Numerical simulation of the third body in fretting problems

Wear 270 (2011) 876–887

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Numerical simulation of the third body in fretting problems Stéphanie Basseville a,∗ , Eva Héripré b , Georges Cailletaud c a

LISV, Universitˇıe de Versailles Saint-Quentin, 45 avenue des Etats-Unis, 78000 Versailles, France Laboratoire de Mˇıecanique des Solides, Ecole Polytechnique, 91128 Palaiseau Cedex, France c MINES ParisTech, Centre des Matˇıeriaux, CNRS UMR 7633, BP 87 91003 Evry Cedex, France b

a r t i c l e

i n f o

Article history: Received 5 March 2010 Received in revised form 16 February 2011 Accepted 17 February 2011 Available online 24 February 2011 Keywords: Finite elements Third body Fretting Dang Van’s fatigue model Wear

a b s t r a c t This study is devoted to the computation of realistic stress and strain fields at a local scale in fretting. Models are proposed to improve surface and volume modelling, by taking into account the heterogeneity of stress fields due to the irregular interface. This gives a new view toward damage mechanisms. The surface heterogeneity which is considered here, results from the third body trapped in the contact zone. This third body is known to drastically influence the contact conditions. The competition between wear and crack initiation is investigated with respect to local stress fields. The first model is used to study the evolutions of particles and the contact stress according to the loading conditions. Then, Dang Van’s multiaxial fatigue model is used to predict crack initiation during the fretting test. This criterion may highlight the presence of microcracking everywhere in the contact zone. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Damage prediction in contact areas submitted to fretting is an serious issue, since it is often a limiting factor for the design of industrial components. Most of the papers in the literature consider a comprehensive approach of this problem, and use pure macroscopic models to evaluate the stress and strain fields at the contact surfaces [1,2]. This does not take into account the fact that, in contact problems, the volume of material enduring critical load levels is remarkably small, i.e. much less than a cubic millimeter. This size may be comparable to the size of several grains only in metallic alloys and/or particles that could be trapped in the contact zone. Thus, they probably have an impact in the prediction of the fretting damage, and they must be introduced in the modelling. The aim of this paper is to propose solutions to take into account the heterogeneous character of the material, and to show that the mechanism change can be captured by simple critical variables at a local scale in the FE computation to point out the consequences for life predictions. In fretting contacts, wear and fatigue mechanisms interact, so one has to consider these two phenomena simultaneously. At low-displacement amplitudes in partial slip, fatigue dominates, whereas wear prevails at large displacement amplitudes in gross slip [3]. The concept of fretting map is used to define

∗ Corresponding author. Tel.: +33 1 39 25 30 15. E-mail addresses: [email protected] (S. Basseville), [email protected] (E. Héripré), [email protected] (G. Cailletaud). 0043-1648/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2011.02.016

the conditions for which each damage mechanism is predominant as described in section 2 for cylinder–plate tests performed on titanium alloys. Some wear mechanisms, such as adhesion and oxidation, might promote fatigue in partial slip [4]. In the gross slip conditions, wear can prevent fatigue damage to occur by eliminating surface cracks at a faster rate than their growth [5]. Due to the complexity of the sliding surface interaction, literature presents many models about these damage mechanisms. More than one hundred wear models are listed in [6]. Nevertheless, Archard’s law [7] often inspires wear modelling and it has been adapted to fretting wear in finite element (FE) analysis [8–10]. In this study, the wear model is based upon the Oqvist’s approach [8] as described in Section 4. The model explicitly includes the third body which is known to drastically affect the contact conditions. Oqvist’s model has been adapted to represent the matter transfer towards the third body. A literature review on fretting fatigue modelling can be found in [4]. These last years, applications of multiaxial fatigue parameters [11,12], fracture mechanics [13] and fatigue mechanics [2,14] to fretting contact problems have been developed. Some other authors have studied the role of material modelling in fretting fatigue prediction [1,15,16] In our study, it has been decided to use Dang Van’s multiaxial fatigue model to predict crack initiation during a cylinder–plate fretting test. The definition of this model is recalled in Section 3. The important problem of the description of the third body is considered here. It is known to drastically influence the contact conditions. An attempt to develop an original description of this material is then presented in Section 4. It allows to predict the cap-

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3. Material modelling

Fig. 1. Material response fretting map as determined by [20]. The loading conditions of the computations performed in our study are indicated by black points in the diagram.

ture of the matter inside the contact or its ejection in relation with the loading conditions (Section 5). Its influence on wear and fatigue life prediction is evaluated here and compared to FE computation without considering the third body.

2. A material response fretting map The concept of fretting map, already approached in the past [17] has been introduced by [18,19] to underline the relation between the transition from partial slip to gross slip and the transition from crack initiation to wear. It has been recently applied to titanium alloy (Ti6Al4V) which is chosen as an example in this study [20]. The corresponding response is drawn for 106 cycles (Fig. 1) of a cylinder/plate fretting test. The cylinder radius is equal to 10 mm. Depending on the displacement amplitude and the normal load, one of these responses is observed: no damage, fatigue cracking, fatigue cracking and wear or wear only. These four domains are indicated in the material response fretting map (Fig. 1). By means of linear elastic computations, the field can be splitted into a partial slip and a total slip domain. In elastic–plastic computations, an additional domain can be found where the relative displacements between the bodies are not accommodated by contact sliding but by near surface material flow. The horizontal boundary indicated in Fig. 1 marks the onset of large plastic flow. In the partial slip domain, no damage is obtained in experiment with low normal forces (133 MPa). At higher normal forces (266 MPa, 333 MPa), cracks have been detected in tests with sufficiently high displacement amplitude. In the total slip regime, wear is the only damage mechanism for low normal forces (133 MPa). At higher normal forces (266 MPa, 333 MPa), wear and cracking are both detected for lower displacement amplitudes, and only wear for higher displacement amplitudes. Adhesion is the main source of wear at low normal forces so that the strain fields predicted with models using a von Mises criterion are not relevant to fully explain the wear process. When a microstructural scale material model is introduced, larger local strains will be produced, due to a series of localization problems that cannot be observed with classical plasticity models. This effect is of increasing importance with increasing normal force – then plasticity related wear may be predominant, and may hide the phenomena linked to adhesion. The deepest cracks in fretting tests are obtained in the “partial slip regime” where wear is also observed.

TA6V is a near ␣ titanium base alloy. In the present state, it is made of 50% lamellar packs of HCP ␣-phase and bcc ␤-phase and 50% ␣ grains with a grain size of 30 ␮m. On the whole the ␤-phase does not constitute more than a few percent in this alloy. A crystallographic texture is present at two scales. The material exhibits a macro texture. A preferential orientation of all the grains is present at a macroscale. Additionally, a micro texture has been observed. The domains of neighbouring grains have a very similar orientation constituting super grains [21]. Some qualitative information about the macro texture of the material can be found in [22]. The main texture is caused by a preferential orientation of the c-axis of the hexagonal lattice in the normal direction (ND) of the specimen whereby uniaxial tests with an imposed deformation in the “rolling direction” (RD) were made. The elastic constants are respectively 119 GPa for the Young’s modulus and  = 0.29 for Poisson’s ratio. The elastic limit is Rp0.2 = 850 MPa and the tensile strength Rm is around 1000 MPa. For cyclic material modeling a phenomenological “von Mises” approach [2] was applied. All material parameters were identified using cyclic fatigue tests under deformation control at Rε = εmin /εmax = − 1. As a first approach, we restrict ourselves to elastic computations. The applications will show that the von Mises stress remains below or just above the yield stress even in the contact area. The classical version of Dang Van’s criterion introduces a combination of the actual shear stress and hydrostatic pressure, P. For  , the circle that encloses a given facet, characterised by its normal n the shear path described during one cycle is defined. The shear n (t) that is used in the criterion is the distance between the current point of the shear path and the center of this circle. For a given time t, one of the facet presents the highest value of the linear combination of n (t) and P(t), called equivalent Dang Van’s stress,  DV (t). The path in the P– plane should remain in the half space where this stress remains smaller than a critical value, that is nothing but the fatigue limit in pure shear,  0 . DV (t) = max(n (t) + bP(t))

(1)

f () = maxDV (t) − 0

(2)

 n

t

The material parameters b and  0 are constants that have to be calibrated from two independent tests, either pure tension and shear fatigue, or two tensile tests with different R ratios (R =  min / max ). In the present application, an alternative formulation is used. The Tresca type shear is replaced by a von Mises evaluation. The load path is considered in the tensorial stress space, and one search for the hypersphere that encloses the full path. Its center is a deviatoric tensor. The actual shear at time t does not introduce any facet definition. It is just computed as the distance in the deviatoric stress space between the current point of the cycle and the previously defined center, denoted by J*. Following the previous case, a linear combination is introduced between J* and the trace I1 of the stress tensor, I1 (t) = Trace( )(t). The computation of the criterion is much ∼

faster in this case, since there is no facet. As a drawback, there is no more information concerning the orientation of the crack initation plane. ∗ DV (t) = (1 − b∗ )J ∗ (t) + b∗ I1 (t) ∗

f () =

∗ maxDV (t) − 0∗ t

(3) (4)

With the shape chosen in Eq. (3), crack initiation occurs when the critical stress reaches the fatigue limit  0 in pure tension under reverse loading R = − 1. As an example, the maximum values of ∗ (t) are respectively: DV

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Fig. 2. SE micrographs of cylinder and plate contact surfaces at the same contact spot after 100 fretting cycles. One micrograph is horizontally mirror-inverted. (a) SE micrograph of the plate contact surface. (b) SE micrograph of the cylinder contact surface [24].

•  max for a tension test at R = − 1, since J∗ (t) = (t) and I1 (t) = (t); • (1 + b∗ ) max /2 for a tension test at R = 0, since J∗ (t) = (t)/2 and I1 (t) = (t); √ √ • (1 − b∗ )max 3 for a pure shear test, since J ∗ (t) = (t) 3 and I1 (t) = 0. The fatigue limit in each case can be found accordingly. The parameter values for our material are respectively 523 MPa for 0∗ and 0.395 for b*, providing a limit of 523 MPa in tension at R = − 1, 750 MPa in tension at R = 0, and 500 MPa in pure shear.

ticles, thus describing the evolution of the thickness of the debris layer and its movement. In the following, the contact–adhesion aspects are privileged to better understand the ejection conditions according to the damage mechanisms. A simple input consists in assuming that the development of an efficient wear process is related to the ejection of the debris. When the debris are captured in the contact zone, the coefficient of friction is decreasing and wear is also decreasing [23]. The purpose of this section is to investigate the conditions of the ejection in relation with the gross slip regime.

4. Third body modelling

4.1. FE mesh and boundary conditions

One of the phenomena that controls wear is the concept of a “third body” introduced in the 90s by [23]. The third body is formed by fragments of material along the interfaces (Fig. 2). Several approaches have been proposed. A first possibility consists in considering the particles as a granular material [25]. By using this kind of model, the physical detachment of particles (seen as a consequence of degradation of materials) can be explained, as well as the movement of these particles within the contact and their ejection. Other authors consider the debris as a layer structure with its intrinsic mechanical properties [26]. A modified version of Archard’s equation [7] is used to describe the migration of wear par-

A cylinder–plate fretting wear test including an explicit modelling of the third body is represented in a 2D plane strain FE computation (Fig. 3). The cylinder has a radius of 10 mm in order to be consistent with the fretting map. The third body is explicitely modelled in the contact area by several rectangular particles which sizes are 5 ␮m × 1 ␮m, spaced by 2.5 ␮m, as shown in Fig. 3. The number of particles depends on the loading conditions in order to cover the whole contact area (37 particles for P = 100 N/mm, 64 particles for P = 260 N/mm, 78 particles for P = 333 N/mm). To follow the particles evolution, we numbered them starting from the left. The contact zone is defined when the particles are in contact with the

Fig. 3. (a) FE mesh of a cylinder–plate contact model, (b) zoom of the contact area, and (c) boundary conditions.

S. Basseville et al. / Wear 270 (2011) 876–887

Nodes of the target surface

Nodes of the impactor surface

Nodes of the target wear box

Nodes of the impactor wear box

Target wear box

Impactor wear box

879

Fig. 4. Schematic definition of the impactor/target surfaces and of the impactor/target wear boxes (according to the case study, 37–78 debris are present in the contact area).

cylinder and the plate. Far from the contact zone, free domains with plane strain 3-node triangular full integration elements are used, the mesh is refined towards the contact area. In order to combine a high precision with reasonable computation times, the proximity of the mesh in the contact zone is paved with regular domains consisting of plain strain rectangular 4-node full integration elements with a size of 1.1 ␮m × 0.8 ␮m. The mesh of the particles is composed of the same type of elements which size is 1 ␮m × 0.5 ␮m.

(a)

During the wear process, the debris will interact. Their impact is just regulated by a contact condition between the particles. A warning distance allows the detection of the contact particles and activates the contact algorithm. The loading is applied in two steps: first, a normal reaction is imposed on the top of the half cylinder while the plate is kept fixed by locking its bottom and side nodes. Second, an horizontal cyclic displacement (U1) is imposed to the bottom and side nodes of the

(b)

Fig. 5. Illustration of the remeshing operation in the vicinity of one particule (a). Impactor wearbox before wear; initial geometry of the debris (b). Impactor wearbox after wear; updated shape of the debris.

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0.04

0.035

pression, cyl cycle 5, cyl cycle 10, cyl cycle 20, cyl cycle 30, cyl pression, pl cycle 5, pl cycle 10, pl cycle 20, pl cycle 30, pl

0.03

y [mm]

0.025

0.02

0.015

0.01

0.005

0

-0.005 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.2

0.3

0.4

x [mm] 0.008

cycle 5, cyl cycle 10, cyl cycle 20, cyl cycle 30, cyl cycle 5, pl cycle 10, pl cycle 20, pl cycle 30, pl

0.006

y [mm]

0.004

0.002

0

? ?

-0.002

-0.004 -0.4

-0.3

-0.2

-0.1

0

? ?

0.1

x [mm] Fig. 6. Wear profile of the cylinder and the plate including the third body: (a) partial slip and (b) wear. The symbols allow to locate the position of the debris between the first and second body (partial slip:  – –  symbols; wear:  – –  symbols for cycle 5 and 䊉 – – 䊉 for cycle 20.

plate (Fig. 3c). The time for one complete cycle is always equal to 8 s, thus the displacement velocity differs for the various loading conditions, nevertheless this does not produce any effect on the result, that is time independent. The loading conditions are chosen in such a way that they cause fretting according to the map shown in Fig. 1. The material model for cylinder and plate is chosen linear elastic. The elastic constants are E = 119 GPa and  = 0.29. They correspond to the elastic properties of Ti6Al4V alloy. The friction behaviour is introduced by means of Coulomb’s law [27]. The coefficient of friction is set to 0.8, which represents the friction behaviour of a dry Ti6Al4V–Ti6Al4V contact. A large number of cycles is applied to the system. During the loading, the geometry of the debris is updated, as a result of wear on the cylinder and the plate. The amount of matter lost by them is added to the debris.

4.2. Numerical modelling of the wear process 4.2.1. Strategy to update mesh geometry Wear computation is introduced in the finite element code ZSeT/ZéBuLoN [28]. It consists in a mesh manipulation that is applied after each loading increment. The contact treatment is modelled by an impactor/target technique and solved by means of a classical flexibility method [29,30]. According to this technique, the contact reactions are at first computed in a local contact algorithm and then added to the global problem. The friction behaviour is introduced by the Coulomb’s law between the debris particle and the first body. Among other variables, the contact pressure and the work of the tangent forces are computed at the impactor nodes.

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881

0.2 43 40 37

Horizontal displacement [mm]

0.15 0.1

Cylinder

0.05

1

0 -0.05 -0.1

2

22

1

Plate

7 13

Particles

42

43

19 -0.15

(b) Initial position and numbering of particles

25 31

-0.2 0

1

2

3

4

Number of cycles

(a) Wear: P=100 N/mm, δ = ± 30 µm. Fig. 7. Illustration of the displacement of some debris: wear mechanism P = 100 N/mm, ı = ± 30 ␮m. The right number of curves corresponds to the number of particles.

The plate and the cylinder are taken as impactor surfaces, meanwhile upper and lower edges of the particles are considered as targets, as shown in Fig. 4. Wear computation could be performed using a local wear criterion defined at each node i after each load increment. The wear criterion used in this study is described in the next section. To allow the conservation of matter, the modelling of wear requires an uptaded mesh geometry according to two different procedures, one for the impacted surfaces (see Section 4.2.3), and an other one for the debris (see Section 4.2.4). The adaptive meshing algorithm is implemented via a in-house program within ZSeT/ZéBuLoN.

x, an incremental slip s and the pressure p at specific time t, is used:

4.2.2. Wear criterion The wear model used is a modified version of the Archard model [7], as described in [8] for modelling fretting wear. The Archard wear equation is given by:

 w = −kw (p s )n  h i i  = −kw ewi n i

V = KPH ı

(5)

where V is total wear volume, ı is the sliding distance, K is the wear coefficient, P is the normal load and H the hardness of the material. To simulate the evolution of wear along the contact surface during a cycle, a first option is to calculate the local wear. In the finite element model, Eq. (5) is expressed in terms of local wear depth increment, dh, for an increment of local slip, ds: dh = kw p(x) ds

(8)

The incremental wear h depends on the type of surface (impactor or target) and is more precisely described in the next two sections. 4.2.3. Wear of the impactor • At each impactor node, the nodes of the impactor surfaces are moved according to: (9)

where hwi is the distance between node i before the load increment and its position after the increment,  is the coefficient of friction, pi is the pressure computed at node i and si is the slide increment of node i. The direction of the vector which defines wear  is assumed to be perpendicular to the surface. Consequently, wear n is represented by a vertical translation of each external node, from a depth proportional to the surface energy density ewi . The energy per width unit Ew dissipated on the whole surface s can then be determined by the following expression



Ew =

ew ds

(10)

L

(6)

where p is the local contact pressure, K/H is replaced by k and represents the local wear coefficient. The k material parameter is calibrated by integrating the model on the whole wear scar and adjusty the result to experimental measurement [31]. A similar model is proposed by Fouvry et al. [5]. It refers to a “dissipated energy approach”, since the critical variable is now computed as the energy dissipated by the tangential component of the contact reaction. dh = kw p(x) ds

h(x, t) = kw p(x, t)s(x, t)

(7)

In both cases, the authors propose to update the mesh geometry after each computed fretting cycle. In the present study, a modified local “dissipated energy approach”, defining the incremental wear h for a specific point

where L is the imprint lenght and will be used to model the mass transfer in Section 4.2.4. The wear coefficient kw is set to 40 × 10−8 mm2 /mJ. This is the highest acceptable value that does not produce any perturbations linked to wear geometry updating in the contact zone of the cylinder/plate model. With the chosen value a maximum wear height of about 0.05 ␮m per updating step is obtained, which is close to the results reported by [9]. In order to preserve mesh quality, an additional rule is also considered to move the nodes under the surface. • An impactor wear box is defined as shown in Fig. 4. It contains nodes near the impactor contact surfaces. The position of these nodes are then modified by the following expression: hw(i) = kw ((1 − ˛)ew(i) + ˛ew(i+1) )

(11)

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where the parameter ˛ ∈ [0, 1] takes into account the wear box node position with respect to the contact surface. The translation of these nodes is assumed to be perpendicular to the impactor surface. This operation allows to preserve a good element shape near the contact zone (Fig. 5). 4.2.4. Wear of the debris The last step consists in taking into account the evolution of the debris. The key idea is that there is a transfer of matter from the bulk material toward the debris (from the impactors towards the targets in algorithmic terms). The number of debris is fixed during the simulation. The constitutive equations used for the debris particles are just elastic. In fact, since the debris at a given time are nothing but a mixture of former debris and newly pasted material, it would be very difficult to justify a given constitutive equation. Moreover, knowing the stress and strain fields in the debris is not the key point of our simulation. The important phenomenon regarding the physical aspect of the problem is the way the shape will be updated during the wear process. If a local rule is used for estimating the wear effect, the criterion reaches its maximum at the corners of the debris, due to the stress concentration. A catastrophic process follows, where the transfer of matter due to wear increases the stress concentration, and produces more and more wear. The resulting shape of the debris is totally unrealistic, since the matter accumulates on both ends and transforms the initial rectangle into a sort of “bone shaped” figure. A better view to the real process allows to realize that the extracted matter is spread on the debris and that the criterion must be taken in average on each particle. For each of them, the energy dissipated on the whole contact surface is computed during one increment. The obtained value, Ew , is then used to compute a smoothed local ∗ , whose distribution follows a parabolic shape, energy density, ew with a maximum value in the middle of the debris, and lower values on both ends. In the local frame of one debris (0 < x < L), the expression is ∗ ew =6



Ew x x 1− L L2



(12)

∗ on the particule lenght is equal Obsiously, the integral of ew to ew . In each point of the debris, the mesh update is proportional to the energy density. The nodes of the target surfaces are then moved according to:

 ∗ = kw e∗ (i)n  h w w(i)

0.05 0

Horizontal displacement [mm]

882

-0.05 -0.1 -0.15 -0.2 Cracking No damage Wear Cracking + Wear Plastic shearing

-0.25 -0.3 0

12

24

36

Number of cycles Fig. 8. Illustration of central debris displacement for various mechanisms (plastic shearing, P = 333 N/mm, ı = ± 26 ␮m; cracking, P = 260 N/mm, ı = ± 10 ␮m; wear, P = 100 N/mm, ı = ± 30 ␮m; cracking + wear, P = 260 N/mm, ı = ± 20 ␮m; no damage, P = 260 N/mm, ı = ± 5 ␮m).

ing the fretting test when the third body is present in the interface. Fig. 6 shows the wear profiles of the cylinder and the plate after several cycles of fretting for partial slip and for wear loading. The loading conditions were defined in Fig. 1: for partial slip conditions (respectively for wear conditions), we choose P = 260 N/mm and ı = 5 ␮m (respectively P = 100 N/mm and ı = 30 ␮m). For partial slip, wear profile is the same during the various cycles. It is symmetrical with respect to the plane (x = 0) and is similar to the wear profile between a cylinder and a plate (Fig. 6a). This result is due to the fact that the particules remains trapped in the contact area and move with a series of successive stick-slip periods, see Sections 5.2 and 5.3. Conversely, Fig. 6b presents the updated geometry of contact’s plate and cylinder during the first thirty cycles. The wear profiles illustrate the inclusion of the particules in the wear modelling, more precisely, the mass transfer of healthy body to the third body. For this type of loading, the geometry is not symmetrical about x = 0. The wear depends on the movement of the particles in the contact area: the particules move, can be collected and can leave the contact, see Sections 5.2 and 5.3.

(13)

The resulting shape is then parabolic, as shown in Fig. 5b. Thus, this model allows the matter transfer from the first and the second body (cylinder and plate) toward the third body (particles) and ensures the conservation of matter:





h∗ ds

hds = Lcylinder ∪ Lplate

(14)

∪ Lparticules

where Lcylinder (respectively Lplate , Lparticules ) are the contact surface of the cylinder (respectively the contact surface of the plate, the contact surface of the particules). This procedure is repeated until the final number of fretting cycles has been completed. 5. Results and discussion 5.1. Geometry update This first section aims at illustrating the strategy of update geometry contact surfaces of the cylinder and the plan dur-

Fig. 9. Illustration of a series of successive stick–slip contact periods during one cycle: wear mechanism P = 100 N/mm, ı = ± 30 ␮m.

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Fig. 10. Position of the particles during the fretting test according to the mechanism. The horizontal continuous (respectively dotted) line represents the initial contact (respectively the contact after the particule ejection). The right number of curves corresponds to the number of particles.

5.2. Evolution of the third body Series of FE analyses have been performed according to the various conditions defined in Fig. 1. The output of the computations is first the definition of the working conditions that provide particle trapping and the kinetics of the particles when they are able to slip. Table 1 summarizes the various regimes. The particles remain captured in the contact area in the case of partial

Table 1 Particle ejection according to the loading conditions – results of the FE computations. Load (N/mm)

P = 25 P = 100 P = 200 P = 260 P = 333

Displacement ı (␮m) 5

10

15

20

26

30

Eject Capture Capture Capture Capture

Eject Eject Capture Capture Capture

Eject Eject Eject Eject Eject

Eject Eject Eject Eject Eject

Eject Eject Eject Eject Eject

Eject Eject Eject Eject Eject

slip, for small displacement amplitudes and a large normal force. These numerical results meet experimental studies on this subject. For instance, scanning electron microscope analyses of surface damage obtained by [32] outlines two evolutions for the particles: the particles can remain inside the contact where they play a role of velocity accomodation, or they can be ejected out of the contact, causing direct interaction between the first bodies. Fig. 7(a) shows typical debris displacement, for a loading producing wear (P = 100 N/mm, ı = 30 ␮m). The debris collide during the ejection process, then they remain stuck together, so that the distance between the curves remains constant through the history. Fig. 8 shows the normal displacement of the central fragment according to damage mechanism type. For the sake of clarity, only one point is saved for each cycle, so that the detail of the local displacement during a cycle is not shown. The ejection velocity depends on the loading conditions. The fastest ejection is obtained for the case of wear. This fretting

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Fig. 11. Evolution of the contact pressure during a fretting test in condition of partial slip (P = 260 N/mm, ı = ±5 ␮m) for a zero horizontal displacement.

phenomenon is observed experimentally by [33,24]. The authors hightlight the role of sliding amplitude in the ejection process (Fig. 9). Fig. 10 illustrates the evolution of some particles during the early cycles for the different loading conditions. The ejection is observed for the cases of the mixed regime wear/cracking, plastic shearing, and pure wear under a low normal force. The flow of particles depends on displacement amplitude. The particles move, can be collected, are ejected and can leave the contact on both sides. The ejection is the result of a series of successive stick–slip contact periods (Fig. 10). Whatever the initial position of the particle, the adhesion contact periods precede the slip contact periods. The only change is the time interval which depends on the initial position of the particle. The adhesion contact periods are longer for particles located inside the contact. Fig. 9 illustrates the stick–slip phenomenom. Indeed, for the positive load, the particles are trapped between the cylinder and the plane (A–B, A–C) before sliding into positive direction (B–D, C–D). The negative load is then applied and particles are blocked again between the cylinder and the plane (D–E, D–F, D–G) before sliping into the negative direction (E–H, F–H, G–H). Then, a positive load is applied and leads to the same phenomenon (stick: H–I, H–J, H–k and positive slip: I–L, J–L, K–L). These successive stick-slip contact periods keep repeting during cycles until the final ejection of particles. In the regime of partial slip and crack initiation, the particle remains trapped in the contact zone. 5.3. Evolution of the contact stress Although this study does not reflect the physical reality, it allows to view the evolution of particles according to the loading conditions (see Section 5.2). On the other hand, it can highlight the high level stresses due to the presence of third body in the contact zone. The spacing of these particles implies a non-smooth contact stress which is characterized by the presence of high level and vanishing stresses. Hovewer, in the following, for the sake of clarity, the average stress  22 , in the vertical direction, is considered. For the cylinder and the plate, the stresses are averaged in the contact area, whereas the average is performed on each particle for the third body.

Fig. 11 shows the evolution of the contact stresses  22 for the three bodies when the case of partial slip. The contact stresses  22 do not change during the fretting test. This result is correlated with the capture of debris under the contact area for partial slip. The profile average stresses  22 in the contact zone is parabolic, similar to Hertz theory but differs from Hertz theory result. Indeed, there is an increase of the contact area and a decrease of the contact stress. Note that the difference of maximum stresses between the plate/cylinder and the particle corresponds to the ratio between the plate/cylinder contact area and the particle contact area. In contrast, Fig. 12 shows the evolution of average contact stresses  22 during few cycles in the case of the gross slip for a zero horizontal displacement. Fig. 12a shows the stresses distribution after the normal load P. At the initial state, the contact area is 273 ␮m and all the 37 particles are in contact. The profile of the contact average stresses  22 is parabolic, similar to Hertz theory. After a few cycles, the particles move. Complex stress are observed inducing an evolution of the contact zone. It corresponds to the displacement of these particles in the interface (Fig. 12b–e). Indeed, after one cycle, particles move and seven of them leave the contact area. The contact size does not change. Only the position of the particles are modified, so that the stress distribution is no longer symmetrical in relation to the center of the contact area (Fig. 12b). After 5 cycles, particules move outwards from the contact area and a new particle leaves the contact zone (Fig. 12c). Near the contact center, the stress becomes equal to zero. The cylinder and the plate are no longer in contact in this zone. On both sides of this area, stress peaks appear and correspond to the presence of particles. At this state, the contact size is 351 ␮m. This is the result of the particle displacement before the final ejection. After 10 cycles, 23 particles are ejected and the contact size is 471 ␮m, as shown in Fig. 12d. A peak stress is observed on the right contact area due to a captured particle. The ejection of all the debris is obtained after 20 cycles (Fig. 12f). The contact zone has decreased (303 ␮m) but is still larger than Hertz contact (198 ␮m). This difference is explained by the geometry change of the plate and the cylinder by wear. Consequently, the influence of the third body into the contact zone is highlighted for the case of total slip. The dependence between the stress distribution and the particle position (Fig. 12) highlights a high heterogeneity that could not be taken into account without an explicit modelling of the third body. This heterogeneity can influence the crack initiation prediction. This subject is the aim of the next part of this section. 5.4. Prediction of fatigue crack initiation Dang Van’s multiaxial model is used to predict crack initiation for the various loading conditions. It is applied to each integration point of the plate mesh. ∗ (Eq. (3)) at the plate In Fig. 13, the relevant equivalent stress DV surface is shown for five fretting conditions after the first cycle. The triangle figures the end of the contact zone. Considering the parameters of Dang Van’s criterion (cf. Section 3), the results obtained with a normal force of 260 N/mm and a 5 ␮m-displacement do not predict crack initiation (Fig. 13a). The result is similar to that in cracking regime (Fig. 13b) but the maximum value is very close to the HCF limit of Ti6Al4V. This implies the possibility of crack initiation. Thus, crack initiation is also predicted for a normal force of 260 N/mm and a 20 ␮m displacement. Under wear conditions, Dang Van’s contour plot changes, but the highest values are still found at the end of the contact zone. Moreover, this is accompanied by heterogeneities of the stress field throughout the contact area due to the presence of debris. However, no crack

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Fig. 12. Evolution of the contact stress  22 during the fretting test for a zero horizontal displacement (P = 100 N/mm, ı = ±30 ␮m).

initiation has been observed in corresponding experiments [20]. The same remarks apply for the plastic shearing regime. A plastic behaviour should help to highlight high plastic deformation for these loading conditions. As a consequence, it can be thought that microcracking is present everywhere in the contact zone and can activate the wear phenomenon by cutting little pieces of material. These results can be compared to the model without third body ∗ is symmet(Fig. 14). As a result, the relevant equivalent stress DV rical with respect to the center of the contact zone. For various loading conditions, these results of crack prediction are qualitatively identical to those obtained with the particles: with a normal force of 260 N/mm and a 5 or 10 ␮m displacement, the results

do not predict crack initiation (Fig. 14a and b); crack prediction is obtained under cracking-wear, wear and plastic-shearing conditions (Fig. 14c–e). However, the initiation crack location is only obtained at the end of the contact zone. Consequently, the prediction provided by the fatigue model is rather different, as shown in Figs. 13 and 14. Since the debris particles have similar mechanical behaviour to the first bodies, the presence of the third body induces the critical area for fatigue spreads out on the whole surface. Moreover, there is no significant plastic deformation around the contact region. As a consequence, it can be though that microcracking is present everywhere in the contact and can activate the wear phenomenon by cutting little

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. . Fig. 13. Dang Van equivalent stress computed for the fretting cycle under various loading conditions: , limits of the contact zone; ., center of the contact.

Fig. 14. Dang Van parameter computed for 50 fretting cycles under various loading conditions without explicit third body modelling.

material elements. Note that the appearance of several cracks could be observed in a test of fretting in the absence of particles [34].

6. Concluding remarks The purpose of this study was to introduce improved models of the contact surface in fatigue fretting conditions. For this purpose, the third body is explicitely represented and its effect on the stress fields and on the resulting wear process are investigated. The simulations show that the debris are ejected from the contact area in gross and mixed slip due to a series of stick–slip mechanisms. On the contrary, the fragments remain trapped in the contact area for partial slip. In contrast with the smooth solution obtained when using perfect surfaces without a third body, the presence of debris introduces a series of local overstress in the contact area. As a result,

the heterogeneity of the stress field is larger, and critical points can be found everywhere in the contact area, instead of at both ends, like for the perfect case. This observation may lead to a renewed description of the wear process, that may result from multicracking in the contact area. Several aspects of fretting damage are neglected in the present computations. Among these aspects are the cyclic material response, finite strains just under the contact zone, adhesive wear and possible oxidation effects. Some of them are to be investigated in future works. • First, the improvement is to consider the detachment of material and consequently, the particle creation input to the third body. • On the other hand, the choice of the material behaviour is rather important. Indeed, the influence of plasticity on the plate and cylinder wear could be considered. Plastic behaviour is a source

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