A major issue in fracture mechanics is to model the nucleation of a crack in a sound material. There are two difficulties: the first one is to propose a law able to predict that nucleation; the second is a purely numerical issue. It is indeed difficult to compute with a good accuracy all the mechanical quantities like the energy release rate associated with a crack of small length which appears at the tip of a notch. The classical finite element method leads to inaccurate results because of the overlap of two singularities which cannot be correctly captured by this method: one due to the tip of the notch, the other due to the tip of the crack. A specific method of approximation based on asymptotic expansions is preferable as it is developed in analog situations with localized defects. The first chapter of the thesis is devoted to the presentation of this Matched Asymptotic Method (shortly, the MAM) in the case of a defect (which includes the case of a crack) located at the tip of a notch in the simplified context of antiplane linear elasticity. The main goal of the thesis is to use these asymptotic methods to predict the nucleation or the propagation of defects (like cracks) near those singular points. The second chapter of the thesis will be devoted to this task. This requires, of course, to overcome the first issue by introducing a criterion for nucleation. This delicate issue has not received a definitive answer at the present time and it was considered for a long time as a problem which could not be solved in the framework of Griffith theory of fracture. The main invoked reason is that the release of energy due to a small crack tends to zero when the length of the crack tends to zero. Therefore, if one follows the Griffith criterion which stands that the crack can propagate only when the energy release rate reaches a critical value characteristic of the material, no nucleation is possible because the energy release rate vanishes when there is no preexisting crack. This "drawback" of Griffith's theory was one of the motivations which led Francfort and Marigo to replace the Griffith criterion by a principle of least energy. It turns out that this principle of global minimization of the energy is really able to predict the nucleation of cracks in a sound body. However, the nucleation is necessarily brutal in the sense that a crack of finite length suddenly appears at a critical loading. Moreover the system has to cross over an energy barrier which can be high when the minimum is "far". Another way to overcome the issue of the crack nucleation is to leave the pure Griffith setting by considering cohesive cracks. Indeed, since any cohesive force model contains a critical stress, it becomes possible to nucleate crack without invoking global energy minimization. Accordingly, we propose to revisit the problem of nucleation of a crack at the tip of a notch by comparing the three criteria. One of our goal is to use the MAM to obtain semi-analytical expressions for the critical loading at which a crack appears and the length of the nucleated crack. Specifically, the thesis is organized as follows. Chapter 1 is devoted to the description of the MAM on a generic anti-plane linear elastic problem where the body contains a defect near the tip of a notch. We first decompose the solution into two expansions: one, the outer expansion, valid far enough from the tip of the notch, the other, the inner expansion, valid in a neighborhood of the tip of the notch. These expansions contain a sequence of inner and outer terms which are solutions of inner and outer problems and which are interdependent by the matching conditions. Moreover each term contains a regular and a singular part. We explain how all the terms and the coefficients entering in their singular and regular parts are sequentially determined. The chapter finishes by an example where the exact solution is obtained in a closed form and hence where we can verify the relevance of the MAM. In Chapter 2, the MAM is applied to the case where the defect is a crack. Its main goal is to compute with a good accuracy the energy release rate associated with a crack of small length near the tip of the notch. Indeed, it is a real issue in the case of a genuine notch (by opposition to a crack) because the energy release rate starts from 0 when the length of the nucleated crack is 0, then is rapidly increasing with the length of the crack before reaching a maximum and finally is decreasing. Accordingly, after the setting of the problem, one first explains how one computes the energy release rate by the FEM and why the numerical results are less accurate when the crack length is small. Then, one uses the MAM to compute the energy release rate for small values of the crack length and one shows, as it was expected, that the smaller the size of the defect, the more accurate is the approximation by the MAM at a certain order. It even appears that one can obtain very accurate results by computing a small number of terms in the matched asymptotic expansions. We discuss also the influence of the angle of the notch on the accuracy of the results, this angle playing an important role in the process of nucleation (because, in particular, the length at which the maximum of the energy release rate is reached depends on the angle of the notch). It turns out that when the notch is sufficiently sharp, i.e. sufficiently close to a crack, it suffices to calculate the first two non trivial terms of the expansion of the energy release rate to capture with a very good accuracy the dependence of the energy release rate on the crack length. Then a cohesive model, the so-called Dugdale model, is considered in the last section of the chapter. Combining the MAM with the G method allows us to calculate in an almost closed form the nucleation and the evolution of the crack, namely the relations between the external load and the lengths of the non-cohesive zone and the cohesive zone. Specifically, it turns out that the inner problem can be seen as an Hilbert problem which can be solved with the help of complex potentials. Thus, the access to the solution is reduced to a few quadratures which are computed numerically. One obtains so an analytical expression of the critical load at which a "macroscopic" crack will appear in the body after an unstable stage of propagation of the nucleated crack. The order of magnitude of that critical load is directly associated with the power of the singularity of the solution before nucleation which is itself a known function of the angle of the notch. Chapter 3 proposes a generalization of all the previous results in the plane elasticity setting. Specifically, the goal is still to study the nucleation of non cohesive or cohesive cracks at the angle of a notch in the case of a linearly elastic isotropic material but now by considering plane displacements. Moreover, we will consider as well pure mode I situation as mixed modes cases. In the first part of the chapter we use the global minimization principle in the case of a non cohesive crack. In the second part we consider Dugdale cohesive force model. In both cases the MAM is used to compensate the non accuracy of the finite element method. All the derived results can be seen as simple generalizations of those developed in the antiplane case. Indeed, from a conceptual and qualitative viewpoint, we obtain essentially the same types of properties. However, from a technical point of view, the MAM is more difficult to apply in plane elasticity because the sequence of singularities can be obtained only by solving transcendental equations. Therefore, the numerical procedure becomes more expansive. Moreover, from the analytical point of view, the calculations become much more intricate and consequently a part of these calculations are given in the appendix.
Domain decomposition and multi-scale computations of singularities in mechanical structures