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In this thesis we analyse the high-order in time discontinuous Galerkin nite element method (DGFEM) for second-order in time linear abstract wave equations. Our abstract approximation analysis is a generalisation of the approach introduced by Claes Johnson (in Comput. Methods Appl. Mech. Engrg., 107:117-129, 1993), writing the second order problem as a system of fi rst order problems. We consider abstract spatial (time independent) operators, highorder in time basis functions when discretising in time; we also prove approximation results in case of linear constraints, e.g. non-homogeneous boundary data. We take the two steps approximation approach i.e. using high-order in time DGFEM; the discretisation approach in time introduced by D Schötzau (PhD thesis, Swiss Federal institute of technology, Zürich, 1999) to fi rst obtain the semidiscrete scheme and then conformal spatial discretisation to obtain the fully-discrete formulation. We have shown solvability, unconditional stability and conditional a priori error estimates within our abstract framework for the fully discretized problem. The skew-symmetric spatial forms arising in our abstract framework for the semi- and fully-discrete schemes do not full ll the underlying assumptions in D. Schötzau's work. But the semi-discrete and fully discrete forms satisfy an Inf-sup condition, essential for our proofs; in this sense our approach is also a generalisation of D. Schötzau's work. All estimates are given in a norm in space and time which is weaker than the Hilbert norm belonging to our abstract function spaces, a typical complication in evolution problems. To the best of the author's knowledge, with the approximation approach we used, these stability and a priori error estimates with their abstract structure have not been shown before for the abstract variational formulation used in this thesis. Finally we apply our abstract framework to the acoustic and an elasto-dynamic linear equations with non-homogeneous Dirichlet boundary data.

Ce mémoire s'inscrit dans l'étude du comportement tribologique d'un couple de matériaux ferromagnétique acier XC48 en contact mécanique et magnétique à sec. Le but consiste à identifier les effets liés à la présence ou à l'absence d’un champ magnétique au niveau de l'interface de contact statique et de contact glissant. La première étape a été d'établir des équations de contraintes qui régissent la réponse linéaire, ponctuelle et surfacique des solides sous chargements donnés, en utilisant des modèles développés dans la théorie d'élasticité. Elles ont ensuite été simulées sous Mathématica pour obtenir des résultats analytiques des champs de contraintes en sub-surfaces. Dans la deuxième étape, des essais par contact sec effectués au moyen d'un tribomètre du type pion-disque nous révèlent une augmentation du coefficient de frottement avec le champ magnétique sur le tribocontact. L'analyse en diffraction des rayons X montre que les débris d'usures lors des essais avec champ magnétique comportent la présence d'oxyde de fer. La caractérisation en microscopie électronique à balayage et en profilométrie du matériau usé met en évidence l'état de la sub-surface avec modification de la structure des grains et une présence d'usure douce due aux essais avec champ magnétique. La mesure en photoélasticimétrie montre l'influence du champ magnétique sur les champs de contraintes. En faisant varier les paramètres cinématiques de l'essai sur notre modèle numérique, nous observons une augmentation des champs contraintes. Ces paramètres mécaniques, magnétiques et physico-chimiques sont complexes et l'étude contribue à comprendre ces problématiques liées au frottement sec. / This thesis is about the study of the tribological behavior of a couple of ferromagnetic materials XC48 steel in mechanical and magnetic dry contact. It consists in identifying the generated effects of the presence or absence of a magnetic field at the static and sliding contact interface. The first step was to establish constraint equations which govern the linear, punctual and surface response of the solids under given loadings, by using models developed in the theory of elasticity. They were then simulated under Mathematica to obtain analytical results of stress fields in sub-surfaces. In the second phase, the dry contact tests carried out by means of a standard pin-on-disc tribometer reveal an increase in the coefficient of friction with the magnetic field on the tribological contact. The analysis by X-ray diffraction shows that the wear debris during the tests with magnetic field comprises the presence of iron oxide. The characterization by scanning electron microscopy and profilometry of worn material shows the state of the sub-surface with modification of the grain structure and presence of soft wear from testing with magnetic field. The photoelasticity measurements show the influence of the magnetic field on the stress fields. By varying the kinematic parameters of the test on our numerical model, we observe an increase in stress fields. These mechanical, magnetic and physicochemical parameters are complex and the study contributes to understanding these problems of dry friction.