A FEM fretting map modeling: Effect of surface wear on crack nucleation

Wear 330-331 (2015) 145–159

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A FEM fretting map modeling: Effect of surface wear on crack nucleation S. Garcin, S. Fouvry n, S. Heredia Ecole Centrale de Lyon, LTDS, Ecully, France

art ic l e i nf o

a b s t r a c t

Article history: Received 15 September 2014 Received in revised form 28 December 2014 Accepted 8 January 2015

Much research demonstrated that the fretting sliding condition greatly influences fretting damage. Small displacement amplitudes, inducing partial slip, favor cracking, whereas large dissipative sliding gross slip amplitudes favor wear. Considering a Ti–6Al–4V/Ti–6Al–4V cylinder/plane contact, this typical evolution was quantified by plotting the evolution of maximum crack length versus displacement amplitude. Under partial slip, the crack nucleated above a critical tangential loading, related to a threshold δCN_PS displacement amplitude. Above the sliding transition (δt), although tangential loading remained high, crack length decreased to zero at the gross slip threshold δCN_GS, due to surface wear extension which reduced contact stress and removed incipient nucleated cracks. This fretting damage evolution was simulated using an FEM code, enabling synergic modeling of wear and crack phenomena. The crack nucleation risk was quantified using an SWT parameter combined with a linear cumulative damage law. Surface wear evolution was simulated by a local friction energy density wear approach. The three displacement values, δt, δCN_PS and δCN_GS, were shown to be accurately predicted if, respectively, the FEM simulation takes account of the tangential accommodation of the test system, the damage law is calibrated using reverse analysis of experimental partial slip crack nucleation results, and the energy wear rate is determined from the wear volume analysis in gross slip regions next to the sliding transition. This very good correlation enabled “Material Response Fretting Map” modeling and optimization of palliative coating strategy. & 2015 Elsevier B.V. All rights reserved.

Keywords: FEM Ti–6Al–4V Wear simulation Fretting cracking simulation Fretting map

1. Introduction Fretting occurs when two bodies in contact undergo smallamplitude oscillations. It is observed in mechanical assemblies such as keyway-shaft couplings, blade disk contacts, etc. [1,2]. Fretting involves two sliding conditions, depending on the displacement amplitude [3–5]: partial slip, which involves an inner stick zone, and larger amplitude gross slip, inducing a full sliding response in the interface. In terms of evolution over time, three fretting regimes are usually distinguished: a stabilized Partial Slip Regime (PSR), a stabilized Gross Slip Regime (GSR), and a Mixed Fretting Regime (MFR) when the sliding condition evolves from partial to gross slip and reciprocally. The friction coefficient of metal interfaces usually tends to increase, so that only two stabilized sliding conditions need to be considered: Stabilized Partial Slip (PS) and Stabilized Gross Slip (GS). The transition between the two is defined by the so-called δnt displacement transition. Using ‘fretting map’ approaches, Vingsbo et al. [6] and Vincent et al. [7] showed that the evolution of fretting damage

n

Corresponding author. E-mail address: [email protected] (S. Fouvry).

http://dx.doi.org/10.1016/j.wear.2015.01.013 0043-1648/& 2015 Elsevier B.V. All rights reserved.

strongly depends on the sliding condition (Fig. 1). Under partial slip, above a threshold displacement amplitude δnCNðPSÞ , that is related to critical contact stress, a crack can nucleate and propagate up to a maximum value observed at the sliding transition, where tangential force is maximal. Above the gross slip transition, although the tangential load is still very high, the cracking risk tends to decrease, as does the maximum crack length. Waterhouse et al. first suggested that this non-monotonic evolution was induced by competition between wear and cracking processes [2]. Under partial slip, friction dissipation (i.e., the area of the Q–δ fretting loop) is very small and surface wear effects are negligible. On the other hand, above δnt , friction dissipation causes significant surface wear, affecting the cracking process in two ways: - by extending the contact area and modifying contact geometry, it significantly reduces the pressure profile and contact stress; - by removing the top surface, it progressively removes the incipient crack nucleated on the surface. The larger the displacement amplitude, the faster the wear extension and the reduction in contact stress, and the slower the cracking process. Hence, above a threshold gross slip displacement amplitude δnCNðGSÞ , the wear process is fast enough to fully eliminate

146

S. Garcin et al. / Wear 330-331 (2015) 145–159

αV (mm3/J)

Nomenclature a (mm) aH (mm) bp (mm) bpCN (mm) C S (mm/N) d (mm) D DðiÞ Dmax E (GPa) ESAP (GPa) Ed (J) f F n (N) F t (N) F t n ( 7N) GS GS% h (x) (mm) L (mm) Nc PS pmax (MPa) P (N/mm) Q (N/mm) Q*(7N/mm) Q nCN ( 7N/mm) R (mm) v V (mm3) V p (mm3) V c (mm3) WS (mm²)

worn contact radius, Hertzian contact radius, projected crack length measured from cross section expertise, projected crack length related to the crack nucleation condition, tangential compliance of the test system, FEM mesh size, accumulated damage, cumulated damage at the ith numerical fretting cycle, maximum cumulated damage value in the interface, elastic modulus, adjusted elastic modulus used in the SAP FEM layer to satisfyδnt;th ¼ δnt , friction energy dissipated during a fretting cycle, tangential force ratio (f ¼Q*/P), normal force imposed in the contact, tangential force imposed in the contact, tangential force amplitude imposed in the contact, stabilized gross slip condition, gross slip sliding ratio (i.e. relative proportion of gross slip cycles during a test), wear depth at the x position of the interface, transverse width of the contact (i.e. cylinder pad), fretting cycle at the crack nucleation condition, stabilized partial slip condition, maximum contact pressure, normal force per unit of length (P ¼Fn/L), tangential force per unit of length (Q¼ Ft/L), tangential force amplitude per unit of length (Q* ¼Ft*/L), partial slip tangential force amplitude related to crack nucleation condition, radius of the cylinder shape of the pad, Poisson's coefficient, total wear volume, wear volume of the plane, wear volume of the cylinder pad, worn surface of the total 2Deq profile,

Grec letters α (mm3/J)

friction energy wear coefficient used for FEM computations,

the cracking phenomena. In this case, only wear damage is observed. Hence, the Material Response Fretting Map concept developed by Vincent et al., [7] can be formalized using the following displacement amplitude analysis (Fig. 1a and b): - δn o δnCNðPSÞ : partial slip non-damage domain (I). - δnCNðPSÞ r δn rδnt : partial slip cracking domain (II). - δnt r δn rδnCNðGSÞ : gross slip combined wear and cracking domain (III). - δn ZδnCNðGSÞ : gross slip full wear domain (IV). The typical evolution of fretting damage can be transposed to the fretting fatigue endurance analysis (Fig. 1c). Fretting fatigue endurance

βðiÞ

δ (mm) δC (mm) δS (mm) δn (7mm) δnC ( 7mm) δnC;t ( 7mm) δng (7mm) δng;C ( 7mm) δnS (7mm) δnt (7mm) δnt;th ( 7mm) δnCNðGSÞ (7mm) δnCNðGSÞ;th ( 7mm) δnCNðPSÞ (7mm) δnCNðPSÞ;th ( 7mm) δngCN (7 mm) δngCN;th (7mm) ΔhðiÞ ðxÞ (mm) ΔDðiÞ Γ (MPa) Γ max (MPa) Γ C (MPa) φ (mJ/mm²) φðiÞ ðxÞ (mJ/mm²) m ΣEd (J)

friction energy wear coefficient defined from wear volume analysis, acceleration factor related to the ith numerical fretting cycle (i.e. number of (experimental) fretting cycles simulated during the ith numerical fretting cycle), measured displacement, contact displacement, displacement accommodated by the test system, measured displacement amplitude, contact displacement amplitude, contact PS/GS transition displacement amplitude, measured sliding amplitude, contact sliding amplitude, displacement amplitude accommodated by the test system, measured PS/GS transition displacement amplitude, FEM transition displacement amplitude, measured gross slip displacement amplitude above which crack nucleation is prevented, FEM gross slip displacement amplitude above which crack nucleation is prevented, measured partial slip displacement amplitude above which crack nucleation is activated, FEM partial slip displacement amplitude above which crack nucleation is activated, measured sliding amplitude defining the gross slip cracking domain (III), FEM sliding amplitude defining the gross slip cracking domain (III), increment of wear depth during the ith numerical fretting cycle at the x position, increment of damage generated during the ith numerical fretting cycle, SWT's fatigue parameter, maximum SWT's parameter generated in the interface during a fretting cycle, Threshold SWT parameter related to the bpCN crack nucleation condition at Nc, friction energy density dissipated in the interface during a fretting cycle, friction energy density dissipated during the ith numerical fretting cycle at the x position, coefficient of friction, Accumulated friction energy dissipated during a test.

decreases in the partial slip domain to a minimum value at the sliding transition and then increases in the following gross slip region when surface wear reduces the cracking rate. The sliding transition was explicitly described decades ago by Mindlin and Cattaneo for a Hertzian sphere/plane configuration [4,5]. FEM simulation can predict the sliding transition for more complex geometries. Prediction of partial slip crack nucleation was achieved by neglecting surface wear processes and applying multiaxial fatigue criteria, as in the Dang Van approach for infinite endurance conditions [8] or using SWT criteria for finite endurance situations [9]. Predictions were improved taking account of the severe stress gradients imposed by the contact stress, using a non-local process volume stress averaging strategy [10] or an equivalent critical distance approach [11].

S. Garcin et al. / Wear 330-331 (2015) 145–159

Fretting wear analysis was first addressed using the Archard wear law, which expresses the increase of wear volume as a function of accumulated Archard's work (ΣW), defined as the product of normal force and total sliding distance multiplied by the so-called Archard wear coefficient (i.e., V ¼KV .ΣW) [12]. This approach was improved by considering the accumulated friction energy dissipated in the interface (ΣEd), which takes account of the friction effect (i.e., V ¼αV .ΣEd) [13].

Partial Slip (stabilized)

normal force, Fn (N)

* δ CN (PS )

no damage (I)

cracking (II)

δ*t

Gross Slip (stabilized)

cracking & wear (III)

δ*CN(GS ) wear (IV)

displacement amplitude, δ* (±μm)

bp(μm) projected crack length wear volume V(μm3)

* δCN (PS)

* δCN (GS)

Fretting Fatigue Endurance (N cycles)

δ * (±μm)

fatigue endurance

147

These global descriptions of wear were translated to local descriptions using numerical approaches (FEM, boundary elements, etc.) or semi-analytical approaches [14–17]. These simulations showed that worn surface geometry evolved in a very few cycles to flat homogeneous pressure and shear profiles. This tendency was demonstrated to be a “natural evolution” of the interface, satisfying a constant wear depth rate over the whole fretted interface [17]. Crack nucleation in intermediate wear and cracking fretting regimes (II) was more recently modeled by Leen and co-authors [18–21]. This involved more complex modeling strategies to consider the change of surface wear and the activation of cracking process simultaneously using a cumulative damage description. The authors applied the Archard wear law and predicted fretting fatigue endurance using an SWT formalism. The results enabled global fretting fatigue endurance to be describe from partial to gross slip conditions (i.e., increased fretting fatigue endurance in the gross slip domain). However, these models were mainly developed using fatigue data found in literature, and did not take account of very severe contact stress gradient effects and wear data derived from the literature or obtained for different contact configurations. The objective of the present study was to palliate these limitations by combining complete experimental analysis of a Ti–6Al–4V/Ti– 6Al–4V cylinder/plane fretting interface with a representative FEM model. Dedicated fretting tests under partial slip conditions were performed to calibrate crack nucleation rate, and wear rate was directly determined from gross slip experiments. Combining a couple cracking and wear model, the δnt , δnCNðPSÞ and δnCNðGSÞ displacement amplitudes were predicted and Material Response Fretting Maps simulated.

2. Experiments 2.1. Plain fretting set-up

δ * (±μm) Fig. 1. Schematic illustration of fretting damage: (a) plain fretting Material Response Fretting Map, (b) cracking and wear evolution versus applied displacement for a given normal load (plain fretting); (c) related fretting fatigue endurance evolution as a function of displacement amplitude for a given normal load and fatigue stress (the crack nucleation process is assumed to be mainly controlled by the fretting stress).

X

10 mm

plane

extensometer

Plain fretting tests were carried out using a tension-compression MTS hydraulic system [22]. Normal force (Fn) was held constant while tangential force (Ft) and displacement (δ) were recorded (Fig. 2). For the present 2D cylinder/plane configuration, contact stress and damage are better described using forces per unit length [3], respectively: P ¼ F n =L and Q ¼ F t =L

ð1Þ

with L (mm), the transverse width of the cylinder/plane contact.

δ(t)

(Ti-6Al-4V) 10 mm Ra= 0.4 μm 10 mm

plane Ti-6Al-4V

counterbody Ti-6Al-4V pad

tangential force, Ft(N)

δ*

fretting scar

Fn 10 mm

cylinder shape R= 40 mm Ra= 0.4 μm

δg∗

cyl. pad (Ti-6Al-4V)

10 mm

friction energy

fixed

Ed (J)

fretting scar L= 5mm

10 mm

P= Fn/L Q= Ft/L

mobile

Ft (t)

partial slip

Ft *

tangential displacement δ (μm) gross slip

Fig. 2. (a) Design of the fretting specimens, (b) diagram of the fretting machine (c) partial and gross slip fretting cycles (determination of the quantitative variables used to analyze fretting damage).

148

S. Garcin et al. / Wear 330-331 (2015) 145–159

3D wear volume analysis

2D equivalent wear analysis

Plane scar Vp (μm3) (Volume integrated below the reference plane)

X

L Vc (μm3)

Cylinder scar X

Averaging procedure over L (3D => 2D)

x

2D(plane)

Averaging procedure over L (3D => 2D)

+

2D(cylinder) =

total equivalent 2D profile : 2Deq Total Wear Vol. V = Vc + Vp

L

Total Wear Vol. V = WS. L

2a

Worn surf. WS

Fig. 3. Schematic illustration of the methodology used to quantify wear damage: 3D surface profiles ¼ 4 extraction of mean plane and cylinder 2D wear profiles ¼ 4 computation of total equivalent 2Deq wear profile summing the plane and cylinder 2D profiles.

The fretting loop was plotted and the corresponding Ft*, Q* and δ* amplitude values were extracted. Friction energy Ed (J) was determined by integrating the δ–Ft loop. Small partial slip displacement amplitudes resulted in closed non-dissipative fretting loops, whereas larger gross slip displacement amplitudes led to quadratic dissipative fretting loops, favoring wear damage. After the test, the mean values of δ* and Q* were computed, with the accumulated friction energy ΣEd integrating the dissipated friction over the entire test duration. As underlined in [13], the displacement analysis in fretting interfaces is quite complex. The measured displacement (δ) includes the real contact displacement (δC) and the test system accommodation (δS) induced by the elastic deformation of the fretting test apparatus subjected to tangential loading: δ ¼ δC þ δS

ð2Þ

deduce: δng ¼ δngC

ð8Þ

Note that the sliding amplitude can be directly approximated from the fretting loop by determining the residual displacement when the tangential force is zero: δng  δðwhen Q ¼ F t =L ¼ 0Þ

ð9Þ

These aspects will be carefully considered in the following FEM modeling, so as to achieve pertinent correlations between simulated and experimental results. Note that just as for the force variables, the friction dissipated energy Ed is not affected by the elastic disturbance induced by the fretting frame and can directly be extracted from the measured (δ–Ft) fretting loop integration. 2.2. Materials and contact conditions

with δS ¼ C S  F t

ð3Þ

with CS the tangential compliance of the test system. Considering the amplitude condition, this implies: δn ¼ δnC þ δnS with δnS ¼ C S  F t n

ð4Þ

Contact sliding is moreover different from contact displacement. Its estimation depends on the sliding condition. Under partial slip conditions, most of the contact displacement is accommodated by elastic deformation of the interface, so that overall sliding amplitude (δngC ) is negligible. If

δnC r δnC;t ði:e: δn o δnt Þ; δngC ¼ δng ¼ 0 n

ð5Þ

n

with δC;t and δt , the contact and measured gross slip displacement transition amplitudes respectively and, δngC and δng defining the contact and measured sliding amplitudes. Under gross slip conditions, the contact combines a partial slip elastic accommodation activated during each reversal movement and a full sliding sequence during the tangential force plateau (i.e. Q* ¼m  P). Hence, assuming a given contact configuration and a constant friction value (μ), it follows for the contact displacement variables: if

δnC 4 δnC;t ; δnC ¼ δnC;t þ δngC

ð6Þ

The specimens (plane and cylinder) used in the study were composed of a 60% alpha–40% beta titanium alloy (Ti–6Al–4V) widely used in aeronautics, especially for fan blades and disks. Its mechanical properties are listed in Table 1 [22]. The plane consisted of 10  10  10 mm3 cubes and the cylinder pad consisted of equivalent 10 mm  10 mm  10 mm cubes with a face machined to achieve R¼ 40 mm cylinder radius and L¼ 5 mm transverse width, satisfying a 2D plane strain hypothesis. Each plane and cylinder surface subject to fretting was polished to achieve a low Ra¼ 0.4 mm. All specimens were cleaned with ethanol prior to testing. The normal force was fixed at Fn ¼1950 N, which corresponds to a linear normal force of about P¼ 390 N/mm. Assuming Hertzian contact conditions, this leads to a maximum pressure of pmax ¼450 MPa and a Hertzian radius of aH ¼0.55 mm. The contact pressure was chosen sufficiently low as to assume an elastic description whatever the applied sliding condition. All tests were performed at a frequency of 10 Hz. Partial-to-gross slip fretting damage was investigated over 105 fretting cycles, varying displacement amplitude from 71 to 780 mm. To determine crack nucleation endurance, several partial slip experiments were performed at 25  104, 50  104, 75  104 and 105 fretting cycles. 2.3. Fretting scar examination

Regarding the measured variables, this leads to: if

δn 4 δnt ;

δn ¼ δnt þ δng

ð7Þ

Combining equations (5) and (6) and considering the constant friction response during the gross slip sliding sequence, we can

2.3.1. Fretting wear examination After each test, the plane and cylinder specimens were ultrasonically cleaned with ethanol for 20 min to remove all dust and wear debris. Then 3D surface profiles were performed to

S. Garcin et al. / Wear 330-331 (2015) 145–159

inner part of the contact

Table 1 Mechanical properties of studied Ti–6Al–4V alloy.

Ti–6Al–4V

149

Elastic Young's modulus, E (GPa)

Poisson coefficient, v

Vickers hardness, Hv0.3

Plastic yield σY0.2% (MPa)

120

0.3

360

880

contact border (right side)

bp (μm) determine the wear volume generated on the plane (Vp) and cylinder (Vc), so as to determine total wear volume (Fig. 3):

40 μm

ð10Þ

Fig. 4. Illustration of the cross-section to quantify fretting cracking damage (bp: maximum projected crack length).

partial slip (μ≠ f)

1.2

gross slip (μ=f)

1.0

, GS%

FEM wear simulation consisted in comparing surface wear profiles. For the comparison to be relevant to the experiments, the following strategy was applied (Fig. 3): an equivalent 2D wear profile was extracted for each scar, averaging their 3D profiles over the transverse contact width; the equivalent total wear profile was then computed, summing the 2D cylinder and plane profiles. This equivalent wear profile analysis gave a global overview of the local wear damage in the interface and allowed direct comparison with FEM wear simulations. It also limited discrepancy: wear profiles can be significantly disturbed by mutual metal transfer between the two contact surfaces. For instance, metal transfer from the plane to the cylinder will generate a dip in the plane profile and a peak in the cylinder profile. By superimposing the cylinder and plane profiles, such effects are reduced. Moreover, the integrated worn surface WS of the 2Deq profile provided a physical illustration of the worn material volume removed from the interface. Multiplying this value by the transverse contact width yields the total wear volume (V) of the interface.

0.8

μ = 0.9

0.6

f = Q*/P

V ¼ Vp þVc

0.4

δ*t = ± 23μm

0.2

0.0 0

10

20

30

40

50

60

70

80

displacement amplitude, δ* (±μm)

1.2

200 180

1.0

160

PS

GS

140

0.8

120 0.6

100 80

0.4

60

bpCN = 10μm

δ *t = 23μm

40

0.2

20 0.0

0 0

3. Experimental results 3.1. Influence of displacement amplitude on friction and fretting damage A first set of experiments was performed at 105 cycles, varying the displacement amplitude from 71 to 780 mm to quantify the effect of displacement amplitude on fretting damage. Fig. 5a plots the evolution of the mean tangential force ratio and GS% criterion as a function of displacement amplitude [24]. The GS% ratio represents the proportion of gross slip cycles during a fretting test: if it equals 0, a pure partial slip regime is operating, whereas if it equals 1, a pure gross slip regime is activated; the mixed fretting regime is the intermediate domain between 0 and 1. Using this definition, the PS/GS sliding transition δnt was arbitrarily fixed at GS%¼0.5 which, for the studied conditions, yielded δnt E 723 mm. Above δnt , tangential force amplitude stabilized, allowing determination of a representative friction coefficient μ¼ 0.9.

total wear volume, V(mm3)

projected crack length, bp [μm]

2.3.2. Fretting crack examination After examination of surface wears, the plane specimens were cut along their median axis and polished in order to examine the cross section of the fretting crack. The fretting crack pattern displayed symmetrical slant cracks on either side of the fretting scar. The cracks systematically nucleated on the top surface and propagated toward the inner part of the interface, leading to a typical V-shaped crack pattern. The longest cracks were located at the contact border in partial slip conditions, but usually inside the fretted interface when gross slip was prevalent. To simplify analysis, fretting crack damage was quantified by measuring the maximum projected crack length (bp) over the whole fretting interface (Fig. 4) [23].

10

20

* δ CN (PS ) = ±5μm

30

40

50

60

70

80

* δCN ( GS ) = ±45μm

displacement amplitude, δ* (±μm) Fig. 5. Influence of fretting displacement amplitude (δ*) on sliding condition and damage evolution (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, L ¼5 mm , P¼ 390 N/mm, N ¼105 cycles): (a) evolution of tangential force ratio f ¼Q*/P and gross slip ratio GS% (f and GS% are non-dimensional variables); (b) evolution of fretting damage (bp: maximum projected crack length; V: total wear volume; PS : Stabilized Partial Slip; GS: Stabilized Gross Slip).

Under partial slip conditions, the tangential force ratio f¼ Q*/P is not equal to the friction coefficient, due to the elastic contribution of the stick zone. Fig. 5b compares the evolution of fretting damage (i.e., bp, maximum crack length, and V, total wear volume) as a function of the displacement amplitude. The results corresponded perfectly to the previous description presented in Section 1. Maximum projected crack length (bp) increased continuously with tangential loading in the partial slip condition up to a maximum value at the δnt sliding

S. Garcin et al. / Wear 330-331 (2015) 145–159

transition. It then showed a sharp decrease in the gross slip region, due to surface wear stress field accommodation. The crack nucleation condition was related to a threshold crack length bpCN. This value must be small enough to describe the crack nucleation process but large enough to limit scatter. The present study arbitrarily set bpCN at 10 mm, consistent with our optical facilities and corresponding to the radius of the alpha grain size, allowing micro-structural description of the crack nucleation process. Hence, on this definition, the two crack nucleation displacement thresholds were δnCNðPSÞ ¼ 75 mm for partial slip condition and δnCNðGSÞ ¼ 745 mm for gross slip condition (Fig. 5b). 3.2. Determination of energy wear rate

maximum projected crack length, b (μm)

150

50

experiments

45

25000 cycles 50000 cycles 75000 cycles 100000 cycles

40 35 30 25 20 15 10 5 0 0

V ¼ αV  ΣEd

ð11Þ

with αV ¼1.43  10–5 mm3/J 3.3. Quantification of crack nucleation endurance A common strategy to predict fretting cracking endurance is to apply SWT endurance formalisms [9,25]. However, this strategy is limited in two ways:

50

To eliminate these difficulties, an original strategy was adopted, consisting in determining crack nucleation rate by reverse analysis of partial slip fretting crack nucleation experiments. Fig. 7a plots

total wear volume, V (mm3)

0.80

150

200

V = 1.43 10 -5 × ΣEd

0.40

±45μm ±40μm ±35μm 0

5000

10000

15000

0.825

R²=0.9

250 200 150 100 50 0 25000

50000

75000

100000

fretting cycles, Nc (cycles) Fig. 7. Reverse identification of crack nucleation rate under partial slip condition (Ti–6Al–4V/Ti–6Al–4 V, cylinder/plane, R¼ 40 mm, L ¼ 5 mm P¼ 390 N/mm, δn o δnt E 723 mm): (a) plotting of bp versus tangential force amplitude for various test durations: identification of the Q nCN (bpCN ¼10 mm) crack nucleation conditions from a linear approximation (Table 2); (b) evolution of the corresponding threshold Q nCN (bpCN ¼ 10 mm) crack nucleation tangential force versus test duration.

the evolution of projected crack length versus partial slip tangential force amplitude for various test durations (i.e., N ¼25  103, 50  103, 75  103 and 105 cycles). For crack nucleation bpCN ¼ 10 mm, the evolution of Q nCN crack nucleation thresholds versus fretting cycles was determined (Table 2), and plotted in Fig. 7b. Crack nucleation rate was then expressed using a basic power law expression: ð12Þ

ð13Þ

4. Contact damage modeling

±30μm

0.00

400

with Q n0 ¼ 1.62  106, n ¼0.825, where Q n0 constant corresponds to the tangential force amplitude inducing a crack nucleation after a single fretting cycle.

R² = 0.54

0.20

350

300

which can be inversed as:
n n Q CN Nc ¼ Q n0

δ* = ±50μm

300

* QCN 1.62 106 (Nc)

350

Q nCN ¼ Q n0  ðNcÞ  n

0.60

250

400

0

- the crack nucleation process generated in fretting contacts is affected by severe stress gradients, which are not considered using such fatigue data; - this fretting crack analysis considers the incipient crack nucleation process (bpCN ¼10 mm), whereas fatigue data are related to the failure condition; fatigue crack nucleation endurance may be extrapolated by subtracting the crack propagation period from the failure endurance, but extrapolation is affected by many complex parameters, such as long-crack and above all short-crack propagation rates.

100

tangential force amplitude, Q*(N/mm)

tangantial force, Q*CN (N/mm)

The friction energy wear law used to quantify surface wear degradation consisted in relating total wear volume to the accumulated friction energy dissipated in the interface. To optimize prediction, the wear law was directly derived from the mixed cracking and wear gross slip sliding conditions (i.e., 723 rδ* r 750 mm). Fig. 6 confirms an almost linear evolution, allowing a representative (αV) energy wear coefficient to be determined [13]:

20000

25000

4.1. FEM modeling of surface wear

dissipated friction energy, ΣEd (J) Fig. 6. Evolution of total wear volume versus accumulated friction energy (Ti–6Al– 4 V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, L ¼ 5 mm, P¼ 390 N/mm, N ¼ 105 cycles, δnt E 7 23 mm o δn r 50 mm); determination of the friction energy wear coefficient αV ¼1.43  105 mm3/J.

A numerical program was developed to predict contact durability according to the experimental formulation previously determined. To do so, a finite element model was first created to represent the cylinder-on-plane configuration (Fig. 8). A “wear

S. Garcin et al. / Wear 330-331 (2015) 145–159

box” was designed around each side of the contact, illustrated in Fig. 3, using two-dimensional linear plane strain elements with four nodes. Fine wear box elements were chosen: 40 mm width and 40 mm depth. Embedment was imposed on the bottom and side surfaces of the plane, while the cylinder's top surface was subjected to both a uniformly distributed normal pressure and tangential fretting oscillations. Frictional contact conditions were described according to the Lagrange multiplier approach using an isotropic coefficient of friction (μ¼0.9, experimentally determined). Considering the Hertzian hypothesis, the maximum Von Mises stress generated in the interface at the sliding transition was found to be around 720 MPa which is lower than the Ti–6Al64V's yield stress allowing all FEM simulations to be computed in the elastic regime. 4.1.1. Correlation between experimental and simulated fretting displacement FEM analysis did not consider the experimental accommodation induced by the fretting test. To compare the experimental results, an additional System Accommodation Part (SAP) was added to the top of the cylinder pad modeled in the FEM to reproduce the experimental system accommodation displacement (δS ) (Fig. 8). The geometry of this SAP layer was fixed, and the elastic modulus of this part (ESAP) was adjusted through an iterative procedure in order to check that the simulated gross slip transition corresponded to the experimental value: δnt;th ðESAP Þ ¼ δnt

151

and δng sliding amplitudes. This supports the proposed SAP strategy to establish a simple correlation between experimental and simulated displacements. Obviously, different ESAP elastic moduli must be established if different test systems or specimen holders are used. 4.1.2. Surface wear modeling A numerical program, equivalent to the strategy developed in [13–15] was implemented to extract the surface shear, pressure and slip profiles to model the local surface wear processes, whilst the loading paths were extracted at the integration point (IP) to compute the cracking risk. The tangential shear stress distribution and the relative slip between each contacting node were used to obtain the local value of the dissipated energy density φðiÞ ðxÞ (in mJ mm  2) for each X-coordinate during the ith FEM iteration. According to the global–local equivalence of wear processes that had been ascertained in a previous study [26], a local description of the global wear law (2) was introduced using relation (8): ΔhðiÞ ðxÞ ¼ βðiÞ  α  φðiÞ ðxÞ

ð15Þ

with Δh(i)(x) being the local wear depth generated during the ith numerical iteration, α being the energy wear coefficient used for

δ*

tangential force 2000 Ft (N) 1500

ð14Þ

1000

The ESAP modulus was then held constant for successive simulations. Fig. 9 compares the experimental and simulated fretting gross slip cycles achieved at δ¼ 7 35 mm. A very good correlation was observed, providing equivalent fretting loop areas

500

-40

-30

0 -10 0 -500

-20

Table 2 Determination of the crack nucleation rate extracted from the reverse partial slip fretting crack nucleation investigation.

δ*g 10

25,000

50,000

75,000

100,000

Q nCN (N/mm) SWT Γ (MPa)

340 3.337

262 2.384

164 1.208

104 0.660

30

40

-1000 -1500

Nc

20

displacement, δ (μm) FEM Exp.

-2000 Fig. 9. Comparison between experimental and FEM fretting cycles (Ti–6Al–4V/Ti– 6Al–4V, cylinder/plane, R ¼ 40 mm, L ¼ 5 mm P ¼390 N/mm, μ ¼0.9, δn ¼ 735 mm).

RF1

WearBox System Accommodation Part (SAP)

RF2

Δh δ(t)

In depth distribution of node translation to simulate wear

Fig. 8. Illustration of the FEM model of the Ti–6Al–4V/Ti–6Al–4V cylinder/plane interface, integration of a System Accommodation Part (SAP) and illustration of the progressive node translation strategy used to simulated wear on both the plane and cylinder surface.

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S. Garcin et al. / Wear 330-331 (2015) 145–159

the FEM simulation and β(i) and an acceleration factor related to the ith numerical iteration [26]. For each node, the program computed the local wear depth associated with the value of the dissipated energy and the modified wear box's y coordinate. It is noteworthy that the “wear box” model is a very convenient approach for the modification of surface geometry, since initial coordinates are known and only the y value changes at each iteration. The program allows modeling of unilateral wear on either a plane or a cylinder, by activating only the plane wear box or only the cylinder wear box respectively. Bilateral wear, with similar or dissimilar wear rates, can also be simulated. In this investigation a homogeneous bilateral wear strategy was applied [26]. The total number of “iterations” (i.e., simulated fretting cycles) could be reduced by introducing an acceleration factor β(i). This accelerator β(i) corresponds in fact to the number of (experimental) fretting cycles simulated during the ith FEM numerical fretting cycle. The total number of fretting cycles is therefore given by: N¼

M X

βðiÞ

ð16Þ

i¼1

with M, the total number of FEM iterations. The β(i) parameter was optimized using a procedure developed in [26] to accelerate wear simulation while maintaining stable and continuous friction energy density profiles and continuous wear profiles. The β(i) acceleration parameter, defined as a function of the ratio between the maximum ΔhðiÞ wear increment and mesh size, was computed after each numerical iteration. As maximum wear depth rate decreases with surface wear extension, the β(i) parameter can be increased, reducing total computation time. To limit surface mesh distortion and enable simulation of very deep wear profiles, the ΔhðiÞ ðxÞ translation was not only applied on the top surface nodes but proportionally translated to the subsurface nodes. Hence stable and fast wear surface simulations could be achieved (Fig. 8). Combining all these aspects, the evolution of the total wear profile was compared for different fretting cycles. In-depth and axial wear extension of the fretted interface were clearly observed, leading to a conventional U-shaped wear morphology (Fig. 10). The corresponding friction energy profiles and related pressure profiles were also compared. As previously pointed out [14,15], surface wear extension tends to drastically reduce and flatten the initial Hertzian elliptic pressure profile. An almost constant pressure profile was observed at 5  104 cycles. This evolution is consistent with the assumption that the worn contact geometry evolves such that the wear depth rate is constant within the interface (i.e., flat φ(x) friction energy density profile) when the steady state wear regime is reached [17]. Fig. 11 compares the experimental and simulated 2Deq profiles obtained at δn ¼ 740 mm after 105 cycles. As expected, total worn surfaces (i.e., wear volumes) were nearly equivalent. The experimental integrated worn surface (WS) was slightly smaller than the simulated one, as the wear volume measured at δn ¼ 740 mm was slightly smaller than the global trend used to determine wear rate (Fig. 6). FEM simulation provided a correct description of global wear volume, but with some differences in surface profile morphology. The model tends to underestimate wear depth extension and overestimate axial extension. To interpret this difference, it must be kept in mind that Ti–6Al–4V/Ti–6Al–4V interfaces are characterized by significant transfers and adherent third body layers. Such a structure embedded in the center of the interface modifies normal and shear load transfer, favoring friction dissipation in the center of the fretting scar (Fig. 12). This can explain the faster

depth extension of wear compared to the axial one. This third body structure is partly removed during ultrasonic cleaning, which can impair the estimation of real wear volume, usually defined as the volume of worn material ejected from the interface during the fretting test. This suggests that more advanced FEM wear simulations, including the presence of a third body layer, are required to achieve more realistic fretting wear simulation. However, this approach, already developed in [27], is very complex and leads to significant numerical distortions. Hence, the present investigation assumed that the wear model was sufficient to capture the effect of surface wear on cracking processes under gross slip fretting. 4.2. FEM crack modeling 4.2.1. Application of the SWT multiaxial fatigue criterion under fretting wear gross slip conditions After each surface node translation, the stress path at the integration points at the center of each element of the plane wear box was computed, recorded and transposed to a multiaxial fatigue stress analysis. The SWT criterion, which consists in computing a stress-strain parameter defined as the maximum value of the product of normal stress and normal strain applied to the material, was considered [28]. All θ plane orientations were investigated (i.e., θA [  901,þ 90]) to obtain the maximum value of the Γ parameter and thus quantify crack nucleation risk at the M position of the integration point of the plane wear box:   Δεðθ; M Þ ΓðMÞ ¼ max σ max ðθ; M Þ  ð17Þ θ 2 with θ the angle between the normal of the studied material plane and the X-axis of the plane specimen surface, with σ max ðθ; M Þ ¼ max ðσðt; θ; MÞÞ

ð18Þ

t

and Δεðθ; M Þ ¼ max ðεðt; θ; MÞÞ  min ðεðt; θ; MÞÞ t

t

ð19Þ

where σ andε are the normal stress and strain on the θ-oriented plane respectively: the larger the Γ parameter, the larger the crack nucleation risk. Fig. 10 compares the evolution of surface Γ profiles (i.e., Γ determined at the first layer of the integration points) as a function of test duration and worn surface profile evolution. The maximum value of the Γ parameter (Γ max ) was systematically located at the contact borders. However, the axial location of Γ max followed the contact radius wear extension and sharply decreased due to a reduction of the pressure and shear profiles induced by surface wear. Hence, in contrast to partial slip conditions, where a constant Γ profile could be assumed during the whole test duration, a cumulative damage approach was required to predict crack nucleation risk under gross slip conditions. 4.2.2. Determination of the Γ C –Nc endurance chart To achieve this cumulative damage approach, a Γ C –Nc endurance chart must be drawn up. As mentioned previously, a reverse computation strategy was used to model the Γ C threshold for each Q nCN experimental crack nucleation condition (Table 2). The corresponding Γ C –Nc crack nucleation endurance chart is plotted in Fig. 13, and again approximated by a basic power law expression: Γ C ¼ Γ 0  Nc  m

ð20Þ 5

with m ¼ 1.14 and Γ0 ¼3.97  10 MPa for bpCN ¼ 10 mm,where Γ0 corresponds to the SWT parameter value inducing a crack nucleation after a single fretting cycle. Using this expression, we can

S. Garcin et al. / Wear 330-331 (2015) 145–159

Total surface wear profile axial position, X (mm) 0 wear profile, h (mm)

friction energydensity,

25

ϕ (mJ/mm²) cycles 1000 10 000 50 000 100 000

15

-0.005

10

cycles 1000 10 000 50 000 100 000

-0.01 -0.015

500 cycles 1000 10 000 50 000 100 000

30

20

-2-1012

-0.02

153

5 0 -1 0 1 axial position, X (mm) flattening of φ (x) => φ = Cst

-2

surface wear extension

4

pressure

SWT parameter, Γ(MPa)

3.5

p(MPa) 400

3

cycles 1000 10 000 50 000 100 000

2.5

300

2 200

2

1.5 1

100

0.5 0

0 -1 0 1 axial position, X (mm)

-2

-2

2

-1

01

2

axial position, X (mm)

lateral extension & flattening of pressure profile

lateral translation & reduction of SWT parameter

Fig. 10. Finite Element simulation of surface wear and SWT crack nucleation parameter distribution for a representative gross slip condition (Ti–6Al–4V/Ti–6Al–4V, cylinder/ plane, R¼ 40 mm, P¼ 390 N/mm, μ¼ 0.9, δn ¼ 740 mm, α ¼ αV ¼ 1.43  10–5 mm3/J): the reduction in pressure and shear stress intensity due to surface contact area extension significantly reduces the SWT crack nucleation parameter.

0.01 0.005

SWT parameter is expressed by:

axial position, X(mm)

ΔDðiÞ ¼

0 -2.5

-2

-1.5

-1

-0.5 0 -0.005

0.5

1

1.5

2

2.5

wear depth, h (mm)

-0.02

D¼

FEM simulation Experiment

-0.03 -0.035 -0.04

Fig. 11. Comparison between experimental and simulated total wear (Ti–6Al–4V/ Ti–6Al–4V, cylinder/plane, R ¼40 mm, P ¼390 N/mm, 105 cycles, μ¼ 0.9, δn ¼ 7 40 mm, α ¼ αV ¼1.43  10–5 mm3/J).

deduce Nc endurance for a given Γ value by applying:
m1 Γ0 Γ

M X

ΔDðiÞ

ð23Þ

i¼1

-0.045

Nc ¼

ð22Þ

The cumulative damage generated over N fretting cycles (i.e., M FEM iterations) was expressed by:

-0.01 -0.015

-0.025


m1 βðiÞ Γ0 with NcðiÞ ¼ NcðiÞ Γ ðiÞ

ð21Þ

4.2.3. Determination of the cumulative damage law As in [20], a linear Miner's cumulative damage law was considered. The increment of damage induced by the ith FEM fretting iteration associated to βðiÞ fretting cycles enduring the Γ ðiÞ

Crack nucleation is assumed to occur when D¼ 1. During wear simulation, the integration points progressively shifted below the initial surface position. In contrast to the Γ computation, integration of cumulative damage requires an adequate transfer field strategy. The following numerical scheme was implemented: after each ith numerical fretting cycle simulation, the damage value on the (i 1)th iteration was extrapolated from the previous (i 1)th integration point position to the new ith integration point position, leading to a Dnði  1Þ damage transfer field (Fig. 14). The actual damage field distribution was then computed by summing the ith increment of damage: DðiÞ ¼ Dnði  1Þ þ ΔDðiÞ

ð24Þ

This procedure was repeated from the beginning to the end of the test. Using this strategy, the cumulative damage field was updated and correlated to the actual position of the integration points, allowing direct comparison with the distribution of Γ. The surface

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S. Garcin et al. / Wear 330-331 (2015) 145–159

ϕ(x)

Wear without transfer and embedded third body (FEM hypothesis)

transfers and third body modifying the load carrying capacity (eliminated before the 3D surface profiles)

2D eq. surface profile defined from the 3D measurements performed after ultrasonic cleaning

ϕ(x)

Wear with transfers and embedded third body (Ti contact configuration)

Fig. 12. Illustration of transfer and third body effects on pressure field distribution and related evolution of surface wear assuming equivalent apparent wear volumes.

progressive shift of the X(Dmax) position between the initial and final contact radius where the Γ max value was systematically observed (Fig. 10). This typical evolution is illustrated in Fig. 17, where the X positions of the Γ max and Dmax damage values are plotted versus the simulated fretting cycles. Both parameters were located at the same contact border position at the first cycle, but a significant difference was observed with surface wear extension.

SWT parameter, ΓC (MPa)

4.5 4 3.5

Γc = 3.97 10 5 (NC )−1.14

3

R²=0.9

2.5 2 1.5

5. Comparison between experiments and modeling: results and discussion

1 0.5

5.1. Prediction of partial slip and gross slip crack nucleation displacement amplitudes

0 0

50000

100000

150000

fretting cycles, Nc Fig. 13. Evolution of Γ C values related to the Q nCN crack nucleation conditions (Fig. 7, Table 2) versus the corresponding Nc crack nucleation endurance (Ti–6Al– 4V/Ti–6Al–4V, cylinder/plane, R ¼40 mm, P ¼390 N/mm, μ¼0.9).

distribution of Γ and D used in the following analysis correspond to the line connecting the first layer of integration points of the deformed worn plane specimen surface. It corresponds to the following Y position: Y ¼ hðXÞ þd=2

ð25Þ

with h(X) the plane wear depth at the axial X position, and d the vertical dimension of the mesh size. 4.2.4. Influence of surface wear extension on fretting damage distribution Figs. 15, 16 and 17 compare surface and subsurface damage distributions simulated for δn ¼40 mm. The damage law used for this computation is related to Eqs. 20–24; the FEM local energy wear rate was assumed to be equivalent to the macroscopic rate (i.e., α ¼αV ¼1.43  10–5 mm3/J). As expected, the accumulated damage increased continuously and asymptotically, due to the reduction in Γ. The maximum damage Dmax was systematically observed on the top surface, just as for the Γ max parameter. However, except for the first cycle, Dmax was no longer found at the contact borders but rather inside the lateral zones between the initial and final contact radii (Fig. 16b). Computing Dmax involves integrating the Γ ðiÞ surface profiles from the beginning to the end of the simulated test, which implies a

Combined modeling of wear and cracking damage was used to simulate cumulative damage evolution as a function of displacement amplitude. Contact geometry and loading conditions similar to those applied in the experimental analysis were simulated (Fig. 18). As expected, a non-monotonic evolution of the Dmax variable was observed. In the partial slip domain, Dmax increased with the displacement value from zero to a maximum at the sliding transition. However, above the sliding transition, when surface wear was activated, Dmax decreased asymptotically. Using this representation, both partial slip δnCNðPSÞ and gross slip δnCNðGSÞ crack nucleation displacement amplitudes can be estimated by determining the intersection with the crack nucleation condition (DCN ¼1). This approach gave the following FEM predicted amplitudes: δnCNðPSÞ;th ¼ 78 mm and δnCNðGSÞ;th ¼ 7 45 mm. Despite significant differences between simulated and experimental surface profiles and the simplicity of the model, the coincidence between experimental and numerical results (i.e., δnCNðPSÞ ¼ 7 5 mm and δnCNðGSÞ ¼ 745 mm) was surprisingly good. FEM overestimation of contact area extension was expected to underestimate contact stress and therefore the δnCNðGSÞ displacement amplitude. This effect seemed not so strong, presumably because the Dmax variable located inside the fretted interface, is less affected by the contact border Γ max fluctuations than expected. 5.2. Modeling of fretting maps The previous numerical computation of δnt;th , δnCNðPSÞ;th and δnCNðGSÞ;th was extended to various normal force conditions to establish the

S. Garcin et al. / Wear 330-331 (2015) 145–159

damage field D

(i-1)th iteration Integration Point (IP) : Node

155

Interpolation of the (i-1)th damage field to the new ith position of the integration points

Computation of the Γ (i) distribution Γ

( i−1)

(i)

ith iteration => Increment of surface wear shifted position of integration points

Increment of damage deneratedduring the i th iteration

D

D*(i−1) transferred to

( i−1)

(i-1)th damage field

the (i)th damage field

ΔD(i)

Updated damage field at the new ith position of integration points Fig. 14. Illustration of the transfer field methodology used to compute the cumulative damage generated in the plane specimen during the fretting wear process.

worn area

cumulated 1.2 damage, D threshold crack nucleation

D% =D×100

1

cycles

0.8

100000 50000 10000 1000

0.6 0.4 0.2 0 -2

Axial position, X (mm)

-1

0

1

2

axial position, X (mm) Fig. 15. (a) Illustration of the cumulated D damage subsurface distribution; (b) surface distribution of the cumulated damage parameter as a function of the simulated fretting cycles. (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R ¼40 mm, P ¼390 N/mm, μ¼ 0.9, δn ¼ 7 40 mm; wear law: α ¼ αV ¼ 1.43  10–5 mm3/J; crack nucleation law: m¼ 1.14 and Γ0 ¼ 3.97  105 MPa).

corresponding 105-cycle numerical Material Response Fretting Map (Fig. 19). The boundary between the non-damage (I) and cracking domains (II) in the partial slip region displayed an almost vertical evolution, consistent with previous analytical computations and experimental results [22,23]. This confirmed that, under partial slip, when no wear is generated, the cracking process was controlled by a threshold cyclic tangential force amplitude, itself related to a threshold displacement amplitude. The evolution of the sliding transition was also consistent with literature. It displayed a nearly linear evolution, involving a contact displacement contribution and, above all, the test system accommodation, which was found to be proportional to the normal force [13]. From Eq. (4), we deduce:

displayed a very strong dependence on normal force: the greater the normal force, the larger the mixed wear and cracking domain. This typical evolution may be explained by synergic interactions between surface wear, contact stress and crack nucleation rate. An increase in normal force induces a proportional increase in gross slip contact stress (i.e., Q* ¼mP). However, it also increases the friction energy density dissipated in the interface and the pressure and shear flattening rate. The positive slope of the δnCNðGSÞ;th evolution suggests that the effect of the normal force on the increase in contact stress is predominant. Both δnCNðGSÞ;th and δnt;th displacement amplitudes simulated by FEM evolves with the applied normal load. In order to quantify the effect of normal loading on gross cracking domain (III), the following sliding variable is introduced (Fig. 19):

δnt ¼ δnC;t þ m  P  L  C S

δngCN;th ¼ δnCNðGSÞ;th  δnt;th

ð26Þ

In contrast to the partial slip crack nucleation boundary, the δnCNðGSÞ;th defining the transition between the mixed wear and cracking domain (II) and the full fretting wear domain (III)

ð27Þ

The δngCN sliding amplitude and related δngCN;th FEM computed value define the displacement range above the transition boundary where crack nucleation can be observed. It indirectly quantifies

S. Garcin et al. / Wear 330-331 (2015) 145–159

4

δ*= 23 μm

3.5

SWT parameter, Γmax (MPa)

3

δ* = 30 μm

δ*

δ* = 50 μm 2

1

0 -3

-2

1.2

4.0

Damage, D

-1

0

1

2

axial position, X (mm)

-1

δ*

3

1.0

3.0

0.8

2.5

0.6

2.0 1.5

0.4

1.0

crack nucleation (Nc)

0.5

damage , D max

156

0.2 0.0

0.0 0

20000 40000 60000 80000 100000

fretting cycles -2

Wear depth, h(mm)x100

-3

1.6

X(Γmax)= a (worn contact radius) 1.4

X(Dmax)

1.2 1.4

1st cycle 105

3.0

cycles

1.2

2.5

1.0

2.0

0.8

1.5

0.6

105

cycles

1.0

0.4

0.5

0.2

0.0

0.0

Damage, D

SWT parameter, Γ (MPa)

3.5

lateral position, X (mm)

1.6

4.0

1.0 0.8 0.6

aH (Hertzian contact radius)

0.4 0.2 0.0 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

40000

60000

80000

100000

fretting cycles

2

axial position, X (mm) Fig. 16. Comparison between D, Γ and h surface wear profiles (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, P¼390 N/mm, μ¼0.9; wear law: α ¼ αV ¼ 1.43  10–5 mm3/J; crack nucleation law: m¼ 1.14 and Γ0 ¼ 3.97  105 MPa): (a) comparison between D surface distribution and wear profiles after 105 cycles for different displacement amplitudes from partial slip transition (no wear) to large gross slip condition; (b) comparison between D damage and Γ SWT profiles under gross slip condition (δn ¼ 740 mm).

the extension of the gross cracking region so called mixed cracking and wear domain (III). Fig. 20 suggests a significant increase of the mixed cracking and wear domain with the applied normal load. This extension can be formalized using a power law expression. For the studied interface, this leads to: δngCN;th ¼ K P  P mP

20000

ð28Þ

where KP ¼3.5  10  8, and mP ¼3.4 Obviously, different KP, mP variables must be considered if various contact geometries or materials are investigated. However, this very simple expression provides a fast description of the normal load effect on gross slip cracking response. Another aspect concerns the effect of test duration. Fig. 21 confirms previous experimental findings [13,23] that an increase in the number of loading cycles promotes a symmetrical extension of cracking domains in both partial slip (II) and gross slip (III) domains. It also drastically reduces the normal load threshold below which no crack nucleation should be observed.

Fig. 17. (a) Evolution of the maximum damage Dmax and maximum SWT parameter Γ max generated in the interface as a function of the simulated fretting cycles; (b) evolution of the related X axial positions of Dmax and Γ max variables as a function of simulated fretting cycles (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, P ¼390 N/mm, δn ¼ 7 40 mm, μ¼ 0.9; wear law: α ¼ αV ¼ 1.43  10– 5 mm3/J; crack nucleation law: m¼ 1.14 and Γ0 ¼3.97  105 MPa).

A more interesting aspect concerns the effect of wear rate on fretting cracking response (Fig. 22). The partial slip crack nucleation boundary remained unchanged. Friction dissipation was so small that a variation in the energy wear rate could not influence the crack nucleation process. On the other hand, the gross slip crack nucleation boundary displayed significant evolution. An increase in the friction energy wear coefficient proportionally increases the wear volume. Alternatively, assuming constant friction and fatigue responses, the cracking risk under the gross slip condition is drastically reduced (i.e., reduction of the mixed cracking and wear domain III compared to the full wear response domain IV in Fig. 22). Indeed, the larger the energy wear coefficient, the faster the wear contact area extension, the faster the contact stress reduction and the smaller the cracking damage. This “competition” between wear and cracking processes is adequately monitored in many industrial problems, such as turbine engine dovetail blade/disk fretting interfaces, where “sacrificial” high-wear coatings are adopted to eliminate the danger of cracking. This tendency can be formalized through a parametric analysis of the influence of wear rate. Fig. 23 plots the evolution of the

S. Garcin et al. / Wear 330-331 (2015) 145–159

partial slip

3.5

30

gross slip

25

* -8 3 .4 δgCN ,th = 3.5 × 10 × P

μm)

20

2.0

CN,th (±

3.0

15

1.5

δ g*

maximum cumulated damage, Dmax

4.0

157

2.5

R²=0.99

10

1.0

5

δ *t , th = δ *t = 23 μm

0.5

0

0.0 0 5 10 15 20 * δCN (PS ),th = ±8μm

25

30

35

40

45

50

55

0

60

100

200

* δCN ( GS ),th = ±45μm

displacement amplitude, δ* (μm) 5

Fig. 18. Evolution of computed maximum damage Dmax after 10 fretting cycles as a function of displacement amplitude: estimation of δnCNðPSÞ partial slip and δnCNðGSÞ gross slip crack nucleation displacement amplitudes (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, P¼ 390 N/mm, μ¼ 0.9; crack nucleation law, m¼ 1.14 and Γ0 ¼ 3.97  105 MPa; wear law: α¼ αV ¼ 1.43  10  5 mm3/J).

450

PS

GS cracking & wear (III)

350

cracking (II)

no damage (I)

300 250 200

δ *gCN , th

normal force, P(N.mm-1)

400

normal force, P(N.mm-1)

δ *t, th

* δ CN (PS ), th

* δCN (PS ),th

δ *t, th

* δ CN ( GS ), th

wear (IV)

(IV) Wear

250 200 150

N, cycles 50 000 100 000

100

100

0 0

105 cycles 20

30

40

10

20

* δ CN (PS ), th

30

* δCN (GS),th

40

50

displacement amplitude, δ* (± μm) Fig. 21. FEM simulation of the fretting RCFM map: influence of test duration. (Ti– 6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, P¼ 390 N/mm, μ¼0.9; crack nucleation law: m¼ 1.14 and Γ0 ¼3.97  105 MPa; wear law α ¼ αV ¼ 1.43  10–5 mm3/J).

0 10

* δCN (GS),th

N.D. (I)

300

50

0

500

(III)

(II)

350

150

50

400

Fig. 20. FEM simulation of sliding crack nucleation amplitude (i.e. gross slip condition) versus the applied normal (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, P¼ 390 N/mm, μ ¼0.9, 105 cycles; wear law: α ¼αV ¼ 1.43  10–5 mm3/J; crack nucleation law: m¼ 1.14 and Γ0 ¼3.97  105 MPa).

400

450

300

P (N/mm)

50

displacement amplitude, δ* (± μm) Fig. 19. FEM simulation of the MRFM fretting map (running condition fretting map and material response fretting map) at 105 cycles; (Ti–6Al–4V/Ti–6Al–4V, cylinder/ plane, R¼ 40 mm, P ¼390 N/mm, μ¼ 0.9; wear: α ¼ αV ¼ 1.43  10–5 mm3/J; crack nucleation law: m¼ 1.14 and Γ0 ¼3.97  105 MPa).

simulated δngCN;th sliding amplitude as a function of the inverse value of the applied FEM energy wear coefficient (α) for a given P ¼250 N/mm normal loading condition. A linear increase is observed, which suggests that the δngCN sliding amplitude, and therefore the gross slip cracking domain (III), is inversely proportional to the wear rate of the material. The following inverse function is extrapolated:

gross slip cracking domain can be kept within a harmless narrow interval. Future experimental research will be undertaken to confirm this conclusion, although it is already in line with the methods used in practice in industry to limit problems associated with fretting cracking.

6. Conclusion

ð29Þ

Combining an experimental investigation of Ti–6Al–4V/ Ti–6Al–4V cylinder/plane fretting damage evolution with a Matlab–Python–Abaqus FEM code allowing synergic modeling of wear and cracking phenomena, the following points emerged:

with δngCN;thðαV Þ ¼5.8 mm and Kα ¼5.8  αV ¼8.3  10  5 mm (mm3/J). As a conclusion, the possibility of formalizing the competition between cracking and wear processes and the related simulation of Material Response Fretting Maps appears to offer a powerful tool to predict damage evolution and optimize palliative fretting strategies. By adjusting the gross slip wear rate, the dangerous

- Experimental investigation confirmed a non-monotonic evolution of the projected crack length generated in fretting interfaces with the applied displacement amplitude. It first displayed an increase in the partial slip domain due to a rise in tangential stress, but decreased above the gross slip sliding transition due to extension of the worn surface area, a related

δngCN;th ¼ δngCN;thðαV Þ 

α  V

α

¼ Kα 

1 α

158

S. Garcin et al. / Wear 330-331 (2015) 145–159

450

* δCN ( PS ), th δ * t,th

350

No damage (I)

normal force, P(N.mm-1)

400

300 250 200

Cracking (II)

 crack nucleation rate is formalized through reverse analysis of Cracking & Wear (III)

* δCN ( GS ), th

α Wear (IV)

150

α FEM αV αV / 2

100 50

* δCN ( PS ), th

* δ CN ( GS ), th

0 0

10

20

30

40

50

60

displacement amplitude, δ*(± μm) Fig. 22. FEM simulation of the RCFM fretting map after 105 fretting cycle: influence of the α friction energy wear coefficient used for the computations (Ti–6Al–4V/Ti–6Al–4V, cylinder/plane, R¼ 40 mm, P¼390 N/mm, μ¼0.9; m¼1.14 and Γ0 ¼ 3.97  105 MPa, αV ¼1.43  10–5 mm3/J).

20

δ *gCN,th = 5.8 ×

18

αV 8.3 × 10 -5 = α α



partial slip fretting crack nucleation conditions obtained for different test durations; and an energy wear law is defined from wear data obtained in the mixed cracking and wear region located just above the gross slip sliding transition boundary.

A FEM parametric investigation was undertaken and concluded that an increase in loading cycles increases cracking risks in the partial (II) and gross slip (III) domains. Moreover, an increase in wear rate did not modify the partial slip crack nucleation boundary but significantly reduced the gross slip cracking risk region. From this numerical investigation it can be concluded that, for the studied cylinder/plane interface, the gross slip crack nucleation domain is inversely proportional to the wear rate. This analysis underlines the potential interest of such numerical investigation to optimize palliative coatings to limit the fretting cracking problem. Other aspects, such as fatigue properties, contact geometry, variable loading conditions and fretting fatigue, will be investigated in future research. Micro-structural aspects could also be considered, although recent modeling studied suggested wide scatter of crack nucleation locations compared experimental results [21,29–31]. References

R²=0.999

16 14 12 10 8 6 4 2 0 0

1

2

3

4

Fig. 23. Evolution of the FEM simulation sliding crack nucleation amplitude (δngCN;th ) as a function of the applied α FEM energy wear coefficient (Ti–6Al–4V/Ti–6Al–4V, cylinder/ plane, R¼40 mm, P¼250 N/mm, μ¼ 0.9, 105 cycles, αV ¼1.43  10–5 mm3/J; crack nucleation law m¼1.14 and Γ0 ¼ 3.97  105 MPa).

reduction in pressure and shear surface stress fields, and progressive elimination of incipient cracks by wear. For a given test duration, the crack nucleation domain could be bracketed by determining the partial slip δnCNðPSÞ displacement amplitude below which no crack can nucleate and the gross slip δnCNðGSÞ displacement amplitude above which cracks are potentially erased by surface wear. - To quantify this damage “competition” process, an FEM code, allowing coupled modeling of wear and cracking phenomena, was developed to simulate the evolution of the cumulative cracking damage, taking account of the evolution of surface wear. Despite some differences between experimental and simulated wear profiles, this numerical analysis predicted δnt , δnCNðPSÞ and δnCNðGSÞ amplitudes well if:  test system compliance was taken into account by including a “SAP” elastic layer in the elastic properties of the whole FEM model, allowing simulation of test apparatus interference on experimental displacement variables;

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