Rough Surface Contact Model Research Articles

The Mechanism of Layer Stacked Clamping (LSC) for Polishing Ultra-Thin Sapphire Wafer.

Double-sides polishing technology has the advantages of high flatness and parallelism, and high polishing efficiency. It is the preferred polishing method for the preparation of ultra-thin sapphire wafer. However, the clamping method is a fundamental problem which is currently difficult to solve. In this paper, a layer stacked clamping (LSC) method of ultra-thin sapphire wafer which was used on double-sides processing was proposed and the clamping mechanism of layer stacked clamping (LSC) was studied. Based on the rough surface contact model of fractal theory, combining the theory of van der Waals force and capillary force, the adhesion model of the rough surfaces was constructed, and the reliability of the model was verified through experiments. Research has found that after displacement between the two surfaces the main force of the adhesion force is capillary force. The capillary force decreases with the increasing of surface roughness, droplet volume, and contact angle. For an ultra-thin sapphire wafer with a diameter of 50.8 mm and a thickness of 0.17 mm, more than 1.4 N of normal adhesion force can be generated through the LSC method. Through the double-sides polishing experiment using the LSC method, an ultra-thin sapphire wafer with an average surface roughness (Ra) of 1.52 nm and a flatness (PV) of 0.968 μm was obtained.

Mixed elastohydrodynamic lubrication analysis of line contact with Non-Gaussian surface roughness

An approach for mixed elastohydrodynamic lubrication (EHL) analysis of line contact with non-Gaussian rough surface is proposed, and a minimum film thickness formula is derived based on the simultaneous solutions. The Patir and Cheng's average flow model and Kogut and Etsion's rough surface contact model is used to obtain hydrodynamic pressure and asperity pressure. The effect of non-Gaussian rough surface on pressure, film thickness and asperity-to-total load ratio are studied. It is observed that the influence of kurtosis on pressure and film thickness is slight. Both skewness and kurtosis have strong effect on asperity-to-total load ratio. Lastly, a minimum film thickness is derived and expresses as a function of dimensionless load, speed, material, standard deviation and skewness.

Simulating tribological characteristics of Palm Methyl Ester (PME) lubricated contact

The study simulates the lubricant Stribeck curve for Palm Methyl Ester (PME) by coupling modified Reynolds solution with Greenwood and Tripp’s rough surface contact model. The predicted lubricant Stribeck curves for PME is validated with measured data from a pin-on-disc tribometer. The Reynolds equation is modified to accommodate for the lubricant properties of PME (e.g. viscosity and density), which is mathematically described using Gibbs energy additivity approach. Solving the modified Reynolds equation would then provide the fluid film formation behaviour of PME, such as the contact pressure and film thickness. These fluid film parameters are then used as the input to determine the boundary and viscous friction of the investigated lubricated contact using Greenwood and Tripp’s rough surface contact model. The proposed mathematical solution correlates well with experimental data and is shown to be capable of predicting lubricant Stribeck, capturing the frictional behavior for the whole range of lubrication regimes. The findings of the present study prepare for a mathematical foundation to further explore the use of biodiesel as an alternative biodegradable lubricant to mineral-based ones.

A fractal model of contact between rough surfaces for a complete loading–unloading process

The mechanical properties of contact between rough surfaces play an important role in the reliability of the electromechanical system. In order to improve the design accuracy of precision instruments, an elastic-plastic contact model for three-dimensional rough surfaces based on the fractal theory is developed for a complete loading–unloading process based on the Majumdar and Bhushan model. The truncation size distribution functions of asperities for different values of asperity level in the loading process are given. Relationships between true contact area and total contact load in the complete loading–unloading process are obtained according to the truncation size distribution functions of asperities. The results show the range of asperity levels has significant effects on contact mechanical behaviors of fractal rough surfaces. When the first six levels of asperities do not exceed the critical elastic level, the fractal rough surfaces exhibit elastic behavior in a complete contact process, and the load–area relationships in the loading and unloading processes are coincident approximately. When the critical elastic level is less than the minimum level of asperity, the inelastic deformation begins to appear in fractal rough surfaces and the true contact area during the unloading process is always greater than the true area during the loading process for a given total contact load. In comparison with the K-K-E model, the present model is proved to be reasonable.

A comparison of nanoscale measurements with the theoretical models of real and nominal contact areas

In this study, an experimental method is proposed to measure the real area of contact between an alumina sphere and an Al surface based on the adhesive transfer of the Au film and the scanning electron microscope in the back-scattered mode. A thin film of Au is sputtered on the alumina sphere before the indentation with the Al surface. After indentation, the interfaces of the alumina sphere and Al surface are observed by the scanning electron microscope. The contact area can be identified based on both the distributions of the alumina and Au on the alumina sphere and Al surface, respectively. The measured contact area at different nominal pressures are compared to predictions made by several popular theoretical elastic-plastic rough surface contact models.

Exact Spectral Moments and Differentiability of the Weierstrass-Mandelbrot Fractal Function

Abstract Fractal mathematics using the Weierstrass-Mandelbrot (WM) function has spread to many fields of science and engineering. One of these is the fractal characterization of rough surfaces, which has gained ample acceptance in the area of contact mechanics. That is, a single mathematical expression (the WM function) contains characteristics that mimic the appearance of roughness. Moreover, the “roughness” is “similar” across large dimension scales ranging from macro to nano. The field of contact mechanics is largely divided into two schools of thought: (1) the roughness of real surfaces is essentially random, for which stochastic treatment is appropriate, and (2) surface roughness can be reduced to fractal mathematics using fractal parameters. Under certain mathematical constraints, the WM function is either stochastic or deterministic. The latter has the appeal that it contains no randomness, so fractal mathematics may offer closed-form solutions. Spectral moments of rough surfaces still apply to both approaches, as these represent physical metrology properties of the surface standard deviation, slope, and curvature. In essence, spectral moments provide a means of data reduction so that other physical processes can subsequently be applied. It is well known, for example, that the contact model of rough surfaces, by Greenwood and Williamson (GW), depends on parameters that are direct outcomes of these moments. Despite the vast amount of publications on the WM function dedicated to surfaces, two papers stand out as originators, where the others mostly rework their results. These two papers, however, contain some omissions and approximations that may lead to gross errors in the estimation of the spectral moments. The current work revisits these papers and adds information, but departs in the mathematical treatment to derive exact expressions for the said moments. Moreover, it is said that the WM function is nondifferential. That is also revisited herein, as another approach to derive the spectral moments depends on such derivatives. First, the complete mathematical treatment of the WM function is made, then the spectral moments are derived to yield exact forms, and finally, examples are given where the physical meanings of the approximate and exact moments are discussed and their values are compared. Numerical procedures will be introduced for both, and the effectiveness of the computational effort is discussed. One numerical procedure is particularly effective for any digitized signal, whether that originates from analytical functions (e.g., WM) or real surface measurements.

Predictive tool for frictional performance of piston ring-pack/liner conjunction

Fundamental understanding of piston ring-pack lubrication is essential in reducing engine friction. This is because a substantial portion of engine frictional losses come from piston-ring assembly. Hence, this study investigates the tribological impact of different piston ring profiles towards engine in-cylinder friction. Mathematical models are derived from Reynolds equation by using Reynolds’ boundary conditions to generate the contact pressure distribution along the complete piston ring-pack/liner conjunction. The predicted minimum film thickness is then used to predict the friction generated between the piston ring-pack and the engine cylinder liner. The engine in-cylinder friction is predicted using Greenwood and Williamson’s rough surface contact model. The model considers both the boundary friction and the viscous friction components. These mathematical models are integrated to simulate the total engine in-cylinder friction originating from the studied piston ring-pack for a complete engine cycle. The predicted minimum film thickness and frictional properties from the current models are shown to correlate reasonably with the published data. Hence, the proposed mathematical approach prepares a simplistic platform in predicting frictional losses of piston ring-pack/liner conjunction, allowing for an improved fundamental understanding of the parasitic losses in an internal combustion engine.

A statistical model of elastic-plastic contact between rough surfaces

In a mechanical interface, the contact surface topography has an important influence on the contact stiffness. In the contact processes of asperities, elastic-plastic change can lead to discontinuity and lack of smoothness at a critical contact point. The result is a large difference between the elastic-plastic deformation and the actual asperity deformation. Based on Hertz contact theory, the heights of asperities on a rough surface obey a Gaussian distribution. To take into consideration the continuity of elastic-plastic asperity deformation, we divide the elastic-plastic deformation into three stages: pre-elastic-plastic, mid-elastic-plastic, and post-elastic-plastic deformation. This establishes an elastic-plastic contact model of asperity at a continuous critical point. The contact model of a single asperity fits well with the Kogut–Etsion model and the Zhao–Maietta–Chang model, and the variation trend is consistent. At a lower plastic index, the present model coincides with classical models of contact area and contact load. At a higher plastic index, the simulation results of the present model differ from the Greenwood–Williamson model and the Chang–Etsion–Bogy model but are similar to results from the Kogut–Etsion and Zhao–Maietta–Chang models. This study provides a more accurate microscopic contact model for rough surfaces and a theoretical framework for interface design and analysis.

The elastic-plastic contact behavior of rough surfaces with hard coatings

Abstract An elastic-plastic coated rough surface contact model is presented, which incorporates existing single coated asperity contact models in a GW-based statistical multi-coated-asperity surface model. Effects of the coating thickness and its material properties as well as the substrate surface roughness on the mode of the coated surface contact deformation and its contact behavior are investigated through a qualitative parametric analysis. An equivalent plasticity index for the coated surface contact is defined to determine the mode of the coated surface contact deformation, which mainly depends on the coating material properties and the substrate surface roughness. It is found that a coated surface with a thicker, stiffer, harder coating and a rougher substrate has a higher separation and smaller real contact area under a given load.

Numerical Running-In Method for Modifying Cylindrical Roller Profile Under Mixed Lubrication of Finite Line Contacts

A numerical method for modifying cylindrical roller profile was proposed to smooth axial pressure distributions of finite line contacts under the mixed lubrication regime. The mixed lubrication model, in which the Reynolds equation modified by Patir and Cheng has been solved with implementing the rough surface contact model of Kogut and Etsion for the stochastic solution of hydrodynamic pressure and asperity-contact pressure, was established and it is validated by the comparison between simulation results and experiments. Some common roller profiles were carried into the mixed lubrication model and obvious increment of pressure appears near the roller ends or at the central contact area. A numerical running-in method was developed to smooth pressure shapes and the crown drop of roller profile was modified gradually implementing Archard's wear law, where a higher asperity-contact pressure leads to a larger crown drop on a roller profile. The results of the numerical running-in method indicated that pressure distributions of finite line contacts are uniform if the optimized roller profile is employed.

Normal Contact Model for Elastic and Plastic Mechanics of Rough Surfaces

An elastic and plastic mechanical model is proposed to characterize the normal contact of rough surfaces. The contact process is considered as three separated contact regimes, i.e., the fully elastic, mixed elastic–plastic and fully plastic ones. The Hertzian elastic contact solutions and the classical fully plastic contact model of Abbott and Firestone are used to model the contact behaviors of fully elastically deformed asperities and fully plastically deformed ones, respectively. For the mixed elastic–plastic regime, an improved Hermit interpolation method is applied to enforce the continuity and smoothness at the critical contact interference, and to decrease the interpolation waving effect by transforming the physical quantities of contact load into logarithmic coordinates. The contact model of rough surfaces is formulated by using the Greenwood and Williamson’s statistics analysis. Comparisons with the CEB, ZMC and KE models are then performed to examine the effects of plasticity index and mean separation. Agreeing well with the ZMC model, our results show that the total contact load of rough surfaces decreases with the increase in mean separation, and the difference between the prediction results of elastic models and the plastic ones also decreases, indicating that the contact behaviors mainly dominated by plastically deformed asperities have been changed to be controlled by elastically deformed ones. Larger plasticity index represents more plastically deformed asperities, inducing normal contact behaviors mainly dominated by plastic solutions.

Theoretical and Finite Element Analysis of Static Friction Between Multi-Scale Rough Surfaces

The current work considers the multi-scale nature of roughness in a new model that predicts the static friction coefficient. This work is based upon a previous rough surface contact model, which used stacked elastic–plastic 3-D sinusoids to model the asperities at multiple scales of roughness. A deterministic model of a three-dimensional deformable rough surface pressed against a rigid flat surface is also carried out using the finite element method (FEM). The accuracy of the deterministic FEM model is also considered. At the beginning of contact, which is surface-point contact, the asperities or peaks are isolated, sharp, and the contact areas consist of an inadequate number of elements and sources of error. In this range of contact, the results are not presented as real or accurate. As the normal load increases, the number of the contact elements become larger, and thus, the results become more accurate. That is, the deterministic FEM results are most accurate at high loads. Spectral interpolation is used to smooth the geometry in between the original measured nodes. The effects of normal load and plasticity index on static friction are then analyzed. The results predicted by the theoretical model are also compared to other existing rough surface friction contact models and the FEM results. They are in a good qualitative agreement, especially for higher loads and higher plasticity indices. The FEM model also has significant error, but it is more accurate at higher loads where the proposed multi-scale static friction model and FEM model are in better agreement.

Frictional characteristics of sheet metals with superimposed ultrasonic vibrations

The forming performance of sheet metals in the deep-drawing process with ultrasonic vibrations can be improved by the surface effect between the sheet metal and the die. A sheet metal friction test with ultrasonic vibrations is performed to explore the cause of the surface effect. The frictional characteristics are investigated, and the corresponding friction expressions are established based on the contact mechanics and the elastic—plastic contact model for rough surfaces. Friction is caused by the elastic—plastic deformation of contacting asperities under normal loads. The actual contacting region between two surfaces increases with normal loads, whereas the normal distance decreases. The normal distance between the contacting surfaces is changed, asperities generate a tangential deformation with ultrasonic vibrations, and the friction coefficient is eventually altered. Ultrasonic vibrations are applied on a 40Cr steel punch at the frequency of 20 kHz and the amplitude of 4.2 μm. In the friction tests, the punch is perpendicular to the surface of the magnesium alloy AZ31B sheet metals and is sliding with a relative velocity of 1 mm/s. The test results show that the friction coefficient is decreased by approximately 40% and the theoretical values are in accordance with the test values; Ultrasonic vibrations can clearly reduce wear and improve the surface quality of parts.

Numerical Prediction of Surface Wear and Roughness Parameters During Running-In for Line Contacts Under Mixed Lubrication

A stochastic model for predicting the evolutions of wear profile and surface height probability density function (PDF) of initial line contacts during running-in under mixed lubrication condition is presented. A numerical approach was developed on the basis of stochastic solution of mixed lubrication, which combined the Patir and Cheng's average flow model for calculation of the hydrodynamic pressure and the Kogut and Etsion's (KE) rough surface contact model for calculation of the asperity contact pressure. The total friction force was assumed to be the sum of the boundary friction at the contact asperities and the integration of viscous shear stress in the hydrodynamic region. The wear depth on the contact region was estimated according to the modified Archard's wear model using the asperity contact pressure. Sugimura's wear model was modified and used to link the wear particle size distribution and the variation of surface height PDF during wear. In the wear process, the variations of profile and surface height PDF of initial line contacts were calculated step by step in time, and the pressure distribution, friction coefficient, and wear rate were updated consequently. The effect of size distribution of wear particles on the wear process was numerically investigated, and the simulation results showed that the lubrication condition in which small wear particles are generated from the asperity contact region is beneficial to reduce friction coefficient and wear rate, and leads to a better steady mixed lubrication condition.

Thermal Conduction in Deforming Isotropic and Anisotropic Granular Porous Media with Rough Grain Surface

Resistance to the heat flow in solid–solid contact areas plays a fundamental role in heat transfer in unconsolidated porous materials. In the present work, we study thermal conduction in granular porous media that undergo deformation due to an external compressing pressure. The media’s grains have rough surface, with the roughness profile following the statistics of self-affine fractals that have been shown to be abundant in natural porous media. We utilize a fractal contact model of rough surfaces in order to estimate the deformation of the contact areas, which is a function of roughness fractal parameters, the grains’ Young modulus, and the compressing pressure. For porous media saturated by a single fluid, the effects of various factors, such as the porosity, the grains’ overlap (consolidation), and shapes (circular vs. elliptical), are all studied. Increasing the compressing pressure enhances heat transfer due to deformation of the rough surface of the gains. The thermal conductivity of the medium is strongly affected by the porosity, when the grains’ conductivity is much larger than that of the fluid that saturates the pore space. Furthermore, we show that thermal anisotropy is a decreasing function of roughness deformation. In other words, granular media with rougher grains exhibit larger anisotropy as measured by the ratio of the directional thermal conductivities. Whereas in one type of granular media the anisotropy eventually vanishes at very high compressing pressure, it persists in a second model of anisotropic media that we study.

A new elliptical microcontact model considering elastoplastic deformation

A new elliptical microcontact model considering elastoplastic deformation is proposed to overcome the shortcomings of the existing elastoplastic microcontact model. A new low-order interpolation function is used to describe the relationship between the normal deformation of asperity and the mean contact pressure in the elastoplastic deformation stage, and a smooth, continuous, monotonic change curve of the mean contact pressure is obtained. Then, the contact of rough surfaces is studied based on the new elastoplastic elliptical microcontact model and the height distribution of asperity. The calculated results are compared with those obtained from the existing rough surface contact models and the comparisons show that: (1) the change curve for the average contact pressure in the new model is smooth, continuous, and monotonic; (2) the calculation results of the new model are consistent with the numerical results based on the finite element method; (3) the new model is helpful to study the rough surface contact analysis of the elliptical micro-convex body.

Statistical contact model of rough surfaces: The role of surface tension

The sizes of most asperities on actual rough surfaces range from nanometers to microns. Recent investigations reveal that surface tension plays an important role in micro-/nano-sized contact. Therefore, in this paper, we address the influence of surface tension on the elastic contact of rough surfaces. Nayak's model is employed to describe rough surface with stochastic distribution of both heights and curvatures. As a fundament, we present the general relations of contact area and load as the function of indent depth for a single micro-/nano-sized asperity by accounting for surface tension. Based on the refined relations, we derive the general relations concerned in the contact of rough surfaces. Compared with the conventional multi-asperity contact models based on the Hertzian contact theory, the existence of surface tension decreases the real contact area and requires higher load to generate a given indent depth. It is found that the area of intimate contact is still approximately proportional to the external load, but the proportionality depends on the surface tension.

A Review of Elastic–Plastic Contact Mechanics

In typical metallic contacts, stresses are very high and result in yielding of the material. Therefore, the study of contacts which include simultaneous elastic and plastic deformation is of critical importance. This work reviews the current state-of-the-art in the modeling of single asperity elastic–plastic contact and, in some instances, makes comparisons to original findings of the authors. Several different geometries are considered, including cylindrical, spherical, sinusoidal or wavy, and axisymmetric sinusoidal. As evidenced by the reviewed literature, it is clear that the average pressure during heavily loaded elastic–plastic contact is not governed by the conventional hardness to yield strength ratio of approximately three, but rather varies according to the boundary conditions and deformed geometry. For spherical contact, the differences between flattening and indentation contacts are also reviewed. In addition, this paper summarizes work on tangentially loaded contacts up to the initiation of sliding. As discussed briefly, the single asperity contact models can be incorporated into existing rough surface contact model frameworks. Depending on the size of a contact, the material properties can also effectively change, and this topic is introduced as well. In the concluding discussion, an argument is made for the value of studying hardening and other failure mechanisms, such as fracture as well as the influence of adhesion on elastic–plastic contact.

A New Contact Model for Multilayered Solids with Rough Surfaces

A new numerical model is proposed to investigate the normal contact of multilayered solids with rough surfaces. The Hankel transform and the transfer matrix technique are used to solve the problem of the deformation of a multilayered solid. Then, the normal contact of an asperity is solved with Abel transform. Using this solution, an asperity-based contact model of rough surfaces is developed considering interactions between asperities. Numerical results are presented and compared to finite element calculations. The present model provides good results. The effects of interactions and the solid layers properties are discussed.

A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces

A revision of the elastoplastic contact model of fractal rough surfaces, initiated by Majumdar and Bushan, is further developed in this paper. According to the revised contact model, the critical contact areas of a single asperity are scale dependent. With an increase in contact load and contact area, a transition from elastic, elastoplastic to full plastic deformation takes place in this order. The relation between size distribution function for all level asperities and size distribution function for each level asperity is given. The total contact load, real contact area, contact load of each level asperity and contact area of each level asperity are obtained. The revised model shows that mechanical property of rough surface depends on asperity levels. The minimum level asperities and sequential five levels asperities are major contribution to rough surface. When asperity levels nmin and nmin+5 are less than elastic critical level nec, the rough surface appears to have elastic property under contact load. When asperity level nmin is greater than the second elastoplastic critical level npc, the rough surface appears to be carrying inelastic properties because the ratio of elastic deformation to inelastic deformation is very small.