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Path-following methods for describing unstable structural responses induced by strain-softening are discussed. The main ingredients of the formalisms introduced by Riks and Crisfield for arc-length methods for geometrical non-linearities are presented. A link between two ways (monolithic and partitioned) of solving the resulting augmented equilibrium problem is discussed based on the Sherman–Morrison formula. The original monolithic approach assumes that the path-following constraint equation is differentiable with respect to the unknown displacement field and load factor. However, when dealing with material non-linearities, it is often preferred to consider constraint equations controlling the maximum of a field defined on the computational domain (e.g., a scalar strain measure, the rate of variation of an internal variable of the constitutive model). In that case, differentiability cannot be guaranteed due to the presence of the maximum operator. This makes only the partitioned formulation usable. Several path-following constraint equations from the literature are presented, and the corresponding implementations in the finite element method are discussed. The different formulations are compared based on a simple two-dimensional test case of damage localization in a beam submitted to tension. A test case involving multiple snap-backs is illustrated, finally, to show the robustness of the considered formulations.

The appearance of snapping phenomena is frequent in structural analyses involving nonlinearities such as the propagation of localized damage zones, or cracks, as encountered in quasi-brittle materials. In the presence of snapping effects, the standard Newton-based algorithms cannot predict the equilibrium curves correctly by usual displacement or force control. This problem can be circumvented by using path-following techniques. This study proposes a numerical approach for nonlinear structural analysis based on the Boundary Element Method (BEM) coupled with a partitioned path-following strategy. In this strategy, the acting loads are divided into two groups: (a) those perfectly known and (b) those in which only the direction is known, while the magnitude (or load factor/multiplier) is determined to respect the equilibrium of the boundary fields (indirectly controlled) and an additional problem equation, the path-following constraint equation. The resulting nonlinear system is solved using an incremental iterative scheme. For each iteration, the corrections to the boundary fields are obtained in a partitioned manner. First, the portion coming from the unbalanced boundary fields is calculated. Then the correction resulting from a unit load in the direction of the load to be indirectly controlled is obtained. Finally, the load factor is calculated independently using the path-following constraint equation, which can be linear, nonlinear, differentiable, or not. In the numerical tests, the Control of the Nodal Displacement Increment (CNDI) is chosen as path-following constraint equation for the sake of illustration. The results show that the proposed approach can efficiently capture the equilibrium curve even in the presence of severe snap-backs. The approach is designed to be implemented in existing BEM codes straightforwardly, provided that the influence matrices can be accessed. Moreover, no additional pre-conditioning technique is required.

International audience; Damaging, cracking, and strain localization mechanisms often lead to unstable structural responses characterized by snap-backs(i.e., the force and the displacement decrease simultaneously). The usual nonlinear Newton-based solution algorithms with displacement/force control cannot capture the complete equilibrium curve in a numerical context. This shortcoming can be circumventedusing path-following formulations. They rely on two hypotheses: (i) the decomposition of loading into two parts, one knownand the other unknown (only its direction is imposed); (ii) the use of path-following equations so that the unknown load (forceor displacement) contribution can be determined indirectly. Collecting the essential concepts and the formalism of path-followingarc-length methods, we design a unique framework into the Cast3M toolbox, capable of receiving existing and new path-followingequations without significant modifications. Three path-following constraints were chosen to demonstrate its applicability: a constrainton the combination of the displacement increment at a given set of nodes, a constraint on the maximum strain incrementover the computational domain, and a constraint on the maximum elastic predictor of the damage/plastic criterion function over thecomputational domain. Two- and three- dimensional strain localization simulations show that the proposed framework behaves ina stable and convergent manner, even in the presence of multiple severe snap-back instabilities. The proposed study proves helpfulbecause the user can focus on developing new path-following equations into the Cast3M toolbox. The developments discussed inthis manuscript will be made available to the users/developers community with Cast3M 2021 (release date: April 2021).

International audience; Non-linear finite element modelling of complex structures made of composites , such as reinforced concrete, remains a challenge because, until now, the only way to consider the important phenomenon of sliding between the reinforcements and the brittle matrix of the composite has been to mesh the reinforcements and their interfaces explicitly. This method is accurate but so expensive in terms of computational resources that only critical small elements of composites structures are modelled using it. To get around this limit, a method avoiding the meshing of composite reinforcements is proposed. It consists in treating the sliding between reinforcements and matrix with a differential formulation that provides the deformation of reinforcements directly as a continuous field superimposed to the displacement field of the matrix. The method needs a minor modification of the finite element code, which can take advantage of its analogy with the anisotropic thermal formulation. After the analytical presentation of the method, two theoretical cases of study are given to confront the results obtained with this method without meshing of reinforcements, with reference results obtained using a complete mesh of the matrix, reinforcements and interfaces.

The work presented is a part of the french ANR (Agence Nationale pour la Recherche) project MACENA (Maitrise du Confinement en Accident), its main objective is to better present the role of concrete heterogeneities in RC structures in the cracking process. This paper aims to develop and use the size effect method (WL2) applicable to RC structures proposed by Sellier and Millard 2014 [1]. The originality of the method lies on introducing a weighting function defined in the direction of the maximum principal stress using a scale length. In this work, an inverse analysis of the method allows to identify this scale length using experimental test series of concrete specimens under tensile load and 3 point bending beams. The approach is then applied to predict the sensitivity of the mechanical behavior of a reinforced concrete tie under tensile load. The method is applied in the elastic phase and allows providing the structural tensile strength corresponding to the first crack which is affected by size effect and plays a key role because cracked and uncracked structures behave in severe environment in a very different way. In FE model, correlated random fields on the tensile strength of the concrete can be generated using the identified scale length to characterize the autocorrelation length.

International audience; Geological disposal of radioactive wastes relies on a multiple barrier system to provide long-term containment and isolation of the wastes. The excavation of the repository creates openings and disturbed zones in the host rock formations that need to be properly sealed. Bentonite-based materials are being considered worldwide as a preferred type of sealing material, since they possess desirable characteristics such as low permeability, high sorption capability, and swelling potential allowing them to close internal cracks and gaps at interfaces with other materials. The French Institute for Radiation Protection and Nuclear Safety (IRSN) has led an experimental program consisting of a series of laboratory and large in situ experiments to assess the hydromechanical behaviour of bentonite seals. The experiments consisted of the forced re-saturation of pre-fabricated blocks of bentonite-sand mixture, with technological voids between bentonite seals and the walls of the steel cell (in the laboratory tests) and between bentonite seals and the host rock (in the in situ experiment). Relative humidity and total stress were monitored during both tests. The Canadian Nuclear Safety Commission (CNSC) collaborated with Geofirma Engineering, IRSN, and Commissariat à l’énergie atomique (CEA) to develop a mathematical model to simulate the experiments. The model was developed within the framework of poromechanics, with the inclusion of partial saturation characteristics and swelling potential to simulate the behaviour of the bentonite-based material. The model results were in good agreement with the experimental measurements for relative humidity and swelling stresses. The model also predicted the closure of technological voids and gaps due to swelling. Although swelling into the technological voids leads to an increase in permeability, that permeability remains low and insignificant from a safety perspective. © 2017, Canadian Science Publishing. All rights reserved.

This paper proposes a refined Weibull effective volume (WEV) approach in order to model the size effect in concrete structures. Both the Highly Stressed Volume (HSV) and Weibull Weakest Link and Localization (WL2) methods are presented. An analytical probabilistic approach to running WL2 is then developed, knowing that this method takes into account the energetic-statistical size effect. The approach employed recognizes the inelastic phase before the peak load; it provides an analytical and fast estimation of the structural tensile strength at various scales. This approach depends on a scale length, which accounts for the spatial randomness of the concrete tensile strength, and is identified on a series of concrete specimens under uniaxial tension by use of an inverse analysis. The Weibull modulus estimation is also discussed herein. Moreover, it is shown that the analytical probabilistic approach to WL2 implementation yields the size effect prediction in both average and dispersion for various experimental series, from laboratory tests to large massive structures. The experimental test series discussed in this paper consists of concrete specimens under uniaxial tension and 3-point bending loading.

Comprehension and quantification of quasi-brittle materials behavior requires complex experiments when focusing on cyclic or multi-axial loadings. As an alternative, virtual testing, which can be computed using lattice discrete elements models (LDEM), is particularly interesting. LDEM already provide a physical description of the quasi-brittle materials behavior, but further attention has to be paid to numerical integration. LDEM are explicitly integrated, such integration has been proven in the literature to be accurate when cracking is involved, by means of efficient schemes such as the ``Saw-tooth'' algorithm. In order to extend the range of application of LDEM to more complex loading paths, such as compressive or cyclic loadings, involving contact and friction mechanisms, qualitativeness as well as quantitativeness of explicit integration has to be assessed anew. We hereby propose an implicit quasi-static integration scheme for LDEM based on specific non-linearities encountered in quasi-brittle materials, namely contact and fracture, to circumvent expected stability and accuracy issues. Efficiency of both schemes is investigated by means of simulations of a uniaxial cyclic test and a compression test.

International audience; At the Mont Terri Underground Research Laboratory (Switzerland), a field-scale investigation has been conducted in order to investigate the hydro-mechanical and chemical perturbations induced in the argillaceous formation by forced ventilation through a tunnel. This experiment has been selected to be used for processing model development and validation in the international project DECOVALEX-2011. The conceptual and mathematical representation of the engineered void, which itself forms a major part of the experiment and is not simply a boundary condition, is the subject of this paper. A variety of approaches have been examined by the contributors to DECOVALEX and a summary of their findings is presented here. Two major aspects are discussed. Firstly, the approaches for the treatment of the surface condition at the porous media/tunnel interface are examined, with two equivalent but differing formulations successfully demonstrated. Secondly, approaches for representing the tunnel with associated air and water vapour movement, when coupled with the hydro-mechanical (HM) representation of the porous medium, are also examined. It is clearly demonstrated that, for the experimental conditions of the ventilation experiment (VE) that abstracted physical and empirical models of the tunnel, can be used successfully to represent the hydraulic behaviour of the tunnel and the hydraulic interaction between the tunnel and the surrounding rock mass.

International audience; We address Hydro-Mechanical (H-M) feed-back coupling in a fluid-filled fractured rock, whereby the hydraulic conductivity of the fractured rock is modified by its deformation. We present an upscaled description of this coupling based on previous work on tensorial macro-permeability (Kij) of fractured rock. The H-M feed-back coupling effects are expressed for the 2D case with fractures as straight line cracks, and for the 3D case with fractures as planar disc cracks. We assume at first that the deformable rock matrix remains impervious: feed-back effects are then due solely to crack deformation, while the effect of matrix deformation on Kij is neglected. The resulting tensorial H-M sensitivity coefficients, (K′ijkl) and (δ′kl), are expressed explicitly. Extensions of this work are indicated in the conclusive section, e.g.: strain of the permeable porous matrix; thermal expansion; and combining all types of H-M couplings (stress/pressure/fluid production coupling, as well as feed-back coupling).

International audience; In this paper, a comparative modelling exercise from the DECOVALEX-2015 project is presented. The exercise is based on in situ experiments, performed at the Tournemire Underground Research Laboratory (URL), run by the IRSN (Institut de Radioprotection et de Sûreté Nucléaire), in France. These experiments aim at identifying conditions (e.g. technical specifications, design, construction, and defects) that will affect the long-term performance of swelling clay-based sealing systems, which is of key importance for the safety of underground nuclear waste disposal facilities. A number of materials are being considered as seals; the current work focusses on a 70/30 MX80 bentonite–sand mixture initially compacted at a dry density of 1.94 Mg/m3. The performance of the sealing plug involves at least three different important components, which are the hydro-mechanical behaviour of the bentonite–sand core, the overall permeability of the surrounding argillite, and the influence of the technological gap between the core and the argillite. Two particular tests have been selected for a comparative modelling exercise the WT-1 test, which was designed to study the rock mass permeability, and the PT-A1 test, which aimed at quantifying the evolution of the hydro-mechanical field within the bentonite–sand core. A number of independent teams have worked towards modelling these experiments, using different codes and input parameters calibrated on additional small-scale laboratory experiments. Their results are compared and discussed. © 2017, Springer-Verlag Berlin Heidelberg.

International audience; A comparative modelling exercise involving several independent teams from the DECOVALEX-2015 project is presented in this paper. The exercise is based on various laboratory experiments that have been carried out in the framework of a French research programme called SEALEX and conducted by the IRSN. The programme focuses on the long-term performance of swelling clay-based sealing systems that provide an important contribution to the safety of underground nuclear waste disposal facilities. A number of materials are being considered in the sealing systems; the current work focuses on a 70/30 MX80 bentonite–sand mixture compacted at dry densities between 1.67 and 1.97 Mg/m3. The improved understanding of the full set of hydro-mechanical processes affecting the behaviour of an in situ sealing system requires both experiments ranging from small-scale laboratory tests to full-scale field emplacement studies and coupled hydro-mechanical models that are able to explain the observations in the experiments. The approach was to build models of increasing complexity starting for the simplest laboratory experiments and building towards the full-scale in situ experiments. Following this approach, two sets of small-scale laboratory experiments have been performed and modelled. The first set of experiments involves characterizing the hydro-mechanical behaviour of the bentonite–sand mixture by means of (1) water retention tests under both constant volume and free swell conditions, (2) infiltration test under constant volume condition, and (3) swelling and compression tests under suction control conditions. The second, more complex, experiment is a 1/10th scale mock-up of a larger-scale in situ experiment. Modelling of the full-scale experiment is described in a companion paper. A number of independent teams have worked towards modelling these experiments using different conceptual models, codes, and input parameters. Their results are compared and discussed. This exercise has enabled an improved modelling of the bentonite–sand mixture behaviour, in particular accounting for the dependence of its retention curve on the dry density. Moreover, it has shown the importance of the technological voids on the short-term behaviour of the sealing system. © 2016, Springer-Verlag Berlin Heidelberg.

The aim of this article is to present the modelling approach developed in order to predict the effects of corrosion of the reinforcements induced by carbon dioxide penetration. The final objective of the project is to propose a phenomenological and coupled model taking into account all the processes involved in the degradation of reinforced concrete. In this context, the CORDOBA model considers the coupling between different processes involved in the cracking induced corrosion phenomenon. More precisely, hydric behaviour of concrete is coupled with oxygen transport and corrosion reaction, as well as with mechanical damage of concrete. The effect of a pre-existing crack on the development of corrosion is also analysed. Cet article presente la demarche de modelisation mise en œuvre pour predire les effets de la corrosion des armatures par carbonatation du beton arme. L’objectif final du projet est de disposer d’un modele phenomenologique couple prenant en compte tous les processus mis en jeu dans la degrada...

Most reinforced concrete structures are damaged due to corrosion of reinforcements in concrete. In normal conditions the pH near the reinforcements is around 12–13 which means that steel is in a passive state. But aggressive species, such as chloride ions or carbon dioxide, may penetrate into concrete and promote active corrosion. As a consequence (hydro)oxides are produced leading to degradation of concrete structures. For instance cracking of the concrete is generated due to the pressure induced by rust. In this paper, we study the inception and the propagation of cracking on reinforced mortar plates with rebars located either in the middle or at the corner. Additional experiments have been performed on cylindrical specimens to determine the local effect of rust pressure at the interface rust/mortar. The specimens have been subjected to imposed current density in order to enhance the corrosion and digital image intercorrelation has been used to determine displacement fields. The experiments have been compared to numerical modelling.

This paper gives a review of several years of research on archaeological analogues in order to predict the long term behaviour of the steel canisters or the reinforced concrete structures involved in disposal or interim storage of nuclear wastes in France. This article aims at showing the specific methodology, the complementariness with different other approaches and the complete integration of the research on analogues in the frame of research programs on long term prediction. Archaeological sites on which field measurement can be performed were referenced and described. A significant number of artefacts collected from these sites was selected for study. Detailed chemical and microstructural characterisation of the artefacts were undertaken by a combination of microbeam analytical techniques (μRaman, μXRD, μRaman spectroscopy,…). Hypotheses on the corrosion mechanisms were then tested using specific isotopic markers during re-corroding experiments on analogues. Specific parameters were measured, allowing crucial steps in modelling long-term corrosion of steel.

The prediction of concrete cracking due to corrosion in atmospheric/carbonated conditions is a major issue for the evaluation of the durability of structures and the choice of maintenance policies. Because of the complexity of the phenomenon, a fully predictive approach is still missing. The proposed work can be considered as one step in this direction. It deals with a modelling study achieved at the Commissariat a l'Energie Atomique (CEA) with the CAST3M finite elements software. Model is constituted of three components: (1) concrete hydric behaviour, (2) rebar corrosion and (3) mechanical consequences on concrete (mainly concrete cracking). Actual developments consider analogies between rebar corrosion mechanisms and atmospheric corrosion ones, assuming that corrosion processes are influenced by the relative humidity evolution of atmosphere and/or of concrete.

The prediction of long-term behaviour of reinforced concrete structures involved in the nuclear industry requires a phenomenological modelling of the rebars corrosion processes. Previous analytical characterisation of archaeological artefacts allowed to identify a typical layout constituted of four layers (the metal, the dense product layer, the transformed medium and the binder). Additional experiments leaded to identify the long-term corrosion mechanisms. Following these results, this paper proposes an analytical model of long-term corrosion of rebars embedded in concrete. This modelling is considering the kinetic of oxygen diffusion through the system and its consumption at the metal/dense product layer interface as a function of concrete water saturation degree. Corrosion products thicknesses estimated with the model are then compared to corrosion product thicknesses measured on archaeological artefacts. A parametric study is performed and demonstrates that the oxygen diffusivity and the kinetic constant of the cathodic reaction affect in a wide range the model results.

Pour etudier le comportement thermo-hydro-mecanique et chimique (THMC) de l'argilite de l'Est, un modele base sur la thermodynamique des milieux poreux est propose. Une reaction de dissolution est etudiee et deux analyses de tendance sur l'argilite sont effectuees a l'aide du code de calcul Cast3M. L'echantillon etudie est une galette d'argilite. L'influence d'une reaction chimique sur la succion est mise en exergue. L'article se termine par une discussion sur les evolutions futures du modele.

International audience; A continuum model for quasi-brittle materials able to reproduce mechanical phenomena related to cyclic loading is proposed. Specific care is taken to reproduce progressive stiffness recovery and hysteresis effects caused respectively by crack closure and friction. A virtual testing approach is set up to analyse the evolution of microscopic quantities during uni-axial cyclic tests and to justify an original and efficient modelling of these phenomena. Thus , a reg-ularised formulation of the homogenised multiple contact problem induced by the non-simultaneous closure of microscopic cracks is presented. The proposed continuum model is validated by means of member-scale simulations of reversely loaded reinforced concrete shear walls .

Coulomb criterion is often used in plane strain, due to the simplicity of its formulation. This criterion is sometimes compared to the less singular Drucker-Prager criterion. This technical note deals with the correspondence between both criteria in plane strain and highlights the limits of such a comparison. It is shown that an analogy exists between both criteria, leading to the same limit loading parameters in case of associated flow rule. Nevertheless, even in this case, the elastoplastic evolution problems are not identical. This difference in the evolution problem does not allow to compare both “materials” in case of non-associated flow rule.

The cooling towers are reinforced concrete structures affected by different ageing mechanisms. The material data, the loading actions and the environmental conditions are subject to important fluctuations. It is necessary to have efficient tools allowing to evaluate and to supervise the reliability level of these structures, by taking account for these fluctuations as well as the available measurements on the structure. We propose herewith to show the capability of the reliability methods to evaluate the failure risk of the reinforced concrete shells in function of the amplification of the deformation measured in-situ.

WOS:000343606400009; International audience; The random nature of geomaterial tensile strength, combined with a softening behaviour, leads to a strong and systematic dependency of the first crack stress value on the loaded volume. This problem is so important that the predictive capability of efficient damage models can be occulted by the phenomenon. The method proposed here allows this aspect to be efficiently considered in a non-linear finite element context thanks to a variant of the Weibull theory, in which the classical weakest link theory is rearranged in a non-local form permitting a second gradient implementation.

International audience; In the framework of the THERESA European project, numerical modelling of coupled thermo-hydro-mechanical (THM) and thermo-hydro-mechanical-chemical (THMC) behaviour of buffer (bentonite) and buffer-rock interfaces for deep underground nuclear waste repositories has been undertaken, with focus on the performance assessments. A major step of the project was the analysis of a large scale test, called the Canister Retrieval test, which has been performed in Aspö Hard Rock Laboratory. It consists in a full scale test of the emplacement of a canister with the surrounding buffer material. A deposition hole was first bored, and then the canister with heaters was installed together with bentonite blocks. The gap between the rock and the bentonite blocks was filled with bentonite pellets. The whole set was artificially wetted from its external boundary in order to accelerate the expected natural rehydration by the surrounding rock. The evolution of the THM processes was recorded over 5 years. Before analysing the whole CRT experiment, a preliminary simpler problem has been defined, which consisted in modelling a disc of buffer at canister mid-height. Thanks to the available experimental recorded measurements, it has been possible to numerically investigate the respective influence of the various THM parameters involved in the modelling of the physical processes. The theoretical model is based on one hand on the Richard's approximation for the flow calculation, and on the other hand on a Biot's type model for the hydro-mechanical behaviour. It has revealed the large influence of the liquid relative permeability, which is unfortunately in general not directly available from experiments and must be determined through inverse analysis techniques. Then, in a second stage, the whole CRT experiment has been analysed. For simplicity reasons, an axisymetrical model has been adopted, although the presence of a neighbouring experiment did influence the CRT results. The comparisons of the most relevant features of the test, between predicted and measured values, are presented and discussed in the paper. © 2013 Elsevier B.V. All rights reserved.

International audience; The SEALEX experiment, performed in the Tournemire Underground Research Laboratory, run by IRSN in France, aims at identifying conditions that will affect the performance of swelling clay-based sealing systems. Current experiments focus on a 70/30 MX80 bentonite-sand mixture compacted at dry densities between 1.67 Mg/m3 and 1.97 Mg/m3. Prior to the in-situ tests, a 1/10th scale mock-up has been tested and proposed as modelling benchmark, in the framework of the international project DECOVALEX. Three independent research teams have worked towards modelling the three phases of this experiment, using different codes and input parameters. In the paper, blind predictions and model calibrations of the three phases are compared to the experimental measurements and discussed.

International audience; In order to investigate the hydro-mechanical (HM) and chemical perturbations induced in an argillaceous formation by forced ventilation during the operational period of a nuclear waste repository, a specific experiment has been performed in a tunnel, at Mont Terri Underground Research Laboratory (URL) in Switzerland. This experiment has been selected in the international project DECOVALEX for model validation and the numerical simulation of this ventilation experiment (VE) is the object of the present paper. Since the argillaceous rock exhibits anisotropic properties, particular attention is given to the evaluation of the effects of various anisotropic features on the predicted results. In situ measurements such as relative humidity (RH), global water mass extracted, pore water pressure, water content, and relative displacements are compared to predictions using both isotropic and anisotropic parameters. Water permeability anisotropy is shown to be the most influencing parameter by far, whereas in situ stress anisotropy has an effect only during the excavation phase. The anisotropy for mechanical parameterization has also some influence, in particular through HM couplings. These HM couplings have the potential to be very significant in terms of providing confidence in describing the experimental observation, and should be considered for further investigation.

A numerical investigation of the desaturation process at the argillaceous Tournemire site has been carried out. This desaturation is initialized by the contact of the saturated rock with the ambient air in excavated openings. The used hydraulical model is based on the Richards' approximation for unsaturated one phase flow coupled with the deformation of a porous medium with anisotropic linearly elastic behavior. In relation to the extent of the desaturated zone around an excavated opening, the intrinsic permeability and the relative permeability have been identified to be the most important model parameters. The mechanical deformation process itself, the seasonal influences and the tunnel lining are less important for the formation of the desaturated zone. The comparison with measured saturation values bares some difficulties but indicates the principle capacity of the applied finite element codes to simulate the desaturation process. The consideration of seasonal changes in humidity in the ambient air results in a constantly recurring desaturation-resaturation cycle in the near field of the openings. This seasonally influenced zone amount 1-2 m and is nearly independent from time and from a variation of model parameters within a reasonable range. The possibility of material weakening in this zone is of special interest, since claystone is a potential host rock for the disposal of radioactive waste. © 2008 Springer-Verlag.

One of the most current degradations in reinforced concrete structures is related to the corrosion of the reinforcements. The corrosion products during active corrosion induce a mechanical pressure on the surrounding concrete that leads to cover cracking along the rebar. The objective of this work is to study the cracking of concrete due to the corrosion of the reinforcements. The phenomenon of corrosion/cracking is studied in experiments through tests of accelerated corrosion on plate and cylindrical specimens. A CCD camera is used to take images every hour and the pictures are analyzed by using the intercorrelation image technique (Correli LMT ) to derive the displacement and strain field. Thus the date of appearance of the first through crack is detected and the cinematic crack initiations are observed during the test. A finite element model that allows prediction of the mechanical consequences of the corrosion of steel in reinforced concrete structures is proposed. From the comparison between the test results and numerical simulations, it may be concluded that the model is validated in term of strains up to the moment when the crack becomes visible and in terms of crack pattern.

Cracking of concrete due to corrosion is a very common pathology that can be observed on various structures. Corrosion is initiated by the penetration of chlorides or by the reactive transport of carbon dioxide in most cases. The consequences of corrosions are (i) the reduction of the resistive section of reinforcements (ii) the creation of expansive products (commonly denoted rust) (iii) the fragilization of steel and finally (iv) the cracking of concrete [1, 2, 3, 4].

International audience; This paper aims to develop a multiaxial concrete model for implementation in finite element software dedicated to the analysis of structures in fire. The need for proper concrete model remains a challenging task in structural fire engineering because of the complexity of the concrete mechanical behavior characterization and the severe requirements for the material models raised by the development of performance-based design. A fully three-dimensional model is developed based on the combination of elastoplasticity and damage theories. The state of damage in concrete, assumed isotropic, is modeled by means of a fourth order damage tensor to capture the unilateral effect. The concrete model comprises a limited number of parameters that can be identified by three simple tests at ambient temperature. At high temperatures, a generic transient creep model is included to take into account explicitly the effect of transient creep strain. The numerical implementation of the concrete model in a finite element software is presented and a series of numerical simulations are conducted for validation. The concrete behavior is accurately captured in a large range of temperature and stress states. A limitation appears when modeling the concrete post-peak behavior in highly confined stress states, due to the coupling assumption between damage and plasticity, but the considered levels of triaxial confinement are unusual stress states in structural concrete.

Resume Nous presentons dans cette Note une experimentation en laboratoire d'un effet mecanique (fissuration du beton) de la corrosion acceleree sur des plaques de beton avec armature metallique. Ces plaques permettent de suivre l'evolution de la deformation du beton au cours de l'essai de corrosion grâce a l'enregistrement d'images numeriques a differents instants, images traitees par correlation numerique. La corrosion est acceleree par imposition de densite de courant electrique. L'avantage de ces essais, rapides, est de permettre de suivre la cinetique de la fissuration, l'inconvenient est que la representativite de ce type d'essai est toujours discutee. Deux positions d'armatures ont ete imaginees. La cinematique de la fissuration a ete obtenue et comparee a une modelisation dans laquelle la croissance de la couche d'oxyde est simulee par un chargement thermique, le beton etant elastique endommageable. Pour citer cet article : Q.T. Nguyen et al., C. R. Mecanique 335 (2007).

Macroscopic mechanical behavior models are developed for their light computational costs, allowing the simulation of large structural elements, and the precise description of mechanical phenomena observed by the material at lower scales. Such constitutive models are here developed in the seismic solicitation framework, therefore implying cyclic alternate loadings at the material scale, and applied to civil engineering buildings, often made of concrete, or more generally of quasi-brittle materials. To date, macroscopic models applicable to structural computations, while representing the cyclic mechanical behavior are rare. In consequence of the intricacy of the fracture processes to homogenize, macroscopic constitutive models either do not present sufficient robustness or miss on important phenomena. One of the limitations to the resolution of this issue is the lack of experimental data. Indeed, because of the complexity of the experiments to set up, few results on alternate cyclic tests on concrete are available in the literature.A virtual testing approach has therefore been established on a microscopic model of the material, able to provide results needed to the formulation and the calibration of a macroscopic model. In the microscopic model, the material is considered as structure itself, it is developed so as to only necessitate a reduced amount of results from controlled experimental tests, in order to be used. The microscopic model, a lattice discrete element model, has been developed on the basis of an existing lattice model and extended to the simulation of multi-axial and cyclic loadings. The microscopic model has then been validated as a virtual testing tool and used to establish equations of the macroscopic model, on the basis of damage and plasticity theories. The consistency of the proposed constitutive relation, embedding progressive unilateral effect, has been achieved using non-linear elasticity. The macroscopic model has finally been calibrated, entirely with the microscopic model, and employed to simulate the response of a reinforced concrete wall under alternate shear loading. This simulation has served to showcase the numerical robustness of the proposed model, as well as the significant contribution of the uni-axial alternate behavior of concrete to the structural damping of such structures., Les modèles de comportement mécanique, dits macroscopiques, sont développés à la fois pour leur légèreté, permettant le calcul d’éléments structuraux pouvant atteindre d’importantes dimensions, et pour leur finesse de représentation des phénomènes mécaniques observés par le matériau à des échelles plus fines. Le développement de tels modèles est ici effectué dans le cadre de la sollicitation sismique, donc des chargements cycliques alternés, appliquée à des ouvrages en matériaux quasi-fragiles, et plus précisément en béton. À ce jour, les modèles macroscopiques, effectivement applicables au calcul de structures, et représentatifs du comportement cyclique du béton sont encore rares. En conséquence de la complexité du problème de fissuration à homogénéiser, les modèles macroscopiques existants affichent une robustesse limitée ou ne permettent pas de reproduire l’ensemble des phénomènes mécaniques observés par le matériau. Une des barrières à la résolution de ces deux problématiques est le manque de données expérimentales relatives aux phénomènes à modéliser. En effet, en cause de la difficulté technique de les réaliser, peu de résultats d’essais cycliques alternés sur du béton sont disponibles dans la littérature. Une démarche d’expérimentation numérique a donc été élaborée sur la base d’un modèle fin du matériau, dit microscopique, capable de fournir les résultats nécessaires à la formulation et à l’identification d’un modèle macroscopique. Dans le modèle microscopique le matériau est considéré comme une structure à part entière, il a été développé afin de ne nécessiter qu’une quantité réduite de résultats d’essais, maîtrisés, pour être mis en oeuvre. Le modèle microscopique, un modèle particulaire lattice, a été développé sur la base d’un modèle lattice existant, enrichi pour être en mesure de simuler le comportement des matériaux quasi-fragiles sous chargements multi-axiaux et cycliques. Le modèle microscopique a alors été validé en tant qu’outil d’expérimentation numérique, et exploité afin d’établir les équations constitutives du modèle macroscopique fondées sur les théories de l’endommagement et de la plasticité. La régularité de la relation de comportement proposée, intégrant un effet unilatéral progressif, a notamment été garantie par l’utilisation d’un modèle d’élasticité non-linéaire. Le modèle macroscopique a finalement été calibré, entièrement, à l’aide du modèle microscopique, et mis à l’oeuvre dans la simulation de la réponse d’un voile en béton armé soumis à un chargement de cisaillement cyclique alterné. Cette simulation a permis de mettre en avant la robustesse numérique du modèle développé, ainsi que la contribution significative du comportement uni-axial cyclique alterné du béton à l’amortissement de telles structures.

This thesis is a contribution to the modeling of the spatial variability of tensile strength of concrete structures, at different scales, and its influence on concrete cracking pattern. Particularly, a size effect law and random fields are used through two approaches:On the one hand, an analytical probabilistic approach of the Weakest Link and Localization (WL2) method is proposed. This method estimates the distribution of the tensile strength, at different scales, accounting for the stress redistributions around the weakest point.It depends on a scale length, whose identification is discussed. This scale length accounts for spatial randomness of the concrete tensile strengthOn the other hand, another contribution of this thesis is the development of a Stochastic Finite Element (SFE) method, used to model both size effect and the spatial variability of the tensile strength. The method consists first on defining a random field, using the mean tensile strength estimated from the analytical approach of WL2. Then, discretized autocorrelated random field realizations are generated. Moreover, the choice of autocorrelation parameters, used to define the random fields, is discussed.The applicability of both methods is evaluated using various experimental series exhibiting particularly statistical size effect. Furthermore, the SFE method is used to complete the simplified FE model of a 1/3 mock-up of a double-wall containment building. The uncertainties on the tensile strength, at this scale, are modeled using independent autocorrelated random field at each scale. Uncertainties propagation, at initial state, shows its pertinence in the estimation of crack positions., Cette thèse contribue à la modélisation de la variabilité spatiale de la résistance à la traction des structures en béton, à différentes échelles, et son influence sur la fissuration du béton. En particulier, une loi d'effet d'échelle et des champs aléatoires sont utilisés à l'aide de deux approches:D'une part, une approche analytique probabiliste de la méthode Weakest Linkand Localization (WL2) est proposée. Cette méthode estime la distribution de la résistance à la traction, à différentes échelles, en tenant compte des redistributions des contraintes autour du point le plus faible. Cela dépend d'une longueur d'échelle, dont l'identification est discutée. Cette longueur d'échelle explique le caractère aléatoire de la résistance à la traction du béton.D'autre part, une autre contribution de cette thèse est le développement d'une méthode Élément Fini Stochastique (EFS), utilisée pour modéliser l'effet d'échelle et la variabilité spatiale de la résistance à la traction. La méthode consiste d'abord à définir un champ aléatoire, en utilisant la résistance à la traction réduite estimée à partir de l'approche analytique de WL2. Ensuite, des réalisations de champs aléatoires autocorrélées discrétisées sont générées. En outre, le choix des paramètres d'autocorrélations, utilisés pour définir les champs aléatoires, est discuté.L'applicabilité des deux méthodes est évaluée à l'aide de différentes séries expérimentales présentant des effets d'échelle particulièrement statistique. En outre, la méthode EFS est utilisée pour compléter le modèle EF simplifié de la maquette d'enceinte à double paroi VeRCoRs (échelle 1/3). Les incertitudes sur la résistance à la traction, à cette échelle, sont modélisées à l'aide d'un champ aléatoire autocorrelé indépendant à chaque levée. La propagation des incertitudes, à l'état initial, montre sa pertinence dans l'estimation des positions de fissures.