Formulation variationnelle de l'endommagement anisotrope dans les matériaux quasi-fragiles et régularisation par l'approche Lip-field

When submitted to mechanical loadings, quasi-brittle materials experience gradual damage. This process stems from the development of micro-cracks inside a finite volume of matter. These micro-cracks are by nature anisotropic, oriented by the mechanical loading. Complex material laws, often using a tensorial damage description, are thus needed to take into account this anisotropy.Furthermore, the growing interest for formulations that allow simple and robust numerical implementations lead to a massive development of energetical variational approaches of damage and fracture these last few years. Such formulations trade off a restriction of the modeling possibilities due to a strict mathematical framework for the possibility to perform structure-scale computatons. They are for now mostly restricted to isotropic damage, using a scalar variable.The major objective of this thesis was to bridge the gap between these two conceptions in order to exploit both the finesse of an anisotropic modelling of damage and the numerical efficiency of energetical variational approaches. Theoretical developments were conducted in order to shed light on the conceptual locks for this type of formulations. A first anisotropic damage model was developped in a Lip-field framework, using a novel concept of cumulated damage. It also stands as one of the few anisotropic damage models formulated in an energetical variationnal framework. It serve as both a proof of concept and a support for the analysis of the advantages and disadvantages of such an approach. This formulation was implemented in an ad-hoc finite elements code so as to perform simple test cases to further the study of the formulation.Finally, damage models - both isotropic and anisotropic - suffer from localization. A mathematical analysis method of this phenonmenon able to take into account such complex models was proposed.