Michele Scaraggi

Adhesive contact of rough surfaces: Comparison between numerical calculations and analytical theories

The authors have employed a numerical procedure to analyse the adhesive contact between a soft elastic layer and a rough rigid substrate. The solution to the problem, which belongs to the class of the free boundary problems, is obtained by calculating Green’s function which links the pressure distribution to the normal displacements at the interface. The problem is then formulated in the form of a Fredholm integral equation of the first kind with a logarithmic kernel. The boundaries of the contact area are calculated by requiring the energy of the system to be stationary. This methodology has been employed to study the adhesive contact between an elastic semi-infinite solid and a randomly rough rigid profile with a self-affine fractal geometry. We show that, even in the presence of adhesion, the true contact area still linearly depends on the applied load. The numerical results are then critically compared with the predictions of an extended version of Persson’s contact mechanics theory, which is able to handle anisotropic surfaces, as 1D interfaces. It is shown that, for any given load, Persson’s theory underestimates the contact area by about 50% in comparison with our numerical calculations. We find that this discrepancy is larger than for 2D rough surfaces in the case of adhesionless contact. We argue that this increased difference might be explained, at least partially, by considering that Persson’s theory is a mean-field theory in spirit, so it should work better for 2D rough surfaces rather than for 1D rough surfaces. We also observe that the predicted value of separation is in agreement with our numerical results as well as the exponents of the power spectral density of the contact pressure distribution and of the elastic displacement of the solid. Therefore, we conclude that Persson’s theory captures almost exactly the main qualitative behaviour of the rough contact phenomena.

On the transition from boundary lubrication to hydrodynamic lubrication in soft contacts.

We consider the contact between elastically soft solids with randomly rough surfaces in sliding contact in a fluid, which is assumed to be Newtonian with constant (pressure-independent) viscosity. We discuss the nature of the transition from boundary lubrication at low sliding velocity, where direct solid-solid contact occurs, to hydrodynamic lubrication at high sliding velocity, where the solids are separated by a thin fluid film. We consider both hydrophilic and hydrophobic systems, and cylinder-on-flat and sphere-on-flat sliding configurations. We show that, for elastically soft solids such as rubber, including cavitation or not results in nearly the same friction.

Lubrication in soft rough contacts: A novel homogenized approach. Part I - Theory

We study the lubricated steady sliding contact between rough surfaces of (elastically) soft solids. A novel mean field theory of mixed lubrication is presented, which takes into account the coupled effect of asperity–asperity and asperity–fluid interactions. We calculate the fluid flow factors, and discuss the nature of the transition from the boundary lubrication regime, where the normal load is supported by the asperity–asperity interactions (sometimes mediated by boundary films), to the hydrodynamic regime, where a thin fluid film prevents direct contact between the mating surfaces.

Friction and universal contact area law for randomly rough viscoelastic contacts

We present accurate numerical results for the friction force and the contact area for a viscoelastic solid (rubber) in sliding contact with hard, randomly rough substrates. The rough surfaces are self-affine fractal with roughness over several decades in length scales. We calculate the contribution to the friction from the pulsating deformations induced by the substrate asperities. We also calculate how the area of real contact, A(v, p), depends on the sliding speed v and on the nominal contact pressure p, and we show how the contact area for any sliding speed can be obtained from a universal master curve A(p). The numerical results are found to be in good agreement with the predictions of an analytical contact mechanics theory.

Textured Surface Hydrodynamic Lubrication: Discussion

We discuss on a recently presented theory of textured surface hydrodynamic lubrication (Scaraggi, Phys Rev E, 2012). The model, based on the Bruggeman effective medium approach, allows to analytically evaluate the effects of a generic texture shape, distribution, and area density on the macroscopic hydrodynamic characteristics of the contact, such as friction and supported load. In this study, we apply the cited theory to practical cases, and in particular we derive the flow and shear stress tensors for two limiting conditions, i.e., for isotropic (circular inclusion in isotropic medium) and perfectly anisotropic (infinite slit inclusion) flow conductivities. These results are then used to perform near-optimum design calculations for the simplest case of one- and two-dimensional thrust-bearing geometries. Finally, a comparison with published results is presented and discussed. The developed theory may be a very useful tool in the process of evaluating the lubrication performances of sliding microtextured surfaces and for the near-optimum design of a textured pair, where texturing could be achieved by both physical (e.g., microstructuring) and chemical surface manipulation.

Lubricated sliding dynamics: Flow factors and Stribeck curve

We study the fluid flow at the interface between elastic solids with randomly rough surfaces. We derive (approximate) analytical expressions for the fluid flow factors which enter in the equation describing the fluid flow, and for the frictional shear stress factors which enter in the equation for the frictional shear stress. Numerical results for a rubber cylinder with surface roughness sliding on a flat lubricated substrate, under “low” and “high” pressure conditions, are presented and discussed. Finally we discuss the role of the fluid-induced elastic deformations of the surface roughness profile.

Lubrication in soft rough contacts: A novel homogenized approach. Part II - Discussion

In M. Scaraggi, G. Carbone, B. N. J. Persson and D. Dini, Soft Matter, 2011, 7, DOI: 10.1039/c1sm05128h (Ref. 1), we have presented a novel mean field theory of mixed lubrication for soft contacts. This theory has been developed within the framework of homogenization techniques. It is based on a perturbation second order accurate expansion; a rigorous averaging process is applied to obtain the average flow equations without the need of any preliminary fully deterministic calculations of the mixed-lubrication problem. The theory allows to calculate the flow factors and the shear stress factors taking into account the coupled effect of asperity–asperity and asperity–fluid interactions occurring at the interface. In this second paper we present the main results of the theory and we discuss the transition from the boundary lubrication regime, where the normal load is supported through asperity–asperity interactions (sometimes mediated by boundary films), to the hydrodynamic regime, where a thin fluid film prevents the surfaces from being in direct contact. We show that fluid-induced asperities flattening, as well as local percolation effects and roughness anisotropic deformation all occur at the contact interface. Finally we describe the origin of the macroscopic friction laws (the so called Stribeck curves) in terms of local average shear stresses.

Time-Dependent Fluid Squeeze-Out Between Soft Elastic Solids with Randomly Rough Surfaces

We study the time dependency of the interfacial separation and of the area of real contact between a soft elastic cylinder and a rigid solid with a randomly rough surface, squeezed with normal approach in a fluid. This problem is relevant for biological, as well as for bio-medical, seals and tires applications. An ad-hoc numerical scheme is developed to solve the transient mixed-EHD homogenized lubrication problem. We show that, for the soft contact case, the transition from the EHD to the boundary regime can be much more efficiently studied within a simplified (Grubin-like) problem formulation then with the full numerical scheme. This is not only of great conceptional importance, but also of practical importance as the latter calculation is much simpler and faster than the full scheme calculation.

General contact mechanics theory for randomly rough surfaces with application to rubber friction

We generalize the Persson contact mechanics and rubber friction theory to the case where both surfaces have surface roughness. The solids can be rigid, elastic, or viscoelastic and can be homogeneous or layered. We calculate the contact area, the viscoelastic contribution to the friction force, and the average interface separation as a function of the sliding speed and the nominal contact pressure. We illustrate the theory with numerical results for the classical case of a rubber block sliding on a road surface. We find that with increasing sliding speed, the influence of the roughness on the rubber block decreases to the extent that only the roughness of the stiff counter face needs to be considered.

Optimal Textures for Increasing the Load Support in a Thrust Bearing Pad Geometry

We present the calculation results of optimal texture geometries which maximize the supported load for a three-dimensional thrust bearing pad. By making use of the recently developed mean field theory of texture hydrodynamics, we develop an efficient multigrid optimization procedure based on the sequential genetic and conjugate gradient optimization. We show that our model allows to determine optimal solutions based on a two-scale hierarchy of structures, and the existence of particularly effective optimal geometries is presented and discussed.