In order to model concrete cracking, a phase-field model is retained which converges toward cohesive fracture when the regularisation length goes to zero. The damage threshold in stress-space is adapted from experimental failure surface. In addition, damage in compression is introduced to bound the compressive stresses. The model is deliberately kept as simple as possible to preserve robustness. Application to a reinforced concrete beam subjected to four-point bending demonstrates the model’s predictive capabilities, even under conditions of distributed multi-cracking.To achieve this level of accuracy, the phase-field model operates at a descriptive scale where each individual crack is explicitly represented. While effective for a limited number of structural cracks, this approach may become impractical for large structures experiencing generalised distributed cracking induced, for instance, by reinforcement bars. Therefore, a companion model is derived by means of a basic uniaxial homogenisation scheme, under the assumption of diffuse damage. The diffuse model relies on the a priori knowledge of the average crack spacing. It operates at a coarser scale while preserving average characteristics of the phase-field model.Because the diffuse model is purely local, it is susceptible to localisation which contradicts the assumption of diffuse damage and must therefore be avoided. A crude localisation limiter is retained, which relies on a coarse mesh. Application to the same beam provides consistent, though less accurate, results for the limit load, global response, and crack extent.

Ductile fracture behavior in ferritic steels is investigated using two complementary experimental databases. The first database involves a wide range of cracked and uncracked specimen geometries tested at various temperatures for a A533 (18MND5) steel, enabling a detailed analysis of the effect of stress states, particularly stress triaxiality and the Lode parameter, on damage nucleation and growth. The second database, for a WB36 (15NiCuMoNb5) steel, includes both laboratory-scale specimens and full-scale structural tests on precracked pipes at various temperatures. A gradient-enhanced energy GTN (Gurson-Tvergaard-Needleman) model incorporating a Lode-parameter-dependent nucleation function is employed to simulate ductile damage across different stress states. The model is first calibrated and validated on the A533 dataset. It is then applied to the WB36 dataset to assess its transferability from specimens to structural components. The results confirm the model's ability to accurately capture damage evolution and crack propagation, demonstrating its robustness and relevance for structural integrity assessments.

Tests on cracked CT and SENT specimens (152 tests) extracted from a nuclear pressure vessel steel (18MND5) were conducted between C and C together with tests on smooth and notched bars. Tests on tensile bars were used to calibrate a temperature-dependent hardening law. Tests on cracked specimens that exhibited brittle failure only were analyzed using the Beremin model. Special care was taken to evaluate local stresses using enhanced mixed finite elements. The Beremin model was able to represent the entire database provided the reference failure stress increases with temperature. Finally, the results were also interpreted using the Master Curve approach, which was adapted to account for different stress states using the -parameter.

Modélisation numérique en mécanique fortement non linéaire traite des avancées récentes sur le traitement numérique des phénomènes de contact/frottement et d’endommagement.Bien que distincts sur le plan physique, ces phénomènes entraînent tous deux une forte non-linéarité du problème mécanique qui limite la régularité du problème, dorénavant non différentiable. En outre, ceci implique deux conséquences directes : d’une part, les caractéristiques mathématiques du problème s’écartent des formes bien maîtrisées et requièrent notamment des schémas de discrétisation innovants ; et d’autre part, la faible régularité complique singulièrement la résolution robuste et performante des systèmes algébriques de grande taille correspondants.Par ailleurs, ni l’unicité, ni même l’existence de solutions ne restent assurées, ce qui se traduit par des points de bifurcation, des charges limites et des instabilités structurelles, toujours délicats à franchir sur le plan numérique.

Ductile tearing of a full size precracked pipe is experimentally investigated. In order to model and interpret the test, the pipe material is characterized using smooth and notched tensile bars and precracked C(T) specimens. This experimental database is used to fit the parameters of the non local Gurson-Tvergaard-Needleman (GTN) proposed in Zhang et al. (2018) and Chen et al. (2020). The model is used in finite element simulations using specific elements allowing for the control of strain/damage localization as well as volumetric locking. Mesh size independence is checked on notched tensile bars. The model is then able to represent the early stages of crack propagation in the pipe. In particular, experimentally observed crack branching is reproduced, whereas this appeared much more difficult to obtain using a local GTN model.

This study aims at investigating a non-local Gurson-Tvergaard-Needleman (GTN) ductile damage model at finite strains within the framework of small-scale yielding. This model solves the problems of spurious strain localization and volumetric locking. The model is applied to simulate large crack propagation under small-scale yielding and plane-strain mode I conditions. A new method to extract crack length from the porosity field is introduced. Besides, purely numerical parameters are introduced to help convergence. An adequate range is exhibited for each of them so that their impact on the J−Δa crack growth resistance curves remains negligible. A parametric study is performed for several values of the material properties in order to estimate their influence on the crack growth resistance. It is found that the relation between the non-local intrinsic length implicitly introduced by the hardening gradient terms and the width of the damage/strain localization band is quasi-linear; crack tip blunting, crack initiation and large crack propagation can be well captured with the modified GTN model; the numerical formulation is robust; wide ranges for material plasticity and damage parameters can be used in a reliable way so that toughness at crack initiation as well as ductile tearing behavior can be thoroughly studied.

This study aims at investigating the properties of a non-local locking-free GTN ductile damage model [1] at finite strain within the framework of small-scale yielding. This model solves the problems of spurious localization and volumetric locking. The former is achieved by introducing the gradient of the hardening variable into the Helmholtz free energy (gradient plasticity). On a numerical ground, this results in spatial gradients of state variables within the constitutive relations: a decomposition-coordination techniques [2] is used to treat the corresponding term. Regarding the volumetric locking resulting from plastic incompressibility prior to damage, the Hu-Washizu mixed variational principle [3] is put in practice. An additional penalty term is also introduced into the corresponding Lagrangian in order to ensure coercivity. Finally, a new 5-field finite element is derived from the non-local locking-free variational formulation, in combination with the set of constitutive relations. The model is applied to simulate large crack propagation under small-scale yielding and plane-strain mode I conditions. A new way to extract the crack length from the porosity field is introduced. Besides, purely numerical parameters are introduced to help convergence. An adequate range is exhibited for each of them so that their impact on the J − Δa crack growth resistance curves be negligible. Finally, a parametric study is performed for several values of the material properties in order to estimate their influence on the crack growth resistance. Here are the main findings: 1. A linear relationship between the non-local intrinsic length implicitly introduced by the hardening gradient terms and the width of the damage/strain localization band is established. 2. Crack tip blunting, crack initiation and large crack propagation can be well captured with the modified GTN model. 3. The numerical formulation is robust; wide ranges for the material plasticity and damage parameters can be used in a reliable way so that toughness at crack initiation as well as ductile tearing behaviour can be thoroughly studied.

With the life extension of NPPs world-wide, new challenges have emerged in engineering calculations. These challenges often stem from the difficulty to demonstrate an adequate margin for some key components, which have gradually been ageing during the operation of the plant. In particular, the Reactor Pressure Vessel (RPV) is impacted by the irradiation, and the risk of brittle fracture under severe cold shocks must be assessed. Over the past decades, the RSE-M code [1], which is used in France and internationally for in-service inspection, has been developing methods using a conventional approach to brittle fracture. Analyses are typically performed either using tabulated indices to evaluate analytically the stress intensity factor, or using more advanced approaches which require more complex and time-consuming FEA calculations. Recently, the ongoing trend has been to rely on the latter to demonstrate an adequate margin on the RPV for potential operation beyond 40 years: the question today is whether these existing methods will still provide adequate margins after 50 or 60 years of operation. In parallel to the conventional approach, a significant amount work has been performed over the past 20 years in France to adapt the historic Griffith energy release-rate approach [2] to engineering space. The work was initiated by Francfort and Marigo [3] who set up a new elastic fracture theory, extended from the Griffith approach. Within EDF R&D, Lorentz et al. [4] and Wadier et al. [5] have then relied on some of their ideas and applied them to the easier case of the propagation onset of a preexisting crack along a given crack path. Several ingredients are involved in this reduced formulation: the application of an energy minimization principle, the definition of a specific damage model and the use of a notch to represent the crack. Among other advantages, the Gp method has been developed as a true engineering approach, i.e. not relying on difficult and time-consuming models to set up. It is hence easy to implement in a FE software as a post processing of a mechanical calculation. The method has also been applied to various test cases and has shown the potential to increase margins. The drawbacks are that the method is likely restricted to 2D cases for practical reasons. The paper also provides an overview of the methods implemented in the EDF open source tool code_aster with a specific focus on the G(p) approach.

A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete under tensile loading. Here, the target scale is the individual crack. In order to deal with localised damage, the model is inherently nonlocal: the gradient of the damage field is explicitly involved in the constitutive equations; it is parameterised by a nonlocal length scale which is interpreted as the width of the process zone. The model is defined so that its predictions are close to those of a cohesive law for vanishing nonlocal length scales. Therefore, the current model is plainly consistent with cohesive zone model analyses: the nonlocal length scale appears as a small parameter which does not need any specific identification. And four parameters—among which the tensile strength and the fracture energy—enable to adjust the softening cohesive response. Besides, a special attention has been paid to the shape of the initial damage surface and to the relation between damage and stiffness. The damage surface takes into account not only the contrast between tensile and compressive strengths but also experimental evidences regarding its shape in multiaxial tension. And the damage–stiffness relation is defined so as to describe important phenomena such as the stiffness recovery with crack closure and the sustainability of compressive loads by damaged structures. Finally, several comparisons with experimental data (global force/opening responses, size dependency, curved crack paths, crack opening profiles) enable to validate qualitatively and quantitatively the pertinence of the constitutive law in 2D and 3D.

SummaryThe major goal of this work is to develop a robust modelling strategy for the simulation of ductile damage development including crack initiation and subsequent propagation. For that purpose, a Gurson‐type model is used. This model class, as many other damage models, leads to significant material softening and must be used within a large deformation framework due to the ductile character of the materials. This leads to 2 main difficulties that should be dealt with carefully: mesh dependency and volumetric locking. In this work, a logarithmic finite strain framework is adopted in which the Gurson‐Tvergaard‐Needleman constitutive law is reformulated. Then a nonlocal formulation with regularisation of hardening variable is applied so as to solve mesh dependency and strain localization problem. In addition, the nonlocal model is combined with mixed “displacement‐pressure‐volume variation” elements to avoid volumetric locking. Thereby, a mesh‐independent and locking‐free finite strain framework suitable for the modelling of ductile rupture is established. Attention is paid to mathematical properties and numerical performance of the model. Finally, the model parameters are identified on an experimental database for a nuclear piping steel. Simulations of standard test specimens (notched tensile bars and compact tension and single edge notched tensile cracked specimens) are performed and compared to experimental results.

estimation of the overall leakage of leads to .

A nonlocal damage model is presented which is based on the explicit introduction of the gradient of the damage field. The model is designed as nonlocal from the beginning so that to ensure its convergence towards a cohesive zone model for vanishing nonlocal length scales. In particular it alleviates the sensitive task of identifying the nonlocal length scale. Moreover, the model relies on a damage surface representative of concrete, even for bi-tension stress states that are characteristic of crack propagation. Finally, the physical validity of this new constitutive relation is assessed through comparisons with two experimental settings: a CLWL-DCB specimen submitted to mixed-mode loading and leading to crack branching and the Brokenshire torsion test resulting in non-planar 3D crack propagation.

This paper presents an approach to predict ductile fracture of real-life structures. It relies on Rousselier’s constitutive model to describe plastic void growth, a specific finite strain formulation that preserves energetic properties and a non local theory to deal with strain localisation. It is finally applied to the computation of a notched specimen.

Most structures exposed to severe loadings may suffer damage, possibly resulting in safety risk or economic loss. In complement to experimental studies and design codes and standards, advanced numerical simulation of damage appears as a promising approach to describe such critical regimes. However, damage modelling still faces several difficulties. In the case of ductile damage, characterized by a large amount of plastic strain, spurious strain localization and plastic volumetric locking are observed and should be dealt with.