)tensile tensilestress, stress, Figure 15．Stress distribution of Case 5 below the contact surface, (a) Mises stress, (b) tensile stress, andand (c) shear stress. (c) shear stress. and (c) shear stress.

4.2. Randomly Distributed and Manually Placed Inclusions After observing the preliminary analysis of the effects of random phenomena on inclusion, we artificially placed four inclusions below the contact surface of the specimen for each case shown in Figure 11. The geometric center is zero, and the abscissas corresponding to the four inclusions are 0.43, 0, 0.21 and 0.47. These four points correspond to the two peaks of shear stress, the geometric center and the contact edge point. As for the vertical position, their original position is 100 µm below the specimen surface. In order to analyze the effect of inclusion on the contact surface stress distribution, the size and shape of inclusions are corresponding to each case.

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After observing the preliminary analysis of the effects of random phenomena on inclusion, we artificially four Type inclusions below the contact surface of the specimen for each case shown in 4.2.1. Effect ofplaced Inclusions Figure 11. The geometric center is zero, and the abscissas corresponding to the four inclusions are As commonly known, there are many kinds of inclusions inside metallic materials depending on 0.43, 0, 0.21 and 0.47. These four points correspond to the two peaks of shear stress, the geometric the process of smelting impurities. Here, we consider two different inclusions to compare case 1 and center and the contact edge point. As for the vertical position, their original position is 100 µ m below case 3 (Table 2). the specimen surface. In order to analyze the effect of inclusion on the contact surface stress As we can see in Figure 16a, in both of homogenous and heterogeneous specimens, the von-Mises distribution, the size and shape of inclusions are corresponding to each case. stress peak is near the contact edge, and it changes very sharply near the edge of the contact. The presence of inclusions makes the peak of the von-Mises stress increase significantly, and Al2 CuMg 4.2.1. Effect of Inclusions Type has a more obvious effect than Al2 O3 . This means that in the heavy load condition, the presence of As commonly known, are many of inclusions metallic materials depending inclusions may accelerate the there distortion of thekinds contact edge of theinside fretting fatigue contact member. on the process of smelting impurities. Here, we consider two different inclusions to compare case 1 The effect of inclusion on the surface tensile stress is not obvious. However, the inclusion will affect case and 3 (Table 2).stresses. theand normal shear

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Figure16. Misesstress, stress,(b)(b)tensile tensile stress, normal stress, shear stress distribution Figure 16. (a)(a)Mises stress, (c)(c) normal stress, (d)(d) shear stress distribution onon thethe contact surface for different inclusion materials cases. contact surface for different inclusion materials cases.

4.2.2. Effect of Distance Surface As we can see from in Figure 16a, in both of homogenous and heterogeneous specimens, the von-Mises stress peak is near the edge, it changes veryorsharply the edge of the The volume ratio (cases 2, 3 and 4)contact is not easy to and measure for a single severalnear inclusions, so here contact. The presence of inclusions makes the peak of the von-Mises stress increase significantly, and we refer to the volume ratio as the particle crowding, which is reflected here as the distance between 2CuMg has more obvious effect than Al2for O3.case This2,means that case in the heavy load between condition, theAl inclusion andathe contact surface. Thus, here case 3 and 4, the distance thethe presence of inclusions may accelerate the distortion of the contact edge of the fretting fatigue contact center of particle to the surface is 120 µm, 100 µm and 80 µm, respectively. The results are shown in member. Figure 17. The effect of inclusion on the surface tensile stress is not obvious. However, the inclusion will affect the normal and shear In the experimental study [48],stresses. the authors believe that the distance between the inclusions or the distance from the surface of the defect is important for fatigue damage. From our results, it is clear 4.2.2. Effect Distance from Surface from Figure 17ofthat the distance from the surface has a more significant effect on the shear stress and normalThe stress. The smaller the distance greaterfor thea peak of the shear stress so volume ratio (cases 2, 3 andfrom 4) isthe notsurface, easy tothe measure singlevalue or several inclusions, generated. However, it can also be seen that the influence of the depth on the surface contact stress here we refer to the volume ratio as the particle crowding, which is reflected here as the distance distribution is different thethe inclusions at different lateral between the inclusionfor and contact surface. Thus, herepositions. for case 2, case 3 and case 4, the distance

between the center of particle to the surface is 120 µ m, 100 µ m and 80 µ m, respectively. The results are shown in Figure 17.

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-40stress, (d) shear stress distribution on the Figure 17. (a) Mises stress, (b) tensile stress, (c) normal -100 -60 contact surface for different distance cases. -150

-80

In the experimental study [48], the authors believe that the distance between the inclusions or -100 -200 (c) is important for fatigue damage. From our the distance from the surface of the defect (d)results, it is -120 clear from Figure 17 has a more effect on -1.0 -0.5that the 0.0 distance 0.5 from 1.0 the surface -1.0 -0.5 significant 0.0 0.5 1.0 the shear X(mm) stress and normal stress. The smaller the distance from the surface, theX(mm) greater the peak value of the shear stress generated. However, it can also(c) be(c) seen that the(d) influence of the depth ononthe surface Figure Mises stress, tensile stress, normal stress, (d) shear stress distribution on Figure 17.17. (a)(a) Mises stress, (b)(b) tensile stress, normal stress, shear stress distribution thethe contact stress distribution is different for the inclusions at different lateral positions. contact surface forfor different distance cases. contact surface different distance cases.

Tensile stress(MPa)

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4.2.3. Effect Inclusion Size 4.2.3. Effect of Inclusion Size In theofexperimental study [48], the authors believe that the distance between the inclusions or theFrom distance from the surface ofthat the defect is important for fatigue From our results, it is Figure 18,18, wewe can seesee the effect of of thethe inclusion size isdamage. more pronounced than the From Figure can that the effect inclusion size is more pronounced than the clear from Figure 17 that the distance from the surface has a more significant effect on the shear influence of of thethe inclusion type onon thethe surface stress distribution (Figure 16). influence inclusion type surface stress distribution (Figure16). stress and normal stress. The smaller the distance from the surface, the greater the peak value of the shear stress generated. However, it can also be seen 300 that the influence of the depth on the surface 300 Case6-65μm contact stress250 distribution is different for the inclusions at differentCase6-65μm lateral positions. Case3-44μm Case5-23μm

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From Figure 18, we can see that the effect of the 0inclusion size is more pronounced than the 50 influence of the inclusion type on the surface stress distribution (Figure16). -100 (a)

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Figure 18. (a) -50 Mises stress, (b)Case5-23μm tensile stress, (c) normal stress, (d) shear Case5-23μm stress distribution on the -20 -40 contact surface for different inclusion size cases. -100 -60 -80 -100 -120 -1.0

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Figure the Metals 2018, 8, 83618. (a) Mises stress, (b) tensile stress, (c) normal stress, (d) shear stress distribution on16 of 20 contact surface for different inclusion size cases.

4.2.4. Effect of of Inclusion Shape 4.2.4. Effect Inclusion Shape Here wewe consider thethe effect of different aspect ratios of inclusions. As shown in Figure 19, there is Here consider effect of different aspect ratios of inclusions. As shown in Figure 19, there noissignificant difference betweenbetween inclusioninclusion shape cases. Forcases. all cases, of the inclusion no significant difference shape Forthe alleffect cases, effectparameters of inclusion onparameters the tensile stress very modest. on theistensile stress is very modest.

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Figure 19.19. (a)(a) Mises stress, (b)(b) tensile stress, (c) (c) normal stress, (d)(d) shear stress distribution on on thethe Figure Mises stress, tensile stress, normal stress, shear stress distribution contact surface for different inclusion shape cases. contact surface for different inclusion shape cases.

ForFor thethe current work, wewe considered different stresses forfor analysis in in order to to visualize thethe current work, considered different stresses analysis order visualize contribution of each stress. The local stress and stress concentration effects will have a direct impact contribution of each stress. The local stress and stress concentration effects will have a direct impact onon thethe fretting fatigue lifelife of of thethe specimen. The next article shall contain thethe application of of damage fretting fatigue specimen. The next article shall contain application damage parameters (critical plane approach) to compute crack initiation lives, where we shall present thethe parameters (critical plane approach) to compute crack initiation lives, where we shall present comparison of of numerical lives with experimental lives. comparison numerical lives with experimental lives. 5. Conclusions 5. Conclusions In this study, in order to consider the effect of inclusions in aluminum alloy 2024-T3 on the fretting In this study, in order to consider the effect of inclusions in aluminum alloy 2024-T3 on the fatigue stress distribution, a numerical research method combining RVE with a finite element method fretting fatigue stress distribution, a numerical research method combining RVE with a finite is conducted. Based on the research results, we can draw the following conclusions. element method is conducted. Based on the research results, we can draw the following conclusions. For the cases considered in this study, with the same inclusion material to volume ratio, the shape For the cases considered in this study, with the same inclusion material to volume ratio, the and size of the inclusions have little effect on the macroscopic material properties. However, shape and size of the inclusions have little effect on the macroscopic material properties. However, the material properties of inclusion and volume ratio have a significant impact on macro elastic the material properties of inclusion and volume ratio have a significant impact on macro elastic modulus. The macroscopic Poisson’s ratio has hardly changed with the change of different parameters. modulus. The macroscopic Poisson's ratio has hardly changed with the change of different Due to the randomness of the inclusion distribution, the fatigue cracking nucleation depends parameters. on the most dangerous inclusion. The von-Mises stress peak very likely increases and transfers from Due to the randomness of the inclusion distribution, the fatigue cracking nucleation depends on the contact surface to the interior of the specimen. However, various parameters of inclusions have the most dangerous inclusion. The von-Mises stress peak very likely increases and transfers from the little effect on the peak value and position of the tensile stress; it remains almost same as in the contact surface to the interior of the specimen. However, various parameters of inclusions have little homogeneous material. Among the materials with inclusions, the peak value and position of the shear effect on the peak value and position of the tensile stress; it remains almost same as in the stress are consistent with the homogeneous material, but sometimes it is also transferred to the inside homogeneous material. Among the materials with inclusions, the peak value and position of the of the specimen. Moreover, in all cases, higher shear stress occurs in many places inside the sample, shear stress are consistent with the homogeneous material, but sometimes it is also transferred to the which can result in multiple cracking points inside the sample as well as at the contact surface.

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Due to the presence of hard inclusions, the equivalent elastic modulus of the material becomes larger, resulting in a situation where the stress peak in heterogeneous material is sometimes even lower than that of the homogeneous material. A void compared to inclusion will cause larger stress and randomness of stress distribution. Therefore, the influence of microstructure on macroscopic material properties should be considered in engineering. Inclusion in high stress areas is more likely to cause stress concentration leading to peak transfer. Stress coupling between adjacent inclusions will form a high stress band, which has an important effect on the direction of crack growth. The size of the inclusions and the distance from the surface have a more significant effect on the surface stress distribution than the type and shape of inclusions. The effect of inclusion parameters on the tensile stress is insignificant. Author Contributions: Q.D. carried out the research work and wrote the first draft of the manuscript. N.B. supported the finite element simulations and analysis of the results. X.Y. and M.A.W. provided advice on the entire work and supervised all research work. Funding: This research was funded by [National Natural Science Foundation of China] grant number [11372138 and 11572157] and by the Research Foundation-Flanders (FWO), The Luxembourg National Research Fund (FNR) and Slovenian Research Agency (ARRS) in the framework of the FWO Lead Agency project: G018916N ‘Multi-analysis of fretting fatigue using physical and virtual experiments’]. The APC was funded by the Research Foundation-Flanders (FWO). Acknowledgments: The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 11372138 and No.11572157), the Research Foundation-Flanders (FWO), The Luxembourg National Research Fund (FNR) and Slovenian Research Agency (ARRS) in the framework of the FWO Lead Agency project: G018916N ‘Multi-analysis of fretting fatigue using physical and virtual experiments’. Conflicts of Interest: The authors declare no conflict of interest.

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