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This book conveys, in a self-contained manner, the fundamental concepts for classifying types of contact, the essential mathematical methods for the formulation of contact problems, and the numerical methods required for their solution. In addition to the methodologies, it covers a broad range of applications, including contact problems in mechanical engineering, microelectronics and nanomechanics. All chapters provide both substantial background on the theory and numerical methods, and in-depth treatments of cutting-edge research topics and applications.

The book is primarily intended for doctoral students of applied mathematics, mechanics, engineering and physics with a strong interest in the theoretical modelling, numerical simulation and experimental characterization of contact problems in technology. It will also benefit researchers in the above mentioned and neighbouring fields working in academia or at private research and development centres who are interested in a concise yet comprehensive overview of contact mechanics, from its fundamental mathematical background, to the computational methods and the experimental techniques currently available for the solution of contact problems.

### Chapter 1. Fundamentals of Elastic Contacts

Abstract
Contacts are classified into the fundamental types and their characteristics briefly explored. The formulation of incomplete contacts using a half-plane formulation is then developed, and used to obtain the plane solution for a Hertzian contact, while providing the framework for many other geometries. Williams’ solution for a sharp infinite elastic wedge is described in detail, and it is shown how this may be applied to advantage in understanding complete contacts and especially their near-edge properties. We then go back to look at incomplete contacts and analyse how they respond when there is interfacial friction present and they remain stationary, but a partial slip state evolves. The chapter concludes by reviewing the other possible types of contact.
David Hills, Hendrik Andresen

### Chapter 2. Contact Problems Involving Friction

Abstract
The Coulomb friction law is simple to apply in the formulation of elastic contact problems, but it is also a rich source of unexpected physical phenomena, including ranges of unstable dynamic response, history-dependence, ‘wedging’ and mathematical problems of existence and uniqueness of solution. We first explore the implications of the law in the context of simple discrete systems and demonstrate the importance of interaction [coupling] between the normal and tangential contact problems, particularly in problems of periodic loading. The discussion is then extended to problems of the elastic continuum, and to cases where elastodynamic effects must be included [for example, the interaction of a seismic disturbance with a frictional interface]. It is shown that finite element formulations of elastodynamic problems with Coulomb friction are inherently ill-posed and alternative friction laws that avoid this difficulty are discussed.
J. R. Barber

### Chapter 3. Nonequilibrium Molecular Dynamics Simulations of Tribological Systems

Abstract
Nonequilibrium molecular dynamics (NEMD) simulations are increasingly being used to investigate the nanoscale behaviour of tribological systems. This chapter focuses on the application of classical NEMD simulations of liquid lubricants and additives confined between solid surfaces. Ab initio NEMD, which can be used to accurately model tribochemsitry, and coupled computational fluid dynamics (CFD)-NEMD are also introduced. Specific example systems and recommendations for future research are provided.
James P. Ewen, Eduardo Ramos Fernández, Edward R. Smith, Daniele Dini

### Chapter 4. Computational Methods for Contact Problems with Roughness

Abstract
This chapter provides a self-consistent introduction to computational methods for the solution of contact problems between bodies separated by rough interfaces. Both frictional and frictionless contact problems are examined. The mathematical formulation of the boundary element method is presented first, with details on the possible algorithmic implementation strategies and their computational efficiency. In the second part of the chapter, the fundamentals of the finite element method for the solution of contact problems are presented, along with an overview on the different strategies available in the literature to accurately discretize the multiscale features of roughness. A synopsis of the major advantages and disadvantages provided by the computational methods based on the boundary element method or the finite element method concludes the chapter, illustrating also perspective research directions.
Marco Paggi, Alberto Bemporad, José Reinoso

### Chapter 5. Emergent Properties from Contact Between Rough Interfaces

Abstract
Interface phenomena at the micro- and nanoscales are of paramount importance in nature and technology. Real surfaces present roughness over multiple scales, and understanding the role of roughness in surface physics (heat and electric transfer, hydrophobic properties), surface chemistry (chemical reactions) and tribology (stress transfer, adhesion, lubrication) is a very active research topic. This chapter focuses on the key research question of how nonlinear interactions between contact patches induced by roughness across different length scales influence the emergent physico-mechanical properties of an interface. Special attention is given to the scaling of the real area of contact with the applied normal load, the dependency of the thermal and electric contact conductance on the normal pressure, the evolution of the free volume network between rough surfaces in contact, the role of adhesion and also the evolution of partial slip in frictional contacts.
Marco Paggi

### Chapter 6. Modelling Flows in Lubrication

Abstract
This chapter introduces the reader to lubrication theory and describes the governing equations, models and methods that can be used to simulate various types of lubricated systems. It starts with an introduction to the tribological contact and to the different lubrication regimes. The basis for the classical lubrication theory is then given and thereafter follows a presentation of how to obtain the Reynolds equation by means of scaling and asymptotic analysis of the Navier–Stokes equations. After having obtained the Reynolds equation, a quite elaborate presentation of cavitation algorithms is given. It includes discretisation and presents the analytical solution for a pocket bearing as a benchmark model problem. Then, the concept of homogenisation of surface roughness is introduced. This starts from the simplest iso-viscous and incompressible case, expands to include compressibility with a constant bulk modulus constitutive relation and then also addresses the case of ideal gases. Thereafter, the relation between homogenised coefficients and the Patir and Cheng flow factors is described and finally it is shown how to incorporate the effect of mixed lubrication into the model.
Andreas Almqvist, Francesc Pérez-Ràfols

### Chapter 7. Contact Mechanics of Rubber and Soft Matter

Abstract
This chapter reviews recent advances made in the treatment of contact problems involving soft materials often characterized by non-linearly elastic material properties, such as rubber and soft biological tissues. Starting from the fundamental formulation developed to solve viscoelastic contact mechanics, the treatment of complex problems involving surface roughness, layered materials, and reciprocating contacts in dry contacts is presented in increased order of complexity. The reader is then introduced to the study of lubricated contacts, with a discussion of the interplay between viscoelastic effects in the solids and the viscosity marking the lubricant behavior. Experimental validations that cover various aspects of the work are also presented.
Carmine Putignano, Daniele Dini