Elastic-plastic halfspace simulation

Due to the roughness of technical surfaces only the surface peaks are in contact for moderate contact pressures. Thus, the real contact area is smaller than the apparent contact area. Contact forces can only occur in the real contact area. Consequently it is necessary to determine the deformation of surface asperities in order to analyse the tribological properties of surfaces. The real contact area is usually small in initial contact. This leads to large contact pressures which in turn lead to the plastic deformation of surface roughness peaks. Therefore an elastic-plastic model is necessary. The halfspace model seems to be beneficial because there is only a system of equations on a surface mesh to be solved and not on a volume mesh like in the Finite-Element-Method. This leads to a much smaller system of equations which should allow reasonable calculation times even for large contact surfaces. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Accurate modeling of contact using cubic splines

In this paper, we develop and implement a new method for the accurate representation of contact surfaces. This approach overcomes the difficulties arising from the use of traditional node-to-linear surface contact algorithms. In our proposed method, contact surfaces were modeled accurately using C¹-continuous cubic splines, which interpolate the finite element nodes. In this case, the unit normal vectors are defined uniquely at any point on the contact surfaces. These splines preserve the local deformation of the nodes on each flexible contact surface. Consequently, a consistent linearization of the kinematic contact constraints, based on the spline interpolation, was derived. Moreover, the gap between two contact surfaces was modeled accurately using an efficient surface-to-surface contact search algorithm. Since the continuity of the splines is not affected by the number of nodes, accurate stress distribution can be obtained with less finite elements at the contact surface than that using the traditional linear discretization of the contact surface. Two numerical examples are used to illustrate the advantages of the proposed representation. They show a significant improvement in accuracy compared to traditional piecewise element-based surface interpolation. This approach overcomes the problem of mismatch in a finite element mesh. This is very useful, since most realistic engineering problems involve contact areas that are not known a priori.     

Investigations of the influence of geometrical irregularities and roughness on the normal contact stiffness of joints

Conducting experiments is necessary to determine the normal contact stiffness of rough surfaces. In compression tests the interface of specimen pairs is loaded in normal direction and the deflection is measured by extensometers with strain gages. The results can be used for example to describe the contact behaviour of joints in finite element analyses. It is shown that geometrical irregularities which are superposed by the surface roughness have a large influence on the elastic deformation of a joint and on the determination of the contact stiffness. They can be characterized by a skewed and a waved form of the surface with wavelengths which are larger than the surface roughness. In the experimental setup several different surfaces are compressed. In addition, the influence of the size of the contact area on the deformation of a joint is analysed. The results of the normal contact tests are compared with numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence for a highly nonlinear temperature-dependent system modeling nonlocal adhesive contact

In this paper we analyze a new temperature-dependent model for adhesive contact that encompasses nonlocal adhesive forces and damage effects, as well as nonlocal heat flux contributions on the contact surface. The related PDE system combines heat equations, in the bulk domain and on the contact surface, with mechanical force balances, including micro-forces, that result in the equation for the displacements and in the flow rule for the damage-type internal variable describing the state of the adhesive bonds. Nonlocal effects are accounted for by terms featuring integral operators on the contact surface.The analysis of this system poses several difficulties due to its overall highly nonlinear character, and in particular to the presence of quadratic terms, in the rates of the strain tensor and of the internal variable, that appear in the bulk and surface heat equations. Another major challenge is related to proving strict positivity for the bulk and surface temperatures.We tackle these issues by very careful estimates that enable us to prove the existence of global-in-time solutions and could be useful in other contexts. All calculations are rigorously rendered on an accurately devised time discretization scheme in which the limit passage is carried out via variational techniques.     

On the contact geometry of nodal sets

In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.

     

A characterization of hemispheres

We prove that for constant contact angle γ=0, a capillary surface over a convex domain has no umbilical points unless that the surface is a hemisphere. The method involves the comparison of a lower hemisphere with the given surface at a second-ordered contact point and it is based on an argument of Alexandrov.     

The axisymmetric contact problem taking into account transient heat production due to sliding friction

The contact problem of the sliding of a solid heat insulator with a plane surface along the boundary of an axisymmetric elastic body is considered, taking into account heat release and the thermal distortion of the boundary of the deformable body due to friction. It is assumed that the shear stresses have no effect on the value of the contact pressures, which enables the problem to be investigated in an axisymmetric formulation. The solution is constructed in two stages: first the form of the thermally distorted surface is determined using known expressions, obtained by Carslaw and Jaeger and also by Barber, and then the contact condition is considered taking into account the elastic displacements and distortion of the form of the surface due to heating, and the integral equation of the problem for determining the unknown contact pressures is derived. The latter equation is solved numerically by approximating the unknown contact pressures by a piecewise-constant function.     

Friction properties of highly elastic materials at high contact pressures

A newly designed high-pressure tribometer has been employed to investigate the pressure dependence of the friction force of SKN-40 crosslinked butadiene-nitrile rubber in contact with a steel surface on the pressure range to 1200 kgf/cm² (20°C). Over the entire range of contact pressures the friction process is molecular-kinetic in nature and characterized by a linear dependence of the friction force on the logarithm of the sliding velocity. In the region of normal pressures up to 200 kgf/cm², where the effect of pressure on the friction force reduces to the formation of the actual contact area, the friction constant (proportionality factor relating the friction force and the actual contact area) is practically independent of the pressure. At pressures above 200–300 kgf/cm² the increase in the friction force at fixed actual contact area is attributable to the effect of pressure on the friction constant. The nature of this effect is related not with an increase in the chain-surface interaction energy (the activation energy does not increase), but with an increase in the forces of adhesion owing to the greater number of polymer chain-steel surface contacts on the actual contact area (increase in contact density).Lenin Moscow State Pedagogical Institute, Laboratory for Problems of Polymer Physics. Translated from Mekhanika Polimerov, No. 1, pp. 140–146, January–February, 1971.     

Finite element and boundary element contact stress analysis with remeshing technique

The fundamental part of the contact stress problem solution using a finite element method is to locate possible contact areas reliably and efficiently. In this research, a remeshing technique is introduced to determine the contact region in a given accuracy. In the proposed iterative method, the meshes near the contact surface are modified so that the edge of the contact region is also an element’s edge. This approach overcomes the problem of surface representation at the transition point from contact to non-contact region. The remeshing technique is efficiently employed to adapt the mesh for more precise representation of the contact region. The method is applied to both finite element and boundary element methods. Overlapping of the meshes in the contact region is prevented by the inclusion of displacement and force constraints using the Lagrange multipliers technique. Since the method modifies the mesh only on the contacting and neighbouring region, the solution to the matrix system is very close to the previous one in each iteration. Both direct and iterative solver performances on BEM and FEM analyses are also investigated for the proposed incremental technique. The biconjugate gradient method and LU with Cholesky decomposition are used for solving the equation systems. Two numerical examples whose analytical solutions exist are used to illustrate the advantages of the proposed method. They show a significant improvement in accuracy compared to the solutions with fixed meshes.     

Determination of a Constitutive Friction Law Using an Elastic-Plastic Half-Space Model

Friction influences metal forming processes both in economic and technical terms. A precise understanding of friction is inevitable as friction restricts the potential of the product design. Friction depends on the occurring contact loads which is especially true for sheet-bulk metal forming (SBMF) as the incorporated contact loads appear in a very broad range. Numerical simulations, which are verified experimentally, are carried out to analyse contact interactions which typically appear in SBMF. On the one hand the multi-scale character of rough surfaces requires a very fine resolution of the contact area, on the other hand a large contact area is necessary to be representative. A half-space model is chosen for the contact analysis, because it only depends on the two-dimensional surface boundary which consumes less computing capacity than the finite element method (FEM) with its three-dimensional volume approach for the same surface resolution and area. The outcome is a constitutive friction law (CFL) consisting of two equations. The law is implemented into the framework of FEM to see the impact in a typical SBMF-process which aims to form metal sheet into cups with integrated gearing teeth. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A viscoelastic frictionless contact problem with adhesion

We consider a model for the quasistatic, bilateral, adhesive and frictionless contact between a viscoelastic body and a rigid foundation. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and an integro-differential equation for the bonding field. The existence of a unique weak solution for the problem is established by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space. We also consider the problem describing the bilateral contact between two viscoelastic bodies, and establish similar results.     

An Applicable Boundary Condition at the Moving Contact Line in Case of a Free Surface Reorientation after Step Reduction in Gravity

In this study the sudden change in the acceleration level on a free surface in a right circular cylinder is investigated numerically. The intention to describe the .uid motion encounters the difficulty modelling the moving contact‐line problem. In the numerical simulations the Navier slip boundary condition and a model for the dynamic contact angle are used. The model for the dynamic contact angle γ_d includes the dependence on the Capillary number as well as the contact angle hysteresis. We achieved very good agreement with experimental data.     

Friction of polymeric materials

The case of contact of polymer and metallic surface has been examined. A formula is proposed for calculating the contact area as a function of time of holding the compression surface under load and for calculating the coefficient of friction, the dependence of Young's modulus and shear strength on temperature, deformation rate, and average force on the contact surface being taken into account. The nature of deformation of irregularities on the surfaces of polymeric materials under the action of normal and tangential loading has been examined.See [1] for Communication 1.Institute for Polymer Mechanics, Academy of Sciences of the Latvian SSSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 809–815, September–October, 1972.     

Solvability of some two-dimensional quasistationary free boundary problems for the Navier-Stokes equations with moving contact points

The paper is concerned with some quasistationary two-dimensional free boundary problems of viscous flow with moving contact points and with contact angle equal to π. A typical example of such a flow is filling a capillary tube in the presence of surface tension. The proof of the solvability of these problems is based on the analysis (made by the author and V. V. Pukhnachëv about 10 years ago) of the asymptotic formulas for the solutions of the Navier-Stokes equations in a neigborhood of contact points. Bibliography: 10 titles.     

Thermoelastic rolling contact problem with temperature dependent friction

The paper is concerned with the numerical solution of a thermoelastic rolling contact problem with wear. The friction between the bodies is governed by Coulomb law. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The friction coefficient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The original thermoelastic contact problem described by the hyperbolic inequality governing the displacement and the parabolic equation governing the heat flow is transformed into elliptic inequality and elliptic equation, respectively. In order to solve numerically this system we decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Numerical examples showing the influence of the temperature dependent friction coefficient on the temperature distribution and the length of the contact zone are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simple method for spherical indentation of an elastic thin layer with surface tension bonded to a rigid substrate

An alternative method is proposed to solve the spherical indentation problem of an elastic thin layer with surface tension bonded to a rigid substrate. Based on the Kerr model, we establish a simple modified governing equation incorporating the surface tension effects for describing the relationship between the pressure and downward deflection of the impressed surface of the layer. This modified governing equation holds both inside and outside the contact zone, making it possible to analyze the whole layer by a unified differential equation. Numerical results are presented for the contact pressure inside the contact zone, the surface deflection of the elastic layer and the load-contact zone width relation to illustrate the present method. The validity and accuracy of the present method are demonstrated by comparing our results with those available in the existing literature.     

Surface Deformation due to Shear and Ploughing in a Halfspace

Sheet and bulk metal forming are widely used manufacturing methods. The interaction between worktool and workpiece in such a process causes friction which has a remarkable impact on the expended energy of the process. Therefore the influence of friction is important. Friction can be split into shearing and ploughing [1]. Ploughing is the plastic deformation of a soft surface by a hard contact partner. Shear forces are only transferred in the real contact area where material contact occurs. The investigation of the contribution of both ploughing and shearing to the total friction resistance is done with the use of an elasto-plastic halfspace model. The multiscale character of surfaces demands a fine discretization, which results in numerical effort. While a finite element method takes into account both surface and bulk of the contact partners, the halfspace model only regards the contact surfaces and thereby consumes less computing capacity. In order to identify the friction resistance, two rough surfaces get into contact. After full application of the normal load, the surfaces are moved relatively to each other. New asperities of the contact surfaces get into contact and are plastically deformed. These deformations are used to estimate the ploughing effect in dependency on the relative displacement. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Excitation of Rolling Tires from the Road Surface Texture

Rolling tires are excited from the contact with the rough road surface to vibrations, which cause rolling noise. A two scale approach is suggested, where at the macro–scale the vibration of the rolling tire structure is modeled by quite detailed finite element methods. The road surface is described using measured textures. A fine resolution finite element discretization of the tread rubber is performed in order to resolve the asperity contact. The material properties are described by a non–linear viscoelastic rubber model. The tread patch is enforced to approach the rough surface in a transient dynamics manner. From these investigations an enveloping surface profile is reconstructed to be used for the excitation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic antiplane contact problem with adhesion

We study a mechanical problem modeling the antiplane shear deformation of a linearly elastic body in adhesive contact with a foundation. The material is assumed to be homogeneous and isotropic and the process is quasistatic. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding is included. We establish the existence of a unique weak solution for the problem, by construction of an appropriate mapping which is shown to be a contraction on a Banach space.