CONTACT MODEL FOR A ROUGH SURFACE

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1 23 Paper presented at Bucharest, Romania CONTACT MODEL FOR A ROUGH SURFACE Sorin CĂNĂNĂU Polytechnic University of Bucharest, Dep. of Machine Elements & Tribology, ROMANIA s_cananau@yahoo.com ABSTRACT The paper presents a model for the determination of the contact characteristics, pressure distribution and subsurface stress field at the contact cylinder- plane. The first part of the paper presents the geometrical models used to determine the stress field. There are three models (at different scale) identified. In the second part there is performed an analysis with the first model, using a FEM cod and procedure in order to identify the stress filed. Then, for the second order profile and the second model of contact we present a model for determination of the asperity pressure distribution assuming the elastic-perfectly plastic deformation. The numerically generated rough surface is the base for the contact algorithm. The results show differences in comparison wit results predicted by Hertz theory. The maximum shear stress developed is significantly higher then predicted by theory. Keywords: contact mechanics, profile, subsurface stress. 1. INTRODUCTION Surface roughness is recognized as an important factor for the operational behavior of machine ele-ments. This factor is important because it influences the dry or lubricated contacts, friction and wear phenomena or surface failure. The problem of elastic contact of rough surfaces was studied from many points of view, but a numerical treatment in a contact zone requires a detailed discretization of actual topography [1]. There are some specific problems in discussion: a)-estimation of a reasonable portion of the assembly of bodies in contact; b)-the need to make the model as manageable as possible without introduction of nonrealistic simplifications; c)-to obtain a good estimation of the forces that are transferred in a contact zone; d) the connection between the shape of the roughness profile, pressure distribution and the subsurface stress field. The main target in this paper is the development of a contact model which includes surface roughness and to identify subsequent surface stress field in comparative rough model contacts. 2. MULTI-SCALE CONTACT ROUGH MODELS Regarding the models for real contact area there are various solution proposals. The first problem is the geometry of the asperities: some of these solutions transform the real rough surface in a smooth equivalent one with asperities; others made the assumption that the asperities are different known geometrical models such spheres, Greenwood and Williamson [2], Greenwood and Tripp [3], elliptical paraboloids, Bush et al. [4], pyramidal forms et al. The second problem concerns the distribution of the asperities. There is a model based on the concept of deterministic distributions of asperities, Archard, [5], or a model based on a statistical distribution as in [3], [6]. The later development uses a random process theory as in Whithouse and Archard [5], Bush et al. [4], Nayak [7]. In order to study the deformation of a rough surface we propose a multi-scale elasto-plastic model deformation [8, 9]. This model incorporates the Hertzian contact model. But the key of all these models is the consideration of the nature of microcontact model. Some of them assume that the contact is purely elastic, some consider that the contact is

2 24 THE ANNALS OF UNIVERSIT DUNĂREA DE JOS OF GALAŢI elasto-plastic and the last category takes into consideration the plastic asperity deformations. 2. CONTACT MODEL In this paper we assume that up to a certain value of contact deformation the materials deform elastically according Hertz theory. From this value, the plastic deformation starts. In the intermediate regime the contact width and pressure distribution will be computed using numerical simulation and we use the model proposed by Chang [8]. This model incorporates the Hertzian contact model. First of all we consider a real profile (2D model) for contact. At this scale a profilometer was used to measure the profile from an arbitrary machined metal sample of a hobbed gear. The physical model of contact is the model of the contact between one cylinder with given geometry (radius R 1 ) and a plane. Also we know the materials properties (oung s modulus E 1,2, Poisson ratio υ 1,2. We suppose that the surface roughness has approximately the same value and, consequently, we can study the contact model between a smooth plane and a cylinder with a composite roughness y o (x). I assume that the deformations of the asperities are elastic or plastic. The deformations of the model are: (a) deformations of the asperities at a micro scale; (b) deformation of the elastic body. The deformation of the rough surface is shown in figure 1 and is given by following equation: 2 x d( x) = y( x) + yp + yo( x) 2R (1) where d(x) is the deformation of the roughness profile, y o (x) is the roughness height measured from the mean line, y(x) is the distance of the deformed plan from the referenced plane, y p is the composite approach distance and x 2 /(2R) is the distance of the point on the circle to the rigid plan. In equation (1) d(x) represent only the elastic or plastic deformation of the asperity. If the y P approach is given, the macro geometry is known (the circle) and the distribution of the asperities could be y x ), the deformation d(x) is a determinate ( ( ) o function of distance y(x). The sample length for contact in multi-scale model was chosen 1.5 mm. The equivalent radius of curvature was R=20mm. The yield strength of the cylinder material was considered M =1300 N/mm 2. From this first order scale we are going to develop two contact models. The idea in this paper is to obtain a second or third order profile using a numerical procedure. This profile is used for contact between a rough profile and a rigid plane and is function of: a) the original form of the un-deformed profile; b) the roughness of the profile (density of asperities); c) the nature of the materials (oung modulus, E 1, 2, Poisson ratio ν 1, 2 ). In the first step of contact model we achieve a set of values of rough profile with asperities. This set includes positive values for asperities and negative values for valleys. With an original procedure realized in Java language we transform the original profile into a second order profile (fig. 2). The third order multi-scale profile is obtained using a six range polynomial interpolation of the previous model (.2). A detailed Finite Element model will be very difficult to manage will be very large and thus is highly impractical Fig. 2. Multi-scale contact profile models. Fig. 1. Contact deformation model. In a second order profile with find a profile which can be used for a non-linear contact, with elastic and plastic deformations. With a third order multi-scale profile it could be supposed that the contact model is

3 25 much appropriate to Hertz contact model. In this paper we intend to find the differences regarding the state of sub-surface stress field resulting in these two contact models. Using a FEM code a series of numerical simulations were conducted in order to determine the effect of the roughness profiles, the pressure distribution and the resulting subsurface stress field. RESULTS AND VALIDATION OF THE MODELS First of all we study the contact model using the third order multi-scale profile. Contact between this profile and a rigid plan is discontinuous, but the characteristics of the contact are the same as in Hertzian model if we take into account a wavelength. This wavelength can be described when, in the contact process, the stress and the engineering strain in the normal direction due to the contact between one asperity and the plan does not influence the behaviour of the contact for the neighbor asperity. The theory of Hertz remains the foundation of this contact model. The theory applies to normal contact between two elastic bodies that are smooth and can be described locally with orthogonal radii of curvature. For the validation of this model, a large region of the contact model was taken into account and was discretized with linear izoparametric elements. The bulk material in both domains was chosen in a generic analogy with linear elastic steel properties. In the first attempt to validation, the value of oung s modulus E, for both domains was the same, E= N/mm 2. In the next steps we increase the oung s modulus value for rigid plane to ten times. The entire model comprises 3057 elements including GAP master-slave contact elements. The contact pressure was calculated as a function of the applied normal load, without any tangential forces. The applied load line was 3857 n/mm and the resulting maximum Hertzian pressure was 1175 N/mm 2. The FE contact analysis model is shown in figure 3. Fig. 3. FE analysis for third order multi-scale profile contact. It is important to observe two things: first, the very good approximation of the third order profile used in FE analysis (see fig. 2); second, the results concerning the stress field distribution. We consider that every round asperity and the plane follows the Hertz theory assumptions. The results follow those predicted by the Hertzian theory. The location of the maximum shear stress is, with a very good approximation, at z=0.78bh, for every individual contact along wavelength, as we can see in figure 4. Even if we are going to work with a model with a dense distribution of asperities (a shorter wavelength), the results remain in the same domain. The results can be seen in figure 5 for a model with a density of five times than the one used in figure 4. Fig. 4. Stress field at the contact for third order multi-scale profile contact Fig. 5. Stress field at the contact for third order multiscale profile contact with a dense distribution. In the second step we study the contact model using the second order multi-scale profile (fig. 2). Contact between this profile and a rigid plan is discontinuous, and the characteristic of the contact are unknown. In this model the asperities are assumed to follow the elastic perfectly plastic deformation model. After the discretization of the second order profile (and the consequent region) we make the assumption that some initial nodes are in contact. A linear set of equations, which correlates the pressure with the deformation, is set up to be solved. The aim is to find the pressure values in the contact points. The pressures are determined by inverting the influence matrix and multiplying it by the deformation matrix. The pressure deformation model relationship follow the function pm d = f (2) σ d where pm is the mean pressure and will examinee for the contact of a given profile with a smooth plane.

4 26 THE ANNALS OF UNIVERSIT DUNĂREA DE JOS OF GALAŢI σ is uni-axial yield strength, dy is the deformation at onset of yielding. As we said, up to a certain value dy (a certain contact deformation) the materials deforms elastically according the Hertzian theory. From this value the plastic deformation starts. But in the inner Hertzian comportment we have analytical relations according to which [10] 2 d phm = Eech (3) 3π r when the maximum pressure ph0 reaches a critical value p which is calculated function of plasticity HO coefficient K and the yield limit of the material σ we are in the plastic deformations domain. p = K σ (4) HO Fig. 6. Plasticity coefficient. Regarding the values of coefficient K according to Chang [11] there is a dependence of materials properties. An empirical expression for this calculus gives a linear relation on material s Poisson ratio υ. For a common set of values (common steels) with values for Poisson s ratio υ from υ=0.27 to υ=0.33 the coefficient K varies from K=1.594 to K= Thus, we can consider a linear function K(υ). Regarding the equations (2), (3), (4) and figure 6 we can deduce for the plastic deformation domain: pho = 1.506σ (5) With this assumption we can compute the corresponding upper limit of the elastic deformation, d (eq. 2) Because our model supposes the deformation being only elastic or plastic, it may consider that when the deformation exceeds a larger value dp the material behaves as a plastic one. In this case the pressure is considered constant and according to [12] it could be supposed: pm = 2.8σ (6) In the multi-scale contact model, using the second order contact profile, a semi-infinite body subjected to pressure load along the discontinuous contact line is taken into account (fig. 7). The resulting deformation can be calculated by integrating equation (7): ( ν ) 2 lb 21 1 x d( x) = ln p() l d() l (7) π E 1 x τ la Fig. 7. Semi-infinite body subject to pressure. Equation (7) allow for the determination of the pressure distribution for a roughness profile if we know all the geometrical data and the material properties. According to elastic-perfectly plastic model, we consider the asperity will deform elastically to a pressure up to 2.8σ after that value we consider a perfectly plastic deformation. The resulting stress components could be computed using the Timoshenko [13] set of equations for normal and shear stress. The maximum shear stress τ max it is computed as follows: a) τ max 2 σx σz 2 = τ xz 2 + (8) b) Fig. 8. Subsurface stress field at the contact cylinder with roughness profile and rigid plane. In figure 8a and b there are shown some results; there is the subsurface stress field for contact with a profile corresponding to a standard deviation of a roughness height at 0.5 µm. Some of these asperities

5 27 deform elastically, but the subsurface stress field deviates from the Hertzian one. In figure 8b, the corresponding standard deviation of the profile model is 0.8 µm and the results show even more deviations from the Hertz theory. The location of the maximum shear stress shifts from that predicted by Hertz theory and is close to the surface. About 30% of the asperities are deformed plastically. It is obvious that roughness alters the subsurface stress field. DISCUSSION A multi-scale contact model was performed. The first model is the contact model using the third order multiscale profile. This model incorporates the Hertzian contact model. Contact between this profile and a rigid plan is discontinuous, but the characterristics of the contact are the same as in Hertzian model if we take into account a wavelength. The results show that every round asperity and the plane follow the Hertz theory assumptions. Using the second order model contact profile we were able to find that some contacting asperities are elastically deformed, some are plastically deformed. However the contact problem with no fixed contact region is nonlinear. The dimension of contact region is almost proportional to two thirds of normal load applied, but for high Hertzian pressure the changes are significant. In this case even the maximum shear stress change the position relative to the contact line and appears very close to the surface. REFERENCES 1. Creţu, S., 2001, Mecanica contactului, vol. I, Editura Gh.Asachi, Iasi. 2. Greenwood J.A., Williamson J.P.B., 1966, Contact of nominally flat surfaces, Proc. R. Soc. London Ser. A, 295, pp Greenwood J.A., Tripp J.H., 1970, The Contact of two nominally flat rough surfaces, Proc. Inst. Mech. Eng., 185, pp Bush A.W., Gibson R.D., Thomas T.R., 1975, The elastic contact of a rough surface, Wear 35, pp Whitehouse D.J., Archard J.F., 1970, The properties of random surfaces of significance in their contacts, Proc. R. Soc., London Ser. A, pp Tudor A., 1990, Contactul real al suprafetelor de frecare, Editura Academiei Romane. 7. Nayak P.R., 1971, Random process model of rough surfaces, ASME J. Lbr. Technol. 93, pp Chang W.R., Etsoin I., Bogy D.B., 1987, An elasticplastic model for the contact of rough surfaces, ASME J. Tribol. 110, pp Jackson, R.L., Streator, J.L., 2006, A multi-scale model for contact between rough surfaces, Wear, 261, pp Mihailidis A., Bakolas V., Drivakos N., 2001, Subsurface stress field of a dry line contact, Wear, 249, pp Chang W.R., 1986, Contact, Adhesion and Static Friction of Metallic Rough Surfaces, University of California, Berkely, CA. 12. Tabor D., 1951, The Hardness of Metals, Oxford University Press. 13. Timoshenco S., Goodier J.N., 1970, Theory of Elasticity, 3rd Edition, McGrow Hill.