ERROR ANALYSIS

Author: RODRÍGUEZ ARÓS ÁNGEL DANIEL.
Year: 2004.
University: SANTIAGO DE COMPOSTELA [www.usc.es].
Place of defense: FACULTADE DE MATEMÁTICAS.
Place of preparation: FACULTADE DE MATEMÁTICAS.

Summary: The phenomena of contact involved deformable bodies abound in industrial processes, for example in the automotive sector, the steel or the construction industry, but also in everyday life, the examples are many and varied, as noted simply citing contact with the brake wheel, tire with a purse or wire with "brackets" dental orthodontics. In all these cases, friction is one of the phenomena of tribológicos most important partners in contact. For this reason there is a vast literature devoted to the study of as different forms of contact and friction for a wide variety of materials. With big boost from the 60's, classical mechanics modeling these phenomena through strongly nonlinear differential inclusions and addresses her study with arguments of the theory of maximal monotonous operators, which leads to formulations based on Variational inequations. Problems without friction constitute a first approach in studying problems more realistic than they have in mind. Moreover, research in solid mechanics of deformable has been enriched in recent decades with overcoming the classical models in elasticity, which had been studied in great detail over the twentieth century. Their inability to describe real events, such as hardening, relaxation, or irreversible deformation aging has become necessary to study more complex models that will allow us to face and give accomplished response to these difficulties. This paper seeks to make a contribution in this area, making the mathematical analysis and numerical several problems of contact for a particular class of materials, visco-elastic memory long. The main feature of these materials, from the mechanical point of view, is that the evolution of tensions s deformation depends not only on what is happening "now" in the material, but it also depends (to a greater or lesser extent) what it has happened before ", that is, its history or memory. In nature, different types of polymers, rubber, wood pulp and have a mechanical behavior that fits this description. The contents presented are the results of research into various problems in contact with viscoelasticidad long memory, and without friction and mostly published or are awaiting publication in international journals of specialty. The formulations contain Variational that can be put into two different classes deinecuaciones variations evolutionary. The first type, which we call "inequations with Variational term comprehensive type Voltera" is associated with the problems of contact without friction. The second type, that we call "inequations Variational integro-diferenciales Volterra," problems in contact with friction, too. His study is based mainly on arguments of the theory of operators monotonous in Banach spaces and fixed point. Therefore, the objective of this paper is twofold. On the one hand, we study the existence and uniqueness of solutions, the properties and numerical approximation of two classes deinecuaciones Variational evolving, but at the same time, these theoretical results are applied to variational analysis and numerical 8 a vari 660 ada range of problems contact with and without friction solid visco-elastic memory for long. Among these we can highlight the problem of contact unilaterial without friction between a body and a foundation rigid or deformable (Signorini) (normal response), the problem of contact between two deformable bodies conditions with no interpretation or friction, the issue of bilateral contact type Tresca and friction between a deformable body and a rigid foundation, the problem unilaterally without friction contact between deformable body and a semi-rigid foundation, and some others. In general, for each problem of contact, the interaction has been the basis for the mechanical model covered in their theoretical aspects and practical study of the existence and uniqueness of solutions weak, its numerical approximation, analysis of the error, and eventually the implementation of a computer algorithm to perform numerical simulations. Thus, in each chapter or section arise and study the problem of contact as differential, as it is rational, the approximate problem with discretizan space and the problem with approximate spatial and temporal discretization.