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The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton-Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausium-Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

International audience; Integral non-local (INL) formulations on the internal variables of the constitutive model [1] are widely used to regularize Continuum Damage computations, in the presence of stress softening for instance. According to this approach the thermodynamic variable driving the damage evolution process at a given material point is computed by weighted averaging of the corresponding local field over the entire domain. Averaging is performed through a non-local weighting function, such that the higher the Cartesian distance between two material points, the lower is their interaction. A main drawback of this assumption consists, however, in nonphysical interactions between material points across damaged bands, cracks and holes. From a numerical viewpoint, this induces some parasite effects, such as damage spreading over a large damaged band, damage diffusion near notches and free-edges, etc. Enhancements of the initial methods were proposed by allowing non-local interactions to evolve depending on mechanical fields (e.g. damage, stress, strain). The Eikonal Non-Local (ENL) formulation [2, 3] provides a novel interpretation of damage dependent evolving non-local interactions. According to this approach, one considers that damage induces a curvature of the Riemannian space in which interaction distances are computed. As a consequence, non-local interactions between material points are controlled by geodesic distances approximating the viscosity solution of an isotropic time-independent Eikonal equation with a damage dependent Riemannian metric function. Geodesic distances are then used to define the kernel of weighting function to be used in integral non-local averaging. This mathematical and physical framework allows for directly modeling evolving interactions throughout the damage process, from damage growth to strain local-ization. In this contribution, a simple isotropic ENL damage model and its numerical implementation [3] are presented first. Two-dimensional strain localization problems are then simulated in order to present the main features of the proposed formulation. Attention is mainly focused on non-local interactions across damaged bands and near notches and/or free-edges, to show that parasite effects classically associated with standard INL formulations are avoided, or at least strongly reduced.

International audience; Experimental study of non-linear air flow through a single concrete crack characterised geometrically by X-ray microtomography

International audience

Une structure en béton doit assurer des fonctions structurales qui vont au delà de la simple résistance. Dans ce cadre, la fissuration du béton armé joue un rôle primordial sur la durabilité, l’étanchéité et même la sûreté des structures. La structure poreuse du béton rend naturellement possible la pénétration au cours du temps d’espèces délétères. En outre, sous l’effet des chargements mécaniques et des conditions environnementales au sens large, le béton se fissure. Les fissures constituent, elles aussi, des voies préférentielles pour la pénétration de fluides ou d’agents agressifs et ajoutent de manière significative leur contribution à la dégradation des performances structurelles. Dans la thèse une stratégie de modélisation macroscopique probabiliste du couplage entre fissuration et transferts de fluides dans les structures en béton est présentée. Le béton est modélisé comme un milieu poreux saturé d’eau tandis que la fissuration (mécanique) est modélisée au travers d’une approche numérique probabiliste tenant compte de l’hétérogénéité naturelle du matériau et des effets d’échelle qu’elle induit. L’hypothèse physique de base du modèle de fissuration est que chaque élément fini peut être considéré comme représentatif d’un volume de matière hétérogène dont le comportement est géré par son degré d’hétérogénéité, défini comme le rapport entre le volume élémentaire et un volume représentatif de l’hétérogénéité du matériau. Dans la formulation développée, les propriétés mécaniques du matériau sont considérées comme des variables aléatoires (non corrélés) distribuées dans les éléments du maillage selon des distributions statistiques validées expérimentalement. Une approche par analyse inverse permet d’accéder aux paramètres de fonctions de distribution qui, selon les hypothèses du modèle, varient en fonction de la dimension des éléments finis. Le couplage fissuration-transfert est traité de manière faible, sous l’hypothèse d’absence d’interaction entre les deux processus (à savoir que la fissuration de l’élément fini, d’origine mécanique, induit une variation locale de sa perméabilité). L’utilisation d’une loi de Poiseuille modifiée et adaptée expérimentalement selon un protocole développé dans le cadre de la thèse permet de mettre en relation une telle variation avec l’ouverture de fissure et de prendre en compte, de manière macroscopique, les principales causes d’écart entre l’écoulement idéalisé, représenté par la loi de Poiseuille (loi cubique), et l’écoulement dans des fissures réelles. Une approche de type Monte-Carlo permet de valider les résultats des simulations mécaniques et hydriques. Les capacités de la stratégie de modélisation proposée en termes de prédiction des débits d’eau en milieu fissuré sont explorées au travers de la simulation d’essais de perméabilité sous charge sur des éprouvettes cylindriques soumises à du fendage. Ces essais sont utilisés dans le cadre du protocole expérimental. Une première validation à l’échelle d’un élément structurel multifissuré est présentée. Elle consiste en la simulation d’un essai (récemment proposé dans la littérature) développé pour l’étude de l’impact de la fissuration sur les propriétés de transfert de tirants en béton armé. Concrete durability is strongly affected by the flow of fluids, gas and pollutants in its porous matrix. The presence of cracks weakens the resistance of concrete porous matrix and constitutes preferential flow paths for aggressive components. In the thesis, a probabilistic numerical modeling strategy for modeling fluids transfers in cracked concrete structures is presented. The concrete is modeled in the framework of water saturated porous media. Its (mechanical) cracking is modeled by means of a macroscopic probabilistic approach, explicitly taking into account material heterogeneity as well as size effects. The main assumption of the model, developed in the frame of the the Finite Element Method, is to consider a finite element volume as a volume of heterogeneous material and to assume that physical mechanisms influencing the cracking processes remain the same whatever the scale of observation. At the scale of the finite element, mechanical properties are then functions of its own volume. To describe the heterogeneity of the material, these mechanical properties are considered as uncorrelated random variables distributed over the finite element mesh. Characteristics of statistical distribution laws are directly depending on the degree of heterogeneity of the finite element (the ratio between its volume and the volume of the coarsest aggregate) and of the quality of the cement paste. An inverse analysis approach allows to find their parameters as functions of the elementary volume. A weak coupling between cracking and fluid transfers is considered, under the assumption of no interaction between the two processes (i.e. the mechanically produced cracking of a finite element induces a local variation of its permeability tensor). An experimentally adapted Poiseuille (cubic) law, based on an original experimental protocol, allows to relate this permeability variation to the crack aperture and to macroscopically take into account the influence of crack roughness, aperture variation and tortuosity. A Monte-Carlo like approach is used in order to statistically validate mechanical and hydraulic simulations. The coupling strategy is validated in two phases, both at the scale of a laboratory specimen and at the scale of a multi-cracked structural element.

L'objectif de cet article est de présenter une étude sur la relation entre les propriétés géométriques d'une fissure dans un béton saturé et sa transmissivité hydraulique. Un essai de fendage classique est adapté et amélioré de manière à permettre le suivi d'une fissure unique et le débit d'eau traversant cette fissure. La description complète de la géométrie de la fissure est obtenue dans un premier temps grâce à la technique de corrélation numérique d'images (DIC en anglais). L'ouverture de fissure est alors déterminée sur la hauteur de chaque face de l'éprouvette et une relation statistique est établie entre l'ouverture déterminée par la mesure de déplacement à mi hauteur et la surface déterminée par DIC. Cette relation est ensuite utilisée, lors des essais suivants de perméabilité sous charge, pour estimer la transmissivité de la fissure en reliant le flux d'eau traversant la fissure et sa géométrie. Enfin, ces résultats expérimentaux sont comparés à la valeur théorique de transmissivité hydraulique.

The aim of this paper is to present a study on the relationship between crack geometrical properties (width and area) and water permeability evolution in a cracking saturated concrete sample. The tensile splitting (Brazilian) test is enhanced and adapted so that the monitoring of a single crack opening of the specimen and the water-flux through the crack is possible. The complete description of the crack geometry is firstly obtained. The crack opening displacement (CODm) at mid-height of the sample is derived from LVDTs measurements while the crack path on the whole height is obtained by means of the digital imaging correlation (DIC) technique. For both sides of the sample the complete description of the displacement field is obtained. The COD all along the path of the crack is then computed and a statistical relationship between the computed mean crack area and measured CODm is developed. This relationship is used, during the second series of tests, to relate at each time step of the test the mass flux of water through the cracked samples (under loading) and the mean crack area through the measured CODm. Finally, some considerations about the feasibility of a test for the real-time monitoring of permeability in cracking concrete are presented.

Eighteen academic, research and industrial European institutions from Slovenia (GEN, JSI), Croatia (APOSS), Italy (ENEA, UNIPI), France (CEA, BRGM, IRSN, EDF, Framatome – ex Areva NP), Austria (NUCCON), Poland (NCBJ, WUT), Germany (KIT, Framatome - ex. Areva), Finland (VTT), The Netherlands (TU Delft, NRG), United Kingdom (EDF Energy) formed a consortium and applied to the H2020-Euratom call. The main ambitions of the consortium are to fill some gaps identified in existing external events probabilistic safety analyses (PSA) and to improve parts of the existing methodologies by 3 points: (1) to adapt most up to date frameworks and methodologies already existing or under development outside of nuclear community; (2) to use knowledge and experience on recent national and international projects; (3) to develop demonstration cases at the real NPP scale. Interactions are envisaged with on-going international initiatives and with the International Advisory Board, which will follow and discuss the project results with the aim to propose recommendations for future regulations. The main expected results are the development of an integrated risk framework for safety analyses and the development of a decision-making tool for demonstration of nuclear facility management. The integrated risk framework consists of: · Scenarios comprising single or multiple external hazards. Hazards can be combined or cascading and include earthquake, flooding, extreme weather and others, · The physical and functional fragilities and interdependencies between systems/equipment are taken into account, · Human factors are taken into account and may play important role during severe accidents. · A support decision-making tool will be developed to demonstrate nuclear facility management during severe accidents due to external natural events. The project is structured into seven work packages (WP): · WP1: External hazards characterisation, · WP2: Fragility assessment of main NPPs critical elements, · WP3: Integration and safety analysis, · WP4: Applying & comparing various safety assessment approaches on a virtual reactor, · WP5: Supporting tool for severe accident management, · WP6: Dissemination, recommendation, and training, · WP7: Project management and coordination. The NARSIS project started in autumn 2017, with the duration of 4 years.