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Development of a contact compliance method to detect the crack propagation under fretting
ARTICLE IN PRESS
Tribology International 39 (2006) 1262–1270 www.elsevier.com/locate/triboint
Development of a contact compliance method to detect the crack propagation under fretting K. Elleucha,, H. Proudhonb, C. Meunierb, S. Fouvryb a
LASEM, De´partement de Ge´nie des Mate´riaux, Ecole Nationale d’Inge´nieurs de Sfax, B.P. W-3018, Sfax, Tunisia b LTDS, CNRS, UMR 5513, Ecole Centrale de Lyon, France Available online 20 March 2006
Abstract Partial slip fretting conditions are classically known to favor contact crack nucleation and crack propagation. Considered as a plague for modern industries, numerous theoretical researches have been conducted during the past two decades to predict such fretting damage. However, a review of last few years critically outlines the need of precise and in-situ experiments to qualify and quantify the given models. To palliate such aspect, an original approach which consists in following the contact stiffness evolution as an indicator of the fretting cracking phenomena, has been developed. Applied for an aluminium/steel contact, it demonstrates that the incipient crack propagation is related to a discontinuous decrease of the contact stiffness. Based on this online analysis, a fretting cracking endurance parameter has been extrapolated to develop fast and low cost fretting cracking endurance chart. A FEM analysis has been performed in an attempt to formalize the given experiments. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fretting; Cracking; Contact stiffness; Crack arrest; Aluminium alloy
1. Introduction Cracking induced by fretting loadings has been deeply investigated during the past two decades. The crack nucleation phenomena have been formalized applying multiaxial fatigue criteria combining averaged stress volume approaches to take into account the contact stress gradient and the size effect [1]. Alternatively, this stressmaterial length dimension approach has been more recently captured through the development of the ‘‘notch similitude’’ concept [2–5]. The crack propagation induced by combined contact fretting and macroscopic fatigue stressing has been quantified by different numerical approaches like distributed dislocation methods [6–8] and weight function formalism [9]. These methodologies consist to estimate the stress intensity factors at the tip of the crack enduring complex fretting fatigue loadings and to predict the specimen endurance using modified crack propagation Paris laws [3].
Corresponding author. Tel.: +216 74 274 088; fax: +216 74 275 095.
E-mail address: [email protected] (K. Elleuch). 0301-679X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2006.02.015
If most of these models have been well established, a critical aspect concerns currently the experimental validation of such approaches. Indeed to calibrate the different models, the following questions need to be answered:
How many cycles are required to activate the incipient crack nucleation? What is the incipient crack propagation kinetics and when a fretting fatigue crack leaving the contact stress field domain will be fully controlled by the macroscopic fatigue stress field?
Numerous other experimental aspects are required to definitively establish pertinent fretting cracking modelling. Heavy in situ tomography X observations are expected to provide such fundamental scientific materials [10]. However, the current limitations of the tomography X like the specimen size (1 mm2 for the section) clearly restrict the investigation of the contact size effect, which has been demonstrated to be a dominating factor of fretting fatigue damages. Alternative approaches still need to be investigated.
ARTICLE IN PRESS K. Elleuch et al. / Tribology International 39 (2006) 1262–1270
Exploiting the transparent properties of polymers Chateauminois and co-authors [11] have carefully quantified the crack nucleation process and incipient crack propagation of PMMA polymer submitted to fretting wear test loadings. They also investigate the evolution of the contact stiffness evolution as a function of the contact cracking extension. Unfortunately, such a direct observation is not possible in the present aluminium/steel contact. However, it appears interesting by combining finest fretting cycle analyses and coupling FEM modelling, to evaluate how the contact stiffness analysis could be adapted to achieve an undirected in situ observation of fretting cracking. 2. Experiments 2.1. Fretting wear tester A schematic diagram of the tester is shown in Fig. 1. The system used has already been described elsewhere [12]. In particular, it permits the online acquisition of the tangential force (Q) evolution versus the imposed displacement (d). Thus, the fretting loop Qd can be plotted (Fig. 2(a)). A dynamic overview of the fretting test can be illustrated via the fretting log, which is a superimposition of fretting cycles (Fig. 2(b)). Some quantitative fretting parameters can also be extracted like: the tangential force and displacement amplitude (Q*, d*), the dissipated energy (Ed), the cycle opening (d0) abusively associated under gross slip conditions to the sliding amplitude (dg) and finally the measured contact Stiffness (K) defined as the slope of the fretting cycle at the loading amplitude points (Fig. 2(a)) (i.e. slope of the loading path when Q ¼ Q or
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negative slope of the unloading path when Q ¼ þQ ) [13]. To calculate this slope, only the quarter of the fretting cycle was considered (i.e. 250 loading-amplitude points). 2.2. Material specimens Flat samples of A357 aluminium alloy (30 20 10 mm3) have been manufactured from a casting bar. The chemical composition of the A357 is shown in Table 1. The surfaces of these specimens were polished with abrasive paper, finished with diamond paste, until a 0.1 mm roughness (Ra) was reached, and then ultrasonically cleaned in pure alcohol. The counter body is the AISI 52100 steel. It consists of a cylinder (20 20 mm2) with a machined spherical end (R ¼ 100 mm). The tested surface, i.e. the spherical part, was polished with a suitable device until 0.1 mm roughness (Ra) was reached. Table 2 summarizes the principal mechanical properties of the tested materials. These materials are used in order to simulate fretting damage encountered in some aeronautical application. Especially, when steel bearings are brought into contact with aluminium crankcase [14].
Fig. 2. (a) Identification of the fretting loading parameters from the fretting cycle analysis and (b) illustration of a fretting log.
Table 1 Chemical composition of A357 aluminium alloy
Fig. 1. Fretting wear test apparatus: (a) schematic and (b) picture of the setup.
A357
% Si
% Mg
% Fe
% Cu
%Ni
6.5–7.5
0.4–0.7
0.2
0.1
0.05
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100
Table 2 Mechanical properties of tested materials E (GPa)
n
Rm (Mpa)
Rp0.2 (MPa)
Hv (GPa)
sd (MPa)
A357 52100 steel
70 210
0.36 0.3
330 2200
200 2000
1.13 8.40
60 800
K (N/µm)
Materials
δ* = 8µm δ* = 7µm
80 60 40
Nc 20
2.3. Experimental conditions
0 1
10
(a)
102 103 Fretting cycles
104
105
6 δ* = 8µm δ* = 7µm
5
Nc
4 δo (µm)
The applied normal load is fixed at 230 N, corresponding to 250 MPa maximum Hertzian pressure for the studied flat on sphere contact configuration. Tests have been done with a frequency of 5 Hz up to 105 cycles. The imposed displacement amplitude (d*) ranged from 3 to 20 mm in order to cover partial and mixed regime conditions [15] (under mixed fretting regime the contact endures gross slip sliding before stabilizing under partial slip condition). All tests were carried out under laboratory-controlled conditions (temperature, 2372 1C; relative humidity, 40710%).
3 2 1
3. Experimental analysis of the contact stiffness evolution
0 1
Fig. 3(a) compares the evolution of the measured contact stiffness as a function of the fretting cycles for two different displacement amplitudes. The observed contact stiffness evolutions are characterized by an initial sharp increase of the contact stiffness followed by a smoother progression. However, if the smaller displacement amplitude condition (d ¼ 7 mm) maintains a constant evolution a discontinuity is observed for the largest amplitude (d ¼ 8 mm). The discontinuity has been identified at a given number of cycles, Nc, after which there is a decrease of the contact stiffness. In the same time, an increase of the cycle opening value (d0) is also observed (Fig. 3(b)). It appears interesting to relate the symmetrical evolutions of the contact stiffness and the cycle opening. These two parameters appear really pertinent to quantify the interface contact evolution and potentially to detect fretting cracking phenomenon. The given study mainly focuses on the contact stiffness parameter due its most extended application in the contact mechanics research field. However, displaying a higher amplitude evolution, it suggests that the aperture variable is potentially more interesting to characterize the interface evolution. More investigations are needed to implement the application of this latter variable. To interpret the initial increase of the contact stiffness (i.e. symmetric decrease of the cycle aperture) it appears essential to consider some basic theoretical aspects. The elastic formalism of the sphere/flat contact stiffness (KCth), first introduced by Mindlin for a full sticking contact (i.e. condition which is presumably verified at the cycle amplitudes) is expressed by [13,16] K Cth ¼ aG ,
(1)
10
(b)
102 103 Fretting cycles
104
105
Fig. 3. Fretting contact responses versus the fretting cycles as a function of the applied displacement amplitude: (a) evolution of the measured contact stiffness and (b) evolution of the cycle opening.
where G* (Pa) is defined as an effective elastic shear modulus expressed by G ¼
8G 1 G 2 Ei with G ¼ G1 ð2 n2 Þ þ G 2 ð2 n1 Þ 2ð1 þ ni Þ
(2)
and a the sphere/plane Hertzian contact radius expressed by 3PR 1=3 a¼ , (3) 4E where P and R are, respectively, the normal force and the sphere radius whereas E* is defined as the effective contact modulus expressed by 1 1 n21 1 n22 ¼ þ , E E1 E2
(4)
where n1 and n2 are Poisson coefficient of the plane (1) and the sphere (2), respectively, E1 and E2 are elastic modulus of the plane (1) and the sphere (2), respectively, G1 and G2 are shear modulus of the plane (1) and the sphere (2), respectively. Note that even verifying perfect contact conditions, the measured contact stiffness is not equal to theoretical estimation. Indeed, the fretting wear apparatus system compliance must be considered. The measured stiffness (K) is in fact a contribution of two stiffness: the contact (KC)
ARTICLE IN PRESS K. Elleuch et al. / Tribology International 39 (2006) 1262–1270
and the experimental system (KS) stiffness [17]. 1 1 1 ¼ þ . K KC KS
(5)
Specific attentions have been done in previous works to extract the system perturbation, to quantify the sliding transition and identify the local friction coefficient operating through the sliding annulus under partial slip conditions [17]. However, in the present study, only the stiffness fluctuations are required to be detected and do not request a precise and difficult determination of the exact contact stiffness. Indeed, because the system stiffness is assumed constant, the observed fluctuations are only function of the contact stiffness variations. Considering the theoretical expression given in Eq. (1) it is assumed that the contact stiffness is only function of the material elastic properties and the apparent contact area. It is not directly dependent on the tangential force loading or the friction coefficient. Therefore, to interpret the experimental evolutions, it appears fundamental to consider the fretting sliding history imposed through the interface as well as the interface morphology. Indeed, native surfaces display specific roughness and the real contact area is in fact smaller than the theoretical Hertzian estimation. As illustrated in Fig. 4, the contact is running under gross slip condition during the first 1000 cycles. During this period the interface is extensively modified with a critical increase of the metal interaction promoting and increase of the friction coefficient but also an extension of the real
Fig. 4. Evolution of the sliding behaviour as a function of the fretting cycles (Q*): tangential force amplitude; A ¼ Ed/(4Pd*): energy sliding criterion [14], if A40:2 the contact is running under gross slip, if Ao0:2 the contact is under partial slip condition): (a) d ¼ 7 mm and (b) d ¼ 8 mm.
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contact area. Moreover during this short period a significant wear of the interface is observed which again increases the operating contact area. The smoother increase of the contact stiffness observed after the partial slip stabilization is less easily explained. A stable evolution should be expected as partial slip condition should not activate any critical interface modification. This stabilization is in fact verified by the cycle opening variable but not by the contact stiffness. Hence other aspects like a possible modification of the interfacial third body rheology and possible hardening plastic mechanisms should be considered. One interesting conclusion of this investigation is the monotonous increase of the contact stiffness with the partial slip stabilization. The incoming question is then how to explain the discontinuous decrease observed for the highest loading condition? Post mortem analyses have been conducted on these fretting scars in order to interpret the reason of these discontinuities. In particular, cross section investigations have shown some interesting results (Fig. 5). For d ¼ 7 mm, very short cracks sited at the contact border have been observed. These cracks have never cross the first encountered grain boundary. Hence, their maximal length does not exceed a critical length, Lc, which corresponds to the grain size (LcE40 mm). This result was confirmed even under very long fretting test (d ¼ 7 mm, N ¼ 106 cycles). For d ¼ 8 mm, cracks are longer; their maximal length can reach 450 mm crossing several grain boundaries. It is interesting to mention that very short cracks have been identified even for the smallest loading situations. Therefore, it can be concluded that the stiffness discontinuity can be related to an incipient crack propagation activated when the loading condition (i.e. tangential force amplitude) permits the incipient crack to overpass the first and strongest barrier: the grain boundary. Hence, Miller approach can be seriously considered at least for the studied aluminium alloy [18]: ‘‘in polycrystalline materials, there is no phase of crack initiation since it is immediate. Then cracks can propagate or stop depending on loading conditions’’. The transition appears very abrupt. When the crack overpasses the first grain boundary, the successive grains are rapidly passed away and the initial few micron micro-cracks reach a stabilized in 300–500 in few thousand cycles. The introduction of such long cracks significantly increases the contact compliance and inversely decreases the contact stiffness. The present studied fretting test conditions are displacement controlled. The drop of the tangential amplitude associated to a contact stiffness decrease can, therefore, be explained and quantified. Therefore the contact stiffness discontinuity (Nc) be related to the activation of the incipient crack propagation and this methodology could be extended to development fast and non expensive fretting cracking endurance charts (Fig. 6). As previously demonstrated by Mindlin [13,16], a close relationships exists between d* and Q* under partial slip
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Fig. 5. Cross-section observation of a cracked fretting scar (d ¼ 9 mm, P ¼ 230 N, f ¼ 5 Hz and N ¼ 105 cycles).
70
200
60 50
150
40
100
30 20
50
10
0
0 0
(a)
2
4 6 8 10 12 displacement amplitude δ* (µm)
14
100
350
90
maximum crack length I (µm)
300
80
250
70 2
200
60 50
150
40 30
100 Qc*
50
Lc Lc 1
0 0
100 200 300 tangential force amplitude Q* (N)
20 10
fretting cycles (Nc x 1000)
Q* (N)
; L (µm)
80 250
fretting cycles (Nc x 1000)
90
300
(b)
evolution of maximal crack length versus maximal tangential force (Q*). Two domains can be identified:
100
350
0 400
Fig. 6. Introduction of fretting cracking endurance charts (defined from the online contact stiffness analysis, a). Evolution of the maximum crack length (L), related tangential force amplitude (Q*) and contact endurance (Nc) as a function of the applied displacement amplitude (d*). (b) Evolution of the maximum crack length (L) and contact endurance (Nc) as a function of the maximum tangential force amplitude (Q*).
These two regions are clearly identified when defining the critical length Lc and a critical tangential force Qc. When Q* is bigger than a threshold tangential force amplitude QcE305 N there is a discontinuity of the contact stiffness and simultaneously the crack propagates. On the other hand, below this critical tangential force amplitude, the incipient crack is limited to a grain dimension (LoLc). For such condition, the crack extension is not sufficient to influence the contact compliance inducing a constant evolution of K. Therefore, by considering the contact compliance as a pertinent parameter to detect the crack propagation condition, ‘‘fretting cracking wholer curves’’ could been defined which consist to report the critical number of cycles (Nc) defined from the contact stiffness discontinuity (i.e associates to the crack propagation condition: L4Lc) as a function of the applied tangential force amplitude (Fig. 6(b)). To evaluate the dynamic evolution of the crack progress, a constant cracking condition has been applied (d ¼ 10 mm) and an interrupt tests procedure has been applied (Fig. 7). This experimental procedure has permitted to outline following aspects:
elastic conditions. However, in order to avoid any experimental artefacts, the given analyses focuses on the tangential force amplitude Q* instead of d*. Fig. 6(b) shows
domain 1: crack arrest domain, domain 2: incipient crack propagation domain.
Up to N1 cycles, max(Q*)oQc, there is no discontinuity in the K variation and cross section investigation has confirmed that maximal crack length (L1) is smaller than Lc. Up to N2 cycles, max(Q*)4Qc, there is a decrease of K and crack length (L2) is bigger than Lc. Up to N3 cycles, crack propagation progress and maximal crack length (L3) longer than L2 has been detected. This condition is also associated to a critical
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N2 N2
350
N4 N4
300
Q* (N)
250 200
N3 N3
150 N1 N1
100 50 0 1
10
100
1000 N cycles
10000
100000
Fig. 7. Illustration of the fretting test methodology to bracket the threshold tangential force amplitude marking the incipient crack propagation condition: four interrupted test have been carried out until N1, N2, N3 and N4 cycles with the same loading condition (d ¼ 10 mm, P ¼ 230 N).
drop of the contact stiffness and a significant decreased of tangential force amplitude. When performing the final test up to N4 cycles, a similar crack length defined at N3 cycles has been found (i.e. Lmax ¼ L4 ¼ L3 ). In fact, the crack has reached the maximum possible crack extension associated to the studied loading conditions. Formalized through a conventional crack arrest description this maximum crack length is in fact directly connected to the contact stress field distribution defined from contact size and the imposed cyclic tangential loading [8,19].
This analysis suggests that two crack arrest conditions must be considered. A first one associated to a metallurgical aspect marking the interaction of the incipient crack with the first and strongest barrier: the grain boundary (L ¼ Lc ) (i.e detected via a contact stiffness analysis). A second which is basically dominated by the subsurface decrease of the contact stress field which can be explained and quantified through a classical crack arrest description via a local crack tip stress intensity factor analysis (L ¼ Lmax ) [8,19].
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The cylinder (R ¼ 49 mm) on flat configuration, associated respectively to the 52100 and A357 elastic properties (Table 2) has been meshed. The contact is supposed elastic so only the Young modulus and Poisson coefficients variables have been considered. The loading conditions have been adjusted to achieve similar Hertzian pressure Po ¼ 250 MPa, (i.e. P ¼ 160 N=mm) inducing a semi width contact a ¼ 400 mm. Perpendicular cracks, located symmetrically at the contact borders have been designed. Six crack lengths have been considered to evaluate the contact stiffness evolution (25, 50, 75, 100, 200, 275 mm) (Fig. 8). CPE4 quadratic elements have been used for the mesh with a specific refinement near the crack tips (Fig. 8). Fig. 9 depicts the distribution of Von Mises equivalent stress in the plane after applying the normal load and two fretting cycles. Fig. 10 compares the computed fretting cycles with and without cracks at the contact borders. A very smooth decrease of the contact stiffness and an equivalent increase of the cycle opening are detected. To rationalize the analysis, the calculated numerical contact stiffness have been normalized versus the value defined for a contact without any crack (K0). Symmetrically, the respective crack length has been normalized versus the contact semi width (a). Fig. 11 reveals a linear decrease of the normalized contact stiffness versus the normalized crack length. No stiffness discontinuity is observed which infers that the experimental stiffness drop should be related to a critical variation of the crack propagation rate. Indeed, passing the first grain boundary, the crack propagation should be very fast promoting by consequence the observed contact stiffness discontinuity. A second aspect concerns the amplitude of the variation. In spite of the very large computed crack length range, the computed relative contact stiffness drop remains inferior to 5%, which is significantly smaller than the 20% variation observed for the experiments (Fig. 3(a)). Moreover, after Eq. (5), the introduction of constant system compliance even tends to minor the measured variation compared to the contact real
4. FEM modelling of the contact stiffness evolution versus the crack length The contact stiffness analysis appears as a pertinent approach to detect fretting cracking phenomena. However the previous analysis outlines that a crack must display a significant length to be identified via the contact stiffness parameter. In order to evaluate the resolution of this methodology, a parametric FEM analysis of the contact stiffness versus the crack length has been conducted. Due to the numerical cost of a complete 3D fretting modelling, an equivalent elastic 2D cylinder/plane fretting contact configuration has been defined (ABAQUS).
Fig. 8. Illustration of the FEM crack mesh at the fretting contact borders.
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S, Mises (Ave. Crit.: 75%)
2
ODB: cyl_plan_mesh_fissure_L100_ele_3696.odb
ABAQUS/Standard 6.3-1
Thu Jun 17 13:21
1 Step: DEPL2+Q Increment 10: Step Time = 2.000 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: +1.000e+00
3
Fig. 9. Distribution of the Von Mises stress in the plane submitted to fretting at a displacement amplitude (+d*).
1.00
1000
500
0 -30
-20
-10
0 -500
-1000
10 20 displacement amplitude δ* (µm)
30
crack length = 275 µm no cracks
-1500
normalized contact stiffness K/Ko
tangential force Q (N)
1500
The given FEM computations consider a basic 2D description whereas the experimental analysis has been conducted on a 3D sphere/flat configuration. The model considers vertical cracks whereas a precise cross-section observation confirms the presence of inclined cracks (Fig. 5). The model only considers first body interactions whereas surface scar observations are promoting the
0.98
0.97
0
0.2
0.4 0.6 normalized crack length L/a
0.8
1
Fig. 11. Normalized evolution of the contact stiffness as a function of the crack length (K0): contact stiffness without crack, a: Hertzian contact radius (400 mm).
Fig. 10. Evolution of the FEM fretting cycle as a function of the crack length.
one. A critical question is then how to interpret such a difference. Different aspects could be considered:
0.99
idea that a interfacial third body, displaying a specific rheology should be intercalate to better formalize the real contact stiffness. Finally, the present FEM analysis is based on a monocrack description whereas more in depth observations (Fig. 12) suggest a multicrack phenomenon which potentially can significantly increase the contact compliance.
These different aspects are currently investigated by converging the experimental approach (2D cylinder/flat fretting test) and an inclined multicrack fretting contact FEM modelling. However, in addition to the critical
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Fig. 12. Cross section observation of a typical fretting scar associated to the domain 2 defined in Fig. 6(b) (d ¼ 10 mm, Q ¼ 330 N, P ¼ 230 N, f ¼ 5 Hz and N ¼ 105 cycles). A multi-cracking can be observed at the contact borders and close to the inner stick domain.
numerical difficulties infers by such combined contact and crack non linearity, the development of such approach is still mainly limited by the absence of specific third body stiffness interfacial descriptions. 5. Conclusion The objective of this work was to study and compare how the contact stiffness analysis can be used to detect fretting cracking phenomena. The following aspects have been highlighted:
A significant increase of the contact stiffness has been observed during the initial fretting loading. Without involving any cracking phenomena, this evolution has been associated to a transient evolution of the contact interface (i.e. wear and contact area extension) promoted by the incipient gross slip period and potentially by a reohologic ‘‘hardening’’ of the interface during the stabilized partial slip condition. Depending on the contact loading conditions and test duration a contact stiffness drop has been observed. Cross section observations confirm that such contact stiffness discontinuity is related to a multi-grain crack propagation. By marking such discontinuity a fretting cracking life time parameter (Nc) has been deduced. It permits the non-expensive development of ‘‘fretting cracking’’ endurance charts. More in depth investigations have shown that the stiffness discontinuity is associated to a critical crack length Lc ¼ 40 mm equivalent to the studied grain size. For shorter cracks no stiffness discontinuities has been observed. Above this threshold, the propagation is very fast. The sudden transition tends to confirm a microstructural crack arrest description at least for the studied aluminium alloy. Hence depending on the contact loading conditions, the incipient crack is either stopped at the first and strongest barrier (i.e. the grain boundary) or will propagate until reaching the macroscopic crack arrest condition controlled by the very sharp decrease of the stress gradient below the surface. Based on this online stiffness investigation a multi-scale crack arrest description (i.e. microstructural and contact stress dimension) can be expertized and potentially quantified.
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A simple 2D FEM modelling confirmed that the introduction of cracks through the interface promotes a decrease of the contact compliance. A normalized representation has been introduced confirming a continuous and linear decrease of the contact stiffness with the crack length. It demonstrates that the experimental stiffness discontinuity can only be explained by a crack propagation rate discontinuity. A significant difference between the experimental and computed stiffness variation amplitudes has been established. It suggests that a simple pressure and contact width equivalence combined with a basic perpendicular crack orientation modelling is not sufficient to fully represent the complexity of a real fretting cracking contact. Finest correlations between experiments and modelling descriptions are required involving for instance perfect correlation between the geometry, the crack orientations and taking into account multicracking phenomena. It also outlines the necessity to better integrate the potential impact of the third body interface on the global contact stiffness description.
Finally, this research, which is presently processed for a fretting wear test condition in order to isolate the fretting contact impact, can adequately be extended to any fretting fatigue test configuration insuring that the fretting cycles (i.e. evolution of the tangential force versus the applied displacement) are correctly recorded. It is also expected that such approach could be optimized for industrial components to provide future online fretting cracking damage detections.
References [1] Fouvry S, Elleuch K, Simeon G. Prediction of crack initiation under partial slip fretting conditions. J Strain Anal ImechE 2002;37(6): 549–64. [2] Giannakopoulos AE, Lindley TC, Suresh S. Aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and a life-prediction methodology for fretting-fatigue. Acta Mater 1998;46(9):2955–68. [3] Vallellano C, Dominguez J, Navarro A. On the estimation of fatigue failure under fretting conditions using notch methodologies. Fatigue Fract Eng 2003;M26(5):469–78. [4] Ciavarella M, Macina G. A note on the crack analogue model for fretting fatigue. Int J Solids Struct 2003;40:807–25. [5] Mugadu A, Hills DA. Characterising the process zone in complete fretting contacts using plain fatigue sharp V-notch specimens. J Mech Phys Solids 2002;50(7):1417–29. [6] Hills DA, Kelly PA, Dai DN, Korsunsky AM. In:Gladwell GML. Solution of crack problems. Dordrecht: Kluwer; 1996. [7] Dubourg MC, Lamacq V. Stage II crack propagation direction determination under fretting fatigue loading: a new approach in accordance with experimental observations. ASTM STP 2000;1367: 436–50. [8] Araujo JA, Nowell D. The effect of rapidly varying contact stress fields on fretting fatigue. Int J Fatigue 2002;24:763–75. [9] Kimura T, Sato K. Stress intensity factors KI and KII of oblique thickness cracks in a semi-infinite body under fretting fatigue conditions. ASTM STP 2003;1425:76–88.
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[10] Proudhon H, Buffie`re JY, Fouvry S. Characterization of fretting fatigue damage using synchrotron X-ray micro-tomography. Tribol Int. [submitted]. [11] Chateauminois A, Kharrat M, Krichen A. Analysis of fretting damage in polymers by means of fretting maps. ASTM STP 2000;1367:352–66. [12] Elleuch K, Fouvry S, Kapsa Ph. Fretting maps for anodised aluminium alloys. Thin solid films 2003;426(1&2):271–80. [13] Johnson KL. Contact mechanics. Cambridge: Cambridge Univ. Press; 1985. [14] Elleuch K, Mezlini S, Fouvry S, Kapsa P. Friction damage of aluminium alloys. Ind Lubric Tribol 2003;55(6):279–86.
[15] Zhou ZR, Fayeulle S, Vincent L. Cracking behaviour of various aluminium alloys during fretting wear. Wear 1992;155:317–30. [16] Mindlin RD, Deresiewicz H. Elastic spheres in contact under varying oblique forces. ASME Trans E J Appl Mech 1953;20: 327–44. [17] Fouvry S, Kapsa Ph, Vincent L. Analysis of sliding behaviour for fretting loadings: determination of transition criteria. Wear 1995;185:35–46. [18] Miller KJ. The three thresholds for fatigue crack propagation. ASTM STP 1997;1296:267–86. [19] Hills DA, Nowell D. Mechanics of fretting fatigue. Dordrecht: Kluwer; 1994.
Tribology International 39 (2006) 1262–1270 www.elsevier.com/locate/triboint
Development of a contact compliance method to detect the crack propagation under fretting K. Elleucha,, H. Proudhonb, C. Meunierb, S. Fouvryb a
LASEM, De´partement de Ge´nie des Mate´riaux, Ecole Nationale d’Inge´nieurs de Sfax, B.P. W-3018, Sfax, Tunisia b LTDS, CNRS, UMR 5513, Ecole Centrale de Lyon, France Available online 20 March 2006
Abstract Partial slip fretting conditions are classically known to favor contact crack nucleation and crack propagation. Considered as a plague for modern industries, numerous theoretical researches have been conducted during the past two decades to predict such fretting damage. However, a review of last few years critically outlines the need of precise and in-situ experiments to qualify and quantify the given models. To palliate such aspect, an original approach which consists in following the contact stiffness evolution as an indicator of the fretting cracking phenomena, has been developed. Applied for an aluminium/steel contact, it demonstrates that the incipient crack propagation is related to a discontinuous decrease of the contact stiffness. Based on this online analysis, a fretting cracking endurance parameter has been extrapolated to develop fast and low cost fretting cracking endurance chart. A FEM analysis has been performed in an attempt to formalize the given experiments. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fretting; Cracking; Contact stiffness; Crack arrest; Aluminium alloy
1. Introduction Cracking induced by fretting loadings has been deeply investigated during the past two decades. The crack nucleation phenomena have been formalized applying multiaxial fatigue criteria combining averaged stress volume approaches to take into account the contact stress gradient and the size effect [1]. Alternatively, this stressmaterial length dimension approach has been more recently captured through the development of the ‘‘notch similitude’’ concept [2–5]. The crack propagation induced by combined contact fretting and macroscopic fatigue stressing has been quantified by different numerical approaches like distributed dislocation methods [6–8] and weight function formalism [9]. These methodologies consist to estimate the stress intensity factors at the tip of the crack enduring complex fretting fatigue loadings and to predict the specimen endurance using modified crack propagation Paris laws [3].
Corresponding author. Tel.: +216 74 274 088; fax: +216 74 275 095.
E-mail address: [email protected] (K. Elleuch). 0301-679X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2006.02.015
If most of these models have been well established, a critical aspect concerns currently the experimental validation of such approaches. Indeed to calibrate the different models, the following questions need to be answered:
How many cycles are required to activate the incipient crack nucleation? What is the incipient crack propagation kinetics and when a fretting fatigue crack leaving the contact stress field domain will be fully controlled by the macroscopic fatigue stress field?
Numerous other experimental aspects are required to definitively establish pertinent fretting cracking modelling. Heavy in situ tomography X observations are expected to provide such fundamental scientific materials [10]. However, the current limitations of the tomography X like the specimen size (1 mm2 for the section) clearly restrict the investigation of the contact size effect, which has been demonstrated to be a dominating factor of fretting fatigue damages. Alternative approaches still need to be investigated.
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Exploiting the transparent properties of polymers Chateauminois and co-authors [11] have carefully quantified the crack nucleation process and incipient crack propagation of PMMA polymer submitted to fretting wear test loadings. They also investigate the evolution of the contact stiffness evolution as a function of the contact cracking extension. Unfortunately, such a direct observation is not possible in the present aluminium/steel contact. However, it appears interesting by combining finest fretting cycle analyses and coupling FEM modelling, to evaluate how the contact stiffness analysis could be adapted to achieve an undirected in situ observation of fretting cracking. 2. Experiments 2.1. Fretting wear tester A schematic diagram of the tester is shown in Fig. 1. The system used has already been described elsewhere [12]. In particular, it permits the online acquisition of the tangential force (Q) evolution versus the imposed displacement (d). Thus, the fretting loop Qd can be plotted (Fig. 2(a)). A dynamic overview of the fretting test can be illustrated via the fretting log, which is a superimposition of fretting cycles (Fig. 2(b)). Some quantitative fretting parameters can also be extracted like: the tangential force and displacement amplitude (Q*, d*), the dissipated energy (Ed), the cycle opening (d0) abusively associated under gross slip conditions to the sliding amplitude (dg) and finally the measured contact Stiffness (K) defined as the slope of the fretting cycle at the loading amplitude points (Fig. 2(a)) (i.e. slope of the loading path when Q ¼ Q or
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negative slope of the unloading path when Q ¼ þQ ) [13]. To calculate this slope, only the quarter of the fretting cycle was considered (i.e. 250 loading-amplitude points). 2.2. Material specimens Flat samples of A357 aluminium alloy (30 20 10 mm3) have been manufactured from a casting bar. The chemical composition of the A357 is shown in Table 1. The surfaces of these specimens were polished with abrasive paper, finished with diamond paste, until a 0.1 mm roughness (Ra) was reached, and then ultrasonically cleaned in pure alcohol. The counter body is the AISI 52100 steel. It consists of a cylinder (20 20 mm2) with a machined spherical end (R ¼ 100 mm). The tested surface, i.e. the spherical part, was polished with a suitable device until 0.1 mm roughness (Ra) was reached. Table 2 summarizes the principal mechanical properties of the tested materials. These materials are used in order to simulate fretting damage encountered in some aeronautical application. Especially, when steel bearings are brought into contact with aluminium crankcase [14].
Fig. 2. (a) Identification of the fretting loading parameters from the fretting cycle analysis and (b) illustration of a fretting log.
Table 1 Chemical composition of A357 aluminium alloy
Fig. 1. Fretting wear test apparatus: (a) schematic and (b) picture of the setup.
A357
% Si
% Mg
% Fe
% Cu
%Ni
6.5–7.5
0.4–0.7
0.2
0.1
0.05
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Table 2 Mechanical properties of tested materials E (GPa)
n
Rm (Mpa)
Rp0.2 (MPa)
Hv (GPa)
sd (MPa)
A357 52100 steel
70 210
0.36 0.3
330 2200
200 2000
1.13 8.40
60 800
K (N/µm)
Materials
δ* = 8µm δ* = 7µm
80 60 40
Nc 20
2.3. Experimental conditions
0 1
10
(a)
102 103 Fretting cycles
104
105
6 δ* = 8µm δ* = 7µm
5
Nc
4 δo (µm)
The applied normal load is fixed at 230 N, corresponding to 250 MPa maximum Hertzian pressure for the studied flat on sphere contact configuration. Tests have been done with a frequency of 5 Hz up to 105 cycles. The imposed displacement amplitude (d*) ranged from 3 to 20 mm in order to cover partial and mixed regime conditions [15] (under mixed fretting regime the contact endures gross slip sliding before stabilizing under partial slip condition). All tests were carried out under laboratory-controlled conditions (temperature, 2372 1C; relative humidity, 40710%).
3 2 1
3. Experimental analysis of the contact stiffness evolution
0 1
Fig. 3(a) compares the evolution of the measured contact stiffness as a function of the fretting cycles for two different displacement amplitudes. The observed contact stiffness evolutions are characterized by an initial sharp increase of the contact stiffness followed by a smoother progression. However, if the smaller displacement amplitude condition (d ¼ 7 mm) maintains a constant evolution a discontinuity is observed for the largest amplitude (d ¼ 8 mm). The discontinuity has been identified at a given number of cycles, Nc, after which there is a decrease of the contact stiffness. In the same time, an increase of the cycle opening value (d0) is also observed (Fig. 3(b)). It appears interesting to relate the symmetrical evolutions of the contact stiffness and the cycle opening. These two parameters appear really pertinent to quantify the interface contact evolution and potentially to detect fretting cracking phenomenon. The given study mainly focuses on the contact stiffness parameter due its most extended application in the contact mechanics research field. However, displaying a higher amplitude evolution, it suggests that the aperture variable is potentially more interesting to characterize the interface evolution. More investigations are needed to implement the application of this latter variable. To interpret the initial increase of the contact stiffness (i.e. symmetric decrease of the cycle aperture) it appears essential to consider some basic theoretical aspects. The elastic formalism of the sphere/flat contact stiffness (KCth), first introduced by Mindlin for a full sticking contact (i.e. condition which is presumably verified at the cycle amplitudes) is expressed by [13,16] K Cth ¼ aG ,
(1)
10
(b)
102 103 Fretting cycles
104
105
Fig. 3. Fretting contact responses versus the fretting cycles as a function of the applied displacement amplitude: (a) evolution of the measured contact stiffness and (b) evolution of the cycle opening.
where G* (Pa) is defined as an effective elastic shear modulus expressed by G ¼
8G 1 G 2 Ei with G ¼ G1 ð2 n2 Þ þ G 2 ð2 n1 Þ 2ð1 þ ni Þ
(2)
and a the sphere/plane Hertzian contact radius expressed by 3PR 1=3 a¼ , (3) 4E where P and R are, respectively, the normal force and the sphere radius whereas E* is defined as the effective contact modulus expressed by 1 1 n21 1 n22 ¼ þ , E E1 E2
(4)
where n1 and n2 are Poisson coefficient of the plane (1) and the sphere (2), respectively, E1 and E2 are elastic modulus of the plane (1) and the sphere (2), respectively, G1 and G2 are shear modulus of the plane (1) and the sphere (2), respectively. Note that even verifying perfect contact conditions, the measured contact stiffness is not equal to theoretical estimation. Indeed, the fretting wear apparatus system compliance must be considered. The measured stiffness (K) is in fact a contribution of two stiffness: the contact (KC)
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and the experimental system (KS) stiffness [17]. 1 1 1 ¼ þ . K KC KS
(5)
Specific attentions have been done in previous works to extract the system perturbation, to quantify the sliding transition and identify the local friction coefficient operating through the sliding annulus under partial slip conditions [17]. However, in the present study, only the stiffness fluctuations are required to be detected and do not request a precise and difficult determination of the exact contact stiffness. Indeed, because the system stiffness is assumed constant, the observed fluctuations are only function of the contact stiffness variations. Considering the theoretical expression given in Eq. (1) it is assumed that the contact stiffness is only function of the material elastic properties and the apparent contact area. It is not directly dependent on the tangential force loading or the friction coefficient. Therefore, to interpret the experimental evolutions, it appears fundamental to consider the fretting sliding history imposed through the interface as well as the interface morphology. Indeed, native surfaces display specific roughness and the real contact area is in fact smaller than the theoretical Hertzian estimation. As illustrated in Fig. 4, the contact is running under gross slip condition during the first 1000 cycles. During this period the interface is extensively modified with a critical increase of the metal interaction promoting and increase of the friction coefficient but also an extension of the real
Fig. 4. Evolution of the sliding behaviour as a function of the fretting cycles (Q*): tangential force amplitude; A ¼ Ed/(4Pd*): energy sliding criterion [14], if A40:2 the contact is running under gross slip, if Ao0:2 the contact is under partial slip condition): (a) d ¼ 7 mm and (b) d ¼ 8 mm.
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contact area. Moreover during this short period a significant wear of the interface is observed which again increases the operating contact area. The smoother increase of the contact stiffness observed after the partial slip stabilization is less easily explained. A stable evolution should be expected as partial slip condition should not activate any critical interface modification. This stabilization is in fact verified by the cycle opening variable but not by the contact stiffness. Hence other aspects like a possible modification of the interfacial third body rheology and possible hardening plastic mechanisms should be considered. One interesting conclusion of this investigation is the monotonous increase of the contact stiffness with the partial slip stabilization. The incoming question is then how to explain the discontinuous decrease observed for the highest loading condition? Post mortem analyses have been conducted on these fretting scars in order to interpret the reason of these discontinuities. In particular, cross section investigations have shown some interesting results (Fig. 5). For d ¼ 7 mm, very short cracks sited at the contact border have been observed. These cracks have never cross the first encountered grain boundary. Hence, their maximal length does not exceed a critical length, Lc, which corresponds to the grain size (LcE40 mm). This result was confirmed even under very long fretting test (d ¼ 7 mm, N ¼ 106 cycles). For d ¼ 8 mm, cracks are longer; their maximal length can reach 450 mm crossing several grain boundaries. It is interesting to mention that very short cracks have been identified even for the smallest loading situations. Therefore, it can be concluded that the stiffness discontinuity can be related to an incipient crack propagation activated when the loading condition (i.e. tangential force amplitude) permits the incipient crack to overpass the first and strongest barrier: the grain boundary. Hence, Miller approach can be seriously considered at least for the studied aluminium alloy [18]: ‘‘in polycrystalline materials, there is no phase of crack initiation since it is immediate. Then cracks can propagate or stop depending on loading conditions’’. The transition appears very abrupt. When the crack overpasses the first grain boundary, the successive grains are rapidly passed away and the initial few micron micro-cracks reach a stabilized in 300–500 in few thousand cycles. The introduction of such long cracks significantly increases the contact compliance and inversely decreases the contact stiffness. The present studied fretting test conditions are displacement controlled. The drop of the tangential amplitude associated to a contact stiffness decrease can, therefore, be explained and quantified. Therefore the contact stiffness discontinuity (Nc) be related to the activation of the incipient crack propagation and this methodology could be extended to development fast and non expensive fretting cracking endurance charts (Fig. 6). As previously demonstrated by Mindlin [13,16], a close relationships exists between d* and Q* under partial slip
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Fig. 5. Cross-section observation of a cracked fretting scar (d ¼ 9 mm, P ¼ 230 N, f ¼ 5 Hz and N ¼ 105 cycles).
70
200
60 50
150
40
100
30 20
50
10
0
0 0
(a)
2
4 6 8 10 12 displacement amplitude δ* (µm)
14
100
350
90
maximum crack length I (µm)
300
80
250
70 2
200
60 50
150
40 30
100 Qc*
50
Lc Lc 1
0 0
100 200 300 tangential force amplitude Q* (N)
20 10
fretting cycles (Nc x 1000)
Q* (N)
; L (µm)
80 250
fretting cycles (Nc x 1000)
90
300
(b)
evolution of maximal crack length versus maximal tangential force (Q*). Two domains can be identified:
100
350
0 400
Fig. 6. Introduction of fretting cracking endurance charts (defined from the online contact stiffness analysis, a). Evolution of the maximum crack length (L), related tangential force amplitude (Q*) and contact endurance (Nc) as a function of the applied displacement amplitude (d*). (b) Evolution of the maximum crack length (L) and contact endurance (Nc) as a function of the maximum tangential force amplitude (Q*).
These two regions are clearly identified when defining the critical length Lc and a critical tangential force Qc. When Q* is bigger than a threshold tangential force amplitude QcE305 N there is a discontinuity of the contact stiffness and simultaneously the crack propagates. On the other hand, below this critical tangential force amplitude, the incipient crack is limited to a grain dimension (LoLc). For such condition, the crack extension is not sufficient to influence the contact compliance inducing a constant evolution of K. Therefore, by considering the contact compliance as a pertinent parameter to detect the crack propagation condition, ‘‘fretting cracking wholer curves’’ could been defined which consist to report the critical number of cycles (Nc) defined from the contact stiffness discontinuity (i.e associates to the crack propagation condition: L4Lc) as a function of the applied tangential force amplitude (Fig. 6(b)). To evaluate the dynamic evolution of the crack progress, a constant cracking condition has been applied (d ¼ 10 mm) and an interrupt tests procedure has been applied (Fig. 7). This experimental procedure has permitted to outline following aspects:
elastic conditions. However, in order to avoid any experimental artefacts, the given analyses focuses on the tangential force amplitude Q* instead of d*. Fig. 6(b) shows
domain 1: crack arrest domain, domain 2: incipient crack propagation domain.
Up to N1 cycles, max(Q*)oQc, there is no discontinuity in the K variation and cross section investigation has confirmed that maximal crack length (L1) is smaller than Lc. Up to N2 cycles, max(Q*)4Qc, there is a decrease of K and crack length (L2) is bigger than Lc. Up to N3 cycles, crack propagation progress and maximal crack length (L3) longer than L2 has been detected. This condition is also associated to a critical
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N2 N2
350
N4 N4
300
Q* (N)
250 200
N3 N3
150 N1 N1
100 50 0 1
10
100
1000 N cycles
10000
100000
Fig. 7. Illustration of the fretting test methodology to bracket the threshold tangential force amplitude marking the incipient crack propagation condition: four interrupted test have been carried out until N1, N2, N3 and N4 cycles with the same loading condition (d ¼ 10 mm, P ¼ 230 N).
drop of the contact stiffness and a significant decreased of tangential force amplitude. When performing the final test up to N4 cycles, a similar crack length defined at N3 cycles has been found (i.e. Lmax ¼ L4 ¼ L3 ). In fact, the crack has reached the maximum possible crack extension associated to the studied loading conditions. Formalized through a conventional crack arrest description this maximum crack length is in fact directly connected to the contact stress field distribution defined from contact size and the imposed cyclic tangential loading [8,19].
This analysis suggests that two crack arrest conditions must be considered. A first one associated to a metallurgical aspect marking the interaction of the incipient crack with the first and strongest barrier: the grain boundary (L ¼ Lc ) (i.e detected via a contact stiffness analysis). A second which is basically dominated by the subsurface decrease of the contact stress field which can be explained and quantified through a classical crack arrest description via a local crack tip stress intensity factor analysis (L ¼ Lmax ) [8,19].
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The cylinder (R ¼ 49 mm) on flat configuration, associated respectively to the 52100 and A357 elastic properties (Table 2) has been meshed. The contact is supposed elastic so only the Young modulus and Poisson coefficients variables have been considered. The loading conditions have been adjusted to achieve similar Hertzian pressure Po ¼ 250 MPa, (i.e. P ¼ 160 N=mm) inducing a semi width contact a ¼ 400 mm. Perpendicular cracks, located symmetrically at the contact borders have been designed. Six crack lengths have been considered to evaluate the contact stiffness evolution (25, 50, 75, 100, 200, 275 mm) (Fig. 8). CPE4 quadratic elements have been used for the mesh with a specific refinement near the crack tips (Fig. 8). Fig. 9 depicts the distribution of Von Mises equivalent stress in the plane after applying the normal load and two fretting cycles. Fig. 10 compares the computed fretting cycles with and without cracks at the contact borders. A very smooth decrease of the contact stiffness and an equivalent increase of the cycle opening are detected. To rationalize the analysis, the calculated numerical contact stiffness have been normalized versus the value defined for a contact without any crack (K0). Symmetrically, the respective crack length has been normalized versus the contact semi width (a). Fig. 11 reveals a linear decrease of the normalized contact stiffness versus the normalized crack length. No stiffness discontinuity is observed which infers that the experimental stiffness drop should be related to a critical variation of the crack propagation rate. Indeed, passing the first grain boundary, the crack propagation should be very fast promoting by consequence the observed contact stiffness discontinuity. A second aspect concerns the amplitude of the variation. In spite of the very large computed crack length range, the computed relative contact stiffness drop remains inferior to 5%, which is significantly smaller than the 20% variation observed for the experiments (Fig. 3(a)). Moreover, after Eq. (5), the introduction of constant system compliance even tends to minor the measured variation compared to the contact real
4. FEM modelling of the contact stiffness evolution versus the crack length The contact stiffness analysis appears as a pertinent approach to detect fretting cracking phenomena. However the previous analysis outlines that a crack must display a significant length to be identified via the contact stiffness parameter. In order to evaluate the resolution of this methodology, a parametric FEM analysis of the contact stiffness versus the crack length has been conducted. Due to the numerical cost of a complete 3D fretting modelling, an equivalent elastic 2D cylinder/plane fretting contact configuration has been defined (ABAQUS).
Fig. 8. Illustration of the FEM crack mesh at the fretting contact borders.
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S, Mises (Ave. Crit.: 75%)
2
ODB: cyl_plan_mesh_fissure_L100_ele_3696.odb
ABAQUS/Standard 6.3-1
Thu Jun 17 13:21
1 Step: DEPL2+Q Increment 10: Step Time = 2.000 Primary Var: S, Mises Deformed Var: U Deformation Scale Factor: +1.000e+00
3
Fig. 9. Distribution of the Von Mises stress in the plane submitted to fretting at a displacement amplitude (+d*).
1.00
1000
500
0 -30
-20
-10
0 -500
-1000
10 20 displacement amplitude δ* (µm)
30
crack length = 275 µm no cracks
-1500
normalized contact stiffness K/Ko
tangential force Q (N)
1500
The given FEM computations consider a basic 2D description whereas the experimental analysis has been conducted on a 3D sphere/flat configuration. The model considers vertical cracks whereas a precise cross-section observation confirms the presence of inclined cracks (Fig. 5). The model only considers first body interactions whereas surface scar observations are promoting the
0.98
0.97
0
0.2
0.4 0.6 normalized crack length L/a
0.8
1
Fig. 11. Normalized evolution of the contact stiffness as a function of the crack length (K0): contact stiffness without crack, a: Hertzian contact radius (400 mm).
Fig. 10. Evolution of the FEM fretting cycle as a function of the crack length.
one. A critical question is then how to interpret such a difference. Different aspects could be considered:
0.99
idea that a interfacial third body, displaying a specific rheology should be intercalate to better formalize the real contact stiffness. Finally, the present FEM analysis is based on a monocrack description whereas more in depth observations (Fig. 12) suggest a multicrack phenomenon which potentially can significantly increase the contact compliance.
These different aspects are currently investigated by converging the experimental approach (2D cylinder/flat fretting test) and an inclined multicrack fretting contact FEM modelling. However, in addition to the critical
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Fig. 12. Cross section observation of a typical fretting scar associated to the domain 2 defined in Fig. 6(b) (d ¼ 10 mm, Q ¼ 330 N, P ¼ 230 N, f ¼ 5 Hz and N ¼ 105 cycles). A multi-cracking can be observed at the contact borders and close to the inner stick domain.
numerical difficulties infers by such combined contact and crack non linearity, the development of such approach is still mainly limited by the absence of specific third body stiffness interfacial descriptions. 5. Conclusion The objective of this work was to study and compare how the contact stiffness analysis can be used to detect fretting cracking phenomena. The following aspects have been highlighted:
A significant increase of the contact stiffness has been observed during the initial fretting loading. Without involving any cracking phenomena, this evolution has been associated to a transient evolution of the contact interface (i.e. wear and contact area extension) promoted by the incipient gross slip period and potentially by a reohologic ‘‘hardening’’ of the interface during the stabilized partial slip condition. Depending on the contact loading conditions and test duration a contact stiffness drop has been observed. Cross section observations confirm that such contact stiffness discontinuity is related to a multi-grain crack propagation. By marking such discontinuity a fretting cracking life time parameter (Nc) has been deduced. It permits the non-expensive development of ‘‘fretting cracking’’ endurance charts. More in depth investigations have shown that the stiffness discontinuity is associated to a critical crack length Lc ¼ 40 mm equivalent to the studied grain size. For shorter cracks no stiffness discontinuities has been observed. Above this threshold, the propagation is very fast. The sudden transition tends to confirm a microstructural crack arrest description at least for the studied aluminium alloy. Hence depending on the contact loading conditions, the incipient crack is either stopped at the first and strongest barrier (i.e. the grain boundary) or will propagate until reaching the macroscopic crack arrest condition controlled by the very sharp decrease of the stress gradient below the surface. Based on this online stiffness investigation a multi-scale crack arrest description (i.e. microstructural and contact stress dimension) can be expertized and potentially quantified.
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A simple 2D FEM modelling confirmed that the introduction of cracks through the interface promotes a decrease of the contact compliance. A normalized representation has been introduced confirming a continuous and linear decrease of the contact stiffness with the crack length. It demonstrates that the experimental stiffness discontinuity can only be explained by a crack propagation rate discontinuity. A significant difference between the experimental and computed stiffness variation amplitudes has been established. It suggests that a simple pressure and contact width equivalence combined with a basic perpendicular crack orientation modelling is not sufficient to fully represent the complexity of a real fretting cracking contact. Finest correlations between experiments and modelling descriptions are required involving for instance perfect correlation between the geometry, the crack orientations and taking into account multicracking phenomena. It also outlines the necessity to better integrate the potential impact of the third body interface on the global contact stiffness description.
Finally, this research, which is presently processed for a fretting wear test condition in order to isolate the fretting contact impact, can adequately be extended to any fretting fatigue test configuration insuring that the fretting cycles (i.e. evolution of the tangential force versus the applied displacement) are correctly recorded. It is also expected that such approach could be optimized for industrial components to provide future online fretting cracking damage detections.
References [1] Fouvry S, Elleuch K, Simeon G. Prediction of crack initiation under partial slip fretting conditions. J Strain Anal ImechE 2002;37(6): 549–64. [2] Giannakopoulos AE, Lindley TC, Suresh S. Aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and a life-prediction methodology for fretting-fatigue. Acta Mater 1998;46(9):2955–68. [3] Vallellano C, Dominguez J, Navarro A. On the estimation of fatigue failure under fretting conditions using notch methodologies. Fatigue Fract Eng 2003;M26(5):469–78. [4] Ciavarella M, Macina G. A note on the crack analogue model for fretting fatigue. Int J Solids Struct 2003;40:807–25. [5] Mugadu A, Hills DA. Characterising the process zone in complete fretting contacts using plain fatigue sharp V-notch specimens. J Mech Phys Solids 2002;50(7):1417–29. [6] Hills DA, Kelly PA, Dai DN, Korsunsky AM. In:Gladwell GML. Solution of crack problems. Dordrecht: Kluwer; 1996. [7] Dubourg MC, Lamacq V. Stage II crack propagation direction determination under fretting fatigue loading: a new approach in accordance with experimental observations. ASTM STP 2000;1367: 436–50. [8] Araujo JA, Nowell D. The effect of rapidly varying contact stress fields on fretting fatigue. Int J Fatigue 2002;24:763–75. [9] Kimura T, Sato K. Stress intensity factors KI and KII of oblique thickness cracks in a semi-infinite body under fretting fatigue conditions. ASTM STP 2003;1425:76–88.
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[10] Proudhon H, Buffie`re JY, Fouvry S. Characterization of fretting fatigue damage using synchrotron X-ray micro-tomography. Tribol Int. [submitted]. [11] Chateauminois A, Kharrat M, Krichen A. Analysis of fretting damage in polymers by means of fretting maps. ASTM STP 2000;1367:352–66. [12] Elleuch K, Fouvry S, Kapsa Ph. Fretting maps for anodised aluminium alloys. Thin solid films 2003;426(1&2):271–80. [13] Johnson KL. Contact mechanics. Cambridge: Cambridge Univ. Press; 1985. [14] Elleuch K, Mezlini S, Fouvry S, Kapsa P. Friction damage of aluminium alloys. Ind Lubric Tribol 2003;55(6):279–86.
[15] Zhou ZR, Fayeulle S, Vincent L. Cracking behaviour of various aluminium alloys during fretting wear. Wear 1992;155:317–30. [16] Mindlin RD, Deresiewicz H. Elastic spheres in contact under varying oblique forces. ASME Trans E J Appl Mech 1953;20: 327–44. [17] Fouvry S, Kapsa Ph, Vincent L. Analysis of sliding behaviour for fretting loadings: determination of transition criteria. Wear 1995;185:35–46. [18] Miller KJ. The three thresholds for fatigue crack propagation. ASTM STP 1997;1296:267–86. [19] Hills DA, Nowell D. Mechanics of fretting fatigue. Dordrecht: Kluwer; 1994.