Docteur De L'

Fanshi Meng

Thèse Dirigée Par

Marc Marc Fivel
email: marc.fivel@grenoble-inp.fr

Emilie Ferrie

M Nicolas

M Alexander Hartmaier

M Xavier Feaugas

M Christian Robertson

M Christophe Depres

Emilie Mme

Ferrie

Christophe Depres

THÈSE Pour obtenir le grade de

Keywords: early stages of fatigue, discrete dislocation dynamic, Cu, Persistent Slip Band (PSB), irreversibility, extrusions and intrusions les stades précurseurs de l'endommagement par fatigue, dynamique des dislocations discrètes, Cu, Bandes de Glissement Intensif (BGI), irréversibilité, extrusions et intrusions Discrete dislocation dynamics, Fatigue simulation, Cu single crystal, Persistent slip band, Cyclic irreversibility 2010 MSC: 00-01, 99-00

Meso-scale simulations of fatigue in FCC metals

Fatigue is one of the main failure mechanism for metallic components. The early stages of fatigue refer to the period before the initiation and propagation of fatigue cracks, and accounts for up to 90% of fatigue life. Therefore, the understanding of fatigue damage mechanisms at the early stages is a key issue to improve the operational lifetime of components. Experimental studies have shed light on the importance of the Persistent Slip Band (PSB) as the sites of plasticity localization and subsequent initiation of cracks. The purpose of this thesis is to contribute to the understanding of the formation of fatigue microstructure at the scale of dislocations using 3D Discrete Dislocation Dynamics (DDD). First, simulations of single slip on Cu single crystal are realized. The formation process of the dislocation related microstructures inside PSBs and the surface roughness evolution are elucidated. Under a large strain (> 10 -3 ), a transformation process of the homogeneously distributed dislocations into the organized PSBs is observed, which can be well explained from the diminution of the shear stress on the cross-slip plane. The stability of the numerical PSB is verified with a decreasing loading and is found to be comparable to experimental results. Besides, the comparison between Cu and AISI 316L austenitic stainless steel confirms the importance of cross-slip probability to the distribution and number of PSBs. Simulations of different double slip combinations are also realized to identify the effect of dislocation interactions on cyclic behavior. In the end, the preliminary cyclic simulations for bi-crystals and aggregates are launched with a newly developed DDD code for poly-crystals.

Acknowledgment

When I write this acknowledgement, I still have one day more to finish my quarantine at Canton. Tomorrow is Christmas and I hope they will not extend the quarantine one more week due to the new virus found in UK. . . First of all, I'd like to express my sincere gratitude to my two supervisors Marc FIVEL and Emilie FERRIE, who has provide me constant supports during my PhD career. Most importantly, during these 4 years, they have taught me and set an example about how to do a research, how to understand a question deeper, and how to interpret the complicated simulation results, which has changed the way I think and live. I was also touched and inspired by their motivations and enthusiasm for academic. It is worth mentioning that all this starts from the internship that Emilie offered at the last year of my master. I am always happy that I got this chance.

A similar gratitude belongs to Christophe DEPRRS, who often helped me at difficult times. It's a pleasure to read your thesis, which has given me lots of inspirations. I was always amazed by the way you discussed and interpreted the results in the thesis. I enjoy very much the week working with you at Annecy. I will send you the new code later :) I haven't forgot it.

I'd like to thank Dr.YUAN, who has ended his defense in July. It's very funny that we had the same grades at the exam for the university entrance and then we do a PhD in the same field. It is nice to discuss with you for dislocations and mechanics. Maybe some day, we can write a paper together.

I also want to thank all the members of GPM2 and SIMaP. Thanks to Robin, Alexis and Etienne, who has organized the group entre-aide numeric and introduced me to the world of numerical simulations at the first year. Margarita and Elena, who also study dislocations, I wish you two good luck for the redaction and defense. I'd like to thank specially the members of salle verte, Nicolas, Arthur, Oguz, Guillaume and Aatreya for the hottest summer in Grenoble. Now you have a new office with an air conditioner, I am so jealous. I want to thank all PhD students Kaoutar, Pauline, Zoe, Alexis, Thibault, Jessica . . . . for all the organization of ski-weekend, hiking, restaurants and bars. I should not forger Gurminder for the shots of chartreuse and beers, I wish you will enjoy the beers in Ireland. I'd like to thank my friends in Grenoble, Dawei, Sunrise, Xiaoming and Lingling. I enjoy the time spent with you in Grenoble. I wish you good luck for the new challenges.

Special thanks to my dear Maeve, who has accompanied me and helped me during this four years journey. Without you, I cannot get my PhD and we still have lots things to do and a big world to explore together. Of course, I cannot forget my cat Quatorze, who has bring us lots of happiness. I wish I can take you back to China in January.

Because of Covid, the world in 2020 is not what it was 4 years ago. No bise, no bar and no traveling, my Christmas wish is that everything can return to normal in 2021. i

General introduction

Fatigue damage is one of the main origin of structure failures in different industrial fields. For example, in aviation industry, the crash of the comet airliner is a typical accident caused by fatigue. The repeated pressurization of the aircraft cabin is reported as the cause of the propagation of micro-cracks around the rivet holes, which are used to fix the aerial window on the cabin roof. During the flights, the loading condition is complicated and the dynamic loading is prevailing. The dynamic loading exposes many aircraft components such as propeller blades and the flap actuator rods to possible fatigue failure [1]. Fatigue is also and surely will be a common cause of failure in industrial machinery, such as pumps. Fractures of motor shafts and pump shafts caused by fatigue have been mentioned [2]. Even for the diffuser fixing screws [3] (used to fasten the diffuser disc to the stage casting) which do not endure apparent alternating stress, fatigue failure is evidenced from the fracture surface. Here, the cyclic load on the screws is coming from the vibration of the diffuser disc induced by the rapid change in fluid velocity and pressure during operation. Fatigue failure is difficult to be detected. A stress lower than the yield stress of monotonic loading can still result in a fatigue failure. Fatigue failure occurs in typical scenario but also in deceptive conditions such as the case of screws. A mistake during the inspection, maintenance or repair can lead to a sudden accident or breakdown caused by fatigue. All these factors contribute to the importance of fatigue.

Several methods have been proposed to prevent fatigue failure such as polishing of the material surface [4] or improving mechanical design to avoid stress concentration points [5]. It has been reported that aircraft failures caused by fatigue are nowadays at the lowest level [4]. A better understanding of fatigue damage mechanism has obviously helped the improvement of operation life. However, due to the complexity of the underlying physics in terms of time and space scales, a complete understanding of the fatigue failure mechanism is still not available.

The fatigue damage process is generally divided into two stages [6]: the initiation stage and the propagation stage of cracks. The initiation stage (also called the early stages) of fatigue can represent about 90% of the fatigue life, which is hence important to the fatigue mechanism understanding. The fatigue behaviors of Face-Centered Cubic (FCC) metals at the early stages have been deeply investigated because of the relatively uniformed fatigue behaviors and wide applications in industry [7]. One remarkable observation of FCC metals during experiments of the early stages in fatigue is the formation of Persistent Slip Bands (PSBs) [8].

Therefore, the necessary condition to understand fatigue is to identify the formation mechanisms of PSBs, which is highly induced by the dislocation movements. However, dislocation motion in fatigue is crucial but always difficult to be traced with experimental methods. On the other hand, numerical simulations offer new possibilities. The recent iii General introduction 3D Discrete Dislocation Dynamics (DDD) simulation [9] is a mesoscopic simulation method which aims at relating the dislocation activity to the mechanical response of micrometric volumes. DDD possesses special advantages in studying the collective behaviors of massive dislocation populations because most of the physics of the dislocation mechanisms are taken into account, as well as reasonable constitutive rules or empirical laws when the continuum elastic framework is not sufficient. When both in-situ and post-mortem experiments can only provide limited information, DDD sheds new light on the understanding of fatigue damage at the mesoscopic scale.

The first application of DDD to fatigue was realized by Christophe DEPRES [10] using TRIDIS to study the fatigue behavior of AISI 316L austenitic stainless steel. The objective of his thesis was to investigate thermal fatigue mechanism in piping system of Pressurized Water Reactors (PWRs). Following the work pioneered by Christophe DEPRES, the objective of this new thesis work is to continue the research of fatigue damage using DDD method. The recent development of TRIDIS has improved the simulation capacity, which enables more simulation cycles and larger simulated volume sizes. In addition, the newly developed TRIDIS code for poly-crystals favors further investigations. In this thesis work, Cu is chosen as the major material to be studied because its fatigue behavior is representative of FCC metals and has not been investigated by DDD. The abundant publications on Cu can contribute to the comparison between simulation results and experimental observations.

The manuscript is divided into six chapters:

-Chapter 1 starts with a general introduction of fatigue. A detailed review is given on the early stages of the fatigue behavior of Cu, including the microstructure evolution and the surface roughness. The effect of crystal orientation and grain boundary is also presented. In the numerical section, application of the Finite Element Method (FEM) to fatigue is summarized to compare with DDD method;

-Chapter 2 concerns the methodology part. The model of discrete dislocation dynamics is explained precisely as well as the special settings dedicated to fatigue problems. The former applications of DDD to fatigue problems are reviewed. The motivation and open questions of this thesis work is summarized at the end of this chapter;

-Chapter 3 is the first chapter of simulation results. In single slip loading conditions, a PSB has been constructed in a single crystal of Cu. The main objective of this chapter is to elucidate the formation process of a PSB. The dislocation movements inside the grain are explained as well as the formation process of elementary dislocation structures such as dipoles, prismatic loops and pile-ups. The surface relief at the surface is computed and it is shown how its evolution can be correlate to the PSB mechanism;

-Chapter 4 deals with the effect of the applied plastic strain amplitude on the PSB formation (number, size, dispersion). The relation between the PSB volume fraction and the applied plastic strain is identified. A special attention has been paid to the interaction between PSBs, through their induced stress field. The stability of the numerical PSBs is examined by conducting a simulation with a sudden drop of the applied plastic strain amplitude; iv General introduction -Chapter 5 presents the simulation results in double slip loading conditions.

Similarly to what has been done in Chapter 3 and 4 in the case of single slip loading conditions, this chapter investigates the effect of double glide loading conditions where a slip system and its associate deviate slip systems are simultaneously activated;

-Chapter 6 presents a first application of the newly developed poly-crystal version of TRIDIS to fatigue simulations. A polycrystalline aggregate made of 7 grains is loaded in fatigue and results focus on the effect of the grain texture on the fatigue behavior.

The thesis ends with general conclusions and perspectives for future studies.

As mentioned in the general introduction, the objective of this work is to contribute to a better understanding of the fatigue degradation mechanism. This chapter starts with a general presentation of the basic concepts of fatigue such as the load characteristics and the different stages of fatigue damage. Then, the detail of dislocation related microstructure evolution (veins, PSBs and cells) from experiments under cyclic loading is presented. The related surface roughness evolution (extrusions and intrusions) is reviewed as well. Because of the numerous publications, Cu is chosen as the mainly reviewed metal in this chapter. Afterwards, to elucidate the difference in the fatigue behavior of FCC metals, slip modes in different metals and alloys are discussed. Then, since crystal orientation and grain boundary play an important role in the fatigue of poly-crystals, the effect of these two factors on the dislocation microstructure variation is demonstrated. In the end, applications of the Crystal Plasticity Finite Element Method (CPFEM) to investigate the fatigue behavior is reviewed shortly. Special attention is paid to the simulation of microstructure formation at the dislocation scale and the limitations of the CPFEM method.

Basic terms of fatigue damage

During the history of engineering design, thanks to a better understanding of the failure mechanism, much progress has been made in order to improve the reliability of components. Mechanical failure can originate from different loading conditions such as excessive deflection, buckling, ductile fracture, brittle fracture, impact, creep, stress corrosion cracking, and different types of fatigue [11]. Among all of these different mechanisms, most of them are induced by monotonic mechanical loading. Whereas, fatigue damage is special for the repetitive character of its loading, which adds more difficulties for a deep understanding. Metal fatigue has been of interest for about 170 years both for scientists and engineers. This interest dates back to the development of the steam engine, mechanical transport, and the extensive use of mechanical devices.

The characteristics of cyclic loading play a key role in fatigue damage. Its frequency and amplitude can lead to different micro-mechanical behaviors. In Figure 1.1, the basic parameters used to define a typical fatigue cycle are shown:

Chapter 1. Bibliography -σ max and σ min stand for the maximum and minimum values of the stress during the cyclic loading;

-ε max and ε min represent the maximum and minimum values of the total strain during the cyclic loading. Note that in this work, the cyclic loading is controlled by the plastic strain ε p and the detail of the simulations settings is explained in chapter 2;

-The amplitude of stress and total strain are defined respectively as: ∆σ/2 = (σ maxσ min )/2 and ∆ε/2 = (ε maxε min )/2; -Plastic strain amplitude is defined as: ∆ε p = ∆ε/2 -∆σ/2E, E is Young's modulus;

-The average stress and strain are defined respectively as: σ m = (σ max + σ min )/2 and ε m = (ε max + ε min )/2;

-The load ratio: R = σ min /σ max defines the shape of the cycle so that R = -1 is symmetric cycle (σ m = 0).

(a) (b) Compared with other types of damage mechanisms, fatigue damage is one of the most difficult to detect and prevent. Restriction in the deformation amplitude of deformation is a common-used way to prevent mechanical failure of monotonic loading. However, this becomes useless for cyclic loadings because a single cycle generally does not cause any apparent damage. When a material is subjected to a fatigue loading, the damage would first accumulate, then appears as a crack. For engineers and researchers on fatigue, prediction of an operation lifetime is an essential task. In 1870, Wöhler [13] measured the deformation on railway tracks. He related the deformation resulting from the passage of trains with failure. This work inspired his successor Spangenberg [14] to establish the famous Wöhler diagram, also called the S-N diagram, shown in Figure 1.2. It demonstrates the relation between loading stress amplitude ∆σ/2 and the number of loading cycles to failure N f . A simple uni-axial cycling is usually used to obtain the S-N diagram. As shown in the S-N diagram (Figure 1.2), the number of cycles for failure varies with the imposed stress amplitude, which signifies that the micro-mechanism behind can be different. According to different mechanical responses, fatigue problem is divided into three main categories:

Basic terms of fatigue damage

-Oligocyclic fatigue or Low Cycle Fatigue (LCF): enduring a strong imposed loading, the fatigue crack emerges quickly and the fracture takes place after a small number of cycles (N f ≈ 10 4 ∼ 10 5 );

-Polycyclic fatigue or High Cycle Fatigue (HCF): the loading is relatively weaker; in this case, fracture occurs after a large number of cycles (N f ≈ 10 6 ∼ 10 7 ); -Very High Cycle Fatigue (VHCF) or Ultra High Cycle or gigacyclic fatigue (UHCF): under an extremely small deformation, the loading cycle tends to be infinite.

These three categories of fatigue are shown in Figure 1.3. A horizontal asymptote can be observed when N f is very large. The intersection of this horizontal line to the longitudinal coordinate axis is conventionally regarded as the fatigue limit stress, noted as σ F L . If the fatigue loading is below this limit, the endurance is theoretically unlimited, which means that there is no nucleation of cracks.

The fatigue damage is a progressive process. A series of physical phenomenon takes place in a chronological sequence and eventually leads to failure. As shown in Figure 1.4, the crack size a increases with the number of loading cycles N and encounters different mechanical barriers. The successive stages of the fatigue damage process can be summarized as follows:

(1) Nucleation of cracks: normally the micro-cracks nucleate within the Persistent Slip Bands (PSBs, presented in detail in Section 1.3.1.2);

Chapter 1. Bibliography (2) Propagation of cracks across several grains: at this stage, it is called a short crack, and the grain boundaries play a role of barrier [17] because the propagation velocity decreases sharply near the grain boundary;

(3) Propagation of a macroscopic crack (a > 0.5 ∼ 1mm): it is then called a long crack and the behavior is described by fracture mechanics;

(4) Failure of a component. (3) propagation of macro-crack; (4) failure [16].

Moreover, the stages before the initiation of cracks, which are also called early stages of fatigue degradation, account for up to 90% of fatigue lifetime. This highlights the importance of the early stages of fatigue degradation. In fact, as mentioned before, the early stages of fatigue are the main focus of this thesis. After the introduction of a general understanding of the fatigue problem, in the next section, the dislocation microstructure evolution during the early stages of fatigue is presented, focusing on Cu.

1.3. Microstructure evolution during the early stages of fatigue

Microstructure evolution during the early stages of fatigue

The investigation of fatigue damage mechanism focuses on from macroscopic-scale to microscopic-scale. The first research connecting fatigue damage to microstructure dates back to the study of [START_REF] Ewing | The fracture of metals under repeated alternations of stress[END_REF] [18] who reported the fatigue crack propagation along the trace of active slip planes visible at the surface of specimens. After the fast development of the characterization methods in material science such as Transmission Electron Microscopy (TEM), Scanning Electron Microscopy (SEM), Atomic-Force Microscopy (AFM), numerous highly qualified images of microstructure under fatigue loading are published, that gives valuable insight into the fatigue mechanisms.

Because of their relative simplicity, the Face-Centered Cubic (FCC) metals have been studied for a long time so was the dislocation microstructure developed in fatigue. Cu is probably the most documented material among FCC metals, and the publications are abundant. In this section, a brief review of the microstructure evolution in Cu under cyclic loading at the early stages of degradation is presented and explained in detail.

Stress-strain curve, saturation state and associated dislocation microstructures

The dislocation microstructure observed in the bulk of a crystal under cyclic loading depends on many factors such as the mode of loading (uni-axial, bi-axial or multi-axial), the loading amplitude, the loading rate, the temperature and crystal orientation. Among all these factors, for FCC materials, the relation between the crystal orientation and the loading direction is the most important one because it has an effect on the activation of different slip systems. The simplest case is the single slip situation, meaning that only one slip system is activated.

As shown in Figure 1.5, for well-annealed single crystals of Cu in single slip situation [8] under fully reversed loading, a rapid hardening can be observed during the first few cycles. Then the hardening rate decreases with continuous cyclic loading until a saturation state is reached after N s cycles.

The evolution of the resolved shear stress value at saturation, τ s with the different imposed resolved shear strain amplitude, γ pl for Cu single crystals is represented in Figure 1.6, which was firstly measured by Mughrabi [8] (called Cyclic Stress-Strain Curve or CSS curve). The relation between the microstructure and the imposed shear strain amplitude can be briefly summarized with the help of this curve.

There are mainly three regimes (A, B, C) in the schematic diagram (plotted in Figure 1.6) which stand for different hardening behaviors. The hardening here refers to the saturation stress evolution with the applied plastic strain amplitude, which is different from the rapid cyclic hardening shown in Figure 1.5. At low plastic strain amplitude (γ pl < γ pl,AB ), in regime A, the increase of the saturation shear stress τ s is caused by the formation of veins, dislocation bundles or loop patche microstructures. These microstructures are constructed by multiplication of dislocations on the active slip systems, also called the primary dislocations. In regime B, a plateau appears, where the saturation stress, τ * s is independent of the applied plastic strain amplitude. A type of slip band, where the plastic strain localizes, appears through all the bulk. It corresponds Figure 1.5 Typical evolution of the resolved shear stress, τ R as a function of the imposed resolved shear plastic strain amplitude, γ R under fatigue loading for Cu single crystals in single slip mode [7].

1.3. Microstructure evolution during the early stages of fatigue to the so-called PSBs [19], which play an important role in the nucleation of cracks. For higher plastic strain amplitudes, (γ pl > γ pl,BC ), an increment of the saturation stress, τ * s occurs in regime C, which is induced by the activation of the secondary slip system. The corresponding microstructure transforms into dislocation cells or labyrinths. Figure 1.6 A schematic diagram for the relation between stress-strain saturation state and dislocation microstructure [7].

The general idea for the microstructure evolution under fatigue loading is simple: the microstructure needs to evolve in order to accommodate the imposed plastic strain amplitude. The underlying mechanisms are nevertheless complex. In the following section, the experimental observations of the fatigue microstructures (veins, PSBs and labyrinth) are presented in detail.

Dislocation vein structure

In regime A, when the applied strain is relatively low, for single-slip oriented Cu single crystals, the vein structure can be observed during the saturation state. The forming process and phenomenon can be described as follows:

(1) At the beginning of loading, existing dislocation sources generate dislocations on the primary glide system [20], and more dislocation sources are produced with the straining;

(2) For fully reversed cyclic loading, a nearly equal amount of positive and negative dislocations are generated [21]. The force between dislocations of the different sign is strong and attractive. For edge dislocations, the strong attraction effect can trap two non-coplanar dislocations and form a dislocation dipole. But for screw dislocation, the possibility of cross-slip leads to a mutual annihilation of the screw dipoles, especially for a material with high Stacking Fault Energy (SFE). Another mechanism for the formation of dislocation dipoles proposed by Laird is the doublecross-slip mechanism [22], presented in Figure 1.7;

Chapter 1. Bibliography (3) A group of dipoles can be considered as dipole arrays, which is very common during the early stages of fatigue damage;

(4) The highly packed grid structure developed by dislocation dipoles or multi-poles is referred to as veins, bundles or loop patches. Figure 1.8 shows a typical vein structure observed in Cu [22,24]. The veins have a high dislocation density (10 15 m -2 corresponding to a mean dislocation spacing of 30 nm) and are separated by channels, where the dislocation density is three orders of magnitude lower.

The vein structure gives rise to the hardening evidenced in the first stages by preventing the dislocation movement on the slip planes. For a given applied strain amplitude, a dynamic equilibrium of the dislocation arrangement is built-up in the vein structure. This dynamic equilibrium allows the accommodation of the imposed plastic deformation without changing the microstructure. However, a limit of the capacity to accommodate the imposed plastic strain exists. If the plastic stain amplitude is further increased, the vein structure is not able to accommodate the plastic deformation and veins have to transform into the PSB structure to accommodate more deformation.

Dislocation structure of PSBs

Even though the vein structure of dislocations is different from the dislocations in the rest of the grain (the matrix), the distribution is quite homogeneous (as shown in Figure 1.8(a)). At the surface also, the slip traces printed by the dislocations are homogeneous with veins in bulk. However, when the plateau in region B is reached, numerous slip bands are observed on the surface of the specimen. In Figure 1.9 [25], a series of photographs of the same area of a (1 2 3) Cu single cristal surface shows the evolution of slip traces with the number of cyles (2500 to 125000). With the increase in the number of cycles, the slip traces formed at the begining of the loading (2500 ∼ 20000 cycles) develop and new ones form as well. After 20000 cycles, the slip traces are more concentrated without significant multiplications. These concentrated slip traces 1.3. Microstructure evolution during the early stages of fatigue (a) A TEM (Transmission Electron Microscopy) image of vein structures made of dislocations in the primary glide system for Cu single crystals in the saturation state at 77.4K [24].

(b) 3D view of vein structure in Cu for γ pl = 2.6 × 10 -5 [22]. Chapter 1. Bibliography construct a slip band. These bands are called persistent because they always reappear at the same location during continuous cyclic loading even after the electro-polishing of the surface [7]. This observation proves that a special microstructure is formed under the specimen surface. Figure 1.9 Evolution of the slip traces on the (1 2 3) surface for a Cu single crystal under γ pl = 1.5 × 10 -3 [25] (the number of cycles is in the unit of 1000).

Each PSB consists of several thousands of slip planes (≈ 5000 at 20 • C), which form a lamellar structure crossing the whole crystal. Within the PSB, the so-called wall/channel structure can be observed. Figure 1.10(b,c) shows a direct but simplified explanation of the wall structure. PSB walls mostly consist of firmly packed edge dislocation dipoles and are normal to the primary Burgers vector b. Inside the channels, screw dislocations glide back and forth for carrying the plastic deformation. When the PSB is observed along the edge dislocation line direction [1 2 1] in Figure 1.10(c), a ladder type structure appears, which has been observed in numerous studies. Compared with the vein structure, the dislocation density in the PSB walls is still higher and more importantly, the spatial distribution of PSBs is not as homogeneous as for veins.

In fact, the PSBs are surrounded by veins as shown in Figure 1.11. Figure 1.11(a) is a TEM observation from Polák et al [28] and Figure 1.11(b) is the corresponding illustration taken from the Ph.D. thesis of Phung [15]. In the surrounding matrix, veinlike structure accounts for 50% of the volume. The parallel wall structure occupies about 10% of the volume of PSBs [29]. The dipole wall width is about 0.03 ∼ 0.25µm and the distance between two walls is about 1.0µm for Cu [30] and 0.5µm for AISI 316L austenitic stainless steel [15]. The dislocation density in wall structures or veins Chapter 1. Bibliography is between 10 11 ∼ 10 22 m -2 which is a three order magnitude larger than the dislocation density in channels.

(a) TEM image close to the free surface from copper cycled at 83K. Three PSBs are identified with evidence of the ladder shaped structure. A, B, C marks indicate the corresponding surface roughness. The matrix surrounding the PSBs presents the vein structure [28].

(b) Illustration of the ladder structure inside PSBs and the surrounding matrix-veins [15]. 1.12(a) shows 3D PSB in Cu. In the plane (1 2 1) plane (Figure 1.12(b)), the ladder structure appears and is perpendicular to the primary Burgers vector b, which is the same geometric relation as in Figure 1.10(c).

If γ pl < γ pl,AB , vein structures predominate in the matrix. When the imposed strain amplitude reaches the threshold value for the formation of PSBs, the plastic deformation gets localized in PSBs. As mentioned in [31], this change in the dislocation organization means that the PSBs have a higher capacity to endure plastic deformation than the vein structure. With the increases of the applied plastic shear strain amplitudes γ pl , the volume fraction of PSBs increases to accommodate the increasing plastic deformation. The augmentation of PSB volume fraction ensures the stability of the saturation shear stress. This is the reason for the plateau observed in Figure 1. 6.

The veins can be considered as the precursor of PSBs and they all contain many edge dislocation dipoles. Figure 1.13 presents the microstructure transformation process: dislocation dipoles or multi-poles are produced during the early stages of the cyclic loading; the dislocation dipoles gather together in veins; more dislocation dipoles aggregate to construct the walls in PSBs and the deformation becomes heterogeneous.

To explain the dislocation movements during the saturation state, the deformation is believed to be accomplished by the back and forth motion (flip-flop) of the dislocation loops [32] in the veins and PSBs. The screw dislocations move reversibly in the channel and the edge dislocations move from one dipole bundle to another [33]. Then, it has also been proposed that, the balance between multiplication and annihilation within the PSBs is responsible for further accumulation of plastic deformation whereas the local screw and edge dislocation densities remain unchanged [34].

With the increasing imposed plastic strain amplitude, the dislocation structure changes. The fatigue microstructure evolution under an increasing applied strain is such 1.3. Microstructure evolution during the early stages of fatigue (a) A three-dimension reconstruction of the PSB structure from TEM images for Cu fatigued at 20 • C at γ pl = 1.5 × 10 -3 [30].

(b) TEM image of the dislocation structure in a Cu crystal with γ pl = 1.5 × 10 -3 at room temperature, viewed from the [1 2 1] direction. The matrix-vein structure is noted as M. The wall structures (ladder shaped) and the screw dislocations in the channel are clearly seen [30]. Chapter 1. Bibliography that a dislocation structure that enables the accommodation of a higher deformation is built. The labyrinth dislocation microstructure formed for large imposed strain amplitudes is based on the same idea: it is built as soon as PSBs cannot accommodate any longer the imposed plastic strain amplitude.

The labyrinth microstructure

We still consider the situation of a Cu single crystal. For extremely large applied strain, single slip is not able to accommodate the deformation so that a second slip system is activated: progressively, labyrinth or cell structure appear for γ pl > 2 × 10 -3 [35]. Figure 1.14 shows a typical labyrinth structure. Dislocation bundles with different Burgers vectors ( b 1 and b 2 indicate the primary slip system and the conjugate slip system respectively) interlock to each other. For a crystal oriented for double slip, the labyrinth structure can also be observed even for low applied strain values. Figure 1.14 Labyrinth microstructure (view from (0 1 0)) for a Cu single crystal fatigued under γ pl = 5 × 10 -3 [35].

In summary, with the increasing imposed plastic strain amplitude, the microstructure formed during the saturation state transforms from veins to PSB and finally to labyrinth. Finally, double slip or multi-slip occurs eventually to accommodate the plastic deformation.

Nucleation and evolution of PSBs at the micrometer scale

Normally when it comes to PSBs, the scale refers to the bulk materials, where the PSBs are first observed. Nevertheless, the existence of PSBs at the micro-meter scale has always been a question remained to be discovered. Recently, Steven Lavenstein and Jaafar El-Awady [36] has designed a high-frequency micro-fatigue experiment to investigate the PSB formation in a very confined sample. The sample is sized to have rectangular cross sections with nominal dimensions of 12µm by 13µm and gauge lengths of 27µm and tested in single slip mode. At the same time, an in-situ SEM is installed to observe the deformation during the fatigue test.

Microstructure evolution during the early stages of fatigue

The main experimental results are summarized in Figure 1.15. The figures in the top left present the successive micrographs obtained from the SEM, which shows the initiation and propagation of PSBs at the micro-sized sample. The proposed PSB formation mechanism is presented in the bottom left. The evidence of PSB structurethe dislocation walls is presented in the right part. In this article, the author proposed that the localization of PSBs can be identified at the first few cycles from the slip traces and then the PSBs are developed. This statement is very important for the DDD simulations because due to the computational cost, dozens of cycles can be simulated. These experimental results indicated that it was not necessary to realize too many cycles, which favors the rationality of DDD simulations.

Surface relief and crack initiation

The vein structures are homogeneously distributed in the crystal and the corresponding surface slip traces are uniformly distributed as well. When the plasticity localization starts with the build-up of PSBs, at the same time, the surface roughness starts to localize at the intersection of PSBs and the free surface. It is believed that the roughness is caused by the irreversible slip of dislocations within the PSBs. Surface roughness is mainly represented by microscopic hills and valleys when the PSBs emerge at the surface. The microscopic hills and valleys are called intrusions and extrusions. The surface relief may lead to stress concentration and augmentation of the local dislocation density, which is of considerable importance for crack nucleation. Many studies have reported crack nucleation at the PSB/surface intersection [27,37].

Intrusions and extrusions

Experimental observations

The first observed phenomenon of surface roughness has been reported by Forsyth [6] on Aluminum-Copper alloy, which showed that thin ribbons of the metal emerge at the specimen surface from the PSBs. Similar extrusions were also observed on Cu [38]. The taper-sectioning method introduced by Wood [39], combined with experimentally imaging techniques, allows us to obtain high-resolution images of the surface morphology of fatigued crystals. This method has been widely used for Cu [40,41] and enabled to identify the main characteristics of extrusions and intrusions that are summarized hereafter:

-Intrusions, extrusions and protrusions cover the PSB surface. A protrusion is a large extrusion containing small extrusions and intrusions and includes some matrix and PSB lamellae [41] (see protrusion in Figure 1.17);

-The extrusion has a width of 1 ∼ 2µm and a height of 2 ∼ 3µm;

-The protrusion height augments along with the width of the PSBs and in the direction of the primary Burgers vector;

-Optical interferometry shows that at the PSB-matrix interface, the local strain is higher than the average value.

Morphology of extrusions and intrusions

The morphology of the surface roughness, also noted as Persistent Slip Marking (PSM), is easily influenced by other factors like the material, the loading amplitude etc. Figure 1.17 summarizes the different forms of PSM observed. Three types of PSMs are shown: "tongue-like" extrusions & intrusions, "ribbon-like" extrusions & intrusions and protrusions. The first two extrusions & intrusions are much thinner so that the corresponding PSMs are called individual PSMs while the protrusions are called macro-PSMs. The image quality of PSM has greatly improved with the newly developed characterization methods. In [42], J. Man and al. provide an exhaustive list of the latest developments in microscopic techniques for the study of surface relief. The morphology of extrusions and intrusions varies. Figure 1.18 shows the representative experimental images of the PSMs' morphology (even though Martensite is not the material we will study). Depending the extrusion dimension, they are divided into "tongue-like" (Figure 1.18(a)), "ribbon-like" (Figure 1.18(b)), "band-like" (Figure 1.18(d)) and "cord-like" (Figure 1.18(d)). The secondary extrusions can also be observed in (Figure 1.18(a)), which is caused by the secondary activated slip systems. The reason for the different morphology surface is still not clear. In chapter 3, we will explain the formation mechanism for "tongue-like" and "ribbon-like" extrusions.

Model of Polák

When it comes to the extrusions and intrusions, the model of Polák [42] should be mentioned. On the basis of the point defect diffusion and the matter redistribution in the PSBs or between the PSBs and the matrix, Polák has constructed a complete theory to explain the morphology of the extrusions, and the crack initiation. The detail of the model is presented in Figure 1.19. Firstly, the vacancies are supposed to be homogeneously generated within the whole PSB, including the dislocation-rich walls and the dislocation-poor channels. However, the annihilation rate is considered as a function of sink density, and edge dislocations and edge dipoles are preferred sinks for vacancies. As the walls contain a higher density of edge dislocations and edge dipoles than that in channels, more vacancies are annihilated in the walls. Consequently ], the vacancy concentration reaches its maximal value in the middle of channels and decreases towards the walls (see A -A and B -B in Figure 1. 19(a)). This heterogeneous distribution results in a flux of vacancies from the channels to the walls or from the PSBs to the matrix. On the other hand, the flux of vacancies causes a flow of the mass in the opposite direction: from the walls to the channels or from the matrix to the PSBs. Based on this interpretation, Polák has explained two morphology of extrusions and the mechanism for the crack initiation. In the case of "tongue-like" extrusions (see . With the continuously cyclic loading, the "tongue-like" extrusions appear periodically and the distance between "tongue-like" extrusions are highly related to the distance between walls. Figure 1.19(c) presents the case for a "ribbon-like" extrusion, because of the mass flux from the matrix to the PSB, mass accumulates in the PSBs and decreases in the thin layer at the interface between the PSBs and the matrix. The redistribution of mass results in a compressive stress in the PSB lamella, which can only be released by gliding dislocations in the Burgers vector direction. This process leads to a "ribbon-like" extrusion with a width of the corresponding PSB. At the interface between the PSB and the matrix, the tensile stress rises and is relaxed by dislocation. Under this mechanism, a narrow intrusion appears and grows progressively to a crack.

Crack initiation along PSBs

The roughnesses induced at the PSB surface are privileged nucleation sites for fatigue cracks mainly because the surface irregularities induce stress concentration. As indicated in the macro-PSM in Figure 1.17, the first possible nucleation site is the interface between the matrix and the PSB. This interface is a plane of discontinuity for the dislocation density [40,44]. This reveals that the strain is not homogeneous and it is localized in the PSB-matrix interface. Figure 1.20 shows a crack initiated at the interface for a microspecimen of Cu single crystal [45]. Meanwhile, the difference between an intrusion and Chapter 1. Bibliography a crack is sometimes difficult to distinguish. In [46], Hunshe and Neumann proposed a criterion to distinguish an intrusion from crack nuclei if the vertex angle is higher than 30 • ; otherwise, it is considered as a crack. As shown in Figure 1. 16, we can see intrusion-like cracks located between two extrusions. It is worth mentioning that the initial surface roughness obviously leads to crack nucleation and removing the roughness by electropolishing is found useful to increase the fatigue life [19]. Figure 1.20 Cracks observed at the interface between an extrusion and the matrix for a micro-scale Cu specimen [45].

In this section, we have reviewed the typical microstructure formed inside the crystal and at the surface during cyclic loading. The majority of the content is from the literature on Cu. In the following section, we shall discuss the similarities and differences in the behavior of FCC metals.

Specificity of other FCC metals/alloys

The cyclic behaviors of different FCC single crystals present many common features, such as the PSB formation, surface roughness and initiation of cracks. But when it comes to the detailed analysis of fatigue characteristics, differences can still be observed concerning the dislocation structures, PSB distributions, and hysteresis curves. One of the critical factors in fatigue behavior is the slip mode. According to experimental results for various FCC metals [47] [23] [48] [49], two typical slip modes have been observed: wavy slip and planar slip.

Cu is the best-known and typical of wavy-slip material. In single slip loading conditions, the main characteristics of wavy-slip materials include the saturation of the stress in the cyclic hardening and the saturation plateau observed in the CSS curve as discussed in Section 1.3. Nickel and silver are also identified as wavy-slip materials [23].

On the other hand, planar slip is much more commonly seen in some copper alloys, such as copper-aluminum alloys and copper-zinc alloys. Planar slip materials are characterized by the absence of the saturation plateau in the CSS curve and the absence of PSB dislocation structures as reported for Cu-31wt% Zn single crystals [50]. The same conclusion has also been drawn by Abel [51] who has investigated the impact of an increase in the Aluminum content from 2 to 16 at% on the cyclic deformation behavior 1.4. Specificity of other FCC metals/alloys for Cu-Al single crystal. It is found that for an Al concentration lower than 4at%, the cyclic saturation behavior in the hardening curve of Cu-Al alloys was similar to that of pure copper, while the saturation behavior was not observed when the Al concentration is more than 11at%. Additionally, Yan and Laird [52] observed a fluctuating appearance of the saturation behavior in Cu-16 at% Al single crystal. For Cu-30wt% Zn, only a short saturation period could be observed before the fracture.

The wavy slip and planar slip are only two extreme situations. An intermediate slip mode owning both the characteristics of wavy slip and planar slip is possible. A detailed comparison of two types of the slip mode is given in the following section.

Distinguishing features for planar slip and wavy slip

Wavy slip is always characterized by the easy cross-slip [49] [47]. Easy crossslip contributes notably to the elimination of screw dislocations and promotes the construction of multi-pole edge dislocation vein structure. Vein structure is considered to be metastable [20] and can be transformed into the PSB-ladder structures, which is mentioned above with the saturation behavior together as one of the main characteristics for the wavy slip. Consequently, the ease of cross-slip can be considered as a potential criterion to distinguish the slip modes. The first possible criterion is the SFE proposed by Lukas and Klesnil [50]. Because from the micro-scale point of view, for FCC metals, a perfect screw dislocation is dissociated into two partial dislocations separated by a stacking fault, which is considered as an extended dislocation. Before cross-slip, the two dissociated dislocations need to transform into a perfect dislocation. As this is difficult to realize along the entire dislocation line, what often happens is that a local constriction is formed and then moves to the cross-slip plane and dissociates again to two partial dislocations. High stacking fault energy eases the formation of the constriction, which consequently helps the cross-slip. It is thought that a low SFE leads to planar slip and high SFE leads to wavy slip. Based on this conclusion, several studies addressed the influence of the SFE on the slip mode by adding solute elements in Cu to reduce the SFE.

In the study of Abel [51], concerning the cyclic behavior of Cu-Al alloys with various Al contents (2 at%, 5 at%, 7 at%, 11 at%, and 16 at%), it was found that, the cyclic hardening rates decreased and the saturation stresses increased with decreasing the SFE. Furthermore, for poly-crystalline Cu-16 at% Al, Feltner and Laird reported that the CSS curve of this alloy depends strongly on the initial condition and the deformation history, which is regarded as the signature of planar slip. Owing to its low SFE, Cu-16 at% Al single crystal is considered as a model planar slip alloy. For planar slip materials, compared with wavy slip materials, the incorporation of solute elements in alloys induces more diversities in the fatigue behaviors.

Starting from the dislocation structures in FCC metals or alloys, under low and intermediate applied deformation amplitude, wavy slip materials including pure metal and dilute solid solution present a more three-dimensional arrangement of dislocations. The arrangements consist of dipole bundles (shown in Figure 1.21(a)) or dislocation walls [7], as shown in Figure 1.12. It has also been said that for wavy slip, the distribution of dislocations are more heterogeneous because high dislocation density regions co-exist with dislocations free regions. While for planar slip material, the distribution is much more uniform in the slip plane. Dislocations come out in the form of extended pile-ups and multi-poles. Hundreds of dislocations can be observed in one plane and the activated slip planes are well separated from each other. This behavior results in the formation of multi-poles aligning along the intersection line between the primary plane and the cross-slip plane, as shown in Figure 1 The cross-slip path in the two types of slip modes is quite different. Figure 1.22 depicts well the effect of the presence of an obstacle in the crystal but it can also be used to help the understanding of pure metals. For planar slip, the first dislocation gets through the obstacle under a large resistance and breaks the obstacles. This is a kind of softening effect so that the following dislocations produced by the activated source shall choose the same itinerary to advance and push the first one to overcome the obstacle resistance. This process leads to an increase in the slip velocity as well as the accumulated plastic deformation. The deformation is highly localized in this slip plane, which contains many dislocations from emitted the same dislocation source. Known as glide plane softening, this phenomenon also explains the strain burst observed in planar slip materials [51] [53]. On the opposite, for wavy slip, the cross-slip events are more like independent individual events and not a collective mechanism as in the planar slip case. The cross-slip can occur on any plane and thus spread in a larger width.

The comparison of the surface roughening of the two slip modes is not so clear because it is difficult to quantify the difference. Generally, it is concluded that for wavy slip, the surface morphology is fine and always consists of extrusions and intrusions, while for planar slip the surface morphology is more coarse.

From a point of view of energy, Hong [48] has concluded that the activation energy for low SFE alloys to surmount the aggregation barrier is higher and additionally, the driving force for clustering is lower in low SFE alloys. In Figure 1.23, the evolution of the free energy for the clustering of wavy slip and planar slip is presented schematically.

The free energy varies with the dislocation arrangements. The curve on the left refers to the well-separated dislocation situations while the one on the right is associated with the clustered dislocations (dislocation walls or dipole arrays). The aggregation energy barrier of dislocations for wavy slip is lower than that of planar slip and the aggregation form of dislocations in wavy slip material is more stable than that of planar slip material.

Important factors determining the slip mode

We have seen that the slip mode is an important factor in the cyclic behavior of FCC metals. The proposition of a criterion for the wavy-to-planar-slip transition is thus very critical for the understanding of the basic deformation mechanism and also the fatigue failure mechanism. In the case of FCC metals, the dislocation distributions, and the dislocation microstructures highly depend on the cross-slip behavior. As a result, the suggestion of a possible criterion always originates from the consideration of the intensity or the easiness of cross-slip. As mentioned above, SFE is the first criterion proposed by Lukas and Klesnil [50] and it can be explained as follows: high SFE contributes to the cross-slip and frequent cross-slip destroys the planar arrays, conversely for low SFE, planar arrays are easier to be constructed. Indeed in low SFE alloys, the change of the plane during cross-slip is more difficult due to the large dissociations of the dislocation lines which makes the constriction difficult. Thus, the SFE criterion seems to be appropriate for pure metals [54] [49].

However, some alloys with relatively high SFE present the planar slip characteristics [55]. This anomaly was attributed to the impact of solute atoms on the cross-slip. Mughrabi and Wang [56] have proposed a criterion based on the yield stress and the SFE. Hong [48] has discussed the friction stress effect on the slip mode. The basic idea is that the Shockley partials of the dissociated screw dislocations have edge components and during the cross-slip process, they are impeded by the friction stress induced by the solute atoms. Hong [48] has also proposed that the long SF generally hinders the clustering of dislocations owing to the "geometrical nature" of the stacking fault. Based on this hypothesis, a model combining the friction stress and the stacking fault energy has been developed. The prediction made by the model for the transition from wavy slip to planar slip agrees well with experimental observations on Cu-based alloys. Moreover, the model has concluded that the planar slip is also promoted by the increase of the shear modulus, the atomic size misfit and the solute content. Besides, Karnthaler and Schügerl [57], Clement [58] and Steffen [55] have proposed that in alloys, the short-range-order (SRO) influences more the slip mode than the SFE.

In the recent study of [47], Wang proposed the ratio e/a (free-electron-to-atom ratio) to be the criterion that determines the slip mode, because all the important factors (SFE, friction stress or SRO) are directly or indirectly dependent on the e/a ratio. The author found that, as shown in Figure 1.24, after plotting e/a as a function of the SFE for Cu-based alloys (e/a can be changed by the variation of solute concentration), there are always a high-slope regime and a low-slope regime. In the high-slope regime, the slip mode is wavy slip; otherwise it is a planar slip and between the two regimes, a mixed mode is observed.

In this section, the slip modes have been reviewed considering the following points: typical alloys, distinguishing features and important influencing parameters. Cu is the most typical pure metals for wavy slip and Cu-based alloys with a high concentration in solute atoms (Cu-16 at% Al alloy and Cu-30% wt Zn alloy) are typical planar slip materials. As mentioned, the dislocation microstructure of planar slip shows 1.5. Crystal orientation effect Figure 1.24 Evolution of the SFE with the e/a ratio of Cu-Zn, Cu-Al and Cu-Mn alloy systems [47]. more planarity characterized by dipole arrays and wavy slip constructs a more threedimensional structure where dislocations prefer to aggregate in the form of veins or walls. The saturation phenomenon in macroscopic behavior is also a remarkable difference. In the next chapters concerned with the DDD simulations, as Cu is a high SFE material and AISI 316L austenitic stainless steels is a relatively low SFE material, the comparison between Cu and 316L will be realized as well to check these differences.

Crystal orientation effect

As presented in Section 1.3, most of the studies on the early stages of fatigue damage concern single slip loading state which leads to veins or PSBs dislocation structures. When the imposed plastic strain is very large, the activation of secondary slip systems can lead to the formation of labyrinth structures. Therefore, it can be inferred that the number of activated slip systems plays an important role in the fatigue deformation and it is determined by the crystal orientation.

The influence of the crystal orientation on the microstructure formed in Cu and other FCC metals (Ni and Ag) has been summarized by Li [23]. Considering the typical saturation plateau (Figure 1.25(a)), a similar plateau has been observed in single slip, conjugate double slip, and [0 1 1] multi-slip oriented Cu single crystals, while the coplanar double slip, critical double slip and [0 0 1] multi-slip oriented single crystals do not display any plateau on the CSS curve and the saturation stress is higher than for the other loading conditions. For the dislocation patterns, some common features can be observed, as shown in Figure 1 J. Nellessen [59] has carried out fatigue tests on AISI 316L poly-crystalline stainless steel and used Electron Back Scatter Diffraction (EBSD) and Digital Image Correlation (DIC) technics to identify the relation between crystal orientation, local cumulative plastic strain and dislocation structures. Even after a fatigue test of only 100 cycles, the orientation effect is obvious. To quantify the complexity of the structures, the author has classified the microstructure into 8 classes (classes 1 to 3 refer to individual dislocations; 4 and 5 stand for dislocation clusters; 6 stands for the pre-veins; 7 and 8 refer to the prewalls structure, the details are presented in Fig. 12 of [59]). The general conclusions have been summarized in Figure 1.26. Figure 1.26(a) depicts the relationship between the structure complexity, the crystal orientation, and the product of the number of loading cycles and the local strain amplitude (blue stands for classes 1 and 2, green stands for classes 3 and 4; orange stands for classes 5 and 6; red stands for 7 and 8). It can be seen that the complexity globally increases with the product of the number of loading cycles and by the strain amplitude which actually corresponds to the absolute value of the deformation accumulated from the beginning of the fatigue test. As for the orientation, it can be concluded that the complexity growth is different due to the different slip system activation. Figure 1.26(b) gives a more direct stereographic representation of the different microstructures developed in the standard triangle.

Crystal orientation influences strongly not only the dislocation microstructure but also the surface roughness morphology. Bayerlein [60] has investigated the "tonguelike" and "ribbon-like" extrusions on the surface for different orientations and concluded that, as shown in Figure 1.27, "tongue-like" extrusions are more apparent in grains with orientation in zone I, where the coplanar interaction dominates in the PSBs. "ribbonlike" extrusion is more frequently observed for single slip situation. Figure 1.27 Distribution of "ribbon-like" and "tongue-like" extrusions for different crystal orientations [60].

Chapter 1. Bibliography

Grain boundary effect

The studies on poly-crystals show that the fatigue behavior is affected by more factors than for single crystals. Since a poly-crystal is an aggregation of single crystals, the orientation effect mentioned in Section 1.5 is an important factor. Another critical factor is the grain boundaries, which can change the mechanical behavior. In the following sections, we will firstly review the difference in the fatigue behavior of Cu single crystals and Cu Poly-crystals. Then the behavior of two bi-crystals of Cu with a special shape is presented to elucidate the grain boundary effect. Finally, the initiation of cracks at the grain boundary is discussed.

Cyclic stress-strain behavior of Cu poly-crystals

There exists abundant literature on the fatigue behavior of poly-crystals, similarly to what is available for single crystals. The existence of the plateau in the CSS curve for Cu poly-crystals has been discussed for a long time [61][62][START_REF] Mughrabi | Plateaus in the cyclic stress-strain curves of single-and polycrystalline metals[END_REF] and as the PSBs cannot cross the entire poly-crystal, the plateau should be much shorter than for single crystal. PSBs have been observed by Winter [START_REF] Winter | Dislocation microstructures in fatigued copper polycrystals[END_REF] and Pohl [START_REF] Pohl | Persistent slip bands in the interior of a fatigued low carbon steel[END_REF] in poly-crystalline copper and ferritic steel. It is found that the labyrinth and cell structures are more frequently observed in poly-crystals than in single crystals. From this observation, it is inferred that doubleslip or multiple slip play an important role in cyclic deformation of poly-crystals. The activation of multiple slip also contributes to the increase in the saturation stress. The CSS curve of a Cu poly-crystal obtained by Wang and Mughrabi [START_REF] Mughrabi | Deformation of polycrystals: mechanisms and microstructures[END_REF][START_REF] Zhang | Grain boundary effects on cyclic deformation and fatigue damage[END_REF] is presented in Figure 1.28. It can be seen that the curve could be divided into four regions: in region I, they considered that the dislocation poles and veins had been constructed in most of the grains, which led to the increase of the saturation stress with the strain amplitude; the PSBs were thought to be constructed in region II, which causes a slight softening and the linear relationship between log(σ s ) and log(ε pl ); region III, is interpreted as the result of multiple slip behaviors; the region IV was rarely studied, but the behavior is attributed to other deformation modes such as texture effects, grain size effects or grain boundaries.

Fatigue crack nucleation on single crystals always occurs at the surface of PSB because of the extrusions and intrusions constructed by the slip irreversibility as mentioned in Figure 1.3.2. On the other hand, for Cu poly-crystals, grain boundaries (GBs) have also been identified as preferential sites for crack initiations, probably because of dislocation pile-ups against the grain boundary [START_REF] Mughrabi | Fatigue mechanisms: advances in quantitative measurement of physical damage[END_REF].

Nano-crystalline and ultrafine-grained materials have attracted increasing attention in the recent years. With the decrease of the grain size, the grain boundary volume fraction increases which augments its effect on the material behavior. Furthermore, for each grain, the behavior is not simply an average over the aggregate because of the restrain effect produced by the grain boundaries. In this next section, the effect of the grain boundary is introduced in the case of a particular bi-crystal with parallel grain boundaries. For more information, Z.F.Zhang has published an article to review the grain boundary effect on fatigue damage [START_REF] Zhang | Grain boundary effects on cyclic deformation and fatigue damage[END_REF].

Grain boundary affected zone

Bi-crystal can be considered as the simplest poly-crystal and is a good choice to eliminate the interference from other possible parameters. To investigate the effect of large-angle GBs (Grain boundaries), a particular bi-crystal sample has been invented [START_REF] Zhang | Comparison of cyclic deformation behavior between copper bicrystals and their component crystals[END_REF][START_REF] Zhang | Effects of grain boundaries on cyclic deformation behavior of copper bicrystals and columnar crystals[END_REF]. As shown in Figure 1.29(a), the sample named RB (Real Bi-crystal) is made of a grown bi-crystal with a GB parallel to the stress axis. Such a sample could be machined from a larger poly-crystal line piece of material by Focus Ion Beam (FIB). The second sample CB (see Figure 1.29(b)) is a combined bi-crystal without GBs, in which G1 G2 are of the same orientations of G1 ([ 6 7 9]) and G2 ([ 1 4 5]) in RB (along the loading direction). According to their orientations, the two crystals G1 and G2 are dominated by the single slip situation. After the cyclic hardening of G1 and G2, the saturation state is reached for all the cases, however, the cyclic hardening rates and the saturation stresses increase with the following order in the configurations: G2, CB, RB and G1. The CSS curves for these two Bi-crystals are shown in Figure 1.30(a), for the same cyclic strain amplitude, the saturation stress in RB is larger than that of CB. Evidently, the original reason is the GB. The shear stress in the PSB for RB and CB is presented in Figure 1.30(b,c), it is believed that the existence of the grain boundary in CB has significantly increased the shear stress, especially near the grain boundary in a zone called the Grain Boundary Affected Zone (GBAZ). The existence of GBAZ can be confirmed by the slip traces on the surface at grain boundary, as seen in Figure 1.31.

Regarding the orientation effect, for CB sample, only the primary slip system is observed in G1 and G2. That is to say, the cyclic deformation is accommodated by the PSBs in the two crystals. However, for RB (as shown in Figure 1.31), near the grain boundary, secondary slip lines, which formed a cross-weaved structure are observed. There are located in the GBAZ caused by the strain incompatibility near the GBs. The volume fraction V GBAZ and mean width W GBAZ have been measured and increase with the applied strain amplitude. In other researches [START_REF] Šittner | Observation and interpretation of grain boundary compatibility effects in Fe-3.3 wt% Si bicrystals[END_REF][START_REF] Paidar | Slip in a {112} bicrystal with asymmetrical orientations of the loading axis[END_REF][START_REF] Chuang | β brass bicrystal stress-strain relations[END_REF][START_REF] Lee | Simultaneous hardening and softening in a nonisoaxial β brass bicrystal[END_REF], it has been found that the slip deformation near grain boundaries is very complicated due to the strain incompatibility, which can cause more activated slip systems (the number can be different).

Fatigue cracking at grain boundaries

Two main nucleation mechanisms of fatigue cracks have been identified experimentally:

(1) fatigue cracking at the surface of PSB as mentioned in Figure 1.3.2; (2) fatigue cracking at the GBs due to the accumulation of dislocation pile-ups [START_REF] Mughrabi | Fatigue mechanisms: advances in quantitative measurement of physical damage[END_REF][START_REF] Liu | Crystallographic features of intergranular crack initiation in fatigued copper polycrystals[END_REF][START_REF] Smith | Nucleation of grain-boundary cavities during hightemperature creep[END_REF].

The final cracking position is a result of the competition between the two mechanisms [START_REF] Zhang | Grain boundary effects on cyclic deformation and fatigue damage[END_REF]. For large-angle GBs (when the misorientation angle is bigger than 15 o ), plastic strain incompatibility is more pronounced. The incompatibility can activate secondary slips, impede the formation of PSB and lead to the accumulation of dislocation pile-ups. While for low-angle GBs (when the misorientation angle is smaller than 15 • ), the PSBs can more easily cross the GBs without creating too much incompatibility so that the major cracking mechanism is caused by the slip irreversibility at the PSB surface. In addition, sometimes, a mixed situation that involves both GB and PSB are observed.

Macroscopic simulations

The bibliography review presented above is mainly about experimental observations. Modeling and simulations of fatigue behavior have also been a popular research field. However, most of them focus on the modeling of cracks at the continuum level. Kiran [START_REF] Solanki | Finite element analysis of plasticityinduced fatigue crack closure: an overview[END_REF] has reviewed the application of Finite Element Analysis on fatigue crack closure. Different XFEM (Extended Finite Element Method) models have also been utilized for the investigation of crack propagation. [START_REF] Pathak | Fatigue crack growth simulations of 3-D problems using XFEM[END_REF][START_REF] Bansal | A parallel and efficient multi-split XFEM for 3-D analysis of heterogeneous materials[END_REF]. The minimum total energy principle has also been used to investigate the growth of multiple cracks in quasi-static state [START_REF] Sutula | Minimum energy multiple crack propagation. Part I: Theory and state of the art review[END_REF]. As for the early stages of fatigue, the publications are more scarce. Here, we just give a brief introduction. Phenomenological models are often applied to describe the degradation at the early stages of fatigue.

Pirondi [START_REF] Pirondi | Simulation of failure under cyclic plastic loading by damage models[END_REF] has applied two micromechanical models (the porosity evolution model and continuum damage mechanics) to investigate the macroscopic response and the local damage accumulation and compared with round notch bars (RNBs) experiments under cyclic plastic loading. The simulations show good agreements with the experimental results, especially for the potential crack nucleation location and the number of cycles for failure. Manonukul [START_REF] Manonukul | High-and low-cycle fatigue crack initiation using polycrystal plasticity[END_REF] has applied a polycrystalline plasticity FE model to simulate a poly-crystal aggregate, which contains the information of grain orientations and the grain morphology. This model has predicted the development of a 10µm width PSB within an individual grain and the crack initiation at grain triple points. J. Pontes [START_REF] Pontès | On dislocation patterning: Multiple slip effects in the rate equation approach[END_REF] has coupled nonlinear evolution equations considering the reaction-diffusion effect of the mobile and immobile dislocation densities to simulate the dislocation patterns (labyrinth structure) in multiple slip situation, the results are presented in Figure 1.32. Recently, dislocation-based CPFEM models at the sub-micrometer length have been proposed, which contain more information on the related physical parameters and the 1.7. Macroscopic simulations underlying mechanism. For the classic single slip situation, Grilli [START_REF] Grilli | Crystal plasticity finite element modelling of low cycle fatigue in fcc metals[END_REF] has developed a CPFEM model at a length scale smaller than the size of the dislocation structure (veins) so that the microstructure feature can be presented. The internal variable of this model is a dislocation density vector, which components refer to straight or curved, positive or negative, screw or edge dislocation densities, as shown in Figure 1. 33(a). By this detailed dislocation decomposition, the simulation of the microstructure becomes possible. In addition, the author has added a new constitutive equation describing the multiplication law at length smaller than 100nm [START_REF] Arsenlis | Modeling the evolution of crystallographic dislocation density in crystal plasticity[END_REF] and introduced the double cross-slip mechanism. The simulation result of the edge dislocation density distribution after 100 cycles is presented in Figure 1. 33(b), where the element size is 200nm and the simulated box size is 12µm. It can be seen that the obtained pattern is very similar to the vein structure.

(a) (b) As for multiple slip situation, Grilli [START_REF] Grilli | Multiple slip dislocation patterning in a dislocation-based crystal plasticity finite element method[END_REF] has also used a CPFEM model considering the junction formation and performed simulations on a 316 poly-crystal. Figure 1.34 presents one of the simulation results. Figure 1.34(a) shows the mesh setting for the poly-crystal with orientation close to [3 29 1] and the color of the mesh refers to the detailed orientation obtained from EBSD [59]. The dislocation density distribution after 50 cycles of loading is presented in Figure 1.34(b). The separation of the dislocation walls by regions with a weaker dislocation density can be simulated.

The CPFEM model has also been used in other respect. Irastorza-Landa [START_REF] Irastorza-Landa | Laue micro-diffraction and crystal plasticity finite element simulations to reveal a vein structure in fatigued Cu[END_REF] has coupled CPFEM with the experimental data obtained from transmission Laue diffraction to calculate the lattice rotation evolution caused by the vein structure. Mayama has applied a CPFEM model to investigate the Geometrically Necessary Dislocation structure for an FCC bi-crystal under cyclic loading [START_REF] Mayama | Geometrically necessary dislocation structure organization in FCC bicrystal subjected to cyclic plasticity[END_REF]. In addition, to consult the implementation of CPFEM model, the article of Meric [START_REF] Me´ric | Single crystal modeling for structural calculations: Part 2-finite element implementation[END_REF] is recommended.

Conclusions

In this chapter, we have reviewed mainly the literature concerning the dislocation behavior at the early stages of fatigue degradation. Owing to the abundant publications, Cu is chosen firstly as the typical investigated metal. The evolution of dislocation related microstructures from veins, PSB to labyrinth has been described in detail considering their main characteristics, such as the microscopic images, formation process and dislocation moving mode. The surface roughness (formation of extrusions and intrusions) at the surface of PSBs is presented attentively as well for its potential as sites for crack nucleation. Afterward, the investigated material is extended to FCC metals and alloys. To elucidate the different fatigue behaviors in these FCC materials, the research of slip mode variation and possible determining factors is synthesized. To obtain more information on the fatigue behavior of poly-crystals, the orientation effect on the dislocation patterns and surface roughness is presented. Then, the grain boundary affected zone is proposed to demonstrate the grain boundary effect of fatigue behavior. Finally, the application of CPFEM to fatigue problem investigation is briefly reviewed.

What should be retained from this chapter:

-This work focuses mainly on the early stages of fatigue degradation, which means before the initiation and propagation of cracks;

-As DDD simulates the dislocation movements, the objective of the full review on the dislocation microstructure evolution is to help the analysis of the forthcoming simulation results;

-Cu is chosen as the major metal to simulate, due to the abundant publications, and the comparison between Cu and AISI 316L austenitic stainless steel will be realized in the following simulations.

Chapter 2

Methodology of 3D Discrete Dislocation Dynamics

Introduction

This chapter aims to give a general presentation about 3D Discrete Dislocation Dynamic (DDD) models and their applications to fatigue problems. The presentation starts with a brief history of DDD. Then, the fundamental ideas of the model regarding the discretization of space and dislocation lines, the calculation of the dislocation velocity from the stress field and the local dislocation interactions are introduced in detail. The origin of the material parameters especially for Cu is also identified. Later on, the special settings required for the simulations of cyclic loading are given (quasi-static loading and virtual space are explicated). At the end of this chapter, a brief review of the exiting applications of DDD to fatigue and the research work of Christophe DEPRES is reviewed. Based on the review, the motivation and open questions for this thesis are summarized at the end.

Discrete dislocation dynamic 2.2.1 Historical retrospective

The movements of dislocations play a vital role in the physics of plastic deformation for crystalline materials, especially for metals. Thus, the computations of plastic deformations at the macroscopic scale based on the physics resulting from the collective dislocation movements at the mesoscopic scale seem to be essential. The idea of 3D discrete dislocation dynamics was firstly proposed by Ladislas KUBIN, Gilles CANOVA and Yves BRECHET [START_REF] Kubin | Dislocation microstructures and plastic flow: a 3D simulation[END_REF]. The first code called Micromegas has been developed at the laboratory LEM of ONERA (Office National d'Etudes et de Recherches Aérospatiales) in Chatillon [START_REF] Devincre | Simulations dynamiques des dislocations a une echelle mesoscopique: une etude de la deformation plastique[END_REF].

At the same time, based on Micromegas, under the direction of Yves BRECHET (LTPCM) and Gilles CANOVA (GPM2), another branch of DDD code development was realized in GPM2 laboratory in Grenoble, initiated by the PhD thesis of Marc VERDIER [START_REF] Verdier | Une étude de la restauration dans des alliages Al-Mg[END_REF] and Marc FIVEL [START_REF] Fivel | Etudes numériques à différentes échelles de la déformation plastique des monocristaux de structure CFC[END_REF]. The DDD code developed in GPM2 is named TRIDIS (http://www.numodis.fr/tridis/), which is the code utilized in this PhD thesis. TRIDIS has been devoted to several PhD thesis in the past 20 years and consequently, the code has Chapter 2. Methodology of 3D Discrete Dislocation Dynamics been optimized and extended with new functions or post-processings. The development of TRIDIS can be well presented with the progress of different PhD projects. A selection of PhD theses related to TRIDIS is listed hereunder:

-Chan Sun SHIN [START_REF] Shin | 3D discrete dislocation dynamics applied to dislocation-precipitate interactions[END_REF] in 2004 studied the interactions between dislocations and precipitates and has added some parallel calculation algorithms;

-Christophe DEPRES [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF] in 2004 applied DDD to investigate the fatigue behavior of AISI 316L stainless steel and developed the post-processing method needed to calculate the surface deformation from the dislocations; -Juliette CHEVY [START_REF] Chevy | Viscoplasticité et hétérogénéités de déformation du monocristal de glace: expériences et simulations[END_REF] in 2008 has simulated the torsion deformation of single crystals of ice (HCP crystallography) using TRIDIS ;

-Hyung-Jun CHANG [START_REF] Chang | Analysis of nano indentation size effect based on dislocation dynamics and crystal plasticity[END_REF] in 2009 investigated the nano-indentation of Cu using molecular dynamics, crystal plasticity finite element method and also dislocation dynamics. Later, he developed the polycrystalline version of TRIDIS now included in the code;

-Julien CHAUSSIDON [START_REF] Chaussidon | Etude multiéchelle de la Plasticité du Fer-a à basse température[END_REF] in 2007 and Daniel GARCIA-RODRIGUEZ [START_REF] Garcia Rodriguez | Optimisation d'un code de dynamique des dislocations pour l'étude de la plasticité des aciers ferritiques[END_REF] in 2011 explored and developed the DDD model for BCC crystals. Applications were performed in the case of ferritic laths of α-iron loaded in bending mode;

-Thomas NOGARET [START_REF] Nogaret | Approche multi-échelle des mécanismes de plasticité des aciers austénitiques irradiés[END_REF] in 2007 and Gururaj KADIRI [START_REF] Kadiri | Dislocation dynamics simulations of strain localization in irradiated steels[END_REF] in 2013 studied the deformation behaviour of irradiated metals with a specific focus on clear channeling;

-Gyu-Seok KIM in 2011 [START_REF] Kim | Small volume investigation of slip and twinning in magnesium single crystals[END_REF] has simulated the compression of micro-pillars of Mg single crystals, which is of HCP crystal structure;

-Siwen GAO [START_REF] Gao | 3D discrete dislocation dynamics study on fundamental creep mechanisms in single crystal superalloys[END_REF] in 2016 coupled DDD code with a Fast Fourier Transform (FFT) solver which computed the diffusion of vacancies and studied the creep behavior of Ni-based superalloys. The climbing of dislocations was also taken into consideration;

-Hareesh TUMMALA in 2016 [104] has investigated the effect of grain shape on the kinematic hardening and he also studied thin film plasticity using the polycrystalline version of TRIDIS ;

-The PhD thesis of Elena JOVER CARRASCO is still in progress. This thesis is about the interactions of cracks with dislocations, realized by the coupling between DDD and the X-FEM method.

From the description above, we can see that DDD modeling covers almost all different fields of plastic deformation (traction, compression, torsion, bending, fatigue, nanoindentation, strain-localization). The simulation scale of DDD is between molecular dynamics (microscopic) and finite element methods (macroscopic). Figure 2.1 shows a typical application of these three simulation methods acting at different length scales to study the nano-indentation problem [START_REF] Chang | Analysis of nano indentation size effect based on dislocation dynamics and crystal plasticity[END_REF]. Besides, the coupling of these three simulations method is also very common and effective for mechanic studies.

Discrete dislocation dynamic

It is worth mentioning that the work presented in this PhD is highly related to the one of Christophe DEPRES. In this work, we will study the fatigue behavior of Cu will then be compared with AISI 316L stainless steel, thus benefiting from the well improved computational capacity and post-processing codes developed by Christophe DEPRES. The PhD theses of Hyung-Jun CHANG and Hareesh TUMMALA have also provided an opportunity for us to investigate the fatigue behavior of poly-crystalline Cu.

Discretization of space and dislocations

The space for numerical simulations is discretized in a 3D lattice of points (as shown in Figure 2.2a), upon which the dislocations are attached. The structure of these points corresponds to the face-centered cubic (CFC) crystal lattice. The minimal distance (pointed out in Figure 2.2(a) in the discretized structure is called discretization unity, noted xl. During a DDD simulation, dislocations move on a large FCC lattice with lattice parameter xl (typically xl = 10b). Dislocations then start and end on this meso-FCC lattice. One example of the placement for one edge dislocation and one screw dislocation is also illustrated in Figure 2.2(b). Note also that the dislocation segments will move on the meso-lattice via integer values of displacements along the [1 1 0] and [1 1 2] vectors. This implies that there exist two sublattices in the discrete space visualized in Figure 2.2(a,b) as black and red points. To make sure that dislocations could annihilate during a simulation, one has to enforce that each introduced dislocation starts and ends on the same sub-lattice.

Additionally, xl determines the minimal distance between two existing dislocation segments. Physically, xl should be taken as large as possible to save computation time but small enough to account accurately for physics of plasticity. In practice xl is chosen to be the minimum annihilation distance of two dislocations of opposing signs. Essmann and Mughrabi in 1979 [34] measured the minimal distance y c between two edge dislocations of opposing signs in a fatigued Cu after neutron irradiation to fix the dislocation positions. The value y c was found to be equal to 1.5nm. As the value is obtained from the distance between edge dislocations which are located in the {1 1 1} slip plane, we have xl = √ 3y c = 2.6nm ≈ 10b. Based on these experimental results, the value of xl is commonly taken as 10 times the Burgers vector magnitude, b. Note that xl can be deceased for simulations of atomistic processes like in the case of irradiation studies where dislocations and irradiated defects should be simulated simultaneously. In our studies, we will use xl = 10b so that, for a cube with a side length equal to 10µm, there are about 3906 3 cubes (for b = 2.56 Ȧ) of xl side length. Not only is the space discretized but also the dislocations lines. The line of each dislocation is divided into a series of connected edge dislocation segments and screw dislocation segments (shown in Figure 2.3). That is why the model is also called the "edge-screw" model proposed by Devincre and Condat [START_REF] Devincre | Model validation of a 3D simulation of dislocation dynamics: discretization and line tension effects[END_REF]. Edge dislocations and screw dislocations are two theoretical elementary dislocations and their physical characters have been well studied. In terms of internal stress field, any mixed-character dislocation line can be represented by the superposition of its decomposition into edge and screw segments. Hence, by the discretizations and combinations of edge and screw dislocations, the behavior of a mixed dislocation can be described. There are 12 slip systems in FCC metals. In order to describe dislocations on every different slip system, 12 vectors for edge dislocations (in the direction of 1 1 2 ) and 6 vectors for screw dislocations (in the direction of 1 1 0 ) are required. For a screw dislocation, considering the cross-slip direction, one screw dislocation vector can belong to 2 slip systems. The coordinates of the vectors for screw l screw , edge dislocations l edge and the normal vector n of the slip plane are listed in Table 2.1, as well as the Schmid & Boas slip system nomenclature (Letter = Slip plane; Number = Burgers vector). In this 2.2. Discrete dislocation dynamic model, an edge dislocation moves along the line direction of the screw segments, which is also the direction of the Burgers vector b. Meanwhile, for screw segments, they can move in two edge directions. The slip plane normal vector n is then defined as the cross product of the edge direction and the screw direction. From the descriptions above, it can be noted that only 18 vectors are needed to depict lines and glide directions of all the dislocations and they are also noted as the subscripts in Table 2

Mobility laws for dislocation segments

Kinetic law for velocity calculation As described above, the dislocation segments move on the slip planes located in the discretized space. Each segment moves in the direction perpendicular to its line and is contained in its slip plane. The velocity v of a segment is calculated from the local stress field, named effective stress τ ef f . As Eq.(2.1), in the case of FCC crystals, the velocity possesses a linear relationship with τ ef f and the value of the Burgers vector magnitude b. The term B in Eq.(2.1) is the phonon drag coefficient, which depends on the material and temperature.

v = τ ef f * b B (2.1)
In addition, the speed of sound in the material is set as a limit for the segment velocity in case of a too brutal movement.

Effective stress on the dislocation segments

The effective stress τ ef f is the major impetus to the motion of dislocations segments. As shown in Eq.(2.2), it is composed of two parts: (1) the shear stress τ P K resolved on the segment slip plane and the line tension τ lt contribute to the advancement of dislocations; (2) the Peierls stress τ P eierls (also noted as the friction stress τ f r ) leads to the blockage of dislocation motions.

τ ef f = τ pk + τ lt -sign(τ P K + τ lt ) * τ f r (2.2)
the friction stress τ f r comes from the mechanism of dislocation motions in the crystal lattice. Dislocations advance in the crystal lattice by breaking and rebuilding the bonds close to the dislocation core, which produces a kind of resistance. This resistance is modeled by a shear stress criterion and that is the friction stress τ f r . τ f r is always in the opposing direction of the dislocation movement, besides, the movement of dislocation segments can be activated if and only if the absolute value of the sum of τ pk and τ lt is bigger than τ f r . In FCC metals, τ f r is in the order of 10 -5 µ and is calculated by τ f r = α f r * µ, where µ is the shear modulus of the material and α f r is called the coefficient of friction stress. The expression of the Peach Koehler force dF pk , demonstrated by Peach and Koehler in 1950 [START_REF] Peach | The forces exerted on dislocations and the stress fields produced by them[END_REF] enables to calculate the force applied on the dislocation segments from any arbitrary stress fields σ σ σ. For a dislocation segment with a line vector dl and a Burgers vector b, embedded by a stress field σ σ σ, the Peach Koehler force dF pk is expressed as follows:

dF pk = (σ σ σ * b) × dl (2.3)
To get the resolved shear stress on slip plane τ pk , we project the Peach Koehler force dF pk along the glide direction g and normalize the stress by the Burgers vector magnitude b.

:

τ pk = 1 b dF pk * g (2.4)
In DDD, the stress field is determined by two terms:

σ σ σ = σ σ σ app + σ σ σ int (2.5)
where σ σ σ app , called the applied stress field, indicates the external mechanical loading. The applied stress can be homogeneously applied to each dislocation segment or calculated by the finite element method (FEM) considering the mechanical loading and the boundary conditions. The term σ σ σ int is the internal stress field. This term refers to the stress field produced by the dislocations inside the crystal. The computation of σ σ σ int is very crucial in the DDD code. Assuming that N dislocation segments exist, in a time step dt, for every dislocation segments, the value of the stress tensors σ σ σ int produced by N dislocation segments should be calculated. That is to say, the stress tensor σ σ σ int is calculated N 2 times in one time step. When many dislocations are generated, this process can largely slow down the simulation. Because of this, researches have been realized to ameliorate the algorithm. All the simulations performed in this thesis use the box method as introduced in [9] and shared memory parallel calculations based on OpenMP library. In addition, the image stress field σ σ σ image , that could arise from the presence of a free surface will be neglected in our simulations.

The line tension τ lt reflects the fact that in the absence of loading forces, a dislocation line wants to minimize its length and straighten to a straight line. This is due to the self stress field that every dislocation precedes per unit length. In TRIDIS, the Eq.(2.6) proposed by Foreman [START_REF] Forest | Strain gradient crystal plasticity: thermomechanical formulations and applications[END_REF] is used to calculate the line tension.

τ lt = µb 4π(1 -ν)R (1 -2ν + 3νcos 2 θ)(ln( L 2b ) -νcos(2θ)) (2.6)
where L is the length of the dislocation segment, ν is the Poisson's ratio. The notations of R and θ are indicated in Figure 2.5. R is the radius of the circle defined by the 3 center points of dislocations including the targeted segment and its two neighboring segments. θ is the angle between the Burgers vector b and the line direction. These two parameters indicate the relative position of the moving dislocation with the neighboring dislocations. The global deformation increment dε ε ε p during a time step dt is calculated from the collective movements of all the dislocations. We suppose that there are N dislocation segments inside the crystal. For dislocation i, as indicated in Eq.(2.1), the velocity v i can be calculated. According to Philibert's equations [START_REF] Philibert | Glissement des dislocations et frottement de réseau[END_REF], the dislocation velocity begins with an acceleration regime and then converges to a constant speed. The transient period is very short and never exceeds 10 -11 s which is between 1 and 2 order of magnitudes lower than the typical time step used in TRIDIS simulations. Consequently, in our DDD simulations, the velocity v i is always considered as a constant during a time step dt given by the expression of Eq.(2.1). Therefore, following an explicit time integration algorithm, the displacement ∆d i of the dislocation i is calculated by the product of dt and v i :

Line direction R

∆d i = dt.v i (2.7)
then, the shear plastic deformation γ i p associated with the dislocation i is computed from the area dA i = l i .v i .dt scanned by the dislocation of length l i :

γ i p = b. dA i V (2.8)
where V is the volume of the studied crystal. The increment of deformation dε ε ε i p related to the dislocation i is given as follows:

dε ε ε i p = 1 2 ( b s ⊗ n s + n s ⊗ b s ).γ i p (2.9)
where s identif ies the slip system to which the dislocation belongs and b s , n s are the corresponding Burgers vector and the normal vector to the slip plane respectively. As a result, the global deformation increment dε ε ε p is the sum of all the increment related to the N dislocations and we have:

dε ε ε p = N i=1 dε ε ε i p (2.10)
From the equations above, the basic numerical calculation procedure can be established. As soon as the external stress is applied to the system, the dislocations can begin to move and accumulate plastic deformation. The amount of accumulated deformation can be calculated and depending on the loading mode, we can increase or decrease the loading value, just like in experiments. In DDD the natural loading mode is obviously the case of stress loading: for each time step the stress increment is added to the stress at the previous step and dislocations move accordingly, leading to plastic deformation. In the case of a strain control mode, the stress needs to be continuously adjusted so that the desired strain rate is obtained. We will see in Section 2.6.1 that in the case of the fatigue simulations developed here, we have enforced a quasi-static loading mode which is a mixture between these two loading conditions.

Local interactions

When a dislocation moves in the crystal, it is possible to meet the other dislocations. In order to minimize the local energy, different interactions exist. The local interactions 2.2. Discrete dislocation dynamic play a key role in the plastic behavior because they can significantly change the movement of the dislocations. Dislocation reactions lead to modifications of the dislocation microstructure, the internal stress distributions and the global mechanical response, and are at the origin of cyclic hardening and softening. The principal local interaction rules are presented as follows:

-Annihilation/recombination As illustrated in Figure 2.6, if a dislocation of the opposed sign is detected in the area swept during a time step, the moving dislocation will annihilate with this dislocation segment and the other part will recombine to form new dislocation arms.

(a) (b) -Junction formation

According to the minimal energy criterion (Frank's law), when two dislocations k and m approach, if the following relation is satisfied, a junction will be formed:

b 2 m + b 2 n > (b m + b n ) 2 (2.11)
In this equation, the left part refers to the energy addition of two separate dislocation segments and the right part, to the energy after the junction formation.

This expression stands for any configuration of the junction. There exists coplanar junctions (same plane normal), collinear (same Burgers vector), Hirth lock, Glissile junction and Lomer-Cottrel lock. The type of the junction is determined by the geometry position and the Burgers vector of the two encountering dislocations. This part will be explained in detail in the chapter on double slip loading conditions (chapter 5).

-Cross-slip

In our DDD simulations, the model of [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications[END_REF] has been applied for calculating the cross-slip probability. For a dislocation segment i, the cross-slip probability law is presented as:

P i = β l l 0 dt t 0 exp( τ dev -τ III kT V act ) (2.12)
where β = 10 5 is a normalization coefficient to assure the cross slip probability P i ∈ [0, 1], l l 0 is the ratio between the length of screw dislocation segment l and the reference length l 0 (l 0 = 1µm [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications[END_REF]) and δt t 0 is the ratio between the simulation time step δt and the reference time step t 0 (t 0 = 1s [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications[END_REF]). Inside the exponential function, the entire expression represents the activation energy for cross slip: (τ devτ III ) is the comparison between the absolute value of the resolved shear stress on the deviate plane, τ dev and the reference stress value, τ III , V act is the activation volume, k, T represents the Boltzman constant and temperature respectively.

(a) (b) (c) (d) [-1 0 1] (1 - 1 1)
(1 1 1)

Treatment for cross-slip

Cross slip is the most fundamental dislocation moving mechanism because it provides another possibility to bypass obstacles and to relax the internal stress field. In experiments, many observed dislocation microstructures and macroscopic responses are related to continuing, reciprocating and brutal cross-slip events, such as the prismatic dislocation loops [START_REF] Déprés | Low-strain fatigue in AISI 316L steel surface grains: a three-dimensional discrete dislocation dynamics modelling of the early cycles I. Dislocation microstructures and mechanical behaviour[END_REF] and the strain burst observed in fatigue tests [53]. Similarly, for DDD simulations, the role of cross-slip is also crucial. Firstly, cross-slip helps the formation of many dislocation microstructure elements. Secondly, cross-slip can lead to the avalanche of dislocation generation, which can significantly slow down the simulation process. Therefore, the correct simulation of cross-slip determines the quality of DDD simulations.

In order to well demonstrate the cross-slip situation, a series of criteria has been proposed and added in TRIDIS. The criteria are composed of the following parts:

-The first criterion is to ensure the dislocation segment characteristics. In practice, the effective line character is estimated from the line connecting the middle points of the two neighboring segments attached to the screw segment. If the angle between this line and the Burgers vector is smaller than 10°, the segment is considered to be in a screw part of the dislocation line. This criterion eliminates the screw segments located in the edge parts of dislocation loops;

Parameters identifications

-The second criterion is to verify the position of the dislocation segments. If the dislocation is out of the grain (for example, in the virtual space, mentioned in Section 2.6.2), the cross-slip mechanism will not be evaluated;

-Then there is a minimum length for cross-slip, that is to say, the cross-slip can only happen for a dislocation segment larger than a given value. This criterion largely limits the number of cross-slips so that the dislocation generation avalanche can be avoided;

-For pinned dislocation segments (typically Frank-Read), the cross-slip is also forbidden;

-Before we calculate the probability of cross-slip, the value of shear stress on the primary slip plane and the cross-slip plane are compared. Cross-slip happens when the cross-slip shear stress is bigger. The idea here is to demonstrate that there is a competition between the shear stress on the two planes to determine the slip path. In the majority of cases, the primary shear stress value is also amplified by a coefficient before the comparison. As an example in pure shear stress loading mode, the applied primary shear stress is three times bigger than the cross-slip shear stress. As a result, the occurrence of cross-slip is mainly influenced by the internal stress produced by the dislocations, which has a physical meaning. It is worth noting that this criterion is in fact very strong. many dislocation segments are blocked, due to this criterion. Sometimes, we need to decrease the primary shear stress to weaken this criterion;

-The last criterion is the calculation of probability using the Eq.(2.12). A random number P ∈ [0, 1] is generated and the cross-slip probability P c is calculated by Eq.(2.12). If P is bigger than P c , cross-slip happens, otherwise there will be no cross-slip and the slip plane of the moving screw segment remains unchanged.

Parameters identifications

As presented in Section 2.2.3, to run the DDD simulations, several parameters need to be identified. In this thesis, we have mainly simulated the mechanic behavior of pure metal Cu and AISI 316L stainless steel. The parameters for Cu are listed in Table 2.2. Cu has been fully investigated by TRIDIS in different situations. Therefore, the parameters for Cu come from precedent researches [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications[END_REF]. The parameters for AISI 316L stainless steel which are from the work of Christophe DEPRES [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF][START_REF] Depres | Low-strain fatigue in 316L steel surface grains: a three dimension discrete dislocation dynamics modelling of the early cycles. Part 2: Persistent slip markings and micro-crack nucleation[END_REF][START_REF] Déprés | Low-strain fatigue in AISI 316L steel surface grains: a three-dimensional discrete dislocation dynamics modelling of the early cycles I. Dislocation microstructures and mechanical behaviour[END_REF], are listed in Table 2.2. The parameters shown here are all the physical parameters. There are also some numerical parameters that can highly influence the simulation convergence. The numerical parameters are shown in the following chapters.

In this table, the values of Stacking Fault Energy (SFE) γ for the two materials are also presented. It should be mentioned that SFE is not explicitly introduced in the DDD model, but it plays an important role in calculating the value for τ III and V act .

Based on the article of Bonneville [START_REF] Bonneville | A study of cross-slip activation parameters in pure copper[END_REF], The activation volume V act for cross-slip is proportional to V 0 , which is evaluated as follows: where d 0 is the dissociation width at zero stress (V Cu 0 = 4.00864 × 10 -30 m 3 and V 316L 0 = 2.425 × 10 -29 m 3 ). Bonneville has also measured the activation volume of Cu V Cu activation = 300b 3 . As a result, we have 3 . τ III is the stress controlling the cross-slip intensity at the third stage of monotonic traction in the single slip situation. A semi-empirical linear formula is by Table 2.14 to calculate the value given in Table 2.2:

V 0 = µb 3 d 0 γ (2.13)
V 316L activation = V 316L 0 V Cu 0 V Cu activation = 1800b
τ III µ = A 1 (1 - b d 0 ) 2 γ µb -A 2 γ 2 µb µb 3 kT = A 1 α 1 -A 2 α 2 .T (2.

14)

where A 1 = 5.59 and A 2 = 1.28 × 10 6 are independent of the simulated materials [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF].

Stress field computation from dislocation segments

Compared with continuous models like FEM, one of the advantages that DDD possesses is that the stress field induced by the dislocations is explicitly considered. The stress fields generated by edge and screw dislocations can be calculated from isotropic elastic theory [115]. Li [START_REF] Li | Stress field of a dislocation segment[END_REF] has proposed a compact formula which gives the stress field produced by a semi-infinite dislocation segment. The situation is shown in Figure 2.8: the dislocation segment is along the z-axes starting from the origin O and directed towards the positive direction of z-axes. The formula of Li is identical to the classic formula inferred from elastic theory but is more applicable in numerical simulations. That is why it has been introduced in TRIDIS. The formula of Li to calculate the stress field at the point P (x, y, z) or r = (x, y, z) is presented as follows: The formulas above are for a dislocation of infinite length in the positive z-direction. For a dislocation [SE] of finite length L, the internal stress σ σ σ int produced by the dislocation at the point E is calculated by the difference from the stress fields of two semi-infinite dislocations:

σ Li xx = -b x y -b y x r(r -z) -x 2 (b x y -b y x)(2r -z) r 3 (r -z) 2 (2.15) σ Li yy = b x y + b y x r(r -z) -y 2 (b x y -b y x)(2r -z) r 3 (r -z) 2 (2.16) σ Li zz = z(b x y -b y x) r 3 - 2ν(b x y -b y x) r(r -z) (2.17) σ Li yz = y(b x y -b y x) r 3 - νb x r + (1 -ν)b z x r(r -z) (2.18) σ Li xz = x(b x y -b y x) r 3 + νb y r - (1 -ν)b z y r(r -z) (2.19)
σ σ σ int ( r) = σ σ σ Li ( r) -σ σ σ Li ( r -L k) (2.21)
where k is the unit vector in the z-direction. This formula can be applied to any points P in the space without causing divergence (except when r = 0), even for z < 0 (we can inverse the sign of the dislocation).

Displacement field calculations from dislocations

The implementation of the following model in TRIDIS is realized in the thesis work of Christophe DEPRES [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF]. An article [START_REF] Fivel | An easy implementation of displacement calculations in 3D discrete dislocation dynamics codes[END_REF] is also available for more detail. As the displacement field produced by dislocations is crucial to the free surface relief calculation, to help the understanding in the chapters of simulation results, the basic model and fundamental calculation cases are presented precisely in this section.

Calculation of the displacement field

Unlike the abundant research on dislocation stress field calculation, the displacement field of dislocations has been discussed much less. In the textbook of Hirth and Lothe [START_REF] Hirth | Theory of dislocations[END_REF], the following equation has been proposed to calculate the displacement ( U ) at point R induced by a dislocation loop of surface A:

u m ( r) = - A b i c ijkl ∂ ∂x l g mk ( r -r)dA j (2.22)
where c ijkl is the elastic module tensor, and g ij is the Green tensor, the other notations are illustrated in Figure 2.9(a). By the surface integration of Eq.(2.22), we can get the formula for an arbitrary dislocation loop to calculate the displacement at any point located in an infinite medium:

u( r) = - b 4π Ω - 1 4π C b × d l R - 1 8π(1 -ν) ∇ C b × R R .d l (2.23)
where R = rr, R = R . Then the integration on the dislocation line C gives the final displacement vector u( r). The Eq.(2.23) above comprises the calculation of solid angles and surface integration. It can be non-trivial when the dislocation geometry is very complex. One of the simplest dislocation loop geometry is the triangle, as shown in Figure 2.9(b). By applying the Eq.(2.23), a simplified expression for calculating the displacement induced by a triangle dislocation loop was proposed by Barnett [START_REF] Barnett | The displacement field of a triangular dislocation loop[END_REF]. The displacement is expressed as follows:

u( r) = - b 4π Ω + F ab + F bc + F ca (2.24)
the angle solid Ω is computed as:

Ω = -sign( R i . n) 4 arctan( tan( s 2 ) tan( s -a 2 ) tan( s -b 2 ) tan( s -c 2 )) (2.25)
and the F ab , F bc , and F ca are given as follows:

F ij = - 1 -2ν 8π(1 -ν) ( b × t ij )ln( R j + R j . t ij R i + R i . t ij ) + 1 8π(1 -ν) ( b. n ij )( R j R j - R i R i ) × n ij (2.26) where                          s = a+b+c
R i = r i -r t ij = r j -r i r j -r i n ij = R i × R j R i . R j (2.27)
Eq.(2.27) above facilitates the calculation of the dislocation displacement field because any irregular dislocation loop can be considered as a composition of triangular dislocation loops. According to the superposition principle in the linear elasticity framework, the calculation of the displacement field from a dislocation loop can always be subdivided into several calculations based on a set of triangular dislocation loops and finally added together. The decomposition of irregular dislocation loops is easy to achieve in the "screw and edge" dislocation model. In the following part, we will give some practical examples for the different types of loops.

Decomposition of co-planar dislocation loops

The decomposition of the coplanar dislocation loop into triangle dislocation loops is the simplest case. Figure 2.10(a) shows the decomposition for a convex coplanar dislocation loop. The closure point C p is chosen at the interior of the loop and the triangle is constructed based on C p and the extremities of the dislocation segments. The two sides of each triangle have been defined two times in opposing directions so that in the final superposition, the total value of the two sides is neutralized and only the baseline effect is taken into consideration.

For a concave dislocation loop, the method above can still be utilized. The closure point can be located outside the loop and, as shown in Figure 2.10(b), the obtained triangles have different orientations. Hence, the extra surface effects are eliminated in the final summation. In fact, the position of the closure point C p can be selected arbitrarily and the final result is not influenced by its position. The closure point should be unique for all the segments forming the loop. In TRIDIS, in order to accelerate the simulation, the closure point C p is calculated by a vector u and the original point O. The vector u is not parallel to any slip plane of the 12 slip systems in the discretized space. As a consequence, all the slip planes in the discretized space must have an intersection point with the straight line originated from point O in the direction u. Afterward, this intersection point is chosen as the closure point for the dislocation loops belonging to this slip plane. Therefore, all the dislocation segments belonging to the same plane share the same closure point. This process can increase the computation efficiency, since the same automatic closure procedure could be applied to each segment without any need to identify the loop it belongs to.

Decomposition of non-coplanar dislocation loops

Due to cross-slip, a dislocation loop is not always lying in the same slip plane. In this case, if we keep the same calculation method presented above, some dislocation surfaces can be missed in the integration (as shown in Figure 2.11(a)). To solve the problem, the algorithm used to divide the loop into triangles has been modified. The purpose is to construct two triangles for every dislocation segments. The method used here is to build the two triangles from the two extremities of the segment and the projection points of the closure point C p in the line passing the extremities directing by the Burgers vector b. The process is illustrated in Figure 2.11(b.1). the dislocation segment SE is directed from S to E, with Burgers vector b. C p is the closure point. P 1 and P 2 are the projection points of S and E in the direction b onto a line passing by C p and normal to b. The oriented triangles used for the integration are P 2 SE and P 1 SP 2 . With this modification, the missing part is automatically filled (Figure 2.11(b.2)).

Decomposition of prismatic dislocation loops

A prismatic loop is a closed loop for which the habit plane is perpendicular to the Burgers vector. Therefore the prismatic loop is composed of edge segments sharing a given Burgers vector. In DDD simulations, prismatic loops can be generated by multiple dislocation cross-slips and line reconnections. We will see that prismatic loops are numerous in fatigue simulations and they are at the origin of the formation of extrusions and intrusions at the surface. According to the orientation of the dislocation segments forming the loop, prismatic loops can be either of vacancy type or interstitial type. In DDD simulations, a single prismatic loop cannot be introduced since is not a conservative process and it is equivalent to the introduction or removal of atoms. Under this conservative principle, the prismatic loop should always exist in couples: one interstitial type prismatic loop should match to a vacancy type prismatic loop.

In this case, the precedent calculating method can be applied to a pair of prismatic loops. As shown in Figure 2.12(a), any pair of prismatic loops is equivalent to 4 glissile loops depicted by the blue rectangles. Their induced displacement fields are equivalent. Figure 2.12(b) shows the effect of the automatic closure procedure on the pair of prismatic loops. The displacements produced by a pair of prismatic loops on a plane located between the two loops are represented in Figure 2.12(c). A rectangular step with the same shape as the pair of prismatic loops and of height b is formed.

To end this section, the simulated shape of the free surface after the passage of the half of a dislocation loop is presented in Figure 2.13(b). The step profile was obtained by DDD (using the dedicated displacements field calculation method) in the same configuration. 2.6. Applications for fatigue problems

Applications for fatigue problems

Most DDD simulations deal with monotonic loadings, such as the classical uniaxial tensile test simulations. A cyclic loading cannot be considered as a simple change of the loading sign in a monotonic loading. Special consideration needs to be added to the loading method.

Quasi-static cyclic loading

The applied stress field σ σ σ app is imposed homogeneously and directly on dislocations. The applied stress field at step k is given as follows:

σ σ σ k app = σ σ σ k-1 app + λ * (-1) a * ∆σ σ σ app (2.28)
where σ σ σ 0 app = 0 0 0, a = 0 so that (-1) a = 1 for loading, and a = 1 for unloading so that (-1) a = -1. λ is the coefficient to control the quasi-static loading state, as explained hereafter. ∆σ σ σ app is the increment of the stress field, given as follows in our simulations:

∆σ σ σ app =        σ xx σ xy σ xz σ xy σ yy σ yz σ xz σ yz σ zz        * σ * δt (2.29)
In Eq.(2.29), the ratio between different components of the stress tensor is fixed, which indicates the loading direction is always the same, in other words, the loading is proportional and the Schmid factors for the different slip systems are invariant, where σ is the stress rate. As we have described above, the simulation process is totally explicit. For this reason, the choice of the time step δt is of great importance. If δt is too small, the computational time can be very long. On the other hand, if δt is too big, the local interaction cannot be well simulated. Based on several test simulations, a compromise should hence be found and δt is chosen as 2 × 10 -10 s.

In addition, in order to better control dislocation motions so that dislocations can move in a quasi-static manner, λ is introduced. λ at step k is determined by the increment of deformation ∆ε V M,k-1 p at step k -1. When ∆ε V M,k-1 p is smaller than ∆ε V M,min p , the dislocations are considered in an equilibrium position so that the amplitude of the applied stress should be increased and λ = 1 (In the ideal situation, ∆ε V M,min p should be zero, but the vibration of dislocations cannot be avoided, the value of ∆ V M,min p is chosen by different value before running a large scale simulation). Reversely, if ∆ε V M,k-1 p is bigger than ∆ε V M,max p , the increment of deformation at the last step is thought to be too brutal, which may lead to dislocation avalanches. In this situation, the applied stress is reduced by choosing λ = -0.5. The value of ∆ V M,max p depends on the simulation. If the increment of deformation ∆ε V M,k-1 p is between the two critical values, the applied stress stays unchanged and λ = 0 so that the dislocations can keep moving under the same stress field to find the equilibrium position. The choice of λ at step k is summarized as follows:

λ =    1 if ∆ε V M,k-1 p ≤ ∆ε V M,min p 0 else -0.5 if ∆ε V M,max p ≤ ∆ε V M,k-1 p (2.30)
There are two main advantages of quasi static loading. First, physically, it ensures that, before increasing the stress in the loading part of the fatigue cycle, the dislocations have all reached an equilibrium position where the total effective shear stress at each segment is nearly zero. This signifies that the small loading velocity in simulations corresponds well to the fatigue experimental situation. Otherwise, in the case of high strain rate, the influence of the elastodynamic stress field should be considered [START_REF] Cui | Computational 3dimensional dislocation elastodynamics[END_REF].

Secondly, in numeric, this quasi-static loading method enables to control the behavior of cross-slips and dislocation generation. The extra generation of short dislocation debris may slow down the simulation. Compared with constant stress rate loading, the quasistatic loading method can be considered as a compromise between simulation time and precision.

Virtual space

A virtual space is introduced relatively to the real space, where the grain is located. In the real space, facets are introduced in order to confine the dislocation in a grain. In the code TRIDIS, a very strong stress value (considered as infinite) is imposed as the facet strength [START_REF] Shin | 3D discrete dislocation dynamics applied to dislocation-precipitate interactions[END_REF] so that the dislocation cannot cross the grain boundaries. In experimental fatigue tests, the surface roughness is formed by the passage of dislocations through the free surface. In order to simulate this physical process, the concept of virtual space is put up. As shown in Figure 2.14(a), first, the facet between the crystal grain and the virtual space corresponds to the free surface through which the dislocations can pass without impediments. Once a dislocation crosses the free surface and enters the virtual space, the shear stress on the edge segments is multiplied by 10 and the shear stress on the screw segments is fixed to zero. This setting is introduced to accelerate the movements of the edge dislocations and fix the position of the screw dislocations in the virtual space. The accelerated edge dislocations should arrive at the border of the virtual space where a trap zone is established. In the trap zone, the effective stress is forced to zero so that the trapped dislocation cannot come back to the crystal when the loading direction is reversed. This setting tries to mimic reality, since in experiments, the departed dislocations are not able to return and the formed surface steps are not influenced by the departed dislocations. The simulation situation for a dislocation loop is shown in Figure 2.14(b), one part of an edge dislocation is in the trap zone while another part is in the crystal. Note that this virtual space procedure is only relevant for surface grain simulations where the Burgers vector is close to the normal to the free surface. For any other situation, one should apply the trapping algorithm.

A review on the application of DDD to fatigue

Due to the very special characteristics of fatigue problems in terms of spatial and temporal scales, there is still no full understanding of the fatigue mechanism. Experimentally, the fast development of characterization methods has led to a better understanding of the formation of fatigue microstructures in the bulk and at the surface of the crystals. However, due to the complexity of dislocation movements, the information on the microstructure evolution obtained from both in-situ and post-mortem experiments is quite limited and idealized and it remains difficult to observe the dynamic process.

Discrete dislocation dynamics is based on physical constitutive or empirical laws. The local interaction between dislocation-dislocation and dislocation-obstacle are also included in the simulation. Compared with other simulation methods, DDD can well reproduce the movements of dislocations under different loadings with acceptable computation time. The development of computer science enables large scale simulations. Apart from the numerical dislocation microstructure, the macroscopic response, the internal stress distribution and the surface roughness can also be simulated. DDD simulations help to reveal some meaningful hidden information related to dislocations. Finally, the fatigue damage process is influenced by many factors, and it is quite easy in DDD to investigate the effect of one individual parameter whereas it could be complicated or even impossible to do the same experimentally. This has motivated several groups to the realization of DDD simulations of fatigue as presented below.

Application of 2D DDD

The simulations using DDD in fatigue start with a 2D framework. 2D dislocation dynamics was applied in 2000 by Yang [START_REF] Yang | Simulation and observation of dislocation pattern evolution in the early stages of fatigue in a copper single crystal[END_REF] to investigate the dislocation patterning evolution in the early stages of fatigue for Cu single crystal. The simulation is quite simplified. Only edge dislocations have been considered. The cross-slip has also been forbidden because in the single slip situation, it is mainly the primary slip system that determines the microstructure. As shown in Figure 2.15(a), the author has observed the randomly distributed dislocation sources evolving to a regular matrix-wall structure and when the cyclic loading continuing, the matrix-wall (not the wall in PSB) structure Chapter 2. Methodology of 3D Discrete Dislocation Dynamics presented a tendency to construct a matrix-vein structure. The simulation results fit well with the experimental observations obtained by ECCI (Electron Channeling Contrast Imaging) technique, shown in Figure 2.15(b). The author thinks it is the unbalanced force exerted from screw dislocation segments between the matrix and the wall that plays a role in the evolution from the matrix-wall to the matrix-vein configuration [START_REF] Yang | Simulation and observation of dislocation pattern evolution in the early stages of fatigue in a copper single crystal[END_REF][START_REF] Li | The early stages of fatigue and evolution of persistent slip bands in a copper single crystal[END_REF].

Application of nodal DDD model

Recently, nodal dislocation dynamics models have been employed to investigate the dislocation behavior under cyclic loading by Ahmed M. Hussein, Jaafar A. El-Awady [START_REF] Hussein | Quantifying dislocation microstructure evolution and cyclic hardening in fatigued face-centered cubic single crystals[END_REF] in USA and by Taymour El-Achkar, Daniel Weygand [START_REF] El-Achkar | Analysis of dislocation microstructure characteristics of surface grains under fatigue loading by discrete dislocation dynamics[END_REF] in Germany.

In the research work of Hussein et al [START_REF] Hussein | Quantifying dislocation microstructure evolution and cyclic hardening in fatigued face-centered cubic single crystals[END_REF], a multi-slip simulation has been run on a single crystal of Ni subjected to cyclic loading. The crystal size and the initial dislocation density have been adjusted to identify their effect. By contrast with what was observed in different simulation groups, they found that the crystal strength increases with crystal size and decreases with the increase of initial dislocation density. Similarly, the dislocation density raises with the increasing crystal size and increasing initial dislocation density. In a word, a higher crystal size and a higher initial dislocation density can both contribute to the formation of the dislocation microstructure, as it can be clearly seen in Figure 2.16. The authors have also applied the box-counting process to investigate the size variation of formed microstructure and pointed out that, after the initial formation, the dislocation wall thickness and the dislocation cell size do not change with the subsequent cyclic loading.

Chapter 2. Methodology of 3D Discrete Dislocation Dynamics

In the publication of T El-Achkar [START_REF] El-Achkar | Analysis of dislocation microstructure characteristics of surface grains under fatigue loading by discrete dislocation dynamics[END_REF], the author puts more attention to the description of the dislocation structure, made of prismatic loops, dislocation debris and dislocation dipoles. Graph analysis is applied to quantify the evolution of the dislocation microstructure with the cycling. Two different plastic deformations have been imposed. For ε pl = 0.01%, 16 cycles have been simulated while for ε pl = 0.001%, 16 cycles have been completed. The simulation shows that the plastic slips are mainly accommodated by the long dislocations, whereas short dislocations work more as a barrier to the mobile dislocations. As shown in Figure 2.17, obviously the dislocation microstructure has been constructed under larger deformation while no clear structure formation has been found for small strain amplitude. The dislocation density augments with the loading cycles in these two cases but the value reached for large deformation is almost three times higher than for the smaller strain. The edge dislocations have taken up most of the dislocation population and the screw dislocations are the fewest and their dislocation densities fluctuate around a constant value. This phenomenon is more evident for large deformation.

Figure 2.17 Dislocation microstructure simulated for two ranges of the imposed plastic strain amplitude (the colors indicate the different components, the black one is the largest) [START_REF] El-Achkar | Analysis of dislocation microstructure characteristics of surface grains under fatigue loading by discrete dislocation dynamics[END_REF].

According to a former publication [127], the prismatic loops are divided into two groups: the 1 st order prismatic loop and the 2 nd order, as is presented in Figure 2.18(a). It shows (in Figure 2.18(b)) that the number of loops for 1 st order prismatic loops and the 2 nd order as well as dislocation debris varies similarly during the simulation except at the early cycles, 1 st order prismatic loops are more numerous. And for both imposed strain amplitude, elongated loops are more commonly seen and they become shorter when the loading continuing. Moreover, the author has also compared their numbers of vacancy type prismatic loops and interstitial type prismatic loops, the author states that the number equals for large deformations but for smaller deformation amplitude, at the end, there are more vacancy loops than interstitial loops, as shown in Figure 2.19.

Application to nano-scale fatigue simulation

Owing to the dynamical behaviors of dislocations, the mechanical response at micro and submicron scales for metallic material and the associated microstructure can be quite different from bulk materials. Discrete dislocation dynamics simulation is then an Chapter 2. Methodology of 3D Discrete Dislocation Dynamics ideal tool to investigate the underlying mechanism. M. Stricker [START_REF] Stricker | Irreversibility of dislocation motion under cyclic loading due to strain gradients[END_REF] and J Senger [START_REF] Senger | Dislocation microstructure evolution in cyclically twisted microsamples: a discrete dislocation dynamics simulation[END_REF] have realized DDD simulations to study the microstructure evolution under cyclic torsion loading. M. Stricker has introduced a new possible irreversible mechanism: the intrinsic asymmetry between the backward and forward motion. In other words, during the loading and unloading stages, the dislocation motion and the related plastic deformation are not totally reversed, as shown in Figure 2.20. They suggest a natural asymmetry can be introduced by the dislocation glide and the dislocation source itself. Even for a dislocation pile-up, the local boundary conditions are not symmetrical during the reverse loading. The dislocation source can be considered as an obstacle for the dislocation motions. As mentioned above, the Bauschinger effect has been studied by discrete dislocation methods, especially for thin Cu films as demonstrated by the 3D simulations proposed by Lesar et al. [START_REF] Zhou | Dislocation dynamics simulations of the Bauschinger effect in metallic thin films[END_REF] and the 2D simulations of Van der Giessen et al. [START_REF] Shishvan | Bauschinger effect in unpassivated freestanding thin films[END_REF]. The purpose is still to investigate the role played by the dynamic behavior of the discrete dislocations. Caizhi [START_REF] Zhou | Dislocation dynamics simulations of the Bauschinger effect in metallic thin films[END_REF] has studied two different systems: with or without a passivation layer (the surface being either considered as an impenetrable obstacle or a free surface). Simulations are realized in a 3×3×1 polycrystal. The grains are all oriented in the same direction with a grain size of 500nm. For the grain boundary, the author has proposed three different treatments: free passage, five times the active force of the Frank-Read source or infinity. The author interpreted that the Bauschinger effect is much more pronounced in passivated films than in bulk materials because the hardening process is totally different: for bulk metals, it is the cell structures or microstructure formed from dislocation reactions that determine the hardening process; for passivized thin films, dislocation pileups are the main contributors to the hardening process. The cell structure is relatively stable during unloading while dislocation pile-ups tend to relax in a larger degree, which leads to a more pronounced Bauschinger effect. Siamak [START_REF] Shishvan | Bauschinger effect in unpassivated freestanding thin films[END_REF] has used 2D dislocation dynamics to investigate freestanding polycrystalline films. The author concluded that it is generally the inhomogeneity that caused the Bauschinger effect. This inhomogeneity can be induced by dislocation pile-ups against grain boundaries; constraints at the grain level by neighboring grains; variations in the grain orientation, dispersion of the grain size. Erel [START_REF] Erel | Generation and interaction mechanisms of prismatic dislocation loops in FCC metals[END_REF] has used DDD to study the motion of prismatic loops and the interaction between prismatic loops and single dislocations, at the end of his publication, the author 2.8. The motivation and open questions to investigate in this thesis work has proposed a tripolar transport mechanism which can contribute to the clustering of prismatic loops near the Persistent Slip Bands (PSBs).

The research works of Christophe DEPRES

The first extensive study that applied DDD to fatigue problem is the PhD work of Christophe DEPRES [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF]. In his thesis, he has simulated the cyclic behavior in single slip of AISI 316L surface grain. During about twenty cycles simulation, the formation of PSBs and the related surface roughness formation have been observed. The formation mechanism of the PSB and the growth of surface extrusions are explained in his thesis. Moreover, based on the obtained numerical PSBs, he has proposed a crack initiation criterion when the stored elastic energy, extrusion height and the stress concentration coincide with each other, as shown in Figure 2.21. Later on, C. Depres has pursued his study in the field of crack propagation. In the case of a single crystal, he has tested the influence of a pre-existing crack on the dislocation density evolution and the corresponding Crack Tip Slip Displacement (CTSD) variation under cyclic loading. To investigate the indirect transmission mechanism of the crack propagation in bi-crystals, in the latest work, a parametric study has been realized to elucidate the effect of the crack position on the extrusion growth at the neighboring grain. Full detail of the crack propagation study is available in [START_REF] Déprés | An extensive 3D dislocation dynamics investigation of stage-I fatigue crack propagation[END_REF].

One of the question that we want to answer is the cycle number that can be simulated. Firstly, from the article of [36], to identify the plasticity localization, it is enough to have only a few cycles simulation. This argument can be applied to the low cycle fatigue where the deformation during one cycle is much bigger. For high cycle fatigue, we need to optimize the model to introduce the irreversibility. For example, as the cross-slip is very important for the irreversibility introduction. A better cross-slip model should help the simulation for the high cycle fatigue, which means more simulation cycles.

The motivation and open questions to investigate in this thesis work

After this literature survey on the early stages of fatigue, the presentation of the simulation methods used and the former work of Christophe DEPRES, the motivation and the open questions addressed in this thesis are summarized here. First, even though Cu is the most experimentally studied metal, the application of DDD to Cu is rare.

Chapter 2. Methodology of 3D Discrete Dislocation Dynamics

Definitely, with the help of abundant research publications, it should be inspirational for a better deformation mechanism understanding to apply DDD to study the behavior of Cu, which is the origin of the study motivation. The simulation model of C. Depres will be reused because of its advantage in single slip simulations.

On one hand, there are still some open questions to answer, such as the further evolution of the PSB structure after twenty cycles, the different behaviors of Cu and AISI 316L, the influence of the applied cyclic strain amplitude and the stability of the numerical PSB. On the other hand, the progress of computational capacity has offered new opportunities for more complicated simulations such as the surface roughness variation calculation, double slips and the polycrystalline version of the code.

Based on these backgrounds and motivations, the simulations results are presented to answer these questions in the following chapters.

Conclusions

In this Chapter, we have firstly reviewed the brief history of DDD and the code TRIDIS that will be used in this thesis. The "screw-edge" DDD model is presented in detail, including the discretization of space and dislocation lines, the mobility law, the local dislocation interactions and the treatment of cross-slip. As Cu is the main material in following chapters, the parameters of Cu are identified as well as those of 316L stainless steel. Then, the expressions used for stress field and displacement field calculations are explained, especially the related implantation algorithm in TRIDIS. In order to adapt TRIDIS to fatigue specific configurations, the quasi-static cyclic loading has been added to increase the computation stability and a virtual space concept is proposed to facilitate the surface roughness calculation. In the end, a short review of the application of DDD to fatigue problems and the research work of Christophe DEPRES is presented. Inspired by former literature review. The motivations and the questions to investigate are summarized in the last section (Section 2.8).

What should be retained from this chapter:

-The special advantage that DDD is able simulate the collective behavior of dislocations under cyclic loading which can surely help the understanding of deformation mechanisms at the early stages of fatigue degradation;

-As a classic material studied in fatigue experiments, the investigation of Cu by DDD is rare, which also provides motivation for the work of this thesis, the parameters used for Cu are shown in Table 2.2;

-Quasi-static cyclic loading and virtual space are two specific numerical settings dedicated to fatigue problems.

Chapter 3

Formation process of one individual PSB

Introduction

In this chapter, we mainly focus on the mechanical behavior of one individual PSB using DDD simulation method. We will consider an individual PSB built up in a hexagonal surface grain under a relatively low fatigue loading amplitude (ε V M p,applied = 1.0 × 10 -4 ). The collective movements of dislocations will be tracked. Meanwhile, the evolution of the surface roughness will be evidenced by computing the dislocation displacement field at the free surface. Furthermore, to better understand the formation mechanism of the PSB, we will investigate in detail the formation process of the elementary microstructures in the PSB, such as the prismatic loops, the dislocation dipoles and the dislocation pile-ups. The advantage of one single PSB simulation is a better understanding of the dislocation movements without the influence of other PSBs.

Simulation settings

Grain geometry and boundary conditions

The DDD simulation speed is mainly limited by the number of dislocation segments, which is the reason why only small grains are considered. However, for this Chapter, we will only allow one single PSB to form, and for that a relatively large grain can be chosen. The grain geometry is presented in Figure 3.1(a). The simulated grain is a cylinder (of axis [8 1 -6]) with an hexagonal basis and a truncated top face, of which the normal vector is [9.9 -4. 45 1]. The hexagonal base is inscribed in a circle of D g = 20µm diameter. Its maximum and minimum heights are 30µm and 10µm respectively, as indicated in Figure 3.1(a). The boundary conditions for this grain correspond to the situation of a surface grain in a fatigue test specimen. Dislocations can only escape from the free surface which is the top surface (colored in dark blue) where the surface roughness will be calculated. Other surfaces (like the six lateral surfaces and the bottom surface) are all impenetrable facets with infinite strength.

Geometry, boundary conditions and loading

A single Frank-Read dislocation source belonging to the system B4 is put at the center of the grain as initial conditions. The length of the source is 0.8217µm = 10xl and is a pure edge dislocation segment. The brown plane in Figure 3.1(b) is the primary slip plane B4, with the normal vector n primary = [1 1 1] and the Burgers vector b primary = [ 1 0 1], which is almost perpendicular to the free surface. The green plane is the deviate slip plane D4, with the normal vector n deviate = [1 1 1] and Burgers vector b deviate = [ 1 0 1], same as for the primary plane. The applied loading is a pure shear stress on system B4, as indicated by the red arrows of Figure 3.1(b). In the crystal coordinate system x = [1 0 0], y = [0 1 0] and z = [0 0 1], the applied stress field increment ∆σ σ σ app is given as follow, (also defined in Section 2.6.1):

∆σ σ σ app =        0.08423 -0.03747 -0.08336 -0.03747 0.16668 -0.37080 -0.08336 -0.37080 0.82490        (3.1)
The shear stresses resolved on the 12 slip systems using Eq.(2.3) are listed in Table 3.1. These values represent the activation possibility and can be considered as Schmid factors. The Schmid factor of the primary slip system B4 and cross-slip system D4 is 0.50 and 0.17 respectively. Since the simulated grain is large, the DDD simulation is controlled by a small plastic strain amplitude ε V M p,applied = 1.0 × 10 -4 . Considering a large number of dislocations in the simulated grain, the product of the stress rate σ and the time step δt is small enough to avoid numerical divergence. The numerical parameters used in the simulation are presented in Table 3.2.

As the simulations are dedicated to the understanding of the formation process of a single PSB in a large hexagonal surface grain under a small applied fatigue strain 64 3.3. Evolution of the dislocation microstructure (ε V M p,applied = 1.0 × 10 -4 ), the numerical parameters, and especially the stress rate and time step should be small enough to capture accurately the dislocation interactions. The same simulation settings have been used for both Cu and AISI 316L austenitic stainless steel. Cu is selected as the main material to investigate. Then a comparison between 316L and Cu will be conducted to identify the influence of some material parameters.

Evolution of the dislocation microstructure

In this section, the results of the simulation of a Cu single crystal are presented, including the dislocation localization process to build up a PSB, the evolution of the dislocation density and the evolution of the dislocations spatial distribution. A comparison between the numerical PSB structure and experimental PSB structures is also given at the end.

Localization of dislocations

The evolution of the arrangement of dislocations at cycles 1, 4, 8 and 12 is presented in Figure 3.2, at the moment of zero global deformation. Figure 3.2(a) shows the microstructures developed on the primary slip system (the direction of observation is [1 1 1]). After the 1 st loading cycle, only a few dislocations and isolated loops remain in the grain. At the 4 th cycle, more dislocations are stored. As noted in the figure, two dislocation dipoles are formed. Dislocation dipoles can impede the movement of dislocation loops so that one part of the dislocation loop remains from one cycle to the next one. At the 8 th cycle, the number of stored dislocations increases and most of them are in the form of dipoles or tangles and dislocation debris appears at the center of the microstructure (in the form of minor dislocation spots). At the last cycle (the 12 th cycle), the stored dislocations become denser, the arrangement of dipole arrays more compact and the formed tangles more complex.

Figure 3.2(b) presents the same microstructure but with an observing direction normal to the deviate plane [1 1 1]. Between the 1 st and the 4 th cycle, the formed microstructure is quite loose. The distance between each line is quite large. After the 4 th cycle, the PSB gets denser and denser as the number of cycles increases. The thickness of the PSB increases due to the occurrence of many cross-slip events. But at long distance, there is no dislocations. This is mainly caused by the low cross-slip probability and the small deformation value, which only drives the dislocation to move

Evolution of the dislocation microstructure

in the short distance. As a result, the PSB is constructed near the initial dislocation source.

From the descriptions above, the constructed microstructure is composed of the dislocations on the consecutive primary slip planes and these dislocations are connected by the cross-slipped dislocations on the deviate slip planes. The whole 3D microstructure can be considered as a slip band. We also call it a Persistent Slip Band (PSB) because the name PSB in experiments is related to the fact that after polishing and , with further loading, the PSB remains at the same position [8,19,31,[START_REF] Woods | Low-amplitude fatigue of copper and copper-5 at.% aluminium single crystals[END_REF], as mentioned in chapter 1. The numerical PSB shows a similar permanent character in the simulation, as will be demonstrated later in Section 4.3.1 of chapter 4.

Dislocation density evolution

Because the initial condition is restricted to a single dislocation source, only the primary dislocations and dislocations on the cross-slip planes are allowed to be generated. Figure 3.3 shows the evolution of the dislocation densities during the first 12.5 cycles for pure Cu. During the first six cycles, the primary dislocation density fluctuates with the increases and decreases of strain. Meanwhile, the deviate dislocation density remains almost equal to zero. In the second period from the 6 th to the end of the simulation, the dislocation densities for the two slip systems increase with the loading cycles. Although the dislocation density fluctuation with the applied plastic deformation in one cycle is still evident, the storage of dislocations becomes the main tendency and follows a linear trend with the loading cycles (see the straight line in Figure 3.3). Moreover, most of the stored dislocations are primary dislocations and at the 12 th cycle, their density is 3 times bigger than the density of deviate dislocations.

The dislocation density is almost reversible before the 6 th cycle. After the 6 th cycle, dislocations are stored. In the following sections, the first 6 cycles can be defined as the reversible period and the following cycles after the 6 th cycle as the storage period.

Quantification of the dislocation distribution evolution 3.3.3.1 The calculation method

A PSB has a given thickness, which means that the dislocation tangles and dipoles lie on different planes (see Figure 3.3). To quantify the dislocation distribution inside a PSB, we have developed a method that can measure the spatial dislocation distribution. As shown in Figure 3.4, we consider dislocations of the same slip system but located on two distinct slip planes. The distances (noted as P osition1 and P osition2 in blue) of the slip planes to a reference slip plane are measured. Then, the total length of dislocations (noted as L1 and L2 ) contained in each slip plane is also computed. This gives the distribution histogram plotted in Figure 3.4. This method can help us to obtain more precise dislocation related information, such as the thickness of the PSB, the distribution of the positive and negative dislocations or the distribution of edge or screw dislocations.

Evolution of primary and deviate dislocation distributions

With the method described above, the dislocation distribution evolution in the PSB presented in Figure 3.2 for primary and deviate dislocations is now analyzed.

The position of the slip plane is measured along the [1 1 1] direction of the normal to the primary system (x-axis of the graphs in Figure 3. 5(a-d)) and along [1 -1 1] direction of the normal to the deviate planes in Figure 3.5(e-g). We start the analysis with the distribution of the primary dislocations shown in Figure 3. 5(a-d). It should also be noted that the distribution is calculated when the global deformation is zero. Positive and negative dislocations are represented in blue and red respectively. Notably, the interval in the x-axis is from 0µm to 2.5µm, over a distance of 2.5µm. It is worthwhile mentioning again that the choice of the reference slip plane is not that important. The absolute value indicates the position of dislocations but the relative distance is more important, as it shows the PSB thickness.

At the 1 st cycle (Figure 3.5(a)), only 4 slip planes have stored dislocations at zero deformation. The distance between these 4 slip planes is 2µm, which is quite small 3.3. Evolution of the dislocation microstructure compared to the grain diameter D g = 20µm. At the 6 th cycle (Figure 3.5(b)), at the end of the reversible period, more dislocations are stored in more slip planes and the dislocation distribution is almost uniform in each slip plane, but the spread distance is still 2µm. This distance can be regarded as the maximum mobility distance for the dislocations under this deformation amplitude. In the dislocation storage period, at the 9 th cycle (Figure 3.5(c)), dislocations are more gathered to construct a slip band at the position of 1.0µm. At around 0.65µm, dislocations also show a tendency to gather. At the 12 th cycle (Figure 3.5(d)), the dislocation lengths in the slip band located at 1.0µm are still increasing. However, the slip band width remains almost unchanged (0.15µm) since the 9 th cycle. The dislocations at 0.65µm have also constructed a small slip band with fewer dislocations but almost the same thickness (0.15µm). In addition, whatever the cycle, the length of positive dislocations is almost equal to the negative dislocation length for each slip plane.

In Figure 3.5(e-h), the dislocation distribution evolution for the deviate slip system is presented. and measured along the PSB width ( about 30µm, which is much larger than the PSB thickness) As the bin is smaller, the length of positive and negative dislocations is added, and not displayed separately as for the primary slip plane. At the 1 st cycle (Figure 3.5(e)), the first stored deviate dislocation appears at the position of 5µm and 23µm near the two sides of the grain boundary. The explanation for the appearance of first deviate dislocations near the grain boundary will be given in Section 4.3.2. Then, at the 6 th cycle (Figure 3.5(f)), a few deviate dislocations also begin to be blocked in the central part of the PSB. For the dislocation storage period, as the loading continues (9 th and 12 th cycle in Figure 3.5(g,h)), the deviate dislocation lengths in all the slip planes significantly augment and the arrangement is much denser. In addition, the distribution of dislocations in the central part presents an M -like profile (noted in a black line), while at the two sides of the grain boundary, there is a zone (indicated in Figure 3.5(h)) where no deviate dislocation is stored. This signifies that the cross-slip probability in the PSB is not uniform. In Section 3.4.2, we will provide an explanation of the presence of this "deviate dislocation free zone" near the grain boundary.

To summarize, the primary dislocations gather to form a PSB. Even though the dislocation length in each slip plane is increasing with the loading cycles, the thickness of the formed PSB remains stable. The deviate dislocations are firstly generated at the two sides of the PSB and then are stored in the central part of PSB and a deviate dislocation free zone can be observed close to the grain boundary. The stored positive and negative dislocation lengths are almost identical for primary and deviate dislocations.

Comparison with experimental data

As presented in chapter 1, the PSBs of FCC alloys have been extensively observed in experiments using different techniques. In this section after a description of the numerical dislocation microstructure, experimental data will be compared to the dislocation structure simulated by DDD.

Numerical dislocation microstructure

The detail of the dislocation arrangements in the PSB at the 11 th cycle for Cu is presented when the total equivalent strain is zero (Figure 3 In contrast, most of the dislocation dipoles and prismatic loops show more permanence because their location seems to remain constant whatever the strain value. The low mobility of these structures suggests that they contribute little to the accommodation of the plastic deformation. However, dislocation pile-ups are found to build up near the grain boundaries at the maximum strain and show a relatively higher mobility. From this point, it can be inferred that during one cycle of deformation, most of the plastic deformation is accommodated by the pile-ups. These observations suggest that the build-up and build-down of dislocation pile-ups correspond to the fluctuation of reversible dislocations while the permanence of dislocation dipoles and prismatic loops refers to dislocation storage.

Comparison with experimental PSB structures

Experimental studies [23,52,[START_REF] Kroupa | Dislocation dipoles and dislocation loops[END_REF] have reported the existence and stability of dislocation dipoles under cyclic loading for FCC metals. Typical dipole arrays observed in Cu-16 at% Al alloys at the plastic strain amplitude ε pl = 3.0 × 10 -3 are shown in Figure 3.7(b). This image is quite similar to the dislocation dipole observed at the two sides of the numerical PSB (Figure 3.7(a)).

In experiments, dislocation dipoles also exist in several different aggregation forms such as dipole bundles [49], veins [24] and ladders [START_REF] Mughrabi | Strength of metals and alloys[END_REF]. However, in the DDD simulation, the aggregation seems to be weak. The aggregation mechanism is still not clear and the DDD model is simplified. A more physical DDD model should be considered, for example by adding the climbing mechanism, for further study of the aggregation effect.

To summarize, in this section, we have investigated the dislocation structure evolution in a surface grain of D g = 20µm cyclically loaded at ε V M p,applied = 1.0 × 10 -4 . After 6 cycles of a reversible period, a PSB is constructed with the storage of dislocations. The thickness of the PSB is found to be stable after the built-up. In addition, the simulated PSB is comparable with the experimental PSBs. In the next section, we will focus on the formation process of the elementary dislocation structures constituting the PSB.

Formation process of the elementary dislocation microstructures in the PSB

In the previous section, we have shown that the simulated PSB is composed of different elementary dislocation microstructures including dislocation dipoles, prismatic loops, debris and dislocation pile-ups. In this section, we will try to explain the formation process for these microstructures.

Dislocation movement in the first cycle of the loading

In the first cycle of the simulation, dislocations are emitted from the dislocation source and multiply back and forth via the famous Frank-Read mechanism. The dislocation density remains low so that the microstructure analysis is easier. We propose to use this simplified situation to investigate how the elementary dislocation structures form. In order to accelerate the formation process, the applied deformation amplitude is now fixed to ε V M p,applied = 1.5 × 10 -4 which is slightly higher than than the one used in Section 3.2.2 (ε V M p,applied = 1.0 × 10 -4 ). The dislocation movement is shown in Figure 3.8 and Figure 3.9 for the first half cycle and the second half cycle, respectively. The dislocation movements are presented in the primary and the deviate slip planes (as indicated in Figure 3.8). We start the analysis from the first half cycle. As the plastic deformation increases from 0 to ε V M p,applied = 0.75 × 10 -4 (Figure 3.8(a)), the Frank-Read source is activated to produce multiple 3.4. Formation process of the elementary dislocation microstructures in the PSB dislocation loops in the same primary slip plane. The first dislocation loop or the front part of dislocation loops in the dislocation pile-up is driven by the applied stress and the internal stress produced by the subsequent dislocation loops. It is then balanced by the blocking stress applied on the grain boundary as described in Section 2.2.3. The other dislocation loops are driven by the strong applied shear stress as well as the internal stress produced by the subsequently generated dislocation loops but impeded by the formerly produced dislocation loops. Moreover, in order to accumulate plastic deformation, the applied shear stress τ applied,shear is slightly higher than the total internal stress τ int produced by the dislocations. Owing to the quasi-static loading, the dislocation segments are positioned in a pseudo equilibrium position and, the total effective stress should hence be zero at each dislocation position. If we neglect the very small line tension, the segments can be considered to be balanced by the friction stress τ f r :

0 < |τ applied,shear | -|τ int | < |τ f r | (3.2)
The friction stress τ f r is the critical stress to overcome the dislocation movement. For different dislocation loops, as long as the stress value difference between τ applied,shear and τ int is smaller than τ f r , the dislocation segment cannot move, which is the socalled equilibrium state. As the stress increases, the maximal deformation is reached (Figure 3.8(b)). More dislocation loops are generated and piled at the grain boundary. The augmentation of the dislocation density in the pile-up produces a very strong internal shear stress in the cross-slip plane, which is released by cross-slips at the two sides of PSB. Deviate dislocations are generated, as visualized by the dark blue segments plotted in Figure 3.8(b) and in the zoomed region. Dislocations located at the two sides of the PSB have opposite signs and hence cross-slip occurs in two opposite directions (see orange arrow in Figure 3.8(b)). Then the stress begins to decrease, the dislocation loops in the pile-up begin to move backward (as indicated by the red arrow in Figure 3.8(c)). In this case, the internal stress on the primary slip plane is larger than the applied stress. We still have:

0 < |τ int | -|τ applied,shear | < |τ f r | (3.3)
It can be stated that even though the dislocations are still in an equilibrium state, the internal stress is pushing them to move backward. In other words, the internal dislocations are forced by the external dislocations to move backwards. Some dislocation loops disappear by self-annihilation. When the deformation goes back to zero (Figure 3.8(d)), most of the primary dislocation loops in the pile-ups have disappeared or departed from the free surface but some edge dislocation dipoles and screw dislocation dipoles stay in the grain. The formation mechanism will be explained in Section 3.4.2.2. For the deviate dislocations (zooms in Figure 3.8(d)), because of the pure shear stress applied on the primary slip system, some of them have cross-slipped back to the primary slip plane and continued to move towards the central part (this is the double-cross-slip mechanisms). A few deviate dislocations have reversed firstly their slip direction and moved in the deviate slip plane to another side of the slip bands, then some of these dislocations move back as well to the primary slip plane. The process described above contributes to generating more dislocation sources.

The other half-cycle is shown in Figure 3.9. While the absolute value of the applied stress increases again, its value becomes once again larger than the internal stress and In this way, the construction of several pile-ups leads to the augmentation of the PSB thickness. It should be noted that, at the maximal deformation, some dislocations are still stored in the form of dislocation dipoles. Then the value of applied stress decreases, the internal stress begins to push back the dislocation loops (see the red arrows in Figure 3.9(c)). After the accomplishment of one complete cycle (Figure 3.9(d)), the total deformation goes back to zero but many dislocation tangles and dipoles are stored in the grain. As the applied strain amplitude is larger, the storage process is accelerated, compared with the 6 reversible cycles with ε V M p,applied = 1.0 × 10 -4 in Section 3.3.2. Here, dislocations can be stored in the first cycle. We can also observe the dipole arrays near the grain boundary.

(b) (a) (c) (d) [1 1 1] [-1 0 1] [1 1 1] [-1 0 1]
This detailed analysis of the first cycle of deformation gives a general idea on how dislocations move under cyclic loading. It can be concluded that dislocation pile-ups are constructed or deconstructed as the deformation increases or decreases. During this process, dipole arrays and debris is stored in the crystal. The storage process will be described in the next section.

Formation mechanism of dipole arrays

As shown in Figure 3.6, on the two sides of the PSB, near the grain boundaries, there are two channels made of dipole arrays. After a literature review on the dipole array formation mechanisms, a possible scenario for the dipole array formation in the PSB channels is proposed.

Dipole formation mechanism in the literature

A dislocation dipole refers to a stable microstructure formed by two dislocations of opposite sign. Fouriel et al [START_REF] Fourie | Production of dislocation loops by a combined climb and glide mechanism[END_REF] has proposed a dipole formation mechanism, which is based on dislocation climb, and is hence valid at high temperature. At low temperatures, at which dipoles are also observed [139,[START_REF] Fourie | Elongated dislocation loops and the stress-strain properties of copper single crystals[END_REF], another dipole formation mechanism has been proposed by Johnston and Gilman [START_REF] Johnston | Dislocation multiplication in lithium fluoride crystals[END_REF]. This formation process is shown in Figure 3.10. We consider a moving screw dislocation segment XX and a jog formed by a cross-slip at P (see Figure 3.10(a)). With the continuing movement of the screw dislocation beyond point P , a trail Y is left (see Figure 3.10(b)). According to Johnston and Gilman, if the size of Y is of a few atomic spacing, the trail can become a row of vacancies or interstitial; if the size of Y is bigger, the trail can become a dislocation dipole for AB and CD, as presented in Figure 3.10(c); when the size of Y is too large, the dipole cannot remain stable because the attractive stress is decreasing so that the dislocations AB and CD can be separated. Then, AB and CD act as dislocation sources with one point pinned. We can see that this mechanism involves only one single dislocation.

Teleman et al [START_REF] Tetelman | Dislocation dipole formation in deformed crystals[END_REF] proposed another mechanism with two dislocations. As indicated in Figure 3.11, two dislocations (M M and N N in Figure 3.11(a)) of the same slip system but of opposing signs move on different slip planes and cross each other at point O (the distance between the two slip planes is Y , as shown in Figure 3.11(b)). Then, to lower the energy of the system, the parts P P and RR of the two dislocations reorient to be parallel (presented in Figure 3.11(c) from two view directions : plane perspective and side). The authors also think that P P and RR adjust their distance X in the slip plane to be equal to the distance between their slip planes Y to lower the energy (Figure 3.11(d)). The distance X can depend on the internal stress produced by the surrounding dislocations. After that, a dislocation dipole (P P and RR ) is built-up.

We have presented the two well-accepted dislocation formation mechanisms [START_REF] Johnston | Dislocation multiplication in lithium fluoride crystals[END_REF][START_REF] Tetelman | Dislocation dipole formation in deformed crystals[END_REF]. Other mechanisms [START_REF] Erel | Generation and interaction mechanisms of prismatic dislocation loops in FCC metals[END_REF][START_REF] Veyssière | Local dislocation reactions, selforganization and hardening in single slip[END_REF] have been proposed based on these two with some variance or details. In the following section, we will present how the dipole arrays are constructed in the DDD simulations.

Dipole formation observed in DDD simulations

From Figure 3.8(d), it is observed that the first dipole is constructed when the applied stress is decreasing from the maximal value. At that moment the dislocation loops in the pile-up are drawing back. This process is illustrated precisely in Figure 3.12. With the increasing applied stress, a dislocation pile-up is constructed near the grain boundary (shown in Figure 3.12(a), v refers to the velocity direction of the dislocation). Because of the strong applied stress, dislocation loops are largely stored in the pile-up. The storage of dislocation loops induces an increase of the stress on the deviate slip system, which could trigger cross-slips in some part of the screw dislocation segment embedded in the pile-up (the cross-slipped dislocation is noted in orange in Figure 3.12(b)). These segments move in the cross-slip plane, but, as the deviate shear stress is not exactly the same value for all of the cross-slipped dislocations, the moving distance in the crossslip plane is thus different. Then the applied stress begins to decrease, as shown in Figure 3.12(c). Most of the cross-slipped portions of the dislocation move back to the primary slip plane and keep advancing in the drawing back direction. By this doublecross-slip process, edge dislocation jogs are left in the cross-slip plane (noted in orange in Figure 3.12(c)). Under the applied primary shear stress, the primary edge dislocations of opposing signs (noted in black in Figure 3.12(c,d)) move in the opposite direction on the primary slip planes. By this way, the edge dislocations may encounter the other edge dislocations of opposing sign (red box of Figure 3.12(d)). As the cross-slip distances are different, the two approaching edge dislocations are in fact in different slip planes, which makes the dipole formation possible. The formed dislocation dipoles are well presented in Figure 3.12(d) from the side view. The process described here is quite similar to the mechanism of Johnston and Gilman [START_REF] Johnston | Dislocation multiplication in lithium fluoride crystals[END_REF]. It should be noted that the cross-slip occurs because of the strong deviate shear stress. The change of the direction of dislocation movement, as observed in Figure 3.12(c), is due to the reverse loading.

In the description above, we explained the dipole formation mechanism considering one dislocation line in the pile-up. As there are many dislocation lines in the pile-up, many dislocation dipoles can hence be constructed. The density of dipoles increases with the number of loading cycles. This explains why two channels without deviate dislocations are constructed at the two sides of PSBs near the grain boundaries and how the dipole arrays are constructed near the grain boundary. The dislocation dipoles also exist in the central part of the PSB. It is believed that 3.4. Formation process of the elementary dislocation microstructures in the PSB the dipoles are formed in the same way as for the dipoles at the two sides of the PSB. But in the channel at the middle of the PSB, there are many dislocation prismatic loops. In the following part, we will discuss the possible formation process of prismatic loops.

Prismatic loops formation

There are mainly two kinds of dislocation loops: shear loops (also called glissile loops) and prismatic loops. The shear loops refer to the dislocation loops constructed by a dislocation moving in the same slip plane, while the prismatic loop definition is such that all the dislocations in this loop share the same Burgers vector b but all the segments are perpendicular to b. In other words, a prismatic loop is made of edge segments lying in the primary and deviate slip systems.

The model of Li and Laird [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF][START_REF] Li | Cyclic response and dislocation structures of AISI 316L stainless steel. Part 1: single crystals fatigued at intermediate strain amplitude[END_REF] explains the formation of dislocation prismatic loops in fatigue. The process is described in Figure 3.13. When a screw dislocation meets an array of dislocation dipoles, collinear junctions are produced and the dislocation arms are reconnected. As the screw dislocation passes the dipoles, owing to the two colinear reactions, one side of the dipole can be connected. Then, if the same process is reversed, some special microstructure can be constructed , such a dislocation helices (Figure 3.13(b)), or prismatic loops (Figure 3.13(c)). Can Erel [START_REF] Erel | Generation and interaction mechanisms of prismatic dislocation loops in FCC metals[END_REF] has divided the mechanism of prismatic formation in two categories: (1) Super-Jog-Drag-Truncation (SJDT), it refers to the truncation of dipoles by the forest dislocation; (2) Offset-Double-Cross-Slip (ODCS) for a dipole, two cross slips take place on offset conjugate planes then two sides of the dislocation dipoles annihilate to construct a dislocation prismatic loop. To conclude and for an easier understanding, prismatic loops can be considered as a dislocation dipole cut off two times by collinear interactions.

Formation of dislocation debris

Based on this mechanism for the formation of prismatic loops, one question arises: how can cross-slip happen in the middle of the grain to cut the dislocation dipoles? In fact, in the mechanism described in Figure 3.12, cross-slip occurs generally at the grain boundary. Concerning the cross-slip in the middle of grain, even though it is not easy to observe, some clues can still be found from the formation of dislocation debris.

In Section 3.4.2, the formation of edge dislocation dipoles has been discussed. But for screw dislocations, the screw dislocation dipole is not stable because of the annihilation by cross-slips. Figure 3.14 shows how the annihilation process occurs in DDD simulations. Two dislocation loops from the different sides of PSBs are approaching each other (Figure 3.14(a,b)). These dislocations have opposed signs. Due to their strong attractive stress, cross-slip happens when they encounter (colored in dark blue in Figure 3.14(c)). The following phenomenon is the annihilation of these two screw dislocations. However, the two cross-slip dislocations have different lengths. As a result, one part of the cross-slipped segment is left in the grain after the annihilation (the minuscule remaining cross-slipped dislocation marked in dark blue in Figure 3.14(d)). The applied stress on the deviate dislocations is much smaller than that of the primary dislocations. Therefore, the cross-slipped dislocation possesses a relatively lower mobility. As a result, the minuscule dislocation segment is stable. This process can be confirmed by the article of Kuhlmann-Wilsdorf [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue[END_REF] and Antonopoulos [START_REF] Antonopoulos | Weak-beam study of dislocation structures in fatigued copper[END_REF]: as shown in Figure 3 3.4. Formation process of the elementary dislocation microstructures in the PSB the evidence of screw cross-slip is found from the screw dislocation traces between the dipole walls.

Furthermore, as the loading continues, the most important is that the screw dislocation debris could be a potential dislocation source for the deviate dislocations, which may favor the truncation of dislocation dipoles into prismatic loops. In this section, we have discussed precisely the moving mechanism of different dislocation microstructures. The dislocation movements in the first cycle have generally demonstrated how the dislocation loops move and cross-slips occur in the pile-ups during the loading and unloading sequences. After reviewing the dipole formation mechanisms in the literature, we have proposed a mechanism that can describe the formation of dipole arrays near the grain boundary from cross-slip in the pile-ups. The formation of prismatic loops in the middle of the PSB is considered as a result of a double truncation of dipole arrays by the collinear interactions from deviate dislocations. At the end, the formation of screw dislocation debris observed in the simulation is presented. The remaining screw debris are considered as potential sources of deviate dislocations, which are then responsible for the truncation of the dislocation dipoles to construct prismatic loops.

Surface roughness at the PSB location

As mentioned in Section 1.3.2, one of the distinguishing features of PSBs is the surface relief formed at the surface. Considering this phenomenon, the surface relief at the PSB is calculated using the model of Barnett [121] (Section 2.5.1). In order to get rid of the influence of the macroscopic deformation, we only present the surface relief when the global deformation is zero.

The displacements produced by the PSB are calculated on a 12.8µm×12.8µm square plane located at the free surface above the PSB. The plane is discretized in 3000 × 3000 calculation points. In the following sections, only surface displacements and reliefs near the PSB are considered because calculations have shown that displacements away from the PSB are negligible.

Extrusions and intrusions

We start with the analysis of the surface roughness calculated at the last cycle of the simulation. The results are presented in Figure 3.16(a). Longitudinal line profiles of the PSB along with L1, L2 and L3 are displayed in Figure 3.16(c), and transverse line profiles along with C1 and C2 are shown in Figure 3. 16(d). Locally, at the sites of PSB, intrusions and extrusions are formed as in experimental observations. We will also call it extrusions or intrusions (Marked in Figure 3. 16(c,d)). Along C1 and C2 (Figure 3.16(d)), the arrangement of extrusions and intrusions is presented. This morphology can be seen more clearly in the 3D view (Figure 3.16(b)). The small extrusions and intrusions are constructed on the long extrusions and intrusions. In the following section, the evolution of the surface relief at the PSB during the 12 cycles is presented.

Evolution of extrusions and intrusions

The growth of the extrusion and intrusion heights is always a difficult point in the experimental studies because the in-situ observation is still difficult for fatigue tests. The surface morphology for each cycle at zero deformation is presented in Figure 3.17. From the 1 st cycle to the 5 th cycle, the surface is nearly smooth. Only insignificant deformation lines caused by the individual dislocation loops are observed. However, at the 6 th cycle, as noted in the zoom of Figure 3.17, one extrusion and two intrusions with an elongated shape appear. Then, from the 7 th cycle to the 10 th cycle, more and more extrusions are constructed, especially at the 10 th cycle, 4 long extrusions have emerged at the surface and nearly crossed all the PSB. Finally, at the last two cycles, some cube-shaped small extrusions are produced, as shown in the zoom of Figure 3.17

Global evolution

To quantify the evolution of the extrusion and intrusion heights, the maximal extrusion and intrusion height measured on the probed plane is plotted in Figure 3.18 as a function of the loading cycles. The height is given in Burgers vector unit. For both extrusion height and intrusion height, during the first 6 cycles (the reversible period), they remain at about 0.5 ∼ 1b. This can be explained by the passage of the individual dislocation loops. Afterward, in the last 6 cycles, the heights increase almost linearly from 1b to 4b for intrusion and 6b for extrusion, respectively. In the end, the sum of the height reaches the value of 10b, which is equal to 2.5nm. This value gives us an order of magnitude for the surface roughness obtained in DDD simulations.

Local evolution

After having an idea about the global evolution of the PSB, we will now focus on its local growth, which has been already discussed in many experimental researches. Locally, the growth of extrusions and intrusions is realized layer by layer. To be more precise, the first or second layer is always a long extrusion/intrusion. After that, smaller extrusions/intrusions emerge on top of the pre-existing relief.

It must be mentioned that the extrusions and intrusions are highly related to the stored dislocations. For example, when the internal microstructure is still in the reversible period (before the 6 th cycle), the surface extrusion and intrusion height are very small; but when dislocations are starting to be stored in the PSB, the surface extrusions and intrusions begin to grow at the surface. In the next section, we will explain the relation between the surface roughness and the internal dislocation microstructure.

Relation between the internal microstructure and the surface relief

The model for calculating the displacement field from dislocations has been explained in detail in Section 2.5.1. One of the remarkable characteristics of dislocations is the one Burgers vector step left at the free surface after the departure of one dislocation loop. By using Barnett's model [START_REF] Barnett | The displacement field of a triangular dislocation loop[END_REF], the passage of one dislocation loop leaves a step at the surface. To be more precise, the one Burgers vector step displacement is realized by the combination of one half Burgers vector displacement at the two sides of the dislocation loop. The displacement is not constant for the two sides and beyond 2µm. In fact, it decreases progressively with the distance to the dislocation loop. The model presented below will not take into account this decrease but this simplification will not affect the analysis.

Figure 3.20(a) describes a simplified scenario to explain the formation of a surface step. A simplified dislocation loop is represented by two edge dislocations of opposing signs while the diminution of the displacement with the distance to the dislocation loop is neglected. Figure 3.20(a.2) is depicted the case of an opposite sign dislocation pushed toward the surface, for example when the load is reversed. Then, we try to regroup the two dislocations together to see the surface deformation. As shown in Figure 3.20(b.1), for the first combination, the two dislocation loops construct a net positive displacement on the surface and the height is one Burgers vector. For another combination (Figure 3.20(b.2)) where the loop positions are exchanged, a net negative displacement is created on the surface. The depth of which is also one Burgers vector. Thus, the basic formation mechanism for extrusion and intrusion is elucidated: extrusions and intrusions correspond to surface crossing of the different dislocation dipole types present under the free surface. In addition, if a third dislocation loop is added in the extrusion situation (Figure 3.20(b.1)), which is the case of the three dislocation loops (shown in Figure 3.20(c)), it can be seen that the pre-existing extrusion is declined or elevated together by half a Burgers vector, but the shape of the extrusion and intrusion can not be eliminated. The case is shown in Figure 3.20(c.1) gives us a possible observation that an extrusion is formed just beside an intrusion and in the case of Figure 3.20(c.2), the increasing of extrusion is realized by the construction of a smaller extrusion above a bigger one. In summary, the extrusions and intrusions are constructed because of the dipoles inside the grain and can be elevated or declined by the extra individual dislocation loops.

In the paragraphs above, we have discussed the relation between the internal In experiments, it is called "ribbon-like" extrusion or intrusion [60]. Then, as shown in Figure 3.21(c), if the dipole changes to a prismatic loop by a double collinear reaction, the long extrusion or intrusion can be also cut to smaller extrusions or intrusions. In experiments, it is called "tongue-like" extrusion or intrusion [60]. The corresponding prismatic loops can be considered as vacancy type prismatic loops or interstitial type prismatic loops for extrusions and intrusions respectively. To confirm this relation between the surface roughness morphology and the internal dislocations, a simulation result of chapter 4 is shown in Figure 3.21(a). An individual "ribbon-like" extrusion, one "tongue-like" extrusion and three "tongue-like" intrusions are observed on the surface. As indicated by arrows, under this long extrusion, two dislocations are tangled together, and a dislocation dipole is properly constructed in the central part. For the other "tongue-like" extrusions or intrusions, the prismatic loops remaining beneath the surface are visible. Consequently, the mechanism interpreted in Figure 3.21 should be reasonable.

It can be concluded that, the dislocation dipoles build up the basis of "ribbon-like" extrusions and intrusions, while the prismatic loops build up "tongue-like" extrusions and intrusions, which contributes mainly to the increase of the surface roughness. In this section, we have mentioned interstitial or vacancy dipoles but no detailed information is given. In the next part, we shall discuss more the dipole types.

Volume of extrusions and intrusions

In Figure 3.21, we have mentioned vacancy type or interstitial type dipoles beneath the surface for extrusions and intrusions respectively. In fact, experimental observations of the Cu surface after fatigue tests show that more extrusions are produced than intrusions [START_REF] Essmann | A model of extrusions and intrusions in fatigued metals I. Point-defect production and the growth of extrusions[END_REF]. In reference [START_REF] Mott | A theory of the origin of fatigue cracks[END_REF], Mott suggested that the observed extrusion has the same size as the vacancy cluster present under the surface. With the same logic, the intrusion can be inferred to have the same volume as the volume induced by the displacements of the interstitial clusters present in the crystal. On the other hand, there exists a transformation from the vacancy clusters or interstitial clusters to the dipoles. To demonstrate the transformation process, Steffen [START_REF] Brinckmann | On the formation of vacancies by edge dislocation dipole annihilation in fatigued copper[END_REF] has used molecular dynamic to investigate the stable configuration of dipoles. The stable configuration of the dipole for different distances of the two dislocations is calculated and shown in Figure 3.22. He has used two different methods of energy minimization: Global Microconvergence Integrator (GLOK) and Fast Inertial Relaxation Engine (FIRE). The simulations are initiated from whether a dipole or a vacancy (also named as void in Figure 3.22) and finished by the final transformation to a stable configuration (dipoles or a vacancy). The final stable configurations for different distances between two dislocations are presented in Figure 3.22. We can see that for a vacancy type dipole, when the distance is small, the final configuration is a void; when the distance increases, a dipole configuration is more stable than a void. As for an interstitial type, except at 1d 111 , dipole is often a more stable configuration. The objective of reviewing this article is to show the existence of transformation between dipoles and vacancies or interstitials.

In our DDD simulations, we did not considered point defects (vacancies or interstitials) but we can use the extrusion or intrusion volume to characterize and quantify the evolution of internal vacancy-type and interstitial-type dipoles or prismatic loops. To calculate the extrusion/intrusion volume, the initial free surface is meshed into regular squared elements by a mesh as presented in Figure 3.23. For each element in the grid, the corresponding extrusion or intrusion volume is calculated in the following way: (1) the displacements of the four points h 1 , h 2 , h 3 , h 4 are calculated and defined as shown in Figure 3.23(a) and the average value h = h 1 +h 2 +h 3 +h 4 4 is computed; (2) the volume is then calculated by the multiplication of the average height and the square area dV = h * dS and is integrated on the whole surface. It should be noted that the volume we calculated here is not totally the same as in experiments. In the simulations, the initial position of the free surface before the deformation is considered as the reference position. The volume of extrusion and intrusion is calculated considering this reference position. However, in experiments, the initial surface position of the sample cannot be determined. The experimentally observed extrusions and intrusions are measured on the basis of the relative displacement of the sample surface. So the extrusion and intrusion volume calculated in the simulation can be considered as the absolute volume of extrusion and intrusion. We calculate separately the volume of extrusions and intrusions. Figure 3.23(b) presents the evolution of extrusion and intrusion volumes with the number of loading cycles. During the reversible period of the loading (from the 1 st cycle to the 6 th cycle), the intrusion volume is a little bigger than that of extrusions. When the dislocations begin to accumulate (after the 6 th cycle), the volume of extrusions starts to increase and the volume of intrusions is reduced to a very small value. In the period of dislocation storage, most of the surface roughness is extrusions. This phenomenon indicates that under the free surface more vacancy-type dipoles and prismatic loops are stored than interstitial-type. Now a question arises: how is the distribution of dipoles or prismatic loops under the surface along the vertical axis, i.e. along the axial direction of the grain?

To understand this distribution, we have introduced 8 virtual surfaces of the same dimension as calculation planes parallel to the free surface and located at different positions between the free surface and the bottom of the grain (Figure 3.23(c)) at the 12 th cycle. Each calculation will give the quantity of the vacancy-type and interstitialtype dipoles or prismatic loops beneath the calculation plane. The results are presented in Figure 3. 23(d). The evolution of both extrusion and intrusion volume shows a linear trend along the axial direction. This linear relationship means that the distribution of these two types of dipoles in the crystal is homogeneous along the grain axis. Thus, there is no preference for these two types of dipoles stored in the grain.

Since experimentally, the volume of extrusions is found to be larger than the volume of intrusions, it implies that the quantity of vacancy dipoles or vacancy prismatic loops is larger than that of interstitial dipoles or interstitial prismatic loops. Considering that the generation of dislocation loops is conservative, the number of vacancy dipoles or prismatic loops should be equal to that of interstitial dipoles or prismatic loops in the simulation box. We may infer that the vacancy dipoles have a tendency to stay in the grain while the interstitial dipoles prefer to move to the virtual space (cf. Section 2.6.2). DDD simulations show that the distribution of vacancy and interstitial dipoles are both homogeneous along the grain axis. However, Figure 3. 23(d) shows that more extrusions are created than intrusions as observed in the experiments, but this tendency shall be validated by realizing more simulations.

In this section, we have discussed the extrusion and intrusion volume variation. It should be kept in mind that an extrusion at the surface is always associated with a vacancy dipole stored in bulk, while the intrusion on the surface refers to an interstitial dipole beneath. On the basis of this finding, the extrusion and intrusion volume can be used to quantify the number of the two dipole types existing beneath the calculating plane of the surface displacement.

Possible limits for the maximal height of extrusions or intrusions

During the advancement of the PhD project, the height of the surface relief has been a concern for us. As shown in the previous section, the total height of extrusions and intrusions in our simulation is about 2.5nm (see Figure 3.18). This value is far smaller than data from the experimental studies ( 1µm for single crystal Cu in [7]).

To understand the possible reason for the discrepancy, an extreme situation is simulated. Numerous prismatic loops of the same type are introduced in a surface grain and we propose to compute how many will survive in the absence of the applied loading. It is expected that, because of the internal stress, the prismatic loops of the same type will repel each other so that many will escape the grain from the free surface. As a result, there should exist a limit in the number of prismatic loops that could be piled up in a surface grain.

The simulation scenario is presented in Figure 3. The simulation has been performed with different numbers of prismatic loops (from 10 to 100) and results are summarized in Table 3.3. It is found that the possible limited number for prismatic loops of the size L = 4µm is about 30. The effect of the prismatic loop size is also investigated. For smaller sizes, more prismatic loops can be stored in 3.5. Surface roughness at the PSB location the 12µm layer. For example, for a prismatic loop in the size of L = 0.5µm, the limit number is 40. Indeed, the distance between two opposing dislocations of the same slip system in a smaller prismatic loop is also smaller so that the stress field produced by these two dislocations can be neutralized by each other, which reduces the interaction between dislocations.

L = 0.5µm YES YES YES YES NO NO L = 1µm YES YES YES YES NO NO L = 2µm YES YES YES NO NO NO L = 3µm YES YES YES NO NO NO L = 4µm YES YES YES NO NO NO
As discussed before, this simulation presents an extreme case. About 35 prismatic loops of the same type can be stored stably in a 12µm layer. As a result, the maximum height of extrusion or intrusion is about 35 × b ≈ 9nm. Compared to the experimental results ( 1µm for single crystal Cu in [7] and 250nm for the static extrusion growth in austenite 316 [START_REF] Man | Extrusions and intrusions in fatigued metals. Part 2. AFM and EBSD study of the early growth of extrusions and intrusions in 316L steel fatigued at room temperature[END_REF]), this value is quite small. There are two main possible explanations. The first one is the size of the grain: in experiments, the grain size is much larger than 12µm grain can lead to more prismatic loop storage and thus a higher extrusion or intrusion. Let's then estimate the grain's dimension that could lead to a surface extrusion of 1.0µm as evidenced in Cu. The number of vacancy prismatic loops stored beneath the extrusion should equal 1.×10 3 0.256 ≈ 3906 loops. Assuming a loop size of 1µm, the equilibrium distance between the loops as measured by DDD is 12×10 3 40 = 300nm (see Table 3.3). Thus, the grain size should be of the order of 1.17mm which is possible. Another reason is that, the extrusion or intrusion height calculated here is only from dislocations. The effect of vacancy or interstitial diffusion has not been considered in our work and may change the extrusion height. As an example, the diffusion of vacancy is the foundation of the model of Polák [START_REF] Man | Extrusions and intrusions in fatigued metals. Part 2. AFM and EBSD study of the early growth of extrusions and intrusions in 316L steel fatigued at room temperature[END_REF]. Our simulations tend to prove that Polák's model of diffusion assisted extrusion growth is realistic since dislocations alone cannot lead to the extrusion heights as measured experimentally.

This section was dedicated to study the surface relief growth at the surface of a PSB. Under cyclic loading, the free surface on the PSB has the similar morphology as observed in experiments. The relation between the extrusion & intrusion morphology and the dislocations in bulk is elucidated in detail. By calculating the volume of extrusion and intrusion, the number of vacancy type dipoles stored in the crystal is found to exceed the number of interstitial dipoles. Finally, by considering an extreme situation of prismatic loop pilling, we could demonstrate that dislocation activity without considering diffusion mechanism will never be able to explain the large extrusion heights as measured experimentally.

Comparison between Cu and AISI 316L austenitic stainless steel

The comparison between different materials has been experimentally studied for many years [23]. To compare with the literature, simulations with the same geometry and loading parameters have been conducted on Cu and AISI 316L austenitic stainless steel to compare the PSB differences. The strain amplitude is ε V M p,applied = 1.0 × 10 -4 , and the grain diameter and height are D g = h = 20µm.

First, Figure 3.25 shows the evolution of primary and deviate dislocation densities for both Cu and 316L with the loading cycles. Two main regimes can be observed as noted in Section 3.3.2: the reversible period and the storage period. However, obviously, the reversible cycles for 316L are longer (40 cycles) than that of Cu (6 cycles). For the storage cycles, the essential difference is the accumulation velocity for the dislocations. Generally, we can find that it takes 6 cycles for the dislocation density of Cu to increase from 1 × 10 11 m -2 to 4 × 10 11 m -2 . In contrast, the dislocation density in 316L increases from 1 × 10 11 m -2 to about 3 × 10 11 m -2 during about 10 cycles. A linear fit of the evolution of the dislocation density gives a slope value of 4.55 × 10 10 m -2 .s -1 for Cu and 2.26 × 10 10 m -2 .s -1 for 316L. This shows that the storage of dislocations in Cu is faster than that in 316L.

Cu 316L

(b) (a) In terms of the microstructure, since the mechanism of dislocation motion is similar, the formed microstructures for these two materials are quite similar and are composed of dislocation dipoles, prismatic loops and pile ups. The biggest difference is the thickness of the formed PSB. Figure 3.26 presents the PSB structure and the distribution in the traversal direction in the last cycle of the simulation for Cu and 316L. At this cycle, the PSB has been well constructed. As we have mentioned in Section 3.3.1, the PSB thickness is changeless after its formation so that, although the number of cycles simulated for Cu (12 cycles) is much smaller than for 316L (50 cycles), the PSB microstructures can be compared without lack of generality. It is found that the PSB formed in Cu is thicker (e = 0.15µm) than the PSB formed in 316L (e = 0.1µm). Since the thickness is smaller, the arrangement of dislocations in the PSB for 316L is more compact than that for In summary, the accumulation of dislocations in Cu starts earlier and evolves quicker than in 316L. Moreover, the constructed PSB in Cu is relatively thicker and Cu has a tendency to construct a second one. In the next chapter in Section 4.8, we will further discuss and investigate again the differences between Cu and 316L and give a general explanation.

Introduction of fatigue irreversibility

The irreversibility is considered as the cause of fatigue damage or failure. In other words, the permanent or the topological changes during the cyclical loading are caused by the accumulation of irreversibility. In DDD simulations, the origin of irreversibility can be listed as follow:

-The first irreversible factor is the variation of the dislocation pile-up arrangement during the cyclic loading. For the same deformation value of every cycle, the dislocation loops in the pile-ups are not at the same position as they were in the former cycles. This phenomenon has been reported in reference [START_REF] Stricker | Irreversibility of dislocation motion under cyclic loading due to strain gradients[END_REF].

If the movement of dislocations was completely reversible for every cycle, the dislocation should always be located at the same position for a given amount of the accumulated plastic strain. The change of the dislocation segment position may lead to internal stress field variations, which may cause cross-slips;

-The second one is the departure of dislocations from the free surface. Since a single free surface is introduced only in one side of the simulation box, only one part of the dislocation pile-ups can leave the grain, which may cause an asymmetry of the internal stress field;

-The third one is the cross-slip. The occurrence of cross-slip has destroyed the quasireversible movement of dislocation pile-ups. The produced deviate dislocations cannot move back when the load is reversed. The main consequence of this irreversibility is the dislocation source generation;

-An important irreversibility factor is the dipole formation. The dipole structure is stable enough so that the dislocation segments can stay inside the grain after one cycle loading. That is the first storage of dislocations. The storage of dislocation dipole will significantly change the internal stress distribution to a more heterogeneous state;

-The internal stress field heterogeneity will lead to irreversibility. Dislocation velocity will be strongly affected by the local stresses and different parts of the same dislocation loop will move differently. This phenomenon can result in the formation of dislocation tangles, which also favors the dislocation storage;

-One last important factor is the colinear reaction. This reaction can cause the formation of prismatic loops and debris, which are also the stable microstructures for dislocations.

The irreversibility factors listed above generally happen in the presented order and they can also interact with each other to generate more irreversibility.

Conclusions

The main purpose of this chapter is to present and discuss DDD simulation results when only a single PSB is constructed. A simulation of single slip is realized on a 20µm size Cu hexagonal grain under the deformation ε V M p,applied = 1.0 × 10 -4 . The mechanism of formation of the PSB is presented. The results are discussed and analyzed in two aspects: in bulk and at the sample free surface.

Inside the crystal, first of all, the accumulation and localization of the dislocations lead to the formation of an organized dislocation microstructure which is a PSB. The regular microstructure is composed of dislocation dipoles, prismatic dislocation loops, dislocation debris and reversible dislocation pile-ups. The mechanisms of motion of these elementary microstructures are analyzed. A formation mechanism of dipole arrays near the grain boundary from the cross slip in pile-ups in the simulation is proposed.

At the surface, with the formation of the PSB, the surface relief is growing at the same time. Extrusions and intrusions can be observed. The growth of extrusions or intrusions is realized by the pile-up of smaller extrusions on the long extrusion. From a simplified illustration, the relation between the shape of extrusions and intrusions and internal dislocations is elucidated. By calculating the extrusion and intrusion volumes at 3.8. Conclusions the free surface, it is found that more vacancy-type dipoles or prismatic loops are stored in bulk than the interstitial-type. In the end, we have done a simulation to assess the stability of prismatic loops piling up in a fixed layer, which explains the physical limit of the extrusion height measured by dislocation activity, without the consideration of the vacancy diffusion.

To complete the chapter, a comparison is made between AISI 316L austenitic stainless steel and Cu to check the material parameter effects. Moreover, we also listed the possible sources of irreversibility occurring during the simulation in a chronological way. In the next chapter, we shall introduce more PSBs to investigate their interaction effects.

What should be retained from this chapter:

-PSB is composed of dislocation dipoles, prismatic dislocation loops, dislocation debris and reversible dislocation pile-ups. The typical dipole arrays near the grain boundary are constructed by the cross slip of dislocations in the pile-ups;

-"Ribbon-like" extrusions & intrusions at the surface are related to long dislocation dipoles, while "tongue-like" extrusions & intrusions are related to the prismatic loops. The vacancy-type dipoles or prismatic loops stored in the grain are more numerous than the interstitial-type dipoles or prismatic loops, which explains that the extrusion volume is higher than the intrusion volume;

-Compared with experimental observations, the height of extrusions obtained in DDD is quite small, which may be caused by the grain size or the lack of consideration of vacancy or interstitial diffusion mechanisms.

Chapter 4

PSB multiplication process

Introduction

In the previous chapter, the formation process of one PSB is presented. We got a general idea about how dislocations move in the crystal and lead to the formation of a PSB via a complex mechanism involving several sources of irreversibility. To gain more insights on the fatigue problem, the multiplication process of the PSBs and the interaction between several PSBs should now be explored. To reach the aim, simulations on a smaller grain of 5µm diameter under different cyclic plastic strain amplitude are realized so that more PSBs can be generated. Besides the microstructure formation and the surface roughness evolution that we have discussed in chapter 3, we will investigate how dislocations move under a larger plastic strain amplitude and how the internal stresses are modified by the formed PSBs. Moreover, the stability of simulated PSBs and the effect of the initial conditions is tested by a simulation with a reduced amplitude. In the end, we discuss the macroscopic behavior obtained in the simulation and compare in more detail the differences between Cu and 316L.

Most of the results presented in this Chapter have been reported in a paper submitted to International Journal of Fatigue (see the end of this thesis).

Simulation settings

The simulation setting follows the same logic as presented in chapter 3 but the grain size and the geometry have been changed. The crystal is a cylindrical surface grain, shown in Figure 4.1(a). The orientation of the grain is [2 2 1] with the diameter D g = 5µm and height h = 5µm. As usual, the top surface is considered as a free surface where dislocations can pass but cannot come back, while the other surfaces are all closed for the passage of dislocations. Only one Frank-Read dislocation source of the 8 th slip system (see Table 4.1) is put inside the crystal as the initial dislocation configuration. Slip system 8 (1 1 1)[1 1 0] will be called the primary slip system.

The simulated situation is still single-slip with the applied stress field increment:

∆σ σ σ app =        0.577 0.
-0.288 0.

-0.577 -0.288 -0.288 -0.288 0.

       (4.1)
The relative shear stresses resolved on the 12 slip systems with respect to the shear applied on the primary slip system τ system τ primary is shown in Table 4.1. Schmid Factor 0.50 0.17 0.50 0.17 0.67 0.67 0.33 1.00 0.17 0.50 0.17 0.50 The different applied plastic strain amplitudes are: ε V M p,applied = 0.6, 0.8, 1.0, 1.2, 1.5 × 10 -3 . The numerical parameters for the simulation are presented in Table 4.2. Since the deformation is larger in chapter 3, the product of the stress rate σ and the time step δt could be increased to accelerate the simulation. After the first two cycles, multiple dislocation loops are generated from the initial Frank-Read source in order to accommodate the imposed plastic strain. In this multiplication process, the dislocations are homogeneously distributed. It should be noted that dislocations quickly spread in almost the entire crystal through the double cross-slip mechanism, as described in [START_REF] Carstensen | Structural evolution and mechanisms of fatigue in polycrystalline brass[END_REF]. As can be seen in Figure 4.2(a), the distribution remains homogeneous until the 5 th cycle. However, from the 10 th cycle, localization of the dislocations can be observed. Starting from the 10 th cycle, some dislocations gather to form adjoining slip planes connected by dislocations of the cross-slip system. The built-up microstructure compares well to the macro slip bands as observed in experiments [7,[START_REF] Mughrabi | Strength of metals and alloys[END_REF]. Most of the plastic strain is confined in these bands. Like in chapter 3, the name PSB will be used to describe this organized microstructure. The dislocation arrangements inside the PSBs are relatively much denser than everywhere else in the grain but randomly distributed dislocations are still visible.

With the increase of the number of cycles, more PSBs are constructed (see Figure 4.2(a)).

After the 15 th cycle, besides the already formed two PSBs, the embryo of the third one appears on the right side of the grain. From the 20 th cycle, four PSBs are about to be formed but the frontiers between the PSBs are still not very clear because of the existence of many reversible dislocations. After the 26 th cycle, four PSBs are now well constructed and easy to be distinguished directly from the 3D picture. Finally, it is found that the positions and thicknesses of these four PSBs stay almost invariant from the 26 th cycle to the 50 th cycle similarly to what is observed experimentally in [7,[START_REF] Mughrabi | Strength of metals and alloys[END_REF] and in the simulation of Section 3.3.3. In order to quantify the organization of the dislocation microstructure during the cyclic loading, the dislocation densities on the primary and cross-slip slip systems (or deviate slip system) are plotted as functions of the loading cycles in Figure 4.2(b). During one cycle loading, the dislocation densities for both slip systems fluctuate with the increasing or decreasing of the strain. Thus, the dislocation density reaches two maximum per cycle, one for the maximum positive strain and a second one for the minimum negative plastic strain. The plot shows that the dislocation population can be subdivided in two parts: ρ dis = ρ dis,stored + ρ dis,rever . The stored dislocation ρ dis,stored is plotted as a dashed line in Figure 4.2(b) and corresponds to the envelope of the minimum of the curve; the evolution of the reversible dislocation ρ dis,rever corresponds to the fluctuations within each cycle.

The reversible dislocations mainly contribute to the accommodation of the plastic deformation during one cycle. The stored dislocations have a lower mobility. They accumulate inside the crystal and contribute to the formation of the microstructures. The evolution of the stored dislocation densities matches well with the formation of the PSBs. When the four PSBs are still under construction (before cycle 26), the stored dislocation density increases and the amplitude of the reversible dislocation density is quite low. Once the PSBs are all formed (after cycle 26), the stored dislocation density remains constant and the amplitude of the reversible dislocation density reaches a maximum. The microstructure constructed by a limited number of stored dislocations is described hereafter.

The dislocations stored in the PSBs shows the same patterning as discussed in chapter 3 in the case of a single PSB. As a reminder, the detailed structure of the dislocation arrangement within a PSB at the 26 th cycle (visualized in the green box in p,applied = 0, dislocations exist in the form of dipoles, prismatic loops and tangles. Multiple dislocation dipoles are built-up by the attraction of two edge dislocations of opposed signs and are arranged in regular arrays. In the center of the PSB, due to the back and forth motion of dislocations by cross-slip during the loading and unloading stages, dislocation dipoles are cut into pieces by line multiple recombinations which create dislocation prismatic loops. Tangled dislocations are also formed from the recombination process on the side of the prismatic loops. When the applied plastic strain amplitude is increased, most of the dislocation lines or tangles are destroyed or extended by the stronger applied shear stress at maximal deformation. In contrast, dislocation dipoles and prismatic loops show more permanence because most of them stay almost pinned by the stress. The low mobility of these structures suggests that they contribute little to the accommodation of the plastic deformation. Nevertheless, dislocation pile-ups are found to be built up near the grain boundaries at the maximal deformation. These observations show that the build-up and build-down of the dislocation pile-ups correspond to the fluctuation of reversible dislocations while the persistence of dislocation dipoles and prismatic loops refers to the storage of dislocations.

Quantification of the dislocation distribution

As introduced in chapter 3, with the same method, the distribution histogram of the primary dislocations length at zero deformation is calculated and presented in Figure 4.3. At the 2 nd cycle, the dislocations concentrate in the central part of the grain, and the spreading is close to a Gaussian distribution. However, no evident localization of dislocations can be observed. Then, the dislocations move further towards the sides of the grain. At the 5 th cycle, in the right part of the histogram, the distribution of dislocations is quite uniform, while in the left, a weak localization of dislocations can be seen at the position 1µm. More localization can be observed at the 10 th cycle: at positions 1µm and 2.4µm. The two distribution pics correspond to the two embryones PSBs observed in Figure 4.2. At the 20 th cycle, two localizations are now visible on the left (at 0.4µm and at 1µm) but on the right side (3µm ∼ 4µm) the localization is less pronounced. After the 26 th cycle, four pics are built-up which correspond to the four constructed PSBs. From the 26 th cycle to the 50 th cycle, as we have seen in Figure 4.2, the positions and the thickness of the four PSBs are almost fixed. However, for each PSB, the dislocation length distribution in each slip plane has changed, which may indicate that even though the PSB position and thickness are almost changeless, the dislocations may multiple and change their positions in the PSB from cycle to cycle.

Dislocation movement mechanism in the crystal

In the discussions above, we have seen that the dislocation structure started from one single dislocation source, then spread homogeneously over all the crystal and finally gather at some positions to construct PSBs. So here a question comes: how do the dislocations travel in the crystal? In fact, as observed in the simulation and mentioned in publications [START_REF] Carstensen | Structural evolution and mechanisms of fatigue in polycrystalline brass[END_REF], with the cyclic loading, dislocations move in the crystal through the double-cross-slip mechanism. As shown in Figure 4.4, the double-cross-slip consists of two times cross-slip occurrences: the first time occurs when moving from the primary plane to the deviate plane; then the second time takes place when moving back from the deviate plane to the primary plane. The slip on the primary slip planes is dominant but it is the slip on the deviate planes that is more crucial for the movement in the transversal direction. Short range slips on the deviate system can strengthen the thickness of one PSB, while long range slips allow dislocations to find a new stable position to construct a new PSB. It can be inferred that the dynamics of cross-slip plays a critical role in the dislocation distributions, thus in the PSBs patterning.

Cross slip

Initial Condition

Final Cycle

Short Range

Long Range

Initial dislocation source In Section 3.4, we have seen that cross-slip happens at the extreme side of dislocation pile-ups. In Figure 4.5, the shear stress on the cross-slip plane produced by half of the dislocation pile-ups is presented. Colors indicate the stress magnitude that may trigger cross-slip events. The darkest blue indicating the strongest shear stress value is located at the end of the pile-ups, which explains why the cross-slip occurs here. Each orange arrow is calculated as the combination of the primary slip direction vector and the cross-slip direction vector with weights of the shear stress value on the two slip planes respectively. By this method, the obtained vectors can be considered as indications of the direction of motion for dislocations near the pile-ups. For most positions, the direction of motion is aligned with the primary slip plane. The arrows incline towards the cross-slip direction when approaching the pile-up. It can be concluded that the internal stress field produced by pile-ups plays an important role in driving the dislocations to move in short distances and long distances.

In this section, we have seen that the simulation results are much similar to the results in chapter 3. However, there are two points that should be noted: the first one is that four PSBs have been constructed in this simulation and the dislocation distribution changed from a homogeneous state to a heterogeneous state, which has not been observed in chapter 3; the second one is the dislocation density shows a saturation state after the 26 th cycle. The detailed analysis of the saturation state will be discussed in Section 4.5. Analysis of the stress field induced by a dislocation pile-up has revealed how double cross-slip mechanisms could propagate dislocations.

Surface roughness evolution 4.4.1 Extrusions and intrusions

As described in chapter 3, the formation of several PSBs also leads to the roughness growth at the surface. The whole configuration of the free surface at the 26 th cycle, when the total plastic deformation is zero, is presented in Figure 4.6(a) and the color indicates the value of the displacement projected along with the normal direction to the surface. From the profile line in Figure 4.6(c), extrusions and intrusions are clearly illustrated, which are the typical surface morphologies observed experimentally [42,44,[START_REF] Man | Extrusions and intrusions in fatigued metals. Part 2. AFM and EBSD study of the early growth of extrusions and intrusions in 316L steel fatigued at room temperature[END_REF]. The evolution of maximal extrusion and intrusion heights as a function of the loading cycles are plotted in Figure 4.6(d)). The two values increase during the formation process of the PSBs but after the 26 th cycle, the height values simply fluctuate around a constant. Figure 4.6(b) presents the local evolution of extrusions from cycle 22 to cycle 26. At the 22 nd and 24 th cycle, there are only two layers of extrusions. Then, one layer is constructed again at the 26 th cycle. The growth of extrusions is induced by the passage of prismatic loops across the free surface [START_REF] Déprés | Low-strain fatigue in 316L steel surface grains: a three dimension discrete dislocation dynamics modelling of the early cycles. Part 2: Persistent slip markings and micro-crack nucleation[END_REF]: vacancy prismatic loops lead to intrusions Chapter 4. PSB multiplication process and interstitial prismatic loops induce extrusions. Each prismatic loop contributes to one layer of extrusion or intrusion with a magnitude of one Burgers vector. The extrusions and intrusions growth is realized by the addition of these layers. It should be noted that the dimension of the layer depends also on the size of the prismatic loops. In other words, the height and morphology of the extrusion and intrusion are influenced by the character and shape of the prismatic dislocation loops crossing the free surface.

Analysis of the saturation state

A saturation state is reached after about 26 cycles as evidenced in Figure 4.2(b). In this so-called saturation regime, even if the plastic deformation loading continues, many characteristics remain stable including the PSB number, PSB locations, PSB thicknesses, dislocation densities, and maximal height of surface extrusions and intrusions. The stability of the dislocation density suggests that a dynamic equilibrium in dislocation movements is reached, which implies that the number of dislocations created within a cycle is equal to the ones annihilated or escaping at the free surface. Moreover, constant PSBs number and thickness imply that the dislocation-related microstructures do not change significantly in the saturation regime, that is to say the microstructure is stable. As the maximal height of extrusion and intrusion is correlated to the crossing of prismatic loops, the slight fluctuation of these two values demonstrates that the flux of prismatic loops nucleation inside the PSB is almost constant during the saturation regime. In other words, the dislocation nucleation kinetics has reached a permanent regime.

Analysis of the saturation state

Experimental evidence

It should be emphasized that the saturation phenomenon has been widely observed in fatigue experiments conducted on FCC metals. The most famous experiments are the ones of Mughrabi on the single slip fatigued single crystal of Cu [7,8]: under the cyclic loading, PSBs are formed and results in a rapid hardening. The observation of specimen surfaces has demonstrated that the PSBs are stable once they are well constructed: the position, the thickness and the volume fraction remain almost unchanged, which corresponds to the simulation outcome (See Figure 4.2). This characteristic seems to be a constant among FCC metals. Pure Ni single crystal has shown the same behavior as Cu in the research work of Mecke [START_REF] Mecke | Saturation dislocation structures in cyclically deformed nickel single crystals of different orientations[END_REF]. Huang et al. [START_REF] Huang | Plastic behavior of a nickelbased alloy under monotonic-tension and low-cycle-fatigue loading[END_REF] have used in situ neutron diffraction to measure the dislocation density in a Ni-based super-alloy and the result is presented in Figure 4.7. The same tendency was observed: dislocation density increases with the cycles and saturates when the PSBs formed. Essmann [34] has used a simple model of dislocation accumulation as a function of the plastic strain to study its importance and concluded that the annihilation of dislocations can lead to a natural limit of the dislocation density under cyclic loadings. The saturation of the dislocation density in our simulation suggests the same idea that an upper limit of stored dislocations exists for a given value of the imposed plastic strain amplitude. Regarding the surface roughness, it is generally admitted that the height of extrusions or intrusions continuously increases and will ultimately lead to the initiation of cracks at room temperature. Compared with the model of Polák [42,[START_REF] Man | Extrusions and intrusions in fatigued metals. Part 2. AFM and EBSD study of the early growth of extrusions and intrusions in 316L steel fatigued at room temperature[END_REF] for the formation of surface extrusions, the growth of extrusions in our simulation can be considered as the so-called rapid growth of static extrusions because it is induced by the dislocation motion and the crossing of prismatic loops. While in their model, the second stage of extrusion or intrusion growth is called stable growth [42,[START_REF] Man | Extrusions and intrusions in fatigued metals. Part 2. AFM and EBSD study of the early growth of extrusions and intrusions in 316L steel fatigued at room temperature[END_REF], this part of growth is mainly accounted Chapter 4. PSB multiplication process by the diffusion of vacancies. As the effect of vacancy diffusion is not included in our model, that may be the reason for the saturation of extrusion and intrusion heights in the DDD simulation. Moreover, the saturation in the PSB surface roughness has been reported by Mughrabi [155] shown in Figure 4.8: the average extrusion height of poly-crystal Cu is observed to saturate with the number of loading cycles especially in low temperature and with vacuum environment. The low temperature has prevented the diffusion of vacancies while the vacuum environment can reduce the irreversibility introduced by the oxidation at the surface. These experimental conditions match well the assumptions made in the simulations which may be another explanation for the observed saturation phenomenon on the surface.

Free surface deformation during one cycle

In experiments, the slip-path of dislocations within PSBs has attracted many research interests such as the study comparing the rhythmical and randomly oriented flip-flop in loop patches [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue II. Friction stress and back stress as inferred from an analysis of hysteresis loops[END_REF]. Generally, it is thought that dislocations move in the PSBs in an energy-favored way to accommodate plastic deformation [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue[END_REF][START_REF] Glazov | Self-organized dislocation structures (SODS) in fatigued metals[END_REF]. One of the models is the gliding of screw dislocations in the channels and bowing out of edge dislocations from the walls [START_REF] Brown | Dislocation bowing and passing in persistent slip bands[END_REF]. In order to understand the dislocation slip-path required to accommodate plastic deformation during the saturation regime of the simulation, we have shown in Figure 4.9 the dislocation distribution evolution and the related surface deformation during the 40 th cycle. Snapshots and dislocation length analyses are performed at four instants of the cycle corresponding to the zero and maximal plastic deformations. First of all, during one cycle, the cyclic shear deformation is well simulated as evidenced by the changes in the surface shape during the cycle. The dislocation density also fluctuates with the variation of the applied deformation. It is mainly the dislocations in the two sides of the PSBs that multiply to accommodate the plastic deformation and the sign of the dislocation loops alternates with the direction of the applied load. Combined with 4.5. Analysis of the saturation state the observation in Figure 4.2(c), it can be inferred that in our simulation, the plastic deformation is mainly accommodated by the construction and destruction of pile-ups which located at the interfaces between the PSBs and the matrix so that during the cyclic loading, the dislocation dipoles and prismatic loops located in the center of the PSBs are more stable and do not impede pile-ups movements.

Internal stress evolution

Dislocations move on slip planes (primary and cross-slip planes), driven by the corresponding shear stresses. According to the geometric relation between the simulated grain and the slip plane orientations (see Figure 4.1), dislocations which slip on the primary slip system mainly lead to the formation of pile-ups, dipoles in the PSB and dislocations crossing the free surface, while the movements in the transverse direction of the crystal are contributed by the dislocations slipping on the cross-slip plane. The localization of dislocations in the PSBs is the critical consequence of these transverse movements. To further explore the reasons for the dislocation localization, the distribution of the effective shear stress resolved on the cross-slip plane, including both the internal stress produced by dislocations and the applied stress, is presented in Figure 4.10 for different cycles. As indicated in the illustration, the plane used for the calculation is perpendicular to both the primary slip plane and the cross-slip plane, i.e. the plane normal is aligned with the Burgers vector shared by the two slip systems n n n = [1 1 0]. At the second cycle, the dislocations are distributed homogeneously enough but as indicated by the red box, most of the dislocations are bearing a very strong cross-slip shear stress, which means they are not stable. Under the effect of the shear stress, dislocations will move in the cross-slip plane (in the transverse direction) during loading and unloading. Let recall here that the positive or negative moving direction is determined by the sign of the shear stress and also the sign of the dislocations. The shear stress on the cross-slip plane provides indirect information on the dislocation ability to interact so that from the 2 nd cycle to the 10 th cycle, dislocations show a tendency to repeal each other. That is to say, dislocations repeal each other to move towards the two opposite sides of the grain respectively to reduce the stress on the cross-slip plane, which enables the formation of two PSBs. As the dislocations are more isolated, the shear stress and hence the interactions between the dislocations in the two PSBs get weaker. The continuing cyclic loading from the 10 th cycle to the 26 th cycle allows more dislocations to move away from the two PSBs and be pushed towards the grain outer surfaces, which contributes to the formation of four PSBs. At the same time, it can be observed that some dislocations are stored in the center of the PSBs (dislocation dipoles and prismatic loops) and the shear stress resolved on the cross-slip system becomes smaller. Finally, in the saturation regime, the cross-slip shear stress in the matrix between two PSBs is much reduced, which explains why the dislocations cannot further move in the matrix to form another PSB. This can be considered as a key mechanism to clear the dislocations in the matrix. As a result, from the 26 th to the 50 th cycle, the number of PSBs and their positions remain constant. We can conclude that the stress state induces a self-organization of dislocations that lead to the localization and PSB formations. And reciprocally, this localization largely reduces the dislocation interactions and hence the shear stress on the cross-slip planes. It 4.6. Influence of the applied strain amplitude stabilizes the dislocation microstructure so that the dislocations can move in a so-called low-energy mode with less stress interaction to accommodate the plastic deformation, which explains the saturation state from 26 th cycle to the 50 th cycle.

Influence of the applied strain amplitude

The influence of the imposed plastic deformation on the fatigue behavior for FCC metals has been widely studied [31,[START_REF] Cheng | Mechanisms of fatigue hardening in copper single crystals: The effects of strain amplitude and orientation[END_REF][START_REF] Abel | Fatigue of copper single crystals at low constant plastic strain amplitudes[END_REF]. The plastic deformation amplitude is thought to have a significant effect on the fatigue behavior, especially on the microstructure, the macroscopic response and the hardening rate. For Copper single crystals, a larger deformation will create more PSBs [31] and a very brutal raise of the applied plastic strain amplitude is able to activate more slip systems and will eventually modify the PSB microstructure into cells [START_REF] Gong | Cyclic deformation behavior and dislocation structures of [001] copper single crystals-I Cyclic stress-strain response and surface feature[END_REF]. In order to depict the effect of the imposed plastic strain amplitude, a simulation campaign is conducted on the same geometry (cylindrical grain D = H = 5µm) under five different strain amplitudes (ε V M p,applied = 0.6 × 10 -3 , 0.8 × 10 -3 , 1.0 × 10 -3 , 1.2 × 10 -3 and 1.5 × 10 -3 ). Simulations are performed both for Cu and 316L. Figure 4.11 shows the evolution of the total dislocation densities for each of the five applied strain values for Cu. Dislocation density increases along with ε V M p,applied . Regardless of the value of the applied strain amplitude, and as seen in the previous simulation, after about 20 cycles of loading, the saturation regime is systematically reached. In fact, this stands not only for the dislocation densities, but also for the PSB height and the surface roughness.

The comparisons between the fatigue behaviors of Cu and 316L will be discussed in the last section.

Dislocation density

Relation between strain amplitude and number of PSBs

The dislocation microstructures obtained in the saturation regime (corresponding to the 40 th cycle) for Cu are compared in Figure 4.12 for the different values of ε V M p,applied and Chapter 4. PSB multiplication process when ε V M p = 0. From the 3D snapshots, it can be intuitively seen that a larger strain amplitude leads to more storage of dislocations as well as more construction of PSBs. It is worthwhile mentioning that, instead of increasing the individual PSB thickness to accommodate the plastic deformation, more PSBs are constructed as if each PSB could only accommodate a finite amount of plastic strain.

Cu

PSB volume fraction evolution

Based on the linear relationship between the PSB volume fraction and the applied deformation for Cu in single slip mode, observed in experiments, Winter [31] has proposed a two-phases model to describe the relation between the applied deformation and the formed PSBs. In his model, the plastic deformation is accommodated by the two phases separately: the first phase is the PSB, which is soft and active; the second one is the matrix, which is hard and almost inactive. The capacity of accommodation for the plastic deformation in these two phases are fixed. The relation is given as follows:

ε V M p,applied = ε p,P SB * f + ε p,M atrix * (1 -f ) (4.2)
where f is the volume fraction of PSBs in the grain and ε p,P SB , ε p,M atrix are the plastic strain in the PSBs and the matrix respectively. This model has been applied successfully in many FCC metals such as Cu and Ni [31,[START_REF] Mecke | Saturation dislocation structures in cyclically deformed nickel single crystals of different orientations[END_REF].

In order to verify if the evolution of the PSB volume fraction (f ) with the imposed plastic strain amplitude obtained in our simulations follows Winter's model, the PSB thicknesses are calculated using the method described in Figure 4.14. Firstly, the thickness is measured at maximal deformation, when many dislocation pile-ups are built-up so that the frontiers between PSBs are easier to identify. Figure 4.14(a) shows a typical distribution of dislocations at maximal deformation for 316L (cylinder grain with D = 5µm and H = 5µm, under the applied deformation ε V M p,applied = 0.6 × 10 -3 ) taken at cycle 40. For this situation, a single PSB is formed inside the grain. PSB line profiles, measured along with five lines regularly spaced on the free surface and crossing the PSB, are plotted in Figure 4.14. Because of the non-zero deformation, the profiles are globally inclined. Note that the decrease of the profiles on the extremities comes from the numerical method used to calculate the surface displacement which is based on the calculations of the solid angle of the dislocation loops (cf Section 2.5.1). The biggest steps visible on the profiles correspond to the local shear strain induced by dislocation pile-ups in PSBs. By means of this property, as shown in Figure 4.14(b), the thickness of PSB is estimated from the length in the profile where the slope is maximum. The distribution of 4.6. Influence of the applied strain amplitude the total dislocation length along with the position of the slip plane (Figure 4.14(c)) at maximal deformation and zero deformation is considered as a validation of the thickness measurement and helps to get rid of the influence of isolated reversible dislocations.

The volume fraction of PSBs is then calculated as the ratio between the total PSB thicknesses and the grain diameter D. The relation found between the PSB volume fraction and the imposed plastic strain amplitude for Cu and 316L is illustrated in Figure 4.13. Since the matrix in our simulation contains almost no dislocations, ε p,M atrix is supposed to be zero so that the fitting function should write ε V M p,applied = ε p,P SB * f . As it can be seen in Figure 4.13, such a linear relationship matches well the evolution of the PSB thickness with strain measured from our DDD simulations. The slope of the fitting linear equation gives the value of ε p,P SB = 2.9×10 -3 for Cu. Moreover, this value is near to the experimental value 8 × 10 -3 , at least for the order of magnitude, which confirms the validity of the simulation results. The relatively smaller value is attributed to the relatively low dislocation density in the dipole arrays obtained in DDD. In experiments, the dislocation walls have a higher dislocation density. Consequently, the PSBs have a higher capacity to accommodate the plastic deformation.s

In brief, the linear relationship between the volume fraction of PSBs (f ) and the applied plastic strain is confirmed by the simulations. The formed microstructure in the PSBs has a fixed capacity to accommodate plastic deformation. The PSBs adjust their volume fractions according to the imposed plastic strain amplitude. Winter has constructed its model by analogy with the liquid-vapor system but there is one limitation: the transformation from vapor to liquid or from liquid to vapor is perfectly reversible while the constructed PSB cannot have such a high level of reversibility. Winter has confirmed this limitation by realizing experiments with a decreasing loading amplitude and observed surface slip lines of PSB. He concluded that few PSBs constructed at a high deformation amplitude survive when the amplitude is decreased but becomes inactive, which proves the irreversibility of the PSB microstructure. In order to investigate the irreversible character of PSBs formed in our DDD simulations, a two steps fatigue loading is applied on the Cu single crystal (D = H = 5µm). During the first 25 cycles, a plastic strain amplitude of ε V M p,applied = 1.2 × 10 -3 is applied and then reduced to ε V M p,applied = 0.6 × 10 -3 . The obtained stress-strain curve is plotted in Figure 4.15(a). During the second part of the loading, when the applied strain is reduced by a factor 2, the dislocation configuration has been changed by the prior construction of the PSBs during the first 25 cycles and as the applied deformation decreases, the initial microstructure contains more PSBs than necessary to accommodate the plastic deformation. As shown in This simulation setting offers us not only a possible way to verify the stability of the PSBs but also a routine to investigate the loading history effect. As shown in Figure 4.16, to reach the same final deformation state, we can follow two paths: the first one is a constant loading (ε V M p,applied = 0.6 × 10 -3 ) during 60 cycles, until the saturation state is reached; the second path starts with the same initial condition, but with a bigger deformation loading for 25 cycles to the saturation state (ε V M p,applied = 1.2 × 10 -3 ) and then the loading strain amplitude is changed to ε V M p,applied = 0.6 × 10 -3 until the new saturation state is reached (35 more cycles). Both paths imply a total of 60 cycles and comparisons can be made to check the effect of the loading history.

Initial condition

Microstructure changes induced by the strain drop

Figure 4.17 presents the changes in the dislocation microstructure when the applied strain amplitude is reduced. It can be seen that, the number of PSBs in the last cycle of the larger deformation is about six, including two smaller PSBs and four thicker PSBs. The thicknesses of the PSBs are clearly evidenced by the dislocation distribution histogram. After the drop in the strain amplitude, with the decreasing of the dislocation densities, the thickness of each PSB also decreases. On one hand, the four thicker PSBs stay at the same position; on the other hand, the two thinner PSBs have almost disappeared and only some debris remains.

Effect of the loading history

The comparison of the microstructure obtained with the two loading paths is presented in Figure 4.18. It is observed that the larger strain amplitude has dispersed the dislocation density: the dislocations in Figure 4.18(b) occupy a larger volume than the ones in Figure 4.18(a). More PSBs are constructed in the reduced loading path than in the constant loading. It also should be noted that the final dislocation density after the loading paths of loading shows no significant difference, which seems not to be influenced by the load history.

Comparison with experimental results

Winter's model has been cited in many publications but the fatigue test with a decreasing loading is rarely tried in practice. Blochwitz [START_REF] Blochwitz | A method for the determination of the cyclic stress-strain curve of presistent slip bands in fatigued single crystals[END_REF] has tested the evolution of PSBs with decreasing loading for Ni and he has observed the changes induced on the PSBs. Chapter 4. PSB multiplication process macro-bands developed at the same sites after a decrease of the strain amplitude to ε p = 1.0 × 10 -3 until another saturation regime. By comparing the 'old' and the 'new' bands, the authors concluded that the positions of macro-bands at the higher and lower amplitude correspond well to each other, however, at the lower amplitude, the intensity of the bands is smaller than at higher amplitude.

(a) (b) Our DDD simulation results show an excellent agreement with these experimental results, especially, for the four thicker PSBs which stay at the same positions, but reduces in thickness after reduction of the plastic strain amplitude. Similarly, the debris observed at the lower applied strain remains at the same location as the PSBs formed during the high plastic strain loading.

Comparison between Cu and 316L

As mentioned in Section 3.6, the fatigue behaviors of Cu and AISI 316L austenitic stainless steel are different. For larger applied strain amplitudes, the two behaviors are also different. For example, as shown in Figure 4.20 for ε V M p = 0.6 × 10 -3 , Cu has formed two PSBs while 316L has only constructed one PSB occupying less volume than Cu. A more quantitative comparison is plotted in Figure 4.21(a) for different applied strain amplitudes. Systematically, for a given strain amplitude, Cu builds up a larger PSBs and occupies more volume fraction. The analysis of the volume fraction of PSBs with respect to the plastic strain amplitude plotted in Figure 4. 21(a) shows that Cu has a higher capacity to accommodate the plastic deformation. In addition, the individual thickness of each well-formed PSB is presented in Figure 4.21(b). It can be seen that, under the same deformation, Cu is more likely to construct a thicker PSB. The distribution of the PSBs thicknesses in Cu is more dispersed than in 316L. In fact, the dislocation density is almost identical at the saturation state. However, the dislocations in Cu occupy more space. That is to say, the dislocation arrangement in Cu is less compact than in 316L.

The linear relationship between the PSB volume fraction and the applied plastic strain amplitude is demonstrated in 316L as well. The slope of the fitting equation gives the value of ε p,P SB for Cu and 316L. The capacity of PSBs to accommodate plastic deformation for 316 (ε p,P SB = 5.055 × 10 -3 ) is bigger than that of Cu (ε p,P SB = 2.945 × 10 -3 ).

The results of this section agree with the comparison made in Section 3.6. We can conclude that firstly, under the same strain amplitude, in the same simulation condition,

Chapter 4. PSB multiplication process

Cu can constructs more PSBs and occupy more volume fraction; secondly, the capacity of the PSBs to accommodate the plastic deformation ε p,P SB of 316L is bigger than that of Cu; and the arrangements of dislocations in Cu are less compact than in 316L.

Origin of the differences between Cu and 316L

To understand the observed results, the PSB formation mechanism should be investigated. In fact, as observed in the simulation and mentioned in [START_REF] Carstensen | Structural evolution and mechanisms of fatigue in polycrystalline brass[END_REF], dislocation glide on the primary slip plane is dominant and dislocation glide on the deviate plane is minor but crucial for the deformation spreading in the transverse direction and for the formation of PSBs. Finally dislocation multiplication occurs in the crystal through a double cross-slip mechanism which consists of two consecutive cross-slip occurrences: from primary plane to deviate plane, then from deviate plane to primary plane. It can be inferred that the character of cross-slip plays a critical role in the dislocation distributions.

The probabilities for cross-slip in our model for Cu and 316L are plotted in Figure 4. 22(a). Evidently, under the same shear stress τ dev , Cu has a higher probability of cross-slip. Cross-slip is triggered for a lower stress value for Cu than for 316L. All simulations for Cu and 316L starts with a single initial dislocation source. The primary microstructure to be constructed in the first cycle is one individual dislocation pile-up. The first cross-slip occurs in the pile-up for Cu when the applied plastic deformation is ε V M p = 0.25 × 10 -3 while in 316L, the first cross-slip happens at ε V M p = 0.65 × 10 -3 : it reflects also a higher possibility of cross-slip in Cu.

During the whole simulation, a good indicator of the cross-slip activity is the number of slip plane changes in the microstructure. The evolution of this parameter with the number of the cycle is plotted in Figure 4. 22(b). It is shown that the number of slip plane changes is increasing with the number of loading cycles. And for larger deformation, slip plane changes are more frequent. As expected, we observed that generally, for a given deformation, the number of slip plane changes in Cu is more important than in 316L. Even though this parameter doesn't distinguish what is cross-slip events and what is line recombinations, it proves fairly well that Cu presents the characteristic to trigger more cross-slip events than 316L.

The relatively higher cross-slip frequency of Cu has several consequences. Primarily, the mobility of dislocations is significantly enhanced. Because of this, dislocations in Cu can explore a larger volume which explains a higher possibility of Cu to form wide PSBs. Secondly, the easiness of cross-slip promotes the creation of more dislocation sources via the Köhler mechanism. More initial dislocation sources lead to more pile-ups as well as more PSBs to share the accommodation of the plastic deformation. Since the total plastic strain is fixed, the average plastic deformation accommodated by every pile-ups is smaller, which means that in Cu, fewer dislocation loops are contained in the piled-up zones. The number of pile-ups in one PSB is limited to a maximum of 2 per PSB (on the 2 sides). Thus, for Cu, the reduced number of dislocation loops in pile-ups limits the capacity of PSBs to accommodate the imposed strain so that more PSBs are required. It can be concluded that cross-slip easiness can be considered as the origin of the differences in the fatigue behaviors of Cu and 316L.

The cross-slip ability of 316L and Cu can also be deduced from the analysis of the shear stress resolved on the cross-slip system during the saturation regime (see Cu, the dislocations are well stabilized in the grain and the shear stress in the cross-slip system is very small. However, from the distribution of shear stresses in 316L, it can be seen that the shear stress on the cross-slip system is still very strong, at least bigger than the one of Cu, which means the dislocations in PSBs of 316L are likely to cross slip. Furthermore, dislocations in 316L have not moved to another place for self-stabilization.

It is because in 316L, cross-slip requires much larger shear stress, even though the shear stress near the PSBs is really high, dislocations in the PSB cannot slip on the cross-slip plane to seek for a more stable position. Since the cross-slip probability is correlated to the stacking fault energy (SFE), the slip mode in fatigue is strongly determined by the SFE as reported in literature [23,[47][48][49]. As explained in chapter 1, for high SFE materials, like Cu, the slip mode is called wavy slip, while for low SFE, like Cu-16 at% Al alloy, the slip mode is called planar slip. The dislocation microstructure of planar slip shows dislocations confined in planes marked by dipole arrays whereas wavy slip materials prefer to construct a more three-Chapter 4. PSB multiplication process dimensional microstructure where dislocations aggregate to form veins or walls. Even though veins or wall have not been simulated in this work, our simulation results for Cu and 316L corresponds well to these experimental observations: since Cu has a higher SFE (80 M J/m 2 ) than 316L (30 M J/m 2 ), Cu should construct a more 3D structure, which is what we have obtained in our simulations: Cu constructs PSBs occupying more volume fraction than 316L. The 3D character is not evidenced by our simulations which were performed in single slip loading conditions, but surely a different loading state such as a double glide configuration, would lead to a more three-dimensional geometry for Cu than for 316L.

Macroscopic response

In all the precedent sections, we have mainly discussed the dislocation behaviors. Apart from the dislocation behaviors, the macroscopic response is also an important aspect. In this section, the macroscopic response is presented and the relationship with the internal dislocation movements is discussed.

Figure 4.24 presents the macroscopic response of Cu in a cylindrical grain (D g = h = 5µm) under ε V M p,applied = 10 -3 . Firstly the stress-strain hysteresis loop is illustrated in Figure 4. 24(b). The stress in the first cycle is significantly higher than the other cycles, which is caused by the initial single dislocation source. As we have mentioned in Section 4.3.1, during the first cycle, many dislocation loops are produced and stored in a single pile-up, which leads to a strong internal stress to impede the dislocation motion. Because of the high internal stress, the applied stress should also be relatively higher to drive the dislocations. Due to the increasing deviate shear stress produced by dislocation loops in the pile-up, cross-slips occur and lead to many new dislocation sources for potential subsequent pile-ups. The first cycle is in fact an artificial cycle and the purpose is to produce more dislocation sources. In the next cycles, the evolution of the tress is much smoother and the shape of the hysteresis loop stays the same.

In Figure 4.24(a), the evolution of the applied stress with the loading cycles is presented. Evidently, during the first ten cycles, the maximal applied stress decreases with the number of cycles. This phenomenon is called cyclic softening. However, for Cu single crystals, the classic experimental studies have rather confirmed a cyclic hardening behavior before reaching the saturation regime [7]. In the following part, we will analyze the reason for the cyclical softening simulated by DDD.

Cyclic hardening and cyclic softening

In fact, the cyclic softening is caused by the increasing number of dislocation sources and the related pile-ups. At the beginning of the cycling, from cycle to cycle, more dislocation sources are generated and we have seen that the total number of reversible dislocation loops in the pile-up needed to accommodate the plastic deformation is limited and consequently, each pile-up contains fewer dislocation loops. Thus the internal stress in the pile-up to impede the dislocation movement is reduced. As a result, the required applied stress is also decreased. This explication is confirmed by a simple test simulation realized under the same strain amplitude (ε V M p = 1. × 10 -3 ) but with different numbers of initial dislocation sources (1, 3 or 10 sources). The evolutions of the applied stress with the number of cycles are presented in all these three simulations. However, for the simulation with more initial dislocation sources, the cyclic softening rate is smaller. This is because the need to increase the number of sources during the cycling is less important when more sources are already present initially so that the cyclic softening is less significant. This observation can prove the inference above. The cyclic softening is also observed when the applied strain amplitude is suddenly increased during the cycling (ε V M p = 0.6 × 10 -3 for the first 25 cycles and then the deformation changes to ε V M p = 1.2 × 10 -3 ). As shown in Figure 4.26(b), the cyclic softening is also observed in the second deformation stage. It is caused by the same mechanism. Even though at the first loading stage, dislocation sources are already largely produced, when the strain amplitude increases, the former dislocation sources or pile-ups are insufficient. More dislocation sources are therefore generated, which lead to the cyclic softening.

initial source 3 initial sources 10 initial sources

The only cyclic hardening behavior observed in our simulations was for the situation introduced in Section 4.7, where the strain amplitude was suddenly reduced after 25 123 Chapter 4. PSB multiplication process cycles (ε V M p = 1.2 × 10 -3 for the first 25 cycles and then the strain amplitude changes to ε V M p = 0.6 × 10 -3 ). As shown in Figure 4.26(a), a slight cyclic hardening can be observed during the second loading stage. Contrary to the simulation with the increased strain amplitude, here instead of insufficiency, there is a surplus of dislocation sources. We could think that this would decrease the applied stress and induces a softening which is not the case. Indeed, the possibility of dislocation source activation should be taken into account. With the decrease of the strain amplitude, the applied stress is reduced as well. Considering this effect, some dislocation sources may not be activated anymore. Indeed, the system prefers to transfer the part of plastic deformation which ought to be accommodated by these inactivated dislocation sources to the other already activated sources. The applied stress can still increase a little. This period lasts several cycles at the beginning. It should be noted that this effect is not significant, as shown in Figure 4.26(a). On the other hand, with the reduced deformation, the dislocations which depart from the crystal are more than the generated dislocations. The departed dislocations may be impeded by dislocation dipoles, dislocation tangles, and prismatic loops. These blockages can cause the cyclic hardening as well. Finally we can also point out that as the crystal is a surface grain surrounded by other grains, the closed grain boundary contributes to the formation of many pile-ups constructed from dislocation sources. The increasing of these dislocations sources or pileups can lead to cyclical softening behavior. While with the decreasing cyclical loading, the inaction of dislocation sources and the effect of impediment produced by dislocation dipoles, tangles, or prismatic loops can induce cyclic hardening.

Effect of the friction stress

At the beginning of this thesis work, the value of the friction stress was taken from the classical values retained for Cu: the coefficient of friction stress α is chosen to be 4.10. Effect of the friction stress 3 so that the friction stress τ f r = µ * 10 -5 * α = 1.26M P a. However using this value, the movements of dislocation in the fatigue simulations are completely reversible. As a result, dislocations cannot be stored in the crystal with the cycling. After several tests, it is found that a larger friction stress could contribute to the storage of dislocations. That is also the reason why the α f r in the simulation taken as 50 as shown in Figure 2.2. In this section, we have tested different values of α f r to identify its effect on the macroscopic response and the microstructure evolution.

Effect on the macroscopic response

We have realized the same simulation of a Cu cylindrical grain (D g = h = 5µm) under ε V M p,applied = 10 -3 but with different values of α f r . The macroscopic strain-stress curve for different values of α f r is presented in Figure 4. 27(a). For values between 35 and 60, the shape of the curve does not show significant variations. While, for the value of α f r larger than 60, a larger stress is required to get the same amount of deformation which comes from Eq.(2.5). If we compare the maximal stress value with experiments [7], (as shown in

Effect on the dislocation distribution

The value of the friction stress can be regarded as a kind of threshold which controls or limits the movements of dislocations. As a result, it should have an effect on the dislocation distribution. In Figure 4.28, the dislocation distributions obtained for different α f r at the 5 th cycle are presented. When α f r is between 35 and 60, the distributions are similar : dislocations have occupied almost the same volume of the crystal and the distribution is quite homogeneous. The of α f r is more effective starting from α f r = 80. The distribution becomes more centralized and the dislocations tend to stay closer. The phenomenon is easy to be observed at α f r = 200 where a condensed slip band appears. The phenomenon is caused by the threshold effect of the friction stress on the cross-slip. A larger friction stress has limited the moving distance along with the cross-slip direction. The friction stress also affects the microstructure developed on the primary slip system. Figure 4.29(a) shows the dislocation microstructure in the PSB viewed along with the normal to the primary slip system at zero plastic strain. For the value of α = 200, the arrangement of dislocations is more dense and compact and most of the dislocations are decomposed in small segments. Figure 4. 29(b) shows the evolution of the number of dislocation segments with the loading cycle for different values of α f r . It can be observed that with a larger α f r more dislocation segments are produced. As a result, simulation time can be significantly increased by a larger friction stress. In the following paragraph, we shall explain why more dislocations are stored in a point of view of dipole formation.

Effect on the dipole formation

In the discussion above, we have seen that a larger value of α f r contributes to the storage of dislocations. As we have seen in Section 3.3.4, most of the dislocations are stored in the form of edge dipoles formed by the internal stress produced by these two opposed-sign dislocations. [START_REF] Kroupa | Dislocation dipoles and dislocation loops[END_REF], he has analyzed in detail how an external stress field affects the dipole stability in 2D situation. For two edge dislocations of opposed signs with an infinite length in direction Z, the force on the plane of XOY produced by the dislocation at the origin O applied at another dislocation can be expressed as follows:

F = - Gb 2 2π(1 -ν) x(x 2 -y 2 ) (x 2 + y 2 ) 2 x - Gb 2 2π(1 -ν) y(3x 2 + y 2 ) (x 2 + y 2 ) 2 y (4.3)
where G and ν are shear modulus and Poisson ratio respectively. We consider the situation (Figure 4.30(a)) where the normal vector to the slip plane along with the ydirection, the distance between the two slip planes of these two dislocations is h, and (b) (a) edge dislocations can only move on the slip plane (x-direction). Hence, it is the shear component F x of the force F that determines the dislocation movement, shown as follow:

F x = - Gb 2 2π(1 -ν) x(x 2 -h 2 ) (x 2 + h 2 ) 2 (4.4)
The evolution of F x with the distance in the direction of x is plotted in Figure 4.30(b). For x > 0, when x > h, the force is negative, which signifies the attraction between the two dislocations; at x = h the force is zero and then turns to be positive for x < h, which means the repulsion. As a result, the dislocation should be stable at the point x = h as well at the point x = -h. When x = 0, the force is also zero but it is a metastable state. From this analysis, it can be seen that the case without external stress, where there are two stable points x = ±h is the principal mechanism for the dipole formation. Now if we consider that there is an fixed external stress field F x,external in the direction of x-axis applied homogeneously on the dislocations, the stable point should be different. As shown in Figure 4.30(c), for x > 0, when the internal force produced by the dislocation F x is smaller than F x,external , it is the attraction and it turns to the repulsion for F x < F x,external . Hence, there is a shift of the stable point as shown in the X stable of Figure 4.30(c).

To simulate the effect of the friction stress, we can consider a homogeneous stress field with a varying magnitude within -F x,external ∼ F x,external . That is too say, when the internal stress value F x is among this interval, the total stress is zero or it is a stable point. As shown in Figure 4. 30(d), by this assumption, the stable position is not only a point but all the interval highlighted in blue can be a potential stable position, which significantly increases the dipole formation possibility. An easier formation of dipoles contributes to the dislocation storage.

It should be noted that, if there is no external stress field, the potential stable position is only some points. The DDD simulation is explicit. As a result, it is rare or even hard for dislocations to arrive exactly at the stable position. It means that there is no storage of dislocations and hence, no introduction of irreversibility and no build-up of microstructure. That is also the reason why a small friction stress can lead to a totally reversible deformation. Another important points is that, in the DDD simulation, the stress condition is more complex than this simplified configuration of 2 infinite dislocations, but the analysis gives us a general idea about the mechanism behind the friction stress effect and demonstrates that a larger friction stress increases the irreversibility.

(a)

X stable =h X stable =-h X stable X stable - (b) (c) (d) h x y O b

Conclusions

In this chapter, we have conducted simulations of a Cu cylindrical grain (D g = h = 5µm) under different strain amplitudes. Different from the case of a single PSB formation studied in chapter 3, several PSBs are constructed by strain localization. A saturation state is reached and PSBs are formed, during which the PSB volume fractions, PSB positions and the height of extrusions and intrusions at the surface are stable. The saturation regime corresponds to the experimental results in Cu. The linear relationship proposed by Winter between the applied deformation amplitude and the PSB volume fractions is redemonstrated in the simulations. By reducing the applied strain amplitude, the reversibility of formed numerical PSBs is verified. It is stated that the position 4.11. Conclusions of formed PSBs stays unchangeable, which corresponds well to the model of Winter. Compared with similar simulations realized on AISI 316L austenitic stainless steel, Cu constructs more PSBs and has a higher tendency to build up thicker PSBs, while the capacity of PSB in Cu to accommodate plastic deformation is found to be weaker than in AISI 316L. Then, the differences of cyclic behaviors between Cu and 316L are thought to be induced by the cross-slip capacity difference. The easiness of cross-slip in Cu leads to a higher mobility of dislocations. Higher mobility in Cu facilitates the PSBs construction and the augmentation of the PSB thickness, on the other hand, it reduces the number of dislocation loops in the pile-ups, which is a signature of the plastic accommodation capacity. We have also explained the cyclical hardening and softening behavior observed in the simulation as well as the related microstructure mechanism. In the last section, the effect of the friction stress on the mechanical behavior and on the microstructure formation are also discussed.

What should be retained from this chapter:

-The saturation of dislocation density, PSB numbers, PSB thickness and surface roughness is in good agreement with the experimental observations. The linear relationship between the applied strain amplitude and the PSB volume fractions are redemonstrated. These results prove the rationality of DDD model in fatigue modeling;

-The stability of the formed numerical PSB is verified by a simulation with a reduced loading and corresponds well to the experimental results. However, when the plastic strain amplitude is too small, the movement of dislocations is fully reversible, which is not the case in experiments. The DDD model can still be optimized to introduce irreversibilities, especially for cyclic loading situation;

-The comparison between Cu and 316L shows also good agreement to the criterion of SFE to distinguish wavy slip mode and planar slip mode; -Cyclic softening is often observed in the these simulations and the cyclic hardening is difficult to obtain. This is caused by the pile-ups constructed at the closed grain boundary. A better description of the grain boundary should be introduced in the simulation for further studies.

Chapter 5

Fatigue behavior in double slip loading condition

Introduction

All the simulations that have been presented before concern the single glide situation, which is the most classic one studied in fatigue. Then a question arises: what will happen if two slip systems are activated simultaneously ? In this chapter, the simulation results of the double slip situation are presented. Simulation results are compared based on the interaction characteristics of the two activated slip systems.

Interactions between dislocations

The mechanical behavior of metals is highly affected by the dislocations and the encountered obstacles. One of the important obstacles is the forest interactions which account for junctions or locks constructed after the interaction between two dislocations. As presented in chapter 2, there are 12 slip systems in FCC metals. Due to the high symmetry in the crystal lattice structure, there are mainly five types of interaction:

-Coplanar junction: the interaction between coplanar dislocation systems (same normal to the slip systems). This kind of interaction occurs for the dislocations sharing the same slip planes. The Thompson tetrahedra plotted in Figure 2.4 shows that each {1 1 1} plane contains three 1 1 0 Burgers vectors;

-Hirth lock: this interaction takes place for two perpendicular slip systems (when the Burgers vectors are orthogonal). As evidenced in the Thompson tetrahedra, each Burgers vector admits a perpendicular relationship with a unique different Burgers vector. Thus, each slip system can form a Hirth lock with two other slip systems (the two systems which share the identified perpendicular Burgers vector);

-Lomer-cottrell lock: it is constructed when the final resulted Burgers vector of the two encountered dislocations contained in one of the two incoming planes. The junction line can then move in the plane containing the Burgers vector. (shown in Figure 5.1(a)). In FCC, a given slip system can form a glissile junction with 4 other slip systems;

-Glissile junction: it occurs when the final resulted Burgers vector of the two encountered dislocations is the line direction of the intersection between the two slip planes, and the junction is glissile (shown in Figure 5.1(b));

-Collinear junction: this interaction happens when the two encountered dislocations share the same Burgers vector but the slip planes are different. That is the case for a dislocation system and its corresponding cross-slip system (as for single slip in chapter 3 and chapter 4, the illustration is shown in Figure 5.1(c)). The junctions and locks affect the macroscopic response. After the formation of one junction or one lock, to compensate the plastic strain augmentation, the blocked dislocations need to be reactivated, which demands a stronger applied stress. On the other hand, the increases of the dislocation densities, favor the possibility of junction formation, which contributes to a larger applied stress. This dislocation strengthening mechanism has been well demonstrated in the Taylor-like relation:

τ i c = µb( j a ij ρ j ) 1 2 (5.1)
where µ is the isotropic shear modulus, ρ j is the dislocation density of slip system j and a ij are the so-called interaction matrix coefficients, which are dimensionless. The interaction matrix a ij signifies the strengthening effect of the interaction between the slip systems i and j. The detailed interaction matrix is shown in Table 5.1:

In this table, a 1∼6 refer to the five possible slip systems interactions listed above, noted as self (a 1 ), coplanar (a 2 ), Hirth lock (a 3 ), Glissile junction (a 4 ), lomer-cottrell lock (a 5 ) and collinear (a 6 ) respectively. This matrix provides the information on the 5.3. Double slip simulation settings type of interaction between two slip systems, which can facilitate the choice of two slip systems for the following simulation settings.

Double slip simulation settings

As we have discussed above, the interaction between slip systems is critical for the macroscopic response and there exist 6 types of slip system interactions. In the precedent chapters, the single slip situation was considered to consist of a pure shear on a given slip system. The only possible interactions were between the primary slip system and its cross-slip system. It should be pointed out that the number of cross-slip dislocations is quite small since the applied shear stress on the deviate slip system was just a third of that on the primary. Hence, the interaction between the two dislocation families is relatively weak. In the case of double slip situation, we want to apply the same shear stress on two selected slip systems.

The selection of the two slip systems can be realized from the interaction matrix. In order to make the activation possibility equal, the shear stress or the Schmid factor should be the same. Table 5.2 recalls the Schmid factors induced on the different slip systems by the stress tensor presented in Eq.(4.1). Schmid Factor 0.50 0.17 0.50 0.17 0.67 0.67 0.33 1.00 0.17 0.50 0.17 0.50

From this table, we can see that the slip systems 1, 3, 10 and 12 have the Schmid Factor: 0.5. By combining two of these slip systems, we see that, we can form 4 out of the 6 possible interactions: the interaction between S1 and S3 is Lomer-Cottrell lock; the interaction between S1 and S10 is the Glissile junction; the interaction between S1 and S12 is the Hirth lock; the interaction between S10 and S12 is the coplanar interaction. To control the number of slip systems during the cyclic loading, only two dislocation sources of the combined slip system are introduced inside the crystal as initial conditions so that the generation of dislocations from other slip systems are excluded.

Another important setting is the crystal orientation because the free surface orientation is crucial for the formation of the surface roughness. In the single slip shown in Figure 5.2(a), the free surface has the same orientation as the Burgers vector b 1 of the primary slip system. Consequently, a step can be well constructed when a dislocation loop passes through the surface. As for the double slip situations, in order to construct a similar geometric relation, the crystal orientation is chosen as the sum of the two slip systems ( b crystal = b 1 + b 2 , as presented in Figure 5.2(b)) so that displacement steps can be constructed for the two systems. The crystal orientations by different double slip combinations are summarized in Table 5.4. In order to study the collinear interaction the following stress field increment is used ∆σ σ σ app =        0.583 -0.417 -0.333 -0.417 0.583 -0.333 -0.333 -0.333 1.333

       (5.2)
and the corresponding Schmid factors are presented in Table 5.3. Slips systems S7 and S8 with Burgers vector [1 1 0] have the same Schmid factors and will be selected for the study of the colinear interaction. Note that these 2 slip systems were already investigated in chapter 4 so that we can use the same geometry for the surface orientation as recalled in Table 5.4.

Dislocation evolution inside the crystal

For all the simulations, the shape of the crystal is the same as in chapter 4, a cylindrical grain with h = 5µm. Since the slip planes are not perpendicular to the free surface, to facilitate the departure of dislocation loops into the virtual space, the diameter of the grain is chosen to be larger D g = 7µm. An example of the crystal shape and slip plane geometry is shown in Figure 5.3. In addition, to avoid numerical divergence, a small plastic strain amplitude is applied: ε V M p,applied = 0.6 × 10 -3 .

Slip planes

Two dislocation sources Table 5.4 Summary for system combinations of double slips.

System combination S1 S3 S1 S10 S1 S12 S10 S12 S7 S8 Interaction Lomer Glissile Hirth Coplanaire Collinear Crystal orientation [1 1 0] [ 1 1 0] [0 0 1] [ 1 1 3] [2 2 1]
Now, the five possible combinations of double slip systems are identified (only the self interaction is not regarded since it would be meaningless in fatigue simulations) and the orientation is set. At this stage, it should be noted that in the edge-screw DDD model used here, the formation and expansion of glissile junctions cannot be taken into account in a consistent manner but simulations will nevertheless be conducted to check the effect of the internal stress interactions.

Dislocation evolution inside the crystal

Dislocation density evolution

The evolutions of the dislocation densities for the five double slip combinations are shown in Figure 5.4 respectively: (a) for the lomer-cottrel lock double slip (S1 and S3);

(b) for the Glissile junction double slip (S1 and S10); (c) for the Hirth lock (S1 and S12); (d) for the coplanaire double slip (S10 and S12); (e) for the collinear junction double slip (S7 and S8). To differentiate the slip system number, in Figure 5.4 the same slip system is noted with the same color in the five subfigures. In addition, to demonstrate the relation between the primary slip system and its related cross-slip system, the dislocation density of the cross-slip slip system is systematically plotted in the same but lighter color as the primary slip system (except for the collinear interaction, S7 and S8 which are noted in red and black respectively).

A dozen of cycles has been reached for the five double slip configurations, which is less than the single slip simulations. The main reason lies in the higher possibility of numerical divergences introduced by the stronger interactions between the different slip systems.

Generally, in all the five simulations, the dislocation densities of the activated slip systems augment with the loading cycles. For each configuration, the evolution of the dislocation densities of the two main interaction slip systems follows the same trend. While for the related cross-slip system, the dislocation density shows the same tendency of evolution as its related primary slip system.

For the first four double slip simulations (except for the collinear double slip), the dislocation density of the cross-slip system is nearly a third of its primary dislocation density. This ratio is the same as in Section 3.3.2, which indicates that the double slip behaves like two single slips. This is also evidenced in the Schmid factors listed in Table 5.3: the ratio of the shear stresses on the primary and deviate slip systems are systematically equal to 0.5/0.17 which is close to the 1/3 ratio of the single slip configuration. As for the saturation behavior, which has been commonly seen in single slip simulations, the perfect dislocation density saturation can be observed in the collinear interaction case (Figure 5.4(e)). The saturation can also be observed in other situations but is less obvious.

For the Lomer-Cottrell double slip configuration (Figure 5.4(a)), the dislocation density of S3 saturates after about five cycles as well as its cross-slip system (S4), while the S1 dislocation density shows a short saturation plateau from 10 th cycle to 12 th cycle but continues to increase afterward. In Figure 5.4(b), a temporary saturation regime is visible for the two systems from the 3 rd cycle to the 5 th cycle, then the dislocation densities increase again. In the Hirth configuration (Figure 5.4(c)), the saturation phenomenon occurs in S12 and its related cross-slip system, while the saturation behavior has not been observed for S1. The saturation behavior has not been traced in the coplanar configuration (Figure 5.4(d)), even though the dislocation density increases very slowly. Compared with the other four interactions, the coplanar configuration is special because the generated dislocations of the two slip systems stay in the same slip planes, which leads to a high possibility of impediment. In the next section, we will focus on the dislocation microstructure corresponding to these different configurations.

Dislocation distribution inside the crystal

The dislocation distributions for the five double slip combination are presented in Figure 5.5. Dislocations of the same Burgers vector have the same color. The blue signifies the dislocations from a smaller slip system number while the red refers to the dislocations belonging to the slip system with a bigger system number. For example, ( in Figure 5.5(a), the blue lines refer to the dislocations of the S1 slip system (or the cross-slip system S2) and the red lines are the dislocations of the S3 slip system (or the cross-slip system S4). Based on Figure 5.5(a,b,c), the distribution of the dislocations can be considered as the sum of two single slip distributions. As an example, in Figure 5.5(a), both slip systems S1 and S3 have formed 3 well separated PSBs. The heterogeneous PSB formation is the typical feature of a single slip. For Figure 5.5(b,c), we can observe the formation of individual PSBs for each slip system and the geometric relation is different. But it should be noted that, the dislocation interactions mainly take place at the overlapping zone of PSBs. The coplanar case is the limit case where the overlapping area could be either zero or the entire PSB. When the slip bands of the two systems meet each other, the dislocations are tangled tightly in the encountered band (the PSB in the middle of Figure 5.5(d)). For the collinear interaction configuration, it can be considered as an extension of the single slip loading mode. The main difference is that the cross-slip effect has been largely enhanced so that the slip bands are cut more frequently. As a result, the distribution of the dislocations is more homogeneous than in the single slip mode and a cell structure, consisting of the surrounding primary dislocations and cross-slip dislocations, is built-up.

In this section, from the dislocation density evolution and the dislocation distribution, it can be inferred that for the Lomer-Cottrell, glissile and Hirth configurations, the dislocation movement can be regarded as the combination of two single slips and the dislocation interaction is confined at the intersection zone of the PSBs. The single slip loading mode is an elementary slip mode. The collinear configuration can be considered as the extension of a single slip. As for coplanar case, the special point is that two slip systems share the same slip plane, which means the intersection zone could be as large as the entire slip band. Chapter 5. Fatigue behavior in double slip loading condition

Surface configuration

The surface roughness for the five double slip configurations is calculated with the same method as for the single slip situation. Figure 5.8 presents the evolution of the maximal extrusion and intrusion height with the loading cycles, as well as the addition of these two values. For all the five configurations, the maximal height of extrusions and intrusions shows a general tendency to increase. The evolution of extrusions and intrusions height in Lomer-Cottrell and collinear configurations shows a saturation plateau. The maximal height of extrusion is generally higher or equal to the maximal intrusion height. The predominance of extrusions has also been pointed out in chapter 3 and chapter 4. Moreover, compared with the other four double slip configurations, the maximal heights of the extrusion and intrusion in the junction configuration is much smaller, which may be related to the technical difficulty to construct a glissile junction in our DDD model. The order of magnitude of the maximal height is also important. The total value of extrusion and intrusion height in these double slip simulations (except for the glissile junction) is about 8 Burgers vector. This value is of the same order of magnitude as in single slip situation, which means that the dislocation interactions have little influence on the maximum height of extrusions or intrusions.

From these observations, it can be confirmed that, the growth of extrusions is still quite limited even though dislocation interactions have been taken into account.

The maximum height of extrusions and intrusions can not fully describe the surface morphology. The surface relief and the profile line at the sites of the slip bands for the five configurations at the final cycle are presented in Figure 5.6 and Figure 5.7. In Figure 5.6(a), for the Lomer-Cottrell case, extrusions and intrusions are well constructed on the surface of the slip bands and most of the extrusions demonstrate the characteristics of "tongue-like" extrusions. Let recall that the "tongue-like" extrusions are induced by the small dislocation loops like prismatic loops. Thus, the formation of these "tonguelike" extrusions should be highly related to small dislocation loops blocked by the Lomer-Cottrell junctions, especially inside the interaction zone. For Figure 5.6(b), the extrusions and intrusions are also constructed but their structure is straighter due to the failure in the reaction of the glissile junctions. In Figure 5.6(c), as expected, we can see two perpendicular slip bands, which is the geometric relation for the two slip systems of the Hirth lock. As only two individual bands are formed, the intersection zone is insignificant and the surface morphology is similar to the single slip situation. The case of the coplanar interaction is special (Figure 5.7(a)), as the dislocations share the same slip plane, as expected, only one slip band is constructed. The slip band is relatively larger than the "tongue-like" extrusions and the decreasing or increasing slope of the extrusions is smother than for the other situations. This phenomenon should be contributed by the compact dislocation arrangement in the single slip band. As for the collinear double slip (Figure 5.7(b)), the surface morphology and profile also resemble the surface in single slip situation. However, because of the enhanced cross-slip events, the long extrusions and intrusions (or "ribbon-like" extrusions and intrusions) are subdivided into small "tongue-like" extrusions and intrusions.

In this section, we have compared the surface relief for the five double slip configurations. The main result that should be noted is the tendency to construct more "tongue-like" extrusions for the Lomer-Cottrell configuration. These results motivate a literature review of similar experimental investigations as proposed in the next section. Chapter 5. Fatigue behavior in double slip loading condition

Comparison with experimental results

Compared with the extensively studied single slip situation, the experimental research on double slips is rather limited. The first paper on the double slip fatigue behavior was published by Jin [START_REF] Jin | Cyclic deformation of copper single crystals oriented for double slip[END_REF] in 1984 who has investigated Cu single crystals subjected to uniaxial cyclic loading in three orientations: [ 1 1 2], [0 1 2] and [ 1 2 2]. Bo Gong [START_REF] Gong | Cyclic deformation response and dislocation structure of Cu single crystals oriented for double slip[END_REF] and Li [START_REF] Li | Dislocation structures in fatigued critical and conjugate double-slip-oriented copper single crystals[END_REF] have also studied the double slip for Cu single crystal in other double slip orientations. Later on, J. Nellessen [59] In these studies, the double-slip configurations have often been characterized into three cases: the conjugate double slip, the critical double slip and the coplanar slip (the stereographic triangle is shown in Figure 5.9). These three combinations correspond to the Lomer-Cottrell, Hirth and coplanar configurations studied in our DDD simulations respectively. In the following part, we shall compare our simulation results with these experimental studies. two slip systems is marked in the red box. This geometric relation corresponds to the PSB geometric relation in Figure 5.5(a). Hence, the simulation results match well with the experimental observations. In [START_REF] Li | Dislocation structures in fatigued critical and conjugate double-slip-oriented copper single crystals[END_REF], Li has stated that due to the high symmetry, the two slip systems tend to glide and multiply in separated zones and in each zone, a single-slip-type microstructure (PSB, ladders) is constructed and intersected with the PSBs of the other system to build-up the Lomer-Cottrell locks. Definitely, the simulation results can be considered as a numerical validation for this proposition.

Hirth (critical double slip) configuration

Since the two Burgers vectors are perpendicular to each other, the constructed jogs are not strong [START_REF] Nellessen | Effects of strain amplitude, cycle number and orientation on low cycle fatigue microstructures in austenitic stainless steel studied by electron channelling contrast imaging[END_REF]. As a result, the slip bands of each slip system occupy the region independently as the double slip for Lomer-cottrel lock [START_REF] Jin | Cyclic deformation of copper single crystals oriented for double slip[END_REF]. This is good agreement with the Figure 5.5(c) where two slip bands were simulated by DDD.

Coplanar double slip configuration

The main character of coplanar double slip is that the slip band planes are the same for the two slip systems. As a result, the slip band lies along one direction on the surface. Figure 5.11(a) shows the backscattered electron (BSE) image for an Aluminum sample in coplanar double slip mode [START_REF] Nellessen | Low cycle fatigue in aluminum single and bi-crystals: On the influence of crystal orientation[END_REF]. Three slip bands appear on the surface oriented in the same direction. The slip band in the middle shows a very good morphology similarity with the simulation results shown in Figure 5.7(b). Another point that has been argued is the dislocation patterns in the slip band. Even though the dislocations are distributed in one slip plane, compared with single slip mode, two slip systems have been activated. In the publication of Jin [START_REF] Jin | Cyclic deformation of copper single crystals oriented for double slip[END_REF], a uniform cell structure has been observed in Figure 5.7(b). The wall of the cell structure is mainly composed of dislocations from [ 1 0 1] (noted by A and C in Figure 5.7(b)) and dislocations from [ 1 1 0] (noted B and D in Figure 5.7(b)). The wall structure is similar to PSB ladders. They are all compact dislocation dipole aggregates. Figure 5.7(c) presents the dislocation pattern obtained in the simulation for coplanar double slip. The red and blues dislocations refer to the two coplanar dislocation systems. For each slip system, the ladder structure has been constructed separately (in the red and blue box). However, considering the cell structure, even though, the intersection of the ladders from the two slip system can be seen, the cell structure is not evident, maybe, more simulation cycles are needed. In addition, in Figure 5.7(c), the ladders from the two slip systems are more dense than the dislocation dipole arrays formed in single slip mode. This phenomenon may infer that the occurrence of coplanar dislocations or dislocations from other slip systems can help the aggregation of dislocation dipoles. The collinear interaction double slip and glissile double slip are difficult to be designed in experiments and thus have been rarely studied. But in the thermal fatigue test developed at CEA (CYTHIA) [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF], the thermal fatigue loading is found to be bi-axial on the surface of the pipe inducing a double slip on colinear slip systems. The TEM observation (shown in Figure 5.12(a)) revealed that the generated dislocations were mainly from one slip system and its cross-slip system. The TEM image shows good From the comparisons above, our simulation results show some agreement with the experimental observations. In the next section, we will discuss the influence of dislocation interactions on the macroscopic responses.

Back stress and friction stress

The macroscopic response is a signature of the microstructure evolution inside the crystal. Especially in experimental studies, as the observation of the microstructure evolution is limited, the macroscopic response is considered as a possible path to explore the microstructure characteristics.

Regarding the stress-strain curve, one can distinguish 2 important stress terms: the back stress and the friction stress. The back stress τ b refers to the yield surface translation in the stress space, corresponding to the kinetic hardening [START_REF] Feaugas | On the origin of the tensile flow stress in the stainless steel AISI 316L at 300 K: back stress and effective stress[END_REF]. While the friction stress τ f means the expansion or contraction of the elastic domain, corresponding to the isotropic hardening [START_REF] Beranger | Low cycle fatigue behavior of an α+ β titanium alloy: Ti6246[END_REF]. These two stress components are highly related to the internal microstructures. The long range stress is considered as the main reason for the back stress, which may be caused by dislocation pile-ups, the immobilized dislocation at surface and the dislocation multipoles [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue II. Friction stress and back stress as inferred from an analysis of hysteresis loops[END_REF]. The friction stress is considered to be caused by the different barriers during the motion of the dislocations [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue II. Friction stress and back stress as inferred from an analysis of hysteresis loops[END_REF].

Calculation of friction stress and back stress by Cottrell method is suggested by Kuhlmann-Wilsdorf and Laird [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue II. Friction stress and back stress as inferred from an analysis of hysteresis loops[END_REF]. The following equations are proposed:

τ f = τ E + τ S 2 (5.3)
τ b = τ E -τ S 2 (5.4)
where τ E and τ S refer to the maximal applied stress and yield stress respectively (as shown in Figure 5.13). The friction stress is independent of the loading direction. It is opposed to the direction of the dislocation motions. On the other hand, the sign of the back stress changes during each half cycle. The back stress reaches its maximal value at the maximal deformation. At the maximal deformation, the applied stress needs to overcome the impediment of both friction stress and back stress:

τ applied = τ f + τ b (5.5)
where the applied stress can be replaced by the maximal stress τ E :

τ E = τ f + τ b (5.6)
Afterward, as the applied stress decreases, the dislocation motion is driven by the back stress. At the yield point, where the dislocations begin to move, we have:

τ b = τ f + τ applied (5.7)
where the applied stress τ applied can be replaced by the yield stress τ S :

τ b = τ f + τ S (5.8)
By combining the Eq.(5.6) and Eq.(5.8), the relations in Eq.( 5.3) and Eq.( 5.4) can be obtained. The relation between the different stress terms in the Cottrell method is summarized in Figure 5.13. With the application of the Cottrell method, we have measured the maximal stress and yield stress for the five double slip configurations and for the single slip situation. In order to eliminate the influence of the Schmid factor, the shear stresses resolved on the primary system are used for the computation of τ E , τ S , τ f and τ b and results are plotted in Figure 5.14. The yield stress τ S is measured when the resolved shear strain is 4 × 10 -5 smaller than the maximal strain. For maximal stress tau E (Figure 5.14(a)), the 5.7. Back stress and friction stress coplanar double slip configuration shows the largest value than the other double slips. The collinear and Lomer double slips present almost the same value for the maximal stress. The same phenomenon can be seen for the yield stress (Figure 5.14(b)) and for the friction stress (Figure 5.14(c)). For the back stress (Figure 5.14(d)), the coplanar doubleslip configuration leads to a larger value at the beginning than the other configurations and then decreases to almost the same value. According to these four figures, it can be seen that, the behavior of the coplanar double slip is the most distinct one. This may be caused from the strong impediment of coplanar interactions in the overlapping planes.. The other three double slip configurations can all be considered as the combination of two single slips and dislocation interactions at the intersection zone. As a result, they all present similar stress evolution tendencies, which are close to the single slip situation.

The back stress is mainly related to the long range interaction. In the simulation, the long range stress is induced by the dislocation pile-ups at the grain boundary. The decreases of back stress can be explained by the generation of more dislocation sources. More dislocation sources can lead to more dislocation pile-ups with fewer number of dislocation loops which will decrease the intensity of the long-range back stress. A possible explanation for the higher back stress measured in the coplanar configuration comes from the difficulty to generate efficient dislocation sources in the slip bands. That is to say, because of the strong interactions existing in the shared slip planes, the dislocation sources in the coplanar double slip configuration are more difficult to be generated. As a result, more dislocation loops are stored in the pile-ups leading to a high intensity of long range stress. The difficulty for double slip configurations to generate new dislocation sources can be ranked as coplanar, Hirth, glissile, collinear and Lomer-Cottrell configurations.

The friction stress indicates the barrier effect on the dislocation motion. The coplanar double slip configuration shows a stronger barrier effect in the dislocation motion. The Hirth lock configuration shows the second stronger effect. Then, we have the glissile double slip, Lomer-Cottrell double slip and collinear double slip. The reason why the coplanar double slip has the strongest barrier effect should again be related to the number of interacting dislocations. Since the dislocations from two slip systems move on the same plane, they have a much higher possibility to meet each and to get pinned. For the other three double slips configurations, the glissile junction cannot be well simulated so that the higher value may be an artifact of the numerical problem.

In this section, we have analyzed the macroscopic response for the four double slip configurations and the related possible microstructure mechanism. Compared with experimental results [START_REF] Kuhlmann-Wilsdorf | Dislocation behavior in fatigue II. Friction stress and back stress as inferred from an analysis of hysteresis loops[END_REF][START_REF] Yang | Back stress strengthening and strain hardening in gradient structure[END_REF][START_REF] Xu | A study of the back stress and the friction stress behaviors of Ti-6Al-4V alloy during low cycle fatigue at room temperature[END_REF], the simulation results have not presented obvious hardening or softening evolution, except for the coplanar double slip situation. There are two possible reasons: the first one is the insufficient number of interacting dislocations, more fixed dislocation impediment should be introduced to create more interactions; another one is that the modeling of DDD may have accelerated the fatigue loading process and the description of dislocation should be more precise.

Influence on the shape of hysteresis loop

The dependence of the normalized hysteresis loop shape on the applied strain amplitude has been proposed by Mughrabi [8]. As shown in Figure 5.15(a), at low strain amplitude, the loop shape is acute and slim and with the increase of strain (a) ( 5.7. Back stress and friction stress amplitude, the shape becomes more rounded or rectangular. We have tried to calculate the normalized hysteresis loop evolution under different strain amplitudes in the single slip situation. However, the normalized shape hysteresis remains the same for different strain amplitudes, which signifies that for the single slip mode, the mechanisms of dislocation motion stays the same for different strain amplitudes. This conclusion is reasonable because we have seen that the main effect of a larger strain amplitude is the increases of the PSB volume fraction but the PSB structure and the dislocation motion mode stays the same.

Afterward, the simulations of the double slip configurations provide an opportunity to test the effect of dislocation interactions on the hysteresis loop shape. The normalized hysteresis loops corresponding to the five double slips and single slip are presented in Figure 5. 15(b). With the dislocation interaction effect, the shape of the loop has changed. Due to the weak interaction strength, the Hirth configuration has the similar normalized loop as the single slip situation. The collinear double slip configuration shows the strongest effect to make the loop shape to be more rectangular. The dislocation interactions of the different configurations tend to flatten the hysteresis loop, and this effect, from less to more pronounced, can be ranked as collinear, Lomer-Cottrell, glissile, coplanar and then Hirth. In the single-slip simulation, the activation of other slip systems has not been considered.

In experiments at low plastic strain amplitude, only the primary dislocations accommodate the plastic deformation. As the simulations have shown that, it can be considered as an evidence to prove that at higher strain amplitude in experiments, dislocations from other slip systems have also been activated to affect the plastic deformation process so that the shape of the normalized hysteresis loop can be modified. Chapter 5. Fatigue behavior in double slip loading condition

Conclusions

In this chapter, we have applied DDD to study the double slip behaviors. Inspired by the different possible slip system interactions in FCC, we have selected 5 different configurations. The dislocation density evolution, the dislocation distribution and the surface roughness at the free surface have been discussed and compared. Then, the simulation results have been compared to experimental observations. In order to highlight the similarities and differences, we then have focused on the macroscopic behavior. The friction stress and back stress are calculated using Cotrell method to illustrate the relation between the macroscopic behavior and the dislocation microstructures. The effect of the type of dislocation interaction on the shape of the normalized hysteresis loop was also analyzed.

What should be retained from this chapter:

-Except for the coplanar case, all the double-slip configurations can be considered as a combination of two single slips with a zone of interaction. The dislocations from the two slip systems occupy the individual region to construct individual PSBs and these results are also evidenced in experiments;

-The coplanar double slip shows a significantly different behavior compared with other four configurations, which may be induced by the strong coplanar interactions taking place in the shared planes;

-The surface relief for the Lomer-Cottrell configuration shows a strong tendency to construct more "tong-like" extrusions;

-The type of dislocation interactions has an effect on the flattening of the normalized stress-strain hysteresis loop.

Chapter 6

Simulations of poly-crystals

Introduction

The simulated domain of the DDD simulations presented in the previous Chapters is now extended to treat poly-crystals. The changes made in the code to account for polycrystals are first presented. To demonstrate the capacity of the code, a fatigue simulation on a polycrystalline aggregate of seven grains is then presented. Finally, we will present the simulations realized in a bi-crystal.

Poly-crystal version of TRIDIS

Compared with the single crystal case, DDD simulations of poly-crystals are more complicated. The complexity mainly lies in two aspects: the crystal orientations and the grain boundaries. As an integration of single crystals, each grain of a polycrystalline aggregate is oriented differently. As a result, for a given loading direction, the slip system activation varies from grain to grain. The grain boundary also plays an important role for the mechanical behavior. It is not only because the grain boundary is a strong barrier for the dislocation motions but also various phenomenon occurs at the grain boundary, such as the grain boundary gliding and the absorption, emission and transmission of dislocations. These mechanisms all affect the mechanical behaviors. The poly-crystal version of TRIDIS is a pioneering DDD code developed by Hyung-Jun Chang to treat simulations of poly-crystals. It has been used in the Ph.D. of H. Tummala to treat the case of thin films of Palladium. The "edge-screw" model presented in chapter 2 is retained in each grain. However, in each grain, the line direction and moving direction are changed by the lattice rotation. As a consequence, the dislocation coordinates of each segment (Start and End) are no more defined by integers in the large FCC lattice (see Section 2.2.2), but are now floating numbers expressed in the laboratory reference frame.

The lattice rotation in the simulation is characterized by the three Euler angles (ψ, θ, φ). As defined in Figure 6

R θ =        1 0 0 0 cos(θ) -sin(θ) 0 sin(θ) cos(θ)        (6.2) R Φ =        cos(Φ) -sin(Φ) 0 sin(Φ) cos(Φ) 0 0 0 1        (6.3)
where R ψ , R θ , R Φ refer to the rotation matrices of ψ, θ, Φ, respectively. Consequently, the total rotation matrix R from the crystal to the laboratory framework is expressed as:

R = R ψ R θ R Φ =        cos(Φ)cos(ψ) -cos(θ)sin(Φ)sin(ψ) -cos(ψ)sin(Φ) -cos(Φ)cons(θ)sin(ψ) sin(ψ)sin(θ) cos(Φ)sin(ψ) + cos(ψ)cos(θ)sin(Φ) -sin(Φ)sin(ψ) -cos(Φ)cons(ψ)cos(θ) -cos(ψ)sin(θ) sin(Φ)sin(θ) cos(Φ)sin(θ) cos(θ)        (6.4
) For each grain, three Euler angles uniquely define the rotation matrix R. This can be computed at the beginning of the simulation. For all the model parameters in the form of a vector, such as the Burgers vector b, the moving direction g and the line direction dl, the new vectors in the rotated grain are obtained by b = R b, g = R g and dl = R dl. It is worth mentioning that the applied stress is given in the global reference, as the results, the Peach Koehler force dF pk and the shear stress τ pk are calculated as follows:

dF pk = (σ σ σ * R b) × R dl (6.5)
τ pk = 1 b dF pk * R g (6.6)
The movement of each dislocation segment can now be realized in the rotated crystal.

Since the number of grains has increased in the code, more dislocations are expected in the simulation. To accelerate the simulation, parallel calculations are used for the computation of elastic interactions between the dislocations. The parallel algorithm is realized by the OpenMP package in Fortran.

The rotation of crystal lattice in a grain and the parallel calculation are the two main aspects that have been improved in the new code. It should be mentioned that, the grain boundary is still treated as a totally closed barrier for dislocations. Each grain is closed (there is no free surface) and the dislocations cannot pass through the grain boundaries. The simulation of transmission or production of dislocations at the grain boundary is still under development [104]. Moreover, the stress that a dislocation segment endured is calculated also from all the dislocations in the crystal, including the dislocations in the other grains. For the loading method, the quasi-static fatigue loading presented in Section 2.6.1 is also implanted in the new code and is used for simulations presented hereafter.

Simulations of an aggregate 6.3.1 Simulation settings

To demonstrate the capacity of the newly developed code, a fatigue simulation on a polycrystalline aggregate made of 7 grains is realized. The geometry of the aggregate is given in Figure 6.2. The aggregate is composed of 7 grains. Each grain is taken as an hexagonal prism with the height h = 5µm and and edge length l = 2.5µm. The orientation of the aggregate surface is [0 0 1]. The euler angle of the lattice in each grain is set randomly. The initial dislocation microstructure consists of 60 dislocation sources of 2µm length (5 for each slip system) randomly introduced inside each grain. Different from the initial condition in single slip mode, the slip system activation is unknown since it may depend on the dislocation activity in the neighboring grains so that all the 12 systems will be considered. The material simulated is Cu. The increment of applied stress field is given as: ∆σ σ σ app =        0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0

       (6.7)
which indicates the simulated situation is a traction-compression along the Z-direction. The numerical parameters are summarized in Table 6.2. The criterion of quasi-static fatigue loading (see Section 2.6.1) for the loading is given as: ∆ε V M,max p = 1.0 × 10 -5

Chapter 6. Simulations of poly-crystals and ∆ε V M,min p = 3.5 × 10 -7 . The simulation is controlled by the plastic strain amplitude (ε V M p,applied = 2.25 × 10 -4 ). The evolution of the applied stress with loading cycles is plotted in Figure 6.3(a). About 30 cycles have been simulated. It can be seen that the stress value fluctuates with the loading steps. No significant cyclic hardening or cyclic softening phenomenon is observed. The stress value evolution is almost the same in each cycle, which can be considered reversible. This is because the number of initial dislocation sources is excessive considering the relatively small cyclic deformation ε V M p,applied = 2.25 × 10 -4 , consequently most of the dislocations move in the reversible mode: the construction and destruction of dislocation pile-ups (as we have observed in the single crystal simulation [START_REF] Fanshi | 3D discrete dislocation dynamic investigations of persistent slip band formation in FCC metals under cyclical deformation[END_REF]). In the point of view of macroscopic response, Figure 6.3(a) shows that no irreversibility has been introduced in the grain during the cyclic loading. However, Figure 6.3(b) presents that the poly-crystal's total dislocation density still increases with the loading cycle. The linear augmentation of the dislocation density indicates that even though most of the dislocation movements are reversible, dislocation dipoles, dislocation tangles, and dislocation junction may still be constructed inside the crystal to contribute to the dislocation storage. The total dislocation density evolution in each grain is presented in Figure 6.4(a) to demonstrate more details. The curves show that the dislocation densities in each grain augment very slowly and show a tendency to saturate. Besides, the dislocation density in grain 7 is much larger than that in the other grains. Although it starts to evolve in the same manner as in the other grains, at about the 6000000 simulation steps, it begins to increase brutally. After about 2000000 more simulation steps, the dislocation density in grain 7 also shows a sign of saturation. The dislocation density saturation is typical in fatigue, as reported in experimental studies [34,[START_REF] Huang | Plastic behavior of a nickelbased alloy under monotonic-tension and low-cycle-fatigue loading[END_REF]. The saturation phenomenon has also been reported in the single slip mode and more detailed analysis can be found in [START_REF] Fanshi | 3D discrete dislocation dynamic investigations of persistent slip band formation in FCC metals under cyclical deformation[END_REF]. The different dislocation evolution in each grain should be attributed to the different orientations in each grain. The plastic deformation accommodated by each grain should also be different due to the different orientations. The Von Mises plastic strain at the maximal deformation of the last cycle is calculated for each grain and presented in Figure 6.4(b). The Von Mises plastic strain of the poly-crystal is ε V M p,aggregate = 2.25×10 -4 plotted as the blue line in Figure 6.4(b), then for each grain, we have ε V M p,grain1 = 2.38 × 10 -4 , ε V M p,grain2 = 1.75 × 10 -4 , ε V M p,grain3 = 2.25 × 10 -4 , ε V M p,grain4 = 2.21 × 10 -4 , ε V M p,grain5 = 1.70 × 10 -4 , ε V M p,grain6 = 2.71 × 10 -4 , ε V M p,grain7 = 3.03 × 10 -4 . It is worth mentioning that the amount of plastic strain accumulated in each grain is calculated by Eq.(??) from the dislocations contained inside the grain and the volume of the grain, then the Von Mises Chapter 6. Simulations of poly-crystals equivalent plastic strain is calculated from the strain tensor. As observed in experiments, the different orientation has led to a heterogeneous plastic strain deformation in each grain.

(b) (a)

(a) (b)

Dislocation distributions in each grain

The dislocation distribution at the maximal deformation of the final cycle is shown in Figure 6.5. As the initial dislocation sources contain the 12 slip systems, we can see the generation of dislocations from all the slip systems in each grain. However, for some grains, some dislocation slip systems begin to show their predominance. For example, in grain 7, the dislocations in light blue are dominant (they are the dislocations of the S1 slip system) and in grain 3, there are more dislocations in amaranth (the S12 slip system).

In order to manifest better the slip activation mode in each grain, the vector of the loading direction ( Z direction in the global laboratory framework) in the local crystal reference of each grain is calculated by R -1 Z, where R -1 is the inverse matrix of the rotation matrix given by Euler angles of the corresponding grain. The obtained vector is then located in the stereographic projection map of FCC crystals to demonstrate the activation mode, as shown in Figure 6.6(a). It can be seen that all of the grains are in the single slip mode, but some of the grains (grain 4, grain 7, grain 5, and grain 6) are near the intersection line, which may show the double slip characteristics.

To confirm the slip activation mode in each grain, we have introduced a calculation plane at the middle of the poly-crystal to compute the dislocation displacements using Barnett's formula [START_REF] Barnett | The displacement field of a triangular dislocation loop[END_REF]. The calculated plane has the same orientation as the (0 0 1) normal to the aggregate. As slip bands are composed of pile-ups, the surface steps printed on the calculation plane indicate the plastic localization in each grain. In Figure 6.6(b), the slip band distributions are presented. Most of the bands are very thin, like slip traces (grain 1, grain 2, grain 5, grain 6) because, as we have mentioned, they are only made of reversible individual pile-ups, which may eventually form thick slip bands with further cyclic loadings. For grain 3, grain 4 and grain 7, some thick slip bands can be observed, which indicate the storage of dislocations in these grains. In addition, the slip bands present well the activation mode. Grain 1, grain 2, grain 3 and grain 4 are dominated by the single slip while grain 5, grain 6 and grain 7 also show the slip traces of secondary slip.

Surrounding grain effect

This simulation provides us an opportunity to verify the activation status of each grain. That is to say, with the existence of the neighboring grains, if the Schmid's law is still validated strictly in each grain. The Schmid factors of 12 slip systems in each grain are calculated by Eq.(6.5) and Eq.(6.6). The results are presented in Table 6.6. Additionally, the dislocation density evolutions of each slip system in each grain are plotted in Figure 6.7. From Figure 6.7(a,b,e,f), the dislocation density varies up and down in one cycle loading but no significant augmentation can be observed. As we have mentioned, this is because of the perfectly reversible dislocation movements in grain 1, grain 2 grain 5, and grain 6 during the construction and destruction of pile-ups with the loading and unloading. The reversible dislocation density evolution also confirms why in Figure 6.6(b), the slip bands are relatively thinner in these grains. Moreover, it can be seen that in these four grains, the Schmid's law is well respected. It is always the system with the largest Schmid factor that has the largest dislocation density. For example, the S4 for grain 1, the S2 for grain 2, the S9 for grain 5, and the S10 for grain 6. Even for the system with secondarily largest Schmid factor, the Schmid law works still well, such as the S9 for grain 1, the S11 for grain 2 and the S4, S5 in grain 5. When it comes to grain 3, grain 4 and grain 7, the evolutions of dislocation density are not reversible and storage of dislocations is more significant. From Figure 6.7(d), in grain 4, S12 and S1 have the two largest dislocation density and S12 demonstrates an obvious storage, which corresponds well to the order of Schmid factor (S1: f=0.48, S12: f=0.49). For grain 7 (Figure 6.7(g)), the S12, S1 and S2 have the three largest dislocation densities. S12 and S1 match well with the two largest Schmid factors (shown in Table 6.6) but S2 has a smaller Schmid factor than S1, S12 and also S3, S4, S9 and S10, which is not in good agreement with the order of Schmid factors. Nevertheless, the S2 is the cross-slip system of S1, which may be the possible reason for the relatively higher dislocation density. As for grain 3 (Figure 6.7(c)), S12 and S8 have the largest dislocation density, which presents a typical double slip characteristic. S8 has the second highest dislocation density but the value of its Schmid factor (f=0.28) is the fourth one in the 12 slip systems. S1 has the second (f=0.40) and very close to the primary slip system S12 (f=0.48) but its dislocation density is not significant and the same for S4 (f=0.36). The phenomenon observed in grain 3 and grain 7 indicates that the activation of slip systems can also be influenced by the dislocations in the surrounding grains, especially for the activation of the secondary or the third slip systems. The main reason should Chapter 6. Simulations of poly-crystals be attributed to the accumulated pile-ups from the neighboring grains blocked by grain boundaries. Even though the stress field produced by the blocked dislocation pile-ups of the neighboring grains is not significant, the cumulative effect from all the neighboring grains can not be neglected. The total stress field produced from the dislocation pile-ups at the neighboring grains can activate new slip systems. In experiments, T.

(a) (b) (c) (d) (e) (f) (g)
Benjamin Britton [174,[START_REF] Britton | Stress fields and geometrically necessary dislocation density distributions near the head of a blocked slip band[END_REF] has investigated the stress field produced ahead of the dislocation pile-ups blocked at the grain boundaries by the analysis of HR-EBSD. The phenomenon described in his article is quite similar to the reason interpreted here for the simulations.The experimental confirmation of the stress field produced by the pile-ups crossing the grain boundary in his article favors our arguments.

To gain more insights, the misorientation of each grain compared to other grains are calculated using the MTEX package of Matlab and summarized in Table 6. 4 (the underlined values are the misorientation of the surrounding grains, for example, grain 2 is surrounded by grain 1, 3, 7). To unify the surrounding grain condition, we only considered grains from number 2 to number 7 because they all have 3 surrounding grains. The average values of the surrounding grain misorientations are also presented in Table 6.4 but the value shows no direct relation with the phenomenon in grain 3 and grain 7. To be more precise, the difference of the maximal misorientation and minimal orientation in the three surrounding grains is computed and presented in the last row of Table 6.4. The values of the differences in grain 3 (21.92 • ) and grain 7 (20.43 • ) are relatively higher than the other grains. As a result, we conclude here that the effect of the surrounding grains or the strength of the stress field produced by the pile-ups of the neighboring grains on the slip activation may be indicated by the large difference of misorientations in the surrounding grains. We have managed to conduct a simulation of this polycrystalline aggregate during about 30 cycles and owing to the parallel algorithm, it took about 24 hours. To help the following analysis, the Schmid factors (or the resolved shear stress) for each slip system of each grain are presented here: 6.4. Simulations of bi-crystals

Simulations of bi-crystals

Another possibility of simulations offered by the poly-crystal version of TRIDIS is to investigate the interaction of dislocations in two neighbour grains. In other words, we want to construct PSBs in two neighboring grains, and change the orientation of the lattice in one grain to see if it has an influence on the PSB construction or the total macroscopic response. With the change of the lattice orientation, at the same time, we want to keep the Schmid factor equal in the two grains to ensure the activation possibility for each slip system is the same in the two grains.

Simulation settings

In the experiment of thermal fatigue test in CEA (CYTHIA) [START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF], a grain of the orientation [2 2 1] in the piping system is proved to endure an equibiaxed tractioncompression (as indicated in Figure 6.8). The related stress field is a biaxial tensor presented in Eq.(6.8) and the stress tensor in the global reference is shown in Eq.(6.9). Furthermore, as the loading directions are perpendicular to [2 2 1], the rotation of the crystal lattice around the axis [2 2 1] can not change the Schmid factor. The Schmid factors (or the resolved shear stress) for the 12 slip systems are shown in Table 6.6. The 7 th system and 8 th system are equally activated in a colinear double slip configuration. Chapter 6. Simulations of poly-crystals Based on these considerations, the bi-crystal shown in Figure 6.9 is constructed. The bi-crystal consists of two hexagonal based cylindrical grains grains with height h = 5µm and edge length l = 2.5µm. Two dislocation sources of the 8 th slip system in opposite sign are put inside as initial dislocation configuration. The placement of two dislocations in opposite sign is to facilitate the cross-slip for the PSBs formation. The orientation of the bi-crystal is [1 1 0], which is the direction of the Burgers vector for slip systems 7 and 8. For grain 1, the crystal lattice is fixed but for grain 2, the crystal lattice is rotated around the axis [2 2 1] from 0 • to 150 • .

∆σ σ σ app =        1 0 0 0 1 0 0 0 0        { * . * .[2 2
The Euler angles for the corresponding rotation angle are presented in Table 6.7. Through the rotation of the crystal lattice in grain 2, the angle between the formed PSBs in the two grains should be changed as well, which is the parameter we want to investigate. Indeed, if the 2 grains do not interact, they should construct PSBs with a relative angle equal to the misorientation angle. If interactions are involved, the angle between the PSBs should be different from the misorientation angle. The numerical parameters are the same as in Table 6.2. The simulation is controlled by the plastic strain amplitude (ε V M p,applied = 2.25 × 10 -4 ). The dislocation density evolution with the number of loading steps is shown in Figure 6.10. In Figure 6.10(a), the total dislocation density computed in the two grains is presented. The dislocation density increases with the loading cycles, indicating dislocation storage in the bi-crystal. It can also be found that, in general, the growth rate increases with the rotation angle and a larger rotation angle can lead to more stored dislocations. The storage of dislocations in the two grains is not identical. In Figure 6.10(b) and Figure 6.10(c), the dislocation density evolution in the two grains is presented separately. For misorientation angles equal to 0 • and 30 • , dislocations are preferentially stored in grain 2, while for rotation angles between 60 • and 150 • , more dislocations are stored in grain 1. The Schmid factor is set to be equal in the two grains, but a competition to accommodate the plastic strain is observed.

Dislocation distributions

In Figure 6.11, the dislocation distribution evolution is presented for the rotation angle equal to 0 • (Figure 6.11(a)) and 90 • (Figure 6.11(b)). In Figure 6.11(a), at the initial condition, the dislocation sources in the two grains are parallel. After one loading cycle, the dislocation distribution is similar in the two grains: a dislocation pile up is constructed in the center and cross-slip occurs near the grain boundary. However, from the 5 th cycle, more dislocations are stored in the grain 2. The dislocation arrangement in the two grains becomes more dense from the 5 th cycle to the 14 th cycle, while at the same time, the dislocations stored in grain 1 remain almost unchanged. As for the rotation angle equal to 90 • (Figure 6 In experiments on poly-crystals, the inequality of dislocation activation is often observed and is thought to be related to the different values of the Schmid factors. Here, the Schmid factors are equal in all these bi-crystal simulations. However, even for the case with a null misorientation, the inequality of dislocation distribution in the two grains occurs. The reason may be due to the cross-slip probability. At the beginning of the loading, the dislocations in the two grains constructed the same pile-up structure. But owing to a randomly higher probability of cross-slip in one of the grains, a dislocation segment begins to cross-slip in this grain, which can reduce the internal stress. Then, dislocations in the grain with the first cross-slip are more easily to be pushed to move by the applied stress. As a result, more dislocations are produced in this grain. Note also that in reality the elastic relaxation induced by the sudden slip activity in a grain could also contribute to the slip destabilization. Using DDD, this could be simulated using a coupling with the FEM.

Macroscopic response

The evolution of the maximal applied stress with the loading cycles is shown in Figure 6.12. The cyclic softening occurs for all the six rotation angles. The reason is the same as for the single crystal simulations: the increasing number of pile-ups decreases the applied stress required to drive the dislocations. Furthermore, after about ten cycles, when the value of the applied stress is stable, it can be seen that for a larger rotation angle, the value of the applied stress is more likely to be higher, which may be caused by the geometric relation between the PSBs and the grain boundary in grain 2.

In this section, we have investigated the effect of grain misorientation in bi-crystals. Differences of the dislocation distributions in the two grains can be observed, which Figure 6.12 Evolution of the maximal applied stress with loading cycles. may be caused by statistical triggering of cross-slip. The evolution of the dislocation densities and the maximal stress value indicate that the rotation of grain 2 does not have a significant effect on these parameters.

Conclusions

In this chapter, we have performed preliminary simulations of fatigue in poly-crystals using the newly developed poly-crystal version of TRIDIS. We have firstly introduced the changes made in the basic version of the code. To demonstrate the capacity of the code, we have presented the simulation of an aggregate composed of 7 grains. The dislocation density evolution, the dislocation distribution evolution and the formation of slip bands are discussed. Then, simulations of bi-crystals with different misorientation angles have been launched. It is found that, even if it is not significant, a larger rotation angle contributes to a higher total dislocation density and a larger maximal stress. Differences in the dislocation distributions in the two grains are observed even though the Schmid factors are the same for the slip systems in the two grains.

What should be retained from this chapter:

-The newly developed code of DDD for poly-crystals provides new possibilities for future simulations;

-The simulation of the aggregate of 7 grains is remarkable because it is the largest physical scale reached at the moment in the DDD simulations of fatigue. From the analysis of the dislocation density of each system in each grain, it is found that the Schmid's law can predict the most activated slip system (primary system). However, the surrounding grains may have an influence on the activation of the secondary slip system.

-Simulations of bi-crystals with different misorientation angles show that grain interactions exist even without slip transmission at the grain boundary.

Conclusions

The aim of this thesis is to investigate the fatigue damage mechanism of FCC metals using 3D Discrete Dislocation Dynamics simulation method. The distinguished feature of DDD is the ability to precisely simulate the collective behavior of dislocations. The mesoscopic simulation scale of DDD is between FEM and Molecular Dynamic (MD), which allows for a compromise between the physics and acceptable computation time.

As the fatigue damage phenomenon is highly related to the dislocation movements, the application of DDD to fatigue problem is helpful for the understanding of the underlying mechanisms. The main period that is simulated in this thesis work is the early stages of fatigue degradation, referring to the plastic deformation before the initiation and propagation of cracks. The mainly studied material is Cu which is also the most investigated material in experimental researches and also a reference material for FCC metals. The various publications available in the literature facilitate the comparisons between simulation results and experimental observations, which have been done throughout this thesis work. The common thread of this thesis work is the extension of the physical scale: from one PSB to several PSBs, from one activated slip system to two activated slip systems, from one single crystal grain to two crystal grains or an aggregate. The fatigue damage scenario is also a multi-scale physical phenomenon. Consequently, a complete understanding in the different scales is necessary. Under a low amplitude cyclic loading of 10 -4 in a configuration of single slip, the DDD code was able to reproduce the accumulation and localization of dislocations into the formation of a PSB, composed of dislocation dipoles, prismatic dislocation loops, dislocation debris and reversible dislocation pile-ups. Dislocation dipole array is a typical structure observed in experiments for FCC metals under fatigue loading. From the dynamic observations of the simulations, a possible dipole array formation mechanism is proposed, based on the cross-slips of the dislocations in the pile-ups near the grain boundary. At the surface of PSB, the surface roughness is growing at the same time as the formation of the dislocation structures inside the grain. The surface roughness refers to extrusions and intrusions. There are two forms of extrusions and intrusions obtained from the simulations: "tongue-like" and "ribbon-like". The conservative slip of dislocations allows to consider that "tongue-like" extrusions & intrusions are associated with the internal prismatic loops, while "ribbon-like" extrusions & intrusions are associated with the internal dislocation dipoles. By calculating the absolute volume of extrusions and intrusions, the volume of extrusions is found to be larger than that of the intrusions, which can be considered as an indication that more vacancy-type dipoles or prismatic loops are stored in the grain than the interstitial-type.

By reducing the grain size to 5µm, it was possible to simulate one hundred cycles simulation. It is found that after about 25 cycles, a saturation state can be reached. In the saturation state, the PSB volume fractions, the PSB positions and the height of extrusions and intrusions at the surface remain stable, which fits well with experimental observations in Cu. The saturation state can be well explained from the reduction of the shear stress resolved on the cross-slip system caused by the formation of several PSBs. The shear stress on the cross-slip system can largely weaken the interactions between dislocations, which facilitate the dislocation movements. In addition, the linear relation proposed by Winter between the applied deformation amplitude and the PSB volume fraction is retrieved by the simulations. The stability of the formed numerical PSB is verified by reducing the applied strain amplitude. It is stated that the position of the formed PSBs stays unchanged but the PSBs become less pronounced, which corresponds well to experimental results.

To complete the analysis, a comparison is done between AISI 316L austenitic stainless steel and Cu to check the effect of material parameters such as the stacking fault energy. It is found that Cu constructs more PSBs and has a higher tendency to build up thicker PSBs. Then, the differences of cyclic behaviors in simulations between Cu and 316L are thought to be induced by the cross-slip possibility difference. The obtained results here indicate that Cu shows more the characteristics of a wavy slip material, as observed in experiments.

In a more general point view, the irreversibility is introduced firstly by the asymmetry of the pile-ups during the loading and unloading stages, then the partial departure of dislocation loops from the free surface is another source of irreversibility. The most important introduction of irreversibility is the cross-slip and the dislocation dipole formation. The heterogeneity of the stress filed can also bring some irreversibility by the build-up of dislocation tangles. The formation of dislocation junctions is also a important factor such as the colinear reaction in single slip situation.

Single slip is the most fundamental slip situation. The different combinations of single slips can lead to complex deformation modes. The simultaneous activation of two slip systems, also called double slip is then investigated by DDD. Considering the different interactions between the FCC slip systems, five different double slip configurations are tested: Lomer-Cottrell lock double slip, glissile junction double slip, Hirth lock double slip, coplanar double slip and collinear double slip. It has been found that, except the coplanar double slip, the other double slips can be considered as the combination of two single slips with a zone of interaction. Dislocations from the two involved slip systems occupy distinct region of the crystal where they form PSBs and evidences can be found in experiments. The coplanar double slip shows a different behavior compared with the other four double slips, which may be induced by the strong coplanar interactions. At the free surface, the morphology of Lomer-Cottrell double slip configuration shows a strong tendency to construct more "tong-like" extrusions. We have also compared the shape of the normalized hysteresis loop, which shows the relation with different dislocation interactions.

At the end of thesis work, the newly developed DDD code for poly-crystal is tested. A preliminary simulation of an aggregate made of 7 grains is presented, to demonstrate the code capabilities. Finally, simulations of bi-crystals with a varying misorientation angle have been performed and the effect of the grain interactions has been evidenced. The new poly-crystal version of the DDD offers more possibilities to explore in future studies.

Perspectives

Evidently, the newly developed TRIDIS code for poly-crystals opens up promising perspectives. The primary critical task is to optimize the role of grain boundaries. Grain boundary is not only a barrier for dislocations, it is also a site for the transmission and generation of dislocations. Another important point is to add the free surface done in the single crystal version. With a more physical grain boundary and a free surface, we can start investigating the fatigue behavior of bi-crystals with different orientation combinations, as mentioned in many articles [START_REF] Zhang | Grain boundary effects on cyclic deformation and fatigue damage[END_REF][START_REF] Nellessen | Low cycle fatigue in aluminum single and bi-crystals: On the influence of crystal orientation[END_REF]. In terms of simulations of aggregates, one idea is to realize fatigue simulations on a real aggregate constructed from EBSD data, so that the effect of the real grain shape can be addressed. It should indeed be pointed out that for the moment, the simulated grain shape was always taken as a regular polyedra or cylinder. An introduction of an irregular grain shape is also a challenge for the new TRIDIS code. For the fatigue behavior of nano-crystalline or gradient nano-grained material, the grain size and grain boundary are critical to determine the mechanical behavior. In these fields, the application of the poly-crystal version of DDD should be very helpful. It is worth mentioning also that the newly developed TRIDIS code should also be tested with monotonic loadings to assess its reliability and optimize the numerical settings, which saves more time than cyclic loading.

As mentioned in chapter 4, compared with experimental observations, the irreversibility is more difficult to be introduced in DDD simulations. In this thesis, the friction stress was increased to add more irreversibility. The application of nodal model for DDD can be also a possible way to contribute the irreversibility, because of a more physical description of dislocation interactions in the nodal model. In recent publications [START_REF] Hussein | Quantifying dislocation microstructure evolution and cyclic hardening in fatigued face-centered cubic single crystals[END_REF][START_REF] El Achkar | Analysis of dislocation microstructure characteristics of surface grains under cyclic loading by discrete dislocation dynamics[END_REF], the nodal model is used to study the fatigue behavior, but only in the multi-slip situation. The most classic situation -the single slip has not been simulated yet, which may be due to the difficulty to introduce irreversibility. The irreversibility is crucial for the storage of dislocations, which finally leads to the formation the organized fatigue microstructure. In that respect, cross-slip mechanism is a key process to induce irreversibility. In the thesis of Margarita Longsworth (in progress), a more precise cross-slip model have been implanted in numodis , the nodal DDD code developed in collaboration with CEA. This new model [177][START_REF] Malka-Markovitz | Cross-slip in face-centered cubic metals: a general Escaig stress-dependent activation energy line tension model[END_REF][START_REF] Santos-Güemes | Influence of the stress state on the cross-slip free energy barrier in Al: An atomistic investigation[END_REF] gives a more physical description of cross-slip by the consideration of multiple stress components. Afterwards, this new model will be applied to simulate the Very High Cycle Fatigue (VHCF) behaviors. To have more storage of dislocations, another possible research direction is to investigate the effect of precipitations or forest dislocations on the fatigue behavior. Moreover, the coupling between DDD and FEM can also be a choice to impose a more consistent stress field, which may increase the storage of dislocations. In my master thesis, we have developed the coupling method between DDD and a FEM crystal plasticity model. As shown in Figure 6.13, this method is applied on an real aggregate obtained by EBSD: the central grain where cracks are observed is treated with DDD and the surrounding grains are treated with an elasto-visco-plastic FEM model. Due to the low computation capacity, we only managed to reach a dozen of cycles under a small deformation. The extension or shrinkage of dislocation loops were observed but no obvious storage of dislocations. For the future exploration, this aggregate mesh can still be used under a larger deformation. The difficulty lies on how to improve simulation stability and reduce numerical divergence. In addition, for the moment, most DDD simulations of fatigue focus on the early stages of fatigue. The interaction between cracks and dislocations is also important for the propagation of cracks. In the thesis of Elena JOVER CARRASCO (in progress), the coupling between XFEM and DDD is used to simulate the crack propagation under monotonic loading. An example is presented in Figure 6.14, the propagation of a crack is calculated based on the G-THETA algorithm. An implementation of cyclic loading in such a coupling should be interesting to study the crack propagation in fatigue situation.

Free surfaces of the crack Dislocations

Introduction

Fatigue damage is one of the main origin of structure failure in long terms, especially for aerospace and power plant industry [1,2]. For example, in nuclear industry, materials at the mixing zone of hot fluid and cold fluid in the piping system of nuclear plant always endure thermal fatigue shocks due to the temperature gradients, which may ultimately lead to leakage of cooling water (Accident Civaux [3]). Simular failure 5 problems are observed in airplane components or in any structural materials subjected to repetitive loads [4]. Investigations of the fatigue damage mechanism is thus a major concern for industry and academic research. Since the first systematic fatigue research conducted by August Wöhler [5] on railway axles, fatigue damage mechanism has been investigated for more than 100 years. However, due to the complexity of the underlying physics in terms of time and space scales, a complete understanding of the fatigue failure 10 mechanism is still not available. The fatigue damage process is generally divided into two stages [6]: the initiation and the propagation of cracks. The initiation stage represents about 90% of the fatigue life, which is also called the early stage of fatigue damage. The work presented here exclusively focuses on the early stage of fatigue damage.

One remarkable observation during experiments of the early stages in fatigue damage is the formation of slip bands, which was firstly evidenced by Ewing [7] at the surface of a Swedish iron. In the last fifty years, the fatigue behaviors for FCC (face-centered cubic) metals have been deeply investigated [8,9,10] because of the relatively uniformed fatigue behaviors and wide applications in industry. The development of persistent slip bands (PSBs) [8,11,12,13] is considered as a characteristic feature for FCC metals at early stages of fatigue damage. Different aspects concerning PSBs have been studied extensively such as the dislocation microstructures within the PSBs [4,8] , the surface roughness evolution [14,15] on the PSB's surface and the dislocation motion mechanism [16]. The recent publication of Lavenstein et al. [43] has reported dynamic observations of the formation of PSBs at a micro-scale. In situ SEM observations of Ni microcrystals oriented for single slip showed that the PSBs nucleate from inside the crystal and gradually grow in the sample until they reach the surfaces. It was also concluded that the location of the PSBs corresponds to the positions of the slip traces evidenced during the first 10 cycles.

Among the FCC metals, Cu single crystal is the most studied material and commonly considered as a reference for academic study of FCC metals. The evolution of the dislocation microstructure with the applied strain amplitude of Cu in single slip situation has been well explained by Mughrabi [8]. Through his famous CSS (cyclic stress-strain) curve: for small plastic strain amplitude, a matrix and vein structure is formed; for larger strain amplitudes, PSBs are formed; for even larger applied deformation, the activation of a second slip system [17] leads to the formation of dislocation cells or labyrinth structures. The formation of PSBs is always accompanied by a saturation phenomenon [8,4] regarding both the macroscopic response and the PSBs volume fractions and positions.

On the other hand, fatigue behavior is affected by many factors. Since the applied plastic strain amplitude influences the dislocation behaviors, it has a strong effect on the PSB volume fraction [12], the hardening rate [18] and the shape of the macroscopic response [19]. Winter [12] has proposed a model relating the volume fraction of PSB with the applied strain amplitude through a linear relationship. Even though the fatigue behaviors of most of the FCC metals are quite similar, slight differences exist between them. By comparing copper-aluminum alloy with pure Cu, Feltner and Laird [20] found different sensitivities to the initial conditions, which allows the authors to bring out the influence of the stacking fault energy (SFE) on the cyclic slip mode [21]. Later on, short-range order (SRO) [22,23] in alloys and annihilation distance y s [24] were also pointed out to characterize the slip mode. Despite these studies, no unified criterion of the slip mode transition has been proposed yet.

It can be seen that dislocation motion in fatigue damage is crucial but always difficult to be traced with experimental methods. On the other hand, numerical simulations offer new possibilities. The recent 3D Discrete Dislocation Dynamics (DDD) simulation model is a mesoscopic simulation method which aims at relating the dislocation activity to the mechanical response of micrometric volumes. Proposed in the early 1990s [25,26], DDD possesses special advantages in studying the collective behavior of massive dislocation populations because most of the physics of mechanisms for dislocations are taken into account, as well as reasonable constitutive rules or empirical laws when continuum elastic framework is not sufficient. When both in situ and post-mortem experiments can only provide limited information, DDD shed new light on the understanding of fatigue damage at the mesoscopic scale.

Considering the computer capacity, the application of DDD to fatigue started with 2D simulations. In a very simplified 2D DDD model, Yang et al. [27] have observed that randomly distributed dislocations could evolve to a regular matrix-wall structure in Cu. Deshpande et al. [28] studied the interaction between dislocations and pre-existing cracks by 2D DDD. Brinckmann and Van der Giessen [29] also used 2D DDD to investigate the stress concentration during the early fatigue cycles. In three dimensions, the most extensive DDD investigations of fatigue behaviors were conducted by Déprés et al. [30,31] (see also [32] for a review of this work) who used a 3D DDD edge-screw model to study the Low Cycle fatigue behavior of AISI 316L stainless steel in single slip mode. The main result of this study is the prediction of the number of cycles required to initiate a crack and its dependence on material parameters. More recently, they have investigated the propagation of fatigue cracks inside a grain [33] and into the neighbor grains [34].

Later on, more attention was paid to multiple slip like Hussein and El-Awady who used a nodal DDD model to investigate the microstructure characters developed in a fatigued Ni grain in multiple-slip situation [35] and the surface roughness of Ni single crystals [36]. They pointed out specifically the influence of the initial condition and grain size on the macroscopic response and localization of plasticity at the surface. El Achkar and Weygand [37] recently used graph analysis to quantify the evolution of the dislocation microstructure in terms of prismatic loops and dipoles developed in a surface grain of Al in multiple slip fatigue loading conditions. Erel et al. [38] have artificially constructed a wall structure in 3D DDD which mimics the microstructure configuration of a typical PSB observed experimentally to study the movement of dislocations within the wall and the relationship with cyclic hardening and saturation stress. Several researchers have also applied DDD to study the irreversibility under cyclic torsion loading [39,40], as well as the Bauschinger effect on thin films [41,42].

Surprisingly, although DDD has already been applied to study the fatigue behavior of FCC metals, the cyclical mechanical behavior of Cu during single slip has never been studied in details despite the fact that Cu is certainly the most studied metal in classic experimental fatigue research and well documented. Moreover, the localization mechanism associated to the formation of multiple persistent slip bands has never been simulated. Finally, no DDD simulation was able to reproduce the saturation regime evidenced experimentally. Therefore, in this article, we shall use DDD model to simulate the fatigue behavior of Cu in single slip mode with the aim of elucidating the formation and propagation mechanisms at the origin of the PSBs. The paper is written as follows. Next section is a brief presentation of the 3D DDD model and its adaptation to treat fatigue simulations of a surface grain. Then simulations results are exposed in the next section, starting with a careful examination of the PSB formation and multiplication mechanisms, followed by a one to one comparison between Cu and ASI316L stainless steels behaviors.

Simulation methods

Screw-edge model

The DDD model retained for this study is the edge-screw model TRIDIS [26] previously used in the modeling of Low Cycle Fatigue of AISI 316L stainless steels [32]. In this model, dislocation lines are divided into connected discrete segments of edge and screw character. The velocity of the dislocation segments is calculated by a linear mobility law with the effective shear stress τ ef f , resolved on the dislocation slip system:

v = τ ef f * b B ( 1
)
where v is the velocity of the dislocation segment, b is the length of the Burgers vector and B is a phonon drag coefficient. τ ef f is a combination of different stress fields acting on the dislocation segment:

τ ef f = τ -sign(τ ) * τ f r (2)
τ = {[ 1 b (σ σ σ app + σ σ σ int )b b b] × l l l} * g g g + τ lt (3)
In these two equations, b b b is the Burgers vector, l l l is the line vector of the segment and g g g is the glide direction vector for the movement direction. The effective stress field τ ef f is the shear stress resolved on the dislocation segment slip system obtained from the superposition of different stress fields: σ σ σ app , the applied stress field that results from directly applied loading; σ σ σ int , the internal stress field produced by the other dislocations contained in the crystal. τ lt is the line tension correction and τ f r is the lattice friction stress also called Pierls stress which refers to the impediment caused by the crystal lattice on the dislocation movements. Note that in the present simulations we will neglect the image stress field that shall arise from the interaction between dislocations and the grain free surface since [33] has shown that they could be neglected in fatigue simulation of surface grains.

When the effective shear stress τ ef f is bigger than the friction stress τ f r , the dislocation i is able to move. Then the displacement ∆d i is calculated in an explicit way by multiplying the time step δt and velocity v i : ∆d i = v i * δt. The associated shear strain is then calculated as: ∆γ i p = b * ∆A i V , where ∆A i = l i * ∆d i , ∆A i is the area swept by the dislocation segment i and l i is the length of the segment i. V is the volume of studied grain. The increment of the total plastic deformation ∆ε p is defined as the sum of the contribution of each dislocation ∆ε i p :

∆ε i p = 1 2b (b b b (s) ⊗ n n n (s) + n n n (s) ⊗ b b b (s) )∆γ i p (4) ∆ε p = N i=1 ∆ε i p ( 5
)
where s is the slip system of the dislocation segment and n n n s and b b b s are the slip plane normal and Burgers vector of this slip system, respectively. An equivalent Von Mises plastic strain is calculated from the second invariant of the plastic strain tensor as:

ε V M p = 2 3
3(ε 2 xx + ε 2 yy + ε 2 zz ) 2 + 3(ε 2 xy + ε 2 xz + ε 2 yz ) 4 (6)
where ε ij are the components of the plastic strain tensor.

During every time step of the calculation δt, the following interactions are taken into consideration: annihilation and combinations of dislocations, junction formations, cross-slips and line reconnections. In TRIDIS [26], the cross-slip probability for the screw segments is computed from the following expression adapted from [44] :

P i = β l l 0 δt t 0 exp( τ dev -τ III kT V act ) (7)
where β = 10 5 is a normalization coefficient to assure that the cross-slip probability P i ∈ [0, 1], l l0 is the ratio between the length of the considered screw dislocation segment l and the reference length l 0 (l 0 = 1µm [26]) and δt t0 is the ration between the simulation time step δt and the reference time step t 0 (t 0 = 1s [26]). Inside the exponential function, the entire formula represents the activation energy for cross-slip: (τ devτ III ) is the comparison between the absolute value of the shear stress resolved on the deviate plane, τ dev and a reference stress value, τ III corresponding to the shear stress measured at the beginning of Stage III in single crystals tensile tests; V act is the activation volume, k and T represent the Boltzman constant and the temperature respectively.

Quasi-static single shear loading

The applied stress field σ σ σ app is imposed as a homogeneous stress tensor, identical for each dislocation segment. The applied stress field at step k is given as follows:

σ σ σ k app = σ σ σ k-1 app + λ * (-1) a * ∆σ σ σ app (8)
where σ σ σ 0 app = 0 0 0, a = 0 for loading so that (-1) a = 1, and a = 1 for unloading so that (-1) a = -1. λ is the coefficient to control the quasi-static loading rate. ∆σ σ σ app is the stress increment at each time step of the proportional applied loading. The stress tensor is chosen so that it induces a pure shear stress on a 2 (1 11 ) [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications[END_REF] slip system. When expressed in the crystal coordinate system, it writes as:

∆σ σ σ app =        0.577 0.
-0.288 0.

-0.577 -0.288 -0.288 -0.288 0.

       * σ * δt (9)
In Eq 9, σ is the stress rate ( σ = 5 × 10 9 M P a/s). Let recall here that the time integration algorithm is purely explicit. For this reason, the choice of the time step δt is of great importance. If δt is too small, the computational time can be very long. On the other hand, if δt is too big, the local interaction cannot be well simulated. According to validation tests, δt is chosen as δt = 2 × 10 -10 s. λ at step k is computed from the increment of the equivalent plastic strain at step k-1, ∆ε V M,k-1 p . When ∆ε V M,k-1 p is smaller than the value ∆ε V M,min p , the dislocations are considered in an equilibrium position so that the amplitude of the applied stress should be increased and λ = 1. In the ideal situation, ∆ε V M,min p should be zero, but small vibration of dislocations cannot be avoided, so we have chosen ∆ε V M,min p = 10 -7 . If ∆ε V M,k-1 p is bigger than the value ∆ε V M,max p (set to 10 -5 ), the increment of deformation at last step is judged to be too high and may lead to dislocation avalanches. In this situation, the applied stress is reduced by choosing λ = -0.5. Otherwise, if the increment of deformation ∆ε V M,k-1 p is between the two values, the applied stress remains unchanged and λ = 0 so that the dislocations can keep on moving under the same applied stress field to find the equilibrium position. The value of λ at step k is summarized as follows:

λ =    1 if ∆ε V M,k-1 p ≤ ∆ε V M,min p -0.5 if ∆ε V M,k-1 p ≥ ∆ε V M,max p 0 else ( 10
)
The load is reversed when the equivalent cumulated plastic strain reaches the desired maximum imposed strain amplitude. In the next simulations, the load is always symmetric (R = 1) so that an imposed plastic strain amplitude ε V M p,applied = 10 -3 signifies that the plastic strain will vary between -10 -3 and +10 -3 .

Simulation settings

The grain geometry and the initial dislocation configuration are selected so that it ensures strain localization on a particular slip system and simulations could be performed in a reasonable CPU time. These informations are provided by the previous work of Déprés et al [30] who demonstrated two key features. Firstly, strain localization is possible on slip systems admitting a low angle between the Burgers vector and the normal to the free surface. Secondly, the PSB microstructure can be obtained from a single initial Frank-Read source, i.e. without the need to include all the 12 possible slip systems.

In our case, the simulated situation is a [2 2 1] surface grain as shown in Fig 1 . For sake of simplicity the grain geometry is defined as an opened cylinder of D = 5µm diameter and H = 5µm height. Its top surface (as shown in yellow in Fig 1) is considered as a free surface where dislocations can escape and leave the crystal, while the other surfaces are all impenetrable for the passage of dislocations. A single Frank-Read dislocation source is introduced inside the crystal on the a 2 (1 1 1)[1 1 0] slip system, refered as the primary slip system. The applied stress field defined by Eq 9 induces a pure shear stress of the primary slip system. Although DDD simulations will only concern two families of dislocations, namely a 2 (1 1 1)[1 1 0] and a 2 (1 1 1)[1 1 0], the ratio between the shear stress for each of the total 12 systems and the shear stress on the primary slip system τ system τ primary are given in Tab 1 for information. The green plane drawn in Fig 1 is a typical slip plane of the primary slip system #8. The blue plane is a slip trace of system #7 which corresponds to the conjugate slip plane of the primary system. System #7 will be later called the deviate slip plane since dislocation can cross-slip from system #8 to system #7 during the fatigue simulations.

Table 1: Slip systems in FCC crystals and activation factor induced on each system by the stress tensor given in Eq 9. Primary slip system corresponds to slip system #8 and deviate slip system is system #7. τ system τ primary 0.50 0.17 0.50 0.17 0.67 0.67 0.33 1.00 0.17 0.50 0.17 0.50

Slip

Most of our simulations are realized for Cu but for sake of comparison, some complementary simulations are also performed for a second material having a higher activation volume, i.e. a lower stacking fault energy.

In that objective, a comparison has been made between Cu and austenitic stainless steel AISI 316L. All the parameters used in the DDD simulations for these two materials are listed in Tab 2 and taken from [31,26].

Spatial dislocation distributions measurement

Since the simulated situation is single slip, most of the dislocations are primary dislocations slipping on system #8. For the analysis of Persistent Slip Bands formation, it is important to quantify the distribution of the dislocation positions with time. The proposed method is illustrated in After the first two cycles, multiple dislocation loops have been generated from the initial Frank-Read source in order to accommodate the imposed plastic strain. During this multiplication process, the dislocations are homogeneously distributed in the entire grain through the double cross-slip mechanism as also evidenced in [45]. As can be seen in Fig 3(a), the distribution remains homogeneous until the 5 th cycle. However, from the 10 th cycle, a localization of dislocations can be observed. Starting from the 10 th cycle, some dislocations gather to form adjoining slip planes connected by dislocations of the cross-slip system. The built-up microstructure compares well to the macro slip bands as observed in experiments [4,46]. Most of the plastic strain is confined in these bands. In this paper, the name PSB will be used to describe this organized microstructure although at this stage this dislocation organization obviously corresponds more to the embryo of the PSBs that will form later in the cycling. According to Lavenstein et al. [43], the slip traces observed during the first cycles coincide with the PSBs. These simulated bands can therefore be considered as the premises of the PSBs. The dislocation arrangements inside the PSBs are relatively much denser than everywhere else in the grain but randomly distributed dislocations are still visible. With the increase of the number of cycles, more PSBs are constructed (see Fig 3(a)). After the 15 th cycle, besides the two already formed PSBs, the embryo of the third one appears at the right side of the grain. From the 20 th cycle, four PSBs are about to be formed but the frontiers between the PSBs are still not very clear because of the existence of many reversible dislocations. After the 26 th cycle, four PSBs have been well constructed and easy to be distinguished directly from the 3D picture. The width of the formed PSB is about 1µm, which is of the same order of magnitude as observed in the micro-sample fatigue experiment of Lavenstein et al. [43]. Finally, it is found that the positions and thicknesses of these four PSBs stay almost invariant from the 26 th cycle to the 50 th cycle which could then be qualified as a saturation regime similar to what was observed experimentally in [4,46]. It is worth mentioning also that the distribution process of the numerical PSBs presented here resembles to the in-situ SEM observations printed in Figure 1 of Lavenstein et al. 2020 [43]. In order to quantify the organization of the dislocation microstructure during the cyclic loading, the dislocation densities on the primary and cross-slip slip systems (or deviate slip system) are plotted as functions of the loading cycles in Fig 3(b). During one cycle loading, the dislocation densities for both slip systems fluctuate with the increasing or decreasing of the strain. Thus, the dislocation density reaches two maximum per cycle, one for the maximum positive strain and a second one for the minimum negative plastic strain. The plot shows that the dislocation population can be subdivided in two parts: ρ dis = ρ dis,stored +ρ dis,rever . The stored dislocation ρ dis,stored is plotted as a dashed line in Fig 3(b) and corresponds to the envelope of the minimum of the curve; the evolution of the reversible dislocation ρ dis,rever corresponds to the fluctuations within each cycle.

The reversible dislocations mainly contribute to the accommodation of the plastic deformation during one cycle. The stored dislocations have lower mobility. They accumulate inside the crystal and contribute to the formation of the microstructures. The evolution of the stored dislocation densities matches well with the formation of the PSBs. When the four PSBs are still under construction (before cycle 26), the stored dislocation density is still increasing and the amplitude of the reversible dislocation density is quite low.

Once the PSBs are all formed (after cycle 26), the stored dislocation density remains constant and the amplitude of the reversible dislocation density reaches a maximum. The microstructure constructed by a limited number of stored dislocations (at cycle 26) described hereafter.

The detailed structure of the dislocation arrangement within a PSB at the 26 th cycle (visualized in the when it is maximal. When ε V M p,applied = 0, dislocations exist in the form of dipoles, prismatic loops and tangles. Multiple dislocation dipoles are built-up by the attraction of two edge dislocations of opposed sign and are arranged in a regular array. In the center of the PSB, due to the back and forth motion of dislocations by cross-slip during the loading and unloading sequences, dislocation dipoles are cut into pieces by line multiple recombinations which create dislocation prismatic loops. This cutting mechanism has already been described in [30,47,9]). Tangled dislocations are also formed from the recombination process on the side of the prismatic loops. The simulated dislocation microstructures are quite similar to the experimental results. The existence and persistence of dislocation dipoles under cyclic loading have been reported in several experimental studies [48,49]. When the applied plastic strain amplitude is increased, most of the dislocation lines or tangles are destroyed or extended by the stronger applied shear stress at maximal deformation. In contrast, dislocation dipoles and prismatic loops show more permanence because most of them seldom move or are broken down by the stress. The low mobility of these structures suggests that they contribute little to the accommodation of the plastic deformation. Nevertheless, dislocation pileups are found to be built up near the grain boundaries at maximal deformation. Dislocation pile-ups have a relatively higher mobility since they can easily unstack when the load is reversed. From this point, it can be inferred that during one cycle of deformation, most of the plastic deformation is accommodated by the pile-up dislocations. These observations also show that the build-up and build-down of the dislocation pile-ups correspond to the fluctuation of reversible dislocations while the persistence of dislocation dipoles and prismatic loops refers to the storage of dislocations.

In experiments, dislocation dipoles exist also in several different aggregation forms such as dipole bundles [50], veins [51] and ladders [46] but the formation mechanism is still not clear and under investigation [48].

Due to the limitations of the model in term of computational power, the formation of the ladder and vein structure are not accessible by the DDD simulations but multipoles are clearly evidenced in the dislocation organization.

It is worth mentioning that the saturation state normally appears after 1000 cycles in experiments whereas in the DDD simulation, the saturation regime is observed after about 25 cycles. This difference suggests that the DDD simulation may accelerate the process of microstructure formation under cyclic loading. A possible reason could come from the physical mechanisms which are not accounted for in our DDD simulations such as forest hardening, point defect interactions, diffusion mechanism, dislocation/grain boundary interactions and the presence of the grain neighborhood. Note also that the relative small size of the simulated volume contributes to the accelerated organization of the dislocation microstructure. Nevertheless, the in-situ fatigue tests on micro-samples [43], have demonstrated that the strain localization that will ultimately lead to PSBs starts within the first 10 cycles. This statement can be considered as a favorable argument for the limited number of cycles reached in DDD simulation. Finally, despite these restrictions, it is remarkable that DDD did reproduce the main characteristics of dislocation self-organization in fatigue.

Growth of the surface roughness at the surface

Another important aspect of the early stages of fatigue is the growth of roughness on the surface at its intercept with the PSBs. In DDD, the plastic steps associated with the dislocation shear can be computed using Barnett formula optimized for DDD codes [52]. The procedure allows to compute the actual shape of the free surface during the cycling.

The whole configuration of the free surface at the 26 th cycle, when the total plastic deformation is zero, is presented in The growth of extrusions is induced by the passage of prismatic loops across the free surface [31]: vacancy prismatic loops lead to intrusions and interstitial prismatic loops induce extrusions. Each prismatic loop contributes to one layer of extrusion or intrusion with a magnitude of one Burgers vector. The extrusions and intrusions growth is realized by the accumulation of these layers. It should be noted that the dimension of the layer depends also on the size of the prismatic loops. In other words, the height and morphology of the extrusion and intrusion are influenced by the character and shape of the prismatic dislocation loops crossing the free surface.

Analysis on the saturation regime

As said before, a saturation state is reached after about 26 cycles as evidenced in Fig 3 . In this so-called saturation regime, even if the plastic deformation loading continues, many characteristics remain stable including the PSB number, PSB locations, PSB thicknesses, dislocation densities, and maximal height of surface extrusions and intrusions. It should also be pointed out that the saturation regime has not been reported by former DDD simulations of fatigue.

The stability of the dislocation density suggests that a dynamic equilibrium in the dislocation movements is reached, which implies that the number of dislocations created within a cycle is equal to the ones annihilated or escaping at the free surface. Moreover, the consistency of PSBs' number and thickness imply that the dislocation-related microstructures do not change significantly in the saturation regime, that is to say the microstructure is stable. As the maximal height of extrusion and intrusion is correlated to the crossing of prismatic loops, the slight fluctuation of these two values demonstrates that the flux of prismatic loops nucleation inside the PSB is almost constant during the saturation regime. In other words, the dislocation nucleation kinetics has reached a permanent regime.

It should be emphasized that saturation phenomenon has been widely observed in fatigue experiments Huang et al. [55] has used in situ neutron diffraction to detect the dislocation density in a Ni-based super-alloy. The saturation of the dislocation density was observed: dislocation density increases with the cycles and saturates when the PSBs are formed. Essmann and Mughrabi [56] have used a simple model of dislocation accumulation as a function of the plastic strain to study the importance of dislocation annihilation and concluded that the annihilation of dislocations can lead to a natural limit of the dislocation density under cyclic loadings. The saturation of the dislocation density in our simulation suggests the same idea that an upper limit of stored dislocations exists for a given value of the imposed plastic strain amplitude.

Regarding the surface roughness, it is generally admitted that the height of extrusions or intrusions continuously increases and will ultimately lead to the initiation of cracks at room temperature. Compared with the model of Polák [15,14] for the formation of surface extrusions, the growth of extrusions in our simulation can be considered as the so-called rapid growth of static extrusions because it is induced by the dislocation motion and the crossing of prismatic loops. In their model, the second stage of extrusion or intrusion growth is called stable growth [15,14] and is associated to the diffusion of vacancies. Since the effect of vacancy diffusion is not included in our model, that may be the reason for the saturation of extrusion and intrusion heights in the DDD simulation. Moreover, the saturation in the PSB surface roughness has been reported by Mughrabi [57] especially at low temperature and with vacuum environment.

The low temperature has prevented the diffusion of vacancies while the vacuum environment can reduce the irreversibility introduced by the oxidation at the surface. These experimental conditions match well the assumptions made in the DDD simulations which may be another explanation for the observed saturation phenomenon on the surface. In experiments, the slip-path of dislocations within PSBs has attracted many research interests such as the study comparing the rhythmical and randomly oriented flip-flop in loop patches [58]. Generally, it is thought that dislocations move in the PSBs in an energy-favored way to accommodate plastic deformation [59,60]. One of the models is the gliding of screw dislocations in the channels and bowing out of edge dislocations from the walls [16]. In order to understand the dislocation slip-path required to accommodate the plastic deformation during the saturation regime of the simulation, we have shown in Fig 5 the dislocation distribution evolution and the related surface deformation during the 40 th cycle. Snapshots and dislocation length analyses are performed at four instants of the cycle corresponding to the zero and maximal plastic deformations. First of all, during one cycle, the cyclic shear deformation is well simulated as evidenced by the changes in the surface shape during the cycle. The dislocation density also fluctuates with the variation of the applied deformation. It is mainly the dislocations in the two sides of the PSBs that multiply to accommodate the plastic deformation and the sign of increasing dislocations alternates with the direction of the applied load. Combined with the observation made in Fig 3(c), it can be inferred that in the simulation, the plastic deformation is mainly accommodated by the construction and destruction of pile-ups which are located at the interfaces between the PSBs and the matrix so that during the cyclic loading, the dislocation dipoles and prismatic loops located in the center of the PSBs are more stable and do not impede the pile-ups movements.

Dislocations move on slip planes (primary and cross-slip planes), driven by the corresponding shear stresses. According to the geometric relation between the simulated grain and the slip plane orientations (see Fig 1), dislocations which slip on the primary slip system mainly lead to the formation of pile-ups, dipoles in the PSB and dislocation crossing the free surface, while the movements in the transverse direction of the crystal is contributed by the dislocations slipping on the cross-slip plane. The localization of dislocations in the PSBs is the critical consequence of these transverse movements. To further explore the reasons for the dislocation localization, the distribution of the effective shear stress resolved on the cross-slip plane, including both the internal stress produced by dislocations and the applied stress, is presented in Fig 6 for different cycles. As indicated in the illustration, the plane used for the calculation is perpendicular to both the primary slip plane and the cross-slip plane, i.e. the plane normal is aligned with the Burgers vector shared by the two slip systems n n n = [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications[END_REF]. At the second cycle, the dislocations are distributed homogeneously enough but as indicated by the red box, most of the dislocations are bearing a very strong cross-slip shear stress, which means they are not in a stable position. Under the effect of the shear stress, dislocations will move in the cross-slip plane (in the transverse direction) during loading and unloading. Let recall here that the positive or negative moving direction is determined by the sign of the shear stress and also the sign of the dislocations. The shear stress on the cross-slip plane provides indirect information on the dislocation ability to interact so that from the 2 nd cycle to the 10 th cycle, dislocations show a tendency to repeal each other. That is to say, dislocations move towards the two opposite sides of the grain respectively to reduce the stress on the cross-slip plane, which enables the formation of two PSBs. As the dislocations are more isolated, the shear stress and hence interactions between the dislocations in the two PSBs get weaker. The continuing cyclic loading from 10 th cycle to the 26 th cycle allows more dislocations to move away from the two PSBs and be pushed away towards the grain outer surfaces, which contributes to the formation of four PSBs. At the same time, it can be observed that some dislocations are stored in the center of the PSBs (dislocation dipoles and prismatic loops) and the shear stress resolved on the cross-slip system becomes smaller. Finally, in the saturation regime, the cross-slip shear stress in the matrix between two PSBs is much reduced, which explains why the dislocations cannot further move in the matrix to form another PSB.

This can be considered as a key mechanism to clear the dislocations in the matrix. As a result, from the 26 th to the 50 th cycle, the number of PSBs and their positions remain constant. We can conclude that the stress state induces a self-organization of the dislocations that lead to the localization and PSB formations. And reciprocally, this localization largely reduces the dislocation interactions and hence the shear stress on the cross-slip planes. It stabilizes the dislocation microstructure so that the dislocations can move in a so-called low-energy mode with less stress interactions to accommodate the plastic deformation, which explains the saturation state from the 26 th cycle to the 50 th cycle.

Influence of the imposed plastic strain amplitude

The influence of the imposed plastic deformation on the fatigue behavior for FCC metals has been widely studied in experiments [18,12,19]. The plastic deformation amplitude is thought to have a significant effect on the fatigue behavior, especially on the microstructure, the macroscopic response and the hardening rate. For Copper single crystals, a larger deformation will eventually create more PSBs [12] and a very brutal raise of the applied plastic strain amplitude is able to activate more slip systems and will modify the PSB microstructure into cells [17]. In order to depict the effect of the imposed plastic strain amplitude, a simulation campaign is conducted on the same geometry (cylindrical grain D = H = 5µm) under five different strain amplitudes (ε V M p,applied = 0.6 × 10 -3 , 0.8 × 10 -3 , 1.0 × 10 -3 , 1.2 × 10 -3 and 1.5 × 10 -3 ). Simulations are performed both for Cu and 316L. Fig 7 shows the evolution of the total dislocations densities for each of the five applied strain values for Cu. Dislocation density increases along with ε V M p,applied . Regardless of the value of the applied strain amplitude, and as seen in the previous simulation, after about 20 cycles of loading, the saturation regime is systematically reached. In fact, this stands not only for the dislocation densities, but also for the PSB height and the surface roughness. p,applied and when ε V M p = 0. From the 3D snapshots, it can be intuitively seen that a larger strain amplitude leads to more storage of dislocations as well as more construction of PSBs. It is worthwhile mentioning that for a given material, instead of increasing the individual PSB's thickness to accommodate the plastic deformation, more PSBs are constructed as if each PSB could only accommodate a finite amount of plastic strain. Comparing with Cu, under the same deformation, 316L has constructed fewer PSBs.

Based on the linear relationship between the PSB volume fraction and the applied deformation for Cu in single slip mode, observed in experiments, Winter [12] has proposed a two-phases model to describe the relationship between the applied deformation and the formed PSBs. In his model, the plastic deformation is accommodated by the two phases separately: the first phase is the PSB, which is soft and active; the second one is the matrix, which is hard and almost inactive. The capacity of accommodation for the plastic deformation in these two phases are fixed. The relation is given as follows:

ε V M p,applied = ε p,P SB * f + ε p,M atrix * (1 -f ) ( 11
)
where f is the volume fraction of PSBs in the grain and ε p,P SB , ε p,M atrix are the plastic strain in the PSBs and the matrix respectively. This model has been applied successfully in many FCC metals such as Cu and Ni [12,54].

In order to verify if the evolution of the PSB volume fraction (f ) with the imposed plastic strain amplitude obtained in our simulations follows Winter's model, the PSB thicknesses are calculated using a method The volume fraction of PSBs is then calculated as the ratio between the total PSB thicknesses and the grain diameter D. The relation found between the PSB volume fraction and the imposed plastic strain amplitude for Cu and 316L is illustrated in Fig 10 . Since the matrix in our simulation contains almost no dislocations, ε p,M atrix is supposed to be zero so that the fitting function should write ε V M p,applied = ε p,P SB * f . As it can be seen Fig 10, such a linear relationship matches well the evolution of the PSB thickness with strain measured from our DDD simulations. The slope of the fitting linear equation gives the value of ε p,P SB for Cu and 316L. It is concluded that the capacity of a PSB to accommodate plastic strain for 316 (ε p,P SB = 5.055 × 10 -3 ) is bigger than that of Cu (ε p,P SB = 2.945 × 10 -3 ). The comparison between the behaviors of Cu and 316L is discussed in the last section.

316L Cu

In brief, for the first time, the linear relationship between the volume fraction of PSBs (f ) and the applied plastic strain is confirmed by DDD simulations. Such a relationship means that the formed microstructure in the PSBs has a fixed capacity to accommodate plastic deformation. Consequently, the PSBs adjust their volume fractions according to the imposed plastic strain amplitude.

Stability of the PSBs

Winter has constructed its model by analogy with the liquid-vapor system but there is one limitation: the transformation from vapor to liquid or from liquid to vapor is perfectly reversible while the constructed PSB cannot have such a high level of reversibility. Winter has confirmed this limitation by realizing experiments with a decreasing loading amplitude and observed surface slip lines of PSB. He concluded that few PSBs constructed at a high deformation amplitude survive when the amplitude is decreased but becomes inactive, which proves the irreversibility of the PSB microstructure. In order to investigate the irreversible character of PSBs formed in our DDD simulations, a two steps fatigue loading is applied on the Cu single crystal (D = H = 5µm). During the first 25 cycles, a plastic strain amplitude of ε V M p,applied = 1.2 × 10 -3 is applied and then reduced to ε V M p,applied = 0.6 × 10 -3 . The obtained stress-strain curve is plotted in An interesting feature concerns the evolution of the PSB microstructure distribution. In Fig 11(c), the number of PSBs in the last cycle before the strain drop is about 6 (two smaller PSBs and four thicker PSBs). The thicknesses of the PSBs are clearly presented in the dislocation distribution diagrams. After 35 cycles continuing the lower deformation loading: the dislocation density decreases and the thickness of each PSB also becomes thinner. On one hand, the four thicker PSBs stay at the same position after the reduction of loading, on the other hand, the two thinner PSBs have almost disappeared and only a few debris remains.

Winter's model has been cited in many publications but the fatigue test with a decreasing loading is rarely tried in practice. Blochwitz [10] has tested the evolution of PSBs with a decreasing loading for Ni and he has observed the changes induced on the PSBs. ) displays the macro-bands developed at the same sites after a decrease of the strain amplitude to ε p = 1.0 × 10 -3 until another saturation regime. By comparing the "old" and the "new" bands, the authors concluded that the positions of macro-bands at the higher and lower amplitude correspond well to each other, however, at the lower amplitude, the intensity of the bands is smaller than at higher amplitude. Our DDD simulations could reproduce these experimental results, especially, for the four thicker PSBs which stay at the same positions, but reduces in thickness after reduction of the plastic strain amplitude. Similarly, the debris observed at the lower applied strain remain at the same location as the PSBs formed during the high plastic strain loading. Even though the thickness scale of the experimental PSBs (∼ 10µm from Fig 12) and the numerical PSBs (∼ 1µm from Fig 3) in DDD is different, the interpreted similarity can be considered to be qualitative.

Comparisons between Cu and 316L

We have already mentioned in the discussions above that the fatigue behaviors of Cu and AISI 316L austenitic stainless steel are different. In Fig 8, under the same deformation amplitude, evidently, the distribution of dislocations for Cu and 316L is different. As an example, for ε V M p = 0.6 × 10 -3 , Cu has formed two PSBs while 316L has only constructed one PSB occupying less volume than Cu. The simulations have shown that this is a general trend: for a given plastic strain amplitude, Cu builds up more PSBs which occupy a larger volume fraction of the grain than 316L. From the fitting equation of Fig 10, we have concluded that 316L's PSBs have a higher capacity to accommodate the plastic deformation. A more quantitative comparison is now plotted in Fig 13 in terms of the thicknesses of the PSBs for the two materials. It can be seen that under the same strain amplitude, Cu is more likely to construct a thicker PSB. The distribution of individual PSB thickness in Cu is more dispersed than that in 316L. In fact, the dislocation density in the two materials is almost identical in the saturation regime. However, the dislocation arrangement in Cu is less compact than in 316L.

To understand the observed results, the PSB formation mechanism should be further investigated. As observed in the simulation and mentioned in [45], dislocation glide on the primary slip plane is dominant and dislocation glide on the deviate plane is minor but crucial for the deformation spreading in the transverse direction and for the formation of PSBs. Finally dislocation multiplication occurs in the crystal through a double cross-slip mechanism which consists of two consecutive cross-slip occurrences: from primary plane to deviate plane, then from deviate plane to primary plane. It can be inferred that the character of cross-slip plays a critical role in the dislocation distributions.

The probabilities for cross-slip in our model for Cu and 316L are plotted in Fig 14(a). Evidently, under the same shear stress τ dev , Cu has a higher probability of cross-slip. As indicated by the value given to τ III in Tab 2, cross-slip is triggered for a lower stress value for Cu (32MPa) than for 316L (56MPa) but this should be scaled by the elastic constant of the investigated materials. The simple ratio R cs = τ III µ can be defined to compare the cross-slip activities in the two materials. It is remarkable that this ratio is comparable for both materials: R Cu cs /R 316L cs = 1.03 while the fatigue behavior is different. All simulations for Cu and 316L starts with a single initial dislocation source. The primary microstructure to be constructed in the first cycle is one individual dislocation pile-up. The first cross-slip occurs in the pile-up for Cu when the applied plastic deformation is ε V M p = 0.25 × 10 -3 while in 316L, the first cross-slip happens at ε V M p = 0.65 × 10 -3 : it reflects also a higher possibility of cross-slip in Cu. During the whole simulation, a good indicator of the cross-slip activity is the number of slip plane changes in the microstructure. The evolution of this parameter with the number of cycles is plotted in Fig 14(b).

It is shown that the number of slip plane changes is increasing with the number of loading cycles. And for larger deformation, slip plane changes are more frequent. As expected, we observed that generally, for a given deformation, the number of slip plane changes in Cu is more important than in 316L. Even though this parameter doesn't distinguish what is cross-slip events and what is line recombinations, it proves fairly well that Cu presents the characteristic to trigger more cross-slip events than 316L.

The relatively higher cross-slip frequency of Cu has several consequences. Primarily, the mobility of dislocations is significantly enhanced. Because of this, dislocations in Cu can explore a larger volume which explains a higher possibility for Cu to form wide PSBs. Secondly, the easiness of cross-slip promotes the creation of more dislocation sources via the Koehler mechanism. More initial dislocation sources lead to more pile-ups as well as more PSBs to share the accommodation of the plastic deformation. Since the total plastic strain is fixed, the average plastic deformation accommodated by every pile-ups is smaller, which means that in Cu, fewer dislocation loops are contained in the piled-up zones. The number of pile-ups in one PSB is limited to a maximum of 2 per PSB (on the 2 sides). Thus, for Cu, the reduced number of dislocation loops in pile-ups limits the capacity of PSBs to accommodate the imposed strain so that more PSBs are required. It can be concluded that cross-slip easiness can be considered as the origin of the differences in the fatigue behaviors of Cu and 316L.

The cross-slip ability of 316L and Cu can also be deduced from the analysis of the shear stress resolved on the cross-slip system during the saturation regime (see Fig 15). As previously discussed in Fig 6, after the formation of four PSBs in Cu, the dislocations are well stabilized in the grain and the shear stress in the cross-slip system is very small. However, from the distribution of shear stresses on 316L, it can be seen that the shear stress on the cross-slip plane is still really high, at least bigger than the one of Cu, which means the dislocations in PSBs of 316L are likely to cross slip. Furthermore, dislocations in 316L have not Since the cross-slip probability is correlated to the stacking fault energy (SFE), the slip mode in fatigue is strongly determined by the SFE as reported in literatures [48,50,61,62]. For high SFE materials, like Cu, the slip mode is called wavy slip, while for low SFE, like Cu-16 at% Al alloy, the slip mode is called planar slip. The dislocation microstructure of planar slip shows dislocations confined in planes marked by dipole arrays whereas wavy slip materials prefer to construct a more three-dimensional microstructure where dislocations aggregate to form veins or walls. Even though veins or wall have not been simulated in this work, our simulation results for Cu and 316L correspond well to these experimental observations: since Cu has a higher SFE (80 M J/m 2 ) than 316L (30 M J/m 2 ), Cu should construct a more 3D structure, which is what we have obtained in our simulations: Cu constructs PSBs occupying more volume fraction than 316L. The 3D character is not evidenced by our simulations which were performed in single slip loading conditions, but surely a different loading state such as a double glide configuration, would lead to a more three-dimensional geometry for Cu than for 316L.

Concluding remarks

Three dimensional Discrete Dislocation Dynamic simulations have been conducted on a cylindrical surface grain of Cu to investigate the formation of PSBs and the involved mechanisms. A detailed analysis of the PSB formation process, the influence of the imposed plastic strain amplitude and the comparison between Cu and 316L behaviors, have led to the following conclusions : Localization of dislocations is observed in the simulations and results in the formation of slip bands, which are constituted mainly by dislocation dipoles and prismatic loops. These slip bands correspond to embryo of PSBs. Dislocation pile-ups are evidenced at the interface between the PSBs and the matrix. These pile-up dislocations accommodate most of the imposed plastic deformation. The mechanism of formation of the PSBs is highly related to the shear stress distributions in the cross-slip plane.

After a few cycles, a saturation regime is reached: the average dislocation densities, the PSB volume fraction, PSB location as well as extrusions and intrusions heights are stable. This saturation phenomenon is also observed experimentally in Copper surface grains.

The effect of the imposed plastic strain amplitude on the PSB volume fractions verifies the Winter two-phases model. To the best of the author's knowledge, this is the first time that this model is validated by DDD simulations.

By reducing the imposed strain amplitude, the degree of reversibility of formed numerical PSBs is verified. It is stated that the position of formed PSBs remains unchanged, which is also predicted by Winter's model.

When comparing similar simulations realized on two materials chosen for their difference in their stacking fault energy (Cu and 316L), it is found that Cu constructs more PSBs and has a higher tendency to build up thicker PSBs. Also, the capacity of the PSBs in Cu to accommodate plastic deformation is found to be weaker than in AISI 316L. The differences of cyclic behaviors are thought to be induced by a different cross slip ability. The easiness of cross-slip in Cu leads to a higher mobility of the dislocations which, on one hand, facilitates the PSBs construction and augmentation of PSB's thickness, and on the other hand, reduces the dislocations loops localized in the pile-ups, which reduces the plastic accommodation capacity.

Although most of the main characteristics of fatigue at the early stages are well reproduced by the simulations presented here, DDD application to fatigue still has some weakness. In experiments, the observation is realized at the surface of samples where the oxidation may contribute to the slip irreversibility of the bands, which can deepen the slip lines at the surface and eventually modify the extrusion growth rate. In our DDD simulations, the oxidation process at the free surface has not been considered such as many other factors like precipitation and forest hardening from the other slip systems, which can all contribute to the irreversibility in dislocation motion.

For forthcoming simulations of fatigue, the irreversibility in DDD simulations should be reconsidered. Because in a monotonic loading process, the irreversibility is not as important as in cyclic loading situation, it has often been neglected in DDD modeling of monotonic simulations. In fatigue simulations, the irreversibility is introduced mainly by cross-slip, dipole formations and line recombinations but if the deformation is too small, the dislocations cannot be stored due to the lack of irreversibility. This result is different from the experimental observations, in which the irreversibility can always be brought even in the case of high-cycle fatigue. The augmentation of the friction stress is a possible way to augment the irreversibility based on arguments from solid solution contents. The add of precipitations or secondary slip systems can also be a solution. To conclude, the current DDD model can still be optimized to better simulate dislocation interactions, typically for high cycle fatigue simulations. Fatigue is one of the main failure mechanism for metallic components. The early stages of fatigue refer to the period before the initiation and propagation of fatigue cracks, and accounts for up to 90% of fatigue life. Therefore, the understanding of fatigue damage mechanisms at the early stages is a key issue to improve the operational lifetime of components. Experimental studies have shed light on the importance of the Persistent Slip Band (PSB) as the sites of plasticity localization and subsequent initiation of cracks. The purpose of this thesis is to contribute to the understanding of the formation of fatigue microstructure at the scale of dislocations using 3D Discrete Dislocation Dynamics (DDD). First, simulations of single slip on Cu single crystal are realized. The formation process of the dislocation related microstructures inside PSBs and the surface roughness evolution are elucidated. Under a large strain (> 10 -3 ), a transformation process of the homogeneously distributed dislocations into the organized PSBs is observed, which can be well explained from the diminution of the shear stress on the cross-slip plane. The stability of the numerical PSB is verified with a decreasing loading and is found to be comparable to experimental results. Besides, the comparison between Cu and AISI 316L austenitic stainless steel confirms the importance of cross-slip probability to the distribution and number of PSBs. Simulations of different double slip combinations are also realized to identify the effect of dislocation interactions on cyclic behavior. In the end, the preliminary cyclic simulations for bi-crystals and aggregates are launched with a newly developed DDD code for poly-crystals. Keywords: early stages of fatigue, discrete dislocation dynamic, Cu, Persistent Slip Band (PSB), irreversibility, extrusions and intrusions Simulations à l'échelle mésoscopique du comportement en fatigue de métaux CFC La fatigue est l'une des principaux mécanismes de défaillance des composants métalliques. les stades précurseurs de l'endommagement par fatigue concernent la période avant l'initiation et la propagation des fissures de fatigue et représente jusqu'à 90 % de la durée de vie en fatigue. De fait, la compréhension des mécanismes de l'endommagement par fatigue aux stades précurseurs est un enjeu clé pour améliorer la durée de vie opérationnelle des composants. Des études expérimentales ont montré l'importance des Bandes de Glissement Intensif (BGIs) en tant que sites de localisation de la plasticité et d'amorçage des fissures. Le but de cette thèse est de contribuer à la compréhension de la formation de microstructure de fatigue à l'échelle des dislocations avec l'aide de la dynamique de dislocations discrètes en 3D (DDD). Tout d'abord, des simulations de glissement simple sur monocristal de Cu sont réalisées. Le processus de formation des microstructures liées à l'organisation des dislocations à l'intérieur des PSB et l'évolution de la rugosité développée en surface sont élucidés. Pour un chargement de fatigue en cisaillement simple sous une amplitude de déformation imposée importante (> 10 -3 ), on observe une réorganisation progressive des dislocations dont la répartition initialement homogène dans le grain se transforme en des microstructures organisées en BGIs. Le processus est expliqué à partir de calculs des contraintes internes sur le système dévié. La stabilité des BGIs simulées est vérifiée en diminuant subitement l'amplitude du chargement après les avoir construites. Les simulations se comparent bien avec les observations expérimentales de la littérature. En outre, la comparaison entre Cu et l'acier inoxydable austénitique 316L confirme l'importance de la probabilité de glissement dévié pour la distribution et le nombre de BGIs. Des simulations de différentes combinaisons de glissements doubles sont également réalisées pour identifier l'effet des interactions de dislocations sur le comportement cyclique. Finalement, des simulations cycliques de bi-cristaux et d'agrégats polycristallins sont réalisées grâce au nouveau code DDD dédié aux poly-cristaux.

Mots clés : les stades précurseurs de l'endommagement par fatigue, dynamique des dislocations discrètes, Cu, Bandes de Glissement Intensif (BGI), irréversibilité, extrusions et intrusions

Figure 1 . 1

 11 Figure 1.1 Schematic representation of the common parameters of a fatigue loading cycle [12]: (a) variation of stress as a function of time; (b) macroscopic response during one cycle also refereed to as the hysteresis loop.

Figure 1 .

 1 Figure 1.2 S-N diagram which presents the influence of the imposed stress on the number of cycles before failure, N f [15].

Figure 1 . 3

 13 Figure 1.3 Schematic presentation for different categories of fatigue [16].

Figure 1 . 4

 14 Figure 1.4 The crack size (or crack advance) as a function of loading cycle: (1) nucleation of crack; (2) propagation of micro-crack; (3) propagation of macro-crack; (4) failure [16].

Figure 1 . 7

 17 Figure 1.7 Schematic presentation of the dipole formation mechanism [23]: (a) cross-slip of an extended screw dislocation from one slip plane to another; (b) extended dislocations now spread out in two parallel slip planes under the action of applied shear stress resulting in two edge dislocation dipoles.

Figure 1 . 8

 18 Figure 1.8 TEM observations of the vein microstructure.

Figure 1 .

 1 Figure 1.10 Graphical representation of the dislocation arrangement in FCC metals: (a) veins structure in the matrix; (b) an amplified view for the dipolar walls and dislocation movement in the channel; (c) a simplified sketch of PSB geometry observed in Cu at 20 • C [26, 27].

Figure 1 .

 1 Figure 1.11 TEM image and simplified illustration of the PSB ladder and surrounding veins structure in the matrix.

Figure

  Figure1.12(a) shows 3D PSB in Cu. In the plane (1 2 1) plane (Figure1.12(b)), the ladder structure appears and is perpendicular to the primary Burgers vector b, which is the same geometric relation as in Figure1.10(c).If γ pl < γ pl,AB , vein structures predominate in the matrix. When the imposed strain amplitude reaches the threshold value for the formation of PSBs, the plastic deformation gets localized in PSBs. As mentioned in[31], this change in the dislocation organization means that the PSBs have a higher capacity to endure plastic deformation than the vein structure. With the increases of the applied plastic shear strain amplitudes γ pl , the volume fraction of PSBs increases to accommodate the increasing plastic deformation. The augmentation of PSB volume fraction ensures the stability of the saturation shear stress. This is the reason for the plateau observed in Figure1.6.The veins can be considered as the precursor of PSBs and they all contain many edge dislocation dipoles. Figure1.13 presents the microstructure transformation process: dislocation dipoles or multi-poles are produced during the early stages of the cyclic loading; the dislocation dipoles gather together in veins; more dislocation dipoles aggregate to construct the walls in PSBs and the deformation becomes heterogeneous.To explain the dislocation movements during the saturation state, the deformation is believed to be accomplished by the back and forth motion (flip-flop) of the dislocation loops[32] in the veins and PSBs. The screw dislocations move reversibly in the channel and the edge dislocations move from one dipole bundle to another[33]. Then, it has also been proposed that, the balance between multiplication and annihilation within the PSBs is responsible for further accumulation of plastic deformation whereas the local screw and edge dislocation densities remain unchanged[34].With the increasing imposed plastic strain amplitude, the dislocation structure changes. The fatigue microstructure evolution under an increasing applied strain is such

Figure 1 .

 1 Figure 1.12 TEM observations of the PSB structure [30].

Figure 1 .

 1 Figure 1.13 Illustration of the transformation process from dislocation dipole to veins and finally to PSBs: (a) dipole segment; (b) vein structure; (c) PSB ladder [23].

Figure 1 .

 1 Figure 1.15 Micrographs of the PSBs at the surface of the micro-sized sample (top left); illustration of the PSB formation (bottom left); observation of the wall structure to identify the PSB formation.[36]

Figure 1 .

 1 Figure 1.16 shows the typical intrusions and extrusions along with the slip bands for a Cu poly-crystal [28].

Figure 1 .

 1 Figure 1.16 Extrusions and intrusions observed at the surface of PSB for a Cu polycrystal [28].

1. 3 .

 3 Figure 1.17 Illustration of the different forms of the PSM morphology in fatigued FCC metals [42].

Figure 1 .

 1 Figure 1.18 Different morphology of extrusions at the 500 th cycle in the martensitic steel [43].

Figure 1 .

 1 Figure 1.19(b)), the reverse flux of mass in the PSBs results in an augmentation of mass in the channels and its diminution in the walls. The redundant mass can only be relaxed in the y-direction (noted in Figure 1.19(b)). With the continuously cyclic loading, the "tongue-like" extrusions appear periodically and the distance between "tongue-like" extrusions are highly related to the distance between walls. Figure1.19(c) presents the case for a "ribbon-like" extrusion, because of the mass flux from the matrix to the PSB, mass accumulates in the PSBs and decreases in the thin layer at the interface between the PSBs and the matrix. The redistribution of mass results in a compressive stress in the PSB lamella, which can only be released by gliding dislocations in the Burgers vector direction. This process leads to a "ribbon-like" extrusion with a width of the corresponding PSB. At the interface between the PSB and the matrix, the tensile stress rises and is relaxed by dislocation. Under this mechanism, a narrow intrusion appears and grows progressively to a crack.

Figure 1 .

 1 Figure 1.19 Illustration for the model of Polák [42]: (a) flux of vacancies marked as white arrows and evolution of the concentration of vacancies C v along with an individual PSB along A-A and B -B directions; (b) the formation process of "tongue-like" extrusions; (c) the formation process of "ribbon-like" extrusions and initiation of an intrusion or crack.

  .21(b).

  (a) Example for a microstructure in wavy slip (Ni), edge dislocations aggregate together to form dipole bundles [49]. (b) Example for a microstructure with planar slip (Cu-10 at% Al), dislocations forms extended planar pile-ups and multipole dislocations.

Figure 1 .

 1 Figure 1.21 A direct comparison of microstructure between planar slip and wavy slip [49].

1. 4 .Figure 1 .

 41 Figure 1.22 Slip path for wavy slip and planar slip [49].

Figure 1 .

 1 Figure 1.23 Free energy changes for low energy dislocation structure, wavy slip on the left, planar slip on the right.[48]

  Figure 1.25 CSS curves and the corresponding dislocation patterns for single Cu of different orientations [23].

Figure 1 .

 1 Figure 1.26 (a) The dislocation trace evolution with the increasing local cumulative strain for different orientations; (b) final dislocation structure for different orientations after 100 cycles loading [59].

1. 6 .

 6 Figure 1.28 The CSS curve for poly-crystal Cu [66, 67].

Figure 1 .

 1 Figure 1.29 Bi-crystal specimens for the fatigue test: (a) bi-crystal RB with a real grown parallel GB; (b) Combined bi-crystal CB without parallel GBs.

Figure 1 .Figure 1 .

 11 Figure 1.30 The CSS curves for RB and CB samples (a), and the increase with the applied strain amplitude distribution in the PSB of CB (b) and in the PSB of RB (c).

Figure 1 .

 1 Figure 1.32 Evolution of the dislocation density distribution with random initial distribution and the final labyrinth structure (Primary slip directions is the box diagonals) [83].

Figure 1 .

 1 Figure 1.33 (a) The classification of the dislocation density for a dislocation loop; (b) the edge dislocation distribution after 100 cycle loading [84].

Figure 1 .

 1 Figure 1.34 (a) the mesh setting for the simulation of a crystal with the orientation close to [3 29 1]; (b) the dislocation density after 50 cycles of simulation [86].

Figure 2 . 1

 21 Figure 2.1 Illustration of simulation results using three different simulation methods at different length scales for the nano-indentation problem [97].

Chapter 2 .Figure 2 . 2

 222 Figure 2.2 The discretization of space: (a) illustration of the discretized space for FCC crystal structure; (b) placement example of the screw and edge dislocations in an FCC unit cell. (The black points refer to the atom position at the corner of the cube, while the red points refer to the atom position at the face center of the cube)

Figure 2 . 3

 23 Figure 2.3 Discretization of a dislocation line: (a) dislocation line before discretization; (b) dislocation line after the discretization into edge dislocations (in green) and screw dislocations (in blue).

Figure 2 .

 2 Figure 2.4 presents the tetrahedron of Thompson [106] in the 3D coordinate system of the crystal lattice and the expansion view. Thompson tetrahedron represents well the geometric relations between the 18 vectors. The four triangles in different colors correspond to the four {1 1 1} slip planes. The intersection lines of different planes or the edges of the tetrahedron are the direction vectors of the 1 1 0 screw dislocations. The 3 central lines in each triangle plane are the direction vectors of the edge dislocations. In this 3D representation of the FCC directions, the symbol [abc signifies that vector [abc] is oriented in the direction indicated by the symbol .

Figure 2 . 4

 24 Figure 2.4 Illustration of the different vectors used in TRIDIS in the Thompson tetrahedra representation [93, 106].

Chapter 2 .

 2 Methodology of 3D Discrete Dislocation Dynamics

Figure 2 . 5

 25 Figure 2.5 Illustration for line tension correction in a symmetrical case [104, 108].

Figure 2 . 6

 26 Figure 2.6 Process of dislocation annihilation: (a) meeting of two dislocations of the opposed sign; (b) annihilation of the dislocation meeting parts and formation of two different arms.

Figure 2 . 7

 27 Figure 2.7 Double cross-slip process for a screw dislocation: (a) the dislocation slips on the primary slip plane; (b),(c) the dislocation changes to the cross-slip plane and begins to move; (d) the cross-slipped dislocation slip again and move back to the primary slip plane.

σ

  Li xy = b x xb y y r(rz) xy (b x yb y x)(2rz) r 3 (rz) 2 (2.20) where r = x 2 + y 2 + z 2 , b = (b x , b y , b z ) and it should be noted that the unity for the tensor is µ 4π(1-ν) . µ and ν are the shear modulus and Poisson coefficient respectively of the simulated material.

Figure 2 . 8

 28 Figure 2.8 Illustration for the calculation of the stress field at point P from the dislocation segments [SE] of finite length L and Burgers vector b [117].

Figure 2 . 9

 29 Figure2.9 Illustration of the calculation procedure to evaluate the displacement field of a closed dislocation loop: (a) loop of any arbitrary shape[START_REF] Hirth | Theory of dislocations[END_REF]; (b) case of a triangular loop.[START_REF] Fivel | An easy implementation of displacement calculations in 3D discrete dislocation dynamics codes[END_REF]

2 a

 2 = arccos ( r b -r).( rc-r) r br . rc-r b = arccos ( ra-r).( rc-r) ra-r . rc-r c = arccos ( ra-r).( r b -r) ra-r . r br

Figure 2 .

 2 Figure 2.10 Division of a random dislocation loop into triangles: (a) a convex dislocation loop; (b) a concave dislocation loop.[START_REF] Fivel | An easy implementation of displacement calculations in 3D discrete dislocation dynamics codes[END_REF]

  (a) The missing triangle using the former algorithm for non-coplanar loop.[START_REF] Fivel | An easy implementation of displacement calculations in 3D discrete dislocation dynamics codes[END_REF] (b) Optimized algorithm for the division of a non-coplanar dislocation loop in triangles: (b.1) graphic expression for the algorithm; (b.2) application in a complete non-coplanar dislocation loop.[START_REF] Fivel | An easy implementation of displacement calculations in 3D discrete dislocation dynamics codes[END_REF]

Figure 2 .

 2 Figure 2.11 Comparison between the two division methods for non-coplanar dislocation loop.

Figure 2 .Figure 2 .

 22 Figure 2.12 Division in triangles for the prismatic dislocation loop [118]: (a) Schematic representation of the set of prismatic loops as a superposition of 4 glissile loops; (b) presentation of the triangles formed by the calculation algorithm; (c) calculation result on a plane in the middle of the two prismatic loops;

Figure 2 .

 2 Figure 2.14 The virtual space simulation setting: (a) the simplified illustration; (b) the real situation in the simulation.

  (a) Dislocation configuration evolution observed in 2D discrete dislocation dynamics: from random distribution dislocation sources to matrix-wall structure and then to the matrix-vein structure. (b) Dislocation pattern observed by ECCI technique: matrixwall structure in the left; matrix-vein structure in the right.

Figure 2 .

 2 Figure 2.15 Evolution of dislocation patterning observed from 2D DDD simulation and from ECCI technique respectively [123].

  (a) The final dislocation microstructure for simulations set with the same initial dislocation density ρ dis initiation = 1.0 × 10 12 m -2 but different crystal sizes D = 1.0µm for the left one, D = 5.0µm for the right one. (b) The final dislocation microstructure for simulations set with same crystal size D = 2.0µm but different initial dislocation density ρ dis initiation = 1.0 × 10 11 m -2 for the left one, ρ dis initiation = 1.0 × 10 13 m -2 for the right one.

Figure 2 .

 2 Figure 2.16 The final dislocation microstructure for DDD simulation with different crystal sizes and initial dislocation densities [125].

  (a) Prismatic loops' order and dislocation debris (The color indicates the normal plane and Lomer junction is presented in yellow). (b) Evolution of prismatic loop numbers and dislocation debris as a of the loading cycles.

Figure 2 .

 2 Figure 2.18 Illustration of the order of prismatic loops and of their evolution with the loading cycles [126].

Figure 2 .

 2 Figure 2.19 Comparison between the number of vacancy prismatic loops and interstitial prismatic loops[START_REF] El-Achkar | Analysis of dislocation microstructure characteristics of surface grains under fatigue loading by discrete dislocation dynamics[END_REF].

Figure 2 .

 2 Figure 2.20 Dislocation position around the neutral line before bending, at bending and after bending with M B,max the bending moment [128].

Figure 2 .

 2 Figure 2.21 Schematic illustration of the crack initiation proposed by Christophe DEPRES, which includes the extrusions height, the stored elastic energy and the stress concentration [134].

Figure 3 . 1

 31 Figure 3.1 Simulation settings: (a) grain geometry, the boundary conditions and initial condition of the simulated grain; (b) the geometric illustration for the primary and deviate slip plane, as well as the loading direction.

Figure 3 . 2

 32 Figure 3.2 The evolution of the dislocation related microstructure with the loading cycles: (a) on the primary plane; (b) on the deviate plane.

Figure 3 . 3

 33 Figure 3.3 Evolution of the dislocation density in function of cycles for the primary and deviate slip systems.

Plane 1 2 Figure 3 . 4

 1234 Figure 3.4 Illustration of the process of measuring dislocation distributions.

Figure 3 . 5

 35 Figure 3.5 Quantification for the evolution of primary dislocation distributions and deviate dislocation distributions.

3. 3 .

 3 Evolution of the dislocation microstructure are stored in the form of dipoles, prismatic loops and tangles. At the two sides of the PSB, near the grain boundaries, two channels containing a few dislocations can be clearly seen. In these channels, dislocations are arranged in columns and aggregate to form dislocation dipoles arrays. In the central part, a channel also exists: many dislocation dipoles are arranged, but combined with many dislocation debris. Most of the dislocation dipoles in the central part of PSB have been transformed into prismatic loops by the cutting of dislocation lines gliding on the deviate slip system. The formation of dipoles and tangles helps to store dislocations in the grain. The main reason for the tangle formation is the different mobility of the dislocations and the collinear interactions in different parts of the PSB. With the increasing applied plastic deformation (Figure3.6(b)), most of the dislocation lines or tangles are destroyed or extended by the strong applied shear stress at the maximum strain.

Figure 3 . 6

 36 Figure 3.6 Numerical microstructure of a PSB at the 11 th cycle: (a) for ε V M p = 0; (b) for ε V M p

Figure 3

 3 Figure 3.7 (a) Dipole arrays observed in DDD simulation at the right side of PSB; (b) dipole arrays observed in Cu-16 at% Al alloys at a plastic strain amplitude ε pl = 3.0×10 -3 [52].

Figure 3 . 8

 38 Figure 3.8 Dislocation movement in the PSB during first half cycle of the first cycle, presented in two view directions: (a) dislocation loops are generated from one dislocation source and move on the same slip plane and a reversible pile-up is constructed; (b) cross-slips occur at the two sides of the pile-up and in the opposing direction, due to the opposite dislocation sign; (c) as the applied stress decreases, the dislocation loops begin to move backward; (d) when the global deformation returns to zero, dislocation dipoles have been built-up. (the red arrows indicate the direction of dislocation motion in the primary slip plane while the orange arrows indicate the direction of dislocation motion in the cross-slip planes)

Figure 3 . 9

 39 Figure 3.9 Dislocation movement in the PSB during the second half cycle of the first cycle, viewed from two directions: (a) driven by the increasing applied stress, dislocation loops begin to move again towards the grain boundary; (b) at the point of maximal deformation, dislocation pile-ups are constructed again but over a larger thickness, with dislocation dipoles stored inside; (c) the reduction of the applied stress leads to the collapse of dislocation pile-ups; (d) more dislocation dipoles are stored when the global deformation returns back to zero. (the red arrow indicates the direction of dislocation motion on the primary slip plane)

Figure 3 .

 3 Figure 3.10 Dislocation dipole formation mechanism proposed by Johnston and Gilman [141]: (a) XX is a screw dislocation and a cross-slip occurs at the point P ; (b) the XX passes the point P and a jog of height Y is formed; (c) if the size of Y is intermediate, a dislocation dipole can be constructed.

Figure 3 .

 3 Figure 3.11 Dislocation dipole formation mechanism proposed by Teleman [142]: (a) two dislocations M M and N N on two parallel slip planes at a distance Y are moving towards each other; (b) the two dislocations cross each other at the point O and the projection intersection angle is 2θ 1 ; (c) the closest parts of the two dislocations P P and RR reorient to be parallel to lower their energy; (d) to lower again their energy, P P and RR can adjust their distance in the slip direction such that X = Y .

Figure 3 .

 3 Figure 3.12 Formation process of dislocation dipole arrays observed near the grain boundary: (a) driven by the primary shear stress, dislocations gather at the grain boundary in the form of pile-up (we focus on the second dislocation counted from the right); (b) the continuously increasing deviate shear stress leads to cross-slips at different positions along this dislocation line; (c) the applied stress begins to be drawn back, at the same moment, the cross-slipped dislocations move back to the primary slip plane and continue moving, which creates two primary edge dislocation tails of opposing signs; (d) the created edge dislocations keep moving and encounter the edge dislocations which enables the formation of many dislocation dipoles; (e) seen from the side view of (d), the dipole structure is more clear to distinguish. ( v is the velocity of the dislocation)

  Figure 3.13 Mechanism of prismatic loop formation proposed by[START_REF] Li | Cyclic response and dislocation structures of AISI 316L stainless steel. Part 1: single crystals fatigued at intermediate strain amplitude[END_REF]: (a) recombination between edge dislocation dipoles and a passing screw dislocation gliding on the deviate slip system; (b) during the second time combination, if the passing screw dislocations are of the same sign as the precedent one, a helices-dislocation is constructed; (c) during the second time combination, if the passing screw dislocations are in the different sign to the precedent one, a prismatic dislocation loop is constructed.[START_REF] Déprés | Modélisation physique des stades précurseurs de l'endommagement en fatigue dans l'acier inoxydable austénitique 316[END_REF]

Figure 3 .

 3 Figure 3.14 Formation of screw dislocation debris in the central part of the PSB: (a) two screw dislocations of opposing signs, coming from the two sides of the PSB, meet each other in the central part of the PSB; (b) because of their opposing signs, the two screw dislocations move in the opposing direction under the same applied stress; (c) the strong internal attraction stress results in a cross-slip; (d) the annihilation happens on the cross-slip plane when the two screw dislocations meet and screw dislocation debris is left on the deviate slip plane because of the different screw dislocation lengths.

Figure 3 .

 3 Figure 3.15 The dislocation traces left after the annihilation: TEM bright field (a) and dark field (b) images of the dislocation traces between the dislocation walls of the PSB show the evidence of screw dislocation cross-slip[145, 146].

3. 5 .Figure 3 .

 53 Figure 3.16 The configuration of free surface at the final cycle: (a) position of the calculating plane for the surface roughness on the free surface; (b) configuration of the free surface at the sites of PSB in 3D; (c) three profile lines in the direction crossing the PSB; (d) two profile lines along the PSB. (the color indicates the value of the displacement projected in the normal direction of the surface)

  .

Figure 3 .

 3 Figure 3.17 Evolution of the free surface relief from the 1 st cycle to the 12 th cycle.

3. 5 .

 5 Figure 3.18 Evolution of maximal intrusion height and maximal extrusion height in function of the loading cycle.

Figure 3 .

 3 19 presents an example of the local extrusion growth. It can be seen that, the growth of extrusions consists of the formation of a long extrusion (at the 4 th cycle in Figure3.19(b)). At the 8 th cycle (Figure3.19(c)), on the top of a long extrusion, two more layers of extrusions are built. At the 12 th cycle (Figure3.19(d)) the long extrusions are sub-divided into middle sized extrusions, and more minor extrusions are constructed on the top. The augmentation of the extrusion height is a sort of pilling up mechanism and the same mechanism can be inferred for the intrusions but in an opposing direction.

Figure 3 .

 3 Figure 3.19 The evolution of a local extrusion with the number of loading cycles.

3. 5 .Figure 3 .

 53 Figure 3.20 Simplified illustration for the relation between surface steps and internal dislocation loops: (a) surface step formation by the passage of one dislocation loop of opposing sign; (b) surface extrusion and intrusion formation because dipoles of different types inside the crystal; (c) by the consideration of a 3 rd dislocation loop, extrusion can be declined or elevated entirely.

Figure 3 .

 3 Figure 3.21 Illustration of differences between the dipoles or prismatic loops in bulk of the crystal and the surface morphology: (a) surface relief and the related dislocation microstructure in bulk; (b) dislocations dipoles and the related "ribbon-like" extrusions or intrusions; (c) dislocations prismatic loops (dash = vacancy type, continuous = interstitial type) and the related "tongue-like" extrusions or intrusions.

3. 5 .

 5 Surface roughness at the PSB location

Figure 3 .

 3 Figure 3.22 MD simulations of the stable dislocation configuration [149] after the energy minimization (two minimization way GLOK and FIRE) for different positions of the initial dislocations.

3. 5 .Figure 3 .

 53 Figure 3.23 Evolution of extrusion volume and intrusion volume: (a) method to calculate the volume on an element of the free surface; (b) evolution of total extrusion volume and total intrusion volume at the free surface with the loading cycles; (c) illustration for the positions of the calculation planes utilized to calculate the corresponding extrusion and intrusion volume in different layers; (d) the evolution of extrusion and intrusion volume with the position of the calculation plane at the 12 th cycle.

  24(a). The prismatic loops of the same type have a square shape of 4µm × 4µm and are piled up in a layer of 12µm height. The dislocations are tightly placed in the layer. Results are shown in Figure3.24(b,c,d). The prismatic loops closer to the top surface move further and further while the prismatic loops at the bottom move and are stopped by the boundary.

Figure 3 .

 3 Figure 3.24 Simulation to test the limit of the number of prismatic loops in a fixed layer: (a) 100 interstitial prismatic loops of 4µm size are inserted in a layer of 12µm, as the initial condition; (b,c,d) without the applied stress, driven by the internal stress, dislocation loops are repealing each other.

Figure 3 .

 3 Figure 3.25 Comparison of dislocation density evolution for Cu and 316L.

3. 7 .Figure 3 .

 73 Figure 3.26 Comparison of dislocation distribution and the PSB thickness for Cu and 316L in the last cycle of simulation (12 th cycle for Cu, 50 th cycle for 316L).

Figure 4 .

 4 Figure 4.1(b) shows the relative orientations of the primary and deviate slip planes. The green plane is the primary plane (system 8 (1 1 1)[1 1 0]) and the blue plane corresponds to the deviate slip system 7: (1 1 1)[1 1 0].

Figure 4 . 1

 41 Figure 4.1 Simulation settings: (a) grain geometry and initial dislocation configuration; (b) the geometric illustration of the primary and deviate slip planes and loading direction.

4. 3 .

 3 Evolution of the microstructures distributions computed at the maximal deformation are shown in Figure 4.2(a) for different cycle numbers.

Figure 4 .

 4 Figure 4.2(a)) is presented in Figure4.2(c) when the VonMises equivalent plastic strain is zero and when it is maximal. When ε V M p,applied = 0, dislocations exist in the form of dipoles, prismatic loops and tangles. Multiple dislocation dipoles are built-up by the attraction of two edge dislocations of opposed signs and are arranged in regular arrays. In the center of the PSB, due to the back and forth motion of dislocations by cross-slip during the loading and unloading stages, dislocation dipoles are cut into pieces by line multiple recombinations which create dislocation prismatic loops. Tangled dislocations are also formed from the recombination process on the side of the prismatic loops. When the applied plastic strain amplitude is increased, most of the dislocation lines or tangles are destroyed or extended by the stronger applied shear stress at maximal deformation. In contrast, dislocation dipoles and prismatic loops show more permanence because most of them stay almost pinned by the stress. The low mobility of these structures suggests that they contribute little to the accommodation of the plastic deformation. Nevertheless, dislocation pile-ups are found to be built up near the grain boundaries at the maximal deformation. These observations show that the build-up and build-down of the dislocation pile-ups correspond to the fluctuation of reversible dislocations while the persistence of dislocation dipoles and prismatic loops refers to the storage of dislocations.

Figure 4 . 2

 42 Figure 4.2 Simulation results for a cylindrical Cu grain (D = H = 5µm) at ε V M p,applied = 10 -3 : (a) Evolution of the dislocation microstructure with the increasing cycles at maximal deformation; (b) dislocation density evolution for primary and deviate slip systems with the number of loading cycles; (c) snapshots of the dislocation microstructure inside a PSB formed at cycle 26 (see green box) observed at zero and maximal deformation.

Figure 4 . 3

 43 Figure 4.3 Distribution histogram evolution of the primary dislocation lengths for the simulation in a cylindrical grain of Cu (D g = h = 5µm under ε V M p,applied = 10 -3 ).

Figure 4 . 4

 44 Figure 4.4 Illustration of the double cross-slip mechanism to explain the dislocation moving mechanism in the crystal.

Figure 4 . 5

 45 Figure 4.5 Mapping of the shear stress resolved on the deviate slip system induced by dislocations piled-up on the primary slip system.

Figure 4 . 6

 46 Figure 4.6 Surface relief on the free surface of the grain when the global deformation is zero for simulation of a Cu cylindrical grain (D = H = 5µm) under (ε V M p,applied = 10 -3 ): (a) Global configuration of the surface with the corresponding dislocation microstructure at cycle 26; (b) Zoom on the local growth of one extrusion during 4 cycles; (c) Line profile at cycle 26 plotted along with the diameter (blue line); (d) Evolution of the maximal extrusions/intrusions heights with loading cycles;

Figure 4 . 7

 47 Figure 4.7 Dislocation density evolution detected by neutron diffraction for Ni-based alloys [154].

Figure 4 . 8

 48 Figure 4.8 Average extrusion height evolution in Cu poly-crystals [155].

Figure 4 . 9

 49 Figure 4.9 Surface deformation and evolution of the dislocation distributions at the 50 th simulated cycle for different values of the plastic strain cumulated during the loading cycle (D = H = 5µm) under ε V M p,applied = 10 -3 (the deformation has been amplified 300 times for clarity).

Figure 4 .

 4 Figure 4.10 Evolution of the effective shear stress distribution on the cross-slip system inside the grain with the formation of PSBs for simulation on Cu (D = 5µm and H = 5µm) under ε V M p,applied = 10 -3 .

Figure 4 .

 4 Figure 4.11 Evolution of the total dislocation density with the number of loading cycles for different applied strain amplitudes (Cu, cylindrical grain D = 5µm and H = 5µm).

Figure 4 .

 4 Figure 4.12 Dislocation distribution for Cu under different plastic deformation.

Figure 4 .

 4 Figure 4.13 Relation between the PSB volume fraction and the applied strain amplitude for Cu.

Figure 4 . 4 . 7 .

 447 Figure 4.14 Illustration of PSB thickness measurement: (a) Dislocations microstructure (PSB) inside the crystal with five profile lines at the free surface; (b) PSB line profiles at maximal strain; (c) dislocation distribution histograms at the maximal and zero strain (316L, cylindrical grain, ε p,applied = 0.6 × 10 -3 , D = 5µm, H = 5µm, at the 40 th cycle).

Figure 4 .

 4 15(b), the dislocation density drops down sharply at the 26 th cycle and decreases gradually in the subsequent 15 cycles to attain a new saturation regime.

Figure 4 .

 4 Figure 4.15 (a) Stress-strain curve for the simulation with the decreasing applied deformation, the first 25 cycles are applied with ε V Mp,applied = 1.2 × 10 -3 then the loading amplitude is reduced to ε V M p,applied = 0.6 × 10 -3 ; (b) dislocation density evolution for primary and deviate slip systems with the loading cycles.

Figure 4 .

 4 Figure 4.16 Illustration of the two loading paths used for comparisons.

Figure 4 .

 4 19 shows the main experimental results of this publication. The position of macro-bands developed at the plastic amplitude ε p = 3.0 × 10 -3 in the saturation regime can be observed in Figure 4.19(a) while Figure 4.19(b) displays the

Figure 4 .Figure 4 .

 44 Figure 4.17 Dislocation distribution comparison before and after the deformation reduction.

Figure 4 .

 4 Figure 4.19 Comparison of formed macro-band position: (a) the positions at high amplitude ε p = 3.0 × 10 -2 ; (b) the position at lower amplitude ε p = 1.0 × 10 -2 [162].

4. 8 .

 8 Comparison between Cu and 316L 316L Cu

Figure 4 .

 4 Figure 4.20 Dislocation distributions at the 40 th cycle for different applied strain amplitudes in Cu and 316L (cylindrical grain, D g = 5µm, H = 5µm).

Figure 4 .

 4 Figure 4.21 (a) The linear relation between the volume fraction f and the applied deformation amplitude ε V M p,applied for both Cu and 316L; (b) thickness for individual PSBs under different applied deformation for Cu and 316L (cylindrical grain D = 5µm, H = 5µm).

4. 8 .

 8 Comparison between Cu and 316L

Figure 4 .

 4 Figure 4.22 (a) Probability of cross-slip in the model of DDD for Cu and 316L; (b) Evolution of the number of slip plane change for Cu and 316L under different applied deformation with loading cycles.

Figure 4 .

 4 Figure 4.23). As previously discussed in Figure4.10, after the formation of four PSBs in Cu, the dislocations are well stabilized in the grain and the shear stress in the cross-slip system is very small. However, from the distribution of shear stresses in 316L, it can be seen that the shear stress on the cross-slip system is still very strong, at least bigger than the one of Cu, which means the dislocations in PSBs of 316L are likely to cross slip. Furthermore, dislocations in 316L have not moved to another place for self-stabilization. It is because in 316L, cross-slip requires much larger shear stress, even though the shear stress near the PSBs is really high, dislocations in the PSB cannot slip on the cross-slip plane to seek for a more stable position.

Figure 4 .

 4 Figure 4.23 Comparison of shear stress on the cross-slip system in Cu and 316L at the 50 th cycle (Cylindrical grain, D = 5µm, H = 5µm, ε V M p

Figure 4 . 25 .Figure 4 .

 4254 Figure 4.24 Macroscopic response of Cu in a cylindrical grain (D g = h = 5µm) under ε V M p,applied = 10 -3 : (a) evolution of applied stress with the loading cycles; (b) hysteresis stress-strain curve.

Figure 4 .= 1 .

 41 Figure 4.25 Evolution of applied stress with loading cycles for the different number of initial dislocation sources ε V M p

Figure 4 .= 1 . 2 ×

 412 Figure 4.26 Evolution of the applied stress with the loading cycles: (a) simulation with a decreasing cyclical loading, ε V M p

Figure 4 .Figure 4 .

 44 Figure 4.27 Effect of α f r on the macroscopic response: (a) the strain-stress hysteresis loop for different values of α f r ; (b) maximal applied stress for different α f r compared with experimental value [7].

Figure 4 .

 4 Figure 4.28 Dislocation distribution at the 5 th cycle for different values of α f r in the transversal direction.

Figure 4 .

 4 Figure 4.29 (a) Dislocation distribution at the 5 th cycle for values of α f r = 50 and 200 viewed along with the primary slip plane normal at zero deformation; (b) evolution of the number of dislocation segments with the number of cycles for different values of α f r .

Figure 4 .

 4 Figure 4.30 Evolution of component F x of the force between two edge dislocations of opposite signs in three cases (the scenario is presented in (a)): (b) no external stress field; (c) an external stress field constant value exists; (d) an external stress field of varying amplitude is applied.

Figure 5 . 1

 51 Figure 5.1 Dislocation interactions in FCC metals: (a) lomer-cottrell lock; (b) Glissile junction (c,d) Collinear junction; (e,g) Illustration of multi-junction formation [163].

Figure 5 . 2

 52 Figure 5.2 Schematic representation of the geometry and slip systems simulated in the single (a) and double slip (b) cases.

Figure 5 . 3

 53 Figure 5.3 Initial condition and geometry relation between slip planes and crystal orientation.

Figure 5 . 4

 54 Figure 5.4 Dislocation density evolution with the loading cycle for the following double slip configurations: (a) Lomer-Cottrell lock interaction (S1 and S3); (b) Glissile interaction (S1 and S10); (c) Hirth interaction (S1 and S12); (d) coplanar interaction (S10 and S12);(e) collinear interaction (S7 and S8).

Figure 5 . 5

 55 Figure 5.5 Dislocation distribution at the final cycle for: (a) Lomer-Cottrell lock interaction (S1 and S3); (b) Glissile interaction (S1 and S10); (c) Hirth interaction (S1 and S12); (d) coplanar interaction (S10 and S12);(e) collinear interaction (S7 and S8).

Figure 5 . 6

 56 Figure 5.6 Surface relief and profile along the black line indicated on each sample surface for: (a) Lomer-Cottrell lock interaction (S1 and S3); (b) Glissile interaction (S1 and S10); (c) Hirth interaction (S1 and S12).

Chapter 5 .

 5 Fatigue behavior in double slip loading condition

Figure 5 . 7

 57 Figure 5.7 Surface relief and profile along the black line indicated on each sample surface for: (d) coplanar interaction (S10 and S12); (e) collinear interaction (S7 and S8).

Figure 5 . 8

 58 Figure 5.8 Evolution of maximal extrusion and intrusion height: (a) Lomer-Cottrell lock interaction (S1 and S3); (b) Glissile interaction (S1 and S10); (c) Hirth interaction (S1 and S12); (d) coplanar interaction (S10 and S12); (e) collinear interaction (S7 and S8).

  has combined Electron Backscatter Diffraction (EBSD), Digital Image Correlation (DIC) and Electron Channelling Contrast Imaging (ECCI) to identify the effect of the grain orientation on the dislocation patterns in grains oriented for double slips ([1 1 2], [1 2 2] and [0 1 2]) for an austenitic stainless steel polycrystal. More recently, Nellessen [167] has investigated the fatigue behavior of Aluminum bi-crystals in double slip orientations [ 2 3 2], [ 3 3 2] and [1 3 0].

Figure 5 . 9

 59 Figure 5.9 Stereographic triangle indicating the double slip modes for different orientations of the loading axis [166].

Figure 5 .

 5 Figure 5.10 Experimental observations in the Lomer-Cottrell lock double slip situation: (a) slip traces for the two slip systems [168]; (b) intersection zone of the two ladder structures [166]; (c) dislocation distributions obtained by the simulation.

Figure 5 .

 5 Figure 5.11 Experimental observations and simulation results in the coplanar double slip mode: (a) slip bands formation at the free surface [167]; (b) the dislocation patterns inside the crystal at the PSB [164]; (c) dislocation distribution obtained by simulation in the slip band at the final cycle.

5. 7 .

 7 Back stress and friction stress agreement with the simulated collinear dislocation microstructure in Figure5.12(c).

Figure 5 .

 5 Figure 5.12 Experimental observations and simulation results in the colinear double slip mode: (a) TEM observation on the surface of a pipe in CYTHIA thermal fatigue test [95]; (b) dislocation distribution obtained by the simulation at the final cycle.

Figure 5 .

 5 Figure 5.13 Illustration of the stress decomposition in Cottrell model: τ f is the friction stress and τ b the back stress.

Figure 5 .

 5 Figure 5.14 Macroscopic response for the different double slip configurations: (a) peak stress evolution τ E ; (b) yield stress τ S ; (c) friction stress τ f ; (d) back stress τ b .

Figure 5 .

 5 Figure 5.15 (a) Evolution of the normalized hysteresis loop for Cu single crystal under different strain amplitudes [8]; (b) normalized hysteresis loops simulated in Cu for the double-slip and single slip configurations.

  .1, the transformation of the reference (O, X, Y, Z) to (O , X , Y , Z ) is obtained by successively applying: (1) the rotation of the angle ψ around the axis OZ ((O, X, Y, Z) to (O, U, V, Z)); (2) the rotation of the angle θ around the axis OU ((O, U, V, Z) to (O, U, W, Z )); (3) the rotation of the angle Φ around the axis OZ ((O, U, W, Z ) to (O , X , Y , Z )).

Figure 6 . 1

 61 Figure 6.1 Illustration of Euler angles (ψ, θ, Φ) to definer the rotation of a solid.

Figure 6

 6 Figure 6.2 (a) Geometry of the polycrystalline aggregate; (b) illustration of the initial condition and the grain numbering.

Figure 6 .

 6 Figure 6.3 (a) Evolution of the applied stress with the loading step; (b) evolution of the total dislocation density in the poly-crystal with the loading step.

Figure 6 .

 6 Figure 6.4 (a) Total dislocation density evolution of each grain with the number of calculation steps; (b) distribution of the Von Mises plastic strain in each grain (the blue line represents the total plastic strain of the poly-crystal aggregate).

Figure 6 . 5

 65 Figure 6.5 Dislocation distribution of the final cycle at the maximal strain amplitude.

Figure 6 .

 6 Figure 6.6 (a) The location of the loading direction in the stereographic projection map of each grain; (b) the slip band localization in each grain.

Figure 6 . 7

 67 Figure 6.7 Dislocation density evolution of each slip system in each grain: (a) grain 1; (b) grain 2; (c) grain 3; (d) grain 4; (e) grain 5; (f) grain 6; (g) grain 7.

Figure 6 . 8

 68 Figure 6.8 Indication for the equibiaxed traction-compression loading in CYTHIA thermal fatigue experiment [95].

Figure 6

 6 Figure 6.9 (a) Shape of the grains for the bi-crystal used in DDD simulation; (b) top view along [1 1 0] of the two grains.

  .11(b)), at the initial condition, the dislocation sources of the two grains are perpendicular, because the axis of rotation [2 2 1] is approximately parallel to the crystal orientation [1 1 0], it looks like a rotation of 90 • around [1 1 0]. Then, the majority of the dislocations are stored in the grain 1. For grain 2, from the first cycle to the 10 th cycle, there is only one constructed pile-up. The first cross-slip

Figure 6 .

 6 Figure 6.10 Dislocation density evolution for different misorientation angles: (a) total dislocation density evolution; (b) dislocation density evolution in grain 1; (c) dislocation density evolution in grain 2.

Figure 6 .

 6 Figure 6.11 Dislocation distribution evolution in the case of rotation angle equal to 0 • and 90 • .

+Figure 6 .

 6 Figure 6.13 Coupled simulations between DDD and FEM (Cast3M) on a polycrystalline aggregate obtained from EBSD.

Figure 6 .

 6 Figure 6.14 Coupling simulation between DDD and XFEM to study the propagation of a crack.

Figure 1 :

 1 Figure 1: Illustration of the grain shape, the relative positions of the slip systems, the initial condition, and the loading mode used in the DDD simulations.

Fig 2 . 2 Plane 1

 221 Let assume that the two labeled planes (Plane 1 and Plane 2) in Fig 2 are two primary slip planes. The total length of dislocation lying on each slip plane, noted as L1 and L2, is calculated. The evolution of the total dislocation length on every slip plane of the grain is plotted with the relative plane position calculated along the direction of the normal to the slip plane (as shown inFig 2). With this method, the spatial distribution of the primary dislocations along the transverse direction is calculated and the obtained histograms give information on the dislocation density distribution in the grain: peaks stand for slip bands. If needed, the quantitative evolution of positive or negative dislocations is a possible outcome of this method.P o si ti o n 1Po si tio n 2 L1: Total length of dislocations on the primary slip plane length of dislocations on the primary slip plane Plane 2

Figure 2 :

 2 Figure 2: Illustration of the method used to quantify the dislocation distributions.

3. Results and discussions 3 . 1 .

 31 Formation process and distribution evolution of the slip bands This section presents the simulation of Cu in a cylindrical grain shown in Fig 1 (D = H = 5µm) under ε V M p,applied = 10 -3 applied plastic strain amplitude. The dislocation distributions computed at maximal deformation is shown in Fig 3(a) for different cycle numbers.

Figure 3 :

 3 Figure 3: Simulation results for a cylindrical Cu grain (D = H = 5µm) at ε V M p,applied = 10 -3 : (a) Evolution of the dislocation microstructure with the increasing cycles at maximal deformation; (b) dislocation density evolution for primary and deviate slip systems with loading cycles; (c) snapshots of the dislocation microstructure inside a PSB formed at cycle 26 (see green box) observed at zero and maximal deformation.

  green box in Fig 3(a)) is presented in Fig 3(c) when the VonMises equivalent plastic strain is zero and

  Fig 4(a) and the colors indicate the value of displacement projected along the normal to the surface. From the profile line in Fig 4(c), extrusions and intrusions are clearly illustrated, which are the typical surface morphologies observed experimentally [15, 14, 53]. The evolution of maximal extrusion and intrusion heights as a function of the loading cycles are plotted in Fig 4(d)). The two values increase during the formation process of the PSBs but after the 26 th cycle, the height values simply fluctuate around a constant. Fig 4(b) presents the local evolution of extrusions from cycle 22 to cycle 26. At the 22 nd and 24 th cycle, there are only two layers of extrusions. Then, one layer is constructed again at the 26 th cycle.

Figure 4 :

 4 Figure 4: Roughness on the free surface of the grain when the global deformation is zero for simulation of a Cu cylindrical grain (D = H = 5µm) under (ε V M p,applied = 10 -3 ): (a) global configuration of the surface with the corresponding dislocation microstructure at cycle 26; (b) zoom on the local growth of one extrusion during 4 cycles; (c) line profile at cycle 26 plotted along the diameter (blue line); (d) evolution of the maximal extrusions/intrusions heights with loading cycles.

  conducted on FCC metals. The most famous experiments are the ones of Mughrabi on the single slip fatigued single crystal of Cu[4, 8]: under the cyclic loading, PSBs are formed and result in a rapid hardening. The observation of specimen surfaces has demonstrated that the PSBs are stable once they are well constructed: the position, the thickness and the volume fraction remain almost unchanged, which corresponds to the simulation outcome (See Fig3). This characteristic seems to be a constant among FCC metals. Pure Ni single crystal has shown the same behavior as Cu in the research work of Mecke[54]. Lavenstein et al. have also confirmed the saturation of the number of PSBs with the cycling[43].

Figure 5 :

 5 Figure 5: Surface deformation and evolution of dislocation distributions at the 50 th simulated cycle for different value of stress and strain in the loading cycle (D = H = 5µm) under ε V M p,applied = 10 -3 (the deformation has been amplified 300 times for clarity).

Figure 6 :

 6 Figure 6: Evolution of the shear stress distribution on the cross-slip plane inside the grain with the formation of PSBs for simulation of Cu (D = 5µm and H = 5µm) under ε V M p,applied = 10 -3 .

Figure 7 :

 7 Figure 7: Evolution of the total dislocation density with the number of loading cycles for different applied strain amplitudes (Cu, cylindrical grain D = 5µm and H = 5µm). The dislocation microstructures obtained in the saturation regime (corresponding to the 40 th cycle) for Cu and 316L are compared in Fig 8 for the different values of ε V M p,applied and when ε V M

Figure 8 :

 8 Figure 8: Comparison of the dislocation microstructures of Cu and 316L at the cycle 40 for the different imposed plastic strain amplitudes (cylindrical grain, D = H = 5µm).

  Fig 11(a). During the second part of the loading, when the applied strain is reduced by a factor 2, the dislocation organization mechanism is affected by the prior construction of the PSBs during the first 25 cycles and as the applied deformation decreases, the initial microstructure contains more PSBs than necessary to accommodate the plastic deformation. As shown inFig 11(b), the dislocation density drops down shapely at the 26 th cycle and decreases gradually in the subsequent 15 cycles to attain a new saturation regime. It is found that the decrease of a factor 2 in the applied strain amplitude leads to a decrease of a factor 2 in the dislocation density measured in the respective saturation regimes.

Figure 11 :

 11 Figure 11: (a) Stress-strain curve for the simulation with decreasing applied deformation, the first 25 cycles are applied with ε V M p,applied = 1.2 × 10 -3 then the loading amplitude is reduced to ε V M p,applied = 0.6 × 10 -3 ; (b) dislocation density evolution for primary and deviate slip systems with the loading cycles; (c) the dislocation distribution evolution when the applied deformation is reduced from ε V M p,applied = 1.2 × 10 -3 ) at the 25 th cycle to ε V M p,applied = 0.6 × 10 -3 at the 60 th cycle and the corresponding distribution histogram (Cu, cylindrical grain, D = 5µm, H = 5µm).

  Fig 12 shows the main experimental results of this publication. The position of macro-bands developed at the plastic amplitude ε p = 3.0 × 10 -3 in the saturation regime can be observed in Fig 12(a) while Fig 12(b

Figure 12 :

 12 Figure 12: Comparison of formed macro-band position: (a) the positions at high amplitude εp = 3.0 × 10 -2 ; (b) the positions at lower amplitude εp = 1.0 × 10 -2 from [10] with permissions.

Figure 13 :

 13 Figure 13: Thicknesses of each PSB under different imposed strain amplitude for Cu and 316L (cylindrical grain, D = H = 5µm).

Figure 14 :

 14 Figure 14: (a) Probability of cross-slip in the DDD model for Cu and 316L; (b) evolution of the count of slip plane changes for Cu and 316L under different imposed plastic strain amplitude with the loading cycles.

Figure 15 :

 15 Figure 15: Comparison of the shear stress resolved on the cross-slip plane in Cu and 316L at the 50 th cycle (Cylindrical grain, D = 5µm, H = 5µm, ε V M p = 0.6 × 10 -3 ).

  539,CET4-601) Fortran (熟练), Python (熟练) Matlab (掌握), C (掌握) LabVIEW, LaTeX, Arduino 软件使用 Cast3M (开源有限元软件), Gmesh (网格工具), bConverged (有限元求解器), Paraview (后处理), OpenMP, Fluent (CFD), Ansys (FEM), SolidWorks (CAD), Git, Office 操作系统 Linux, Windows 教育背景 工作及项目经历 fanshi.meng@outlook.com (+33) 06 95 92 67 76 微信: 158 2058 6128 力学模拟科研助理(博士生) 2016. 11 -2020. 10, SIMaP实验室, 格勒诺布尔, 法国 面心立方金属疲劳行为的实验和多尺度模拟研究 -使用Fortran开发位错动力学软件TRIDIS,使之能够进行疲劳荷载下的模拟运算,运用在位 错尺度研究铜单晶和奥氏体不锈钢316L的疲劳行为; -开发TRIDIS 软件前后处理工具(Matlab和Python); -分析模拟结果中位错分布特点,从而从微观角度解释对应的宏观力学行为,例如周期性硬 化以及软化,疲劳荷载下饱和状态的出现; -通过模拟结果和实验现象进行对比分析,阐释塑性集中导致的常驻滑移带形成机制,可能 的裂纹产生原因,以及晶体取向对宏观疲劳行为的影响; -使用OpenMP进行并行算法优化,提升仿真速度,使之能适用于大型多晶运算; -开展了目前尺度最大的多晶位错动力学运算,阐释了滑移系和周边晶粒的相互作用。 [主要成果已经投递International Journal of Fatigue期刊,审阅中,三篇文章仍在准备,博 士期间参加两次国际会议(Fatigue2018,TMS2020)并做口头报告,以及一次法国会议] 力学模拟科研实习生 2015. 7 -2015. 9; 2015. 12 -2016. 5, SIMaP实验室, 格勒诺布尔, 法国 使用有限元方法研究奥氏体不锈钢316L的循环力学行为 -将晶体塑性有限元(CPFEM)本构模型植入到Cast3M的UMAT中并模拟不同取向下奥氏体 316L单晶以及由EBSD重构的多晶体的疲劳行为; -分析应力集中以及位错密度随周期变化,推出裂纹可能的产生位置,结果与实验相吻合; -开发位错动力学和有限元的耦合运算(使用Python和Cast3M),并应用在多晶体上。 核工程实习生 2014. 7 -2014. 8, 大亚湾核电站,深圳 -核电站准入许可及核安全培训; -通过实地考察和模拟机操作学习核电站运行原理和规范。 团队科研项目助理 2013. 10 -2014. 5, 团队合作完成 (评分 94/100, 排名 5/25), 中山大学,珠海 -与法国格勒诺布尔综合理工合作,团队由一名法方导师,一名中方导师,三名同学组成; -通过湿化学方法和高温煅烧 (1000℃),制备不同配比的镧系元素氧化物粉末 Ce 1- x Ln x O (2-x/2) (Ln = Nd, Sm),再使用XRD、IR、TG等表征方法和Rietveld精修计算其晶体 结构变化,从而类比核燃料中锕系元素氧化物的结构变化。 [主要结果已发表在核心期刊《核化学与放射化学》(第一作者)] 格勒诺布尔阿尔卑斯大学(上交2020世界大学排名99) 2016-2020 -博士(力学与材料科学), 导师: Marc FIVEL(教授),Emilie FERRIE (副教授) -工学学士(核工程与核技术), GPA 3.4/5, 排名15/107 -五年数值仿真软件的使用与开发经验,具备良好编程能力; -熟悉材料失效(蠕变,断裂), 流体力学及传热、传质等工程问题; -simulations of fatigue in FCC metals

Table 2 .

 2 .1. 1 Slip systems in FCC structure and the notation for each system 101 1 011 4 011 4 110 7 110 7 110 10 110 10 01 113 01 113 101 16 101 16

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	l screw 101 1 l edge 12 12 121 3 2 11 5 21 16 11 28 112 9 1 12 11 11 212 2 11 14 211 15 121 17 12 118 n 111 1 11 11 1 1 11 111 11 1 1 11 1 11 111 111 11 1 1 11
	Schmid & Boas B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3

Table 2 .

 2 2 Parameters used in the simulations for Cu and AISI 316L.

		Shear	Young's	Friction	Phonon drag			Burgers	Staking
	Parameters	modulus µ	modulus ν	stress coefficient α f r (10 -5 )	coefficient B	Vact	τ III	vector b	fault energy γ
	Cu AISI 316L	42000 M P a 81000 M P a	0.324 0.27	50 50	1.5 × 10 -5 0.71 × 10 -5	300b 3 1800b 3	30 M P a 56 M P a	2.56 Ȧ 2.54 Ȧ	80 M J/m 2 30 M J/m 2

Table 3 .

 3 1 Schmid factor for each slip system.

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	Notation	B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3
	Schmid Factor	0.50 0.17 0.32 0.22 0.25 -0.12 0.05 -0.18 -0.25 -0.29 -0.19 -0.47

Table 3 .

 3 2 Numerical parameters applied in the simulation.

	σ(M P a.s -1 )	δt (s)	Discretization length (xl) Minimum length for cross-slip (xl)
	1.5 × 10 8	2 × 10 -10	600	400

Table 3 .

 3 3 Stability indication of the maximum number of prismatic loops of a given size that could stay in a 12µm surface grain ("YES" corresponds to a stable loop).

	Loop number 10	20	30	40	50 100

Table 4 .

 4 1 The value of the Schmid factor for each slip system

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	Notation	B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3

Table 4 .

 4

		2 Numerical parameters applied in the simulation
	σ(M P a.s -1 )	δt (s)	Discretization length (xl) Minimum length for cross-slip (xl)
	5 × 10 9	2 × 10 -10	600	100
	4.3 Evolution of the microstructures

4.3.1 Distribution evolution of the formed PSBs

Table 5 .

 5 1 Interaction Matrix of FCC crystals with the following interactions: a 1 -self interaction, a 2 -coplanar, a 3 -Hirth, a 4 -glissile, a 5 -Lomer-Cottrell, a 6 -collinear. a 2 a 2 a 6 a 4 a 4 a 3 a 4 a 5 a 3 a 5 a 4 A3 a 1 a 2 a 4 a 3 a 5 a 4 a 6 a 4 a 5 a 3 a 4 A6 a 1 a 4 a 5 a 3 a 5 a 4 a 3 a 4 a 4 a 6 B2 a 1 a 2 a 2 a 3 a 5 a 4 a 3 a 4 a 5 B4 a 1 a 2 a 5 a 3 a 4 a 4 a 6 a 4 B5 a 1 a 4 a 4 a 6 a 5 a 4 a 3 C1 Symmetry a 1 a 2 a 2 a 6 a 4 a 4 C3 a 1 a 2 a 4 a 3 a 5 C5 a 1 a 4 a 5 a 3

	A2 A3 A6 B2 B4 B5 C1 C3 C5 D1 D4 D6
	A2 a 1 D1	a 1 a 2 a 2
	D4	a 1 a 2
	D6	a 1

Table 5 .

 5 2 The value of the Schmid factor for each slip system in single slip situation.

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	Notation	B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3

Table 5 .

 5 3 The value of the Schmid factor for each slip system

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	Notation	B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3
	Schmid Factor 0.34 0.27 0.61 0.00 0.00 0.00 0.27 0.27 0.34 0.27 0.61 0.00

Table 6 .

 6 1 Euler angles of the grains in the polycrystalline aggregate

	Number of the grain 1	2	3	4	5	6	7
	ψ θ Φ	0 • 18 • -29 • 29 • 18 • 29 • 29 • -50 • 25 • 38 • -18 • 11 • 70 • 11 • 75 • 29 • 29 • 8 • 105 • 62 • 11 •

Table 6 .

 6 2 Numerical parameters applied in the simulation of the aggregate

	σ(M P a.s -1 )	δt (s)	Discretization length (xl) Minimum length for cross-slip (xl)
	5 × 10 9	1 × 10 -10	500	350

Table 6 .

 6 3 Schmid factors of each slip system in each grain.

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	Notation	B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3
	Grain 1	0.35 0.31 0.24 0.49 0.11 0.05 0.18 0.11 0.47 0.26 0.29 0.37
	Grain 2	0.09 0.48 0.28 0.19 0.02 0.12 0.29 0.18 0.11 0.36 0.40 0.17
	Grain 3	0.40 0.17 0.11 0.36 0.11 0.01 0.18 0.28 0.28 0.19 0.09 0.48
	Grain 4	0.48 0.25 0.30 0.35 0.13 0.06 0.09 0.17 0.34 0.31 0.24 0.49
	Grain 5	0.09 0.12 0.09 0.39 0.37 0.10 0.27 0.01 0.47 0.01 0.20 0.02
	Grain 6	0.26 0.41 0.45 0.27 0.03 0.07 0.14 0.18 0.23 0.48 0.37 0.29
	Grain 7	0.46 0.32 0.36 0.39 0.07 0.04 0.07 0.10 0.39 0.36 0.31 0.47

Table 6 .

 6 4 Misorientation of each grain with respect to each grain (the underlined values signify the surrounding grains).

	Grain number	1	2	3	4	5	6	7
	1 2 3 4 5 6 7 Average Max-Min	0 • 47.69 • 24.73 • 28.95 • 43.69 • 41.91 • 37.40 •	47.69 • 0 • 46.65 • 39.29 • 38.94 • 26.88 • 53.62 • 49.32 • 6.97 •	24.73 • 46.65 • 0 • 31.10 • 19.64 • 45.79 • 39.72 • 34.16 • 21.92 •	28.95 • 39.29 • 31.10 • 0 • 39.45 • 18.12 • 31.66 • 33.17 • 10.50 •	43.69 • 38.94 • 19.65 • 39.45 • 0 • 38.40 • 32.25 • 43.69 • 11.44 •	41.91 • 26.88 • 45.79 • 18.12 • 38.40 • 0 • 33.18 • 37.82 • 8.73 •	37.40 • 53.61 • 39.72 • 31.65 • 32.25 • 33.18 • 0 • 41.09 • 20.43 •

Table 6 .

 6 5 Schmid factors of each slip system in each grain.

	Slip system	1	2	3	4	5	6	7	8	9	10	11	12
	Notation	B4	D4	C1	D1	B5	C5	D6	A6	B2	A2	C3	A3
	Grain 1	0.35 0.31 0.24 0.49 0.11 0.05 0.18 0.11 0.47 0.26 0.29 0.37
	Grain 2	0.09 0.48 0.28 0.19 0.02 0.12 0.29 0.18 0.11 0.36 0.40 0.17
	Grain 3	0.40 0.17 0.11 0.36 0.11 0.01 0.18 0.28 0.28 0.19 0.09 0.48
	Grain 4	0.48 0.25 0.30 0.35 0.13 0.06 0.09 0.17 0.34 0.31 0.24 0.49
	Grain 5	0.09 0.12 0.09 0.39 0.37 0.10 0.27 0.01 0.47 0.01 0.20 0.02
	Grain 6	0.26 0.41 0.45 0.27 0.03 0.07 0.14 0.18 0.23 0.48 0.37 0.29
	Grain 7	0.46 0.32 0.36 0.39 0.07 0.04 0.07 0.10 0.39 0.36 0.31 0.47

Table 6 .

 6 6 Schmid factors (or the resolved shear stress) in the equibiaxed tractioncompression loading.

	(6.8)
	1]}

Table 6 .

 6 7 Euler angle of the crystal lattice in grain 2 for different rotation angles around [2 2 1] axis.

	Simulation number 1	2	3	4	5	6
	Rotation	0 •	30 •	60 •	90 •	120 •	150 •
	ψ	0 • 50.1039 • 55.8934 • 63.4349 •	75 •	96.206 •
	θ	0 • 28.2475 •	56.251 •	83.6206 • 109.471 •	131.2 •
	Φ	0 • -39.896 • -34.107 • -26.565 •	-15 •	-353.79398 •
	6.4.2 Simulation results
	Dislocation density evolution

Table 2 :

 2 Parameters used in the simulations for Cu and AISI 316L

	Parameters	Shear modulus µ MPa	Poisson ratio ν	Friction stress τ f r /µ	Drag coefficient B Pa.s	V act	τ III MPa
	Cu AISI 316L	42000 81000	0.324 0.27	50 × 10 -5 50 × 10 -5	1.5 × 10 -5 0.71 × 10 -5	300b 3 1800b 3	30

This section presents the simulation of Cu in a cylindrical grain shown in Figure4.1(a) (D g = h = 5µm) under ε V M p,applied = 10 -3 applied plastic strain amplitude. The dislocation

Acknowledgments

The authors gratefully acknowledge the 4-year PhD scholarship support from China Scholarship Council (CSC).

P S B t h i c k n e s s

316L:

Cu: