Computational Materials Science 25 (2002) 54–60 www.elsevier.com/locate/commatsci Numerical simulation of wear-mechanism maps A. Cantizano a a,* , A. Carnicero a, G. Zavarise b Instituto de Investigaci on Tecnol ogica (IIT), Universidad Pontiﬁcia Comillas de Madrid. C/Alberto Aguilera, 23, 28015, Madrid, Spain b Dipartimento di Ingegneria Strutturale e Geotecnica, Politecnico di Torino, C.so Duca degli Abruzzi, 24, 10129 Torino, Italy Abstract Wear-mechanism maps for diﬀerent materials, actually steel on steel, are being modeled with FEM. A microthermomechanical approach has been used in order to model accurately the macroscopic phenomena of wear. A plastic law for the normal micromechanical contact of asperities has been implemented in FEAP and a slight modiﬁcation, based on experimental results, is proposed. For the three mechanisms modeled, good correlation between the numerical results of wear and those found in literature has been obtained for a pin-on-disk conﬁguration. The ﬂash temperatures reached in the contact interface have been also studied and fair good agreement with literature is achieved. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 46.30.P; 73.40.J; 02.60 Keywords: Wear model; Contact; Friction; Metal-to-metal; Micromechanics; Finite elements 1. Introduction Wear is an inevitable phenomenon generated whenever two rough surfaces slide. It is a very complicated process due to the many variables that play an important role and therefore, diﬀerent approaches have been used in order to ﬁnd useful equations that could describe the regimes of wear, as can be seen in . A model of the wear phenomena depends on the thermo-mechanical characteristics of the two surfaces sliding. Thus, an accurate model for contact based on the real geometry of the surfaces is needed. These models have been developed either using a theoretical analysis of microscopi- * Corresponding author. cally rough surfaces as in  or in , or can be based on experimental data as is proposed in . There are many diﬀerent ways of representing the wear data, usually as a tabulation of the wear rates of the dominant regimes considered. However, these presentations can be considered restrictive because they usually cover a relatively narrow range of sliding conditions related to the mechanical system which is being studied. A much more complete analysis can be found in the wearmechanism maps proposed and developed by , where a wider range of sliding conditions can be included, showing how the mechanisms interface under the possible various operating situations. Thus, the dominant mechanism for any speciﬁc condition can be investigated. The process of wear has been widely modeled using the ﬁnite elements method, although these approaches have studied only one wear 0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 4 9 - 5 A. Cantizano et al. / Computational Materials Science 25 (2002) 54–60 55 Nomenclature A Ao Ar C nominal area of contact Arrhenius constant for oxidation real area of contact constant used in the model for mildoxidational wear F normal load F~, Fnorm normalized load ðF~ ¼ F =AHo Þ Fs seizure load H hardness of sliding surface Ho room-temperature hardness Kox thermal conductivity of oxide P normal pressure Qo activation energy for oxidation R molar gas constant Tb bulk temperature Tf ﬂash temperature Tmox melting temperature of oxide W volume lost per unit area of surface W~ , Wnorm normalized wear rate ðW~ ¼ W =AÞ Zc critical thickness of oxide ﬁlm mechanism. With the ﬁnite element developed here, diﬀerent wear mechanisms are implemented so that depending on the operating conditions–– normal force and sliding velocity––the predominant wear process is activated following the wear-mechanism map proposed in . The treatment of the contact surfaces have never taken into account the thermo-microscopical behavior of a real surface when modeling wear with ﬁnite elements. In the present paper, the micromechanical contact law allow a more realistic simulation of what is actually happening in the contact interface during the wearing out process. The ﬂash temperatures achieved in the contact interface when two surfaces slide highly inﬂuence the wear rates. The heat generated at the surfaces enters through the true contact area which is a function of the radius of asperity that is considered constant for any loading condition, as is explained . In their other work, , the calculation of the ﬂash temperatures is improved because they depend on a radius of a contact-junction which can be made up of many unit contacting asperities for a c1 , c2 d fm kA lf m q ra rj ro v v~, vnorm a l r thermal diﬀusivity experimental constants current mean plane distance volume fraction of molten material removed during sliding Archard’s dimensionless wear coeﬃcient equivalent linear diﬀusion distance for ﬂash heating mean absolute asperity slope rate of heat input per unit area radius of an asperity radius of a contact-junction radius of the pin sliding velocity normalized velocity ð~ v ¼ vro =aÞ heat distribution coeﬃcient coeﬃcient of friction RMS surface roughness that load. The micromechanical law used here allows a good approach to the ﬂash temperature calculation, where the radius of asperity depends on the load and much better results are obtained when comparing with the improved method proposed in . The model implemented assumes a constant statistical characterization of the contacting surfaces. The wear achieved in every pass of the pin over the disk is considered to regenerate the roughness of the pin. This approximation is necessary because the wear-mechanism map that is modeled does not study the wear-rate transitions related to the mechanisms , and is accurate enough to obtain good results for a steady-state wear process. 2. Modeling The importance in modeling, accurately, wearmechanism maps for any pair of materials using the ﬁnite elements method is the ability to obtain 56 A. Cantizano et al. / Computational Materials Science 25 (2002) 54–60 precisely the amount of material worn out for any sliding situation and for any geometry of a mechanical system. In this way, a complete precise study of the diﬀerent types of wear acting on every part of the mechanical system can be done. A law for the normal microscopical contact is required in order to reproduce the real behavior of the contact interface between two rough surfaces. In the present work, a plastic law for the behavior of the asperities in contact, based on a statistical characterization of the surfaces, has been implemented. Also a slight modiﬁcation, based on experimental results, has been proposed. With the characteristics of the mechanical system deﬁned, the procedure followed by the ﬁnite element in order to calculate the material worn out is described in Fig. 1. The normal contact force obtained from the law mentioned above is introduced in the classic constitutive law of friction in order to verify the sliding condition. The tangential contact force is, therefore, calculated and also the heat generation due to friction, with which the bulk temperature achieved can be obtained. Then, the ﬂash temperatures are calculated taking into account the radius of the contacting asperities under deformation. With these magnitudes, the thermomechanical behavior at a microscopic scale is completely deﬁned. Thus, according to the normal force and the sliding velocity, a predominant wear mechanism is activated following the wear-mechanism maps developed in , and the volume of material worn out calculated. 2.1. Micromechanical contact law All surfaces are rough, as was said in . The shape and distribution of the asperities depend on the method of production of the surfaces. The basic hypothesis of the model consider the asperities to be randomly distributed over the apparent contact area and having a Gaussian distribution of the asperity heights. As a result of the deformation, there is a number of circular contact spots with a determined radius. Following the work initiated by , an implicit expression (Eq. (1)) for the contact hardness and the applied pressure, considering plastic behavior of the asperities––Eq. (3), n ¼ 1––was proposed in , ( c " 2 ) #c2 P P 4 106 r 2 2P 2P ¼ exp erfc1 H c1 m H H ð1Þ In , a useful explicit expression (Eq. (2)) was found, by approximating the inverse of the complementary error function of Eq. (1) in the range of values where the diﬀerence between both expressions is negligible ð6 108 < P =H < 4 103 Þ. " #1=ð1þ0:0711c2 Þ P P c2 ¼ ð2Þ H c1 1617646:152r m Eq. (2) is used in [12,13] in order to ﬁnd an important correlation between the current mean plane distance and the apparent mechanical pressure which is required for the numerical implementation in ﬁnite elements. Finally, taking into account the mechanical relationship based on experiments,  presented the following relationship between the true ðAr Þ and the nominal area of contact ðAÞ, Fig. 1. Flow-chart for the simulation of wear. Ar ¼ A P H n ; n¼ 5 6 ð3Þ A. Cantizano et al. / Computational Materials Science 25 (2002) 54–60 A more general expression of the correlation between the current mean plane distance and the apparent mechanical pressure, as a function of n, is proposed: h i c 2 1þ0:0711nc2 1617646:152r 2 exp d ð1:363rÞ2 P ¼ c1 m 5:589ð1þ0:0711nc2 Þ=n ð4Þ 2.2. Friction The classical law of friction, but where the coeﬃcient of friction depends on the sliding velocity, as in , l ¼ 0:78 0:13 log10 ð~ vÞ 57 W~ ¼ kA F~ ð8Þ In the ﬁrst two mechanisms described above, the value of the real area of contact is critical, Eq. (7), and determines the ﬂash temperatures reached in the contact interface, Eq. (6), and thus the volume of the material that has been worn out. These three important mechanisms of wear, for a steel on steel contact pair are widely described in . The transition between these three regimes mainly depends on the sliding velocity for most of the operating conditions. The inﬂuence of the normal force applied has to be taken into account only for quite high normal forces or extremely low. ð5Þ has been implemented. A modiﬁcation of Eq. (5) is proposed in , which takes into account the local melting of the asperities at high velocities but is not considered in this work. 2.3. Mechanisms of wear When steel surfaces slide at speeds above 1 m/s, the ﬂash temperatures reached cause oxidation and therefore, the wear debris consists mainly of iron oxides. At low velocities, the oxide ﬁlm formed on the sliding surface is cold and brittle. The splitting oﬀ of this ﬁlm is called mild-oxidational wear. The normalized amount of material removed W~ is deﬁned as 2 C Ao r o Qo F~ W~ ¼ ð6Þ exp Zc a RTf v~ At higher velocities (v > 10 m/s), the oxide ﬁlm formed is thicker and more continuous, hotter and more plastic, which implies a change in the wear process. This is called severe-oxidational wear and can be calculated as fm Kox ðTmox Tb Þ Ar ~ W ¼ aq ð7Þ lf v A When the sliding velocities are very small (v < 0:1 m/s), the surface heating can be considered negligible. Then the frictional force deforms the metal surface and causes the removal of slivers of metal from both surfaces. In this case, the wear rate follows the well-known Archard’s law : 3. Results and discussion Three diﬀerent mechanisms of wear have been simulated with a pin-on-disk conﬁguration, as shown in , with an enhanced version of the ﬁnite element program FEAP developed by R.L. Taylor and J.C. Simo and documented in  (Fig. 2). The radius of the pin ðro Þ is chosen to be 1.5 mm and its length of 9 mm ð6ro Þ, and these values allow us to compare the numerical results with those obtained in . The operating conditions simulated are: F~ v~ Mildoxidational wear 3 7.1 10 14.1 103 5.0 103 1.6 103 5.0 103 1.5 103 1.65 102 1.65 102 1.65 102 1.65 102 3.3 102 3.3 102 Severeoxidational wear 7.1 103 15.6 103 8.7 103 2.1 103 14.7 103 1.65 104 1.65 104 1.65 104 8.24 104 8.24 104 Delamination wear 7.1 103 14.2 103 5.0 103 1.5 103 5.0 103 1.65 101 1.65 101 1.65 101 1.65 101 1.65 58 A. Cantizano et al. / Computational Materials Science 25 (2002) 54–60 Fig. 2. Model of a pin-on-disk conﬁguration in sliding conditions. The results are presented to show the ability of the contact element for predicting accurate values of wear for several conditions of operation and its possible application for real problems (Fig. 3). Several numerical simulations have been made in order to obtain normalized wear rates ðW~ Þ, for diﬀerent values of normalized applied force ðF~Þ and normalized sliding velocity ð~ vÞ. In Fig. 3 the normalized wear rates can be seen for the three mechanisms of wear implemented. The severe oxidational wear ð~ v > 104 Þ and also the delamination mechanism of wear ð~ v < 10Þ are well predicted by the numerical results. The Fig. 3. Normalized wear rates for three mechanisms: severe oxidational, mild oxidational and delamination wear. mechanism of mild-oxidational wear presents a worse correlation and it is mainly due to the difference in the ﬂash temperatures calculated. These temperatures highly depend on the radius of asperity ðra Þ that was considered a constant of 105 m in the calculations made in . But here, the radius of asperity was allowed to increase as the load was applied. In order to obtain a value of ra approximate to the constant given above, after loading, diﬀerent microscopical characterizations of the surfaces have been chosen for the several values of F~ to guarantee the same value of ra for every test. This artiﬁcial characterization of the surfaces might be a reason for the discrepancy found, although with a constitutive law that allows the deformation of the asperities in the contact interface a much more realistic behavior of the contact should be achieved. In , the calculation of the ﬂash temperatures is improved and a radius of a contact-junction ðrj Þ is also taken into account (Eqs. (9)–(11)). The micromechanical law is able to obtain ﬂash temperatures quite similar in this case, specially for n ¼ 5=6, as is shown in Fig. 4. The real area of contact, in , is expressed as Ar ¼ Nj prj2 ð9Þ and the following mechanical relationship is deﬁned, Fig. 4. Flash temperatures for the mild-oxidational mechanism of wear. A. Cantizano et al. / Computational Materials Science 25 (2002) 54–60 Ar F ¼ A Fs ð10Þ where Fs is the load when seizure occurs Fs ¼ Ho A ð1 þ 12l2 Þ 1=2 ð11Þ and for which the real area of contact equals the nominal area. For the range of velocity considered Eq. (5) and the load applied, Eq. (10) is much more similar to Eq. (3), specially when n ¼ 5=6, than the classical plastic relationship used in , Ar F ¼ A Ho ð12Þ The value of rj is fairly well predicted by the actual radius of asperity, after loading, calculated with the microscopical law for every F~, which is slightly higher than the constant ra . The ﬂash temperatures, obtained as in , for the mechanism of mild-oxidation wear are represented in Fig. 4. As can be seen in Fig. 4, better results for the ﬂash temperatures are obtained when n ¼ 5=6 than for n ¼ 1, but only for the sliding conditions considered. 4. Conclusions A new way of modeling steady-state wear has been proposed based on a micromechanical law for the normal contact that takes into account a statistical characterization of the rugosity of the surfaces that slide together. This realistic treatment of the surface allows the calculation of the ﬂash temperatures achieved in the contact interface, that play a very important role in the wear rates. Other micromechanical laws can be treated in the same way in order to obtain accurately these temperatures. Three mechanisms of wear have been modeled using the ﬁnite elements method, with a pin-ondisk conﬁguration. The numerical results predict quite well the mechanisms except for the mild oxidational, where the ﬂash temperatures calculated diﬀer. An improved formulation for the ﬂash temperatures was tested and a much better agree- 59 ment with the values obtained with the model, was found. A ﬁnite element with a whole wear-mechanism map can be build and be used for the study of any geometry and operating condition of a mechanical system where diﬀerent mechanisms of wear can interface at the same time. Also an elastic law of micromechanical contact can be implemented to take into account the elastic steady state for situations with repeated sliding conditions. Acknowledgements The authors wish to acknowledge the interest and the support of Prof. Franco Bonollo from the Universita di Padova at Vicenza during the study of the wear-mechanism maps. References  J.F. Archard, Contact rubbing ﬂat surfaces, Journal of Applied Physics 24 (8) (1953) 981–988.  M.F. Ashby, J. Abulawi, H.S. Kong, Temperature maps for frictional heating in dry sliding, Tribology Transactions 34 (4) (1991) 577–587.  M.G. Cooper, B.B. Mikic, M.M. Yovanovich, Thermal contact conductance, International Journal of Heat and Mass Transfer 12 (1969) 279–300.  J.A. Greenwood, J.B.P. Williamson, Contact of nominally ﬂat surfaces, Proceedings of the Royal Society of London A 295 (1966) 300–319.  S.C. Lim, M.F. Ashby, Wear-mechanism maps, Acta Metallurgica 35 (1) (1987) 1–24.  S.C. Lim, M.F. Ashby, J.H. 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