Numerical methods for contact between two joined quarter spaces and a rigid sphere

AbstractQuarter space problems have many useful applications wherever an edge is involved, and solution to the related contact problem requires extension to the classical Hertz theory. However, theoretical exploration of such a problem is limited, due to the complexity of the involved boundary conditions. The present study proposes a novel numerical approach to compute the elastic field of two quarter spaces, joined so that their top surfaces occupy the same plane, and indented by a rigid sphere with friction. In view of the equivalent inclusion method, the joined quarter spaces may be converted to a homogeneous half space with properly established eigenstrains, which are analyzed by our recent half space-inclusion solution using a three-dimensional fast Fourier transform algorithm. Benchmarked with finite element analysis the present method of solution demonstrates both accuracy and efficiency. A number of interesting parametric studies are also provided to illustrate the effects of material combinations, contact location and friction coefficient showing the deviation of the solution from Hertz theory

Fundamental relations for frictional and adhesive nonoindentation tests

Fundamental relations for depth-sensing nanoindentation are derived for indenters of various shapes and for various boundary conditions within the contact region. In particular, it is shown that some uncertainties in nanoindentation measurements, which are sometimes attributed to properties of the material, can be explained and quantitatively described by properly accounting for geometric deviation of the indenter tip from its nominal geometry

The JKR-type adhesive contact problems for transversely isotropic elastic solids

The JKR (Johnson, Kendall, and Roberts) and Boussinesq–Kendall models describe adhesive frictionless contact between two isotropic elastic spheres or between a flat end punch and an elastic isotropic half-space. Here adhesive contact is studied for transversely isotropic materials in the framework of the JKR theory. The theory is extended to much more general shapes of contacting axisymmetric solids, namely the distance between the solids is described by a monomial (power-law) function of an arbitrary degree d⩾1d⩾1. The classic JKR and Boussinesq–Kendall models can be considered as two particular cases of these problems, when the degree of the punch d is equal to two or it goes to infinity, respectively. It is shown that the formulae for extended JKR contact model for transversely isotropic materials have the same mathematical form as the corresponding formulae for isotropic materials; however the effective elastic contact moduli have different expression for different materials. The dimensionless relations between the actual force, displacements and contact radius are given in explicit form. Connections of the problems to nanoindentation of transversely isotropic materials are discussed

Self-similarity in abrasion of metals by nanosharp asperities of hard carbon containing films

Feodor M. Borodich Technological Institute, Northwestern University, Evanston, Illinois 60208-3109 Stephen J. Harris Physical and Environmental Sciences Dept., MD #3083, Ford Research Laboratory, Dearborn, Michigan 48121-2053 Leon M. Keer Technological Institute, Northwestern University, Evanston, Illinois 60208-3109 We discuss a remarkably simple power-law relationship observed between the abrasion rate of an initially spherical slider by hard carbon-containing films and the number of sliding cycles n to which the film has been subjected. The power-law relationship is valid up to 4 orders of magnitude in n. We model this phenomenon by connecting it with nanocharacteristics of the coatings. It is proposed to explain the phenomenon by a statistically self-similar variation of the pattern of relatively sharp nanometer-scale asperities of the films

Self-similarity in abrasiveness of hard carbon-containing coatings

The abrasiveness of hard carbon-containing thin films such as diamond-like carbon (DLC) and boron carbide (nominally B4C) towards steel is considered here. First, a remarkably simple experimentally observed power-law relationship between the abrasion rate of the coatings and the number of cycles is described. This relationship remains valid over at least 4 orders of magnitude of the number of cycles, with very little experimental scatter. Then possible models of wear are discussed. It is assumed that the dominant mechanism of steel wear is its mechanical abrasion by nano-scale asperities on the coating that have relatively large attack angles, i.e. by the so-called sharp asperities. Wear of coating is assumed to be mainly due to physical/chemical processes. Finally, models of the abrasion process for two basic cases are presented, namely a coated ball on a flat steel disk and a steel ball on a coated flat disk. The nominal contact region can be considered as constant in the former case, while in the latter case, the size of the region may be enlarged due to wear of the steel. These models of the abrasion process are based on the assumption of self-similar changes of the distribution function characterizing the statistical properties of patterns of scattered surface sharp asperities. It is shown that the power-law relationship for abrasion rate follows from the models

Analytical study of fundamental nanoindentation test relations for indenters of non-ideal shapes

Nanoindentation techniques provide a unique opportunity to obtain mechanical properties of materials of very small volumes. The load–displacement and load–area curves are the basis for nanoindentation tests, and their interpretation is usually based on the main assumptions of the Hertz contact theory and formulae obtained for ideally shaped indenters. However, real indenters have some deviation from their nominal shapes leading researchers to develop empirical 'area functions' to relate the apparent contact area to depth. We argue that for both axisymmetric and three-dimensional cases, the indenter shape near the tip can be well approximated by monomial functions of radius. In this case problems obey the self-similar laws. Using Borodich's similarity considerations of three-dimensional contact problems and the corresponding formulae, fundamental relations are derived for depth of indentation, size of the contact region, load, hardness, and contact area, which are valid for both elastic and non-elastic, isotropic and anisotropic materials. For loading the formulae depend on the material hardening exponent and the degree of the monomial function of the shape. These formulae are especially important for shallow indentation (usually less than 100 nm) where the tip bluntness is of the same order as the indentation depth. Uncertainties in nanoindentation measurements that arise from geometric deviation of the indenter tip from its nominal geometry are explained and quantitatively described

WTC2005-64085 MODELS OF FRICTION AND WEAR OF DLC FILMS

ABSTRACT Wear and friction of DLC (diamond-like carbon) covered counterparts are under consideration. Experiments showed that abrasion is the leading wear mechanism at the beginning of the wear process. However, the ability of the surface to wear away the counterpart reduces very rapidly, often as a power law function of the cycle numbers. This phenomenon was explained assuming that the initial abrasiveness of a coating is determined by the number of the nano-sharp asperities that were in contact with the counterpart, i.e. by the number of the sharp asperities within the nominal region of contact. On this basis, a model of abrasive wear was developed, using the concept of statistical self-similarity of distribution of the nano-sharp asperities within the current contact region. After the sharp asperities were blunted or removed from the contact region, the wear is related either to phase transformations or to chemical mechanisms. Recent experimental studies of dry sliding between two hydrogenated DLC coated counterparts in low oxygen environment showed that adsorbates have considerable influence on friction and the friction coefficient increases with the increasing of the time interval between contacts. The observed friction phenomena are assumed caused by a reaction between the adsorbate and carbon atoms of the coatings, and when the slider passes a point on the track, it removes mechanically some adsorbate from the surface. The mechanical action leads to re-exposure of the surface to gases in the environment. We assume that there is a transient short-life high temperature field at the vicinities of contacting protuberances that may cause various transformations of the surface. We suppose that first an adsorbate molecule becomes physically adsorbed to the surface and then chemisorbtion may occur between the carbon atoms of the coating and the 'sticky' oxygen atoms. The atoms or molecules of adsorbate interact with the conterpart. Our modeling established a direct connection between this kind of molecular friction and gradual wear. Using the new adsorption-desorption model, the numerical analyses of the friction coefficient were compared with experimental DLC friction results. INTRODUCTION Carbon-containing thin films such as DLC and boron carbide have the ability to enhance the fatigue resistance of heavily loaded steel component

WTC2005-64085 MODELS OF FRICTION AND WEAR OF DLC FILMS

ABSTRACT Wear and friction of DLC (diamond-like carbon) covered counterparts are under consideration. Experiments showed that abrasion is the leading wear mechanism at the beginning of the wear process. However, the ability of the surface to wear away the counterpart reduces very rapidly, often as a power law function of the cycle numbers. This phenomenon was explained assuming that the initial abrasiveness of a coating is determined by the number of the nano-sharp asperities that were in contact with the counterpart, i.e. by the number of the sharp asperities within the nominal region of contact. On this basis, a model of abrasive wear was developed, using the concept of statistical self-similarity of distribution of the nano-sharp asperities within the current contact region. After the sharp asperities were blunted or removed from the contact region, the wear is related either to phase transformations or to chemical mechanisms. Recent experimental studies of dry sliding between two hydrogenated DLC coated counterparts in low oxygen environment showed that adsorbates have considerable influence on friction and the friction coefficient increases with the increasing of the time interval between contacts. The observed friction phenomena are assumed caused by a reaction between the adsorbate and carbon atoms of the coatings, and when the slider passes a point on the track, it removes mechanically some adsorbate from the surface. The mechanical action leads to re-exposure of the surface to gases in the environment. We assume that there is a transient short-life high temperature field at the vicinities of contacting protuberances that may cause various transformations of the surface. We suppose that first an adsorbate molecule becomes physically adsorbed to the surface and then chemisorbtion may occur between the carbon atoms of the coating and the 'sticky' oxygen atoms. The atoms or molecules of adsorbate interact with the conterpart. Our modeling established a direct connection between this kind of molecular friction and gradual wear. Using the new adsorption-desorption model, the numerical analyses of the friction coefficient were compared with experimental DLC friction results. INTRODUCTION Carbon-containing thin films such as DLC and boron carbide have the ability to enhance the fatigue resistance of heavily loaded steel component