In a three-body (3-body) contact tribology system, which contains (1) surface 1, (2) surface 2, and (3) particles and lubricant (Figure 1), the normal load (F _t ) is the sum of the load of fluid lubricant (F _f ) and load of solid (F _s ): F t = F f + F s . (1)

This study applies the 3-body microcontact model to analyze the real contact areas and loads of solid. Here, the load for solid includes contact between surface 1 and 2 (F _ss ), as well as between particle and surface 1 (F _sa ). Similarly, the real contact area of solids (A _t ) also comprises the contact area between surface 1 and 2 (A _ss ) and between the particle and surface 1 (A _sa ). F s = F s s + F s a (2) A t = A s s + A s a (3)

When particles are present between the relative motions of rough surfaces in mechanical components, it is assumed that the particles, due to work hardening during operation, do not deform when in contact with surface 1 or 2. The 3-body dry contact model can be employed to calculate the relationship between the real contact area, surface separation, and the normal load. The normal load of solid, F _s , and total contact area of solid, A _t , are given by the following equations for the 3-body micro-contact model (Horng et al., 2022): F s = F s s − s a + F s a = F s s × 1 − π H s 1 η a H s 1 + H s 2 ∫ X min X max x 2 ϕ a x d x + F s a = F s s × 1 − π H s 1 η a H s 1 + H s 2 ∫ X min X max x 2 ϕ a x d x + π H s 1 H s 2 η a A n H s 2 + H s 1 9 π 2 4 H s 1 2 E s a 2 + H s 2 2 E s s 2 ∫ d − h e d x 2 ϕ a x d x + ∫ d x max x 2 ϕ a x d x , (4) A t = A s s − s a + A s a = A s s × 1 − π H s 1 η a H s 1 + H s 2 ∫ d x max x 2 ϕ a x d x + A s a = A s s × 1 − π H s 1 η a H s 1 + H s 2 ∫ d x max x 2 ϕ a x d x + π H s 2 η a A n H s 2 + H s 1 9 π 2 4 H s 1 2 E s a 2 + H s 2 2 E s s 2 ∫ d − h e d x 2 ϕ a x d x + ∫ d x max x 2 ϕ a x d x , (5) where F _ss and A _ss are calculated according to the previous 2-body contact model (Zhao et al., 2000). F _ss-sa and A _ss-sa are the contact load of surface 1 with surface 2 and the contact area between surface 1 with surface 2 minus the particle effects, respectively. d is mean separation between two surfaces. x is mean diameter. H _s1 and H _s2 are the hardness of surfaces 1 and 2, respectively; η _a is particle density, x _max is maximum particle size, x _min is minimum particle size. A _n is the apparent contact area at the interface; E _sa and E _ss are the equivalent elastic modulus between particle and surface, and the equivalent elastic modulus between surfaces 1 and 2, respectively; h _e is the maximum distance of the particle deducted from the contact interference with surface 1 and the contact interference with surface 2. The probability density function of summit heights, ϕ(z), and particle size, ϕ _a (x), are given as follows: ϕ z = 1 2 π × σ s exp − 0.5 z σ s 2 , ϕ a x = 1 2 π × σ a exp − 0.5 x σ a 2 , (6) where σ _s is the standard deviation of the summit height and σ _a is the standard deviation of the particle size.

The separation of relative surfaces can not only obtain the load percentage of solids, but also distinguish the lubrication regime of the contact interface. According to Masjedi and Khonsari (2015), the central oil film thickness is expressed thus: H c = h c R x = 3.672 W − 0.045 κ 0.18 U 0.663 κ 0.025 G 0.502 κ 0.064 1 − 0.573 e − 0.74 κ × 1 + 0.025 σ * 1.248 V 0.119 W − 0.133 U − 0.884 G − 0.977 κ 0.081 , (7) where the W, U, G, σ ^* , V, and κ are dimensionless load, speed, material, surface roughness, surface hardness, and ellipticity parameter, respectively. The above dimensionless parameters are as follows. W = F t E s s R x 2 , U = μ 0 u E s s R x , G = α E s s , σ * = σ R x , V = ν h E s s , κ = a / b , (8) where R _x is the curvature radii of the summit in x directions, μ ₀ is the ambient lubricant viscosity, u is the rolling speed, α is the pressure–viscosity coefficient, σ is the equivalent root mean square roughness of surfaces 1 and 2, v _h is the Vickers hardness of softer material, and a and b are the contact half-length and half-width of surface, respectively. This study combines the calculation of lubricant film thickness with the 3-body micro-contact model and constructs a 3-body lubrication contact model based on the concept of force balance. The analysis results provide information on the film thickness, contact area of the solid, and fluid-borne and solid-borne loads under 3-body contact conditions.

The fluid- and solid-borne loads obtained from this theoretical analysis can be expressed under 3-body lubrication contact conditions based on the relationship in Eq. 1: L s = F s F t × 100 % , L f = F f F t × 100 % . (9)

In accordance with Zhu et al. (2012), Zhu and Wang (2013) and Masjedi and Khonsari (2015), the research indicates that the load percentage of solid (L _s ) can be utilized to assess the lubrication regime at the contact interface, defined as follows. When L _s is greater than 90%, the lubrication regime is in the boundary lubrication (B.L.). When L _s is less than 3%, the lubrication regime is in the full-film lubrication (F.L.) (Horng et al., 2022). When L _s falls between 3% and 90%, the lubrication regime is in the mixed lubrication (M.L.).

When lubricants contain added micro/nano particles, or when components generate debris after a certain period of operation, a 3-body tribology system will be formed at the contact interface. To elucidate the role of the third particle at the 3-body contact interface, this paper utilizes the contact area ratio of the third particle to determine the three contact modes at the 3-body contact interface. The contact area ratio of particle and surface, A _sa,t ^*, and contact area ratio of surface 1 and surface 2, A _ss,t ^*, are defined as follows: A s a , t ∗ = A s a A t , A s s , t ∗ = A s s A t . (10)

Under different conditions, the relationship between the three contact modes and the A _sa,t ^* are observed (Horng et al., 2023), as shown in Figure 2. Figure 2A shows the surface-to-surface 3-body contact mode (s-s mode). The particles with relatively smaller sizes and the majority of the normal load are borne by the contact between surfaces 1 and 2, expressed as A _t ≅ A _ss and A _sa,t ^*< 0.05. Figure 2B shows the particle-to-surface 3-body contact mode (p-s mode). The particle sizes and concentrations are relatively larger. The normal load is almost borne by the contact points between the particles and surface 1, expressed as A _t ≅ A _sa and A _sa,t ^* > 0.95. Figure 2C shows the hybrid 3-body contact mode (hybrid mode). The load is simultaneously supported by the surface-to-surface contact and particle-to-surface contact, expressed as A _t = A _sa + A _ss , 0.05 < A _sa,t ^* < 0.95. If no particles are at the lubricated contact interface, then only the contact between surfaces 1 and 2 occurs, expressed as A _t = A _ss and is referred to as the 2-body contact mode (2-body mode). Therefore, these three 3-body contact modes, in the presence of lubricating oil, belong to mixed and boundary lubrication regimes.

Figure 3 shows dimensionless area vs dimensionless normal load at various particle diameters: (A) contact area ratio of particle (A _sa,t ^*); (B) dimensionless total contact area (A _t ^*) under dry contact condition for σ = 100 nm, η _a = 10¹¹ m ^-2. This explains the evolution of three contact modes occurring at contact interfaces of particles larger than 300 nm as the dimensionless normal load increases: the p-s mode first transitions to the hybrid mode, and then to the s-s mode. As depicted in Figure 3A, x _a = 1,000 nm, A _sa,t ^* = 1.0 indicates that the contact interface is in the p-s mode. x _a = 0 nm, A _sa,t ^* = 0 indicates that the contact interface is in the s-s mode, representing the traditional 2-body contact value. Figure 3B shows that the A _t ^*-F _t ^* relationship forms a straight line of maximum value for x _a = 0 nm (2-body contact) and a line of minimum value for x _a = 1,000 nm. These two lines form upper and lower boundary lines; when particles are present at the interface, their corresponding A _t ^* values are all smaller than that of a pure 2-body contact surface. The larger the particles, the smaller the A _t ^*. As F _t ^* increases, the A _t ^* of the 3-body contact interface gradually rises between the approximate upper and lower limit lines to the upper limit value (2-body contact interface). Therefore, Figure 3 shows that the transition of the contact mode from p-s to hybrid as the normal load increases as the same particle size and slope of A _t ^* increases. With the transition of the contact mode from hybrid to s-s, A _t ^* also gradually approaches the curve of x _a = 0 nm. The evolution of the three contact modes at the 3-body contact interface under different operating conditions has been detailed in previous study (Chern et al., 2022).

In the situation of 3-body lubrication contact interfaces, their contact behavior differs from dry contact. Figure 4 shows contact characteristics vs dimensionless normal load at various kinematic viscosities (v) of the lubricant (A) contact area ratio of particle, A _sa,t ^* (B) dimensionless contact area, A _t ^* (C) load percentage of solid, L _s (D) film thickness, H _c for σ = 300 nm, and η _a = 10¹¹ m ^-2, x _a = 500 nm. Figure 3 indicates that the contact mode of the interface can be determined by A _sa,t ^*. Therefore, as shown in Figure 4A, when v < 90 cSt, there will be, with the increase of dimensionless normal load, an evolution three contact modes at the interface. However, when v > 90 cSt, it will only transition from p-s to hybrid mode. Figure 4B shows that when v < 50 cSt, the value of A _t ^* almost overlaps with that of 3-body dry contact, indicating that it has little impact on the interface contact characteristics. When v < 70 cSt, as the load increases to a certain critical value (F _t ^* = 1.5 × 10⁻⁵), the slope of A _t ^* with respect to the dimensionless normal load will decrease, and the critical load decreases with increasing viscosity. This is due to higher viscosity resulting in a smaller load required generating hydrodynamic pressure. At the same dimensionless normal load, as the viscosity increases, the hydrodynamic pressure increases, resulting in a decrease in A _t ^*. The reason for this phenomenon is explained in Figure 4C.

Figure 4C shows that for the interface of v < 50 cSt and dry contact, L _s is 100% under all dimensionless normal loads, indicating that the contact interface is in the boundary lubrication regime. When v > 70 cSt, L _s decreases with the increasing F _t ^*. L _s decreasing to less than 90% indicates a transition in the lubrication regime from boundary to mixed. A further increase in the load, causing L _s to decrease to less than 3%, will transition the lubrication regime from mixed to full-film. Additionally, L _s decreases with increasing viscosity at the same dimensionless normal load. The higher the viscosity, the smaller the dimensionless normal load required for the transition from boundary to mixed lubrication and from mixed to full-film lubrication. Corresponding to Figures 4B,C, when the interface at lubricated condition and L _s = 100%, its A _t ^* coincides with dry contact. When L _s decreases with increasing load to 90% (the lubrication regime transitions from boundary to mixed), its critical dimensionless load is approximately the same as the critical load where the slope of A _t ^* decreases. As L _s decreases with the increasing dimensionless normal load, although the slope of A _t ^* decreases, the value of A _t ^* continues to increase. This indicates that the real solid-borne load continues to increase as the rate at which it increases with normal load decreases, while the rate at which the load borne by fluid increases with normal load increases. Therefore, the decrease in the slope of A _t ^* is primarily caused by the transition in lubrication regime induced by the lubricant rather than by changes in the solid contact mode. When v = 90 and 120 cSt, L _s decreases from 100% to 90% (transitioning from boundary to mixed lubrication), their A _t ^* decreases by approximately 14.9% and 21.0% compared to the 3-body dry contact. When L _s decreases to 50%, A _t ^* decreased by approximately 53.7% and 65.0% compared to the 3-body dry contact.

Figure 4D shows that the H _c value of the 3-body dry contact is significantly larger than that of the 2-body dry contact under medium to low dimensionless normal loads. The interface transitions into the hybrid mode with increasing dimensionless normal load, resulting in a decrease in the H _c value until it reaches a similar value to the 2-body contact. When the lubricant is present at the interface and under lower dimensionless normal loads, the curves of H _c at different viscosities overlap with that of the 3-body dry contact, indicating that the system is currently in the boundary lubrication regime. The H _c gradually deviates from and becomes larger than the curve of the 3-body dry contact as the dimensionless normal load increases, indicating that the lubricant at the contact interface forms a fluid film, leading the interface into a mixed lubrication regime. The results in Figure 4 demonstrate the evolution of the lubricated interface with increasing load, taking v = 90 cSt as an example: from boundary lubrication in the p-s 3-body contact mode → boundary lubrication in the hybrid 3-body contact mode → mixed lubrication in the hybrid 3-body contact mode → mixed lubrication in the s-s 3-body contact mode → full-film lubrication in the s-s 3-body contact mode (if the load further increases). Our analysis provides references for selecting lubricant viscosity under different operating conditions, and third particle sizes are provided for various types of components.

Figure 5 illustrates the load percentage of solid versus dimensionless normal load at various equivalent root mean square surface roughness (σ) under the same operating conditions as Figure 4. Figure 5A shows that for σ = 100 nm, when v = 10 cSt, the L _s approaches 90% at higher dimensionless normal loads. For Figure 5B with σ = 500 nm, a viscosity greater than 70 cSt is required to transition the lubrication regime of contact interface from boundary to mixed. The integration of Figure 4 and Figure 5 reveals that as the equivalent root mean square surface roughness of the component decreases, a lower viscosity lubricant is sufficient to transition the 3-body contact interface from a boundary to mixed lubrication regime.

According to the results from Figure 4, when v = 90 cSt, the contact interface potentially encompasses three contact modes and three lubrication regimes (if the load further increases) with increasing load. Therefore, Figure 6 selects v = 90 cSt and considers the differences between lubricated contact and dry contact under different particle sizes. It is particularly important to note that when lubricating oil forms a film between the interfaces, the sequence of effects on A _t ^* due to changes in particle size is the opposite for lubricated contact compared to dry contact, which is explained as follows. The dry contact curve (dashed line) in Figure 6A shows that A _t ^* decreases with increase in the third particle size. This is because larger particles are more likely to separate the surfaces, with the load only borne by a few larger particles, resulting in A _t ^* being smaller than when x _a = 0 nm. Figure 4A shows that as A _t ^* of the lubricated interface begins to decrease and move away from A _t ^* of dry contact, the interface enters the mixed lubrication regime. Therefore, as shown in Figure 6A, after the interface enters the mixed lubrication regime, A _t ^* increases with the particle size at higher loads (approximately F _t ^* > 5.96 × 10⁻⁵). As depicted in Figure 6B, L _s of dry contact (dashed line) remains 100% for all particle sizes. In lubricated contact, L _s decreases as the dimensionless normal load increases, and smaller particles require only a smaller load to transition the contact interface to the mixed lubrication regime. At a dimensionless normal load, L _s increases with particle size. When the interface of all particle sizes enters mixed lubrication (approximately F _t ^* > 10⁻⁴), a larger L _s implies more solid contact, resulting in Figure 6A transitioning to A _t ^* increasing with particle size at higher loads. Figure 6C illustrates that the A _sa,t ^* of oil lubrication and dry contact almost overlap for L _s = 100%. The A _sa,t ^* of lubrication contact is larger than dry contact when the interface enters mixed lubrication. This difference arises because the oil film supports the surfaces, reducing the contact of surface asperities as the load is shared by the oil film and most of the particles. This is also a reason why nanoparticles are added to lubricants to help surfaces asperities share the load and reduce surface wear.

The concentration of wear debris on the interface or concentration of added micro/nano particles in the lubricant will both affect the contact characteristics. Therefore, Figure 7 discusses the effect of particle density (η _a ) on contact characteristics. Figure 7A shows that the A _t ^* of lubricated contact is similar to that of dry contact under small and medium dimensionless normal loads, and it decreases with increasing particle density. When F _t ^* > 5.0 × 10⁻⁵, A _t ^* transitions to increase with increasing particle density. It can be seen from Figure 7B that this change is caused by the transition of the lubrication regime from boundary to mixed. Figures 7B, C demonstrate that L _s and A _sa,t ^* decrease with increasing dimensionless normal load, while they increase with increasing particle density.

Combining the results from Figures 4 and 7 in lubricated contact interfaces, whether it is water content, particle size, or concentration causing lubricant degradation, particularly with larger particle sizes and concentrations, it tends to transition the interface into 3-body boundary lubrication p-s mode. These result in a reduced total real contact area and increased particle contact pressure and temperature, leading to pitting and wear or causing excessive interface gaps; this can result in unstable precision during component operation. When particle sizes are too small and particle densities are too low, insufficient lubrication at the interface can lead to it being in the s-s mode. This results in excessive contact of surface asperities on the components, leading to even more wear. The total real contact area increases with increasing particle size and density when the lubricated interface is in a 3-body mixed lubrication regime. It is noteworthy that this exhibits a contrasting trend compared to previous studies conducted under dry contact (Chern et al., 2022) and boundary lubrication conditions. Additionally, in lubricated contact analysis, the percentage of solid load decreases with increased total normal load and exhibits a similar trend to the analysis of Masjedi and Khonsari (2015).

Figure 8 discusses the influence of surface roughness on interface contact characteristics, particularly the turning point of contact area (TPCA) phenomenon. Figure 8A shows that for all equivalent root mean square surface roughness, the A _t ^* of lubricated contact overlaps with that of dry contact at smaller loads. The reason for this is evident from Figure 8B, where it can be observed that the L _s values are 100% for all cases. Under dry contact conditions, when the interface is in the p-s mode, A _t ^* increases with increased surface roughness. However, as the interface mode transitions to near s-s mode with increasing load, A _t ^* decreases with increased surface roughness. Under lubricated contact conditions, when the interface is in mixed lubrication and approaching the s-s mode, the trend is opposite to that observed when in the s-s mode under dry contact conditions (Figure 8A∼C). Additionally, from Figure 8C for σ = 500 nm, it is apparent that A _sa,t ^* exhibits a rise followed by a decrease as the load increases. This phenomenon is called “turning point of contact area” (TPCA) (Chern et al., 2022). The operating conditions under which TPCA occurs will be explored in Figures 9–13.

Figure 8 shows that the TPCA phenomenon occurs at x _a = 500 nm and σ = 500 nm (x _a /σ = 1.0). Therefore, Figure 9 investigates the range of x _a /σ values where TPCA occurs for dry contact with σ = 300 nm. Figure 9A shows that the TPCA phenomenon occurs for particle sizes ranging from approximately 250–400 nm. Figure 9B illustrates that when the interface experiences TPCA, A _t ^* exhibits a variation characterized by an initial decrease followed by an increase with increasing dimensionless normal load, as indicated by the red dashed line in the figure. If the interface does not experience TPCA, it will enter the p-s contact mode at low dimensionless normal loads (e.g., x _a = 350 nm). The results from Figure 9 indicate that the occurrence of the TPCA phenomenon occurs approximately within the range of x _a /σ = 0.83–1.33 under σ = 300 nm and η _a = 10¹¹ m ^-2. This phenomenon helps maintain the contact mode of the interface primarily in the hybrid mode, and the load is shared between the particles and the surface. This prevents the transition of the contact mode to the p-s mode, which would result in a relatively small A _t ^* and consequent excessively high real contact pressures in the contact area.

Figure 10 investigates the influence of different surface roughness on the x _a /σ range for the occurrence of TPCA under dry contact conditions. Figure 10A shows that the TPCA phenomenon occurs approximately in the range of x _a /σ = 0.80–1.20 when σ = 100 nm; for σ = 500 nm, it occurs in the range of x _a /σ = 0.80–1.30 (Figure 10B). From Figures 9 and 10, it is apparent that TPCA occurs approximately in the range of x _a /σ = 0.8–1.3, and the critical dimensionless normal load for TPCA (F _TPCA ^* ) decreases with the increase of x _a /σ. Although the range of x _a /σ values for TPCA occurrence is the same, the F _TPCA ^* varies for differing surface roughness and particle sizes even when the x _a /σ values are the same. Therefore, to achieve optimal load sharing between particles and surfaces in actual situations, it is still necessary to coordinate surface roughness to control the size of debris or micro/nanoparticles.

Figure 11 investigates the influence of lubricant kinematic viscosity on TPCA when x _a = 300 nm and σ = 300 nm (x _a /σ = 1.0). Under these conditions, the contact characteristic curves for v = 25 and 50 cSt overlap with dry contact and are therefore not shown in there. Figure 11A shows that when v < 90 cSt, F _TPCA ^* is almost the same (F _TPCA ^* = 3.27 × 10⁻⁶). L _s can be observed as the interface is in the boundary lubrication regime at this critical load (Figure 11B), indicating that viscosity has little effect on TPCA at this stage. The F _TPCA ^* = 1.49 × 10⁻⁵ when v = 120 cSt; the interface has entered mixed lubrication at this viscosity. These results indicate that when the interface enters the mixed lubrication regime, a larger dimensionless normal load is required to induce TPCA. Furthermore, it can be observed from Figure 11A that viscosity does not affect the maximum value of A _sa,t ^*. The maximum value of A _sa,t ^* remains approximately 0.20 across all viscosities. The reason for this can be observed from Figure 11B, where at v = 120 cSt with F _TPCA ^* = 1.49 × 10⁻⁵, L _s = 21.1%. The real dimensionless normal load borne by the solid (1.49 × 10⁻⁵ × 21.1% = 3.14 × 10⁻⁶) is comparable with that of dry contact (F _TPCA ^* = 3.27 × 10⁻⁶). Furthermore, as depicted in Figure 11C, different viscosities and total loads result in different H _c values (for v = 70 cSt, H _c = 1.15 × 10^-4-2.23 × 10⁻⁴; for v = 120 cSt, H _c = 1.63 × 10^-4-2.34 × 10⁻⁴). However, it is evident that A _sa,t ^* values remain on the same curve. This illustrates that the occurrence of the TPCA phenomenon is primarily determined by the real load borne by the solid. As long as the total load increases to make the real load borne by the solid equivalent to that in dry contact, the increase in lubricant viscosity will still lead to the occurrence of TPCA at the interface.

Figure 12 shows the contact characteristics versus dimensionless normal load at various viscosities of lubricant (A) A _sa,t ^* (B) L _s for σ = 100 nm, x _a = 100 nm (C) A _sa,t ^* for σ = 500 nm, x _a = 500 nm. Figure 12A demonstrates that as the viscosity increases, larger loads are required for the occurrence of TPCA, like in Figure 11. However, A _sa,t ^* equals 0 after the viscosity reaches 50 cSt, so the TPCA phenomenon is no longer observed. This is primarily evident from Figure 12B, where L _s drops to 0% after viscosity reaches 50 cSt, indicating that the interface has entered the full-film lubrication regime with no solid contact at all. In Figure 12C for σ = 500 nm and x _a = 500 nm, it can be clearly observed that regardless of the viscosity variation, the values of A _sa,t ^* almost overlap, and the F _TPCA ^* values are the same. The reason for this is evident from the L _s values under the same conditions in Figure 5B, where it can be observed that when the dimensionless normal load is less than 6 × 10⁻⁵, the L _s values are all 100% for all viscosities. Therefore, viscosity does not affect the occurrence of TPCA under these conditions.

Influence of particle density on TPCA at x_a/σ = 1.0. In the analyses of Figures 13A, 13B, and 13C, different lubricant kinematic viscosities of 25, 120, and 150 cSt are, respectively, employed, primarily considering the more significant variations in A_sa,t* caused by lubricant viscosity under different surface roughness conditions. Figure 13A shows that for σ = 100 nm, η_a has little effect on the F_TPCA* value, which remained approximately 4 × 10^–6 for different viscosities. Figure 13B shows that for σ = 300 nm, the F_TPCA* value increases from 1.5 × 10^–5 to 3.0 × 10^–5 when η_a increases to 5 × 10¹¹ m^–2. Figure 13C clearly shows that for σ = 500 nm, the F_TPCA* value increases with increasing η_a . The results from Panel indicate that as σ increases, the trend of F_TPCA* increases with increasing η_a becoming more pronounced. The analysis results from Figures 9–13 collectively indicate that F_TPCA* decreases with increasing x_a but increases with increasing v and η_a . Under the same x_a/σ ratio, both A_sa,t* and F_TPCA* values increase with increasing σ.