Elastic Contact Mechanics Research Articles

Mechanical properties of WC coatings evaluated using instrumented indentation technique

Instrumented indentation tests have been conducted on the surface and cross-section of the two WC based coatings obtained by means of high velocity oxygen fuel thermal spray techniques in order to investigate their mechanical properties. The yield strength of the coatings has been determined by means of the Zeng and Chiu model. In addition, the absolute hardness of the coatings has been determined by means of the modified proportional specimen resistance model. Additionally, conventional spherical indentation tests were carried out on the surface of the coatings, aimed at determining the yield strength by means of the equations employed in classical elastic contact mechanics. The analysis indicates that the yield strength determined from the Zeng and Chiu model is of 3·5 and 3·1 GPa for the WC–10Co–4Cr and WC–12Co coatings respectively. The absolute hardness of the WC–10Co–4Cr coating was found to be of 11·1 GPa, whereas that of the WC–12Co was of 9·3 GPa.

Effect of Pile-Up on the Mechanical Characteristics of Steel by Depth Sensing Indentation

Mechanical properties by depth sensing indentation are derived from the indentation load-displacement data used a micromechanical model developed by Oliver & Pharr (O&P). However, O&P analysis on the indentation unloading curve is developed from a purely elastic contact mechanics (sink-in). The applicability of O&P analysis is limited by the materials pile-up. However, when it does, the contact area is larger than that predicted by elastic contact theory (material sinks-in during purely elastic contact), and both hardness H and Youngs modulus E are overestimated, because their evaluation depends on the contact area deduced from the load-displacement data. H can be overestimated by up to 60 % and E by up to 30 % depending on the extent of pile-up [1,2]. It is therefore important to determine the effect of pile-up on obtained mechanical characteristics of the material by depth sensing indentation. The work experimentally analyses the effect of pile-up height on mechanical characteristics H and E, which are determined by O&P analysis. Pile-up height was measured by atomic force microscopy (AFM).

An efficient approach for numerical treatment of some inequalities in solid mechanics on examples of Kuhn–Tucker and Signorini–Fichera conditions

A computationally efficient solution scheme is presented for the mechanical problems whose formulations include the Kuhn–Tucker or Signorini–Fichera conditions. It is proposed to reformulate these problems replacing inequalities in these conditions by equations with respect to new unknowns. The solutions of the modified problems have simple physical meanings and determine uniquely the unknowns of the original problems. The approach avoids application of multi-valued operators (inclusions or inequalities) in formulation of the problems. Hence, the modified formulations are suitable for numerical analysis using established powerful mathematical methods and corresponding solvers developed for solving systems of non-linear equations.To demonstrate the advantages of the proposed approach, it is applied for solving problems in two different areas: constitutive modeling of single-crystal plasticity and mixed boundary value problems of elastic contact mechanics with free boundaries. The original formulations of these problems contain respectively the Kuhn–Tucker and Signorini–Fichera conditions. A problem of the former area is integrated using an implicit integration scheme based on the return-mapping algorithm. The derived integration scheme is free of any update procedure for identification of active slip systems. A problem of the latter area is reduced to solution of non-linear integral boundary equations (NBIEs). Numerical examples demonstrate stability and efficiency of the solution procedures and reflect the mathematical similarities between the both non-linear problems.

Deterministic contact mechanics model applied to electrode interfaces in polymer electrolyte fuel cells and interfacial water accumulation

An elastic deterministic contact mechanics model is applied to the compressed micro-porous (MPL) and catalyst layer (CL) interfaces in polymer electrolyte fuel cells (PEFCs) to elucidate the interfacial morphology. The model employs measured two-dimensional surface profiles and computes local surface deformation and interfacial gap, average contact resistance, and percent contact area as a function of compression pressure. Here, we apply the model to one interface having a MPL with cracks and one with a crack-free MPL. The void size distributions and water retention curves for the two sets of CL|MPL interfaces under compression are also computed. The CL|MPL interfaces with cracks are observed to have higher roughness, resulting in twice the interfacial average gap compared to the non-cracked interface at a given level of compression. The results indicate that the interfacial contact resistance is roughly the same for cracked or non-cracked interfaces due to cracks occupying low percentage of overall area. However, the cracked CL|MPL interface yields higher liquid saturation levels at all capillary pressures, resulting in an order of magnitude higher water storage capacity compared to the smooth interface. The van Genuchten water retention curve correlation for log-normal void size distributions is found to fit non-cracked CL|MPL interfaces well.

Viscoelastic-adhesive contact modeling: Application to the characterization of the viscoelastic behavior of materials

In contacts involving viscoelastic materials, both the time-dependent modulus of the material and the adhesive interaction between contact surfaces play important roles. This work aims to derive viscoelastic-adhesive contact models for realistic contact analyses and develop a model-based-empirical method to determine localized properties (such as the time dependent creep compliance function) of a viscoelastic material considering surface adhesion and utilizing data from contact creep tests. The reported work is based on the viscoelastic fracture mechanics theory by Schapery (Schapery, 1989) and elastic contact mechanics models (Maugis, 1992; Johnson et al., 1971; Derjaguin et al., 1975). The Particle Swarm Optimization method (Kennedy and Eberhart, 1995) is utilized to find solutions to the nonlinear equations of this model. Contact creep tests of PDMS samples were designed and performed to validate the model. A multiple-decade power-law creep compliance function was obtained for a group of PDMS samples. The results show that the obtained creep compliance is a function of crosslink density of PDMS, independent of indenter tip size and the applied load.

An analytical calculation of the fluid load support fraction in a biphasic material: An alternative technique

BackgroundThe fluid load support fraction (WF/WT) can be used to define the mechanical contribution of the interstitial fluid (WF) to the total force (WT) in the deformation of cartilage. Traditionally, WF/WT is calculated using complex experimental setups or time-consuming micromechanical poroelastic Finite Element (FE) simulations. AimTo define and validate a fast and efficient technique to predict WF/WT using an analytical approach that can be applied without micromechanical detail or experimental measurement. MethodologyPoroelastic FE simulations defined accurate values of WF/WT for a range of loading configurations and were used to validate subsequent predictions. The analytical prediction of WF/WT used elastic contact mechanics to calculate WF, and viscoelastic FE representation to calculate WT. Subsequently, these independent calculations of WF and WT provided values of WF/WT that were compared with the poroelastic FE calculations. Results and discussionThe analytical prediction of WF/WT proved effective and suitably accurate (mean difference S<0.05). This technique demonstrated how WF and WT can be determined independently, without a biphasic constitutive model. Here we used viscoelasticity to calculate WT as an example, however, WT could be measured experimentally or predicted computationally.

Elastic contact mechanics: Percolation of the contact area and fluid squeeze-out

The dynamics of fluid flow at the interface between elastic solids with rough surfaces depends sensitively on the area of real contact, in particular close to the percolation threshold, where an irregular network of narrow flow channels prevails. In this paper, numerical simulation and experimental results for the contact between elastic solids with isotropic and anisotropic surface roughness are compared with the predictions of a theory based on the Persson contact mechanics theory and the Bruggeman effective medium theory. The theory predictions are in good agreement with the experimental and numerical simulation results and the (small) deviation can be understood as a finite-size effect. The fluid squeeze-out at the interface between elastic solids with randomly rough surfaces is studied. We present results for such high contact pressures that the area of real contact percolates, giving rise to sealed-off domains with pressurized fluid at the interface. The theoretical predictions are compared to experimental data for a simple model system (a rubber block squeezed against a flat glass plate), and for prefilled syringes, where the rubber plunger stopper is lubricated by a high-viscosity silicon oil to ensure functionality of the delivery device. For the latter system we compare the breakloose (or static) friction, as a function of the time of stationary contact, to the theory prediction.

Numerical Analysis Method of Adhesive Contact Mechanics between a Sphere and a Flat Considering Lennard-Jones Surface Forces of Noncontacting Rough Surfaces

A new numerical analysis method for elastic adhesive contact mechanics between a sphere and a flat with a sub-nanometer roughness is presented by taking account of Lennard-Jones (LJ) surface forces. In contrast to conventional theories, the LJ forces of contacting asperities and all noncontacting rough surfaces are taken into account. Elastic deformation of mean height surfaces is also taken into account. Convergent solutions obtained by a simple under-relaxation iteration method are shown for two contact models: 2-mm radius glass slider and magnetic disk with relatively large roughness, and 20-mm radius head model contacting with a magnetic disk with low roughness. Comparison of this new theory with previous theory considering LJ surface forces of mean height surfaces and modified Chang-Etsion-Bogy theory with and without deformation of mean height surfaces are presented. It is found that the present theory gives the largest external adhesion force and that this theory tends to approach the mean height surface theory as roughness height decreases to zero.

Coupling of dynamics and contact mechanics of artificial hip joints in a pendulum model

To date, fully coupled dynamics and contact mechanics analysis is still limited by expensive computational cost and long computing time and has not been addressed comprehensively, particularly in the hip joint. To understand the influence of different parameters on the biomechanics of the total hip replacement (THR) and improve its design, two numerical approaches were developed and implemented in finite element models to investigate the coupling between the dynamics response and the contact mechanics for three different THR configurations, metal-on-polyethylene (MOP), metal-on-metal (MOM), and ceramic-on-ceramic (COC). The dynamic force and the contact pressure distribution at the bearing surfaces from the two methods were predicted and compared. The influences of various parameters (motion angle, load applied in the pendulum, friction coefficient, geometry, and material properties) were subsequently investigated. From the comparisons, the decoupled method, based on the rigid-body dynamics and the quasi-static elastic contact mechanics, was adequate to predict the performance of the THRs efficiently. The load had the greatest influence on the dynamics/contact mechanics among other factors.

サブナノメートルの粗さをもつ球面と平面のLennard-Jones表面力による接触特性の数値解析法(機械要素,潤滑,設計,生産加工,生産システムなど)

A numerical analysis method for elastic contact mechanics between a spherical slider and a flat disk with a sub-nanometer roughness is presented by taking account of Lennard-Jones(LJ) surface forces. In contrast to conventional theories, the elastic deformations and LJ surface forces of both mean surfaces and contacting asperities are taken into account. Calculated results for 2-mm radius glass slider are shown and compared with experimental dynamic indentation characteristics. Convergent solutions obtained by a simple under-relaxation iteration method are shown for low roughness contacts with sub-nanometer mean surface distance and asperity height of 2-mm radius glass slider and 20-mm radius head slider contacting with a magnetic disk. It is found that the increase in adhesion force with a decrease in mean surface distance can be suppressed by deceasing asperity radius and increasing asperity density and Young's modulus.

Numerical Analysis of Contact Mechanics between a Spherical Slider and a Flat Disk with Low Roughness Considering Lennard–Jones Surface Forces

Although analytical and numerical analyses of the contact mechanics of a completely smooth sphere–flat contact have been done, the analysis of a realistic sphere–flat contact with a surface roughness whose mean height planes have a spacing greater than the atomic equilibrium distance has not been done thoroughly. This paper is a fundamental study of the elastic contact mechanics due to Lennard–Jones (LJ) intermolecular surface forces between a spherical slider and a flat disk with low roughness whose height is larger than equilibrium distance z 0. First, neglecting the effect of the attractive force at contacting asperities, adhesion contact characteristics of a 2-mm-radius glass slider with a magnetic disk are presented in relation to the asperity spacing σ between mean height planes. Results showed that the contact behavior at a small asperity spacing of ∼0.5 nm cannot be predicted either by the Johnson–Kendall–Roberts or Derjaguin–Muller– Toporov theories. Second, contact characteristics of a 1-μm-radius sphere on a flat disk are presented to examine how LJ attractive force at contacting asperities can be evaluated. It was found that the adhesion force of contacting asperity is a function of separation in general, but it becomes almost constant when σ = ∼z 0. A simple equation to evaluate the LJ attractive pressure of contacting asperities is presented for the rough contact analysis. Third, numerical calculation methods for a sphere–flat contact including LJ attractive forces between the mating mean height planes and contacting asperities are presented. Then, adhesion characteristics of a 2-mm-radius glass slider and magnetic disk are calculated and compared with the previous experimental results of dynamic contact test. It is shown that the calculated LJ adhesion force is much smaller than the experimental adhesion force, justifying that the adhesion force observed at the separation of contact is caused by meniscus force rather than by vdW force.

Message from the Scientific Organizers

This volume of Tribology Letters contains articles that were presented at the 428th Wilhelm and Else Heraeus Seminar ‘‘Physics of Tribology—Understanding Friction and Wear in technical systems’’. The seminar took place at the ‘‘Physikzentrum’’ in Bad Honnef, Germany from 23rd to 25th of March, 2009 with 60 participants from 14 countries. The purpose of the seminar was to present and discuss the physics of friction and wear in a multi-scale approach from the atomic level up to the size of an engine component. Within this topical setting, several contributions dealt with plastic deformations and third-body formation caused by sliding friction and by manufacturing (grinding). Insights into how nanocrystalline microstructures in bcc and fcc metals are generated as an effect of cyclic straining is steadily being gained. In addition, the importance of on-line wear measurement and ultra-low wear rates was discussed in several talks. Another important topic for macroscopic systems was the use of novel water based lubricants and of polyols. On the simulation side, several presentations dealt with elastic contact mechanics. A new theory now allows the exact calculation of the contact area even between millimeter-sized technical (e.g., aluminum–silicon alloys) surfaces. A novel reactive potential allows more realistic MD simulations of diamond-like carbon (DLC). On the atomic scale the experimental focus was on friction force microscopy. New experimental results showed a link between atomic friction measured on graphene and electron–phonon coupling for the first time. Another talk showed that the widely used Prandtl–Tomlinson model is over-simplifying the fast motion of the AFM tip apex. The poster session also allowed younger researchers to present their work. The poster prize was awarded to a contribution from Julich which dealt with experiments on rubber sealing and was an impressive example of intelligent research on a small budget. Overall the seminar showed how atomic-scale experiments and simulations are able to stimulate engineering tribology and vice versa. It therefore seems that future workshops of this kind may help to fertilize all disciplines of tribology. At this point we would again like to thank the Wilhelm and Else Heraeus foundation for funding, as well as their perfect organization of the seminar.

A Scaled Bilateral Control System for Experimental One-dimensional Teleoperated Nanomanipulation

In this work, one-dimensional experimental teleoperated nanomanipulation with force and visual feedback is demonstrated by using an atomic force microscope on the slave side and a haptic device on the master side. Three main topics are addressed that are especially relevant for force feedback in teleoperation at the nanoscale: transparency, impedance reflection according to human perception and stability. To further assist the operator, the Maugis—Dugdale elastic contact mechanics model is utilized to provide an augmented reality visual interface. As a result, a robust platform to perform teleoperated nanomanipulation on a broad range of materials is achieved. The performance of the resulting system is demonstrated on experimental results for vertical tip positioning tasks on glass, polystyrene and silicone rubber substrates with force and visual feedback. Experimental results indicate that softer substrates yield higher positioning errors and lower reaching speeds. Finally, one-dimensional nanoindentation of a flat polymethylmethacrylate (PMMA) surface using the proposed teleoperation system is demonstrated.

球面と平面のLennard-Jones表面力による吸着・接触特性の数値解析(機械要素,潤滑,工作,生産管理など)

A numerical method for analyzing the elastic contact mechanics due to Lennard-Jones (LJ) surface forces between a spherical slider and a flat disk is presented by taking account of roughness contact effect. First, neglecting the effect of attractive force at asperity contact, various features of elastic contact of 2-mm radius slider are presented in relation to the distance between mean roughness heights. It is revealed that the contact behaviors at a small mean roughness height distance can not be predicted either by JKR or DMT theory. A simple equation to evaluate LJ adhesion force after contact is presented. Next, contact characteristics of 1-μm radius sphere and disk are presented to examine how LJ attractive force at asperity contact can be evaluated. It is found that the adhesion force can be approximated by JKR or DMT theory if the surface energy can be related to Hamaker constant properly. Then, a simple equation to evaluate LJ attractive force at asperity contact is presented for the roughness contact analysis.

Hydromechanical properties of Fontainebleau sandstone: Experimental determination and micromechanical modeling

We measured ultrasonic velocity, hydraulic permeability, and specific storage capacity of eight blocks of Fontainebleau sandstone, a well‐sorted, medium‐grained, almost pure quartz‐sandstone of Oligocene age, covering a range in connected porosity from 3% to 10% and varying significantly in pore geometry for a given porosity. Ultrasonic P‐wave velocity measured on water‐saturated samples covers the full range predicted by variational Hashin‐Shtrikman bounds. Permeability and specific storage were determined by the linear pressurization method at different effective pressures up to 180 MPa and room temperature. The permeability of tested samples varies from 10−13 m2 down to 10−20 m2, depending primarily on porosity and pore geometry, and subordinately on effective pressure. Specific storage capacity always exceeds the contribution of pore fluid compression. The excess corresponds to the contribution of pore deformability controlled by pore geometry and matrix material properties. Since the sandstone samples are composed of a single solid component, quartz, we were able to calculate various poroelastic parameters (Biot‐Willis and Skempton coefficients, drained and undrained bulk moduli) crucial for estimating hydromechanical coupling in fluid‐saturated aggregates. The micromechanical modeling of the effective pressure sensitivity of the hydraulic transport and poroelastic parameters by the contiguity model of Takei (1998) yields good agreement between static and dynamic elastic parameters for saturated samples, partial agreement between the pressure dependence of effective elastic parameters and Hertzian contact mechanics, and an explanation for pressure‐insensitive permeability of samples with the highest porosity.

Estimation of contact area of nanoparticles in chains using continuum elastic contact mechanics

The key challenge to understand the nanomechanical behavior of nanoparticle chain aggregates (NCA) is strongly dependent on accurate determination of the contact area between individual particles. Experimental determination of the contacting radius is extremely difficult and often not reliable. We present calculations based on elastic contact mechanics models, that predict the contact radii of nanoparticles. The model considers important parameters, such as the particle size, the elastic property of the materials and the effect of humidity that intervenes between the contacts. Calculations carried out for four selected materials show that there is a large difference between the radii of contacts formed in vacuum and in ambient conditions. In vacuum, the surface energy of the material is higher compared to that in ambient condition and the contact formation is dominated by the strong interactions between the surfaces. In ambient condition, the contact is dominated by the Laplace pressure due to the condensed water meniscus between the contacts. Calculations presented in this article show that the contact radius between nanoparticles depends on materials properties and the environment.

Modeling of chemical–mechanical polishing considering thermal coupling effects

The removal of surface material in the conventional chemical–mechanical polishing (CMP) process is the result of synergetic effects of two dominant mechanisms: a mechanical process due to the abrasion of particles in the slurry, and a chemical process due to the reactions between the wafer and the chemicals in the slurry. In the overall material removal mechanism, in particular for metal layers, the mechanical and chemical effects are not independent, but are strongly coupled. Many models do not account for these coupling effects and cannot explain the non-Prestonian behavior that occurs when the material removal rate is a nonlinear function of the input areal power density. To address this deficiency and coupling effects, we propose a new integrated thermo-chemical–mechanical model that considers the synergistic effects of both the mechanical and chemical removal processes using the heat transfer mechanism as a bridge between them. In the modeling process, the material removal model is developed based on elastic and plastic contact mechanics and the dominant chemical reactions at the wafer surface. The temperature variation of the CMP system is treated as the coupling factor. The mechanical abrasion by the abrasive particles causes friction, which generates frictional heat on the contacting interfacial area. This heat plays a key role in accelerating the overall chemical reaction for the material removal. We performed a computer simulation with the proposed model using known parameters, and compared the results with other data to ensure its validity.

Mechanical Characterization of Polymers on a Nanometer Scale through Nanoindentation. A Study on Pile-up and Viscoelasticity

The analysis of nanoindentation force curves collected on polymers through the common Oliver and Pharr procedure does not lead to a correct evaluation of Young's modulus. In particular, the estimated elastic modulus is several times larger than the correct one, thus compromising the possibility of a nanomechanical characterization of polymers. Pile-up or viscoelasticity is usually blamed for this failure, and a deep analysis of their influences is attempted in this work. Piling-up can be minimized by indenting on a true nanometer scale, i.e., at penetration depth smaller than 200 nm. On the other side, it is common knowledge that fast indentations minimize the effect of viscoelasticity. However, changing the indentation time in a broad range of contact time (fractions of second up to hundreds of seconds) did not allow the correct estimation of Young's modulus for the polymers used in this work. The final result is that the Oliver and Pharr procedure as well as any other procedure analyzing the unloading curve with elastic contact mechanics models cannot be employed to measure Young's modulus of polymers because its application is incorrect from a theoretical point of view, unless the analysis is limited to the very first nanometers of penetration depth when the contact is perfectly elastic. Viscoelastic contact mechanics models should instead be employed to characterize these materials.

2947 スピン成分を伴う油固化高面圧トラクション試験と粘着理論による近似(S57-4 機構・トラクション・ベルト,S57 伝動装置の基礎と応用)

Under high contact pressure of traction-type CVT of automobiles, traction oil seems to behave like solid lubricants and elastic contact mechanics (creep theory) seems to be appropriate for evaluation of traction curves similar to that of railway wheels. In the present study, a traction testing machine up to 3GPa was established, in which a sphere and a v-notch-shape roller contacted at two points to generate spin component. Obtained high-pressure traction curves with small spin were lower than the roughly calculated ones based on the creep theory. However, the curves with large spin were almost consistent with them. The modified calculation curves with small spin considering small skew angle 0.2° from pure roll traction experiments, were almost consistent with the above experimental curves. It seems to be impossible to avoid the small skew caused by fitting space of the bearing and eccentric rotation of the coupling even in any traction testers.

Correction of Random Surface Roughness on Colloidal Probes in Measuring Adhesion

Atomic force microscopes (AFM) are commonly used to measure adhesion at nanoscale between two surfaces. To avoid uncertainties in the contact areas between the tip and the surface, colloidal probes have been used for adhesion measurements. We measured adhesion between glass spheres and silicon (100) surface using colloidal probes of different radii under controlled conditions (relative humidity of < 3%, temperature of 25 +/- 1 degrees C). Results showed that the adhesion forces did not correlate with the radii of the spheres as suggested by elastic contact mechanics theories. Surface roughness and random surface features were found on the surfaces of the colloidal probes. We evaluated various roughness parameters, Rumpf and Rabinovich models, and a load-bearing area correction model in an attempt to correct for the roughness effects on adhesion, but the results were unsatisfactory. We developed a new multiscale contact model taking into account elastic as well as plastic deformation in a successive contacting mode. The new model was able to correct for most of the surface roughness features except for surface ridges with sharp angular features, limited by the spherical asperity assumption made in the model.