Roughness Signature Of Tribological Contact Calculated By A New Method .
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017Roughness Signature of Tribological Contact Calculated by a NewMethod of Peaks Curvature Radius Estimation on Fractal SurfacesM. Bigerelle*1,2, J.M. Nianga3, D. Najjar4, A. Iost4, C. Hubert1,2, K.J. Kubiak1,5,61Université de Valenciennes, Laboratoires TEMPO / LAMIH, 59313 Valenciennes Cedex 9, France,2PRES Lille Nord de France,3Equipe Mécanique des Structures, HEI, 13 Rue de Toul, 59046 Lille Cedex, France,4LML, UMR CNRS 8107, F-59650 Villeneuve d'Ascq, Arts et Métiers ParisTech, France,5University of Leeds, School of Mechanical Engineering (iETSI), Leeds LS2 9JT, United Kingdom,6University of Liverpool, School of Engineering, Liverpool L69 3GH United Kingdom.krzysztof@kubiak.co.ukAbstractThis paper proposes a new method of roughness peaks curvature radii calculationand its application to tribological contact analysis as characteristic signature oftribological contact. This method is introduced via the classical approach of thecalculation of radius of asperity. In fact, the proposed approach provides ageneralization to fractal profiles of the Nowicki's method [Nowicki B. Wear Vol.102,p.161-176, 1985] by introducing a fractal concept of curvature radii of surfaces,depending on the observation scale and also numerically depending on horizontallines intercepted by the studied profile. It is then established the increasing of thedispersion of the measures of that lines with that of the corresponding radii and thedependence of calculated radii on the fractal dimension of the studied curve.Consequently, the notion of peak is mathematically reformulated. The efficiency ofthe proposed method was tested via simulations of fractal curves such as thosedescribed by Brownian motions. A new fractal function allowing the modelling of alarge number of physical phenomena was also introduced, and one of the greatapplications developed in this paper consists in detecting the scale on which themeasurement system introduces a smoothing artifact on the data measurement. Newmethodology is applied to analysis of tribological contact in metal forming process.Keywords: Roughness, friction, curvature radius, fractal, drawing process, surfacemetrology.-1-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.0171IntroductionIn general, the physical responses are due to interactions between physicalprocesses and some surface characteristic parameters, such as the geometricalones, and in particular, the curvature radius from which an estimate could be easilyobtained for periodic or stochastic surfaces. However, although its importance is veryoften underestimated, this one appears in the mathematical formulation of numerousphysical models: in optics, it represents a threshold, under which the reflected beamon a surface could not be modelled by the Kirchhoff method, in tribology, it plays animportant role in the determination of the contact pressure.Generally, the calculation of the radius of curvature requires rather smooth curves ofstudied surfaces, however it is not always the case when dealing with fractalsurfaces, as it was shown in Mandelbrot's works [1, 2]. Furthermore, as all the metricparameters relative to a fractal curve depend on the scale of measurement, it thusbecomes particularly difficult to give a sense to the notion of local radius of curvaturefor fractal surfaces. In tribological contact fractal surface are often used to avoidsensibility to scale of measurement [3, 4, 5]. However, some methods thanks toFourier analysis were proposed with the aim of the estimation of the radius ofcurvature [6, 7]. Let us note moreover that, for special classes of surfaces, for whichthe spectrum could be related to the fractal dimension, the curvature radius could beestimated. However, restrictive conditions of surfaces, as the self-affinity, as well asthe existence of artifacts in the Fast Fourier Transform, make that method uncertain.Furthermore, the representation of the radius of curvature in the Fourier space hasnot been extensively studied. Numerous questions arise then: What really the radiusof curvature for a fractal curve means? Does it possess a geometrical meaning?What is the interest of its eventual estimation? What could we deduct, from physicalmodels, based on the consideration of such a parameter? All these questions showthe necessity to give a geometrical formulation of the curvature radius for a fractalcurve. Consequently, in this paper, we suggest establishing, at the same time, thedependence of the radius of curvature on the scale under which the studied surfaceis observed, as well as, its relation to the fractal dimension of that surface. So, as theproperties of the fractal curve are defined from the fractal dimension, the regular nonfractal surface is then influenced by the fractal dimension of the studied real surface.-2-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017Consequently, even under a formulation supposing regular surfaces, as the physicalformulation evoked above, we could not neglect the fractal dimension. To calculatethis last one, there are various numerical methods, still not giving the same result,when they are applied to a modelled surface with known fractal dimension. Due tothat inconsistency, the method which we should choose is the one presenting thesame properties as those of the curves used in the physical model. So, if we want toestimate the influence of the fractal dimension of a surface on the estimation of theradius of curvature, we have to calculate this one, from the last one, with the samescale used for the computation.This paper is divided into two parts:In Part I we first review classical methods of calculation of the radius of curvature,and in particular, that of Nowicki's [8] relative to the regular curves, and for which, weproceed to an adaptation, before its extension to fractal curves. Then, we introduce anew calculation approach for the fractal dimension of surfaces. Its accuracy is testedon fractal curves with known fractal dimension, and some mathematical properties ofthe radius of curvature are stated.In Part II the proposed method is applied to analysis of tribological contact in metalforming process. Variation of peaks curvature radius before and after the process isrevealing detailed topographical signature of different parts of tribological contact.Therefore, the history or contact conditions can be analysed and different zonesinside the contact area can be distinguished. We also show that proposed methodcould be coupled with an inverse methodology to obtain simulated profiles presentingthe same morphology as experimental curves measured by tactical profilometer onsurfaces obtained by polishing. Next presented application is an analysis of artifactsintroduced by radius of tip during measurement of surface by a stylus profiler.2Part I – Mathematical model of Curvature Radius of a rough surface2.1Model of Curvature Radius2.2A Fractal definition of the Curvature Radii of a SurfaceLet Γ be the profile of a given rough surface. Γ can be considered as the graph of acontinuous function z, defined by-3-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017z : [a, b ] IRt z(t )(a b),(1)and parameterized by the real variable t, where the parameterization is introduced bythe function γ, defined as follows:γ : IR Πt γ (t ) (t , z(t ))(2)(t,z(t)) represents any point of the real plane Π. The curvature χ(t) of theparameterized curve led by γ, is then defined by:χ (t ) z ''(t )1 z ' 2 (t )(3)3/2Therefore, the curvature radius r(t) of the profile Γ at the location t, can be(1 zwritten such as: r (t ) '2(t ))32(4)z ''(t )with z ''(t ) 0 for any t [a, b ]. Consequently, the mean curvature radius of thatprofile, on the interval [a, b ], is then given by: r 1r (t ) dtb a t [ a, b](5)The derivative functions in Eq.3 and Eq.4 are generally estimated by the finitedifferences method, which is far from being stable. Whitehouse [9] then proposed abetter estimation, by using the polynomial interpolation. However, it was established,without using the fractal concept, therefore this method can not be used [10]. Othermethods based on the Fourier analysis [11] could be used, but they present theweakness not to be consistent in numerical calculation. For those reasons, LonguetHiggins [12] proposed a statistical method based, respectively, on the distributions ofthe maxima and crossings of the mean level, for a random surface, but this methodsupposes the curvature to be statistically independent of the scanning scale.Whitehouse and Archard [13] then proposed a method using the autocorrelationfunction to estimate the curvature statistics. That method was subsequently modifiedby Sales and Thomas, who used the truncated autocorrelation function, according tothe Maclaurin series expansions [14] . Moalic et al. [10] proposed the application ofthe finite differences method on the modified autocorrelation function, in order toestimate the repartition of the curvature of the profile. However, the authors foundthat errors increase with increasing of the wave number. Using now the fractal-4-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017approach, Ganti and Bhushan [15] found that the curvature of the profile follows apower-law χ (t ) fh , where fh is the Nyquist frequency of the surface, related to theresolution of the instrument, and where the parameter is its fractal dimension. Thistheory supposes that the spectrum of the surface follows a power law P ( f ) f 2 5 ,with fh 1/L, and where L is the scanning length. However, as noticed by Gallant etal. [16] in the context of the estimation of the spectrum, the effect of the smoothingdue to the measurement, provides yet another factor which limits the size of thefrequency fh . However, the condition P ( f ) f 2 5 restricts strongly the use of thismethod, as we could establish it, in the case of a white noise, for which P ( f ) isconstant.The common point of all these methods is that they are based on statistical,differential or fractal properties, which could be indirectly related to the radius ofcurvature of the studied profile.Profile heightAlyIntercept mBrclxOlxCProfile length Profile lengthFigure 1: Definition of l x and l y used to calculate the local curvature radius rc ( l x ) .However, contrary to the previous ones, the method proposed by Nowicki [8] allowsthe study of the surface roughness, by introducing a parameter, directly measuredfrom the surface: the so-called, radius of asperity (Figure 1), defined as follows:rc l x2 8l y(6)with l y 0.1Rmax or l y 0.05Rmax , and where Rmax is the maximal range amplitude ofthe profile. This method consists in finding the radius rc of a circle of canter O,-5-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017passing by a crest A, and by two other points B and C of the profile; these two lastones being distant of l x . The distances which separate the line (B, C), respectivelyfrom A and from O, are l y and r- l y The parameter l y is considered sufficiently small,so that the segment [A, O] is supposed to be perpendicular to [B, C], in its middle I. Ifwe now apply Pythagoras' theorem to the triangle OBI, it then follows:(lx)/ 2 ( rc l y ) rc 222(7)Consequently, we get:rc ly2 l x28l y(8)Eq. (8) is obtained, assuming that l y is sufficiently small, and the following conditionis satisfied:ly lx(9)Nevertheless, some remarks can be drawn for such a method:I) The techniques to detect the peaks are not well defined. So, when l y value is fixed,Nowicki's method determines all local peaks (in a discretized case, if zi 1 zi andzi zi 1 , then zi corresponds to a peak). In what precedes, it is assumed a uniformpartition of the interval [a, b], with a grid t0 . t i 2 t i 1 . tN , and withzi z(t i ), i 1,2,., N. Then, as l y values are fixed, there exists for each peak, aunique value of l x giving the following discretized set {zi q ,., zi 1, zi , zi 1, zi 2 ,., zi p } .The peak is retained if zi q . zi 2 zi 1 zi and zi zi 1 zi 2 . zi p . This localradius curvature will be named Euclidian Radii Curvature with the following notationrɶi (t i ) . On the other hand, the analytical method supposes that zi is the maximal peakof the non-discretized surface, and implies p q, if the peak gets a perfect circularshape. Reciprocally, if p q, this last one does not get such a shape.II) The threshold used to estimate l y α Rmax does not have any theoreticaljustification. Indeed, trying to determine the parameter radius of the crest, we havel y lim α Rt . However, on discretized curves and for l y sufficiently small, the choiceα 0-6-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017of l x becomes indeterminate, due to the fact that l x k δ x , where δ x is the length ofthe sampling interval. Furthermore, because of the stochastic aspect of the profile,the three points A, B and C become more and more aligned, when the value of l x isdecreasing, implying a dramatic increasing of the rc variance estimator.III) On experimental profiles, a smoothing effect is realized under a characteristiclength [17]. For example, if the profile is recorded by a tactile profiler, the recordingsurface is then seen smoother at a length of the same order of magnitude than the tipcurvature radius. Consequently, for l y sufficiently small, one has to record thecurvature radius of the measurement artefacts, and the curvature radii will then bewrongly increasing.IV) If the surface contains some noise (white or pink), there exists a great probabilityfor the Nowicki's algorithm to detect false peaks. Finally, the Radii of curvature geterroneous.V) If the profile is the result of the combination of different processes acting at variousscales, the radii of curvature so obtained are different. However, it becomes evidentthat the detection of peaks becomes uncertain and will so favour smaller peaks.VI) For physical surfaces possessing a fractal aspect [18, 19, 20, 21], the calculationof rc has no physical sense and the Nowicki's method will lead to different values ofthat parameter, depending on the sampling rate. Furthermore, it is noticed that thedecreasing of the sampling rate will decrease rɶc . This confirms that rɶc calculated bythe Nowicki's method have no sense if we postulate that zi q . zi 2 zi 1 zi andzi zi 1 zi 2 . zi p (see appendix A for more detailed justification).2.2.1 Theoretical relation in proposed methodFor fractal curves, rc depends on the scale at which the observation is made. And, aswe postulate that the curvature radii could be defined at a given scale, the Nowicki'smethod has then to be reformulated. We will conserve the notion of l x withoutimposing any property to the points of the profile that are related to it, since α cannotbe fixed without introducing an artefact. For these reasons, we choose to calculate l xby the following method:-7-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.0171) We choose a horizontal straight line at the level h that crosses the profile, and ecisely,zi h, j 1, (z i 1, z i 2 , , zi j ) m, zi j 1 m, l x x i j 1 x i where m is a number ofintercepting horizontal lines used in algorithm that are uniformly randomly chosen.2) For each l x , the local maximal peak (maximum value of profile) is obtained whichgives l y . More precisely,zi h, j 1, (zi 1, zi 2 , , zi j ) m, z i j 1 m, l y sup(zk ) hk {i 1,i 2, i j }3) rc is then computed from Eq.6, and this process is repeated for all the otherelements of the set of l x values.4) Another horizontal straight line is chosen randomly and the steps 1 to 3 arerepeated.The detailed algorithm is presented in Figure 2.-8-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017Figure 2: Algorythm used to calculate the Curvature Radius Estimation on FractalSurface.-9-
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017Theorem 1: If l x exists, for all non-constant continuous function funiformlyHölderian, anti-Hölderian, and defined on a real interval [a, b ], one gets, for the fractaldimension of the graph Gf of f : (Gf ) limsup ( log rc ( l x ) log l x )(10)l x 0Rationalization of the approach: As f is Hölderian in t, with exponent H ( 0 H 1 ),there exists a positive constant c, such that, for any t ' :( )f (t ) f t ' c t t 'H(11)Eq.11 follows Hölderian form: [22]v (t , ε ) supt ′,t ′′ [t ε ,t ε ]f (t ′′) f (t ′) c (t )ε H(12)Then if f is uniformly then the constant c is independent of t, by integration over thedomain of definition T, v (T , ε ) Tsupt ′,t ′′ [t ε ,t ε ]f (t ′′) f (t ′) dt cε Hwith (Gf ) 2 H(12)(13)On the other hand, as f is uniformly anti-Hölderian too, with the same exponent,there exists a positive constant c ', independent of t such that, for any ε :v (T , ε ) c ′ε H(14)With (Gf ) 2 H(15)Taking (12) and (14) into account, we can then write, for ε taking the particular valuel x ( t, h )2at the level h that crosses the profile (the term 1.2 is due to the fact thatv ( t , ε ) is defining on a 2ε interval from Eq.12): l ( t, h ) l x ( t, h ) l x ( t, h ) c (t ) x v t, c′ (t ) 2 2 2 Hsupt ′,t ′′ [t ε ,t ε ]H(16)f (t ′′) f (t ′) from Eq.12 is the local range of the function and is identified withthe height l y ( t , h ) of the peak of width l x (t ,h ), localized in t according to ourdefinition. l ( t, h ) Then v ( t , ε ) l y x, t and one gets:2 - 10 -
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017 l ( t, h ) l x (t, h ) l (t, h ) c (t ) x,t c′ (t ) x ly 2 2 2 HH(17)But, f being uniformly Hölderian, then c and c ' do not depend on t, and therefore [22]: (Gf ) 2 H(18) And summing the same l x (t , h ) l x with h values defined in h inf f (t ) , sup f (t ) andt [T ] t [T ] t [0,T ] , defining a 2D domain Ω(l x ) with a A(Ω(l x )) area (in case of discreet set ofmeasured profile points A(Ω(l x )) is the size of this set), then : l ( t, h ) l x ( t, h ) l ( t, h ) , t d ω c ′ xc x d ω l y dω2 2 ω Ω( l x ) ω Ω( l x )ω Ω( l x ) 2 HHand from Eq.12 and Eq.14 l ( t, h ) l l c A ( Ω ( l x ) ) x l y x,t d ω c′ A ( Ω ( l x )) x 2 2 ω Ω ( l x ) 2 HHthat can be rewritten by l x ( t, h ) 1 l l the practical forms c x ,t dω c′ x l y A ( Ω ( l x ) ) ω Ω( l x ) 2 2 2 HwhereH(19) l x ( t, h ) 1ly l 2 , t d ω represents the mean of all highest peaks ofA ( Ω ( l x ) ) ω Ω( x)l x width and is noted l y (l x ,T ) .Then from the definition of the fractal dimension related to the holder exponent [22], itcan be noted that H log l y log l xand therefore from Eq. 18 we can write (Gf ,T ) 2 log l y ( l x ,T ) log l x(20)Now if l y (l x ,T ) l x then l y ( l x ,T ) l x2 8rc ( l x ,T ) and we obtain the final result (Gf ,T ) 2 ( log rc ( l x ,T ) log l x ) .(21)Remarks.Experimentally, the fractal dimension (Gf ) is obtained as a slope, by fitting in a loglog plot the discretized data(log l ,log r ( l ) )xcxperformed by our algorithm. If theregression line fit well, the experimental data then allow writing:- 11 -
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.017rc ( l x ) α l x (22)So, the relation between lx and ly becomes:l y ( l x ) β l x2 ,β 1 8α(23)An interesting properties linked to the fractal concept is the box counting method.More precisely, the box counting is a method of gathering data for analyzing complexpatterns by breaking a dataset, object, image, etc. into smaller and smaller pieces,typically "box"-shaped, and analyzing the pieces at each smaller scale. When boxcounting is done to determine a fractal dimension known as the box countingdimension, the information recorded is usually either yes or no as to whether or notthe box contained the curve or not. In our cases, we will apply similar methodcounting the circles. We will count the number of cases N ( rc ( l x ) ) when the radius ofrc ( l x ) is met for a given l x . This expression allows us to quantify the density of peaksof the surface that is fundamental in tribology (contact mechanic, wear, ). However,the density of peaks depends also on the scale. Intuitively, for a fixed macroscopicarea, the number of peaks will decrease when their radius will increase. It becomesthen obvious to find the scaling law of this decrease. On the other hand, introducingthe number N (rc (l x , T )) of cases where a radius rc (l x ,T ) on the profile length is metthrough the above algorithm (of this is the same number of count that l y (l x ,T ) i.e.N (rc (l x ,T )) N (I y (l x ,T )) ). We have found the following results: log N (l y (l x ,T )) (Gf ,T ) lim sup log l xl x 0 (24) log N (rc (l x ,T )) and therefore (Gf ,T ) lim sup l x 0 log l x (25)The same reasoning as for Eq.22 and Eq.23 applied to Eq. (24) and Eq. (25), allowsus to obtain the following power laws:N ( rc ( l x ) ) α ' l x (26)N ( l y ( l x ) ) α ' l x (27)- 12 -
Published in Tribology International, Elsevier, Vol.65 (2013) 013.03.0172.2.2 Properties1) The fractal dimension is unchanged if f is multiplied by a given factor.2) The fractal dimension is unchanged, through an γ homothetic transform (i.e.z′ f (γ t ) ) of the parameter t.3) If m and m' denote respectively the numbers of z and z' intercepts, with m' km,then N ( rc ( z′, x ) ) kα ′ l x .4)Ifwedonotimposethezi q . zi 2 zi 1 ziconditionsandzi zi 1 zi 2 . zi p , the Nowicki's method is then a particular case of ourmethod, by taking different values l y 0.1 Rm .2.3Analysis of discretisation error by simulation of Brownian profileA Brownian profile with fractal dimension 1.5 is generated by an algorithmic process(Figure 3) and discretized in 107 points. An advantage of this type of profile is toavoid the arithmetic error due to the floating point and inherent to fractal functions asKnopp or Weierstrass.1.1(a)0.70.50.3(b)0.65profile heightProfile height0.90.60.550.50.1-0.10.4502.5e65e67.5e601e7
introduced by radius of tip during measurement of surface by a stylus profiler. 2 Part I - Mathematical model of Curvature Radius of a rough surface 2.1 Model of Curvature Radius 2.2 A Fractal definition of the Curvature Radii of a Surface Let Γ be the profile of a given rough surface. Γ can be considered as the graph of a
Three standardized (DIN 4768/ISO 4287) roughness parameters: average roughness (Ra), average roughness depth (Rz), and maximum roughness depth (Rt); and two derived numbers: peak roughness (Pr) and peak index (Pi) were calculated and compared statisti-cally. Ra is the arithmet
have neglected the effect of changing displacement height. In this experiment two surface roughness changes are investigated, both consisting of a change from upstream 3D Lego type roughness elements to downstream 2D bar type roughness. The downstream surface roughness bar height to canyon width ratio is different in each roughness change case
surface roughness can be seen in Figures 3 through 5 as examples of the observations. Fig. 4. Example of a surface with relatively average surface roughness. (0.335 μm) Fig. 3. Example of a surface with relatively low surface roughness. (0.141 μm) Table 1. 3D Surface roughness parameters Parameters Name Definition Comments Sa Average Roughness
account by modeling the roughness profile as a series of circle segments in the rolling direction, as shown in Fig. 12)3. The roughness profile radius was decided so that the average roughness of the roughness profile model was 3 μm Ra. The other basic analytical condi-tions except the roll surface roughness model are the same as in Chapter 2.
programmed surface roughness and non-programmed surface region are given, respectively. The AFM surface roughness value in the non-programmed surface roughness region in Fig. 3a showed a similar value as the first roughness value measured in the programmed surface roughness region (y -60mm). It means there is no Cr film at that location. This
roughness declined with the discharge energy, associated with the current simultaneous reduction and the discharge time. In addition, the electrode roughness also shows influence on the machined surface roughness. Keywords: Electrical Discharge Machining, Optimization, Surface Roughness, Electrode Roughness Influence.
Introduction Types of surfaces Nomenclature of surface texture Surface roughness value Machining symbols Roughness grades and symbols Surface roughness indication Direction of lay Roughness symbol on drawing Surface roughness for various machining processes SURFACE QUALITY & MACHINIG SYMBOLS 2 Engineering components are manufactured by
known as Arithmetic Average (AA), Center Line Average (CLA). The average roughness is the area between the roughness profile and its mean line, or the integral of the absolute value of the roughness profile height over the evaluation length 2) Rz: Rz is the arithmetic mean value of the single roughness depths of consecutive sampling lengths.
