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Publication# Modèle numérique du comportement non-linéaire d'ouvrages massifs en béton non armé

Résumé

Linear Elastic Fracture Mechanics (LEFM) are used to study crack propagation in unreinforced concrete of massive structures. This approach is well suited to the nonlinear analysis of structures of large dimensions, taking into account the size effect, i.e., the influence of the dimension of the structure on its behaviour at rupture. The Finite Element method is used to calculate the structure and the hypothesis of plane strain is made. The use of a criterion of crack propagation, based on LEFM requires the calculation of stress intensity factors. The evaluation of the latter is performed by the mean of a surface integral defined around the tip of the crack studied. It has been shown in this work that this integral is derived from the path integral J. The use of the surface integral has also been extended to the cases where body forces (gravity, inertie) act, or when the edges of the crack are subjected to pressure. This method is precise and numerically efficient. A smeared crack model is used in order to avoid continuous remeshing during crack propagation. But, as it has been shown in this research, classical smeared crack elements do not give satisfactory results when the crack is submitted to shear loading, a new finite element, using smeared cracking but with discontinuous shape functions has been developed. The model which combines LEFM and the smeared crack approach has been applied to different classical problems of fracture mechanics. It leads to good results under static or dynamic loading. As an example, the nonlinear behaviour of the vertical cross section of a gravity dam during an earthquake has been calculated and the crack pattern identified. Many further developments could be done starting from this original approach, in order to simulate more closely the real behaviour of structures, taking into account friction in the cracks and the three-dimensional development of then.

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In this study, the tensile and fracture properties and the microstructure of the reduced-activation tempered martensitic steel Eurofer97 have been investigated. This technical alloy is a 9%Cr steel developed within the European fusion material research program. To a lesser extend, the plastic flow properties of a equiaxed ferritic Fe9%Cr model alloy were also studied for comparison with those of the tempered martensitic structure. The main objectives of this work are described hereafter: to correlate the microstructural features with the plastic flow properties measured by the tensile tests for both the Eurofer97 steel and the model alloy. The correlation established should be reflected in a physically-based model of plastic flow. to study the fracture properties of the Eurofer97 steel in details in the lower ductile-to-brittle transition region. to calculate with finite element modeling the stress fields at the crack tip. This information is further used in conjunction with a local approach model for quasi-cleavage to reconstruct the fracture toughness-temperature curve. Tensile tests were carried out at different imposed nominal strain rate at several temperatures from 77K up to 473K, on both the Eurofer97 steel and the Fe9%Cr model alloy. The temperature dependence of the yield stress was precisely determined. As expected for body-centered cubic (BCC) materials, a strong increase of the yield stress by decreasing temperature, below 200 K, was observed. At higher temperatures, the temperature dependence of the yield stress was found much weaker, being associated with the temperature dependence of the shear modulus. Efforts were made to analyze in details the post-yield behavior (strain-hardening) as a function of temperature. The post-yield behavior was modeled using the Kocks phenomenological model based on the competition of storage and annihilation of dislocations. While this model was originally developed for face-centered cubic (FCC) metals where the rate-controlling mechanism of dislocation motion is the dislocation-dislocation interaction, we used is model in the high temperature domain (T>200K) of BCC materials, to model the strain-hardening evolution of the Eurofer97 steel and the model alloy. The values obtained for the key parameters of the model, namely the dislocation mean free path and annihilation coefficient, were found consistent with the microstructural features. The parameters temperature-dependence observed was also consistent with the physics of two basic mechanisms of dislocation storage and annihilation (dynamical recovery). For the low temperature domain, the strain-hardening model was modified to account for the strong Peierls lattice friction, based on an original idea of Rauch. Transmission electron microscopy observations were done to characterize the undeformed microstructures and their evolution with strain. A clear correlation was established between the stress-dependence of the strain-hardening and the microstructures. Fracture toughness tests on the Eurofer97 steel were performed with 0.2T C(T) and 0.4T C(T) specimens in the lower transition. Five temperatures were selected at which many tests were repeated to determine the amplitude of the inherent scatter of quasi-cleavage. The temperature were chosen in such a way that the measure fracture toughness remain below 150 MPa m1/2, which correspond for the 0.4T C(T) at a M value larger than 70 at the highest temperature. Such a M value for 0.4T C(T) specimens is known to ensure enough constraint. An attempt to analyze the experimental data in the framework of the master-curve approach by following the ASTM E-1921 standard was done. It was clearly demonstrated that the assumed shape of the toughness-temperature curve as described in the ASTM E-1921 standard for the reactor pressure vessel steels is not adequate for the tempered martensitic Eurofer97 steel, which present a particularly steep transition. Our Eurofer97 fracture database was then compared to the existing one on another similar steel, the F82H. Differences in the amplitude of the scatter of both steels was found while the lower bound of the toughness-temperature curve, describing the 1% failure probability was shown to be the same. 2D finite element simulations of the compact tension specimens were performed at various temperatures using the constitutive laws determined previously. The stress field around the crack tip were calculated and used to determine a local criterion of quasi-cleavage. The criterion is defined by the attainment of a critical stress encompassing a critical area. The lower bound of the toughness-temperature curve was then successfully reconstructed by using this local criterion. Finally, the relationship between the critical area and the applied stress intensity factor for the C(T) specimen was shown to follow a power law whose coefficients are dependent on the real dimensions of the specimen. Such a relationship allows scaling the C(T) toughness data from one size to another in case of in-plane constraint loss.

Fabian Barras, Michael Ludovic Brun, Jean-François Molinari, Roozbeh Rezakhani

Numerous laboratory experiments have demonstrated the dependence of the friction coefficient on the interfacial slip rate and the contact history, a behavior generically called rate and state friction. Although numerical models have been widely used for analyzing rate and state friction, in general they consider infinite elastic domains surrounding the sliding interface and rely on boundary integral formulations. Much less work has been dedicated to modeling finite size systems to account for interactions with boundaries. This paper investigates rate and state frictional interfaces in the context of finite size systems with the finite element method in explicit dynamics. It is shown that due to the highly non-linear nature of rate and state friction and its sensitivity to numerical noise, the time integration step to achieve an accurate steady state solution is orders of magnitude smaller compared to the stable time step required in boundary integral formulations. We provide evidence that the noise, which is source of instability in the finite element solution, originates from internal discretization nodes. We then investigate the long term behavior of the sliding interface for two different friction laws: a velocity weakening law, for which the friction monotonously decreases with increasing sliding velocity, and a velocity weakening-strengthening law, for which the friction coefficient first decreases but then increases above a critical velocity. We show that for both friction laws at finite times, that is before wave reflections from the boundaries come back to the sliding interface, a temporary steady state sliding is reached, with a well-defined stress drop at the interface. This stress drop gives rise to a stress concentration and leads to an analogy between friction and fracture. However, at longer times, that is after multiple wave reflections, the stress drop is essentially zero, resulting in losing the analogy with fracture mechanics. Finally, the simulations with applied constant traction boundary conditions reveal that velocity weakening is unstable at long time scales, as it results in an acceleration of the sliding blocks. On the other hand, velocity weakening-strengthening reaches a steady state sliding configuration. (C) 2020 Elsevier Ltd. All rights reserved.

A critical aspect in the design of tubular bridges is the fatigue performance of the structural joints. Economic viability depends on it. Lower fatigue strength for joints with thicker failing members was observed in welded joints typical to the bridge application. Different approaches to this phenomenon, called size effect, have been suggested, all based on the thickness correction for welded plate joints first proposed by Gurney. For the welded tubular joints, few studies on the size effects have been carried out; most of the existing investigations refer to geometries typical to petroleum industry offshore structures. In contrast to offshore structures, bridge structures have different absolute sizes and different member proportions (in particular lower chord radius to thickness ratios, γ). Tubular joints are far more complex than welded plate joints, multiple parameters are needed to describe the geometry (α, β, γ, τ, ζ) and there are several load scenarios. For these reasons, the fatigue behaviour analysis of such joints is a complex task. Current design recommendations combine the use of the structural (hot-spot) stress at the weld toe with a correction factor to take into account the wall thickness of the failing member. This approach oversimplifies the problem and can be very penalising, in particular for joints composed of thicker tubes, as is commonly the case for bridges. Furthermore, the truss member sizes that result from static design are likely to fall out of the validity range of current recommendations. This thesis focuses on a case of commonly used tubular joints: welded steel K-joint made out of circular hollow section (CHS). The main goals of this research are to understand the fatigue behaviour of as-welded CHS K-joints and to clarify the influences/effects of the different geometric parameters on their fatigue strength. In order to carry out a thorough study on the geometric size effects in CHS K-joints for bridges, fatigue tests were conducted for large-scale specimens with crack depth measurements and an advanced 3-D crack propagation model was developed. The first chapters of this thesis provide an introduction and a brief review of the main concepts in tubular joint fatigue and size effects on fatigue behaviour. The experimental tests of two tubular trusses under fatigue loading are then outlined. Crack growth in selected truss joints is monitored using the Alternating Current Potential Drop (ACPD) system. An advanced 3-D modelling of welded K-joint with surface crack is implemented using the boundary element method (BEM). A crack propagation model, based on Linear Elastic Fracture Mechanics (LEFM), is then developed using a step-wise incremental crack growth strategy. This model allows for fatigue strength and life estimations. Furthermore, it considers the influence of all geometric parameters that define CHS K-joints in a realistic way. The validation of the crack propagation model is made by comparisons with experimental data at different levels (i.e. member and joint strains and stresses, ACPD crack growth data). A parametric study is then carried out on joint geometries typical for a bridge application (low chord radius to thickness ratio) considering three basic load cases. Examples of results are shown and analysed on a "geometry cause"/"effect over the stress intensity factor and fatigue strength" basis. Parametric study results are then analysed, highlighting the case where the joint is proportionally scaled. The geometry correction factor, Y, is introduced as a function of the relative crack depth that is common to homothetic joints. The influence of the absolute size of the joint, also known as thickness effect, is determined for the three basic load cases. Parametric results are finally explored bringing to light the effect of non-proportional scaling. It is shown that size correction factors for fatigue strength can be expressed as a function of the non-dimensional geometrical parameters β, γ and τ, chord thickness, T, and different load cases. A new fatigue design method is proposed for welded (CHS) K-joints, based on LEFM and accounting for geometric size effects.