This review is concerned with the basic physical meaning of hardness. It is shown that indentation hardness of ductile materials is essentially a measure of their plastic properties. With brittle solids the high hydrostatic pressures around the deformed region are often sufficient to inhibit brittle fracture. Under these conditions both indentation and scratch hardness are essentially a measure of the plastic rather than the brittle properties of the solid. This provides a simple physical basis for the Mohs scratch hardness scale

https://iopscience.iop.org/article/10.1088/0034-6683/1/3/I01/pdf

The hardness of solids

A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring 'sinh law' fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus $G$, the zero-rate viscosity $\eta $ and a reference stress $\tau _{0}$. The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.

Shear behaviour of elastohydrodynamic oil films

Experiments have shown that nanoindentation unloading curves obtained with Berkovich triangular pyramidal indenters are usually welldescribed by the power-law relation P = α(h − hf)m, where hf is the final depth after complete unloading and α and m are material constants. However, the power-law exponent is not fixed at an integral value, as would be the case for elastic contact by a conical indenter (m = 2) or a flat circular punch (m = 1), but varies from material to material in the range m = 1.2–1.6. A simple model is developed based on observations from finite element simulations of indentation of elastic–plastic materials by a rigid cone that provides a physical explanation for the behavior. The model, which is based on the concept of an indenter with an “effective shape” whose geometry is determined by the shape of the plastic hardness impression formed during indentation, provides a means by which the material constants in the power law relation can be related to more fundamental material properties such as the elastic modulus and hardness. Simple arguments are presented from which the effective indenter shape can be derived from the pressure distribution under the indenter.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0884291400062129

Understanding nanoindentation unloading curves

A new approach for analyzing indentation plasticity and for determining indentation stress fields is presented. The analysis permits relations to be established between material properties (notably hardness, yield strength, and elastic modulus) and the dimensions of the indentation and plastic zone. The predictions are demonstrated to correlate with observations performed on a wide range of materials. The indentation stress fields are computed along trajectories pertinent to three dominant indentation crack systems: radial, median, and lateral cracks. The peak load and residual tensile stresses are shown to be consistent with observed trends in indentation fracture.

The response of solids to elastic/plastic indentation. I. Stresses and residual stresses

This book draws upon the science of tribology to understand, predict and improve abrasive machining processes. Pulling together information on how abrasives work, the authors, who are renowned experts in abrasive technology, demonstrate how tribology can be applied as a tool to improve abrasive machining processes. Each of the main elements of the abrasive machining system are looked at, and the tribological factors that control the efficiency and quality of the processes are described. Since grinding is by far the most commonly employed abrasive machining process, it is dealt with in particular detail. Solutions are posed to many of the most commonly experienced industrial problems, such as poor accuracy, poor surface quality, rapid wheel wear, vibrations, work-piece burn and high process costs. This practical approach makes this book an essential tool for practicing engineers. Uses the science of tribology to improve understanding and of abrasive machining processes in order to increase performance, productivity and surface quality of final products. A comprehensive reference on how abrasives work, covering kinematics, heat transfer, thermal stresses, molecular dynamics, fluids and the tribology of lubricants. Authoritative and ground-breaking in its first edition, the 2nd edition includes 30 per cent new and updated material, including new topics such as CMP (Chemical Mechanical Polishing) and precision machining for micro-and nano-scale applications.

Tribology of Abrasive Machining Processes

It is shown that the mechanism and extent of the deformation of materials by wedges depends upon the angle of the indenter, and on the elasticity of the indented material as well as on its yield point. The theory for a plastic rigid solid applies only when the angle of the wedge is acute and the indented materials are of low elasticity. At more acute angles, the mechanism of deformation has been shown by Mulhearn to become one of radial compression. It is shown here that this mode of deformation is also favoured when more elastic materials are used. It is also shown that the indentation pressure approximates to that for the expansion of a semicylindrical cavity in an infinite elastic medium with definite yield point and takes the form $P/Y = m \ln (E/Y)+C$ where m and C are constants, E is the Young modulus, Y the yield point in simple extension and P the flow pressure. The value of C depends upon the angle of the wedge. The result is analogous to that obtained by Marsh using a 136$^\circ$ pyramidal indenter, but elastic effects are more important in wedge indentation than when indenting with a pyramid. For blunt wedges and elastic materials, elastic effects predominate in importance; the process can then be regarded as that of the indentation of an ideally elastic solid by a rigid wedge. The pressure distribution on the indentation has been determined. It is shown that elastic theory satisfactorily predicts this distribution of pressure, as well as its mean value. For the indentation of highly elastic materials by acute angled wedges, the elastic and plastic deformations become comparable and the mode of deformation is very complex.

https://www.jstor.org/stable/pdfplus/2416241.pdf

The indentation of materials by wedges

The wear of diamond when rubbing against materials softer than itself is investigated. It is shown that the rate of wear depends primarily on the hardness of the second member, varying as hardness to the power 3.5 for hardnesses above 200 V.p.n. When the hardness of the second member is high enough, the tensile stresses acting on the diamond reach the fracture stress, and the wear quickly becomes visible. Below this point, the rate of wear is small and there is an appreciable delay before its onset can be detected. For metals, such as copper, silver and gold, which are well below the critical hardness, the scale of the wear process becomes extremely small and the relation between wear rate and hardness changes. It is thought that in these conditions the rate of wear becomes affected by residual effects arising from the stressing of the surface during the initial processes of polishing.

The Wear of Diamond

An experimental study has been made of the elastohydrodynamic behaviour of a range of fluids, including some of widely different pressure coefficient of viscosity. It is shown that the complex pattern of elasto-hydrodynamic behaviour observed at high pressure is due predominantly to the interplay between two non-Newtonian effects. The first is viscoelasticity. This has the effect that, at small degrees of shear, all the fluids behave as viscous liquids at low pressure and as elastic solids at high. The transition occurs when the viscosity attains 10 5 Pa s. The second is the non-linearity at high degrees of shear between the shear stress and the rate of shear. The consequence of this is that at sufficiently high rates of shear the fluid loses any elastic character and behaves as a non-Newtonian liquid. It is shown that the behaviour of the real system resembles that of a Maxwell viscoelastic model into which Eyring’s elementary expression for viscosity is incorporated to describe the non-Newtonian character of the liquid component. At high shear stresses the value of the traction coefficient {T/W ) is given by the expression T/W = α ¯ T 0 - T 0 /T 0 / p ln ( T 0 /2 n 0 s), where α ¯ is the pressure coefficient of viscosity, T 0 a characteristicshearstress, p the pressure, n 0 the viscosity at atmospheric pressure and s the rate of shear. The values of α ¯ which are appropriate relate to the actual pressures applied. These pressures are considerably in excess of those obtainable in conventional high pressure viscometry but it is shown that the appropriate value of α ¯ can be derived satisfactorily from the elastohydrodynamic experiments themselves. The fluid giving the highest value of the traction coefficient proved to be one whose pressure coefficient of viscosity was considerably greater at high pressure than at low. The results suggest that with high polymers the structural unit responsible for flow is considerably smaller than the polymer molecule itself. In contrast, certain types of molecules having branch chains appear to become entangled making the unit for flow appreciably larger than the individual molecule.

Elastohydrodynamic lubrication at high pressures II. Non-Newtonian behaviour

Film thickness and traction have been measured in a two-disk machine over a range of rolling and sliding speeds, by using two mineral oils which have previously been studied at lower pressures. The results show: (1) That the lubricant film thickness is correctly given by 'classical' elastohydrodynamic theory, even when the behaviour of the lubricant in the high pressure regions is quite different from the Newtonian viscous behaviour postulated in the theory. (2) That over the range of pressure 0.7-2.5 GPa both oils behave as elastic solids with a well defined shear modulus. (3) That the elastic compliance of the disks may be comparable with or exceed that in the oil film and must be taken into account in the calculation of the shear modulus of the oils. (4) That at these high pressures the elastic properties, not the viscous properties, of the oil determine the traction when the shear is small. (5) That there is an elastic limit at a critical shear stress above which the shear increases more rapidly than the stress. The magnitude of this critical stress increases with the pressure. (6) That the transition from viscous to elastic behaviour takes place over a relatively narrow range of pressure above which the elastic behaviour becomes insensitive to the rolling speed at which the tests are performed.

Elastohydrodynamic Lubrication at High Pressures

The traction in pure rolling has been measured at constant temperature over a range of load and speed. Measurements have been made at controlled temperatures of 10, 20 and 30 0C on a paraffinic and a naphthenic based mineral oil. The results form a systematic pattern and provide a fuller picture of the variation of the enhanced viscosity of the oil with load and rolling speed. than has been available hitherto. It is shown that the viscosity tends towards a limiting value when the rolling speed is low. The results are compared with recent measurements by Hamilton & Moore of the variation of the pressure distribution with load and speed. Both sets of results appear to suggest that the response of viscosity to the application of pressure is not instantaneous, but is delayed with a characteristic relaxation time. However, quantitative examination shows that this hypothesis is untenable. It is then shown that the variation of viscosity with rolling speed originates from the variation of the pressure distribution with speed. In the region of highest pressure the response of viscosity to pressure is normal. In the inlet and outlet zones, the shear stresses are very high and in these regions the oils exhibit non-Newtonian behaviour.

W. Hirst

Papers

The indentation of materials by wedges

The Wear of Diamond

Elastohydrodynamic lubrication at high pressures II. Non-Newtonian behaviour

Elastohydrodynamic Lubrication at High Pressures

Frictional traction in elastohydrodynamic lubrication