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

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

A novel nonlinear contact stiffness model of concrete–steel joint based on the fractal contact theory

The heavy-duty machine tool is usually assumed in the concrete foundation, in which the machine tool-foundation joints have a critical effect on the working accuracy and life of heavy-duty machine tool. This paper proposed a novel contact stiffness model of concrete–steel joint based on the fractal theory. The topography of contact surface exist in concrete–steel joint has a fractal feature and can be described by fractal parameters. Asperities are considered as elastic, plastic deformation in micro-scale. However, the asperities of concrete surface will be crushed when the stress is larger than their yield limit. Then, the force balance of contact surfaces will be broken. Here, an iteration model is proposed to describe the contact state of concrete–steel joint. Because the contact asperities cover a very small proportion (less than 1%), the load on crushed asperities is assumed carried by other no contact asperities. This process will be repeated again and again until the crushed asperities are not being produced under external load. After that, the real contact area, contact stiffness of the concrete–steel joint can be calculated by integrating the asperities of contact surfaces. Nonlinear relationships between contact stiffness and load, fractal roughness parameter G, fractal dimension D can be revealed based on the presented model. An experimental setup with concrete–steel test-specimens is designed to validate the proposed model. Results indicate that the theoretical vibration mode shapes agree well with the experimental variation mode shapes. The errors between theoretical and experimental natural frequencies are less than 4.18%. The presented model can be used to predict the contact stiffness of concrete–steel joint, which will provide a theoretical basis for optimizing the connection characteristic of machine tool-concrete foundation.     

Phenomenological continuous contact–impact modelling for multibody simulations of pedestrian–vehicle contact interactions based on experimental data

Multibody modelling of pedestrian collisions requires the definition of contact–impact between the pedestrian and the vehicle. An examination of relevant impact test data reveals large rate-dependent components of the reaction force, permanent indentation, and concomitant energy loss. Contact–impact models previously used in simulations of pedestrian impacts typically have not adequately modelled one, two or all three of these phenomena. This paper presents a phenomenological contact–impact model based on the Hunt–Crossley model of impact, which includes rate-dependent damping, and is extended to include permanent indentation. The proposed model suitably characterises impact test data in a form that can also be implemented in the multibody simulation code MADYMO (TASS-Safe, Netherlands). The proposed contact–impact model was used to characterise the impact between a legform and the bumper of a vehicle, based on two impact tests conducted at different impact speeds. A single contact–impact definition in MADYMO closely reproduced the dynamics of both tests. The proposed model may be suitable in a wide range of impact conditions where the impact is modelled using multibody techniques and it is practicable to conduct impact tests as part of the modelling process.     

Effect of wheelset flexibility on wheel–rail contact behavior and a specific coupling of wheel–rail contact to flexible wheelset

The fexibility of a train's wheelset can have a large effect on vehicle–track dynamic responses in the medium to high frequency range.To investigate the effects of wheelset bending and axial deformation of the wheel web,a specifi coupling of wheel–rail contact with a fexible wheelset is presented and integrated into a conventional vehicle–track dynamic system model.Both conventional and the proposed dynamic system models are used to carry out numerical analyses on the effects of wheelset bending and axial deformation of the wheel web on wheel–rail rolling contact behaviors.Excitations with various irregularities and speeds were considered.The irregularities included measured track irregularity and harmonic irregularities with two different wavelengths.The speeds ranged from 200 to400km/h.The results show that the proposed model can characterize the effects of fexible wheelset deformation on the wheel–rail rolling contact behavior very well.     

From Boussinesq–Love contact to impact between hyperelastic bodies

This article is devoted to the modeling of finite deformations of hyperelastic bodies under contact/impact conditions. A total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. A first order algorithm is applied to integrate the equations of motion. The contact problem is solved by the bi-potential method. For the finite element implementation, an explicit expression of the tangent operator for the $\mathbf{3}\acute{\mathbf{e}}$ hyperelastic model is derived. The classical Boussinesq–Love contact problem is first investigated numerically. A second example concerns the impact between two hyperelastic bodies in three-dimension. To cite this article: Z.-Q. Feng, C. Vallée, C. R. Mecanique 335 (2007).     

Slipping–rolling transitions of a body with two contact points

Nonlinear Dynamics - In this paper, the general kinematics and dynamics of a rigid body is analysed, which is in contact with two rigid surfaces in the presence of dry friction. Due to the rolling...     

Oblique interaction of alfvén and contact discontinuities in magnetohydrodynamics

A general method of solving problems of the interaction of stationary discontinuities is proposed. The problem of the oblique incidence of an Alfvén plane-polarized discontinuity on a contact discontinuity is examined in the general formulation. A solution is constructed numerically over the entire range of variation of the governing parameters. A number of effects associated with the magnetohydrodynamic nature of the interaction are explored. For example, the formation in space of sectors in which the density falls by several orders (almost to a vacuum) is detected. The solutions obtained are of interest, for example, for investigating the interaction between Alfvén discontinuities in the solar wind and the magnetopause, plasmopause and other inhomogeneities whose boundary can be approximated by a contact discontinuity [13–15].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 131–142, January–February, 1990.     

A continuous–discontinuous cellular automaton method for regular frictional contact problems

In the present paper, a continuous–discontinuous cellular automaton method is developed to model the discontinuous problems caused by regular frictional contact, in which level set method, discontinuous enriched shape function, discontinuous cellular automaton method and contact friction theory are combined, by which an analysis from continuity to discontinuity can be achieved, and no assembled stiffness matrix but only cell stiffness is needed in the whole calculation, because of the use of discontinuous cellular automaton method. In the present method, level set method is used to track the discontinuous surface, and discontinuous enriched shape function is employed to describe the discontinuity of displacement and stress. Contact friction theory is applied to construct the Coulomb frictional contact model of discontinuous surfaces; besides, combined with discontinuous cellular automaton method, a new mixed iteration method is proposed to obtain the solution of the problem, and no assembled stiffness matrix is constructed. And the frictional contact iterations are done simultaneously with the updating of cellular automaton, in which the contact states and contact areas can be previously obtained in the cellular automaton updating process, and the efficiency can get much higher. Finally, some numerical examples are given to show that the present method is effective and accurate and can be further extended into some practical engineering.     

A 3D FEM–BEM rolling contact formulation for unstructured meshes

This work presents a new formulation for solving 3D steady-state rolling contact problems. The convective terms for computing the tangential slip velocities involved in the rolling problem, are evaluated using a new approximation inspired in numerical fluid dynamics techniques for unstructured meshes. Moreover, the elastic influence coefficients of the surface points in contact are approached by means of the finite element method (FEM) and/or the boundary element method (BEM). The contact problem is based on an Augmented Lagrangian Formulation and the use of projection functions to establish the contact restrictions. Finally, the resulting nonlinear equations set is solved using the generalized Newton method with line search (GNMls), presenting some acceleration strategies as: a new and more simplified projection operator, which makes it possible to obtain a quasi-complementarity of the contact variables, reducing the number of contact problem unknowns, and using iterative solvers. The presented methodology is validated solving some rolling contact problems and analyzed for some unstructured mesh examples.     

Estimation of a three-dimensional tyre footprint using dynamic soil–tyre contact pressures

A method for estimating the three-dimensional (3D) footprint of a 16.9R38 pneumatic tyre was developed. The method was based on measured values of contact pressure at the soil–tyre interface and wheel contact length determined from the contact pressures and the depths and widths of ruts formed in the soil. The 3D footprint was investigated in an area of the field where the pressure sensors of the tyre passed in a soft clay soil. The tyre was instrumented with six miniature pressure sensors, three on the lug face and the remaining three on the under-tread region between two lugs. The instrumented tyre was run at a constant forward speed of 0.27 m/s and 23% slip on a soft soil, 0.48 MPa cone index, 25.6% d.b. moisture content for four wheel load and tyre pressure combination treatments. The 3D footprint assessment derived from soil–tyre interface stress used in this research is a unique methodology, which could precisely relate the trend profile of the 3D footprint to the measured rut depth. The tyre–soil interface contact pressure distributions results showed that as inflation pressure increased the soil strength increased significantly near the centre of the tyre as a compaction increase sensed with the cone penetrometer.     

Nonlinear elastic wave propagation in a phononic material with periodic solid–solid contact interface

Phononic materials enable enhanced dynamic properties, and offer the ability to engineer the material response. In this work we study the wave propagation in such a structure when introduced with nonlinearity. Our system is comprised of pre-compressed material with periodic solid–solid contacts, which exhibit a quadratic nonlinearity for small displacements. We suggest a new approach to modeling this system, where we discretize the unit cell in order to derive an approximate analytical solution using a perturbation method, which we are then able to easily validate numerically. With these methods, we study the band structure in the system and the second harmonic generation originating from the nonlinearity. We qualitatively analyze the second harmonic response of the system in terms of the single-crack response with linear band structure considerations. Significant band structure manipulation by changing system parameters is demonstrated, including possible in-situ tuning. The system also exhibits effective frequency doubling, i.e. the transmitted wave is primarily comprised of the second harmonic wave, for a certain range of frequencies. We demonstrate very high robustness to disorder in the system, in terms of band structure and second harmonic generation. These results have possible applications as frequency-converting devices, tunable engineered materials, and in non-destructive evaluation.     

Sliding contact conditions using the master–slave approach with application on geometrically non-linear beams

Frictionless sliding conditions between two bodies are usually defined using either the method of Lagrangian multipliers or by prescribing an artificial (penalty) stiffness which resists the penetration at the contact point. Both of these methods impose the condition that the contact force should be normal to the contact surface, with the Lagrangian multiplier or the penalty parameter serving as a measure of this force. In this work, an alternative approach is undertaken: the frictionless sliding condition is defined through a relationship between nodal parameters of the virtual displacements of a discretised principle of virtual work. This method, which does not involve additional force parameters or degrees of freedom, is known as the master–slave or the minimum-set method and is particularly convenient for displacement-based finite-element implementation. The method is analysed in detail in context of bilateral sliding constraints characteristic of prismatic and cylindrical joints in flexible beam assemblies undergoing large overall motion. Two numerical examples are presented and assessed against the results in the literature.     

Propagation of slip pulse along frictionless contact interface with local separation between two piezoelectric solids

The Stroh formalism of piezoelectric materials,Fourier analysis and singular integral equation technique were used to investigate the existence of a pulse at the fric- tionless interface in presence of local separation between two contact piezoelectric solids. The two solids were combined together by uniaxial tractions and laid in the electric field. The problem was cast into a set of Cauchy singular integral equations,from which the closed-form solutions were derived.The numerical discussion on the existence of such a slip pulse was presented.The results show that such a slip pulse,which has square root singularities at both ends of the local separation zone,can propagate in most material combinations.And the existence of such a slip pulse will not be affected by the applied mechanical and electric fields in some special material combinations.     

Determination of the global responses characteristics of a piecewise smooth dynamical system with contact

In this paper the global response characteristics of a piecewise smooth dynamical system with contact, which is specifically used to describe the rotor/stator rubbing systems, is studied analytically. A method to derive the global response characteristics of the model is proposed by studying each piece of the equations corresponding to different phases of the rotor motion, i.e., the phase without rubbing, the phase with rubbing and the phase of self-excited backward whirl. After solving the typical responses in each phase and deriving the corresponding existence boundaries in the parameter space, an overall picture of the global response characteristics of the model is obtained. As is shown, five types of the coexistences of the different rotor responses and deep insights into the interactive effect of parameters on the dynamic behavior of the model are gained.     

Algorithm and implementation of soil–tire contact analysis code based on dynamic FE–DE method

One of the fundamental problems in terramechanics is soil–tire system. Past achievements on this topic can be observed in various literatures. Fast development on CPU power of PC system enables us to apply numerical methods to this basic subject. Among others, finite element method (FEM) has been applied to simple problems of soil–tire system not only in 2D but also in 3D approach. However, it is noted that the current FEM technology cannot handle “singular” boundary conditions with sufficient accuracy of analysis. Typical example of this limitation can be seen in an application to traction tire–soil contact problems, where the contact point of tire lug tip behaves as the singular point of stress field. On the other hand, distinct or discrete element method (DEM) has in essence the capability of analyze microscopic deformation (or flow) of soil as many researchers have already been demonstrated. It is noted that DEM suffers large calculation time that is consumed not only at contact check between particulate elements but also at incremental time step. In our present study, we try to combine both merit of FEM and DEM together in order to analyze the soil–tire system interaction, where, for example, a tire and deep soil layer are modeled as FEM and soil surface layer as DEM. We propose simple algorithm of this FE–DE coupled method and sample program is developed that can solve some basic terramechanics problems in order to verify our idea. The obtained result shows qualitatively sufficient accuracy.     

An efficient formulation based on the Lagrangian method for contact–impact analysis of flexible multi-body system

In this paper,an efficien formulation based on the Lagrangian method is presented to investigate the contact–impact problems of f exible multi-body systems.Generally,the penalty method and the Hertz contact law are the most commonly used methods in engineering applications.However,these methods are highly dependent on various non-physical parameters,which have great effects on the simulation results.Moreover,a tremendous number of degrees of freedom in the contact–impact problems will influenc thenumericalefficien ysignificantl.Withtheconsideration of these two problems,a formulation combining the component mode synthesis method and the Lagrangian method is presented to investigate the contact–impact problems in fl xible multi-body system numerically.Meanwhile,the finit element meshing laws of the contact bodies will be studied preliminarily.A numerical example with experimental verificatio will certify the reliability of the presented formulationincontact–impactanalysis.Furthermore,aseries of numerical investigations explain how great the influenc of the finit element meshing has on the simulation results.Finally the limitations of the element size in different regions are summarized to satisfy both the accuracy and efficien y.     

Delamination of an elastic film from an elastic–plastic substrate during adhesive contact loading and unloading

Adhesive contact between a rigid sphere and an elastic film on an elastic–perfectly plastic substrate was examined in the context of finite element simulation results. Surface adhesion was modeled by nonlinear springs obeying a force-displacement relationship governed by the Lennard–Jones potential. A bilinear cohesive zone law with prescribed cohesive strength and work of adhesion was used to simulate crack initiation and growth at the film/substrate interface. It is shown that the unloading response consists of five sequential stages: elastic recovery, interface damage (crack) initiation, damage evolution (delamination), film elastic bending, and abrupt surface separation (jump-out), with plastic deformation in the substrate occurring only during damage initiation. Substrate plasticity produces partial closure of the cohesive zone upon full unloading (jump-out), residual tensile stresses at the front of the crack tip, and irreversible downward bending of the elastic film. Finite element simulations illustrate the effects of minimum surface separation (i.e., maximum compressive surface force), work of adhesion and cohesive strength of the film/substrate interface, substrate yield strength, and initial crack size on the evolution of the surface force, residual deflection of the elastic film, film-substrate separation (debonding), crack-tip opening displacement, and contact instabilities (jump-in and jump-out) during a full load–unload cycle. The results of this study provide insight into the interdependence of contact instabilities and interfacial damage (cracking) encountered in layered media during adhesive contact loading and unloading.     

Frictional moving contact over the surface between a rigid punch and piezomagnetic materials – Terfenol-D as example

A general theory of the frictional moving contact of piezomagnetic materials indented by a flat or cylindrical punch is set up. The rigid punch moves at a constant speed and the Coulomb friction law applies inside the contact region. Terfenol-D with high magnetostriction and coupling is chosen. Employing the Galilean transformation and Fourier transform, Cauchy integral equations of the second kind are obtained and solved exactly. Closed-form expressions of physical quantities on the surface in terms of elementary functions are given. Numerical analyses are conducted to reveal the effects of the friction coefficient and moving speed of the punch on various surface stresses and magnetic induction. The singularity, discontinuity and spike of the surface magnetic induction may be important factors to explain why surface damage occurs for piezomagnetic materials.     

Adhesive contact on power-law graded elastic solids: The JKR–DMT transition using a double-Hertz model

A cohesive zone model of axisymmetric adhesive contact between a rigid sphere and a power-law graded elastic half-space is established by extending the double-Hertz model of Greenwood and Johnson (1998). Closed-form solutions are obtained analytically for the surface stress, deformation fields and equilibrium relations among applied load, indentation depth, inner and outer radii of the cohesive zone, which include the corresponding solutions for homogeneous isotropic materials and the Gibson solid as special cases. These solutions provide a continuous transition between JKR and DMT type contact models through a generalized Tabor parameter μ$\mu$. Our analysis reveals that the magnitude of the pull-off force ranges from (3+k)πRΔγ/2$(3+k)\pi R\mathrm{\Delta }\gamma /2$ to 2πRΔγ$2\pi R\mathrm{\Delta }\gamma$, where k$k$, R$R$ and Δγ$\mathrm{\Delta }\gamma$ denote the gradient exponent of the elastic modulus for the half-space, the radius of the sphere and the work of adhesion, respectively. Interestingly, the pull-off force for the Gibson solid is found to be identically equal to 2πRΔγ,$2\pi R\mathrm{\Delta }\gamma ,$ independent of the corresponding Tabor parameter. The obtained analytical solutions are validated with finite element simulations.