Contact Mechanics Of Rough Surfaces Research Articles

A two-scale stochastic model for the contact mechanics of rough surfaces including a wide span of length scales

Two-scale models are an effective tool to study mechanical problems in which features in a wide span of length scales must be considered. In the context of contact mechanics, however, this cannot be directly used due to a lack of scale separation, which results in a lack of representativeness of the local-scale results. In this work, a model is proposed that circumvents this issue by incorporating stochastically the variability of the local scale results into the global-scale problem. Several test cases are performed to show the performance of the model.

Effect of particle roughness on the bulk deformation using coupled boundary element and discrete element methods

Particles slide and roll on each other when a granular medium is sheared. Consequently, the tribological properties, such as inter-particle friction and adhesion, play a major role in influencing their bulk failure and rheology. Although the influence of roughness on adhesion and friction of contacting surfaces is known, the incorporation of the surface roughness in the numerical modelling of granular materials has received little attention. In this study, the boundary element method (BEM), which is widely used for simulating the mechanics of interacting surfaces, is coupled with discrete element method (DEM) and the bulk deformation of granular materials is analysed. A BEM code, developed in-house, is employed to calculate the normal force–displacement behaviour for rough contact deformations, based on which a contact model is proposed. This is an efficient and relatively fast method of calculating the contact mechanics of rough surfaces. The resulting model is then implemented in the simulations by DEM to determine the effect of micro-scale surface roughness on the bulk compression of granular materials. This study highlights the importance of the effect of surface characteristics on contact behaviour of particles for the case of shallow footing and provides an efficient approach for modelling the flow behaviour of a large number of rough particles.

Quantifying uncertainties in contact mechanics of rough surfaces using the multilevel Monte Carlo method

We quantify the effect of uncertainties on quantities of interest related to contact mechanics of rough surfaces. Specifically, we consider the problem of frictionless non adhesive normal contact between two semi infinite linear elastic solids subject to uncertainties. These uncertainties may for example originate from an incomplete surface description. To account for surface uncertainties, we model a rough surface as a suitable Gaussian random field whose covariance function encodes the surface’s roughness, which is experimentally measurable. Then, we introduce the multilevel Monte Carlo method which is a computationally efficient sampling method for the computation of the expectation and higher statistical moments of uncertain system output’s, such as those derived from contact simulations. In particular, we consider two different quantities of interest, namely the contact area and the number of contact clusters, and show via numerical experiments that the multilevel Monte Carlo method offers significant computational gains compared to an approximation via a classic Monte Carlo sampling.

Adhesive wear and particle emission: Numerical approach based on asperity-free formulation of Rabinowicz criterion

In 1953 Archard formulated his general law of wear stating that the amount of worn material is proportional to the normal force and the sliding distance, and is inversely proportional to the hardness of the material. Five years later in 1958, Rabinowicz suggested a criterion determining the minimum size of wear particles. Both concepts became very popular due to their simplicity and robustness, but did not give thorough explanation of the mechanisms involved. It wasn’t until almost 60 years later in 2016 that Aghababaei, Warner and Molinari (AWM) used quasi-molecular simulations to confirm the Rabinowicz criterion. One of the central quantities remained the “asperity size”. Because real surfaces have roughness on many length scales, this size is often ill-defined. The present paper is devoted to two main points: First, we generalize the Rabinowicz-AWM criterion by introducing an “asperity-free” wear criterion, applicable even to fractal roughness. Second, we combine our generalized Rabinowicz criterion with the numerical contact mechanics of rough surfaces and formulate on this basis a deterministic wear model. We identify two types of wear: one leading to the formation of a modified topography which does not wear further and one showing continuously proceeding wear. In the latter case we observe regimes of least wear, mild wear and severe wear which have a clear microscopic interpretation. The worn volume in the region of mild wear occurs typically to be a power law of the normal force with an exponent not necessarily equal to one. The method provides the worn surface topography after an initial settling phase as well as the size distribution of wear particles. We analyse different laws of interface interaction and the corresponding wear laws. A comprehensive parameter study remains a task for future research.

Boundary interactions of rough non-Gaussian surfaces

Surface topography is important as it influences contact load-carrying capacity and operational efficiency through generated friction, as well as wear. As a result, a plethora of machining processes and surface finishing techniques have been developed. These processes yield topographies, which are often non-Gaussian, with roughness parameters that alter hierarchically according to their interaction heights. They are also subject to change through processes of rapid initial running-in wear as well as any subsequent gradual wear and embedding. The stochastic nature of the topography makes for complexity of contact mechanics of rough surfaces, which was first addressed by the pioneering work of Greenwood and Williamson, which among other issues is commemorated by this contribution. It is shown that their seminal contribution, based on idealised Gaussian topography and mean representation of asperity geometry should be extended for practical applications where surfaces are often non-Gaussian, requiring the inclusion of surface-specific data which also evolve through process of wear. The paper highlights a process dealing with practical engineering surfaces from laboratory-based testing using a sliding tribometer to accelerated fired engine testing for high performance applications of cross-hatched honed cylinder liners. Such an approach has not hitherto been reported in literature.

A new strategy for the relative movement of rough surfaces in contact using a boundary element method

Contact mechanics of rough surfaces is becoming increasingly important in understanding the real behaviours of machine elements in contact. Due to the complicated physical, chemical and mechanical phenomena occurring at surfaces, especially in boundary lubrication, a multiphysics numerical model is essential to capture the behaviour. Boundary Element Method is a well-known numerical approach to model such a problem because of several advantages. Firstly, it is far faster than Finite Element Method since only the boundaries of the solids are discretised. In addition, there is no problem of remeshing the contacting bodies due to plastic deformation and wear. Conventional Boundary Element models simulate movement of contacting surfaces by shifting matrices of numbers in one direction. In this new proposed approach, the big matrices of surfaces are cut into small matrices which indicate the part of surfaces that are in contact. The influence matrix is also cut into a smaller square matrix corresponding to the size of surface matrices. This approach enables matrix implications in smaller sizes than the original big surfaces and reduces the computational time.

A multiscale method for frictionless contact mechanics of rough surfaces

An efficient methodology is proposed for the analysis of frictionless contact between rough surfaces. The surface is described by parabolic asperities which deform according to Hertz Theory. The problem is solved considering interactions between elliptic contact zones. Such analysis provides interface laws that are incorporated into a macroscopic numerical model where contact surfaces are flat. This operation is done by means of Love solution for elastic half spaces and the penalty method.A numerical example of this multiscale method is presented to show its robustness. In comparison with a purely numerical model where roughness is explicitly described, the proposed strategy provides good results and saves a considerable amount of time.

Rough contacts near full contact with a very simple asperity model

Recently, asperity theories in contact mechanics of rough surfaces have been extended in the range near full contact by elaborating on the idea of searching a corrective solution to the tensile regions of the full contact solution. The full contact solution defines a “surface” on which asperity theory can be applied. It is here shown that a very simple prediction can be made for the non-contact area, improving the accuracy near full contact with respect to the other simple solution available, that of Persson.

A Model for the Prediction of Thermal Response of Bone in Surgical Drilling

This paper develops a mathematical model for predicting the thermal response in the surgical drilling of bone. The model accounts for the bone, chip, and drill bit interactions by providing a detailed account of events within a cylindrical control volume enveloping the drill, the cut bone chip within the drill bit flute, and the solid bone. Lumped parameter approach divides the control volume into a number of cells, and cells within the subvolumes representing the drill solid, the bone chip, and the bone solid are allowed to interact. The contact mechanics of rough surfaces is used to model chip–flute and chip–bone frictional interaction. In this way, not only the quantification of friction due to sliding contact of chip–flute and chip–bone rough surface contact is treated but also the contact thermal resistances between the rubbing surfaces are included in the model. A mixed combination of constant and adaptive mesh is employed to permit the simulation of the heat transfer as the drill bit penetrates deeper into the bone during a drilling process. Using the model, the effect of various parameters on the temperature rise in bone, drill, and the chip is investigated. It is found that maximum temperature within the bone occurs at the location adjacent to the corner of the drill-tip and drill body. The results of the model are found to agree favorably with the experimental measurements reported within the existing literature on surgical drilling.

A new efficient numerical method for contact mechanics of rough surfaces

In this work, a numerical method has been developed to investigate the adhesionless contact mechanics between rough surfaces. To solve the elastic problem a boundary elements approach is used with self-equilibrated square elements. The domain of integration is discretized developing an “intelligent” adaptive mesh and obtaining a considerable memory saving. The numerical convergence of the method has been verified by comparing the results with the Hertzian solution and by checking the stress probability distribution at the contact interface. The methodology has been then utilized to analyse the contact between an elastic flat substrate and a periodic numerically generated self-affine fractal rigid surface. The fractal surface has been generated by employing spectral methods. The results of our investigation supports the findings of some analytical theories ( Persson, 2001) and numerical findings ( Yang et al., 2006; Hyun et al., 2004; Carbone and Bottiglione, 2008; Campana and Muser, 2007) in terms of linearity between contact area and load and stress probability distributions.

Contact mechanics of rough surfaces: a comparison between theories

In this paper the authors briefly review the most important theories of contact mechanics between randomly rough surfaces. The multiasperity contact theories are here shown under a unifying point of view and compared with the more recent approach developed by Persson. In particular we focus on the relation between the applied load and separation, which is fundamental in many engineering applications as seals, lubrication and wear. We found that the difference between the multiasperity contact models and Persson’s theory is not only quantitative, as in case of area-load linearity, but also qualitative. A brief comparison with existing experiments and numerical calculations shows that Persson’s predictions should be more accurate.

Influence of Surface Roughness, Material and Climate Conditions on the Friction of Human Skin

The gliding comfort and performance of personal care and wellness products is strongly influenced by the sliding friction behaviour of human skin. In the open literature, most of the results on skin friction are related to the performance of cosmetic products or to the slip and grip properties of surfaces. Experiments were usually carried out on the forearm or the fingertips. The influence of the surface roughness and the material of engineering surfaces have received little attention so far, especially not in sliding contact with the skin of the cheek, or under different climate conditions. A custom-built rotating ring device was used to study the influence of the probe surface roughness (R a = 0.1–10 μm), the probe material (metals, plastics), the climate conditions (21–29 °C, 37–92% RH) and skin hydration on the frictional behaviour of the skin on the cheek and the forearm. The amplitude of the surface roughness has a dominant influence on the friction behaviour: the smoother the surface, the higher the friction. Differences can be as large as a factor 5–10, especially in the range R a < 1 μm. The probe material itself has no significant influence; except for PFTE which reduces the friction by approximately 25% compared to the other materials. In a humid climate, the skin becomes hydrated and the friction is twice as high as in a dry climate. The effect of skin hydration is smaller on the cheek than on the forearm, probably due to the presence of beard stubbles. A simple friction model for human skin is presented, based on adhesion friction, contact mechanics of rough surfaces and the interfacial shear stress of thin organic films. The model explains the effects of the probe surface roughness and skin compliance. Quantitative application of the model indicates that the biomechanical indentation and shearing behaviour of the stratum corneum is influenced by the same physical process, i.e. the intercellular bonding strength of the corneocytes.

Asperity contact theories: Do they predict linearity between contact area and load?

During the last few years, the scientific community has been debating about which theory of contact between rough surfaces can be considered as the most accurate. The authors have been attracted by such a discussion and in this paper try to give their personal thought and contribution to this debate. We present a critical analysis of the principal contact theories of rough surfaces. We focus on the multiasperity contact models (which are all based on the original idea of Greenwood and Williamson (GW) [1966. Proc. R. Soc. London A 295, 300]), and also briefly discuss a relatively recent contact theory developed by Persson [2001. J. Chem. Phys. 115, 3840]. For small loads both asperity contact models and Persson's theory predict a linear relation between the area of true contact and the applied external load, but the two theories differ for the constant of proportionality. However, this is not the only difference between the two approaches. Indeed, we show that the fully calculated predictions of asperity contact models very rapidly deviates from the predicted linear relation already for very small and in many cases unrealistic vanishing applied loads and contact areas. Moreover, this deviation becomes more and more important as the PSD breadth parameter α (as defined by Nayak) increases. Therefore, the asymptotic linear relation of multiasperity contact theories turns out to be only an academic result. On the contrary, Persson's theory is not affected by α and shows a linear behavior between contact area and load up to 10–15% of the nominal contact area, i.e. for physical reasonable loads. The authors also prove that, at high separation, all multiasperity contact models, which take into account the influence of summit curvature variation as a function of summit height, necessarily converge to a (slightly) improved version of the GW model, which, therefore, remains one of the most important milestones in the field of contact mechanics of rough surfaces.

A “re-vitalized” Greenwood and Williamson model of elastic contact between fractal surfaces

Greenwood and Williamson in 1966 (GW) proposed a theory of elastic contact mechanics of rough surfaces which is today the foundation of many theories in tribology (friction, adhesion, thermal and electrical conductance, wear, etc.). However, the theory has periodically received criticisms for the “resolution-dependence” of the asperity features. Greenwood himself has recently concluded that: “The introduction by Greenwood and Williamson in 1966 of the definition of a ‘peak’ as a point higher than its neighbours on a profile sampled at a finite sampling interval was, in retrospect, a mistake, although it is possible that it was a necessary mistake” [Greenwood and Wu, 2001. Surface roughness and contact: an apology. Meccanica 36 (6), 617–630]. We propose a “discrete” version of the GW model, keeping the approximation of a surface by quadratic functions near summits, where the summit arrangement is found from numerical realizations or real surfaces scans. The contact is then solved either summing the Hertzian relationships, or considering interaction effects to the first-order in a very efficient algorithm. We conduct experiments on Weierstrass–Mandelbrot fractal surfaces, concluding that: (1) the real contact area–load relationship is well captured by the original GW theoretical model, once the correct mean radius is used. The relationship is robust and shows relatively little scatter; (2) the conductance–load relationship is vice versa only approximately given by the original GW theoretical model. Significant deviations from linearity and significant scatter seem to be found, particularly at low fractal dimensions; (3) the load, area and conductance dependences with separation show significant dependence on the actual phase arrangements, and hence significant scatter at large separations. Effect of interaction is seen strongly at low separations, where scatter is minimal. The discrete GW model permits to include these effects, except when the asperity description breaks down. Refinements of the original GW theory using the full random process theory (such as that by Bush Gibson and Thomas, BGT) result only in small improvements with a significant additional complication. However, the BGT relationship between contact area and load at low loads is more accurate than the more recent theory by Persson. The distribution derived from the original GW theory has been obtained, and shown to be closer to the numerical results than that predicted by the Persson model, even if the area error is removed. It is concluded that the original GW theory deserves the general success received so far, since the resolution-dependence of geometrical features is an intrinsic feature of “fractals” but not a problem for the GW theory, when interaction effects are included.

Contact mechanics of rough surfaces in tribology: multiple asperity contact

Contact modeling of two rough surfaces under normal approach and with relative motion is carried out to predict real area of contact and surface and subsurface stresses affecting friction and wear of an interface. When two macroscopically flat bodies with microroughness come in contact, the contact occurs at multiple asperities of arbitrary shapes, and varying sizes and heights. Deformation at the asperity contacts can be either elastic and/or elastic-plastic. If a thin liquid film is present at the interface, attractive meniscus forces may affect friction and wear. Historically, statistical models have been used to predict contact parameters, and these generally require many assumptions about asperity geometry and height distributions. With the advent of computer technology, numerical contact models of 3-D rough surfaces have been developed, particularly in the past decade, which can simulate digitized rough surfaces with no assumptions concerning the roughness distribution. In this article, a comprehensive review of modeling of multiple-asperity contacts in dry and wet conditions is presented. Contact models for homogeneous and layered, elastic and elastic-plastic solids with and without tangential loading are presented. The models reviewed in this paper fall into two groups: (a) analytical solutions for surfaces with well-defined height distributions and asperity geometry and (b) numerical solutions for real surfaces with asperities of arbitrary shape and varying size and height distributions. Implications of these models in friction and wear studies are discussed.

Contact Mechanics of Rough Surfaces in Tribology: Single Asperity Contact

Contact modeling of two rough surfaces under normal approach and with relative motion is carried out to predict the real area of contact which affects friction and wear of an interface. The contact of two macroscopically flat bodies with microroughness is reduced to the contact at multiple asperities of arbitrary shapes. Most of deformation at the asperity contact can be either elastic or elastic-plastic. In this paper, a comprehensive review of modeling of a single asperity contact or an indentation problem is presented. Contact analyses for a spherical asperity/indenter on homogeneous and layered, elastic and elastic-plastic solids with and without tangential loading are presented. The analyses reviewed in this paper fall into two groups: (a) analytical solutions, primarily for elastic solids and (b) finite element solutions, primarily for elastic-plastic problems and layered solids. Implications of these analyses in friction and wear are discussed.