The dissertation focuses on the analysis, through combined analytical and numerical techniques, of the partial differential equations arising from a new approach to modeling brittle fracture, based on extension of continuum mechanics to the nanoscale. The main part of this work deals with the analysis of several fracture models. Integral transform methods are used to reduce the problem to a Cauchy singular, linear integro-differential equation. It is shown that ascribing constant surface tension to the fracture surfaces and using the appropriate crack surface boundary condition, given by the jump momentum balance, leads to a sharp crack opening profile at the crack tip, in contrast to the classical theory of brittle fracture. However, such a model still predicts singular crack tip stress. For this reason a modified model is studied, where the surface excess property is responsive to the curvature of the fracture surfaces. It is shown that curvature-dependent surface tension, together with boundary conditions in the form of the jump momentum balance, leads to bounded stresses and a cusp-like opening profile at the crack tip. Further, an alternative approach, based on asymptotic analysis, which is suitable to apply in cases when the model includes a mutual body force correction term, is considered. The nonlinear nonlocal problem, resulting from the proposed model, is simplified which allows us to approximate the crack opening profile and derive asymptotic forms for the cleavage stress in a neighborhood of the crack tip. Finally, two possible fracture criteria, in the context of the new theory, are discussed. The first one is an energy based fracture criterion. Classically the energy release rate arises due to singular fields, whereas in the case of the modeling approach adopted here, a notion analogous to the energy release rate arises through a different mechanism, associated to the rate of working of the surface excess properties at the crack tip. Due to the fact that the proposed modeling approach allows us to fully resolve the stress in a neighborhood of the crack tip, without the customary singularity, a second fracture criterion, based on crack tip stress, is possible.

L‟acétonitrile (ACN) peut être considéré comme le solvant de référence utilisé dans les électrolytes pour supercondensateurs car, industriellement, il est le plus utilisé. Il présente en effet de nombreux avantages tels qu‟une viscosité faible et une permittivité élevée, conduisant à une excellente conductivité en présence d‟un sel. Il est cependant hautement volatile, très inflammable et toxique quand il brûle (dégagement de HCN). Ainsi, dans le but de réduire sa pression de vapeur, et donc le risque d‟inflammabilité, il peut être mélangé à un liquide ionique (LI) qui lui est non volatile. Le but de ce travail est de remplacer l‟électrolyte classique des supercondensateurs à base de ACN et de sel de tétraethylammonium tétrafluoroborate (ACN + 1M Et4N+BF4-) par un mélange LI/ACN. Pour ce faire, l‟étude physico-chimique, électrochimique et thermodynamique des mélanges LI/ACN est entreprise. Dans un premier temps, les liquides ioniques ont été synthétisés et caractérisés par différentes techniques d‟analyses physico-chimiques. Ensuite des mélanges LI/ACN ont été formulés. Ces mélanges ont subi des tests de sécurité (tests préliminaires d‟inflammabilité, mesures de point flash) afin de trouver le mélange optimal. Les phénomènes de transport dans ces mélanges ont été aussi étudiés afin de comprendre leur comportement en température. Par ailleurs, l‟étude électrochimique menée sur ces mélanges a montré qu‟il n‟y avait pas de dégradation des performances électrochimiques par comparaison à l‟électrolyte classique. Enfin l‟étude de l‟équilibre liquide vapeur à partir de modèles thermodynamiques semi-prédictifs comme UNIQUAC ou prédictifs comme Cosmo-RS a permis de déterminer les grandeurs d‟excès. / Acetonitrile (ACN) is the most popular solvent in electrolytes designed for use in supercapacitors. It presents many advantages such as a low viscosity and a high permittivity, leading to excellent conductivities in the presence of salts. However it is highly flammable and very toxic when burning (release of HCN ). Thus, in order to reduce its vapor pressure and hence its flammability we propose to mix it with an ionic liquid (IL). As ILs are non volatile compounds, the vapor pressure of the mixture will be reduced (Raoult‟s law). In addition many other benefits may be expected from these mixtures. The aim of this work is to replace the conventional ACN and Et4NBF4 based electrolyte for supercapacitors by a IL/ACN mixture. Thus, the physico-chemical, electrochemical and thermodynamical studies of IL/ACN mixtures have been undertaken. Synthesized and commercial ILs are characterized by mean of different physico-chemical analysis. Then IL/ACN mixtures were formulated. These mixtures were tested for safety (preliminary flammability tests, flash point measurements) and the optimal mixture determined. Transport phenomena in these mixtures were also studied to understand their behavior in response to temperature. Furthermore, the electrochemical study conducted on these mixtures showed that there was no degradation of the electrochemical performances as compared to the conventional electrolyte. Finally the study of vapor liquid equilibrium from semi-predictive thermodynamic models like UNIQUAC or predictive models like Cosmo-RS allowed us to determine the excess properties.