Evaluating the safety of unreinforced concrete structures, such as concrete dams, requires an accurate prediction of cracking. Developing a suitable constitutive material model and a reliable computational procedure for analysing cracking processes in concrete has been a challenging and demanding task. Although many analytical methods based on fracture mechanics have been proposed for concrete dams in the last few decades, they have not yet become part of standard design procedures. Few of the current research findings are being implemented by practising engineers when evaluating dam safety. This research is focused on the development of a suitable crack modelling and analysis method for the prediction and study of fracturing in concrete gravity dams, and consequently, for the evaluation of dam safety against cracking. The research aims to contribute to the continuing research efforts into mastering the mechanics of cracking in concrete dams. An analytical method for the purpose of establishing a crack constitutive model and implementing the model for the fracture analysis of concrete structures, in particular massive concrete gravity dams under static loading conditions, has been developed, verified and applied in the safety evaluation of a concrete gravity dam. The constitutive material model is based on non-linear fracture mechanics and assumes a bilinear softening response. The crack model has various improved features: (1) an enhanced mode I bilinear strain-softening approach has been put forward; (2) a new formula for bilinear softening parameters has been developed and their relation with linear softening has been outlined; (3) the influence of bilinear softening parameters on the cracking response has been studied; and (4) an enhanced modification to the shear retention factor which depends on the crack normal strain is included. The material model has been incorporated into a finite element analysis using a smeared crack approach. A sub-program was specially coded for this research. The validity of the proposed cracking model and the computational procedure developed for the purpose of analyzing the tensile fracture behaviour of concrete structures has been confirmed by verification on various concrete structures, including beams, a dam model and actual gravity dams. The crack modelling technique developed was successfully used in evaluating the safety of an existing concrete gravity dam in South Africa and adequately predicted the cracking response of the dam structure under static loadings. The main conclusions drawn are as follows: <ul><li>Both mode I and mode II fracture have been modelled successfully.</li> <li>The proposed bilinear softening model remains relatively simple to implement but significantly improves on predicting the softening response of “small-scale” concrete structures.</li> <li>Both plane stress and plane strain crack analyses have been considered and can be confidently adopted in two-dimensional applications.</li> <li>The proposed method is mesh objective.</li> <li>The crack modelling method developed can correctly predict the crack propagation trajectory and the structural behaviour with regard to fracturing in concrete structures.</li> <li>If not considering shear stress concentration near the tip of a crack, constitutive crack analysis normally indicates a higher safety factor and a higher Imminent Failure Flood (IFF) than the classical methods in the analysis of concrete gravity dams for safety evaluation.</li></ul> / Thesis (PhD(Civil Engineering))--University of Pretoria, 2007. / Civil Engineering / PhD / unrestricted

Cracks form in all materials. This project was about investigating the crackpropagation in 3D-printed PLA. Both simulations and experiments has beenperformed to get the results. The simulations used a already written code to solvethe problem with FEM. It turned out to be much harder then anticipated to getthe simulations to run as they should. Both simulations and experiments wheresupposed to be executed with 3 point bending, but because of convergence problemsin the simulations, simplifications where made. When comparing the simulationsand experiments, no trustworthy conclusions could be made. There are a lot ofsources of error in this project. But with a bit more time and experience with theused software, it should be able to get some good results which one could use tosimulate other material cracking.

La fatigue induite par chargement cyclique est un mode d'endommagement majeur des métaux. Elle se caractérise par des effets environnementaux et de grandes dispersions de la durée de vie qui doivent être mieux comprises. Les matériaux analysés sont de type cfc : aluminium, cuivre, nickel et argent. Le comportement de marches naturellement créées en surface par le glissement cyclique de dislocations est examiné par simulations en dynamique moléculaire sous vide et sous environnement oxygène pour le cuivre et le nickel. Un phénomène de reconstruction est observé sur les marches en surface, qui peut induire une forte irréversibilité. Trois mécanismes de reconstruction des marches apparues en surface sont observés et décrits. L’irréversibilité de ces marches est ensuite analysé. Elles sont irréversibles pour des chargements expérimentaux, sauf arrivée de dislocations de signe opposé sur un plan de glissement directement voisin.Avec arrivée de dislocations sur des plans non voisins, l'irréversibilité s’accumule cycle par cycle et il est possible de reproduire l’apparition de fissures en surface dont la profondeur augmente graduellement.Un environnement oxygène modifie la surface (début d’oxydation) mais pas l’irréversibilité parce que l’oxygène n’a pas d'influence majeure sur les différents mécanismes liés à l’évolution du relief.Une estimation grossière de l'irréversibilité est faite pour des dislocations coin pures dans une bande de glissement persistante pour les matériaux dits ondulés. On obtient un facteur d’irréversibilité entre 0,5 et 0,75 pour le cuivre, sous vide et sous l’environnement oxygène, en accord avec des mesures récentes en microscopie à force atomique.La propagation de fissures est simulée en environnement inerte. Les fissures peuvent se propager à cause de l'irréversibilité des dislocations générées, liée à leurs interactions allant jusqu’à la création de jonctions. / Fatigue is one of the major damage mechanisms of metals. It is characterized by strong environmental effects and wide lifetime dispersions which must be better understood. Different face centred cubic metals, Al, Cu, Ni, and Ag are analyzed. The mechanical behaviour of surface steps naturally created by the glide of dislocations subjected to cyclic loading is examined using molecular dynamics simulations in vacuum and in air for Cu and Ni. An atomistic reconstruction phenomenon is observed at these surface steps which can induce strong irreversibility. Three different mechanisms of reconstruction are defined. Surface slip irreversibility under cyclic loading is analyzed. All surface steps are intrinsically irreversible under usual fatigue laboratory loading amplitude without the arrival of opposite sign dislocations on direct neighbor plane.With opposite sign dislocations on non direct neighbour planes, irreversibility cumulates cycle by cycle and a micro-notch is produced whose depth gradually increases.Oxygen environment affects the surface (first stage of oxidation) but does not lead to higher irreversibility as it has no major influence on the different mechanisms linked to surface relief evolution.A rough estimation of surface irreversibility is carried out for pure edge dislocations in persistent slip bands in so-called wavy materials. It gives an irreversibility fraction between 0.5 and 0.75 in copper in vacuum and in air, in agreement with recent atomic force microscopy measurements.Crack propagation mechanisms are simulated in inert environment. Cracks can propagate owing to the irreversibility of generated dislocations because of their mutual interactions up to the formation of dislocation junctions.

This Ph.D. thesis was written under the supervision of Assoc. prof. Pavel Hutař, Ph.D., and Assoc. prof. Luboš Náhlík, Ph.D. The thesis is focused on the effect of plasticity induced crack closure, its characteristic and ways of numerical modelling. Premature fatigue crack closure has a significant effect on the fatigue crack propagation rate and therefore on the residual lifetime of a structure. A three-dimensional numerical model allows a detailed look at the stress and strain distribution along the crack front, and particularly it allows a local description of parameters along the crack front which governs the fatigue crack propagation rate. In the first part of the thesis, the study is focused on the influence of a singular stress field at the vicinity of the free surface on the crack front curvature without crack closure being involved. In the second part, a numerical model in 2D of plasticity induced crack closure was created and verified by experimental results. In the final part, a 3D numerical model is used to describe the influence of the crack closure on its fatigue propagation rate and explains typical crack front curvature. The suggested technique allows quantitative accuracy improvement of numerical simulation of the fatigue crack propagation and therefore, more reliable estimation of the residual lifetime of the cracked structure.

This Ph.D. thesis deals with methodology for determination of residual fatigue lifetime of railway axles based on damage tolerance approach. This approach accepts an existence of potential defect, which could lead to fatigue failure of whole axle. The behavior of crack in railway axle is described by approaches of linear elastic fracture mechanics. There are plenty of factors, which more or less influence determined residual fatigue lifetime. The aim of this thesis is to quantify effects of these factors. The first part of Ph.D. thesis represents overview of studied problems relating to fatigue damage of railway axles. This part is focused on parameters, which influence fatigue crack growth in railway axle materials. The second part of thesis shows procedure for determination of residual fatigue lifetime, which was developed at the Institute of Physics of Materials of the Academy of Sciences of the Czech Republic. The main aim of this thesis was to improve current procedure for more precise estimation of residual fatigue lifetime. Significant part of this work is determination of significance of studied factors, which influencing calculated residual fatigue lifetime of railway axles (e.g. effect of threshold value, load spectrum, retardation effects, residual stress, axle geometry, material of axle etc.). The procedures described and results obtained can be also used for determination of residual fatigue lifetime of general mechanical component (not only railway axles). Therefore, results obtained in this Ph.D. thesis can be used e.g. for assessment of regular inspection intervals of cyclically loaded general mechanical parts.

Stålbroars approximativa livslängd bestäms av stålets utmattningshållfasthet, då utmattning är en av de främsta anledningarna till att livslängden begränsas. I Sverige existerar ett antal broar som närmar sig slutet av sin livslängd, samtidigt som behovet för kapacitet och kraven på broarna ökar. Flertalet av dessa broar är i behov av upprustning. Däremot är det inte möjligt ur vare sig ett ekonomiskt- eller miljöperspektiv att byta ut alla broar, och därför behöver de broar som är mest kritiska prioriteras. Vid utmattningsdimensionering av stålbroar beaktas hela spänningsvidden, oavsett om spänningarna är i drag eller tryck. En spricka propagerar endast vid dragspänningar, vilket innebär att tryck- spänningar egentligen inte bör vägas in i samband med dimensionering. Detta innebär att en del stålbroar skulle kunna ha en längre livslängd än vad den traditionella dimensioneringen ger. Spänningsintensitetsfaktorn 𝐾 används inom brottmekaniken för att förutspå spänningsintensiteten i närheten av sprickspetsen, och appliceras till linjärelastiska material. Det finita elementprogrammet Abaqus användes när brodetaljen modellerades och analyserades. Brodetaljen representerar en balk med en påsvetsad anslutningsplåt, som utsätts för trafiklasten på en bro och en temperaturlast för att simulera egenspänningar. Detaljen representerar problematiken med utmattning i stålkonstruktionsdelar. Motivet för denna studie är att inga sprickor har hittats under inspektioner av liknande detaljer, det till trots att en del stålbroar teoretiskt sett förbrukat sin livslängd. Studien genomfördes med en mer avancerad modell än vad som vanligtvis skapas för bedömning av utmattning, med syftet att modellera verkligheten mer korrekt. Resultaten visar hur egenspänningarna bidrar till dragspänningar, vilket leder till sprickpropagering i modellen. Vid spricklängden 9,5 mm övergår spänningarna från drag till tryck, och då upphör spricktillväxten. Resultaten visar även att utmattningssprickor kan växa i stålkonstruktionsdelar som i huvudsak utsätts för nominella tryckspänningar, ifall höga egenspänningar uppstår vid anslutningsplåten. / Steel bridges estimated service life is determined by the fatigue strength of the steel, since fatigue is one of the main reasons for limiting the service life. In Sweden there is a number of bridges that approach the end of their service life, while the need of increasing the capacity and demands on bridges grows. The majority of these bridges is in need of reparation. On the other hand, it is not possible either from a financial- or environmental perspective to replace all bridges, and therefore the bridges that are most critical needs priority. In the case of fatigue design calculation of steel bridges, the entire stress range is taken into account, regardless of whether the stresses are in tension or pressure. A crack propagates only at tensile stresses, which means that pressure should not really be considered in the design calculations. This means that some steel bridges could have a longer life span than the traditional design calculation gives. The stress intensity factor K is used within the fracture mechanism to predict the stress intensity near crack tip, and is applied to linear elastic materials. The finite element program Abaqus was used when the bridge detail was modeled and analyzed. The bridge detail represents a beam with a welded connection plate, which is exposed to traffic load at the bridge and a temperature load to simulate residual stresses. The detail represents the problem of fatigue in steel structural parts. The motive for this study is that no cracks have been found during inspections of similar details, despite the fact that some steel bridges theoretically have consumed their longevity. The study is conducted with a more advanced model than usually created for assessment of fatigue, with the purpose of modeling the reality more correctly. The results show how the residual stresses cause tensile stresses, which leads to crack propagation in the model. At a crack length of 9,5 mm, the stresses change from tension to compression, and then the crack growth ceases. The results also indicate that fatigue cracking can grow in steel structural parts that are mainly exposed to compressive nominal stresses, if tensile residual stresses appear at the connection plate.

This dissertation presents a comprehensive investigation into the mechanical properties of fiber-reinforced concrete (FRC), focusing on fracture and flexural toughness properties, the impact of fiber orientation and distribution, and the evaluation of flexural models for predicting the behavior of functionally graded FRC. It embarks on a critical investigation aimed at bridging a significant gap in the understanding of FRC materials' behavior, particularly in terms of fracture and flexural performance. Across five distinct manuscripts, this work employs a variety of experimental methodologies, including three-point bend tests, four-point bend tests, digital image correlation, X-ray computed tomography, and the implementation of the two parameter fracture model and then size effect fracture method to explore the effects of different casting techniques – namely, conventional casting and pump-driven extrusion – on the performance of FRC. The core hypothesis tested throughout these studies suggests that the extrusion process, by aligning fibers parallel to tensile stresses, significantly enhances the concrete's ductility, post-peak behavior, and overall fracture and flexural properties. This hypothesis was corroborated across various experiments, which demonstrated that fiber alignment via extrusion not only enhances the concrete's mechanical properties but also leads to more effective crack propagation control, increased toughness, and enhanced residual strengths. The research encompasses a series of systematic investigations into the effects of fiber alignment on the mechanical properties of FRC, revealing that the extrusion process significantly enhances fracture and flexural properties and maintains residual strength after peak stress. Utilizing both extrusion-based and conventional casting methods with varying dosages of polyvinyl alcohol fibers, the study demonstrates notable improvements in fracture properties, deflection at failure, and equivalent flexural strength ratio for extrusion-based specimens compared to their conventionally cast counterparts. Moreover, the dissertation explores the impact of casting methods and fiber orientation on fracture energy, offering a size-dependent improvement in extrusion-based methods. The strategic distribution of steel fibers, employing an innovative targeted fiber injection for creating Functionally Graded FRC (FG-FRC), is shown to significantly enhance the structural integrity and resilience of the material. The analysis of flexural models applied to FG-FRC specimens, proposing a novel functionally graded factor to improve model predictability, further advances the understanding of the predictability and reliability of these models in assessing FRC's structural behavior. This dissertation advances academic knowledge in the field of FRC casting and offers significant implications for the construction industry, demonstrating a profound understanding of the challenges and opportunities in extrusion-based FRC casting. Through its innovative approach and detailed investigations, this work contributes significantly to the advancement of the FRC casting field, paving the way for the development of more resilient and efficient construction materials. / Doctor of Philosophy / This research explores the enhancement of concrete's strength and flexibility through the incorporation of individual fibers, with a special focus on the integration and alignment of these fibers. The study examines how concrete can be made more resilient by mixing in fibers in specific ways. A variety of tests, including bending beams and employing advanced imaging techniques, were conducted to observe the effects of mixing fibers using traditional methods versus a novel extrusion-based technique that aligns the fibers in the desired direction in the concrete. The research hypothesized that this innovative alignment method would improve the concrete's ductility and enhance its ability to resist crack propagation. The findings confirmed this hypothesis, revealing that aligned fibers significantly improve concrete's bending capacity, reduce sensitivity to cracking, and retain residual strength even after cracking. Further investigation into varying methods of fiber addition, such as a targeted approach for placing fibers in strategic locations, demonstrated a marked enhancement in the material's ductility. Additionally, the study evaluated mathematical models for predicting the behavior of fiber-reinforced concrete, aiming to improve the understanding and reliability of these models for practical construction applications. In short, the research underscores that adjusting the method of fiber integration into concrete can lead to the development of structures that are both stronger and more durable. This advancement holds promising implications for the future of construction, offering pathways to create more resilient and efficient building materials.

Les phénomènes liés au comportement à la rupture de la glace sous impact sont fréquents dans le génie civil, pour les structures offshore, et les processus de dégivrage. Pour réduire les dommages causés par l'impact de la glace et optimiser la conception des structures ou des machines, l'étude sur le comportement à la rupture dynamique de la glace sous impact est nécessaire. Ces travaux de thèse portent donc sur la propagation dynamique des fissures dans la glace sous impact. Une série d'expériences d'impact est réalisée avec un dispositif de barres de Hopkinson. La température est contrôlée par une chambre de refroidissement. Le processus dynamique de la rupture de la glace est enregistré avec une caméra à grande vitesse et ensuite analysé par des méthodes d'analyse d'images. La méthode des éléments finis étendus complète cette analyse pour évaluer la ténacité dynamique. Au premier abord, le comportement dynamique de la glace sous impact est étudié avec des échantillons cylindriques afin d'établir la relation contrainte-déformation dynamique qui sera utilisée dans les simulations numériques plus tard. Nous avons observé de multi-fissuration dans les expériences sur les échantillons cylindriques mais son étude est trop difficile à mener. Pour mieux comprendre la propagation des fissures dans la glace, des échantillons rectangulaires avec une pré-fissure sont employés. En ajustant la vitesse d'impact on aboutit à la rupture des spécimens avec une fissure principale à partir de la pré-fissure. L'histoire de la propagation de fissure et de sa vitesse sont évaluées par analyse d'images basée sur les niveaux de gris et par corrélation d'images. La vitesse de propagation de la fissure principale est identifiée dans la plage de 450 à 610 m/s ce qui confirme les résultats précédents. Elle varie légèrement au cours de la propagation, dans un premier temps elle augmente et se maintient constante ensuite et diminue à la fin. Les paramètres obtenus expérimentalement, tels que la vitesse d'impact et la vitesse de propagation de fissure, sont utilisés pour la simulation avec la méthode des éléments finis étendus. La ténacité d'initiation dynamique et la ténacité dynamique en propagation de fissure sont déterminées lorsque la simulation correspond aux expériences. Les résultats indiquent que la ténacité dynamique en propagation de fissure est linéaire vis à vis de la vitesse de propagation et semble indépendante de la température dans l'intervalle -15 à -1 degrés. / The phenomena relating to the fracture behaviour of ice under impact loading are common in civil engineering, for offshore structures, and de-ice processes. To reduce the damage caused by ice impact and to optimize the design of structures or machines, the investigation on the dynamic fracture behaviour of ice under impact loading is needed. This work focuses on the dynamic crack propagation in ice under impact loading. A series of impact experiments is conducted with the Split Hopkinson Pressure Bar. The temperature is controlled by a cooling chamber. The dynamic process of the ice fracture is recorded with a high speed camera and then analysed by image methods. The extended finite element method is complementary to evaluate dynamic fracture toughness at the onset and during the propagation. The dynamic behaviour of ice under impact loading is firstly investigated with cylindrical specimen in order to obtain the dynamic stress-strain relation which will be used in later simulation. We observed multiple cracks in the experiments on the cylindrical specimens but their study is too complicated. To better understand the crack propagation in ice, a rectangular specimen with a pre-crack is employed. By controlling the impact velocity, the specimen fractures with a main crack starting from the pre-crack. The crack propagation history and velocity are evaluated by image analysis based on grey-scale and digital image correlation. The main crack propagation velocity is identified in the range of 450 to 610 m/s which confirms the previous results. It slightly varies during the propagation, first increases and keeps constant and then decreases. The experimentally obtained parameters, such as impact velocity and crack propagation velocity, are used for simulations with the extended finite element method. The dynamic crack initiation toughness and dynamic crack growth toughness are determined when the simulation fits the experiments. The results indicate that the dynamic crack growth toughness is linearly associated with crack propagation velocity and seems temperature independent in the range -15 to -1 degrees.

Tato disertační práce vychází z výzkumu v rámci francouzsko-českého programu doktorátu pod dvojím vedením na pracovišti Institut français de mécanique avancée v Clermont-Ferrand a na Ústavu fyziky materiálu AV v Brně. Úvodní výzkumný úkol na brněnském pracovišti se zabýval numerickou analýzou pole napětí v okolí čela trhliny v tenké kovové fólii. Zvláštní pozornost byla zaměřena na vliv speciálního typu singularity v průsečíku čela trhliny s volným povrchem. Těžiště disertační práce spočívá v numerickém modelování a stochastické analýze problémů šíření trhlin se složitou geometrií v dvojrozměrném prostoru. Při analýze těchto problémů se dříve zřídka používaly numerické metody, a to z důvodu vysoké náročnosti na výpočtový čas. V této disertaci je ukázáno, že aplikací moderních metod numerické mechaniky a vhodných technik v analýze spolehlivosti lze tyto problémy řešit s pomocí numerických metod i na PC. Ve spolehlivostní analýze byla využita lineární aproximační metoda FORM. Pro rychlost šíření trhlin se vycházelo z Parisova-Erdoganova vztahu. Pro parametry tohoto vztahu byl použit dvourozměrný statistický model, který postihuje vysokou citlivost na korelaci obou parametrů. Mechanická odezva byla počítána rozšířenou metodou konečných prvků (XFEM), která eliminuje výpočetní náročnost a numerický šum související se změnou sítě v klasické metodě konečných prvků. Prostřednictvím přímé diferenciace bylo odvozeno několik vztahů pro derivace funkce odezvy, čímž se dosáhlo lepší numerické stability a konvergence spolehlivostní analýzy a výrazného zkrácení doby výpočtu. Problém zatížení s proměnou amplitudou byl řešen aplikací transformace zatížení metodou PREFFAS. Využití distribuce výpočtů v síti PC umožnilo další zrychlení analýzy.

La fatigue de contact est un des modes de défaillance prédominants des composants tels que les engrenages ou les roulements. Les mécanismes d’initiation de fissures associés à ce mode de défaillance sont fortement liés à la microstructure du matériau. Cependant, la plupart des modèles utilisés pour prédire la durée de vie se situent à l’échelle macroscopique. Un modèle basé sur une représentation de type Voronoi des grains (échelle mésoscopique) est développé afin d’analyser les mécanismes d’initiation. Le concept d’endommagement est appliqué aux joints de grain modélisés par la méthode des zones cohésives. L’objectif de ce modèle est (i) de contribuer à une meilleure compréhension de l’influence de paramètres tels que ceux liés aux conditions de contact (rugosité, lubrification) ou aux matériaux (présence d’inclusions ou gradients de propriétés et contraintes résiduelles générés par les traitements de surface…) sur les mécanismes d’initiation et (ii) de fournir une estimation de la durée de vie jusqu’à cette initiation. Un premier modèle 2D isotrope a permis de mettre en place l’approche proposée et d’analyser le comportement numérique des éléments cohésifs : influence de la valeur des raideurs cohésives et apparition de singularités aux jonctions triples. Cette singularité semble inévitable, mais l’approche consistant à considérer le joint de grain comme une unique entité, et donc à utiliser des valeurs moyennes le long du joint de grain permet de s’affranchir de cette singularité. La représentativité du modèle a ensuite été améliorée par la modélisation de l’anisotropie cristalline. Un modèle de type élasticité cubique a été utilisé pour modéliser le comportement des grains. Enfin, une analyse approfondie de l’application du concept d’endommagement aux joints de grains a permis de proposer une nouvelle formulation entraînant une influence plus réaliste de cet endommagement sur le cisaillement intergranulaire et conduisant à une durée de vie estimée (apparition des premières micro-fissures) d’un ordre de grandeur comparable à celles données par l’expérience. / Contact fatigue is the predominant mode of failure of components subjected to a repeated contact pressure, like rolling element bearings or gears. This phenomenon is known as rolling contact fatigue (RCF). A large number of models have been developed to predict RCF, but there is today no complete predictive life model, and understanding RCF failure mechanism remains a significant challenge. RCF failure mechanisms are known to be very sensitive to a large number of parameters linked to contact conditions (roughness, lubrication) or materials (inclusions, gradients properties, residual stresses…). To improve knowledge about the influence of these parameters on failure mechanisms and life, a numerical model is developed to simulate the progressive damage of a component subject to rolling contact fatigue. Mechanisms associated with the initiation stage of failure process are located at a scale lower than the macroscopic scale. The proposed approach is to develop a grain level model (mesoscopic scale) in order to focus on initiation mechanisms. A Voronoi tessellation is used to represent the material microstructure. The progressive deterioration is simulated by applying the concept of damage mechanics at grain boundaries represented by cohesive elements. This approach has been first applied to a 2D isotropic model. The numerical behaviour of cohesive elements has been investigated: the influence of cohesive stiffness has been analysed and singularities at the triple junctions has been highlighted. The representativeness of the original model was improved by modelling crystal anisotropy. A cubic elasticity model was used to represent the behaviour of grains. Finally, a thorough analysis of the application of the damage concept at grain boundaries highlighted that the initial formulation results in a very low influence of the damage on the intergranular shear stress. A new formulation leading to a direct influence of the damage on the intergranular shear stress has been proposed. This new formulation has resulted in (i) a change in the distribution of micro-cracks, with coalescence between the different micro-cracks, and (ii) a large increase in the RCF life estimated by the model. The order of magnitude of the number of cycles corresponding to the first micro-cracks is comparable to that given by experiments.