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In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by $$d_\mathrm{CBM}$$ . Explicitly, we consider the following two set-ups, either on a contact manifold $$W \times S^1$$ where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of $$d_\mathrm{CBM}$$ are different in these two cases. In the former case the inputs are (contact) star-shaped domains of $$W \times S^1$$ which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance $$d_\mathrm{CBM}$$ defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of $$d_\mathrm{CBM}$$ . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

Traditional physics-based contact models have been widely used for describing various contact phenomena such as robotic grasping and assembly. However, difficulties in carrying out contact parameter identification as well as the relatively low measurement accuracy due to complex contact geometry and surface uncertainties are the limiting factors of the physics-based contact modeling methods. In this paper, we present a novel hybrid contact modeling (HCM) method as an endeavor to discover models that can more accurately simulate practical contact scenarios than traditional physics-based contact models. The proposed method is implemented by combining a physics-based contact model and a data-driven error model. This approach is validated by using simulations of a bouncing ball, a flat-shot, and a three-dimensional (3D) peg-in-hole. The results demonstrate the feasibility and consistent performance of the HCM method.

Wet adhesion is widely adopted in biological adhesion systems in nature, and it is beneficial to design new materials with desired properties based on the underlying physics of wet adhesion. The aim of this work is to develop a design criterion to regulate the wet adhesion. The effects of different contact shapes (flat and sphere) and morphologies of the substrate (smooth, microstructure and nanostructure) on the adhesion force are investigated. Combining with the theoretical models, the dominated factors in the separation process and isolating the viscous contributions from the capillary interactions are evaluated. The results demonstrate that the adhesion mechanisms depend significantly on the capillary numbers of the interstitial liquid and the contact geometry, and the ratio of capillary force to viscous force is a key to regulate the wet adhesion mechanism. These findings can not only explain some phenomena of wet adhesion to organisms, but also provide some inspirations to design new adhesion technology for robotic fingers that can grasp objects in wet environments.

We improve the previous result “A complete Ricci soliton whose metric g is K-contact and the soliton vector field X is strictly contact, is compact Sasakian Einstein” and show that, if a complete Ricci soliton (M, g, X) whose metric g is a contact metric and the soliton vector field X is strictly contact, then X is an infinitesimal automorphism and g is Einstein. Finally, for a Ricci soliton with X as the Reeb vector field, we show that (M, g) is compact Einstein and and Sasakian.

In the present paper, we initiate the study of $$*$$ - $$\eta $$ -Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a $$*$$ - $$\eta $$ -Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $$*$$ - $$\eta $$ -Ricci soliton. Next, we deliberate $$*$$ - $$\eta $$ -Ricci soliton admitting $$(\kappa ,\mu )^\prime $$ -almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$ . Finally we present some examples to decorate the existence of $$*$$ - $$\eta $$ -Ricci soliton, gradient almost $$*$$ - $$\eta $$ -Ricci soliton on Kenmotsu manifold.

A two-parametric geometry optimization problem for a symmetrical punch with compound curvature in frictionless contact with and an elastic half-plane is considered. An almost constant pressure distribution is sought for by means of least-square method. Another sub-optimal solution, which equalizes the three local maximum peaks (one at the center, and two at either side of the contact interval), is constructed using an asymptotic modeling approach and compared with the exact solution known in the literature.

We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of $$ \eta $$ -Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field $$\xi $$ . Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided $$m\ne 1$$ . We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is $$ \eta $$ -Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.

Harmonic wear is one of the common wear phenomena of high-speed train wheels. The change of wheel/rail rolling contact geometry leads to unstable characteristics of wheel/rail contact, which directly affects the quality and safety of high-speed train operation. Because wheelset lateral displacement directly affected the wheel/rail contact geometry, in this paper, the UM software was used to establish a high-speed train vehicle-track coupled dynamics simulation model of the CRH2 (China Railway High-speed 2) head car. Lateral displacement of harmonic wear wheel was calculated under different wear condition to analyze the influence of harmonic wear wheels on wheelset lateral displacement. Then according to macroscopic dimension change of the wheel profile from lateral and radial directions of high-speed train induced by harmonic wear wheels, wheel profiles under different wear condition were selected. And wheel/rail contact geometry under different harmonic wear stages was calculated to discuss the influence of harmonic wear wheel on wheel/rail contact geometry. The results show that harmonic order and wave depth of harmonic wear wheel have small influence on the lateral displacement of wheelset, but the influence of wave depth is greater than harmonic order. The average difference of lateral displacement between harmonic wear wheel and no harmonic wear wheel increases with the increase of wave depth and order. In a harmonic wave length, the maximum value of both equivalent conicity and contact angle appear in deepest wave depth. The greater the wave depth, the more obvious the fluctuations, and equivalent conicity changes are less obvious.

Weyl semimetal TaAs, congenially accommodating the massless Weyl fermions, furnishes a platform to observe a spontaneous breaking of either the time-reversal or the inversion symmetry and the concurrent genesis of pairs of Weyl nodes with significant topological durability. Former experimental analysis, which reveals that the near-zero spin-polarization of bulk TaAs, experiences a boost in proximity of point-contacts of non-magnetic metals along with the associated tip-induced superconductivity, provides the impetus to study the large-area stacked interfaces of TaAs with noble metals like Au and Ag. The primary outcomes of the present work can be listed as follows: (1) First-principles calculations on the interfacial systems have manifested an increment of the interface-induced spin-polarization and contact-induced transport spin-polarization of TaAs in proximity of noble metals; (2) In contrast to the single interface, for vertically stacked cases, the broken inversion symmetry of the system introduces a z-directional band-dispersion, resulting in an energetically separated series of non-degenerate band crossings. The simultaneous presence of such band-crossings and spin-polarization indicated the coexistence of both broken time reversal and inversion symmetries for metal-semimetal stacked interfaces; (3) quantum transport calculations on different device geometries reveal the importance of contact geometry for spin-transport in TaAs devices. Lateral contacts are found to be more effective in obtaining a uniform spin transport and larger transport spin polarization; (4) the phonon dispersion behaviour of TaAs displays a closure of band-gap with the associated increase of phonon-density of states for the acoustic modes in proximity of lateral contacts of noble metals.

To investigate the effects of the contact geometry, interfacial friction, and substrate recovery on the behavior of polymer scratching using a conical tip, an analytical model is proposed. The normal stress acting on the contact surface between the tip and the substrate is described as a function of the included angle θ, representing the angle between two planes across the axis of the conical tip, and the attack angle β, representing the angle between the conical surface and the substrate material surface. The effects of the rear contact geometry on the scratch friction between the tip and substrate, represented by recovery angle φ, owing to the instantaneous elastic recovery of the polymer substrate, are also introduced. Validated by the experimental and numerical results from the literature, the proposed analytical model can describe well the scratch coefficient of friction (SCOF), which is defined as the ratio of the tangential force to the normal force. Meaningful guidance is provided to understand the scratch friction behavior.

The friction properties of a dilute solution of block copolymer additives in poly-α-olefin (PAO) confined between mica surfaces were investigated using the surface forces apparatus. Friction measurements were made for poly(11-acrylamidoundecanoic acid)-block-poly(alkyl methacrylate)-block-poly(11-acrylamidoundecanoic acid) with an alkyl chain length of C12 (A5L946A5) in the poly(alkyl methacrylate) blocks. The results were compared to those for a PAO solution of a similar triblock copolymer with an alkyl chain length of C18 (A5S992A5) examined in our previous work (Yamada et al., Langmuir 2015). The friction coefficient μ for A5L946A5/PAO was approximately 0.5, which was one or two orders of magnitude larger than that for A5S992A5/PAO (μ of 0.002–0.039). Detailed analysis of the dynamic thickness and contact geometry enabled us to investigate the difference in the friction reduction abilities of the two additives. A5L946A5 has a high affinity to PAO and the molecules have expanded conformation in PAO; extensive chain entanglement effect results in the formation of a thick and viscous lubricant layer, which gives rise to high friction. On the other hand, A5S992A5 has low affinity to PAO, and the shrunken A5S992A5 molecules adsorbed on the surfaces form a rigid slip plane at the interface, which gives rise to low friction. Our results provide insights into the oil/additive combinations used for low friction in oil-based lubrication. The friction properties of a dilute solution of block copolymer additives in a base lubricant oil confined between mica surfaces were investigated using the surface forces apparatus. Friction force, sliding film thickness, and contact geometry were evaluated in detail, and the results were compared to those of the solutions of a different block copolymer additive having similar molecular structure. The relationship between the chemical structure of additives, confined structures between surfaces, and friction mechanisms were discussed, which shed new light of the design of low-friction surfaces in oil-based lubrication.

In the last two decades, significant progress has been made on developing new nanoscale mechanical property measurement techniques including instrumented indentation and atomic force microscopy based techniques. The changes in the tip-sample contact mechanics during measurements uniquely modify the displacement and force sensed by a measurement sensor and much effort is dedicated to correctly retrieve the sample mechanical properties from the measured signal. It turns out that in many cases, for the sake of simplicity, a simple contact mechanics model is adopted by overlooking the complexity of the actual contact geometry. In this work, a newly developed matrix formulation is used to solve the stress and strain equations for samples with edge geometries. Such sample geometries are often encountered in today’s nanoscale integrated electronics in the form of high-aspect-ratio fins with widths in the range of tens of nanometers. In the matrix formulation, the fin geometries can be easily modeled as adjacent overlapped half-spaces and the contact problem can be solved by a numerical implementation of the conjugate gradient method. This method is very versatile in terms of contact geometry and contact interaction, either non-adhesive or adhesive. The discussion will incorporate a few model examples that are relevant for the nanoscale mechanics investigated by intermittent contact resonance AFM (ICR-AFM) on low-k dielectric fins of high-aspect-ratio. In such ICR-AFM measurements, distinct dependence of the contact stiffness was observed as a function of the applied force and distance from the edges of the fins. These dependences were correctly predicted by the model and used to retrieve the mechanical changes undergone by fins during fabrication and processing.

An appropriate clearance is required between mating gears for effective power transmission. In the present work, the effect of radial misalignment on mild wear of spur gear is modeled and simulated. The changes in the gear tooth contact geometry due to progressive wear have also been accounted in the model. The simulation results show that in the pinion dedendum region, pressure angle increases, while the contact pressure and wear depth decrease with increase in number of wear cycles. As far as the effect of radial misalignment is concerned, contact pressure decreases and wear depth increases with the increase in misalignment. In the pinion addendum region, pressure angle and wear depth increase with the increase in wear cycles, while the contact pressure and wear depth decrease with the increase in misalignment. The pressure angle and pitch point change from initial 20° (23rd pitch point) to 20.95° (26th pitch point) and 21.86° (28th pitch point) due to 0.5- and 1-mm radial misalignment, respectively. The wear depths after 50,000 wear cycles are − 1.32 × 10−2 μm, − 3.33 × 10−3 μm and − 4.39 × 10−4 μm at 23rd, 26th and 28th pitch points, respectively. The effect of radial misalignment on backlash, pressure angle, pitch point, contact ratio, double tooth contact region and speed ratio is also discussed.

A metamodel considering material plasticity is presented for computationally efficient prediction of wheel–rail normal contact in railway switches and crossings (S&C). The metamodel is inspired by the contact theory of Hertz, and for a given material, it computes the size of the contact patch and the maximum contact pressure as a function of the normal force and the local curvatures of the bodies in contact. The model is calibrated based on finite element (FE) simulations with an elasto-plastic material model and is demonstrated for rail steel grade R350HT. The error of simplifying the contact geometry is discussed and quantified. For a moderate difference in contact curvature between wheel and rail, the metamodel is able to accurately predict the size of the contact patch and the maximum contact pressure. The accuracy is worse when there is a small difference in contact curvature, where the influence of variation in curvature within the contact patch becomes more significant. However, it is shown that such conditions lead to contact stresses that contribute less to accumulated plastic deformation. The metamodel allows for a vast reduction of computational effort compared to the original FE model as it is given in analytical form.

Foot–ground contact models are an important part of forward dynamic biomechanic models, particularly those used to model gait, and have many challenges associated with them. Contact models can dramatically increase the complexity of the multibody system equations, especially if the contact surface is relatively large or conforming. Since foot–ground contact has a large potential contact area, creating a computationally efficient model is challenging. This is particularly problematic in predictive simulations, which may determine optimal performance by running a model simulation thousands of times. An ideal contact model must find a balance between accuracy for large, conforming surfaces, and computational efficiency. Volumetric contact modelling is explored as a computationally efficient model for foot–ground contact. Previous foot models have used volumetric contact before, but were limited to 2D motion and approximated the surfaces as spheres or 2D shapes. The model presented here improves on current work by using ellipsoid contact geometry and considering 3D motion and geometry. A gait experiment was used to parametrise and validate the model. The model ran over 100 times faster than real-time (in an inverse simulation at 128 fps) and matched experimental normal force and centre of pressure location with less than 7% root-mean-square error. In most gait studies, only the net reaction forces, centre of pressure, and body motions are recorded and used to identify parameters. In this study, contact pressure was also recorded and used as a part of the identification, which was found to increase parameter optimisation time from 10 to 164 s (due to the additional time needed to calculate the pressure distribution) but helped the results converge to a more realistic model. The model matched experimental pressures with 33–45% root-mean-square error, though some of this was due to measurement errors. The same parametrisation was done with friction included in the foot model. It was determined that the velocity-based friction model that was used was inappropriate for use in an inverse-dynamics simulation. Attempting to optimise the model to match experimental friction resulted in a poor match to the experimental friction forces, inaccurate values for the coefficient of friction, and a poorer match to the experimental normal force.

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and $$(\kappa ,\mu )$$ -contact manifolds. First, we prove that a complete K-contact metric satisfying $$\mathcal {L}^{*}_g(\lambda )=0$$ is Einstein and is isometric to a unit sphere $$S^{2n+1}$$ . Next, we prove that if a non-Sasakian $$(\kappa ,\mu )$$ -contact metric satisfies $$\mathcal {L}^{*}_g(\lambda )=0$$ , then $$ M^{3} $$ is flat, and for $$n > 1$$ , $$M^{2n+1}$$ is locally isometric to the product of a Euclidean space $$E^{n+1}$$ and a sphere $$S^n(4)$$ of constant curvature $$+\,4$$ .

We introduce a novel systematic construction for integrable ( $$3+1$$ )-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of ( $$3+1$$ )-dimensional integrable dispersionless systems associated with the Lax pairs which are polynomial and rational in the spectral parameter.

The strength of an adhesive contact between two bodies can strongly depend on the macroscopic and microscopic shape of the surfaces. In the past, the influence of roughness has been investigated thoroughly. However, even in the presence of perfectly smooth surfaces, geometry can come into play in form of the macroscopic shape of the contacting region. Here we present numerical and experimental results for contacts of rigid punches with flat but oddly shaped face contacting a soft, adhesive counterpart. When it is carefully pulled off, we find that in contrast to circular shapes, detachment occurs not instantaneously but detachment fronts start at pointed corners and travel inwards, until the final configuration is reached which for macroscopically isotropic shapes is almost circular. For elongated indenters, the final shape resembles the original one with rounded corners. We describe the influence of the shape of the stamp both experimentally and numerically. Numerical simulations are performed using a new formulation of the boundary element method for simulation of adhesive contacts suggested by Pohrt and Popov. It is based on a local, mesh dependent detachment criterion which is derived from the Griffith principle of balance of released elastic energy and the work of adhesion. The validation of the suggested method is made both by comparison with known analytical solutions and with experiments. The method is applied for simulating the detachment of flat-ended indenters with square, triangle or rectangular shape of cross-section as well as shapes with various kinds of faults and to “brushes”. The method is extended for describing power-law gradient media.

The application of polymers in power-transmitting machine elements, e.g., gears, is limited by moderate thermo-mechanical properties and the detrimental accumulation of contact heat, even with external lubrication. Hence, polymer rolling–sliding elements are often prone to thermo-mechanical overload or abrasive wear. Diamond-like carbon (DLC) coatings are well known from steel applications for enhancing wear resistance and reducing friction. Since preliminary results indicate promising results for such coatings for polymers as well, their influence on the behavior of lubricated polymer contacts is investigated by numerical simulation. For polymer–steel contacts, the mechanical and thermophysical properties of coating and polymer are varied. The contact geometry is dominated by a local conformity, in which most of the deformation is related to the polymer. The DLC coatings affect film thickness and hydrodynamic pressure only little even for untypical high coating thicknesses. In contrast, the contact temperature decreases already for very thin coatings due to enhanced heat removal. Hence, DLC coatings can act as a thermal barrier protecting the polymer from detrimental heat and protecting the polymer from abrasive wear.

This paper is a survey on bubbling phenomena occurring in some geometric problems. We present here a few problems from conformal geometry, gauge theory and contact geometry and we give the main ideas of the proofs and important results. We focus in particular on the Yamabe type problems and the Weinstein conjecture, where A. Bahri made a huge contribution by introducing new methods in variational theory. Open image in new window

Legendrian submanifolds in Sasakian space forms play an important role in contact geometry. Defever et al. (Boll Unione Mat Ital B 7(11):365–374, 1997) established the first Chen inequality for C-totally real submanifolds in Sasakian space forms. In this article, we improve this first Chen inequality for Legendrian submanifolds in Sasakian space forms.

The initial stage of contact formation during field-activated/assisted sintering of powders is investigated experimentally and numerically for spherical particles of nickel, steel, and copper. The results of experimental studies of the contact resistance and the neck formation process due to melting/sintering in two- to three-particle systems under a single high-current pulse from a capacitor discharge are compared with results of fully coupled thermo-electrical finite-element simulations. The impact of particle size, contact geometry, and electrical load on the temperature and temperature gradient distribution during the pulse discharge is analyzed numerically for nickel spheres. The influence of the material itself is compared for nickel, steel, and copper systems.

Ball screws are robust and economic linear positioning systems that are widely employed in high speed and high precision machines. Due to precision and stability requirements, the preload force has been considered one of the main parameters in order to define the axial rigidity and the maximum axial load capacity of ball screw feed drives. In high speed motions, thermal effects are also considerably relevant regarding positioning precision and dynamic stability of the machine. The temperature increase and the thermal gradient between the screw, the balls and the nuts led to a variation in the contact geometry and consequently in the preload force. This paper presents an experimental analysis of the preload variation due to the temperature increase. The study has been performed for several initial preload forces and linear speeds in a preloaded double nut ball screw. The heating of the system results in a decrease of the preload force, where a maximum decrease of 60 % has been observed for a temperature increase of 28 K in the test bed of the study. Regarding the consequences of the loss of rigidity, the first eigenvalue decreased by only 5 % for a preload drop of 33 %.

A comprehensive thermal model is developed to analyse the heat transfer mechanism in form grinding of involute gears. The model analyses the thermal effects of a transient non-uniform heat source, moving with a three-dimensional curved shape, on the involute tooth profile. The variation of the contact geometry along the tooth profile is considered and analysed in detail, including the local grinding depth, equivalent wheel diameter and local contact arc length as well as the variation of the three-dimensional heat flux distribution and the heat partitioning to different thermal parts on the curved tooth profile. Based on the model established, the grinding temperature distribution patterns along the involute contact profile under different Peclet numbers, numbers of teeth and gear modules have been investigated. Along the tooth profile, the temperature increases significantly with the rolling angle. Grinding temperatures along the tooth profile of a straight spur gear, made of 20CrMnTi low-carbon steel, are measured using a special test rig designed with thermocouples imbedded into workpiece at different measurement positions. The results demonstrate that the calculated temperature distribution along the tooth profile has a rather good agreement with the measured result. This research provides a new method to predict grinding temperature in form grinding mode of precision gears.

Thermal damage control in high efficiency form grinding of precision gears is critical for obtaining the required gear surface integrity and ensuring the contact strength in service life. Complex thermal models have been developed to determine the variation of burn thresholds of power under different grinding parameters in form grinding mode of gears. The thermal models analyse in detail the variation of local contact geometry between the grinding wheel and gear surface, and the variation of heat partitioning to different thermal parts including the workpiece, wheel, grinding fluid and grinding chips. The burn thresholds of power in the form grinding of carbonised steel gears have been determined by relating the grinding temperature with the Barkhausen Noise and micro-hardness measurements for the ground surfaces. The models and the determined burn thresholds can be used for presetting the parameters and conditions of grinding process and monitoring the grinding process to avoid the occurrence of thermal damages to the gear surfaces.

A considerable portion of the power loss in a valve train mechanism is due to the losses in the cam follower mechanism. The surface of cam and follower experience counterformal contact with a varying nature that makes it a difficult component in an engine to model. Transient contact geometry and entraining may cause the inlet boundary reversal and short lived cessation of entraining motion which can lead to mixed regime of lubrication. Thermal effects and side leakage intensify the adverse governing tribological condition. This paper presents a thermo-elastohydrodynamic model with considering roughness of surfaces for a finite length cam-follower mechanism. Temperature variation, friction coefficient, and film thickness are the outputs of this model which can be used to evaluate the performance of this mechanical element. The results are validated by comparison to other published data. The results show that thermal effects, side leakage and roughness of contacting surfaces play an important role in the tribological performance of the mechanism. A friction analysis considering a comprehensive asperity interaction model is conducted to investigate the proposed lubrication model capability.

Friction between two surfaces is due to nano- and micro-asperities at the interface that establish true contact and are responsible for the energy dissipation. To understand the friction mechanism, often single-asperity model experiments are conducted in atomic-force microscopes. Here, we show that the conventional interpretation of the typical results of such experiments, based on a simple mass-spring model, hides a fundamental contradiction. Via an estimate of the order of magnitude of the dissipative forces required to produce atomic-scale patterns in the stick-slip motion of a frictional nano-contact, we find that the energy dissipation must be dominated by a very small, highly dynamic mass at the very end of the asperity. Our conclusion casts new light on the behavior of sliding surfaces and invites us to speculate about new ways to control friction by manipulation of the contact geometry.

A new continuum damage mechanics model for fretting fatigue life prediction is established. In this model, the damage evolution rate is described by two kinds of quantities. One is associated with the cyclic stress characteristics obtained by the finite element (FE) analysis, and the other is associated with the material fatigue property identified from the fatigue test data of standard specimens. The wear is modeled by the energy wear law to simulate the contact geometry evolution. A two-dimensional (2D) plane strain FE implementation of the damage mechanics model and the energy wear model is presented in the platform of ABAQUS to simulate the evolutions of the fatigue damage and the wear scar. The effect of the specimen thickness is also investigated. The predicted results of the crack initiation site and the fretting fatigue life agree well with available experimental data. Comparisons are made with the critical plane Smith-Watson-Topper (SWT) method.

Recently there had been a great deal of activity associated with various schemes of designing both analytic and experimental methods describing knotted structures in electrodynamics and in hydrodynamics. The majority of works in electrodynamics were inspired by the influential paper by Ranada (Lett Math Phys 18:97–106, 1989) and its subsequent refinements. In this work and in its companion we analyze Ranada’s results using methods of contact geometry and topology. Not only our analysis allows us to reproduce his major results but in addition, it provides opportunities for considerably extending the catalog of the known/obtained knot types. In addition, it allows to reinterpret both the electric and magnetic charges purely geometrically thus opening the possibility of treatment of masses and charges in Yang–Mills and gravitational fields purely geometrically.

Wheel/rail relationship is a fundamental problem of railway system. Wear of wheel profiles has great effect on vehicle performance. Thus, it is important not just for the analysis of wear characteristics but for its prediction. Actual wheel profiles of the high-speed trains on service were measured in the high-speed line and the wear characteristics were analyzed which came to the following results. The wear location was centralized from −15 mm to 25 mm. The maximum wear value appeared at the area of 5 mm from tread center far from wheel flange and it was less than 1.5 mm. Then, wheel wear was fitted to get the polynomial functions on different locations and operation mileages. A binary numerical prediction model was raised to predict wheel wear. The prediction model was proved by vehicle system dynamics and wheel/rail contact geometry. The results show that the prediction model can reflect wear characteristics of measured profiles and vehicle performances.

The purpose of this study is to propose a coupled train model with a multibody system considering three-dimensional wheel/rail contact geometry and to assess the safety of high-speed railway vehicles. Railway vehicles are becoming lighter and are running at increased speeds, so it is necessary to discuss their safety when subjected to a crosswind. In this paper, we use the proposed model in a situation presupposing strong crosswinds in order to analyze the behavior of coupled trains under this condition. We discuss the modeling and formulation of a coupled train as well as wheel unloading ratios. Our results show that vibration of a car body is largest in the wheel unloading ratios of the car body when the frequency of the car body is close to the second-mode natural frequency of the vehicle system.

On a Riemannian manifold, any parallel form is preserved by the flow of any Killing vector field with constant magnitude. As a consequence, on a 2n+1-dimensional K-contact manifold, there are no nontrivial parallel forms except of degrees 0 and 2n+1. Flat contact metrics on 3-manifolds are characterized by reducible holonomy.

Using ABAQUS/Explicit, the effects of contact conditions between the particle and substrate, including tangential friction, normal constraint, and contact geometry on the plastic deformation during the cold-spray process are studied. It was found that the onset of shear instability, an event often used to indicate the establishment of bonding, is very sensitive to the choice of contact conditions. This suggests that the onset of shear instability does not serve as an accurate means to identify the plasticity threshold responsible for bonding. On the other hand, it is demonstrated that the evolution of the overall equivalent plastic strain (i.e., $$\overline{\text{PEEQ}}$$ ) and the overall von Mises stress, being linearly proportional to each other, are both independent of contact conditions. Furthermore, it is shown that an energy value, defined as the product of the $$\overline{\text{PEEQ}}$$ and the von Mises stress integrated over all particle elements, can quantitatively represent the energy dissipated via plastic deformation while being independent of contact conditions. The $$\overline{\text{PEEQ}}$$ and associated energy value as defined may provide robust tools to assess the plasticity and the consequent bonding during cold-spray.

Most of the previous studies on fretting fatigue have been accomplished under constant contact load condition, and a less attention has been paid to cyclic contact load. That is the case in many engineering applications such as bearing cap bolted joints of the V-type engines and dovetail joint of turbines. In this study, the effect of contact geometry on fretting fatigue behavior of Al7075-T6 under cyclic normal contact loading is investigated by experiment, numerical simulation and analysis. Two contact types including flat-on-flat and cylinder-on-flat are considered in this study. The crack initiated at a lower cycle for cylinder-on-flat contacts. It was found that in comparison with the constant contact loading condition, the cyclic contact loadings have more damaging effect on fretting fatigue life, particularly for cylinder-on-flat contact. With the increase in the pad width, fretting fatigue life increased for flat and cylindrical pads, particularly for high cycle fatigue (HCF) conditions. Examination of the fretting scars was performed using optical microscopy. Crack propagation life was determined by numerical simulation using the commercial FE codes ABAQUS® and FRANC2D/L®. Crack initiation life was calculated by a combination of numerical simulation and experiment.

Graphene was grown by low-pressure chemical vapour deposition on polished steel samples using ethanol as a precursor at different pressures: 200 and 400 mTorr. The samples were characterized by Raman spectroscopy, and the Raman maps clearly show the formation of an inhomogeneous graphene coating with surface regions covered with single-layer, bilayer, and multilayer graphene, the last one covering most of the surface area of the sample when the sample was prepared at 400 mTorr. Scanning electron microscopy revealed the partial coverage of the sample surface when the graphene coating was prepared at a pressure of 200 mTorr. Friction measurements were taken using a micro-tribometer in a ball-on-flat contact geometry with a stainless steel ball as the counterbody. An important reduction in the coefficient of friction was verified even with partial coverage of the steel surface, and it was attributed to the formation of nanographitic structures in the wear track region and partially transferred to the counterbody.

Safety and economy of railway traffic is enormously influenced by the contact stress variation caused by wheel rail contact profile changes. A change in designed surface topology may result from wear that brings in a wide change in contact geometry and stresses. To study the influence of interacting wheel and rail profile topology of standard rail UIC60, the standard wheel profile as per Indian Railway standards are considered in this paper. Rail profile radii, wheel profile radii and wheel profile taper are chosen for six different values. The analytical formulation is based on Timoshenko’s approach and Finite Element Method (FEM) based simulation of the problems is undertaken. With these tools, distribution of contact zones, contact stress and contact pressure for different configuration of the wheel and rail profiles are obtained. The mesh density in contact region is found to have a direct influence on the accuracy of the solution [1]. To standardize the analysis of the contact region, mesh with an element size of 1 mm for all the configurations are chosen. Using stress response obtained through FEM analysis and multiaxial fatigue crack initiation model, the effects of vertical loading on fatigue crack initiation life are investigated. This may allow a direct design application for railways in particular.

The main aim of this paper is to study paraholomorpic Sasakian metric and Killing vector field with respect to the Sasakian metric in the cotangent bundle.

We provide some examples of harmonic and minimal unit vector fields as normalized gradients of isoparametric functions coming from a K-contact geometric setting.

During sliding low elastic modulus, materials can form ridges and distinct zones of contact vice the commonly observed Schallamach wave phenomena. It was shown that at low velocities (0.001 and 0.01 m/s), the contact geometry retained a single region of contact and at higher velocities (0.1 and 1 m/s) the contact separated into two distinct zones. It was also shown that this phenomenon was independent of the applied normal load.

We discuss parabolic contact geometries carrying a smooth system of symmetries. We show that there is a symmetric space such that the parabolic geometry $$ \left( {\mathcal{G}\to M,\omega } \right) $$ is a fibre bundle over this symmetric space if and only if the base manifold M is a homogeneous reexion space. We investigate the conditions under which there is an invariant geometric structure on this symmetric space induced by the parabolic contact geometry.

The multi-sphere method and JKR model are used in the discrete element method simulation to investigate the effect of the particle size, aspect ratios, and cohesiveness on the packing structure, characterized by porosity, radial distribution function (RDF), coordination number and contact geometry. In the absence of cohesive force, the porosity is nearly invariable with fixed aspect ratio, regardless of the size of the particles. In contrast, as surface energy increases, the porosity increases with decreasing particle size and increasing aspect ratio. The RDF results show that the number of peaks for different aspect ratios changes and show trends similar to the relaxation algorithm, expected for the finer particles. In the case of finer, cohesive particles, the most novel outcome of contact analysis is the existence of single contact, attributed to the formation of a cage structure, which has not been previously reported. The peak position and the width of the contact distributions are affected by higher surface energy because fewer contacts are required to achieve the mechanical equilibrium. Another interesting observation is that higher porosity does not always imply fewer contacts for particles with non-zero aspect ratios and high surface energies. The analysis of the distribution of the contact vector angles is found to better explain increased porosity in spite of higher coordination numbers. The results presented shed light on the packing density and structure, revealing features not easily discerned via experiments, and confirming the important role of the cohesion and aspect ratio in packing.

We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization., 9 pages. v2: Added references, improved proof of Proposition 3.3. v3: Expanded introduction, clarifying remarks, some changes of sign conventions. Main results are unchanged. v4: Final version, implementing changes suggested by referees

Accurately characterizing the three-dimensional geometric contacts between the crust of the Chinese mainland and adjacent regions is important for understanding the dynamics of this part of Asia from the viewpoint of global plate systems. In this paper, a method is introduced to investigate the geometric contacts between the Eurasian and Indian plates at the Burma arc subduction zone using earthquake source parameters based on the Slab1.0 model of Hayes et al. (2009, 2010). The distribution of earthquake focus depths positioned in 166 sections along the Burma Arc subduction zone boundary has been investigated. Linear plane fitting and curved surface fitting has been performed on each section. Three-dimensional geometric contacts and the extent of subduction are defined quantitatively. Finally, the focal depth distribution is outlined for six typical sections along the Burma arc subduction zone, combining focal mechanisms with background knowledge of geologic structure. Possible dynamic interaction patterns are presented and discussed. This paper provides an elementary method for studying the geometric contact of the Chinese mainland crust with adjacent plates and serves as a global reference for dynamic interactions between plates and related geodynamic investigations.

The present work studies the different effects in the contact surfaces subjected to fretting movements and electric current flow on an austenitic stainless steel. The main purpose is to establish a relationship between the energy dissipation by friction and the energy dissipation by the Joule effect as well as the synergetic effects. As this type of contact is often used in a large number of small connectors, especially on the interface of printed circuit boards, there is a significant interest in studying the entire problem. Austenitic stainless steels have some properties, mainly the low oxidation rate, that, allied with relatively low cost and availability, qualify this type of material for this specific task. The fretting tests consisted in reciprocating movements on a sphere-on-flat-surface contact geometry. Both specimens were made of AISI 304 stainless steel. The electric current was kept constant during the entire test at various set values, ranging from 0 mA (no current), corresponding to no energy dissipation by the Joule effect, to 120 mA. The addition of energy in the contact by the Joule effect increases the oxidation of the contact surfaces, inducing wear acceleration in both specimens by synergetic effect. A synergetic model was introduced in order to quantify individual contributions to total wear volume, which is composed by mechanical wear, a Joule effect parcel of wear and an incremental factor of degradation due to the combined effect.

In this work, nanoindentation and nanoscratching experiments are combined with atomic force microscopy to investigate the relationships between contact geometry, apparent friction, and deformation modes of two grades of Kevlar® (Dupont) fiber—Kevlar KM2 and Kevlar 49. Changes in the relative angle between the scratching probe and the fiber surface, often termed as the attack angle, result in changes in deformation mode, which correlate with the changes in the apparent friction. As attack angle increases, the observed deformation modes of the fiber surface change from a smoothing of the surface, often termed as ironing, to fibrillation, in which the fibrils break and coalesce in front of the progressing probe. A mixture of these two modes occurs at intermediate attack angles. When fibrillation occurs, material pile-up forms in front of the progressing probe. This pile-up introduces an additional component to the frictional response that is largely responsible for an increase in apparent friction with an increasing attack angle and/or scratch length. The level of friction associated with fibrillation is measured to be up to approximately three times higher than previously reported for Kevlar yarn–yarn friction. Fibrillation of Kevlar KM2 occurs at larger attack angles as compared to Kevlar 49, which is believed to be related to a near-surface region of reduced modulus and hardness previously observed in KM2 fibers. A detailed discussion of the measured response is given based on the interactions between the scratching probe and the fibrillar network and the resulting deformation mechanisms.

The irregularity is a key factor affecting the wheel-rail contact geometry relationship. In this paper, we calculated the wheel-rail contact points at typical sections and obtained the longitudinal variation of the wheel-rail geometry relationship with the trace line method. The profile of the key rail sections was matched by cubic spline curve, and the shape interpolation was realized in non-controlling sections. The results show that the roll angles at each typical section increases gradually with the enlargement of track alignment irregularity. When the flange contact occurs, the roll angle increases dramatically. Proper track alignment irregularity towards the switch rail improves the structure irregularity of the turnout.

This paper presents a new method for modelling and simulation of the dynamic behaviour of the wheel-rail contact. The proposed dynamic wheel-rail contact model comprises wheel-rail contact geometry, normal contact problem, tangential contact problem and wheelset dynamic behaviour on the track. This two-degree of freedom model takes into account the lateral displacement of the wheelset and the yaw angle. Single wheel tread rail contact is considered for all simulations and Kalker's linear theory and heuristic non-linear creep models are employed. The second order differential equations are reduced to first order and the forward velocity of the wheelset is increased until the wheelset critical velocity is reached. This approach does not require solving mathematical equations in order to estimate the critical velocity of the dynamic wheel-rail contact model. The mathematical model is implemented in Matlab using numerical differentiation method. The simulated results compare well with the estimated results based on classical theory related to the dynamic behaviour of rail-wheel contact so the model is validated.

On a 2n+1-dimensional K-contact manifold, there are no nontrivial parallel forms except of degrees 0 and 2n+1.

In this work, a friction and wear simulator was used to reproduce the Anterior-Posterior (AP) sliding and the Flexion-Extension (FE) rotation generated in the knee joint during human gait cycle. We chose to simplify the contact geometry between the Total Knee Arthroplasty (TKA) femoral component and tibial insert. A 304L stainless steel cylinder which replaces the femoral component was loaded onto a flat High Density Polyethylene (HDPE) block which replaces the tibial insert. The tribological behavior of the considered contact was analyzed by tracking the number of cycles, the friction coefficient, the roughness of the wear track on HDPE, the HDPE weight loss and the damage mechanisms. The friction coefficient shows a gradual increase with the number of cycles for both AP and FE kinematics. The evolution of friction coefficient with the number of cycles is not affected by the value of the imposed normal load in the case of AP sliding. For the FE rotation, decreased friction coefficient is obtained when the imposed normal load increases. For both considered AP and FE kinematics, the roughness of the wear track on the HDPE is not affected by the imposed normal load. It shows a progressive decrease when the number of cycles increases. The wear of HDPE obeys the Archard law and the wear coefficient increases with the normal force. For a given value of normal load, the obtained wear coefficient for the AP sliding is larger than that obtained for FE rotation. A predominant adhesive wear mechanism was identified for both AP and FE kinematics. Under the same normal load, damage development in terms of plastic deformation, micro-cracking and debonding is more pronounced for the AP sliding if compared with the FE rotation. For a given kinematics, the damage severity increases with the normal load. This finding is in good agreement with the predicted values of the wear coefficient according to the Archard law.

An instrument has been developed that allows in situ optical analysis and tribological measurements for contacts between solid bodies; an interferometric optical analysis can be used to measure and observe contact size, contact geometry, near contact topography, tribofilm formation, tribofilm motion, tribofilm thickness, wear debris formation, and wear debris morphology. The optical arrangement is in such a way that a 0th order interference fringe highlights the real contact area of contact, while near contact regions are height-mapped with higher order Newton’s rings interference fringes. Images synchronized with force and position measurements allow for the potential to test and validate models for contact mechanics, adhesion, and sliding. The contact and friction measurement between a rough rubber sphere and a polished glass counterface were studied over a range of loads from 1 to 50 mN.