Elastohydrodynamic lubrication of compliant circular contacts near glass-transition temperature

Citation: Krupka, J.; Dockal, K.; Krupka, I.; Hartl, M. Elastohydrodynamic Lubrication of Compliant Circular Contacts near Glass-Transition Temperature. Lubricants 2022,10, 155. https:// doi.org/10.3390/lubricants10070155 Received: 14 March 2022 Accepted: 9 May 2022 Published: 13 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). lubricants Article Elastohydrodynamic Lubrication of Compliant Circular Contacts near Glass-Transition Temperature Jiri Krupka * , Krystof Dockal , Ivan Krupka and Martin Hartl Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic; [email protected] (K.D.); [email protected].cz (I.K.); [email protected].cz (M.H.) *Correspondence: jiri.kr[email protected] Abstract: Lubrication of polymer materials nowadays represents a subject of interest in many engineering applications, such as bearings or gears, to utilize them in the areas where conventional metal materials have so far dominated. However, material properties of polymers are strongly dependent on temperature that delimits a lubrication process and leads to manifestations of viscoelastic behavior of polymers. An understanding of mechanisms, which are responsible for formation of film thickness near the glass-transition temperature, is necessary to prevent initialization of failure modes and to increase the durability of polymer engineering components. Optical chromatic interferometry was applied to investigate development of film thickness and changes in contact geometry of compliant circular contacts operated in the elastohydrodynamic lubrication regime (EHL). Film thickness was compared with soft EHL prediction models, differences in contact geometry were assessed and their contribution to film thickness development were evaluated. Qualitatively good agreement of experimental results of central film thickness and soft EHL predictions was observed; however, minimum film thickness shows significant discrepancies. Outcomes and findings confirm the operation of the compliant circular contact in Isoviscous-elastic regime of EHL and the main influence of temperature and load to thermomechanical response of amorphous polymer PMMA. Keywords: elastohydrodynamic lubrication; Isoviscous-elastic regime; compliant circular contact; film thickness; glass-transition temperature; amorphous polymer; optical chromatic interferometry 1. Introduction The last several decades have witnessed a fundamental evolution of plastic materials in many applications and the former materials, such as steel, non-ferrous metals, glass, paper, or wood, were consecutively replaced mostly by plastics. The main development of plastics started after WW2 due to research performed in defense-industry sector in worldwide scale. One of the main efforts was to expand the use of thermoplastics—polymers in engineering applications, especially in military and automotive sector where, at that time, metals dominated. In the last 30 years, many new engineering polymers with improved material properties were developed [ 1 ]; thus, the process of replacement of metal machine engineering components, e.g., gears, bearings, or cams, was accelerated. Polymer gears can work under lubricated conditions [ 2 , 3 ] and dry conditions [ 4 , 5 ] without presence of liquid or grease lubricant. For polymer gears, there is a current effort to expand their use in high performance transmitting mechanisms for torque transfer that was not possible in the past. This was related to their weak mechanical and tribological properties [ 6 – 8 ] and manufacturing accuracy in comparison with steel gears. Due to this, the presence of lubricant is especially important, and its effect can influence the durability of polymer gears. At present, the advantages of polymer gears include low weight, low costs, resistance to wear, resistance to chemicals and oils, and the ability to damp vibrations and impacts. Lubricants 2022,10, 155. https://doi.org/10.3390/lubricants10070155 https://www.mdpi.com/journal/lubricants Lubricants 2022,10, 155 2 of 18 On the other hand, the main disadvantages of polymer gears are low mechanical properties which are strongly dependent on temperature, load, and frequency of loading in comparison with their steel equivalents, as was demonstrated in studies [ 9 – 12 ]. Another disadvantage may be a certain ecological burden on the environment, but the society-wide benefits of polymer gears are indisputable. A low level of mechanical properties, especially of Young’s modulus E, allows significant deformations of non-conformal contact surfaces. Increase of temperature and load W leads to the enlargement of the area of contact A. Deformation of contact surfaces can lead up to plastic deformation connected with the creep of the polymer material [6,7]. Failure modes represent a condition in which a machine component is unable to safely perform its function. For polymer gears, two failure modes are typical; the first one is plastic deformation of teeth due to overloading. The second one is plastic deformation of teeth as a result of melting wear, typically after the temperature reaches or exceeds the glass-transition temperature T g of polymer [ 10 , 11 ]. In dependence on operation conditions, these phenomena can occur together. In accordance with this, it is necessary to ensure operation conditions in full-film lubrication, which helps to dissipate heat from the contact region resulting from contact pressure and shear stress between surfaces; thus, it reduces friction and a probability of initialization of failure modes [6,7]. In the case where the film thickness of the lubricant ensures a full separation of contact surfaces, and the main role is played by hydrodynamics of the liquid, elastic deformation of contact surfaces, and the change in dynamic viscosity η with contact pressure, the lubrication regime is described as elastohydrodynamic lubrication—EHL [ 13 – 15 ]. This regime is suitable especially for operation of steel machine components such as gears and cams [ 16 ]. In relation to low mechanical properties of polymers, the lubrication regime is specifically called the Isoviscous-elastic regime of EHL or also “soft” EHL, as was stated by Johnson [13]. The Isoviscous-elastic (I-E) regime occurs between non-conformal surfaces where the contact pressure is sufficient to induce the elastic deformation of contact, but not to induce a considerable change in the dynamic viscosity η of lubricant [ 13 , 16 – 20 ]. This is different from the Piezoviscous-elastic regime (P-E) of EHL, where this change is significant [ 21 – 23 ]. Distribution of elastohydrodynamic contact pressure include the center region as well as the inlet, and outlet regions where pressure deviates from the Hertzian elliptical profile with maximum pressure p 0 [ 16 , 22 ]. Based on the results in [ 15 , 18 ], the transition region between these regimes also represents the operation area of polymers lubricated by mineral oils [ 13 ]; however, the relevant knowledge in the theoretical [ 13 , 21 ] and experimental field is still missing. In the field of soft EHL lubrication, many articles focused on numerical solution [ 13 , 15 – 20 , 22 , 24 , 25 ] of central h c and minimum h m film thickness as well as on elastohydrodynamic contact pressure distribution, often compared with Hertzian theory [16,22,26] , have been published. Hamrock and Dowson [ 19 ], and, later, Hooke [ 20 ], compiled primary equations for prediction of film thickness in soft EHL contacts. The most important parameters have been identified and described by the power exponents of dimensionless parameters U , W , and G [ 19 , 27 ]. Other prediction models based on numerical solution [ 15 , 21 ] and experimental approaches [ 26 , 28 ] were developed as well. The experimental approach most commonly includes two optical detection methods as was demonstrated by Myant et al. [ 29 ] where a laser induced fluorescence method (LIF) was applied. The other method is the optical chromatic interferometry that was also successfully implemented for detection of film thickness in soft contacts by Marx et al. [28]. A necessary condition for the use of both methods is the application of at least one transparent material (e.g., optical glass) that allows for the light to enter the area of contact and thus a detection of film thickness of lubricant. The method of optical chromatic interferometry [ 30 ] is more precise than the fluorescence method [ 29 , 31 ] and optical monochromatic interferometry [ 32 , 33 ], but it is limited by a measurable range of thicknesses (approx. up to 800 nm) and necessity of high reflectivity of surface as was demonstrated by Hartl et al. [ 30 ]. Lubricants 2022,10, 155 3 of 18 This can be ensured by applying coatings onto the surface of polymer that provide the desired properties. Otherwise, it is possible to use the method of fluorescence or optical monochromatic interferometry where the limit for detection of film thickness is up to several micrometers. The application of optical methods is limited by material properties of polymers. Light transmittance, and optical smoothness of surface are essential requirements. This is not the case of polymers, such as PA 66, POM, or PEEK, commonly used to design gears [ 2 , 10 ]. The internal structure of these polymers is semi-crystalline or even crystalline that transmits only part of the light or no light, respectively [ 34 ]. On the other hand, amorphous polymers dispose of excellent light transmittance; nevertheless, many of them are significantly different in respect to mechanical properties in comparison with semicrystalline polymers [ 34 ]. However, materials such as PC and PMMA represent a group of amorphous polymers [ 35 – 38 ] which dispose of a high light transmittance and simultaneously provide similar mechanical properties as above-listed engineering polymers for design of gears [2,10]. What is common to all mentioned polymers is their low heat resistance which restricts their operation range. Glass-transition temperature T g is usually given as the temperature interval beyond which the phenomena of melting wear are initiated due to rearrangement of atoms in the internal polymer structure as result of viscoelastic or plastic deformation of polymer material [2,10,11]. Interval of glass-transition temperature for PMMA was stated by Ali et al. [ 38 ] as T g∈ (110, 120 ◦ C), and by Mathiesen [ 39 ] as T g∈ (105, 110 ◦ C). However, the question remains how the lubricated contact can change in size near the T g and how this can influence the entire lubrication process. The main aim of this paper is to obtain new knowledge in the field of lubrication of soft EHL circular contacts near the glass-transition temperature T g where the experiments will be performed. Engineering polymers commonly used for the design of gears will be substituted by the amorphous polymer PMMA in respect to the purposes of experiments. The present paper is focused on development of central and minimum film thickness during the lubrication process and on monitoring the changes of contact geometry using optical chromatic interferometry and a rotary tribometer in the ball-on-disc configuration. 2. Materials and Methods 2.1. Material of Experimental Specimens The main requirements for the polymer material were light transmittance over 90% and similar mechanical properties, especially of Young’s modulus Ecompared to PA 66, POM, or PEEK semi-crystalline polymers [ 2 , 10 ]. Both requirements are fulfilled by amorphous polymer PMMA. We used a specimen of 12 mm thick PMMA of disc-shape with semitransparent thin chromium layer vapor-deposited on the bottom side and the antireflexive layer on the top side of the disc which allows for optical chromatic interferometry. A more detailed description is given in subchapter 2.4. A counterpart is a bearing steel ball 100Cr6 (AISI 52100) of 25.4 mm (1 0 inch) in diameter and its material characteristics are described at ambient temperature (24 ◦ C) in Table 1, similarly to the first specimen. With regard to [38,39], the value of Tgis considered as 105 ◦C. Table 1. Material properties of specimens. Specimens Specification Young’s Modulus, EPoisson’s Ratio, υGlass-Transition Temperature, TgRMS Roughness, Rq PMMA disc Polymethylmethacrylate 3.3 GPa 0.39 105 ◦CRq< 0.005 µm Steel ball 100Cr6 206 GPa 0.30 - Rq< 0.01 µm Lubricants 2022,10, 155 4 of 18 2.2. Lubricant Properties The lubricant used during experiments was FVA3 [ 40 , 41 ]—reference mineral oil corresponding to the viscous index ISO VG 100 (SAE 30). The viscous index of FVA3 corresponds to lubricants commonly used for, e.g., heavily loaded gearboxes, air compressors and hydraulic systems, etc. FVA3 lubricant was supplied by Gear Research center (FZG) in Germany, and its properties are described in Table 2according to [ 41 ]. The term of FVA3 as “reference” lubricant is connected to its longtime use within Research Association for Drive Technology—Forschungsvereinigung Antriebstechnik in Germany. Table 2. FVA3 lubricant properties [41]. Oil Kinematic Viscosity at 40 ◦C, υ40 Kinematic Viscosity at 100 ◦C, υ100 Density at 15 ◦C, ρ15 FVA3 95.0 mm2/s 10.7 mm2/s 864 kg/m3 2.3. Experimental Apparatus Experiments were conducted on a rotary optical tribometer in ball-on-disc configuration, see Figure 1, which is the main part of EHL rig (Brno University of Technology, Czech Republic). Both specimens (steel ball and PMMA disc) are fixed to shafts and powered by a pair of servomotors controlled by frequency converters. The steel ball is placed in the oil reservoir equipped with heating cells and lubricant is distributed into contact by rolling movement of the steel ball. Temperature is monitored by several thermocouples placed inside the oil reservoir (oil temperature), and one of them is manually positioned near the contact where lubricant enters the contact (inlet temperature T), as Figure 1shows. The PMMA disc is forced down to the steel ball in normal direction, while both specimens can rotate independently, and their shafts are perpendicular to each other. Figure 1. Rotary optical tribometer in ball-on-disc configuration. The contact is loaded through a polymer disc, which, together with the moveable load, is placed on a double reversible load lever, see Figure 1. The rotary optical tribometer is equipped with CMOS camera (Point Grey BFLY-PGE-23S6C) which captures the contact region in high pixel resolution, and with a powerful Nikon halogen lamp to provide a light source. Due to the expected higher elastic deformation of soft EHL contacts opposite Lubricants 2022,10, 155 5 of 18 to the hard EHL contacts, the Nikon CF Plan 5 × /0.13 objective with high depth of field was chosen. Temperature is controlled by PLC unit of EHL rig that allows to record and regulate heating process. 2.4. Experimental Conditions and Methods Before the experiments, both specimens were properly cleaned by isopropyl alcohol. Additionally, the surface of the steel ball was polished by pastes to decrease the RMS roughness R q and increase its reflectivity. After the cleaning process, the 3D optical profilometer (Bruker) was used for surface texture analysis, see results in Table 1. Nevertheless, in the case of the PMMA disc, it is an optically smooth surface. The specimens were placed to the tribometer and fixed, and the oil reservoir was filled with 50 mL of FVA3 lubricant, as illustrated in Figure 1. In respect to glass-transition temperature T g of PMMA (Table 1.), film thickness was measured in the temperature interval from 90 ◦ C to 110 ◦ C, where a disruption of polymer internal structure and acceleration of flow of individual atoms are expected. Consequently, viscoelastic, or plastic deformation occurs, and this can bring about unexpected phenomena in the lubrication process influencing film thickness. The range of entrainment speed U, load W, and other experimental conditions are given in Table 3. Table 3. Experimental conditions. Parameter Value Entrainment speed, U 0.2–1.2 m/s Load, W 20, 50 N Maximal contact pressure (24 ◦C), p045, 62 MPa Dynamic viscosity of FVA3 at atm., η00.013–0.008 Pa s Inlet temperature, T90–110 ◦C The optical chromatic interferometry [ 30 ] in connection with the EHL rig performs as a powerful tool for measurement and evaluation of film thickness. Therefore, the bottom side of transparent PMMA disc was coated with semi-transparent chromium layer which ensures improved interference. The chromium layer divides light from the light source into several beams. The first of beam, C I , is reflected by the chromium layer onto the disc surface while the second beam, C II , passes through the chromium layer over the lubricant and is reflected from the surface of steel ball. Each of these beams travels a different distance while their phases are mutually shifted resulting in phase shift ϕ . The different distance is calculated based on the interference of light. Both beams interfere with each other and indicate the magnitude of lubricant film thickness. This is schematically illustrated in detail A in Figure 2. Evaluation of film thickness is based on CIELAB color-film thickness calibration between monochromatic and chromatic interferograms deposed on each other via Newton’s fringes. Film thickness is measured, calibrated, and evaluated in software Achiles (ver.4.0.117, Radek Poliscuk, Brno University of Technology, Brno, Czechia), and this is illustrated in detail B in Figure 2. A more detailed description of optical chromatic interferometry method could be found in [30]. Lubricants 2022,10, 155 6 of 18 Figure 2. Process of film thickness evaluation in compliant contacts. ( A ) optical interference principle; (B) calibration principle. 3. Results 3.1. Static Contacts—Change of Contact Geometry Film thickness measurement was preceded by acquisition interferograms of contact in steady state condition (static contacts) at different inlet temperatures Tand loads W. Figure 3 demonstrates a different increase of contact size as a result of rising inlet temperature in dependence on load. However, for load W = 20 N, this increase is more noticeable than for W = 50 N, whereas, for the latter, an additional deformation occurs of originally circular contact after the temperature reached 110 ◦ C (and thus exceeds T g , Table 1), see Figure 3f. Ellipticity kof static contacts is predominantly close to the circular contact (k= 1) and belongs to the interval where k∈(1.01, 1.06). Lubricants 2022, 10, x FOR PEER REVIEW 7 of 19 Figure 3. Size of static contacts in dependence on W and T. (a) 20 N, 90 °C; (b) 20 N, 100 °C; (c) 20 N, 110 °C; (d) 50 N, 90 °C; (e) 50 N, 100 °C; (f) 50 N, 110 °C. Interferograms of static contacts were post-processed, and dimensions of contact were identified and averaged for both loads. Young’s modulus E 1 of PMMA, based on semi-contact radius a (see Nomenclature) and reduced Young’s modulus E’, was evaluated, see Figure 4. Development of Young’s modulus E 1 of PMMA shows a gradual decrease in the whole temperature range, especially close to the frequently stated glass-transition temperature of PMMA (black dash-line, T g ≈ 105 °C, [39]). The red rectangle in Figure 4 represents the area of interest from 90 °C to 110 °C where static contacts were evaluated and compared with the results of Boubimba et al. [42] from the DMA experiment for PMMA. On the one hand, the results from DMA experiments [42] demonstrate a similar development, but, on the other hand, they differ in values 18% on average. Obtained values of E 1 from Figure 4 were used in the next step to evaluate film thickness prediction models [19,20,28]. Figure 3. Size of static contacts in dependence on W and T. ( a ) 20 N, 90 ◦ C; ( b ) 20 N, 100 ◦ C; ( c ) 20 N, 110 ◦C; (d) 50 N, 90 ◦C; (e) 50 N, 100 ◦C; (f) 50 N, 110 ◦C. Lubricants 2022,10, 155 7 of 18 Interferograms of static contacts were post-processed, and dimensions of contact were identified and averaged for both loads. Young’s modulus E 1 of PMMA, based on semicontact radius a(see Nomenclature) and reduced Young’s modulus E 0 , was evaluated, see Figure 4. Development of Young’s modulus E 1 of PMMA shows a gradual decrease in the whole temperature range, especially close to the frequently stated glass-transition temperature of PMMA (black dash-line, T g≈ 105 ◦ C, [ 39 ]). The red rectangle in Figure 4 represents the area of interest from 90 ◦ C to 110 ◦ C where static contacts were evaluated and compared with the results of Boubimba et al. [ 42 ] from the DMA experiment for PMMA. On the one hand, the results from DMA experiments [ 42 ] demonstrate a similar development, but, on the other hand, they differ in values 18% on average. Obtained values of E 1 from Figure 4were used in the next step to evaluate film thickness prediction models [ 19 , 20 , 28 ]. Figure 4. Young’s modulus E1of PMMA versus inlet temp. T. 3.2. Film Thickness of Lubricant After acquisition of static contacts, central film thickness h c was measured at different loads (20 N and 50 N), inlet temperatures (90 ◦ C, 100 ◦ C and 110 ◦ C), and entrainment speeds (0.2–1.2 m/s), see the results in Figure 5. These results demonstrated that load W participates in development of the central film thickness h c only marginally, up to the glass-transition temperature T g , and, with the increasing load, the central film thickness decreases, see Figure 5a. Influence of load is more significant above the T g , where, at 110 ◦C , a different development of the central film thickness between W = 20 N and W=50N was observed. An increase of load W causes the unexpected increase of central film thickness h c in respect to the EHL theory, which is highest at 110 ◦C, see Figure 5b. The effect of temperature turned out to be more substantial in the case of contact geometry changes than for major changes in central film thickness development. Nevertheless, the results showed a similar profile of central film thickness h c with increasing temperature (especially for W = 50 N), where central thickness differences were observed only up to 50 nm. In respect to this, the phenomenon of the central film thickness h c increase with temperature and the load above glass-transition temperature T g was observed. This can implicate a minor influence of lubricant viscosity η and a major influence of Young’s modulus Eon formation of central film thickness h c. Moreover, for entrainment speed U, approximately 1.0–1.2 m/s, only slight changes were observed in film thickness evolution in this speed range, see Figure 5b. Lubricants 2022,10, 155 8 of 18 Figure 5. Central film thickness hcversus entrainment speed U: (a)W=20Nand(b)W=50N. In the next step, experimental data were compared with Hamrock and Dowson (H&D) [ 19 ] prediction model, see Figure 6a for 90 ◦ C and Figure 6b for 110 ◦ C, where a different agreement was obtained in respect to the applied load and inlet temperature. Figure 6a demonstrates similar development of central film thickness h c almost independent of the applied load W where only a minor difference occurs. In the case of W = 20 N, experimental data differs from the prediction model by about 15% in average, while for W = 50 N, the difference is double that of W = 20 N, and the deviation from prediction increases with entrainment speed. Figure 6. Comparison of experimental results with H&D [ 19 ] prediction model: ( a ) 90 ◦ C and (b) 110 ◦C. Dependence of central film thickness h c on the entrainment speed U is described by the power exponent of dimensionless speed U which is equal to 0.64 in H&D [ 19 ] prediction. However, the experimental results for W = 20 N and W = 50 N showed only the values of 0.59 and 0.61, respectively, which corresponds with higher central film thickness, see Figure 6a. Simultaneously, film thickness deviation points to the influence of lubricant dynamic viscosity ηin this region. Lubricants 2022,10, 155 9 of 18 The results obtained above the glass-transition temperature T g in Figure 6b showed a very good agreement between the experimental data and H&D [ 19 ] prediction model where the difference is less than 5% on average for W = 20 N. Opposite to this, for W=50N , fluctuation of central film thickness h c with entrainment speed U was observed, and, similarly as in Figure 6a, experimental data do not correspond with the prediction model and differ even more, see Figure 6b. The deviation is linked with the deflection of power exponent of dimensionless entrainment speed U where, for W = 20 N, the results showed the value of 0.61, while for W = 50 N, it was only 0.58. This again confirms the dependence of the central film thickness h c on temperature and load; after the T g is exceeded, the significance of the load W is strongly enhanced. 3.3. Running Contact—Change of Contact Geometry In the last step, interferograms of running contacts were post-processed, the variations of contact geometry were evaluated in respect to the applied load W and inlet temperature T, and the most important influences were identified and summarized. Evolution of contact geometry is illustrated by interferograms of running contacts in Figure 7A for load W=20N , and in Figure 7B for load W = 50 N, respectively. Each figure represents the influence of inlet temperature T(in vertical direction), entrainment speed U (in horizontal direction), and load W separates Figure 7where some of mentioned effects conflict with each other. Figure 7C represents an assignment of film thickness to the color spectrum. Lubricants 2022, 10, x FOR PEER REVIEW 10 of 19 Figure 7a for load W = 20 N, and in Figure 7b for load W = 50 N, respectively. Each figure represents the influence of inlet temperature T (in vertical direction), entrainment speed U (in horizontal direction), and load W separates Figure 7 where some of mentioned effects conflict with each other. Figure 7c represents an assignment of film thickness to the color spectrum. (A) (B) (C) Figure 7. Interferograms of running contact at different temperatures and entrainment speeds: (a) W = 20 N, (b) W = 50 N, and(c) assignment of film thickness to the color spectrum. From Figure 7a, it is evident how the contact is deformed, and, at the same time, it is enlarged in respect to the increasing inlet temperature (vertical direction). Ellipticity of initially circular contact (k = 1) gradually increases with inlet temperature T. This was observed as shortening of contact in the direction of entrainment speed U (y-axis) and, simultaneously, the contact extension in perpendicular direction to the entrainment speed (x-axis). Results showed that the entrainment speed contributes to the increase of k only at 110 °C, while, at lower temperatures, ellipticity was almost constant. This change is most noticeable in the last row of Figure 7a, where, apart from the temperature, an increase of entrainment speed U causes gradual deformation from the initially circular to wide elliptical contact. At higher loads (W = 50 N), see Figure 7b, results were partially different. It is interesting that, below Tg, the behavior of contact was very similar to previous results, but after the inlet temperature exceeded Tg, the contact was reduced in both axes and the ellipticity of contact rapidly increased up to k = 1.3. Below Tg, an increase of entrainment speed U led to a reduction of contact size in both axes; however, above Tg, the increase of entrainment speed caused the enlargement of contact size, see Figure 7b. The results summarized in Figures 8 and 9 represent mean values of ellipticity of contact k and the size of area of contact A at different temperatures and loads calculated from the whole range of entrainment speeds. The results in Figure 8 once again confirm y Figure 7. Interferograms of running contact at different temperatures and entrainment speeds: (A)W=20N,(B)W=50N,and(C) assignment of film thickness to the color spectrum. Lubricants 2022,10, 155 16 of 18 Author Contributions: Conceptualization, J.K. and K.D.; Methodology, J.K.; Validation, J.K. and K.D.; Formal analysis, J.K. and K.D.; Investigation, J.K.; visualization, J.K. and K.D.; Writing—original draft preparation, J.K.; Supervision, I.K.; Project administration, M.H. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Czech Science Foundation (GACR), grant number 18-26849J. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data concerned have been obtained through the research of the author and reported exclusively in this article. Acknowledgments: The authors thank to our colleagues from Gear Research center (FZG) and Technical University Munich (TUM) in Germany for donation in providing the FVA3 reference lubricant for experiments. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. Nomenclature AArea of contact, (mm2) aSemi-contact radius, (µm) a3= 3WR0/2E0 CI, CII Light beam I and II, (-) E Young’s modulus, (GPa) E1, E2Young’s modulus of solid 1 and 2, (GPa) ELElastic part of E (Storage modulus), (MPa) ESViscous part of E (Loss modulus), (MPa) E0Reduced Young’s modulus, (GPa) 2/E0= (1 −υ12)/E1+ (1 −υ22)/E2 G Dimensionless parameter of material, (-) G =E0α geParameter of elasticity, (-) ge=W8/3/U2 gvParameter of viscosity, (-) gv=G W3/U2 hmMinimum film thickness, (nm) hm,exit Minimum film thickness at contact exit, (nm) hm,side Minimum film thickness on side lobes, (nm) hcCentral film thickness, (nm) hcp Predicted central film thickness, (nm) kParameter of ellipticity, (-) P Parameter of pressure, (-) p0Maximum contact pressure, (MPa) p0= 3W/2πa2 R1, R2Curvature radius of solid 1 and 2, (mm) R0Reduced curvature radius, (mm) 1/R0= 1/R1+ 1/R2 RqRoot mean square roughness (RMS), (µm) Rt/ReParameter of ellipticity by Greenwood, (-) S Speed parameter, (-) TInlet temperature, (◦C) TgGlass-transition temperature, (◦C) U Entrainment speed, (m/s) U = (U1+ U2)/2 U Dimensionless speed parameter, (-) U =Uη0/E0R0 U1, U2Surface speed of solid 1 and 2, (m/s) W Normal load, (N) W Dimensionless load parameter, (-) W =W/E0R02 αPressure-viscosity coefficient, (GPa−1) η0Dynamic viscosity at atm., (Pa s) υ1,υ2Poisson’s ratio of solid 1 and 2, (-) υ40 Kinematic viscosity at 40 ◦C, (mm2/s) υ100 Kinematic viscosity at 100 ◦C, (mm2/s) Lubricants 2022,10, 155 17 of 18 ρ15 Density of lubricant at 15 ◦C, (kg/m3) ϕPhase shift, (-) Appendix A Table A1. Experimental data deviations of hcfrom prediction models hcp. Ratio hc/hcp. Exp. A B C D E F Exp. cond. 90 ◦C, 20 N 90 ◦C, 50 N 100 ◦C, 20 N 100 ◦C, 50 N 110 ◦C, 20 N 110 ◦C, 50 N H&D, [19] 13.7% 13.7% 6.4% 18.1% 4.0% 4.9% Hooke, [20]−7.3% −0.3% −18.2% −9.4% −25.7% −25.2% Marx, [28]−37.0% −36.0% −52.3% −49.2% −63.8% −59.9% References 1. Gilbert, M. Chapter 1—Plastics Materials: Introduction and Historical Development. In Brydson’s Plastics Materials, 8th ed.; Gilbert, M., Ed.; Butterworth-Heinemann: Amsterdam, The Netherlands, 2017; pp. 1–18, ISBN 978-0-323-35824-8. 2. Dearn, K.D.; Hoskins, T.J.; Andrei, L.; Walton, D. Lubrication regimes in high-performance polymer spur gears. Adv. Tribol. 2013 , 2013, 987251. [CrossRef] 3. Kunishima, T.; Nagai, Y.; Kurokawa, T.; Bouvard, G.; Abry, J.C.; Fridrici, V.; Kapsa, P. Tribological behavior of glass fiber reinforced-PA66 in contact with carbon steel under high contact pressure, sliding and grease lubricated conditions. 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