Elastoplastic contact of rough surfaces: a line contact model for boundary regime of lubrication

In this study, the shearing of the lubricant film in the contact conjunction (Fig. 2) is predicted through a simultaneous one-dimensional solution of Elrod’s equation and the elastic film shape. The length-to-width ratio of the experimental sliding strip is large, thus a one-dimensional solution of Elrod’s equation is assumed in the direction of the contact face-width, x. Ignoring any side leakage of the lubricant film normal to the direction of the entraining motion, then [4]: where d(θρ _c h)/dt refers to the squeeze film term, making for a transient analysis of the studied problem. The instantaneous contact kinematics is due to the speed of entraining motion, u _av and the squeeze film velocity, dh/dt. The time history of any formed lubricant film is, therefore, retained. The inclusion of this term also takes into account the vertical floatation of the ring strip, as described in Sect. 3.

To include the effect of cavitation at the trailing edge of the contact, Elrod [4] defined the contact pressure, p as a result of a fluid film, comprising some liquid lubricant content, θ. The implication in Elrods definition is that some fraction of the lubricant film content may be as a result of vapour or gaseous medium below the lubricant vaporisation/cavitation pressure, p _c . Therefore, with β being the lubricant bulk modulus. The term g is the switching term and can be expressed as: when 0 < θ < 1, the switching term g = 0, suggesting a two-phase flow below cavitation pressure.

The elastic film shape, h is:

With the initial contact profile, s(x) being defined as:

The lubricant viscosity–pressure variation can be predicted using the Roelands’ equation [36]: where \({\alpha } = ( \ln \eta _o + 9.67 ) \{ [ 1 + p/(1.98 \times 10^8 )]^Z - 1\}/p\) and \(Z = \alpha _o/[5.1 \times 10^{ - 9}( \ln \eta _o + 9.67)]. \)

Lubricant density variation with contact pressure, p is given by Dowson and Higginson [37] as:

The combined solution of Eqs. (1–7), described above, determines the contact pressure distribution, p(x, t) at any instant time. The solution clearly shows that along the cavitation region (g = 0), the Poiseuille flow term on the left hand side of the Eq. (1) diminishes. Therefore, the mass flow rate through the cavitation region is a balance between the Couette flow as a result of lubricant film entrainment and any mutual approach or separation of contiguous surfaces.

Engineering surfaces are microscopically rough, consisting of peaks (also known as asperities) and valleys. During contact, both surfaces of the studied conjunction are rough. Viscous friction dominates when the lubricant film is thick. However, when the lubricant film is thin and comparable to the surface roughness, asperity interaction can no longer be avoided. Hence, this would lead to boundary friction. As a result, the total friction for an element of apparent contact area, dA _app (=L.dx) is as: with df _b referring to the boundary friction component and df _ν as the viscous friction component, the viscous friction, df _ν for a Newtonian fluid can be computed as: where dA _app is the apparent contact area, dA _a is the actual contact area and τ = ηu _av /h(x).

Boundary friction arises from shearing of a very thin film, which prevails along contacting asperity tips. This molecular thin film is non-Newtonian and can be predicted using the classical Eyring thermal-activation model [38]. Therefore, the boundary friction, df _b can be expressed as: where τ _o referring to the Eyring shear stress of the lubricant, κ the pressure coefficient for boundary shear strength of the bounding surfaces and dW _act is the share of elemental contact load carried by the asperities.

Greenwood and Tripp [13] described the load carried by the asperities along a rough surface contact, dW _act and the actual contact area, dA _act as where the term ϕ _o refers to the asperity distribution and is described as follow:

The terms dP _o and dA _o are with w _p being the asperity interference as shown in Fig. 3.

In Eqs. (14) and (15), dP _a and dA _a refer to the single asperity pair load carrying capacity and the actual asperity-pair contact area. Assuming that the asperity contact is fully elastic, Greenwood and Tripp [13] applied the Hertzian theory to compute these two parameters. They also considered an elastic-fully plastic model to describe plastic deformation of asperities in contact. However, in this study, the terms dP _a and dA _a are predicted based on the approach proposed by Chong et al. [28] using the elastoplasticity model of Jackson and Green [23]. The model considers the elastoplasticity transition of contact deformation instead of assuming an elastic then fully plastic asperity deformation.

According to Chong et al. [28], when the normalised deflection d δ/δ _c is less than the deflection transition from elastic to elastoplastic, δ _t , Hertzian theory applies for both terms of dP _a and dA _a (Fig. 4). when d δ/δ _c  ≥ δ _t , the equations proposed by Jackson and Green [23] are applied to describe the load carrying capacity, dP _a and the actual contact area, dA _a of the asperity interaction.

The coefficients used in Eqs. (16) to (19) are explained and given in Appendix (Table 1).

From the modified Greenwood and Tripp model described above, the total friction, f _tot along the contact can be computed as

Figure 3 also shows that under certain conditions, asperities might come into contact along the shoulders. The asperities given by Greenwood and Tripps [13] approach and also in this paper are assumed to be hemispherical, having similar curvature radii. Based on the assumption, initial contact will occur midway between the centres of opposing asperities. For the misaligned asperities, which might interact along the shoulders, the normal forces will not be acting vertically and a tangential component will exist. However, according to Greenwood and Tripp [13], the slopes of asperities along physical rough surfaces are so small that the errors are miniscule and can be ignored. Hence, this leads to the assumption that all asperities will be normally loaded.

The integrals in Eqs. (11) and (12) of the modified Greenwood and Tripp friction model are improper integrals. Therefore, to solve these equations using the Gauss–Legendre quadrature, the limits of the integral are modified as follow where \(z = d+\bar{z}/(1-\bar{z}). \)

The integrals in Eqs. (14) and (15) describing both terms dP _o and dA _o have infinite intervals. Hence, these are solved numerically using the Gauss–Laguerre quadrature. The choice for solving the modified Greenwood and Tripp friction model numerically instead of finding an exact solution is to ensure flexibility in the use of desired single asperity pair interaction models in order to compute dP _a and dA _a .

The experimental techniques used comprise of a purpose built friction measurement rig. The reason for building this rig is twofold. The first aspect is for the validation of the presented friction. The second area of interest is to study various surface textures and their effect on friction reduction [35]. Other studies using similar rigs are also available in literature, such as the work by Ryk and Etsion [40]. All the experimental results shown in this paper correspond to the validation of the model only.

Figure 5 presents the test rig as a series of schematic, not to scale drawings. Figure 5a is a top view of the aforementioned rig. It attempts to describe the test bed in its most generic form. The optical table where the rig is placed is shown, together with the control mechanism (CPU). The control and data acquisition systems are both written within the Labview environment. A National Instruments DAQ-card is used to send and receive data from the CPU to the rig. The encoder placed at the end of the lead screw serves two main functions: (1) acts as a feedback to the control system and (2) synchronizes the data acquisition, saving friction and speed on an encoder count base. A proportional-integral-derivative (PID) controller was introduced in an attempt to minimize response times. The direction of motion is controlled by a digital channel.

The speed is accurately captured with a two-beam laser vibrometer and the friction values are measured with the aid of two load sensors. These are located at both ends of the substrate plate and pre-loaded to eliminate any possible excess clearances. The test sample is rigidly clamped onto the substrate plate. The reciprocating slider part holds the flat test ring.

Figure 5b shows a side view of the assembly. In this image, the test sample and substrate plates are presented in a clearer manner, together with their corresponding load sensors. Also, the loading mechanism is introduced. To further understand this function, one should refer to Fig. 5c, where a detailed view of the reciprocating slider/ring holder is shown. The test ring is attached to the floating ring holder. The ring holder is essentially a cylinder inserted onto the reciprocating slider. It is only allowed to float in the vertical direction; all other degrees of freedom are constrained. In this manner, a load can be applied, retaining the fundamental squeeze action.

The bottom plate tested is made of EN 14 steel and ground to an Rq value of 0.29 μm, with a flatness of 1 μm along the test area (26 × 35 mm²). The counterpart sliding strip is made of hardened 440 C stainless steel, with a hardness of 54 Rockwell C. A parabolic profile of crown height radius 31 mm was then cylindrically ground on the face-width of the strip, resembling that of a compression ring in an internal combustion engine piston. The roughness value of the strips profile was measured to be 0.18 μm (Rq). The overall size of the strip is set to 1 × 31 × 17 mm, where the profile was introduced along the 1 × 31 mm edge (contact strip).

Before proceeding to predict friction for a line contact slider bearing problem, the study first looks at the elastoplastic single asperity friction model. Figure 7a shows the actual contact area variation as a function of contact load for a single asperity contact. It is observed that with the initiation of plasticity, within the asperity contact (dδ/δ _c  > δ _t ), an increase in contact load produces a larger contact area as compared with an assumed elastic contact. This contact characteristic reflects the decrease in load carrying capacity of the asperity contact once plasticity is initiated.

By distributing the single asperity contact along a rough surface using Greenwood and Tripp’s approach for rough surfaces, Fig. 7b illustrates the change in the actual contact area with an increase in the applied load. The contact area predicted for the rough surface contact is larger for the elastoplastic model than that of an elastic model, which is based on Hertzian theory. This shows that the elastic deformation assumption underestimates the contact area for a given applied load when plasticity is initiated. Hence, this would produce inaccuracies in the prediction of the frictional characteristics along the rough surface contact. Figure 7c shows the contact area bearing ratio (actual contact area/apparent contact area) with respect to decreasing surface separation. Again, an elastic approach seems to be misleading, since it over-estimates the separation of bodies; leading to an under-estimation of friction.

To predict friction for an element of a rough surface, an experimental value for the term κ, referring to the pressure coefficient for boundary shear strength of the bounding surfaces is required (Eq. 10). This is obtained empirically using the AFM by applying the lateral force microscopy (LFM) approach. The tip radius for the silicon nitride tip (DNP-10) is small (≈60 nm) as compared to the surface roughness of the test plate (≈0.33 μm). Hence, this reduces the influence of surface roughness (friction arising predominantly as a result of boundary interaction) when measuring the friction using the AFM, leading to a more accurate measurement for the κ term.

In this study, the slider rig represents a realistic contact while the AFM tip resembles an asperity contact. Hence, the Veeco 3.5 nano scope AFM with a silicon nitride tip (DNP-10) is used to measure the pressure coefficient, κ on the test plate. Calibration of the tip is conducted using a silicon wafer by adopting the approach proposed by Buenviaje et al. [41]. To take into account the influence of the lubricant shear along asperity interactions, it is appropriate to measure the pressure coefficient of the boundary shear strength in a lubricated condition using the fluid imaging approach for the AFM. The friction force obtained from the AFM measurements at different applied loads are plotted in Fig. 8. The slope of the friction-load curve gives the value of κ required to predict the boundary friction. One thing to note is that the intersection of the friction-load curve is neglected when defining the pressure coefficient, κ. This is because the intersection magnitude in nano-metric range will be minute compared with the applied load and is mathematically deemed to be negligible.

Using the measured contact parameters in Appendix (Table 1), the sliding speed profiles for the simulated contact as a function of encoder count, under three separate sliding conditions are shown in Fig. 9. These are used as input for the contact kinematics in the numerical model, simulating the real time experiment conditions of the contact.

Elrod’s equation is solved numerically with an initially assumed elastic Greenwood and Tripp friction model. Figure 10a shows the pressure distribution of the contact peaks at different magnitudes for varying sliding speed profiles (along location X in Fig. 9). The reduction in peak pressure is a result of the increasing squeeze film effect with higher entraining motion of the contact, leading to a thicker lubricant film formation, hence reducing boundary friction of the contact, which is reflected in Fig. 10b, c. In them, the boundary shear component decreases significantly, while viscous shear increases with entraining speed of the contact. One thing to note is that in the cavitation region, rupture of fluid film occurs, leading to a discontinuity in viscous shear (Fig. 10b).

By plotting the friction coefficient as a function of the Stribeck oil film parameter, λ _s in Fig. 11, the simulated sliding speed profiles are observed to shift from mixed towards boundary lubrication regime as the speed decreases. The shift towards boundary lubrication is expected, because the decreasing sliding speed reduces the entrainment of lubricant into the contact, leading to a thinner lubricant film. By varying the speed, it is noted that a thicker oil film, enhancing the effect of viscous friction, may govern the conjunctional behaviour. Therefore, a much slower speed profile is preferred, where the fluid film formation is assessed by means of Elrod’s cavitation algorithm, retaining the effect of elastoplasticity. The validation for the new modified Greenwood and Tripp model, presented here, is solely conducted on speed profile A, where a boundary regime of lubrication is clearly attained.

Figure 12a compares the predicted friction using the modified Elrod and the Reynolds’ solution, based on the sliding speed profile A, with the measured friction values. It is observed that by assuming elastic deformation of asperities, the predicted friction by both methods (the modified Elrod and Reynolds’ solution) is lower than that measured. By plotting the friction coefficient as a function of the Stribeck oil film parameter, λ _s , the experiment conducted using the precision slider mechanism is shown to be well into the boundary lubrication regime (Fig. 12b). This shows that boundary interactions of asperities dominate the underlying friction mechanism in the studied conjunction. Figure 12c plots the friction coefficient against the Stribeck number, comparing the numerical results with the experimental measurement.

Figure 12b clearly shows that the contact resides in boundary lubrication regime and it seems crucial to be able to evaluate the influence of elastoplastic deformation of the interacting asperities. Considering only the modified Elrod’s solution and comparing the numerical prediction to the measured data, Fig. 13a shows that the friction force given by the elastoplastic model correlates reasonably well with the experimental values (particularly if compared to the elastic model). This indicates that elastoplastic deformation of interacting asperities plays a significant role in the frictional characteristics of the studied tribological conjunction. Figure 13b illustrates the friction coefficient in the function of the Stribeck number comparing the experimental result with the numerical ones. The boundary shear stress for both elastic and elastoplastic asperity deformation are shown in Fig. 13c. The viscous shear predicted for both friction models is the same, because the minimum film thickness given along location X remains unaltered. However, when elastoplasticity is considered, the boundary shear predicted increases when compared with the case of an elastic contact friction model. This is because in the operating condition investigated here, plasticity is initiated, giving a larger contact area, leading to higher contact friction. This observation is reinforced through the measured topography for the unworn and worn surfaces (Fig. 14). It is noted that in the worn surface, the asperity peaks and valleys have been smoothened, indicating a high boundary interaction of contiguous solids.