Many civil engineering problems lead to coupled multi-physics formulations: among them are the classical poro-mechanical and non-local problems, and more recently the multiple chemical-mechanical coupling problems (Alkali-aggregate reactions, delayed ettringite reaction, decalcification, etc.). Each new phenomenon to be considered leads finite element code developers to question the best strategy to adopt to solve these increasingly complex mechanical-multiphysics problems. There is always a choice between a strongly coupled implementation of all the equations, requiring significant development time, or a segregation of the physics requiring less implementation time but more computation time to ensure compatibility of the equations via an iterative algorithm. The aim of our work is to propose criteria for choosing between these two strategies. The article recalls the algorithms of the methods and provides an illustration allowing its effectiveness to be verified through a comparison between the solution of a classic linear poromechanics problem obtained by a fully coupled method and the solution of this same problem obtained by a segregated algorithm. The interest of this classical example lies in the fact that the fully coupled formulation of the problem is linear, whereas it is non-linear in a segregated context. This case study permits to have an exact reference numerical solution of the coupled problem, and simultaneously a main problem with a non-linear segregated sub-problem to test the segregated algorithm numerically. The article concludes with a discussion of the drawbacks and advantages of the segregated method, both in terms of implementation time and computation time.

Modeling concrete at elevated temperatures is essential to understanding the behavior of structural elements during fire, particularly with respect to spalling. To accurately predict temperatures and pore pressures, models must be validated against experimental data. However, most models in the literature focus on replicating experimental outcomes and often rely on input parameters sourced from the literature or determined by empirical tuning. To explore this further, a study of five models was conducted as part of the activities of the RILEM Technical Committee 256-SPF. On the theoretical side, state-of-the-art formulations are reviewed and similarities and differences between implementations are discussed. Using input parameters from various test reports, simulations of temperatures and pore pressures were performed and compared with test results for two types of concrete. While all of the models gave satisfactory results, they did so only when permeability values were applied that were significantly lower than those obtained from the standard tests. Since this trend was consistent across all models, it suggests that the permeability of concrete under heating conditions differs from that measured in standard material tests. As noted by some researchers, gas permeability in concrete is altered by the presence of water, probably due to swelling and rehydration. Identifying an accurate permeability value for these conditions remains an open research challenge.

The lifespan of a reinforced concrete (RC) structure can be greatly influenced by the spatial variability of its material characteristics which, in particular, explains the observed or measured reduction of the tensile strength at first crack when the volume under tension increases. This paper discusses the ability of accounting for the spatial variability of the tensile strength of concrete in RC structures using a stochastic finite element (SFE) method based on random field simulations. In this work, the generation of random fields on the concrete tensile strength aims at computing the force corresponding to the first crack occurrence, and the reduced tensile strength of the structure. The method can be applied in particular to large-sized structures, which show a pronounced size effect, for different types of loading. The method consists of, first, estimating the mean of the random field, using the analytical approach of the weakest link and localization method (WL2A). Then, the discretized random field is defined on a particular 2D or 3D grid, and it is finally projected on the finite element mesh of the studied structure. The study of the parameters that influence the prediction of the cumulative density functions (CDFs) of the rupture force or the tensile strength is highlighted using experimental series of concrete beams having different volumes and subjected to 4-point bending loading. Moreover, the SFE method is applied to a RC tie-beam under tensile loading, characterized by a weak stress gradient, which complicates the prediction of crack positions.

The recommendation is based on the co-authors’ work organized by the RILEM TC 256-SPF “Spalling of concrete due to fire: testing and modelling”. The Committee has defined two types of screening tests for characterization of concrete propensity to fire spalling: Material screening tests and Product screening tests. Definitions of both types of tests are given in the paper. The following recommendations apply to Material screening tests. The material screening tests described in these recommendations are a set of minimum requirements to test concrete spalling propensity (for example, the minimal specimen size). This document covers the aspects of concrete characterization, specimen geometries, storage conditions, test methods and measured parameters.

The recommendation is based on the co-authors’ work organized by the RILEM TC 256-SPF “Spalling of concrete due to fire: testing and modelling”. It aims to provide useful information, guidance and best practices in fire spalling assessment to laboratories that perform large-scale tests based on fire resistance test standards. It provides guidance on the spalling observation techniques during testing, as well as post-test spalling quantification/assessment methods. This document is intended to be used in conjunction with the fire resistance test standards, e.g. EN 1363-1 and ISO 834-1.

In this paper, a novel approach is introduced for handling orthotropic plasticity coupled with damage, using second gradient formulations based on several directional diffusion tensors.This method automatically identifies the crack location and opening through the curvature of the non-local profiles of principal plastic strains.The paper includes an application demonstrating the numerical objectivity of the obtained solution.

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.

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.

Despite the increase in computational power, the accurate modelling of crack openings in reinforced concrete remains an open problem for structural elements of complex shapes. Since the national CEOS.fr project dedicated to the control of cracking for large structures, it is accepted that the statistical scaling effect on the tensile strength of concrete and the realism of the concrete steel slip law are two essential ingredients to achieve a good level of accuracy in the prediction of crack spacing and openings. The implementation of these two aspects in finite element codes faces two problems: on the one hand, the statistical scaling effect depends on the dimension of the tensioned zone, which varies with cracking, and on the other hand, the steel-concrete slip is generally longer than the size of the finite elements. In this context, proposing homogenized reinforced concrete finite elements is problematic. The most common methods consist in using random mechanical property fields to integrate the statistical aspect of the tensile strength, and the slip is explicitly modelled by joint elements, preventing the use of homogenized elements. However, it is possible to avoid both random draws and explicit modelling of concrete-steel interfaces. The proposed method generalizes the notion of phase fields to address both problems simultaneously. A first phase field integrates the scale effect, and another one the steel-concrete slip. The FE code Castem (CEA) has been modified to allow such generalizations. After having given the theoretical bases of the methods used, we will comment two implementations.

Damage, cracking, and strain localization mechanisms often lead to unstable structural responses characterized by snap-backs (i.e., force and displacement decrease simultaneously). Standard nonlinear Newton-based solution algorithms with displacement/force control cannot capture the equilibrium curve in its entirety. This can be overcome using path-following formulations. A general (i.e., valid for any finite element code) implementation framework can be designed and applied to the Cast3M software by collecting the essential concepts and the formalism of partitioned path-following arc-length algorithms. Thanks to these developments, Cast3M is now capable of processing path-following equations without any major modifications. Three path-following constraints were selected to demonstrate the applicability of this framework: a first one on the combination of the displacement increment at a given set of nodes, a second one on the maximum strain increment over the computational domain, and a third one on the maximum elastic predictor of the damage/plastic criterion function over the computational domain. Two- and three-dimensional strain localization simulations show that the proposed framework behaves in a stable and convergent manner, even when multiple severe snap-back instabilities are present. Users of Cast3M shall find the proposed study helpful in that it allows them to focus on developing new path-following equations for the software. Cast3M is developed by the French Alternative Energies and Atomic Energy Commission (CEA) and freely available for research purposes. The developments discussed in this paper have been made available to the user/developer community along with Cast3M 2021 (release date: June 2021).

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 formula 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.

This chapter presents the most commonly used approach to analyse the thermo-mechanical behaviour of concrete structures subjected to high temperatures as in the case of fire loading. Prescriptions of the Eurocode are detailed for the thermal as well as the mechanical analysis. Finally, recommendations from two national (Austrian and German) guidelines give some improvements for underground infrastructure.

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: feedback 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 (Phi'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).

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.

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.

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.

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/m 3 . 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.

The durability of the cooling towers of nuclear power plants (NPP) is an up-to-date issue: some of them can be affected by some cracks.Cracks formation is accompanied by some damages at the steel-concrete interface.These load-induced damages accelerate the diffusion of atmospheric carbon dioxide in the concrete and along the damaged steel-concrete interface.Carbonation at the interface induces steel corrosion which could cause the development of further cracks in the structure, thus, threaten its durability.To understand the effect of cracking, both in terms of corrosion initiation and propagation, an accurate experimental procedure leading to cracks and steel-interface damage representative of those existing on the cooling towers should be found.The objective of this paper is to characterize numerically and experimentally the damage induced at steel/concrete interface by the applied load during three point bending test.This would be determinant parameter for carbonation and then corrosion in RC structures as cooling towers of NPP.