Modelling Localisation and Spatial Scaling of Constitutive Behaviour: a Kinematically Enriched Continuum Approach

It is well known that classical constitutive models fail to capture the post-peak material behaviour, due to localisation of deformation. In such cases the concept of Representative Volume Element (RVE) on which classical continuum models rest ceases to exist and hence the smearing out of local inhomogeneities over the whole RVE is no longer correct. This paper presents a new approach to capturing localised failure in quasi-brittle materials, focusing on the kinematic enrichment of the constitutive model to describe correctly the behaviour of a volume element with an embedded localisation band. The resulting models possess an intrinsic length scale which in this case is the width of the embedded localisation band. The behaviour therefore scales with both the width of the localisation band and the size of the volume on which the model is defined. As a consequence, size effects are automatically captured in addition to the model capability in capturing behaviour at the scale of the localisation zone.Comment: Proceedings of Asian-Pacific Conference on Fracture and Strength 2014 and the International Conference on Structural Integrity and Failure, 9-12 December, Sydney, Australi

Mechanical damage, chemical damage and permeability in quasi-brittle cementitious materials

International audienceThe serviceability of concrete structures is a coupled problem in which fracture and damage are coupled with several environmental attacks. In this paper, we start with the description of chemo-mechanical damage and the case of calcium leaching which is relevant to waste containment vessels. The second example of implementation of continuum damage models discussed deals with coupled damage permeability effects. In the case of diffuse damage, the material permeability is controlled by the decrease of average stiffness due to micro-cracking. After a macro-crack has formed, permeability is controlled by a power function of the crack opening (Poiseuille flow). A relationship between permeability and damage, consistent with the two above asymptotic cases, is defined.La durabilité de structures en béton est un problème dans lequel la rupture et l'endommagement sont couplés aux différentes attaques de l'environnement. Nous commençons par illustrer un tel couplage avec la lixiviation et la modélisation chimiomécanique de l'endommagement dans les bétons. Les couplages entre l'endommagement et les propriétés de transport du béton sont le deuxième exemple traité. Dans le cas d'un endommagement diffus, la perméabilité du matériau est contrôlée par la diminution de raideur moyenne due à la microfissuration. Lorsqu'une macrofissure s'est formée, la perméabilité est contrôlée par une fonction puissance de l'ouverture de fissure (écoulement de Poiseuille). Une relation entre la perméabilité et l'endommagement est définie, en accord avec ces deux cas limites

Boundary effect on weight function in nonlocal damage model

International audienceSome insights on boundary effects in nonlocal damage modelling are addressed. Interaction stresses that are at the origin of nonlocality are expected to vanish at the boundary of a solid, in the normal direction to this boundary. Existing models do not account for such an effect. We introduce tentative modifications of the classical nonlocal damage model aimed at accounting for this boundary layer effect in a continuum modelling setting. Computations show that some nonnegligible differences may be observed between the classical and modified formulations. In a one dimensional spalling test, only the modified formulation provides a spall of finite nonzero thickness, whereas spalls smaller than the internal length cannot be obtained according to the original formulation. For the same set of model parameters, including the internal length, the fracture energy derived from the size effect test method is also very different according to both approaches. Parameters in the size effect laws for notched and unnotched specimens, obtained from computation of geometrically similar bending beams, are more consistent with the modified nonlocal model compared to the original nonlocal formulation

Hydraulic behaviour of a representative structural volume for containment buildings

For particular structures like containment buildings of nuclear power plants, the study of the hydraulic behaviour is of great concern. These structures are indeed the third barrier used to protect the environment in case of accidents. The evolution of the leaking rate through the porous medium is closely related to the changes in the permeability during the ageing process of the structure. It is thus essential to know the relation between concrete degradation and the transfer property when the consequences of a mechanical loading on the hydraulic behaviour have to be evaluated. A chained approach is designed for this purpose. The mechanical behaviour is described by an elastic plastic damage formulation, where damage is responsible for the softening evolution while plasticity accounts for the development of irreversible strains. The drying process is evaluated according to a non-linear equation of diffusion. From the knowledge of the damage and the degree of saturation, a relation is proposed to calculate the permeability of concrete. Finally, the non-homogeneous distribution of the hydraulic conductivity is included in the hydraulic problem which is in fact the association of the mass balance equation for gas phase and Darcy law. From this methodology, it is shown how an indicator for the hydraulic flows can be deduced.Peer ReviewedPostprint (author’s final draft

Estimation of crack opening from a two-dimensional continuum-based finite element computation

International audienceDamage models are capable of representing crack initiation and mimicking crack propagation within a continuum framework. Thus, in principle, they do not describe crack openings. In durability analyses of concrete structures however, transfer properties are a key issue controlled by crack propagation and crack opening. We extend here a one-dimensional approach for estimating a crack opening from a continuum-based finite element calculation to two-dimensional cases. The technique operates in the case of mode I cracking described in a continuum setting by a nonlocal isotropic damage model. We used the global tracking method to compute the idealized crack location as a post-treatment procedure. The original one-dimensional problem devised in Dufour et al. [4] is recovered as profiles of deformation orthogonal to the idealized crack direction are computed. An estimate of the crack opening and an error indicator are computed by comparing finite element deformation profiles and theoretical profiles corresponding to a displacement discontinuity. Two estimates have been considered: In the strong approach, the maxima of the profiles are assumed to be equal; in the weak approach, the integrals of each profile are set equal. Two-dimensional numerical calculations show that the weak estimates perform better than do the strong ones. Error indicators, defined as the distance between the numerical and theoretical profiles, are less than a few percentages. In the case of a three-point bending, test results are in good agreement with experimental data, with an error lower than 10% for widely opened crack (>40μm)

Experimental and modeling investigations of adsorption-induced swelling and damage in microporous materials

International audienceThe purpose of this work is to achieve a better understanding of the coupling between adsorption and swelling in microporous materials. This is typically of utmost importance in the enhancement of non-conventional reservoirs or in the valorization of CO 2 geological storage. We consider here the case of fully saturated porous solids with pores down to the nanometer size (≤ 2nm). Hardened cement paste, tight rocks, activated carbon or coal are among those materials. Experimentally, different authors tried to combine gas adsorption results and volumetric swelling data, especially for bituminous coal. However, most results in the literature are not complete in a sense that the adsorption experiments and the swelling experiments were not performed on the exact same coal sample. Other authors present simultaneous in-situ adsorption and swelling results but the volumetric strain is extrapolated from a local measurement on the surface sample or by monitoring the two-dimensional silhouette expansion. Only elastic and reversible swellings are reported in the literature. Theoretically, most continuum approaches to swelling upon adsorption of gas rely on a coupling between the adsorption isotherms and the mechanical deformation. A new poromechanical framework has been recently proposed to express the swelling increment as a function of the increment of bulk pressure with constant porosity. However, this framework has to be extended to take into account the porosity evolution upon swelling. This paper aims at presenting a new experimental setup where both adsorption and strain are measured in-situ and simultaneously and where the full-field swelling is monitored by digital image correlation. Permanent strain and damage are observed. On the other hand, we present an extended poromechanical framework where the porosity is variable upon swelling. A new incremental nonlinear scheme is proposed where the poromechanical properties are updated at each incremental pressure step, depending on the porosity changes. Interactions between swelling and the adsorption isotherms are examined and a correction to the classical Gibbs formalism is proposed. Predicted swellings are compared with results from the literature

Etudes numériques et expérimentales des corrélations induites durant la rupture des matériaux quasi-fragiles

La dégradation des matériaux quasi-fragiles met en jeu la création et la propagation de microfissures qui interagissent et coalescent afin de former une macro-fissure. Ces phénomènes sont localisés au sein de la zone d'élaboration (Fracture Process Zone - FPZ). Cette communication vise à préciser la description de l'évolution de la FPZ à l'aide d'une analyse statistique du processus d'endommagement. L'analyse statistique est basée sur l'utilisation de fonction de Ripley qui ont été développée initialement pour caractériser des motifs particuliers en écologie spatiale. On montre qu'une longueur de corrélation peut être extraite à partir de ces analyses par fonction de Ripley. Une comparaison numérique et expérimentale de l'évolution de ces longueurs de corrélation extraites est enfin proposée

Modèle pour l'estimation des propriétés de transfert des milieux poreux

L'accident de Fukushima a démontré qu'aujourd'hui encore, malgré les progrès déjà réalisés dans le domaine de la sûreté nucléaire, une interruption prolongée du circuit de refroidissement primaire du réacteur (plusieurs semaines) était possible. La France s'est donc vue contrainte de réévaluer le niveau de sûreté de ses centrales. Plus spécifiquement, le cas le plus défavorable qu'EDF considérait jusqu'alors, qui consistait en un arrêt total du système de refroidissement primaire de 24h, à été porté à deux semaines. Ce changement d'échelle temporelle a introduit des problématiques de fluage, d'évolution hygrométrique du béton ainsi que de flux de vapeur ? celle-ci transportant les radionucléides ? jusque-là laissées de côté. De ces considérations est né le projet MACENA (MAîtrise du Confinement d'une ENceinte en Accident), qui vise à évaluer le confinement d'une enceinte soumise à une température de 180°C et à une pression de 5bar pendant deux semaines. Dans sa thèse, Khaddour[1] a étudié l'impact de l'endommagement sur les propriétés de transport du béton. Il a développé un modèle capable de déterminer les perméabilités apparente et intrinsèque d'un milieu poreux à partir de sa distribution de taille de pores et des propriétés physiques du fluide considéré[2]. Ce modèle emploie une description classique du flux macroscopique au sens de Darcy avec une description microscopique combinant des écoulements de Poiseuille et Knudsen. Le milieu poreux est ensuite reconstitué suivant une approche hiérarchique cohérente avec la technique de porosimétrie par intrusion de mercure. Les résultats obtenus sont concordants avec l'approche de Klinkenberg[3] en ce qui concerne la détermination expérimentale de la perméabilité apparente. Ne comprenant aucun paramètre d'ajustement, ce modèle permet de reproduire l'effet de l'endommagement sur les perméabilités apparentes et intrinsèques de mortiers sollicités en compression. L'idée est naturellement venue d'étendre ce modèle à des milieux poreux partiellement saturés, d'une importance capitale pour notre étude. En effet, un milieu poreux partiellement saturé présente des perméabilités relatives aux phases gazeuse et liquide qui le parcourent inférieures à sa perméabilité intrinsèque. Ce sujet a déjà été traité par le biais de lois phénoménologiques dont l'une des plus largement répandues est celle de Van Genuchten[4]. L'approche présentée ici se base sur le modèle de Khaddour afin de s'efforcer d'apporter un aspect prédictif. Le modèle original a été adapté à l'aide d'une série de règles complémentaires : ? La saturation à l'intérieur des pores suit la loi de Kelvin ; ? La prise en compte des pores dans le calcul des perméabilités relatives dépend de leur état : saturé de gaz, partiellement saturé ou saturé de liquide ; ? Le réseau hiérarchique original est modifié en redistribuant les plus petits pores. En effet, de par leur volume relativement important et leur faible taille, il est probable que leur répartition spatiale soit telle qu'une fois saturés en eau, ils paralysent totalement le flux de gaz dans l'échantillon, et cet effet a une grande importance pour le calcul de la perméabilité relative au gaz. Les résultats obtenus sont comparés avec des données expérimentales issues d'une campagne d'essais réalisée sur un béton du projet MACENA, et l'influence de l'endommagement dans les perméabilités relatives est caractérisée. Références: [1]KHADDOUR, Fadi (2014). ?Amélioration de la production de gaz des «Tight Gas Reservoirs»?. Thèse de doct. Pau. [2]KHADDOUR, Fadi, David GRÉGOIRE et Gilles PIJAUDIER-CABOT (2015). ?Capillary bundle model for the computation of the apparent permeability from pore size distributions?.In:European journal of environmental and civil engineering 19.2,p.168?183. [3]KLINKENBERG,LJ et al. (1941). ?The permeability of porous media to liquids and gases?.In:Drilling and production practice.American Petroleum Institute. [4]VAN GENUCHTEN,M Th (1980). ?A closed-form equation for predicting the hydraulic conductivity of unsaturated soils?.In:Soil science society of America journal 44.5,p.892?898

Simulation numérique de fracturation hydraulique à l'échelle mésoscopique à l'aide d'un modèle lattice

Characterizing the path of the hydraulic fracture in a heterogeneous medium and the change of the transport properties of this medium is one of the challenges of current research on hydraulic fracturing. A 2D lattice hydro-mechanical model is used to describe hydraulic fracturing. First, we present the propagation of fracture restricted to the mechanical case. The influence of a natural joint crossed by the fracture is analyzed with the help of a parametric study. Then, the hydro-mechanical coupling is introduced. Results concerning characteristic length and hydraulic fracture behavior are discussed. The size of the area concerned by the change of transport properties, the stimulated reservoir volume, with respect to permeability, is analyzed. This parameter is important in order to evaluate the efficiency of the stimulation by hydraulic fracture in the context of tight hydrocarbon reservoirs