Roman Pohrt

Partial-slip frictional response of rough surfaces

If two elastic bodies with rough surfaces are first pressed against each other and then loaded tangentially, sliding will occur at the boundary of the contact area while the inner parts may still stick. With increasing tangential force, the sliding parts will expand while the sticking parts shrink and finally vanish. In this paper, we study the fractions of the contact area, tangential force and tangential stiffness, associated with the sticking portion of the contact area, as a function of the total applied tangential force up to the onset of full sliding. For the numerical analysis randomly rough, fractal surfaces are used, with the Hurst exponent H ranging from 0.1 to 0.9. Numerical simulations by boundary element method are compared with an analytical analysis in the framework of the Greenwood and Williamson (GW) model. In both cases, a universal linear dependency between the real contact area fraction in stick condition and the applied tangential force is found, regardless of the Hurst exponent of the rough surfaces. Regarding the dependence of the differential tangential stiffness on the tangential force, a linear relation is found in the GW case. For randomly rough surfaces, a nonlinear relation depending on H is derived.

Contact stiffness of randomly rough surfaces.

We investigate the contact stiffness of an elastic half-space and a rigid indenter with randomly rough surface having a power spectrum , where q is the wave vector. The range of is studied covering a wide range of roughness types from white noise to smooth single asperities. At low forces, the contact stiffness is in all cases a power law function of the normal force with an exponent α. For H > 2, the simple Hertzian behavior is observed . In the range of 0 < H < 2, the Pohrt-Popov behavior is valid (). For H < 0, a power law with a constant power of approximately 0.9 is observed, while the exact value depends on the number of modes used to produce the rough surface. Interpretation of the three regions is given both in the frame of the three dimensional contact mechanics and the method of dimensionality reduction (MDR). The influence of the long wavelength roll-off is investigated and discussed.

Complete boundary element formulation for normal and tangential contact problems

The boundary element method as a numerical tool in contact mechanics is widely used and allows for surface roughness to be investigated with very fine grids. However, for every two grid points, influence coefficients have to be employed for every force-displacement combination. In this paper, we derive the matrixes of influence coefficients for the deformation of an elastic half space, starting from the classical solutions of Boussinesq and Cerruti. We show how to overcome complexity problems by using FFT-based fast convolution. A comprehensive algorithm is given for solving the case of dry Coulomb friction with partial slip. The resulting computer program can be used effectively in iterative schemes also in similar problems, such as mixed lubrication and notably improves the applicability of the boundary element method in contact mechanics.

Contact Mechanics of Rough Spheres: Crossover from Fractal to Hertzian Behavior

We investigate the normal contact stiffness in a contact of a rough sphere with an elastic half-space using 3D boundary element calculations. For small normal forces, it is found that the stiffness behaves according to the law of Pohrt/Popov for nominally flat self-affine surfaces, while for higher normal forces, there is a transition to Hertzian behavior. A new analytical model is derived describing the contact behavior at any force.

Investigation of the dry normal contact between fractal rough surfaces using the reduction method, comparison to 3D simulations

The reduction method was applied to solving the elastic contact problem of fractal rough surfaces having different fractal dimensions. These surfaces are of particular interest, as contact spots influence each other on large and small scales, depending on the power spectrum. Results were compared to previous boundary-element method simulations that had been conducted with 3D surfaces, taking into account all interdependencies between contact spots. Excellent agreement in the contact stiffness was found. Even though no interactions are allowed in the reduced model, the same power-law behavior can be observed as in the 3D case, due to the transformation of the power spectrum. Theoretical considerations about the power-law for different fractal dimensions are given.

Relaxation damping in oscillating contacts.

If a contact of two purely elastic bodies with no sliding (infinite coefficient of friction) is subjected to superimposed oscillations in the normal and tangential directions, then a specific damping appears, that is not dependent on friction or dissipation in the material. We call this effect “relaxation damping”. The rate of energy dissipation due to relaxation damping is calculated in a closed analytic form for arbitrary axially-symmetric contacts. In the case of equal frequency of normal and tangential oscillations, the dissipated energy per cycle is proportional to the square of the amplitude of tangential oscillation and to the absolute value of the amplitude of normal oscillation, and is dependent on the phase shift between both oscillations. In the case of low frequency tangential oscillations with superimposed high frequency normal oscillations, the dissipation is proportional to the ratio of the frequencies. Generalization of the results for macroscopically planar, randomly rough surfaces as well as for the case of finite friction is discussed.

Johnson-Kendall-Roberts adhesive contact for a toroidal indenter.

The unilateral axisymmetric frictionless adhesive contact problem for a toroidal indenter and an elastic half-space is considered in the framework of the Johnson–Kendall–Roberts theory. In the case of a semi-fixed annular contact area, when one of the contact radii is fixed, while the other varies during indentation, we obtain the asymptotic solution of the adhesive contact problem based on the solution of the corresponding unilateral non-adhesive contact problem. In particular, the adhesive contact problem for Barber’s concave indenter is considered in detail. In the case when both contact radii are variable, we construct the leading-order asymptotic solution for a narrow annular contact area. It is found that for a v-shaped generalized toroidal indenter, the pull-off force is independent of the elastic properties of the indented solid.

General procedure for solution of contact problems under dynamic normal and tangential loading based on the known solution of normal contact problem

In this article, we show that the normal contact problem between two elastic bodies in the half-space approximation can always be transformed to an equivalent problem of the indentation of a profile into an elastic Winkler foundation. Once determined, the equivalent profile can also be used for tangential contact problems and arbitrary superimposed normal and tangential loading histories as well as for treating of contact problems with linearly viscoelastic bodies. In the case of axis-symmetric shapes, the equivalent profile is given by the method of dimensionality reduction integral transformation. For all other shapes, the profile is deduced from the solution of the elastic contact normal problem, which can be obtained numerically or experimentally.

Normalkontakt mit rauen Oberflächen

Zusatzlich zu den geometrisch streng definierten Fallen, die in Kap. 3 mit der Reduktionsmethode abgebildet wurden, mochten wir uns nun der Frage widmen, ob auch raue Oberflachen in einem reduzierten Modell dargestellt werden konnen.

SIMULATION OF FRICTIONAL DISSIPATION UNDER BIAXIAL TANGENTIAL LOADING WITH THE METHOD OF DIMENSIONALITY REDUCTION

The paper is concerned with the contact between the elastic bodies subjected to a constant normal load and a varying tangential loading in two directions of the contact plane. For uni-axial in-plane loading, the Cattaneo-Mindlin superposition principle can be applied even if the normal load is not constant but varies as well. However, this is generally not the case if the contact is periodically loaded in two perpendicular in-plane directions. The applicability of the Cattaneo-Mindlin superposition principle guarantees the applicability of the method of dimensionality reduction (MDR) which in the case of a uni-axial in-plane loading has the same accuracy as the Cattaneo-Mindlin theory. In the present paper we investigate whether it is possible to generalize the procedure used in the MDR for bi-axial in-plane loading. By comparison of the MDR-results with a complete three-dimensional numeric solution, we arrive at the conclusion that the exact mapping is not possible. However, the inaccuracy of the MDR solution is on the same order of magnitude as the inaccuracy of the Cattaneo-Mindlin theory itself. This means that the MDR can be also used as a good approximation for bi-axial in-plane loading.