In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.

This paper is devoted to an analytical, numerical, and experimental analysis of adhesive contacts subjected to tangential motion. In particular, it addresses the phenomenon of instable, jerky movement of the boundary of the adhesive contact zone and its dependence on the surface roughness. We argue that the "adhesion instabilities" with instable movements of the contact boundary cause energy dissipation similarly to the elastic instabilities mechanism. This leads to different effective works of adhesion when the contact area expands and contracts. This effect is interpreted in terms of "friction" to the movement of the contact boundary. We consider two main contributions to friction: (a) boundary line contribution and (b) area contribution. In normal and rolling contacts, the only contribution is due to the boundary friction, while in sliding both contributions may be present. The boundary contribution prevails in very small, smooth, and hard contacts (as e.g., diamond-like-carbon (DLC) coatings), while the area contribution is prevailing in large soft contacts. Simulations suggest that the friction due to adhesion instabilities is governed by "Johnson parameter". Experiments suggest that for soft bodies like rubber, the stresses in the contact area can be characterized by a constant critical value. Experiments were carried out using a setup allowing for observing the contact area with a camera placed under a soft transparent rubber layer. Soft contacts show a great variety of instabilities when sliding with low velocity - depending on the indentation depth and the shape of the contacting bodies. These instabilities can be classified as "microscopic" caused by the roughness or chemical inhomogeneity of the surfaces and "macroscopic" which appear also in smooth contacts. The latter may be related to interface waves which are observed in large contacts or at small indentation depths. Numerical simulations were performed using the Boundary Element Method (BEM).

In 1953 Archard formulated his general law of wear stating that the amount of worn material is proportional to the normal force and the sliding distance, and is inversely proportional to the hardness of the material. Five years later in 1958, Rabinowicz suggested a criterion determining the minimum size of wear particles. Both concepts became very popular due to their simplicity and robustness, but did not give thorough explanation of the mechanisms involved. It wasn’t until almost 60 years later in 2016 that Aghababaei, Warner and Molinari (AWM) used quasi-molecular simulations to confirm the Rabinowicz criterion. One of the central quantities remained the “asperity size”. Because real surfaces have roughness on many length scales, this size is often ill-defined. The present paper is devoted to two main points: First, we generalize the Rabinowicz-AWM criterion by introducing an “asperity-free” wear criterion, applicable even to fractal roughness. Second, we combine our generalized Rabinowicz criterion with the numerical contact mechanics of rough surfaces and formulate on this basis a deterministic wear model. We identify two types of wear: one leading to the formation of a modified topography which does not wear further and one showing continuously proceeding wear. In the latter case we observe regimes of least wear, mild wear and severe wear which have a clear microscopic interpretation. The worn volume in the region of mild wear occurs typically to be a power law of the normal force with an exponent not necessarily equal to one. The method provides the worn surface topography after an initial settling phase as well as the size distribution of wear particles. We analyse different laws of interface interaction and the corresponding wear laws. A comprehensive parameter study remains a task for future research.

60 years ago, in 1958, Ernest Rabinowicz published a 5 page paper titled “The effect of size on the looseness of wear fragments” where he suggested a criterion determining the minimum size of wear particles. The criterion of Rabinowicz is based on the consideration of the interplay of elastic energy stored in “asperities” and the work of separation needed for detaching a wear particle. He was probably the first researcher who explicitly emphasized the role of adhesion in friction and wear. In a recent paper in Nature Communications, Aghababaei, Warner and Molinari confirmed the criterion of Rabinowicz by means of quasi-molecular dynamics and illustrated the exact mechanism of the transition from plastic smoothing to formation of wear debris. This latter paper promoted the criterion of Rabinowicz to a new paradigm for current studies of adhesive wear. The size arguments of Rabinowicz can be applied in the same form also to many other problems, such as brittle-ductile transition during indentation, cutting of materials or ultimate strength of nano-composites.

The present paper is devoted to a theoretical analysis of sliding friction under the influence of in-plane oscillations perpendicular to the sliding direction. Contrary to previous studies of this mode of active control of friction, we consider the influence of the stiffness of the tribological contact in detail and show that the contact stiffness plays a central role for small oscillation amplitudes. In the present paper we consider the case of a displacement-controlled system, where the contact stiffness is small compared to the stiffness of the measuring system. It is shown that in this case the macroscopic coefficient of friction is a function of two dimensionless parameters—a dimensionless sliding velocity and dimensionless oscillation amplitude. In the limit of very large oscillation amplitudes, known solutions previously reported in the literature are reproduced. The region of small amplitudes is described for the first time in this paper.

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 influence of out-of-plane oscillations on friction is a well-known phenomenon that has been studied extensively with various experimental methods, e.g., pin-on-disk tribometers. However, existing theoretical models have yet achieved only qualitative correspondence with experiment. Here we argue that this may be due to the system dynamics (mass and tangential stiffness) of the pin or other system components being neglected. This paper builds on the results of a previous study [19] by taking the stiffness and resulting dynamics of the system into account. The main governing parameters determining macroscopic friction, including a dimensionless oscillation amplitude, a dimensionless sliding velocity and the relation between three characteristic frequencies (that of externally excited oscillation and two natural oscillation frequencies associated with the contact stiffness and the system stiffness) are identified. In the limiting cases of a very soft system and a very stiff system, our results reproduce the results of previous studies. In between these two limiting cases there is also a resonant case, which is studied here for the first time. The resonant case is notable in that it lacks a critical sliding velocity, above which oscillations no longer reduce friction. Results obtained for the resonant case are qualitatively supported by experiments.

Computer simulations have been an integral part of the technical development process for a long time now. Industrial tribology is one of the last fields in which computer simulations have, until now, played no significant role. This is primarily due to the fact that investigating tribological phenomena requires considering all spatial scales from the macroscopic shape of the contact system down to the micro-scales. In the present paper, we give an overview of the previous work on the so-called method of reduction of dimensionality (MRD), which in our opinion, gives a key for the linking of the micro- and macro-scales in tribological simulations.

MRD in contact mechanics is based on the mapping of some classes of three-dimensional contact problems onto one-dimensional contacts with elastic foundations. The equivalence of three-dimensional systems to those of one-dimension is valid for relations of the indentation depth and the contact force and in some cases for the contact area. For arbitrary bodies of revolution, MRD is exact and provides a sort of “pocket edition” of contact mechanics, giving the possibility of deriving any result of classical contact mechanics with or without adhesion in a very simple way.

A tangential contact problem with and without creep can also be mapped exactly to a one-dimensional system. It can be shown that the reduction method is applicable to contacts of linear visco-elastic bodies as well as to thermal effects in contacts. The method was further validated for randomly rough self-affine surfaces through comparison with direct 3D simulations.

MRD means a huge reduction of computational time for the simulation of contact and friction between rough surfaces accounting for complicated rheology and adhesion. In MRD, not only is the dimension of the space reduced from three to one, but the resulting degrees of freedom are independent (like normal modes in the theory of oscillations). Because of this independence, the method is predestinated for parallel calculation on graphic cards, which brings further acceleration. The method opens completely new possibilities in combining microscopic contact mechanics with the simulation of macroscopic system dynamics without determining the “law of friction” as an intermediate step.