La modélisation numérique de la multi-fissuration et son influence sur les ouvrages du Génie Civil reste un sujet ouvert et nécessite le développement de nouveaux outils numériques de plus en plus performants. L’approche retenue dans cette thèse est basée sur l’utilisation des concepts des équations intégrales de Galerkin accélérées par la méthode multipôle rapide. Les méthodes intégrales sont bien connues pour leur souplesse à définir des géométries complexes en 3D. Elles restent également très performantes en mécanique de la rupture, lors de la détermination des champs singuliers au voisinage des fissures. La Méthode Multipôle Rapide, quant à elle, permet via une judicieuse reformulation des fonctions fondamentales propres aux formulations intégrales, de réduire considérablement le coût des calculs. La mise en œuvre de la FM-SGBEM a permis de pallier les difficultés rencontrées lors de la phase de résolution et ce lorsqu’on traite de domaines de grandes tailles par équations intégrales de Galerkin pures. Les présents travaux, viennent en partie optimiser et renforcer cette phase dans les environnements numériques existants. D’autre part, des adaptations et des développements théoriques des formulations FM-SGBEM pour prendre en compte le caractère hétérogène des domaines en Génie Civil qui en découlent, ont fait l’objet d’une large partie des travaux développés dans cette thèse. La modélisation du phénomène de propagation de fissures par fatigue a également été étudiée avec succès. Enfin, une application sur une structure de chaussée souple a permis de valider les modèles ainsi développés en propagation de fissures par fatigue dans des structures hétérogènes. De réelles perspectives d’optimisations et de développements de cet outil numérique restent envisagées. / The modeling of cracks and its influence on the understanding of the behaviors of the civil engineering structures is an open topic since many decades. To take into consideration complex configurations, it is necessary to construct more robust and more efficient algorithms. In this work, the approach Galerkin of the boundary integral equations (Symmetric Galerkin Boundary Element Method) coupled with the Fast Multipole Method (FMM) has been adopted. The boundary analysis are well-known for the flexibility to treat sophisticated geometries (unbounded/semi-unbounded) whilst reducing the problem dimension or for the good accuracy when dealing with the singularities. By coupling with the FMM, all the bottle-necks of the traditional BEM due to the fully-populated matrices or the slow evaluations of the integrals have been reduced, thus making the FM-SGBEM an attractive alternative for problems in fracture mechanics. In this work, the existing single-region formulations have been extended to multi-region configurations along with several types of solicitations. Many efforts have also been spent to improve the efficiency of the numerical algorithms. Fatigue crack propagations have been implemented and some practical simulations have been considered. The obtained results have validated the numerical program and have also opened many perspectives of further developments for the code.

A systematic procedure is followed to develop singularity-reduced integral equations for modeling cracks in two-dimensional, linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations (IEs) for "generalized displacement" and "generalized stress"; these serve as the basis for the development of a Symmetric Galerkin Boundary Element Method (SGBEM). The implementation is carried out to allow treatment of general mixed boundary conditions, an arbitrary number of cracks, and multi-region domains (in which regions having different material properties are bonded together). Finally, a procedure for calculation of T-stress, the constant term in the asymptotic series expansion of crack-tip stress field, is developed for anisotropic elastic media. The pair of weak-form boundary IEs that is derived (one for generalized displacement and the other one for generalized stress) are completely regularized in the sense that all kernels that appear are (at most) weakly-singular. This feature allows standard Co elements to be utilized in the SGBEM, and such elements are employed everywhere except at the crack tip. A special crack-tip element is developed to properly model the asymptotic behavior of the relative crack-face displacements. This special element contains "extra" degrees of freedom that allow the generalized stress intensity factors to be directly obtained from the solution of the governing system of discretized equations. It should be noted that while the integral equations contain only weakly-singular kernels (and so are integrable in the usual sense) there remains a need to devise special integration techniques to accurately evaluate these integrals as part of the numerical implementation. Various examples for crack problems are treated to illustrate the accuracy and versatility of the proposed procedure for both unbounded and finite domains and for both single-region and multi-region problems. It is found that highly accurate fracture data can be obtained using relatively course meshes. Finally, this dissertation addresses the development of a numerical procedure to calculate T-stress for crack problems in general anisotropic elastic media. T-stress is obtained from the sum of crack-face displacements which are computed via a (regularized) integral equation of the boundary data. Two approaches for computing the derivative of the sum of crack-face displacements are proposed: one uses numerical differentiation, and the other one uses a weak-form integral equation. Various examples are examined to demonstrate that highly accurate results are obtained by means of both approaches. / text