Advanced friction modeling for sheet metal forming

Transcription

1 Avance friction moeling for sheet metal forming Authors J.Hol a, M.V. Ci Alfaro b, M.B. e Rooij c, T. Meiners a Materials innovation institute (M2i) b Tata Steel Research, Development & Technology c University of Twente, Faculty of Engineering Technology, Laboratory for Surface Technology an Tribology University of Twente, Faculty of Engineering Technology, chair of Forming Technology - Please Note Care has been taken to ensure that the information herein is accurate, but Tata Steel an its subsiiary companies o not accept responsibility for errors or for information which is foun to be misleaing. Suggestions for or escriptions of the en use or applications of proucts or methos of working are for information only an Tata Steel an its subsiiaries accept no liability in respect thereof. Before using proucts supplie or manufacture by Tata Steel the customer shoul satisfy themselves of their suitability All rawings, calculations an avisory services are provie subject to Tata Steel Stanar Conitions available on request.

2 Avance friction moeling for sheet metal forming J.Hol a,, M.V. Ci Alfaro b, M.B. e Rooij c, T. Meiners a Materials innovation institute (M2i) - P.O. box GA Delft - The Netherlans b Tata Steel Research, Development&Technology - P.O. box CA IJmuien - The Netherlans c University of Twente, Faculty of Engineering Technology, Laboratory for Surface Technology an Tribology - P.O. box AE Enschee - The Netherlans University of Twente, Faculty of Engineering Technology, chair of Forming Technology - P.O. box AE Enschee - The Netherlans Abstract The Coulomb friction moel is frequently use for sheet metal forming simulations. This moel incorporates a constant coefficient of friction an oes not take the influence of important parameters such as contact pressure or eformation of the sheet material into account. This article presents a more avance friction moel for large-scale forming simulations base on the surface changes on the micro-scale. When two surfaces are in contact, the surface texture of a material changes ue to the combination of normal loaing an stretching. Consequently, shear stresses between contacting surfaces, cause by the ahesion an ploughing effect between contacting asperities, will change when the surface texture changes. A friction moel has been evelope which accounts for the change of the surface texture on the micro-scale an its influence on the friction behavior on the macro-scale. This friction moel has been implemente in a finite element coe an applie to a full-scale sheet metal forming simulation. Results showe a realistic istribution of the coefficient of friction epening on the local process conitions. Keywors: friction moeling, friction mechanisms, asperity contact, flattening, real contact area, ploughing, ahesion 1. Introuction The automotive inustry makes extensive use of Finite Element (FE) software for formability analyses to reuce the cost an lea time of new vehicle programs. In this respect, FE analysis serves as a stepping stone to optimize manufacturing processes. However, an accurate forming analysis of an automotive part can only be mae if, among others, the material behavior an friction conitions are moele accurately. For material moels, significant improvements have been mae over recent ecaes. However, in the majority of simulations a simple Coulomb friction moel is still use. This moel oes not incorporate the influence of important parameters on the contact behavior, such as pressure, punch spee or eformation of the sheet material. Consequently, even using the latest material moels, it is still cumbersome to preict the raw-in an springback of a blank uring the forming process correctly. To better unerstan contact an friction conitions uring lubricate sheet metal forming (SMF) processes, experimental an theoretical stuies have been performe. At the microscopic level, friction is ue to ahesion between contacting asperities [1, 2], the ploughing effect between asperities [1, 2] an the appearance of hyroynamic friction stresses [3, 4]. Ploughing effects between asperities an ahesion effects between bounary layers are the main factors causing friction in the bounary lubrication regime. If the contact pressure is carrie by the asperities an lubricant flow - as in the mixe lubri- Corresponing author. Tel.: ; fax: aress: j.hol@m2i.nl (J.Hol) cation regime - or fully carrie by the lubricant - as in the hyroynamic lubrication regime - hyroynamic shear stresses will contribute or even preominate. This article will focus on the two friction mechanisms present in the bounary layer regime: ploughing an ahesion. Wilson [1] evelope a moel which treate the effect of ahesion an ploughing separately. A more avance moel was evelope by Challen & Oxley [2] which takes the combine effect of ploughing an ahesion on the coefficient of friction into account. Challen & Oxley performe a slip-line analysis on the eformation of a soft flat material by a har wege-shape asperity an erive expressions for the coefficient of friction an wear rates. Westeneng [5] extene the moel of Challen & Oxley to escribe friction conitions between multiple tool asperities an a flat workpiece material. Their moel consiers the flattene plateaus of the workpiece asperities as soft an perfectly flat, an the surface texture of the tool as har an rough. The amount of ploughing an ahesion epens on the real area of contact. Hence, the coefficient of friction will change if the real area of contact changes. The real area of contact epens on the various flattening an roughening mechanisms of the eforming asperities. The three ominating flattening mechanisms uring SMF processes are: (1) flattening ue to normal loaing [6]; (2) flattening ue to stretching [7, 8]; an (3) flattening ue to sliing [9]. Flattening increases the real area of contact, resulting in a higher coefficient of friction. Roughening of asperities, observe uring stretching of the eforme material [10], tens to ecrease the real area of contact Preprint submitte to WEAR November 15, 2010

3 resulting in a lower coefficient of friction. The two mechanisms outline in this article are flattening ue to normal loaing an flattening ue to stretching. Future work is planne on moeling the roughening effect an the influence of sliing on the flattening behavior. A major research area within the fiel of friction moeling is focuse on eveloping moels to preict the flattening behavior of asperities ue to normal loaing. Most of these moels are base on the pioneering work of Greenwoo & Williamson [6] which evelope a stochastic moel base on contact between a flat tool an rough workpiece surface. Their moel mathematically escribes contact between two surfaces base on the assumption that summits of the rough surface are spherical, that summits only eform elastically an that the surface texture can be escribe by a istribution function. Over recent ecaes, moifications have been mae to this moel to account for arbitrary shape asperities, plastically eforming asperities an the interaction between asperities. Zhao an Chang [11] evelope a moel to escribe interactions between asperities on the micro-scale using the Saint-Venant principle an Love s formula (elastic interaction between asperities is assume). This moel is integrate into the elastic-plastic contact moel of Zhao, Maietaa an Chang (ZMC-moel) [12] which inclues a transition regime from elastic to fully plastic eformation of asperities. Jeng an Wang [13] extene the ZMCmoel for elliptical contact situations an Pugliese et al. [14] for parabolic profile approximations. Pullen & Williamson [15] evelope an ieal plastic contact moel base on the conservation of volume uring plastic eformation an the assumption that isplace material reappears as a uniform rise in the non-contacting surface. Conservation of energy is use to obtain relations between the contact pressure, separation an real area of contact as a function of the surface height istribution. The moel of Pullen an Williamson inspire Westeneng [5] to erive an ieal-plastic an nonlinear-plastic contact moel base on the conservation of volume an energy. Westeneng moele the asperities as bars which can represent arbitrarily shape asperities. The moels inclue a persistence parameter, work-harening parameters an are able to escribe the interaction between asperities. A further increase of the real area of contact coul occur if, uring normal loaing, a bulk strain is applie to the material. The effective harness of the asperities can be largely reuce if a bulk strain is present in the unerlying material [7]. Wilson & Sheu [7] evelope an analytical upperboun moel to escribe the flattening behavior using wege-shape asperities with a constant angle, Figure 1a. The length of the asperities is much greater than the with of the asperities. Therefore, a plane-strain state transverse to the asperities (x-irection) an a plane-stress state in the irection of the asperities (y-irection) is assume since the stress in this irection might be neglecte. The semi-empirical relation of Wilson & Sheu provies a relation between the effective harness, the real area of contact an a non-imensional strain rate. Sutcliffe [8] extene the moel of Wilson & Sheu to escribe a plane-strain situation in the irection of the asperities (strain in y-irection equals zero, Figure 1b). A slip-line analysis is performe to escribe the flattening of transverse wege-shape asperities. Westeneng [5] evelope a strain moel which escribes the influence of strain on a surface geometry using arbitrary shape asperities. His metho is base on volume an energy conservation laws an assumes that the crushe asperities cause a constant rise of the non-contacting asperities. The moel is applicable to both plane-strain an plane-stress situations, epening on the efinition of the non-imensional strain rate [5]. In this article, a friction moel is propose which inclues various friction mechanisms: flattening ue to normal loaing, flattening ue to straining, ploughing an ahesion. Existing moels have been use to escribe these mechanisms: - To escribe the flattening behavior of asperities ue to normal loaing the contact moel of Westeneng [5] has been use. His contact moel inclue flattening parameters which are not inclue in other loaing moels. Therefore, the moel of Westeneng will likely have better preicting capabilities in escribing the flattening behavior of F N F N z y z y ε y = 0 x ε x = 0 (a) ε y,σ y = 0 x ε x (b) Figure 1: Representation plane-stress moel of Wilson an Sheu (a) an plane-strain moel of Sutcliffe (b) 2

4 1. Input step: Process variables an material characteristics 2. Flattening ue to normal loaing (Section 2.3.1) 3. Flattening ue to stretching (Section 2.3.2) 4. Inentation moel (Section 2.4) 5. Calculation shear stresses (Section 2.4) 6. Calculation coefficient of friction (Section 2.4) Figure 2: Solution proceure asperities than other moels. - The flattening behavior of asperities ue to straining has been escribe by the strain moel of Westeneng [5]. The strain moel of Westeneng inclues flattening parameters which are not accounte for in other moels. Especially the possibility to escribe arbitrarily shape asperities makes the strain moel preferable to others. - The influence of ploughing an ahesion on the coefficient of friction has been escribe by the extene moel of Challen & Oxley [2, 5]. The ability to escribe contact problems between multiple asperities makes this extene version favorable. An overview of the friction moel is presente in this article an the translation from micro to macro moeling is outline. The theoretical backgroun of the moels use to escribe the various friction mechanisms is escribe an the implementation in FE coes is iscusse. The flattening moels are valiate by means of FE simulations on the micro-scale an the applicability of the friction moel in FE coes is proven by a full-scale sheet metal forming simulation. 2. Theoretical backgroun 2.1. Unifie friction theory A friction moel, to be use in finite element coes, has been evelope to couple the various micro friction moels, Figure 3 2. The friction moel starts with efining the process variables an material characteristics (step 1). Process variables are the nominal contact pressure an strain in the material which are calculate by the FE coe. The contact force carrie by the asperities equals the total nominal contact force since hyroynamic friction stresses will not be accounte for. Significant material characteristics are the harness of the asperities an the surface properties of the tool an workpiece material. Once the input parameters are known, the real area of contact is calculate base on the moels accounting for flattening ue to normal loaing (step 2) an flattening ue to stretching (step 3). The amount of inentation of the harer tool asperities into the softer workpiece asperities can be calculate if the real area of contact an the contact pressure carrie by the asperities are known (step 4). After that, shear stresses ue to ploughing an ahesion effects between asperities (step 5) an the coefficient of friction (step 6) are calculate. It is note that in reality flattening ue to normal loaing an flattening ue to stretching will appear simultaneously uring sheet metal forming, as well as the combination between flattening (step 2 an 3) an sliing (step 4 an 5). Nevertheless, it has been assume that the various mechanisms act inepenently of each other in this research. Friction moels encompassing micro-mechanisms are generally regare as too cumbersome to be use in large-scale FE simulations. Therefore, translation techniques are necessary to translate microscopic contact behavior to macroscopic contact behavior. Using stochastic methos, rough surfaces are escribe on the micro-scale by their statistical parameters (mean raius of asperities, asperity ensity an the surface height istribution). Statistical parameters can be use if assume that the surface texture is isotropic an can be represente by 2- imensional correlate ranom noise. It is assume that these restrictions are true for the workpiece an tool material, which makes the use of statistical parameters esirable in making the translation from micro to macro contact moeling. Assuming that the surface height istribution on the micro-scale represents the surface texture on the macro-scale, it is possible to escribe contact problems that occur uring large-scale FE analyses of sheet metal forming processes [5] Characterization of rough surfaces A iscrete surface height istribution of the tool an workpiece material is obtaine from surface profiles (Figure 3a). However, a continuous function is esirable to eliminate the nee for integrating iscrete functions uring the solution proceure of the friction moel. Various methos exist to escribe iscrete signals by continuous functions. The Gauss istribution function can be use if it is assume that the surface height istribution is symmetric an approximates a normal istribution function. However, the initial surface height istribution is usually asymmetric an will become even more asymmetric if there is flattening of contacting- an rising of non-contacting asperities. The asymmetric Weibull istribution function is a more flexible criterion but can only approximate smooth surface height istributions. A more avance metho to escribe iscrete signals can be achieve by using a Fourier series. A

5 6 4 Surface profile Real istribution Fourier fit half range z(µm) 0 φ(z)(µm 1 ) x(mm) z(µm) (a) (b) Figure 3: Surface profile (a) an corresponing surface height istribution (b) Fourier series makes it possible to escribe non-smooth asymmetric istribution functions from which the accuracy of the evaluation epens on the number of expansions use. The results iscusse in this article are obtaine by evaluating the surface height istribution functions φ(z) by a half range sine Fourier function [16], given by: φ(z)= with: b n = 2 L n=1 L 0 ( nπ ) b n sin L z ( nπ ) f (z) sin L z in which n represents the number of expansions, L the evaluation omain an f (z) the iscrete form of the surface height istribution. In Figure 3b, the iscrete surface height istribution from the workpiece material (Figure 3a) is evaluate by a Fourier function using 15 expansions Flattening mechanisms Two flattening mechanisms have been implemente in the friction moel to calculate the real area of contact of the workpiece: flattening ue to normal loaing an flattening ue to stretching. The moels of Westeneng are use for this purpose [5]. Westeneng assume the tool as rigi an perfectly flat, which inents into a soft an rough workpiece material. This assumption is vali since the ifference in harness an length scales between the tool an workpiece material is significant in the case of sheet metal forming processes. Westeneng moele the asperities of the rough surface by bars which can represent arbitrarily shape asperities, Figure 4. Westeneng introuce 3 stochastic variables as presente in Figure 4: The normalize surface height istribution function of the asperities of the (1) (2) 4 rough surface φ(z), the uniform rise of the non-contacting surface U (base on volume conservation) an the separation between the tool surface an the mean plane of the asperities of the rough surface Flattening ue to normal loaing Using the normalize surface height istribution φ(z), the amount of flattening of the contacting asperities an the rise of the non-contacting asperities U can be calculate base on energy an volume conservation. Contact between a flat har smooth surface an a soft rough surface is assume without sliing an bulk eformation. Only plastic eformation of asperities is assume without work-harening effects. The amount of external energy must equal the internal energy in orer to account for energy conservation. The amount of external energy is escribe by the energy neee to inent contacting asperities. The internal energy is escribe by the energy absorbe by the inente asperities an the energy require to lift up the non-contacting asperities. The inentation variables use to escribe Westeneng s moel are epicte in Figure 5. A istinction is mae between asperities in contact with the inenter, asperities which will come into contact ue to the rise of asperities an asperities which will not come into contact with the inenter. The amount of inentation is escribe by the variable z while the rise of asperities is escribe by the variable u. The number of asperities in contact with the inenter is inicate by the counter N with corresponing inentation heights of z i (i=1, 2,..., N). The number of asperities coming into contact with the inenter ue to a rise of non-contacting asperities is escribe by the counter N with inentation heights of z j ( j = 1, 2,..., N ). Hence, the total number of asperities in contact with the inenter after applying the normal loa equals N+ N. Asperities which will not come into contact uring the loa step are inicate by the counter N with corresponing rising heights of u l (l=1, 2,..., N ).

6 Inente asperities Rise non-contacting Surface U Tool surface Mean plane z φ(z) Workpiece asperities U Figure 4: A rough soft surface inente by a smooth rigi surface The total amount of asperities M equals N+ N + N. The amount of external energy epens on the total number of asperities in contact with the inenter (N+ N ). Normally, the non-contacting asperities woul rise with a istance u l. But ue to the presence of the inenter some of the asperities are restricte to rise with a istance of u j. A certain amount of external energy is require to prevent a rise of z j = u l u j. The energy require to inent contacting asperities is given by: W ext = N N F Ni z i + F N j z j = i=1 j=1 Equation 3 can be rewritten as: W ext =ζ 1 F N z k with ζ 1 = N+N k=1 N+N k=1 F Nk z k (3) F Nk z k z k F N (4) In which F N represents the total force an z k the maximum inentation height.ζ 1 is calle the energy factor since it is influence by the amount of external energy require to inent the rough surface. V z i V F Ni N N N z j u j u l Figure 5: Zoom-in on inente an rising asperities 5 The amount of internal energy is escribe by the summation of the energy absorbe by the inente asperities W intab an the energy require to rise the non-contacting asperities W intri over all asperities, Equation 5. W int = W intab + W intri (5) The asperities are escribe by bars having the same area A. The maximum pressure an asperity can carry before eformation occur equals the harness H of the material since ieal plasticity is assume. Therefore, the absorbe energy W intab over N+ N inente asperities can be escribe as: W intab = N+N k=1 H A z k (6) W intab can also be written as Equation 7 using A r = (N+ N ) A representing the real area of contact anζ 2 representing a shape factor. N+N W intab =ζ 2 HA r z k with ζ 2 = (7) (N+ N ) z k k=1 z k W intri is escribe by the sum of energy require to rise N asperities which comes in contact with the inenter after application of the normal loa an N asperities which o not come into contact with the inenter, Equation 8: N N W intri =η u j H A+ u l H A (8) j=1 l=1 Equation 8 inclue a persistence parameter η which escribes the amount of energy require to lift up the non-contacting asperities. A value ofη=0 means that no energy is neee to rise the asperities, a value ofη=1 implies that a maximum amount of energy is neee to rise the asperities. Equation 8 can be simplifie by using volume conservation: W intri =ηh N z i A (9) i=1

7 which can also be written as: W intri =ηζ 3 H (A r N A) z k with ζ 3 = N z i i=1 N z k (10) ζ 3 is calle the shape factor since it is influence by the shape of the surface. Balancing the total internal energy (Equation 5) to the external energy (Equation 4) gives: constant rise of asperities U are unknowns in Equation 16. An equation can be obtaine from volume conservation, Equation 17. Using stochastic parameters, the equation for volume consistency can be written as Equation 18: N N N z i A= u l A+ u j A i=1 } {{ } V l=1 } {{ } V j=1 } {{ } V (17) F N H =ζ 2 ζ 1 A r + ζ 3 ζ 1 η (A r N A) (11) The parametersζ 1,ζ 2 anζ 3 can be expresse as a function of each other. Appenix A shows the erivation of the expressions given in Equation 12. ζ 2 = ξ α ζ 1 ζ 3 =αζ 2 χ=ξζ 1 χ (12) α represent the ratio of the real to the nominal area of contact,ξ can be regare as a energy factor anχas a shape factor.α, as well asξ anχare variables which epen on the statistical parameters U (the constant rise of asperities) an (the separation between the tool surface an the mean plane of the asperities of the rough surface). In aition, ξ is a function of the normal forces acting on the asperities F N (z). It shoul be note that a constant rise of asperities has been assume in the erivation ofα,ξanχ, which correspons to the experimental results of Pullen & Williamson [15]. Statistical parameters have been introuce by using the following stochastic variables, with M the total amount of asperities: F N = M N+ N = M an: { z k = F N (z)φ(z) z (13) z k z k + U φ (z) z= Mα (14) for z k > for U z k < (15) In which the surface height istribution of the rough surface φ (z), to be use for the translation from micro to macro friction moeling, has been introuce. Substituting Equation 12 into Equation 11 gives: P nom H =ξ+ηχξ α φ (z) z =ξ 1+ηχ φ (z) z (16) Equation 16 gives the relation between the nominal contact pressure P nom (efine as F N /A nom ), the separation an the constant rise of the non contacting asperities U. Another equation is require to compute P nom since the separation an the 6 A nom (z )φ(z) z= A nom z 1 u (z)φ(z) z+ A nom z 1 ( z)φ(z) z (18) where z 1 represents the initial height of an asperity which just comes into contact with the inenter after applying the normal loa, (z 1 + u(z 1 )=). Taken a constant rise U of the noncontacting asperities into account, equation 18 becomes: U (1 α)= (z )φ(z) z (19) The amount of flattening of contacting asperities an the rise of non-contacting asperities U ue to normal loaing can be calculate by solving Equation 16 an Equation 19 simultaneously Flattening ue to stretching Besies an ieal-plastic contact moel for normal loaing, Westeneng erive an analytical contact moel to escribe the influence of strain on eforming, arbitrary shape, asperities. The effective harness of asperities can be largely reuce if a bulk strain is present in the unerlying material. As a result, more inentation of contacting asperities will occur. The outcome of the ieal-plastic loa moel, escribe in Section 2.3.1, will be use as an input. The subscript S has been use to inicate the variables which are strain epenent. For example, U S correspons to the rise of non-contacting asperities an S to the inentation of contacting asperities ue to straining. Analogous to the loaing moel, the stretching moel consiers contact between a flat har surface an a soft rough surface. Only plastic material behavior is assume without workharening effects. The erivation of the stretching moel starts on single asperity scale, where the change of the fraction of the real contact areaᾱ S as a function of the nominal strainεis erive. ᾱ S ε can be written as a function of the inentation spee of an the strain rate in the unerlying the inenting asperity ω S t S bulk material ε t s : ᾱ S ε = ᾱ S ω S t S ω S t S ε (20)

8 ω S represents the inentation istance of an asperity. The increase of the inentation istance ω S can be written as: ω S = U S + S = (U S S ) (21) Hence, the first term in Equation 20 can be written as: ᾱ S ᾱ S = ω S (U S S ) (22) The secon term, which represents the velocity of the inenting asperity, is etermine by the ownwar velocity of the inenting asperity v a an the upwar velocity of the rising asperities v b, Equation 23. The thir term represents the strain in the bulk material of the asperities an can be written as: ω S t S = v a + v b t S ε = 1 ε Substituting Equations 22 an 23 into Equation 20 gives: ᾱ S ε = v a+ v b ᾱ S ε (U S S ) A non-imensional strain rate can be efine as [7]: E= (23) (24) εl v a + v b (25) with l representing the mean half spacing between asperities: l= 1 2 Qα S (26) The efinition for l, with Q representing the asperity ensity, is approximately true for surfaces with no particular roughness istribution [5]. Equation 24 can now be written as a function of the nonimensional strain rate: ᾱ S ε = l ᾱ S E (U S S ) (27) It is assume that the fraction of the real contact area for one asperityᾱ S equals the total fraction of contact areaα S. Therefore, the stochastic form of the real contact area (Equation 28) can be use to solve the ifferential equation in Equation 27: α S = S U S φ (z) z (28) To calculate the change ofα S, the value of U S an S nees to be solve simultaneously whileεis incrementally increase. Base on volume conservation (Equation 31) an the efinition of the fraction of real contact area (Equation 28) U S an S can be obtaine. U S (1 α S )= S U S (z S )φ(z) z (31) The implementation of Equations 30, 31 an 28 into FE coes is shown in Section Calculation of shear stress The moel of Challen & Oxley [2, 17] takes the combining effect of ploughing an ahesion between a wege-shape asperity an a flat surface into account. Westeneng [5] extene the moel of Challen & Oxley to escribe friction conitions between a flat workpiece material an multiple tool asperities, Figure 6b. He assume that the flattene peaks of the asperities are soft an perfectly flat an the surface of the tool material is rough an rigi. As mentione earlier, the ifference in harness between the tool an workpiece material an the ifference in length scales between the two surfaces is significant in the case of a sheet metal forming process. Therefore, it is vali to make a subivision in two length scales using a rigi tool an a soft workpiece. The macro-scale moel of Challen & Oxley has been implemente in the friction moel to escribe friction conitions between the tool an workpiece material Shear stresses single asperity contact The moel of Challen & Oxley has been use to calculate the coefficient of friction between a har asperity an a soft flat surface. The moel takes the combining effect of ploughing an ahesion between contacting surfaces into account. They performe a slip-line fiel analyses to escribe contact conitions of wege-shape asperities. The attack angleθof the asperity can be escribe by a constant angle [2] or a varying angle if a spherical shape asperity is assume [17], Figure 6c. In case of spherical shape asperities, the attack angleθis small if the amount of inentationωis small an grows if the amount of inentation increases. This conforms more to reality then using a constant attack angle. The amount of inentation ω an the attack angleθfor a spherical-shape asperity is efine as: ω= s δ (32) α S (U S S ) = (U S S ) =φ ( S U S ) S U S φ (z) z Substituting Equation 29 into Equation 27 yiels: ᾱ S ε = l E φ ( S U S ) (30) θ= 1 (29) 2 α=arctan ω (33) ω (2βt ω) in whichδrepresents the istance between the mean plane of tool asperities an the flat plateaus of the workpiece anβ t the mean raius of tool asperities. The attack angleθis chosen half of the maximum attack angleαwhich can be regare as the mean attack angle. 7

9 Tool surface Mean plane F Nasp δ s δ ω β t α Workpiece surface φ t (s) θ r Bounary layer (a) (b) (c) Figure 6: Inentation tool asperities Challen & Oxley [17] euce slip-line fiels assuming a plane-strain eformation state in the irection of the asperity an ieal-plastic material behavior. The following set of equations has been foun for the coefficient of frictionµ asp for a ploughing, wege-shape, asperity: µ asp = A sinθ+cos (arccos f C θ) A cosθ+sin (arccos f C θ) (34) with: A=1+ π 2 + arccos f sinθ C 2θ 2 arcsin (35) 1 fc f C is calle the friction factor efine asτ/k, withτescribing the shear stress in the bounary layer an k the shear strength of the softer material. The coefficient of friction epens on the attack angleθ, an thus epens on the normal loa F Nasp. The friction force acting on one asperity is escribe by: F wasp =µ asp F Nasp (36) The normal loa one asperity can carry epens on the amount of inentation ω. A relation between the raius of the contact length r (Figure 6c) an the inentationωcan be efine as: r= 2ωβ t ω 2 (37) The normal loa becomes: Shear stress in a multi asperity contact Westeneng escribe the translation from friction forces occurring at single asperity contacts to the total friction force at multiple asperities by: F w =ρ t α S A nom s max δ F wasp φ t (s) s (40) In which the stochastic variable escribe in Equation 13 has been use.ρ t represents the asperity ensity of the tool surface, α S the ratio of the real to the nominal area of contact of the workpiece, A nom the nominal contact area,φ t the normalize surface height istribution function of the tool surface an F wasp the friction force occurring at a single asperity. The bouns of the integral are escribe by s max, the maximum height of the tool asperities, an δ, the separation between the workpiece surface an the mean plane of the tool asperities (Figure 6b). The friction force F w can be calculate once the amount of separationδis known.δ can be obtaine base on force equilibrium for the normal loa by solving Equation 41: 0=Hα S A nom s max δ φ t (s) s P nom A nom (41) The integral in Equation 41 represents the fraction of the nominal contact area of the tool penetrating into the workpiece material. If the shear stresses are known from Equation 40 the coefficient of friction can finally be obtaine by: F Nasp = AH=π ( 2ωβ t ω 2) H (38) Assuming that only half of the contacting area carries the loa uring ploughing an that only small inentations takes place (ω<<β t ), F Nasp can be written as: F Nasp =πωβ t H (39) From which the friction force F wasp acting on one asperity can be calculate. 8 µ= F w F N (42) 3. Implementation The friction moel escribe in Section 2.1 has been implemente into the in-house implicit FE coe Dieka, evelope at the University of Twente. The friction moel is calle for every noe in contact uring a FE simulation. If a noe is in contact, the nominal contact pressure an strain in the bulk material is

10 P nom, H,ηanφ(z) 0=ξ 1+ηχ φ (z) z P nom H 0= (z + U)φ(z) z U α, an U Figure 7: Calculation scheme loa step calle from the source coe. The iscrete surface height istribution is escribe by a continuous function using the half range Fourier serie (Section 2.2). Then, the fraction of real contact area ue to normal loaing an stretching, shear stresses ue to ploughing an ahesion an the coefficient of friction are being calculate by the equations escribe in Section 2. This section escribes the implementation of these equations into the FE coe in more etail Implementation of flattening ue to normal loaing The amount of flattening of contacting asperities an the rise of the non-contacting asperities U ue to normal loaing can be calculate by solving Equation 16 an Equation 19 simultaneously. The solution scheme is schematically shown in Figure 7. An expression for F N (z) must be substitute into the expression forξ (see Appenix A, Equation A.3) in orer to solve the system of Equations presente in Figure 7. It is assume that the eforming asperities can be represente by plastically eforming spheres since an expression for F N (z) oes not exist for plastically eforming bars. Assuming that only small inentations occur ( z<<β w ), the following expression for F N (z) can be substitute (see also Equation 38): F N (z)=2πβ w H z (43) orer to get values for the amount of flattening of contacting asperities S an the rise of the non-contacting asperities U S, as presente in Figure 8. The friction moel can be use to solve plane-strain or plane-stress situations, epening on the efinition of the nonimensional strain rate E. Wilson & Sheu [7] evelope a efinition for E base on the flattening behavior of wege-shape asperities using a constant angle. They assume a plane-stress state an a nominal strain in the longituinal irection of the asperities (y-irection) an a plane-strain state transverse to the irection of the asperities (x-irection), Figure 1a. Base on an upperboun analysis, Wilson & Sheu provie a relation between the effective harness H ef f, the real area of contactαan the non-imensional strain rate E. They also propose a semiempirical relation for E as a function of H anα S, fitte on the results obtaine by the upperboun-metho [7]: E= 2 H ef f f 2 (α S ) f 1 (α S ) with H ef f = P nom α S k (45) in which f 1 (α S ) an f 2 (α S ) are fitting parameters. The parameter ( k) represents the shear strength of the surface material H/3 3. Sutcliffe [8] extene the moel of Wilson & Sheu to escribe plane-strain situations. In their moel a plane-strain state in the longituinal irection of the asperities (y-irection) an a nominal strain transverse to the orientation of the asperities (x-irection) is assume, see Figure 1b. A slip-line analysis has been performe to escribe the flattening of wege-shape asperities. Sutcliffe provie a relation between the effective harness H ef f, real area of contactα S an non-imensional strain rate E. The relation between the non-imensional strain rate an the fan angleγ, base on results obtaine by the slipline analysis, has been presente in their article. The fan angle represents a characteristic slip-line angle which is boune by 0 γ π/2 ue to geometrical conitions. Base on these results, Westeneng [5] propose a semi empirical relation for the non-imensional strain rate as a function of the fan angle: E= with: exp (1.47γ) (46) with: { z= z z + U for z> for U z< (44) anβ w representing the mean raius of asperities. Using these expressions the set of equations presente in Figure 7 can be solve by using the secon orer Newton Raphson scheme Implementation of flattening ue to normal loaing an stretching Equation 30 has to be solve to calculate the change of the fraction of real contact areaα S as a function of the nominal strain ε. Equation 28 an 31 must be solve simultaneously in 9 γ= H ef f 4 (1 α S ) (47) The calculation scheme presente in Figure 8 can be solve by implementing one of the two efinitions of the nonimensional strain rate (Equation 45 or 46). The ifferential equation can be solve by applying the Euler metho while the root of the inner two equations can be foun by the secon orer Newton Raphson metho Implementation calculation scheme for the coefficient of friction The amount of inentation of the harer tool asperities into the softer workpiece asperities (Equation 41) can be calculate

11 U, anαfrom loa moel α S ε = l E φ ( S U S ) P nom anε in α S = α S + α S 0= φ (z) z α S 0= S U S S U S (z S U S )φ(z) z U S S an U S ifε+ε<ε in Figure 8: Calculation scheme strain step, P nom anε in are coming in from the FE coe mente in the friction moel to escribe the shear factor f C : τ (p)=3.94p 0.81 (48) During ploughing, the contact pressure p equals the effective harness H ef f of the softer material since ieal plasticity is assume. The shear strength k is relate to the harness H by the relation k=h/3 3. Substituting Equation 48 an the efinition of k into the expression for f C gives: f C = τ k = 20.47H 0.19 ef f (49) if the real area of contact is known from the two flattening mechanisms. Shear stresses ue to ploughing an ahesion effects between asperities an the coefficient of friction can be calculate if the amount of inentation is known, as presente in Figure 9. An expression for the shear factor f C is necessary (Equation 34) in orer to solve the system of Equations presente in Figure 9. f C is efine asτ/k withτescribing the friction force in the bounary layer an k the shear strength of the softer material. The moel of Timsit & Pelow [5, 18] has been impleφ t (s),α S, P nom an H ef f 0=H ef f α S A nom s max δ F w =ρ t α S A nom φ t (s) s P nom A nom s max δ δ µ= F w F N F wasp φ t (s) s Figure 9: Calculation scheme friction step 10 Timsit & Pelow performe experiments to obtain an empirical relation between the normal stress an the shear strength of a stearic aci film eposite on aluminum. The relation is applicable for contact pressures in between 70MPa an 740MPa, which are likely to occur on a micro scale uring eep rawing processes. 4. Valiation of flattening mechanisms The flattening moels propose by Westeneng are use to etermine the real area of contact between the tool an workpiece material. FE simulations on the micro-scale have been performe in orer to valiate these moels. Two sets of simulations have been performe for this purpose. In the first analysis, a two-imensional rough surface of 4mm long was inente by a perfectly flat an rigi tool, as shown in Figure 10a. The secon analysis was focuse on inenting a rough surface by a normal loa incluing a bulk strain in the unerlying material, as shown in Figure 10b. The roughness istribution use in the FE simulation equals the istribution measure for DC04 lowcarbon steel. The surface was moele by 4 noe 2D planestrain elements. The yiel surface was escribe by the Von

12 Tool Tool Workpiece (VM stresses) (a) Workpiece (VM stresses) (b) Figure 10: Schematic view analysis 1 (a) an analysis 2 (b) Mises yiel criterion using the Naai harening relation to escribe work-harening effects. Material parameters use for the FE simulation are liste in Appenix B. Contact between the tool an the rough surface was escribe by the penalty metho using a penalty stiffness of 1 N/mm. The surface height istribution use for the analytical moel correspons to the roughness istribution of the FE simulation. A fixe harness of 450 MPa (3σ y ) was use in the analytical moel since a yiel strength of 150MPa was use for the FE simulation. The amount of inentation of the rough surface - as well as the evelopment of the real area of contact - has been tracke uring the simulation an compare with the analytical solution. The analytical solution an the FE solution for elastic ieal-plastic an elastic non-linear plastic material behavior is presente in Figure 11 for analysis 1. The material behavior of the elastic ieal-plastic FE simulation is comparable to the material behavior assume in the analytical moel (ieal-plastic). The material behavior of the elastic nonlinear-plastic FE simulation correspons more to the actual material behavior than that of the elastic ieal-plastic FE simulation. The analytical moel uses a persistence parameter η to escribe the amount of energy require to lift up the noncontacting asperities (see Equation 8). A value ofη=0 means that no energy is neee to rise the asperities, whereas a value ofη=1 implies that a maximum amount of energy is neee to rise the asperities. Since the exact value of this parameter is not known, ifferent calculations have been performe in orer to obtain a precise value for this parameter. A higher value for the persistence parameter η results a lower amount of inentation (Figure 11a) an a smaller value of the real contact area (Figure 11b). Both the inentation an the evelopment of the real contact area calculate by the analytical solution using a value ofη=1 correspon very well to the elastic ieal-plastic FE solution. The analytical moel eviates from the more realistic elastic nonlinear-plastic FE simulation, since work-harening effects are not accounte for. The flattening of the asperities will be lower ue to work-harening effects, which in turn result in a lower amount of inentation an real area of contact (Figure 11). Combine normal loaing an stretching the unerlying bulk material ecreases the effective harness [7]. A lower harness results in an increase of the real area of contact. Both the analytical an the FE results of analysis 2, where a rough surface has been inente by a nominal loa an a bulk strain has been applie to the unerlying material, are presente in Figure 12. For the FE solution, only the evelopment of the real area of contact is shown, since the eformation of the asperities is ifficult to separate from the eformation of the unerlying bulk material. It can be conclue from Figure 12b that work-harening effects have a large influence on the flattening behavior of the asperities. A ifference of 20% in the real area of contact is obtaine at the en of the simulation between the results of the elastic ieal-plastic an the elastic nonlinear-plastic simulation. The non-imensional strain rate E, use in the analytical moel, can be escribe by the efinition given by Wilson & Sheu (Equation 45) or the efinition of Sutcliffe (Equation 46). The results of the FE simulation presente in Figure 12b are base on a plane-strain eformation moe in the longituinal irection of the asperities. Therefore, the efinition propose by Sutcliffe has been use. The ensity of workpiece asperities (in mm 2 ) is an unknown parameter in the analytical strain moel. Various methoologies exist to extrue the asperity ensity from the surface profile. Results obtaine by these methoologies are highly epenent on the chosen metho an the resolution of the use roughness measurement evice. Future work is planne to etermine this parameter using the most suitable metho. Until then, the asperity ensity will be taken as an unknown parameter. Calculations have been performe using realistic values for the asperity ensity for DC04 to show the importance of this parameter, see Figure 12. From Figure 12, it can be conclue that the asperity ensity of the workpiece has a significant influence on the evelopment of the real area of 11

13 Inentation (µm) η=0.0 FEM: elastic ieal-plastic FEM: elastic nonlinear-plastic η=0.5 η=1.0 Fraction of real contact area (-) FEM: elastic ieal-plastic FEM: elastic nonlinear-plastic Analytical solution η=0.0 η=0.5 η=1.0 Analytical solution Nominal contact pressure (MPa) Nominal contact pressure (MPa) (a) (b) Figure 11: Results analysis 1: Amount of inentation rough surface (a) an evelopment of real area of contact (b) asp/mm FEM: elastic ieal-plastic Inentation workpiece (µm) asp/mm asp/mm 2 Fraction of real contact area (-) FEM: elastic nonlinear-plastic Analytical solution 3000 asp/mm asp/mm asp/mm 2 Analytical Strain (-) Strain (-) (a) (b) Figure 12: Results analysis 2: Amount of inentation rough surface (a) an evelopment of real area of contact (b) contact. The amount of inentation of the workpiece asperities will be lower if a higher value of the asperity ensity is use, Figure 12a. Consequently, a lower amount of inentation results in a lower amount of real area of contact, Figure 12b. The tren of the graphs correspons well to the flattening behavior obtaine by the FE simulations. Using an asperity ensity of 5000 asp/mm 2 it is possible to escribe the results of the elastic ieal-plastic FE solution (which has comparable material characteristics) precisely Application The cross-ie prouct is a test piece esigne at Renault which approximates process conitions of complex automotive parts. The cross-ie prouct is use to test the numerical performance of the evelope friction moel in a large-scale FE simulation. In this article, the focus is on the numerical performance an feasibility of the friction moel that has been evelope. To valiate this moel, an experimental test proceure nees to be evelope an execute. Due to symmetry of the cross-ie prouct only a quarter of the workpiece was moele. The workpiece was meshe by 9000 triangular iscrete Kirchhoff shell elements using 3 integration points in plane an 5 integration points in thickness

14 irection. The yiel surface was escribe by the Vegter moel [19] using the Bergström-Van Liempt harening relation [20] to escribe harening behavior. Material parameters were use from DC04 low carbon steel, a typical forming steel use for SMF processes. A list of material parameters is liste in Appenix C. Contact between the tools an the workpiece was escribe by a penalty metho using a penalty stiffness of 200 N/mm. The coefficient of friction use in the contact algorithm was calculate on the basis of the friction moel presente in this article. Roughness parameters are given in Table 1, an the normalize surface height istribution functions that were use for the tool an workpiece material are presente in Figure 13. The simulation was performe by prescribing the isplacement of the punch until a total isplacement of 50 mm was reache. The punch spee was set to 5 cm/sec an the force applie to the blankholer was 50 kn. Two simulations have been performe in orer to quantify the iniviual contributions of the two flattening mechanisms. The first simulation only accounte for the influence of normal loaing on the coefficient of friction, Figure 14. The secon simulation uses both flattening moels to etermine the coefficient of friction, Figure 15. Results shown in both Figure 14 an 15 are from the punch sie of the sheet. The gray areas represent the non-contacting areas. If only flattening ue to static loaing is assume (Figure 14), rather low values for the ratio of the real to the nominal area of contact are obtaine. This results in friction coefficients that vary between 0.13 an Higher values are obtaine in high-pressure regions: the contact area of the punch raius (region A) an the thickene area of the blankholer region (region B). Lower values occur in low-pressure regions: the blankholer region an the top area of the punch. Results look reasonable, but it shoul be note that only one of the two flattening mechanisms was taken into account uring the simulation. If the secon flattening mechanism is taken into account (flattening ue to stretching), higher values for the real area of contact are obtaine (Figure 15). The higher contact ratios result in higher values of the coefficient of friction, i.e. between 0.13 an It can be observe from Figure 15b that higher values of the coefficient of friction occur at regions where high strains occur (region C, D an E). Region C is purely stretche, region D is compresse which causes thickening of the material an region E is stretche over the ie raius. On the other han, low values of the coefficient of friction can be observe in low-strain regions. Overall it can be conclue that the istribution of the coefficients of friction lies within the range of expectation. The increase in calculation time is also promising. An increase of 60% for the first simulation an 200% for the secon simulation was obtaine compare to the calculation time require to perform a Coulomb base FE simulation of the cross-ie prouct. During the implementation of the friction moel into FE software some important assumptions ha to be mae: - Full recovery of asperities is assume uring unloaing of the workpiece rough surface. The amount of recovery of inente asperities is escribe by the amount of elastic spring-back in reality. Due to elastic spring-back, the amount of recovery will be smaller than in case of full recovery. This will result a higher real area of contact at lower loas. Consequently, the coefficient of friction will be smaller ue to a smaller amount of inentation of tool asperities into the softer workpiece asperities. A realistic unloaing moel is require to escribe this effect. - A efinition of the non-imensional strain rate is require to calculate the amount of flattening ue to bulk straining. Various efinitions exist to escribe this variable, but most of them are base on a plane-strain or a plane-stress assumption. The plane-strain efinition of Sutcliffe (Section 3.2), taken the equivalent plastic strain as a strain measure, has been use for the application iscusse in this section. However, strains an stresses occur in ifferent irections uring sheet metal forming in reality. This gives rise to the question if Sutcliffe s moel is still applicable an how the strains shoul be accounte for. - The moel of Timsit & Pelow [5, 18] has been implemente in the friction moel to escribe the shear factor f C (Section 3.3). Timsit & Pelow performe experiments to obtain an empirical relation between the normal stress an the shear strength of a stearic aci film eposite on aluminum. The applicability of Timsit & Pelow s moel Roughness parameter Value Unit Harness workpiece (H) 1400 MPa Persistence parameter (η) 1 Density workp. asp. (ρ work ) mm 2 Density tool. asp. (ρ tool ) mm 2 Raius tool. asp. (β tool ) mm Nr. of Fourier expansions 10 Non-im. strain rate (E) Sutcliffe φ(µm 1 ) workpiece φ workpiece φ tool φ(µm 1 ) tool Table 1: Roughness parameters z(µm) Figure 13: Tool an workpiece istribution 13

15 5.2% A 0.0% B 0.13 (a) (b) Figure 14: Development ratio of real to apparent area of contact (a) an coefficient of friction (b) for normal loaing only (gray represents the non-contacting area) 29.4% C 0.19 E 0.0% 0.13 (a) D (b) Figure 15: Development ratio of real to apparent area of contact (a) an coefficient of friction (b) for normal loaing only (gray represents the non-contacting area) must be checke when using other metal-lubricant combinations. In an upcoming article, the valiity of the friction moel will be shown - base on experimental research an large scale FE simulations. 6. Conclusion A friction moel that can be use in large-scale FE simulations is evelope. The friction moel inclues two flattening mechanisms to etermine the real area of contact at a microscopic level. The real area of contact is use to etermine the 14 influence of ploughing an ahesion effects between contacting asperities on the coefficient of friction. A statistical approach is aapte to translate the microscopic moels to a macroscopic level. The friction moel has been valiate by means of FE simulations at a micro-scale. An excellent comparison between the analytical an the FE simulation is obtaine in case of inenting a rough surface by a normal loa. It was also foun that work-harening effects o not play a significant role in the case of pure normal loaing. If a nominal strain is applie to the bulk material, the effect of work-harening becomes much more significant. A ifference of 20% in real contact area, compare