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This application-oriented book introduces the associations between contact mechanics and friction and with it offers a deeper understanding of tribology. It deals.

**Table of contents**

- Contact Mechanics and Friction : Physical Principles and Applications
- The Mathematical Sciences in 2025
- Contact Mechanics and Friction: Physical Principles and Applications
- Physical Principles and Applications
- Get PDF Contact Mechanics and Friction: Physical Principles and Applications

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## Contact Mechanics and Friction : Physical Principles and Applications

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Add to Cart Add to Cart. Add to Wishlist Add to Wishlist. The book was expanded by the addition of a chapter on frictional problems in ear- quake research. Additionally, Chapter 15 was supplemented by a section on elasto-hydrodynamics. The problem sections of several chapters were enriched by the addition of new examples. This book would not have been possible without the active support of J. Gray, who translated it from the German edition. Fujimasa I. The book contains a structured collection of the complete solutions of all essential axisymmetric contact problems.

Based on a systematic distinction regarding the type of contact, the regime of friction and the contact geometry, a multitude of technically relevant contact problems from mechanical engineering, the automotive industry and medical engineering are discussed. The book takes into account adhesive effects which allow access to contact-mechanical questions about micro- and nano-electromechanical systems. Solutions are always obtained with the simplest available method — usually with the method of dimensionality reduction MDR or approaches which use the solution of the non-adhesive normal contact problem to solve the respective contact problem.

To aid engineers in design decisions, Friction and Wear of Materials evaluates the properties of materials which, under specified conditions, cause one material to function better as a bearing material than another. Featured also are thorough treatments of lubricants and the sizes and shapes of wear particles. This updated Second Edition includes new material on erosive wear, impact wear, and friction.

Valentin L. This application-oriented book introduces the associations between contact mechanics and friction and with it offers a deeper understanding of tribology. It deals with the associated phenomena of contact, adhesion, capillary forces, friction, lubrication, and wear from one consistent viewpoint.

The author goes into 1 methods of rough estimation of tribological quantities, 2 methods for analytical calculations which attempt to minimize the necessary complexity, 3 the crossover into numerical simulation methods. With these methods the author conveys a consistent view of tribological processes in various scales of magnitude from nanotribology to earthquake research.

Also, system dynamic aspects of tribological systems, such as squeal and its suppression as well as other types of instabilities and spatial patterns are investigated.

This book contains problems and worked solutions for individual chapters in which the reader can apply the theory to practical situations and deepen the understanding of the material. James R. Contact Mechanics. Springer, This book describes the solution of contact problems with an emphasis on idealized mainly linear elastic problems that can be treated with elementary analytical methods. General physical and mathematical features of these solutions are highlighted. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer.

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The paper is devoted to a qualitative analysis of friction of elastomers from the point of view of scales contributing to the force of friction. We argue that — contrary to widespread opinion — friction between a randomly rough self-affine fractal surface and an elastomer is not a multiscale phenomenon, but is governed mostly by the interplay of only two scales — as a rule the largest and the smallest scales of roughness of the contacting bodies.

The hypothesis of two-scale character of elastomer friction is illustrated by computer simulations in the framework of the paradigm of Greenwood, Tabor and Grosch using a simplified one-dimensional model. Already Coulomb knew that the coefficient of sliding friction depends on sliding velocity and normal force and that static friction depends explicitly on time 2.

### The Mathematical Sciences in 2025

However, even years after Coulomb no generalized laws of friction exist that could be reliably used in engineering practice. It is often argued that the reason for this lies in the multi-scale nature of friction, and that all scales necessarily have to be included into consideration to achieve a realistic model of friction.

In the present paper we will argue that this is not the case: In friction there exist characteristic scales which provide the main contribution to the quantities of interest. Therefore, as in most areas of physics, the most productive approach is searching for the relevant scales and studying them. The present work is devoted to an analysis of the contribution to friction of different scales for one class of materials — elastomers with linear rheology.

Since Greenwood and Tabor 3 , and especially after the classic work of Grosch 4 , it has been widely accepted that elastomer friction is mostly due to internal dissipative losses in the material that are caused by deformation through surface asperities of the counter body. In the present paper, we follow the above paradigm of Greenwood-Tabor-Grosch and do not discuss the adhesive contribution to friction. The force of friction typically increases with velocity, reaches a plateau and decreases again, as schematically shown in Fig.

The plateau is normally of most practical interest.

## Contact Mechanics and Friction: Physical Principles and Applications

Physically, the behavior of an elastomer in this range is dominated by the loss modulus of the elastomer 5 , and the elastomer behaves roughly speaking as a simple fluid. In particular, the relaxation of the elastomer after indentation or ploughing by an asperity is very slow. Practically all micro contacts therefore will be in one-sided contacts as shown in Fig.

It is easy to understand that the coefficient of friction is then roughly equal to the one-sided average of the local slope of the surface profile in the contact region. This simplified picture is valid if the contact between elastomer and the rigid body is friction free on scales smaller than that of the asperities. Despite the apparent simplicity of this physical picture, already at this point an interesting and non-trivial question arises: What is the average slope of the profile? Let us start the discussion of this question with an estimation of the rms value of the gradient of the surface profile over the whole surface.

In this limited range, the spectral power density of typical fractal surfaces is known to be a power-function of the wave-vector q : where H is the Hurst-Exponent and q 0 is some reference wave-vector 8. For Hurst exponents smaller than one, the resulting integral diverges at the upper limit of integration.

This means that for a true fractal surface without an upper cut-off wave vector , the surface gradient would be infinitely large.

### Physical Principles and Applications

In practice, of course, there is always some upper cut-off wave vector q max , and the surface gradient is determined by one or two orders of magnitude of wave vectors at and below q max. In other words, for typical fractal rough surfaces, the friction force is determined by the roughness components with the largest wave-vectors or the smallest scale of the system.

One can say that understanding friction is equivalent to understanding the nature of this smallest relevant scale. The conclusion that the surface gradient is mainly determined by the smallest space scales follows from very general scaling considerations of self-affine fractals and is not limited to the spectral representation of rough surfaces. However, for the sake of simplicity and transparency of argumentation, we will confine ourselves to consideration of randomly rough surfaces according to definition 6. Of course, the above estimation is oversimplified in the sense that it is the surface slope in the contact region and not over the whole surface which is determining the coefficient of friction at the plateau.

However, as has been shown already by Archard in th 9 , the main effect of changing normal force is the number of asperities coming into contact, while the local conditions in the real contact area, including the surface gradient, depend only weakly on the normal force. Thus, the average gradient over the whole surface is already a good estimation for the gradient in the real contact area. However, the mentioned relatively weak dependence on the normal force is exactly what we would like to discuss in more detail.

The actual surface gradient is a function of the current contact configuration e. In the next section we will argue that the governing parameter for the contact configuration is the indentation depth d. The indentation depth, in turn is connected with the normal force over the contact stiffness, which is dependent practically only on the large wavelength components of roughness or on the macroscopic form of the indenter We will thus come to the following hypothesis: while the friction force is almost entirely dependent on the smallest-scale roughness, its weak dependence on the normal force is related only to the large scale roughness.

We then will substantiate this two-scale hypothesis with a numerical simulation of the force of friction between an elastomer and a randomly rough fractal surface using a simplified one-dimensional model. If a rigid body of an arbitrary shape is pressed against a homogeneous elastic half-space then the resulting contact configuration is only a function of the indentation depth d. At a given indentation depth, the contact configuration does not depend on the elastic properties of the medium, and will be the same even for indentation of a viscous fluid or of any linearly viscoelastic material.

## Get PDF Contact Mechanics and Friction: Physical Principles and Applications

This general behavior was recognized by Lee 12 and Radok 13 and was verified numerically for fractal rough surfaces Further, the contact configuration at a given depth remains approximately invariant for media with thin coatings 15 and for multi-layered systems, provided the difference of elastic properties of the different layers is not too large In 17 , it was argued that this is equally valid for media which are heterogeneous in the lateral direction along the contact plane.

Along with the contact configuration, all contact properties including the real contact area, the contact length, the contact stiffness, as well as the rms value of the surface gradient in the contact area will be unambiguous functions of the indentation depth. Note, that this is equally valid for tangential contact. This result does not depend on the form of the body and is valid for arbitrary bodies of revolution and even for randomly rough fractal surfaces In practice, however, the controlled and measured quantity is normally not the indentation depth but the normal force.