Cyril Sola
email: cyril.sola@airbus.com

Bruno Castani

Laurent Michel

Ed Eric Lachaud

Arnaud Delabie

Emmanuel Mermoz

Bearing fatigue of composite laminates: Damage monitoring and fatigue life prediction

Keywords:

a b s t r a c t

In hybrid composite/metallic structures, loads can be transmitted from one part to the other through localized contact pressures, i.e., bearing. Such structures may be rotating structures, which can accu mulate as many as 10 9 load cycles during their service life. Designing safe hybrid rotating structures thus requires a sound understanding of how composite joints degrade under bearing fatigue. Pin bearing fatigue tests were run under load control. Damage mechanisms were investigated using computed to mography, and the pin displacement was closely monitored thanks to a green LED micrometer. Building upon the gathered experimental evidences, several damage indicators were then analysed. In particular, hysteresis losses were found to give interesting insights into the fatigue phenomenon, suggesting the existence of a fatigue limit in the very high cycle fatigue (VHCF) regime.

The bearing behaviour of composite joints has been the object of a large number of studies over more than four decades. However, most of these have focused on the strength of bolted assemblies [1e7] and few have dealt with the intrinsic bearing behaviour [8e10]. More recently, some papers investigated the influence of manufacturing flaws on the bearing behaviour, but, again, the scope was mainly limited to bolted joints [11e13]. As new technological opportunities arise [START_REF] Handschuh | Hybrid gear preliminary results application of composites to dynamic mechanical components[END_REF][START_REF] Black | New aerocomposites niche: helicopter transmission gears?[END_REF], it is necessary to enhance our knowl edge on the consequences of a bearing loading mode, especially for rotating structures subjected to a large number of fatigue cycles, potentially greater than 10 9 .

Several studies have shown that the accumulation of permanent hole elongation is the consequence of bearing fatigue damage, and that this elongation can be used as a damage metric [16e19].

Although it has no real influence on the residual strength [START_REF] Grant | Bearing fatigue and hole elongation in composite bolted joints[END_REF], an excessively large hole elongation can ultimately promote, in pinned or bolted joints, a transition from bearing to more catastrophic failure modes such as shear out or cleavage: it should thus be avoided. Several empirical failure criteria based on hole elongation have been proposed. However, depending on the layup or other factors, the critical permanent hole elongation ranges from 4% of the hole diameter [START_REF] Persson | Effects of hole machining defects on strength and fatigue life of composite laminates[END_REF][START_REF] Wei | The effect of interference fit size on the fatigue life of bolted joints in composite laminates[END_REF] to as much as 10% [START_REF] Counts | Bolt bearing fatigue of polymer matrix composites at elevated temperature[END_REF]. Furthermore, it is not known whether hole elongation build up can be stopped or not, i.e., whether there is a practical fatigue limit (or endurance limit) or not. In most bearing fatigue tests reported in the literature, run out was considered when the fatigue life exceeded 10 6 cycles. In this fatigue domain, the S N curve had no asymptote [21e23]. Another question that arises is whether the residual stiffness evolves significantly in fatigue, which could prove critical for stiffness driven applications. In particular, Smith and Pascoe [START_REF] Smith | Fatigue of bolted joints in (0/90) CFRP laminates[END_REF] suggested that, for crossply laminates, very long lives would imply a very low residual stiffness.

The first objective of this study was to understand the different fatigue damage mechanisms involved when CFRP laminates are subjected to bearing loads. A simple test set up and coupon were selected with the aim of focusing on this peculiar loading mode. The main macroscopic consequence of damage was an incremental collapse behaviour, or ratcheting, which explains why permanent hole elongation can be used to monitor the fatigue damage evo lution in pinned or bolted joints, just as under monotonic loads. The second part of the paper investigates the capabilities of this damage metric regarding lifetime prediction. It also considers other damage indicators, such as residual stiffness and hysteresis losses. In the last part of the article, the analysis of the experimental trends and damage metrics are used to assess the possible existence of a fa tigue limit.

Materials and methods

Half hole bearing tests were chosen. This set up enabled friction to be minimized so as to prevent heat build up. This test is similar to the pinned bearing test which essentially differs from the bolt bearing one in that there is no out of plane restraint. The pin, or indenter, was the reamer that had been used to ream the holes in the composite coupons. Half hole bearing tests were successfully used in a previous study focusing on the bearing behaviour under monotonic loads [START_REF] Sola | On the role of kinking in the bearing failure of composite laminates[END_REF] (see Fig. 1 and Fig. 2).

Fatigue tests were performed at 10 Hz, a frequency sufficiently low to avoid excessive elevation of the temperature. Temperature measurements obtained with an infrared camera showed that the increase in the surface temperature of the coupons was lower than 10 C, even after thousands of cycles.

The coupons were made of carbon fibre prepreg with a UD fabric or 4 harness satin weave fabric. The chosen layup was quasi isotropic and symmetric, with [90,45,0, 45] ns ply sequences. The number of sequences n was adapted so that the UD and woven laminates would have equivalent thicknesses, close to 4 mm.

Tests were load controlled as, with displacement control, obtaining a clear failure1 would have been close to impossible because of the expected build up of permanent hole elongation, that could ultimately have led the coupon to not being loaded any longer. Load ratios (R F min /F max ) of 10 and 3 were selected in order to assess the mean load influence. This choice implied that the coupons were never completely unloaded; a small preload was always present. Therefore, debris that were generated as a result of wear were not evacuated and remained trapped between the pin and the hole. This was initially a concern, as Grant and al [START_REF] Grant | Bearing fatigue and hole elongation in composite bolted joints[END_REF]. had shown that the presence of debris could affect fatigue life. However, the amount of debris was later found to be negligible.

For a given material, three coupons were tested to failure for each chosen load amplitude. A minimum of three load amplitudes was selected to generate each of the four S N curves corresponding to the two materials and the two load ratios. The maximum abso lute load in a cycle jF min j was in the range ½0:65,jF ultimate j; 0:93,jF ultimate j for the woven fabric laminates and ½0:7,jF ultimate ; 0:94,jF ultimate j for the UD fabric laminates. The initial test matrix is shown in Table 1. A few extra data points were added to this matrix later.

The pin displacement was measured, without contact, with a Keyence LS7030 green LED micrometer (see Fig. 3). Displacement and force were both sampled at 200 Hz, so 20 lines of data were available for each fatigue cycle.

Under load control, fatigue failure was sudden. Hence, there was no need to define a fatigue failure criterion as a certain amount of permanent hole elongation or stiffness decrease.

Damage mechanisms

Incremental collapse: a macroscopic consequence of damage

Upon fatigue loading, a permanent hole elongation gradually accumulates. In the current study, this damage metric was esti mated as shown on Fig. 4. It can be noted that this damage metric is closely related to the fatigue modulus of Hwang and Han [START_REF] Hwang | Fatigue of composites fatigue modulus concept and life prediction[END_REF].

The increase in permanent hole elongation followed three stages. The first stage was probably a run in stage; the permanent hole elongation increased quickly. The second stage was charac terized by a linear increase of the permanent hole elongation. Finally, the hole elongation increased exponentially as final failure occurred. Interestingly, the permanent hole elongation evolution curve followed the same trend as the well known damage evolu tion curve of a unidirectional laminate under tensile loading, where the transition from stage 1 to stage 2 coincides with the so called "Critical Damage State" (CDS) [START_REF] Wu | A fatigue damage model of composite materials[END_REF]. When plotted as a function of the normalized number of cycles N/N f , where N f is the number of cycles to failure, the permanent hole elongation appeared to have a critical threshold value around 50 mm 60 mm, i.e., slightly less than 1% of the hole diameter. Once this value was exceeded, failure occurred in a matter of a few cycles, typically less than 10% of the total fatigue life, though some coupons had a much larger residual life after reaching the threshold (Fig. 5).

It should be noted that this critical value is very close to that reported by Seike et al. [START_REF] Seike | Bearing damage evolution of a pinned joint in CFRP laminates under repeated tensile loading[END_REF], despite the different test set up and materials (but the similar pin diameter and quasi isotropic layups). The permanent hole elongation curve has the same shape irre spective of the load ratio. When expressed as a fraction of the total fatigue life, the three stages have constant durations. The first stage typically represents less than 10% of the fatigue life, 80% is accounted for by the second stage, and the last 10% corresponds to the final failure stage.

Ply level damage: the role of kinking

Each coupon has its own permanent hole elongation curve, but for any such curve, three characteristic points can be defined, as shown on Fig. 6. For woven and UD fabric laminates, three tests short test duration enabled the evolution of the permanent hole elongation to be monitored closely and tests to be stopped at the most appropriate moment. It should be noted that, at this load level, damage could initiate at the first fatigue cycle, as static damage typically starts at about 70% of the static failure load [START_REF] Sola | On the role of kinking in the bearing failure of composite laminates[END_REF]. Just after the run in phase, kink bands were found in all the 0 plies and close to the hole surface. Locally bent packs of fibres and inter fibre cracks probably paved the way for kink band initiation upon further loading (Fig. 7). At this stage, delaminated areas were almost non existent. Therefore, most of the initial permanent hole elongation could be attributed to these local damage mechanisms in the load bearing plies, at, or in the close vicinity of, the hole surface.

At the second characteristic point, coupons were highly damaged. Kink bands were found in both 0 and ±45 plies. Shear driven compressive failure was also noticed (Fig. 8), especially in ±45 plies. Delaminations were essentially located in the outer ply sequences. It appeared that delaminations initiated mainly as a result of wedge mechanisms triggered by kinking. The incremental collapse behaviour was most probably connected with the pro gressive development of kink bands throughout the whole course of the damage process. Kink bands obviously engendered residual strains, particularly when they collapsed upon repeated loading, which seems to have happened for many of them, especially in the last stages of the fatigue life (characteristic point n 3, Fig. 9). The complete fatigue failure scenario in the so called bearing plane, i.e., the plane dividing the coupon width into two equal parts, is shown in Fig. 10. No clear difference was noted between woven and UD fabric laminates: kink bands and delaminations explained the final failure in both cases.

Finally, the second stage of damage evolution, delimited by the first and second characteristic points, was characterized by the stable development of kink bands, which tended to more and more offset from the hole surface. As long as these kink bands did not trigger out of plane cracks, the increase in permanent hole elon gation was small because the greater presence of kink bands at a certain distance from the hole probably only marginally affected the local behaviour at the pin/hole interface, where damage initi ated before the first characteristic point and clearly affected the contact surface.

To investigate the nature of fatigue damage when fatigue loads did not exceed the static initiation load of kink bands, fatigue tests were performed at low load levels (with jF min j 0:54,jF ultimate j) and stopped after 10 6 cycles. Although matrix cracks were present, especially in the outer ply sequences, no proper kink bands were noticed (although some packs of fibres had apparently bent under load close to the hole surface). Permanent hole elongation was found to be insignificant (lower than 10 mm). However, it increased slightly but steadily at the very beginning of the fatigue test (before 10 5 cycles), remaining almost constant thereafter, which suggested a long residual life.

Therefore, it is very clear that permanent hole elongation can be used as a damage metric, but a question arises: is it the only pos sibility, and how would it compare to other damage metrics, should they exist? Residual stiffness and hysteresis losses, which are necessarily related to the amount of fatigue damage, come to mind. Since fatigue damage develops so early in the fatigue life, clear trends should emerge. The purpose of the following section is to investigate these points.

Evaluation of different damage parameters

Stiffness as a damage parameter

Residual stiffness has long been used to monitor damage evo lution in composite laminates. When it comes to bearing, the study by Smith and Pascoe [START_REF] Smith | Fatigue of bolted joints in (0/90) CFRP laminates[END_REF] promoted this damage metric as a valu able alternative to permanent hole elongation measurements, which they deemed difficult to carry out. However, their study dealt with cross ply laminates, which do not behave like quasi isotropic laminates. In addition, the coupons they used were not half hole specimens, and this is also a significant difference with respect to the current study.

For the quasi isotropic laminates of our study, the residual stiffness was not a very interesting damage metric as it did not change significantly with increasing cycle count. It increased at the very beginning of the fatigue life, and then decreased progressively for most of the lifetime up to the final failure (Fig. 11). In most cases, after reaching its peak value, stiffness did not decrease by more than 4%, and in some cases, the residual stiffness could even remain almost constant for the whole fatigue life. The initial increase in the residual stiffness was attributed to a run in phenomenon, during which the composite contact surface was assumed to adapt itself to the pin surface by wearing slightly, ultimately resulting in an increased contact surface. High initial stiffness was not positively correlated with longer fatigue lives. Should such an effect exist, it would be difficult to demonstrate as it could be masked by slight variations in the gauge length of the coupon, which was not care fully controlled, as it was considered a non critical parameter.

Hole elongation as a damage parameter

Being characterized by a three stage evolution, the permanent hole elongation evolution can be described by the Wu and Yao model [START_REF] Wu | A fatigue damage model of composite materials[END_REF], which was originally designed to describe damage evolution in a tension loaded unidirectional laminate. lt can be assumed that the permanent hole elongation v is directly propor tional to damage D, so equation (1) holds.

(1) lt should be noted that the parameter k can be chosen arbitrarily, as it only acts as a scaling parameter, transforming the adimen sional damage variable into a length. The main interest of the previous equation is that it can be used to predict the fatigue life if v after N fatigue cycles is known. lt could also be used to generate master curves that would prove very useful if a FE damage model were to be developed, by making it possible to check the credibility of the simulated permanent hole elongation curve, irrespective of the number of cydes to failure.

(2)

Of course, to do this, the parameters a and b should preferably only depend on the test configuration (load amplitude, R ratio, etc.), through a mathematical expression that needs to be defined. ln this regard, it can be considered that the parameter ais related to the load through a power law. This parameter mainly controls the shape of the curve in the initial run in stage. A lower a means a higher initial increase in the permanent hole elongation. lt has been noticed that a tends to be smaller as fatigue life becomes longer. Therefore, long fatigue lives seem to be positively correlated with a high initial increase of the permanent hole elongation.

a A ( IFminl ) 8 lfultimatel (3)
The parameter b does not seem to be related to any test parameter. However, as b mainly controls the shape of the evolu tion curve in the vicinity of Nt, it can be expressed as a function of a critical permanent hole elongation value V, reached at some spe cific N/Nt ratio close to 0.9. However, since this critical value is not reached at exactly the same N/Ntratio for ail fatigue tests, variations around this ratio should be considered. Therefore, only an envelope of the true fatigue behaviour can be obtained. As a result, the predictive abilities of the model are considerably reduced. Even if the parameters a and b are made dependent from the number of cydes to failure Nt, as originally suggested by Wu and Yao, it is not possible to find a law that enables accurate values of a and b to be obtained for a given test configuration, without having to perform curve fitting.

b (4)

A more interesting approach is to evaluate the derivative of equation ( 1) for N/Nt 0.5, which corresponds to the permanent hole elongation accumulation rate at mid life, i.e., in the second stage of damage development. Since the permanent hole elonga tion increases almost linearly for the whole of stage 2, it can be considered that, for this stage, dv/dN is constant. Therefore, if the permanent hole elongation rate is estimated experimentally at the beginning of stage 2, it can be directly identified to dv/dN evaluated for N/Nt 0.5. lt was found that dv/dN evaluated for N/Nt 0.5 could be related to Nt through an inverse power law (Fig. 12 ).

This result means that the fatigue life can be easily predicted, as long as the permanent hole elongation accumulation rate is known once stage 2 is reached. A proper fatigue life estimation is, however, difficult when fatigue lives are greater than 10 6 cycles. ln this particular case, the permanent hole elongation accumulation rate is extremely low in stage 2, which makes its experimental estimation tricky. Moreover, unless a very large number of cycles have been performed, it is not trivial to determine whether stage 2 has been reached or not. Should this not be the case, the life prediction would be overly conservative. Thus, this model is best suited to the anal ysis of fatigue data with Nt ranging from 1<>2 to 10 6 cycles. The ca pabilities of the method are shown on Fig. 13. The predictions were made using fatigue data limited to N � 0.33 • Nt, i.e., limited to stage 1 and the beginning of stage 2. Finally, it has to be acknowledged This investigation showed that hysteresis never completely vanished. However, two different trends emerged, so the experi mental data points could be fitted by a function that was a sum of two power laws (Fig. 16). This probably means there was a shift in the way energy was dissipated. Dissipation might have occurred essentially through viscous mechanisms at low load levels, and damage would then play a more and more important rote as the alternating load increased. Referring to an S N curve (see Fig. 17 for example), the change in trend can clearly be associated with the frontier between the HCF and the VHCF(Nf� 1ü8) fatigue regimes. The existence of a fatigue limit still remains to be debated.

Does a bearing fatigue limit exist?

Although no true fatigue limit really exists for metallic materials [START_REF] Bathias | There is no infinite fatigue life in metallic materials, Fatigue and Fracture of Engineering[END_REF], it is well known that, formost of them, the SN curve becomes asymptotic in the VHCF fatigue regime. Assessing the existence of such asymptotic behaviour in composite materials is extremely difficult, as the self heating phenomenon usually imposes test 0. 45 ,._, ----------------�-- are pooled together, so as to increase the number of data points.3 a Weibull equation fits the experimental data points fairly well (Fig. 19). Although this equation im plies a fatigue limit, the true existence of a fatigue limit for R 10 is difficult to infer from this analysis alone, even though it seems to be rational for R 3, especially when the two run out tests are considered. lt is inter esting to note that the Weibull equation suggests the existence of different coupon families, with coupons having the highest statie strength also having the highest fatigue strength (see the dotted lines on Fig. 19). This fact is consistent with the SLERA (Stress Life Equal Rank Assumption) theory [START_REF] Barnard | The Strength-Life Equal Rank Assumption and its application to the fatigue life prediction of composite materials[END_REF], which states that statie and fatigue strengths are correlated, making it a sensible argument in favour of the Wei bull equation.

A third argument is the fact that, under monotonie loading, kinking only starts when the bearing Joad reaches 70-75% of the failure Joad. Therefore, when the applied fatigue Joad If mi ni is close to this threshold, 4 the fatigue life is very long, because kinking has to be triggered and has to affect a certain material volume before fatigue failure can occur. The suspected fatigue limit is such that the corresponding lf mi nl Joad is below the Joad at which kink bands initiate in statie, for both R ratios. Hence, since fatigue life can be very long when If mini is slightly above this threshold (Nf > 10 7 for R 3), it can be expected that, when it is below, no fatigue failure would occur before a much higher number of cycles, supporting the idea that a practical fatigue limit might exist. Although fatigue failure seemingly occurred below the kink band initiation threshold on some occasions, this might not have actually been the case. ln fact, in most fatigue tests, the peak Joad lfminl exceeded the threshold, even at low Joad levels, the scatter in statie strength being rather large when evaluated on a large number of coupons (> 10). The possibility should also be considered that, even if the peak fatigue Joad exoeeds the statie initiation threshold, the Joad amplitude might be too small to significantly damage the coupons. This is what appeared to happen with R 3, as no fatigue failure was obtained for peak loads falling into the kink bands statie 3 Which is reasonable, as these points fell within similar scatter bands. initiation Joad band.

The last argument refers to the change in trend in the evolution of the mean hysteresis Joss as a function of the alternating Joad, which was observed in Fig. 16.

Conclusions

The fatigue life of quasi isotropie laminates subjected to bearing loads can be predieted using damage parameters such as the per manent hole elongation accumulation rate or the cumulative hys teresis energy. Using such damage metries, clear trends can be obtained. The permanent hole elongation is difficult to measure and, as is the case with cumulative hysteresis losses, fatigue data need to be post processed However, it relates directly to the amount of damage, whereas hysteresis energy encompasses not only damage but also other dissipative phenomena such as visco elastiàty. Both damage metries can be used to prediet the fatigue life based on a limited amount of fatigue data, typieally accounting for Jess than 30% of the total fatigue life. To achieve an in service predietion of the residual fatigue life, a monitoring system should be implemented. The measurement of a given damage metric, either directly or indirectly, would then be possible. But such a system is still utopie, espeàally for rotating structures or compo nents. A more down to earth application of these damage metries would be to evaluate the capabilities of a damage mode!. lndeed, those would be useful to assess the consistency of the predieted damage rates or hysteresis losses, whieh would be key if the mode! in question were to be used to predict the fatigue life under spectral loadings. Finally, even though the existence of a fatigue limit has not been proven, the ditferent analyses strongly suggest that a practical fatigue limit does seem to exist. No fatigue failure would occur for alternating loads lower than about 20% of the statie failure Joad, since, at this Joad level, the expected fatigue lives are clearly located in cycle ranges that are not susceptible of being reached in any known industrial application. lt has been shown that this practieal fatigue limit is dosely related to the statie initiation Joad of kink bands, in full consistency with the fact that the latter drive the fatigue failure process.

Fig. 1 .

 1 Fig. 1. Half-hole bearing test set-up.

Fig. 2 .Fig. 3 .-Fig. 4 .

 234 Fig. 2. Coupon geometry. Both UD-fabric and woven fabric coupons were about 4 mm thick.

Fig. 5 .Fig. 6 .

 56 Fig. 5. lncrease in the permanent hole elongation. The percentages correspond to the ratio lfmrn l/lfuttnoc,I, where lfu1t!moc, I was estimated using 12 static tests. 0.8 fil 0.7 � § 0.6

Fig. 7 .Fig. 8 .

 78 Fig. 7. Local damage mechanisms that develop during stage 1 explain the initial increase in permanent hole elongation. At this stage, kink bands are rarely fully developed; therefore, they can be referred to as "incipient" kink bands.

Fig. 9 .Fig. 10 .

 910 Fig. 9. At stage 3, kink bands appear to ultimately collapse as fibre debris are generated in the bands due to repeated loading.

Fig. 11 .

 11 Fig. 11. Stiffness variations for six UD-fabric coupons with R 10. Stiffness was calculated for each individual �le. which explains the slight oscillations.

Fig. 12 .Fig. 16 .

 1216 Fig. 12. The permanent hole elongation accumulation rate, evaluated for N/Nt 0.5, can be obtained through an inverse power law, where the independent variable is the number of cycles to failure. Therefore, the fatigue life of a given coupon can be pre dicted as long as the stag,e 2 permanent hole elongation accumulation rate is known.

Fig. 17 .

 17 Fig. 17. Based on the knowledge of the mean hysteresis loss for low load levels and using the equations linking the critical cumulative hysteresis to the number of cycles to failure, it is possible to estimate the number of cycles to failure of coupons expected to fail in the VHCF regjme (points with err or bars� Points without err or bars correspond to coupons cyclically loaded until fatigue failure. The fitted S-N curves do not take the err or bars into account. as errors are asymmetric.

NJ = 10 4 NJ = 10 12 -Nt = 10 1s - Nt = 10 6 R = 3 • 1 Fig. 18 .

 41263118 Fig. 18. Should the S-N curves cross, a knee point would appear in the CLD diagram (circled areai translating the fact that. despite lower mean and altemating loads. the fatigue life for R 10 could end up being the same as for R 3.

Fig. 19 .

 19 Fig. 19. The S-N curves might have a horizontal asymptote in the VHCF regime. If they were not asymptotic nor superposed, they would cross. resulting in an absurd CID diagram (Fig. 18 ). The arrows indicate runout.

4

  ln fatigue, this threshold corresponds to lf o1 ,l/lf utimar ,I <': 0.31 (R 10) or 023 (R 3), see Fig. 19.

Characterized by a sharp load drop.