Abrasive wear between rough surfaces in deep drawing

Wear 256 (2004) 639–646

Abrasive wear between rough surfaces in deep drawing M.A. Masen∗ , M.B. de Rooij Laboratory of Surface Technology and Tribology, Department of Engineering Technology, University of Twente, The Netherlands

Abstract In tribology, many surface contact models are based on the assumption that surfaces are composed of a collection of small asperities of which the tips are equally sized and spherically shaped and have some kind of statistical height distribution. This approach was used in 1966 by Greenwood and Williamson and was successfully followed by many researchers during the following decades. The statistical representation of surface topography enables calculation of contact forces and asperity deformations with reasonable accuracy using well established equations. Although this approach has proven to be suitable for static contact situations, alternative representations of the surface topography are required when modelling abrasive wear. In the current work an elastoplastic contact model is developed in which a representation of the surface topography is obtained by best fit approximations of the micro-contacts, obtained from real, measured surface height data. In this deterministic surface representation, the tips of the contacting asperities are assumed to have an ellipsoidal shape. Given the material parameters and contact conditions, the load and deformation of a single asperity can be computed. Subsequently, the wear induced by each individual asperity is obtained by inserting its size and shape and the conditions into a “single asperity micro-abrasion model”. By summing the contributions of all individual asperities, the total abrasive wear volume is obtained. The results of the developed abrasive wear model are compared with results obtained using a statistical approach. © 2003 Elsevier B.V. All rights reserved. Keywords: Abrasive wear; Rough surfaces; Contact; Sheet metal forming

1. Introduction A recent development in sheet metal forming industry is the application of tailored blanks: blanks prepared from separate pieces of sheet material, welded together prior to the forming process. The pieces of sheet material can have different properties, enabling weight reduction of components, corrosion resistance using cathodic protection or the possibility to create local weak spots, resulting in well defined deformation zones in the event of an accident (e.g. crash zones in automotive applications). However, a major drawback in the utilisation of these pre-welded or tailored blanks is that the weld between the pieces of sheet exhibits increased hardness and roughness, leading to possible abrasive wear of the sheet metal forming tool [1]. In this work, a model is developed describing the abrasive wear of deep drawing dies caused by the application of tailored blanks. The model is set up on the assumption that the macro-scale contact of two surfaces equals a collection of independently operating single asperity contacts. As a start, the contact between a single roughness asperity of the weld and the relatively soft and smooth tool surface is ∗ Corresponding author. E-mail address: [email protected] (M.A. Masen).

0043-1648/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2003.10.006

analysed. Subsequently the contact between multiple asperities and the tool surface is analysed, assuming both surfaces to be nominally flat.

2. Single asperity contact It is common knowledge in contact mechanics that the real contact area between two rough surfaces covers only a fraction of the apparent contact area, i.e. when two surfaces are in contact, the load is carried by a number of relatively small contact spots. The macro-scale contact between two rough surfaces can be seen as a summation of these micro-scale asperity contacts. 2.1. Circular single asperity contact The elastic contact between two solids is described by the Hertz theory [2]. In this theory, the contact between two curved solids is “scaled” to the contact between a sphere and a flat, resulting in relatively simple equations. The contact area and the contact load as a function of the penetration depth ω for the contact between a single spherical asperity and a flat are given by Eqs. (1) and (2). After a critical penetration depth the elastic Hertz theory is not applicable as

640

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646

Nomenclature a A An Ar A A C Dp Ei E E(e) ftr F H K(e) L n N p0 R Ra Rmaj Rmin Rq R1 R2 s V Vi

contact radius (m) contact area (m2 ) nominal or apparent contact area (m2 ) real contact area (m2 ) area of front view of wear groove (m2 ) area of front view of shoulders (m2 ) shape factor according to [15] asperity degree of penetration (–) Young’s modulus of solid i (Pa) reduced Young’s modulus (Pa) elliptic integral of the second kind relative contact shear strength (–) contact load (N) hardness (Pa) elliptic integral of the first kind sliding distance (m) asperity density (m−2 ) number of deforming asperities (–) pressure (Pa) asperity radius (m) CLA surface roughness parameter (m) major asperity radius (ellipsoid) minor asperity radius (ellipsoid) RMS surface roughness parameter (m) semi-major axis of ellipse (m) semi-minor axis of ellipse (m) asperity height (m) volumetric wear (m3 ) volumetric wear due to ith asperity (m3 )

Greek letters α fraction of asperities αx dimensionless radius of contact spot (–) βHK asperity degree of wear (–) γ dimensionless normal approach (–) θ asperity attack angle (–) κ bulk shear strength of soft surface (Pa) µ coefficient of friction (–) νi Poisson ratio of solid i (–) ξ angle between major axis of micro-contact and moving direction (rad) σ asperity standard deviation (m) τ shear strength of the interface (Pa) φ(s) asperity height distribution (m) ω asperity penetration depth (m) ωcr critical asperity penetration depth, elastic limit (m) Ω wear resistance (N/m2 ) Subscripts and superscripts asp indicates single asperity contact c cutting mode det calculated using deterministic model

el p pl stat w x y

calculated assuming pure elastic contact ploughing mode calculated assuming pure plastic contact calculated using statistic model wedge formation mode in x-direction in y-direction

either the asperity or the flat starts to deform plastically. The contact between surface asperities operating in the plastic regime is, amongst others, discussed by Johnson [3] and Francis [5]. In this regime the relations for contact area and load as a function of the asperity penetration are, in their simplest form, based on the truncation of a spherical cap (Eqs. (3) and (4)): asp

(1)

asp

(2)

asp

(3)

asp

(4)

Ael = πRω Fel = 23 E R1/2 ω3/2 Apl = 2πRω Fpl = 2πRωH

In Eq. (2), E denotes the reduced Young’s modulus as defined by −1
1 − ν12 1 − ν22 + (5) E = 2 E1 E2 Eqs. (1) and (3) are shown in Fig. 1, in which a sudden jump in the contact area can be observed at the critical penetration depth ωcr . Such a discontinuity is implausible and suggests the existence of an interim regime between the elastic and plastic asymptotes [3,5]. Chang et al. [6] included volume conservation in the plastic regime, providing a more gradual transition from the elastic into the plastic regime. Zhao et al. [7] discussed an interim elastoplastic regime in which the relation between contact area and asperity penetration depth is given by a third order polynomial template. The intermediate stage in the asperity penetration process is often neglected (see e.g. [8]). Because of the nature of the deep drawing

Fig. 1. Relation between contact area and penetration depth for an elastic–perfectly plastic single asperity contact.

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646

641

process, this intermediate stage is not of importance in this study. 2.2. Elliptical single asperity contact When the surface roughness is anisotropic, the asperities on the surface should be described by ellipsoids instead of spheres (see also Bush et al. [9] and Halling and Nuri [10]). The contact area and the load carried by an elastic deforming ellipsoidal asperity in contact with a smooth solid as a function of the asperity penetration ω are described by Eqs. (6) and (7), in which Rx and Ry denote the radii of the ellipsoid, αx , αy the dimensionless radii of the contact spot and γ the dimensionless normal approach, as defined by Reussner [11]: asp

R x R y αx α y ω Rx + R y γ   3/2   Rx Ry 1/2 8 ω = E 9 γ Rx + R y

Ael = 2π asp

Fel

(6)

(7)

The maximum penetration depth for elastic deformation ωcr is given by   Rx R y 0.4H 2 (8) ωcr = 18K(e)E(e) Rx + R y E In the fully plastic regime the contact area between an asperity and a flat is, in a similar way as in the spherical case, given by 9. In this regime, the contact pressure by definition equals the hardness of the softest material, resulting in Eq. (10) for the load carried by an ellipsoidal shaped asperity:  asp Apl = 2πω Rx Ry (9)  asp (10) Fpl = Apl H = 2πω Rx Ry H 2.3. Wear caused by a single asperity Challen and Oxley [4] modelled the sliding contact between a rigid plastic wedge and a softer deforming surface using slip-line models. They presented three possible wear regimes: ploughing, cutting and wedge formation. In a later work [12], they extended their model to rigid cylindrical asperities, but according to Bressan et al. [13] this extension might lead to theoretically unacceptable solutions. Based on the validity of the slip-line models used by Challen and Oxley, as well as on experimental results, Kato and Hokkirigawa [14] presented a wear mode diagram depicting the wear mode in which a single scratching asperity operates as a function of the dimensionless shear strength ftr and the asperity attack angle θ. The dimensionless shear strength is defined as the quotient of the shear strength of the interface (τ) and the bulk shear strength of the soft surface (κ): τ ftr = (11) κ

Fig. 2. Single asperity.

In subsequent work [15–17], the asperity attack angle on the y-axis of the wear mode diagram was replaced by the degree of penetration, defined as the amount of penetration into the counter surface divided by the radius of the contact spot between the asperity and the counter surface in moving direction (Eq. (12), see Fig. 2): ω Dp = (12) ax Based on experimental data [14], the translation from asperity attack angle θ to Dp is given by   θ Dp = 0.8 tan (13) 2 Inserting Eq. (13) into the approximate transitions between the three wear modes as derived by Challen and Oxley [4], gives Eqs. (14) and (15): Dppl→wf = 0.8 tan( 41 arccos ftr ) Dppl,wf→cu =

0.8 tan[(1/8)(3π + arccos ftr )]

(14) (15)

The wear mode diagram, as schematically shown in Fig. 3, can be used to determine the wear mode in which a single asperity operates. For each wear mode, the ratio of material removed from the surface to material transferred to the shoulders of the wear groove was determined [15,18]. This ratio is expressed in terms of the parameter βHK as defined in Eq. (16) and Fig. 4: βHK =

A − A A

(16)

Fig. 3. Schematic representation of the wear mode diagram, after Hokkirigawa and Kato.

642

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646

possible to apply these equations to a macro-contact with multiple asperity contacts. Usually, this is done using a statistic approach as discussed below. 3.1. Statistic contact model

Fig. 4. Model of a wear groove.

For the ploughing mode all material is transferred from the groove to the shoulders (βHKp = 0), resulting in no volume loss, and hence zero wear. For the cutting mode, the value falls within the range 0.8 < βHKc < 0.95 and for wedge formation βHKw ranges from 0.2 to 0.8. Because the asperities are assumed to be ellipsoidal, the front view of a typical wear groove as illustrated in Fig. 4 is an elliptic segment with radii R1 = Rmaj sin ξ + Rmin cos ξ and R2 = s. In this equation, ξ is the orientation of the asperity with respect to the moving direction and s is the asperity height. Eq. (17) gives the geometric relation for an ellipse:  2  2 y z + =1 (17) R1 R2 Substituting z = R2 − ω as y = ay , as shown in Fig. 4 gives Eq. (18). According to Harris and Stocker [19], the area A of this segment can be approximated by Eq. (19): R1  ay = 2R2 ω − ω2 (18) R2 A ≈

4 ω3 ay ω + 3 4ay

(19)

Combining these equations results in the following equation for the area of the front view of a wear groove: A ≈

ω2 32R21 R2 + 16R21 ω − 3R22 ω  12 R1 R2 2R2 ω − ω2

(20)

Now, the volumetric wear caused by the sliding of a single ellipsoidal asperity over a distance L can be described by V = A βHK L ≈

ω2 12

32R21 R2

+ 16R21 ω

 R1 R2 2R2 ω

− 3R22 ω βHK L − ω2

(21)

3. Multiple asperity contacts Once the relations for the contact between a single hard asperity and a smooth, relatively soft solid are known, it is

The most commonly used statistic contact model is the GW-model, as developed by Greenwood and Williamson in 1966 [20]. It gives a description of the contact between two rough surfaces based on the random nature of surface roughness: rough surfaces can be represented by three parameters: the asperity height distribution φ(s) (often assumed to be Gaussian with standard deviation σ), the average asperity radius Ravg and the asperity density n. Ignoring both bulk deformation and any possible interaction between neighbouring asperities, this surface representation enables describing the elastic contact between two rough surfaces using the Hertz theory as described before. During the past three decades this model has shown to provide reasonable accuracy (see e.g. [21,22]). Extensions of the GW-model to include plastic deformation and elliptical contacts are described by Johnson [3], Pullen and Williamson [23], Francis [5], McCool [21], Chang et al. [6], Halling and Nuri [10], Ogilvy [24], de Rooij [25] and Horng [26]. In statistical models, often a truncated Gaussian distribution of asperity heights is assumed, with a maximum asperity height smax = 3σ. When n denotes the asperity density and An the apparent contact area, the number of asperities in contact with a smooth counter surface as a function of the penetration depth ω is given by  smax 1 2 2 N = nAn √ e−(s /2σ ) ds (22) σ 2π h When the indentation depth of an asperity into a soft surface is less than the critical indentation depth ωel , the asperity operates in the elastic regime. However, when the penetration depth is exceeded, the asperity enters the plastic regime. The number of asperities in a multiple asperity contact operating in the elastic, respectively, the plastic regime are given by Eqs. (23) and (24):  h+ωel nAn 2 2 Nel = √ e−(s /2σ ) ds (23) σ 2π h  smax nAn 2 2 Npl = √ e−(s /2σ ) ds (24) σ 2π h+ωel Combining Eqs. (23) and (24) with the equations for single asperity contact (Eqs. (7) and (10)) gives for the real contact area Ar and the total contact load F: Ar = Ael + Apl

 h+ωel Rx Ry nAn 2 2 e−(s /2σ ) ds √ Rx + Ry σ 2π h  smax  nAn 2 2 + 2π Rx Ry √ e−(s /2σ ) ds σ 2π h+ωel

=π

(25)

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646

643

F = Fel + Fpl    h+ωel Rx Ry 1/2 2 nAn 2 2 = E √ e−(s /2σ ) ds 3 Rx + R y σ 2π h  smax  nAn 2 2 + 2πH Rx Ry √ e−(s /2σ ) ds (26) σ 2π h+ωel 3.2. Deterministic contact model When two surfaces are in contact, the load is carried by a number of relatively small micro-contacts. In the described statistic model, these micro-contacts are assumed to be the summits on the rough surface, calculated using a five-point or nine-point criterion as described by Greenwood [27]. However, a typical micro-contact in the sliding contact between rough surfaces is larger than one single summit, and the determination of summits depends on the resolution of the surface height measurement. A more scale independent measure of the properties of the micro-contacts can be obtained using a deterministic approach. Surface height data of a rough surface can be measured using an optical interference microscope as described by de Rooij [25]. Fig. 5 shows a typical resulting surface height representation of an area covering 103 ␮m × 84 ␮m. From the surface height data of a micro-contact, the size and shape of the ellipsoidal approximation are obtained by a least squares fit through the measured height data of the contact spot. From this fit, two radii Rx and Ry and the orientation angle ξ of the asperity are obtained. Inserting these parameters and the penetration depth ω into Eqs. (7) or (10), depending on the regime in which the single asperity operates, gives the load carried by one single micro-contact. Simply summing the contributions of all individual micro-contacts gives the load carried by the macro-contact. Repeating this procedure for a sequence of load increments, i.e. simulating that one surface subsides into the other, a force–displacement relation for the rough surface is obtained. Merging of several adjacent asperities into a single micro-contact with increasing load is automatically taken into account because the contact spot fitting procedure is repeated for each load increment and only surface height data that lies within the contact area (coloured dark in Fig. 6(a)) is used as input for the ellipsoid fitting

Fig. 6. Change of fitted asperity radius with changing surface approach.

function. In Fig. 6(b), this is illustrated for the single asperity contact shown in Fig. 6(a). In this two-dimensional case, as the two surfaces approach, the radius of the fitted sphere changes considerably with changing approach. 3.3. Deterministic wear model When the contact parameters are determined using the contact model, the wear volume of each separate asperity is determined using Eq. (21). Because a rough surface is represented as a collection of asperities, the total wear volume can be calculated according to Eq. (27), i.e. simply summing the contributions of all individual micro-contacts: V =

N 

Vi i=1
N 

≈L

i=1

ω2 32R21 R2 + 16R21 ω − 3R22 ω  βHK 12 R1 R2 2R2 ω − ω2



(27)

i

The wear resistance is defined as the reciprocal of the volumetric wear per unit load and unit sliding distance:  −1 V Ω= (28) FL

4. Results

Fig. 5. Optical surface measurement.

As stated before, statistic contact models are often used and accurate results are reported in literature. Hence it is a logical step to compare the results of the developed deterministic contact model with a statistic one. After that, the developed wear model can be compared with the results obtained using a conventional abrasive wear model.

644

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646

Table 1 Statistic properties of the sample surfaces

(m−2 )

Surface A anisotropic

Surface B anisotropic

Surface C isotropic

7.33 × 1.23 × 10−7 5.89 × 10−6 103 756 2395

7.03 × 2.91 × 10−7 3.00 × 10−6 99 725 2297

6.15 × 1010 3.07 × 10−7 3.00 × 10−6 86 634 2010

1010

n Ravg (m) σ (m) Nstat (ω = σ) Nstat (ω = 2σ) Nstat (ω = 3σ)

1010

4.1. Comparing contact models In Table 1, for three numerically generated sample surfaces, some contact parameters calculated using the statistic approach are shown. Each surface representation covers an area of 256 ␮m × 256 ␮m with a resolution of 1 ␮m, similar to an actual surface height measurement obtained using an optical interference microscope. From the asperity density n, the number of micro-contacts Nstat at a given approach can be calculated using Eq. (22), assuming An equals the surface area. Table 2 shows the number of micro-contacts Ndet according to the deterministic model. At low approaches the number of micro-contacts calculated using both methods are about equal, but with increasing approach, Nstat increases faster than Ndet , because in the deterministic contact model several adjacent asperities can merge into one relatively large micro-contact, while in the statistic model the radius of the micro-contacts is assumed to be constant. The data for the asperity radii as tabulated are shown in Fig. 7. The dotted lines depict the average radius calculated using the statistical approach for each surface. The black lines give the band in which the radius lies according to the deterministic model, with the squares showing the average radius. The values for the minimum radius obtained using the deterministic model are of the same order of magnitude as the average radius of the statistic approach. This can be exTable 2 Deterministic surface parameters Surface A

Surface B

Surface C

ω =σ Ndet Rmin (m) Rmax (m) Ravg (m)

52 1.6 × 10−7 3.6 × 10−6 8.0 × 10−7

50 3.6 × 10−7 2.8 × 10−4 1.1 × 10−5

31 2.2 × 10−7 2.0 × 10−6 6.7 × 10−7

ω = 2σ Ndet Rmin (m) Rmax (m) Ravg (m)

459 1.1 × 10−7 3.3 × 10−4 3.3 × 10−6

426 2.7 × 10−7 1.2 × 10−3 1.2 × 10−5

262 1.9 × 10−7 2.9 × 10−5 1.6 × 10−6

ω = 3σ Ndet Rmin (m) Rmax (m) Ravg (m)

814 1.1 × 10−7 1.2 × 10−3 9.8 × 10−6

524 2.9 × 10−7 8.5 × 10−3 5.8 × 10−5

590 1.9 × 10−7 9.0 × 10−4 7.4 × 10−6

Fig. 7. Change of asperity radius Rmin , Ravg and Rmax with increasing approach.

plained as the statistic model calculates the radius of each summit on the surface, assuming the summit to be a single pixel higher than its neighbours (see [25]), while the size of the smallest micro-contacts, which in general are the ones with a small radius, in the deterministic approach is also one pixel. As expected, with increasing surface approach, the minimum asperity radius found remains about the same, while the maximum radius and the average radius increase. In Fig. 8, the relation between contact pressure and contact area for surface A is shown. The statistic and deterministic contact models provide comparable results, although the used values for the number of micro-contacts and radii vary. The same conclusion is valid for surfaces B and C. Because the total wear volume as calculated using Eqs. (21) and (27) depends both on the number of asperities in contact and on the radius of these contacting asperities, the calculated wear volumes of a statistic and a deterministic approach might differ considerably.

Fig. 8. Comparison of results obtained using a statistic and a deterministic contact model.

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646 Table 3 Parameters in deep drawing Etool (N/mm2 ) Htool (N/mm2 ) νtool Esheet (N/mm2 ) Hsheet (N/mm2 ) νsheet E (N/mm2 ) p0 (MPa)

180000 1225 0.3 210000 2700 0.3 213000 300

4.2. Application of the model to deep drawing The developed deterministic contact-wear model is applied to the sample surfaces A, B and C using deep drawing conditions (Table 3). A comparison is made with the results obtained using a modified equation of Mulhearn and Samuels, as defined by Hokkirigawa et al. [15] (Eq. (29)). In this equation, only wear caused by Ω=

CH αc βHKc

645

As the deep drawing process operates in the transition between the boundary and the mixed lubrication regime, the corresponding value of 0.5 for ftr is applied. The results for the three surfaces A, B and C show some difference, with the highly anisotropic surface A causing a relatively low wear resistance. Such a difference is unexpected as the same material parameters (hardness, Young’s modulus, Poisson’s ratio) are used for all three surfaces. No explanation has been found yet, but it might be due to the fact that the transitions between the wear regimes as shown in the wear mode diagram (Fig. 3) are based on spherical asperities, while in the case of anisotropic surface roughness the asperities are ellipsoidal. Most conventional wear models, like the wear model of Archard, assume the wear resistance to be independent of the applied load. The developed model shows a slight dependence of load as the wear resistance of surfaces B and C increases with a factor 2 as the load increases 100-fold. For surface A no influence of the applied load on the wear resistance is found.

(29)

asperities operating in the cutting mode is calculated. αc denotes the fraction of asperities operating in the cutting mode and βHKc is the degree of wear for the cutting mode. A lower bound for the wear resistance is obtained by assuming that all asperities in contact operate in the cutting mode, hence αc = 1. Hokkirigawa et al. [15] define C as a shape factor for the abrasive surface, depending on asperity size and shape and the hardness ratio of both solids. It can be argued that C has a value between 1 and 10. With βHKc = 0.875 and the hardness of the soft abraded material H = 1225 MPa, Eq. (29) becomes 1.4 N m/mm3 < Ω < 14 N m/mm3 Inserting the parameters applicable to industrial deep drawing as mentioned in Table 3 into the developed model, the results shown in Fig. 9 are obtained (solid lines). The dotted lines represent the upper and lower band calculated using Eq. (29).

5. Conclusions The weld of a tailored blank is relatively hard and rough and, when processed in deep drawing, causes abrasive wear of the soft tool. In order to provide a solution for this problem, a deterministic abrasive wear model has been developed for the contact between tailored blank and deep drawing tool. The model assumes a rough surface to be composed of a collection of ellipsoidal shaped asperities. The major and minor axes of each asperity are determined by applying a least squares fitting procedure through the measured surface height data of the micro-contact. As the surfaces are in contact and have a relative displacement, each asperity penetrates into the softer counter surface, removing some material. The amount of material removed per asperity is calculated using the wear mode diagram proposed by Hokkirigawa and Kato. The results obtained using the developed contact model agree with results obtained using a conventional wear model. The applied abrasive wear diagram is based on hemispherical asperities, while the representation used in this model allows asperities to have an ellipsoidal shape. Therefore, more work is needed on experimental validation of the abrasive wear mode diagram to adapt it for ellipsoidal shaped asperities, and eventually ellipsoidal shaped asperities with an orientation not perpendicular to the moving direction. References

Fig. 9. Comparison of wear resistance calculated with two wear models.

[1] M. Vermeulen, Effect of Pre-welded Blanks on the Wear of Deep Drawing Tools, Abstracts of Papers, Proceedings of the First World Tribology Congress, London, 1997, p. 333. ISBN 1 86058 109 9. [2] H. Hertz, Über die Berührung fester elastischer Körper, J. Reine Angew. Math. 92 (1882) 156–171.

646

M.A. Masen, M.B. de Rooij / Wear 256 (2004) 639–646

[3] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. ISBN 0 521 34796 3. [4] J.M. Challen, P.L.B. Oxley, An explanation of the different regimes of friction and wear using asperity deformation models, Wear 53 (1979) 229–243. [5] H.A. Francis, Phenomenological analysis of plastic spherical indentation, J. Eng. Mater. Technol. 98 (1976) 272–281. [6] W.R. Chang, I. Etsion, D.B. Bogy, An elastic–plastic model for the contact of rough surfaces, J. Tribol. 109 (1987) 257– 263. [7] Y. Zhao, D.M. Maietta, L. Chang, An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow, J. Tribol. 122 (2000) 86–93. [8] A. Mihailidis, V. Bakolas, N. Drivakos, Subsurface stress field of a dry line contact, Wear 249 (2001) 546–556. [9] A.W. Bush, R.D. Gibson, G.P. Keogh, Strongly anisotropic rough surfaces, J. Lubr. Technol. 101 (1979) 15–20. [10] J. Halling, K.A. Nuri, Elastic/plastic contact of surfaces considering ellipsoidal asperities of work-hardening multi-phase materials, Tribol. Int. 24 (5) (1991) 311–319. [11] H. Reussner, Druckflächenbelastung und Oberflachenverschiebung im Wälzcontact von Rotationskörpern, Ph.D. Thesis, University of Karlsruhe, Germany, 1977. [12] J.M. Challen, P.L.B. Oxley, Slip-line fields for explaining the mechanics of polishing and related processes, Int. J. Mech. Sci. 26 (6–8) (1979) 403–418. [13] J.D. Bressan, G.M. Genin, J.A. Williams, The influence of pressure, boundary film shear strength and elasticity on the friction between a hard asperity and a deforming softer surface, in: Proceedings of the 25th Leeds-Lyon Symposium on Tribology, Elsevier Tribology Series, vol. 36, 1998, pp. 79–90. [14] K. Kato, K. Hokkirigawa, Abrasive wear diagram, Proceedings of the Eurotrib’85, September 9–12, 1985, Elsevier, Amsterdam.

[15] K. Hokkirigawa, K. Kato, Z.Z. Li, The effect of hardness on the transition of the abrasive wear mechanism of steels, Wear 123 (1988) 241–251. [16] K. Hokkirigawa, K. Kato, An experimental and theoretical investigation of ploughing, Tribol. Int. 21 (1) (1988) 51–57. [17] K. Hokkirigawa, K. Kato, Theoretical estimation of abrasive wear resistance based on microscopic wear mechanism, in: K.C. Ludema (Ed.), Proceedings of the Seventh International Conference on Wear of Materials, American Society of Mechanical Engineers, 1989, pp. 1–8. [18] T. Kayaba, K. Hokkirigawa, K. Kato, Analysis of the abrasive wear mechanism by successive observations of wear processes in a scanning electron microscope, Wear 110 (1986) 419–430. [19] J.W. Harris, H. Stocker, Handbook of Mathematics and Computational Science, Springer, New York, 1998, pp. 92–93. [20] J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proc. Roy. Soc. London A 295 (1966) 300–319. [21] J.I. McCool, Comparison of models for the contact of rough surfaces, Wear 107 (1986) 37–60. [22] Z. Handzel-Powierza, T. Klimczak, A. Polijaniuk, On the experimental verification of the Greenwood–Williamson model for the contact of rough surfaces, Wear 154 (1992) 115–124. [23] J. Pullen, J.B.P. Williamson, On the plastic contact of rough surfaces, Proc. Roy. Soc. London A 327 (1972) 159–173. [24] J.A. Ogilvy, Numerical simulation of elastic–plastic contact between anisotropic rough surfaces, J. Phys. D 25 (1992) 1798–1809. [25] M.B. de Rooij, Tribological aspects of unlubricated deepdrawing processes, Ph.D. Thesis, University of Twente, The Netherlands, 1998. ISBN 90 3651218 2. [26] J.H. Horng, An elliptic elastic–plastic asperity microcontact model for rough surfaces, J. Tribol. 120 (1998) 82–88. [27] J.A. Greenwood, A unified theory of surface roughness, Proc. Roy. Soc. London 393 (1984) 133–157.