Vincent De Greef

Mr Alexandre Tanguy

Mr Simon Hallias

Mr Jean-Christophe

also, Dr. Eva Héripré, who helped me during my thesis with various

Résumé

Les connecteurs clips utilisés pour assembler les joints de riser pour le forage pétrolier offshore subissent un chargement cyclique dû aux vagues. 90% de la durée de service est passée en mode connecté (riser fixé en haut et en bas) qui implique une amplitude modérée, mais une contrainte moyenne élevée (R = 0, 7), alors que 10% est passé en mode déconnecté (riser suspendu à la plateforme) caractérisé par une plus grande amplitude, mais une faible contrainte moyenne (R = 0, 2). Des calculs numériques réalisés avec une loi de comportement elasto-plastique identifiée sur l'acier qui constitue les connecteurs montrent qu'un chargement cyclique de tension biaxiale en phase prévaut dans la zone critique de la structure. Les effets de contrainte moyenne et de biaxialité doivent donc tous deux être pris en compte pour un design approprié. Les critères de fatigue multiaxiale proposés dans la littérature sont basés sur des données de traction et torsion combinées et ne discriminent pas bien l'influence de la tension biaxiale de celle d'une contrainte moyenne. L'objectif de cette étude est donc de caractériser séparément ces deux effets.

Pour étudier les effets de la contrainte moyenne, des essais de fatigue uniaxiale ont été menés sur un acier 2,5% Mo-1% Cr avec différents rapports R. Les durées de la vie, ainsi que la pente de la courbe S-N diminuent avec l'augmentation de R, et la limite d'endurance suit la parabole de Gerber. À faible contrainte moyenne et amplitude de contrainte relativement élevée, les fissures s'amorcent en surface, tandis que pour des rapports R élevés et des amplitudes plus faibles, les fissures s'amorcent à partir de défauts internes ou coupant la surface. Cette transition est analysée à partir de calculs élastoplastiques des champs de contrainte et déformation autour des défauts. Les fissures internes se propagent sous un faible ∆K qui s'avère indépendant de R, ce qui est attribué à la quasi absence d'effets de fermeture.

Pour étudier l'effet d'un taux de biaxialité positif, des essais cycliques de tension et pression interne combinées avec diverses proportions de chaque chargement ont été effectués sur des échantillons tubulaires, à un ratio R fixe (0,25). Un taux de biaxialité modéré (B = 0, 25 et 0,5) a un effet bénéfique sur la vie de fatigue, attribué principalement à un retard de l'initiation des fissures, alors que la tension équibiaxiale a un effet légèrement v nuisible, attribué à un "pseudo effet de taille" (probabilité plus grande qu'une microfissure se propage le long de deux plans principaux équivalents au lieu d'un seul).

De facettes intergranulaires révélatrices de fragilisation par l'hydrogène consécutive à une ségrégation d'impuretés lors du traitement thermique de l'acier ont été observées sur les surfaces de rupture. Les évolutions de la fraction de rupture intergranulaire avec ∆K et la charge biaxialité suggèrent une possible réduction de la vitesse de fissuration à B = 0 et 0.5, mais l'effet néfaste de la tension équibiaxiale n'a pu être attribuée à une accélération de la propagation.

Plusieurs des critères de fatigue les plus populaires échouent à décrire toutes les données de cette étude. En particulier, les critères d'endurance qui comprennent un terme linéaire de contrainte moyenne ou de tension hydrostatique ne parviennent pas à prédire les variations non linéaires de la limite d'endurance de ce matériau avec le rapport R et le rapport de biaxialité. Aussi, un nouveau critère de fatigue a été proposé sur la base de la parabole de Gerber. Il décrit bien l'évolution de la limite d'endurance sous les effets combinés d'une contrainte moyenne positive et d'un taux de biaxialité positif.

Pour étudier l'influence d'un milieu corrosif (solution à 3,5% NaCl) sur les durées de vie en fatigue, des essais de tension et pression interne combinées en diverses proportions ont été conduits sous deux types de conditions: 1) corrosion libre et 2) protection cathodique.

En corrosion libre, l'eau salée réduit fortement les durées de vie et supprime la limite d'endurance. Cela est dû à la formation de piqûres de corrosion, qui favorisent l'amorçage précoce et multiple de fissures. La tension équibiaxiale n'accentue pas l'effet nocif de l'eau salée. Les durées de vie en fatigue uniaxiale et biaxiale sont en effet presque les mêmes, malgré des mécanismes de fissuration différents: décohésion fragile transgranulaire dans le cas uniaxial mais principalement intergranulaire en tension biaxiale.

La protection cathodique annule l'effet néfaste de l'eau salée pour tous les taux de biaxialité, en dépit d'un net accroissement de la fragilisation par l'hydrogène des joints de grains. Les surfaces de rupture deviennent presque complètement intergranulaires indépendamment du taux de biaxialité, tandis qu'à l'air, le taux de rupture intergranulaire ne dépassait pas 45%.

vi

Abstract

The clip connectors used to join the riser tubes for offshore oil drilling undergo cyclic loading due to sea waves. 90% of the service life is spent in the "connected mode" with a high mean stress (R = 0.7) and 10% in the "disconnected mode" with a lower mean stress (R = 0.2). Finite element computations revealed in-phase biaxial tension in the critical areas of the clip connector along with high mean stresses. Thus, both the mean stress effect and the biaxiality effect are needed to be addressed for a proper design of these structures. However, most of the multiaxial fatigue criteria are based on tension-torsion fatigue data and do not discriminate the influence of biaxial tension from that of a mean stress. This study investigates separately these two effects.

For investigating the mean stress effect, uniaxial fatigue tests were run on a 2.5%Cr-1%Mo steel with various R ratios (σ min /σ max ). The fatigue lives, as well as the slope of the S-N curves were found to decrease with increasing R, and the endurance limit to follow Gerber's parabola. At low R ratios and thus relatively high stress ranges, fatigue cracks were found to initiate from the surface, while for high R ratios, and thus low stress ranges, cracks initiated from internal or surface-cutting defects. This transition was analyzed based on elastic-plastic computations of stress and strain fields around the defects. The threshold for internal fatigue crack growth from defects was found to be quite low and independent from the R ratio. This was attributed to a nearly closure-free propagation.

To investigate the effect of positive stress biaxiality, combined cyclic tension and internal pressure tests with various proportions of each loading were run on tubular specimens, at a fixed R ratio (0.25). Moderate stress biaxialities (B = 0.25 and 0.5) had a beneficial effect on fatigue lives, attributed mainly to a retardation of crack initiation, while equibiaxial tension had a slightly detrimental effect, attributed to a "pseudo size effect": higher probability for an incipient crack to grow along two possible planes, compared to a single one.

Intergranular facets associated with temper and hydrogen embrittlement were observed on the fracture surfaces. The evolutions of their surface fraction with ∆K and load vii biaxiality suggested a possible reduction in crack growth rate at moderate biaxialities, but the detrimental effect of equibiaxial tension could not be explained in terms of crack growth rate.

Several popular fatigue criteria failed to describe all fatigue data. Endurance criteria that include a linear mean stress term or contain a hydrostatic tension term fail to predict the variations of the endurance limit of this material with the R ratio and biaxiality ratio.

Thus, a new fatigue criterion based on Gerber's parabola has been proposed. It captures the evolution of the endurance limit under the combined effects of a positive mean stress and positive biaxiality.

To investigate the influence of corrosive medium (3.5% NaCl) on fatigue lives combined cyclic tension and internal pressure tests with various proportions of each loading were run under two types of test conditions: 1) free corrosion and 2) cathodic protection.

In free corrosion, salt water strongly reduced the fatigue lives and suppressed the endurance limit. This was due to the formation of corrosion pits that favor early, multiple crack initiations. The detrimental effect of salt water was not enhanced by equibiaxial tension, which also did not modify the aspect ratio and density of corrosion pits. Fatigue lives in uniaxial and biaxial tension were nearly the same, although the crack growth mechanism was different: transgranular brittle decohesion in uniaxial loading and mostly intergranular in biaxial tension.

Cathodic protection cancelled the detrimental effect of salt water for all biaxialities, in spite of a clear enhancement of H-induced embrittlement of the grain boundaries. The fracture surfaces were nearly fully intergranular, irrespective of load biaxialities, while in air the proportion of intergranular fracture never exceeded 45%.

viii

Introduction Industrial context

World energy consumption has increased by two fold since the energy crises of the 1970s.

More than 80% of this energy is obtained from fossil fuels. This consumption is still increasing day by day. Other than looking for other sources of energy, companies are also trying to increase the efficiency of fuel extraction from known sources. Of all the known sources, oil is of particular concern because of its limited reserves available throughout the world with most of the countries being importer of oil. The oil reserves are exhausting at a faster rate with limited number of substitutes available. Global oil consumption has nearly tripled since 1965, with fastest consumption in developing countries. The present study concerns the oil extraction process. In order to increase the production of oil from oil reserves, efforts have been made for deeper offshore drilling through riser tubes and that too in harsher environments [START_REF] Guesnon | Ultra Deep Water Drilling Riser Design and Relative Technology[END_REF][START_REF] Désamais | Use of High Strength Steel Wires for Flexible Pipe in Low Sour Service Conditions: Impact on Deep Water Applications[END_REF]. The design of offshore structures has to integrate a reliable assessment of their durability under complex cyclic loadings. In order to facilitate deep water drilling the riser tubes have to be connected by means of an assembly called "Clip-Riser TM assembly". A Clip-Riser TM joint is typically 75 feet (22.86 m) long and approximately 133 joints are required for 3028 m (10000 ft) drilling. A riser assembly in drilling process is subjected to quite a complex loading. The forces experienced by a riser tube in working condition are depicted in figure 3. When immersed in sea water, the assembly not only experiences external pressure from sea water but also the internal pressure of fluid flowing through it as well as the weight of the riser assembly, buoyancy force, and sea waves and currents, which induce cyclic bending with a random amplitude. Hence fatigue comes into picture.

A simulated spectrum of sea wave amplitude generated from recorded data over a span of time at a harbor and the corresponding tension load experienced by the riser assembly is shown in figure 4. The high mean load comes from the fact that the riser tubes are in pre-tension while the variation in load is induced by sea waves.

Hence, there is need to characterize and model the fatigue behavior of the steel which constitutes the connectors in representative stress states, which will be shown to correspond to in-phase biaxial tension with high mean stresses. Even though the clip connectors are provided with a protective coating, this coating can get damaged while operating, allowing a direct contact between steel and sea water. The literature shows that a significant depreciation in fatigue life can occur due to a corrosive environment. This depreciation thus needs to be assessed under representative stress states as well.

Scientific questions

The majority of experimental studies devoted to multiaxial fatigue focused on tension/compression and torsion tests. Data concerning biaxial cyclic tension are limited and sometimes contradictory, as it will be shown in the literature review presented in part A of chapter 1. Many multiaxial fatigue criteria are based on tension-torsion and some recent studies [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF] have shown that such criteria do not give good results under biaxial-tension loadings.

The primary aim of the study was thus to formulate and identify an accurate and reliable fatigue criterion, for the type of stress states currently encountered in offshore structures, to be implemented in a finite element code thereafter.

Second, the life debit in corrosion-fatigue is generally evaluated from uniaxial tests, which might underestimate the detrimental effect under biaxial loading. The high hydrostatic xiii stress encountered in offshore structures might enhance any environment-induced reduction of their resistance to fatigue that involves hydrogen ingress into the metal, as discussed in the literature review presented in part B of chapter 1.

Structure of the manuscript

While chapter 1 has been devoted to literature surveys on multiaxial fatigue in air and corrosion fatigue, respectively, chapter 2 focuses on the steel constituting the clip connector, which was characterized in terms of micro-structure and the constitutive equations.

FE simulations of service load conditions on clip connector assembly were done and the stress states in critical areas were analyzed, in order to determine relevant experimental conditions for the fatigue tests, as described in chapter 3.

The mean stress effect under uniaxial loading was then investigated and modeled. The changes in damage mechanisms associated with a change in mean stress were analyzed and has been described in chapter 4.

Then, biaxial fatigue tests were run under repeated tension and internal fluid pressure on tubular specimens, with various proportions of each loading so as to evaluate the influence of load biaxiality on the fatigue life and damage mechanism at fixed mean stress. This part of the work has been discussed in chapter 5.

To re-evaluate the potential life debit due to contact with sea water under representative loading states, uniaxial and biaxial fatigue tests were performed in salt water, with or without cathodic protection, and the results were analyzed in relation with the observed damage mechanisms. This part of the work has been reported in chapter 6.

A synthesis of all results has finally been presented and suggestions for further work has been made.

xiv

Chapter 1

Literature review

Part A: Fatigue in air

While fatigue studies for uniaxial cases (pure bending or pure tension) have been of much focus before 1975, multiaxial fatigue has been key area of interest for researchers for more than 30 years since many structures experiencing fatigue in real operations undergo multiaxial loadings (bending-torsion, tension-torsion or tension-tension).

Damage mechanisms

In general, in high-cycle fatigue, the fatigue damage can be divided into four stages:

• Crack initiation stage (number of cycles to form microcracks, generally on the surface).

• Microcracks growth in strong interaction with microstructural barriers, with an alternation of fast transgranular propagation and arrest periods.

• Macrocracks propagation stage (smooth propagation of engineering size cracks, following the Paris law)

• Unstable propagation leading to fracture

There is no general agreement on the definition of crack initiation. The number of cycles for crack initiation, N i , can be assumed to correspond to the creation of a crack of 1 Figure 1.1: Total fatigue life, N f , as a function of number of cycles for crack initiation, N i , [START_REF] Jacquelin | Crack Initiation Under Low-Cycle Multiaxial Fatigue in Type 316L Stainless Steel[END_REF] length a 0 of about 0.1 mm, comparable to the mean grain size in many metallic materials [START_REF] Forsyth | Fatigue behaviour and its dependence on microstructure[END_REF]. The ratio of number of cycles representing initiation stage (N i ) to total life (N f ) varies depending upon the loading amplitude. It represents up to 90% of the fatigue life in high cycle fatigue, figure 1.1.

Crack initiation

Surface crack initiation

In general, fatigue cracks get initiated from surface of specimens, as explained by [START_REF] Forsyth | Fatigue behaviour and its dependence on microstructure[END_REF], mainly because of:

• Stress concentrations at extrusions and intrusions generated by dislocations' gliding in persistent slip bands (PSBs).

• Environment effects, even just in moist air.

The application of cyclic strains (or stress) close to the endurance limit, produces persistent slip bands (PSBs), which produce surface roughness, leading to initiation of microcracks, figure 1.2a. In particular intrusion sites are likely to initiate cracks because of a stress concentration effect. Initiation is followed by a shear-driven transgranular propagation stage along a slip plane, described as "Stage I" of crack propagation, figure 1.2b.

Since stage I cracks change their orientation at each grain boundary, it is quite rough. Friction between the crack face can significantly hinder their development. An opening stress which reduces friction thus triggers their propagation, while a compressive normal stress tends to slow it down. During stage I, microcracks growth shows repeated accelerations and decelerations due to interactions with grain boundaries [START_REF] Zhang | The Effect of Grain Boundaries on the Development of Plastic Deformation Ahead of Small Fatigue Cracks[END_REF] that can be modeled if crack tip plasticity and microstructure are taken into account [START_REF] Bertolino | Modelling of the scatter in short fatigue cracks growth kinetics in relation with the polycrystalline microstructure[END_REF]. Some microcracks do not propagate further than the first grain boundary because of a large misorientation between neighbouring grains while some can cross microstructural barriers without any deceleration. For "microstructurally-short cracks", continuum fracture mechanics is not the appropriate tool while for "mechanicallyshort cracks", which are longer than about 5 grains but smaller than 1 or 2 mm and less dependent on microstructure, Linear Elastic Fracture Mechanics can be used, although with a different correction for closure effects than for long cracks, as explained below. In push-pull or bending, stage I is limited to a few grains in depth and is followed by "Stage II" crack growth, during which the crack propagates in mode I perpendicular to the first principal stress.

Internal crack initiation

Cracks can also initiate from internal broken or debonded inclusions. This mechanism tends to become predominant in the very high cycle fatigue or giga-cycle fatigue domain and leads to dual S-N curves (figure 1.3(a)) with a plateau between 5 × 10 5 and 10 7 cycles Literature review 3 which has long been considered as an endurance limit. However, the recent development of ultra-high frequency testing machines has shown that internal cracks initiated from defects can lead to fatigue fracture at even lower stress ranges beyond 10 7 to 10 9 cycles. The size, shape and location of the inclusions can affect the fatigue strength: the bigger the inclusions and sharper their edges, the larger is the reduction in fatigue strength, with existence of a threshold size below which no reduction on fatigue strength occurs, figure 1.3(b). For a given size and fatigue life, surface inclusions are more detrimental than internal inclusions [START_REF] Duckworth | The effects of Externally introduced Alumina Particles on the fatigue life of En24 Steel, Clean steel[END_REF]. Tensile or compressive residual stresses are often induced around the inclusions during the solidification process because of a difference in thermal expansion coefficient of matrix (α m ) and inclusions (α i ). If α m > α i , which is the case for Al 2 O 3 inclusions, tensile stresses are induced on the plane perpendicular to loading direction while if α m < α i , which is the case for MnS inclusions, compressive stresses appear. The elastic properties of the inclusions and their mismatch with those of the matrix also affect the stress distribution in the inclusion and around it and hence change the stress concentration factor and the crack initiation site [START_REF] Donnel | Stress concentration factors[END_REF]. Murakami proposed to treat inclusions as mechanically equivalent to small defects and introduced the √ area parameter in an empirical equation in which the endurance limit is computed as a function of the √ area of the projection of the largest inclusion in the stressed volume on the first principal plane [START_REF] Murakami | Metal Fatigue: Effects of small defects and nonmetallic inclusions[END_REF].

Internal crack initiation often leads to the formation of a "fish eye" (figure 1.3(c)) pattern with the defect as the ward and the bright, non oxidized crack grown "under vacuum"

until it reaches the free surface, as the white of the eye. Such patterns were first reported by [START_REF] Naito | Fatigue behavior of carburized steel with internal oxides and nonmartensitic microstructure near the surface[END_REF] in giga-cycle regime. The crack initiated in fish eye quickly adopts a circular shape (iso-SIF contour). Surface residual stresses play an important role in the competition between surface crack initiation and internal crack initiation [START_REF] Li | Effect of surface finishing and loading condition on competing failure mode of clean spring steel in very high cycle fatigue regime[END_REF][START_REF] Li | Surface vs. interior failure behaviors in a structural steel under gigacycle fatigue: Failure analysis and life prediction[END_REF][START_REF] Schijve | Internal fatigue cracks are growing in vacuum[END_REF]. Compressive surface residual stresses tend to favor internal initiation. Fish eye is often accompanied by a special area around the initiation site called Optically Dark Area, ODA [START_REF] Murakami | On the mechanism of fatigue failure in the super long life regime (N > 10 7 ) cycles. Part II: a fractographic investigation[END_REF] or Fine Granular Area, FGA [START_REF] Shiozawa | Subsurface crack initiation and REFERENCES 151 propagation mechanism in high-strength steel in a very high cycle fatigue regime[END_REF][START_REF] Rosenfeld | Crack Initiation Mechanism of Bearing Steel in High Cycle Fatigue[END_REF] or Granular Bright Facet, GBF [START_REF] Sakai | Characteristic SN property of high carbon Chromium bearing steel under axial loading in long life fatigue[END_REF] in very long fatigue life regime [START_REF] Li | Surface vs. interior failure behaviors in a structural steel under gigacycle fatigue: Failure analysis and life prediction[END_REF]. Several mechanisms for crack initiation from these areas have been proposed like hydrogen assisted cracking due to hydrogen trapped by inclusions and released into the metal, reducing the threshold ∆K for fatigue crack growth (Murakami et al., 2000a), segregation of fine carbides and subsequent formation and coalescence of microcracks [START_REF] Shiozawa | Very high cycle fatigue properties of high speed steel developed by basis on internal failure mechanism[END_REF], local refinement of the micro-structure & de-bonding from matrix [START_REF] Rosenfeld | Crack Initiation Mechanism of Bearing Steel in High Cycle Fatigue[END_REF]. The reduction in grain size would reduce roughness-induced closure and thus reduces the threshold ∆K [START_REF] Petit | An overview on the influence of the atmosphere environment on ultra-high-cycle fatigue and ultra-slow fatigue crack propagation[END_REF]. The most convincing explanation for the formation of the ODA was provided by [START_REF] Nakamura | The effect of vacuum-like environment inside sub-surface fatigue crack on the formation of ODA fracture surface in high strength steel[END_REF]. They showed that the repeated contact of bare metallic crack faces in vacuum is responsible for very high adhesion and contact stresses, leading to repeated welding and fracture of the asperities, and to a modification in their aspect after crack propagation. The local refinement of the micro-structure within the ODA would thus be a side effect and would not play any important role in crack growth.

Internal initiation can also occur from an inhomogeneous composition of the material [START_REF] Noyan | High-cycle rotating bending fatigue property in very long-life regime of high strength steels[END_REF][START_REF] Furuya | Specimen size effects on gigacycle fatigue properties of high-strength steel under ultrasonic fatigue testing[END_REF][START_REF] Li | Effect of surface finishing and loading condition on competing failure mode of clean spring steel in very high cycle fatigue regime[END_REF]. The mechanism of such initiation is different from that arising from inclusion or pores. Few studies have been carried out yet to understand the mechanism of such initiations.

The initial crack growth environment after internal initiation is similar to vacuum [START_REF] Nakamura | The effect of vacuum-like environment inside sub-surface fatigue crack on the formation of ODA fracture surface in high strength steel[END_REF] and plays an important role in crack kinetics, usually much slower Literature review 5 than in air [START_REF] Mcevily | Fatigue Crack Tip Deformation Processes as Influenced by the Environment[END_REF][START_REF] Petit | An overview on the influence of the atmosphere environment on ultra-high-cycle fatigue and ultra-slow fatigue crack propagation[END_REF].

Fatigue striations are generally absent under vacuum, due to diffuse plasticity, promoting crack tip blunting, while in air, moisture adsorption and hydrogen release would favor localized crack tip plasticity, the formation of striations and faster crack growth than in vacuum as explained in figure 1. 4 (McEvily and Gonzalez Velazquez, 1992). This probably partly explains why the numerous attempts to evaluate the number of cycles for internal crack growth in gigacycle fatigue by integration of mode I crack growth kinetic data obtained from lab experiments run in air [START_REF] Ogawa | A fracture mechanics approach to interior fatigue crack growth in the very high cycle regime[END_REF] provide a negligible fraction of the fatigue life. It is thus currently admitted that VHCF (Very High Cycle Fatigue) is controlled by internal crack nucleation, which would constitute the major part of life. This conclusion however seems questionable, since it is not based on crack growth kinetics measured in vacuum.

Crack growth

Stage II crack growth is controlled by the range of mode I stress intensity factor at the crack tip. This stage of crack propagation is often characterized by the presence of striations formed by repeated plastic blunting and resharpening of the crack tip, figure 1.5.

Under tension, crack faces are pulled apart generating shear stresses which induce dislocation emission and glide away from the tip, along these planes, figure 1.5(b). This results in blunting of the crack tip into a semi circle ahead of previous crack tip, figure 1.5(c).

At unloading, some but not all the dislocations glide back towards the former crack tip (now located somewhat behind the current one), figure 1.5(d). Due to the difference in (1.1) [START_REF] Elber | Fatigue crack closure under cyclic tension[END_REF] introduced the concept of crack closure to explain the R ratio-dependence of crack growth kinetics. Fatigue crack closure phenomenon can be attributed to three basic mechanisms: plasticity-induced closure [START_REF] Elber | Damage Tolerance in Aircraft Structures[END_REF], roughness-induced closure [START_REF] Suresh | A geometric model for fatigue crack closure induced by fracture surface roughness[END_REF][START_REF] Lin | Effect of grain size and cold work on the near threshold fatigue crack propagation rate and crack closure in iron[END_REF] and oxide-induced closure [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF].

Plasticity induced closure arises due to the existence of a "plastic wake" behind the crack tip with some residual tensile strain restrained by the surrounded elastic material, most prevalent under plane stress conditions. But some studies have shown significant closure effects in plane strain conditions near the crack growth threshold, ∆K th , as shown in figure 1.6(a) [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF]. To explain this, the other two mechanisms were evoked.

Oxide induced closure effects arise from the oxide deposits formed on crack tip surfaces due to active environmental conditions (eg moisture etc.) or repeated friction between the crack faces, with a thickness comparable to the crack tip opening displacement.

The oxide-induced closure induced by repeated contact of crack faces gets reduced by a high mean stress or the presence of non-oxidizing environment (H 2 ), figure 1.6(b).

Literature review 7 [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF].

Roughness induced closure effects arise from the asperities of fracture surfaces comparable to crack opening displacements, and to significant mode II displacements [START_REF] Suresh | A geometric model for fatigue crack closure induced by fracture surface roughness[END_REF]. To take into account closure effects, ∆K is replaced by ∆K eff = K max -K op , where K max is maximum stress intensity factor and K op is the stress intensity factor corresponding to the nominal stress required to open the crack. When the crack growth rates measured with different R ratios are replotted versus ∆K eff , all the data fall onto a single and intrinsic curve, figure 1.7.

"Mechanically short cracks"have been extensively studied since the work published by [START_REF] Pearson | Initiation of fatigue cracks in commercial aluminum alloy and the subsequent propagation of very short cracks[END_REF] in which they are reported to behave differently as compared to long cracks in respect to following two aspects:

• These cracks grow faster than long cracks at a given ∆K (B to D in figure 1.8).

• They grow at ∆K lower than threshold value for long cracks.

This behavior can be explained by lower plasticity-induced crack closure. [START_REF] Journet | A crack closure study to predict the threshold behaviour of small cracks[END_REF] proposed an exponential relationship between K op and the crack length, a:

K op = K 0 (1 -e -ka )) (1.2)
The use of ∆K eff eliminates the growth rate difference between long and short cracks [START_REF] Ravichandran | Effects of crack aspect ratio on the behavior of small surface cracks in fatigue: Part II. Experiments on a titanium (Ti-8al) alloy[END_REF][START_REF] Verreman | Mechanically short crack growth from notches in a mild steel[END_REF]. [START_REF] Lankford | Influence of microstructure on the growth of small fatigue cracks[END_REF] Literature review 9

Effect of biaxial tension

Load biaxiality can affect the proportions of stage I or stage II in total fatigue life. In torsion, the transition from stage I to stage II is postponed and may not even occur:

at large shear stress or strain ranges, the cracks continue to grow in mode II and III until fracture [START_REF] Socie | Multiaxial fatigue damage models[END_REF]. By contrast, biaxial tension has been reported to produce predominantly stage II cracking [START_REF] Havard | Low cycle biaxial fatigue of AISI 1018 steel[END_REF].

In Spheroidal Graphite (SG) cast iron, cracks often initiate from inclusions and this process is affected by load biaxiality because stress concentration (K t ) varies with type of loading, for e.g. K t for 2D circular center hole is 3 under uniaxial loading, 4 under torsion loading and only 2 under equi-biaxial loading [START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF] and for 3D spherical internal holes, the maximum K t varies from 2.04 under uniaxial tension to 1.36 under equibiaxial tension [START_REF] Goodier | Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws[END_REF].

T-stress, higher-order term in the Williams series, affects the distribution of plastic strain around the crack [START_REF] Brown | Mode I fatigue crack growth under biaxial stress at room and elevated temperature[END_REF][START_REF] Mcclung | Closure and Growth of mode I cracks in biaxial fatigue[END_REF]. There is very little difference between the plastic zone size in uniaxial tension (T = 0) and biaxial tension (T > 0), but under shear loading (T < 0), it is larger (figure 1.9a). Negative T-stress has been found

to accelerate crack growth [START_REF] Brown | Mode I fatigue crack growth under biaxial stress at room and elevated temperature[END_REF]. For crack growth under mode I, [START_REF] Cotterell | Slightly curved or kinked cracks[END_REF] straight crack path has been shown to be stable under T < 0 and unstable under T > 0, figure 1.9b [START_REF] Cotterell | Slightly curved or kinked cracks[END_REF].

Fatigue cracks in SG cast iron under biaxial tension have been reported to be more tortuous (with more branching) as compared to that in uniaxial and torsional loading [START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF]. The same effect was reported by [START_REF] Kane | Surface crack and cracks networks in biaxial fatigue[END_REF] for 304L stainless steel (at R = 0), and also the beneficial effect of biaxial tension on fatigue crack growth rate. As a consequence, asperity-induced closure should be more pronounced under biaxial tension.

Fatigue crack growth rates (FCGR) in AL alloy 2024-T3 (range 10 -5 to 10 -2 mm/cycle, R = 0.1) were found to be noticeably sensitive to load biaxiality (accelerating with increase in biaxiality) under both constant amplitude as well as spectrum loading [START_REF] Sunder | Fatigue crack growth under flight spectrum loading with superposed biaxial loading due to fuselage cabin pressure[END_REF]. [START_REF] Hoshide | Stress-ratio effect of fatigue crack propagation in biaxial stress field[END_REF] found that equibiaxial loading does not always give minimum FCGR rather it is also dependent on mean stress (R = 0 or -1). [START_REF] Yuuki | The effect of biaxial stress state and changes of state on fatigue crack growth behaviour[END_REF] reported no biaxiality effect on crack growth rate at low stress levels but noticeable effect at high stress levels. The existence of a threshold stress level (roughly 40% of yield strength) below which no biaxiality effect (at R = -1) exists has been reported by [START_REF] Mcclung | Closure and Growth of mode I cracks in biaxial fatigue[END_REF] in growth range of 10 -6 to 10 -3 mm/cycle. Marquis and [START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF] observed almost no effect of equibiaxial loading on fatigue life as compared to uniaxial one. [START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF] found that the FCGR in aluminum alloy 7075-T6 is same in uniaxial and equibiaxial loading, although the damage mechanism was different. However, [START_REF] Lee | Fatigue behavior of aluminum alloys under biaxial loading[END_REF] reported the beneficial effect of in-phase equibiaxial tension on the same aluminum alloy. [START_REF] Kamaya | Evaluation of equi-biaxial fatigue of stainless steel by the pressurized disc fatigue test[END_REF] reported the beneficial effect of biaxial stress on the fatigue lives of 316 stainless steel.

In summary, many literatures exist concluding that the detrimental effect [START_REF] Bonnand | Investigation of multiaxial fatigue in the context of turboengine disc applications[END_REF][START_REF] Joshi | Fatigue-crack Propagation in a Biaxial-stress Field[END_REF][START_REF] Vansovich | The Effect of Stress State Characteristics on the Surface Fatigue Cracks Growth Rate Taking into Account Plastic Deformations[END_REF][START_REF] Clàudio | Biaxial high-cycle fatigue life assessment of ductile aluminium cruciform specimens[END_REF], beneficial effect [START_REF] Bellett | A Biaxial fatigue specimen for uniaxial loading[END_REF][START_REF] Doquet | Dwell fatigue of a titanium alloy at room temperature under uniaxial or biaxial tension[END_REF][START_REF] Kane | Surface crack and cracks networks in biaxial fatigue[END_REF][START_REF] Lee | Fatigue behavior of aluminum alloys under biaxial loading[END_REF] or neutral effect [START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF][START_REF] Leevers | Crack growth in plastic panels under biaxial stress[END_REF][START_REF] Kitagawa | Fracture mechanics study of fatigue crack growth in high-strength steel sheet under in-plane biaxial loading[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF] on fatigue life and fatigue crack growth in air (depending upon R ratio and loading amplitude) under biaxial tension, leading to no general perception. There is a consensus to attribute the effects of load biaxiality on fatigue crack growth to their influence on plasticity or roughness-induced closure effects.

As a consequence, parameters which take closure effects into account that is either ∆K eff or Crack opening displacement (COD) range, can be used to account for biaxiality effects.

Literature review 11

1.2 Fatigue criteria

Low-cycle fatigue

Lots of low cycle fatigue criteria have been proposed in the literature. Only few and popular ones are discussed here. A very well know criterion is the Smith-Watson-Topper criterion [START_REF] Smith | A stress-strain function for the fatigue of metals[END_REF]:

σ N,max N,a = (σ f ) 2 E (2N ) 2b + σ f f (2N ) b+c (1.3)
The energy term is defined on the plane undergoing the maximum normal strain amplitude. It assumes mode I cracking because of involvement of normal stress and strain terms. The material parameters are identified using the Coffin-Manson curves. [START_REF] Kandil | Biaxial low cycle fatigue fracture of 316 stainless steel at elevated temperatures[END_REF] proposed another form of criterion which is based on their previous study [START_REF] Brown | A theory for fatigue failure under multi-axial stressstrain conditions[END_REF], with the plane of maximum shear strain as the critical plane.

γ a + 2 τ -1 σ -1 -0.5 1 + ν 1 -ν N,a = γ f (2N ) c + τ f G (2N ) b (1.4)
The terms τ f /G & γ f are related to elastic and plastic part of strain respectively. One of the problems resulting from the criteria related to strain amplitudes alone is the nonconsideration of the mean stress and stress amplitude. The KBM criterion is only applicable to proportional loadings without any additional strain hardening effect. To overcome the shortcomings of this model, [START_REF] Socie | Biaxial fatigue of inconel 718 including mean stress effects[END_REF] introduced a term accounting for the mean normal stress, which influences crack opening, with the plane of maximum shear stress amplitude as critical plane:

γ a + N,a + σ N,m E = γ f (2N ) c + τ f G (2N ) b (1.5)
Fatemi and [START_REF] Fatemi | A critical plane to multiaxial fatigue damage including out-of-phase loading[END_REF] made another modification by changing the normal strain term to maximum normal stress, considering the extra hardening due to the phase shift between the stresses.

γ a 1 + k σ N,max σ y = γ f (2N ) c + τ f G (2N ) b (1.6)
The coefficient k is determined by push-pull and torsion tests data. +

High-cycle fatigue

In order to determine the allowable stress limit for safety of mechanical component, many multiaxial fatigue endurance criteria exist in literature which can be divided into four different categories:

(i) Criteria based on a critical plane approach (ii) Criteria based on an integral approach (iii) Criteria based on energy (iv) Criteria based on stress invariants

Critical plane-based criteria

Experimental observations indicate that cracks nucleate and grow on specific planes called critical planes. Depending on the material and loading conditions, these planes can be either maximum shear planes or maximum tensile stress planes. Failure assessment is performed on a plane where the amplitude or value of some stress component or combination of them attains its maximum. This type of approach reflects the physical nature of fatigue damage in their formulation.

The physical basis for such type of models is that the irregular shape of micro-cracks result in interlocking and friction between crack surfaces during cyclic shear. A tensile stress perpendicular to crack surface assists crack growth by reducing these forces. However, when the mean normal stress relative to critical plane is larger than a threshold value, it does not result in further increase of fatigue damage [START_REF] Kaufman | The influence of static mean stresses applied normal to the maximum shear planes in multiaxial fatigue', Biaxial and Multiaxial Fatigue and Fracture[END_REF][START_REF] Susmel | Multiaxial fatigue limits and material sensitivity to non-zero mean stresses normal to the critical planes[END_REF]. This observation is based on the fact that once mean normal stress is high enough to fully open the micro/meso cracks, the crack faces can no longer interact with each other and shear forces are directly transmitted to crack tip favoring the mode II propagation. When the mean normal stress is lower than the threshold value, crack faces can interact, leading to a decrease in fatigue damage because crack growth is retarded by asperities on crack faces.

One of the earliest works based on critical plane approach was by [START_REF] Findley | A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending[END_REF] with damage parameter as a linear combination of maximum shear stress amplitude (τ a ) and maximum normal stress (σ N,max ) acting on the critical plane (Euler angles: φ * , θ * ) which Literature review 13

is the plane of maximizing the damage function.

τ a (φ * , θ * ) + κ f σ N,max (φ * , θ * ) ≤ λ f (1.7)
where κ f and λ f are material parameters given by:

κ f = 2 -w 2 √ w -1 , λ f = σ -1 2 √ w -1 , w = σ -1 τ -1
σ -1 and τ -1 are the fatigue limits in fully reversed push-pull fatigue and torsional fatigue.

This criterion incorrectly predicts the absence of mean shear stress effect on fatigue life, first observed by [START_REF] Gough | Some experiments on the resistance of metals to fatigue under combined stresses[END_REF]. By redefining the critical plane as a plane of maximum shear stress amplitude only, [START_REF] Matake | An explanation of fatigue limit under combined stress[END_REF] proposed another criterion with the same damage parameter as proposed by [START_REF] Findley | A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending[END_REF]. The Matake parameters are deduced from the endurance limits in fully reversed torsion (τ -1 ) and fully reversed bending (σ -1 ).

κ = 2s -1, λ = τ -1 , s = τ -1 σ -1
It correctly predicts the absence of mean shear stress effect in torsion (true in most cases, provided the maximum von-mises stress does not the yield limit) and the effect of mean tensile/compressive stress on fatigue life [START_REF] Papadopoulos | A comparative study of multiaxial high-cycle fatigue criteria for metals[END_REF]. [START_REF] Mcdiarmid | A shear stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction[END_REF] then modified the criterion by using different parameters while using the same damage function and critical plane (plane of maximum shear stress amplitude), employing the concept of type A and type B facets introduced by [START_REF] Brown | A theory for fatigue failure under multi-axial stressstrain conditions[END_REF] under biaxial loading conditions, figure 1.10. Type A facets, arising from negative biaxiality (λ), are cracks for which the maximum shear stress is parallel to the surface plane hence propagating along the surface. Type B facets (more severe than case A), arising from positive biaxiality (λ), are cracks for which maximum shear stress is not parallel to surface hence propagating in inwards direction from surface.

τ a (φ * , θ * ) + τ A,B 2σ u σ N,max (φ * , θ * ) ≤ τ A,B (1.8) τ A,B = τ A or τ B
σ u is the ultimate tensile strength of material. Either of shear endurance limit, τ A or τ B , is used depending upon type of crack potentiality. Case A cracks occur in fully reversed torsion experiments giving τ A = τ -1 with critical plane tangent to the surface of the specimen. This criterion predicts the uniqueness of fatigue torsion limit and mean tensile/compressive stress effect on fatigue limit [START_REF] Papadopoulos | A comparative study of multiaxial high-cycle fatigue criteria for metals[END_REF]. In 1973

Figure 1.10: Planes of maximum shear and crack growth direction under biaxial loading, [START_REF] Brown | A theory for fatigue failure under multi-axial stressstrain conditions[END_REF] Dang-van proposed an endurance criterion using the hydrostatic stress (σ h ) and mesoscopic shear stress (T ), with the critical plane ( n) being the one which maximizes the damage parameter (Dang [START_REF] Van | Sur la résistance a la fatigue des métaux[END_REF].

max n max t (a T + b σ h,max ) ≤ σ -1 (1.9) κ = σ -1 τ -1 , a = κ , b = 3 - 3 2 κ
where σ -1 and τ -1 are endurance limits in completely reversed tension and torsion respectively. This function establishes the Dang Van Endurance Domain but it gives poor results under non-proportional loading [START_REF] Papuga | Two new multiaxial criteria for high cycle fatigue computation[END_REF]. The method has been implemented in a number of commercial softwares and widely used for design purposes in spite of the fact that it is not able to predict multiaxial and mean stress effect simultaneously [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF]. By directly correlating the critical plane with the fracture plane (maximum normal stress amplitude plane), [START_REF] Liu | Multiaxial high-cycle fatigue criterion and life prediction for metals[END_REF] proposed a criterion including mean stress effect. The direction of the critical plane with respect to the fatigue fracture plane is given by parameter α, which is the angle between the two.

The critical plane orientation, α, also depends upon material properties in addition to Literature review 15 stress state. [START_REF] Papuga | Two new multiaxial criteria for high cycle fatigue computation[END_REF] proposed a criteria based on critical plane (PCr criterion) emphasizing the shear stress effect as compared to normal stress (a c & b c are material constants depending upon κ), with the critical plane being the one maximizing the damage parameter.

a c τ 2 a + b c σ a + τ -1 σ 0 σ m ≤ σ -1 ; κ = σ -1 τ -1 (1.10)

Integral-based criteria

In integral approach, the damage parameter is integrated over all possible orientations at a point and then compared with a critical value. In general, results produced by the critical plane approach are better than those obtained by integral approach which are more time consuming [START_REF] Papuga | Two new multiaxial criteria for high cycle fatigue computation[END_REF]. Most of the times, the expressions of such criteria are very complex. Many such criteria exist in literature: [START_REF] Zenner | Eine Festigkeitshypothese für die Dauerfestigkeit bei beliebigen Beanspruchungskombinationen[END_REF]; [START_REF] Papadopoulos | A high-cycle fatigue criterion applied in biaxial and triaxial outof-phase stress conditions[END_REF]; [START_REF] Kenmengne | On a new multiaxial fatigue criterion based on a selective integration approach[END_REF] etc. The criterion proposed by [START_REF] Liu | Berechnung der Dauerschwingfestigkeit bei mehrachsiger Beanspruchung[END_REF] is quite popular, in-spite of complicated expression and involvement of lots of material parameters. Further clarification for the definition of the shear stress amplitude and mean shear stress was provided [START_REF] Zenner | On the fatigue limit of ductile metals under complex multiaxial loading[END_REF] implying the dependence of torsion fatigue limit on superimposed static mean shear stress. Because of its limited performance and time cost and lack of physical meaning, such type of criteria are not the focus of this study.

Energy-based criteria

The criteria based on strain energy per cycle can be classified into three categories:

• Criteria based on the elastic strain energy (high-cycle fatigue)

• Criteria based on the plastic strain energy (low-cycle fatigue)

• Criteria based on sum of elastic and plastic strain energy (low and high-cycle fatigue)

According to most of the authors, either elastic & plastic normal strain energy densities in the plane of maximum normal strain or shear strain energy densities in the plane of maximum shear strain should be considered as the fatigue damage parameter. However, (∆W e ) strain energy densities [START_REF] Ellyin | Cyclic strain energy density as a criterion for multiaxial failure fatigue', Biaxial and multiaxial fatigue[END_REF] application of plastic strain energy density to high cycle fatigue situations is difficult since plastic strains are small. But plastic strain energy per cycle has an advantage of being nearly constant during the fatigue life. Figure 1.11 shows the two energy components during cyclic loading. The plastic strain energy per cycle is the area EGHACDE, the area of hysteresis loop. Elastic strain energy plays a role in fatigue damage in tension only (area ACB). [START_REF] Glinka | A multiaxial fatigue strain energy density parameter related to the critical plane[END_REF] used total strain energy density on the critical plane as a damage parameter. It was further modified so that it can predict fatigue life under multiaxial loading [START_REF] Pan | Fatigue life estimation under multiaxial loadings[END_REF]. The material constants are identified using tensile and fully reversed torsional data. The modified damage parameter can be written as:

W * = ∆γ 2 ∆τ 2 + k 1 k 2 ∆ n 2 ∆σ n 2 ; k 1 = γ f f , k 2 = σ f τ f (1.11)
∆γ, ∆ n and ∆τ , ∆σ n are shear and normal strain/stress range on critical plane, defined as a plane of maximum shear strain amplitude, γ a,max . γ f , f , σ f and τ f are material parameters, identified using pure tension and pure torsion fatigue data. γ f and f are shear and axial fatigue ductility coefficients while σ f and τ f are shear and axial fatigue strength coefficients. For finite life prediction, the criterion can be modified using Manson-Coffin-Basquin equation. [START_REF] Lagoda | A critical plane approach based on energy concepts: application to biaxial random tension-compression high-cycle fatigue regime[END_REF] used maximum normal Literature review 17 strain energy density on critical plane (plane maximizing damage function):

W η (t) = 1 2 η (t)σ η (t) Sgn[σ η (t), η (t)]
(1.12)

The function Sgn[σ η (t), η (t)] includes the effect of positive and negative work done during tension and compression. For finite life prediction, the expression can be written as (with help of Manson-Coffin-Basquin equation):

W max = 1 2 n,max σ n,max = (σ f ) 2 E (2N ) 2b + σ f f (2N ) b+c
These type of criteria can also account for extra hardening effects under non-proportional loadings.

Stress-based criteria

Searching a critical plane which usually takes heavy calculation time, is a drawback of critical plane approaches and integral approaches. This is overcome in stress based criteria. First and second invariants (I 1 , corresponds to hydrostatic stress & J 2 , corresponds to Von Mises stress) of the deviatoric stress tensor are used in damage function. Many such criteria exist in literature but only few of them are discussed here. One of the earliest and most popular criterion was proposed by [START_REF] Crossland | Effect of large hydrostatic pressure on the torsional fatigue strength of an alloy steel[END_REF] and uses the amplitude of second stress invariant (J 2,a ) and maximum hydrostatic stress (σ h,max ) in the damage parameter:

J 2,a + κ σ h,max ≤ λ ; κ = 3 τ -1 σ -1 -1 , λ = τ -1 (1.13)
It predicts mean tensile/compressive load effect on fatigue strength and the absence of mean shear stress effect on fatigue strength. [START_REF] Marin | Proceedings of the International Conference on Fatigue of Metals[END_REF] proposed to use amplitude and mean of second stress invariant (J 2 ), no

hydrostatic stress: 3 J 2,a σ -1 λ + κ J 2,m σ -1 µ ≤ 1 (1.14)
Generally, the values for κ, λ and µ are taken as κ = 1 and λ = µ = 2. This criterion incorrectly predicts a dependence of the fatigue limit on the mean torsion stress and is not able to distinguish the effect of a static tensile stress from that of a static compressive stress because of the squared terms [START_REF] Papadopoulos | A comparative study of multiaxial high-cycle fatigue criteria for metals[END_REF].

Later on in 1959, Sines [START_REF] Sines | Behavior of metals under complex static and alternating stresses[END_REF] proposed to use a combination of J 2 amplitude and mean hydrostatic stress.

J 2,a + κ σ h,m ≤ λ ; κ = 3 τ -1 σ 0 -1 , λ = τ -1 (1.15)
Material parameters κ and λ are identified using fully reversed torsion and repeated bending tests. Sines criterion correctly reproduces the effect of mean static torsion stress on fatigue limit and introduces a linear dependence of the bending limit upon a superimposed static normal stress [START_REF] Papadopoulos | A comparative study of multiaxial high-cycle fatigue criteria for metals[END_REF]. Sines criterion on application under fully reversed bending gives the ratio τ -1 /σ -1 constant (= 1/ √ 3) for all metals which is not true (1/ √ 3 ≤ τ -1 /σ -1 ≤ 1 for brittle metallic alloys and τ -1 /σ -1 ≤ 1/ √ 3 for ductile metallic alloys). Expresion of κ can be modified using Goodman relation [START_REF] Goodman | Mechanics Applied to Engineering[END_REF] (σ a = σ m = σ 0 ), thereby replacing the term σ 0 by σ -1 : [START_REF] Gonçalves | Multiaxial fatigue: a stress based criterion for hard metals[END_REF] used the concept of minimum circumscribed ellipsoid over the load path in five-dimensional deviatoric (d i ) Ilyushin space. It utilizes the maximum principal stress (σ 1,max ) over the load cycle and the distances of the center of the ellipsoid to the faces of any arbitrarily oriented rectangular prism circumscribing the stress path in the deviatoric space (d i ), as a second load parameter.

σ a σ -1 + σ m σ u = 1 (Goodman Line) ; κ = √ 3σ -1 σ u - 1
a g 5 i=1 d 2 i + b g σ 1,max ≤ σ -1 (1.16)
The computation is simple and fast [START_REF] Papuga | Two new multiaxial criteria for high cycle fatigue computation[END_REF]. One of the latest criteria in this category was proposed by [START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF] which include mean stress effect, frequency effect and phase shift effect in multiaxial loading and combination of both stress invariants: I 1 & J 2 ( σD a is the deviator of the amplitude of the stress tensor, σ u is the ultimate strength and I f (I 1,a , I 1,mean ) = I 1,a + I 1,mean for σ u < 750 MPa; and is = titanium alloy, figure 1.12. The bi-linear fit accounts for the saturation of the mean stress effect as discussed earlier. It can be seen that both criteria give poor results with biaxial loading (tension-tension and tension-internal pressure). But it is fair to add that both these criteria were not meant for finite lives and also, testing temperatures were high. [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF] evaluated the pertinence of Dang-Van criterion for mean stress effect and biaxial tension loading in cast hypo-eutectic AlSi alloys, figure 1.13. It shows the loading paths at the endurance limits for different mean stresses under uniaxial bending (purple and green colors), torsion (mustard color) and biaxial tension (red color) on the Dang-van's plot. The Dang-van criterion does not capture the mean stress effect and load biaxiality effect. May be the problem with the criteria involving the hydrostatic stress is their inability to distinguish between the effect of a crack opening stress (normal to crack face) and the effect of a T-stress (parallel to crack face) as both cases gives the same trace, figure 1.14. From above discussion, it can be deduced that none of proposed criteria give satisfactory results for biaxial tension loadings and hence there is a need to find one.

I 1,a + σ -1 τ -1 I 1,mean for σ u > 750 MPa). γ 1 J 2 (t) 2 + γ 2 J 2 2,mean + γ 3 I f (I 1,a , I 1,mean ) ≤ β (1.17) J 2,mean = 1 T T 0 J 2 (t) dt , J 2 (t) = 1 2 σD a (

Specimens for biaxial tension

In order to perform biaxial tension tests various kinds of specimens have been proposed:

Cruciform specimens loaded in two orthogonal directions [START_REF] Bonnand | Investigation of multiaxial fatigue in the context of turboengine disc applications[END_REF][START_REF] Makinde | Development of an apparatus for biaxial testing using cruciform specimens[END_REF][START_REF] Lagoda | A critical plane approach based on energy concepts: application to biaxial random tension-compression high-cycle fatigue regime[END_REF][START_REF] Tanaka | Fatigue crack propagation in biaxial stress fields[END_REF][START_REF] Sunder | Fatigue crack growth under flight spectrum loading with superposed biaxial loading due to fuselage cabin pressure[END_REF][START_REF] Mc Adam | Stress-Strain-Cycle Relationship and Corrosion-Fatigue of Metals[END_REF]Literature review 21 Clung , 1989;[START_REF] Doudard | Determination of an {HCF} criterion by thermal measurements under biaxial cyclic loading[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF][START_REF] Lee | Fatigue behavior of aluminum alloys under biaxial loading[END_REF], tubular specimens loaded in tension and internal pressure [START_REF] Brown | Initiation and growth of cracks in biaxial fatigue[END_REF][START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF][START_REF] Dietmann | Multiaxial fatigue behavior of steels under In-Phase and Out of phase loading including different wave forms and frequencies[END_REF][START_REF] Itoh | Low Cycle Fatigue Damage of Mod.9Cr-1Mo Steel under Non-Proportional Multiaxial Loading[END_REF][START_REF] Varvani-Farahani | Crack growth and closure mechanisms of shear cracks under constant amplitude biaxial straining and periodic compressive overstraining in 1045 steel[END_REF][START_REF] Doquet | Dwell fatigue of a titanium alloy at room temperature under uniaxial or biaxial tension[END_REF] and disc specimens submitted to biaxial bending [START_REF] Zouani | Fatigue Life Parameter for Type 304 Stainless Steel Under Biaxial-Tensile Loading at Elevated Temperature[END_REF][START_REF] Kane | Surface crack and cracks networks in biaxial fatigue[END_REF][START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF][START_REF] Kamaya | Evaluation of equi-biaxial fatigue of stainless steel by the pressurized disc fatigue test[END_REF][START_REF] Joshi | Fatigue-crack Propagation in a Biaxial-stress Field[END_REF]. Tests on cruciform type specimens require specific and expensive multi-axis testing machines with four actuators. Secondly, specimens are large and relatively complex in shape, so that the cost and time to manufacture goes up. Thirdly, the zone in which stress and strain are uniform is usually small and test analysis demands numerical simulations and/or DIC measurements of strain fields.

Biaxial tension tests on tubular specimens require a tension-pressure machine. If only internal pressure is available, then only loading cycles with a positive R ratio can be applied. Negative R ratios become possible only if external pressure can be added. If pressure is applied by a liquid, tests can be performed only at near-ambient temperatures.

For higher temperatures, it is possible to pressurize a gas, but security issues become important. The disc type specimens perform good in fatigue but they are relatively larger [START_REF] Kamaya | Evaluation of equi-biaxial fatigue of stainless steel by the pressurized disc fatigue test[END_REF]. In addition to these, some special specimens have also been proposed to achieve biaxial stress state in uniaxial loading with a suitable geometry, figure 1.15 [START_REF] Sawert | Verhalten der Baustähle bei Wechselnder Mehrachsiger Beanspruchung, 87, Nr 39/40, 2[END_REF][START_REF] Bellett | A Biaxial fatigue specimen for uniaxial loading[END_REF]. In the present study, tubular specimens have been used for tension-internal pressure fatigue loadings.

Part B: Corrosion-fatigue

Environment Assisted Cracking (EAC) was first reported by [START_REF] Haigh | Experiments on the Fatigue of Brasses[END_REF], followed by

Mc [START_REF] Mc Adam | Stress-Strain-Cycle Relationship and Corrosion-Fatigue of Metals[END_REF], who used the term Corrosion fatigue for the first time. The combined action of environment and mechanical loads can lead to Stress Corrosion Cracking (SCC).

Corrosion fatigue is considered as a particular case of SCC where the load is cyclic.

Electrochemical aspects

Aqueous corrosion is the electrochemical degradation of the materials with continuous dissolution (oxidation) of the metal surface in contact with the environment (electrolyte), resulting in a potential difference, ∆E, between the metal (negative) and the electrolyte (positive). This oxidation process is accompanied by the simultaneous reduction of metals cations (M + ) or hydrogen (H 2 ) or dissolved oxygen (O 2 ) in the electrolyte, acting as cathode. Corrosion is thus a consequence of a redox reaction. In aqueous environment, it can be represented by the following redox reactions:

• Anodic Reaction. M → M n+ (aq) + ne - • Water Hydrolysis. H 2 O H + (aq) + OH -(aq) • Cathodic Reactions. O 2 (g) + 2H 2 O + 4e -→ 4OH -(aq) 2H + (aq) + 2e -→ H 2 (g) M + (aq) + nOH -(aq) → M (OH) n (aq)
Cathodic reactions determine the rate of corrosion i.e. the more acidic the medium is (more H + ions), the higher is the corrosion rate. These reactions are charge-transfer processes dependent on the interfacial potential corresponding to what is called the electrode potential of metals (E M/M n+ ).

Polarization

The electrode reactions (anodic or cathodic) are reversible at their equilibrium potential.

The anodic reaction occurs above the equilibrium potential, while the cathodic reaction occurs below the equilibrium potential. A metal electrode corroding in an aqueous so-Literature review 23 until it reaches the passive current density (order of few µAcm -2 ). Region CD is named as "passive region", in which a passive oxide layer forms on the metal surface and protects it from further corrosion. For transition elements (Fe, Cr, Ni etc.) and their alloys, the thickness of this layer can reach a few hundreds ångström ( Å). A sufficient increase in potential leads to the transpassive region (region DE). The sudden increase in current can be, depending upon the alloy/environment combination, due to pitting (localized breakdown of passive film) or evolution of oxygen through an electrolysis of the aqueous media.

Cathodic polarization

A displacement of the electrode potential in the negative direction making it more ca- region BC may represents the reduction of oxygen (as described in section 1.4). Since the reaction rate is controlled by oxygen diffusion, it cannot increase indefinitely. Instead, the current reaches a limiting value, which is a function of the oxygen concentration and its diffusivity. For sufficiently negative potentials, another cathodic reaction comes into picture as illustrated by region CD. This reaction is due to the reduction of other species, typically an evolution of hydrogen(H 2 ) from water reduction. As the potential increases, this reaction becomes dominant. The ingress of hydrogen into the metal is known to enhance fatigue crack growth by the mechanism called "hydrogen embrittlement".

Damage mechanisms

In general, an active environment reacts with the metal surface and initiates damage. • Environmental Attack. Localized corrosion reaction (oxidation of metal surface) leading to formation of pits which serve as crack nucleation sites.

• Crack Initiation. Initiation of cracks from surface pits or localized corrosion on susceptible slip bands or grain boundary precipitates (Inter-granular or Trans-granular SCC).

• Crack Propagation. Propagation of cracks assisted by environment through dissolution based mechanisms, or adsorption based mechanisms, or "hydrogen embrittlement".

• Sudden Fracture. Unstable propagation leading to final fracture.

Crack initiation

Depending upon the initial state of the surface, different initiation mechanisms can occur.

It is known that slip bands, twins, grain boundaries and constituent particles are classical sites. For active metal surfaces, it can occur from localized dissolution of metal (pitting), dealloying or intergranular corrosion (associated with anodic grain boundary precipitates)

Literature review 27 [START_REF] Milella | Fatigue and Corrosion in Metals[END_REF].

as illustrated in figure 1.20. For passive interface, it is the cleavage of passive film followed by active surface initiation. It is also possible that crack initiates as in air (discussed in part A of this chapter), without any influence of corrosion because the frequency of loading or the threshold for SCC could be high.

Persistent slip bands

The persistent slip bands (PSBs) have been observed to react with the corrosive medium.

The presence of mobile dislocations enhances the electrochemical reactions. Consequently, the PSBs are anodic compared to the undeformed material and are preferentially dissolved, which enhances local plastic deformation,and thus contributes to crack initiation. Corrosive environment has been shown to increase the number of PSBs and reduce their inter-spacing distribution [START_REF] Gangloff | Chemistry and Physics of Fracture[END_REF] which can affect crack initiation.

Grain boundary precipitation

Cracks have also been observed to initiate from precipitates at grain boundaries, making it susceptible to inter-granular SCC [START_REF] Strauss | Die Carbidausseheidung beim Gluehen von Nichtrostenden, Unmagnetischen Cr-Ni Stahl[END_REF]. This phenomenon is also known

Passivity breakdown

Crack initiation for passive surfaces takes place by breakdown of passive film. Various mechanisms exist in literature to explain this phenomenon, of which the most popular and commonly observed mechanisms are:

• Film-breaking mechanism.

• Adsorption mechanism.

The film-breaking mechanism proposed by Vetter and Strehblow (1970) The adsorption mechanism proposed by Kolotyrkin (1964) involves the adsorption of aggresive anions onto the oxide surface and subsequent dissolution of metal cations in electrolyte, figure 1.22(b). Exposure of oxide film to aggressive ions, especially the halides (X -) as chloride or fluoride, results in the formation of soluble metal complexes. This leads to thinning of the passive layer and finally, its removal, resulting in enhanced localized dissolution.

Both types of passivity breakdown mechanisms are followed by formation of pits which is the most commonly observed phenomenon for crack initiation in corrosion fatigue.

Corrosion pitting

Corrosion pitting is an important phenomenon of crack initiation in corrosion fatigue involving nucleation of pits, their growth, pit to short crack transition, growth of short cracks in and around the pits and finally, crack growth by coalescence of small cracks.

This mechanism was first proposed by Mc Adam (1926). For a metal electrode, there exist a critical potential below which not pitting occurs called the pitting potential (E p ), point D in figure 1.17. This potential depends upon the concentration of aggressive ions in electrolyte.

The chloride ions have special importance in pitting corrosion because they are ubiquitous and common contaminants in various electrolyte systems and they can reduce the E p significantly. Being a strong base with high diffusivity, chloride ions have tendency to dissolve metal surface and nucleate pits. Corrosion pits almost always initiate at some chemical or physical heterogeneity on the metal surface, such as inclusions, second phase particles, flaws, mechanical damage or dislocation [START_REF] Wang | Effect of pitting corrosion on very high cycle fatigue behavior[END_REF]. A micro-pit of only a few microns can form very quickly after immersion and then grow.

                                 
A schematic view of pit growth is illustrated in figure 1.23(a) [START_REF] Rosenfeld | Crack Initiation Mechanism of Bearing Steel in High Cycle Fatigue[END_REF]. Nucleation of pits increases the current density because of the confinement of current to a small volume inside the pit, while nearby film remains unattacked. The hydrolysis of metal cations confined inside the pit, increases the local pH [START_REF] Butler | Initiation and Growth of Pits on High-Purity Iron and its Alloys with Chromium and Copper in Neutral Chloride Solutions[END_REF]. The variation of pH across the pit for iron in dilute chloride solution is shown in figure 1.23(b).

The local increase in acidity inside the pit forces the aggressive anions (Cl -) from the electrolyte to migrate into the pit so as to maintain neutrality of the solution. These aggressive ions are responsible for the dissolution of metal, forming metal cations. Hence, the process of metal hydrolysis repeats, making the pit grow.

The growth of pit is a time dependent process and its growth law can be expressed as:

c ∝ t b ,
where 'c' is the pit size, 't' is time and 'b' is constant depending upon material and environment. The pit grows until it reaches a critical size and then a transition occurs:

"pit-to-crack" during which short crack initiates in and around the pit, due to the local stress concentration factor. After this transition, a fracture mechanics approach is often used assuming the pit as a short crack [START_REF] Kondo | Prediction of fatigue crack initiation life based on pit growth[END_REF][START_REF] Zhang | Effect of corrosion pits on fatigue life and crack Initiation[END_REF][START_REF] Haddad | Prediction of non propagating cracks[END_REF][START_REF] Schönbauer | Pit-to-crack transition under cyclic loading in 12%Cr steam turbine blade steel[END_REF].

For the "pit-to-crack-growth" transition to occur, two conditions have to be satisfied: (1)

Literature review 31 considering the pit as a semi-elliptical crack, the stress intensity factor has to be greater than the threshold value and ( 2) the fatigue crack growth rate should exceed the pit growth rate. A typical example of crack initiation from a pit is shown in figure 1.24.

Crack growth

A finite threshold stress intensity factor for stress-corrosion crack propagation under sustained load exists, denoted as K I,scc . It depends upon the material, the corrosive medium, pH and temperature.

A typical growth kinetics for SCC is illustrated in figure 1 [START_REF] Austen | Factors affecting the rate of growth of cracks by corrosion-fatigue[END_REF].

Parameters influencing corrosion-fatigue

Various parameters have been known to affect the fatigue life and therefore, the crack growth rate in corrosion by enhancing one mechanism or the other, which are discussed below.

Effect of frequency

Corrosion is a time-dependent process, so the frequency of fatigue tests is an important parameter. Any possible mechanism, dissolution-based or hydrogen-based, depends upon the exposure time of crack surface to the environment. Therefore, a dependency of fatigue crack growth rate (FCGR) and fatigue lives on the frequency is generally expected.

The detrimental effect of a corrosive medium is significantly more important at lower frequencies.

Various studies have shown that the decrease in frequency increases the crack growth rate in corrosive environment [START_REF] Dawson | Corrosion Fatigue Crack Growth of Titanium Alloys in Aqueous Environments[END_REF][START_REF] Vosikovski | Fatigue-Crack Growth in an 1-85 Line-Pipe Steel at Low Cyclic Frequencies in Aqueous Environments[END_REF][START_REF] Barsom | Fracture and Fatigue Control in Structures[END_REF][START_REF] Krishnan | Mechanism of corrosion fatigue in super duplex stainless steel in 3.5 percent NaCl solution[END_REF][START_REF] Berge | Environmental effects on fatigue strength 139 of armour wire for flexible risers[END_REF][START_REF] Knop | Effects of cycle frequency on corrosion-fatigue crack growth in cathodically protected high-strength steels[END_REF]. This is attributed to the fact that at lower frequencies, the crack remains open for a longer duration of time, thereby, allowing more time for corrosion action.

However, for martensitic steels in de-mineralized water at 288℃, an increase in FCGR with decreasing frequency is found only until 0.1 Hz (Van Der rate change at AKsc c increases with decreasing frequency. AKsc c is frequency-dependent, increasing with decreasing frequency (Fig. 7).

In order to relate AKsc c to the sustained-load SCC threshold, Kiscc values were obtained for sheet specimens of the mill-annealed Ti-6-4 and Ti-6-6-2 alloys used in this investigation. For a stress ratio R = 0.1, when Kmax = Kiscc we have AK = 0.gKiscc. Fig. 7 shows that AKsc c approaches 0.9Kiscc as the test frequency is decreased.

At 10 Hz, the highest frequency for which a MN-mAKscc-s/2 value could be identified, AKsc c was about 11 for Ti-6-6-2 and Ti-6-4, which is far below the measured values of the sustained-load SCC threshold for these alloys.

The corrosion fatigue behavior of mill-annealed Ti-6-6-2 in 0.6 M concentration aqueous solutions of LiC1 and KBr was also measured at 1 and 10 Hz. Crack growth behavior in these solutions was similar to NaC1, with minor differences in FCGR and AKsc c values.

Stress wave form effects on FCGR in aqueous 0.6M KBr are shown in Fig. 8. The most important wave form parameters are the rise time (tR) , and the fraction of the stress cycle spent above Kiscc and at Kma x. Below AKscc, the rise time is the most important factor; however, unlike steels 2~ or aluminum alloys,17C F crack growth rates in titanium alloys decrease rather than increase with longer rise times. The FCGR (Fig. 8) are the lowest for haversine wave form at 1 Hz (t R

Fractographic Results

There was a direct correlation between the macroscopic fracture surface roughness and the rate of fatigue crack growth in a corrosive environment. Thus the fracture surface appearance could be changed by changing the frequency or the AK level; distinct fracture bands could be produced on specimens which were tested alternately at different frequencies: I) For "type 1" behavior (e.g., air, and aqueous NazSO 4 at low AK levels), no difference in fracture surface appearance was evident in specimens tested at different frequencies.

2) For "type 2" behavior (distilled water, or meth- 1986; [START_REF] Cullen | Fatigue crack growth rates in pressure vessel and piping steels in LWR environment[END_REF]. This is because at too low frequencies, enough time is provided to material to passivize, blocking any further environmental effect.

A typical example of effect of frequency on behavior of fatigue crack growth rate is shown in figure 1.26(a). The growth kinetics of NiCrMo steel in 3% NaCl solution shows a significant reduction in threshold and an increase in crack growth rate at low frequency (6 cpm = 0.1 Hz) but this effect nearly vanishes at 600 cpm (= 10 Hz). Frequency has also been found to affect the threshold value for crack growth in opposite way, in Ti alloy loaded in 3% NaCl solution, as illustrated in figure 1.26(b): the threshold value for FCGR (∆K th ) increases with decrease in frequency (from 10 Hz to 1 Hz).

Besides, a change in frequency is not always felt instantaneously in terms of crack growth rate as illustrated in figure 1.27(a). A decrease in frequency is felt almost instantaneously through corresponding increase in crack growth rate while an increase in frequency takes some time for crack growth to decrease.

In-spite of the fact that the depreciation of fatigue lives due to corrosive environment tends to vanish at higher frequencies, some studies have shown the existence of a depreciation in fatigue lives at higher frequencies too [START_REF] Cruz | Influence of sea water on the fatigue strength and notch sensitivity of a plasma nitrided B-Mn steel[END_REF][START_REF] Pérez-Mora | Very high cycle fatigue of a high strength steel under sea water corrosion: A strong corrosion and mechanical damage coupling[END_REF][START_REF] Pfennig | Corrosion Fatigue Behavior and S-N-curve of X46Cr13 Exposed to CCS-environment Obtained from Laboratory in-situ-experiments[END_REF]. A drop in fatigue lives of martensitic steel in A3 standard synthetic sea water with 24.53% NaCl solution at 20kHz frequency is illustrated in figure 1.27(b).

Effect of the R ratio

The influence of the stress ratio (R = σ min /σ max ) on fatigue crack growth is often more important than in air, and the detrimental effect of a corrosive environment on fatigue lives is often enhanced by high R ratios [START_REF] Jones | Effect of stress ratio on the cyclic tension corrosion fatigue life of notched steel BS970:976M33 in sea water with cathodic protection[END_REF][START_REF] Bucci | Environment enhanced fatigue and stress corrosion cracking of a titanium alloy plus a simple model for the assessment of environmental influence on fatigue behavior[END_REF][START_REF] Sivaprasad | Corrosion fatigue crack growth behaviour of naval steels[END_REF] (figure 1.28(a)). In corrosive environment, an increase in mean stress promotes SCC that would not have appeared at low R ratios [START_REF] Banford | Crack growth rate testing in reactor pressure vessel steels[END_REF].

Figure 1.28(b) shows the superposition of fatigue and corrosion fatigue for a sinusoidal loading with a time period T. ∆t corresponds to the time for which K I > K I,SCC . An estimate of crack growth can be made using this superposition as:

da dt = ∆a T = ∆a ∆t + f da dN fatigue = 1 ∆t t+∆t t da dt (K I ) dt + f da dN fatigue (1.18)
With an increase in R ratio, the mean stress goes up which increase the time for which In order to isolate the intrinsic R-ratio effect on CFCGR of the two HSLA steels from the crack closure effects, closure free crack growth rates were examined, as a function of

K I > K I,SCC i.e.

Effect of electrode potential

The effect of cathodic electrode potential on fatigue life has been studied widely. [START_REF] Uhlig | Mechanism of Inhibiting Stress Corrosion Cracking of 18-8 Stainless Steel in M gCl 2 by Acetates and Nitrates[END_REF] found that for stainless steel in MgCl 2 solution, the cathodic polarization CYCLES TO FAILURE Fig. 9--Effect of applied potential of 4140 steel, R C 20, stressed above and below the fatigue limit, on fatigue life in aerated 3 pet NaC1, 25~ ber-seal gaskets of the test cell where localized corrosion had initiated, rather than at the reduced section of the test specimen where stress is highest. All specimens after tests in alkaline 3 pct NaC1 solution were relatively free of rust, but with some evidence of random pitting over the specimen surface, accompanied by crevice corrosion at the rubber gaskets. Significantly, pitted specimens did not necessarily fail by fatigue in this particular aqueous environment.

Effect of Cathodic Protection

Specimens of RC 20 steel cathodically protected in aerated 3 pct NaC1 solution were stressed approximately 10 pct above and 10 pct below the fatigue limit. Tests were run to failure maintaining the specimen by means of a potentiostat at a series of constant potentials with reference to a saturated calomel electrode. As the potential was made more active (greater cathodic current), fatigue life increased as shown in Fig. 9. At an applied stress of 53 ksi (below the fatigue limit) failure did not occur within >107 cycles at and CYCLES TO FAILURE Fig. 9--Effect of applied potential of 4140 steel, R C 20, stressed above and below the fatigue limit, on fatigue life in aerated 3 pet NaC1, 25~ ber-seal gaskets of the test cell where localized corrosion had initiated, rather than at the reduced section of the test specimen where stress is highest. All specimens after tests in alkaline 3 pct NaC1 solution were relatively free of rust, but with some evidence of random pitting over the specimen surface, accompanied by crevice corrosion at the rubber gaskets. Significantly, pitted specimens did not necessarily fail by fatigue in this particular aqueous environment.

Effect of Cathodic Protection

Specimens of RC 20 steel cathodically protected in aerated 3 pct NaC1 solution were stressed approximately 10 pct above and 10 pct below the fatigue limit. Tests were run to failure maintaining the specimen by means of a potentiostat at a series of constant potentials with reference to a saturated calomel electrode. As the potential was made more active (greater cathodic current), fatigue life increased as shown in Fig. 9. At an applied stress of 53 ksi (below the fatigue limit) failure did not occur within >107 cycles at and below --0.48 V (S.H.E.). At 65 ksi (above the fatigue limit) maximum life (4.7 x lO s cycles) was observed at and below the same approximate potential. In other words, the specific maximum potential to achieve optimum cathodic protection was not sensitive to applied stress. Maximum life in the latter instance was the same as that observed at the same stress level in deaerated 3 pct NaC1 solution without applied potential.

In the above tests, polarized potentials more active than -0.9 V (S.H.E.) resulted in a distinct odor of gaseous chlorine produced at the anode inside the test cell. Although dissolved chlorine did not appear to affect the results, its formation with accompanying suspected corrosion of the Pt anode was avoided by using 0.5 N NasSO4 for the most active potentials and corresponding highest currents through the cell. Fatigue life in sulfate solutions fell on the line representing data for 3 pet NaC1, confirming that free chlorine had no effect on the reported values. Similar results for RC 37 steel are shown in Fig. 10 at applied stresses 10 pct above and 10 pct below the fatigue limit for which a critical potential of -0.48 V is observed. For RC 44 specimens, tests were carried out only at an applied stress 18 pct below the fatigue limit, leading to results like those depicted in Fig. 10 and to a critical potential between -0.47 and -0.49 V, consistent with the previously observed potentials for RC 20 and 37 steels.

For RC 52 specimens, tests at an applied stress of 141 ksi (90 pct of the fatigue limit) showed increasing life for more active applied potentials, but with maximum life never reaching 107 cycles, was found that this behavior is dependent on the hardness of the metal. For mild steel or low strength steel, a critical potential has been observed (figure 1.29(b)), while for high strength steels the fatigue life increases with cathodic polarization up to a certain level above which it starts decreasing again as illustrated in figure 1.29(c) [START_REF] Uhlig | Corrosion Fatigue of Type 4140 High Strength Steel[END_REF][START_REF] Genel | Effect of cathodic polarisation on corrosion fatigue behaviour of ion nitrided AISI 4140 steel[END_REF]. This phenomenon can be explained by the evolution of hydrogen on metal surface at high cathodic potential, resulting in hydrogen embrittlement.

Effect of pH and Cl -ions

In general, more acidic the medium is, higher is the corrosion rate [START_REF] Maruyama | Fa-REFERENCES 147 tigue strength of 316L-type stainless steel in simulated body fluids[END_REF]. The evolution of fatigue life of carbon steel in 3% NaCl with different pH is illustrated in figure 1.30(a). The fatigue life decreases with decrease in pH. The phenomenon is explained by the presence of a stable passive layer on metal surface in alkaline medium, which hinders the interaction between metal surface and environment. For crack propagation, global pH may not be very significant since a local development of acidity near the crack tip has been observed [START_REF] Butler | Initiation and Growth of Pits on High-Purity Iron and its Alloys with Chromium and Copper in Neutral Chloride Solutions[END_REF].

Cl -ions, being aggressive in nature, are one of the important species from corrosion fatigue point of view. An increase in its concentration leads to significant depreciation in fatigue life as illustrated in figure 1.30(b) [START_REF] Sedriks | Corrosion of Stailess Steels[END_REF][START_REF] Ebara | Corrosion fatigue crack initiation behavior of stainless steels[END_REF]. The concentration [START_REF] Ebara | Corrosion fatigue behavior of 13Cr stainless steel in sodium-chloride aqueous solution and steam environment[END_REF].

Literature review 37 to lead to transgranular crack initiation [START_REF] Perkins | Corrosion fatigue of a 12%Cr low pressure turbine blade steel in simulated service environments[END_REF].

Effect of hydrogen

Many materials have been shown to exhibit cracking associated with a diffusion of hydrogen into the metal lattice resulting in a loss of ductility. This phenomenon has been named as Hydrogen embrittlement, even though this term does not always reflect the real micromechanisms. Damaging effects of hydrogen was first reported by [START_REF] Johnson | On the remarkable changes produced in iron and steel by the action of hydrogen and acids[END_REF] and since then, efforts have been made to understand this phenomenon.

Attempts have been by some researchers to model hydrogen-induced crack growth in finite elements [START_REF] Krom | Modeling hydrogen-induced cracking in steel using a coupled diffusion stress finite element analysis[END_REF][START_REF] Bjerkén | Simulation of corrosion fatigue crack growth under mixed-mode loading[END_REF][START_REF] Olden | Hydrogen diffusion and hydrogen influenced critical stress intensity in an API X70 pipeline steel welded joint -Experiments and FE simulations[END_REF]. Two most popular hydrogen induced damaged mechanisms are:

Hydrogen-enhanced decohesion mechanism. [START_REF] Pfeil | The effect of occluded hydrogen on the tensile strength of iron[END_REF] was probably the first one to demonstrate the decohesion phenomenon due to diffusion of hydrogen. A simple schematic for hydrogen cracking based on its diffusion is illustrated in figure 1.32. Hydrogen diffuses into the region with high hydrostatic stress ahead of crack tip. The accumulation of hydrogen nucleates micro-cracks because of local weakening of inter-atomic bonds and the process repeats, leading to discontinuous cracking in preference to slip. The small atomic size of hydrogen facilitates its solubility and diffusion in materials. [START_REF] Lufrano | Modeling of hydrogen transport and elastically accomodated hydride formation near a crack tip[END_REF] proposed a thermodynamic formulation for hydrogen diffusion flux (J H ) as a function of hydrostatic stress: Literature review 39

J H = - DC RT ∇µ (1.19) µ = µ 0 + RT log e C -
[C]

Figure 1.33: Direct TEM measurements of the influence of hydrogen on dislocation velocity ratio in a carbon steel [START_REF] Birnbaum | Multiscale Phenomena in Plasticity: From Experiments to Phenomenology, Modelling and Materials Engineering[END_REF].

[C] is the carbon conten.

high hydrostatic stresses ahead of crack tip which facilitates hydrogen diffusion.

Hydrogen-enhanced localized plasticity. This mechanism was first proposed by [START_REF] Beachem | A new model for hydrogen assisted cracking (hydrogen embrittlement)[END_REF] and based on the assumption that solute hydrogen atoms facilitates dislocations movement. The localized hydrogen concentration near crack tips, due to hydrostatic stresses, leads to localized plastic deformation and crack grows by localized coalescence of micro-voids. The process is localized plastic failure rather than embrittlement, thereby softening the material in front of the crack allowing ductile fracture. [START_REF] Birnbaum | Multiscale Phenomena in Plasticity: From Experiments to Phenomenology, Modelling and Materials Engineering[END_REF] observed with a TEM the effect of hydrogen on dislocation velocities in a stretched thin foil of carbon steel with different carbon (C) content, as shown in figure 1.33. The introduction of hydrogen enhanced the dislocations mobility and the increase in mobility rose with the increase in carbon content.

Effect of biaxial tension in corrosive medium

Nearly all the corrosion-fatigue studies that have been discussed so far, were done under uniaxial loading and only handful of corrosion fatigue investigation has been carried out to the author's best knowledge [START_REF] Misak | Corrosion fatigue crack growth behavior of 7075-T6 under biaxial tensiontension cyclic loading condition[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF][START_REF] Perel | Biaxial Fatigue Crack Growth Behavior in Aluminum Alloy 5083-H116 Under Ambient Laboratory and Saltwater Environments[END_REF]. These studies investigated the effect of cyclic biaxial tension (B = 0, 1 and 1.5) on the crack growth rate in 2024-T3, 7075-T6 and 5083-H116 aluminum alloys in salt water in high-cycle fatigue regime.

An enhanced detrimental effect of equibiaxial tension was observed in the Paris regime for 7075-T6 alloy but nearly no effect in the near threshold regime (figure 1.34) while for 2024-T3 alloy, a detrimental effect of load biaxiality was observed in the near threshold regime only [START_REF] Misak | Corrosion fatigue crack growth behavior of 7075-T6 under biaxial tensiontension cyclic loading condition[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF]. [START_REF] Perel | Biaxial Fatigue Crack Growth Behavior in Aluminum Alloy 5083-H116 Under Ambient Laboratory and Saltwater Environments[END_REF] also reported a detrimental effect of load biaxiality on crack growth rates in 5083-H116 aluminum alloy.

The influence of load biaxiality on corrosion fatigue crack growth seems thus very dependent on the material and has not been investigated in steels so far and it could be different. One of the objectives of this thesis is to investigate these effects in steels.

Literature review 41

Part C: Synthesis and open questions

There are different crack initiation mechanisms in fatigue and crack initiation is considered to be the most significant part of the fatigue life in high cycle fatigue regime. But, there are still some questions about this fact when crack initiation is internal, partly because of the lack of data on crack growth in vacuum.

Most of the multiaxial fatigue studies were done in uniaxial or in tension-torsion loadings.

Thus, the effects of biaxial tension are not totally clear. Few recent studies have shown that the existing multiaxial fatigue criteria are not suitable under biaxial tension [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF]).

An extensive literature review has shown that biaxial tension loading can have depreciating [START_REF] Bonnand | Investigation of multiaxial fatigue in the context of turboengine disc applications[END_REF][START_REF] Joshi | Fatigue-crack Propagation in a Biaxial-stress Field[END_REF][START_REF] Vansovich | The Effect of Stress State Characteristics on the Surface Fatigue Cracks Growth Rate Taking into Account Plastic Deformations[END_REF][START_REF] Clàudio | Biaxial high-cycle fatigue life assessment of ductile aluminium cruciform specimens[END_REF], beneficial [START_REF] Bellett | A Biaxial fatigue specimen for uniaxial loading[END_REF][START_REF] Doquet | Dwell fatigue of a titanium alloy at room temperature under uniaxial or biaxial tension[END_REF][START_REF] Kane | Surface crack and cracks networks in biaxial fatigue[END_REF][START_REF] Lee | Fatigue behavior of aluminum alloys under biaxial loading[END_REF] or neutral effect [START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF][START_REF] Leevers | Crack growth in plastic panels under biaxial stress[END_REF][START_REF] Kitagawa | Fracture mechanics study of fatigue crack growth in high-strength steel sheet under in-plane biaxial loading[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF] on the fatigue life and the fatigue crack growth (depending upon R ratio and loading amplitude) leading to no general perception. The effect of cyclic biaxial tension on fatigue life and damage mechanisms thus deserves an investigation.

In corrosive environment, various parameters like the frequency, R ratio, pH, dissolved O 2 , Cl -, electrode potential etc. have been observed to affect the fatigue life and FCGR.

Most of the studies in corrosive media have been done under uniaxial loading except few [START_REF] Misak | Corrosion fatigue crack growth behavior of 7075-T6 under biaxial tensiontension cyclic loading condition[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF][START_REF] Perel | Biaxial Fatigue Crack Growth Behavior in Aluminum Alloy 5083-H116 Under Ambient Laboratory and Saltwater Environments[END_REF]. These studies indicated different effects of biaxial-tension in salt water for different aluminum alloys in high-cycle fatigue regime, leading to no general idea. As a result, there is a need to explore this area.

Chapter 2

Material characterization

Clip connectors are made up of F22 low alloy steel whose microstructural and mechanical characterization are described in this chapter. The composition of this steel, according to ASTM A 182, is shown in table 2.1.

The ring rolling process is used to manufacture the clip connector, which is then roughly machined to the delivery dimensions. After that, it is heat-treated, which includes austenitization between 940℃ and 970℃, followed by water-quenching (this producing a martensitic structure) and finally tempering at around 625℃ (this leading to precipitationinduced hardening). At last, the parts are machined to the final dimensions of the clip connector. in the texture relative to that of a textureless material. The maximum relative intensity is 3.48. The crystallograhic texture is thus weak.

Microstructural characterization

However, many special Coincidence Site Lattice (CSL) grain boundaries were present.

The relation between the number of lattice points in the unit cell of a CSL and the number of lattice points in a unit cell of the generating lattice is called Σ (Sigma); it is

Mechanical characterization

Cylindrical specimens with a gage length of 16 mm and a diameter of 8 mm were cut from the clip connector and were used to characterize the tensile and cyclic behaviour of the material. An extensometer with a full range of ±15% was used. The tensile properties of F22 alloy steel identified by uniaxial test (figure 2.7) at constant strain rate, ˙ = 10 -3 , are shown in table 2.4. E and σ u are the Young's modulus and the ultimate tensile strength respectively. ν is the poisson's ratio, σ y,0.2% is the conventional yield strength at 0.2 % offset strain, σ p is the proportional limit, neck is the necking strain or uniform elongation and HV is the Vickers hardness. In order to model the cyclic behaviour of the material for numerical simulations on clip connectors, it was necessary to identify its constitutive equations. For that purpose, four fully reversed strain controlled tests (R = -1) at constant amplitudes of ±0.3%, ±0.75%

(two tests) and ±1.2% were run using sinusoidal wave form. A strain rate of ˙ = 10 -3 s -1

was chosen for all tests. In order to maintain the same strain rate at different strain amplitudes (∆ /2), different loading frequencies (f ) were thus chosen so that:

˙ = 2∆ f = 10 -3 s -1
The stress strain loops for different strain amplitudes are shown in figure 2.8. A progressive increase in the plastic strain range was observed. At ±0.3%, there was very small or no macroscopic plastic strain during the first cycle but the stress strain loop began to

Material characterization 51 open up as the number of cycles went on, figure 2.8(c). For the two higher strain ranges, plasticity occurred during the first ramp of loading with very limited hardening (that is very near to a perfectly plastic behavior) but during subsequent cycles, hardening became non zero. Figure 2.9 shows the variation of stress amplitude with the number of cycles.

It first decreased (depicting cyclic softening of the material) and then, remained constant for most part of the fatigue life until crack growth became unstable.

There are different ways to model cyclic behavior of material, using two types of hardening variables:

• Isotropic Hardening (R)

• Kinematic Hardening (X)

Isotropic hardening accounts for a change in size of the elastic domain while kinematic hardening accounts for the displacement of the elastic domain i.e. Bauschinger effect.

There are various mathematical laws in literature to model these hardening but most commonly used are the models proposed by [START_REF] Voce | The relationship between stress and strain for homogeneous deformation[END_REF] for isotropic hardening and by [START_REF] Armstrong | A Mathematical Representation of the Multiaxial Baushinger Effect[END_REF] for kinematic hardening.

Isotropic Hardening:

R = R 0 + Q(1 -e -b p ) (2.1)
Kinematic hardening:

δX = Cδ p -γXδp (2.2)
where R 0 , Q, b, C and γ are material constants and p and p are the plastic strain and accumulated plastic strain. The constant R 0 represents the yield stress of the material.

Q > 0 denotes hardening and Q < 0 denotes softening. Parameter b accounts for the rate of hardening or softening. The accumulated plastic strain is defined as:

δ ṗ = 2 3 ˙ p : ˙ p
where ':' represents the double inner product. For monotonic tension, | ṗ |=| ˙ p | or δp = δ p , which simplifies the expression of kinematic hardening to: which can be integrated to:

δX = (C -γX)δ p (2.3) R 0 , MPa Q, MPa b C 1 , GPa γ 1 C 2 ,
X = C γ (1 -exp(-γ p )) (2.4)
C/γ is the saturated value of kinematic hardening. The yield criterion, in which Von

Mises criterion can be used if the plastic behavior is isotropic, can be written as:

f (σ -X) -R = 0 (2.5)
Von Mises:

f (σ -X) = 3 2 (σ -X) D : (σ -X) D
where the superscript 'D' denotes the deviatoric part of the stress tensor. Sometimes more than one kinematic or isotropic hardening parameters may be required to model the material behavior. The constants defined in equation 2.1 and in equation 2.3 have to be identified using experimental stress strain loops. The Z-set code, ZéBulon, was used with Leverberg optimization technique [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF]; [START_REF] Moré | The levenberg-marquardt algorithm: Implementation and theory[END_REF]) in order to optimize the variables. In present work, one isotropic hardening variable and two kinematic hardening variables were used. a trendline given by: ∆ 2 (%) = 9.885 N -0.399 f A variable amplitude test was done at constant strain rate ( ˙ = 10 -3 s -1 ) in order to detect a potential memory effect in the material. The loading consists of 15 cycles at each strain amplitude in the order: ±0.2%, ±0.4%, ±0.6%, ±0.8%, ±1.0%, ±0.8%, ±0.6%, ±0.4% and ±0.2%. Figure 2.12 shows the evolution of stress range during the whole test. A comparison was made between the experiment and simulated stress amplitude plotted against cumulative plastic strain, figure 2.13. The constitutive equation, without any extra variable, predicted the experimental results quite nicely. Hence, it can be concluded that the material exhibits no memory effect and can be suitably modeled using the constitutive equations identified earlier. Another important parameter to be analyzed for design of clip connector is the plastic Mathematically, the local stress state at any element in the region below the lugs can be written as:

Material characterization 55

σ I = σ I,mean + σ I,a Sin ωt σ II = σ II,mean + σ II,a Sin ωt σ III ≈ 0 (3.1)
σ I , σ II and σ III are the three principal stresses in numerically decreasing order, and the subscripts 'mean' and 'a' are their corresponding mean and amplitude values.

In order to study the statistics of biaxiality ratio and mean stress in the critical area, a random set of approximately 200 nodes was chosen for both type of loading conditions, figure 3.7. The biaxiality ratio at each node was defined as:

B = σ II,a σ I,a
where σ a,I and σ a,II are as defined in equation 3.1. The R ratio has been defined independently for each principal stress as:

R I = σ I,min σ I,max and R II = σ II,min σ II,max
The statistical distributions of the biaxiality ratio (B) and the two R ratios (R I and R II ) at these nodes under the connected mode and the disconnected mode respectively, are shown in figure 3.8. In the connected mode, the far field R ratio was 0.7 while in the disconnected mode it was 0.2. It can be seen that at most of the points, the R ratio is almost equal to applied one, for both principal stresses. Many points exist with a biaxiality ratio between 0.1 and 0.3 (82% for connected mode and 81% for disconnected mode).

In order to precise the effect of biaxiality, the distribution of load biaxiality was plotted against the corresponding Von-mises equivalent stress range for the two types of loadings, as shown in figure 3.9. The most critical points could be the ones with a high biaxiality ratio and a high Von-mises equivalent stress range values.

Hence, the fatigue behavior of this material was investigated by performing experiments under in-phase biaxial tension loading (combined tension and internal pressure tests) in this range of biaxiality ratio and the mean stress (R ratio).

Experimental results

Fatigue lives

The resulting fatigue lives are reported in table 4.1 and plotted as Whöler curves (S-N curves) to illustrate the effect of mean stress on fatigue life, figure 4.3. For R ≤ 0.25, tests were stopped at 3,000,000 cycles and the corresponding stress amplitude was defined as the endurance limit for that R ratio. For R > 0.5, tests were run until 10,000,000 cycles and no fatigue failure could be observed. The endurance limit as well as slope of the curves tends to drop with increasing R ratio. Horizontal arrows denote the corresponding run-out tests.

In some tests, cracks were found to initiate from defects (internal and surface) in the material, as discussed in section 4.2.2.

Another representation of fatigue test data is the Goodman plot (Amplitude-mean stress plane) shown in figure 4.4. The Goodman plot can be divided into three domains: safe domain, fatigue failure and ductile failure respectively. Ductile failure occurs when the amplitude or the mean stress crosses the line: σ a + σ m = σ u . Fatigue data has been Influence of the mean stress 67 compared with Gerber's parabola [START_REF] Gerber | Calculation of allowable stresses in iron structures[END_REF]:

σ a,R = σ -1 1 - σ m σ u 2 (4.1)
σ a,R denotes the amplitude corresponding to endurance limit at given mean stress, σ m .

σ u and σ -1 denotes ultimate tensile stress and the fatigue strength (at 3 × 10 6 cycles)

under fully reversed push-pull test (R = -1). Gerber's parabola is found to provide an accurate prediction of the endurance limit at different R ratios.

At high mean stresses, although no macroscopic plasticity was observed beyond the first ramp, but the ratcheting did occur. Figure 4.5 shows the measured engineering stressstrain loops for two high R ratios: one for which fatigue failure occurred at 459, 584 cycles (for R = 0.5) and one for which there was no fatigue failure (for R = 0.7). Shakedown was always achieved, well before the fatigue fracture with up to 3% accumulated strain which did not induce significant difference between the engineering stresses and the true stresses. That is why, engineering stresses were used in the whole manuscript.

Damage mechanisms

To study the crack initiation and propagation mechanisms under different mean stress levels, the fracture surfaces of broken specimens were analyzed using a Keyence digital optical Microscope in addition to the Scanning Electron Microscope (SEM).

The analysis of fracture surfaces revealed the presence of four types of fatigue crack initiation mechanisms in this steel:

• Surface crack initiation merely due to local plasticity.

• Surface crack initiation from surface-cutting defects.

• Internal crack initiation from pores. The region within the fish eye appeared to be much smoother as compared to that in air. No striations were found in this region but as soon as the crack propagated in air, striations could be observed easily. This is in accordance with the results available in the literature [START_REF] Mcevily | Fatigue Crack Tip Deformation Processes as Influenced by the Environment[END_REF]. The shape of the fish eye is circular, following the contours of iso-stress intensity factor. SEM observations did not show the presence of any Optically Dark Area (ODA) or any Fine Granular Area (FGA), which is in consistent with the observations made by [START_REF] Li | Surface vs. interior failure behaviors in a structural steel under gigacycle fatigue: Failure analysis and life prediction[END_REF] according to which they were observed only for very high fatigue lives and tend to disappear when the R ratio increases [START_REF] Nakamura | The effect of vacuum-like environment inside sub-surface fatigue crack on the formation of ODA fracture surface in high strength steel[END_REF].

A few detailed observations of fracture surface in and around the fish-eye were done after platinum deposit and Focus Ion Beam (FIB) sectioning, using a FEI Helios 660 dual- be observed outside the fish-eye, a much smoother crack profile appeared inside the fish eye (R p ≈ 0.6 and 1 µm at 128 µm and 186 µm from the center, respectively) without secondary cracks. No modification of the mean grain size was observed below the fish eye surface.

Crack initiation from internal chemical in-homogeneity

In some specimens, cracks had also been found to initiate from material inhomogeneity. SEM images of both fracture surfaces of a specimen in which the crack initiated from an in-homogeneous area are shown in figure 4.12. There was a crater on one of the fracture surfaces while on the other one there was no such crater, rather some irregular accumulation of non-homogeneous material. Chemical analysis was performed using the Energy Dispersive Spectrum (EDS) device available in the SEM at two places indicated in figure 4.12(a,b) and the corresponding spectra obtained in these locations are shown in figure 4.12(c,d). In the non-homogeneous part (spectrum 4), the oxygen and carbon contents were higher than in the matrix (spectrum 6). The chromium and molybdenum contents were also slightly higher than the average. EDS technique has its limitation in quantification of carbon because it is a light element. Hence, the values for carbon content are not reliable but there is no doubt that the carbon content is not uniform in this area.

Influence of the mean stress 75 This oxygen and carbon rich matter behaves like an inclusion with different chemical composition and thus different mechanical properties than the matrix. De-bonding of this heterogeneous part from the matrix initiated a crack leaving a crater on one of the fracture surfaces.

To characterize the location of the defects, an eccentricity parameter, e, has been defined

as:

e = Ω -ω Ω (4.2)
where Ω is radius of the cylinder and ω is the distance from the center of the fish eye to the surface as described in figure 4.13. Eccentricity will be 0 for a defect located at the center of the specimen and close to 1 when it is located near to surface. The size and location of the defects responsible for internal initiation are tabulated in table 4.2. The diameter of the fish eye is represented by D and the major (2a) and minor (2b) axes of the defects are indicated, assuming an elliptical shape.

When e is close to 1, no fish eye exist and ∆K is not reported. All the defects responsible for internal crack initiation were found near the surface (between 165 µm and 600 µm from the surface) and not randomly distributed on the fracture surface as expected for This suggests that machining-induced residual stresses, more or less redistributed by plastic flow during the first half cycle at high R ratio, were present in the specimens and might played a role in the competition between surface and internal crack initiation.

Residual stress analyses

In order to clarify this question, XRD analyses of surface residual stresses were performed at LAMPA Angers, using a PROTO iXRD diffractometer [START_REF] Achouri | Influence of the edge rounding process on the behaviour of blanked parts: Numerical predictions with experimental correlation[END_REF] equipped with a Cr anode (figure 4.14). For this anode material, the penetration depth (63% of absorption) of X-rays in steel is about 5 µm. The peak positions associated with {2 1 1} planes were measured with two scintillation detectors with a 2θ range of 18.5°each.

For each analysis, the irradiated zone was delimited with a circular collimator (0.5 mm diameter). Residual stresses were estimated from peak positions using the Sin 2 ψ method [START_REF] Noyan | High-cycle rotating bending fatigue property in very long-life regime of high strength steels[END_REF]. The X-ray elastic constants (s 1 = -1.28 × 10 -6 MPa -1 and 1 2 s 2 = 5.83×10 -6 MPa -1 ) identified for α-ferrite were used to compute the residual stress state. A preliminary calibration of the device and of the post-treatment parameters done on F22 steel powder (without residual stresses) had shown that the positions of the diffraction peaks are consistent with these elastic constants. For virgin specimens, the analyses were done for three equally-spaced surface points along the periphery, at mid-height of the gage length (denoted by a,b and c in table 4.3) and revealed the presence of significant biaxial compressive residual stresses, scattered between -338 MPa and -407 MPa in the axial direction and between -208 and -345

X-Ray source

MPa in the circumferential direction.

To estimate the in-depth distribution of longitudinal residual stresses, successive material removal operations to a depth of 150 µm were performed using an electrochemical polishing technique with a solution of salted water. An in-depth profile of residual stresses

Influence of the mean stress 79 which is smaller than the depth of all internal crack initiating defects.

As mentioned above, limited "ratcheting", without any detectable cyclic plasticity, was observed and suspected to correspond to a relaxation of residual stresses. Thus, this potential relaxation was examined by measuring the residual stresses on the tested specimens, as far as possible (> 6 mm) from the fracture surface. Based on the superposition of the loading cycle with the maximum axial and circumferential residual stresses found in virgin specimens (-407 and -329 MPa, respectively), the minimum and maximum

Von-Mises equivalent stresses (σ Mises ) over a cycle were computed and compared with the proportional limit (σ p = 600 MPa): If either the min or max equivalent stress exceeded the proportional limit, the residual stresses were expected to relax. The previsions made with this simple criterion are compared with the residual stresses (σ r ) estimated after the fatigue tests for various R ratios and amplitudes (see table 4.4). In spite of a larger scatter in the measurements than on virgin materials, the following trends were found:

• For R = -1, the residual stresses were expected to relax, but compressive residual stresses exceeding 200 MPa were observed along the axial direction.

• For R = -0.5, 0 and 0.25, relaxation was neither expected nor observed. • For R = 0.5, stress relaxation was expected only for the highest stress range but it seems to be partially relaxed only.

• For R = 0.7, relaxation was expected and indeed observed.

Discussion

Crack initiation

Surprisingly, the significant surface compressive residual stresses that partially or totally persisted after the tests did not inhibit surface crack initiation at R = -1, -0.5 or 0. In addition, when internal crack initiation occurred, the depth of the responsible defects were higher than that of the tensile peak of longitudinal residual stresses, which according to figure 4.15(b) is less than 60 µm. Thus, the transitions from a majority of cyclic plasticity-induced surface crack initiation for R = -1, -0.5 and 0 to a majority of internal crack initiation from a defect at R = 0.25 and then to defect-induced surface crack initiation for R = 0.5 do not seem to be controlled by residual stresses and their relaxation.

Influence of the mean stress 81

The potential role played by the changes in specimen's geometry in these transitions must be examined. For a uniform distribution of defects, the larger the gage volume, higher is the probability to find a large internal defect inside. In the 7 mm diameter specimens used for the tests at R = 0.25, the gage volume and thus this probability were smaller than in the 8 mm diameter specimens used for the tests at lower R ratio by a factor of 0.72. In spite of this, the proportion of internal crack initiation (83%) was higher than in the previous cases. This effect, although it is based on a limited number of tests, thus suggests a mechanical effect of the higher R ratio and of the associated decrease in stress range.

For the tests ran at R = 0.5 and above, with diameter of 6 mm, the gage surface-to-volume ratio was increased by a factor of 1.17 compared to the specimens tested at R = 0.25.

This should increase the probability for a large defect to be located on the surface rather than in the volume in the same proportion. Such a moderate effect cannot explain a change from 83% internal cracks at R = 0.25 to 100% defect-initiated surface cracks at R = 0.5.

Alternative explanations for this transition have been suggested below by a compared analysis of the stress and strain fields around an internal or surface-cutting inclusion and fracture mechanics arguments.

The analytical solution of [START_REF] Goodier | Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws[END_REF] for the stress field around a spherical pore in an infinite elastic medium, yields a stress concentration factor K t of 2.04 under uniaxial tension. For the stress amplitudes corresponding to the endurance limit at R ≥ 0, this is sufficient to induce monotonic plastic flow (2.04σ max > σ p ) around a pore, and in some cases, cyclic plastic flow and ratcheting (2.04∆σ > 2σ yc , where σ yc denotes the cyclic yield stress, substantially lower than σ p in this steel, see table 2.4).

To analyze the stress and strain fields around an internal pore, a 3D finite element unit cell model containing 1/8 th of a spherical pore was developed using quadratic hexahedron type elements (C3D20R) (figure 4.16(a)). Symmetric boundary conditions were applied on the faces adjoining the pore. Elastic-plastic simulations were run for 100 cycles of the loadings leading to internal crack initiation. The constitutive equations identified earlier, were used. For the example at ∆σ = 520 MPa and R = 0.25, the equivalent plastic strain at the edge of pore reached 1.8% at the first peak load and ratcheting up to 2.5% strain was observed. After 100 cycles, the steady-state normal strain range was 0.5% and the local R ratio was close to -1, which is quite different from the far field stress ratio of 0.25, as illustrated in figure 4.16(b),. Influence of the mean stress 83

Similar elastic-plastic F.E. computations were run for a surface-cutting pore. For that purpose, the same mesh was used, but a symmetry condition was removed, to represent the free surface. Due to a plane stress condition, the plastic strains reach higher values at the pore edges than for an internal pore under the same far field loading. At the first peak load, the equivalent plastic strain at the edge of a surface pore is 2.0% as compared to 1.8% for an internal pore and after ratcheting during 100 cycles, it reaches 2.7% instead of 2.5% (figure 4.16(c)). However, the steady-state normal strain amplitude is same in both cases, i.e. 0.3%.

Even though the HCF tests were stress-controlled at a given R ratio, the F.E. computations clearly show that close to a pore, the local loading conditions are quite different and closer to strain-controlled low-cycle fatigue. That is why fatigue data collected in such conditions are considered to provide a reasonable baseline. Based on these fatigue data, the strain amplitude computed close to a pore should lead to crack initiation after a few thousand push-pull cycles only, in air. The lack of moist air and the sharp strain gradient on the one hand, and the higher stress triaxiality around an internal pore, on the other hand, certainly make fatigue crack initiation kinetics different from those in push-pull in air, but considering the above mentioned peak strain and amplitude, it seems quite improbable that crack initiation from internal pores controls the fatigue lives at R = 0.25.

Internal chemical heterogeneities produce smaller stress concentrations than internal pores of similar shape and size [START_REF] Goodier | Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws[END_REF]. However, the data-points for failures initiated from the former or from the latter do not segregate on the Wöhler plot. This supports the idea that crack initiation constitutes a small fraction of the fatigue lives at R = 0.25. For the VHCF regime, there is a consensus about the predominance of the crack initiation stage from internal defects over the crack growth stage, such a conclusion does not seem to hold here.

Simulations were also run at R = 0.5 for ∆σ = 360 MPa for a surface and internal pore respectively. At first peak load, an equivalent plastic strain of 3.9% (and 3.6% resp.) was found at the edge of the pore with ratcheting up to 4.1% (and 3.9% resp.) within 100 cycles. The normal strain amplitude at the 100 th cycle was 0.2% in both cases, and the local R ratio -0.5 (and -0.4 resp.). These local conditions would probably be sufficient for early internal initiation from a pore at this R ratio as well.

Similar simulations ran for an internal pore at R = 0.6 and the highest stress amplitude (154 MPa) used in the experiments yielded an equivalent plastic strain of 4% on the edge of pore at first peak load but no ratchetting, due to the absence of cyclic plasticity.

So, for R ≥ 0.6, fatigue crack initiation seems improbable, in spite of the local stress concentration provided by a surface or internal pore (and even more, a surface or internal inhomogeneity). Anyway, should a crack initiate, it would probably be unable to grow, as discussed in the next section.

Crack growth

Mechanisms

The reduced roughness and the absence of striations on the fracture surfaces within the fish-eye are consistent with the literature on fatigue crack growth in vacuum [START_REF] Stewart | The influence of environment and stress ratio on fatigue crack growth at near threshold stress intensities in low-alloy steels[END_REF][START_REF] Mcevily | Fatigue Crack Tip Deformation Processes as Influenced by the Environment[END_REF]. The latter attribute the absence of striations in stainless steels fatigued in vacuum to the diffuse crack tip plasticity observed in this environment. The relatively large blunting of the tip resulting from such plastic flow would reduce the crack growth rate. By contrast, Hydrogen-Enhanced-Localized-Plasticity (HELP) [START_REF] Beachem | A new model for hydrogen assisted cracking (hydrogen embrittlement)[END_REF][START_REF] Birnbaum | Hydrogen-enhanced localized plasticity-a mechanism for hydrogen-related fracture[END_REF][START_REF] Lynch | 2 -Hydrogen embrittlement (HE) phenomena and mechanisms[END_REF] in moist air would induce coarse, localized crack tip plasticity, producing clear striations and leading to faster crack growth.

Note that all the defects responsible for internal crack initiation were found near the surface and that the border of each fish-eye was tangent to the free surface (as visible in figure 4.9). The frontier of the fish-eye thus corresponds to a sudden ingress of the outside air into the crack.

The presence of intergranular facets on fatigue fracture surfaces formed at low crack growth rates in Ni-Cr-Mo-V and Cr-Mo steels has often been reported as a clear sign of combined temper and hydrogen embrittlement [START_REF] Stewart | The influence of environment and stress ratio on fatigue crack growth at near threshold stress intensities in low-alloy steels[END_REF][START_REF] Islam | Intergranular fracture on fatigue fracture surface of 2.25Cr-1Mo steel at room temperature[END_REF][START_REF] Ritchie | Influence of impurity segregation on temper embrittlement and on slow fatigue crack growth and threshold in 300-M high strength steel[END_REF][START_REF] Hénaff | Environmental influence on the nearthreshold fatigue crack propagation behaviour of a high strength steel[END_REF][START_REF] Liaw | Effect of load ratio and temperature on the near-threshold fatigue crack propagation behaviour in a CrMoV steel[END_REF]. [START_REF] Islam | Intergranular fracture on fatigue fracture surface of 2.25Cr-1Mo steel at room temperature[END_REF] have shown that a preliminary segregation of impurities during the heat treatment is necessary for this phenomenon, but that hydrogen released by a reaction of water vapor with the metal is also necessary and that the latter readily occurs in air at room-temperature and low crack growth rate.

In the present study, the absence of any intergranular or inter-lamellar facet or secondary crack within the fish-eye areas but their presence just outside the fish-eyes (figure 4.17) is consistent with this conclusion. It is also consistent with the lower roughness within the fish-eye than outside, since [START_REF] Liaw | Effect of load ratio and temperature on the near-threshold fatigue crack propagation behaviour in a CrMoV steel[END_REF] underlined that the presence of such Influence of the mean stress 85 the crack grows further from an internal defect with increasing ∆K. However, roughnessinduced crack closure effects (RICC) are certainly less important during crack growth inside the fish-eyes than outside. In addition, oxide-induced crack closure as well as plasticity-induced crack closure (PICC) is absent inside fish-eye. The latter can be expected because of the low ∆K, plane strain/stress state for internal cracks and short crack lengths (although much larger than the former austenite grain size of 7 µm, so that LEFM can reasonably be used). To check that assumption, an axisymmetric FE model of penny shaped crack with a radius typical of that found for internal cracks was developed, with sufficient mesh refinement ahead of the crack tip (less than 1/10 th of the reversed cyclic plastic zone size). Elastic-plastic simulations with periodic node release were run for similar cyclic loadings and R ratios, as those leading to internal cracking in the experiments (without taking into account the residual stresses). The closure level was evaluated at the second node behind the crack tip and no plasticity-induced closure was observed.

Intergranular facets

In summary, in the present study, internal cracks not only seem to grow with very few, if any, closure effects, but also without any assistance from moist air.

Threshold and kinetics

The cracks grown from internal defects were considered as elliptical, their major/minor semi-axes: 'a' and 'b', were measured and the stress intensity factor was computed as [START_REF] Murakami | Stress Intensity Factor Handbook[END_REF]:

∆K I,max = ∆σ √ πb E(k) (4.3) E(k) = π 2 0 (1 -k 2 sin 2 φ) 1/2 dφ k 2 = 1 - b a 2
The maximum value for an elliptical crack occurs at the end point of the minor axis, "b".

The stress intensity factor range is calculated by taking into account only the tensile part of the stress. Both the initial and final ∆K I (considering the defect as a crack in the former case and at the border of the fish eye in the latter) have been calculated and are reported in table 4.5.

The ∆K I values computed with the approximate expression from [START_REF] Murakami | Metal Fatigue: Effects of small defects and nonmetallic inclusions[END_REF] (equation 4.4) were less than 8% different from those computed with equation 4.3.

∆K I = βσ max π √ area (4.4)
where "area" denotes the area of the defect projected on the plane normal to the tensile stress, β = 0.5 for an internal defect and β = 0.65 for a surface defect.

The initial ∆K I is almost constant for all defects observed and lies between 3. In particular, the values found for the two specimens broken at R = -0.5 are close to Influence of the mean stress 87 4.5: Initial and final Stress intensity factor range and estimated number of cycles for crack propagation within the fish eye (N f ish ). Specimens marked (1,2) survived 3 million cycles and were reloaded. In that case, crack growth from the last beach mark was considered.

the specimens broken at R = 0.25. This is consistent with the idea of [START_REF] Stewart | The influence of environment and stress ratio on fatigue crack growth at near threshold stress intensities in low-alloy steels[END_REF] and [START_REF] Vasudevan | Analysis of vacuum fatigue crack growth results and its implications[END_REF], that the threshold ∆K for crack growth in vacuum is much less dependent on the R ratio than in air, although both studies, contrary to the present one, considered only positive R ratios. Anyway, based on the threshold of 7.1 MPa √ m that [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF] obtained in air at R = 0.05, and the usually accepted idea that vacuum makes crack growth slower and the threshold higher, the internal cracks found here at R = -0.5 should not propagate. The fact that these cracks actually grew at such a low ∆K in spite of a lower R ratio suggests either that the reduction in crack closure compensates (or even overcompensates) the lack of environmental assistance to crack growth or that some moist air trapped in the pore assists incipient crack growth. However, no crystallographic, intergranular or interlamellar area, expected in case of environment-assisted crack growth was observed around the defects, but a smooth featureless surface, so that the first explanation is preferred.

An attempt to use the crack growth kinetics measured by [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF] in air or reducing atmosphere at R = 0.75, which probably corresponds to an intrinsic, closure-free crack growth kinetics to evaluate the number of cycles spent in internal crack growth from the defect to the border of fish-eye is made. The curve was thus digitized and integrated stepwise (see appendix Appendix A). The result, denoted by N fish is reported in table 4.5.

It can be seen that N fish does not constitute a negligible fraction of the measured fatigue lives: it reaches 39% for specimen 18.

However, the real growth rates of internal cracks growing without any assistance of moist air is probably significantly lower than suggested by these crack growth data obtained in air.

Beach marks observed on the fracture surface within the fish eye of two specimens (23 and 26) submitted successively to different loading amplitudes provide some information on crack growth kinetics that support this assumption. Specimen 23 survived 3 million cycles at σ a = 300 MPa, R = 0 and was then reloaded at σ a = 260 MPa, R = 0.25.

It failed after only 373,933 cycles. Figure 4.19a shows that a beach mark separates the areas corresponding to those two loading blocks. From the position of the beach mark, it can be deduced that under the second loading, it took 373,933 cycles for the crack to grow by 87 µm. The average crack growth rate is thus 2.9 × 10 -10 m/cycle while ∆K increased from 5.6 to 7.5 MPa √ m so that the mean ∆K is 6.55 MPa √ m. Specimen 26 survived 3 million cycles at σ a = 240 MPa, σ a = 245 MPa and σ a = 250 MPa and finally failed after 1,704,174 cycles at σ a = 260 MPa. For a better analysis of internal crack growth, crack growth kinetics collected in vacuum would thus be more pertinent which is currently not in scope of present thesis.

For R = 0.5, ∆K was found to be 3.7 to 3.9 MPa √ m for surface defects as shown in figure 4.8(b). No threshold data is available for crack growth in air at that R ratio in the paper by [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF], but according to the extensive ∆K threshold versus R database gathered by [START_REF] Bulloch | The influence of mean stress or R ratio on the fatigue crack threshold characteristics of steels-A review[END_REF] for steels with a tempered martensite microstructure, these values are still compatible with crack growth and thus fatigue failure, as observed. Comparatively, according to equation 4.4, ∆K I for a crack initiated from an internal defect of similar size would be between 2.85 and 3 MPa √ m, which is clearly below the scatter-band of threshold ∆K values reported by [START_REF] Bulloch | The influence of mean stress or R ratio on the fatigue crack threshold characteristics of steels-A review[END_REF] for this R ratio in air, and also well below the smallest ∆K computed for any of the internal cracks observed (see figure 4.18).

This explains why at R = 0.5, only cracks initiated from surface defects were observed.

By contrast, the ∆K computed for the cyclic loading conditions used at R = 0.6 or 0.7, assuming the presence of the most critical observed defect in a specimen (which is 199 µm by 66 µm wide), comes out to be around 2 MPa √ m. Comparing it with the 2.8 MPa √ m threshold measured, in the absence of closure by [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF] at R = 0.75 in air and dry H 2 , it seems that such defects would not propagate. This explains why specimens at very high R ratio did not fail from defects by fatigue within 10 7 cycles only.

Concluding remarks

A decrease in the slope of Wöhler curves and an increased scatter in fatigue lives were observed when the R ratio rose. Similar effects have been reported in literature (see for example fig. 6.29 in [START_REF] Murakami | Metal Fatigue: Effects of small defects and nonmetallic inclusions[END_REF]). It can be suggested that for R < 0 and Influence of the mean stress 91 thus relatively high stress ranges, that allow the threshold ∆K for crack growth to be surpassed easily for very small cracks, the crack growth rates and thus the fatigue lives vary significantly with the loading range. By contrast, for higher R ratios and thus smaller stress ranges, the fatigue lives become more and more dependent on the presence of defects large enough to allow the threshold ∆K to be reached. The scatter in fatigue lives increases and the slope of Wöhler curves decreases, which makes the safe fatigue design of structures tricky. It might be reasonable to consider R-ratio dependent safety factors but [START_REF] Murakami | Metal Fatigue: Effects of small defects and nonmetallic inclusions[END_REF] advised not to use such (more or less arbitrary) safety factors in design, rather a lower bound for the endurance limit issued from a statistical analysis of defects size distribution. Although, such an approach is more rational and based on fracture mechanics, it is however not clear if it would capture the R-ratio dependence of the scatter in fatigue lives.

Finally, a tentative map of the mechanisms that control the fatigue strength at 3 million cycles depending on the mean stress or R ratio is shown in figure 4.20. For low mean stresses, surface crack initiation merely due to cyclic plasticity predominates, but when the mean stress rises and the amplitude decreases, the stress concentration provided by a defect becomes necessary for crack initiation. At intermediate R ratios and amplitudes, cracks are able to grow from surface or internal defects, but as the R ratio is further increased (to 0.5 in the present case) and the amplitude decreased, only the cracks grown in air from surface defects which, according to equation 4.4, undergo a higher ∆K than those from internal defects of similar size can cause failure. For even higher R ratios, the amplitude becomes too small to allow any crack to grow, be it in air or in vacuum.

Note that this map would change if a larger number of cycles was chosen to define the endurance limit: the defect-controlled fracture domain would probably extend to lower mean stresses/R ratios. The map, which is plotted for uniaxial loading, would probably change for biaxial loading, because the stress concentration around a defect depends on the far-field stress state [START_REF] Goodier | Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws[END_REF]. For example, the K t around an internal spherical pore is 1.36 under equibiaxial tension instead of 2.04 in uniaxial tension. The defect-controlled fracture domain is then expected to shrink.

Influence of the mean stress 93

Chapter 5

Effect of biaxial tension in air

Combined cyclic tension and internal pressure tests with various proportions of each loading were run to investigate the effect of positive stress biaxiality on fatigue lives and damage mechanisms.

The ability of various fatigue criteria to capture the effects of mean stress and biaxiality was then evaluated. For that purpose, the endurance limit of the material in reversed torsion was also determined. The endurance limits predicted by existing criteria appeared inconsistent with the experimental data, so that a new criterion had to be formulated.

Experimental procedures

Tubular specimens (figure 5.1a) with 25 mm outer diameter, thickness e = 1 mm and R a = 0.2 µm internal/external surface roughness were used for load/pressure-controlled, in-phase tension and internal oil pressure tests run at 10 Hz with R = 0.25 on a triaxial testing machine (±100 kN, ±600 Nm, 1500 bars).

In an infinite tube of inner/outer radius R 1 and R 2 an internal pressure P induces at a point of radius r, a tangential stress (σ θθ ) is given by:

σ θθ (r) = P R 2 1 R 2 2 -R 2 1 + P R 2 1 R 2 2 R 2 2 -R 2 1 1 r 2 (5.1)
which is maximum on the inner surface. The ratio of the inner to outer tangential stress is thus:

σ θθ,max σ θθ,min = 0.5 1 + R 2 R 1 2 (5.2)
For the present geometry, this ratio is equal to 1.08. As it will be shown below, it explains a preference for crack initiation on the inner surface for equibiaxial loading. However, this weak hoop stress gradient was ignored to chose the axial load range ∆F and pressure range ∆P, so as to get the desired biaxiality ratio (B), defined as:

B = ∆σ θθ ∆σ zz = ∆P Rm e ∆F S + ∆P Rm 2e (5.3)
where R m is the mean radius and S is the cross section of the tubes. Strain gage rosettes were used to check the consistency of expected and measured stress states in the gage length within the elastic regime and to determine Poissons ratio, ν, and Youngs moduli E z and E θ along the longitudinal and transverse directions. Less than 3% difference was found between E z (214 GPa) and E θ (220 GPa).

The tests were performed at various biaxiality ratios (0, 0.25, 0.5 and 1) and run until a through crack with oil leakage was formed, or until specimen ran out at 3.10 6 cycles. Where more than 1 specimen ran out, the endurance limit was assumed to be the maximum stress amplitude below which none failed. Tubular specimens (figure 5.1b) with 14 mm outer diameter, 0.8 mm thickness, 16 mm gage length, and R a = 0.2 µm internal/external surface roughness were used for torquecontrolled reversed torsion tests run at 10 Hz, on the same testing machine until the complete separation of the specimens. Where more than 1 specimen ran out after 3.10 6 cycles, the endurance limit was assumed to be the maximum stress amplitude below which none failed.

Scanning Electron Microscopy (SEM) was used for fractographic observations, in order to investigate the damage mechanisms. A digital optical microscope was used to perform topographic measurements along the crack paths.

Experimental results

Cyclic biaxial tension

Biaxial tension fatigue results are reported in table 5.1. Figure 5.2a compares the evolutions of fatigue lives with the amplitude of the maximum principal stress, for different biaxiality ratios, B, including those obtained previously on cylindrical specimens for B = 0. A few uniaxial fatigue tests on tubular specimens at R = 0.25 have shown no significant difference in lives with cylindrical specimens.

In spite of a large scatter, the asymptote of the Wöhler curve for B = 0.25 seems to lie above that for B = 0, suggesting a beneficial effect of that specific stress biaxiality, while the contrary seems to hold for B = 1. The scatter in fatigue lives for B = 0.5 does not allow a firm conclusion, but most fatigue data points for this biaxiality lie above the uniaxial Wöhler curve, and even above the curve for B = 0.25, suggesting a more pronounced beneficial effect. This trend is consistent with results from [START_REF] Bellett | A Biaxial fatigue specimen for uniaxial loading[END_REF] 98 Effect of biaxial tension in air For B= 0, data obtained previously on cylindrical specimens have been added.

Effect of biaxial tension in air 99

Note that in plane stress, Von-Mises equivalent stress range (∆σ eq ) is a non-linear function of load biaxiality, B:

∆σ eq = ∆σ zz √ 1 -B + B 2 (5.4)
It decreases from ∆σ zz for B = 0, to 0.90∆σ zz for B = 0.25 and down to a minimum value of 0.87∆σ zz for B = 0.5, before going up again, to ∆σ zz , for B = 1.

The seemingly beneficial effect of load biaxiality for B = 0.25 or 0.5 could thus be expected to rationalize when the equivalent stress is used to correlate the fatigue lives.

But this is not the case, as shown in figure 5 All biaxial fatigue failures initiated from the surface, and in many cases, due to a surfacecutting defect. The orientation of the crack (longitudinal or transverse) and its origin (internal/external surface of the tube) are reported for each test in table 5.1. For predominant axial loading (B = 0.25 or 0.5), transverse cracks initiated either from the inner or the outer surface of the tubes, while for equibiaxial tension, longitudinal or transverse cracks initiated, and the latter most often from the inner surface of the tubes. This is probably due to the slight gradient in hoop stress, 8% higher on the internal surface than on the external one.

For the cases where the crack initiated from the inner diameter of the tubes (B = 1 only), the gradient in hoop stress (higher on inner diameter) was taken into account and the fatigue data was replotted as shown in figure 5.3. But the conclusion (slight detrimental effect of equibiaxial tension) remained unchanged.

Fractographic observations revealed the presence of a many intergranular facets among areas of transgranular crack growth (figure 5.4a). To evaluate the surface fraction of intergranular facets along the crack path, through the depth, high resolution SEM images were captured with a magnification of 400. The line fraction of intergranular facets intercepted by a 50 µm-long radial segment (so that ∆K I does not change significantly over this segment) was measured at various depths and plotted against ∆K I (figure 5.4b).

100 Effect of biaxial tension in air ∆K I was computed with Newmann and Rajus equation for a surface crack in a plate of finite thickness loaded in tension [START_REF] Newman | Stress-intensity factor equations for cracks in threedimensional finite bodies subjected to tension and bending loads[END_REF], assuming that the crack is semi-circular. The latter assumption is supported by fractographic observations. According to Linear Elastic Fracture Mechanics, a secondary stress parallel to a crack (T Effect of biaxial tension in air 101 A digital optical microscope was used to measure and compare the roughness of fracture surfaces, but no significant difference in peak roughness values along the crack path, from the initiation site to the other surface of the tube, was found between specimens tested at the same axial stress amplitude (270 MPa), but different load biaxiality ratios (figure 5.5).

Reversed torsion

The results of the fully reversed torsion tests are summarized in table 5.2 and the corresponding Wöhler curve is plotted on figure 5.6. Out of nine tests, seven gave rise to transverse, shear-mode failure, and two specimens did not fail until 3 million cycles.

These data allowed an estimation of the endurance limit for torsion τ -1 at 250 MPa, which was used to calibrate a few fatigue criteria, as discussed below. faces were severely worn and so, was very difficult to analyze them under SEM or optical microscope.

Effect of biaxial tension in air 103 5.3 Discussion

Crack initiation

In the high-cycle fatigue regime investigated here, crack initiation, controlled by microscale plasticity, would certainly constitute a large fraction of the fatigue lives for a defect-free material. Since plasticity is related to the equivalent stress, biaxial tension with B = 0.25 (∆σ eq = 0.90∆σ zz ) and B = 0.5 (∆σ eq = 0.87∆σ zz ) could be expected to delay crack initiation, compared to uniaxial and equibiaxial loadings with the same ∆σ zz .

However, the present steel contains metallurgical defects (pores and chemically inhomogeneous areas, as detailed in previous chapter) and were found responsible for many crack initiations (see table 5. On the other hand, for B = 1, a "pseudo size effect" might partly explain the detrimental effect observed for this particular biaxiality. In high cycle fatigue, life is to a large extent, controlled by the number of cycles needed for microcracks to cross the first grain boundaries that they encounter, which depends on the relative misorientations of the grains [START_REF] Zhang | The Effect of Grain Boundaries on the Development of Plastic Deformation Ahead of Small Fatigue Cracks[END_REF]. Fatigue crack initiation thus depends on the presence in the first principal plane, on each side of a grain favourably oriented for cyclic plasticity and crack nucleation, of moderately misoriented grains. Under equibiaxial tension, all planes are equi-probable for crack initiation, so that the probability to find an easy path for crack propagation should be higher.

In summary, for a given axial stress range, crack initiation either from a defect or from the surface, should be postponed for B = 0.25 and even more, so for B = 0.5, compared to B = 0. For B = 1, even though crack initiation from surface-cutting defects should be slightly more difficult than under uniaxial loading, the crack initiation from slip bands should not be affected, and the chance that one of the initiated microcracks propagates instead of being arrested at a grain boundary should be higher.

104 Effect of biaxial tension in air Effect of biaxial tension in air 105

Crack growth

The presence of intergranular facets on fatigue fracture surfaces in Ni-Cr-Mo-V and Cr-Mo steels loaded in air has often been reported as a clear sign of combined temper and hydrogen embrittlement [START_REF] Hénaff | Environmental influence on the nearthreshold fatigue crack propagation behaviour of a high strength steel[END_REF][START_REF] Ritchie | Influence of impurity segregation on temper embrittlement and on slow fatigue crack growth and threshold in 300-M high strength steel[END_REF][START_REF] Islam | Intergranular fracture on fatigue fracture surface of 2.25Cr-1Mo steel at room temperature[END_REF][START_REF] Stewart | The influence of environment and stress ratio on fatigue crack growth at near threshold stress intensities in low-alloy steels[END_REF]. [START_REF] Islam | Intergranular fracture on fatigue fracture surface of 2.25Cr-1Mo steel at room temperature[END_REF] have shown that a preliminary segregation of impurities during the heat treatment is necessary for this phenomenon, but that hydrogen released by a reaction of water vapor with the metal at the crack tip is also necessary.

Similar evolutions of the density of intergranular facets with ∆K as those obtained here, have been reported in the literature, for uniaxial fatigue [START_REF] Zhu | Effect of microstructure on appearance of near-threshold fatigue fracture in Cr-Mo-V steel[END_REF][START_REF] Irving | Measurements of intergranular failure produced during fatigue crack growth in quenched and tempered steels[END_REF][START_REF] Cooke | The slow fatigue crack growth and threshold behaviour of a medium carbon alloy steel in air and vacuum[END_REF]. Some authors considered that the peak density of intergranular facets occurs at the ∆K value for which the reversed plastic zone size is equal to the prior austenite grain size. Below this ∆K, Stage I-like crystallographic fatigue crack growth would favour hydrogen transport towards the grain boundaries by the dislocations emitted at the crack tip, contrary to the more diffuse crack tip plasticity observed above this transition value. In the present case, assuming a plane strain state at the deepest point of a surface crack, the cyclic plastic zone size in uniaxial fatigue would be approximately:

CP Z = 1 3π ∆K 2σ y 2
(5.5)

For ∆K = 10 MPa √ m and σ y = 600 MPa, equation 5.5 yields 6.7 µm, which is close to the former austenite grain size. A tensile stress parallel to a crack, like that generated here on transverse cracks by internal pressure, can reduce the plastic zone size, for a given ∆K [START_REF] Brown | Mode I fatigue crack growth under biaxial stress at room and elevated temperature[END_REF]. However, elastic-plastic finite element computations have shown that in the present case, the influence of load biaxiality on the plastic zone size is limited and cannot explain the shift in the curves observed for B = 0.25 and B = 0.5.

In addition, an explanation based on a modification of the plastic zone size would not be consistent with the absence of shift in the curve for B = 1. Some other authors [START_REF] Irving | Measurements of intergranular failure produced during fatigue crack growth in quenched and tempered steels[END_REF][START_REF] Hénaff | Environmental influence on the nearthreshold fatigue crack propagation behaviour of a high strength steel[END_REF] consider that the surface fraction of intergranular facets is controlled by the crack growth velocity da dt = f da dN , where f is the loading frequency. If the crack growth velocity is too high, hydrogen bulk diffusion to the grain boundary becomes too slow, and if it is too low, the rate of hydrogen production at the crack tip is reduced. According to [START_REF] Irving | Measurements of intergranular failure produced during fatigue crack growth in quenched and tempered steels[END_REF], the peak surface fraction of intergranular facets occurs at the crack velocity for which the hydrogen concentration achieved by diffusion at the point of maximum hydrostatic tension ahead of the crack tip is maximum.

Following this interpretation, the shift of the curves corresponding to B = 0.25 and B = 0.5 towards higher values of ∆K, compared to B = 0, in figure 5.4b, would suggest that a higher ∆K is needed to achieve a given crack growth rate, in other words: crack deceleration, while the absence of shift for B = 1 would suggest an absence of effect on the crack growth rate. The reason for such effects is not clear.

Hydrogen atoms in a metal are attracted towards areas of high hydrostatic tension, according to the relation provided by [START_REF] Oriani | Equilibrium aspects of hydrogen-induced cracking of steels[END_REF].

C(σ) = C 0 exp σ kk VH RT (5.6)
where C(σ) is the equilibrium concentration of hydrogen in stressed metal, C 0 is the equilibrium concentration in the stress-free metal, σ kk is the hydrostatic stress, T is temperature, VH is the molar volume of hydrogen in the metal and R is the gas constant.

Following this idea, [START_REF] Ritchie | Influence of impurity segregation on temper embrittlement and on slow fatigue crack growth and threshold in 300-M high strength steel[END_REF] introduced hydrostatic tension into equations evaluating the reduction of the threshold ∆K and the acceleration of crack growth in the near-threshold regime by H-embrittlement of the grain boundaries.

In order to estimate the effect of cyclic biaxial tension on hydrostatic tension ahead of the deepest point of a surface crack, a 3D finite element model of a 500 µm deep semi-circular crack in a 10 mm (∆K≈ 10 MPa, corresponding to peak of fraction of intergranular facets) thick plate was developed using quadratic hexahedron type elements (C3D20R) (figure 5.9a). An appropriate mesh refinement was adopted ahead of the crack front (less than 1/10 th of the reversed cyclic plastic zone size). For symmetry reasons, only Contrary to [START_REF] Marquis | Long-life multiaxial fatigue of SG cast iron[END_REF] or [START_REF] Kane | Surface crack and cracks networks in biaxial fatigue[END_REF] who reported an increased crack roughness under equibiaxial tension, the present study did not reveal any significant variation in crack roughness with the biaxiality ratio (figure 5.5).

As a consequence, no effect of biaxial tension on asperity-induced closure seems to exist in this material. According to the computations done by [START_REF] Kane | Surface crack and cracks networks in biaxial fatigue[END_REF], the effects on plasticity-induced closure should be very limited as well.

In summary, the improvement of fatigue lives observed for B = 0.25 and 0.5 is probably due to a retardation of crack initiation and maybe also be due to a reduction of the crack growth rate, while for B = 1, the "pseudo size-effect" mentioned above might be responsible for a slight reduction in fatigue resistance of the material.

Fatigue criterion

Three popular HCF strength criteria, calibrated using the fatigue strength (at 3 × 10 6 cycles) measured in fully reversed torsion (τ -1 = 250 MPa) and in push-pull at R = -1 ( σ -1 = 365 MPa), have been tested against the data obtained from the uniaxial fatigue tests and the biaxial fatigue tests.

In Findley's criterion [START_REF] Findley | A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending[END_REF] the damage function is a linear combination of the maximum shear stress amplitude (τ a ) and peak normal stress (σ N,max ) acting on the critical plane, which is the plane maximizing the damage function:

τ a + kσ N,max ≤ τ -1 (5.7)
where k is a material parameter. As shown in figure 5.10, it is too conservative for high R ratios and positive biaxialities.

Matake's criterion [START_REF] Matake | An explanation of fatigue limit under combined stress[END_REF] uses the same damage function but the definition of critical plane is different: it is the plane of maximum shear stress amplitude. As shown in figure 5.11, it is better for biaxial tension, but non-conservative under uniaxial fatigue at R = 0.5.

Dang-Van's criterion (Dang [START_REF] Van | Sur la résistance a la fatigue des métaux[END_REF] is given by:

aT a + bσ h,max ≤ σ -1 (5.8)
Effect of biaxial tension in air 109 where a and b are material constants, T a is the mesoscopic shear stress amplitude, σ h,max is the maximum hydrostatic stress and σ -1 is the fatigue limit in push-pull. As shown in figure 5.12, it is non conservative under uniaxial fatigue at R = 0.5 and too much conservative for biaxial tension (especially for equibiaxial, B = 1). The hydrostatic tension term does not discriminate the effect of a mean stress normal to the crack plane from that of a second tensile stress, parallel to the crack. [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF] reached a similar conclusion. More generally, criteria that predict a linear effect of the mean normal or hydrostatic stress, are not consistent with the fact that uniaxial data follow a non-linear evolution, described by Gerber's parabola, already shown previously.

An alternative criterion, respecting this non-linear dependency, is thus proposed as:

σ eff,a 1 -(σ I,m /σ u ) 2 ≤ σ -1
(5.9)

where the effective stress amplitude, σ eff,a is defined as:

σ eff,a = σ I,a σ eq,a σ I,a

x Tr(σ a ) σ eq,a y

(5.10)

Effect of biaxial tension in air 111 σ I,m and σ I,a are the mean value and amplitude of the first principal stress, respectively, σ eq,a is the amplitude of Von-Mises equivalent stress, σ u is the ultimate tensile stress, and

x and y are material parameters. Tr(σ a ) is the trace of the tensor of stress amplitudes.

Note that this term is not modified by the presence of a mean stress, whose non-linear effect is already taken into account in equation 5.9. It is modified only by the presence of a secondary or tertiary cyclic loading. This criterion, which coincides with the Gerber's parabola under uniaxial loading, can discriminate the distinct effects of a mean stress and positive load biaxiality.

For the present steel, x = 2 and y = 1/5 were found suitable to rationalize both the uniaxial and biaxial tension data (figure 5.13). Data from the torsion tests were not used to fit the criterion, because the latter correspond to shear-mode fracture while the present criterion focuses on opening mode fracture.

An alternative way to compare these criteria could in terms of error in the prediction of endurance limits at various R ratios and biaxiality ratios. The error index has been Chapter 6

Corrosion-fatigue

Combined cyclic tension and internal pressure tests with various proportions of each loading were run to determine the effect of biaxial tension on fatigue lives of this steel in salt water. Only a handful of studies have been done to investigate the effect of biaxial tension in salt water and these studies have shown different effect of load biaxiality on crack growth rates in different aluminum alloys [START_REF] Misak | Corrosion fatigue crack growth behavior of 7075-T6 under biaxial tensiontension cyclic loading condition[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF][START_REF] Perel | Biaxial Fatigue Crack Growth Behavior in Aluminum Alloy 5083-H116 Under Ambient Laboratory and Saltwater Environments[END_REF]. However, the behavior of steels could be different from aluminum alloys, so one of the objectives in this part is to investigate the behavior of this steel under uniaxial and biaxial tension in salt water.

Two types of test conditions were used: 1) free corrosion and 2) cathodic potential. The latter was meant to enhance hydrogen ingress in the metal and to study the corresponding effect on fatigue damage mechanisms and fatigue lives.

Experimental procedures

The experimental setup for corrosion-fatigue tests is shown in figure 6.1. The tubular specimen was surrounded by a transparent reservoir filled with a 35 g/l NaCl solution in distilled water, which was circulated by a peristaltic pump to a 5 litre reservoir. This volume corresponds approximately to 100 mL of solution per cm 2 of exposed specimen.

To avoid any leakage of the solution, a silicone polymer joint was made at the junction between the specimen and the reservoir. An aeration pump was used to continuously bubble air into the reservoir, so as to keep it saturated with oxygen. A mean temperature of 22℃ ± 3℃ and a mean pH of 7.2 ± 0.5 existed in the solution throughout the test. The solution was renewed after each test. To keep the reasonable testing times for fatigue lives between 10 5 and 2 × 10 6 cycles, a frequency of 4 Hz was chosen to evaluate the behavior of this steel in salt water.

Two series of tests were run in salt water: 1) without cathodic protection i.e. free corrosion, for B = 0 and B = 1 and 2) with cathodic protection for B = 0, 0.25 and 1.

An R ratio of 0.25 was fixed for both types of tests.

In order to apply cathodic potential, a potentiostat was used along with a reference electrode (Ag/AgCl electrode filled with saturated KCl solution) and a counter electrode (Platinum electrode). A constant potential of -950 mV/ref was applied to the specimen using this potentiostat, which was sufficiently negative for such steel to provide cathodic protection (i.e. reduces the corrosion rate). However, this potential also enhances the rate of the electrochemical reduction of water, with the risk of hydrogen ingress into the steel, and a potential impact on fatigue behavior. The tests were run until complete separation of the specimen (for B = 0) or until oil leakage (for B = 1 and 0.25) or until 2 × 10 6 cycles for ran-out specimens.

116 Corrosion-fatigue 6.2 Experimental results and discussion

Free corrosion

Fatigue lives

The results of uniaxial and equibiaxial corrosion-fatigue tests are reported in table 6.1. Figure 6.2 compares the evolutions of fatigue lives with the amplitude of the maximum principal stress, for B = 0 and 1 in air and in salt water.

The fatigue lives get dramatically reduced in salt water for both biaxialities, and the endurance limit is vanished. While in air, equibiaxial loading was slightly detrimental, especially in the low cycle fatigue regime, in salt water, no significant difference between B = 0 and 1 can be observed.

Corrosion weight loss and surface pitting

The specimens were weighted before and after the fatigue tests and their weight loss was plotted against the duration of the test (figure 6.3a). A similar increase in weight loss with the exposure time was found for uniaxial and biaxial fatigue.

However, this parameter does not reflect the localized character of corrosion, since the specimens were in fact covered with micro-pits. 3D optical imaging of micro-pits (figure 6.3b) was done to measure their widths and depths and an average of 10 measurements was taken for each specimen. Their average aspect ratio (ratio of depth to surface radius) was found to be 0.33 in uniaxial tension and 0.31 in biaxial tension. Images of the specimens submitted to uniaxial and biaxial tension with same axial stress amplitude (155 MPa) were taken at the same magnification, and the number of pits on a surface of 2000 × 2000µm 2 was counted. The density of pits was approximately 2.7/mm 2 in uniaxial tension, and 3.1/mm 2 in biaxial tension. Thus, biaxial tension does not seem to influence much the aspect ratio and density of pits, as compared to uniaxial tension.

Crack initiation

All fatigue failures in salt water initiated from the outer surface, in contact with the corrosive solution, most often from the bottom of a pit, as shown in figure 6.3b and figure 6.4a.

Remember that in air, a majority of cracks in equibiaxial tension had been found to initiate from the inner diameter of the tubes, which was attributed to the slight gradient in hoop stress, 8% higher on the inner diameter than on the outer diameter. In salt water, the stress concentration provided by the pits was thus high enough to overcome this gradient effect, so that cracks initiated from the outer surface.

An additional difference with fatigue in air was the multiplicity of crack initiations, due to the high density of pits. In some cases, merging of small cracks into a large crack was observed (figure 6.4b). This suggests that salt water in free corrosion makes crack initiation very easy and very fast.

Under uniaxial tension, all the microcracks were perpendicular to the loading direction.

Corrosion-fatigue 119 By contrast, in equibiaxial tension, the orientations of the cracks were random, unlike in air, where only transverse and longitudinal cracks were found. This random orientation of the cracks, illustrated on figure 6.5, confirms the absence of any anisotropy in the specimens.

The surface diameter of pits initiating major cracks, lied between 100 and 1000 µm. No definite correlation with the stress level could be found.

120 Corrosion-fatigue

Crack growth

The fracture surfaces were covered with corrosion products, potentially responsible for crack closure effects. A solution of 3.5 mg of hexamethylene tetramine in 500 mL of hydrochloric acid on a cotton stick was thus used to clean the corroded fracture surfaces before observations.

For uniaxial loading, the fracture surfaces exhibit crystallographic patterns evoking transgranular brittle decohesion (figure 6.6), with a very limited proportion of intergranular facets, while for equibiaxial loading, transgranular cleavage-like areas were also present, but the frequency of intergranular facets was much higher. Many micro-pits could be observed on the broken grain boundaries, near the initiation site, but their frequency decreases in depth, and near the inner surface, the grain boundaries look smooth (figure 6.7).

It cannot be due to an attack by the cleansing solution, otherwise the distribution of pits would be homogeneous. These micro-pits on the fracture surface might be formed by salt water in the same way as on the outer surface, and the reason for their disappearance near the inner surface is probably the reduced exposure time compared to the outer surface.

Even though the crack growth mechanisms for uniaxial and biaxial tension in salt water seem relatively different, the fatigue lives fall within the same scatter band. It suggests either that crack initiation from pits, which does not seem to be influenced by load biaxiality, constituted the major part of the fatigue lives, for the stress range investigated, so that a change in crack growth mechanisms and kinetics would have a limited impact, or that the change from transgranular to intergranular decohesion did not make a significant difference in terms of crack growth rate. The first explanation seems quite inconsistent with the high density of microcracks formed in salt water. The second explanation however is consistent with the literature: [START_REF] Irving | Measurements of intergranular failure produced during fatigue crack growth in quenched and tempered steels[END_REF] wrote: "The addition of up to 35% intergranular fracture does not produce detectable enhancement of crack growth rates". [START_REF] Hénaff | Environmental influence on the nearthreshold fatigue crack propagation behaviour of a high strength steel[END_REF] wrote "the crack growth rates are not affected by variations of the rate of intergranular cracking".

With cathodic potential

Fatigue lives

The results of the tests run at B = 0, 0.25 and 1 in salt water with a cathodic potential of -950 mV/ref are reported in table 6.2. A small number of tests were done at each biaxiality ratio because of a limitation in the number of specimens. to what happened in free corrosion, when cathodic potential was applied the fatigue lives remain unaltered as compared to those in air.

Crack initiation

Figure 6.9 compares the surface conditions of the specimen as seen under optical microscope after the fatigue tests in free corrosion and with cathodic potential. The latter merely suppressed surface pitting and the formation of corrosion deposits. Corrosion-fatigue 123 In the absence of pits, failures initiated from the outer surface or even, in some of the specimens tested at B = 1, from the inner surface, exactly like in air. For B = 0 and 0.25, cracks initiated in a transverse direction, while for B = 1 cracks initiated in longitudinal or transverse directions.

σ

Crack growth

In that case, fractographic observations could be performed directly, without any cleansing treatment. While for the tests run in air, the fraction of intergranular facets was found to depend on ∆K, but never to exceed 45% (see figure 5.4b in chapter 5), for all the tests 124 Corrosion-fatigue brittlement in air that the enhancement of this effect by cathodic protection only had a marginal impact in terms of crack growth rate. The first explanation is much more probable than it was for free corrosion. However, the second explanation is supported by the drastic change in aspect of the fracture surface when an internal crack, initiated from Finally, considering the enhancement in fraction of intergranular facets as observed with cathodic protection under uniaxial tension, it seems logical that an already saturated hydrogen-induced embrittlement phenomenon is not further enhanced by biaxial tension.

126 Corrosion-fatigue

Conclusions

The clip connectors used to join the riser tubes for offshore oil drilling undergo variable amplitude cyclic loading due to sea waves. 90% of the service life is spent in the "connected mode" with a high mean stress (R = 0.7) and 10% in the "disconnected mode" with a lower mean stress (R = 0.2).

Elastic-plastic finite element computations on a FE model of a Clip-Riser TM assembly revealed in-phase biaxial tension in its critical areas, along with high mean stresses. Both the mean stress effect and the biaxiality effect are thus needed to be addressed, for a proper design of these structures. However, most of the multiaxial fatigue criteria are based on tension-torsion fatigue data and do not discriminate the influence of biaxial tension from that of a mean stress.

The influence of mean stress and that of load biaxiality on the fatigue lives and damage mechanisms of 2.5%Cr-1%Mo steel constituting the clip connectors were thus investigated separately.

To investigate the mean stress effect, push-pull fatigue tests were run with various R ratios (σ min /σ max ) ranging from -1 to 0.7. The fatigue lives, as well as the slope of the S-N curves were found to decrease with increasing R, and the endurance limit (defined as fatigue strength at 3 × 10 6 cycles) to follow Gerber's parabola.

At low R ratio and thus relatively high stress ranges, fatigue cracks were found to initiate from the surface, merely due to local cyclic plasticity, while for high R ratios, and thus low stress ranges, cracks initiated due to stress concentrations induced by internal or surfacecutting defects (pores or carbon and oxygen segregation areas), sometimes forming "fisheye" patterns on the fracture surface. At very high R ratio (R ≥ 0.6), no fatigue failures could be observed, within 10 7 cycles. In spite of significant residual stresses within a thin surface layer, their R-ratio and amplitude-dependent relaxation did not seem to control the crack initiation sites. The transition in crack initiation mode associated with an increase in mean stress and correlative decrease in amplitude was thus attributed to stress concentration effects.

Internal cracks were found to propagate for a low ∆K, compared to the threshold values reported in the literature for crack growth in air, and this ∆K seemed not to depend on the R ratio. This was attributed to a quasi absence of crack closure, which might, to some extent, compensate the lack of assistance to crack growth provided by moist air.

To investigate the effect of positive stress biaxiality, combined cyclic tension and internal pressure tests with various stress biaxiality ratios (B = 0, 0.25, 0.5 and 1) were run on tubular specimens, at a fixed R ratio (0.25). The evolution of the fatigue lives as a function of the maximum principal stress amplitude showed a beneficial effect for moderate stress biaxiality (B = 0.25 and 0.5) while equibiaxial tension had a slightly detrimental effect.

The stress concentration around defects was shown to decrease with load biaxiality. In addition, for B = 0.25 and 0.5, the Von-Mises equivalent stress range is reduced, compared to uniaxial loading, this probably leading to a retardation of crack initiation consistent with the beneficial effect observed at those biaxialities.

Intergranular facets due to temper-induced segregation of impurities and hydrogen-induced embrittlement were observed on the fracture surfaces. The evolutions of their surface fraction with ∆K and load biaxiality suggested a possible reduction in crack growth rate at moderate biaxialities, but the detrimental effect of equibiaxial tension could not be explained in terms of crack growth rate based on these data.

The tentative explanation proposed for the latter is a "pseudo size effect" i.e. a higher probability that micro-cracks propagate instead of being arrested at a grain boundary, when there are two equivalent principal planes.

Several popular fatigue criteria failed to describe all fatigue data. Endurance criteria that include a linear mean stress term or hydrostatic tension term fail to predict the variations of the endurance limit of this material with the R ratio and biaxiality ratio. Thus, a new fatigue criterion based on Gerber's parabola was proposed. It captured the evolution of the endurance limit under the combined effects of a positive mean stress and positive biaxiality.

In spite of protective coatings, clip connectors might be in contact with sea water. A part of the study was thus devoted to the influence of salt water on the fatigue behavior of the steel. The purpose was to determine if the effect of salt water on fatigue lives was influenced by load biaxiality, particularly in conditions leading to hydrogen release.

For that purpose, combined cyclic tension and internal pressure tests with various proportions of each loading were run in a 35 g/l NaCl solution, in free corrosion and with -950 mV/ref cathodic protection. The latter was meant to enhance hydrogen release in the solution.

In free corrosion, salt water strongly reduced the fatigue lives and suppressed the endurance limit for B = 0 as well as for B = 1. This was due to the formation of corrosion pits. Biaxial tension did not seem to influence much the aspect ratio and the density of pits, as compared to uniaxial tension, but increased the frequency of intergranular facets on the fracture surfaces. The fatigue lives for uniaxial and biaxial tension however fell within the same scatter band, suggesting that the biaxiality-induced change in crack growth mechanisms did not make any significant difference in terms of crack growth rate.

This observation should not be generalized to other materials, as results in the literature [START_REF] Misak | Corrosion fatigue crack growth behavior of 7075-T6 under biaxial tensiontension cyclic loading condition[END_REF][START_REF] Misak | Biaxial tension fatigue crack growth behavior of 2024-T3 under ambient air and salt water environments[END_REF][START_REF] Perel | Biaxial Fatigue Crack Growth Behavior in Aluminum Alloy 5083-H116 Under Ambient Laboratory and Saltwater Environments[END_REF] indicate different effects of biaxial-tension in salt water for different aluminum alloys.

Cathodic protection suppressed surface pitting and cancelled the detrimental effect of salt water for all biaxialities, in spite of a clear enhancement of H-induced embrittlement of the grain boundaries. The fracture surfaces were nearly fully intergranular along the whole crack path, for all biaxialities, while in air, the fraction of intergranular facets was ∆K-dependent and nether exceeded 45%. The absence of reduction in fatigue lives suggests either that crack initiation constitutes the major part of the fatigue lives, or that the material already undergoes such a substantial hydrogen-induced embrittlement in air that the enhancement of this effect in salt water with cathodic protection has a marginal impact.

The absence of enhancement of hydrogen effects by biaxial tension observed in this medium strength, temper-embrittled steel should not be extrapolated to other materials, especially high purity, high strength steels, nor to thicker structures, in which hydrostatic stress might reach higher values and also, because of existence of its gradient in such structures.

CONCLUSIONS 129

Perspectives

Assumptions regarding the relative proportions of crack initiation and propagation stages in fatigue lives had to be made for the interpretation of the test results in air or salt water.

To check the validity of these assumptions, plastic replicas of the surface of the specimen could be made periodically during interrupted tests and observed with the SEM. This would be possible for the tests run in air and in salt water with cathodic protection, when no corrosion products layer is formed on the surface. It would allow the detection and monitoring of microcracks, provided they do not initiate on the internal surface.

In free corrosion, replication might perturbate the corrosion process and should thus be avoided. However, direct observations with a travelling microscope would be possible, to monitor the evolution of pit size and shape and would allow a better understanding of the pit-to-crack transition.

A fracture mechanics approach was attempted to analyze fatigue fracture from defects, and might also be used for fracture initiated from corrosion pits. However, it was based on crack growth kinetics and thresholds from the literature, mainly measured in air. A characterization of fatigue crack growth kinetics and thresholds in air, vacuum and salt water, with or without cathodic protection, depending on load biaxiality, would be useful.

Sharp gradients in stress and biaxiality ratio were found along the thickness of the clip connector. Cracks might thus initiate at the critical point, but become unable to grow, so that the application of the proposed endurance criterion for safe life design would be too conservative. To deal with this problem, two approaches could be envisaged.

• Use the proposed criterion, and in case it predicts fatigue failure at the critical point, use fracture mechanics to predict crack growth or crack arrest, assuming that the appropriate threshold ∆K has been determined, as suggested above.

• Include gradient effects in a non-local criterion, which might keep the form proposed here, but introduce some averaging of stresses over a critical distance or volume related to some material length scale. Specific experiments on notched specimens would be necessary to calibrate and validate this type of criterion. √ Area max . For calculation of √ Area max it is assumed that for a particular specimen, the size of the defect from which the crack has initiated, is the maximum size in the specimen. Then a function F (x) is defined as (i * 100/(N + 1)) %, where "N" is the total number of specimens (N = 7 for present case) and "i" is the value of the current specimen. This function when plotted against the respective √ Area max , gives a straight line. With help of this chart, one can calculate the maximum size of the defect that can exist in certain volume or number of specimens using the relation T = 1/(1 -F (x)). For 100 specimens (T = 100), the maximum size of defect comes out to be 117.4 µm.

Figure 1 :

 1 Figure 1: Clip-Riser TM connector.

Figure 2 :

 2 Figure 2: Clip riser assembly.Figure 3: Mechanics of clip riser in operation.

Figure 3 :

 3 Figure 2: Clip riser assembly.Figure 3: Mechanics of clip riser in operation.

  xii

Figure 4 :

 4 Figure 4: (a) Simulated Sea wave amplitude with the help of data recorded over a span of time (b) Corresponding variation of load on riser.

  Figure 1.2: (a) Stage I and stage II propagation of cracks (Bathias and Pineau, 2010) (b) Intrusion-extrusion pair on surface

  Figure 1.3: (a) Dual SN curves of high carbon-chromium bearing steel typical of a transition from surface to internal crack initiation when the loading range decreases (Shiozawa et al., 2006) (b) Effect of Inclusion's size and location on fatigue strength (Duckworth and Ineson, 1963) (c) An example of a fish eye pattern.

Figure 1 . 4 :

 14 Figure 1.4: Schematic representation of the blunting process of crack tip as influenced by the environment (air and vacuum), (McEvily and Gonzalez Velazquez, 1992).

Figure 1 . 5 :

 15 Figure 1.5: Stage II crack growth mechanism (a) zero load (b) small tension (c) maximum tension (d) zero load (e) small tension (next cycle),[START_REF] Laird | The influence of Metallurgical Structure on the Mechanisms of Fatigue Crack Propagation[END_REF]

  Figure 1.6: (a) Fatigue crack growth rate in martensitic SA542-2 steel (b) Oxide thickness as a function of crack length in different environment in SA542-3 steel[START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF].

  Figure 1.7: Correlation of crack growth rate with (a) Nominal ∆K (b) Effective ∆K, R is the R-Ratio and η is biaxiality ratio[START_REF] Hoshide | Stress-ratio effect of fatigue crack propagation in biaxial stress field[END_REF]

Figure 1 . 9 :

 19 Figure 1.9: Effect of biaxiality (λ) or T-stress (T) on (a) distribution of plastic strain ahead of the crack tip (McClung, 1989) (b) crack path stability[START_REF] Cotterell | Slightly curved or kinked cracks[END_REF]

Figure 1 . 11 :

 111 Figure1.11: First loading and stabilized cyclic hysteresis loop, plastic (∆W p ) and elastic (∆W e ) strain energy densities[START_REF] Ellyin | Cyclic strain energy density as a criterion for multiaxial failure fatigue', Biaxial and multiaxial fatigue[END_REF]

  Figure1.12: Comparison of Sines and Crossland criteria in multiaxial loading[START_REF] Bonnand | Investigation of multiaxial fatigue in the context of turboengine disc applications[END_REF]

  Figure 1.15: Specimens for biaxial tension tests (a) Cruciform type (b) specimen proposed by Bellett et al. (2010) for biaxial tension under uniaxial loading

Figure 1

 1 Figure 1.16: Schematic polarization curve of metallic corrosion. Subscript 'a' stands for anode, 'c' for cathode and 'corr' for corrosion. 'i' is current, 'E' is equilibrium potential.

Figure 1

 1 Figure 1.18: A schematic of Cathodic Polarization curve.

Figure 1 . 19 :

 119 Figure 1.19: Schematic of modes of corrosion fatigue damage, Schneider (1983).

Figure 1 .

 1 Figure 1.20: Schematic of modes of corrosion fatigue crack initiation on active metal surface,[START_REF] Milella | Fatigue and Corrosion in Metals[END_REF].

  Figure 1.21: (a) Unsensitized 304 Austenitic stainless steel (b) Sensitized 304 Austenitic stainless steel after etching. a a source: http://www.corrosionclinic.com/types of corrosion/intergranular corrosion cracking.htm

  Figure 1.22: Schematic of passivity-breakdown mechanisms.

  Figure 1.23: (a) Development mechanism for corrosion pits, Rosenfeld (1974) (b) Variation of pH across the pit on iron in dilute chloride solution, Butler et al. (1972).

Figure 1 .

 1 Figure 1.24: A typical example of crack initiating from corrosion pit, Lynch (2012).

  .25(a) with three different regimes. In regime I, da/dt sharply increases with K I and is controlled by the creation of fresh, unpassivated metallic surfaces. In regime II, da/dt is controlled by movement of ions towards the crack tip and is independent of K I . In regime III, K I gets closer to the fracture toughness, K I,c and the crack growth rate increases rapidly due to mechanically induced ductile or brittle fracture. Under cyclic loading, depending upon the relative values of K I,scc and ∆K I,threshold , the whole crack growth kinetics curve is shifted upward (figure 1.25(b)) or just the part for which K I,max > K I,scc (figure 1.25(c)).

  Figure 1.25: A typical crack growth kinetics for (a) Stress corrosion cracking (b) True corrosion fatigue (c) Stress corrosion fatigue, modified from[START_REF] Austen | Factors affecting the rate of growth of cracks by corrosion-fatigue[END_REF].

  Fig. 4--Corrosion fatigue crack growth behavior of Ti-6 A1-6 V-2 Sn in aqueous 0.6 M Na2SO 4 at 1 and 10 Hz.

  Fig. 5--Effect of frequency on corrosion fatigue crack growth behavior of Ti-6 A1-4 V in aqueous 0.6 M NaC1.

  Figure 1.26: Effect of frequency (a) on FCGR of 12Ni-5Cr-3Mo steel, Barsom and Rolfe (1977) (b) on FCGR of Ti-6A1-4V in aqueous NaCl solution,[START_REF] Dawson | Corrosion Fatigue Crack Growth of Titanium Alloys in Aqueous Environments[END_REF].

  Figure 1.27: (a) Response of FCGR from changes in load frequency in 4340 steel in water vapours at 4.4 torr, Achilles and Bulloch (1986) (b) SN curves of the R5 martensitic steel under fully reversed tension at 20 kHz, Pérez-Mora et al. (2015).

  Fig. 10. Effect of R-ratio on FCGR behaviour of HSLA-100 steel in air and 3.5% NaCl solution.

Fig. 9 .

 9 Fig. 9. Effect of R-ratio on FCGR behaviour of HSLA-80 steel in air and 3.5% NaCl solution.

  results in increase in time to failure, with a critical potential below which no SCC occurs (figure 1.29(a)). The phenomenon has been explained on the basis of absorption and desorption of aggressive Cl -ions above and below the critical potential. But later on, it                                                                                                                                                                                                                                                                                                                                                                   

  Fig.10--Effect of applied potential of 4140 steel, R C 37, stressed above and below the fatigue limit, on fatigue life in aerated 3 pct NaC1, 25~

  Fig.10--Effect of applied potential of 4140 steel, R C 37, stressed above and below the fatigue limit, on fatigue life in aerated 3 pct NaC1, 25~

  Figure 1.29: Effect of electrode potential on corrosion fatigue (a) 304 Stainless steel in MgCl 2 (Uhlig and Cook, 1969) and AISI 4140 steel in 3% NaCl solution (b) R c = 20 (c) R c = 52 (Uhlig and Lee, 1972). R c is Rockwell Hardness.

  Figure 1.30: (a) Effect of pH on fatigue life of steels under aerated 3% NaCl solution (Duquette and Uhlig, 1969) (b) Effect of Cl -on fatigue life of 13% Cr steel in NaCl environment[START_REF] Ebara | Corrosion fatigue behavior of 13Cr stainless steel in sodium-chloride aqueous solution and steam environment[END_REF].

  Figure 1.31: Effect of oxygen content on fatigue life of 12% Cr steel in de-ionised water, Perkins and Bache (2005).

σFigure 1

 1 Figure1.32: Schematic illustration of Hydrogen Assisted Cracking,[START_REF] Magnin | Advances in corrosion-deformation interactions[END_REF].

Fig. 7 .

 7 Fig. 7. da/dN versus DK I relationships of region I and II for k = 0 in air and salt.

Fig. 8 .

 8 Fig. 8. da/dN versus DK I relationships of region I and II for k = 1 in air and salt.

Figure 1 . 34 :

 134 Figure1.34: Effect of equibiaxial tension on crack growth rate in 7075-T6 aluminum alloys, in air and in salt water[START_REF] Misak | Corrosion fatigue crack growth behavior of 7075-T6 under biaxial tensiontension cyclic loading condition[END_REF].

Figure 2 . 1 :Figure 2 Figure 2

 2122 Figure 2.1: Sample location

Figure 2 .

 2 Figure 2.4 shows the band contrast and the inverse pole figures maps obtained as a result of EBSD analysis. The diffraction band contrast makes the microstructure more visible than optical microscopy. The grains do not exhibit preferential elongation in any particular direction. Pole figures (figure 2.5) represent the intensity of each orientation

Figure 2 . 4 :

 24 Figure 2.4: Band contrast image (a) normal to connector axis (35 H) (b) Parallel to connector axis (35 V). Inverse pole figures for surface (c) normal to connector axis (35 H)(d) Parallel to connector axis (35 V) (e) Color code.

Figure 2 . 5 :

 25 Figure 2.5: Pole figures obtained from EBSD mappings (a)normal to connector axis (35 H)(b) Parallel to connector axis (35 V)

Figure 2 . 8 :Figure 2 . 9 :

 2829 Figure 2.8: Stress strain loops for low-cycle fatigue tests.

  Figure 2.10: Stabilized stress-strain loops comparison for low-cycle fatigue tests (Strain amplitudes (a) ±0.30% (b) ±0.75%(c) ±1.20%) at respective half life.

Figure 3 Figure 3 . 2 :

 332 Figure 3.1: (a) Clip connector model in ABAQUS software (b) Lock ring model (c) Corresponding meshed model using tetrahedral element type C3D10M.

Figure 3 . 3 :Figure 3

 333 Figure 3.3: Stress contours in lock ring in connected mode after 50 cycles at maximum stress (a) overview (b) zoomed in lugs region where stress gradient exist (c) zoomed in contact region where maximum stress exist.

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Evolution of principal and equivalent plastic strains at centroid of contact element after 50 cycles at maximum stress (a) Connected Mode (b) Disconnected Mode.

Figure 3 . 7 :

 37 Figure 3.7: Location of selection of nodes (red dots) in (a) Connected Mode (b) Disconnected Mode.

Finite

  Figure 3.8: Statistical distribution of parameters at randomly selected 200 nodes: R ratio for Maximum principal stress in (a) and (b), R ratio for mid principal stress in (c) and (d), and for biaxiality ratio in (e) and (f).

Figure 4 . 1 :

 41 Figure 4.1: Cylindrical specimen.

Figure 4 . 2 :

 42 Figure 4.2: Experimental device.

Figure 4 . 3 :Figure 4 . 4 :

 4344 Figure 4.3: Whöler curve for high-cycle fatigue tests at different R ratios.

Figure 4 . 5 :

 45 Figure 4.5: Measured engineering stress-strain loops.

°

  Figure 4.9: Fracture surfaces under optical microscopy with "Fish eye" (a) R = -0.5, σ a = 360 MPa, N f = 2, 207, 630 (b) R = -0.5, σ a = 365 MPa, N f = 741, 938.

Figure 4 .

 4 Figure 4.10: Fracture surface under SEM showing a clear difference in aspect of roughness, striations between crack growth in air and in vacuum at R = -0.5, σ a = 360 MPa, N f = 2, 207, 630

  beam microscope. 5 to 10 µm deep, 20 to 100 µm wide sections normal to the fracture surface were cut in a specimen broken after 1.7 million cycles at R = 0.25 at various distances from the internal crack initiation site (figure 4.11). The sections cut 128 µm and 186 µm away from the initiation site (figure 4.11(c) and (d)) are located inside the fish-eye, while the section cut 600 µm away (figure 4.11(a,b)) is outside. While a rough crack profile (peak roughness value, R p ≈ 4 µm) with many tiny secondary cracks could

Figure 4 .

 4 Figure 4.11: SEM images of cross sections made with a FIB after platinum coating at various distances from a fish-eye center on the fracture surface of a specimen broken after 1,704,174 cycles at R = 0.25.

Figure 4 .Figure 4 . 13 :

 4413 Figure 4.12: SEM image of Fracture surfaces (two halves (a) and (b)) and areas chosen for chemical analysis of specimen at R = 0.25, σ a = 270 MPa, N f = 160, 277 and EDS analysis at two locations on the fracture surface (c) Matrix composition, Spectrum 6 (d) Inhomogeneity, Spectrum 4.

Figure 4 .

 4 Figure 4.14: PROTO iXRD diffractometer used to perform residual stress analyses.

Figure 4

 4 Figure 4.15: (a) In-depth profile of initial surface residual stresses and (b) its corresponding profile of Full-width at Half maximum (FWHM) of the X-ray diffraction peaks measured in a virgin cylindrical specimen.

Figure 4 .

 4 Figure 4.16: A 3D Meshed unit cell model and normal stress-strain loops at the edge of a spherical surface and internal pore for ∆σ = 520 MPa, R = 0.25.

Figure 4 . 17 :

 417 Figure 4.17: Presence of Intergranular facets outside the fish-eye for R = -0.5, σ a = 360 MPa and N f = 2, 207, 630 cycles

  the bar superimposed on figure 4.18(a) and the comparison with threshold data in air from[START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF] and[START_REF] Liaw | Effect of load ratio and temperature on the near-threshold fatigue crack propagation behaviour in a CrMoV steel[END_REF] is shown in figure4.18(b)).

  [START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF] investigated fatigue crack growth kinetics and thresholds on a similar 2.25%Cr-1%Mo steel in air, dry helium and dry hydrogen for R = 0.05 and 0.75 (figure 4.18(a)). They obtained ∆K threshold = 7.1 MPa √ m at R = 0.05 in air. In a reducing atmosphere (dry H 2 ), which nearly suppressed oxide-induced closure, the threshold at R = 0.05 dropped to 4.5 MPa √ m. The initial ∆K I values for the crack grown from internal defects are even below this threshold. This seems logical since these internal cracks, which grow in vacuum, are not only free from oxide-induced closure but they are also short and rather smooth cracks and undergo very limited plasticity and roughnessinduced closure.

Figure 4

 4 Figure 4.18: (a) Crack growth kinetics in 2.25Cr-1Mo steel at various R ratios in air and dry Hydrogen (Re-plotted from Suresh et al. (1981)) with some data (red bars) from present study superimposed. (b) Comparison of threshold ∆K values issued from[START_REF] Liaw | Effect of load ratio and temperature on the near-threshold fatigue crack propagation behaviour in a CrMoV steel[END_REF],[START_REF] Suresh | Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behavior[END_REF] and the present study.Influence of the mean stress 89

Figure 4 .

 4 Figure 4.19: Beach marks observed in fish eye for specimens 23 and 26 respectively.

  Figure 4.19b shows that a beach mark separates the areas corresponding to the third and fourth loading blocks. The position of this beach mark gives an average crack growth rate of 1.5 × 10 -10 m/cycle for a mean ∆K of 11.6 MPa √ m. These data, corresponding to internal crack growth at R = 0 and R = 0.25, are superimposed on the kinetics curves re-plotted from Suresh et al. (1981) (figure 4.18(a)) and appears to be on the left side of the kinetics curve obtained in air at R = 0.05.

Figure 4

 4 Figure 4.20: A tentative map of the mechanisms of fatigue failure depending upon the mean stress or R-ratio at 3 million cycles.

Figure 2 .Figure 5 . 1 :

 251 Figure 2. Geometry of specimens used for a) Biaxial tension and c) Torsion fatigue tests (a) (b) Figure 5.1: Geometry of specimens used for a) Biaxial tension and b) Torsion fatigue tests.

Figure 3 :Figure 5 . 2 :

 352 Figure 3: Whöler curves for R = 0.25 and various load biaxiality ratios, B. a) in terms of first principal stress amplitude and b) in terms of Von Mises stress amplitude For B= 0, data obtained previously on cylindrical specimens [16] have been added (a)

  .2b, on which the data-points for B = 0.25 and 0.5, now all lie well below the uniaxial Wöhler curve. By contrast, the relative positions of the uniaxial and equibiaxial Wöhler curves do not change when Von Mises equivalent stress is used instead of the first principal stress. Figure 5.2a and Figure 5.2b both suggest a slight depreciating effect of equibiaxial loading on fatigue lives. Clearly, for all biaxiality ratios, life predictions based on uniaxial fatigue data and Von Mises equivalent stress amplitudes would be non-conservative.

Figure 5

 5 Figure 5.3: Modified Whöler curves including the gradient in hoop stress for data points where crack initiated from inner surface.

Figure 5 :Figure 5 :

 55 Figure 5: Fracture surface with intergranular facets after biaxial tension fatigue at R=0.25, B=1,  zz /2=275MPa.

Figure 5 . 5 :

 55 Figure 5.5: Peak roughness values for various load biaxialities at same stress amplitude (270 MPa).

Figure 5 . 6 :

 56 Figure 5.6: Whöler curve for fully reversed torsion.

  1 and figure 5.7). A finite element model (figure 5.8) was thus used to compute the stress concentration factor, K t , near a surface semi-spherical pore in a unit cell with quadratic hexahedron elements, loaded in biaxial tension. The computed values of K t decrease as load biaxiality increases: K t = 2.04 for B = 0, 1.90 for B = 0.25, 1.81 for B = 0.5, and 1.64 for B = 1. Due to this reduction in stress concentration, biaxial tension should thus reduce fatigue strength as compared to uniaxial tension, and the reduction should increase with increasing biaxiality ratio.

Figure 5 . 7 :Figure 7 .

 577 Figure 5.7: SEM image of crack initiation from surface defect, B = 1, σ a = 245 MPa and N f = 227, 250 cycles.

Figure 5 . 8 :

 58 Figure 5.8: Finite element model and boundary conditions used to evaluate the stress concentration at a surface pore in biaxial tension.

1/ 2

 2 Figure5.9: FE model and results for a semi-elliptical crack in plate for different biaxialities.

Figure 9 :Figure 5 . 10 :Figure 5

 95105 Figure 9: Uniaxial and biaxial data plotted against Findley's criterion.

Figure 11 :Figure 5 . 12 :

 11512 Figure 11: Uniaxial and biaxial fatigue data plotted against Dang-Van's criterion.

Figure 5 . 13 :

 513 Figure 5.13: Existing fatigue data plotted against the proposed criterion.

Figure1

  Figure1 Experimental setup for corrosion fatigue tests

Figure 6 . 1 :

 61 Figure 6.1: Experimental setup for corrosion fatigue tests.

  Wholer curve for uniaxial (B=0) and biaxial tension (B=1) in air and in salt water. cular symbols are for uniaxial tension and squares ones are for equibiaxial tension.

Figure 6 . 2 :

 62 Figure 6.2: Whöler curve for uniaxial (B = 0) and biaxial tension (B = 1) in air and in salt water. Circular symbols are for uniaxial tension and squares ones are for equibiaxial tension.

Figure 3 :

 3 Figure 3: Trend in thickness loss of the specimens with their corresponding exposure time during the fatigue tests.

Figure 4 :Figure 5 :

 45 Figure 4: Corrosion pits formed during corrosion fatigue tests

Figure 6 :Figure 6 . 5 :

 665 Figure 6: Randomly oriented micro-cracks under equibiaxial tension. Axial loading direction

Figure 7 :

 7 Figure 7: SEM image of the fracture surface in uniaxial tension in salt water with axial stress amplitude of 155 MPa.

Figure 6 .

 6 Figure 6.8 shows the evolution of fatigue lives with the maximum principal stress amplitudes for the three biaxiality ratios in comparison with the data obtained in air. Contrary

Figure 6 . 8 :

 68 Figure 6.8: Whöler curves for fatigue data in air and in salt water with cathodic potential for different load biaxialities (B = 0, 0.25 and 1). Hollow symbols represent the ran-out specimens.

  Figure 10: Comparison of surface condition after fatigue tests in salt water

Figure 6 . 9 :

 69 Figure 6.9: Comparison of surface condition after fatigue tests in salt water.

Figure 11 :Figure 6

 116 Figure 11: A typical SEM image of fracture surface (a) in air and (b Fracture surface in cathodic protection was nearly same for

a

  defect and growing in vacuum finally meets the free surface (figure 4.9 and figure 4.10 in chapter 4) and the presence of up to 45% intergranular facets. It is also supported by the work of[START_REF] Huneau | Influence de l'environnement sur le comportement en fatigue d'un acier à haute limite d'élasticité[END_REF] on a martensitic steel tested in fatigue in vacuum, air, salt water and natural sea water with a cathodic potential of -1050 mV/ECS . As can be seen on figure6.11, a large drop in fatigue lives and increase in crack growth rate were observed when the tests were run in air instead of vacuum. The further drop in life and increase in crack growth rate when the tests are run in salt water are comparatively smaller. Note that if these tests had been run at a cathodic potential of -950 mV/ref as in the present study, the hydrogen flux would have been much less, and the detrimental effects even less.

  Figure B.1: Extreme value statistics plot.

                                                

                                                   

Table 2

 2

.1: Nominal composition of F22 steel (weight %), ASTM A 182 standard

Table 2 .

 2 2: Mean intercept grain size on two surfaces and along two orthogonal directions

	HV	Spot Size	WD	Tilt	Pitch	Exposure
	20 kV	4	17 mm	70°0.137 µm	48.7 ms

Table 2

 2

.3: Parameters used for EBSD mappings Material characterization 47

Engineering Stress, Mpa Engineering Strain, % Necking

	800
	700
	600
	500
	400
	300
	200
	100
	0
	0.00%	2.00%	4.00%	6.00%	8.00%	10.00%	12.00%	14.00%
	E (GPa)	ν	σ p (MPa) σ y,0.2% (Mpa) σ u (Mpa) neck HV
	207.4	0.27	600		694	780		6.7% 267

𝜖𝜖̇= 10 -3 𝑠𝑠 -1 Figure 2.7: Engineering stress-strain curve.

Table 2

 2

			First Cycle	800 100th Cycle	200th Cycle			First Cycle	800 100th Cycle	300th Cycle			First Cycle	700 1000 cycle	6000 cycle
				400						400						350
	Stress (MPa)	-1.20%	-0.60%	0	0.00%	0.60%	1.20%	-0.75%	-0.38%	0	0.00%	0.38%	0.75%	-0.30%	-0.15%	0	0.00%	0.15%	0.30%
				-400						-400						-350
				-800		Strain (%)				-800						-700
			(a) a = 1.2%				(b) a = 0.75%			(c) a = 0.3%

.4: Mechanical properties of F22 alloy steel 50 Material characterization

  GPa γ 2

					343.6	-130.7	2.0	350.8	1140.6	12.5	32.1
			700
			Experimental
			Simulated
			350
	Stress, MPa	-0.30%	-0.15%	0 0.00%	0.15%	0.30%
			-350
					Strain, %
			-700

Table 2.5: Coefficients of constitutive equation (one isotropic and two kinematic hardening).

  Table 2.7: Low cycle fatigue tests data. N f is the number of cycles to failure.

	Stress amplitude, MPa	400 500 600 700	0.2%		0.4%		0.6%	0.8%	1.0%	0.8%	0.6%	0.4%	0.2%
		300
	Table 2.6: Predicted and measured stress amplitude at mid lives for LCF tests using 1 isotropic 0 15 30 45 60 75 90 105 120 135
	and 2 kinematic hardening variables.	Number of cycles
					Figure 2.12: Evolution of stress amplitude for variable amplitude test.
								∆ /2 (%) R	˙	f (Hz) N f
					700				0.30	-1 10 -3 0.0833 6701 Experimental
	Strain Amplitude, %	0.2 0.4 0.6 0.8 1 1.2 1.4	Stress Amplitude, MPa	300 400 500 600	0	1.2	0.2	0.75 1.20	515 595 -1 10 -3 0.0208 233 -1 10 -3 0.0333 0.75 0.75 ∆ /2 = 9.885 N -0.399 R² = 0.9875 0.4 0.6 0.8 1 Cumulative plastic strain Coffin-Manson Curve -LCF Simulated	1.2	0.3 1.4 R = -1
			0
				100							1,000	10,000
											No. of Cycles
								Figure 2.11: Coffin-Manson Curve.
	equation was used to do finite element analyses of clip connectors. For high-cycle fatigue
	life prediction, the most important parameter is the steady state stress range (stress range
	at half life). So, large number of cycles under different strain amplitudes i.e. ±0.30%,
	±0.75% & ±1.20% were simulated, and a comparison of measured and predicted steady
	state stress amplitudes is shown in table 2.6. The stress range is underestimated by 10.5%
	for the test with the lowest strain amplitude, ±0.30%, while for the two higher strain
	ranges, the error is within acceptable limits.

These strain-controlled fatigue tests were continued until fracture and the corresponding fatigue lives are reported in table 2.7 in which the stress amplitudes correspond to the values measured at mid-life. The Coffin-Manson curve is shown in figure

2

.11 and follows 54 Material characterization Figure 2.13: Experimental and simulated stress amplitude against cumulative plastic strain.

Table 4

 4

	Test No.	Dia,	Ra,	σa,	σmax,	R	Cycles to failure, N f	Initiation Type
		mm	µm	MPa	MPa
	8	8	< 0.4	435	435	-1	193,620	Surface without defect
	7	8	< 0.4	410	410	-1	367,166	Surface without defect
	9	8	< 0.4	385	385	-1	1,137,624	Surface without defect
	20	8	< 0.4	370	370	-1	521,343	Surface without defect
	40	6	0.2	370	370	-1	> 3, 000, 000	-
	21	8	< 0.4	365	365	-1	> 3, 000, 000	-
	15	8	< 0.4	390	520	-0.5	322,888	Surface without defect
	19	8	< 0.4	378	502	-0.5	390,914	Surface without defect
	14	8	< 0.4	375	500	-0.5	182,856	Surface without defect
	18	8	< 0.4	365	485	-0.5	741,938	Internal Pore with fish-eye
	16	8	< 0.4	360	479	-0.5	2,207,630	Internal Pore with fish-eye
	21	8	< 0.4	350	466	-0.5	> 3, 000, 000	-
	11	8	< 0.4	322	644	0	21,064	Surface without defect
	13	8	< 0.4	318.5	637	0	153,518	Surface without defect
	39-2	6	0.2	317	634	0	575,063	Surface without defect
	10	8	< 0.4	315	630	0	833,695	Surface without defect
	22	8	< 0.4	310	620	0	198,067	Surface without defect
	39-1	6	0.2	305	610	0	> 3, 000, 000	-
	23	8	< 0.4	300	600	0	> 3, 000, 000	-
	29	7	0.2	273	729	0.25	228,359	Surface without defect
	30	7	0.2	270	721	0.25	160,277	Internal Inhomogeneity
	29	7	0.2	265	708	0.25	> 3, 000, 000	-
	26	7	0.2	260	694	0.25	1,704,174	Internal Pore
	23	8	< 0.4	260	694	0.25	373,933	Internal Inhomogeneity
	24	7	0.2	255	681	0.25	300,333	Internal Inhomogeneity
	26	7	0.2	250	668	0.25	> 3, 000, 000	-
	25	7	0.2	250	668	0.25	339,602	Surface Pore
	26	7	0.2	245	654	0.25	> 3, 000, 000	-
	26	7	0.2	240	641	0.25	> 3, 000, 000	-
	37	6	0.2	193	772	0.5	163,919	Machining mark
	33	6	0.2	190	760	0.5	459,584	Machining mark
	38	6	0.2	190	760	0.5	279,630	Machining mark
	36-2	6	0.2	187	748	0.5	> 3, 000, 000	-
	34	6	0.2	185	740	0.5	1,166,933	Surface Pore
	35	6	0.2	180	720	0.5	1,462,890	Surface Pore
	36-1	6	0.2	175	700	0.5	> 10, 000, 000	-
	36-4	6	0.2	154	770	0.6	> 10, 000, 000	-
	36-3	6	0.2	148	740	0.6	> 10, 000, 000	-
	31-3	6	0.2	116.5	777	0.7	> 10, 000, 000	-
	32	6	0.2	115.5	770	0.7	> 10, 000, 000	-
	31-2	6	0.2	115	767	0.7	> 10, 000, 000	-
	31-1	6	0.2	112.5	750	0.7	> 10, 000, 000	-

.1: Test data obtained from high-cycle fatigue tests in air at 10 Hz. σ a is the applied stress amplitude and R a is the measure of surface roughness.

  Table4.2: Size and position of the defects giving rise to internal crack initiation. Specimens marked (1,2) survived 3 million cycles and reloaded. In that case, D corresponds to the crack diameter at the corresponding beach mark on the fracture surface

	Test No. R	σ a	Size of defect	D	ω	e
			MPa 2a, µm 2b, µm µm µm
	16	-0.5	360	135	66	471 277 0.93
	18	-0.5	365	199	66	660 330 0.92
	23-1	0	300	120	60	139 165 0.96
	26-1	0.25 250	304	50	500 514 0.85
	25	0.25 250	95	95	-	95 0.97
	24	0.25 255	127	65	-	104 0.97
	26-2	0.25 260	304	50	1028 514 0.85
	23-2	0.25 260	120	60	355 165 0.96
	30	0.25 270	120	40	-	60 0.98
					Influence of the mean stress 77

tensile fatigue tests, which are supposed to produce a uniform stress field, provided any misalignment and bending stresses are avoided. The absence of such artifacts was verified.

Table 4 .

 4 3: Measured surface residual stresses on virgin specimens 1 and 2.

Table

	Test R	∆σ	Size of defect	∆K I,max ( MPa	√	m) N f	N f ish
			MPa 2a, µm 2b, µm Initial	Final
	16	-0.5	480	135	66	4.1	8.3	2,207,630	160,000
	18	-0.5	487	199	66	4.4	10.0	741,938	150,000
	23-1 0	600	120	60	4.8	6.2	> 3, 000, 000 50,000
	26-1 0.25 500	304	50	4.3	8.9	> 3, 000, 000 110,000
	25	0.25 500	95	95	3.9	-		339,602	-
	24	0.25 510	127	65	4.2	-		300,333	-
	26-2 0.25 520	304	50	9.2	13.3	1,704,174	40,000
	23-2 0.25 520	120	60	5.4	7.5	373,933	70,000
	30	0.25 540	120	40	3.8	-		160,277	-

  who reported a beneficial effect of biaxial tension for B = 0.45 on the fatigue life of two aluminum alloys.

	σ zz,a , MPa B	Cycles to	Initiation type	Crack ori-
		failure, N f		entation
	0	234,213	Inner surface Pore	Transverse
	0	258,366	Outer surface	Transverse
			Inhomogeneity
	0	336,596	Outer surface Pore	Transverse
	0.25	115,531	Outer surface	Transverse
	0.25	114,701	Inner surface	Transverse
	0.25	129,972	Inner surface	Transverse
			Inhomogeneity
	0.25	1,908,883	Inner surface	Transverse
	0.25	219,697	Inner surface	Transverse
			without defect
	0.25	> 3, 000, 000	-	-
	0.25	> 3, 000, 000	-	-
	0.5	302,091	Outer surface	Transverse
			Inhomogeneity
	0.5	> 3, 000, 000	-	-
	0.5	> 3, 000, 000	-	-
	0.5	238,224	Outer surface	Transverse
			Inhomogeneity
	0.5	296,839	Outer surface	Transverse
			Inhomogeneity
	0.5	145,221	Outer surface Pore	Transverse
	0.5	> 3, 000, 000	-	-
	1	65,424	Inner surface	Longitudinal
	1	143,084	Inner surface	Longitudinal
	1	645,832	Outer surface	Longitudinal
	1	160,916	Outer surface	Longitudinal
	1	227,250	Inner surface Pore	Transverse
	1	> 3, 000, 000	-	-
	1	> 3, 000, 000	-	-
			Effect of biaxial tension in air 97

Table 5 .

 5 1: Biaxial cyclic tension data at R = 0.25, σ zz,a is the applied axial stress amplitude and B is the biaxiality ratio.

  The fracture sur-τ a , MPa Cycles to failure, N f Table5.2: Fully-reversed torsion data at 10 Hz, τ a is the applied shear stress amplitude.

		Crack orientation
	15,290	Transverse
	43,946	Transverse
	107,141	Transverse
	174,550	Transverse
	201,773	Transverse
	181,415	Transverse
	> 3, 000, 000	-
	114,677	Transverse
	> 3, 000, 000	-

  Corrosion-fatigue 117 σ zz,a , MPa B Cycles to failure, N f

				Location	Crack orientation
	240	0	146,333	Outer surface	Transverse
	220	0	140,797	Outer surface	Transverse
	210	0	194,558	Outer surface	Transverse
	195	0	363,411	Outer surface	Transverse
	180	0	154,735	Outer surface	Transverse
	155	0	524,377	Outer surface	Transverse
	235	1	155,831	Outer surface	Longitudinal
	220	1	164,753	Outer surface Inclined crack 15°with
					longitudinal direction
	210	1	198,505	Outer surface	Longitudinal
	195	1	266,320	Outer surface Inclined crack 69°with
					longitudinal direction
	180	1	304,076	Outer surface Inclined crack 80°with
					longitudinal direction
	155	1	586,777	Outer surface Inclined crack 58°with
					longitudinal direction

Table 6 .

 6

1: Corrosion fatigue data under uniaxial (B = 0) and equibiaxial tension (B = 1) at R = 0.25, in free corrosive environment. σ zz,a is the applied axial stress amplitude and B is the biaxiality ratio.

Table 6 .

 6 zz,a , MPa B Cycles to failure, N f 2: Corrosion fatigue data for different load biaxialities (B = 0, 0.25, 1) at R = 0.25, with cathodic potential. σ zz,a is the applied axial stress amplitude and B is the biaxiality ratio.

	Location	Crack orientation

Murakami's √ area model

Acknowledgements ix

Acknowledgements

First, I would like to express my sincere thanks to my PhD supervisor (Dr. Doquet) for her consistent support and understanding throughout the work, not only in professional life but also in personal life. She was a great source of motivation for me. I would also like to thank my promoter of this thesis from our industrial partner (IFPEN): Mr.

Emmanuel, Dr. Eléonore and Dr. Jean, who have consistently provided useful advices to this work from time to time, especially for the part devoted to corrosion, thanks to Dr.

Jean Kittel.

I would like to thank all the jury members of my PhD defense, who took out time of their schedule for me and the reporters (Dr. Thierry Palin-Luc and Dr. Franck Morel)

for their useful advices for the improvement of this manuscript.

I would also like to thank my lab (LMS) where I carried out this thesis and my employer (IFPEN) for providing funds for this work.

I would also like to thank the lab technicians especially Mr. Erik Guimbretière, Mr.

, my parents and family members who supported me throughout my life and encouraged me to go through this PhD framework.

Chapter 3

Finite element analysis of clip connector A Finite Element (FE) analysis of the stress and strain state in clip connector has been performed in ABAQUS using the elastic-plastic constitutive equations identified in previous chapter.

The Clip Riser TM assembly model consisting of the lock ring plus the female part and male part is shown in figure 3.1(a). Because of an axial symmetry, only 1/8 th part was modeled to reduce the computation time. An independent model of lock ring is shown in figure 3.1(b). Because of lack of symmetry in the arrangement of lugs in lock ring and female member as well, a model was created in different sections. The mesh was generated manually using 3D modified quadratic tetrahedral elements of type C3D10M with 4 integration points, figure 3.1(c). This type of element is robust for large deformation and contact problems. The mesh was refined in the regions where the lock ring is in contact with the male and female parts of the assembly to ensure good convergence and accuracy of computations.

The load is transmitted to the lock ring through the contacts at lugs. There are two types of algorithm available in ABAQUS for modeling contacts: general contacts and contact pairs. In present case, the general contact algorithm (node to surface) was used since it is less complicated and faster.

Numerical simulations were performed under two different cyclic loading conditions, provided by IFPEN: Connected mode (where the riser tubes are in tension and fixed at both top and bottom) and disconnected mode (where the riser tubes are hanging from top Chapter 4

Influence of the mean stress

Uniaxial load-controlled high-cycle fatigue tests at different R ratios (σ min /σ max ): -1, -0.5, 0, 0.25, 0.5, 0.6 & 0.7 were performed in air to study the fatigue lives and the mean stress effects. As will be shown below, important modifications in the damage mechanisms were found with change in mean stress.

Experimental procedures

Uniaxial tension fatigue tests were performed using a hydraulic machine with maximum load capacity of ±100 kN. Three kinds of cylindrical specimens were used: 8 mm dia & 16 mm gage length with a surface roughness of R a ≤ 0.4 µm for tests at low mean stresses and 7 & 6 mm dia, 15 & 14 mm gage length with R a = 0.2 µm for tests at high mean stresses, figure 4.1. The diameter of specimens was reduced at high mean stresses so as to avoid crack initiation in threaded region. These specimens were cut along the longitudinal direction from the same area of the clip connector as the samples used to characterize the microstructure.

An extensometer with a gage length of 10 mm and a range of ±15% was used, figure 4.2.

In order to avoid fretting-induced crack initiation, small clots of glue were put between the specimen surface and extensometer knives. Sinus loading was applied with a fixed frequency of 10 Hz.

Load-controlled high-cycle fatigue tests were performed at different R ratios (R = σ min /σ max = -1, -0.5, 0, 0.25, 0.5, 0.6, 0.7)

Surface crack initiation without defects

All the fatigue failures at R = -1 and R = 0 and most of the fatigue failures at R = -0.5 initiated from the surface of the specimen without any responsible defect, figure 4.6a.

Such surface initiation could be due to the stress concentration at intrusions and extrusions generated on the surface as discussed in literature review or from machining marks. However the latter mechanism seems less probable since the specimens had a good surface finish with R a = 0.4 or 0.2 µm.

The fracture surfaces were often mated, especially when R < 0. When it was not the case, striations typical of mode I fatigue crack growth could be observed, as shown in figure 4.6b.

To investigate further the surface crack initiation and propagation mechanism, longitudinal sections of the broken specimens were observed carefully under SEM after mechanical polishing and chemical etching (with a 5% Nital solution).

Several secondary micro-cracks were observed in the specimens submitted to low-cycle fatigue. Most of these cracks followed a crystallographic zig-zag path as shown in figure 4.7(a). The cracks were often deflected by grain boundaries or martensite laths inside the grains and sometimes followed the grain boundaries.

Surface initiation

Crack initiation from surface or internal pores

Internal crack initiation from defects became more and more frequent as the R ratio was increased. Internal initiation is generally observed in very high-cycle fatigue regime (> 10 7 cycles), but in the present study it was found to exist for relatively low fatigue lives in order of 10 5 to 10 6 cycles.

In some cases, crack initiated from a pore located very near to the specimen's surface as shown in figure 4.8, while in some others, the crack initiated from a defect located further from the surface, leading to the formation of "Fish eye" pattern, figure 4.9. The 3D topographic analysis of the fracture surface around the initiation site revealed that the crack initiates along a maximum shear plane at an angle of 42°-43°with respect to the loading axis, as shown in figure 4.10(a). But as the crack grew, it became normal to the loading axis. As in case of surface crack initiation, a stage I to stage II transition seems to occur. Where σ Predicted and σ Actual are the predicted and actual values of the endurance limit.

The error (I%) is evaluated for these criteria and the histograms so obtained are shown in figure 5.14. It can be seen that the range of error for Findley's criterion is from -30% to 0%, for Matake's criterion, it is from -15% to 20% and for Dang-van's criterion, it lies between < -30% and 20%. The errors in prediction of endurance limits are widely scattered for these criteria. For the proposed criterion, the error (I) lies between -5% to 5%, which seems to be more accurate than the rest of the criteria.

Effect of biaxial tension in air 113

Constant amplitude fatigue loadings with various R ratios were investigated here, but the actual loading conditions on clip connectors are random variable amplitude. Very few studies investigated variable amplitude multiaxial fatigue, and mainly for combined tension and torsion [START_REF] Marciniak | Fatigue lives of 18G2A and 10HNAP steels under variable amplitude and random non-proportional bending with torsion loading[END_REF][START_REF] Shamsaei | Multiaxial fatigue evaluation using discriminating strain paths[END_REF][START_REF] Kim | Multiaxial fatigue under variable amplitude loads[END_REF].

Variable amplitude in biaxial-tension loading is still an open question.

It might be possible to estimate the crack initiation life from constant amplitude data, using the classical rainflow cycle counting method and Miner's linear damage cumulation method. For crack growth under variable amplitude biaxial tension, the model developed by [START_REF] Hamam | Variable amplitude fatigue crack growth, experimental results and modeling[END_REF] which takes into account the T stress (and thus biaxiality) could probably be used, after a preliminary calibration, based on crack growth kinetics data. The SIF range for the defects that give rise to internal initiation, can be approximated using Murakami's √ area model for internal cracks [START_REF] Murakami | Metal Fatigue: Effects of small defects and nonmetallic inclusions[END_REF] as:

Appendices

The comparison of the exact SIF range computed with equation 4. An extreme value statistics for defects has also been performed using the procedure described in [START_REF] Murakami | Metal Fatigue: Effects of small defects and nonmetallic inclusions[END_REF]. The cumulative probability plot so obtained is shown in figure B.1.

In order to perform this analysis, the data has to be arranged in increasing order of Les durées de vie en fatigue uniaxiale et biaxiale sont presque les mêmes, malgré des mécanismes de fissuration différents. La protection cathodique annule l'effet néfaste de l'eau salée pour tous les taux de biaxialitié, en dépit d'un net accroissement de la fragilisation par l'hydrogène des joints de grains.

Title : Fatigue and corrosion-fatigue in Cr-Mo steel in biaxial tension

Keywords : steel, fatigue, corrosion-fatigue, biaxial tension, mean stress Abstract: The clip connectors used to join the riser tubes for offshore oil drilling undergo cyclic loading with in-phase biaxial tension in its critical areas along with high mean stresses. This study investigates separately the mean stress and biaxiality effects. Uniaxial fatigue tests were run on Cr-Mo steel with various R ratios to investigate the mean stress effect. The fatigue lives were found to decrease with increasing R, and the endurance limit to follow Gerber's parabola.

To study the effect of biaxial tension, combined cyclic tension and internal pressure tests were run on tubular specimens, at a fixed R ratio. Moderate stress biaxialities had a beneficial effect on fatigue lives, while equibiaxial tension had a slightly detrimental effect. Intergranular facets associated with temper and hydrogen embrittlement were observed on the fracture surfaces.

Several popular fatigue criteria failed to describe all fatigue data and a new fatigue criterion has been proposed that captures both mean stress effect and biaxiality effect. To investigate the influence of corrosive medium (3.5% NaCl) combined cyclic tension and internal pressure tests were run under free corrosion and cathodic protection. In free corrosion, salt water strongly reduced the fatigue lives and suppressed the endurance limit. Fatigue lives in uniaxial and biaxial tension were nearly the same, although the crack growth mechanism was different. Cathodic protection cancelled the detrimental effect of salt water for all biaxialities, in spite of a clear enhancement of H-induced embrittlement of the grain boundaries.