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Considerable progress has been achieved recently in modeling and mathematical analysis of various processes involved in contact between deformable bodies. Indeed, a general Mathematical Theory of Contact Mechanics is currently emerging.
Contact processes take place on the surface, and, therefore, are described by boundary conditions. However, these are the boundaries or surfaces of mechanical bodies or structures, and one has to describe the evolution of the mechanical state of the body, as well. The problems, in their classical formulation, consist of the constitutive laws, the equations of motion and the relevant initial...
More is involved in contact than just friction, although it is the main process. Indeed, during a contact process elastic or plastic deformations of the surface asperities may happen. Also, some or all of the following may take place: squeezing of oil or other fluids, breaking of the asperities’ tips and production of debris, motion of the debris, formation or welding of junctions, creeping,...
In this chapter we present a short review of thermodynamic principles and potentials and describe their use in derivation of general thermomechanical conditions and equations, as applied to processes involved in contact. Works on Thermodynamics of Continuua abound, and more specialized applications to contact phenomena can be found in [65,97-99,183] and references therein. We use some notions...
In the previous three chapters various constitutive laws for the behaviour of the material and different contact conditions were described in some detail. In this chapter we take the next step and assemble, in full detail, the relevant equations and conditions for one particular and representative problem into a mathematical model. then we analyze the model in detail. The sections in Part...
Part II provides mathematical formulations of the models, the assumptions that underlie them, their weak or variational formulations and the statements of the results. Each chapter also describes representative proofs which show the mathematical methods used, and indicate the types of problems that can be analyzed using them. These may be skipped on first reading.
Existence results for the problem of quasistatic contact between an elastic material and a reactive foundation were first obtained in [18] and [19]. In both papers the normal compliance contact condition was employed. Additional results were obtained in [25, 199, 200]. Recently, Andersson in [24] succeeded in passing to the normal compliance limit and established the existence of a weak solution...
We describe results for contact problems involving viscoelastic materials. It will be seen that adding viscosity to the models leads to a substantial increase in the regularity or the smoothness of the solutions, and this allows for further analysis. Unlike the case of problems for purely elastic materials, there are numerous results for viscoelastic problems with varying degrees of generality,...
We describe contact problems for viscoplastic materials in which once the stress reaches the so-called yield limit the deformation becomes irreversible. Metals behave as viscoplastic materials, for instance, during metal-forming, and so do some polymers and other materials when hot or under large loads. Such problems are common in industrial processes and deserve considerable attention, both...
The frictional contact problems which have been described up to now contained a constant friction coefficient, although, as was described in Sect. 2.7, in many applications it depends on the slip speed, on the temperature, and on other factors. In this chapter we address this issue and present models of frictional contact when the coefficient of friction depends on the slip, slip rate, or...
In this chapter we present results on models of frictional contact when the wear of the contacting surfaces is taken into account. The importance of the control and minimization of the wear of industrial parts and components cannot be overstated, and therefore, effective models for the prediction of wear in industrial settings are indispensable to the design engineer. We describe three problems...
We describe new results dealing with contact problems for materials that may undergo internal damage, resulting from strains and stresses which lead to the opening and growth of microscopic cracks. The damage measures the deterioration of the strength of the material in that it reduces the load carrying capacity of the body. A novel way to model material damage was proposed in [171, 172],...
The previous part of this monograph makes it clear that the branch of the Mathematical Theory of Contact Mechanics, which deals with quasistatic problems, has been developing and expanding rapidly. Other branches are also undergoing rapid development and in this short chapter, we briefly review a number of recent publications on related dynamic and one-dimensional contact problems with or...
This monograph shows clearly that the branch of the Mathematical Theory of Contact Mechanics which deals with quasistatic processes has made an impressive progress in the last decade. Indeed, from a handful of mathematical results, mainly on static problems and very few on quasistatic and dynamic ones, it has developed into a body of results that encompasses many of the fundamental processes...
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