Maxence Bigerelle
email: maxence.bigerelle@utc.fr

Thomas Mathia

Salima Bouvier

The multi-scale roughness analyses and modeling of abrasion with the grit size effect on ground surfaces

This paper is focused on the study of abrasion using a multi-scale roughness analysis of Extreme Amplitude of Peaks to Valleys (EAPV) on stainless steel (316L) surfaces polished with different grit abrasive papers. In order to examine the different abrasive mechanisms at all scales, roughness measurements are carried out. From the log-log plot of the maximum peaks height or maximum valleys depth amplitude versus the evaluation length, an "apparent" bi-fractal structure is assumed based on the observed break in the plotted curve. Such break may refer to a transition in the abrasive mechanism giving two distinct fractal ranges. However, other mechanisms not completely understood may also explain such break. Using the multiscale approach proposed here, the log-log plot is indeed consistent with a succession of a genuine fractal range (i.e. linear in log-log) and an "ergodic" range above the autocorrelation length, where the "real" amplitude becomes actually stationary, independent of the evaluation length. This means there is no more coupling between the vertical height distribution and the horizontal spread, hence, "the higher scale does not show a fractal structure" above 160 m. In this situation, the continued growth of roughness amplitudes is due to a sampling effect and accurately modeled by the Extreme Value Theory. Then, three regimes of abrasion are recognized. (1) For coarse abrasive particles d > 125 m, where d is the size of the abrasive particles, EAPV seems not to depend on d whatever the scale of observation, (2) for intermediate size particles 10 m < d < 125 m, EAPV decreases with d for all scales which represent the "grit size effect", and (3) for fine particles d < 10 m, EAPV drastically decreases at all scales and becomes independent of the particle size due to the occurrence of the adhesive process. It is shown that regimes 1 and 3 are governed by valleys generation due to respectively cutting process for regime 1 and adhesive process on peaks for regime 3. The regime 2 (where grit size effect is observed) results in peaks arrangement associated to clogging and deterioration of the abrasive surface leading to a lower indentation of the abrasive. The proposed work clearly shows that the bi-fractal structure observed in abrasion can be indeed explained using a single fractal structure.

Introduction

Abrasive machining is one of the conventional manufacturing processes which was extensively studied essentially for technical and economic reasons. The abrasion process is still not well understood and therefore encloses a great number of empirical rules. Besides, such process is also offering very high added value as finishing procedure like polishing, honing, lapping, barreling, grinding etc. and for that reason covered by the secrets. For a large number of situations and as a first approach, it is analogous to conventional machining processes, such as milling or turning, because each of the abrasive particles acts as a miniature cutting tool. However, unlike conventional machining, the grains are much smaller than a cutting tool, and the geometry, the orientation of the individual grains and the acting forces are not well defined even if substantial progress has been achieved recently with some appropriate organized abrasive grits. It is well known that the abrasion process in most of cases can be treated as a stochastic process because the mechanical and the geometrical properties, the movement of the cutting tips are characterized by a large statistical variability. De Pellegrin and Stachowiak [START_REF] De Pellegrin | Evaluating the role of particle distribution and shape in two-body abrasion by statistical simulation[END_REF] modeled abrasion mechanisms with a Monte Carlo-like model. In a same way, Bigerelle et al. [START_REF] Bigerelle | Multiscale functional analysis of wear: a fractal model of the grinding process[END_REF][START_REF] Bigerelle | A numerical method to calculate the Abbott parameters: a wear application[END_REF] have shown that abraded surfaces can be modeled by stochastic fractal functions. Similarly, belt finishing process was recently described using a Monte Carlo model [START_REF] Bigerelle | Mechanical modelling of micro-scale abrasion in superfinish belt grinding[END_REF]. This stochastic aspect is also used to take into account the surface roughness in an abrasion model thanks to a Gaussian approximation of the surface roughness [START_REF] Gorana | Prediction of surface roughness during abrasive flow machining[END_REF]. It is worth noting that a high number of bi-fractal structures met in the literature are based on the observations of the log-log plot (of a given mathematical measure versus a scale of observation) from which the slope is related to the fractal dimension.

The occurrence of a cross over in the log-log plot suggests a bifractal structure. In the present work, it is shown that a log-log plot obtained from abraded surfaces can display a crossover (often linked to a bi-fractal structure) without having a fractal structure for the highest scale. Indeed, the increase of the scale of observation yields to higher probability to meet higher peak amplitudes.

The stochastic process which supports this interpretation assumes severe accumulations of scratches (i.e. damage) which are independent of each other. This regime differs from the fractal structure that supposes the multiscale dependence of tribological mechanisms. In this paper, a methodology is proposed in order to examine whether the peaks or valleys amplitudes of abraded surfaces follow any scaling power law or are mainly based on the damage accumulation.

Materials and methods

Experimental device and surfaces

Rods of 30 mm of stainless steel 316L, were used to characterize the abrasion mechanisms. The chemical composition is given in Table 1. Samples were prepared from a cylinder of 30 mm diameter cut into several discs of 20 mm height. Surface grinding of the studied samples was performed on a specific polishing machine (Planopol-3 and Struers TM Pedemax-2, using different grit silicon carbide (SiC) papers in order to obtain singular abrasion). The silicon carbide, is also known as carborundum which is the synthetic form of SiC and occurs in nature as the extremely rare mineral moissanite. Silicon carbide powder has been massproduced since 1893 and mainly used as an abrasive and today is one of the most used materials for machining. Using the international FEPA 1 standard, the abrasive papers from grit 80 to 4000 which respectively correspond to silicon carbide grain sizes from 200 m to 5 m, were explored for 11 grit abrasive papers (80, 120, 180, 220, 320, 500, 800, 1000, 1200, 2400, and 4000). The correspondence between the different SiC grit papers and the size d of the abrasive particles is given in Table 2. Abrasive process was achieved on an automatic grinding machine at a speed of 300 revolutions per minute. Each grinding step was systematically performed with a new silicon carbide paper under 150 N load during 3 min using distillated, demineralized and filtered water as a cooling medium. Scanning electronic microscopy (HITACHI TM S-520) images of the abraded surfaces are shown in Fig. 1. 1 Federation of European Producers of Abrasives.

Topography characterization

Tactile technique for roughness measurement was selected since a wide range of slopes of the generated morphologies are expected due to the abrasive process of the investigated hard material. The data of the morphological analysis for the explored samples presented in this paper result from systematic metrological investigations using a three dimensional (3D) tactile topometer P10 from TENCOR. The vertical resolution of the used metrological equipment is close to the nanometer while the plane resolution is near 50 nm. The radius of stylus tip curvature is about 2 m checked before the data acquisition. 3D roughness measurements were carried out to ensure isotropic predominance of the examined samples morphologies. Fig. 2 illustrates one of the measured surfaces. Areas of 4 mm × 4 mm are measured using 400 m/s speed and systematically characterized for each surface. Since surfaces are quasi-isotropic, one can proceed with profile analysis as a valuable representation of the examined morphology. The profiles were recorded every 8 m and the points in the profile were recorded every 8 m. Finally, the areas were described by 250,000 points. Since the morphologies of the surfaces were recognized to be quasiisotropic observing the previous 3D measurements, two dimension (2D) high resolution profiles were then recorded for 5 mm using 200 m/s speed and stored for further analysis. Each profile was described by 25,000 points (0.2 m between each point). Fig. 3 represents 2D profiles of surfaces polished with different grit sizes. For each sample, 30 profiles were randomly recorded and a statistical treatment of the results was achieved. The images were stored in order to observe and to analyze the abrasion process and the resulting surface morphologies. This helps to illustrate the fractal aspect of the polishing process. Fig. 4 shows a profile of an abraded surface (polished with grit SiC paper 80) for three spatial zooms (5×, 25×, 250×) located at the origin of the profile acquisition. As can be observed, the structure of the surface presents deep valleys, plateaus and summits due to the grinding process.

Multi-scale roughness characterization

From tribological point of view, there are few metrological and mathematical approaches which can be efficiently used for the characterization of a manufacturing abrasive process, and which can provide a better understanding and modeling of the process [START_REF] Mathia | 3D surface morphology measurements in abrasive machining, 30th scientific school on abrasive machining[END_REF]. Some of them are well established, while others such as fractals, continuous wavelets transform or multi-scale approaches are still under development particularly for the abrasive process application [START_REF]Surface Topography[END_REF]. The average of the minimal amplitude roughness connected with valley depth and maximal amplitude roughness, representing peak height parameters can be used to characterize the surface roughness. Unfortunately, the effect of the evaluation length is not always taken into account although these parameters depend on the observation scale. This scale-dependence was not studied for these two parameters. Only the R t (Peak to Valley roughness parameter) was studied by Dubuc et al. [START_REF] Dubuc | Evaluating the fractal dimension of profiles[END_REF] and was used to characterize the surface roughness [START_REF] Tricot | Courbes et dimension fractale[END_REF] and also the tool machining surface [START_REF] Bigerelle | Roughness characteristic length scales of micro-machined surfaces: a multi-scale modeling[END_REF]. From a mathematical point of view, above a critical length, a loss of the topography memory is observed for the profiles. This loss of memory can be quantified by the autocorrelation function. It is reported in the literature survey that the autocorrelation function can be used to determine the fractal properties of the profiles. The average autocorrelation function of all profiles is plotted in Fig. 5 for grit paper 80. As it can be observed, the autocorrelation decreases until it reaches zero value (in average) for l > 160 m. For longer observation distances, no "memory" (linear correlation) occurs in the profile. The latter becomes a pure random process and must be analyzed with appropriate tools. G,l) and R t (G, l) increase almost logarithmically with the evaluation length l meaning that the amplitudes of peaks and valleys increases with the scale. Indeed, it is worth noting that in the fractal regime, the theory of extreme values is always valid. However, at the scale of the measurement window, when the maximal roughness values are measured, sampling effects do occur which explain the slight deviation from the linear behavior observed in Fig. 7 (see Appendix A for more details on this aspect). Inspection of this figure also indicates that the amplitude of the extreme roughness parameters decreases with an increase of the grit SiC paper.

The different ranges of the multi-scale analysis

Based on a careful examination of the experimental results, three different ranges can be observed. Each range can be correlated to a specific tribological process by a detailed inspection of the interfacial mechanisms of abrasions. With appropriate statistical techniques (minimization of a functional), it can be stated that these three ranges are: Range 1: l ≤ 4 m (log-log linear range), Range 2: 4 m < l < 160 m (fractal range), Range 3: l ≥ 160 m (logarithmic range).

Range 1 (the linear part) is a measurement artefact due to the stylus radius of the profilometer. The second range corresponds to a non linear fractal range (range 2) and the third range represents "an extreme values range" (range 3) that will be analyzed in section below.

Extreme amplitude roughness modeling

The question that should be discussed is "Does the recorded signal exhibit a pure stochastic process above l > 160 m?" Indeed, a new concept was introduced and applied accurately on milling surfaces obtained by single diamond turning [START_REF] Bigerelle | Roughness characteristic length scales of micro-machined surfaces: a multi-scale modeling[END_REF] and Belt finishing [START_REF] Bigerelle | Mechanical modelling of micro-scale abrasion in superfinish belt grinding[END_REF]. Beyond the fractal range, the surface becomes stationary in a statistical sense (ergodicity) meaning that the mathematical expectation of any roughness parameter of the surface remains constant.

However, since the roughness profile signal is given by a collection of heights (i.e. not a continuous function), one can obtain only an estimate of roughness parameters, which indeed is sensitive to the number of distinct values taken into account for the estimation of such parameters. Including sampling effect, fluctuations may occur due to the inherent stochastic process and the magnitude of the extreme values increases with the number of sampling points. Such effect is known as the extreme value distribution (e.g. Gumbel distribution). In the present work, an alternative methodology to the Gumbel distribution is proposed. The surface roughness parameter Y max (G, l) or Y min (G, l) is evaluated at a given observation scale l and the aim is to predict Y max (G, l ′ ) at a higher scale l ′ with l ′ = l + ı, ı > 0. This purpose is of high interest in tribology. Surface integrity is often related to the extreme roughness amplitude and it is impossible to scan surfaces on all the sample size. In order to predict roughness amplitude at higher scales, the evaluation length is supposed to be higher than the fractal regime (l > 160 m). The probability density function of the extreme roughness amplitude values is first modeled using four parameters generalized lambda distribution (GLD) [START_REF] Karian | Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods[END_REF] (see Appendix A.1 for details). Such parameters are estimated by an original method described in [START_REF] Fournier | Estimating the parameters of a generalized lambda distribution[END_REF]. The empirical distribution of the extreme roughness amplitude values are correctly described by the GLD approach, i.e. the maximal and the minimal values are well predicted as verified by a Chi square test. As shown in Fig. 8, the empirical distribution of the extreme roughness amplitude values are correctly modeled by the GLD approach, i.e. the maximum and the minimum values. At the scale of 160 m, the following relations are assumed in order to perform a multi-scale prediction of the maximal, minimal and range amplitude roughness: In order to estimate Y x max (G, 320 m), two values that follow Y x max (G, 160 m) Lambda distributions are generated (using an appropriate random data generator) and a value of Y x max (G, 320 m) is then obtained by taking the maximal values of these two generated data. By widely repeating this procedure, the probability density function of Y max (320 m) can be obtained. This method can be applied at a higher scale l: (G) versus the evaluation length l = k × 160 m for different grit SiC papers G. The curves given in this figure are obtained by averaging the results given in Fig. 7. The proposed model is able to accurately match the experimental data at all scales and for all paper grits, meaning that it is possible to predict the maximum and minimum roughness amplitudes at scales higher than 1000 m by analyzing the roughness at an evaluation length of 160 m. Below this critical value (that is it the initiation (onset) of the ergodic regime), it becomes more and more tricky to predict the extreme amplitude roughness, which seems to confirm that the adopted criterion (i.e. autocorrelation ≈0) for the selection of such critical length is in agreement with the initiation of the stochastic regime.

(l = k × 160 m, k ∈ {2,
Starting from the relation between the size of the abrasive grain and the paper grit according to the FEPA 3 system (Fig. 11), 3 The FEPA system is the same as the ISO 6344 standard. one can plot the maximal and the minimal roughness amplitudes versus the particle abrasive size for different evaluation lengths l (1 m < l < 1024 m) (Fig. 12) which clearly depicts the different stages discussed in this paper.

Discussion of the abrasion mechanisms

As shown in Fig. 10, extreme values model is able to accurately predict the probability to meet a peak or a valley on a surface that is larger than the initial one under the condition that the evaluation length is higher than a critical length. Moreover, it is shown that the self-correlation length is able to define such critical length. This means that the abrasion process that governs the highest peaks and valleys amplitudes is a stochastic process that does not depend on the spatial repartition of the peaks and valleys. Mezghani et al. [START_REF] Mezghani | New criterion of grain size choice for optimal surface texture and tolerance in belt finishing production[END_REF] have shown by a method based on a multi-scale characterization and using the continuous wavelets transform that the abrasion process obtained by belt finishing is located on all scales. This multi-scale approach seems to show that highest peaks or valleys are based on the fact that the roughness is the sum of a large number of independent factors such as the shape of the abrasive particle, the local pressure, the materials properties, the plasticity, the fatigue damage, etc. Therefore, the roughness amplitude can be modeled by a Gaussian law or a lognormal one in the case of abrasion [START_REF] De Pellegrin | Evaluating the role of particle distribution and shape in two-body abrasion by statistical simulation[END_REF]. It can be shown that, if the probability density function follows these two particular PDF, then the maximum of a sample of IID (independent and identically distributed) random variables after proper renormalization converges to an extreme values distribution.

The multi-scale analyses developed in this work can also be used to examine the different steps of abrasion. Inspection of Fig. 12 reveals that the abrasion process can be divided into three regimes. The first regime which corresponds to paper grit equal to 80, the maximal and minimal amplitudes are very high. The second regime, for paper grits lying between [120 and 1200], the maximal and the minimal amplitudes decrease with an increase of the grit of the abrasive paper i.e. the decrease of the size of the abrasive particles. The third regime with paper grits equal to or greater than 2400, a high decrease of roughness is observed. Moreover, the maximal roughness amplitudes remain more pronounced than the minimal ones.

According to a recent work on grit size effect on titanium based alloy TA6V4 [START_REF] Bigerelle | Multiscale characteristic lengths of abraded surfaces: three stages of the grit-size effect[END_REF], it was shown that for regime 1, the wear rate Maximal roughness amplitude (in µm) is independent on the size of the abrasive particle. Applying the Archard's model with a conical shape hypothesis of the particle morphology [START_REF] Rabinowicz | On the mechanism of polishing with abrasives[END_REF], the same wear rate is observed whatever the size of the abrasive particle [START_REF] Sevim | Effect of abrasive particle size on wear resistance in steels[END_REF][START_REF] Giljean | Multiscale analysis of abrasion damage on stainless steel[END_REF][START_REF] Jiang | Modelling of two-body abrasive wear under multiple contact conditions[END_REF].

The second stage is known as the "grit size effect". This means that the decrease of the roughness is principally due to a deviation from the Archard's model. The reason of this scale effect is still under controversy as pointed by Misra and Finnie [START_REF] Misra | On the size effect in abrasive and erosive wear[END_REF]. More precisely, Torrance [START_REF] Torrance | The effect of grit size and asperity blunting on abrasive wear[END_REF] has analyzed this effect and concluded in his bibliographical analysis that the change in the wear rate that leads to loss of roughness [START_REF] Grigoroudis | Modeling low stress abrasive wear[END_REF] is related to three reasons: (i) the decrease of the normal force on the grain, (ii) the flattening of the abrasive grain that is more pronounced for smaller grain size and (iii) finally a sticking of the debris on the abrasive particles for small grit size has a high influence on the abrasion process. The threshold of the grit size effect depends on the wear test, the test apparatus, the materials, the environment [START_REF] Torrance | The effect of grit size and asperity blunting on abrasive wear[END_REF] and the size of the sample [START_REF] Larsen-Badse | Influence of grit diameter and specimen size on wear during sliding abrasion[END_REF]. The literature review indicates that this critical length lies between [50 m and 150 m] and Torrance estimates this critical length around 100 m [START_REF] De Pellegrin | Wear mechanisms and scale effects in two-body abrasion[END_REF]. In the present work, the threshold found by the multi-scale analysis is equal to 125 m (using image analyses of the grain size) which is in agreement with wear test results found in the literature.

The third stage, that is indeed a second "grit size effect", is not often met in the literature, which can be explained by the fact that abrasive particles sizes lower than 10 m (see Table 2) are not usual in tribological analyses of the grit size effect. However, Sasada et al. [START_REF] Sasada | The effect of abrasive grain size on the transition between abrasive and adhesive wear[END_REF] have analyzed a wide range of grain size [3 m-150 m] and their threshold is similar to the one obtained by the present multi-scale analyses: 10 m which corresponds to 2400 grit abrasive paper. They explained that the wear rate becomes high and does not depend on the grain size for stage three. Sasada et al. [START_REF] Sasada | The effect of abrasive grain size on the transition between abrasive and adhesive wear[END_REF] concluded that the wear debris consist of large metal flakes with abrasive particles mixed in. Although abrasive particles are present, the wear is primarily adhesive and the action of the abrasive particle is to promote the removal of the metallic wear debris from the contact region.

Conclusion

From the experimental analyses, it has been shown that Extreme Amplitude of Peaks to Valleys (EAPV) presents three stages (in term of scale) in the abrasive process. Above a critical scale around 200 m, the extreme roughness amplitude parameters are characterized by the extreme values theory. To estimate accurately the maximal peaks to valleys amplitude densities, the generalized lambda distribution and the Monte-Carlo simulation prove efficient, so that EAPV can be predicted. Then, the EAPV can be predicted on ranges higher than the measured one. As consequence, highest peaks or deepest valleys distribution are based on the fact that the roughness of the abraded surface is the sum of a large number of independent factors such as the shape of the abrasive particle, the local pressure, the materials properties, the plasticity, the fatigue damage, etc. This analysis allows identifying and quantifying three regimes of abrasion: for abrasive particle d > 125 m, EAPV is very high with a high wear rate. For 10 m < d < 125 m, EAPV diminishes with d whatever the scale, which represents the "grit size effect". For d < 10 m, EAPV drastically decreases at all scales and is not sensitive to the size of the particle due to an adhesive wear process.

In our statistical approach, the probability density function of maximal peaks and valleys was proposed versus the scale of observation. Therefore, it is possible to introduce these probabilities into models where highest peaks or valleys are critical parameters for surface integrity (crack initiation in fretting, corrosion, etc.) to build up stochastic lifetime prediction. The reliability of the tribological systems and modeling through surface morphology analysis become possible not only for the abrasion as a manufacturing process but also in order to analyze other wear phenomena [START_REF] Sasada | The effect of abrasive grain size on the transition between abrasive and adhesive wear[END_REF][START_REF] Mathia | 3D topography measurements and characterizations of deformable surfaces in context of system's Reliability[END_REF].

Appendix A. Appendix A Some clarifications concerning the non linearity of the extreme value in the fractal stage.

In a large number of papers, a perfect linear stage is never met in a log-log scale. The question is: why does the linearity not perfectly hold? In the proposed work, the authors investigate whether the deviation from the linear behavior can be related to the sampling problem. Inspection of the log-log plot shows that the increase of the R t values diminishes progressively with the profile length which means that if the fractal concept holds (i.e. the linearity in the log-log scale), the R t values are gradually underestimated as the evaluation length increases. A possible explanation of this bias can be related to the total number of data points used for the computation of the R t value. Indeed, the maximum number of data is imposed by the profilometer (80,000 in this study). Therefore, for small evaluation lengths, the R t value will be computed with a limited number of points. The latter will increase with the evaluation length. When the number of points tends to infinity, the evaluation of R t converges to the "true" value. However, when the number of points is limited, the probability to obtain the true maximal range amplitude for a given interval length, diminishes with the number of points. This leads to more and more underestimation of the R t value (at small evaluation length). In order to assess the validity of the previous statement, a perfect Brownian profile ( = 1.5) is simulated (see Fig. A1). In the classical method, the R t is computed using a window of size l where the number of points is equal to l/ x (Fig. A2, left). Besides, R t parameter is computed using a constant number of points (k) (R t k (l)) (Fig. A2, right). For a given interval length, R t values increase with the number of points. However, this increase is less and less significant, which is in agreement with the proposed interpretation of the bias: R t values are more and more underestimated as the number of points used to estimate R t decreases. With the decrease of the size l of the windows, R t values also decrease because of the fractal aspect of the Brownian motion. This sampling effect leads to the occurrence of a non linear stage. If a constant number of points is used in the estimation of R t value for all the evaluation lengths, then a perfect linear relation is observed but the intercept decreases with the number of points used to estimate the R t . This decrease is linked to the extreme values theory.

A.1. The lambda distribution

The generalized lambda distribution (GLD) family is specified in terms of its percentile function (called also the inverse distribution function) with four parameters ( 1 , 2 , 3 and 4 ):

Q X (y; 1 , 2 , 3 , 4 ) = 1 + (y 3 -(1 -y) 4 ) 2 (a1)
The parameters 1 and 2 are, respectively, the location and scale parameters, while 3 and 4 determine respectively the skewness and the kurtosis of the GLD. The probability density function f X (x) can then be easily expressed from the percentile function of the GLD:

f X (x) = 2 ( 3 y 3 -1 + 4 (1 -y) 4 -1 ) (a2)
Obviously, the main problem is to estimate the parameters 1 , 2 , 3 and 4 in order to have the best fitting of the GLD with the experimental frequency distribution (of extreme roughness values in this study). In a first step, empirical moments are calculated from n experimental data x i , i ∈ {1, 2, . . ., n}:

ˆ1 = n i=1 x i n (a3)

Fig. 1 .

 1 Fig. 1. S.E.M. images of 316L samples ground with paper grits: 80, 500, 1200 and 2400.

Fig. 2 .

 2 Fig. 2. 3D representation of the 316L abraded surface with grit paper 80.

Fig. 3 .Fig. 4 .

 34 Fig.3. Recorded profiles of ground surface of 316L alloy with grit SiC papers 80, 120, 180, 320, 500, 800, 1000, 2400, and 4000.

Fig. 5 .

 5 Fig. 5. Mean autocorrelation length of abraded surfaces for grit abrasive paper 80.

3. 1 .Fig. 6 .

 16 Fig. 6. Flowchart of the algorithm for local roughness-amplitude computation. The profile illustrates notations used in the flowchart, d is used for the shift of the window of size l.

Fig. 7 .

 7 Fig. 7. Multiscale roughness values of -Y min (G, l), Ymax(G, l) and Rt(G, l) versus the evaluation length l for recorded profiles of 316L stainless steel polished with five different grit papers G = 80, 220, 800, 1200, 4000.

Fig. 8 .

 8 Fig. 8. Empirical histograms and associated lambda distributions (solid lines) of Ŷ 160 m max and -Ŷ 160 m min PDF functions of ground profiles polished with grit papers 80, 320, 1000, and 4000. The distributions represent all the roughness profiles for different starting positions x.

  Y x,x ′ max (320 m) = max(Y x max (160 m), Y x ′ max (160 m)) Y x,x ′ min (320 m) = min(Y x min (160 m), Y x ′ min (160 m)) for two possible values of x and x ′ .

3 , 4 ,

 34 . . .}) and the values of Y x max (G, k × 160 m) are obtained by taking the maximal value from k values generated from the Y x max (G, 160 m) Lambda distribution. Fig. 9 depicts these PDF 2 functions, obtained from 100,000 Monte Carlo simulations. As shown in the figure, the PDF mode increases with the magnification k. Fig. 10 shows the means of Ŷ

  Fig. 9. Ŷ 160 m,k max and -Ŷ 160 m,k min PDF functions prediction of ground profiles polished with grit papers 80, 320, 1000, and 4000 obtained from 100,000 Monte Carlo simulations for the five magnifications k ∈ {1, 2, 5, 10, 20}. The case k = 1 corresponds to the simulation of the original lambda shown in Fig. 6 (on the right).

Fig. 10 .

 10 Fig. 10. Prediction of the mean of the extreme roughness amplitude parameters -Ŷ 160 m min Ŷ 160 m max R160 m t (predictions are symbols and lines are experimental measurements) versus the evaluation length and the means experimental values Ymax(l), -Y min (l) and Rt (l) for ground surface polished with grit papers 80, 120, 180, 220, 320, 500, 800, 1000, 1200, 2400, and 4000.

Fig. 11 .

 11 Fig. 11. Relation between grit size (in m) and the grit paper (FEPA norm). The fitting equation Grit = 17,474 Paper -1.0245 is obtained by non linear least square method.

Fig. 12 .

 12 Fig.[START_REF] Fournier | Estimating the parameters of a generalized lambda distribution[END_REF]. Mean of the extreme roughness amplitude parameters Ymax(l), -Y min (l) and Rt (l) versus the grit size for different evaluation lengths corresponding to ground surface polished with grit papers 80, 120, 180, 220, 320, 500, 800, 1000, 1200, 2400, and 4000.

Fig. A1 .

 A1 Fig. A1. Simulated profile corresponding to a trace of a perfect Brownian motion.

Log 10 (Fig. A2 .

 10A2 Fig.A2. Classical way to computed fractal dimension (left) and values of Rt k (l) evaluated on a perfect Brownian motion (Fig.1a) versus the evaluation length when Rt k (l) is evaluated with k points on all windows of sizes (l) corresponding to the profile shown in Fig.A1(right).

Table 1

 1 Chemical composition of the 316L stainless steel rod.

	Elements	C	Si	Mn	Ni	Cr	Mo	N	S	P	Cu	Fe
	Wt%	0.008	0.27	1.62	14.58	17.58	2.8	0.060	0.001	0.014	0.070	Bal

Table 2

 2 Correspondence between the grit abrasive paper number and the size of the abrasive particles according to the international FEPA standard.

	Grade	80	120	180	320	500	800	1000	1200	2400	4000
	d (m)	197	127	78	36	30	21	18	14	8	5

PDF probability density function.

where

To calculate 1 , 2 , 3 and 4 , ˛i are estimated by ˆi (Eqs. (a3)-(a6)) and it is necessary to solve a system of four equations highly non linear (Eqs. (a6 ′ )-(a9)). As Eqs. (a12)-(a13) depend only on 3 and 4 and as 3 2 3 = (B -A 2 ) 3/2 and 4 2 4 = (B -A 2 ) 2 , the four equations system become a two equations system with more stable numerical convergence (less numerous local extrema). The solutions amounts to find 3 and 4 by a steepest gradient method on the functional:

and then 2 is calculated from Eq. (a7) and finally 1 from Eq. (a6 ′ ). An algorithm was written and computed using the Statistical Analyses System language to determine the GLD and its related probability density function from the experimental dataset.