Projet «Blanc» N°ANR 12 BS010021 Financé par l'ANR de 2013 à 2016 

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When studying the behavior of mechanical structures, the global response  sufficient for current applications – is usually obtained considering that the material is perfect, ignoring the possible presence of defects. Nevertheless, some applications require finer analyses. This is the case, for instance, when dealing with the behavior until rupture of structures. In that context, the role played by defects is crucial and has to be taken into account in order to realistically describe the behavior till complete failure. Indeed, the presence of porosities or inclusions induces local modifications of the stress field, and this explains that, crack initiation sites are experimentally known to be usually located at the vicinity of those heterogeneities.
Despite the fact that the characteristic sizes of the defects are much smaller than that of the structure, the influence of those defects on the behavior of the structure is not negligible and has to be taken into account particularly when dealing with structural failure. The analysis of the influence of porosities or inclusions needs thus to consider the scale of the defects much smaller than the scale of the structure, while performing computations at the structure scale. This leads to the development and the numerical implementation of an analysis tool based on smalldefect asymptotics. It aims to allow performing the simulation up to rupture of structures, taking into account the variability of the defects, hence the proposed introduction of a stochastic framework.
Nonetheless, depending on the size of the defects compared to the characteristic size of the underlying material, the hypothesis of homogeneity and isotropy – often considered for the computations at the scale of the structure  are not necessarily still valid when the defect scale enters into the modeling process. The difficulties are the same for structures with geometrical singularities generating singularities in the mechanical fields. In this large gradient area, a specific treatment is required to catch the singular behavior. Moreover, if one is interested in priming of defects (cracks or damaged areas), very accurate computations should be performed if the used model takes into account a small (with respect to the size of the structure) characteristic length.
Civil engineering structures are full of situations where adapted methods are useful or essential. Reinforced and/or prestressed concrete structures like fence tank for nuclear reactor, cooling tower, hydroelectric dam, … are the quintessence of the concerned structures: reinforcing steel rods and prestressing cables are periodically distributed on some layers. Their radius is in general small with respect to the width of the walls, their relative distances can be of the same order of size as their radius or, on the converse, as the wall width. Hence, the used asymptotic model has to be adapted to each case. Moreover, the adhesion between the reinforcing steel rods and the concrete is essential to the building durability. It is therefore essential to model in a pertinent way the loss of adhesion by loss of stickiness or by cracking. In addition, since the ratio of stiffness between the reinforcing steel rods and the concrete is large, the modeling of the mechanical behavior has to take into account some small parameters. Up to now, the engineers use simplified models and ask for any rigorous reasoning infirming or validating their approaches.
Regarding numerical simulations on domains presenting several scales, the usual methods of
numerical simulation have to face the problem of mesh refinement. This refinement may
induce a prohibitive cost. Being able to handle this kind of phenomenon is now a major
challenge. One option is to ignore defects provided they are small enough. If this is not the case, it is
necessary to introduce some kind of zoom at the level of the defect. The aim of this project is to propose an alternative
approach. We take the defect into account through its impact on the solution by a localized enrichment of the test and trial functions
space (we adopt a finite element point of view) by the space of functions used to discretize
the boundary value problem. With this strategy we completely avoid adapting the mesh and
we can work with a coarse one. In this process, we also reduce drastically the number of
unknowns: the computation can be done on a laptop. An alternative, the domain
decomposition approach, has the advantage to not require specific code. All the
computations can be done with “classical” finite elements software, the novelty lays in the
implementation of interface conditions between the subdomain. Both approach will be
compared in the ARAMIS project
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