Jacques Besson

Samuel

Renald Brenner

Kostas Danas

Jeffrey Kysar

Christian Niordson

Yi Zhang

Yang Zhang

Tang Gu

Fang Lu

Qi Huang

Youbin Chen

Keywords: ductile fracture, FCC steels, irradiation, intragranular voids, crystal plasticity, strain gradient plasticity 1. Introduction ductile fracture, FCC steels, irradiation, intragranular voids, crystal plasticity, strain gradient plasticity

Après un peu plus de 3 ans de travail, je arrive finalement à la dernière étape de cette aventure scientifique. C'est une aventure pour moi, parce que je découvre le monde de la mécanique en rencontrant des difficultés et proposant des solutions. Cette aventure est inoubliable grâce aux laboratoires qui m'ont accueilli et grâce aux gens avec qui j'ai travaillé. Mes premiers remerciements sont donc dédiés à mes directeurs de

Introduction Résumé

Les aciers austénitiques inoxydables sont largement utilisés dans les réacteurs nucléaires. L'irradiation peut créer des défauts microstructuraux dans les aciers et ainsi modifier leurs propriétés mécaniques. Une diminution de la ténacité à la rupture des aciers en fonction de la dose est observée. La rupture ductile due à la croissance et la coalescence des cavités est un mécanisme dominant dans les aciers irradiés jusqu'à 10 dpa. Des cavités peuvent être créées de manière différente : nucléées à partir des inclusions ou des précipités d'irradiation, ou créées directement par irradiation. Ces cavités se trouvent souvent au sein des grains et elles sont plus petites que les grains. Cette thèse a pour objectif d'étudier la rupture ductile des aciers irradiés due à ces cavités intragranulaires. Dans ce chapitre, une approche multi-échelle est proposée pour la modélisation de la rupture ductile intragranulaire des aciers irradiés. La thèse se décompose en 4 parties. La première partie présente une recherche bibliographique sur les propriétés mécaniques des aciers irradiés et les modèles existant dans la littérature. Dans la deuxième partie, on montre des simulations par éléments finis effectuées sur des cellules unitaires pour étudier la cinétique de la croissance et la coalescence des cavités dans le monocristal. Dans la troisième partie, un modèle élastoviscoplastique en grandes transformations est proposé pour le monocristal poreux. Le modèle est appliqué à la simulation de l'endommagement ductile dans le monocristal et le polycristal. Dans la dernière partie, un modèle micromorphe de plasticité cristalline est proposé et appliqué à la simulation de la croissance et la coalescence des cavités intragranulaires de différentes tailles ainsi qu'aux phénomènes de localisation dans les monocristaux.

Industrial context

Austenitic stainless steels of 300 series are widely used as structural materials in nuclear power plants (see fig. 1.1), especially for reactor vessel internals, due to their good mechanical properties and resistance to corrosion [START_REF] Zinkle | Materials challenges in nuclear energy[END_REF]. These steels are exposed to complex operating conditions such as high temperature (over 300 • C), mechanical loadings, radiation damage and corrosion. Under such conditions, properties of the steels evolve over time. Austenitic stainless steels undergo continual microstructure evolution during neutron irradiation, which leads to drastic modifications of mechanical properties: increase in yield stress, decrease in ductility (uniform elongation) and strain hardening capacity, and decrease of fracture toughness (Pokor et al., 2004a;Pokor et al., 2004b;[START_REF] Zinkle | Materials challenges in nuclear energy[END_REF].

All the modifications in the materials certainly have an influence on the operation of reactors and cause challenges for designing reactors compatible with the changes.

Thus it is very important to understand why these changes occur and what are the consequences on fracture properties of austenitic stainless steels. The modification of mechanical properties are usually related to irradiationinduced microstructural defects, such as Frank loops (a type of dislocation loops) which are often observed in austenitic stainless steels (Pokor et al., 2004c). These defects can interact with dislocations, leading to a change of plastic deformation mechanism. Besides, voids and bubbles can form in the materials during irradiation, which has been shown to be correlated with the change of mechanical properties [START_REF] Neustroev | Very high swelling and embrittlement observed in a Fe-18Cr-10Ni-Ti hexagonal fuel wrapper irradiated in the BOR-60 fast reactor[END_REF][START_REF] Neustroev | Severe embrittlement of neutron irradiated austenitic steels arising from high void swelling[END_REF][START_REF] Margolin | The radiation swelling effect on fracture properties and fracture mechanisms of irradiated austenitic steels. Part I. Ductility and fracture toughness[END_REF].

Fracture toughness, an important parameter for the integrity assessment, displays a decreasing trend with radiation, as reported in the literature (see fig. 1.2 from Zinkle and Was (2013)). Ductile fracture is found to be a dominant fracture mode of austenitic stainless steels after irradiation [START_REF] Little | Fracture mechanics evaluations of neutron irradiated Type 321 austenitic steel[END_REF][START_REF] Fukuya | Fracture behavior of austenitic stainless steels irradiated in PWR[END_REF][START_REF] Margolin | The radiation swelling effect on fracture properties and fracture mechanisms of irradiated austenitic steels. Part I. Ductility and fracture toughness[END_REF], especially at low irradiation dose, while some non-ductile modes, e.g., intragranular and channel fracture, are also reported for doses in the range of 30-80 dpa. However, there is no commonly accepted physical mechanism explaining the reduction in the fracture toughness after irradiation, because of the complexity of the phenomena. It is probably related to a kind of localized plastic deformation mode existing in irradiated materials, to a modification of kinetics of plastic deformation driven void growth and coalescence, or to formation of new voids. The present work will concern the latter two phenomena. To better understand the role of different irradiation-induced microstructural defects in decreasing the fracture toughness of steels, a multi-scale approach is needed to study fracture mechanisms in irradiated austenitic stainless steels.

Objectives

As explained above, fracture toughness of austenitic stainless steels decreases with irradiation, accompanied with change of fracture mechanism. A demand of the nuclear industry is to develop theoretical and numerical tools able to predict the evolution of fracture toughness of austenitic stainless steels with irradiation.

For this purpose, the objective of the present work is to investigate one dominant fracture mechanism in irradiated austenitic stainless steelsductile fracture due to void growth and coalescence. As well known, void is an important notion in the mechanics of ductile fracture. In unirradiated steels, voids can nucleate at inclusions or second phase particles during mechanical loading [START_REF] Pineau | Failure of Metals[END_REF][START_REF] Besson | Continuum Models of Ductile Fracture: A Review[END_REF][START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF][START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF]. Voids can then grow and coalesce in stress fields leading to final failure.

These are known as very important fracture mechanisms. In irradiated steels, voids may have different origins. Besides the same nucleation sources in unirradiated steels, voids can also nucleate from irradiation-induced precipitates [START_REF] Little | Fracture mechanics evaluations of neutron irradiated Type 321 austenitic steel[END_REF].

In addition, irradiation-induced nano-voids may be another origin of voids leading to fracture [START_REF] Margolin | The radiation swelling effect on fracture properties and fracture mechanisms of irradiated austenitic steels. Part I. Ductility and fracture toughness[END_REF]. In another word, the size of voids varies according to their origins: micro-sized for those nucleated from irradiation-induced precipitates and nano-sized for irradiation-induced voids. However, whatever their origins are, voids are very often located inside grains, i.e, intragranular voids, and thus of a size smaller than that of the grains. In addition, growth and coalescence of voids are proved to be controlled by plastic strains. Compared with unirradiated steels, plastic deformation is influenced by irradiation-induced defects in irradiated steels, which may in turn change the evolution of voids.

With consideration of the characteristics mentioned above, the ductile fracture due to void growth and coalescence of unirradiated and irradiated austenitic stainless steels is investigated in the present work at the scale of the grain, i.e., single crystal. and voids are regarded as embedded in a single crystal.

Methodology

For studying void growth and coalescence in single crystals, two approaches are adopted in the present work, as presented in fig. 1.3. The first approach corresponds to finite element simulations on unit cells with periodic boundary conditions, i.e., unit cell simulations, which will be performed using classical continuum theory of crystal plasticity to study the effects of • crystal orientations,

• post-irradiation strain hardening/softening behavior on void growth and coalescence. As the size of voids in irradiated stainless steels varies from nano to micro-scale,

• effects of void size will be investigated in a second step. For this purpose, a strain gradient crystal plasticity model will be derived following the micromorphic approach [START_REF] Germain | The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure[END_REF][START_REF] Forest | Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage[END_REF][START_REF] Aslan | Micromorphic approach to single crystal plasticity and damage[END_REF].

The second approach is to propose an elastoviscoplastic model at finite strains for porous single crystals, which incorporates effects of strain hardening of single crystal matrix and is able to describe void growth to coalescence. This model will be applied to simulate ductile damage evolution in a specimen of single crystal in the presence of a defect representation of a crack. The model can be further applied

Introduction

to the scale of polycrystals. This approach can finally be used to assess effects of post-irradiation strain hardening/softening on fracture toughness on stainless steels.

Organization of the thesis

According to the approaches proposed for studying ductile fracture of irradiated austenitic stainless steels, the present work is organized as follows:

• chapter 2: A literature review is given for experimental observations on mechanical behavior of irradiated stainless steels and modeling tools existing in the literature. Based on this review, a further explanation will be given for the multi-scale approach proposed for this study. • chapter 3: Void growth and coalescence in single crystals are studied with unit cell simulations based on crystal plasticity finite element method. 1) Some parameters influencing the void evolution are investigated, such as crystal orientation, stress triaxiality and initial void volume fracture. 2) Effects of post-irradiation strain hardening/softening behavior are outlined.

• chapter 4: An elastoviscoplastic model for porous single crystals at finite strains is formulated based on the work of [START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF] and assessed numerically with the results of unit cell simulations. The model will be applied to simulate ductile damage in a polycrystalline aggregate under triaxial loadings and a single-edge-notch-tension test on a single crystal specimen of 316L stainless steels.

• chapter 5: Size effects on void growth and coalescence are studied. To this end, an enhanced single crystal plasticity model at finite strains is proposed and implemented in finite element code. Size dependent void growth and coalescence are investigated by unit simulations with the enhanced model for verifying if it is important to consider size effects in the multi-scale modeling of ductile fracture of irradiated austenitic stainless steels.

A literature review is performed in this chapter to first discuss effects of radiation on stainless steels. Irradiation-induced microstructure changes in stainless steels will be first considered, followed by a discuss on modifications of mechanical properties associated with the microstructure evolution. In particular, radiation effects on fracture behaviors of stainless steels will be outlined. It will be shown in the following that ductile fracture is one of the fracture mechanisms in irradiated steels.

Hence, classical methods used to study ductile fracture in metallic materials will be reviewed. In addition, plastic size effects can play a role in ductile fracture and may also an influence on fracture behaviors of irradiated steels due to void growth and coalescence. Thus, a review about size effects on void growth will be performed. Based on the review, the objectives of the thesis will be outlined and the approaches used to study ductile fracture of irradiated steels will be presented as the conclusions of the chapter.

Microstructural effects of radiation damage on stainless steels

Irradiation produces microstructural defects in materials, mainly Frank loops in stainless steels, as a result of particles-matter interaction. The defects interact under irradiation, even without mechanical loading, with grown-in dislocations (already present before irradiation) resulting in dramatic changes of dislocation densities.

Cavities are also produced during irradiation, which can play a role in fracture process. Under mechanical loading, dislocations interact with irradiation-induced defects, which can lead to a particular mode of deformation, where deformation is localized in channels sometimes called "cleared band". These points are outlined in the following.

Frank loops and dislocations

Frank loops correspond to dislocation loops with pure edge character for every segments, which are often nucleated in a displacement cascade. A displacement cascade is a collection of point defects, including vacancies and interstitials, resulting from the interaction of energetic incident particles (irradiation) and lattice atoms.

When the point defects collapse/condense onto a close-packed plane, a Frank loop may be produced. Two types of Frank loops exist: interstitial and vacancy loops.

They correspond respectively to intrinsic and extrinsic stacking fault as shown in fig. 2.1. Interstitial Frank loops are often observed in stainless steels and they can be In annealed metals, dislocation density increases sharply with irradiation dose and eventually approaches a saturation value. In cold-worked metals, however, the dislocation density drops and approaches a saturation value. It is often observed that the saturation values in annealed and cold worked steels are nearly the same.

Inspired by these observations, investigations have been performed for modeling the dislocation evolution during irradiation. Generation and elimination are two competitive mechanisms [START_REF] Wolfer | Dislocation evolution in metals during irradiation[END_REF][START_REF] Stoller | Modeling dislocation evolution in irradiated alloys[END_REF]; Zouari et al., The dose dependent dislocation densities to be used for simulations in the present thesis is taken from the work of Pokor et al., 2004a (see fig. 2.5). The results for SA 304 are considered. Notice that the dislocation density measured for unirradiated SA 304 steels seems lower than usual cases (10 12 m -2 usually reported in the literature, such as in [START_REF] Zinkle | Dose dependence of the microstructural evolution in neutron-irradiated austenitic stainless steel[END_REF] and [START_REF] Lucas | The evolution of mechanical property change in irradiated austenitic stainless steels[END_REF]) and that the dislocation density decreases with irradiation dose, which is different from the dislocation density evolution trend reported by [START_REF] Zouari | Cluster Dynamics Prediction of the Microstructure Evolution of 300-Series Austenitic Stainless Steel under Irradiation: Influence of Helium[END_REF].

Frank loops generated by irradiation modify plastic deformation mechanisms of the steels. Under mechanical loading, Frank loops interact with dislocations and cause a change in the yielding and hardening process. This is usually believed to lead to the mechanical effects of radiation damage presented in section 2.2. Numerous studies have being performed for understanding details about their interactions and also for modeling the irradiation dose dependent mechanical properties. A detailed literature review has already been performed in the thesis work of [START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF].

The author has proposed a single crystal plasticity model for irradiated austenitic stainless steels based on physical mechanism of Frank loop-dislocation interactions. This model will be used in the present work and presented later in section 3.1.

Voids and bubbles

Voids (empty cavities) and bubbles (cavities with gas) are generated by irradiation, whose densities depend on dose and irradiation temperature. The formation of voids is associated with clusters of vacancies absorbing more vacancies than interstitials.

Insoluble gases are formed by transmutation when certain metals are irradiated, which finally leads to the formation of bubbles. Typical cavities observed in stainless steels are shown in fig. 2.6. Voids and bubbles could cause a change in dimensions of materials (called "swelling") and have profound influences on mechanical properties of materials. The irradiation swelling causes a great challenge for designing a reactor to accommodate dimension changes.

It has also been shown that the swelling level is correlated with the modification of mechanical properties of stainless steels. [START_REF] Neustroev | Very high swelling and embrittlement observed in a Fe-18Cr-10Ni-Ti hexagonal fuel wrapper irradiated in the BOR-60 fast reactor[END_REF] showed an almost linear reduction of yield stress and ultimate strength with swelling level in Fe-18Cr-10Ni-0.5Ti stainless steels irradiated in BOR-60 fast neutron reactor (fig. 2.7). [START_REF] Margolin | The radiation swelling effect on fracture properties and fracture mechanisms of irradiated austenitic steels. Part I. Ductility and fracture toughness[END_REF] showed a decrease of the fracture toughness, evaluated by Jintegral, with swelling level in Fe-18Cr-10Ni-Ti stainless steels irradiated in BOR-60 (fig. 2.14). Furthermore, cavities can lead to nano-dimple fracture in relatively high swelling level (see section 2.2.2). Some experimental observations have reported deformed (elongated) cavities located in plastic deformation channels in front of a crack tip (fig. 2.8a) and in a tension specimen (fig. 2.8b). These elongated cavities can probably participate in the process of cracking and final fracture of the steels. As these cavities are potential crack sources, it is important to understand their behaviors in stress fields. Hence, one of the subjects of the present thesis work is to study the mechanical effects of cavities at the grain scale.

Cleared bands

It is well known that plastic strain localization occurs at the grain scale as a result of heterogeneous nature of plastic deformation. A specific plastic strain localization in defect-free channels has been widely observed in irradiated metallic single and polycrystals (see, e.g., [START_REF] Sharp | Deformation of neutron-irradiated copper single crystals[END_REF][START_REF] Tucker | Dislocation channeling in neutron-irradiated niobium[END_REF][START_REF] Sharp | Deformation of neutron irradiated copper alloys[END_REF][START_REF] Singh | Effects of neutron irradiation on microstructure and mechanical properties of pure iron[END_REF][START_REF] Hashimoto | Deformation microstructure of neutron-irradiated pure polycrystalline vanadium[END_REF][START_REF] Hashimoto | Microstructural analysis of deformation in neutron-irradiated fcc materials[END_REF]) since a pioneering work of [START_REF] Greenfield | Effect of Neutron Irradiation on the Plastic Deformation of Copper Single Crystals[END_REF] on irradiated copper single crystals. The channel width is smaller than a few hundred nanometers where irradiation-induced defects (Frank loops in the case of stainless steels) are partially or fully removed by gliding dislocations. Since the number of defects are highly reduced within the channels, they are often called "cleared band" or "clear band" in the literature.

Cleared bands have also been observed in stainless steels [START_REF] Bailat | The effects of irradiation and testing temperature on tensile behaviour of stainless steels[END_REF][START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF][START_REF] Byun | Deformation mode map of irradiated 316 stainless steel in true stress-dose space[END_REF] even with relatively low irradiation damage (0.78 dpa in the study of [START_REF] Byun | Deformation mode map of irradiated 316 stainless steel in true stress-dose space[END_REF], shown in fig. 2.9). In Ni-ion irradiated 304L stainless steels, it was reported that cleared bands are the dominant plastic instability mode at high test temperature (about 280 • C) and slow strain rate conditions [START_REF] Bruemmer | Defect microstructures and deformation mechanisms in irradiated austenitic stainless steels[END_REF][START_REF] Cole | Post-irradiation deformation characteristics of heavy-ion irradiated 304L SS[END_REF]. However, [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] reported that the channeling process in ion irradiated and deformed 316LN is not test temperature dependent. In addition, the observation of [START_REF] Edwards | Initiation and propagation of cleared channels in neutron-irradiated pure copper and a precipitation hardened CuCrZr alloy[END_REF] suggested that cleared bands are initiated at places with high stress concentration such as grain boundaries, inclusions and even previously formed bands. The authors also observed that some channels can even penetrate through grain boundaries. Recently, discrete dislocation dynamics simulations were performed to

Mechanical effects of radiation damage on stainless steels

Neutron irradiation of stainless steels can lead to drastic modifications of mechanical properties, e.g., increase in yield stress, decrease in ductility (uniform elongation), decrease of strain hardening capacity and decrease of fracture toughness [START_REF] Pawel | Effects of low temperature neutron irradiation on deformation behavior of austenitic stainless steels[END_REF]Pokor et al., 2004a;Pokor et al., 2004b). In this section, some effects relevant to ductile fracture of stainless steels are outlined.

Stress-strain relation in tension

Typical stress-strain curves are shown in fig. 2.10 for a solution annealed 304L stainless steels irradiated by neutrons up to 3.4 displacements per atom (dpa) (Pokor et al., 2004a). The behavior is dose dependent. For the unirradiated case (0 dpa), [START_REF] Dai | Mechanical properties and microstructures of copper gold and palladium single crystals irradiated with 600 MeV protons[END_REF] a smooth elastic-plastic transition is observed followed by a plastic regime with small hardening rate before failure. After irradiation, a marked increase of the yield stress and a reduction of uniform elongation is observed. It is also important to notice that the hardening rate is also dose dependent. In addition, a sudden stress drop is observed after yield point at 3.4 dpa. This kind of stress drop, according to Trinkaus and his co-workers (Trinkaus et al., 1997a;Trinkaus et al., 1997b;[START_REF] Singh | Radiation hardening revisited: role of intracascade clustering[END_REF], is related to unpinning process of dislocations from radiation cascade induced Frank loops. After irradiation, a "cloud" of Frank loops decorating grown-in dislocations (which present already before irradiation) is often observed.

The Frank loops play a role of trapping the dislocations and make them immobile. Frank loops and immobile dislocations cannot act as dislocation sources. The yield stress corresponds to the stress needed for unpinning grown-in dislocations from the Frank loops decorating them. Once the dislocations are unpinned, smaller stress is required for the plastic flow. It is worth noticing that the stress drop has also been observed in irradiated single crystals [START_REF] Blewitt | Radiation hardening of copper single crystals[END_REF][START_REF] Dai | Mechanical properties and microstructures of copper gold and palladium single crystals irradiated with 600 MeV protons[END_REF][START_REF] Dai | Mechanical properties and microstructure of 600 MeV proton irradiated copper single crystals[END_REF]). An example is shown in fig. 2.11 for copper single crystals irradiated to 3.9 × 10 -2 dpa. Compared with polycrystalline materials, a marked hardening regime was observed before final failure of single crystals.

The mechanical behavior of irradiated steels under tensile loading depends on many parameters, such as the properties of original materials, irradiation conditions, test conditions, etc. As an example, engineering stress-strain curves are presented in fig. 2.12 for solution annealed 304L stainless steels (SA 304) and cold worked 316 stainless steels (CW 316) irradiated to about 20 dpa (Pokor et al., 2004b). A marked difference between the two types of steels is that a stress drop is observed in SA 304 and not in CW 316.

Fracture mechanisms

Neutron irradiation generally causes a reduction in fracture toughness of stainless steels. The fracture toughness of irradiated stainless steels has been shown to be irradiation dose dependent (figs. 1.2 and 2.13) and swelling dependent (fig. 2

.14).

The decrease of fracture toughness in steels after irradiation is associated with a ). The authors also conducted tensile tests with two strain rates (conventional: 1.1 × 10 -4 s -1 and slow: 6.7 × 10 -8 s -1 ). According to the tensile tests results combined with other results in the literature, the authors concluded that intergranular fracture is promoted under low-temperature high-strain-rate and high-temperature low-strain-rate conditions. However, it is important to note that dimple (ductile) fracture remains one dominant fracture mechanism at high temperature in these steels irradiated up to 38 dpa, i.e., T test = 150 [START_REF] Little | Fracture mechanics evaluations of neutron irradiated Type 321 austenitic steel[END_REF] and finally reach a stabilized value (see fig. 2.13). The author found that the ductile fracture due to void growth and coalescence was the primary fracture mechanism after irradiation and that the ductile dimples were much finer (see fig. 2.17), initiating from irradiation-induced micro-sized TiC precipitates. [START_REF] Neustroev | Severe embrittlement of neutron irradiated austenitic steels arising from high void swelling[END_REF] investigated fracture behavior of a titanium-stabilized stainless steel cut from fuel pin claddings and fuel assembly wrappers of the fast reactor BOR-60. The authors reported a transgranular cup-cone fracture morphology for tensile specimens extracted from assembly wrappers with high swelling (30%). The fracture surface exhibited fine dimples which imply that failure proceeded by micropore coalescence (see fig. Ductile fracture due to void growth and coalescence (see section 2.3) is a dominant fracture mechanism in unirradiated stainless steels. According to the experimental studies presented above, this type of fracture mechanism has been also observed in irradiated stainless steels, however, with a decrease in dimple size with irradiation. These observations motivate the present thesis work.

Ductile fracture in metallic materials

Ductile fracture refers to fracture modes involving extensive plastic deformation before final failure and is widely observed in failure of metallic materials [START_REF] Pineau | Failure of Metals[END_REF][START_REF] Besson | Continuum Models of Ductile Fracture: A Review[END_REF][START_REF] Benzerga | Ductile fracture by void growth to coalescence[END_REF][START_REF] Pineau | Failure of metals I: Brittle and ductile fracture[END_REF].

Ductile fracture is also exhibited by irradiated stainless steels, as presented in section 2.2.2. Since cavities are extensively involved in the fracture processes of irradiated stainless steels, some classical mechanisms in metals and alloys associated with cavities are to be presented in the following.

Experimental observations and micromechanisms

Models for porous isotropic materials and single crystals

Numerous studies have being devoted to the modeling of mechanical behavior of elasto-(visco-)plastic media containing voids. After a pioneering work of McClintock (1968), [START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF] treated the enlargement of a spherical void under remotely uniform stress in a rigid perfectly plastic von Mises material. For high stress triaxiality case, the increase rate of void radius R under a simple tension remote field writes: Ṙ

R = 0.283 exp 3 2 σ ∞ m σ ∞ eq ṗ∞ , ( 2.1)
with the remote von Mises equivalent stress σ ∞ eq , the remote mean stress σ ∞ m , the remote accumulated strain rate ṗ∞ = , on void growth rate. Another seminal work concerning voided isotropic material was conducted by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF]. The author performed limit analysis on a rigid perfectly plastic von Mises matrix containing a void and an effective yield function accounting for void volume fraction f was derived:

σ 2 eq σ 2 0 + 2f cosh 3 2 σ m σ 0 -1 -f 2 = 0, (2.2)
where σ 0 is the tensile yield stress of the von Mises matrix and f is the void volume fraction. One can further formulate a porous plasticity framework combining the yield function with the normality rule and the evolution law of void volume fraction

f based on mass conservation ḟ = (1 -f ) tr ε ∼ p
, where ε ∼ p is plastic strain rate. This model was then extended by Tvergaard and Needleman [START_REF] Tvergaard | On localization in ductile materials containing spherical voids[END_REF][START_REF] Tvergaard | Analysis of the cup-cone fracture in a round tensile bar[END_REF], called GTN model in the literature, to better represent unit cell simulations and experimental results. The Gurson model has also been extended for introducing void shape effects [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids: Case of axisymmetric prolate ellipsoidal cavities[END_REF][START_REF] Gologanu | Approximate models for ductile metals containing nonspherical voids: case of axisymmetric oblate ellipsoidal cavities[END_REF] and matrix anisotropy [START_REF] Benzerga | Plastic potentials for anisotropic porous solids[END_REF]. The same approach has also been used for a matrix obeying other yield criteria such as In the past few years, an increasing number of studies have been devoted to the development of models accounting for void growth in a single crystal matrix. A first yield function derived by [START_REF] Han | A yield function for single crystals containing voids[END_REF] based on the variational homogenisation approach of DeBotton and Castaneda (1995) is written as:

τ s2 τ 0 2 + α 2 45 f σ 2 eq τ 0 2 +2q 1 f cosh    q 2 3 20 σ m τ 0    -1-q 2 1 f 2 = 0, s = 1, 2, • • • , N (2.3)
where N is the number of slip systems, τ s is the resolved shear stress for slip system s, τ 0 is the critical resolved shear stress, and α, q 1 and q 2 are heuristic parameters (as in the GTN model for q 1 and q 2 ). The three parameters were identified by comparison with yield surfaces predicted by unit cell simulations for different crystal orientations and void volume fractions: α = 6.456, q 1 = 1.471 and q 2 = 1.325. Note that the obtained value of q 1 close to 1.5 is in good agreement with the value for GTN model obtained in the literature (see [START_REF] Faleskog | Cell model for nonlinear fracture analysis -I. Micromechanics calibration[END_REF][START_REF] Gao | Cell model for nonlinear fracture analysis -II. Fracture-process calibration and verification[END_REF].

The model is able to accurately represent the numerical limit analysis results. Then, another model was developed by Paux et al., 2015 based on the approach of limit analysis using a regularised form of the Schmid law [START_REF] Arminjon | A regular form of the Schmid law. Application to the ambiguity problem[END_REF][START_REF] Gambin | Refined analysis of elastic-plastic crystals[END_REF]. The yield function writes

   N s=1 |τ s | n 1/n τ 0    + 2qf cosh κ σ m τ 0 -1 -(qf ) 2 = 0, (2.4)
where q is a heuristic parameter and κ = 0.506 is derived semi-analytically. Notice that κ = q 2 3 20 ≈ 0.513 is obtained in Eq. (2.3) with q 2 = 1.325. The two yield functions Eqs. (2.3) and (2.4) share some similarities. However, Eq. (2.3) is a multi-criterion yield function, while Eq. (2.4) is a single criterion yield function. More recently, a more complex model accounting for void shape effects was developed by Mbiakop and his co-workers (Mbiakop et al., 2015a;Mbiakop et al., 2015b) following the variational method of [START_REF] Danas | Numerical modeling of elasto-plastic porous materials with void shape effects at finite deformations[END_REF]. The model was first derived for elliptical voids in two-dimensional case and it was then extended to three-dimensional case for ellipsoidal voids. The proposed model is expressed in the form of the effective stress potential U which is related to the effective strain rate tensor D ∼ by

D ∼ = ∂ U ∂σ ∼ , ( 2.5)
where σ ∼ is the average Cauchy stress of the porous medium. With n denoting the parameter of power law and γs 0 the reference slip rate, U for single crystals with ellipsoidal voids is given by

U = (1 -f ) N s=1 γs 0 (τ s 0 ) -n n + 1 σ ∼ : Ŝ ≈ (mvar),s : σ ∼ (n+1)/2 , ( 2.6)
where Ŝ

≈ (mvar),s = Ŝ ≈ (var),s + (q 2 J -1)J ≈ : Ŝ ≈ (var),s : J ≈ , ( 2.7)
and

q J = 15 f (1 -f )(β n ) 1/n n(f -1/n -1) n/(n+1) with β n = 4 25 6 -n/2 . (2.8)
Here, Ŝ

≈ (var),s = 1 2 E ≈ s + f N Ŝ ≈ * , E ≈ s = 2N ∼ s sym ⊗ N ∼ s sym , J ijkl = 1 3 δ ij δ kl , ( 2.9)
where N ∼ s sym is the symmetric Schmid tensor for slip system s, δ ij is the Kronecker delta, and Ŝ ≈ * is a fourth order microstructural tensor given by

Ŝ ≈ * = Q ≈ -1 - N s=1 1 2 E ∼ s , Q ≈ = lim ρ→∞ lim κ→∞ [S ∼ 0 -1 -S ∼ 0 -1 P ∼ S ∼ 0 -1 ] (2.10) with S ≈ 0 = N s=1 1 2 E ≈ s + 1 2ρ F ≈ s + 1 3κ J ≈ , F ≈ s = K ≈ -E ≈ s , ( 2.11)
where K ≈ and J ≈ are the fourth order shear and hydrostatic projection tensors. P ∼ is given by

P ∼ = 1 4π det Z ≈ |ξ |=1 (S -1 iakb ξ a ξ b ) -1 ξ j ξ l | (ij)(kl) Z ≈ -1 .ξ dS.
(2.12)

The brackets (ij)(kl) denote symmetrization with respect to the corresponding indices, ξ is a unit three-dimensional vector and Z ≈ is a second order tensor describing the shape and orientation of the voids, i.e.,

Z ≈ = w 1 n 1 ⊗ n 1 + w 2 n 2 ⊗ n 2 + n 3 ⊗ n 3 , (2.13)
where w 1 and w 2 are two aspect ratios of the ellipsoidal void and the vectors n i (i = 1, 2, 3) define an orthonormal basis set, which coincides with the principal axes of the ellipsoidal void.

However, to the authors' knowledge, there is currently no model for porous single crystals at finite strains, which is able to describe the void evolution up to coalescence and which is simple enough for finite element implementation in order to carry out structural computations. Those size effects are usually explained by strengthening of materials by strain gradient [START_REF] Fleck | Strain gradient plasticity: Theory and experiment[END_REF]. Strong strain gradient is also expected in the plastic zone at the tip of a crack and in the plastic deformation of polycrystals at the grain boundary because of the mismatch of slip between grains. Therefore, size effects are also observed in these cases (see, e.g., [START_REF] Wei | Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity[END_REF][START_REF] Smyshlyaev | The role of strain gradients in the grain size effect for polycrystals[END_REF], and references therein). Strain gradients result in the storage of geometrically necessary dislocations (GND) [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF], which is regarded as the physical foundation of the size effect. This will be presented in the next section.

Plasticity size effects in metallic materials

Experimental observations

Geometrically necessary dislocations and strain gradient plasticity

Size effects in plasticity are often associated with strain gradients which physically require the storage of GNDs for compatibility reasons [START_REF] Ashby | The deformation of plastically non-homogeneous materials[END_REF]. In contrast to GND, statistically-stored dislocations are responsible for uniform plastic deformation.

A well-known illustration of GND, proposed by [START_REF] Fleck | Strain gradient plasticity: Theory and experiment[END_REF], concerns nonuniform shear of a single crystal beam (fig. 2.27). For a single crystal with a single slip system with normal to the slip plane aligned with the x 2 and slip direction aligned with the x 1 direction, it is shown that a gradient of plastic slip along the x 1 direction causes a density ρ G of GNDs to be stored. However, classical continuum mechanics fails to predict the size effect for lack of length scale in the theory. Thus, tremendous efforts have been devoted to enhance the theory by different approaches. A well-known approach to do so is to include strain gradient hardening. By this way, the so-called "strain gradient plasticity" models were first proposed for isotropic materials.

One candidate idea of modeling [START_REF] Fleck | A reformulation of strain gradient plasticity[END_REF][START_REF] Gudmundson | A unified treatment of strain gradient plasticity[END_REF][START_REF] Gurtin | Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization[END_REF] is assuming that the plastic internal work increment δw p in a volume V follows the form:

δw p = V sδp + m .δ∇p dV, (2.14)
where p is the effective plastic strain (also called "accumulated plastic strain") with its rate ṗ =

Ė2 p = ṗ2 + l 2 * ∇ ṗ.∇ ṗ (2.15)
with the length parameter l 2 * . Another candidate way to include strain gradient hardening is to account for the gradient of the total plastic strain tensor ε ∼ p . Thus, the plastic internal work increment in a volume V writes

δw p = V s ∼ : δε ∼ p + m ∼ . . . δ∇ε ∼ p dV, ( 2.16)
where the generalized stresses s ∼ and m ∼ are respectively a second order and third order tensor. Note that, up to now, there is no common agreement on the formulation of the theories. Detailed discussions on the formulation of strain gradient plasticity can be found in the works of [START_REF] Forest | Elastoviscoplastic constitutive frameworks for generalized continua[END_REF], [START_REF] Gudmundson | A unified treatment of strain gradient plasticity[END_REF], [START_REF] Forest | Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage[END_REF], and [START_REF] Gurtin | Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization[END_REF].

In addition to the models for isotropic plastic materials, strain gradient plasticity models were also proposed for single crystals. They can be classified into three categories. The first category is based on the notion of dislocation density tensor α ∼ [START_REF] Nye | Some geometrical relations in dislocated crystals[END_REF] which is defined as

α ∼ = -curl E ∼ -1 = - 1 J (Curl P ∼ ).F ∼ T , (2.17)
where F ∼ is the deformation gradient, E ∼ and P ∼ are its elastic and plastic part resulting from the multiplicative decomposition F ∼ = E ∼ .P ∼ [START_REF] Lee | Elastic-plastic deformation at finite strains[END_REF][START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF], J is the determinant of F ∼ . 1 The microcurl model introduced by Cordero and his co-workers [START_REF] Cordero | Size effects in generalised continuum crystal plasticity for two-phase laminates[END_REF] and that proposed by Gurtin and his co-workers [START_REF] Gurtin | A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations[END_REF] Numerical implementation of the models for single crystal plasticity is of great challenge especially for finite strain formulation, partly due to the complexity of the implementation is usually restricted to small strain formulation or finite strain formulation with planar double slip. However, the numerical efficiency of the model with a single additional DOF has been shown by [START_REF] Wulfinghoff | Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics[END_REF].

In addition, it is worth noticing that Miehe and his co-workers [START_REF] Miehe | Variational gradient plasticity at finite strains. Part I: Mixed potentials for the evolution and update problems of gradient-extended dissipative solids[END_REF]Miehe et al., 2014b;Miehe et al., 2014a) proposed a variational formulation for strain gradient plasticity models at finite strains and showed its numerical efficiency through 3D simulations.

Size effects on void growth

Experimental observations

Size effects are expected in the plastically driven void growth, when void dimensions are of the same order as the spacing between mobile dislocations and dislocations sources. Even though it lacks of direct evidence, some indications have been shown

by recent experiments that the growth of voids is size dependent. Khraishi et al.

(2001) studied the growth of a machined cylindrical hole in a thin plate made of a superplastically deforming metal. For the hole diameters considered (203, 243 and 635 µm), it was shown that void growth rate was higher for a bigger hole. In these experiments, the size of the plates was kept unchanged. Thus, the porosity, which is generally believed to affect void growth, changed with increasing hole diameter.

However, as the diameter of the hole was far smaller than the width of the plate (6.35 mm), the influence of the porosity was negligible.

Even though few experimental investigations have been devoted to size effects on void growth and coalescence, discrete dislocations dynamics (DDD) and molecular dynamics (MD) have offered a great deal of details about void growth at microand nano-scale. However, in these studies, high strain rate loading conditions are often considered. larger voids and that thus the dislocation mobility is higher. This was believed to be the second reason for the size effect on the void growth rate. Moreover, [START_REF] Segurado | An analysis of the size effect on void growth in single crystals using discrete dislocation dynamics[END_REF] found a size effect on the strain hardening rate. In their simulations, the external boundaries of the single crystal were assumed to be impenetrable. This barrier mimics the effect of grain boundaries or of passivation layers in a microcrystal. When dislocations move towards the external boundaries, they stop at the boundaries and form pile-ups. They generate then back-stresses, reduce the mobility of following dislocations and also the activity of sources. This process results in an increasing flow stress. In this case, the strain hardening rate was found to be higher in the single crystal with smaller void. This effect was explained by two mechanisms. Firstly, dislocation density increases more rapidly as the size decreases, which indicates more dislocations within the pile-ups and leads to a higher strain hardening rate. Secondly, dislocations available in small crystals are limited by crystal size and the applied strain has to be partially absorbed through elastic deformation which causes a higher strain hardening rate. MD simulations were applied for nano-sized voids rather than micro-sized ones, limited by the computationally expensive simulation process. At this scale, the amount of dislocation sources around the void is limited and dislocations nucleating from sources in the matrix are no longer sufficient for its growth. In this case, void growth by dislocation emission at the void surface was widely observed. It was found in the MD simulations that dislocations begin to nucleate at the void surface,

MD and DDD simulations

f v =4.9% f v =19.6%
Incipient strength (GPa) allowing the significant growth of void. Concerning the kinetics of dislocation evolution, there is still a debate [START_REF] Bulatov | Shear impossibility: Comments on "Void growth by dislocation emission" and "Void growth in metals: Atomistic calculations[END_REF]. The evolution of generated dislocations differs in details, which depends on various aspects, such as crystal structure considered (BCC, FCC or HCP) [START_REF] Rudd | Void growth in bcc metals simulated with molecular dynamics using the Finnis-Sinclair potential[END_REF] As a result, atoms are moved away from the void leading to the growth of the void.

L x /a 0 (a) (b)
These two mechanisms are presented in figs. 2.31 and 2.32 [START_REF] Tang | Growth and collapse of nanovoids in tantalum monocrystals[END_REF]. In addition, twinning was also found by [START_REF] Tang | Ductile tensile failure in metals through initiation and growth of nanosized voids[END_REF][START_REF] Rudd | Void growth in bcc metals simulated with molecular dynamics using the Finnis-Sinclair potential[END_REF] Combining the investigations at micro and nano-scale, a map of mechanisms and size effects for plastic void growth can be established, according to [START_REF] Chang | Molecular dynamics modeling and simulation of void growth in two dimensions[END_REF], which is presented in fig. 2.36. It can be summarized as follows. For voids with radius larger than 100 nm, plastic deformation occurs by the nucleation of dislocation in the bulk from dislocation sources. Larger voids grow faster than small one, because the higher number of dislocation sources around the void increases the number of available dislocations, leading to faster void growth. For voids below 50 nm, few dislocation sources are available around the void, voids can only grow by the nucleation of dislocations at the void surface. This mechanism requires higher stress to initiate plastic deformation around the void and the void growth rate is independent of its size. In this regime, the yield stress does not depend on void size. The surface energy density remains constant for the radius in the range from 10 to 50 nm. For voids with radius smaller than 10 nm, the void surface becomes more stable and higher stress is necessary to nucleate dislocations from the void surface. These argument are in agreement with the micromechanics analysis of [START_REF] Ahn | Void growth by dislocation-loop emission[END_REF] based on the interaction between a void and emitted dislocations. They reported that the surface energy plays a significant role only for voids with radius smaller than 100 nm.

Investigations by strain gradient plasticity

Strain gradient plasticity theories have been widely used to study size effects in plasticity, among which size dependent void growth at micron-scale is frequently concerned. In the literature, two modeling approaches can be encountered:

• an isolated void in an infinite medium;

• a void in a finite medium (unit cell).

The first kind of model considers a spherical or cylindrical void embedded in an finite matrix subjected to a uniform remote loading, as shown in fig. 2.37. The model is usually used in analytical analysis to obtain a relation between the void growth rate and the remote loading, as in the well-known work of [START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF]. In contrast to the first type of model, the second accounts for a finite matrix, i.e., unit cell, which can be regarded as a representative volume element (RVE). The RVE is often supposed to be a cube in three dimension (3D) or a square in two dimension (2D) (see fig. In early works, the matrix is assumed to be isotropic in order to simply analysis process. However, as radii of voids are reduced to several microns, whose size is [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF].

smaller than that of the grain, it is more reasonable to consider voids embedded inside a grain whose properties are naturally anisotropic (or at the grain boundary for some special cases). It means that as the size of the void is comparable with that of the grain, the matrix should assume to be anisotropic. This particular aspect attracts an increasing attention in recent years (see, e.g., [START_REF] Kysar | Cylindrical void in a rigid-ideally plastic single crystal. Part I: Anisotropic slip line theory solution for face-centered cubic crystals[END_REF][START_REF] Gan | Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations[END_REF][START_REF] Borg | Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void[END_REF][START_REF] Borg | Size effects on void growth in single crystals with distributed voids[END_REF][START_REF] Hussein | Plasticity size effects in voided crystals[END_REF].

Even though the approach used is different from one study to another, some common results are predicted by strain gradient theories: 1. smaller is stronger; 2. smaller is slower.

It implies that, with a given void volume fraction but a decreasing void size, loading carrying capacity of porous materials increases and void growth rate decreases, which are in agreement with the observations in DDD and MD simulations.

Apart from the very conclusions, some specific issues were addressed in different works, which will be presented in the following. Results obtained by the two models (a void in an infinite matrix and in a finite matrix) are described separately. Besides, works regarding the extension of the Gurson model accounting for the size effect will be exclusively summarized in section 2.5.5.

A void in an infinite matrix

Fleck and Hutchinson [START_REF] Fleck | A phenomenological theory for strain gradient effects in plasticity[END_REF][START_REF] Fleck | Strain gradient plasticity[END_REF][START_REF] Fleck | A reformulation of strain gradient plasticity[END_REF] established in their works the termed phenomenological strain gradient plasticity theory, where size effects in void growth were concerned as an implication of their theories. The problem was solved based on a variational principle established for an infinite matrix containing an isolated spherical void. The matrix is subjected to uniform remote axisymmetric loading specified by σ ∞ 33 = S and σ ∞ 11 = σ ∞ 22 = T . Results for the normalized void growth rate have the functional form V

ε∞ 33 V = F X, l a , n (2.18)
where ε ∞ 33 denotes the remote strain, X the remote stress triaxiality, l the material length parameter, a the radius of the void and n the parameter of power law.

Representative results [START_REF] Fleck | Strain gradient plasticity[END_REF] are given in fig. 2.39, which illustrates the variation of void growth rate with respect to the relative size of the void l/a. A pronounced size effect is predicted by the Toupin-Mindlin theory (SG) 1 which accounts for both the rotation and stretch gradients: the void growth is constrained when the void radius a is of the same order with the material length scale l. However, a much weaker effect is predicted by the couple stress theory (CS)2 that only considers the rotation gradients, which implies that the rotation gradients do not play a significant role in the size dependent void growth.

The problem of an isolated void in an infinite matrix was also investigated by [START_REF] Huang | Mechanism-based strain gradient plasticity-II. Analysis[END_REF] based on the termed mechanism-based strain gradient plasticity of [START_REF] Gao | Mechanism-based strain gradient plasticity-I. Theory[END_REF]. Analyzing a spherical void in the matrix subjected to remote spherically symmetric tension, the authors showed, as presented in fig. 2.40, the remote stress σ ∞ normalized by the yield strength σ Y as a function of the normalized displacement on the void surface u 0 /R 0 , where R 0 is the initial void radius. It was concluded that small voids (large value of l/R 0 ) are less susceptible to growth at a given stress state than larger ones.

Because of the nonlinear nature and complexities of the problem, it is not always possible to derive an analytical result. Instead, numerical approaches, especially the finite element method, offer an efficient way to explore the size effects under different conditions.

Unit cell model

Based on unit cell models, finite element simulations with strain gradient plasticity have been performed for both isotropic materials (see, e.g., [START_REF] Niordson | Void growth to coalescence in a non-local material[END_REF]) and anisotropic single crystals (see [START_REF] Shu | Scale-dependent deformation of porous single crystals[END_REF] shape. Besides, some more implications can be found from this numerical approach, which are outlined in the following. Niordson ( 2008) has studied size effects on both void growth and coalescence in isotropic materials. The strain gradient theory of [START_REF] Fleck | A reformulation of strain gradient plasticity[END_REF] with its extension to finite strains by [START_REF] Niordson | Size-effects in plane strain sheet-necking[END_REF] was utilized. In addition to the common results, the author also showed that, for decreasing void size, the onset of coalescence is significantly delayed and the void volume fraction at the onset of coalescence decreases. Furthermore, it was also found that the size effect is negligible at low stress triaxialities, e.g., T = 0.67.

Given the anisotropy of the grain, a series of studies have been performed to understand size effects in single crystals. [START_REF] Shu | Scale-dependent deformation of porous single crystals[END_REF] applied the model of Shu The normalized void growth rate ε v at ε 22 = 1% as a function of l/a. l is a length scale parameter and a the void radius. [START_REF] Shu | Scale-dependent deformation of porous single crystals[END_REF] and [START_REF] Shu | Strain gradient crystal plasticity: size-dependentdeformation of bicrystals[END_REF], which fits within the framework of [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF] and of [START_REF] Fleck | Strain gradient plasticity[END_REF], to a unit cell with a cylindrical void. Different initial void volume fractions and loading conditions were considered. It was found that the size effect is more significant under biaxial loadings than that at uniaxial loadings. In addition, the effect on the void growth is reduced and that on the macroscopic stress is enhanced with increasing void volume fraction. Note that even though the simulations were performed at small strains, the influence of strain gradients on void shape evolution was not concerned.

Borg and Kysar (2007) analyzed a unit cell with a cylindrical void accounting for 3 in-plane slip systems. The model of [START_REF] Borg | Strain gradient crystal plasticity effects on flow localization[END_REF] based on the principle of virtual power introduced by [START_REF] Gurtin | A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations[END_REF] with the constitutive equations motivated from Fleck and Hutchinson (2001) was used. [START_REF] Borg | Strain gradient crystal plasticity effects on flow localization[END_REF] assumed that plastic slips and plastic slip gradients contribute to plastic work and the plastic dissipation d p per unit volume writes:

d p = N s=1 Q s γs + ξ s l l s .∇ γs + ξ s m m s .∇ γs , ( 2.19)
where

Q s (s = 1, 2, • • • , N
) are the generalized stresses work-conjugate to the plastic slips, ξ s l and ξ s m are the generalized stresses work-conjugate to the slip gradients along the slip direction l s and the slip transverse direction m s . The slip transverse direction vector m s forms a triad with the slip direction vector l s and the normal vector to the slip plane n s . The power of internal forces p i are written as

p i = σ ∼ : D ∼ + N s=1 (Q s -τ s ) γs + N s=1 (ξ s l l s + ξ s m m s ) .∇ γs (2.20)
with the strain rate tensor D ∼ and the resolved shear stress τ s . In the study, the authors focused on deformation and stress fields around the void and void growth was not studied. It has been found that the angular structure of plastic slip around the void is different when using the classical theory and the strain gradient crystal plasticity. Especially, the double slip sectors are bigger with the non-local theory.

The authors also found that the applied stress to activate plastic slip at the void is up to three times higher for smaller voids than for larger ones. [START_REF] Hussein | Plasticity size effects in voided crystals[END_REF] compared the results obtained by the strain gradient crystal plasticity model of [START_REF] Borg | Strain gradient crystal plasticity effects on flow localization[END_REF] with those by DDD. The work was done at small strains considering plane strain conditions with planar slip systems. The internal length scale of the strain gradient theory was fixed according to the DDD simulations. The two theories predicted similar results for two loading conditions: shearing and equi-biaxial straining. For the equi-biaxial straining, the results obtained were consistent with the common conclusion, however, under shearing, an overall stress independent on the void size was found. Note that void growth was not considered in this work. [START_REF] Borg | Size effects on void growth in single crystals with distributed voids[END_REF] analyzed the growth of cylindrical voids under plane strain conditions within finite strain framework. Two levels of initial void volume fractions (12.6%, 3.1%), two crystal orientations and three stress ratios (σ 22 /σ 11 = 0.5, 0.75, 1.0) were considered. The simulations predicted a strong size effect on the overall stress strain response for all the cases considered. However, the effect on void growth is more pronounced for lower initial void volume fraction and higher stress ratios. The authors also found that using the strain gradient theory changes the symmetric structure of plastic slip, and that it makes the deformed void shape smoother.

Effects of surface energy

When material length scale reduces to nano-scale, surface energy can be introduced in continuum mechanics for investigating size effects. Energy associated with the void surface is believed to play a role in void growth when it is nano-sized. To this purpose, the problem of a spherical void of radius R 0 in a finite (spherical) elasto-plastic matrix subjected to uniform remote hydrostatic tensile stress σ m was considered by [START_REF] Huo | A note on the effect of surface energy and void size to void growth[END_REF]. The matrix was supposed to be isotropic and obey a power hardening law. It is solved based on a variational principle accounting for the contribution of surface energy to void growth. Typical results for the initial void volume fraction f = 0.01 are given in fig. 2.44, where the remote stress σ m normalized by the yield strength σ Y is plotted as a function of the actual radius r 0 of the void normalized by its initial value R 0 , i.e., η 0 = r 0 /R 0 . It can be seen that small voids are less susceptible to growth than larger ones. The authors also concluded for typical metals that the effect of the surface energy is negligible for voids larger than 100 nm in size, while may become significant when the void size is of 10 nm.

In addition, some efforts have also been devoted to development of homogenization models for voided materials accounting for surface energy. The details of the models are presented in the next section.

Extension of the Gurson model accounting for void size effects

In the past years, efforts have been dedicated to extend the Gurson model in order to account for effects of void size. All the models predict a reduced void growth rate and an increased loading carrying capacity. The models are summarized in the following.

Model based on Taylor dislocation model

Wen et al. ( 2005) introduced the intrinsic material length l:

l = 18α 2 µ σ 0 2 b (2.21)
with an empirical material constant α around 0.3, the shear modulus µ, the Burgers vector b and the yield stress σ 0 , based on the Taylor dislocation model [START_REF] Taylor | The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical[END_REF], which associates the shear flow stress τ with the dislocation density ρ:

τ = αµb √ ρ, ( 2.22)
Considering an axisymmetric macroscopic strain rate case, the authors obtained the yield surface for a rigid perfectly plastic material containing void given by

1 3 tr Σ ∼ = σ 0 6π Ω dΩ 1 f [2D + λ 2 (1 -3 cos 2 φ)] 1 + 5 2 L f λ 4 1/3 λ 2 + 2Dλ(1 -3 cos 2 φ) + 4D 2 dλ λ (2.23) Σ e = σ 0 6π Ω dΩ 1 f [(1 -3 cos 2 φ)D + λ] 1 + 5 2 L f λ 4 1/3 λ 2 + 2Dλ(1 -3 cos 2 φ) + 4D 2 dλ
(2.24) where f denotes the void volume fraction, D is the ratio of macroscopic volumetric strain rate to the effective strain rate, and λ = r 3 /b 3 with the external radius of the matrix b. Ω dΩ = π 0 sin θ dθ 2π 0 dφ is the integration over spherical angles θ and φ. L, a parameter associated with the intrinsic material length l, is defined by

L = l tr E ∼ a (2.25)
with the trace of the macroscopic strain E ∼ and the radius of the void a. Based on the yield stress, a set of constitutive equations has been established for an elastic-plastic work hardening solid following the approach of [START_REF] Tvergaard | Material Failure by Void Growth to Coalescence[END_REF]. However, this model predicts a very weak size effect on the overall stress-strain curves in uniaxial tension.

Model considering an von Mises interface

σ 2 eq = 3 2 σ ∼ : σ ∼ σ 2 0 (2.26)
with the von Mises equivalent stress σ eq , the stress tensor deviator σ ∼ and the yield stress in simple tension σ 0 , and that the interface obeys a 2D plane stress von Mises-type criterion

3 2 σ ∼ S : σ ∼ S k 2 int (2.27)
with the plane stress deviator tensor σ ∼ S and the yield limit of the interface k int .

A non-dimensional parameter Γ is therefore introduced, characterizing the void size effect:

Γ = k int aσ 0 (2.28)
with the void radius a. Through the limit analysis, the authors obtained a yield surface given by tr

Σ ∼ = σ 0   2   arcsinh(ξ) -arcsinh(f ξ) + Γ 6ξ ξ 2 + 3/5     (2.29)
and The same approach was also used by [START_REF] Monchiet | Combined voids size and shape effects on the macroscopic criterion of ductile nanoporous materials[END_REF] for a matrix containing a spheroidal void.

Σ eq = σ 0   1 + f 2 ξ 2 -f 1 + ξ 2 + Γ 9f 5 ξ 2 + 3/5   , ( 2

Model based on Fleck-Hutchinson strain gradient plasticity theory

Monchiet and Bonnet (2013) extended the Gurson model considering the phenomenological strain gradient plasticity theory of [START_REF] Fleck | Strain gradient plasticity[END_REF] for the solid matrix. Note that the macroscopic strain gradient was not taken into account in the macroscopic dissipation, which was justified by the hypothesis that the size of the representative volume element is large compared with the internal length scales and the size of the voids. With some approximations, an explicit expression of the macroscopic yield surface was obtained, defined by

Φ(Σ ∼ , f, η) = Σ 2 eq σ 2 0 + 2f cosh 3 2η Σ m σ 0 -1 -f 2 = 0, (2.31)
where η is a coefficient related to the material length parameter evaluated by

η = 3 ln(f )   arcsinh α u -1 + u 2 α 2   u=1 u=f 1/3 (2.32) with α = 1 3 2 5 a l 1 . (2.33)
Here, a denotes the void radius and l 1 is the material length parameter in the strain gradient plasticity theory of [START_REF] Fleck | Strain gradient plasticity[END_REF]. A plastic model was also derive based on the yield surface, which predicted a strong size effect on both the stress-strain response and the void growth rate for relatively high stress triaxiality T = 3.

Conclusions

Austenitic stainless steels of 300 series are widely used as structural materials in nuclear power plants, because of their good mechanical properties. However, irradiation produces in the materials different microstructural defects, e.g., Frank loops, voids, etc, resulting in a modification of mechanical properties of the materials.

It has been observed that the fracture toughness of the steels decreases rapidly with radiation dose, especially for the radiation damage less than 10 dpa. In stainless steels irradiated up to 10 dpa, ductile fracture is a dominant fracture mechanisms.

Besides the decrease of fracture toughness with irradiation, a transition from classical dimple fracture to channel fracture and nano-dimple fracture has been observed with increasing dose in stainless steels. The physical foundation of the transition of fracture mechanism is still not well understood.

Dimple fracture, i.e., ductile fracture due to void growth and coalescence, is a dominant fracture mechanism in unirradiated stainless steels. Ductile fracture has being studied in numerous studies. Among various works, FE unit cell simulations are often used to investigate the process of void growth and coalescence under different conditions, e.g., different stress states and initial void volume fractions;

Analytical models are proposed for simulating ductile damage/fracture at structure level. A large number of works were devoted to materials assumed to be isotropic (von Mises materials).

The fracture due to void growth and coalescence remains in stainless steels after irradiation with a trend of decreasing dimple size. These smaller dimples could result from irradiation-induced micro-sized precipitates or nano-sized cavities.

Although voids involved in the ductile fracture process of irradiated steels may have different origins, they are usually intragranular voids, i.e., voids located inside a grain. Few works were devoted to growth and coalescence of intragranular voids, and this problem attracts increasing attentions in recent years. Intragranular voids can be equivalently considered as voids embedded in a single crystal matrix. In this case, effects of the anisotropy of single crystal matrix on void evolution (void growth and coalescence) need to be investigated. In addition, as presented in this section, irradiated stainless steels exhibit a particular strain hardening/softening behavior, whose effects on void evolution is still not clear. Moreover, different approaches were recently adopted for obtaining the effective yield surface of single crystals containing voids. However, there is currently no model for voided single crystals, which is able to describe the void growth up to coalescence and can be used to carry out structural computations.

As mentioned above, voids that are involved in ductile fracture process of irradiated stainless steels have potentially different size (nano to micrometers).

Size effects have been predicted by different approaches for voided materials. In general, it has been found that "smaller is stronger" and "smaller is slower". It implies that, with a given void volume fraction but a decreasing void size, loading carrying capacity of porous materials increases and void growth rate decreases. It is thus necessary to investigate size effects on void growth in FCC single crystals.

Moreover, for taking into account size effects, the Gurson model has been extended through different approaches, such as introducing strain gradient plasticity, interface energy, etc. However, there is currently no ductile fracture model incorporating size effects at single crystal scale.

In the following, ductile fracture due to void growth and coalescence at grain scale will be investigated. FE unit cell simulations will be performed for studying void growth and coalescence in a matrix governed by single crystal plasticity (chapter 3). Effects of some basic aspects, including crystal orientation and stress triaxiality, will be first studied in the case of unirradiated crystals. Effects of postirradiation hardening/softening behavior will assessed in a second step. Following the simulations, an elastoviscoplastic model at finite strains will be proposed for single crystals containing voids (chapter 4). The model incorporates heuristically influences of hardening behavior of single crystal matrix and is able to describe void growth up to coalescence. The model will then be assessed based on the results of the unit cell simulations in the case of unirradiated single crystals. Size effects on void growth and coalescence in single crystals will be investigated (chapter 5).

For this purpose, a strain gradient crystal plasticity model at finite strains will be derived following the micromorphic approach. FE unit cell simulations will be performed with this model for size dependent void growth. This model can be further coupled with the model for porous single crystals, resulting a non-local ductile fracture model for single crystals. The aim of the section is to study mechanical behaviors of voided FCC single crystals, including effective stress-strain relations, void growth and coalescence, by 3D UC simulations. The single crystal plasticity model for the matrix of voided single crystals in described in a first step, followed by the presentation of the FE formulation of UC simulations. The first part of simulations will be devoted to assessment of effects of several parameters, including stress triaxiality, initial void volume fraction and crystal orientation of the matrix, in unirradiated single crystals. The second part of simulations will be devoted to irradiated single crystals. Effects of the post-irradiation hardening law on void growth and coalescence are investigated, by comparison with results obtained by the hardening law for unirradiated single crystals.

Crystal plasticity model for void-free single crystals

FCC single crystals

In the present work, face-centered cubic (FCC) single crystals are considered, since the crystal structure of austenitic stainless steels corresponds to this type. FCC lattice structure, as shown in fig. 3.1, has lattice points on the faces of the cube in addition to the corner lattice points. In FCC single crystals, plastic slip occurs on 12 slip systems, which are specified by the slip direction vector m s and the normal vector n s to the slip plane as summarised in table 3.1 [START_REF] Schmid | [END_REF]. Frank loops are one of irradiation-induced microstructural defects in austenitic stainless steels and will be considered in the hardening laws in the following. In the present work, Frank loops are supposed to remain on the four {111} dislocation slip planes (see table 3.2). For void-free single crystals, a lattice orientation is attributed to each material point. The existence of directors1 associated with lattice orientation allows for the definition of a unique isoclinic intermediate local configuration C i , as recommended by [START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF]. This ensures the uniqueness of the multiplicative decomposition of the deformation gradient F ∼ (see. fig. 3.2) adopted within the finite strain framework:

Kinematics

F ∼ = E ∼ .P ∼ , ( 3.1)
with the elastic part E ∼ and the plastic part P ∼ of the deformation gradient F ∼ . The crystal orientation is the same in the initial local configuration C 0 and the intermediate local configuration C i . The determinants of F ∼ , E ∼ and P ∼ describe the change of volume and of density of a material point:

J = det (F ∼ ) = V V 0 = ρ 0 ρ , ( 3.2
)

J e = det (E ∼ ) = V V i = ρ i ρ , ( 3.3
)

J p = det (P ∼ ) = V i V 0 = ρ 0 ρ i , ( 3.4)
where V 0 , V i and V denote the volume at the reference configuration C 0 , the intermediate configuration C i and the current configuration C; ρ 0 , ρ i and ρ represent the density at the reference configuration C 0 , the intermediate configuration C i and the current configuration C. Note that J p = 1 due to the incompressible plasticity of void-free single crystals undergoing plastic slip.

The velocity gradient L ∼ can be expressed as

L ∼ = Ḟ ∼ .F ∼ -1 = L ∼ e + E ∼ .L ∼ p .E ∼ -1 , ( 3.5)
with the elastic part of the velocity gradient L ∼ e in the current configuration C

L ∼ e = Ė ∼ .E ∼ -1 , (3.6)
and the plastic part of the velocity gradient L ∼ p in the intermediate configuration

C i L ∼ p = Ṗ ∼ .P ∼ -1 . (3.7)

Definition of stresses

According to the approach of [START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF]

Π ∼ e = J e E ∼ -1 .σ ∼ .E ∼ -T , (3.8)
where σ ∼ is the Cauchy stress defined in the current configuration C. The elastic Green-Lagrange strain tensor E ∼ e GL is defined as

E ∼ e GL = 1 2 E ∼ T .E ∼ -1 ∼ . (3.9)
Π ∼ e is related to E ∼ e GL by the elasticity law:

Π ∼ e = C ≈ : E ∼ e GL , ( 3.10)
where C ≈ is the fourth-order anisotropic elasticity tensor, which can be expressed in terms of three parameters C 11 , C 12 and C 44 for cubic elasticity.

In addition, the driving force for single crystal plasticity is known as the Mandel stress M ∼ , which is defined in the intermediate configuration and is workconjugate to L ∼ p :

M ∼ = J e E ∼ T .σ ∼ .E ∼ -T = E ∼ T .E ∼ .Π ∼ e .
(3.11)

Flow rule

For each slip system, s, a yield function can be defined as:

φ s = τ * s -τ s c , with τ * s 0 (3.12)
where τ * s is a scalar stress and τ s c is the critical resolved shear stress (CRSS). For void-free single crystals, the scalar stress τ * s for system s is given by

τ * s = √ τ s τ s = |τ s |, (3.13)
where τ s is the resolved shear stress defined as:

τ s = M ∼ : N ∼ s , ( 3.14)
with the Schmid tensor N ∼ s = m s ⊗ n s (m s is the slip direction vector and n s is the normal vector to the slip plane of slip system s). For each slip system, yielding occurs for φ s 0. The definition of τ * s provides a possibility for formulating the void-free and porous single crystal model in a unified form (see section 4.1).

The plastic strain rate L ∼ p can be defined as

L ∼ p = Ṗ ∼ .P ∼ -1 = N s=1 γs ∂φ s ∂M ∼ = N s=1 γs N ∼ * s , ( 3.15)
with

N ∼ * s N ∼ * s = ∂φ s ∂M ∼ = ∂|τ s | ∂τ s ∂τ s ∂M ∼ = sign (τ s ) N ∼ s (3.16)
and the plastic slip rate γs given by

γs = γref τ * s -τ s c τ ref n , (3.17)
where γref is the reference slip rate and τ ref is a reference resolved shear stress.

Note that

• = • if • > 0, else • = 0.
Here, γs is non-negative by its definition. The sign of plastic slip is given in Eq. (3.15).

Hardening rule

Unirradiated single crystals

For the hardening rule of unirradiated single crystals, an evolution law involving dislocation densities as the only source of hardening is chosen. Following [START_REF] Kubin | Modeling dislocation storage rates and mean free paths in face-centered cubic crystals[END_REF], the slip resistance τ s c , i.e., the critical resolved shear stress, on a particular slip system s is decomposed into a thermal part τ s T and an athermal part τ s A :

τ s c = τ s T + τ s A . (3.18)
τ s T corresponds to lattice friction which is dependent on temperature and assumed to be constant at a given temperature. τ s A represents the athermal contribution of the flow stress. For unirradiated single crystals, only the interaction of dislocation network contribute to τ s A [START_REF] Franciosi | Multislip in f.c.c. crystals a theoretical approach compared with experimental data[END_REF], which is expressed as

τ s A = µb D 12 u a su ρ u D , (3.19)
where b D is the norm of the Burgers vector of dislocations, µ the shear modulus, ρ u D the dislocation density of system u, and a su the matrix of long-range interactions between dislocations. The form of interaction matrix a su is given in appendix A for FCC crystals.

The description of the evolution of dislocation densities is based on two mechanisms: the multiplication and the annihilation of dislocations [START_REF] Mecking | A new aspect of the theory of flow stress of metals[END_REF][START_REF] Essmann | Slip in copper crystals following weak neutron bombardment[END_REF][START_REF] Teodosiu | A finite theory of the elastoviscoplasticity of single crystals[END_REF][START_REF] Tabourot | Generalised constitutive laws for f.c.c. single crystals[END_REF][START_REF] Cheong | Discrete dislocation density modelling of single phase {FCC} polycrystal aggregates[END_REF]. The dislocation density ρ s D follows:

ρs D = 1 b D 1 L s -g c ρ s D γs , ( 3.20)
where L s is the mean free path of the dislocation segment before being stopped at obstacles in the form of forest dislocations and g c is the critical distance controlling the annihilation of dislocations with opposite signs. For unirradiated crystals, L s is written as

L s = L s D = κ 12 u b su ρ u D . (3.21)
The κ parameter is proportional to the number of obstacles crossed by a dislocation before being pinned. The matrix b su describes the interaction between dislocations;

it has the same form as the interaction matrix a su (see appendix A).

Irradiated single crystals

For irradiated single crystals, the slip resistance τ s c is also decomposed, following Eq. (3.18), into the thermal part τ s T , which assumes to be a constant at a given temperature, and the athermal part τ s A . For the athermal contribution of the flow stress τ s A , in addition to the dislocation term (Eq. (3.19)), two more terms are considered to contribute to the athermal slip resistance for the irradiated material:

τ s A = µb D 12 u a su ρ u D dislocation term + α L µb L 4 p φ L ρ p L loop term + τ a exp - γ s γ 0 unpinning term , ( 3.22)
where α L denotes a parameter setting the relative contribution of Frank loops to the hardening, b L the norm of the Burgers vector of Frank loops, φ L the mean diameter of Frank loops, ρ p L the density of Frank loops in the slip plane p, τ a a reference shear stress for dislocation unpinning and γ 0 a coefficient to adjust the avalanche speed after unpinning the dislocations. 1The first term (called the dislocation term in the following) on the right-hand side of Eq. (3.22) is the slip resistance provided by the interaction of dislocation network [START_REF] Franciosi | Multislip in f.c.c. crystals a theoretical approach compared with experimental data[END_REF]. The second term (loop term) accounts for the effect of Frank loops generated by irradiation impeding dislocation motion. Based on Mughrabi's observation [START_REF] Mughrabi | Materials Science and Technology, Plastic Deformation and Fracture of Materials[END_REF] that the junction energy of dislocations and loops is significantly greater than that of thermal activation, the hardening due to Frank loops is considered to contribute to the athermal part. The third term In irradiated stainless steels, the evolution of dislocation densities also follows the form of Eq. (3.20) but with modified mean free path L s . In fact, Frank loops produced by irradiation are not rigid obstacles. Molecular dynamics simulations by [START_REF] Rodney | Dislocation Pinning by Small Interstitial Loops: A Molecular Dynamics Study[END_REF] showed that Frank loops can interact with dislocations, resulting in the unfaulting of the loops. According to [START_REF] Yang | Behavior of Frank-loops under stress environment[END_REF], dislocations can be emitted from the circumference of the loop during the interaction. These results motivate the modification of the mean free path L s in order to include the influence of Frank loops; it is assumed to take the following form:

1 L s = 1 L s D + 1 L L , ( 3.23)
where L s D is still given by Eq. (3.21) and the Frank loop contribution to the mean free path writes

L L = κ k dl 4 p φ L ρ p L , ( 3.24)
where k dl is a coefficient setting the effective interaction between Frank loops and dislocations.

The model proposed by Krishna and his co-workers (Krishna et al., 2010; Krishna and De, 2011) is adopted to describe the evolution of Frank loop densities. It is based on the mechanism of annihilation of Frank loops by dislocations gliding in the loop's plane. The probability and frequency of the annihilation of a Frank loop are taken into account. The decrease rate of Frank loop density ρp L on plane p is

ρp L = -A   s∈plane(p) ρ s D   φ L b D ρ p L -ρ sat L   s∈plane(p) γs   , ( 3.25)
where A denotes the annihilation area and ρ sat L the stabilized effective density of Frank loops. s ∈ plane(p) represents all the dislocation slip systems having the same slip plane as the Frank loop p.

FE formulation of unit cell simulations

The simulations are performed with the FE software Zset [START_REF] Besson | Object-Oriented Programming Applied to the Finite Element Method Part I. General Concepts[END_REF]. Throughout the section, variables at microscopic and macroscopic scale are distinguished. The variables with an overline symbol (e.g., F ∼ ) are used for macroscopic scale at which an effective behavior of the unit cell is observed, while the variables without the overline symbol describe the behavior at the microscopic scale, i.e., at each material point inside the unit cell.

The problem setup is shown in fig. 3.3. It is assumed that the distribution of voids in the single crystal is homogeneous. Thus, a unit cell, i.e., a representative elementary volume, Ω tot 0 can be considered as a cube of length L 0 with a spherical void of radius R 0 at its center (see fig. 3.4a). Notice that L 0 characterizes the average distance between voids. Thus, the initial void volume fraction f 0 is

f 0 = 4 3 π R 3 0 L 3 0 . (3.26)

FE formulation of unit cell simulations

u = F ∼ .X + v , ( 3.27)
where F ∼ denotes the macroscopic deformation gradient, u the displacement vector and v a periodic fluctuation vector which obey the following periodicity condition:

v (x + ) = v (x -) (3.28)
where x + and x -denote the position of homologous nodes on opposite faces of the unit cell. The microscopic first Piola-Kirchhoff stress tensor S ∼ and the associated macroscopic tensor S ∼ are related by

S ∼ = 1 V tot 0 Ω tot 0 S ∼ dV 0 = (1 -f 0 ) 1 V mat 0 Ω mat 0 S ∼ dV 0 , ( 3.29)
where V tot 0 and V mat 0 denote respectively the total volume of the unit cell and the volume of the matrix in the reference configuration. In uniaxial tension, the first Piola-Kirchhoff stress corresponds to the so-called engineering stress.

The macroscopic Cauchy stress tensor σ ∼ is related to the macroscopic first Piola-Kirchhoff stress tensor by:

σ ∼ = 1 J S ∼ .F ∼ T , ( 3.30)
with J = det F ∼ . The Cauchy stress tensor corresponds to the true stress in the uniaxial tension case. The microscopic deformation gradient F ∼ and the macroscopic tensor F ∼ are related by:

F ∼ = 1 V tot 0 Ω tot 0 F ∼ dV 0 . ( 3.31)
Constant macroscopic Cauchy stress triaxiality T is imposed with

σ ∼ =    σ 11 0 0 0 σ 22 0 0 0 σ 33    = σ 11    1 0 0 0 η 2 0 0 0 η 3    = σ 11 η ∼ (3.32)
with

η 2 = σ 22 σ 11 , η 3 = σ 33 σ 11 , 0 η 2 , η 3 1, (3.33)
such that

T = σ m σ eq = 1 + η 2 + η 3 3 1 -η 2 -η 3 -η 2 η 3 + η 2 2 + η 2 3 . (3.34)
In the present study, axisymmetric loadings are mainly considered, which implies η 2 = η 3 = η so that the imposed macroscopic Cauchy stress triaxiality is given by

T = 1 + 2η 3(1 -η) . (3.35)
Moreover, moderate to high stress triaxialities are investigated (see table 3.3), as they correspond to the stress triaxialities encountered in ductile failure process zones, such as in the vicinity of a crack tip. The method used to impose a constant triaxiality with periodic boundary conditions is described in appendix B. 3.4). For the sake of brevity, the crystal orientations are named in the following by the lattice orientation along the main loading direction (axis x 1 ), i.e., [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcclintock | A criterion for ductile fracture by the growth of holes[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF], [210] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF].

Orientation name Crystallographic orientation Number of Schmid factor along x 1 -x 2 -x 3 primary slip systems [100] [100] -[010] -[001] 8 1 √ 6 ≈ 0.408 [110] [110] -[ 110] -[001] 4 1 √ 6 ≈ 0.408 [111] [111] -[ 211] -[0 11] 6 ≈ 0.272 [210] [210] -[ 120] -[001] 2 √ 6 5 ≈ 0.490 [ 125] [ 125] -[1 21] -[210] 1 √ 6 5 ≈ 0.490
Table 3.4: Crystal orientations, number and Schmid factor of primary slip systems activated in uniaxial tension along x 1 .

Results for unirradiated single crystals

The simulations are first performed for unirradiated single crystals. The material parameters used are closely related to those of a solution annealed 304 austenitic stainless steel at 340 • C (see [START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF]).Among the parameters, a su and b su are two matrices (12 × 12 for FCC single crystal), describing the interaction between dislocations. Each of them is constructed by 6 independent parameters:

a 1 , a 2 , • • • , a 6 for a su and b 1 , b 2 , • • • , b 6 for b su (see appendix A).
The matrix a su is assumed to be identical for all s, u = 1, ..., 12 (this is different from the matrix used in the work of [START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF] which is identified by DDD simulations) to avoid some numerical bifurcation problems maybe encountered later in section 4.1; b su is set equal to 0 if s = u and equal to 1 if s = u. Due to the crystal structure and to the considered slip systems, τ s T can be considered as having the same value for all systems. The initial dislocation density is supposed to be equal for every system.

Hence, the initial value of the dislocation density for system s is ρ s The most relevant results could be discussed after choosing the three following orientations: [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] (see table 3.4). In uniaxial tension, [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] is a multiple slip (8 primary slip systems) orientation with mirror symmetry about three coordinate planes; [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] is a multiple slip (6 primary slip systems) orientation with mirror symmetry about the x 1 -x 2 coordinate plane, and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] represents single slip orientation with no mirror symmetry about the coordinate planes. Note that the same primary slip systems are activated in axisymmetric loading as in uniaxial tension along axis x 1 . Schmid factors calculated for the different orientations in the case of uniaxial tension along axis x 1 (see table 3.4) will be used to interpret the results. The influence of the FE discretisation on the overall stress strain behaviour and the void volume fraction evolution is evaluated. The overall Cauchy stress σ ∼ is

D | ini = ρ tot D /

FE discretisation effect

σ ∼ = 1 V tot Ω tot σ ∼ dV = (1 -f ) 1 V mat Ω mat σ ∼ dV, (3.36)
where V tot and V mat denote respectively the total volume of the unit cell Ω tot and the volume of the matrix Ω mat in the current configuration, and the void volume fraction f is calculated by

f = V tot -V mat V tot , ( 3.37)
where the volume of the matrix V mat in the current configuration is calculated by a post-processing of Zset software and the total volume of the unit cell V tot in the current configuration can be obtained as

V tot = det F ∼ V tot 0 , ( 3.38)
with V tot 0 the initial volume of the unit cell. The evolution of normalised macroscopic Cauchy stress component σ 11 /τ s T and the void volume fraction f with respect to the deformation in the main loading direction F 11 -1 are shown in fig. 3.6. It can be seen that the influence of FE discretisation on the overall stress strain behaviour and the evolution of void volume fraction is negligible for both low stress triaxiality (T = 1) and high stress triaxiality (T = 3). Even in the softening regime at T = 3, its influence is weak.

Based on this result, the coarse mesh (n = 5) will be used in the following, except otherwise stated, to obtain the curves of the macroscopic Cauchy stress σ 11 and the void volume fraction f . In particular, the fine mesh (n = 10) will be used to investigate local plastic slip fields in section 3.3 but the coarse mesh (n = 5) will be used later in section 3.4.

Overall stress-strain response

Figure 3.7a shows the overall stress strain response of the unit cell for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation with f 0 = 0.01 and the stress triaxiality T varying from 1 to 3. The macroscopic stress σ 11 is normalised with respect to τ s T . It can be seen that the behaviour consists of a hardening regime followed by a softening regime, resulting from the competition between the strain hardening of matrix, the softening due to void growth and the softening due to void coalescence. The softening occurs earlier at high stress triaxiality compared to that at low stress triaxiality. This can be explained by faster void growth at higher stress triaxiality.

Moreover, a transition from triaxial to uniaxial straining associated with the localization of the plastic flow in the intervoid ligament is observed, which corresponds to the onset of void coalescence according to [START_REF] Koplik | Void growth and coalescence in porous plastic solids[END_REF]. In this work, it is determined by plotting the transverse macroscopic strain F 33 -1 as a function of the longitudinal macroscopic strain F 11 -1. It is observed that F 33 -1 reaches a stabilized value. In this study, by convention, the time step for which the transverse strain F 33 -1 reaches 99% of its stabilized value, is regarded as the onset of coalescence. The corresponding longitudinal The effect of crystallographic orientation on the overall behaviour is presented in fig. 3.8, where the five crystallographic orientations are considered with the initial void volume fraction f 0 = 0.01 and the stress triaxiality T = 2. The [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation shows the hardest response and the orientations [210] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] exhibit the softest response (see fig. 3.8a). The peak stress depends on the orientation and this agrees with the results obtained by [START_REF] Ha | Void growth and coalescence in F.C.C. single crystals[END_REF] for FCC single crystals and by [START_REF] Yerra | Void growth and coalescence in single crystals[END_REF] for BCC single crystals. In particular, the [START_REF] Mcclintock | A criterion for ductile fracture by the growth of holes[END_REF] orientation exhibits the highest peak stress and this is also observed by [START_REF] Ha | Void growth and coalescence in F.C.C. single crystals[END_REF]. As a general result, the softening regime starts earlier for the orientation [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] compared with the other orientations, which implies the fastest void growth for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation, as will be confirmed in fig. 3.10.

For the asymmetric orientations, especially [210] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF], the straining of the unit cell along the loading axis is accompanied by shearing, and the transverse straining of the unit cell may stop only along one transverse axis (x 2 or x 3 axis), which is interpreted as the preferred direction of coalescence. In that case, the onset of coalescence is determined in the "preferred coalescence direction", where void coalescence occurs first. As shown in fig. 3.8b, the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation leads to the earliest onset of coalescence.

Void growth

Figure 3.9 shows the effect of stress triaxiality on the evolution of void volume fraction f for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation with the initial void volume fraction f 0 = 0.01 and the stress triaxiality T varying from 1 to 3. The results confirm the previous analysis: higher stress triaxiality leads to faster void growth, which induces earlier softening of the unit cell. These results are consistent with the observation of Ha and Kim (2010) for FCC single crystals. Onset of coalescence [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] [100]

[110]

[210] in both cases of T = 1 and T = 3, the void growth rate is significantly higher in the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation than the other orientations, which is in good agreement with the previous analysis. A significant result from this calculation is the very limited void growth and the quasi-incompressible response for the [210] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations at low stress triaxiality T = 1. Void aspect ratios are also investigated, allowing to characterise the evolution of the void shape. The evolution of two aspect ratios

W 2 = R 1 R 2 and W 3 = R 1 R 3
are presented in fig. 3.11a for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation and in fig. 3.11b for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation with the initial void volume fraction f 0 = 0.01. R i (i = 1, 2, 3) is the length from the centre of void to the node at the initial pole of the void surface in the x i direction. Two levels of stress triaxiality T = 1 and T = 3 are considered here. For both orientations, the aspect ratios become larger than 1 before void coalescence at T = 1, which implies a void elongation in the main loading direction at low stress triaxiality. For the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation at T = 3, the void aspect ratios decrease without exceeding the value of 1 before the onset of coalescence, which leads to an oblate shape of the void. For the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at T = 3, the void aspect ratio W 2 remains close to 1 and W 3 becomes somewhat larger than 1 before the onset of coalescence. For symmetry reasons, W 2 is equal to W 3 for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation at both high and low stress triaxiality. However, W 3 becomes much larger than W 2 for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] especially at low stress triaxiality, which can be explained by the anisotropy in the transverse plane x 2 -x 3 of the unit cell. It can also be observed that the difference in the evolution of the void aspect ratios between the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation and the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation is much more significant at low stress triaxiality than that at high stress triaxiality.

Field of total accumulated slip

The total accumulated slip γ cum , defined as

γ cum = 12 s=1 γ s , ( 3.39)
is used to display the field of plastic slip around the void, and to explain some results obtained about void growth. The total accumulated slip fields in the middle x 1 -x 2 and x 1 -x 3 cross sections (the definition of the cross sections is shown in fig. 3.4a) of the unit cell are shown for the stress triaxialities T = 1 in fig. 3.12 and T = 3 in fig. 3.13 with the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations and the initial void volume fraction f 0 = 0.01 at F 11 -1 = 0.1. The fine mesh (n = 10, see fig. 3.5b) is used here to obtain the plastic slip field with more accurate local results. Recall that the macroscopic responses obtained with the fine mesh are identical to those with the coarse mesh, as shown in fig. 3.6. For symmetry reasons, the surfaces of the unit cell remain planes and the unit cell keeps its cubic shape during loading for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation. For the other orientations, the unit cell does not remain cubic during loading, as a result of lattice re-orientation and shearing of the mesh. It is observed that for each configuration (crystallographic orientation and stress triaxiality) a different pattern of plastic slip localisation develops. For example, for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation at T = 1 and T = 3, the unit cell shows a symmetric cross shaped localisation zone and the field in the x 1 -x 2 cross section is the same as that in the x 1 -x 3 cross section, as expected from matrix material symmetries. However, such symmetry is not observed for the two other orientations. For almost all six cases, the plastic slip is highly localised around the void, i.e., the red zone, except for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at T = 1, where the localisation around the void is much weaker. A zone where no slip system is activated, i.e., the blue zone, is observed in all cases, except for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at T = 1. In the case of the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at T = 1, the activation of the slip systems is nearly homogeneous in the matrix and it is found that only one slip system is activated almost everywhere in the matrix except in the vicinity of the void. One can conclude that the plastic slip heterogeneity introduced by the void in the matrix is negligible for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at low stress triaxiality, but that the plastic slip heterogeneity will increase with stress triaxiality. Figure 3.12 and fig. 3.13 also show the void shape at F 11 -1 = 0.1. The elongation of the void in the x 1 axes can be observed at T = 1 for the three orientations. For the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation at T = 3, the void evolves into a polygon-like shape. For [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] at T = 3, the void is of lemon-like shape in the x 1 -x 2 cross section and polygon-like in the x 1 -x 2 cross section. For [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] at T = 3, one can see that the cut of the void in the x 1 -x 2 cross section is almost a circle, i.e. R 1 = R 2 remains, however the void is elongated in the x 1 -x 3 cross section. These results are in good agreement with the results shown in fig. 3.11.

Field of lattice rotation

The polar decomposition of the elastic part of deformation gradient E ∼ follows

E ∼ = R ∼ .U ∼ , ( 3.40)
with the rotation tensor R ∼ and the right stretch tensor U ∼ . Neglecting the elastic distortion U ∼ , R ∼ can be interpreted as the lattice rotation. The corresponding rotation angle θ is given by

θ = arccos 1 2 (trace R ∼ -1). (3.41)
The fields of θ (in radian) in the middle x 1 -x 2 and x 1 -x 3 cross sections are shown for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations with the initial void volume fraction f 0 = 0.01 and the stress triaxiality T = 1 in fig. 3.14 and T = 3 in fig. 3.15 at F 11 -1 = 0.1. At T = 1, lattice rotation occurs mainly around the void for [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF], while it is almost homogeneous in the matrix for [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF]. This is consistent with the fields of total accumulated slip observed in section 3.3.4. For [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF], the quasi-homogeneous lattice rotation in the matrix leads to void-free-like macroscopic behaviour of the unit cell, i.e., the unit cell exhibits nearly pure shear, for a single slip orientation, in terms of macroscopic deformation pattern. At T = 3, the lattice rotation around the void is more significant compared with that at T = 1. In particular, the field of θ is no longer quasi-homogeneous for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation and the macroscopic deformation deviates from homogeneous shear.

Slip sectors

The number of activated slip systems is investigated. In all the simulations, plastic deformation always begins in single slip in some regions near the void. with only an orientation deviation between them.

For the different orientations, the multiple slip sectors are shown to be different at T = 1 but similar at T = 3. This observation is consistent with that on the total accumulated slip in section 3.3.4, which shows the different slip localisation modes at T = 1 but the slip localisation covering the void for all the orientations considered at T = 3. These results may be one of possible reasons explaining the observation of significant orientation effects on void growth at T = 1 but reduced effects at T = 3. for BCC single crystals. The difference of the critical strain F 11c -1 between different orientations is significant at low stress triaxiality, but less pronounced at high stress triaxiality.

Void coalescence

Once the critical strain F 11c -1 for the onset of void coalescence has been determined, the corresponding critical void volume fraction f c for the void coalescence can readily be obtained. In fig. 3.17b, the critical void volume fraction Unit cell simulations with larger or smaller initial void volume fractions show similar influences of stress triaxiality and crystallographic orientation on the overall stress strain response and void evolution. For the sake of clarity, these results have not been reported in this section, but they will be used for the assessment of the macroscopic porous single crystal model presented in the next section.

Conclusions

Unit cell simulations have been performed in this section for investigating void growth and coalescence in unirradiated FCC single crystals. Effects of two parameters, i.e., stress triaxiality and crystal orientation, are studied.

The unit cell simulations have shown the dependence of overall stress strain response on the stress triaxiality and the crystallographic orientation. The [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation exhibits harder response compared to the other orientations considered. Void growth rate has been found to increase with stress triaxiality for a given crystal orientation. The void growth also displays an orientation dependence which is more significant at lower stress triaxialities than higher stress triaxialities. In particular, a quasi absence of void growth is observed for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation with small initial void volume fraction at low stress triaxiality, which leads to quasiincompressible overall behaviour. In addition, the critical void volume fraction f c for the onset of coalescence highly depends on crystallographic orientation, while it is almost not influenced by stress triaxiality varying from 1 to 3 for a given crystallographic orientation.

Having understood mechanical behaviors of voided single crystals whose matrix is governed by hardening laws for unirradiated crystals, effects of post-irradiation hardening/softening laws on void growth and coalescence will be studied in the next section.

Results for irradiated single crystals

UC simulations are also carried out for irradiated single crystals with the coarse mesh (n = 5, see fig. 3.5a). Three crystal orientations are considered: [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF]. The material parameters used in the simulations were identified in the work of [START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF] for 304L stainless steels irradiated up to 13 dpa.

The identification of the material parameters was performed by fitting a polycrystalline aggregate FE model response to the tensile test results of Pokor and his co-workers (Pokor et

b 3 b 4 b 5 b 6 0 1 1 1 1 1 0 dpa 13 dpa φ L - 7.3 nm α L - 0.57 k dl - 1.7 × 10 -7 A - 1.25 × 10 -12 m 2 ρ sat L - 10 22 m -3 τ a - 61.2 MPa γ 0 - 5 × 10 -3 ρ s D | ini 8.3 × 10 8 m -2 1.6 × 10 8 m -2 ρ p L | ini - 1.6 × 10 22 m -3

Void-free single crystal: a reference

Simulations are first performed using a single finite element. The results will provide a reference for comparisons with the porous single crystal. Figure 3.18a shows the evolution of the overall first Piola-Kirchhoff stress component S 11 with respect to the strain measure F 11 -1 of the void-free single crystal (dashed lines) for the unirradiated state (0 dpa) and the irradiated state (13 dpa) for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation at T = 1. Recall that the first Piola-Kirchhoff stress corresponds to the engineering stress in uniaxial tension, which can be directly compared with experimental results.

For 0 dpa, a strain hardening regime can be observed, which is related to the interaction among dislocations considered in Eqs. (3.20) and (3.22). With increasing strain, S 11 decreases slightly, since S 11 is evaluated on the undeformed section.

This corresponds to the cross section area reduction of the unit cell due to finite strains. Note that the Cauchy stress σ 11 is increasing in that case due to hardening and to the absence of damage. Compared to the behavior at 0 dpa, the void-free single crystal at 13 dpa displays a marked increase of the yield stress and a sharp softening at the beginning of the plastic regime followed by a slight hardening. The stress-strain curves are also shown in fig. 3.19 for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation and fig. 3.20 for [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF]. The [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation exhibits similar stress-strain response as the [100] orientation. A higher stress level is reached due to the lower Schmid factor of primary slip systems (see table 3.4). However, the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation displays a different response. Recall that the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation has only one primary slip system (s = 1) and one secondary slip system (s = 12). For 0 dpa, a change in the hardening rate can be observed, which is associated with the activation of the secondary slip system leading to greater strain hardening. At 13 dpa and before the activation of the second slip system, results show the initial load drop (unpinning term) followed by a limited hardening. The activation of the secondary system at about A and its components for the activated systems (a) s = 1 and (b) s = 12 in irradiated void-free single crystal for [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] at T = 3.

Irradiation effects on stress-strain response of the unit cell

Figure 3.18a also presents overall stress-strain responses of the voided single crystal (solid lines) for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] orientation at T = 1. For the voided single crystal, the stress level that can be reached is reduced compared to the void-free one at both 0 dpa and 13 dpa. Moreover, strain hardening at 0 dpa is weakened and the strain softening at 13 dpa is enhanced because of void growth. After reaching a certain strain level, the stress decreases dramatically due to void coalescence with a greater void growth rate. This trend is in agreement with the work of Koplik and Needleman (1988) on a von Mises material: the onset of void coalescence is characterised by the transition to a uniaxial straining associated to the localization of the plastic flow in the intervoid ligament and a sharp load drop.

With stress triaxiality increasing up to T = 3 as shown in fig. 3.18b, strain softening begins earlier than at T = 1. Voids also lead to the decrease of the yield stress. This effect is stronger for the irradiated material. The presence of the void induces a plastic strain gradient so that the cell is progressively yielded, which prevents the macroscopic sharp peak load associated with the unpinning term. In addition, the strain hardening regime following initial softening which was observed on the void-free material completely disappears. Instead, the voided single crystal exhibits a strong softening. At the structural level this kind of behavior can lead to strain localization (so called "shear bands") and consequently to a quasi-brittle behavior.

The stress-strains curves are also shown in fig. 3.19 for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation and fig. 3.20 for [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF]. Effects of the voids shown at 0 dpa and 13 dpa for both orientations are similar to those for [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF]. However some differences were observed and are described in the following.

For [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF], the peak stress drops by about 50% at 13 dpa at T = 3 due to the presence of voids. This peak drop is more severe than that observed for [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF]. This effect is related to the low Schmid factor of this orientation. As already outlined above, the presence of voids creates a plastic strain gradient so that the local Schmid factor in areas close to the void boundary is higher than the nominal Schmid factor; this promotes an early yielding of the cell.

The presence of the plastic strain gradient around the void strongly influences the cell behavior for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation. In the voided single crystal, different slip systems are activated in the vicinity of the void and multiple slip occurs whereas single slip prevails in the sound material. As a result, the single to double slip transition is not observed. In addition, latent hardening is locally triggered at the early stages of deformation so that the voided crystal may exhibit a higher flow stress up to a given strain where void growth will induce softening (see fig. 3.20b for 0 dpa).

Irradiation effects on void growth

The evolution of the void volume fraction f under a fixed stress triaxiality T ∈ {1, 1.5, 2, 3} is presented in fig. 3.23. As expected, void growth rate increases with the stress triaxiality for whatever orientation. For both unirradiated and irradiated voided single crystals, the void growth rate is the highest for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation for all considered triaxialities and a quasi-absence of void growth is predicted for 125 with T = 1. As shown in [START_REF] Ling | An elastoviscoplastic model for porous single crystals at finite strains and its assessment based on unit cell simulations[END_REF], this is due to single slip which cannot lead to void growth. For the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation, however, void growth becomes significant and cannot be neglected as the stress triaxiality reaches T = 1.5 as shown in fig. 3.23b. Compared with unirradiated single crystal, a higher growth rate (with respect to the overall strain F 11 ) is systematically observed in the irradiated crystal before the onset of coalescence for all investigated orientations and triaxialities. The effect of the irradiation is particularly strong at moderate stress triaxialities (i.e. T = 1.5 (see fig. 3.23b) and T = 2 (see fig. 3.23c)) for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation. In those cases, the void growth rate at the beginning of plastic regime is multiplied by 2.5 and 1.7 for the irradiated crystal. The onset of void coalescence is determined in the same way as in sections 3.3.2 and 3.3.7: it is determined by plotting the transverse macroscopic strain F 33 -1 as a function of the longitudinal macroscopic strain F 11 -1. It is observed that F 33 -1 reaches a stabilized value. By convention, the time step for which the transverse strain F 33 -1 reaches 99% of its stabilized value, is regarded as the onset of coalescence. The corresponding longitudinal F 11 -1 and porosity are defined as the critical strain F 11c -1 and critical porosity f c at the onset of coalescence. The onset of void coalescence is indicated by a hollow square on each curve. In the void coalescence regime, the void growth rate is fully determined by the coalescence kinematics (uniaxial straining).

Irradiation effects on onset of coalescence

The critical strain F 11c -1 for the onset of coalescence is plotted as a function of stress triaxiality T for the three orientations in fig. 3.24a. The decrease of F 11c -1 with increasing stress triaxiality can be observed for a given orientation. Compared with the unirradiated cases, the onset of coalescence occurs at a smaller strain in the irradiated single crystals. Especially for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation, the decrease of F 11c -1 even reaches about 40% at T = 1.5.

The evolution of critical void volume fraction f c for the onset of coalescence as a function of stress triaxiality T is presented in fig. 3.24b. Considering one crystallographic orientation, f c is almost unchanged for a given irradiation level. The [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation shows a higher f c than the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations. With stress triaxialities varying from 1.5 to 3, similar values of f c are predicted for [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF]. Given one orientation, the value of f c of the irradiated single crystals is lower than that of unirradiated ones.

Irradiation effects on fields of accumulated plastic slip

The fields of γ cum in the middle x 1 -x 2 cross section (fig. 3.4a) are shown in fig. 3.25 for [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] with T = 1.5 (F 11 -1 = 0.35) and T = 3 (F 11 -1 = 0.2). The evolution of the void shape is different at different stress triaxialities. In particular, at a relatively low stress triaxiality T = 1.5, the void becomes elongated along the loading axis x 1 . Moreover, at the same stress triaxiality, a more significant localization of plastic slip is observed in the irradiated case than that in the unirradiated case, especially in the horizontal region near the void. This leads to a larger deformation of the void along the horizontal axis in the irradiated case, resulting in a faster void growth and an earlier void coalescence as shown previously (see fig. which showed an embrittlement after irradiation at the macroscopic level but a large amount of deformation at the failure site showing fine dimples.

The fields of γ cum are presented for [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] the matrix, associated with the anisotropy of the single crystal. Besides, the void displays different shapes in other cross sections, e.g., the x 1 -x 3 cross section, which has been discussed in section 3.3 and is not discussed here. Furthermore, in the irradiated case, again, more significant plastic slip localization can be seen for [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] than that in the unirradiated cases. Larger voids can be observed in the irradiated cases, especially for [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] at T = 1.5 (fig. 3.27a) which is consistent with faster void growth presented in fig. 3.23.

Conclusions

In this section, the void growth and coalescence at constant stress triaxiality in irradiated FCC single crystals are investigated using unit cell FE simulations. A single crystal plasticity model is used accounting for the modification of mechanical properties associated to the irradiation-induced defects. From these unit cell simulations, the following conclusions can be drawn:

• Voids grow at a higher rate in the irradiated single crystal.

• The onset of void coalescence occurs at a smaller value of the overall strain and at a smaller value of the void volume fraction in the irradiated crystals . • More significant plastic strain localization in the region near the void is predicted in the irradiated crystal, correlated with the faster void growth and earlier onset of void coalescence. • Brittle-like overall behavior is predicted at high level of stress triaxiality, i.e. T = 3, in the voided irradiated crystal, while a large amount of plastic deformation is reached in the vicinity of the void. It is worth emphasizing that the numerical results showing faster void growth and earlier onset of void coalescence in irradiated single crystals are in agreement with the experimental observation of decreasing fracture toughness with irradiation. Furthermore, enhanced plastic strain localization predicted for irradiated single crystals is also in good agreement with the observations of [START_REF] Neustroev | Severe embrittlement of neutron irradiated austenitic steels arising from high void swelling[END_REF] in AISI 321 stainless steels irradiated in BOR-60.

Concluding remarks

In this chapter, unit cell simulations have been performed to investigate void growth in FCC single crystals. Effects of stress triaxiality, crystal orientation and post-irradiation strain hardening/softening have been studied.

In order to investigate influences of irradiation on the fracture toughness of stainless steels, a fracture model is needed. As explained above, voids involved in ductile fracture process of irradiated steels are often located within grains, i.e., intragranular voids. Due to this fact, a micromechanics model of ductile fracture model will be proposed at the scale of a grain, i.e., a single crystal, in the next chapter.

The model aims to account for the effects of different parameters (stress triaxiality, crystal orientation and post-irradiation strain hardening/softening) outlined in this chapter. The model will be assessed based on the results of the unit cell simulations and then applied to study ductile fracture in single crystals and polycrystals.

Porous single crystal plasticity model

The porous single crystal plasticity model is formulated within the framework presented in section 3.1. For the sake of brevity, only new notions related to porous single crystals are outlined.

For porous single crystals, it is assumed that, in spite of the presence of pores inside the single crystal volume element and associated inhomogeneous deformation, a unique single crystal lattice orientation can be attributed to each material point.

The definition of a unique isoclinic intermediate local configuration

Ψ s = τ s2 τ * s 2 + α 2 45 f i M 2 eq τ * s 2 + 2q 1 f i cosh    q 2 3 20 M m τ * s    -1 -q 2 1 f 2 i def = 0, τ * s 0, (4.1)
where M m is the mean stress of Mandel stress defined as M m = 1 3 trace M ∼ ; M eq is the equivalent stress of Mandel stress defined as M eq = 3 2 M ∼ : M ∼ with the deviatoric part of Mandel stress M ∼ = M ∼ -M m 1 ∼ ; f i is the void volume fraction in the intermediate configuration and can be calculated as

f i = 1 - 1 -f 0 det (P ∼ ) (4.2)
with the initial void volume fraction f 0 . Note that α, q 1 and q 2 are parameters to be identified. The yield function for each slip system, taking the same form of Eq. (3.12), follows

φ s = τ * s -τ s c (4.3)
with the effective resolved shear stress τ * s defined above. The plastic strain rate L ∼ p can be defined by

L ∼ p = Ṗ ∼ .P ∼ -1 = (1 -f i ) N s=1 γs ∂φ s ∂M ∼ = (1 -f i ) N s=1
γs N ∼ * s , (

with M ∼ still defined by Eq. (3.11) and

N ∼ * s = ∂τ * s ∂M ∼ = - ∂Ψ s ∂τ * s -1 ∂Ψ s ∂M ∼ , ( 4.5)
where

∂Ψ s ∂τ * s = -2 τ s2 τ * s 3 - 4 45 αf i M eq 2 τ * s 3 -2 3 20 q 1 q 2 f i M m τ * s 2 sinh   q 2 3 20 M m τ * s   (4.6)
and

∂Ψ s ∂M ∼ = 2 τ s τ * s 2 N ∼ s + 2 15 αf i 1 τ * s 2 M ∼ + 2 3 3 20 q 1 q 2 f i sinh q 2 3/20M m /τ * s τ * s 1 ∼ . (4.7)
Note that L ∼ p is corrected by the factor 1 -f i resulting from the definition of τ * s in Eq. (4.1) which is obtained by the homogenisation in the matrix of porous single crystal excluding the pores. It corresponds to the vanishing plastic work in the pores [START_REF] Besson | Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms[END_REF][START_REF] Besson | Continuum Models of Ductile Fracture: A Review[END_REF]. When f i = 0, the model of porous single crystals is reduced to that of void-free single crystals.

The plastic slip rate γs , the same as Eq. (3.17), is given by

γs = γref τ * s -τ s c τ ref n , ( 4.8)
where γref is the reference slip rate and τ ref is a reference resolved shear stress.

The hardening rule described in section 3.1.5 holds for porous single crystals.

Numerical assessment

The simulations results presented in section 3.3 for unirradiated single crystals combined with some additional simulations with non-axisymmetric loadings are used to assess the proposed porous single crystal model at finite strains.

Calibration of model parameters

The overall stress strain curves and the evolution of the void volume fraction obtained by the unit cell simulations with axisymmetric loading are used for calibrating the parameters α, q 1 and q 2 in Eq. (4.1) of the macroscopic porous single crystal model. The calibration has been conducted with the material parameters given in table 3.5 which aim to represent the behavior of a solution annealed 304 austenitic stainless steel. The crystallographic orientations, the initial void volume fractions f 0 and the stress triaxialities T considered for the calibration are summarised in table 4.1. 4.1: Crystallographic orientations, initial void volume fractions and stress triaxialities used for the calibration of material parameters α, q 1 and q 2 , see Eq. (4.1).

Test number Crystallographic orientation Initial void volume fractions

The parameters α, q 1 and q 2 have been identified using Levenberg-Marquardt algorithm and taking into account at the same time the stress-strain response and the evolution of void volume fraction by minimizing the cost function:

F = 1 F t 11 F t 11 0 w 1 (σ 11 -σ 11 ) 2 + w 2 (f i -f ) 2 dF 11 , ( 4.9)
where σ 11 denotes the component of Cauchy stress tensor of the porous single crystal model, σ 11 the component of the macroscopic Cauchy stress tensor of the unit cell, f i the void volume fraction of the porous single crystal model defined by Eq. (4.2) and f the void volume fraction determined from the unit cell simulations (see Eq. (3.37)). F t 11 is chosen to be 1.1, i.e., the calibration is done from the beginning of the loading to the strain value F 11 -1 = 0.1 (for the porous single crystal model) and F 11 -1 = 0.1 (for the unit cell). This level of macroscopic deformation corresponds to the regime before void coalescence for all the cases considered. In addition, the weight w 1 and w 2 are chosen in such a way that stress and void volume fraction have the same order of contribution to the cost function.

Optimised values of the parameters are given in table 4.2. Note that the obtained calibrated parameters, accounting for the hardening of the matrix and the evolution of the void volume fraction, are slightly different from those determined by [START_REF] Han | A yield function for single crystals containing voids[END_REF], where only the yield surface was taken into account. Comparison between the predictions of the porous single crystal model and the results obtained by the unit cell will be presented in the next sections. α q 1 q 2 the present work 5. [START_REF] Germain | The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure[END_REF]

Assessment for axisymmetric loading cases

In this part, the porous single crystal model is assessed for axisymmetric loading cases, which were used to calibrate the model parameters.

Figure 4.1 shows the evolution of the normalised Cauchy stress σ 11 /τ s T with respect to the strain F 11 -1 for the porous single crystal model and that of the normalised macroscopic Cauchy stress σ 11 /τ s T with respect to the macroscopic strain F 11 -1 for the unit cell for [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations with f 0 = 0.01 at (a) T = 1 and (b) T = 3. To simplify the notation, the overline for the macroscopic variables of the unit cell will be dropped in the following (e.g., F 11 instead of F 11 ). As void coalescence is not incorporated in the porous single crystal model, the curves are plotted before void coalescence predicted by the unit cell simulations. At low stress triaxiality T = 1, the porous single crystal model successfully predicts the tendency of the orientation effect on the stress strain response. However the model slightly overestimates the strain hardening for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientations and underestimates the strain hardening for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation. This can imply that the model will underestimate the void growth for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientations and overestimate the void growth for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation, which will be confirmed in the next section. At high stress triaxiality T = 3, the prediction of the model for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation is generally in good agreement with the unit cell simulations, while the difference between the model and the unit cell simulation for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation is more significant.

The comparison of the stress strain response is also shown in fig. 4.2 for larger initial void volume fraction f 0 = 0.05. Similar trends can be observed, except for an increased difference between the unit cell and the model for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at low triaxiality T = 1 for this higher initial void volume fraction. It can be observed that the porous single crystal model satisfactorily describes the tendency of the orientation effect on the evolution of the void volume fraction at both low and high stress triaxiality. However, the model underestimates the void growth for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientations, and overestimates the void growth for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at low stress triaxiality T = 1. These discrepancies are consistent with those presented in the previous section. At high stress triaxiality T = 3, the model well predicts the void growth for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations,

Assessment for non-axisymmetric loading cases

The porous single crystal model calibrated from the axisymmetric loading cases is used to predict macroscopic stress-strain behaviour and void volume fraction evolution under non-axisymmetric loadings in this part. The loadings with η 2 = 0.727 and η 3 ∈ {0.4, 0.538, 0.625} are considered for the [START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF], [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations with f 0 = 0.01.

The stress-strain responses predicted by the porous single crystal are compared to those of unit cell simulations in fig. 4.5. In spite of insufficient strain hardening for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations, the model satisfactorily describes, for the non-axisymmetric loadings considered, the hierarchy of stress-strain response with respect to crystallographic orientation. For example, the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation shows the hardest response while the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation has the softest response.

The assessment of the porous single crystal model in terms of void volume fraction evolution is presented in fig. 4.6. The hierarchy of void volume fraction evolution with respect to crystallographic orientation is well predicted by the model.

In addition, for the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation with η 2 = 0.727 and η 3 = 0.4, the unit cell simulation displays low void growth. This situation is poorly captured by the model that predicts significant void growth and in turn insufficient hardening. This

Discussion

The comparisons between the porous single crystal model and the unit cell simulations have been presented. The model with the identified parameters successfully predicts the hierarchy of the macroscopic stress strain behaviour and the evolution of void volume fraction with respect to (i) crystallographic orientation, (ii) stress triaxiality, and (iii) initial void volume fraction. An exhaustive list of initial conditions for the simulations was considered. While the model has been verified for most conditions, specific modeling issues have been highlighted.

The most significant difference between the porous single crystal model and the UC simulations is observed for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation. It has been observed that the porous single crystal model underestimates the strain hardening of the unit cell for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] with f 0 = 0.01 and f 0 = 0.05 at high stress triaxiality T = 3. The overall stress strain response of the unit cell simulation and that predicted by the porous model are replotted in fig. 4.7 for f 0 = 0.01 with the stress strain response of the void-free single crystal at the same stress triaxiality. For the void-free single crystal, the plastic deformation begins with small hardening rate due to the fact that only one slip system is activated. A change of hardening rate occurs when a secondary slip system being activated. For the voided single crystal (the unit cell simulation), on the contrary, the plastic part begins with a significant hardening rate, which is related to an important latent hardening as a result of the multiple slip in the regions near the void as shown in fig. 3.13 and fig. 3.16.

To be more precise, an indicator is proposed as in the case of single slip and, by virtue of normality rule, induces void growth. As the initial void volume fraction increases to f 0 = 0.05, the plastic slip heterogeneity introduced by the void becomes significant even at low stress triaxiality. The porous single crystal deviates significantly from single slip and much more slip systems are activated around the void. As a result, in the unit cell simulation, the strain hardening rate is increased and the void growth is accelerated. As the model only considers a single slip situation, it underestimates the strain hardening and the void growth for the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation with f 0 = 0.05 even at low stress triaxiality T = 1.

I = 1 V V 12 s=1 H | γs | Ḟ11 dV, ( 4

Conclusions

An elastoviscoplastic model is proposed for porous single crystals undergoing finite deformations. The effective resolved shear stress defined in the work of Han et al. (2013) at infinitesimal strains is extended to finite strains and the work hardening of the matrix of porous single crystals is incorporated. To the best knowledge of the author, the proposed macroscopic model is the first model for porous single crystals at finite strains.

The model is calibrated from unit cell simulations for unirradiated single crystals, presented in section 3.3. Compared with the unit cell simulations, the porous single crystal model satisfactorily describes the hierarchy of porous single crystal responses with respect to crystal orientation and void volume fraction. However, the model cannot predict the quasi-incompressible behaviour observed for strongly asymmetric orientations, e.g., [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF], with small initial void volume fraction at low stress triaxiality. The model underestimates the strain hardening for strongly asymmetric orientations with small initial void volume fraction at high stress triaxiality and with large f 0 at all levels of stress triaxiality considered.

Application to simulations of ductile damage in a polycrystalline aggregate

The final goal of the modeling tools developed in this study is their application to evaluate fracture toughness of austenitic strainless steels. A first attempt is done in this section to simulate polycristal aggreggates with the developed porous crystal model. For the sake of simplicity, the stress state ahead of a crack is represented by applying different stress triaxiality levels to a simplified polycristal aggregate.

Problem setup

The FE mesh of a polycrystalline aggregate with 343 cubic grains (

Results

The evolution of overall Cauchy stress as a function of overall strain A inhomogeneous evolution of ductile damage is also observed for T = 1. However, the damage is localized in grains and the damage propagation from grain to grain is not observed at this relatively low level of stress triaxiality.

Conclusions

In this section, the porous single crystal model is applied to simulate ductile damage of polycrystalline aggregates containing intragranular voids. The evolution of void volume fraction is studied by the simulations. A strain softening behavior due to the increase of the porosity inside grains has been predicted. A propagation of ductile damage has been observed at a relatively high stress triaxiality T = 3, while ductile damage is localized inside grains for the stress triaxiality T = 1.

As cracking criterion is not introduced in the simulations, crack propagation and fracture cannot be simulated. This will be done in future works. In addition, more realistic FE meshes, such as Voronoi-type polycrystalline aggregates or meshes obtained from the X-ray tomography, can also be used in future works. The notch opening displacement ∆a, corresponding to the increase of a, is measured using SEM micrographs. The imposed straining is planned to be large enough so that ductile fracture can occur. However, it was not achieved, partly because of the high ductility of the material. Besides, it is important to note that inclusions of ferrite were observed (see fig. 4.14b) including one located in the front of the notch tip (see fig. 4.16), which may have an influence on the experiment results. A 3D FE mesh with five elements in the thickness is created according to the geometry of the specimen to simulate the test (see fig. 4.13). The mesh is refined in the region near the notch tip where a high level of strain is expected. In terms of boundary conditions, the bottom is fixed in three directions; the top is fixed in the directions of x and z axes and the displacement is prescribed along the y-axis. The crystallographic orientation measured by EBSD is considered for the simulation. In absence of the material parameters for the 316L single crystal, those identified for the unirradiated polycrystalline 304L steel at 340 • C (table 3.6) are used with two levels of initial void volume fraction: f 0 = 0 and 0.01. Even though damage/fracture does not occur during the experiment, the simulation with porous single crystal model, as a preliminary study, can offer some implications for further studies.

Application to simulations of the single-edgenotch tension

Experiment and simulation

Results: surface slip traces

During the test, slip traces are observed on the surface of the specimen as a result of dislocation glide, as shown in fig. 4.17 for different normalized notch opening displacements ∆a/a 0 (a 0 is the initial value of a). A few traces of three slip planes can be observed around the notch tip at the beginning of plastic deformation (indicated by black lines in fig. 4 The surface slip traces can be theoretically predicted. According to the Euler angles measured by EBSD, four types of surface slip traces are expected and their angles with respect to the x-axis are calculated and shown in fig. 4.18. One type of slip traces corresponds to dislocations with the same slip plane. In the case of FCC single crystals, there are four types of slip planes and the normal to the planes are respectively: [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF], (1 11), [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and ( 11 1) (see table 4.3). The four types of slip traces are denoted respectively by p1, p2, p3 and p4. Comparing the theoretical predictions and the experimental observations, we conclude that three types, i.e., p1, p2 and p3, are activated. Theoretical and experimental angles of surface slip traces with respect to the x-axis are in good agreement.

To better illustrate the results, accumulated plastic slip for four slip planes are defined: The simulations predict that two types of slip traces, i.e., p1 and p2, are intensively The evolution of loading force is plotted in fig. 4.22 as a function of normalized notch opening displacement ∆a/a 0 for the experiment and the simulations. For the simulations with f 0 = 0 and 0.01, a first hardening regime is observed followed by a second one with a higher hardening rate. The activation of the second hardening regime is related to an increase in the number of activated slip systems with increasing notch opening displacement. In the case of f 0 = 0, a strain softening regime following the hardening regimes is also observed. Compared with the simulations, the experiment exhibits a lower yield stress and a much higher strain hardening rate.

γ p1 = γ 1 + γ 2 + γ 3 , γ p2 = γ 4 + γ 5 + γ 6 , ( 4
The differences between the experiment and the simulations are closely related to the material parameters used for the simulations which are not calibrated for the material of the test. In particular, in the single crystal plasticity models, the yield stress is directly associated with the initial dislocation densities (see Eq. (3.22)).

Hence, the dislocation densities in the single crystal of 316L should be carefully measured/estimated before the test. In addition, the under-estimation of the hardening by the simulations is not well understood. It is probably associated with the interaction matrices of the hardening law, i.e., a su in Eq. (3.22) and b su in Eq. (3.21). The components of the matrix a su were evaluated by DD simulations, while a strong assumption has been made about the matrix b su due to a lack of studies. Further works are needed for accurate calibration of the two matrices.

Conclusions and outlook

The model is applied to simulate the SENT test performed on a single crystal of 316L stainless steel in this section. The prediction of the model about plastic slip activation on the specimen surface and plastic strain filed in front of the notch tip is in good agreement with the experimental observation. However, compared with the experiment, the simulation predicts a notch-opening-displacement relation showing a higher yield stress and a much lower strain hardening rate. These differences are probably related to: 1) the material parameters used for the simulations is identified on polycrystalline 304L stainless steels, not the material used for the SENT test; 2) the ferrite inclusions in the 316L stainless steel single crystal may have an influence on mechanical behaviors of the material, which is not taken into account in the simulation.

Note that this work is not finished yet. The ongoing works concentrate on the improvement of the simulations. Two aspects are being studied:

1. Decrease the initial value of dislocation densities and investigate its influences on strain hardening rate; 2. Introduce inclusions of ferrite by some particles with higher yield stress than the matrix in the FE mesh and study their influences on strain hardening rate;

3. Investigate influences of boundary conditions prescribed in the simulations on the results.

Concluding remarks

In In irradiated stainless steels, ductile fracture can probably occur due to intragranular voids of size varying from nano to micrometers. Plasticity size effects could thus be involved in the particular fracture process associated with void growth and coalescence in irradiated steels. In order to study influence of void size on the ductile fracture process at grain scale, a micromorphic single crystal plasticity model at finite deformation is proposed. The first part of this section is devoted to the formulations of the model. FE implementation of one of the formulations is described in the second part. Then, the model is first applied to study analytically and numerically strain localization in single crystals with single slip, where a material length scale is derived for the model. Finally, UC simulations are performed for investigating size dependent void growth and coalescence in single crystals.

Micromorphic single crystal plasticity at finite deformation

Kinematics of a micromorphic single crystal

We consider a crystalline body occupying the region B 0 in a fixed reference configuration. A point X in B 0 is referred to as a material point. Following the micromorphic approach (see [START_REF] Germain | The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure[END_REF], see also [START_REF] Forest | Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage[END_REF] and Aslan et al. ( 2011)), two types of degrees of freedom (DOF) are attributed to a material point, i.e., the displacement u (X , t) and an additional strain type DOF γ χ (X , t):

DOF = {u , γ χ }. (5.1)
In the present work, the additional DOF, γ χ , is a scalar variable called microslip variable. At time t, the deformed body occupies the region B in space, called the current configuration. The Lagrangian gradients of the degrees of freedom are

H ∼ (x , t) = ∂u ∂X = Grad u , (5.2) K (x , t) = ∂γ χ ∂X = Grad γ χ , (5.3)
where H ∼ is the displacement gradient which is related to the deformation gradient

F ∼ by F ∼ = 1 ∼ + H ∼ ,
and K is referred to as the microslip gradient vector. Besides, the Eulerian gradient of γ χ , which is useful in the following, follows

k = ∂γ χ ∂x = grad γ χ , (5.4)
where x is the current position of material point X at time t. K and k are related to each other by

k = K .F ∼ -1 = F ∼ -T .K . (5.5)
Taking the time derivative of the DOFs gives the generalized velocities { u , γχ }:

u = ∂u (X , t) ∂t , (5.6) γχ = ∂γ χ (X , t) ∂t .
(5.7)

The Eulerian gradient of the generalized velocities is computed as

grad u = Grad u .F ∼ -1 = Ġrad u .F ∼ -1 = Ḟ ∼ .F ∼ -1 , (5.8) grad γχ = Grad γχ .F ∼ -1 = Ġrad γ χ .F ∼ -1 = K .F ∼ -1 , (5.9)
where the commutativity between the Lagrangian gradient and time derivatives is used. Note that Eq. (5.8) represents the classical velocity gradient

L ∼ = Ḟ ∼ .F ∼ -1 .

Principle of virtual power and generalized balance of moment

The principle of virtual power is formulated in the static case following the approach of Germain, 1973. Consider the virtual motion of a subdomain D of the body B specified by the generalized virtual velocity V = { u , γχ }. The principle of virtual power asserts that, given any subdomain D, the virtual power of internal forces P i is equal to the virtual power of external forces P e for any virtual motion V.

The virtual power of internal forces P i is assumed to comprise three parts, which are associated respectively to the strain of the subdomain, the microslip and its gradient: (5.10) where s and m are generalized stresses.

P i = D p i dV = D (σ ∼ : (grad u ) + s γχ + m . grad γχ ) dV, ∀D ⊂ B,
In the absence of body forces, the virtual power of external forces is presumed to arise from a surface traction t related to the macroscopic motion and a generalized surface traction m related to the microslip. P e is thus written in the current configuration as (5.12) Using the divergence theorem, it can be found that where S ∼ is the Boussinesq stress tensor which generates power over Ḟ ∼ in any subdomain D 0 of the body B 0 , M is the generalized stress which generates power over K in D 0 and S is the generalized stress which generates power over γχ in D 0 . In addition,

P c = ∂D p c dS = ∂D (t . u + m γχ ) dS, ∀D ⊂ B. ( 5
D ((div σ ∼ ). u + (div m -s) γχ ) dV + ∂D ((t -σ ∼ .n ) u + (m -m .n ) γχ ) dS = 0, ∀ u , ∀ γχ , ∀D, (5
S ∼ = ρ 0 ρ σ ∼ .F ∼ -T ; (5.20) M = ρ 0 ρ F ∼ -1 .m ; (5.21) S = ρ 0 ρ s. (5.22)
The Neumann boundary conditions then become

T = S ∼ .n 0 , ∀X ∈ ∂D 0 , (5.23) M = M .n 0 , ∀X ∈ ∂D 0 . (5. 24
)
where T is the surface traction measured with respect to ∂D 0 , M is the generalized surface traction which generates power over γχ in D 0 and n 0 is the normal to the surface element of the subdomain D 0 .

Second law of thermodynamics

The where D 0 denotes the considered subdomain in the reference configuration and ρ 0 the reference mass density. Thus Eq. ( 5.25) can be written as

D p i -ρ ψ dV 0. (5.27)
As D is arbitrary, the local form of the entropy imbalance is p i -ρ ψ 0.

(5.28)

Constitutive equations

The goal is to propose a set of constitutive equations accounting for the generalized stress variables and the microslip variables, fulfilling the entropy inequality. To this end, we first consider the multiplicative decomposition of the deformation gradient F ∼ : [START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF]. A cumulative total slip variable γ cum is introduced as

F ∼ = E ∼ .P ∼ , ( 5
γ cum = t 0 12 s=1 | γs | dt. (5.31)
γ cum is related to the microslip variable γ χ via a variable e(X , t) called relative plastic slip and given by e(X , t) = γ cum -γ χ .

(5.32)

The relative plastic slip variable e measures the deviation of γ χ from γ cum and will be associated with an energy cost in the free energy potential in the following.

Considering Eq. (5.29), L ∼ can be expressed as

L ∼ = Ḟ ∼ F ∼ -1 = L ∼ e + E ∼ .L ∼ p .E ∼ -1 (5.33)
with

L ∼ e = Ė ∼ .E ∼ -1 , L ∼ p = Ṗ ∼ .P ∼ -1 . (5.34)
With the consideration of Eqs. (5.32) and (5.33), the internal power density p i in Eq. (5.10) can be further developed as

p i = ρ ρ Π ∼ e : Ė ∼ e GL + ρ ρ Π ∼ M : ( Ṗ ∼ .P ∼ -1 ) + s( γcum -ė) + m . grad γχ . ( 5.35)
with

Π ∼ e = ρ ρ E ∼ -1 .σ ∼ .E ∼ -T , Π ∼ M = ρ ρ E ∼ T .σ ∼ .E ∼ -T = E ∼ T .E ∼ .Π ∼ e , E ∼ e GL = 1 2 E ∼ T .E ∼ -1 ∼ (5.36)
where ρ is the density measured w.r.t. the intermediate configuration

C , E ∼ e GL
denotes the Green-Lagrange elastic strain tensor, Π ∼ e the Piola stress tensor w.r.t.

C and Π ∼ M the Mandel stress tensor. Note that ρ = ρ 0 , because of the plastic incompressibility of the crystalline body. As Eulerian gradient and time derivative do not generally commute, it is more convenient to express m . grad γχ in Eq. (5.35) in terms of Lagrangian gradient. Considering Eqs. (5.9) and (5.21), one obtains

m . grad γχ = m . Grad γχ .F ∼ -1 = F ∼ -1 .m . Grad γχ = ρ ρ 0 ρ 0 ρ F ∼ -1 .m . Grad γχ = ρ ρ 0 M . Grad γχ . (5.37)
Because of the commutativity between Lagrangian gradient and time derivative, one gets

ρ ρ 0 M . Grad γχ = ρ ρ 0 M . Ġrad γ χ = ρ ρ 0 M . K . ( 5.38)
As a result, Eqs. (5.35) and (5.38) yield

p i = ρ ρ Π ∼ e : Ė ∼ e GL + ρ ρ Π ∼ M : ( Ṗ ∼ .P ∼ -1 ) + s( γcum -ė) + ρ ρ 0 M . K (5.39)
In the present model, the vector microslip variable K and the relative plastic slip e are assumed to be state variables in addition to the usually considered termelastic strain E ∼ GL . The free energy density ψ is hence a function of E ∼ GL , K and e:

ψ = ψ E ∼ e
GL , e, K .

(

Thus, ρ ψ yields

ρ ψ = ρ ∂ψ ∂E ∼ e GL : Ė ∼ e GL + ρ ∂ψ ∂e ė + ρ ψ K . K . ( 5.41)
Considering Eqs. (5.28), (5.39) and (5.41) yields

ρ Π ∼ e ρ - ∂ψ ∂E ∼ e GL : Ė ∼ e GL + ρ ρ Π ∼ M : Ṗ ∼ P ∼ -1 -s + ρ ∂ψ ∂e ė+ρ M ρ 0 - ∂ψ ∂K .
K +s γcum 0.

(5.42)

Following the Coleman-Noll procedure, we require that Eq. (5.42) holds for all arbitrarily prescribed Ė ∼ e GL , ė and K . In order to satisfy this requirement, the coefficients associated with the three terms mush vanish, which gives the state laws:

Π ∼ e = ρ ∂ψ ∂E ∼ e GL , ( 5.43
)

s = -ρ ∂ψ ∂e , ( 5.44
)

M = ρ 0 ∂ψ ∂K . ( 5.45)
Note that, as the free energy potential is assumed to depend on the Lagrangian term K , a constitutive relation is thus directly obtained for the work-conjugate pair K and M in the reference configuration C 0 . Furthermore, Eq. (5.42) yields the residual dissipation (also called reduced dissipation) inequality:

d res = ρ ρ Π ∼ M : Ṗ ∼ P ∼ -1 + s γcum 0. (5.46)
Given Eq. (5.30), the term Π ∼ M : Ṗ

∼ P ∼ -1 yields Π ∼ M : Ṗ ∼ P ∼ -1 = Π ∼ M : 12 s=1 γs N ∼ s = 12 s=1 Π ∼ M : N ∼ s γs = 12 s=1
τ s γs , (

with the resolved shear stress τ s = Π ∼ M : N ∼ s in C . Thus, the residual dissipation inequality writes

d res = ρ ρ 12 s=1
τ s γs + s γcum 0.

(

With γ cum defined by Eq. (5.31), Eq. (5.48) can be written as

d res = ρ ρ 12 s=1 τ s γs + s 12 s=1 | γs | 0. (5.49)
Noticing that sign (τ s ) = sign (γ s ), one has τ s γs = |τ s || γs |. Eq. (5.49) can be further written as

d res = ρ ρ 12 s=1 |τ s || γs | + s 12 s=1 | γs | = ρ ρ 12 s=1 |τ s | + ρ ρ s | γs | 0. (5.50)
Considering the term |τ s | + ρ ρ s in Eq. (5.50), we propose to introduce the following yield function:

f s = |τ s | + ρ ρ s -τ s c = |τ s | -τ s c - ρ ρ s 0, , ∀s = 1, 2, ..., N, (5.51)
where τ s c is the critical resolved shear stress (CRSS) for the slip system s. As can be seen from the yield function, the generalized stress s is regarded as a source of isotropic hardening and changes the yield limit of slip systems. Accordingly, a rate-dependent law is chosen for the plastic slip γ s , expressed as

γs = |τ s | -τ s c - ρ ρ s K n sign (τ s ) .
(5.52)

Alternative formulation of constitutive equations

In the previous sections, the constitutive equations are derived by accounting for K , or equivalently k , as an argument of the free energy potential. However, other choices exist. Inspired from the Green-Lagrange elastic tensor E ∼ e GL which is measured in the intermediate configuration C , we push forward K into C , or equivalently pull back k into C , and define K as:

K = P ∼ -T .K = E ∼ T .k . ( 5.53)
The generalized stress M work-conjugate to K in the intermediate configuration C can be found according to the requirement that the work generated by M over the vector microslip variable K on a volume element dV 0 in C 0 is equal to that by M over K on the volume element dV in C . Thus, M .K follows

M .K dV 0 = ρ ρ 0 M .K dV = ρ ρ 0 M .P ∼ T .P ∼ -T .K dV = ρ ρ 0 M .P ∼ T .K dV = M .K dV ,
(5.54) where

M = ρ ρ 0 M .P ∼ T = ρ ρ 0 P ∼ .M . ( 5.55)
Substituting Eq. ( 5.21) into the previous equation leads to

M = ρ ρ E ∼ -1 .m (5.56)
Accordingly, an alternative constitutive formulation can be proposed for which the free energy potential depends on K instead of K , i.e., ψ = ψ E ∼ e GL , e, K . It follows that

ρ ψ = ρ ∂ψ ∂E ∼ e GL : Ė ∼ e GL + ρ ∂ψ ∂e ė + ρ ψ K . K . ( 5.57)
Substitution of Eqs. (5.53) and (5.55) into the power density of internal forces, Eq. (5.39), leads to

p i = ρ ρ Π ∼ e : Ė ∼ e GL + ρ ρ Π ∼ M : ( Ṗ ∼ .P ∼ -1 ) + s( γcum -ė) + ρ ρ 0 M . K (5.58) = ρ ρ Π ∼ e : Ė ∼ e GL + ρ ρ Π ∼ M : ( Ṗ ∼ .P ∼ -1 ) + s( γcum -ė)+ ρ ρ P ∼ -1 .M .( Ṗ ∼ T .K + P ∼ T . K ) (5.59) = ρ ρ Π ∼ e : Ė ∼ e GL + ρ ρ Π ∼ M : ( Ṗ ∼ .P ∼ -1 ) + s( γcum -ė) + ρ ρ P ∼ -1 . Ṗ ∼ .M .K + ρ ρ P ∼ -1 .P ∼ .M . K (5.60) = ρ ρ Π ∼ e : Ė ∼ e GL + ρ ρ Π ∼ M : ( Ṗ ∼ .P ∼ -1 ) + s( γcum -ė) + ρ ρ (K ⊗ M ) : ( Ṗ ∼ .P ∼ -1 ) + ρ ρ M . K . ( 5.61)
Substituting Eqs. (5.57) and (5.61) into Eq. (5.28) writes Then, the residual dissipation inequality writes

p i -ρ ψ = ρ Π ∼ e ρ - ∂ψ ∂E ∼ e GL : Ė ∼ e GL -s + ρ ∂ψ ∂e ė + ρ M ρ - ∂ψ ∂K . K + ρ ρ Π ∼ M + K ⊗ M : Ṗ ∼ P ∼ -1 + s γcum 0. ( 5
d res = ρ ρ Π ∼ M + K ⊗ M : Ṗ ∼ P ∼ -1 + s γcum 0. (5.64)
Considering Eqs. (5.30) and (5.31) yields

d res = ρ ρ Π ∼ M + K ⊗ M : 12 s=1 γs N ∼ s + s 12 s=1 | γs | 0. ( 5.65)
Given the resolved shear stress τ s = Π ∼ M : N ∼ s with N ∼ s = l s ⊗ n s , the previous inequality follows

d res = ρ ρ 12 s=1 τ s + (K .l s )(M .n s ) γs + ρ ρ s| γs | (5.66) = ρ ρ 12 s=1 |τ s | + ρ ρ s | γs | 0, (5.67)
where a generalized resolved shear stress is defined as

τ s = τ s + x s (5.68)
with

x s = (K .l s )(M .n s ). (5.69)
It can be seen that the residual dissipation section 5.1.5 takes the same form as Eq. (5.50), except that τ s is used instead of τ s . Moreover, x s plays the role of a gradient induced back-stress for each slip system. Therefore, a second yield function is thus proposed in the form:

f s = |τ s |-τ s c - ρ ρ s = |τ s +(K .l s )(M .n s )|-τ s c - ρ ρ s 0, ∀s = 1, 2, ..., 12.
(5.70) Compared with the previous formulation of the theory, gradient terms come into play not only through isotropic hardening but also kinematic hardening, as a result of introducing the gradient of microslip measured in the intermediate configuration. Accordingly, the rate-dependent law to determine the plastic slip writes

γs = |τ s | -τ s c - ρ ρ s K n sign (τ s ) .
(5.71)

Hardening laws

The hardening laws describe the evolution of the critical resolved shear stress τ s c with some internal variables. In the present work, the hardening laws based on the evolution of dislocation densities are taken into account. Since the thermodynamics of the evolution of dislocation densities is not known [START_REF] Busso | On the selection of active slip systems in crystal plasticity[END_REF], the hardening laws are not involved in the previous development of the present model. Following [START_REF] Kubin | Modeling dislocation storage rates and mean free paths in face-centered cubic crystals[END_REF], τ s c of the slip system s can be expressed as:

τ s c = τ 0 + µ 12 s=1
a su r u , (

where τ 0 is the thermal component of the CRSS due to the lattice friction, r s denotes adimensional dislocation density (r s = b 2 ρ s with the dislocation density ρ s and the norm of Burgers vector of the dislocation b), µ is the shear modulus, and a su is a matrix for describing long-range interactions between dislocations.

Considering the multiplication and the annihilation of dislocations due to the interactions among them, the evolution of adimensional dislocation densities r s is governed by

ṙs = | γs | √ b su r u κ -G c r s , (5.73)
where κ is proportional to the number of obstacles crossed by a dislocation before being immobilized, G c is the critical distance controlling the annihilation of dislocations with opposite signs, and b su describes the interactions between dislocations. The structure of the matrices a su and b su are given in appendix A for FCC crystals.

Constitutive choices for the free energy potential

Different free energy potentials can be chosen for the two formulations of the constitutive equations. Several potentials as well as resulting regularization equations are proposed in the following. The two formulations are discussed separately.

Formulation with free energy potential depending on K or k

Choices of free energy potential are first discussed for the constitutive formulation presented in section 5.1.4. Choices of free energy potential leads to different state laws, yield functions and regularization functions as shown below.

Model based on the state law M = A ∼ .K

A first quadratic potential is considered in the form:

ρψ = 1 2 ρ ρ E ∼ e GL : Λ ≈ : E ∼ e GL + 1 2 ρ ρ 0 H χ e 2 + 1 2 ρ ρ 0 K .A ∼ .K , ( 5.74)
where Λ ≈ is the fourth rank tensor of elastic moduli, H χ a penalty modulus and A ∼ a second rank tensor of high order moduli. With this potential, the constitutive relations Eqs. (5.43) to (5.45) yield

Π ∼ e = Λ ≈ : E ∼ e GL , (5.75
)

s = - ρ ρ 0 H χ e, ( 5.76)
M = A ∼ .K , ( 5.77)
with e = γ cum -γ χ . Noticing S = ρ 0 ρ s, one can write Eq. (5.76) as

S = -H χ e. ( 5.78)
Substituting Eqs. (5.5) and (5.21) into Eq. (5.77) yields a nonlinear relation between the associated Eulerian vectors:

m = ρ ρ 0 F ∼ .A ∼ .F ∼ T .k . ( 5.79)
Then, we consider the yield function Eq. (5.51). Noticing that ρ = ρ 0 for crystal plasticity and that S = ρ 0 ρ s, one can obtain

f s = |τ s | -τ s c - ρ ρ s = |τ s | -τ s c - ρ 0 ρ s = |τ s | -(τ s c -S) (5.80)
Substituting the balance law Eq. (5.19) into the previous yield function gives

f s = |τ s | -(τ s c -Div M ) . ( 5.81)
In the present work, the model will be applied to crystals with cubic symmetry. It follows that A ∼ must be a spherical tensor, i.e., A ∼ = A1 ∼ . The constitutive relations Eqs. (5.77) and (5.79) thus become

M = A.K , (5.82) m = ρ ρ 0 AB ∼ .k , ( 5.83)
where B ∼ = F ∼ .F ∼ T is the left Cauchy-Green tensor. Taking into account the state law Eq. ( 5.82), one can further get

f s = |τ s | -(τ s c -ADiv (Grad γ χ )) = |τ s | -(τ s c -A X γ χ ) (5.84)
As can be seen in this expression, the term Grad γ χ comes into play as a source of isotropic hardening proportional to the Laplacian of microslip in the spirit of Aifantis model [START_REF] Aifantis | The physics of plastic deformation[END_REF]). Moreover, one can obtain the regularization equation connecting γ χ and γ cum . Combining Eqs. (5.19), (5.32), (5.78) and (5.82) gives

γ χ - A H χ X γ χ = γ cum . ( 5.85)
Here, the regularization equation with a Lagrangian Laplace operator is thus obtained in the reference configuration. Besides, it is also possible to get a regularization function with an Eulerian Laplace operator by choosing a different free energy potential as shown below.

Model based on the state law

m = A ∼ .k
For obtaining an Eulerian Laplace operator in the regularization equation, a second potential is considered in the form: For crystals with cubic symmetric, i.e., A ∼ = A1 ∼ , the constitutive relations Eqs. (5.88) and (5.89) become

ρψ = 1 2 ρ ρ E ∼ e : Λ ≈ : E ∼ e + 1 2 H χ e 2 + 1 2 K .F ∼ -1 A ∼ .F ∼ -T .K . ( 5
M = ρ 0 ρ F ∼ -1 A ∼ .F ∼ -T .K . ( 5
M = ρ 0 ρ AC ∼ -1 .K , ( 5.90
)

m = Ak , ( 5.91)
where C ∼ = F ∼ T .F ∼ is the right Cauchy-Green tensor. Following the same derivation procedure as in the previous section, one can find that the yield function Eq. (5.51) writes

f s = |τ s | -τ s c - ρ ρ A div(grad γ χ ) = |τ s | -τ s c - ρ ρ A x γ χ . (5.92)
and the regularization equation as

γ χ - A H χ x γ χ = γ cum , ( 5.93)
where the Eulerian Laplace operator appears.

Formulation with free energy potential depending on K

In the following, the formulation presented in section 5.1.5 is developed. Three choices of free energy potential are shown.

Model based on the state law

K = A ∼ .M
A first quadratic free energy potential is considered in the form:

ρψ = 1 2 ρ ρ E ∼ e : Λ ≈ : E ∼ e + 1 2 H χ e 2 + 1 2 ρ ρ K .A ∼ .K . ( 5.94)
Substituting the potential into Eqs. (5.44) and (5.63) gives:

s = -H χ e, ( 5.95)
M = A ∼ .K , ( 5.96
)

with e = γ cum -γ χ . With S = ρ 0 ρ s, one has S = - ρ 0 ρ H χ e. ( 5.97)
We express the yield function in terms of γ χ and first consider the Lagrangian treatment of the variables. Substituting Eqs. (5.53) and (5.55) into Eq. (5.96) yields

M = ρ 0 ρ P ∼ -1 .A ∼ P ∼ -T .K (5.98)
Noticing that ρ = ρ 0 for crystal plasticity and that S = ρ 0 ρ s, one can write Eq. (5.70) as

f s = |τ s | -τ s c - ρ ρ s = |τ s | -τ s c - ρ 0 ρ s = |τ s | -(τ s c -S) .
(5.99)

In the case of A ∼ = A1 ∼ , the constitutive equations Eqs. (5.96) and (5.98) become

M = AK , , (5.100)
M = ρ 0 ρ AC ∼ p-1 .K , ( 5.101)
with C ∼ p = P ∼ T .P ∼ . Considering Eqs. (5.19) and (5.101), one can further write the yield function as

f s = |τ s | -τ s c -A Div ( ρ 0 ρ C ∼ p-1 .K ) , (5.102)
with

τ s = τ s + x s and x s = (K .l s )(M .n s ) = (P ∼ -T .K .l s )(AP ∼ -T .K .n s ).
(

Combining Eqs. (5.19), (5.97) and (5.98), one can obtain the regularization function:

γ χ - A H χ ρ ρ 0 Div ( ρ 0 ρ C ∼ p-1 . Grad γ χ ) = γ cum . ( 5.104)
It can be seen that the Lagrangian Laplace operator is not involved in the equation.

For the same formulation, we then consider the Eulerian treatment of the variables to see whether the Eulerian Laplace operator will be obtained. With

Eqs. (5.21) and (5.55), m follows

m = ρ ρ 0 F ∼ .M = ρ ρ E ∼ .M . ( 5.105)
Combining Eqs. (5.5) and (5.53) gives

K = E ∼ T .k . ( 5.106)
Hence, Eq. (5.96) yields

m = ρ ρ E ∼ .A ∼ .E ∼ T .k . ( 5.107)
For crystals with cubic symmetry, the previous equation becomes

m = ρ ρ AB ∼ e .k , (5.108)
with B ∼ e = E ∼ .E ∼ T . The yield function Eq. (5.70) thus writes

f s = |τ s | -τ s c - ρ ρ div ρ ρ AB ∼ e . grad γ χ 0. (5.109)
According to Eqs. (5.15), (5.95) and (5.107), the regularization function writes

γ χ - A H χ div ( ρ ρ B ∼ e . grad γ χ ) = γ cum . ( 5

Model based on the state law

M = A ∼ .K
For involving the Lagrangian Laplace operator in the regularization equation, a second potential is proposed in the form:

ρψ = 1 2 ρ ρ E ∼ e : Λ ≈ : E ∼ e + 1 2 ρ ρ 0 H χ e 2 + 1 2 ρ ρ 0 K .P ∼ .A ∼ .P ∼ T .K , ( 5.111)
According to Eqs. (5.44) and ( 5.63), the potential leads to

s = - ρ ρ 0 H χ e, ( 5.112)
M = ρ ρ 0 P ∼ .A ∼ .P ∼ T .K . ( 5
with B ∼ p = P ∼ .P ∼ T . The yield function can be written as:

f s = |τ s | -(τ s c -A X γ χ ) = |τ s + x s | -(τ s c -A X γ χ ) 0, (5.118)
with

x s = (K .l s )(M .n s ) = (P ∼ -T .K .l s )(AB ∼ p .P ∼ -T .K .n s ). (5.119)
Analogously, the regularization function is found to be

γ χ - A H χ X γ χ = γ cum . ( 5.120)
This regularization function involves the Lagrangian Laplace operator as expected.

Note that this regularization function is the same as Eq. (5.85), but that the yield function is defined with the generalized resolved shear stress τ s instead of τ s .

Model based on the state law

m = A ∼ .k
Based on the idea to obtain a regularization equation with the Eulerian Laplace operator, a third potential considered follows:

ρψ = 1 2 ρ ρ E ∼ e : Λ ≈ : E ∼ e + 1 2 H χ e 2 + 1 2 K .E ∼ -1 .A ∼ .E ∼ -T .K (5.121)
The potential leads to

s = -H χ e, ( 5.122
)

M = E ∼ -1 .A ∼ .E ∼ -T .K . ( 5
For crystals with cubic symmetry, the constitutive relations become

M = AC ∼ e-1 .K , ( 5.125
)

m = Ak , ( 5.126)
with C ∼ e = E ∼ T .E ∼ . The yield function is found to be

f s = |τ s | -τ s c - ρ ρ A x γ χ = |τ s + x s | -τ s c - ρ ρ A x γ χ 0. (5.127)
with

x s = (K .l s )(M .n s ) = (P ∼ -T .K .l s )(AC ∼ e-1 .P ∼ -T .K .n s ). (5.128)
The regularization function can readily be obtained:

γ χ - A H χ x γ χ = γ cum . ( 5.129)
Notice that the obtained regularization function is the same as Eq. (5.93) but the corresponding yield function is defined with the generalized resolved shear stress τ s instead of τ s .

Numerical implementation

Integration of constitutive equations

The formulation of the constitutive equations presented in section 5.1.4 is chosen for the numerical implementation. The free energy potential given by Eq. (5.74) is chosen for the implementation.

The problem of the numerical integration of constitutive equations can be stated as follows: given the initial values of the stress variables S ∼ , S and M and associated internal variables v int to be integrated , if the strain variables F ∼ , γ χ , K

increase to F ∼ + ∆F ∼ , γ χ + ∆γ χ , K + ∆K ,
S ∼ = Jσ ∼ .F ∼ -T = J J e E ∼ .   Λ ≈ 2 (E ∼ T .E ∼ -1 ∼ )   .E ∼ T .F ∼ -T , ( 5.130)
which shows that S ∼ depends on E ∼ and F ∼ . Since the current value of F ∼ is given, the problem of looking for the current value of F ∼ becomes to find that of E ∼ . In order to obtain the equations governing E ∼ , one can combine Eqs. (5.33) and (5.34) and get

Ė ∼ = Ḟ ∼ .F ∼ -1 .E ∼ -E ∼ . Ṗ ∼ .P ∼ -1 . ( 5.131)
Notice that Ṗ ∼ .P ∼ -1 is given in Eq. (5.30) and governed by

Ṗ ∼ .P ∼ -1 = 12 s=1 γs N ∼ s . ( 5.132)
With this relation, Ė ∼ is expressed by

Ė ∼ = Ḟ ∼ .F ∼ -1 .E ∼ -E ∼ .
12 s=1 γs N ∼ s .

(

Recall that γs is determined by Eq. (5.52):

γs = |τ s | -(τ s c -S) K n sign (τ s ) , (5.134)
where S = H χ (γ χ -γ cum ) and τ s c is given by the hardening law. Note that τ s c -S 0 is considered. For γ cum , we consider the rate form of Eq. (5.31)

γcum = 12 s=1 | γs |, (5.135)
Therefore, for calculating the current value of E ∼ , the differential equations that must be integrated are

                                       Ė ∼ = Ḟ ∼ .F ∼ -1 .E ∼ -E ∼ . 12 s=1 γs N ∼ s γs = |τ s | -(τ s c -S) K n sign (τ s ) ṙs D = | γs |   b su r u D κ -G c r s D   γcum = 12 s=1 | γs |, (5.136) (5.137) (5.138) (5.139)
For facilitating the description of the algorithm, the variables to integrate are denoted by v int in the following, i.e., v int = {E ∼ , γ s , r s , γ cum }. The numerical integration of the differential equations is performed with the Newton-Raphson method, which concerns solving the residual equations

{R} = {∆v int } -∆t{ vint }(t + θ∆t) = {0}. (5.140)
For the considered differential equations, the residual equations follow

                                     R E ∼ = ∆E ∼ -∆F ∼ .F ∼ -1 .E ∼ + E ∼ . 12 s=1 ∆γ s N ∼ s R γ s = ∆γ s - |τ s | -(τ s c -S) K n sign (τ s ) ∆t, R r s = ∆r s -|∆γ s | √ b su r u κ -G c r s R γcum = ∆γ cum - 12 s=1 |∆γ s | (5.141) (5.142) (5.143) (5.144)
Within the Newton-Raphson algorithm, the Jacobian [J] needs to be evaluated and writes

[J] = ∂{R} ∂{∆v int } = [1] -∆t ∂{ vint } ∂{∆v int } . ( 5.145)
The Jacobian for the previous residual equation system can be expressed in a partitioned matrix

                 ∂R E ∼ ∂∆E ∼ ∂R E ∼ ∂∆γ p ∂R E ∼ ∂∆r q ∂R E ∼ ∂∆γ cum ∂R γ s ∂∆E ∼ ∂R γ s ∂∆γ p ∂R γ s ∂∆r q ∂R γ s ∂∆γ cum ∂R r s ∂∆E ∼ ∂R r s ∂∆γ p ∂R r s ∂∆r q ∂R r s ∂∆γ cum ∂R γcum ∂∆E ∼ ∂R γcum ∂∆γ p ∂R γcum ∂∆r q ∂R γcum ∂∆γ cum                  .
(5.146)

Analytical expressions for each block is presented in appendix C.

Finite element formulation

The model is implemented in the finite element (FE) code Zset using a total Lagrangian finite element formulation in three dimensions. To this end, we first convert the principle of virtual power, given by Eq. (5.12) and expressed in the current configuration C:

D (σ ∼ : (grad u ) + s γχ + m . grad γχ ) dV = ∂D (t . u + m γχ ) dS, ( 5.147)
to the formulation in the reference configuration C 0

D 0 S ∼ : Ḟ ∼ + S γχ + M . K dV 0 = ∂D 0 (T . u + M γχ ) dS 0 . ( 5.148)
Note that since the FE equations will be solved by an iterative method, the rate terms ( Ḟ ∼ , γχ , K , u ) correspond to their increments (∆F ∼ , ∆γ χ , ∆K , ∆u ). Next, we carry on the FE discretisation on the crystalline body considered.

Tensors and vectors are written in index notation, except otherwise stated. Besides the displacement u i , the microslip γ χ is regarded as an additional nodal degree of freedom. Assuming that the considered crystalline body occupying the domain D 0 in C 0 is discretized by l finite elements, one can write the principle of virtual power (Eq. (5.148)) as

D 0 S ij Ḟij + S γχ + M i Ki dV 0 = ∂D 0 (T i ui + M γχ ) dS 0 (5.149) =⇒ l e=1 D e 0 S ij Ḟij + S γχ + M i Ki dV 0 = l e=1 ∂D e 0 (T i ui + M γχ ) dS 0 . (5.150)
Here, the subdomain D e 0 corresponds to the space occupied by an individual element. The summation is done for the repeated subscripts, e.g., S ij Ḟij denotes

3 i=1 3 j=1 S ij Ḟij ;
for the sake of brevity, the summation operator will not be written for the subscripts.

In general, it is assumed that, within one individual element, u i is interpolated from the displacement values of p nodes and γ χ from the values of q nodes. u i and γ χ within one element are given by

u i = p a=1 u N a u a i (5.151) γ χ = q b=1 χ N b γ b χ , (5.152)
where u N a and χ N b are shape functions respectively for u i and γ χ , and u a i and γ b χ respectively denote the nodal values of u i on node a and those of γ χ on node b. Unlike the subscripts, the summation operators are written explicitly for the superscripts denoting the nodes. The deformation gradient F ij and the Lagrangian gradient of microslip K i are given by

F ij = p a=1 u B a j u a i
(5.153)

K i = q b=1 χ B b i γ b χ (5.154) with u B a j = ∂ u N a ∂X j and χ B b i = ∂ χ N b ∂X i
. Using these relations in Eq. (5.150) leads to Note that, in the terms associated with displacement u i , the summation is done for the subscript i (i = 1, 2, 3) which is not written for the sake of brevity. This is the FE equation for the entire FE mesh and can be solved by Newton's method.

l e=1 D e 0 S ij p a=1 u B a j ˙ u a i + q b=1 S χ N b ˙ γ b χ + M i χ B b i ˙ γ b χ dV 0 = l e=1 ∂D e 0 T i p a=1 u N a ˙ u a i + M q b=1 χ N b ˙ γ b χ dS 0 (5.

and { ˙ γ

b χ } = { ˙ γ e χ } =                    ˙ γ 1 χ ˙ γ 2 χ . . . ˙ γ q χ                    . (5.162)
Here, we drop the superscripts a and b used for summation over the nodes of one element and add a superscript label e, in order to indicate that the vector is for one individual element and to distinguish it from vectors for the entire FE mesh.

Recall that p is the number of nodes possessing displacement degrees of freedom and q is that for microslip γ χ . Voigt's notation is used for writing tensors in the form of vectors and matrices. Especially, the second-order non-symmetric tensor F ∼ and the vector K are arranged in the form:

{F ∼ } =                                  F 11 F 22 F 33 F 12 F 23 F 31 F 21 F 32 F 13                                  and {K } =      K 1 K 2 K 3      .
(5.163) Thus, shape functions u N a i and χ N b can thus be written as

[ u N] =       u N 1 0 0 • • • u N p 0 0 0 u N 1 0 • • • 0 u N p 0 0 0 u N 1 • • • 0 0 u N p       (5.164)
and

[ χ N] = χ N 1 χ N 2 χ N 3 • • • χ N q .
(5.165)

Accordingly, u B a ij and χ B a i can also be written in matrix form denoted by [ u B] and [ χ B]:

[ u B] =                           ∂ u N 1 ∂X 1 0 0 • • • ∂ u N p ∂X 1 0 0 0 ∂ u N 1 ∂X 2 0 • • • 0 ∂ u N p ∂X 2 0 0 0 ∂ u N 1 ∂X 3 • • • 0 0 ∂ u N p ∂X 3 ∂ u N 1 ∂X 2 0 0 • • • ∂ u N p ∂X 2 0 0 0 ∂ u N 1 ∂X 3 0 • • • 0 ∂ u N p ∂X 3 0 0 0 ∂ u N 1 ∂X 1 • • • 0 0 ∂ u N p ∂X 1 0 ∂ u N 1 ∂X 1 0 • • • 0 ∂ u N p ∂X 1 0 0 0 ∂ u N 1 ∂X 2 • • • 0 0 ∂ u N p ∂X 2 ∂ u N 1 ∂X 3 0 0 • • • ∂ u N p ∂X 3 0 0                          
(5.166)

and

[ χ B] =       ∂ χ N 1 ∂X 1 ∂ χ N 2 ∂X 1 ∂ χ N 3 ∂X 1 • • • ∂ χ N q ∂X 1 ∂ χ N 1 ∂X 2 ∂ χ N 2 ∂X 2 ∂ χ N 3 ∂X 2 • • • ∂ χ N q ∂X 2 ∂ χ N 1 ∂X 3 ∂ χ N 2 ∂X 3 ∂ χ N 3 ∂X 3 • • • ∂ χ N q ∂X 3       (5.167)
The interpolation of increment of the displacement ui and that of the microslip γχ in one element thus write

{ u } = [ u N].{ ˙ u e } (5.168) { γχ } = [ χ N].{ ˙ γ e χ } (5.169)
and their Lagrangian gradients follows

{ Ḟ ∼ } = [ u B].{ ˙ u e } (5.170) { K } = [ χ B].{ ˙ γ e χ }. (5.171)
With stress and strain variables expressed with Voigt's notation, Eqs. (5.179)

The reader is referred to [START_REF] Besson | Non-linear mechanics of materials[END_REF] for the description of the assembly procedure. Thus, the global FE equation (Eq. (5.161)) to be solved can be written as

{R int(u) } {R int(γχ) } . { ˙ u} { ˙ γ χ } = {R ext(u) } {R ext(γχ) } . { ˙ u} { ˙ γ χ } .
(5.180)

Since the system is nonlinear, it can be solved by Newton's method which requires the calculation of the Jacobian matrix with respect to the internal reactions [START_REF] Besson | Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms[END_REF]. The Jacobian matrix of an individual element, split into four blocks, writes

[K e (uu) ] [K e (ug) ] [K e (gu) ] [K e (gg) ] =       ∂{R e int(u) } ∂{ u e } ∂{R e int(u) } ∂{ γ e χ } ∂{R e int(γχ) } ∂{ u e } ∂{R e int(γχ) } ∂{ γ e χ }       . (5.181)
Using the assembly operation A, one can calculate the global Jacobian matrix [K]

[K] = A      [K e (uu) ] [K e (ug) ] [K e (gu) ] [K e (gg) ]     
(5.182)

One calculates the blocks for an individual element and obtains the so-called element stiffness matrix:

[K e (uu) ] = ∂{R e int(u) } ∂{ u e } = D 0 [ u B] T . ∂{S ∼ } ∂{F ∼ } . ∂{F ∼ } ∂{ u e } dV 0 = D 0 [ u B] T . ∂{S ∼ } ∂{F ∼ } .[ u B] dV 0 (5.183) [K e (ug) ] = ∂{R e int(u) } ∂{ γ e χ } = D 0 [ u B] T . ∂{S ∼ } ∂{γ χ } . ∂{γ χ } ∂{ γ e χ } + [ u B] T . ∂{S ∼ } ∂{K } . ∂{K } ∂{ γ e χ } dV 0 = D 0 [ u B] T . ∂{S ∼ } ∂{γ χ } .[ χ N] + [ u B] T . ∂{S ∼ } ∂{K } .[ χ B] dV 0 (5.184) [K e (gu) ] = ∂{R e int(γχ) } ∂{ u e } = D 0 [ χ N] T . ∂{S} ∂{F ∼ } . ∂{F ∼ } ∂{ u e } + [ χ B] T . ∂{M } ∂{F ∼ } . ∂{F ∼ } ∂{ u e } dV 0 = D 0 [ χ N] T . ∂{S} ∂{F ∼ } .[ u B] + [ χ B] T . ∂{M } ∂{F ∼ } .[ u B] dV 0 (5.185) [K e (gg) ] = ∂{R e int(γχ) } ∂{ γ e χ } = D 0 [ χ N] T . ∂{S} ∂{γ χ } . ∂{γ χ } ∂{ γ e χ } + [ χ B] T . ∂{M } ∂{γ χ } . ∂{γ χ } ∂{ γ e χ } + [ χ N] T . ∂{S} ∂{K } . ∂{K } ∂{ γ e χ } + [ χ B] T . ∂{M } ∂{K } . ∂{K } ∂{ γ e χ } dV 0 = D 0 [ χ N] T . ∂{S} ∂{γ χ } .[ χ N] + [ χ B] T . ∂{M } ∂{γ χ } .[ χ N] +[ χ N] T . ∂{S} ∂{K } .[ χ B] + [ χ B] T . ∂{M } ∂{K } .[ χ B] dV 0 .
(5.186)

In the element stiffness matrix, one can find nine derivatives which will be evaluated by consistent tangent matrix {J * } in the next section. The consistent tangent matrix {J * } are defined as:

             ∂{∆S ∼ } ∂{∆F ∼ } ∂{∆S ∼ } ∂{∆γ χ } ∂{∆S ∼ } ∂{∆K } ∂{∆S} ∂{∆F ∼ } ∂{∆S} ∂{∆γ χ } ∂{∆S} ∂{∆K } ∂{∆M } ∂{∆F ∼ } ∂{∆M } ∂{∆γ χ } ∂{∆M } ∂{∆K }              .
(5.187)

In addition, it will be shown that ∂{∆S} ∂{∆K } and ∂{∆M } ∂{∆γ χ } vanish for the constitutive model implemented in this work.

Consistent tangent matrix

As presented in sections 5.3.2 and 5.3.3, the tangent operators are necessary for evaluating the element stiffness matrix. In the present work, they are estimated from the incremental form of constitutive equations [START_REF] Besson | Non-linear mechanics of materials[END_REF]. In section 5.3.1, we have shown that the integration of the constitutive equations concerns solving the residual equations (Eqs. (5.141) to (5.144)). It is important to notice that the residual equations is expressed in terms of increments of the integration variables ∆v int and increments of the input variables ∆v IN . The calculation of the consistent tangent matrix is performed after the integration of the constitutive equations, and one has thus vanish residual equations

{R(∆v int , ∆v IN )} = {0} (
{δ∆v int } =   - {∂R} {∂∆v int } -1 {∂R} {∂∆v IN }   {δ∆v IN }, (5.190)
where -{∂R} {∂∆v int }

Convergence improvement of integration of the power law

The integration of Eq. (5.52) displays some difficulties, because of the penalty parameter H χ in Eq. (5.78) which should be of a large value. To improve the convergence, an improved algorithm for Newton's method has been proposed by The algorithm is illustrated in fig. 5.2 and can be stated as follows: in first iterations, the power law is linearized for large values of τ s ; as soon as the slip rate γs calculated during an iteration is smaller than a critical value γL , the original law is integrated. A smooth transition is considered at γL . In another word, the modified law follows γs =

         f s K n sign (τ s ) , if γs < γL n K γ n-1 n L f s -K γ 1 n L + γL , if γs γL (5.195) with f s = |τ s | -(τ s c -J e s) .
( 5.196) Notice that γL should be small enough for ensuring the convergence improvement but should not be too small for obtaining a correct solution of the original power law. In this study, γL = 2 s -1 is chosen. Some simple tests have been performed for showing the convergence improvement by the algorithm, as presented in appendix E.

Application to strain localization in single crystals

Strong strain gradients arise in particular in the presence of localization phenomena.

Such a situation involving strain softening is investigated in this section using the proposed micromorphic model.

Shear banding with single slip

Analytical solutions .199) where κ = ∂u ∂Y . As single slip is considered, we assume

F ∼ = 1 ∼ + κl ⊗ n , ( 5
Ṗ ∼ .P ∼ -1 = 1 ∼ + γl ⊗ n , ( 5.200)
with γ = γ(Y ). P ∼ obeys the following relation in the case of single slip: (5.201) because no lattice rotation takes place under the chosen conditions. Since F ∼ = E ∼ .P ∼ , one obtains

P ∼ = 1 ∼ + γl ⊗ n ,
E ∼ = 1 ∼ + κ e l ⊗ n (5.202)
with κ e = κ -γ, where κ = κ(Y ).

In the following, small elastic strains are assumed, i.e., small κ e , which leads to

C ∼ e = E ∼ T .E ∼ ≈ 1 ∼ + κ e (l ⊗ n + n ⊗ l ).
(5.203) Thus, the Green-Lagrange elastic tensor E ∼ e GL , the Mandel stress Π ∼ M and the resolved shear stress τ have the following approximate forms:

E ∼ e GL ≈ 1 2 κ e (l ⊗ n + n ⊗ l ), (5.204)
Π ∼ M = C ∼ e .Π ∼ e ≈ Π ∼ e , ( 5
M = ADiv K = ADiv ( ∂γ χ ∂X l + ∂γ χ ∂Y n ) = A ∂ 2 γ χ ∂Y 2 .
(5.215)

Combining the balance equation Eq. (5.19) with the previous equation and Eq. ( 5.212) leads to

A ∂ 2 γ χ ∂Y 2 = H χ (γ χ -γ).
(5.216) Substituting Eq. ( 5.214) into the previous equation, one gets

A ∂ 2 γ χ ∂Y 2 - HH χ H + H χ γ χ + HH χ H + H χ (|τ | -τ 0 ) = 0. (5.217)
Recall that τ is spatially constant (cf. Eq. (5.209)). One thus has the differential equation governing γ χ :

A ∂ 2 γ χ ∂Y 2 - HH χ H + H χ γ χ + HH χ H + H χ (|τ | -τ 0 ) = 0.
(5.218)

One trivial solution of Eq. (5.218) is

γ χ = |τ | -τ 0 = constant, (5.219)
which results in a spatially constant shearing strain κ.

As H is negative, a general sinusoidal solution exists in the form:

γ χ = C 1 sin 2π λ c Y + C 2 cos 2π λ c Y + (|τ | -τ 0 ), (5.220)
with the integration constants C 1 and C 2 , and the wave length

λ c = 2π/ H χ |H| A(H χ + H) .
(5.221)

Note that λ c comes into play as a material length scale. Within one period, γ χ increases to its maximum value and then decreases, which represents a shear band. Since a large value is usually taken for the penalty parameter H χ such that

H χ |H|, λ c ∼ A |H| .
Thus, increasing the value of A results in a wider shear band. Another form of periodic solution exists, which represents a periodic array of shear bands separated by nearly constant values.

Analysis with the alternative formulation

In this section, we consider the second formulation of the theory presented in section 5.1.5. The free energy potential given by Eq. (5.111) is considered so that one has the constitutive equations Eqs. (5.211) and (5.212). The yield function (Eq. (5.70)) writes

f = |τ + (K .l )(M .n )| -[τ 0 + Hγ -H χ (γ χ -γ)] .
(5.222)

Noticing K = P ∼ -T .K , M = P ∼ .M and M = AK , one considers P ∼ = 1 ∼ + γl ⊗ n and further obtains: Thus, the yield function is reduced to

K = ∂γ χ ∂X l + ∂γ χ ∂Y -γ ∂γ χ ∂X n , (5.223) M = A ∂γ χ ∂X + γ ∂γ χ ∂Y l + ∂γ χ ∂Y n . ( 5
f = |τ | -[τ 0 + Hγ -H χ (γ χ -γ)] , ( 5

FE solutions

For the FE analysis, a single row of elements with side length L = 1 mm is considered and shown in fig. The periodicity is also enhanced for the γ χ field, i.e., γ χ takes the same value at homologous points on the opposite sides of the strip. Simple shear is imposed by prescribing the overall deformation gradient F 12 .

As for the material parameters, the softening parameter is fixed to be H = -10 3 MPa the shear modulus µ = 104.7 GPa, and τ 0 = 10 MPa. The parameters of power law is chosen as K = 0.1 MPa and n = 15 such that the viscous effect is negligible. Note that, for triggering the localization, the value of τ 0 for a line of elements is set to be τ 0 = 9.9 MPa, which is slightly smaller than that of the others.

The effect of the penalty parameter H χ is first studied. To this end, the higher order modulus is fixed to be A = 5 N, and the value of H χ is changed from 10 3 role of the parameter H χ and shows the necessity of choosing a large value for H χ . However, a too large value of H χ will cause numerical problems. Thus, the value of H χ should not be too large, neither too small.

Strain localization with single slip under uniaxial tension

Strain localization in single crystals has been observed in many studies (see, e.g., [START_REF] Chang | An experimental study of shear localization in aluminum-copper single crystals[END_REF]. It was analyzed in a seminal work of [START_REF] Asaro | Strain localization in ductile single crystals[END_REF] by considering crystals deformed by single slip. The problem was then investigated

Problem setup

Simulations are performed with a plate as shown in fig. 5.8. The plate is meshed by brick elements with reduced integration (20 nodes for the displacement DOFs and 8 nodes for the microslip DOF) and one element is used in the thickness direction.

An in-plane single slip system is defined, whose slip plane has an angle of 56.3°w ith respect to the vertical direction. The crystal orientation is chosen such that both the slip direction and the slip plane normal of the single slip system are parallel to the X-Y plane. This results in plane plastic strain conditions. The boundary conditions are considered as follows:

U X (Y = L 0 ) = 0, U Y (Y = L 0 ) = U (t), (5.229) U X (Y = 0) = 0, U Y (Y = 0) = 0.
(5.230) The simulations are performed with both the conventional and micromorphic model. A linear hardening law is considered with τ c given by:

τ c = τ 0 + Hγ (5.231)
with H > 0. For the simulations, a weak hardening is considered with H = 1 MPa.

For the micromorphic model, H χ = 5 × 10 4 MPa and A = 0.1 N are taken. Strain localization due to the prescribed boundary conditions and lattice rotation will be studied in the following.

Results

Different FE discretizations are considered with n elements for the width and 6n for the height, i.e., totally 6n 2 elements. Four meshes with respectively n = 8, 16, 24, 32, are considered (see fig. 5.8 for a mesh with n = 8). For the simulations with the micromorphic model, a softening regime is also observed. However, the stress level is higher than that obtained by the conventional the slip direction. A major difference between slip and kink bands is that strong lattice curvature occurs at the kink band boundaries.

Some differences are noticed for the two simulations. Compared with the conventional model, the distance between the two kink bands predicted by the micromorphic model is smaller. Besides, the band width (band size along Y direction) predicted by the micromorphic model is larger than that by the conventional model, i.e., the plastic slip field is more diffuse in the simulation with the micromorphic model than with the conventional model. For further confirming the formation of kink bands, lattice rotation needs to be checked. The polar decomposition of the elastic part of deformation gradient (5.233)

E ∼ follows E ∼ = R ∼ .U ∼ , ( 5
The field of θ is shown in fig. 5.12 for the simulation with the conventional model when ∆L/L 0 = 0.012 and ∆L/L 0 = 0.033. The formation of kink bands at ∆L/L 0 = 0.012 is verified, because of relatively high level of θ within the bands. The formation of slip band crossing the kink band at ∆L/L 0 = 0.067 is also checked, because significant plastic slip is observed in the band parallel with the

Application to ductile fracture of single crystals

Void growth and coalescence are known as the mechanisms controlling ductile fracture of metallic materials. In single crystals, lattice orientation has been shown

to have an impact on void growth and coalescence (cf. A cylindrical void of radius R 0 in a square plate of side length L 0 is considered. A typical FE mesh with one element in thickness w 0 is shown in fig. 5.14 for an initial void volume fraction f 0 = πR 2 0 /L 2 0 = 0.1. A biaxial tensile is imposed under plane strain conditions. The boundary conditions applied on the faces of the unit cell are

Unit cell simulation

U X (X = 0) = 0, U X (X = L 0 ) = U 1 (t), (5.234) U Y (Y = 0) = 0, U Y (Y = L 0 ) = U 2 (t), (5.235) U Z (Z = 0) = 0, U Z (Z = w 0 ) = 0. ( 5
σ 11 = F 1 w 0 (L 0 + U 2 ) , σ 22 = F 2 w 0 (L 0 + U 1 )
.

(5.237)

The FCC single crystal with 12 slip systems is considered. The slip systems are specified by the slip direction l s and the normal to the slip plane n s in the crystal

(i = 1, 2, • • • , 6). It is presumed that b 1 = 0 and that b i = 0 for i = 1. The initial values of adimensional dislocation densities r s ini ( s = 1, 2, • • • , 12
) are assumed to be the same. Recall that r s = b 2 ρ s , where ρ s is the dislocation density and b is the norm of Burger's vector of the dislocation. b = 2.54 Å is considered for the FCC single crystal. To study size effects, we change the side length L 0 of the unit cell, while fix the moduli H χ = 5 × 10 4 MPa and A = 10 N. λ c defined by Eq. (5.221) is taken as the material length scale, though H is positive in the present case. λ c is obtained by performing uniaxial tensile simulation on one element with the same parameters. A positive H can thus be estimated according to the evolution of τ s with respect to γ s of one slip system at the beginning of its activation. In this way, λ c is calculated to be λ c = 0.4 mm.

Void growth

Evolution of the void volume fraction is first investigated. The current void volume fraction is calculated by

f = 1 - V mesh V tot , ( 5.238)
where V mesh is the volume of the mesh which can be evaluated by post-processing calculation and V tot is the total volume of the unit cell which follows V tot = (L 0 +U 1 )(L 0 +U 2 )w 0 . The increase in the void volume fraction is equal to ∆f = f -f 0 . The overall stress σ 11 is plotted in fig. 5.15b as a function of the overall strain E 11 . For the simulation with the conventional theory, a hardening regime followed by a softening one is observed, which results from the competition between the strain hardening of the matrix, the softening due to void growth and the softening due to void coalescence. The micromorphic theory predicts a higher stress level as a result of the reduced void growth. The hardening regime is enhanced with the decreasing unit cell size L 0 /λ c . In particular, no softening regime exhibits for the unit cell with L 0 /λ c = 0.5, which is due to extremely low void growth rate. In this case, the voided unit cell behaves like a void-free unit cell. Results for f 0 = 0.1 are shown in fig. 5.16. Compared with the simulations for f 0 = 0.01, the evolution of ∆f /f 0 does not exhibit an obvious transition to the accelerated void growth stage. However, the transition from the uniaxial to biaxial straining can be detected, i.e., the two stages (void growth and coalescence) exist for f 0 = 0.1. Size effect on void growth observed is similar to that for f 0 = 0.01: void growth rate is reduced for small L 0 /λ c . However, the void growth for L 0 /λ c = 0.5 cannot be neglected here, unlike in the case of f 0 = 0.01. In addition, similar size dependent overall stress-strain response is observed as that for f 0 = 0.01. It worth noticing that the unit cell with L 0 /λ c = 0.5 does not show softening regime even with a non-negligible void growth.

Void coalescence

The onset of void coalescence in the present study corresponds to a shift from biaxial to uniaxial straining, which is in good agreement with Koplik and Needleman, 1988.

The onset is determined in the following way: calculate the ratio a

0 = |U 1 |/|U 2 |
at the beginning of the biaxial straining, choose a critical value as a c = 0.1a 0 and find E 11 for the onset as soon as a 0 decreases to a c . The critical value of E 11 for the onset of coalescence thus determined is denoted by E c 11 .

Discussion

Size dependent overall behavior and void growth have been predicted by the micromorphic theory for voided single crystals. Qualitatively similar results have been shown for the unit cell with f 0 = 0.01 and f 0 = 0.1: unit cells of smaller size exhibit reduced void growth rate compared with larger ones; this results in higher hardening rate at the beginning of plastic regime and delays onset of void coalescence.

The impeded void growth in unit cells of small size is associated with the fact that the plastic deformation field is diffused in the micromorphic theory. The cumulative slip γ cum is shown in section 5.5.4 for the unit cells with f 0 = 0.1 and different values of L 0 /λ c at the overall strain E 11 = 0.1. The conventional theory predicts the significant localization of γ cum at four zones at the void, i.e., A, B, C, D shown in section 5.5.4. It is worth noticing that the maximum local γ cum reaches 2.18 with the overall strain E 11 = 0.1. By the micromorphic theory, the γ cum field is diffused especially for the four zones and becomes more and more diffuse as L 0 /λ c decreases from 25 to 0.5. As a result, void growth is slowed down and void coalescence is delayed. The diffused field of γ cum also influences the evolution of void shape, as shown in section 5.5.4 for the unit cells of different sizes with f 0 = 0.1 at the overall strain E 11 = 0.4. The void is shown to be ellipsoid for the small unit cell with L 0 /λ c = 0.5. In this case, since the void coalescence does not occur, the void is well elongated along the X-axis but the deformation of the void along the Y -axis is restricted. The void is also ellipsoid for L 0 /λ c = 5 at E 11 = 0.4 (before the onset of void coalescence for this case). Compared with L 0 /λ c = 0.5, the void is more deformed along the Y -axis because of a less diffuse field of γ cum in the region around the void. For the large unit cell with L 0 /λ c = 25, the lateral deformation of the void is significant at E 11 = 0.4 as a result of void coalescence.

The conventional theory predicts a similar void shape as the micromorphic theory with L 0 /λ c = 25, except that a sharp point shows. This is related to the highly localized plastic strain in the lateral regions. different unit cell sizes. The distribution of γ cum is shown to be smoother at the void surface for unit cells of smaller size than that of larger ones. This is directly related to the reduced void growth for unit cells of small size.

Conclusions

A micromorphic single crystal plasticity at finite deformation is derived by a thermodynamic approach in the present work. The model involves a single additional DOF, i.e., the microslip γ χ . Two types of constitutive formulation are proposed without and with free energy potential depending on the microslip gradient variable K defined in the intermediate configuration. In the formulation involving K , a gradient induced back-stress is found for each slip system. To the author's knowledge, the kinematic hardening is introduced for the first time in the enhanced single crystal plasticity model based on a single additional DOF. The formulation without back-stress in the constitutive equations is implemented in the FE code Zset.

The model is then applied to study analytically and numerically strain localization in single crystals. In shear banding with single slip, the width of shear band predicted analytically by the model is found to be proportional to A |H| with the higher order modulus A and the strain softening modulus H, when a large value is adopted for the penalty parameter H χ . FE simulations are performed for analyzing the influence of A and H χ on the width of shear band. A good agreement is observed between the analytical and numerical results. In simulations for studying strain localization in a crystal under tension, different strain localization modes are predicted by the conventional and the micromorphic model. Smooth stress-strain curves are obtained by the micromorphic model with fine meshes and a mesh convergence is found for the micromorphic model.

The model is then used to investigate growth of a cylindrical void in FCC single crystals by FE simulations. The void growth rate is found to be size dependent: smaller voids grow slower than bigger ones. The size effect on void growth is more pronounced for single crystals with 1% initial porosity than those with 10% initial porosity. In addition, the simulations also predict that onset of void coalescence is delayed for smaller voids. These results are related to a more diffuse field of plastic slip for single crystal with smaller voids. Moreover, a higher strain hardening rate is observed for crystals with bigger voids. The size effect on the overall strain hardening is more obvious for crystals with 10% initial porosity than those with 1% initial porosity. The model will be coupled with the porous single crystal model presented in section 4.1 in a future work, resulting in a non-local model for single crystals.

The coupled model can be applied, in the future, to simulate ductile fracture in single crystals and polycrystals. Effects of crystal orientation and stress triaxiality have been investigated in a first step and some conclusions are drawn as follows:

Conclusions and outlook

Conclusions

• Void growth rate increases with stress triaxiality, which is consistent with the results for von Mises materials in the literature.

• Void growth rate exhibits a crystal orientation dependence, and voids grow at a higher rate with the [START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] orientation than the other orientations considered in this work.

• Crystal orientation effect on void growth is more pronounced at lower stress triaxiality (T = 1).

• Critical void volume fraction for onset of void coalescence exhibits a crystal orientation dependence.

• Critical void volume fraction for onset of void coalescence is almost insensitive to stress triaxiality for a given crystal orientation (at least for the range of stress triaxialities considered T = 1 to 3).

The results are in good agreement with the results for BCC crystals in the literature.

Note that those results hold for unirradiated crystals as well as for irradiated crystals.

However, compared with unirradiated cases, it has been found that:

• voids grow at a higher rate in the irradiated single crystal;

• the onset of void coalescence occurs at a smaller value of the overall strain and at a smaller value of the void volume fraction in the irradiated crystals.

These results are consistent with experimental observations showing that fracture toughness decreases rapidly with dose, especially in the range of 0-10 dpa. Besides, more significant plastic strain localization in the region near the void is predicted in the irradiated crystal, correlated with the faster void growth and earlier onset of void coalescence. In addition, brittle-like overall behavior is predicted at high level of stress triaxiality, i.e. T = 3, in the voided irradiated crystal, while a large amount of plastic deformation is reached in the vicinity of the void. The enhanced plastic strain localization predicted for irradiated single crystals is in good agreement with the observations of [START_REF] Neustroev | Severe embrittlement of neutron irradiated austenitic steels arising from high void swelling[END_REF] in AISI 321 stainless steels irradiated in BOR-60.

An elastoviscoplastic model at finite strains has been proposed for porous single crystals with a future objective to assess fracture toughness of polycrystalline materials. The model incorporates heuristically the influence of hardening behavior of single crystal matrix and is able to describe void growth up to coalescence.

The assessment based on unit cell simulations has shown that the porous single crystal model satisfactorily describes the hierarchy of porous single crystal responses with respect to crystal orientation and void volume fraction. However, the model cannot predict the quasi-incompressible behaviour observed for strongly asymmetric orientations, e.g., [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF], with small initial void volume fraction at low stress triaxiality.

In fact, the [START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation corresponds to single slip orientation for a void-free FCC single crystal. For this type of orientation with small initial void volume fraction at low stress triaxiality, the void does not induce significant plastic slip heterogeneity in the single crystal matrix. As a result, the voided single crystal behaves like a void-free single crystal and exhibits a nearly single slip pattern with quasi-incompressible overall behaviour. No void growth is predicted by the unit cell simulation in this case. In contrast, the porous single crystal model predicts a weak void growth for this case, due to the term 2q 1 f i cosh q 2 3 20 Mm τ * s in Eq. (4.1), which is strictly positive even in the case of single slip and, by virtue of normality rule, induces void growth. Furthermore, the model underestimates the strain hardening for strongly asymmetric orientations with small initial void volume fraction at high stress triaxiality and with large f 0 at all levels of stress triaxiality considered.

In fact, as the stress triaxiality and the initial void volume fraction increase, the plastic slip heterogeneity introduced by the void becomes significant. The porous single crystal deviates significantly from single slip and much more slip systems are activated around the void. As a result, in the unit cell simulation, the strain hardening rate is increased and the void growth is accelerated. As the porous single crystal model only considers a single slip situation for strongly asymmetric orientations, it underestimates the strain hardening and the void growth.

The model has then been applied to simulate the SENT test on a single crystal specimen of 316 stainless steels. The experimental results used in the present work were obtained in another work. The simulations predict that the slip systems of three slip planes are activated, which is in good agreement with the experimental observation. The prediction of a strong strain localization in the region near the notch tip is also in good agreement with the experimental result. However, compared with the experiment, the simulation predicts a force-notch-openingdisplacement relation showing a higher yield stress and a lower strain hardening rate. Note that this work is not finished yet. In order to better simulate the test, the following works will be considered:

• Decrease the initial value of dislocation densities and investigate its influences on strain hardening rate;

• Introduce inclusions of ferrite by some particles with higher yield stress than the matrix in the FE mesh and study their influences on strain hardening rate;

• Investigate influences of boundary conditions prescribed in the simulations on the results.

With the objective to study size effects on void growth and coalescence, a micromorphic single crystal plasticity model at finite strains has been presented in the last part of the work. Two formulations have been proposed, one of which involves the free energy potential depending on the microslip gradient in the intermediate configuration and incorporates a size-dependent back-stress for each plastic slip system. The model has been applied to strain localization in single crystals with single slip. A material length has been found to be proportional to A |H| with the higher order modulus A and the strain softening modulus H, when a large value is adopted for the penalty parameter H χ . Moreover, unit cell simulations have been performed to study size dependent void growth. Some important conclusions are drawn as follows:

• Smaller voids grow slower than bigger ones.

• Size effect on void growth is more pronounced for single crystals with 1% initial porosity than those with 10% initial porosity.

• Onset of void coalescence is delayed for smaller voids.

• A more diffuse field of plastic slip is predicted for single crystal with smaller voids.

• A higher strain hardening rate is observed for crystals with larger voids.

• Size effect on the overall strain hardening is more obvious for crystals with 10% initial porosity than those with 1% initial porosity.

Compared with existing models in the literature, the following advantages of the model are found:

• It is computationally more efficient;

• It regularizes slip shear and kinks bands.

Outlook

Some shortcomings of the elastoviscoplastic model for porous single crystals have been shown in this work. In future works, efforts can be devoted to improve the model, especially accounting for the quasi-incompressible behavior of porous single crystals loaded along single slip orientations. In addition, the elastoviscoplastic model for porous single crystals can also be enhanced using the micromorphic approach for incorporating plastic size effects. Furthermore, in order to assess fracture toughness of polycrystalline materials, the elastoviscoplastic model for porous single crystals can be applied to simulations such as:

• a polycrystalline aggregate with a Voronoi-type mesh in 3D;

• a polycrystalline tensile specimen without or with a notch;

• a compact tension (CT) test of a polycrystalline materials.

These simulations allow to evaluate effects of post-irradiation hardening/softening behavior on fracture toughness of polycrystalline stainless steels. Eventually, the model can be applied to • the prediction of fracture of irradiated internal structure components of nuclear power plants.

For the micromorphic crystal plasticity model, further works are needed for better understanding the model, including:

• Implementing the alternative formulation and comparing the two formulations;

• Comparing different free energy potentials and understanding differences between various regularization operators.

In order to perform unit cell simulations with the micromorphic model for any asymmetric crystal orientations, more numerical tools need to be developed, including:

• Periodic finite element for the micromorphic model;

• Numerical method for imposing constant stress triaxiality with periodic boundary conditions.

In addition, large scale simulations of experiments on polycrystalline specimen, such as SENT tests, CT tests, etc., can be performed for evaluating void size effects on fracture toughness. In future work, localization of plasticity in channels of deformation which is a specific mode of deformation of irradiated FCC materials and its interaction with voids growth and coalescence, which was not considered in the present study, has to be evaluated.

A

Form of the a su and b su matrices

In FCC single crystals, the matrices a su and b su (s, u = 1, 2 (A.1) In the matrix, a 1 corresponds to self hardening, a 2 to coplanar interaction, a 3 to Hirth locks, a 4 to collinear interaction, a 5 to glissile junctions and a 6 to Lomer locks. The matrix b su has the same structure as a su and is not presented here for the sake of brevity.

B Method used for imposing constant stress triaxiality

A special truss element has been developed for imposing constant macroscopic Cauchy stress triaxiality under periodic boundary conditions at finite strains.

This element is aligned with the main loading direction, i.e., x 1 -axis. It has only one degree of freedom { F 11 -1} for the node at the far end of the element and nine degrees of freedom {F ij , i, j = 1, 2, 3} for the node at the near end of the element.

It is connected to the unit cell in such a way that the nine degrees of freedom {F ij , i, j = 1, 2, 3} of the node at the near end correspond to the nine components of the macroscopic deformation gradient of the unit cell, i.e., F ij = F ij , i, j = 1, 2, 3.

Consequently, F ij will be used instead of F ij in the following development for the nine degrees of freedom of the node at the near end of the element.

The element acts as a spring in the main loading direction as follows Equations to be integrated

Ė ∼ = Ḟ ∼ .F ∼ -1 .E ∼ -E ∼ .

Jacobian matrix ∂R int ∂v int

The Jacobian matrix is organized as follows

∂R ∂∆v I =                ∂R E ∼ ∂∆F ∼ ∂R E ∼ ∂∆γ χ ∂R E ∼ ∂∆K ∂R γ s ∂∆F ∼ ∂R γ s ∂∆γ χ ∂R γ s ∂∆K ∂R r s ∂∆F ∼ ∂R r s ∂∆γ χ ∂R r s ∂∆K ∂R γcum ∂∆F ∼ ∂R γcum ∂∆γ χ ∂R γcum ∂∆K                (D.2)
Each block of the matrix ∂R ∂∆v I is calculated respectively in the following. Finally,

• Derivatives of R E ∼ R E ∼ = ∆E ∼ -∆F ∼ .F ∼ -1 .E ∼ + E ∼ .
∂R E ∼ ∂∆F ∼ = -1 ∼ ⊗ E ∼ T .F ∼ -T + (∆F ∼ ⊗E ∼ T ) : (F ∼ -1 ⊗F ∼ -T ) (D.

E

Convergence improvement of the micromorphic model For showing the convergence improvement by the algorithm, simulations of simple shearing on one element are carried out by imposing displacement on all the nodes the element. Only one slip system is defined as shown in fig. E.1. We drop the subscript s of γ s and refer to γ as the plastic slip. A linear hardening law is considered:

τ c = τ 0 + Hγ, (E.1)
with H > 0. As only one slip system is defined, γ cum = γ. Simulations with the micromorphic theory are compared with those with the conventional single crystal plasticity theory. For the problem considered, the plastic slip field is homogeneous. As a result, the result predicted by the micromorphic model should be in agreement with that by the conventional model. Small time steps are prescribed for passing the elastic-plastic transition and then large time steps of the same length are prescribed for steady plastic regime. A critical time step in the plastic regime over which convergence cannot be reached is looked for.

Case 1: H χ = 0 and A = 0 H χ = 0 and A = 0 are first considered with γ χ = 0 imposed on the nodes. As result, the micromorphic model is reduced to the conventional model. Notice that, since all the degrees of freedom are imposed, no iteration is needed for integrating the FE equations. Hence, the consistent tangent matrix is not involved in the solution process. The value of A does not influence the result in this case, since the slip field is homogeneous. H χ = 10 5 which is large enough to ensure γ χ = γ. As in the previous case, we want to imposing all the degrees of freedom in the simulation with the micromorphic model, i.e., {u i , γ χ }, and to exclude influence of consistent tangent matrix on the convergence. However, γ χ is a priori unknown. To this end, the evolution of γ in the simulation with the conventional model is used to prescribe γ χ in the micromorphic model.

As shown in fig. E.3, very small time steps are needed for the simulation with the micromorphic model, while the calculation with the conventional model can even converge with larger time step than that in fig. E.3. Possible explanations are: 1) the Jacobian matrix for integration of the constitutive equations is not correctly derived or implemented in the code; 2) the convergence of the integration of the power law is not good, because a large value of H χ is chosen.

To make sure that the problem is not caused by possible errors in the analytical Jacobian matrix, a perturbation method is used for calculating it.

Abstract

Irradiation causes drastic modifications of mechanical properties of austenitic stainless steels and a decrease in the fracture toughness with irradiation has been observed. Ductile fracture due to void growth and coalescence remains one dominant fracture mechanism for doses in the range of 0-10 dpa. Voids may have different origins: nucleated at inclusions or irradiation-induced precipitates during mechanical loading, or produced directly by irradiation. The present work is to investigate ductile fracture of irradiated steels due to growth and coalescence of intragranulaire voids. Based on continuum crystal plasticity theory, FE simulations are performed on unit cells for studying effects of lattice orientation and stress triaxiality on void growth and coalescence. The influence of post-irradiation hardening/softening on void growth and coalescence is evaluated with a physically based crystal plasticity model. Besides, an elastoviscoplastic model at finite strains is proposed to describe void growth up to coalescence in single crystals, and is assessed based unit cell simulations. The model is then applied to simulate ductile damage in single crystals and polycrystals. As voids in irradiated steels may have different origins, they may have different sizes, which potentially have an influence on ductile fracture process and fracture toughness of irradiated steels. In order to assess the size effect, a micromorphic crystal plasticity model is proposed and applied to simulate growth and coalescence of intragranular voids of different sizes.

Figure 1 . 1 :

 11 Figure 1.1: Schematic of the circuits of a pressurized water reactor and materials of construction (Zinkle and Was, 2013). Stainless steels of 300 series are indicated by red rectangles.

Figure 1 . 2 :

 12 Figure 1.2: Fracture toughness of 304 and 316 stainless steels irradiated in light water reactor (LWR) conditions at 250-350 • C. (Zinkle and Was, 2013)

Figure 1 . 3 :

 13 Figure 1.3: Multi-scale approach for investigating ductile fracture of irradiated stainless steels due to void growth and coalescence.

Figure 2 . 1 :Figure 2 . 2 :

 2122 Figure 2.1: (a) Vacancy Frank loops; (b) interstitial Frank loops.(Was, 2007)

Figure 2 . 3 :

 23 Figure 2.3: Evolution of Frank loop density in austenitic stainless steels irradiated with fission neutrons at 375 to 400 • C (Zinkle et al., 1993).

2 )Figure 2 . 4 :

 224 Figure 2.4: Evolution of dislocation density as a function of irradiation dose in SA and CW 300-series austenitic stainless steels irradiated in a fast breeder reactor at 510 • C. Note that triangle points are for experimental data at 510 • C at and that solid lines with circle points are for simulation results at 470 • C. (Zouari et al., 2012)

Figure 2 . 5 :

 25 Figure 2.5: Evolution of dislocation density for stainless steels irradiated in BOR-60 at 375 • C. (Pokor et al., 2004c)

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure 2.6: Cavities observed in the baffle-former bolts, made of CW 316SS stainless steels, of Tihange 1 PWR reactor. The material is irradiated at T irr = 343 • C to 12.2 dpa.The swelling level is about 0.2%.[START_REF] Edwards | Influence of irradiation temperature and dose gradients on the microstructural evolution in neutron-irradiated 316SS[END_REF]

Figure 2 . 8 :

 28 Figure 2.8: Deformed cavities in irradiated stainless steels: (a) elongated cavities in front of a crack tip of an ion-irradiated 316 stainless steel (Horton et al., 1981); (b) cavities deformed by dislocation channeling observed in a 304 stainless steels irradiated in EBR-II at 370 to 470 • C after a tensile test at 370 • C (Fish et al., 1973).

Figure 2 . 9 :

 29 Figure 2.9: Dislocation channel structure of a 316 stainless steel irradiated by fast neutrons at low temperature to 0.78 dpa and tensile strained to 32%. (Byun et al., 2006)

10 :Figure 2 . 11 :

 10211 Figure 2.11: Tensile test of copper single crystals irradiated with 600 MeV protons to different doses: resolved shear stress vs. plastic slip. The tests are performed at about 22 • C and loading direction is [011].[START_REF] Dai | Mechanical properties and microstructures of copper gold and palladium single crystals irradiated with 600 MeV protons[END_REF]

Figure 2 . 12 :

 212 Figure 2.12: Engineering stress-strain curves for SA 304 and CW 316 steels irradiated at about 330 • C in the BOR-60 reactor. (Pokor et al., 2004b)

Figure 2 . 13 :Figure 2 . 14 :

 213214 Figure 2.13: Dose dependence of dynamic J -integral calculated for impact tests on 321 austenitic stainless steels irradiated at (a) ~230 and (b) ~400 • C.[START_REF] Little | Fracture mechanics evaluations of neutron irradiated Type 321 austenitic steel[END_REF]

Figure 2 . 15 :

 215 Figure 2.15: Typical fracture surface of CW 316 steels irradiated to different doses after impact tests at different at different test temperature. (Fukuya et al., 2008)

  C in impact tests and T test = 320 • C in tensile tests. Dimple fracture has been observed in other studies on stainless steels, especially those irradiated in fast neutron reactors. Solution treated Type 321 austenitic stainless steels irradiated in Dounreay Fast Reactor at about 230 and 400 • C up to 43 dpa were investigated by Little (1986) through impact tests. Fracture toughness evaluated by the dynamic J -integral was found to decrease with irradiation dose

Figure 2 . 16 :Figure 2 . 17 :

 216217 Figure 2.16: SEM micrographs showing the center of fracture surfaces of CW 316 steels irradiated to different doses after impact tests at different test temperature. (Fukuya et al., 2008)

Figure 2 . 18 :Figure 2 . 19 :

 218219 Figure 2.18:Fracture surface with micro-sized dimples of Fe-18Cr-10Ni-Ti stainless steels with a swelling level of 30%.[START_REF] Neustroev | Severe embrittlement of neutron irradiated austenitic steels arising from high void swelling[END_REF]

Figure 2 . 20 :

 220 Figure 2.20: (a) Fracture surface with unstructured zones (Zone 1) and flat facets (Zone 2) of stainless steels irradiated to 130 dpa with 3% swelling; (b) enlarged micrograph showing nano-dimples on the unstructured zone. (Margolin et al., 2016)

  Void nucleation, growth and coalescence are important mechanisms of ductile fracture in metals and alloys. An example of the observations in the literature is presented in fig. 2.21 showing the failure process from void formation to final cracking. Some dimple characteristics are often observed on fracture surface, as shown in

Figure 2 . 21 :Figure 2 . 22 :

 221222 Figure 2.21: Ductile fracture process: transition from void formation to cracking in a notched bar of steel. (see Benzerga (2000) and Benzerga et al. (2004), also Pineau et al. (2016))

Figure 2 . 23 :

 223 Figure 2.23: Ductile fracture micromechanisms: (a) internal necking; (b) void sheeting.[START_REF] Besson | Continuum Models of Ductile Fracture: A Review[END_REF]

Figure 2 . 24 :

 224 Figure 2.24: X-ray tomography showing internal necking in high-purity copper with artificial holes. (Weck et al., 2008)

  result reveals the influence of stress triaxiality, i.e., T = σ ∞ m σ ∞ eq

Figure 2 . 25 :Figure 2 . 26 :

 225226 Figure 2.25: Indentation depth dependence of the hardness measured by nanoindentation of single crystal copper and cold worked polycrystalline copper. (McElhaney et al., 1998)

Figure 2 . 27 :

 227 Figure 2.27: Sketch for non-uniform shear of a single crystal beam showing a gradient of plastic slip along the x 1 direction causes a density ρ G of GNDs to be stored.[START_REF] Fleck | Strain gradient plasticity: Theory and experiment[END_REF]

  belong to this category. Both of them assume a free energy dependence on α ∼ . However, the microcurl model involves directly α ∼ in the power of internal forces, while the Gurtin model relates α ∼ to the scalar dislocation densities and thus α ∼ does not appear explicitly in the derivation of balance equations. The second category is based on the scalar dislocation densities defined on crystallographic slip systems. Statistically stored dislocations (SSD) and geometrically necessary dislocations (GND) are often taken into account in the models of the second category. Various models of this category can be found in the literature, such as Evers et al. (2004), Bayley et al. (2006), Bayley et al. (2007), Ekh et al. (2007), Kuroda and Tvergaard (2008), Ertürk et al. (2009), and Bargmann et al. (2010), among others. An unified formulation of the models is proposed by Svendsen and Bargmann (2010). A third category corresponds to the enhancement of classical crystal plasticity model with the gradient of a single scalar plastic slip variable, which was firstly proposed in the work of Wulfinghoff and Böhlke (2012).

Figure 2 . 28 :Figure 2 . 29 :Figure 2 . 30 :Figure 2 . 31 :

 228229230231 Figure 2.28: (a) Dislocation loops after shearing the void in the crystal of 2 µm at an applied strain of 0.15%. (b) Void shape after multiple shearing by dislocations. (Chang et al., 2015)

  2.29), which was explained as a matter of availability of dislocation sources near the void. As the void size increases, the number of active sources increases. As a result, for a bigger void, a larger amount of dislocations can reach the void surface, which leads to a higher void growth rate. In addition,[START_REF] Huang | Discrete dislocation dynamics modelling of microvoid growth and its intrinsic mechanism in single crystals[END_REF] estimated the driven force of dislocations, i.e., the Peach-Koehler force, as a function of distance to the void surface. It was found that the force is larger at the same distance to the void for

Figure 2 . 32 :

 232 Figure 2.32: Sequence showing formation of shear and prismatic loops in hydrostatic tensile loading at a strain rate of 10 8 s -1 . (Tang et al., 2011)

Figure 2 . 33 :Figure 2 . 34 :

 233234 Figure 2.33: Typical MD simulations results showing size effects on porous single crystals. (a) Stress-strains curves for different void sizes. R denotes radius of the void. (Traiviratana et al., 2008). (b) Yield strength of nano-voided single crystals as a function of cell size L x . a 0 is the lattice constant and f v is the void volume fraction. (Zhao et al., 2009)

Figure 2 . 36 :

 236 Figure 2.36: A map of mechanisms of void growth by plastic deformation: normalized yield strength as a function of void radius. (Chang et al., 2013)

  2.38a). This kind of model is frequently used in finite element analysis to study influences of different aspects on void growth rate, such as effects of stress triaxiality, initial void volume fraction, etc. However, the unit cell can also be of other shapes, especially spherical or spheroidal as considered in the well-known work of Gurson and in those for extending the Gurson model (see fig. 2.38b).

Figure 2 . 37 :

 237 Figure 2.37: Illustration for the model of an isolated void in an infinite medium: (a) a cylindrical void, (b) a spherical void. (Huang and Wang, 2006)

Figure 2 . 38 :

 238 Figure 2.38: Illustration for (a) a cubic unit cell containing an initially spherical void, and (b) a spherical unit cell containing an initially spherical void[START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media[END_REF].

Figure 2 . 39 :Figure 2 . 40 :

 239240 Figure 2.39: Effect of void radius a on normalized void growth rate.[START_REF] Fleck | Strain gradient plasticity[END_REF]

Figure 2 . 41 :

 241 Figure 2.41: Size effects on (a) the overall stress strain response and (b) the evolution of the normalized void volume fraction for a material with f 0 ≈ 5.33 × 10 -3 . Both conventional and size dependent results with R v /l * = 1.0 and R v /l * = 0.5 are shown for three stress ratios ρ. R v denotes the void radius and l * the material length parameter.[START_REF] Niordson | Void growth to coalescence in a non-local material[END_REF]

Figure 2 .

 2 Figure 2.42:The normalized void growth rate ε v at ε 22 = 1% as a function of l/a. l is a length scale parameter and a the void radius.[START_REF] Shu | Scale-dependent deformation of porous single crystals[END_REF]

Figure 2 . 43 :

 243 Figure 2.43: Size effects on the evolution of the relative void growth ((b),(d)) and that of the overall response in terms of the true stress ((a),(c)) with respect to the logarithmic strain. Figs. (a) and (b) are for the initial void volume fraction f 0 = 3.1% and fig. (c) and (d) for f 0 = 12.6%. l denotes a length scale parameter and r 0 the void radius, and κ the stress ratio σ 22 /σ 11 . (Borg et al., 2008)

Figure 2 . figure 2 Figure 3 Figure 4

 2234 Figure 2.

Figure 2 . 44 :

 244 Figure 2.44:The effect of initial void radius R 0 on loading-deformation curves.[START_REF] Huo | A note on the effect of surface energy and void size to void growth[END_REF]

Figure 2 . 45 :

 245 Figure 2.45: Unit cell with the consideration of interface stress for incorporating the void size effect into the Gurson model. (Morin et al., 2015)

. 30 )

 30 where f denote the void volume fraction and the parameter ξ = (2/f )(D m /D eq ) is define from the mean and the equivalent deviatoric parts D m , D eq of the macroscopic strain rate D ∼ . Based on this yield surface, a set of constitutive equations was formulated and implemented in a finite element code by[START_REF] Morin | Numerical assessment, implementation and application of an extended Gurson model accounting for void size effects[END_REF].

Contents 3 . 1

 31 Crystal plasticity model for void-free single crystals . 55 3.1.1 FCC single crystals . . . . . . . . . . . . . . . . . . . . . 55 3.1.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.1.3 Definition of stresses . . . . . . . . . . . . . . . . . . . . 57 3.1.4 Flow rule . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.5 Hardening rule . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 FE formulation of unit cell simulations . . . . . . . . . 61 3.3 Results for unirradiated single crystals . . . . . . . . . 64 3.3.1 FE discretisation effect . . . . . . . . . . . . . . . . . . . 65 3.3.2 Overall stress-strain response . . . . . . . . . . . . . . . 67 3.3.3 Void growth . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3.4 Field of total accumulated slip . . . . . . . . . . . . . . 73 3.3.5 Field of lattice rotation . . . . . . . . . . . . . . . . . . 76 3.3.6 Slip sectors . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.7 Void coalescence . . . . . . . . . . . . . . . . . . . . . . 79 3.3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .82 3.4 Results for irradiated single crystals . . . . . . . . . . . 82 3.4.1 Void-free single crystal: a reference . . . . . . . . . . . . 83 3.4.2 Irradiation effects on stress-strain response of the unit cell 89 3.4.3 Irradiation effects on void growth . . . . . . . . . . . . . 90 3.4.4 Irradiation effects on onset of coalescence . . . . . . . . 91 3.4.5 Irradiation effects on fields of accumulated plastic slip . 92 3.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . 96

Figure 3 . 1 :

 31 Figure 3.1: (a) FCC lattice structure; (b) Plastic slip systems for FCC single crystals.

Figure 3 . 2 :

 32 Figure 3.2: Multiplicative decomposition of the deformation gradient.

(

  unpinning term) is introduced based on the observation of Tanguy et al. (2013) that the hardening effect is under-estimated if only the hardening due to Frank loops is considered as a dispersed barrier hardening. This unpinning term aims at modeling the static ageing effects arising once irradiation defect cascades are present in the material microstructure (Trinkaus et al. (1997a), Trinkaus et al. (1997b), and Singh et al. (1997)). For the unirradiated material, only the dislocation term is considered.

Figure 3 . 3 :

 33 Figure 3.3: Problem setup for unit cell simulations.

Figure 3 . 4 :

 34 Figure 3.4: Unit cell: (a) the geometry, (b) half of a typical FE mesh with f 0 = 0.01.

T 1 Table 3 . 3 :

 133 Values of the stress triaxiality T and corresponding η for axisymmetric loadings used in the simulations. Each unit cell has a different crystal orientation characterized by the lattice orientations aligned with the three coordinate axes x 1 -x 2 -x 3 . Five crystal orientations are considered: [100]-[010]-[001], [110]-[ 110]-[001], [111]-[ 211]-[0 11], [210]-[ 120]-[001] and [ 125]-[1 21]-[210]. They have different symmetry about the coordinate planes and correspond to different number of primary slip systems activated in uniaxial tension (see table

  Before discussing simulation results, a study of the effect of finite element discretisation was first conducted to optimise computation time while keeping sufficient accuracy. Due to the anisotropy of the matrix, complete unit cells are used for the simulations. Two different finite element discretisations are considered as shown in fig. 3.5a and fig. 3.5b for one eighth of the FE mesh. The total number of elements used for the meshes can by calculated by 24 × n 3 , where n = 5 (3000 elements) for the coarse mesh (fig. 3.5a) and n = 10 (24000 elements) for the fine mesh (fig. 3.5b). The variable n is used to characterize the number of elements of the unit cell as shown in fig. 3.5b. The crystallographic orientation [100] is chosen for this study with initial void volume fraction f 0 = 0.01 and two levels of stress triaxiality are considered: T ∈ {1, 3}.

Figure 3 . 5 :

 35 Figure 3.5: Different FE discretisation: (a) the coarse mesh (n = 5) for 1/8 of the full geometry (d) the fine mesh (n = 10) for 1/8 of the full geometry.

Figure 3 . 6 :

 36 Figure 3.6: FE discretisation effect on (a) the overall stress strain behaviour (b) the void volume fraction for the [100] orientation and f 0 = 0.01.

Figure 3 . 7 :

 37 Figure 3.7: Effect of stress triaxiality on the overall behaviour of the unit cell for the crystallographic orientation [100], f 0 = 0.01 and triaxiality T from 1 to 3: (a) overall stress-strain curves and (b) evolution of the transverse strain F 33 -1 vs. longitudinal strain F 11 -1.

F 11 -

 11 1 and porosity are defined as the critical strain F 11c -1 and critical porosity f c at the onset of coalescence. The onset of void coalescence is indicated by a hollow square on each curve. It can be observed that the critical strain F 11c -1 decreases when the stress triaxiality T increases. Void coalescence will be discussed in more details later in this section.

Figure 3 . 8 :

 38 Figure 3.8: Effect of crystallographic orientation on the overall behaviour of the unit cell for f 0 = 0.01, triaxiality T = 2 and different crystallographic orientations: (a) overall stress-strain curves and (b) evolution of the transverse strain vs. longitudinal strain.

Figure 3 . 9 :

 39 Figure 3.9: Evolution of void volume fraction: effect of stress triaxiality for crystallographic orientation [100], f 0 = 0.01 and triaxiality T from 1 to 3.

Figure 3 . 10 :

 310 Figure 3.10: Evolution of void volume fraction: (a) effect of stress triaxiality for crystallographic orientation [100], f 0 = 0.01 and triaxiality T from 1 to 3, (b) effect of crystallographic orientation for f 0 = 0.01 and stress triaxiality T = 1 and (c) effect of crystallographic orientation for f 0 = 0.01 and stress triaxiality T = 3.

Figure 3 . 11 :

 311 Figure 3.11: Evolution of void aspect ratios W 2 and W 3 for (a) the crystallographic orientation [100] and (b) [ 125] with triaxiality T = 1 and T = 3.

Figure 3 . 12 :

 312 Figure 3.12: Fields of accumulated plastic slip γ cum in the x 1 -x 2 (left column) and x 1 -x 3 (right column) cross sections of the fine unit cell mesh (n = 10) for three crystallographic orientations and stress triaxiality T = 1 with the overall deformation F 11 -1 = 0.1. The initial void fraction is f 0 = 0.01.

Figure 3 . 13 :

 313 Figure 3.13: Fields of accumulated plastic slip γ cum in the x 1 -x 2 (left column) and x 1 -x 3 (right column) cross sections of the fine unit cell mesh (n = 10) for three crystallographic orientations and stress triaxiality T = 3 with the overall deformation F 11 -1 = 0.1. The initial void fraction is f 0 = 0.01.

Figure 3 . 14 :Figure 3 . 15 :

 314315 Figure 3.14: Lattice rotation θ (in radian) in the x 1 -x 2 (left column) and x 1 -x 3 (right column) cross sections of the fine unit cell mesh (n = 10) for three crystallographic orientations and stress triaxiality T = 1 with the overall deformation F 11 -1 = 0.1. The initial void fraction is f 0 = 0.01.

  The plastic regions expand and single slip quickly evolves into multiple slip. These results are different from the observation of Kysar et al. (2005) who showed the slip sectors in the regions near the void within which the material points deformed plastically in single slip. The transition of slip sectors from single slip to multiple slip in the present work is closely related to the hardening laws that are used.In fig.3.16, the number of activated slip systems are shown in the elements bounding the void at F 11 -1 = 0.005 for T = 1 and at F 11 -1 = 0.003 for T = 3, before a dramatic change of the void shape. The[START_REF] Lee | On the origin of deformation microstructures in austenitic stainless steel: part I. microstructures[END_REF],[START_REF] Mcelhaney | Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments[END_REF] and[START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations are considered. The slip sectors display regular structures determined by the crystallographic orientation and the stress triaxiality. For the three orientations with the stress triaxiality T = 1, the plastic slip occurs in some parts near the void but the rest remains elastic. The structures of slip sectors, including their geometry and the number of activated systems, vary from one to another. Under the high stress triaxiality T = 3, the slip sectors cover the whole void for the three orientations. Notice that the the slip sectors of the different orientations display very similar structure (both the geometry and the number of activated systems)

Figure 3 .Figure 3 . 16 :

 3316 Figure 3.17a presents the evolution of the critical strain F 11c -1 for the onset of void coalescence with respect to the stress triaxiality T for different orientations with the initial void volume fraction f 0 = 0.01. For the [210] and[START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientations, the simulations are not able to provide coalescence at low stress triaxiality, even though the longitudinal macroscopic strain F 11c -1 reaches 130%. This is related to extremely low void growth rate for these two orientations at low stress triaxiality. For all the orientations, the critical strain F 11c -1 decreases with increasing stress triaxiality T . This result agrees with the observation of[START_REF] Yerra | Void growth and coalescence in single crystals[END_REF]

Figure 3 . 17 :

 317 Figure 3.17: Effect of stress triaxiality and crystallographic orientation on (a) the critical deformation F 11c -1 and (b) the critical void volume fraction f c at the onset of coalescence for various crystallographic orientations and fixed initial void volume fraction f 0 = 0.01.

2 a

 2 be equal for every system. Hence, the initial value of the dislocation density forsystem s is ρ s D | ini = ρ tot D /12,where ρ tot D is the total dislocation density measured in experiments. The same hypothesis is made for the density of Frank loops and each system has an initial value of Frank loop density following ρ p L | ini = ρ tot L /4 with the total Frank loop density being ρ tot L (Renault et al., 2010). b D is the norm of the Burgers vector 1 2 a 110 of dislocations and b L is the norm of the Burgers vector 1 111 of Frank loops. Two cases, unirradiated (0 dpa) and irradiated (13 dpa), are taken into account in the following simulations for emphasizing effects of the post-irradiation hardening/softening behavior on void growth and coalescence. The parameters for the two cases are listed in table 3.6.

  This sharp softening corresponds, according to the observation in the experiments(Pokor et al., 2004a; Pokor et al., 2004b), to the unpinning term in Eq.(3.22). In fig.3.18b, stress-strain curves are presented for T = 3. As expected, the maximum stresses at T = 3 are larger than those at T = 1.

F 4 DFigure 3 . 18 :Figure 3 . 19 :Figure 3 . 20 :Figure 3 . 21 :Figure 3 . 22 :

 4318319320321322 Figure 3.18: Overall stress-strain curves of unit cells with void (f 0 = 0.01, solid lines) and without void (dashed lines) at irradiated and unirradiated state. The crystallographic orientation is [100]-[010]-[001]. (a) T = 1, (b) T = 3.

Figure 3 . 23 :

 323 Figure 3.23: Evolution of void volume fraction for three crystallographic orientations under four stress triaxialities: (a) T = 1.0, (b) T = 1.5, (c) T = 2.0 and (d) T = 3.0. Solid lines are for irradiated materials and dashed lines for unirradiated materials.

Figure 3 . 24 :

 324 Figure 3.24: Influence of stress triaxiality and crystallographic orientation on (a) the critical deformation F 11c -1 and (b) the critical void volume fraction f c at the onset of coalescence for various crystallographic orientations. f 0 = 0.01. Solid lines are for irradiated materials and dashed lines for unirradiated materials.

Figure 3 . 25 :

 325 Figure 3.25: Field of accumulated slip on x 1 -x 2 middle cross section on of the unit cell for [100]. (a) T = 1.5 and F 11 -1 = 0.35, (b) T = 3.0 and F 11 -1 = 0.2.

Figure 3 . 26 :

 326 Figure 3.26: Field of accumulated slip on x 1 -x 2 middle cross section on of the unit cell for [111] with (a) T = 1.5 and (b) T = 3.0. F 11 -1 = 0.14.

Figure 3 . 27 :

 327 Figure 3.27: Field of accumulated slip on x 1 -x 2 middle cross section of the unit cell for [ 125]. (a) T = 1.5 and F 11 -1 = 0.50, (b) T = 3.0 and F 11 -1 = 0.2.

4

  Porous single crystals: an elastoviscoplastic model at finite strainsRésuméUn modèle élastoviscoplastique en grandes transformations est proposé pour le monocristal poreux dans ce chapitre. Le modèle prend en compte l'effet de l'orientation cristallographique, de la triaxialité des contraintes et de la porosité initiale. En comparant avec les simulations de cellules unitaires, on montre que le modèle est capable de décrire qualitativement la croissance de cavité et la réponse mécanique macroscopique du monocristal poreux. Le modèle est ensuite appliqué à la simulation l'endommagement ductile dans un agrégat polycrystallin et dans une éprouvette monocristalline entaillée.

Contents 4 . 1

 41 Porous single crystal plasticity model . . . . . . . . . . 4.2 Numerical assessment . . . . . . . . . . . . . . . . .

4 . 4

 44 Application to simulations of the single-edge-notch tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Experiment and simulation . . . . . . . . . . . . . . . . 4.4.2 Results: surface slip traces . . . . . . . . . . . . . . . . 4.4.3 Results: strain fields at the notch tip . . . . . . . . . . . 4.4.4 Results: force-notch-opening-displacement relation . . . 4.4.5 Conclusions and outlook . . . . . . . . . . . . . . . . . . 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . A first yield function for single crystals containing voids has been derived by Han et al. (2013) based on a variational approach at infinitesimal strains. The model accounts for the effects of initial void volume fraction, stress triaxiality and crystal orientation. In order to describe ductile fracture in single crystals due to void growth and coalescence, it is needed to introduce finite strain framework. In the present section, only void growth is considered and the porous single crystal model is extended to finite strain by introducing flow rule and evolution law for void growth. In this section, an elastoviscoplastic model at finite strains is formulated based on the model of Han et al. (2013). The model also incorporates heuristically strain hardening of single crystal matrix, in the same manner as that proposed by Tvergaard and Needleman (1984). It is then assessed through three-dimensional unit cell finite element simulations based on periodic homogenisation with prescribed constant stress triaxiality. Axisymmetric and non-axisymmetric loadings are taken into account for the assessment. The model is then applied to simulate ductile damage in polycrystalline aggregates and a single-edge-notch tension test on a single crystal.

  C i is allowed by the existence of directors 1 associated with lattice orientation. The uniqueness of the multiplicative decomposition of the deformation gradient F ∼ = E ∼ .P ∼ thus remains. As a result, the kinematics of porous single crystals follows what is described in section 3.1.2, except that det (P ∼ ) = 1 as a result of compressibility of porous single crystals. The initial porous single crystal orientation is taken that of the undeformed single crystal matrix. The same number of slip systems N are attributed to each material point of the porous single crystal as the single crystal matrix. They have the same crystallographic definition as that of the undeformed single crystal matrix. For the flow rule, the yield criterion developed in the work of Han et al. (2013) for porous single crystals is used. The definition of effective resolved shear stress τ * s , derived by Han et al. (2013) in the infinitesimal strain framework, is extended to finite strain framework:

Figure 4 . 1 :

 41 Figure 4.1: Comparison between the unit cell (UC) simulations and the porous single crystal model predictions: evolution of normalised macroscopic stress σ 11 /τ s T with respect to axial strain F 11 -1 for different crystallographic orientations with fixed triaxiality, (a) T = 1 and (b) T = 3, and initial void volume fraction f 0 = 0.01.

Figure 4 .

 4 Figure 4.3 shows the comparison of the evolution of the void volume fraction f for three orientations with f 0 = 0.01 at stress triaxiality (a) T = 1 and (b) T = 3.

Figure 4 . 2 :Figure 4 . 3 :Figure 4 . 4 :

 424344 Figure 4.2: Comparison between the unit cell simulations and the porous single crystal model predictions: evolution of normalised macroscopic stress σ 11 /τ s T with respect to axial strain F 11 -1 for various crystallographic orientations with fixed triaxiality, (a) T = 1 and (b) T = 3, and initial void volume fraction f 0 = 0.05.

4 (T = 3 )

 43 discrepancy has not been observed for the axisymmetric loading cases with this orientation at η 2 = η 3 = 0.4 (T = 1) in fig. 4.1a and fig. 4.3a and at η 2 = η 3 = 0.in fig. 4.1b and fig. 4.3b. This can probably be explained by the potential influence of the third invariant of the macroscopic stress tensor, which is not yet taken into account in the porous single crystal model.

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: Assessment of porous single crystal model for non-axisymmetric cases: evolution of normalised macroscopic stress σ 11 /τ s T with respect to axial strain F 11 -1 for different crystallographic orientations with initial void volume fraction f 0 = 0.01, η 2 = 0.727 and (a) η 3 = 0.625, (b) η 3 = 0.538 and (c) η 3 = 0.4.

20

 20 .10) where the Heaviside function H(x) = 1 if x > 0, else H(x) = 0. This quantity indicates the effective number of activated slip systems depending on crystallographic orientation and the stress triaxiality in both the model and the unit cell. In fig.4.8, the evolution of the indicator with respect to the strain F 11 -1 is presented for the[START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation with f 0 = 0.01. It reveals that the macroscopic porous model predicts single slip pattern for the[START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation at T = 3. However the factor reaches 4 before void coalescence in the unit cell simulation, which significantly deviates from the single slip pattern. This is confirmed by fig. 3.13 and fig. 3.15. Since fewer activated slip systems are predicted by the porous model, strain hardening is underestimated by the model. The enhancement of latent hardening by the high stress triaxiality observed in the unit cell simulation for the single-slip orientation is difficult to be incorporated in the homogenisation model.For the[START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation with small initial void volume fraction f 0 = 0.01 at the low stress triaxiality T = 1, the porous model overestimates the void growth rate as compared to the unit cell simulation (see fig.4.3a). The unit cell shows nearly no void growth in terms of void volume fraction. Notice that the[START_REF] Niordson | Computational strain gradient crystal plasticity[END_REF] orientation corresponds to single slip orientation for a void-free FCC single crystal. For this type of orientation with small initial void volume fraction at low stress triaxiality, the void does not induce significant plastic slip heterogeneity in the single crystal matrix, as shown in section 3.3.4. As a result, the voided single crystal behaves like a void-free single crystal and exhibits a nearly single slip pattern with quasi-incompressible overall behaviour. No void growth is predicted by the unit cell simulation in this case. In contrast, the macroscopic model predicts a weak void growth for this case, due to the term 2q 1 f i cosh q 2 3 Mm τ * s in Eq. (4.1), which is strictly positive even

Figure 4 . 7 :

 47 Figure 4.7: Stress-strain curves of the void-free and the voided single crystals for the [ 125] orientation. T = 3 and f 0 = 0.01.

Figure 4 . 8 :

 48 Figure 4.8: Comparison between the unit cell simulation and the porous single crystal model: evolution of plastic slip indicator I with respect to axial strain F 11 -1 for the [ 125] orientation.

Figure 4 . 9 :

 49 Figure 4.9: (a) FE mesh of the polycrystalline aggregate with 343 cubic grains. Each grain is meshed with 27 elements; (b) Inverse pole figure showing the random distribution of the grain orientations.

F

  11 -1 is shown in fig. 4.10 for different stress triaxialities. A softening regime is predicted at the aggregate scale for the stress triaxialities T = 2 and 3. For T = 1, the simulation diverged before strain softening. The strain softening begins at a smaller overall strain for T = 3 than T = 1. The strain softening observed at the aggregate scale is associated with the evolution of void volume fraction at the grain scale.

Figure 4 . 10 :

 410 Figure 4.10: Overall stress-strain curves for different stress triaxialities.

Figure 4 . 11 :

 411 Figure 4.11: Evolution of void volume fraction f i with the overall strain F -1 for the imposed stress triaxiality T = 3, showing a propagation of ductile damage between grains.

Figure 4 . 12 :

 412 Figure 4.12: Stress triaxiality effects: void volume fraction f i for (a) T = 3 at F 11 -1 = 0.07 and (b) T = 1 at F 11 -1 = 0.38.

AFigure 4 . 13 :Figure 4 . 14 :

 413414 Figure 4.13: The geometry and a 3D FE mesh of the specimen for SENT.

Figure 4 . 15 :

 415 Figure 4.15: SEM micrograph showing gold speckles created on a zone of the specimen surface near the notch tip for the digital image correlation.

Figure 4 . 16 :

 416 Figure 4.16: SEM micrograph showing inclusions of ferrite at the front of the notch tip.

. 11 )Figure 4 . 17 :

 11417 Figure 4.17: SEM micrographs showing surface slip traces with different notch opening displacements: (a) ∆a/a 0 = 0.18; (b)∆a/a 0 = 0.95.

Figure 4 . 18 :Figure 4 . 19 :

 418419 Figure 4.18: Theoretical prediction of slip traces on the free surfaces.

4. 4 . 3

 43 Results: strain fields at the notch tipStrain fields on the surface of the specimen during loading were measured by DIC. The measured field of the strain E 22 = F 22 -1 for the normalized notch opening ∆a/a 0 = 0.95 in the region near the notch tip is shown in fig.4.20a. The obtained result shows a discontinuous strain field which is related to the discrete nature of plastic slip. Furthermore, a strong strain localization is observed in front of the notch tip; three types (p1, p2 and p3) of slip traces display which is in agreement with the observation in section 4.4.2. Besides, slip traces of type p1 and p2 are very intensive even in the region relatively far from the notch tip, while slip traces of type p3 are found only in the region near the notch tip. This is consistent with the FE predictions shown in fig.4.[START_REF] Besson | Non-linear mechanics of materials[END_REF].The damage f i predicted by the porous single crystal model with the initial void volume fraction f 0 = 0.01 is presented in fig.4.21 for ∆a/a 0 = 0.95. The evolution of the damage parameter f i is not homogeneous in the direction of the thickness around the notch. The maximum value of f i , reaching 0.06, is not found on the surface of the specimen. Since the ductile fracture does not occur in the experiment, this simulation result about ductile damage is not compared with the experiment.The results of the simulations, presented in fig.4.20b, succeed in predicting basic characteristics of the strain field observed in the experiment: a strong strain localization in an asymmetric region near the notch tip; an asymmetric mode of notch opening; an activation of plastic slip in the region relatively far from the notch tip.

Figure 4 . 20 :

 420 Figure 4.20: Surface strain filed E 22 for ∆a/a 0 = 0.95: (a) experimental result measured by DIC; (b) simulation predictions.

Figure 4 . 21 :

 421 Figure 4.21: Void volume fraction f i in the region near the notch tip at ∆a/a 0 = 0.95 predicted by the porous single crystal model with the initial porosity f 0 = 0.01

Figure 4 . 22 :

 422 Figure 4.22: Evolution of loading force as a function of normalized notch opening displacement: comparison experiment-simulations.

. 11 )

 11 Applying the principle of virtual power with Eq. (5.10) and Eq. (5.11) yields D (σ ∼ : (grad u ) + s γχ + m . grad γχ ) dV = ∂D (t . u + m γχ ) dS, ∀ u , ∀ γχ , ∀D.

. 13 )

 13 where n is the normal to the surface element dS of the boundary ∂D of the subdomain D in the current configuration. Since this relation must hold for all D, all u and all γχ , one gets the balancesdiv σ ∼ = 0, ∀x ∈ D, (5.14) div m -s = 0, ∀x ∈ D, (5.15) and the Neumann boundary conditions t = σ ∼ .n , ∀x ∈ ∂D, (5.16) m = m .n , ∀x ∈ ∂D. (5.17) Analogously, one can derive the balance laws in the reference configuration: Div S ∼ = 0, ∀X ∈ D 0 , (5.18) Div M -S = 0, ∀X ∈ D 0 , (5.19)

  second law of thermodynamics asserts non-negative energy dissipation rate d for any subdomain D of the crystalline body B in the current configuration, which writes energy density ψ measured per unit mass and the mass density ρ measured in the current configuration. The second term represents the temporal increase in the free energy of subdomain D, which follows ˙ D ρψ dV = ˙ D 0 ρ 0 ψ dV 0 = D 0 ρ 0 ψ dV 0 = D ρ ψ dV, (5.26)

. 29 )

 29 where E ∼ denotes the elastic part and P ∼ the plastic part of the deformation gradient F ∼ . By this decomposition, a local intermediate configuration C is introduced (Mandel, 1973). The reference and current local configuration at X are called C 0 and C respectively. It is assumed that plastic deformation takes place through the slip of dislocations on prescribed slip planes with normal n s along prescribed slip direction l s , and that the evolution of P ∼ is governed by the plastic slip γ s on slip systems s via the relation Ṗ ∼ .P ∼ -1 = N s=1 γs N ∼ s (5.30) with the Schmid tensor N ∼ s = l s ⊗ n s and the total number of slip systems N . The uniqueness of the decomposition (Eq. (5.29)) is obtained by choosing the isoclinic local configuration C where the crystal lattice has the same orientation as in C 0

. 62 )

 62 Following the Coleman-Noll procedure results in the constitutive equations Eqs. (5.43) and (5.44) and another one connecting M and K

. 86 )

 86 By Eq.(5.43), the same constitutive equation as Eq. (5.75) is obtained, while Eqs. (5.44) and (5.45) lead to s = -H χ e,(5.87)

. 88 )

 88 With Eqs. (5.5), (5.21) and (5.88), one can readily find m = A ∼ .k .(5.89)

. 110 )

 110 This function does not involve the Eulerian Laplace operator. It is thus demonstrated that the second formulation based on K with the potential Eq. (5.94) cannot lead to neither Lagrangian nor Eulerian Laplace operator in the regularization equation.

. 113 )

 113 With S = ρ 0 ρ s, one hasS = -H χ e. (5.114) Substituting Eqs. (5.53) and (5.55) into Eq. (5.113) yields M = A ∼ .K . (5.115) For crystals with cubic symmetry, the constitutive relations Eqs. (5.113) and (5.115) become M = AB ∼ p .K , (5.116) M = AK , (

. 123 )

 123 Using Eqs. (5.105) and (5.106),one can obtain m = A ∼ .k .(

  what are the final values of the stress variables and the internal variables v int ? The strain variables are referred to as the input variables (v IN ) and the stress variables as the output variables (v OUT ), the problem is briefly depicted in fig.5.1. For the sake of brevity, the increments of the variables are written in the rate form.

Figure 5 . 1 :

 51 Figure 5.1: Problem setup of the numerical integration of the constitutive equations.

156 ) 0 M χ N b dS 0 .

 15600 According to Eq.(5.156), an internal reaction is associated with each degree of freedom. We thus refer to R a int(u i ,e) as the internal reaction on node a related to u i R a int(u i ,e) = b int(γχ,e) as the internal reaction on node b related to γ χ external reaction is associated with each degree of freedom. We refer to R a int(u i ,e) as the external reaction on node a related tou i R a int(u i ,e) =∂D e 0 (T i u N a ) dS 0 (5.159) and to R b ext(γχ,e) as the external reaction on node b related to γ χ R b ext(γχ,e) = ∂D e

[

  u B] T .{S ∼ } dV 0 , (5.172) {R e int(γχ) } = D e 0 [ χ N] T .{S} + [ χ B] T .{M } dV 0 , (5.173) {R e ext(u) } = ∂D e 0 [ u N] T .{T } dS 0 , (5.174) {R e ext(γχ) } = ∂D e 0 [ χ N] T .{M } dS 0 , (5.175) where [ u B] T is the transpose of the matrix [ u B] and the same notation is used for other matrix. The global FE equation is obtained by applying assembly operator A on internal reactions and external reactions: {R int(u) } = A({R e int(u) }) (5.176) {R int(γχ) } = A({R e int(γχ) }) (5.177) {R ext(u) } = A({R e ext(u) }) (5.178) {R ext(γχ) } = A({R e ext(γχ) }).

  Wulfinghoff and Böhlke, 2013. It is based on the fact that the solution τ s of Eq. (5.52) is located around τ s c -J e s with J e = ρ ρ .

Figure 5 . 2 :

 52 Figure 5.2: Improved algorithm for integrating the power law.

Figure 5 . 3 :

 53 Figure 5.3: Single crystal with a single slip system.

  .205) τ ≈ Π ∼ e : (l ⊗ n ). (5.206) Assuming that µ denotes the shear modulus and substituting Eq. (5.204) into Eq. (5.75), one can obtain Π ∼ e Π ∼ e = τ (l ⊗ n + n ⊗ l ), (5.207)with τ = µκ e . In addition, the Boussinesq stress tensor S ∼ is approximated byS ∼ = Jσ ∼ .F ∼ -T = E ∼ .Π ∼ e .P ∼ -T ≈ Π ∼ e .P ∼ -T = Π ∼ e -τ γl ⊗ l . (5.208) Since Div S ∼ = 0 (see Eq. (5.19)), one has ∂τ ∂Y = 0, (5.209) which makes τ = τ spatially constant. Consider the linear softening law:τ c = τ 0 + Hγ (5.210)with H < 0. In the following, the derivation is carried out first for the formulation presented in section 5.1.3. The free energy potential given by Eq. (5.74) is chosen so that one has the following constitutive equations:K = AM (5.211) and S = -H χ e. (5.212) Given Eq. (5.212), the yield condition (Eq. (5.51)) writes f = |τ | -(τ c -S) = |τ | -[τ 0 + Hγ -H χ (γ χ -γ)] = 0, (5.213) from which one can express γ as γ = |τ | -τ 0 + H χ γ χ H + H χ . (5.214) For the special case of single slip, we assume γ χ = γ χ (Y ). Considering Eq. (5.211), Div M becomes Div

  .224) Substituting the previous equations into Eq. (5.222) results in f = |τ + x s | -[τ 0 + Hγ -H χ (γ χ -γ)] kinematic hardening is found. It can exhibit only in the presence of a gradient of slip in both directions X and Y . Since we assume γ χ = γ χ (Y ), ∂γ χ ∂X = 0.

Figure 5 . 4 :

 54 Figure 5.4: FE mesh for studying shear banding.

5 . 4 .

 54 Periodic boundary conditions are applied for the displacement field u such that u = F ∼ .X + v (5.228)with F ∼ denotes the overall deformation gradient and v a periodic fluctuation vector.

to 10 6 Figure 5 . 5 :

 655 Figure 5.5: Influence of H χ on (a) overall stress-strain curve and (b) distribution of γ χ along Y direction at F 12 = 0.0005. A = 5 N.

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: Influence of A on (a) overall stress-strain curve and (b) distribution of γ χ along Y direction at F 12 = 0.0005. H χ = 10 5 MPa.

by

  Peirce et al. (1982) using FE simulations for double slip. In the present work, we will study strain localization in single crystals using FE simulations with the micromorphic model and consider crystals deformed by single slip. Slip and kink bands are expected in the simulations, which have been investigated in different works (see Forest (1998) for analysis and simulations, and Kysar and Briant (2002), Flouriot et al. (2003), and Sabnis et al. (2012) for experimental observations of slip and kink bands at the crack tip or the notch tip). Mesh dependence of the simulations of slip and kink bands will be studied.

Figure 5 . 8 :

 58 Figure 5.8:A typical FE mesh for simulations of strain localization. The total number of elements used is n × 6n with n for the width and 6n for the height. The mesh with n = 8 shown in the figure. An in-plane single slip system is also shown in the figure, whose slip direction has an angle of 56.3°with respect to the vertical direction.

Figure 5 . 9 :

 59 Figure 5.9: Stress-strain curves obtained by the conventional and micromorphic model with different FE meshes.

  model and it results from the strain gradient hardening taken into account in the micromorphic model. The mesh convergence is observed for the micromorphic model. Moreover, the curves obtained by the micromorphic model with n = 24 and n = 32 are smoother than those obtained by the conventional model. The evolution of plastic slip γ field with ∆L/L 0 is shown in fig. 5.10 for the conventional model and in fig. 5.11 for the micromorphic model. Complex localization modes are observed in the simulations with the conventional and micromorphic model. Both the models predict a formation of two kink bands at the

Figure 5 . 10 :

 510 Figure 5.10: Evolution of plastic slip γ field with ∆L/L 0 predicted by the conventional model with the FE mesh n = 24.

Figure 5 . 11 :

 511 Figure 5.11: Evolution of plastic slip γ field with ∆L/L 0 predicted by the micromorphic model with the FE mesh n = 24.

Figure 5 . 12 :

 512 Figure 5.12: The lattice rotation angle θ predicted by the conventional model for (a) ∆L/L = 0.012 and (b) ∆L/L = 0.033

  .232) with the rotation tensor R ∼ and the right stretch tensor U ∼ . Neglecting the elastic distortion U ∼ , R ∼ can be interpreted as the lattice rotation. The corresponding rotation angle θ is given by θ = arccos 1 2 (trace R ∼ -1).

Figure 5 . 13 :

 513 Figure 5.13: The lattice rotation angle θ predicted by the micromorphic model for (a) ∆L/L = 0.017 and (b) ∆L/L = 0.067

Figure 5 . 14 :

 514 Figure 5.14: Typical FE mesh for the unit cell simulations.

Figure 5 . 15 :

 515 Figure 5.15: (a) Variation of the normalized void volume and (b) overall stress-strain curves for different sizes of the unit cell. f 0 = 0.01.

Figure 5 . 16 :

 516 Figure 5.16: (a) Variation of void volume fraction and (b) overall stress-strain curves for different size of unit cell. f 0 = 0.1.

Figure 5 . 17 :

 517 Figure 5.17: Critical strain E c 11 for onset of coalescence as a function of normalized size of unit cell. f 0 = 0.01 and 0.1.

Figure 5 . 18 :

 518 Figure 5.18: Void shape at E 11 = 0.1 ((a)-(d)) and E 11 = 0.4 ((e)-(h)) for different unit cell size: (a,e) L/λ c = 0.5, (b,f) L/λ c = 5, (c,g) L/λ c = 25 and (d,h) conventional model. f 0 = 0.1. The field of γ cum is shown.

Figure 5 . 19 :

 519 Figure 5.19: γ cum at the void for f 0 = 0.1. E 11 = 0.05.

  In the present work, void growth and coalescence in single crystals have been investigated. These two mechanisms are involved in the ductile fracture of stainless steels. These steels are used as structural materials in the internal structure of the core of nuclear power plants. Ductility and fracture toughness are known to be strongly affected by irradiation in PWR environment. It has been shown that intragranular voids exist in irradiated austenitic stainless steels. The voids potentially have different origins and various sizes, at the nano and micro scale. This work aims to assess the mechanical behavior of voided single crystals.For this purpose, FE unit cell simulations have been performed to investigate void growth and coalescence in FCC single crystals at finite strains. Effects of irradiation have been considered through a physically based crystal plasticity model which takes into account dislocations and irradiation-induced defects (Frank loops).

S 11 =F 22 F 33 -F 23 F 32 F 23 F 31 -F 21 F 33 F 21 F 3 )F 22 F 32 F 23 F 31 -F 21 F 33 F 21 F-F 12 =---

 1122332332233121332132232233121332112 K( F 11 -F 11 ), (B.1) with S 11 the first component of the first Piola-Kirchhoff stress tensor and K the element stiffness.With the macroscopic Cauchy stress σ ∼ taking the form of Eq.(3.32), the stress triaxiality T imposed over the unit cell follows Eq.(3.34).Using Eq. (3.30), the corresponding macroscopic first Piola-Kirchhoff stress tensor S ∼ can be written asS ∼ = Jσ ∼ .F ∼ -T = Jσ 11 η ∼ F ∼ -T , (B.2)If η ∼ and F ∼ are re-written in the matrix form, one has[S ij ] = σ 11 32 -F 22 F 31 η 2 (F 13 F 32 -F 12 F 33 ) η 2 (F 11 F 33 -F 13 F 31 ) η 2 (F 31 F 12 -F 11 F 32 ) η 3 (F 12 F 23 -F 13 F 22 ) η 3 (F 13 F 21 -F 11 F 23 ) η 3 (F 11 F 22 -F 12 F 21 )Because of the connection between the unit cell and the truss element at the near end of the element, one can use the equations (B.1) and (B.3) and obtainS 11 = S 11 . (B.4)Thus, one can getσ 11 = K( F 11 -F 11 ) F 22 F 33 -F 23 F 32 . (B.5)As a result, the macroscopic first Piola-Kirchhoff stress applied over the unit cell via the truss element is equal to[S ij ] = K( F 11 -F 11 ) F 22 F 33 -F 23 F 32    33 -F 23 F 32 -F 22 F 31 η 2 (F 13 F 32 -F 12 F 33 ) η 2 (F 11 F 33 -F 13 F 31 ) η 2 (F 31 F 12 -F 11 F 32 ) η 3 (F 12 F 23 -F 13 F 22 ) η 3 (F 13 F 21 -F 11 F 23 ) η 3 (F 11 F 22 -F 12F 21 ) macroscopic Cauchy stress triaxiality T is imposed as Eq. (3.34).Finally, the boundary value problem, considered in unit cell simulations with prescribed stress triaxiality and periodic boundary conditions, is to search the periodic fluctuation vector v when imposing F 11 such that• the balance of momentum:div σ ∼ = 0 , ∀x ∈ Ω tot (B.7)• and the boundary conditions:-Periodicity:u = F ∼ .x + v , ∀x ∈ Ω tot (B.8) Rotation restriction: F 21 , F 23 = F 32 , F 31 = F 13 (B.9) Rigid translation restriction u (X n ) = 0 , X n is the coordinates of a chosen node of the unit cell (B.10) Loading via the truss element S 11 = K( F 11 -F 11 ) (B.11) Constant macroscopic Cauchy stress triaxiality σ 12 = σ 23 = σ 31 = 0 and σ 22 = η 2 σ 11 , σ 33 = η 3 σ 11 (B.12) are fulfilled. C Implicit integration of the constitutive equations Input variables v I : v I := {F ∼ , γ χ , K } (C.1) Output variables v O : v O := {S ∼ , S, M } (C.2) Variables to integrated v int : v int := {E ∼ , γ s , r s , γ cum } (C.3)

SR

  (|τ s | -(τ s c -S)) sign (τ s ) = |τ s | -(τ s c -S) K n sign (τ s ) (C.5) ṙs D = | γs |   b su r u D κ -G c r s D = H χ (γ χ -γ cum ) (C.15) M = AK (C.16) Residual equations R int R E ∼ = ∆E ∼ -∆F ∼ .F ∼ -1 .E ∼ + E ∼ . γ s = ∆γ s -Φ (|τ s | -(τ s c -S)) sign (τ s ) ∆t (C.18)

D•

  ∂∆γ p = -sign (∆γ s ) δ sp Derivatives of R γcum R γcum = ∆γ cum -

= - 1 ∼ 1 ≈(D. 6 )= - 1 ∼

 1161 ⊗ E ∼ T .F ∼ -T T -(∆F ∼ ⊗E ∼ T ) : (-F ∼ -1 ⊗F ∼ -T ) : ⊗ E ∼ T .F ∼ -T + (∆F ∼ ⊗E ∼ T ) : (F ∼ -1 ⊗F ∼ -T ) (D.7)

  σ ∼ .F ∼ -T ) ⊗ F ∼ -T + J(σ ∼ ⊗1 ∼ ) : (-F ∼ -T ⊗F ∼ -1 ) (D.28)Finally,∂S ∼ ∂F ∼ = J(σ ∼ .F ∼ -T ) ⊗ F ∼ -T + J(σ ∼ ⊗1 ∼ ) : (-F ∼ -T ⊗F ∼ -1 ) (D.

Figure E. 1 :

 1 Figure E.1: Simple glide test on one volume element.

Figure E. 2 :

 2 Figure E.2: Comparison: conventional model and strain gradient model with H χ = 0 and A = 0

Figure E. 3 :

 3 Figure E.3: Comparison: conventional model and strain gradient model with H χ = 10 5 and A = 10

Case 3 :

 3 H χ = 10 5 and A = 10 with the numerical Jacobian matrix

Figure E. 4 :

 4 Figure E.4: Comparison: conventional model and micromorphic model with H χ = 10 5 and A = 10 using perturbation method

Figure E. 5 :

 5 Figure E.5: Comparison: conventional model and micromorphic model with H χ = 10 5 and A = 10 with improved algorithm

Table 2 . 1 :

 21 Degrees of freedom needed for FE implementation of some strain gradient crystal plasticity in the case of FCC single crystals (12 slip systems). Additional DOFs besides displacements are pointed out.

	Models	Additional DOFs	Number of additional DOFs
	(Gurtin, 2002)	γ s	12
	(Cordero et al., 2010)	curl P ∼	9
	(Evers et al., 2004)	ρ GND	18
	(Wulfinghoff and Böhlke, 2012)	γ cum or γ eq	1

models. The number of additional degrees of freedom (DOF) besides displacement for finite element implementation is shown in table 2.1 for some models. Note that, for the model of

[START_REF] Evers | Non-local crystal plasticity model with intrinsic {SSD} and {GND} effects[END_REF]

, 12 edge and 6 screw dislocations are taken into account for the densities GND. Thus, 18 additional DOFs are needed. It can be seen that many additional DOFs are needed for FE implementation. This usually leads to bad computational convergence or high computation time. Therefore, the numerical

1

The curl operator is also denoted by ∇× or rot .

Figure 2.35:

  Evolution of the void volume fraction as a function of strain under biaxial deformation for different void radii.[START_REF] Chang | Molecular dynamics modeling and simulation of void growth in two dimensions[END_REF]

	0.6
		Void radius (nm)			(b)
	0.5	5 10
		15
		30
	0.4	50
	0.3
	v
	0.2
	0.1
	0
	0	0.01	0.02	0.03	0.04
			x
		dislocation nucleation at void surface
		no size effect
		bulk dislocation nucleation
		larger voids grow faster
	no plastic void growth
					, strain rate (Potirniche
	et al., 2006; Rudd, 2009; Tang et al., 2010; Xu et al., 2011; Tang et al., 2014),
	stress triaxiality (Seppälä et al., 2004), crystallographic orientation (Zhu et al., 2007;
	Bringa et al., 2010) and so on. However, it was widely observed that, following

the size effect on void growth rate was quite limited

	(see Potirniche et al., 2006; Traiviratana et al., 2008; Zhao et al., 2009; Xu et al.,
	2011; Tang et al., 2012; Chang et al., 2013; Krasnikov and Mayer, 2015):
	• yield strength increased with decreasing void size;
	• dislocation evolution patterns were influenced by void size.
	It should be noticed that the definition of the yield strength can be different.
	The overall stress needed to nucleate the first dislocation from the void surface
	was often used for the yield strength. As shown in figs. 2.33a and 2.33b, the
	yield strength decreases with increasing void size and asymptotically approaches
	a constant value. The size dependence of yield strength is associated with the
	surface energy of the void. For 2D simulations, Chang et al. (2013) calculated the
	energy γ 2D per unit length at the void perimeter for various void radii presented
	in fig. 2.34a. They argued that the surface energy becomes small for voids with a
	radius below 5 nm, meaning that the void becomes more stable. It is thus more
	difficult to nucleate dislocations at void surface for the very small voids, which
	can be confirmed by higher yield strength for void radius smaller than 5 nm (see
	fig. 2.34b). However, according to Potirniche et al. (2006) and Chang et al. (2013) (see fig. 2.35). No
	explanation is given for this negligible effect.
	to be a possible
	mechanism for void growth. More details concerning physical mechanisms of void
	growth at nano-scale under high strain rate are not discussed here. Nevertheless,
	it should be pointed out that strong size effects were reported by MD simulations

Table 2 . 2 :

 22 ; Borg and Kysar, 2007; Borg et al., 2008; Hussein et al., 2008). Basically, most of the works predicted reduced void growth rate and increased loading carrying capacity with decreasing void size (see, e.g., fig. 2.41 by Niordson (2008) for an isotropic material, fig. 2.42 by Shu (1998) for a single crystal with two planar slip systems, and fig. 2.43 by Borg et al. (2008) for a single crystal with three planar slip systems). These works are summarized in table 2.2 by indicating the material assumption of the matrix and the void Unit cell simulations in the literature on size dependent void growth in isotropic or single crystal matrix.

	Reference	Isotropic or single crystal	Crystal structure	Void shape	Small or finite strains
	(Niordson, 2008)	isotropic	-	spherical (axi-symmetry)	finite strains
	(Shu, 1998)	single crystal	2 planar slip systems	cylindrical	small strains
	(Borg and Kysar, 2007) single crystal	3 planar slip systems	cylindrical	small strains
	(Borg et al., 2008)	single crystal	3 planar slip systems	cylindrical	finite strains
	(Hussein et al., 2008)	single crystal	3 planar slip systems	square (2D)	finite strains

Table 3 . 1 :

 31 Slip systems in FCC single crystals. The labels are given according to the Schmid-Boas convention[START_REF] Schmid | [END_REF].

	p	1	2	3	4
	plane(p) (111) (1 11) ( 111) ( 11 1)

Table 3 . 2 :

 32 Frank loop systems. Plane(p) denotes the normal to the plane of system p.

  [START_REF] Benzerga | Anisotropic ductile fracture: Part I: experiments[END_REF], where ρ tot D is the total dislocation density measured in experiments. b D is the norm of the Burgers vector1 2 a 110 of dislocations. All the material parameters used for the simulations are given in table 3.5.

	C 11	C 12	C 44	τ s T	n
	199 GPa	136 GPa	105 GPa	88 MPa	15
	γref	τ ref	µ	g c	κ
	1.47 × 10 14 s -1	88 MPa	65.6 GPa 2.6 × 10 -9 m	42.8
	a su 0.124	b su (s = u) b su (s = u) 1 0	b D 2.54 Å	ρ s 0 8.33 × 10 8 m -2

Table 3 . 5 :

 35 Material parameters for the unit cell simulations.

  al., 2004a; Pokor et al., 2004b) at 340 • C. Note that, different from in section 3.3, the values of a 1 , a 2 , • • • , a 6 are obtained by discrete dislocation dynamics[START_REF] Monnet | A crystalline plasticity law for austenitic stainless steel[END_REF]. The initial dislocation density is supposed to

	C 11	C 12	C 44	n	τ ref	γref
	199 GPa 136 GPa 105 GPa	15	6.3 MPa 10 -3 s -1
	τ s T	µ	g c	κ	b D	b L
	88 MPa 66 GPa 2.64 nm 42.8	2.54 Å	2.08 Å
	a 1	a 2	a 3	a 4	a 5	a 6
	0.124	0.124	0.07	0.625	0.137	0.122
	b 1	b 2

Table 3 .

 3

6: Material parameters for the simulations

[START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF]

.

  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . Problem setup . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .

	. . .
	4.2.1 Calibration of model parameters . . . . . . . . . . . . .
	4.2.2 Assessment for axisymmetric loading cases . . . . . . .
	4.2.3 Assessment for non-axisymmetric loading cases . . . . .
	4.2.4 4.3 Application to simulations of ductile damage in a
	polycrystalline aggregate . . . . . . . . . . . . . . . . . .
	4.3.1

  1.60 1.19 Han et al. (2013) 6.46 1.47 1.33

Table 4 . 2 :

 42 Identified values of the porous single crystal model parameters, see Eq. (4.1).

  .17a) and they become more pronounced with larger notch opening (fig.4.17b).

	slip trace		p1			p2			p3			p4
	slip system	1	2	3	4	5	6	7	8	9	10	11	12
	n s		(111)			(1 11)			( 111)			( 11 1)
	m s	[ 101]	[0 11]	[ 110]	[ 101]	[011]	[110]	[0 11]	[110]	[101]	[ 110]	[101]	[011]

Table 4 . 3 :

 43 Slip systems and slip traces for FCC single crystals.

  this chapter, an elastoviscoplastic model at finite strains has been formulated porous single crystals based on the work of Han et al. (2013). The model incorporates heuristically strain hardening of single crystal matrix. The model has been shown to be able to predict ductile damage due to void growth to coalescence in single Le modèle est basé sur une variable scalaire et il est donc numériquement efficace. Selon les variables d'état choisies, deux formulations sont présentées et l'une est implémentée dans le code par éléments finis. Le modèle est d'abord utilisé pour étudier la localisation de la déformation dans le monocristal. On montre que le modèle peut régulariser à la fois la bande de glissement et la bande en genou. Le modèle est ensuite appliqué à la croissance de cavité. Les simulations des cellules unitaires montrent que les petites cavités croissent moins vite que les grandes.

	5 Micromorphic single crystal plasticity
	model and its applications
	Poole et al., 1996; Suresh et al., 1999; Stölken and Evans, 1998) and size effects
	in turn are also expected on void growth and coalescence Fleck and Hutchinson,
	2001; Borg et al., 2008. It is hence of importance to investigate the effect of void
	size on the ductile fracture process and fracture toughness at the grain scale. This
	aspect will be treated in the next chapter.

crystals with different orientations. The model has then been applied to simulate ductile damage in polycrystalline aggregates and a single-edge-notch tension test on a single crystal.

In this chapter, effects of post-irradiation hardening/softening behavior is not taken into account. This will be considered in future works. Crack propagation due to ductile fracture at the scale of grains can also be investigated with the proposed model and simulations. To this end, a cracking criterion should be introduced in the future.

The model proposed in this chapter accounts for intragranular voids in polycrystalline materials. As presented in the previous chapters, in irradiated stainless steels, intragranular voids can correspond to irradiation-induced swelling voids or to voids nucleated on irradiation-induced precipitates during plastic straining. The former can lead to nano-sized fracture dimples and their size can be as small as 20-30 nm according to Margolin et al., 2016. The latter, as reported in Little, 1986, can result in fracture dimples of size < 10 µm in irradiated steels, which is smaller than in unirradiated materials (size > 25 µm). In a word, voids of different sizes exist in irradiated steels. Moreover, it is well known that size effects in plasticity exist at the crystal level (see, e.g., Fleck et al., 1994; Stelmashenko et al., 1993; Résumé Dans les aciers irradiés, la rupture ductile peut avoir lieu en raison des cavités de tailles différentes. La taille des cavités peut avoir une influence sur la cinétique de la croissance et la coalescence des cavités. Pour étudier cet effet, un modèle micromorphe de plasticité cristalline en grandes transformations est proposé dans ce chapitre.

  5.188) with imposed ∆v IN and obtained ∆v int . If applying an infinitesimal variation to the increments ∆v IN , one can obtain a new ∆v int according to the requirement of vanish residual equations. The variation of ∆v int resulting from the variation of ∆v IN should make the variation of {R} vanish, which follows

	{δR} =	{∂R} {∂∆v int }	{δ∆v int } +	{∂R} {∂∆v IN }	{δ∆v IN } = {0}.	(5.189)
	Therefore, one can have

  OUT depends on not only v int , but also v IN , i.e.,

						-1 {∂R} {∂∆v IN }	gives an evaluation of	{δ∆v int } {δ∆v IN }	, i.e.,	∂{∆v int } ∂{∆v IN }	.
	Moreover, noticing the constitutive equations Eqs. (5.77), (5.78) and (5.130), one
	can find that v v OUT = v OUT (v int , v IN ),	(5.191)
	which can be written in incremental form as
						∆v OUT = ∆v OUT (∆v int , ∆v IN ).	(5.192)
	The variation of ∆v OUT should follow
					δ∆v OUT =	∂∆v OUT ∂∆v int	δ∆v int +	∂∆v OUT ∂∆v IN	δ∆v IN .	(5.193)
	Substituting Eq. (5.190) into the previous equation leads to
	δ∆v OUT =	  	∂∆v OUT ∂∆v int	  -	∂R ∂∆v int	-1 ∂R ∂∆v IN	  +	∂∆v OUT ∂∆v IN	  	δ∆v I ,	(5.194)
	where	  	∂∆v OUT ∂∆v int	  -	∂R ∂∆v int	-1 ∂R ∂∆v IN	  +	∂∆v OUT ∂∆v IN	  	, denoted by {J * }, is the
	consistent tangent matrix. As can be seen, {J * } is calculated from four matrix:
	∂∆v OUT ∂∆v int numerically. Details concerning calculation of the matrix are given in appendix D. , ∂R ∂∆v int , ∂R ∂∆v IN and ∂∆v OUT ∂∆v IN . Note that the inverse of ∂R is obtained ∂∆v int

  Further studies are needed for this term in the second formulation especially with FEM. This is out of the scope of the present study.

	.227)
	which is the same as in Eq. (5.213).
	As the balance laws Eqs. (5.18) and (5.19) hold for the two formulations, one
	can obtain the same differential equation governing γ χ as Eq. (5.218). It means
	that the two formulations lead to the same solution for the special case of single
	slip. However, it is important to notice that the term A will come into play for the case with ∂γ χ = 0 in shear bands, e.g., pile-ups are ∂γ χ ∂X ∂γ χ in Eq. (5.225) ∂Y ∂X considered.

  [START_REF] Borg | Strain gradient crystal plasticity effects on flow localization[END_REF][START_REF] Borg | Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void[END_REF][START_REF] Borg | Size effects on void growth in single crystals with distributed voids[END_REF] for FE unit cell simulations with strain gradient plasticity, see Hussein et al., 2008 and Chang et al., 2015 for DDD simulations and see Zhao et al., 2009 for MD). In this section, unit cell simulations are carried out with the micromorphic model for predicting size effects on void growth and coalescence in single crystals.

[START_REF] Ling | An elastoviscoplastic model for porous single crystals at finite strains and its assessment based on unit cell simulations[END_REF]

and references therein). As void size decreases to micro-scale, size effects are expected in single crystals, which have been predicted by different approaches (e.g., see

[START_REF] Shu | Scale-dependent deformation of porous single crystals[END_REF]

  .236) Two external forces F 1 and F 2 are associated with U 1 (t) and U 2 (t). The prescribed U 2 is adjusted by one additional element for keeping a constant biaxiality ratio a = σ 22 /σ 11 during the loading, where the overall stresses σ 11 and σ 22 are defined by

Table 5 . 1 :

 51 Slip systems in FCC single crystals coordinate system. They are listed in table 5.1. In the simulation, one crystal orientation is taken into account. The crystal orientation is specified with the lattice directions [100]-[010]-[001] along the coordinate axes X-Y -Z. A cubic elasticity is considered and thus Λ ≈ can be specified by three independent moduli C 11 , C 12 and C 44 . The critical resolved shear stress τ s c of slip systems is determined by Eqs. (5.72) and (5.73). The material parameters used in the simulations are summarized in table 5.2. Note that a su and b su are 12 × 12 matrices and they can be specified by 6 independent parameters a 1 , a 2 , • • • , a 6 and b 1 , b 2 , • • • , b 6 (the structure of the matrices are presented in appendix A). The values of a su are obtained by discrete dislocation dynamics (Kubin et al., 2008; Monnet, 2009), while no study is found in the literature for estimating the values of b i

Table 5 . 2 :

 52 Material parameters for the unit cell simulations (see[START_REF] Han | Modélisation de la fragilisation due au gonflement dans les aciers inoxydables austénitiques irradiés[END_REF]

  , • • • , 12) have respectively 12 × 12 = 144 coefficients. For symmetry reasons, the number of coefficients is reduced to six, i.e., a i and bi with i = 1, 2, • • • ,6 Franciosi, 1985. a su is constructed as follows: 1 a 2 a 2 a 4 a 5 a 5 a 3 a 5 a 6 a 3 a 6 a 5 a 1 a 2 a 5 a 3 a 6 a 5 a 4 a 5 a 6 a 3 a 5 a 1 a 5 a 6 a 3 a 6 a 5 a 3 a 5 a 5 a 4 a 1 a 2 a 2 a 3 a 6 a 5 a 3 a 5 a 6 a 1 a 2 a 6 a 3 a 5 a 5 a 4 a 5 a 1 a 5 a 5 a 4 a 6 a 5 a 3 a 1 a 2 a 2 a 4 a 5 a 5 a 1 a 2 a 5 a 3 a 6 Symmetric a 1 a 5 a 6 a 3 a 1 a 2 a 2 a 1 a 2 a 1

		A2 A3 A6 B2 B4 B5 C1 C3 C5 D1 D4 D6
				A2
	[a su ] =	                       	                       	A3 A6 B2 B4 D6 D4 D1 C5 C3 C1 B5	.

a

  R E ∼ = ∆E ∼ -∆F ∼ .F ∼ -1 .E ∼ + E ∼ .

	• Derivatives of R E ∼ Finally,
		∂R γ s ∂∆E ∼	= -∆tΦ N ∼	s : (1 ∼ ⊗Π ∼ e ) +	1 2	12 s=1 (C ∼ e ⊗1 ∼ ) : Λ ∆γ s N ∼ s ≈	: (1 ∼ ⊗E ∼	(C.22) T + E ∼ T ⊗1 ∼ )
	∂R E ∼ ∂∆E ∼	∂R E ∼ ∂∆E ∼ ∂R γ s with	= 1 ≈ -(∆F ∼ .F ∼ Φ = -1 )⊗1 ≈ + 1 n K |τ s | -(τ s 12 c -S) ∆γ s N ∼ n-1 s ) T K ≈ ⊗( s=1	(C.34) (C.23) (C.35)
	∂R E ∼	∂∆γ p
	∂∆γ p								∂R E ∼ ∂∆γ p = E ∼ .N ∼ ∂R γ s ∂∆γ p = δ sp p	(C.24) (C.36)
	∂R E ∼	∂R γ s
	∂∆r q D ∂∆r q D				∂R γ s			∂R E ∼ D ∂∆r q	= 0	(C.25)
	∂R E ∼
	∂∆γ cum
											∂R E ∼ ∂∆γ cum	= 0	(C.26)
	• Derivatives of R γ ∂R γ s
	∂∆E ∼
		∂R γ s ∂∆E ∼	= -	∂Φ ∂τ s	∂τ s ∂M ∼	:	∂M ∼ ∂C ∼ e :	∂C ∼ ∂E ∼ e	:	∂E ∼ ∂∆E ∼	sign (τ s ) ∆t	(C.28)
	with
	∂Φ ∂τ s =	∂Φ ∂|τ s |	∂|τ s | ∂τ s =	n K		|τ s | -(τ s c -S) K	n-1	sign (τ s ) = Φ sign (τ s )
												(C.29)
	               ∂ C ∼ = N ∼ s e = ∂τ s ∂M ∼ ∂M ∼ ∂C ∼ e ∂C ∼ = 1 ∼ ⊗E ∼ ∂E ∼ ∂E ∼ ∂∆E ∼ = 1 ≈	∂R E ∼ ∂∆E ∼ ∂R γ s ∂∆E ∼ ∂R r s D ∂∆E ∼ ∂R γcum ∂∆E ∼ e . Λ ≈ : 1 2 (C ∼ ∂R γcum ∂R E ∼ ∂∆γ p ∂R γ s ∂∆γ p ∂R r s D ∂∆γ p ∂∆γ p e -1) ∂C ∼ e T + E ∼ T ⊗1 ∼	∂R E ∼ ∂∆r q D ∂R γ s ∂∆r q D ∂R r s D ∂∆r q D ∂R γcum ∂∆r q D = (1 ∼ ⊗Π ∼ eT ) + ∂R E ∼ ∂∆γ cum ∂R γ s ∂∆γ cum ∂R r s D ∂∆γ cum ∂R γcum ∂∆γ cum 1 (C ∼      e ⊗1 ∼ ) : Λ ≈ 2          	(C.30) (C.31) (C.21) (C.32) (C.33)

s R γ s = ∆γ s -Φ (|τ s | -(τ s c -S)) sign (τ s ) ∆t (C.27)

  [START_REF] Bailat | The effects of irradiation and testing temperature on tensile behaviour of stainless steels[END_REF]

	∂R γ s ∂∆K The matrix	∂v O ∂v I	is organized as
	• Derivatives of R r s D	∂R γ s ∂S ∼ ∂∆K ∂S ∼ = 0 D -|∆γ s |  D = ∆r s R r s   ∂v O ∂v I =    ∂γ χ ∂F ∼ ∂S ∂S       ∂γ χ ∂F ∼ ∂M ∂M b su r u ∂M ∂S ∼ ∂K ∂S ∂K D κ -G c r s D ∂F ∼ ∂γ χ ∂K	 	(D.16) (D.17)
	∂R r s
	∂∆F ∼
						∂R r s ∂∆F ∼	= 0	(D.18)
	∂R r s ∂∆γ χ	∂R E ∼ ∂∆γ χ
						∂R r s ∂∆γ χ	∂R E ∼ ∂∆γ χ = 0	= 0	(D.19)	(D.9)
	∂R r s	∂R E ∼
	∂∆K	∂∆K
						∂R r s ∂∆K	∂R E ∼ ∂∆K = 0	= 0	(D.10) (D.20)
	• Derivatives of R γcum • Derivatives of R γ ∂R γ s ∂R γcum	R γcum = ∆γ cum -	12 s=1	|∆γ s |	(D.21)
	∂∆F ∼ ∂∆F ∼
						∂R γ s ∂∆F ∼ ∂R γcum ∂∆F ∼ = 0	= 0	(D.12) (D.22)
	∂R γ s ∂R γcum ∂∆γ χ ∂∆γ χ
						∂R γ s ∂∆γ χ	= -= -Φ H χ sign (τ s ) ∆t ∂Φ ∂S ∂S ∂γ χ ∂γ χ ∂∆γ χ sign (τ s ) ∆t ∂R γcum ∂∆γ χ = 0	(D.13) (D.14) (D.23)
	∂R γcum
	Finally, ∂∆K
						∂R γ s ∂∆γ χ ∂R γcum = -Φ H χ sign (τ s ) ∆t ∂∆K = 0	(D.15) (D.24)

s R γ s = ∆γ s -Φ (|τ s | -(τ s c -S)) sign (τ s ) ∆t (D.11)

  ∼ .Π ∼ e )⊗1 ∼ ] : (1 ∼ ⊗1 ∼ ) (D.[START_REF] Edwards | Influence of irradiation temperature and dose gradients on the microstructural evolution in neutron-irradiated 316SS[END_REF]

	∂S ∂S	∂S ∼
	∂γ χ ∂S ∂K • Derivatives of M ∂M ∂F ∼ ∂r s ∂E ∼ ∂S ∂γ cum • Derivatives of M ∂M ∂E ∼ ∂M ∂γ s	∂S ∂γ χ ∂S ∂K M = AK = H χ = 0 ∂M ∂F ∼ = 0 ∂S ∼ ∂σ ∼ : ∂σ ∼ ∂S ∂r s = 0 ∂E ∼ = J1 ∼ ⊗F ∼ = -1 = -1 J e (E ∼ .Π ∼ e .E ∼ T ) ⊗ E ∼ ∂S = -H χ ∂γ cum + 1 J e (E ∼ ⊗E ∼ ) : ∂Π ∼ e ∂E ∼ + -T + 1 J e M = AK [(E ∂Π ∼ ∂S ∼ ∂E ∼ ∂S ∼ ∂σ ∼ ∂σ ∼ ∂E ∼ 1 J e e ∂E ∼ = ∂Π ∼ e ∂E ∼ e G L : ∂E ∼ e G L ∂E ∼ ∂Π ∼ e ∂E ∼ e G L = Λ ≈ ∂E ∼ ∂M = 0 ∂E ∼ e G L ∂E ∼ = 1 (1 ∼ ⊗E ∼ T + E ∼ T ⊗1 ∼ ) 2	1 ∼ ⊗(Π ∼ e .E ∼	(D.34) (D.35) (D.36) (D.37) (D.54) (D.42) (D.43) T ) T (D.55) (D.56) (D.45) (D.46) (D.57) (D.47)
	∂M ∂γ χ	∂S ∼ ∂γ s			∂M ∂γ s = 0	(D.58)
	∂M ∂r s				∂M ∂γ χ	= 0 ∂S ∼ ∂γ s = 0	(D.38) (D.48)
	∂M ∂K	∂S ∼ ∂r s			∂M ∂r s = 0	(D.59)
	∂M ∂γ cum			∂M ∂K	= A ∂S ∼ ∂r s = 0	29) (D.49) (D.39)
	The matrix	∂S ∼ ∂γ χ ∂v O ∂v int ∂S ∼ ∂γ cum	 is organized as ∂S ∼ ∂S ∼	∂S ∼ ∂γ χ ∂S ∼ ∂γ cum = 0 ∂S ∼ ∂S ∼ = 0 ∂M ∂γ cum = 0		(D.30) (D.50) (D.60)
	∂S ∼ ∂K ∂v O ∂v int • Derivatives of S =          ∂S ∂E ∼ • Derivatives of S The blocks are calculated in the following. ∂K ∂S ∼ ∂E ∼ ∂γ s ∂r s ∂γ cum = 0 ∂S ∂E ∼ ∂S ∂γ s ∂S ∂r s ∂S ∂γ cum ∂M ∂E ∼ ∂M ∂γ s ∂M ∂M S = H χ (γ χ -γ cum )          ∂r s ∂γ cum S = H χ (γ χ -γ cum ) ∂S ∂E ∼ = 0	(D.31) (D.51) (D.40) (D.32) (D.52)
		∂S ∂S
		∂F ∼ ∂γ s
						∂S ∂F ∼ ∂S ∂γ s = 0 = 0	-T	(D.33) (D.41) (D.53)

• Derivatives of S

∼ S ∼ = Jσ ∼ .F ∼ -T = J J e E ∼ .   Λ ≈ 2 (E ∼ T .E ∼ -1 ∼ )   .E ∼ T .F ∼

  RésuméL'irradiation peut modifier les propriétés mécaniques des aciers inoxydables austénitiques. Une diminution de la ténacité à la rupture des aciers en fonction de la dose est observée. La rupture ductile due à la croissance et la coalescence des cavités est toujours un mécanisme dominant dans les aciers irradiés jusqu'à 10 dpa. Des cavités peuvent être créées de manière différente: nucléées à partir des inclusions ou des précipités d'irradiation, ou créées directement par irradiation. Cette thèse a pour objectif d'étudier la rupture ductile des aciers irradiés due à la croissance et la coalescence des cavités intragranulaires. Basées sur la plasticité cristalline, des simulations en éléments finis sont effectuées sur les cellules unitaires pour étudier l'effet de l'orientation cristallographique et de la triaxialité de contraintes sur la croissance et la coalescence des cavités. L'effet de l'écrouissage post-irradiation sur la croissance et la coalescence des cavités est étudié avec un modèle de la plasticité cristalline prenant compte des défauts d'irradiation. En outre, un modèle élasto-visco-plastique en grandes transformations est proposé pour décrire la croissance des cavités dans le monocristal. Le modèle est appliqué à la simulation de l'endommagement ductile dans le monocristal et le polycristal. Des cavités peuvent avoir des tailles différentes et la taille peut avoir une influence sur la ténacité à la rupture des aciers. Afin d'étudier cet effet, un modèle micromorphe de plasticité cristalline est proposé et appliqué à la simulation de la croissance et la coalescence des cavités intragranulaires de différentes tailles ainsi qu'aux phénomènes de localisation dans les monocristaux.

	Mots Clés
	rupture ductile, acier CFC, irradiation, cav-
	ités intragranulaires, plasticité cristalline,
	plasticité à gradient

In the Toupin-Mindlin theory, the yield strength of the solid is taken to depend upon strain and second gradient of displacement.

In the couple stress theory, the yield strength of the solid is taken to depend upon strain and curvature (spatial gradient of the material rotation).

Director vectors are the notion that[START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF] used for define the plastic media. According to Mandel, the plastic deformation is the result of the discontinuity of displacement or rotation between micro-elements which constitute the macroscopic media. Within one micro-element, there is no discontinuity. In order to describe movement of a micro-element, a triplet of director vectors d k (k = 1,

[START_REF] Aifantis | The physics of plastic deformation[END_REF][START_REF] Anoukou | Limit analysis and homogenization of porous materials with Mohr-Coulomb matrix. Part I: Theoretical formulation[END_REF] is attached to the center of each micro-element, whose position indicates the orientation of the micro-element.

A more complex equation taking into account the effect of voids on hardening can be derived as proposed in[START_REF] Tanguy | Dislocations and irradiation defects-based micromechanical modelling for neutron irradiated austenitic stainless steels[END_REF]. However, considering the dose range of interest in this study, the effect of voids on hardening is considered negligible.

√ 6

See the footnote in page 56

Acknowledgements

Details on implementation in FE code

In order to facilitate the numerical implementation in FE code, we rewrite the previous equations in vector and matrix form. The rates of nodal degrees of freedom ˙ u a and ˙ γ b χ are arranged in vector form as

Appendices