A Numerical Study on Thermal Elastohydrodynamic Lubrication of Coated Polymers

To classify polymer–steel TEHL contacts, the hydrodynamic pressure p and the film thickness h are compared in Fig. 2 for a polymer–steel, polymer–polymer and steel–steel material paring at the same line load. The reference material properties of Table 1 were considered.

Hydrodynamic pressure p and film thickness h of the material pairings considered show well-known distributions with almost no second pressure maximum and a constriction with a minimum film thickness. The comparable small hydrodynamic pressure p in the TEHL contacts is responsible for the absence of a second pressure maximum typically known from hard TEHL contacts. In correlation with stiffness of contact pairing, the deformations and contact areas are largest for the polymer–polymer and smallest for the steel–steel contact and, therefore, hydrodynamic pressure is highest for the steel–steel and lowest for polymer–polymer contact. As a consequence of the deformation, the film thickness for the polymer–polymer contact is higher than for the steel–steel contact. The characteristics of pressure and film thickness for the polymer–steel contact lay in between.

To analyze the division of deformation amongst the material pairing, Fig. 3 shows the undeformed and deformed geometry of the rolling elements in the polymer–steel (left), polymer–polymer (middle), and steel–steel contact (right). Note that individual deformations are considered separately in the numerical model. For the illustration, the approach towards the rolling elements according to the load balance is equally applied on the rolling elements.

The polymer–steel contact shows a local surface conformity: the polymer accounts for 98% of total deformation, and steel is nearly undeformed. The central contact zone shows a well-known parallel gap, but it is curved along the steel surface. In contrast, the polymer–polymer and steel–steel contacts show an equal division of deformation of the rolling elements.

In addition, Fig. 4 shows the temperature ϑ for the polymer–steel, polymer–polymer and steel–steel contacts. Note, henceforth the results are referenced in the equivalent system and on the lower surface. Note that, the figure’s axes of steel–steel contact are adjusted for illustration purposes.

The temperature ϑ is highest for the steel–steel contact with a maximum contact temperature of ϑ_max = 90 °C. In comparison, ϑ_max is 55 °C for the polymer–polymer and only 48 °C for the polymer–steel contact. This can be explained by the heat sources in the TEHL contact due to shearing and compression of lubricant and heat removal due to convection and conduction. Higher hydrodynamic pressure p results in higher viscosity, higher shear stress and, consequently, more pronounced heat generation. In contrast, heat removal from the TEHL contact is mainly determined by convection and conduction of the rolling elements. A lower thermal effusivity e typically results in higher contact temperatures. For the steel–steel contact, p and e are highest, resulting in the most pronounced heat generation and removal. As a consequence, the temperature ϑ is concentrated in the middle of the lubricant film and the rolling elements are only sparsely heated. For the polymer–polymer contact, p and e are lowest, so heat generation and removal are weak. Thus, the temperature ϑ is much lower than the steel–steel contact. For the polymer–steel contact, p and therefore heat generation are slightly higher than the polymer–polymer contact. Since the steel shows a high value and the polymer shows a low value of e, thermal insulation is present on the polymer surface. Hence, in spite of the slightly higher heat generation compared to the polymer–polymer contact, the superior heat removal by steel results in the lowest contact temperature ϑ.

The evaluation of the viscosity in the TEHL contacts for the polymer–polymer and polymer–steel contact follows the findings of Maier et al. [25] and shows almost no effect of shear thinning. Hence, the viscosity η and therefore shear stress and coefficient of friction µ depend only on p and ϑ. For the steel–steel contact, shear thinning plays a role with η(T,p,\({\dot{\gamma }}_{fzx}\)) in maximum 25% lower than η(T,p).

Polymers available for gear application and their thermoelasticity feature a wide range of Young’s moduli E. Figure 5 compares hydrodynamic pressure p, film thickness h and temperature ϑ for a polymer’s Young’s moduli of E₁ = {0.8; 2.9; 7.0} GPa. E₁ = 2.9 GPa corresponds to the reference value. A study on the effect of polymer’s Poisson’s ratio varied from ν = 0.25 to 0.50 only had a small effect on the TEHL contact, so it is therefore not explicitly shown here.

In general, the hydrodynamic pressure p increases and film thickness h and contact width decrease with increasing Young’s modulus of the polymer. For E₁ = 7.0 GPa, the maximum pressure p increases by 50% and the central film thickness h_c decreases by 20% compared to E₁ = 2.9 GPa. For E₁ = 0.8 GPa, a reduction of p by 48% and an increase of h_c by 52% is observed. This can be traced back to the increasing stiffness and decreasing deformation of the polymer with increasing Young’s modulus of the polymer.

With increasing E₁, the hydrodynamic pressure p increases, which results in an increase of shear stress and coefficient of friction µ. Hence, the temperature ϑ increases and µ decreases. Overall, the increase of shear stress and µ due to increasing pressure is larger than their decrease due to increasing temperature. For E₁ = 7.0 GPa, µ increases by 17% and ϑ_max by 7%. For E₁ = 0.8 GPa, µ decreases by 14% and ϑ_max by 5%. The temperature distribution shows for E₁ = 0.8 GPa and E₁ = 2.9 GPa higher values in the contact inlet than in the contact zone. This can be traced back to same magnitude of lubricant heat sources due to shearing and compression of lubricant. Hence, the temperature distributions confirm the same magnitude of heat and friction induced by Couette and Poiseuille flow in the TEHL contacts. These relations change with increasing E₁ (cf. E₁ = 7.0 GPa) such that shear heating becomes dominant as known from hard TEHL contacts.

The evaluation of the viscosity shows almost no shear-thinning effect for all considered E₁ (cf. Sect. 4.1.1). This relation is expected to change with further increasing E₁.

Figure 6 shows the elastic strain ε_zz on the surface of the polymer rolling element for various polymer’s Young’s moduli E₁. Note that nearly the entire deformation in the contact is present in the polymer, whereas the steel is almost undeformed (cf. Fig. 3).

The elastic strain ε_zz on the surface of the polymer resembles the distribution of the hydrodynamic pressure p. Hence, ε_zz shows highest absolute values and widths for the lowest E₁ and the lowest absolute values and widths for the highest E₁. For E₁ = 0.8 GPa, the absolute maximum strain is 95% higher than for E₁ = 2.9 GPa. For E₁ = 7.0 GPa, ε_zz is 36% lower. For the lowest E₁ = 0.8 GPa, the maximum absolute elastic strain is ε_zz = 2.9% in a material depth of z ≈ − 0.6 b_H and is a result of pressure. In exemplary stress–strain curves of a tensile test of, e.g., a PMMA material [43], linear elastic behavior is present until ε_zz ~ < 2%. Yield strain is reached at ε_zz ~ > 4.5% [43]. Note that, whereas in tensile tests, material underlies tension, in TEHL contact the material underlies compression. Nonetheless, this indicates a good approximation of linear elastic material behavior in TEHL contacts.

Contact temperatures (Fig. 5) show small increases and penetration depths of lower than 10 µm, therefore the Young’s modulus is weakly influenced and only in the surface near area of the polymer. Since the material areas which underlie strain and stress are much larger than the temperature penetration depth, no significant effect on pressure and film thickness is present. But high surface temperatures and the consequently lower mechanical stiffness can cause thermoplastic deformation and melting [25].

Similar to the Young’s modulus, polymers for gear application feature a wide range of densities ρ, thermal conductivities λ, and specific heat capacities c_p. These properties can be summarized by the thermal effusivity e describing the ability of a material to transport heat. To evaluate the effect of different polymer thermal effusivity e₁ on the steel–polymer TEHL contact, density ρ = [1000…1500] kg/m³, thermal conductivity λ = [0.19…2] W/(m K) and specific heat capacity c_p = [1000…2200] J/(kg K) are varied. This results in considered thermal effusivities e₁ = \(\sqrt{\rho {c}_{p} \lambda }\) from 435 to 2569 J/(K \(\sqrt{\mathrm{s}}\) m²). The thermal effusivity of e₁ = 578 J/(K \(\sqrt{\mathrm{s}}\) m²) corresponds to the reference value. The Young’s modulus and Poisson’s ratio are kept constant as in Table 1. Figure 7 shows the hydrodynamic pressure p, film thickness h and temperature ϑ for different polymer’s thermal effusivities e₁.

The variation of the thermal effusivity of the polymer e₁ has a negligible influence on the hydrodynamic pressure p. The contact temperature ϑ decreases with increasing e₁. A high thermal effusivity e₁ results in a more pronounced heat removal. Hence, the temperature increase in the TEHL contact is smaller. For low e₁, the rolling element acts thermally insulating, so ϑ in the TEHL contact is higher. Additionally, ϑ is localized towards the polymer surface since heat accumulates there. As a consequence of higher temperatures in the TEHL contact, the coefficient of friction slightly decreases. Due to the more pronounced temperature in the contact inlet, the film thickness h decreases slightly by up to 0.2 µm with decreasing e₁.

To analyze the influence of polymer’s thermal effusivity e₁ on the TEHL temperature in more detail, Fig. 8 shows the surface temperature ϑ₁ of the polymer.

The general trend of the polymer surface temperature ϑ₁ shows an increase at the contact inlet to a maximum. This is due to compressive heat generation in the contact inlet, which is in the same order as lubricant shearing (cf. Sect. 4.1.2). After this maximum, the surface temperature decreases as the gradient of the hydrodynamic pressure and, therefore, heat generation due to compression of lubricant decreases. In the contact zone, heat generation due to shearing of the lubricant becomes dominant, resulting in a second maximum of ϑ. Towards the contact outlet, the pressure gradient is negative, so lubricant expansion leads to heat removal. In the contact outlet, the heated polymer surface slowly transfers part of the heat into the lubricant. For low e₁, a high temperature increase with a pronounced temperature maximum in the contact inlet is observed, and vice versa for high e₁. The accumulation of heat in the contact inlet and the temperature level in the contact zone is reduced for high e₁ since heat removal by the polymer is more pronounced.

Figure 9 shows the hydrodynamic pressure p, film thickness h (left) and temperature ϑ in the middle of the lubricant film z = h/2 (right) of the polymer–steel contact for different operating conditions by changing the sum velocity to v_Σ = {4; 12} m/s and the line load to w = {50; 150} N/mm. v_Σ = 8 m/s and w = 100 N/mm correspond to the reference operating condition. The mechanical and thermophysical properties are kept constant as per Table 1.

The hydrodynamic pressure p and contact width increase with increasing line load w, whereas the film thickness decreases. The central film thickness h_c is 8% lower at w = 150 N/mm and 22% higher at w = 50 N/mm compared to w = 100 N/mm. This is in accordance with the analytical formula according to Myers [9]. In contrast to hard TEHL, the formula according to Dowson and Higginson [44] shows for the load variation considered a change in h_c of − 7% and + 12%, respectively. With increasing sum velocity v_Σ, p is almost unaffected and h increases. This is well known from hard TEHL contacts.

The temperature profiles ϑ in the middle of the lubricant film generally corresponds with the temperature of the polymer surface in Fig. 8. In contrast, the temperature ϑ in the contact inlet at z = h/2 shows a higher level and a less pronounced increase than at the polymer surface. This can be traced back to lubricant backflow, which transports preheated lubricant from the contact region to the contact inlet. In the contact outlet, temperature ϑ decreases to a similar temperature ϑ as in the contact inlet. This confirms the same magnitude of heat sources by lubricant’s compression and shearing.

When varying the operating conditions, a dominant influence of the sum velocity v_Σ over the line load w on the temperature ϑ is observed. With increasing line load w, the maximum of ϑ is almost unaffected. However, the trend of ϑ shows varying contact widths and temperature gradients. With increasing v_Σ, sliding velocity v_g increases with the same ratio as SRR is constant. This results in an increase in shear rate, because the increase of h with increasing v_Σ is less pronounced. Hence, shear stress (Eq. 1), and shear heating and therefore temperature ϑ rises. In the contact inlet, high compressive heating contributes to an increase of ϑ, which affects the film thickness h. In the contact zone, ϑ increases more strongly since shear heating is more pronounced due higher sliding velocity v_g with increasing v_Σ.

Based on the presented results, the main findings on uncoated polymer–steel TEHL contacts are summarized:

Figure 11 (left) shows the hydrodynamic pressure p and the film thickness h of coated polymer–steel contacts for coatings with Young’s moduli of E_c = {34; 110; 220; 420} GPa and a constant coating thickness of t_c = 100 µm. Figure 11 (right) illustrates the effect of coating stiffness E_c = {34; 110} GPa and coating thickness t_c on the stiffness expressed by the reduced Young’s modulus E′ of the TEHL contact. Thereby, the extension of the Hertzian theory presented by Liu et al. [47] is used. All thermophysical properties are adopted from Table 1.

The hydrodynamic pressure p shows a pronounced second pressure maximum and becomes more pronounced with increasing Young’s modulus E_c of the coating. Additionally, the contact width slightly decreases and the constriction of the lubricant film thickness is more pronounced. This can be traced back to low contact stiffness of polymer–steel TEHL contacts. In general, the contact stiffness relevant for TEHL contacts depends on Young’s modulus and Poisson’s ratio of the regions which underlie strain and stress. In polymer–steel contacts, these regions are much larger. Hence, the coating’s mechanical properties affect the relevant contact stiffness for polymer–steel contacts only at much higher coating thickness t_c or higher coating stiffness. Hence, the ratio of deformation between coated polymer and steel is not significantly affected by the coating and the deformed geometry of the rolling elements is similar to the uncoated polymer–steel contact (Fig. 2).

A higher thickness of hard coatings generally contributes to a higher stiffness of the contact. For hard TEHL contacts, it was shown that the influence of hard coatings on p is negligible for small coating thickness [18, 20]. With regard to the polymer–steel contact considered, which show low contact pressure and high deformation, the contribution of the coating’s stiffness to the stiffness of the TEHL contact becomes relevant at approximately t_c > 50 µm. This is retraced by the evaluation of the reduced Young’s modulus E′ in Fig. 11 (right). For an uncoated polymer–steel contact E′ is 6.6 GPa. With increasing coating thickness t_c, E′ increases due to higher stiffness of the coating E_c compared to the polymer. At a thickness t_c ≈ 800 µm, the coating with E_c = 110 GPa reaches E′ ≈ 155 GPa, which represents a contact between pure coating material and steel. Hence, the influence of the substrate polymer on the contact stiffness vanishes. The coating with E_c = 34 GPa reaches this point at t_c ≈ 1200 µm with E′ ≈ 62 GPa. For a typical coating thickness of t_c < 8 µm, no noticeable effect on the stiffness of polymer–steel contacts is expected.

Figure 12 (left) shows the temperature ϑ in coated polymer–steel contacts for various coatings with thermal effusivities of e_c = {578; 1156; 2312} J/(K \(\sqrt{\mathrm{s}}\) m²) for a constant coating thickness of t_c = 1 µm. The mechanical properties are adopted from Table 1. e_c = 578 J/(K \(\sqrt{\mathrm{s}}\) m²) represents a coating with thermophysical properties similar to the polymer. Figure 12 (right) shows the effect of coating thickness t_c on the maximum temperatures in the lubricant, coating, polymer, and steel for the coating properties in Table 1.

With increasing thermal effusivity e_c, the lubricant temperature ϑ decreases, and the coefficient of friction µ slightly increases (Fig. 12, left). This is in accordance with the results for uncoated contacts in Fig. 7. The coating shows a decrease in temperature penetration depth with increasing e_c. This results in lower maximum temperatures at the coating–polymer interface. Hence, also very thin coatings t_c = 1 µm with higher thermal effusivity e_c than the polymer can act as thermal barrier protecting the polymer.

By increasing t_c, the maximum temperature in the polymer decreases continuously (Fig. 12, right). For t_c = 100 µm, no further temperature increase is observed. In this case, the heat penetration depth is less than the coating thickness. This results in maximum temperatures in the polymer equal to bulk temperature ϑ_M. In general, the maximum temperatures in the lubricant, coating and steel decrease with t_c up to approximately t_c ≈ 30 µm. This decrease is most pronounced up to t_c ≈ 8 µm. For approximately t_c > 30 µm, these temperatures increase slightly. In general, the coatings considered contribute—by their higher thermal effusivity compared to the polymer—to an improved heat removal (Fig. 12). In contrast, due to increasing coating thickness t_c, pressure (Fig. 10) and, therefore, shear stress and frictional heat increase slightly. Both influences has an inverse effect on temperature. Whereas the effect of the coating stiffness on the TEHL contact appears to be significant at approximately t_c > 50 µm, the influence of the thermophysical properties of the coatings is already present at very low t_c < 8 µm. A transition point is observed at approximately t_c = 30 µm.

Based on the presented results, the main findings on coated polymer–steel TEHL contacts are summarized: