Transition from elastohydrodynamic to mixed lubrication in highly loaded squeeze contacts

Abstract

We analyze the highly loaded strongly non-stationary squeeze process of an oil film sandwiched between an elastic spherical ball and a rigid rough substrate. We show that the coupling between the elastic properties of the contacting solids, the oil rheology, the surface roughness and the applied load determines a wide range of lubrication conditions from fully elastohydrodynamic to mixed and even boundary lubrication. In particular we find that increasing (decreasing) the surface roughness (the applied normal load) speeds up the squeeze process, anticipates and shrinks the time interval during which the transition to mixed lubrication conditions occurs. On the contrary, the initial separation between the approaching bodies only marginally affects the transition time. We also observe that, in mixed lubrication conditions, the highest asperity–asperity contact pressure occurs in the annular region where the separation between solids takes its minimum value. One then concludes that surface damage and wear should nucleate in the outer region of the contact.

Introduction

We present a relatively simple model to study the high loaded squeeze of a thin oil film interposed between two approaching rough stiff solids. The model predicts the all process from elastohydrodynamic lubrication (where the load is supported only by the fluid), to the boundary regime (where direct solid–solid contacts occur). The way asperity–asperity interaction is included in the model is based on an idea presented in Persson and Scaraggi (2009) where the steady sliding contact between soft elastic solids is investigated in presence of light loads. The presence of relatively small loads makes piezo-viscous and piezo-density effects, as well as oil viscoelasticity or shear-thinning phenomena absolutely negligible. However, in many engineering applications, lubrication conditions occur under heavy-loaded contact, as in the case of pin–pulley in continuously variable transmissions (Carbone et al., 2009a, Carbone et al., 2009b). In such conditions one needs to take into account piezo-viscous and piezo-density properties, as well as the real rheology of the lubricant to correctly predict the time-evolution of the hard-elastohydrodynamic squeezing process. In previous papers (Carbone et al., 2009a, Carbone et al., 2009b) we have treated such problems assuming different oil rheologies: (i) Newtonian (Carbone et al., 2009a), (ii) Maxwell viscoelastic, and (iii) Rabinowitsch viscous behaviors (Carbone et al., 2009b), and taking into account the piezo-viscous and piezo-density properties of the lubricant. We showed that in case of heavy loaded squeeze contacts, the spatial pressure distribution strongly differs from the classical Hertzian distribution and is characterized by a non-central annular pressure peak, which first appears in the external region of the contact and after moves toward the center of the pin with rapidly decreasing speed. The very high pressure in the central part of the contact causes a very large increase of viscosity, which in turn makes the oil very ‘stiff’. This stiffened oil then behaves as a rigid punch, which, as well known from the theory of elasticity, produces a very high pressure peak at the periphery of this circular high pressure region, which may give rise to high surface fatigue stress. We also showed that, at high pressures (of GPa order), increasing the load further decreases the squeeze rate as a consequence of the exponential increase of oil viscosity, till to stabilize the fully lubricated EHL conditions for small value of root mean square roughness (e.g. $0.1\mathrm{\mu }\mathrm{m}$ in the case of steel–steel contact, with an applied normal load 1 kN and an equivalent contact radius 0.1 m). However, we neglected the influence of asperity–asperity interaction which may occur because of surface roughness. In this work we make an attempt to overcome this difficulty and we propose a general model to analyze the oil squeeze from (elasto)hydrodynamic lubrication to boundary lubrication conditions, in terms of oil film thickness and pressure spatial distributions as functions of time. We stress that, despite of the strong interest in exploring the effect of roughness on the transition from boundary to hydrodynamic lubrication, at present there is not yet any established theory able to accurately describe that transition, (Dowson, 1998, Zhang, 2005). The main problem which makes the theoretical–numerical treatment so difficult is that all surfaces of practical interest have roughness over a very wide range of length scales, typically from mm to nm. This results in too many degrees of freedom that should be handled by (deterministic) numerical methods. For this reasons, nowadays, two approaches are mainly utilized to overcome the computational difficulties of deterministic methods. The first approach is based on spatial averaging (Patir and Cheng, 1978, Patir and Cheng, 1979, Li, 2000, Harp and Salant, 2001), the other on homogenization techniques (Sun, 1978, Almqvist and Dasht, 2006, Sahlin et al., 2007). Basically, the surface roughness is effectively removed (integrated out), resulting in effective equations of motion of the lubricant in between the elastic solids. Patir and Cheng, 1978, Patir and Cheng, 1979 were the first to adopt the concept of flow factors and to employ them into averaged Reynolds equations. The flow factors were determined by solving the fluid flow equation for a small interfacial rectangular cell of the rough surfaces and averaging over several realizations of the rough surfaces. The basic assumption in this approach is that the surface roughness occurs at length scales much shorter than the nominal contact size. In this case the flow factors described the influence of the surface roughness on the (average) fluid flow in the macroscopic junction. However, in this kind of approaches, the local elastic deformation (at the roughness scale) resulting from the asperity–asperity and asperity–fluid interaction is neglected, which may result in non-negligible errors (as recently shown in Meng et al., 2009), especially when the minimum oil film thickness h_min approaches the root mean square roughness h_rms. This should not surprise, as reducing the interfacial separation increases the number of direct asperity–asperity contacts which then can be neglected no longer. The same limitation also characterizes homogenization techniques as those presented in Almqvist and Dasht (2006) and Sahlin et al. (2007).

Here we propose a mean-field theory where the contact between the asperities is treated statistically by employing Persson's theory of contact mechanics (Persson, 2001, Persson, 2007), which allows to calculate the fraction of real solid–solid contact area as a function of the separation between the approaching solids. Persson's theory is preferred to multiasperity contact models (Greenwood and Williamson, 1966, Bush et al., 1975, Greenwood, 2006), since recent investigations (Carbone and Bottiglione, 2008, Carbone, 2009, Carbone et al., 2009c) have shown that it is more accurate than multiasperity models. In our approach we assume that the largest length scale of the surface roughness is much smaller than the macroscopic size of the contact area. This large separation of length scales, allows to treat, at the macroscopic level, the roughness-induced perturbation of pressure and film thickness spatial distributions through locally averaged solutions. This also will allow us to determine, at any given local position in the macroscopic contact area, the local fraction ${\alpha }_{\mathrm{c}}=A_{\mathrm{c}}/A_0$ of the area in direct solid–solid contact (where A₀ is the local nominal contact area—see Fig. 1), and the areal fraction $1-{\alpha }_{\mathrm{c}}$ where the two approaching solids are separated by the lubricant.