1 Introduction When two bodies come into contact, their surfaces do not engage uniformly across the entire nominal (Hertzian) contact area. Instead, contact occurs at discrete asperities due to surface roughness. Consequently, the real contact area, which transmits the normal force F N from the base to the opposing body, consists of numerous micro-contact regions. The cumulative area of these micro-contacts is significantly smaller than the apparent Hertzian contact area. Due to the equilibrium of forces, the local force acting on each asperity is typically much higher than the equivalent force distributed over a fullsurface contact, as illustrated schematically in Figure 1.1. Analogous to the mean contact pressure, a real contact pressure can be defined as the ratio of the normal force to the sum of all micro-contact areas. This real contact pressure is generally much higher than the apparent pressure can be resolved at a local level. Due to the concentrated forces and small size of the micro-contact areas, Science and Research 5 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Real Contact Area and Pressure Distribution in Mixed Lubricated Rolling Contacts under Consideration of the Real Rheology J. Torben Terwey* submitted: 7.10.2024 accepted: 30.10.2025 (peer review) Presented at GfT Conference 2024 * Dr.-Ing. J. Torben Terwey ORCID: https: / / orcid.org/ 0009-0009-5038-6654 thyssenkrupp rothe erde Germany GmbH Beckumer Straße 87, 59555 Lippstadt, Germany Tribology in machine elements is a major challenge to achieving optimum performance and efficiency. It can be reduced to single component contact, such as between rolling elements and the raceway in a rolling bearing. Contrary to the intuitive notion of full-surface contact between these machine element components, real interactions are characterized by a much more complex interplay. Surface asperities, microscopic peaks and valleys on the contacting surfaces, create a limited “real” contact area comprised of numerous microcontact areas. These microcontact areas collectively transmit the contact force between the contacting components and are responsible for the friction in dry contacts. In practical applications, however, machine elements rarely operate in dry conditions. Therefore, the interplay between these microscopic asperity contacts is further enhanced by the presence of a surrounding lubricant film, introducing the complexities of mixed lubrication. In this regime, force transmission is shared between the asperity contacts and the lubricant reservoirs trapped between them. Lubricant viscosity and density, which are crucial for film formation, are themselves influenced by a variety of factors, including tem- Kurzfassung perature, pressure, and the relative speed of the contacting surfaces. The objective of this work is to obtain an accurate understanding of the real contact area and pressure distribution within these mixed lubricated contacts. A half-space based numerical contact model of the surface asperities is coupled to an analytic film formation model that considers the rheological properties of the lubricant under high pressure and shear conditions. Lubricant properties are derived from high pressure measurements. The application of the proposed model is demonstrated through a practical example: the contact between two discs within a dedicated test apparatus. This allows to validate the ability of the model to predict the real contact area and pressure distribution with a high degree of accuracy. This work was mainly carried out at Leibniz University Hannover. Keywords mixed lubrication, pressure distribution, rolling contacts, numerical model, elastohydrodynamic lubrication (EHL) faces and experimentally obtained surface topographies in contact scenarios involving a sphere and a flat surface. Theoretical and simulation-based approaches for incorporating surface roughness into the formation of pressure profiles in lubricated contacts have also been explored. P ATIR and C HENG (1978) [10] used the Reynolds equation to model fluid flow around asperities, showing how the geometry and distribution of asperities disrupt local flow and pressure formation. One advantage of their approach is the calculation of “flux factors” for representative surface elements, which can be applied across the entire contact surface. However, the approach lacks sufficient local resolution, yielding only integral pressure curves. Extensions of this method to elastohydrodynamic lubrication have been presented by T RIPP and H AMROCK (1985) [11], E FFENDI (1987) [12], Z HU and C HENG (1988) [13], and S ADEGHI and S UI (1989) [14]. Further studies into individual asperities and sinusoidal roughness structures were conducted by C HANG (1995) [15], with extensions to real surface profiles performed by J IANG et al. (1999) [16], H U and Z HU (2000) [17], and R EDLICH et al. (2000) [18]. These works aimed to quantify the area fractions of solid and fluid contacts, force transmission proportions, and resulting lubricant film thickness. W EINHAUER (1996) [19] calculated the frictional contributions of solid-state contacts based on the energetic deformation of asperities. A combination of the flow factor model of R EDLICH and C HENG and the solid-state contact model of G REENWOOD and W ILLIAMSON [1] was proposed by M AKINO et al. (1999) [20]. G ELINCK (1999) [21] employed stochastic models to calculate Stribeck curves, while further work based on flow factor models was conducted by E LROD (1979) [22], T EALE and L EBECK (1980) [23], T RIPP (1983) [24], and R IENÄCKER (1995) [25]. The objective of this study is to develop a deterministic model for calculating the real contact area and pressure distribution in mixed lubricated contacts, incorporating measured non-Newtonian and high-pressure fluid properties. This work was primarily carried out at Leibniz University Hannover, Germany, and published in T ER- WEY (2020) [26]. Science and Research 6 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 the local contact pressures increase significantly. These micro-contact pressures interact with the local deformations of the asperities, which, in turn, influence each other. Unlike the Hertzian contact problem, which can be solved analytically, the relationship between microcontact pressures and asperity deformations requires numerical methods for resolution. The real contact area and the resulting load distribution can be determined using either stochastic or deterministic approaches. For instance, the stochastic model developed by G REENWOOD and W ILLIAMSON (1966) [1] examined the effect of different surface profiles on the real contact area formation. A deterministic model by W EBSTER and S AYLES (1986) [2] incorporated real surface topographies to analyze load distribution and elastic deformation. Further deterministic models, such as those by B AILEY and S AYLES (1991) [3] or S AYLES (1996) [4], focused on calculating pressure distribution in rough contacts, with an emphasis on subsurface stress distributions and their impact on the service life of tribological contacts under pure rolling stress. R EN and L EE (1994) [5] explored rough contacts with a focus on mean gap height and micro-contact pressure, performing numerical studies on a range of artificially generated roughness profiles. G LEß (2009) [6] conducted further studies on pressure and stress distribution in rough rolling contacts. B RECHER et al. (2016) [7] introduced a method to calculate locally resolved contact pressure in large-area rough contacts, emphasizing the feasibility of performing such calculations with limited computational resources. Their method discretizes the contact surface into fine and coarse regions, allowing detailed analysis of small areas while maintaining computational efficiency. This Method of Combined Solutions was also used by T ERWEY et al. (2020) [8] to investigate the real contact surface size under dry and mixed friction conditions. Their study also simulated the effect of varying proportions of solidstate friction on adhesive wear. W ANG and S CHIPPER (2020) [9] examined real contact surfaces and pressure distributions in rough contacts using a semi-analytical elastoplastic half-space model. They analyzed both numerically generated rough sur- 𝐴 𝑖 𝐴 𝑖 𝐴 Ge ge nkörper Grundkörpe r 𝐴 0 𝐹 N 𝐴 𝑖 Base body Counter body Figure 1.1: Hertzian (A 0 ) vs. real (A i ) contact, formed by multiple micro-contacts. [26] 2. Methodology This study investigates the contact behavior between two discs in a two-disc test rig designed to measure frictional characteristics. The simplicity of the setup allows precise control and definition of key tribosystem parameters, including contact geometry, relative and entrainment velocities, and normal force. The formation of the lubrication film, which separates the contact surfaces, is described analytically considering the lubricant’s measured rheological properties. The test rig used in this study has been previously employed in the works of S CHMIDT (1985) [27], M EYER (2010) [28], W ANG (2015) [29], and B ADER (2018) [30]. A schematic of the rig’s structure is shown in Figure 2.1 (left). It consists of two independently driven shafts, each powered by separate electric motors. Mounted on each shaft is a test disc with a diameter of 120 mm. Disc 1 features a crowning radius of 100 mm. The rig enables rotational speeds of up to 2500 rpm, corresponding to a circumferential speed of 15 m/ s. Motor 2 is mounted on a foundation and can be adjusted horizontally and radially via a linear guide. This allows the application of force through a hydraulic cylinder, which can generate a normal load of up to 20 kN, resulting in a Hertzian contact pressure of up to 3000 MPa. The lubricant is externally heated and introduced directly into the contact area via a nozzle, with inlet temperature and flow rate controlled by a valve. The discs used in the tests were ground and polished. Photographs of the test discs are presented in Figure 2.1 (right). The surface condition of the discs is a critical parameter influencing the formation of the lubricating film and the extent of mechanical contact in the mixed friction regime. Consequently, the surfaces were examined using optical methods, and roughness was measured with a tactile surface tester using the tactile section method in both transverse and longitudinal directions. The roughness values measured were combined into a common roughness characteristic using the Euclidean norm (R q = (R 2q,1 + R 2q,2 ) 1/ 2 ). For Disc 1, the roughness R q in the transverse direction was 192 nm, and 108 nm in the longitudinal direction, yielding a combined roughness of 220 nm. Disc 2 exhibited a transverse roughness of 360 nm, and a longitudinal roughness of 25 nm, resulting in a combined roughness of 361 nm. These results are summarized in Figure 2.2 (top). A more detailed examination of the disc surfaces was performed using a Keyence VK-X100 laser microscope, which offers the advantage of capturing a three-dimensional map of the surface, improving the resolution and allowing for direct integration into numerical contact models. The lenses used have a magnification factor of 100, yielding in element widths of 0,139 µm. The scanned surfaces were approximately 570 µm x 428 µm = 0,244 mm 2 . The resolution in depth (z direction) is 0,1 nm. The measured profiles of the discs are illustrated in Figure 2.2 (bottom). Due to the manufacturing process, a wavy surface structure developed along the xaxis. For the calculation of the specific lubrication film thicknesses, the roughness parameters obtained through the tactile method were utilized in this study. These values provided the foundation for determining the surface characteristics relevant to lubrication. In the numerical contact simulations, the three-dimensional surface profiles captured by laser microscopy were incorporated to achieve a more detailed and accurate representation of the surface topography. The two-disc contact is lubricated with two test oils: a synthetic oil based on polyalkylene glycol (“Glycol”), and Fuchs Dexron Titan ATF 6000 SL (“Dexron”). Gly- Science and Research 7 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Figure 2.1: Left: Schematic of the test rig (Leibniz University Hanover) [28]. Right: Test discs. [26] Öle inla s s R60 R50/ R100 Sche ibe 1 Sche ibe 2 Krafts e nsor Hydraulikzylinde r Krafts e nsor Motor 1 Motor 2 Ge häus e Oil inlet Force sensor Hydraulic cylinder Force sensor Disc 2 R60 Disc 1 Housing R50/ R100 Science and Research 8 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Figure 2.2: Top: Quadratic mean roughness measured through tactile surface tester. Bottom: Three-dimensional surface profiles measured with laser microscope. [26] Figure 2.3: Top: Viscosities and Rodermund parameters of the test oils (measured at ITR). Bottom: Pressure and temperature dependence of the dynamic viscosities of the test oils: experiment and Rodermund model. [26] col is predominantly used as a hydraulic and circulating oil, exhibiting a dynamic viscosity of 44.2 mPas at 40 °C and 9.1 mPas at 100 °C, with a viscosity index of 194. Dexron is commonly applied in automatic transmissions, having a viscosity of 22.5 mPas at 40 °C and 5.2 mPas at 100 °C, with a viscosity index of 173. Both serve as bases for various lubricant formulations. To characterize the temperature dependence of both oils’ viscosities, measurements were conducted at ITR TU Clausthal across a range of temperatures. From these measurements, the parameters for the Rodermund equation [31] were derived, and the results are shown in Figure 2.3 (top). Figure 2.3 (bottom) presents a logarithmic plot of viscosity versus temperature for both oils. In this figure, the solid lines correspond to the Rodermund equation approximation, while the plotted diamonds indicate the experimentally measured values. The dotted lines represent for a comparison the Vogel-Barus equation results. 3 Simulation Model 3.1 Dry Contact The simulation of locally resolved contact pressure is performed using the theory of the elasto-plastic halfspace. This theory assumes an infinitely extended body in two spatial directions, implying an ideally smooth surface for the half-space. To account for the actual measured rough surface profiles of both contact partners, these profiles are superimposed to the flat half space, synthesizing a generic combined surface. Despite the inherent inaccuracy caused by the random spatial orientation of the two roughness profiles relative to each other, this study assumes that the resulting combined surface reasonably approximates the contact behavior of the real surfaces. The numerical determination of the locally resolved pressure for large contact surfaces, while considering surface roughness, is highly computationally demanding due to the large number of elements in the influence matrix, which grows quadratically with the number of surface elements. In this work, the Method of Combined Solutions (MOCS) [7] was employed to address this challenge. The implementation of the contact pressure calculation was carried out in MATLAB, following the algorithm outlined in the program flowchart shown in Figure 3.1 (top). The process begins with the input of contact geometries (1), material parameters (2), and external loads (3). The mesh is then defined (4), with the total number of elements determined by the size of the Hertzian contact area and the element width d. The iterative parameters Ω 1 and Ω h are prescribed for the calculation of deformation and pressure based on the elasto-plastic half-space model, according to [32]. After calculating additional parameters for the mesh (5) and the tribosystem (6), the contact area (7) and the Hertzian pressure distribution are analytically determined (8). In order to apply the Method of Combined Solutions, different irregular influence matrices C irr S are required for each calculation step s, which would otherwise require extensive computation. Therefore, following the approach of B RECHER et al. (2016) [7], an irregular global influence matrix C irr ges is calculated once at step (9) of the flow chart. This matrix contains a finely discretized element in the center and is larger than a simple irregular influence matrix. However, it only needs to be computed once and contains all necessary irregular influence matrices, which can be accessed via simple index shifts for the subsequent steps. Once the irregular global influence matrix is determined, the influence matrix for the coarse mesh is calculated (10), followed by the calculation of an initial deformation w H (11). Before starting the iteration loop (16), initial values for the Hertzian pressure, residual, Hertzian solid overlap, and height distribution are set (12-13). The iteration loop for deformation and pressure calculation (16) is repeated n max times (14-15), where n max is chosen to minimize the relative error ϵ. Each iteration step n begins by setting (*) n -1 = (*), where n -1 denotes the previous iteration step (or the initial condition for n = 0). The placeholder (*) represents the pressure p, residual r, gap height h, or solid overlap h 0 . The pressure change Δp from step n - 1 to n is calculated using the residual, parameter Ω 1 , and the matrix C, following [7]. Numerically calculated negative pressures are set to zero according to the Hertz-Signorini-Moreau condition. The yield criterion is satisfied by limiting the pressure to a predefined maximum value. The pressure for step n is the sum of the pressure in step n -1 and the calculated pressure change. The iteration is controlled by adjusting the solid overlap, with the adjustment based on comparing the force derived from the pressure distribution with the applied normal force. For each pressure step n, the deformation and gap height distribution are recalculated. Any non-physical gap heights h i < 0 are set to zero (also under the Hertz-Signorini-Moreau condition), and the residual is calculated for step n. Finally, the values (*) = (*) n are set as initial conditions for the next iteration step n + 1. The deformation and pressure calculations within step (16) are executed in a subprogram where the Method of Combined Solutions is implemented. The corresponding program flowchart is schematically illustrated in Figure 3.1 (bottom). For each iteration step n, the entire contact area is scanned in Z el = M̃ · Ñ steps (a). In each step s ≤ Z el , the corresponding irregular influence matrix is read from the irregular global influence matrix (b). The displacement is then determined using the half-space model for irregular grids (c), with the finely discretized section being written out and temporarily stored (d). Once the partial displacements for each step s are calculated, they are assembled into the total displacement (e), resulting in the final pressure and deformation fields. Science and Research 9 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Science and Research 10 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Figure 3.1: Flowchart. Top: Calculation of contact pressure (dry contact). Bottom: Sub-model, calculation of deformation and pressure (MOCS). [26] ENDE 𝑛 ≤ 𝑛 max 𝑛 = 1 ℎ 0 = 𝑤 H,𝑖 = ge + H + ℎ 0 = H = H Le s e Werte a us Schritt 𝑛 − 1 𝑛− = 𝑛− = 𝑛− = ℎ 0 𝑛 − = ℎ 0 𝑛 = 𝑛− + Δ Be re chne Pre ssung für Schritt 𝑛 ℎ 0 𝑛 = ℎ 0 𝑛 − − 𝐹 − 𝐹 / 𝐹 Be re chne Einfe de rung für Schritt 𝑛 Ände rung und Korre ktur Δ = 𝑛 − / 𝐶 𝑛− 𝑝 𝑖 < 0 = 0 𝑛− 𝑝 𝑖 𝑝 𝑖 = 𝑝 𝑖 Be re chne Kräftegleichgewicht 𝐹 = 𝑝 𝑖 𝑛 𝑖 Be re chne Ve rformung für Schritt 𝑛 𝑛 ℎ 𝑖 < 0 = 0 𝑛 = + 𝑛 + ℎ 0 𝑛 Be re chne Spalthöhe für Schritt 𝑛 𝑛 = − + 𝑛 + ℎ 0 𝑛 Be re chne Re siduum für Schritt 𝑛 Se tze W erte aus Schritt 𝑛 = 𝑛 = 𝑛 = 𝑛 ℎ 0 = ℎ 0 𝑛 𝑛 = 𝑛 + 1 Ve rformungs- und Pre s sungsberechnung MoCS Irre guläre Ges amt- Einflus smatrix be rechnen START Kontaktge ometrie e inle s en Be la stung e inle sen Gitte rde finition e inle s en Triboparameter be re chnen Konta ktflä che be re chnen H ERT Z ‘s che Pressungsverteilung be re chnen Ne tzparameter be re chnen H = g H Ma te ria lpa rameter e inle s en Grobe Einflus sma trix be re chnen (1) (2) (3) (4) (7) (5) (6) (8) (9) (10) (12) (13) (14) (15) (16) (11) Read contact geometry Read material parameter Read loads Read grid definition Calculate grid parameters Calculate tribo parameters Calculate contact area Calculate Hertzian pressure distribution Calculate irregular influence matrix Calculate coarse influence matrix Read parameters from step n-1 Change and correction Calculate pressure for step n Calculate equilibrium of forces Calculate deflection for step n Calculate deformation for step n Calculate gap heigth for step n Calculate residuum for step n Set parameters for step n i = i i i i ges i ≤ 𝑍 el 𝑠 = 1 𝑠 = 𝑠 + 1 ja ne in Ve rformungs- und Pre s sungsberechnung (MOCS) 𝑛 = (a ) (b) (c) (d) (e ) Calculation of deformation and pressure yes no 3.2 Lubricated Contact In lubricated contacts, the normal force acting on the tribosystem is partially absorbed by the pressure generated within the lubricating film. As the film thickness increases, the proportion of the force transmitted through the lubricating film also increases, since fewer asperities make direct contact. When the film becomes thick enough that no asperities come into direct contact, the lubricant alone transmits the load between the two bodies, signifying the regime of full lubrication. However, it is assumed that asperities continue to interact with each other through the lubricant, even when they are not in direct contact. A complete separation of the surfaces, without asperity interaction, is typically assumed to occur at a lubrication film thickness ratio of approximately λ ≥ 3 [33]. Various researchers have proposed ranges for this transition between liquid and mixed friction. H AMROCK et al. (2004) [34] suggest a range of λ = 3 … 5, while L UDEMA (1984) [35] identifies a range of λ = 0.25 … 2.5. G UANGTENG and S PIKES (1997) [36] report a transition at λ = 2, and B ARTEL (2000) [37], based on numerical calculations, proposes a range of λ = 1.95 … 4.09. The calculation process for contact in the mixed friction regime is based on the flowchart for dry contact, with some modifications. In contrast to step (16), the gap height distribution is not iteratively recalculated in each step but is instead coupled with the analytically computed central lubrication film thickness h c according to H AMROCK and D OWSON (1981) [38]. The temperature of the lubricating film is determined by the oil inlet temperature. In each computational step, the arithmetic mean of the gap height distribution (i.e., the mean h¯ of the deformed surface profile) is compared to the analytically calculated film thickness. If the calculated mean gap height h¯ is smaller than the central film thickness h c , a lower deflection is applied in the next iteration step. Conversely, if h¯ is larger, a greater deflection is specified. At the end of the calculation, the sum of the locally acting forces can be used to determine the proportion of the load carried by asperity contact in relation to the total system force (Figure 3.2). Similarly, the area fraction of contacting asperities can be calculated. The hydrodynamic pressure between contacting asperities follows a qualitative pattern similar to Hertzian pressure distribution, but the maximum value corrected from Hertz’s prediction. This maximum value is determined iteratively so that the integrated hydrodynamic pressure over the contact area corresponds to the total force transmitted by the lubricant. If the analytically calculated lubrication film thickness is less than the gap between the deformed surface profiles in dry contact, the lubricant does not contribute to load transmission. 4. Results 4.1 Test conditions The contact pressure can be varied by adjusting the radial load. The mean (Hertzian) contact pressures considered in this work are 850 (1275), 1000 (1500), and 1250 (1875) MPa. The oil supplied to the tribological contact is preheated before being introduced into the test chamber. The target temperature is varied between 40, 60, and 80 °C. The entrainment velocity is varied between 5, 10, and 15 m/ s, resulting in 27 different parameter combinations, each yielding different central lubricant film thicknesses. The specific lubricant film thickness is λ = h c / R q . Since all experiments were conducted with two different oils (Glycol and Dexron), a total of 54 experiments were performed, along with 3 additional dry tests. An overview of all experiments, along with the corresponding specific film thicknesses, is provided in Figure 4.1 (top). Values in the region 2 < λ ≤ 3 are highlighted in light gray, while values in the region λ < 2 are marked in dark gray. Figure 4.1 (bottom) provides a graphical representation of the central specific lubricant film thicknesses as a function of temperature. Values for a combined velocity of 15 m/ s are shown in red, those for 10 m/ s in green, and those for 5 m/ s in blue. Solid lines represent a mean pressure of 850 MPa, dashed lines correspond to 1000 MPa, and dotted lines represent 1250 MPa. The entrainment velocity and the oil inlet temperature have a significantly greater influence on the lubricant film thickness than the contact pressure. On the other hand, within the considered range, pressure variation did not significantly impact the lubrication regime for any velocity-temperature combination. Science and Research 11 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 ℎ c = 2.69 𝑅 𝑈 0.67 𝐺 0.53 𝑊 p−0.067 (1 − 0.61 −0.73𝜅 ) ℎ̅ = 1 𝑍 el ∫ ℎ 𝑖 𝑖 𝐴 𝑖 𝐴 𝑖 𝐴 Ge ge nkörper Grundkörpe r 𝐴 𝑖 ℎ c = ℎ̅ Base body Counter body Figure 3.2: Center plane of the deformed surfaces as the central lubricating film height. [26] pressure distribution for smooth surfaces, as predicted by Hertz’s theory, providing a basis for comparison. It is evident that a significant number of individual roughness asperities exceed the limiting pressure, resulting in the upper portion of the pressure profile being cut off. The ratio of the limiting pressure to the Hertzian pressure is approximately p lim / p Hertz ≈ 3.8, highlighting the localized increase in pressure due to surface roughness. If all locally acting solid-state forces (f i FK ) are summed up for the dry contact, this total sum is identical to the normal force F N applied to the system from the outside. In lubricated contact scenarios (mixed friction), the force transmission is facilitated by a pressurized lubricating film. By specifying a corresponding separation distance between the mid-planes of the roughness profiles, the calculation based on the halfspace model yields a cumulative force significantly lower than the externally applied normal force indicating a reduction in the contribution of force transmitted through solid-state contacts. In this mixed friction scenario, the solid-state force accounts for only 8.8 % of the total force transmission. The difference between the externally applied normal force and the sum of the solid-state forces corresponds precisely to the force transmitted by the lubricating film, which can be calculated accordingly. Science and Research 12 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 4.2 Calculated contact pressure and contact area distributions Contact pressure for dry contact and exemplary Glycol mixed lubrication regime In Figure 4.2 (left), the calculated gap height distribution between the combined deformed surface (black pattern) and the reference half-space plane (zero line) for a dry contact with F = 6850 N respectively p = 1250 MPa is presented in a side view. The gap height g (in the zdirection) is plotted on the ordinate in micrometers (µm), while the abscissa (in the x-direction) corresponds to the long semi-axis of the Hertzian contact ellipse. The data on the abscissa is limited to the Hertzian contact length a Hertz . Figure 4.2 (right) depicts the same load scenario but includes the effect of an elastohydrodynamic lubricating film for Glycol (p = 1250 MPa, T = 60 °C, v = 5 m/ s). The horizontal dotted blue centerline of the deformed profile aligns with the calculated film height, illustrating that the asperities are more widely separated compared to a), thereby reducing the proportion of force transmitted through direct solid contacts. Figure 4.3 represents the contact pressures normalized to Hertzian pressure. The red dashed lines indicate the Figure 4.1: Test matrix (specific film thicknesses). [26] / MPa Glycol Dexron 40 °C 60 °C 80 °C 40 °C 60 °C 80 °C 15 850 4.39 2.72 1.98 3.11 2.04 1.48 1000 4.25 2.63 1.92 3.01 1.97 1.44 1250 4.06 2.51 1.83 2.88 1.89 1.37 10 850 3.34 2.07 1.51 2.37 1.55 1.13 1000 3.24 2.00 1.46 2.29 1.51 1.09 1250 3.09 1.91 1.39 2.20 1.44 1.04 5 850 2.10 1.30 0.94 1.49 0.97 0.71 1000 2.03 1.26 0.91 1.44 0.94 0.68 1250 1.94 1.20 0.87 1.38 0.90 0.65 Temperature T Spec. Film Thickness Temperature T Spec. Film Thickness Contact area In addition to analyzing the force contributions from solid-state and EHL contacts, evaluating the contact area distribution may also be of interest. This can be achieved by visualizing the contact pressure from a top-down perspective. In Figure 4.4, the contact zones for both dry and lubricated conditions are depicted. The abscissa represents the x-direction along the major semi-axis, while the ordinate corresponds to the y-direction along the mi- Science and Research 13 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Figure 4.2: Simulated gap height. Left: dry contact. Right: Mixed lubricated contact (Glycol). (Glycol, T = 60 °C, v = 5 m/ s). [26] a) b) c) d) Figure 4.3: Simulated pressure distribution. 1250 MPa, dry and mixed lubricated contact (Glycol, T = 60 °C, v = 5 m/ s). [26] mixed friction, with an area fraction of 0.01 %, indicating that only a few individual roughness asperities remain in mechanical contact. 4.3 Classification of results into friction regimes In Figure 4.5 the force and surface area proportions for 57 calculations are summarized as a function of the specific film thickness λ. The applied pressure is represented by different symbols: “o” for 850 MPa, “+” for 1000 MPa, and “x” for 1250 MPa. The subdivision of the tribosystem into different friction states based on the specific film thickness, as already discussed, is also reflected in the analysis of the results presented here. For example, the specific film thicknesses at the boundaryto-mixed friction transition are λ = 0.35 / 0.27 / 0.20 for pressures of p = 850 / 1000 / 1250 MPa, respectively. When the analytically determined specific film thickness Science and Research 14 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 nor semi-axis. Both axes are normalized to their respective Hertzian semi-axes. It is evident that, in the case of dry contact, nearly the entire Hertzian contact area is engaged (as indicated by the red dotted line). Additionally, the cyclically repetitive surface waviness is also apparent in the contact pressure pattern. The proportion of the contact area, that is in mechanical contact relative to the total Hertzian contact area, is determined based on the number of contacting roughness asperities for dry contact. In the case of mixed friction, some roughness asperities partially lift off from each other, leading to a reduction in the number of contacting roughness asperities. As a result, the proportion of solid-state contacts relative to the Hertzian contact surface decreases. In addition to the results at a temperature of 60 °C (area fraction: 18.76 %), the area fractions for a lower temperature of 40 °C (0.01 %) and a higher temperature of 80 °C (39.64 %) are also shown. The case at 40 °C is mathematically close to a transition away from significant Temperature T Area share Figure 4.4: Contact surface. Dry contact and contacts in mixed friction (Glycol, T = 60 °C, v = 5 m/ s, p = 1250 MPa. [26] Area Share Spec. Film Thickness Area share Load Share Spec. Film Thickness Load share Figure 4.5: Area (left) and force (right) fraction of solid contacts. The entrainment velocity is represented by the color (blue: 5 m/ s; green: 10 m/ s; red: 15 m/ s). [26] is below these values, the surfaces do not approach each other further, indicating that the system has reached the boundary friction state, where the force and area contributions to load transmission resemble those of dry contact. In Figure 4.5, the regions corresponding to different friction states are shaded as follows: dark gray represents the area of boundary friction, medium gray denotes the mixed friction zone, and light gray highlights the transition zone. As the specific lubricating film height increases, both the force and surface area ratio decline significantly. The end of the true mixed friction regime is distinctly observed at λ=2, beyond which no roughness asperities remain in mechanical contact. Beyond this value, no further roughness asperities make mechanical contact, marking the end of true mixed friction. However, due to assumed continuing interactions between the surface structures via the lubricating film, additional microdeformations of the asperities and the formation of micro-EHL contacts may still occur. Consequently, the actual lubricating film in a real system tends to be thicker than what is predicted by analytical models (which assume smooth contact surfaces). This is accounted for by defining a distinct transition zone in the friction regime classification. Thus, the friction regime based on this study is classified as follows: • 0 < λ ≤ 0.35 boundary friction, • 0.35 < λ ≤ 2 real mixed friction, • 2 < λ ≤ 3 transition zone, • λ > 3 pure fluid friction. 4.4 Analysis of the results The Influence of yield stress The choice of yield stress for the elasto-plastic half space plays a crucial role in influencing the simulation results. In this study, the value of p lim = 7.13 GPa was chosen, which corresponds to a Vickers hardness of 674 HV and a Rockwell hardness of 59 HRC. The conversion from surface hardness to applied pressure follows the approach where the ratio of the force applied during the Vickers hardness test to the residual impression area due to plastic deformation is equivalent to a pressure. To evaluate the effect of different yield stress assumptions on the simulation results, the pressure distribution was simulated for a selected load case under varying ex- Science and Research 15 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 Area share (dry) Figure 4.6: Top: Influence of the yield stress on the surface fraction (850 MPa). Bottom: Area fraction of solid-state contacts as a function of the yield stress. [26] The influence of mesh fineness In addition to the yield stress, the choice of the discrete element width - and consequently the mesh fineness - has a significant impact on the simulation results. In this work, an element width of 2 µm was selected to balance the need for accurate representation of roughness asperities in the micrometer range while avoiding excessive computational demands. To assess the influence of mesh fineness, simulations were conducted for element widths ranging from 0.8 µm to 10 µm on a selected case, and the results were compared based on the area fraction of solid-state contacts. To make the simulations computationally efficient, especially for the smallest element width of 0.8 µm, the analysis was simplified by considering contact between two similar discs instead of the previously modeled two-disc test rig. Both discs were provided with a profiling of 50 mm, which significantly reduced the contact area and made the calculation feasible. The results of these calculations are shown in Figure 4.7 (left). In Figure 4.7 (right), all results are summarized as a function of the element width. Generally, there is a noticeable trend where the area fraction of solid-state contacts increases as the element width increases. A significant jump in results is observed at the step of d = 1.8 μm. The range of results spans from 27.92 % (at d = 1.8 μm) to 50.41 % (at d = 4.5 μm), nearly doubling the area fraction over this range of element widths. The mesh fineness has a significant impact on the simulation results and must be carefully considered when interpreting the outcomes. Variations in element width can lead to substantial differences Science and Research 16 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 DOI 10.24053/ TuS-2025-0025 trusion conditions. Specifically, the contact was modeled at an mean pressure of 850 MPa and for yield stresses of 1000 MPa, along with 1, 2, and 4 times the yield value, as well as without any specification (assuming purely elastic deformation). The results of the contact area formation for these scenarios are illustrated in Figure 4.6 (top). The graphs depict the calculated contact areas for rough surfaces in black, and the corresponding Hertzian contact areas are indicated by a red dashed line. In the case of an yield stress of 1000 MPa, the contact area is larger than the corresponding Hertzian contact area by a factor of 1.64. There are no gaps within the contact surface. Since the yield stress is lower than the mean pressure, a larger contact area is required to sustain the applied force. However, when the extrusion is increased to 2 GPa, the contact area becomes smaller than the corresponding Hertzian contact area (reaching 85.35 % of the Hertzian contact surface). Although some regions outside the Hertzian contact surface remain engaged, gaps start forming within the contact area. As the yield stress increases further, these gaps expand, reducing the solid-state contact area even more. A further increase in yield stress towards infinity shows only marginal changes in the area fraction of solid-state contacts. Figure 4.6 (bottom) plots the results, showing the area fraction of solid contacts as a function of the yield stress. The yield stress has a significant impact on the obtained simulation results. However, for materials typically used in tribologically loaded machine elements, it can be assumed that a yield stress of at least 4000 MPa is consistently applicable. As shown, the deviation in the results for the present case is relatively minor for a yield stress of 4000 MPa. Area share (dry) Element width d Area share y/ b Hertz y/ b Hertz y/ b Hertz x/ a Hertz x/ a Hertz x/ a Hertz x/ a Hertz 0,80 μm 1,00 μm 1,30 μm 1,50 μm 1,80 μm 2,00 μm 2,50 μm 3,00 μm 3,50 μm 4,00 μm 4,50 μm 10,00 μm Figure 4.7: Left: Influence of mesh fineness on the area fraction (850 MPa). Right: Area fraction of solid-state contacts as a function of element width. [26] in the calculated area fractions of solid-state contacts, highlighting the importance of selecting an appropriate mesh resolution for accurate analysis. 5 Conclusion and outlook This study has developed a deterministic model to calculate the real pressure distribution in mixed lubricated contacts, incorporating measured high-pressure fluid properties. The results demonstrate that the proportion of load carried by the lubricant film increases significantly with film thickness, reducing the number of asperities in direct contact. As the specific film thickness increases, the tribosystem transitions through different friction regimes - from boundary to mixed, and ultimately to fluid friction as the asperities lose mechanical contact. The findings highlight the critical role of lubricant viscosity and film thickness in determining the load distribution and contact mechanics within the system, which is relevant for subsequent analyses like friction and wear determination. 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