Final Draft of the original manuscript:

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1 Fnal Draft of the orgnal manuscrpt: Hegadekatte, V.; Kurzenhaeser, S.; Huber, N.; Kraft, O.: A predctve modelng scheme for wear n pn-on-dsc and twn-dsc trbometers In: Trbology Internatonal (2008) Elsever DOI: /j.trbont

2 Elsever Edtoral System(tm) for Trbology Internatonal Manuscrpt Draft Manuscrpt Number: Ttle: A predctve modelng scheme for wear n pn-on-dsc and twn-dsc trbometers Artcle Type: Research Paper Secton/Category: Keywords: modelng, smulaton, fnte element method, contact mechancs, ceramcs, pn-on-dsc, twn-dsc Correspondng Author: Dr. Vshwanath Hegadekatte, Ph.D. Correspondng Author's Insttuton: Unversty of Karlsruhe Frst Author: Vshwanath Hegadekatte, Ph.D. Order of Authors: Vshwanath Hegadekatte, Ph.D.; Sven Kurzenhäuser; Norbert Huber, PhD; Olver Kraft, PhD Manuscrpt Regon of Orgn: Abstract: Study of sldng and rollng/sldng wear n complex mcro-mechancal components s often accomplshed expermentally usng a pn-on-dsc and twn-dsc rollng/sldng trbometer respectvely (conducted wthn the parameter space of the trbo-components). The present paper proposes an approach that nvolves a computatonally effcent ncremental mplementaton of Archard's wear model on the global scale (Global Incremental Wear Model - GIWM) for modelng sldng and slppng wear n such experments. It wll be shown that ths fast smplstc numercal tool can be used to dentfy the wear coeffcent from pn-on-dsc expermental data and also predct the wear depths wthn a lmted range of parameter varaton. Further t wll also be used to study the effect of ntroducng frcton coeffcent nto the wear model and and also to model water lubrcated experments. A smlar tool s presented to model wear due to a defned slp n a twn-

3 dsc rollng/sldng trbometer. The resultng wear depths from ths tool s verfed usng two dfferent fnte element based numercal tools namely, the Wear-Processor, whch s a FE post processor, and a userdefned subroutne UMESHMOTION n the commercal FE package ABAQUS. It wll be shown that the latter two tools have the potental for use n predctng wear and the effectve lfe span of any general trbosystem usng the dentfed wear coeffcent from relevant trbometry data.

4 Manuscrpt A predctve modelng scheme for wear n pn-on-dsc and twn-dsc trbometers V. HEGADEKATTE 1*, S. KURZENHÄUSER 2, N. HUBER 3, 4, O. KRAFT 1, 5 1 Insttut für Zuverlässgket von Bautelen und Systemen, Unverstät Karlsruhe (TH), Kaserstrasse 12, D , Karlsruhe, Germany. 2 Insttut für Werkstoffkunde II, Unverstät Karlsruhe (TH), Kaserstrasse 12, D-76131, Karlsruhe, Germany. 3 Insttut für Werkstoffforschung, GKSS-Forschungszentrum Geesthacht GmbH, Max-Planck-Strasse, D , Geesthacht, Germany. 4 Insttut für Werkstoffphysk und Technologe, Technsche Unverstät Hamburg-Harburg, Essendorfer Strasse 42(M), D-21073, Hamburg, Germany. 5 Insttut für Materalforschung II, Forschungszentrum Karlsruhe GmbH, Hermann von Helmholtz Platz 1, D-76344, Eggensten-Leopoldshafen, Germany. Abstract Study of sldng and rollng/sldng wear n complex mcro-mechancal components s often accomplshed expermentally usng a pn-on-dsc and twn-dsc rollng/sldng trbometer respectvely (conducted wthn the parameter space of the trbo-components). The present paper proposes an approach that nvolves a computatonally effcent ncremental mplementaton of Archard s wear model on the global scale (Global Incremental Wear Model - GIWM) for modelng sldng and slppng wear n such experments. It wll be shown that ths fast smplstc numercal tool can be used to dentfy the wear coeffcent from pn-on-dsc expermental data and also predct the wear depths wthn a lmted range of parameter varaton. Further t wll also be used to study the effect of ntroducng frcton coeffcent nto the wear model and and also to model water lubrcated experments. A smlar tool s presented to model wear due to a defned slp n a twn-dsc rollng/sldng trbometer. The resultng wear depths from ths tool s verfed usng two dfferent fnte element based numercal tools namely, the Wear-Processor, whch s a FE post processor, and a userdefned subroutne UMESHMOTION n the commercal FE package ABAQUS. It wll be shown that the latter two tools have the potental for use n predctng wear and the effectve lfe span of any general trbosystem usng the dentfed wear coeffcent from relevant trbometry data. * Correspondng Address Insttut für Werkstoffphysk und Technologe Technsche Unverstät Hamburg-Harburg Essendorfer Strasse 42(M), D-21073, Hamburg, Germany Tel.: Fax: E-Mal: hegadekatte@tuhh.de 1

5 1 Introducton Of the varous relablty ssues concernng mcro-mechancal components, wear s the least predctable partally due to the mperfect knowledge of the approprate wear rate for the selected materal par whch n turn greatly hnders our ablty to predct the effectve lfe span of components. Often, expermental technques lke pn-on-dsc, twn-dsc, scratch test, AFM etc. are used to characterze the trbologcal propertes of varous materals used for fabrcatng mcro-machnes n order to reduce the dependence on expensve n-stu wear measurements on prototypes of mcro-machnes. These experments attempt to mmc the contact condtons of the trbosystem under study n terms of contact pressure, sldng velocty etc. The specmens should have the same mcrostructure as the mcro-machne tself and the loadng chosen n the experments are such that they mmc the mcro-machne. For example, twn-dsc rollng/sldng trbometer tres to mmc the rollng/sldng contact experenced by mcro-machnes e.g., between the teeth of two matng mcro-gears. Such experments allow for a qualtatve study of the sutablty of a partcular materal combnaton for a gven applcaton and therefore modelng of wear n such experments s necessary n order to predct wear n mcro-machnes tself. Over the past, modelng of wear has been a subject of extensve research [1] n order to derve predctve governng equatons. The modelng of wear found n the lterature [2, 3, 4, 5] can broadly be classfed nto two man categores, namely, () mechanstc models, whch are based on materal falure mechansm e.g., ratchettng theory for wear [6, 7] and () phenomenologcal models, whch often nvolve quanttes that have to be computed usng prncples of contact mechancs e.g., Archard's wear model [8]. Archard s wear model s a smple phenomenologcal model, whch assumes a lnear relatonshp between the volume of materal removed, V, for a gven sldng dstance, s, an appled normal load, F N and the hardness (normal load over projected area) of the softer materal, H. A proportonalty constant, the wear coeffcent, k characterzes the wear resstance of the materal: V F (1) N k. s H Wherever the conventonal Archard s equaton dd not hold, researchers have modfed the model to sut ther specfc cases. One such example s the modfcatons of Archard s equaton to nclude wear of hghly elastc/pseudo elastc materals n [9]. Sarkar has gven an extenton to the Archard's wear model that relates the frcton coeffcent and the volume of materal removed [10]: V F 2 k 1 3 (2) N s H Even though ths model was orgnally ntroduced to study wear n the presence of asperty juncton growth, t wll be shown n ths artcle that ths model when appled on the global scale can favorably descrbe the trends observed n the experments consderng the uncertantes n the measurement. 2

6 Researchers have used both the above categores of wear models n computer smulaton schemes e.g., Ko et al. appled lnnear elastc fracture mechancs and fnte element modelng to predct fatgue wear n steel [11] whch bascally s based on the dea of a mechanstc wear model (The delmnaton theory of wear) proposed by Suh [12, 13]. The ratchettng theory for wear has been used n wear smulaton schemes by [14-16]. [17-19] made qualtatve predcton of the wear of coated samples n a pn-on-dsc trbometer whch showed good qualtatve agreement wth expermental results. On the other hand, a modfcaton of Archard's phenomenologcal wear model where the hardness of the softer materal was allowed to be a functon of temperature was used by Molnar et al. [20] and they also used an elastc-plastc materal model for the contactng bodes. Due to the computatonal expense, only a smple contact problem of a block sldng/oscllatng over a dsc was smulated. As a faster and effcent approach, postprocessng of the fnte element contact results wth Archard's wear model to compute the progress of wear for a gven tme nterval/sldng dstance has started to gan popularty n the recent years as llustrated by the works n [21-28]. [29, 30] have mplemented a remeshng scheme for geometry update n a smlar settng. [31-34] have ncluded a three dmensonal fnte element model and also a re-meshng scheme for smulatng wear and have also shown that ther results compare favorably wth expermental data. The computatonal costs n such fnte element based approaches are manly from to the computaton of the contact stresses, whch requres the soluton of a nonlnear boundary value problem often usng commercal fnte element packages. In case of trbo-systems wth smple geometres, especally trbometers e.g., pn-on-dsc, twn-dsc etc., the estmaton of the contact area can be smple. In such cases, t may not be necessary to solve the contact problem usng fnte elements and nstead wear can be modeled on the global scale lke n the Global Incremental Wear Model (GIWM) to be presented n the followng. The estmaton of the contact area n the GIWM s accomplshed by consderng both the normal elastc dsplacement and wear whch s normal to the contactng surface. From the appled normal load and the estmated contact area, an average contact pressure across the contactng surface s calculated. The average contact pressure (global quantty) s then used n a sutable wear model to calculate the ncrement of wear depth for a pre-determned sldng dstance ncrement. The wear depth s then ntegrated over the sldng dstance to get the tradtonal wear depth over sldng dstance curves. A detaled explanaton on GIWM was presented n [32], where the GIWM was successfully appled to ft and predct predomnantly dsc wear n a pn-ondsc experment. 2 Global Incremental Wear Model for pn wear n a pn-ondsc trbometer The expermental results presented n ths work are from undrectonal sldng tests. A mcro pn-on-dsc trbometer wth a sphercal tpped pn and a dsc of the same materal was used n the tests. Two sets of experments were carred out over a sldng dstance of 3

7 500 m at room temperature, n ambent ar and n water at varous normal loads and a sldng speed of 400 mm/s. The ground dsc specmens had the dmensons of 8 1 mm 2 and the polshed pn specmens had a dameter of mm. The dsc surface had an average surface roughness of R a = 0.11 µm and for the pn specmens R a = 0.07 µm for S3N4 and for WC-Co experment, the pn and dsc had a polshed surface wth an average surface roughness R a = µm and R a = µm respectvely. The parameters used n the experments were based on a system analyss of a mcro-turbne and a -planetary gear tran presented n chapter 1 of [35]. The normal force and the frcton force were contnuously measured wth the help of stran gages durng the tests. The sum of the wear depth on both the pn and the dsc was also contnuously measured capactvely wthn a resoluton of ± 1 µm. In these experments t was observed that the wear on the dsc was below 200 nm for S 3 N 4. For the WC-Co experments, dsc wear was not measurable. At the end of the experment, the maxmum wear depth of pn and (a) (b) Fg. 1: (a) Flow chart of the Global Incremental Wear Model for pn-on-dsc trbometer (b) Results from the GIWM n comparson wth the expermental results from the pnon-dsc trbometer at three dfferent normal loads (200 mn, 400 mn and 800 mn) dsc was measured by whte-lght and contact proflometry respectvely. The dscrepancy between n-stu measured wear from the capactve dsplacement sensor and the wear calculated from the flat crcular contact area of the worn pn dd not exceed ± 250 nm for the WC-Co parng at the end of the experment. For the smulaton, the capactvely measured lnear wear was ftted to the values obtaned for pn wear at the end of the experment, assumng a lnear drft of the capactve devce. The wear depth data as a functon of the sldng dstance s shown n Fg. 1 (a) and (b) for S 3 N 4. 4

8 For the sake of clarty of the present artcle, the GIWM for pn wear wll be explaned n the followng. The flow chart of ths scheme s shown n Fg. 1 (a), where p s the contact pressure, F N s the appled normal load, a s the contact radus due to elastc dsplacement and wear, h s the total dsplacement at the pn tp, R P s the curvature of the pn, h e s the elastc dsplacement, h w s the current wear depth, k D = k/h s the dmensonal wear coeffcent, s s the nterval of the sldng dstance, s max s the maxmum sldng dstance, s the current wear ncrement number and E C s the elastc modulus of the equvalent surface calculated usng the followng equaton (see page 92 of [36]): p 1 (3) d E E E c p d where E p and E d are the Young s modulus of the pn and dsc respectvely and the Posson s ratos of the pn and the dsc s represented by p and d respectvely. The global wear modelng scheme begns wth the computaton of the ntal contact radus a 0 usng the Hertz soluton [37] (4) 3FN RP a 3 0 4E and the elastc deformaton normal to the contact usng the relaton found n [38]: e F (5) N h 1 2E a c C 1 After each ncrement of sldng dstance the current contact radus a s computed from the geometry of the contact based on the sum of the wear depth and the elastc deformaton normal to the contact. The wear depth s ntegrated over the sldng dstance usng the Euler explct method: tll the maxmum sldng dstance s reached. 2.1 Results w w h 1 kd p s h (6) Applcaton of GIWM to unlubrcated pn-on-dsc experment The GIWM was used to ft the data from the 200 mn normal load experment (Fg. 1 (b)), where k D was dentfed to be mm 3 /Nmm. The chosen materal propertes for slcon ntrde were [39]: Young s Modulus, E = N/mm 2 and Posson s Rato, = Later, the dentfed wear coeffcent was used to predct the 400 mn and 800 mn experment. It can be seen from the graph of the ft for 200 mn and predcton for 400 mn n Fg. 1 (b) that the results from the GIWM are n good agreement wth the experments. However, the GIWM over estmates the wear depth for the 800 mn experment as shown 5

9 n the same fgure. As t can be seen from Fg. 1 (b), the wear depth after 500 m of sldng s only slghtly hgher for 800 mn compared to the 400 mn experment. One possble reason for ths behavor could come from the actvaton of a dfferent domnant wear mechansm or formaton of protectve trbologcal layers reducng the wear rate. Such effects are not ncluded n the Archard s wear model and therefore more expermental data would be needed to extend the Archard s wear model for ncluson n GIWM. However, up to the frst 100 m of sldng, the GIWM s stll n good agreement for the 800 mn experment. The GIWM was successful n predctng the results of pn-on-dsc experment when the normal load was doubled Applcaton of GIWM usng Sarkar s wear model GIWM offers a unque possblty to mplement any wear model on the global scale wth relatve ease. In order to nvestgate the effect of the changng frcton coeffcent we use the modfed Archard s wear model gven by Sarkar (see Equaton (2)) to study the evoluton of wear depth for the pn n the slcon ntrde experments dscussed above. The coeffcent of frcton, µ n Equaton (2) was based on an exponental ft as gven n Equaton (7): s / c A e b (7) where s s the sldng dstance and A, b, c are the parameters defnng the change of µ wth ncreasng s. The graph showng the ft and the values for the above parameters from the ft for the 200 mn, 400 mn and 800 mn experment are gven n Fg. 2 (a), (c), (e) and Table 1 respectvely. The GIWM wth Sarkar s modfed Archard s wear model was used to ft the 200 mn normal load experment. The dentfed value of k D usng was mm 3 /Nmm 2 whch was now slghtly lower (approxmately by a factor 1 3 for µ ~ 0.45) when compared to that dentfed from the conventonal Archard s wear model. However, the resultng curves for the wear depth over sldng dstance for the ft (200 mn) was n exact agreement as shown n Fg. 3. Table 1: Values of the parameters from the exponental ft (Equaton 7) of the measured value of the frcton coeffcent as a functon of the sldng dstance for dfferent normal loads Parameter 200 mn 400 mn 800 mn A b c

10 Fg. 2 (b), (d) and (f) shows k D as a functon of µ calculated from the 200 mn, 400 mn and 800 mn experments n comparson to that from Sarkar s modfed Archard s wear model mplemented n GIWM. The effect of ntroducng frcton nto the wear model can be clearly seen for the predcton of the 400 mn and 800 mn experments n Fg. 3. The predcton for 400 mn s an under estmate when compared wth the expermental results. (a) (b) (c) (d) (e) (f) Fg. 2 : Graph showng the exponental ft on the measured µ as a functon of s for (a) 200 mn, (c) 400 mn and (e) 800 mn normal load; Comparson of k D as a functon of µ from the experments on slcon ntrde and from the modfed wear model of Sarkar wth k D = mm 3 /Nmm for (b) 200 mn, (d) 400 mn and (f) 800 mn normal load. Ths under estmaton becomes clear when Fg. 2 (b) and (d) are compared. It can be seen that for the 400 mn the effectve k D values (see Equaton (2)) are on the lower sde of the 7

11 Fg. 3: Results from the Sarkar s wear model (Equaton (2)) mplemented wthn the GIWM for pn wear n comparson wth the expermental results from the pn-on-dsc trbometer at three dfferent normal loads (200 mn, 400 mn and 800 mn). correspondng calculated values from the expermental data whch s n contrast to the 200 mn experment. As n the case of Archard's wear model dscussed earler, the Sarkar s model stll over estmates for the 800 mn experment. Fg. 3 (f) makes ths dfference clear as the effectve k D value s very much hgher than that calculated from the experments. The dfference between the curves n Fg. 3 usng Archard's wear model and Sarkar s model s due to the ncluson of frcton nto the wear model n the latter. However, t can be concluded from Fg. 3 that the effect of ncludng frcton n the wear model has margnal nfluence on the wear behavor for slcon ntrde. However t should be noted that Sarkar s model can satsfactorly descrbe the trends n the experments consderng the scatter n the measured data as shown n Fg. 2 (b), (d) and (f) Applcaton of GIWM to water lubrcated pn-on-dsc experment The GIWM was used to ft and predct another set of experments for the same specmen geometry of the trbometer as explaned before but wth the materals as tungsten carbdecobalt and the experments were water lubrcated wth all the other parameters remanng the same as descrbed before. Fg. 4 shows the wear depth as a functon of the sldng dstance for dfferent appled normal loads. The step of sze approxmately 300 nm n the expermental wear depth curve n Fg. 4 s a result of the measurement uncertanty. As stated earler, the resoluton of the trbometer s 1 µm whle the maxmum wear measured at the hghest load s around 2 µm. The wear coeffcent, k D was dentfed to be mm 3 /Nmm from the ft for the 400 mn experment and was then used to predct the wear depth for hgher loads. The materal propertes used for WC-Co were: Young s Modulus of the pn, E p = N/mm 2, and for the dsc, E d = N/mm 2 and Posson s Rato, = It can be seen n Fg. 4 that the GIWM can predct 8

12 Fg. 4: Wear depth data for pn wear as functon of the sldng dstance n comparson to the results from the GIWM for dfferent normal loads (400 mn, 800 mn, 1200 mn and 1600 mn). (No Error bars for 800 mn are presented as the dscrepancy between contnously measured wear and the wear measured after the experment exceeded 250 nm n one of the experments) the experments wth reasonable accuracy consderng the scatter n the measurement for the entre range of normal loads tested. The dentfed wear coeffcent should be construed as a trbosystem dependent quantty whch ncludes all the effects resultng from e.g., lubrcaton, surface roughness, temperature etc. Thus, we assume that the wear coeffcent dentfed n such a way can be used to predct wear n any general trbosystem as far as the dentfcaton of the wear coeffcent s done from experments whch are conducted wthn the parameter space of the trbosystem tself. 3 Global Incremental Wear Model for dsc wear n a twndsc trbometer When two crcular rotatng bodes (e.g. see Fg. 5 (a)) come n contact wth each other and they have the same tangental velocty at all ponts of contact, then they are sad to exhbt pure rollng contact. In pure rollng there wll be no slp at the contact nterface. However, n realty t s very dffcult to fnd a contact stuaton that exhbts pure rollng. Local sldng wll most lkely take place on a part of the contact e.g., n gears, tooth flanks roll and slde aganst each other at all locatons along the tooth flank except at the ptch pont [40]. Therefore, most rollng contacts are n essence rollng/sldng contacts. Such contacts are often expermentally studed wth twn-dsc rollng/sldng trbometer shown n Fg. 5 (a). The two crcles on top dsc n Fg. 5 (a) and (b) ndcate that the top dsc has a curved surface whereas the bottom dsc has a flat surface. Such an arrangement helps n better 9

13 algnment of the two dscs whle conductng the experments. The two dscs rotate wth veloctes V 1, and V 2 (at the outermost crcumference), such that V 1 V 2. The exstence of slp between the dscs together wth normal load actng on them, results n sldng wear, for whch Archard s wear law s known to be applcable. Such a system can be assumed to be lke the one shown on the rght hand sde of Fg. 5 (b), n whch the bottom dsc s fxed and the top dsc rotates at the slp velocty. Wth ths assumpton, the problem can be reduced from rollng/sldng contact to quas-statc sldng contact. However, ths assumpton s vald only when the bottom flat surfaced dsc does not wear out at all. Snce ths assumpton s vald n the analyss presented n ths paper, the bottom dsc s modeled as an analytcal rgd surface n the fnte element model used by the Wear- Processor and UMESHMOTION. The ncrement of wear depth usng the Archard s wear model n Equaton (6) has to be adopted for the case of twn-dsc trbometer. Equaton (6) wll have to be re-wrtten as n Equaton (8), snce, as the dsc rotates the contact pressure on any surface pont approaches to a maxmum from zero and then gradually approaches to zero. The pont (a) (b) Fg. 5: (a) Schematc of the twn-dsc trbometer (shaded porton n the schematc s modeled wth FE for use n Wear-Processor and UMESHMOTION), (b) Reducton of rollng/sldng contact wth defned slp to sldng contact n the twn-dsc trbometer wears only when t experences pressure whle passng through the contact nterface. Therefore, pressure actng on ths pont, has to be ntegrated along the sldng drecton correspondng to one rotaton for the computaton of the local wear ncrement. Therefore, for one rotaton of the dsc, the wear takng place on ths pont can be wrtten as: 2 (8) h k prd h, j1 D j 0 where r = r(x) s the radus of the dsc at the locaton of the pont (snce the top dsc surface s curved) and s the angle of rotaton. The calculaton of wear depth usng Equaton (8) wll hold for all the nodes lyng along the same crcumference (streamlne). For a gven tme ncrement t j, the wear depth can then be wrtten as: 10

14 h t V V 2 j 1 2 j1 kd j 2r 0 prd h. (9) In ths local form of the Archard s wear model the contact pressure, p s dfferent for each streamlne whch s nherently consdered n fnte element based wear smulaton tools Fg. 6: Flow chart of the Global Incremental Wear Model for twn-dsc rollng/sldng trbometer such as the Wear-Processor [31, 32] or UMESHMOTION. The latter s a user defned 11

15 subroutne n the commercal fnte element package ABAQUS to be dscussed n the followng. For the GIWM as the word global suggests and as dscussed n the prevous secton, an average contact pressure over the entre contact area s assumed. Further. t s assumed that the bottom flat surfaced dsc s rgd and therefore all the wear occurs on the top curved surfaced dsc whle neglgble wear occurs on the bottom dsc. Therefore the top dsc s assumed to have an elastc modulus equal to the contact modulus (see Equaton (3)). The contact area n such a contact s ellptcal. The major axs of the contact ellpse wll be perpendcular to the drecton of rotaton and the mnor axs wll be tangental to the drecton of rotaton of the dscs. The flow chart of the GIWM for dsc wear n a twn dsc rollng/sldng trbometer s shown n Fg. 6, where p s the average contact pressure, F N s the appled normal load, a (x) s the sem major axs length of the contact ellpse (perpendcular to the drecton of rotaton) and a (z) s the sem mnor axs length of the contact ellpse (tangental to the drecton of rotaton), h s the wear depth, r 1(x) and r 1(z) are the radus of curvature of the top dsc and r 2 s the radus of curvature of the bottom dsc, s the correcton factor for a (x) (to be explaned n the followng), u s the elastc dsplacement, h (total) s the total dsplacement of the top dsc whch s the sum of wear depth and elastc deformaton, k D = k/h s the dmensonal wear coeffcent, s s the sldng dstance, s max s the maxmum sldng dstance, s the current wear ncrement number and E C s the elastc modulus of the equvalent surface (see Equaton (3)). The global wear modelng scheme begns wth the computaton of the ntal sem axs lengths of the contact ellpse usng the Hertz soluton [37] for ellptcal contact area and the ntal normal elastc deformaton usng the relaton found n [38] whch s corrected for the ellptcal contact area: F (10) N u0 E a a 2 c ( x)0 ( z)0 Then, the ntal average contact pressure, p over the ellptcal contact area and the total dsplacement, h (total) of the dsc surface s calculated. The wear depth s ntegrated over the sldng dstance, 2a (z) usng the Euler explct method: h k p a (11) h 1 2 D ( z) For one rotaton of the dsc, the sldng dstance ncrement over whch wear takes place s gven by the mnor axs length of the contact ellpse, 2a (z). The current sem major axs length of the contact ellpse, a (x) and the radus of the top dsc, r (x) are calculated from the geometry of the contact whch changes due to wear. The sem mnor axs length of the contact ellpse, a (z) and the average contact pressure, p s computed usng the Hertz soluton [37] for rectangular contact area (assumng a plan stran condton at the center of the contact along the wdth of the dscs) usng: F (12) 4 N r 2a ( eq) 1 ( x) 1 a( z) 1 E C 12

16 and p F N 2a r ( x) 1 4 ( eq) 1 E C (13) respectvely. Snce an ellptcal contact area results from the contact n the twn-dsc trbometer and a rectangular contact area s assumed n Equaton (12), t needs to be corrected by equatng t to the ellptcal contact area. Hence, a correcton factor of = /4 s used n Equaton (12). The updatng of all the parameters and the ntegraton of the wear ncrement over the sldng dstance s contnued tll a defned maxmum sldng dstance s reached. In the absence of relable expermental data for valdatng the GIWM, n secton 3.3 the valdaton wll be carred out usng two separate fnte element based wear smulaton tools. 3.1 Wear-Processor The frst of the fnte element based wear smulaton tools, the Wear-Processor wll only be descrbed n bref here for the sake of condensng the present artcle. It has been descrbed n detal n [31, 32]. The processng of wear begns wth the soluton of a 3D statc contact analyss wth nfntesmal rotaton of the bottom rgd flat surfaced dsc to nclude the asymmetrc effects comng from the frcton between the two slppng dscs (see Fg. 5 (a) and (b)). The soluton of ths boundary value problem s accomplshed wth the commercal fnte element code ABAQUS. One quarter of the top curved surfaced dsc s modeled wth fnte element makng use of the symmetry and the other dsc s modeled as an analytcal rgd surface. The stress feld, the dsplacement feld and the element topology are then extracted from the fnte element results fle. The unt nward surface normal vector at each of the surface nodes s computed based on the element topology by takng the cross product of the four edge vectors that are connected to each of the surface nodes. The contact pressure for each of the surface nodes on the top dsc surface s calculated usng the extracted stress feld and the calculated normal vector. An explct Euler method s used to ntegrate Archard s wear law for each surface node over the sldng dstance usng Equaton (9). The calculated wear from Archard s wear model s used to update the geometry by repostonng the surface nodes wth an effcent re-meshng technque that makes use of the boundary dsplacement method, see [31, 32] for more detals. The obtaned new reference geometry s used to get the updated stress dstrbuton by solvng the contact problem agan, whch n turn s used to compute the updated contact pressure dstrbuton. At the end of each wear ncrement, the total dsplacement (sum of the elastc dsplacement and wear depth) for each of the surface nodes s wrtten to an ABAQUS compatble fle for vewng wth PATRAN (a commercal pre- and post-processor). The procedure s contnued tll a pre defned maxmum sldng dstance s reached. 13

17 It s to be noted that the fnte element model used n the wear smulaton wth the Wear- Processor and the UMESHMOTION was dentcal. The deformable top curved surfaced dsc s not rotated physcally n the contact smulaton, but t s assumed to be rotated for certan tme ncrement n UMESHMOTION and the Wear-Processor. Durng ths tme ncrement, t s assumed that the confguraton changes are neglgble and have mnor effect on the contact soluton. 3.2 UMESHMOTION The second fnte element based wear smulaton tool to be dscussed n ths artcle s the UMESHMOTION, whch s a user-defned subroutne n the commercal FE code ABAQUS. It s ntended for defnng the moton of nodes n an adaptve mesh constrant node set. By defnng the contact surface nodes n the adaptve mesh constrant node set, UMESHMOTION can be coded to shft the surface nodes n the drecton of the local normal by an amount equal to the correspondng local wear. In ths work, t has been specally coded n FORTRAN to smulate wear n a twn-dsc trbometer. A detaled descrpton of the adaptaton of UMESHMOTION for smulatng wear can be found n [41, 42]. Once the equlbrum equatons for the three dmensonal, deformable-rgd contact problem converge, the user-defned subroutne UMESHMOTION s called for each surface node. The UMESHMOTION, whch s specally coded to smulate wear feeds back the local wear ncrement for a gven tme ncrement calculated usng Equaton (9). The adaptve meshng algorthm of ABAQUS apples the local wear ncrement for all surface nodes n two steps. Frst, the surface nodes are swept n the local normal drecton by an amount equal to the correspondng local wear ncrement. The sweepng of the nodes s carred out purely as an Euleran analyss. Thus the geometry s updated. Second, the materal quanttes are re-mapped to the new postons. Ths s accomplshed by advectng the materal quanttes from the old locaton to the new locaton by solvng advecton equatons usng a second order numercal method, called the Lax-Wendroff method. The sweepng of the mesh and the advecton of the materal quanttes cause an equlbrum loss. The equlbrum loss s corrected by solvng the last tme ncrement of the contact problem [44]. In ths way, the contact pressure s updated. The procedure s repeated tll a pre defned maxmum sldng dstance s reached. 3.3 Results The results from the GIWM are compared wth that from the Wear-Processor and the UMESHMOTION dscussed above. The parameters used n the wear smulatons are gven n Table 2. 14

18 Table 2: Parameters used for the wear smulaton usng Wear-Processor and UMESHMOTION Parameter Value Materal ZrO 2 Young s Modulus E t = E b = 152 GPa Posson s Rato t = b = 0.32 Appled Normal Load F N = 0.3 N Frcton Coeffcent = 0.6 Dmensonal Wear Coeffcent k D = mm 3 /Nmm Slp 10 % (a) (b) (c) (d) Fg. 7: (a) Graph of wear depth over the number of rotatons and (b) graph of the pressure drop over number of rotatons usng UMESHMOTION and GIWM. (c) Graph showng the sem major axs length and (d) sem mnor axs length of the contact ellpse as a functon of the number of rotatons usng UMESHMOTION and GIWM. The value of the wear coeffcent was chosen from pn-on-dsc experments for ZrO 2 from [43]. The results from ths wear smulaton are presented n Fg. 7 (a) to (d). It can be seen n Fg. 7 (a) that the wear depth as a functon of the number of rotatons from the GIWM are n good agreement wth that from UMESHMOTION (wthn 16 %). As wear progresses, the curved surface of the top dsc progressvely flattens leadng to a drop n the slope of the wear depth curve because of a drop n the contact pressure (see Fg. 7 (b)) resultng from the ncrease n the contact area. Due to the flattenng of the top dsc, the 15

19 sem major axs length of the contact ellpse contnuously ncreases (see Fg. 7 (c)) whle, the sem mnor axs length of the contact ellpse contnuously decreases (see Fg. 7 (d)) but the resultng contact area ncreases. However, the wear depth curve from the UMESHMOTION n Fg. 7 (a) has a decreasng slope (for larger sldng dstances) than that from GIWM. Further the curve from UMESHMOTION les below that from GIWM whch s dffcult to explan especally consderng the fact that GIWM uses an average pressure for the computaton of the wear depth and UMESHMOTION uses the local pressure to compute local wear (see Fg. 7 (b)) whch s as hgh as 1.5 tmes the average pressure at the contact center. It means that the wear depth curve obtaned from the GIWM forms a lower lmt for wear depth curves obtaned from fnte element based wear smulaton tools for a gven set of ntal parameters. To look further nto ths ssue, the wear depth results from the UMESHMOTION was compared wth that from the Wear-Processor. The smulatons were performed on the same geometry as shown n Fg. 5, but wth a coarser mesh (5256 elements compared to elements) and therefore wth a two order of magntude hgher appled normal load and wear coeffcent. The coarsely meshed model was used n order to reduce the computaton tme for performng ths study. The wear smulaton results usng the Wear-Processor, UMESHMOTION and GIWM are shown n Fg. 8. The wear depth curve from the Wear-Processor and the UMESHMOTION wll have the same ntal slope snce they start wth the same contact pressure (Hertzan) but the curve from GIWM wll have a lower startng slope snce t uses an average contact pressure for the computaton of wear as can be seen n Fg. 8. As wear progresses, the curves from the Wear-Processor and the GIWM wll begn to have the same slope and the accumulated devaton n the early part of sldng remans constant wth further ncrease n the sldng dstance. However, t can also be seen n Fg. 8 that the slope of the wear depth curve obtaned from the UMESHMOTION contnuously decreases as the sldng progresses and shows a trend that t would approach the GIWM curve (also see Fg. 7 (a)). The reason for ths dscrepancy s not clear. It can be due to ether geometry or pressure not beng updated correctly or a combnaton of both. The dfference between the wear depth Fg. 8: Graph of wear depth as a functon the number of rotatons from the Wear- Processor, the UMESHMOTION and the GIWM obtaned from the Wear-Processor and that from the UMESHMOTION s approxmately 11 % (wthn the sldng dstance range tested) and seems to be further wdenng. 16

20 It should be noted at ths pont that whle usng UMESHMOTION, ABAQUS does not solve the complete contact problem to update the contact pressure dstrbuton, but t only solves the last tme ncrement. The result of ths approach s that there s a consderable savng n computatonal tme of the order of one magntude compared to the Wear- Processor. An addtonal test was carred out to check f ABAQUS correctly updates the contact pressure dstrbuton. The wear smulaton usng the UMESHMOTION was nterrupted after 2034 rotatons and dependng on the wear depth dstrbuton at that nstance, the geometry was updated externally by repostonng the surface nodes usng the boundary dsplacement method (see [37, 38] for more detals). Wth the resultng new reference geometry the wear smulaton usng the UMESHMOTION was resumed. If the geometry/pressure was updated correctly, then the resultng pressure dstrbuton on resumpton of the wear smulaton should exactly be the same as the pressure dstrbuton obtaned wthout any form of external geometry correcton. But, t can be seen from Fg. 9 (a) that the pressure updated by ABAQUS does not completely agree wth the correspondng pressure at the same locaton and at the same nstance obtaned when no external geometry update was appled. The dfference s around 7 %. The effect of ths dfference can be seen on the wear depth curve shown n Fg. 9 (b). For comparson, the correspondng curves from the Wear-Processor are also presented n the same graph. However, t should be noted that the curve for the pressure drop as a functon of the number of rotatons s hstory dependent and snce the curve for the pressure drop from the Wear-Processor s obtaned by makng a true update of the geometry, the curves from the UMESHMOTION and Wear-Processor cannot be truly compared. But, as seen from Fg. 9 (b), f the geometry s externally updated, the wear depth curves tend to approach the curve from the Wear-Processor. Thus a frequent external update of the geometry would mnmze, the dfference between the results from the UMESHMOTION and the Wear-Processor. (a) (b) Fg. 9: (a) Graph of pressure drop and (b) wear depth as a functon of the number of rotatons wth and wthout geometry correcton n comparson wth the Wear- Processor The features of the Wear-Processor and the UMESHMOTION nclude the applcaton of a wear model on the local scale, ther ablty to smulate wear on three dmensonal FE models and ther scope for handlng arbtrary geometry of trbosystems that could be 17

21 made of dfferent materals. However, the Wear-Processor s computatonally expensve and therefore has to be used only when t s absolutely necessary for satsfactorly descrbng the evoluton of the worn surface. UMESHMOTION, whch s computatonally less expensve, can be used n stuatons where t s suffcent to smulate only one of the contactng surfaces snce the contact results are avalable for only one of the surfaces n ths case. Ths requres, that n the experments the wear on one of the surfaces s truly neglgble. 4 Conclusons In ths work, the Global Incremental Wear Model, whch represents a computatonally effcent ncremental mplementaton of a sutable wear model on the global scale for modelng sldng and slppng wear, was presented. Ths fast smplstc numercal tool was used to dentfy the wear coeffcent from pn-on-dsc expermental data and also to predct the wear depths wthn a gven range of parameter varaton. The results from the GIWM usng the Archard s wear model showed a good agreement wth the expermental data consderng the uncertantes n the measurement. Therefore t can be concluded that Archard s wear model s vald for the materals and the parameters presented n ths work ncludng the case when the trbometer s lubrcated wth water. An extenton of the Archard s wear model gven by Sarkar was used to study the effect of ntroducng the frcton coeffcent nto the wear model. Ths model favorably descrbed the trends seen n the experments. Ths tool was further extended to model wear due to a defned slp n a twn-dsc rollng/sldng trbometer. The wear depths from ths tool was verfed usng two dfferent fnte element based numercal tools namely, the Wear-Processor, whch s a FE post processor and the second tool s a user-defned subroutne UMESHMOTION n the commercal fnte element package ABAQUS. It was shown that the wear depth results from the GIWM compared favorably wth that from the other two numercal tools, thus verfyng each other. The dfference n the wear depth results from the three numercal tools was wthn 16 %. Tests on UMESHMOTION showed that there was some dscrepancy (of approxmately 11 %) n the results when compared to that from the Wear-Processor. Consderng the typcally larger uncertantes n trbo experments, the accuraces are acceptable GIWM assumes a constant average pressure over the contact area n any sldng dstance ncrement. The worn out surface s assumed to be always flat so that the contact area can be easly estmated. These assumptons n the GIWM lmt ts usage to certan geometres. The GIWM can be used to make a frst guess for the local wear model, whch can then be mplemented n FE based wear smulaton tools. Ths effcent method of wear smulaton can be very handy for trbologsts to quckly nterpret ther measured data for most materal combnatons encountered n practcal applcatons. In the next step, the Wear Processor wll be extended towards the wear smulaton n transent 2D contact problems that are typcal for a mcro planetary gear tran made of ceramcs usng wear coeffcents dentfed from pn-on-dsc and twn-dsc experments. 18

22 Acknowledgement The authors would lke to thank the German Research Foundaton (DFG) for fundng ths work under sub project D4 wthn the scope of the collaboratve research center, SFB 499 Desgn, producton and qualty assurance of molded mcroparts constructed from metals and ceramcs. The authors would also lke to thank Prof. S. Andersson of the Royal Insttute of Technology (KTH), Sweden for co-supervsng the master thess of B. Kanavall. References [1] Zum-Gahr, K. H. (1987). Mcrostructure and wear of materals. Elsever, Amsterdam, The Netherlands. [2] Meng, H. C. (1994). Wear modelng: evaluaton and categorzaton of wear models. PhD thess, Unversty of Mchgan, Ann Arbor, MI, USA. [3] Meng, H. C. & Ludema, K. C. (1995). Wear models and predctve equatons: ther form and content. Wear, , [4] Hsu, S. M., Shen, M. C., & Ruff, A. W. (1997). Wear predcton for metals. Trbol. Int., 30, [5] Blau, P. J. (1997). Ffty years of research on the wear of metals. Trbol. Int., 30, [6] Kapoor, A. & Johnson, K. L. (1994). Plastc ratchetng as a mechansm of metallc wear. Proc. Roy. Soc. Lon. A, 445, [7] Kapoor, A. (1997). Wear by plastc ratchetng. Wear, 212, [8] Archard, J. F. (1953). Contact and rubbng of flat surfaces. J. Appl. Phys., 24, [9] Lu, R. & L, D. Y. (2001). Modfcaton of Archard's equaton by takng account of elastc/pseudoelastc propertes of materals. Wear, 251, [10] Sarkar, A. D. (1980). Frcton and wear. Academc Press, London. [11] Ko, P.L., Iyer, S.S., Vaughan, H., Gadala, M. (2001). Fnte element modellng of crack growth and wear partcle formaton n sldng contact. Wear, 251, [12] Suh, N. P. (1973). The delamnaton theory of wear. Wear, 25, [13] Suh, N. P. (1977). An overvew of the delamnaton theory of wear. Wear, 44, [14] Frankln, F. J., Wdyarta, I., & Kapoor, A. (2001). Computer smulaton of wear and rollng contact fatgue. Wear, 251, [15] Frankln, F. J., Weeda, G.-J., Kapoor, A., & Hensch, E.J.M. (2005). Rollng contact fatgue and wear behavour of the nfrastar two-materal ral. Wear, 258, [16] Staln-Muller, N. & Dang, K. V. (1997). Numercal smulaton of the sldng wear test n relaton to materal propertes. Wear, , [17] Chrstofdes, C., McHugh, P. E., Forn, A., & Pcas, J. A. (2002). Wear of a thn surface coatng: Modelng and expermental nvestgatons. Comput. Mat. Sc., 25, [18] Yan, W., Busso, E. P., & O'Dowd, N. P. (2001). A mcromechancs nvestgaton of sldng wear n coated components. Proc. Roy. Soc. Lon. A, 456, 2387{2407. [19] Yan, W., O'Dowd, N. P., & Busso, E. P. (2002). Numercal study of sldng wear caused by a loaded pn on a rotatng dsc. J. Mech. Phys. Sol., 50, [20] Molnar, J. F., Ortz, M., Radovtzky, R., & Repetto, E. A. (2001). Fnte element modelng of dry sldng wear n metals. Engg. Comput., 18, [21] Podra, P. (1997). FE Wear Smulaton of Sldng Contacts. PhD thess, Royal Insttute of Technology (KTH), Stockholm, Sweden. [22] Podra, P. & Andersson, S. (1999). Smulatng sldng wear wth fnte element method. Trbol. Int., 32, [23] Öqust, M. (2001). Numercal smulatons of mld wear usng updated geometry wth dfferent step sze approaches. Wear, 249, [24] Ko, D. C., Km, D. H., & Km, B. M. (2002). Fnte element analyss for the wear of T-N coated punch n the percng process. Wear, 252,

23 [25] McColl, I. R., Dng, J., & Leen, S. (2004). Fnte element smulaton and expermental valdaton of frettng wear. Wear, 256, [26] Dng, J., Leen, S. B., & McColl, I. (2004). The effect of slp regme on frettng wear-nduced stress evoluton. Int. J. Fatgue, 26, [27] Gonzalez, C., Martn, A., Garrdo, M. A., Gomez, M. T., Rco, A., & Rodrguez, J. (2005). Numercal analyss of pn on dsc tests on Al-L/SC compostes. Wear, 259, [28] Kónya, L., Várad, K., & Fredrch, K. (2005). Fnte element modelng of wear process of a peeksteel sldng par at elevated temperature. Perodca Polytechnca, Mechancal Engneerng, 49, [29] Su, H., Pohl, H., Schomburg, U., Upper, G., & Hene, S. (1999). Wear and frcton of PTFE seals. Wear, 224, [30] Hoffmann, H., Hwang, C., & Ersoy, K. (2005). Advanced wear smulaton n sheet metal formng. Annals of the CIRP, 54, [31] Hegadekatte, V., Huber, N., & Kraft, O. (2005). Fnte element based smulaton of dry sldng wear. Modellng Smul. Mater. Sc. Eng., 13, [32] Hegadekatte, V., Huber, N., & Kraft, O. (2006). Fnte element based smulaton of dry sldng wear. Trbology Letters, 24, [33] Km, N. H., Won, D., Burrs, D., Holtkamp, B., Gessel, G., Swanson, P., & Sawyer, W. G. (2005). Fnte element analyss and experments of metal/metal wear n oscllatory contacts. Wear, 258, [34] Wu, J. S., Hung, J., Shu, C., Chen, J. (2003). The computer smulaton of wear behavor appearng n total hp prosthess. Computer Methods and Programs n Bomedcne, 70, [35] Eds.: Löhe, D., Haußelt, J. H. (2005). Mcro-Engneerng of Metals and Ceramcs, Part I and Part II. Wley-VCH Verlag GmbH, Wenhem, Germany. [36] Johnson, K. L. (1985). Contact Mechancs. Cambrdge Unversty Press, Cambrdge, UK. [37] Hertz, H. (1882). Ueber de beruehrung fester elastscher koerper. J. Rene und Angewandte Mathematk, 92, [38] Olver, W. C. & Pharr, G. M. (1992). An mproved technque for determnng hardness and elastc modulus usng load and dsplacement sensng ndentaton experments. J. Mat. Res., 7, [39] Callster, W. D. (1994). Materal Scence and Engneerng - An Introducton. John Wley and Sons, Inc., New York, USA. [40] Flodn, A. & Andersson, S. (1997). Smulaton of mld wear n spur gears. Wear, 207, [41] Kanavall, B. (2006). Applcaton of user defned subroutne UMESHMOTION n ABAQUS to smulate dry rollng/sldng wear. Master thess, Royal Insttute of Technology (KTH), Stockholm, Sweden.. [42] ABAQUS/Standard 6.5 Example Problems Manual, (Hbbt, Karlsson, & Sorensen, Inc., USA, 2003) [43] Herz, J., Schneder, J., & Zum-Gahr, K. H. (2004). Trbologsche charakterserung von werkstoffen für mkrotechnsche anwendungen. In R. W. Schmtt (Ed.), GFT Trbologe-Fachtagung 2004 Göttngen, Germany. on CD. 20

24 , a E F h E R F a s h c N e c p N w c N e p w e D w w N E a F h h h R a h h h s p k h h a F p s s s s max s 1 1 END YES NO , a E F h E R F a s h c N e c p N w c N e p w e D w w N E a F h h h R a h h h s p k h h a F p s s s s max s 1 1 END YES NO Fgure 1a

25 Fgure 1b h w [µm] Expt. (F N = 200mN) Expt. (F N = 400mN) Expt. (F N = 800mN) GIWM s [m] k D =13.5E-9 mm 3 /Nmm

26 Fgure 2a µ [-] Expt Ft F N =200 mn s [m]

27 Fgure 2b k D [mm 3 /Nmm] 2.0x x x x10-8 Expt. Sarkar (1980) 1.2x10-8 F 1.0x10-8 N =200 mn µ [-]

28 Fgure 2c µ [-] Expt Ft F 0.0 N =400 mn s [m]

29 Fgure 2d k D [mm 3 /Nmm] 2.0x x x x10-8 Expt. Sarkar (1980) 1.2x10-8 F 1.0x10-8 N =400 mn µ [-]

30 Fgure 2e µ [-] Expt Ft F N =800 mn s [m]

31 Fgure 2f k D [mm 3 /Nmm] 1.6x x x x x x10-9 Expt. Sarkar (1980) F N =800 mn µ [-]

32 Fgure 3 50 Expt. (F N = 800 mn) Expt. (F N = 400 mn) Expt. (F N = 200 mn) GIWM (k D =13.5 E-9 mm 3 /Nmm) Sarkar (1980) (k D =10.2 E-9 mm 3 /Nmm) h w [µm] s [m]

33 Fgure 4 h [µm] Experment GIWM F N = 1600 mn (Predcton) F N = 1200 mn (Predcton) F N = 800 mn (Predcton) F N = 400 mn (Ft) s [m]

34 Fgure 5a 2 mm 8 mm r (x) 1 R y y 4 mm z x x z 2 8 mm

35 Fgure 5b V 1 ( V 1 = V 2 ) V V 2 V

36 Fgure 6 Clck here to download hgh resoluton mage

37 Fgure 7a

38 Fgure 7b

39 Fgure 7c

40 Fgure 7d

41 Fgure 8

42 Fgure 9a

43 Fgure 9b

44 Table 1 Table 1: Values of the parameters from the exponental ft (Equaton 7) of the measured value of the frcton coeffcent as a functon of the sldng dstance for dfferent normal loads Parameter 200 mn 400 mn 800 mn A b c

45 Table 2 Table 2: Parameters used for the wear smulaton usng Wear-Processor and UMESHMOTION Parameter Value Materal ZrO 2 Young s Modulus E t = E b = 152 GPa Posson s Rato t = b = 0.32 Appled Normal Load F N = 0.3 N Frcton Coeffcent = 0.6 Dmensonal Wear Coeffcent k D = mm 3 /Nmm Slp 10 %