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Über dieses Buch

This book describes for the first time a simulation method for the fast calculation of contact properties and friction between rough surfaces in a complete form. In contrast to existing simulation methods, the method of dimensionality reduction (MDR) is based on the exact mapping of various types of three-dimensional contact problems onto contacts of one-dimensional foundations. Within the confines of MDR, not only are three dimensional systems reduced to one-dimensional, but also the resulting degrees of freedom are independent from another. Therefore, MDR results in an enormous reduction of the development time for the numerical implementation of contact problems as well as the direct computation time and can ultimately assume a similar role in tribology as FEM has in structure mechanics or CFD methods, in hydrodynamics. Furthermore, it substantially simplifies analytical calculation and presents a sort of “pocket book edition” of the entirety contact mechanics. Measurements of the rheology of bodies in contact as well as their surface topography and adhesive properties are the inputs of the calculations. In particular, it is possible to capture the entire dynamics of a system – beginning with the macroscopic, dynamic contact calculation all the way down to the influence of roughness – in a single numerical simulation model. Accordingly, MDR allows for the unification of the methods of solving contact problems on different scales. The goals of this book are on the one hand, to prove the applicability and reliability of the method and on the other hand, to explain its extremely simple application to those interested.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
The goal of this book is to describe the method of dimensionality reduction in contact mechanics and friction. Contacts between three-dimensional bodies arise in a wide variety of applications. Therefore, their simulation, both analytically and numerically, are of major importance. From a mathematical point of view, contacts are described using integral equations having mixed boundary conditions. Furthermore, the stress distribution, the displacements of the surface, and even the shape of the contact domain are generally not known in such problems. It is, therefore, astounding that a large number of classical contact problems can be mapped to one-dimensional models of contacts with a properly defined linearly elastic foundation (Winkler foundation). This means that the results of the one-dimensional model correspond exactly to those of a three-dimensional model. According to this mapping concept, solving contact-mechanical problems is trivialized is such a way as to require no special knowledge other than the fundamentals of algebra and calculus.
Valentin L. Popov, Markus Heß

Chapter 2. Separation of the Elastic and Inertial Properties in Three-Dimensional Systems

Abstract
For a wide class of “typical tribological systems,” there are a number of properties that allow for the significant simplification of contact problems and, in this way, make fast calculations of multi-scalar systems possible. These simplified properties, which are used by the method of dimensionality reduction, include (1) the ability to separate the elastic and inertial properties in three-dimensional systems and (2) the close analogy between three-dimensional contacts and certain one-dimensional problems. The first of these will be discussed in this chapter, while further chapters are dedicated to the second one.
Valentin L. Popov, Markus Heß

Chapter 3. Normal Contact Problems with Axially-Symmetric Bodies Without Adhesion

Abstract
The method of dimensionality reduction (MDR) is based on the observation that certain types of three-dimensional contacts can be exactly mapped to one-dimensional linearly elastic foundations. The MDR consists essentially of two simple steps: (a) substitution of the three-dimensional continuum by a uniquely defined one-dimensional linearly elastic or viscoelastic foundation (Winkler foundation) and (b) transformation of the three-dimensional profile of the contacting bodies by means of the MDR-transformation. As soon as these two steps are done, the contact problem can be considered to be solved. For axially-symmetric contacts, only a small calculation by hand is required which not exceed elementary calculus and will not be a barrier for any practically-oriented engineer. In spite of its simplicity, all results are exact. The present chapter describes the basic ideas of MDR in its application to normal contacts.
Valentin L. Popov, Markus Heß

Chapter 4. Normal Contact with Adhesion

Abstract
Adhesive contacts of axially-symmetric bodies can also be exactly mapped to a one-dimensional equivalent model. The rule for this mapping was developed in 2011 by Markus Heß. It is based on the basic idea of Johnson, Kendall, and Roberts that the contact with adhesion arises from the contact without adhesion plus a rigid-body translation. Because both parts of the contact problem can be mapped to a one-dimensional equivalent model with a modified geometry, then this is true of the entire problem. To begin, we will formulate the simple rules of application for the adhesive normal contact and refrain from presenting the required evidence. Subsequently, these rules will be explained in more detail, which requires a certain understanding of the theoretical background on adhesion in three-dimensional contacts, which we will also provide. For those not satisfied with these short explanations, the entirety of the necessary evidence may be found in Chap. 17.
Markus Heß, Valentin L. Popov

Chapter 5. Tangential Contact

Abstract
The fundamental property that allows the reduction of three-dimensional contacts to those in one dimension is the proportionality of the incremental stiffness to the diameter of the contact area. This property is exhibited by both normal and tangential contacts. The idea behind dimensionality reduction can, therefore, be directly transferred to tangential contacts. More accurately, the transferability of the method of dimensionality reduction to tangential contacts follows from the ideas and results from Cattaneo, Mindlin, Jäger, and Ciavarella about the close relationship between normal and tangential contacts. In this chapter, we will show how tangential contacts exhibiting Coulomb friction with arbitrarily axially-symmetric profiles can be exactly mapped onto a properly defined one-dimensional elastic foundation.
Markus Heß, Valentin L. Popov

Chapter 6. Rolling Contact

Abstract
Rolling contacts are found in a multitude of engineering applications, such as wheel–rail or tire–street contacts, ball bearings, gears, and various transport mechanisms. In this chapter, it will be shown that the tangentially loaded, three-dimensional rolling contact can also be mapped to a one-dimensional equivalent system. In this way, the method of dimensionality reduction allows complex problems, such as the oscillating rolling contact, to be investigated.
Robbin Wetter, Valentin L. Popov, Markus Heß

Chapter 7. Contacts with Elastomers

Abstract
Rubber and other elastomers play a large role in many tribological applications. They are used where large frictional forces or large deformations are needed. These materials are especially used for tires, transportation rollers, shoes, seals, rubber bands, in electronic devices (e.g., contacts for keyboards) as well as applications for adhesion. When compared to purely elastic contacts, the calculation of elastomer contacts is made more difficult by the fact that they exhibit a time-dependent behavior, which is also normally characterized by a large spectrum of relaxation times. The correct mechanical description must, therefore, take several orders of magnitude in characteristic times into account. The multi-scalar properties of the surface roughness are supplemented here by the multi-scalar character of the relaxation of the material, which makes the numerical simulation of elastomers especially complicated. It is, therefore, important to develop fast simulation methods for the calculation of contact and frictional properties for this class of materials. In this chapter, we will show how the method of dimensionality reduction can be generalized to contacts of elastomers with arbitrary linear rheology.
Silvio Kürschner, Valentin L. Popov, Markus Heß

Chapter 8. Heat Transfer and Heat Generation

Abstract
Thermal conductivity is a decisive parameter for the sizing of heat sinks for semiconductors or for other elements in electronic circuits. Heat production and conductivity may also be an important topic in elastomer friction. In this chapter, it will be shown that also problems of heat and electrical conductivity can be exactly solved within the framework of the method of dimensionality reduction. The mappability is not only limited to the thermal/electrical conductivity or resistance, but also includes local parameters, such as the temperature distribution on the surface.
Markus Heß, Valentin L. Popov

Chapter 9. Adhesion with Elastomers

Abstract
The application of the method of dimensionality reduction to adhesive contacts between elastic bodies is given by the rule of Heß (Eq. 4.1). However, this rule cannot be directly generalized to include contacts between viscoelastic bodies. This can already be seen in the fact that the “separation criterion” from Heß contains the modulus of elasticity. The effective modulus of elasticity of elastomers, however, is dependent on the deformation speed or frequency. Therefore, to be able to transfer the results of Heß to those of viscoelastic media, a better physical understanding of the phenomenon of adhesion is necessary. For this, it is helpful to consider a microscopic picture of an adhesive contact. The fundamentals of this were already described in Chap. 4. At this point, we will generalize these ideas for their application to viscoelastic media.
Valentin L. Popov, Markus Heß

Chapter 10. Normal Contact of Rough Surfaces

Abstract
In addition to the strict geometrically defined cases that were mapped in Chap. 3 with the method of dimensionality reduction, we would now like to devote ourselves to the question of whether rough surfaces can also be handled with the reduction method. The importance of surface roughness for tribological processes was already emphasized by Bowden and Tabor in the 1940s and since that time has become generally accepted. The most important fundamental work dealing with the contact mechanics of rough surfaces was conducted in the 1950s by Archard and in the 1960s by Greenwood and Williamson. However, the contact mechanics of rough surfaces remains even today a current and to some extent, controversial topic. In this chapter, we will show that there exist theoretical as well as the empirical reasons for why the method of dimensionality reduction is also able to be applied to randomly rough surfaces. In this way, the method presents itself as a practical tool for the fast calculation of contact problems.
Roman Pohrt, Valentin L. Popov, Markus Heß

Chapter 11. Frictional Force

Abstract
In the previous chapters, we have seen that within the framework of the method of dimensionality reduction, the relationships between force and displacement can be correctly determined in both the normal and tangential directions. This means that also the energy dissipation, and with it the frictional force, must be mappable. In this chapter, we discuss in detail how the frictional forces between an elastomer and a rigid surface (smooth or rough) can be modeled.
Valentin L. Popov, Markus Heß, Silvio Kürschner

Chapter 12. Frictional Damping

Abstract
Friction is a dissipative process, in which mechanical energy is transformed into heat. This can be both unwanted as well as purposefully taken advantage of. Even at very small amplitudes of tangential oscillations, the small slip displacements at the border of the contact area always lead to energy dissipation. This effect is the physical mechanism of damping in periodically forced frictionally engaged joints, for example, in leaf springs for commercial and transportation vehicles. Similar effects are generally exhibited in all frictionally engaged joints and are, therefore, of great interest. For the investigation of damping caused by dry friction, a dynamic tangential contact is of interest. The exact coincidence of the frictional damping in a true three-dimensional contact and its one-dimensional representation in the framework of the method of dimensionality reduction follows from general theorems concerning tangential contacts. This chapter is an illustration of how the use of the MDR makes dynamic tangential problems simple without loss of exactness.
Elena Teidelt, Valentin L. Popov, Markus Heß

Chapter 13. The Coupling to Macroscopic Dynamics

Abstract
In practical applications, mechanical models are frequently considered in which macroscopic frictional contacts are present. The typical procedure used to describe frictional contacts is to formulate a suitable law of friction, which is then subsequently applied in a macroscopic simulation of the system dynamics. However, it is often difficult to formulate a useable law of friction, as the frictional force is not only a function of the instantaneous state of motion of the system, but is also dependent on the previous history of the system motion. Here, the method of dimensionality reduction opens up a new path: The simulations using the reduction method are carried out so quickly that one can completely forego the previous formulation of the frictional law. The calculation of the contact and frictional forces is then carried out directly within the framework of the macroscopic simulation of the system dynamics. Models that combine both the macrosimulations and microsimulations into one are termed hybrid models in the following. In this chapter, it will be shown how the method of dimensionality reduction is used for the formulation of hybrid models.
Elena Teidelt, Valentin L. Popov, Markus Heß

Chapter 14. Acoustic Emission in Rolling Contacts

Abstract
The dynamics of rolling on rough surfaces is of great interest from the viewpoint of functionality, noise production, and for the diagnostics of many technical systems, such as gears, ball bearings, wheel–rail, tire–road systems, and many others. The main difficulty in analyzing the rolling noise is solving the problem of contact between two elastic rough bodies. In this chapter, we use the method of dimensionality reduction for solving this problem.
Mikhail Popov, Justus Benad, Valentin L. Popov, Markus Heß

Chapter 15. Coupling to the Microscale

Abstract
The application of the method of dimensionality reduction is, of course, limited to the spatial scales for which continuum mechanics can be used. Every practical application using the method will lose its validity even earlier, due to the finite spatial resolution of the surface topography. Therefore, it begs the question of whether the interactions on even smaller scales can be summarized into a microscopic “contact law” or “law of friction,” so that also the properties of the smallest possible scale can be taken into account in the simulation. In this chapter, we explain how the limitations of the finite spatial resolution can be eliminated by the introduction of a “microscopic” nonlinear stiffness.
Valentin L. Popov, Roman Pohrt, Markus Heß

Chapter 16. And Now What?

Abstract
The most valuable aspect of the method of dimensionality reduction is the fact that it is a practical tool for many engineering applications. In several applications, such as the contact mechanics of axially-symmetric bodies, it provides exact results. In the case of the contact of randomly rough surfaces, the method provides asymptotically exact results (for very small and very large forces) and offers in the transitional domain a very good approximation, the accuracy of which exceeds that of the typically available values for material and surface parameters. In this chapter, we would like to discuss several ideas that are also only valid to the extent of “engineering accuracy,” but expand the possibilities of the method of dimensionality or can further simplify its application. At no point in this chapter will a claim to a final and ultimate truth be made. Our goal is much more to make the simulation possible to within a good approximation for instances in which simulations have not yet been possible.
Valentin L. Popov, Markus Heß

Chapter 17. Appendix 1: Exact Solutions in Three Dimensions for the Normal Contact of Axially-Symmetric Bodies

Abstract
In this chapter, the complete proofs will be shown that allow for the exact mapping of frictionless, axially-symmetric contact problems with and without adhesion to one-dimensional contacts. The starting point is the three-dimensional theory for the calculation of axially-symmetric contacts, which we will change step-by-step in a way that the one-dimensional properties may be clearly seen. We assume simply-connected, and therefore, circular contact areas.
Markus Heß, Valentin L. Popov

Chapter 18. Appendix 2: Exact Solutions in Three Dimensions for the Tangential Contact of Axially-Symmetric Bodies

Abstract
In this chapter, the proof of validity of the method of dimensionality to tangential contacts with friction between axially-symmetric bodies is given. The proof is based on the close relationship between normal and tangential contacts found in the works by Cattaneo, Mindlin, Jäger, and Ciavarella.
Markus Heß, Valentin L. Popov

Chapter 19. Appendix 3: Replacing the Material Properties with Radok’s Method of Functional Equations

Abstract
This chapter is devoted to a rigorous proof of the application procedure of the method of dimensionality reduction to contacts with elastomers. The proof is based on Radok’s principle of functional equations. It proceeds from a solution of a similar elastic problem which then is carried over to the original problem by replacing the material properties. We will show in detail how Radok’s method of functional equations is used for the replacement of the material properties.
Silvio Kürschner, Valentin L. Popov, Markus Heß

Chapter 20. Appendix 4: Determining Two-Dimensional Power Spectrums from One-Dimensional Scans

Abstract
The power spectra of rough surfaces are necessary “input parameters” for the calculation of contact and frictional properties. The power spectrum of a randomly rough surface can be determined using the Fourier transform of a measured two-dimensional surface topography. The experimental determination of the entire surface topography, for instance, with an atomic force microscope can, however, be very time intensive. Therefore, it begs the question, whether or not it is possible to determine the entire two-dimensional power spectrum from a limited number of one-dimensional scans. For isotropically randomly rough surfaces, all of the necessary information is, in fact, already contained in the one-dimensional scans of the surface. These can be used to obtain the required surface information more quickly and with less hassle.
Valentin L. Popov, Markus Heß

Backmatter

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