Lucas Prevost

Amaury Fourgeaud

Pierre-Brice Bintein

de beaucoup apprendre durant cette thèse et de conrmer mon intérêt pour la recherche académique. Sylvain Patinet a su me transmettre sa passion pour la recherche et m'a fait découvrir un domaine de recherche passionnant.

Introduction

The research of high yield strength materials is an old but still very active research topic. Indeed, such materials would allow to have smaller and lighter structures performing elastically. In that aspect, metallic glasses constitute a very promising class of materials. First synthesized in the 60's [START_REF] Klement | Non-crystalline Structure in Solidied GoldSilicon Alloys[END_REF][START_REF] Inoue | Recent development and application products of bulk glassy alloys[END_REF], they have the advantage of having a very high yield strength. Even though their toughness is much higher than their conventional silicate cousins [START_REF] Marios | A damage-tolerant glass[END_REF], they were, however, found to be also brittle.

Metallic glasses are obtained by the rapid cooling of a liquid, quickly enough to avoid crystallization [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF]. Metallic glasses are thus solid in a non-equilibrated state. As such, their properties depend on their thermal and mechanical history.

For instance, it was found that increasing the cooling rate used to obtain a metallic glass changes its plastic behavior and can induce a brittle to ductile transition [START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF][START_REF] Li | Eect of cooling rate on plastic deformation of Zr-based bulk metallic glasses[END_REF]. Similarly, deforming a metallic glass beyond its elastic limit was found to induce an anisotropy in the mechanical response upon unloading, the so-called Bauschinger eect [START_REF] Bauschinger | Über die Veränderung der Elastizitätsgrenze und der Festigkeit des Eisens und Stahls durch Strecken und Quetschen, durch Erwärmen und Abkühlen und durch oftmals wiederholte Beanspruchung[END_REF][START_REF] Asaro | Mechanics of Solids and Materials[END_REF].

Furthermore, the size of the system has also a strong inuence in the plasticity of metallic glasses. Indeed, by reducing the size of the sample, a brittle to ductile transition was evidenced [START_REF] Qu | Size-dependent failure of the strongest bulk metallic glass[END_REF]. In metallic glasses, the origin of the brittleness was identied to lie in the formation of shear bands [START_REF] Jörg | Bulk metallic glasses[END_REF].

These shear bands appear to stem from of the localization of the plastic deformation in a persistent band of small width compared to the size of the system. Beyond the specic case of metallic glasses, shear bands can be found in many materials and in most amorphous solids [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. The amorphous solids refer to a wide variety of materials at very dierent scales characterized by their disordered structure [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF]. It goes from metallic glasses whose disorder lies at the atomic scale to colloidal glasses, foams or even granular solids at the macroscopic scale. Despite being made from very dierent constituents and at very dierent scales, the plastic deformation of these dierent amorphous solids share similar phenomenology.

Because of their disordered structure, the full understanding of the mechanical behavior and the formation of amorphous solids remain a highly debated topic. In particular, their lack of structure makes it dicult to study their plastic behavior. Indeed, unlike crystal alloys, amorphous solids do not have a regular structure on a lattice and do not deform plastically through dislocation mechanism. A dislocation is a defect in a crystalline structure. Its motion is the elementary mechanism behind plasticity in crystals. In amorphous solids, an equivalent mechanism is much more complicated to nd as their disordered aspect made it dicult to observe and quantify the plasticity at the microscopic scale. In the 70's, Argon [START_REF] Argon | Plastic deformation in metallic glasses[END_REF] proposed the concept of shear transformation (ST) as the elementary mechanism of plasticity in amorphous solids. It consists in the local rearrangement of a group of few dozens of particles.

The comprehension of the elementary process behind plasticity in amorphous solids helped to understand experiments at larger scale [START_REF] Amon | Hot Spots in an Athermal System[END_REF][START_REF] Le Bouil | Emergence of Cooperativity in Plasticity of Soft Glassy Materials[END_REF][START_REF] Schoenholz | A structural approach to relaxation in glassy liquids[END_REF]. It also contributed to the creation of various models to understand plasticity and the mechanical behavior of amorphous solids [START_REF] Falk | Dynamics of viscoplastic deformation in amorphous solids[END_REF][START_REF] Falk | Simulating the mechanical response of amorphous solids using atomistic methods[END_REF].

To study the plasticity and the mechanical response in amorphous materials, many numerical simulations studies were carried out in parallel to experimental approaches [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. These simulations can be performed at very dierent scales: the atomic scale, the mesoscopic scale and the macroscopic scale, each having their own advantages and limitations.

In simulations at the atomic scale, the particles are linked by an interaction potential. By tuning this potential, dierent amorphous solids can be modeled and most of the phenomenology of the plastic behaviors can be reproduced [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF].

In these simulations, plasticity can then be studied while having a direct access to the structure.

Understanding plasticity and shear banding in metallic glasses is a problem at the interface between mechanics and physics. In this thesis, we will use a physicist approach to study this problem. To do so, we will perform atomistic simulations with a simple model inter-atomic potential: the two-dimensional binary Lennard-Jones potential [START_REF] Shi | Strain Localization and Percolation of Stable Structure in Amorphous Solids[END_REF]. This commonly used potential can describe dierent amorphous solids such as metallic or colloidal glasses.

By using a simple model potential, we hope to understand the main structural reasons behind the generic plastic behaviors that can be observed in many amorphous solids.

To link the local structural details with plasticity in atomistic simulations of amorphous solids, a wide variety of local structural indicators were developed [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF]. In this thesis, we will use a novel structural indicator called the local yield stress [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF]. This indicator measures at which stress a local region will plastify when sheared in a given direction. From this indicator, we will try to have a better understanding of the inuence of the thermal and mechanical history on the structure. We will also apply this indicator to study shear banding in amorphous systems.

In a rst chapter, a bibliography review of the main elements for the comprehension of this manuscript will be presented. The rst section of this chapter will be dedicated to the description of the studied system: the metallic glasses.

Then in a second section, the inuence of the thermal and mechanical history of the metallic glass on the plastic behavior will be detailed. The inuence of the system size will also be addressed. Moreover, a brief overview of the dierent numerical approaches to study plasticity in amorphous solids will be given.

Finally, the third section will cover the shear bands denition, their properties and the mechanisms at play explaining their formation.

The second chapter will be dedicated to the description of the model system we will be studying and of the method used to simulate them. We will then explain how we deform these systems and probe their local properties. A particular attention will be paid to the description of the local yield stress, a key local structural indicator that will be used all along this manuscript.

In the third chapter, we will apply the local yield stress to study the inuence of the thermal history on the structure. This chapter will also concentrate on the link between plasticity and the initial local yield stress eld. Finally, the inuence of the parameters used in the measure of the local yield stress will be discussed.

In the fourth chapter, the local yield stress will be used to study the impact of plastic events on the local structure. We will also use the tensorial aspect of the local yield stress to study the anisotropy induced by plasticity of the shear transformation zones.

Then, in the fth chapter, we will use the local yield stress to understand how plasticity can induce an anisotropy in the mechanical response. In other words, we will study the origin of the Bauschinger eect in model amorphous solids through the measure of the local yield stress. We will also develop a model to reproduce the Bauschinger eect based on the local yield stress distributions.

Finally, in the sixth chapter, we will in a rst part study the inuence of the thermal history and of the system size on the heterogeneity and the persistence of plasticity in model glasses. Then, in a second part, we will try to understand the location of the shear band through the measure of the local yield stress.

The last part will be dedicated to the evolution of the shear band width upon deformation.

This thesis gave rise to various written works and communications: three peer-reviewed articles [START_REF] Barbot | Local yield stress statistics in model amorphous solids[END_REF][START_REF] Barbot | Rejuvenation and shear banding in model amorphous solids[END_REF][START_REF] Patinet | Origin of the Bauschinger Eect in Amorphous Solids[END_REF], an article in progress as well as participation to two international conferences.

Chapter 1

Plasticity and shear band in amorphous solids

The discovery of metallic glasses and the emergence of the study of soft matters in the last decades have lead to a growing interest for the study of their mechanical properties. In particular, the complex phenomenology of plasticity in glass has attracted the interest of physicists. Rather than an exhaustive review of this rapidly growing eld, we will summarize in the following chapter a short selection of the important results which should make easier the reading of this manuscript.

The rst section of this chapter will be dedicated to the description of amorphous solids and in particular of metallic glasses. Then, the second section will concentrate on plasticity in amorphous solids. We will review the plastic behavior and phenomenology at dierent scales. The numerical methods used to study plasticity in amorphous solids will also be presented. Finally, the last section will be dedicated to shear bands, their properties and their causes of nucleation.

Amorphous solids and metallic glasses

This section is dedicated to the description of the amorphous solids and in particular of the metallic glasses. In a rst part, we will describe the dierent amorphous solids and their similarities. Then, the second part will cover a particular example of amorphous solids: the metallic glasses. Finally, the third part will give a brief overview of the glass transition and the associated characteristic temperature scales.

Amorphous solid description

Amorphous solid consists of any non crystalline solid [START_REF] Mahan | Amorphous solid[END_REF]. More precisely, it consists of materials which preserve their solid structure at rest, that can sustain a shear stress and whose constituents are not organized in a lattice. This term can refer to a lot of dierent materials at drastically dierent scales from sub-nanometric to macroscopic. Amorphous solids are composed of two classes [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF]: soft and hard.

Soft amorphous solids are materials mostly composed of liquids. While they remain solid at rest, they ow as a sucient load is applied. As soft amorphous solids, we nd liquid foams, composed of an amorphous structure of liquid bubbles or emulsions where liquid droplets are dispersed in a continuous liquid phase. We also nd colloidal glasses or gels which consist of a dense suspension of colloids (Nicolas et al. [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF]). These materials appear soft when manipulated, and have a relatively low shear moduli (10-100 Pa) compared to hard amorphous solids.

The hard amorphous solids on the other hand mostly consist of polymer, metallic and oxids glasses. These glasses are obtained by a rapid cooling of a glass forming liquid. This contrast with colloidal glasses which are obtained by a rapid increase of the colloids concentration in the suspension [START_REF] Yogesh | Dynamics of Colloidal Glasses and Gels[END_REF].

As we can see on gure 1.1, the hard amorphous solids components are smaller than for the soft ones (from 0.1 nm to 10-100 nm compared with 10 nm to macroscopic size). As the shear modulus increases when the components size reduces, the hard amorphous solids feature a higher shear modulus [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF] (around 10-100GPa for metallic glasses) [START_REF] Lewandowski | Intrinsic plasticity or brittleness of metallic glasses[END_REF]. damping regime of their elementary particles. At the bottom: some popular modeling approaches, arranged according to the length scales of the material for which they are originally developed. From [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF].

Despite drastic dierences in their component length scale and stiness, the hard and soft amorphous solids share many common properties. For instance, hard amorphous solids can ow to some extent without breaking [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF]. As being out of equilibrium solids, their mechanical properties depend on the history of the material. Also, they share similar elementary mechanisms of plasticity.

Due to these similarities, the models used to describe metallic glasses can also applied to colloidal glasses.

In this manuscript, we mostly focus on metallic glasses and study their plastic behavior. Yet, the plastic behaviors which will be observed in this manuscript are not necessarily specic to metallic glasses and can apply to other amorphous solids.

A particular example: the Metallic Glasses

Discovered in the year 1960 [START_REF] Klement | Non-crystalline Structure in Solidied GoldSilicon Alloys[END_REF][START_REF] Inoue | Recent development and application products of bulk glassy alloys[END_REF], the metallic glasses became quickly an active research topic due to their exceptional mechanical properties. Indeed, they have the advantage of possessing a higher yield strength compared with crystalline metals [START_REF] Ashby | Metallic glasses as structural materials[END_REF] (the highest yield strength known for metallic glasses in 5 GPa, compared to 3 GPa for crystalline metals [START_REF] Inoue | Cobalt-based bulk glassy alloy with ultrahigh strength and soft magnetic properties[END_REF]). Nevertheless, unlike silica, which is among the most common used glass former, metals can crystallize easily. As a glass is obtained by reducing a liquid temperature quickly enough to avoid crystallization, a very high cooling rate was required to form the rst metallic glass sample. The rst metallic glass sample was only 10µm in thickness, but it required a cooling rate of 10 5 -10 6 K/s.

The discovery of new glass forming alloys and new quench methods such as melt spinning or additive manufacturing [START_REF] Groza | Materials Processing Handbook[END_REF] allowed to reduce the cooling rate needed to form a metallic glass and thus allowed the creation of larger samples. Indeed, as shown on gure 1.2, in 1960, the critical sample thickness was only 10µm and forty years later some metallic glass samples were approaching 10cm [START_REF] Jörg | Bulk metallic glasses[END_REF]. As the samples size and quality increased, probing their mechanical properties became possible. Chen and Wang in 1970 [START_REF] Chen | Mechanical properties of metallic glasses of pdsingle bond signsi-based alloys[END_REF] were among the rsts to probe the mechanical properties of a melt quenched metallic glass [START_REF] Greer | Shear bands in metallic glasses[END_REF]. They observed a higher yield strength and at the same time a smaller Young modulus than the crystalline metals. While having a high yield strength, metallic glasses were found to be very brittle and so to have a low resistance to plastic deformation [START_REF] Greer | Shear bands in metallic glasses[END_REF]. This particularity of the metallic glasses can be observed in gure 1.3. In this gure, representing the yield strength and the fracture toughness for dierent materials, we can see that most of the metallic glasses (black crosses) have a higher yield strength than the crystalline metals (gray regions) while having a lower fracture toughness [START_REF] Marios | A damage-tolerant glass[END_REF]. We can also remark that oxide glasses have a high yield strength but a very low fracture toughness.

In 1971, Masumoto et al. [START_REF] Masumoto | The mechanical properties of palladium 20 a/o silicon alloy quenched from the liquid state[END_REF] studied the surface of bent metallic glasses.

They observed that instead of being homogeneously deformed, the deformation is concentrated in what they called deformation lines with a width of the order of 20nm. These deformation lines are now known as shear bands and will be studied in this manuscript.

Glass transition brief overview and associated temperatures

While being commonly used and manufactured for thousands of years, glasses are materials which physics and synthesis is still not fully understood. Glasses are solids in a non-equilibrium state obtained by cooling a liquid quickly enough to avoid its crystallization [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF]. As the system is cooled down below the melting point T m , it enters a supercooled liquid state whose viscosity and relaxation time scale diverges as the temperature is reduced. Below a certain temperature, commonly called the glass transition temperature T g , the material is locked in an amorphous metastable state called glass.

However, the determination of T g is empirical: it corresponds to the temperature after which we are not able to measure experimentally the relaxation time scale or the viscosity [START_REF] Frank | Glass Transition Thermodynamics and Kinetics[END_REF]. It is commonly xed at the temperature at which the supercooled liquid viscosity becomes larger than 10 13 kg •m -1 •s -1 or when it relaxation time is typically higher than 10 2 -10 3 s [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF]. Furthermore, T g is not unique and increases with the cooling rate. Indeed, changing the cooling rate changes the glass transition temperature as illustrated on gure 1.4. This introduces an important aspect of the glasses properties. As being out of equilibrium, their properties depend on their thermal history and on their mechanical history. For instance, a lower cooling rate creates a more brittle glass but with a higher yield strength [START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF][START_REF] Li | Eect of cooling rate on plastic deformation of Zr-based bulk metallic glasses[END_REF]. The inuence of the mechanical and thermal history on glasses will be discussed in a later section. T m or for higher cooling rate become a glass at the glass transition temperature T g . This temperature is cooling rate dependent, and a lower glass transition temperature corresponds to a lower cooling rate. From [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF].

As the supercooled liquid is cooled down, the entropy dierence between liquid state and the crystal state reduces. By extrapolating from experimental data, Kauzmann [START_REF] Walter | The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures[END_REF] found that the liquid and crystal state entropy are equal at a non zero temperature, called the Kauzmann Temperature T K , lower than T g .

While having the same entropy, the supercooled liquid state still has a higher enthalpy so the two phases remain structurally distinct. This gives rise to a paradox: if the approximation of the entropy of the liquid state is pushed further, this would lead to a negative entropy at T = 0 which violates thermodynamics third law. One of the theory to solve this paradox is the existence of a rst order phase transition at T K , below which the glass structure remains unchanged [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF].

A similar extrapolation of the behavior of the shear viscosity, of the mean shear-stress and thermal relaxation times, and of the self-diusion constant with the temperature concluded to a divergence of these quantities at a nite temperature close to T K , supporting further the idea of a glass transition at T K [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF].

While the dynamics slowdown is quantied, with a divergence at a nite temperature found by extrapolating the experimental results, the origin of this slowdown is still unclear. Two main categories of theories were developed to explain this behavior: those based on the thermodynamics and explaining that this slowdown is caused by an ideal glass transition; and those based on purely dynamical explanation.

One of the main theories based on thermodynamics is the Random First Order Transition Theory (RFOT) [START_REF] Lubchenko | Theory of Structural Glasses and Supercooled Liquids[END_REF][START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF]. By using mean eld results of spin glasses, it was shown that the Helmholtz Free energy of the described system possesses a metastable minimum at suciently low temperature or high density, corresponding to a rst order transition. It was also shown in this model that the divergence of the relaxation time and the vanishing of the entropy dierence happen at the same temperature, corresponding by denition to the Kauzmann temperature. One of the strength of this theory is also its capacity to predict results about the thermodynamics and kinetic of glass-forming liquids and having good agreement with experimental studies [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF].

On the other hand, without using thermodynamics and by only using the system density-dependent structure factor and its temperature, the Mode Coupling Theory (MCT) [START_REF] David R Reichman | Mode-coupling theory[END_REF] is able to predict the intermediate scattering function evolution with time. This theory also predicts a transition taking place at a nite temperature called Mode Coupling Temperature T M CT corresponding to the temperature at which the particles are locked by their neighbors for an innite time, causing the dynamics of the system to diverge.

As shown gure 1.5, the MCT predicts three main behaviors of the intermediate scattering function: (i) above the melting temperature represented by the curve A, the curve shows a small decay at short time due to inertial motion while having at longer time an exponential decay due to Brownian-motion. (ii)

In the supercooled liquid state (i.e. T < T m ) but above T M CT , the function exhibit an non-monotonic decay instead of an exponential one as shown by the curve B. In this system, the particles are temporary locked their neighbors for a time called the alpha relaxation time τ α . (iii) As the temperature of the system is further lowered below T M CT , the curve C shows an innite plateau caused by the caging of the particles for an innite time.

The Mode Coupling theory is able to predict and explain the intermediate scattering function above T M CT . However, the value of T M CT obtained by tting experimental data are higher than T g , while at T g the relaxation times have not diverged yet [START_REF] Frank | Glass Transition Thermodynamics and Kinetics[END_REF]. T M CT is nonetheless a good indicator of the tem- From [START_REF] Frank | Glass Transition Thermodynamics and Kinetics[END_REF].

perature from which the particles become strongly inuenced and slowed down by their neighbors. Below this temperature, the particles become inuenced by the potential energy landscape [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF].

Indeed, as explained in [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF], at low enough temperature, the particles are trapped in a local minimum of the potential energy landscape. In this limit, the structural relaxations correspond to local rearrangements thermally activated due to the crossing of a potential energy barrier. At high enough temperature, the thermal energy is too high for the system to be trapped in the potential energy landscape. The temperature separating these two regimes is often identied as the mode coupling temperature T M CT [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

As the thermal activation of local rearrangements is not taken into account by most of the mode-coupling theories [START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF], they predict a divergence of the relaxation times at a nite temperature lower than T g which is not the case in experiments.

In conclusion, glasses are solids in an out-of-equilibrium state and their properties are thus inuenced by their mechanical and thermal history. Many attempts to nd a temperature separating the liquid to the glassy state were done.

Among them we can note the mode-coupling temperature which is a good indicator to know if a liquid is inuenced by its potential energy landscape.

In this manuscript, we study how the plastic behavior of glasses is inuenced by their thermal and mechanical history by studying glasses obtained from different parent temperatures and cooling rates (see section 2.2.2). Particular attention is paid around the mode coupling T M CT cross-over temperature range below which the dynamics become inuenced by the potential energy landscape.

Plasticity in amorphous solids

In the previous section, a description of amorphous solids and more particularly metallic glasses was given. This section will now focus on plasticity in these systems.

Plasticity is the ability of certain solids to ow or to change shape permanently when subjected to stresses of intermediate magnitude between those producing reversible deformation, or elastic behavior, and those causing failure of the material [START_REF]The Editors of Encyclopaedia Britannica[END_REF].

In this section, the rst part will cover the plastic behavior of metallic glasses.

More specically, we will rst detail how the plastic behavior is inuenced by the system size and the thermal history of the material. Then, the inuence of the mechanical history will be discussed. The second part will discuss the phenomenology of plasticity in amorphous solids at the microscopic scale. The third part will explain the dierent approaches to study plasticity in amorphous solids through numerical simulations. Finally, the fourth part will give an overview of the dierent structural indicators developed to link structure and plasticity in atomistic simulations of amorphous solids.

Plastic behavior in metallic glasses

1.2.1.1 How system preparation and size inuence plastic behavior

In this part, the question of how plasticity in metallic glasses is inuenced by dierent factors will be addressed. More precisely, we will focus on the inuence of the system size and of the system preparation. Many studies discussed the inuence of temperature on the plastic behavior [START_REF] Raghavan | On factors inuencing the ductile-to-brittle transition in a bulk metallic glass[END_REF][START_REF] Lu | Deformation behavior of the Zr41.2Ti13.8Cu12.5Ni10Be22.5 bulk metallic glass over a wide range of strain-rates and temperatures[END_REF]. As our study will be done at zero temperature, the inuence of the temperature will not be covered here.

Changing the system size for a metallic glass inuences its plastic response [START_REF] Qu | Size-dependent failure of the strongest bulk metallic glass[END_REF]. Many experimental works study this phenomena by compressing a rod composed of metallic glass with dierent diameters [START_REF] Yang | Size eect on stability of shear-band propagation in bulk metallic glasses: an overview[END_REF] or by bending metallic glass ribbons or plates of dierent thickness [START_REF] Conner | Shear bands and cracking of metallic glass plates in bending[END_REF]. From these studies, it is shown that a smaller system exhibit a higher yield strength [START_REF] Wang | Sample size matters for Al88Fe7Gd5 metallic glass: Smaller is stronger[END_REF][START_REF] Ghidelli | Homogeneous ow and size dependent mechanical behavior in highly ductile Zr65Ni35 metallic glass lms[END_REF], and also a higher plastic strain to failure [START_REF] Conner | Shear bands and cracking of metallic glass plates in bending[END_REF][START_REF] Jang | Eects of size on the strength and deformation mechanism in Zr-based metallic glasses[END_REF]. The gure 1.6 for instance shows the yield strength of various metallic glasses as a function of the system size. A clear increase of the yield strength as the system get smaller can be observed from this curve.

For metallic glass rods, when the system diameter is larger than a critical size, unstable shear bands appear and the system fails at small strain [START_REF] Conner | Shear bands and cracking of metallic glass plates in bending[END_REF][START_REF] Jang | Eects of size on the strength and deformation mechanism in Zr-based metallic glasses[END_REF].

At this critical diameter occurs a brittle to ductile transition [START_REF] Yang | Size eect on stability of shear-band propagation in bulk metallic glasses: an overview[END_REF]. This critical diameter is estimated to be around 500nm for a P d 40 Cu 30 N i 10 P 20 glass [START_REF] Yi | Sample size and preparation eects on the tensile ductility of Pd-based metallic glass nanowires[END_REF] and around 400nm for a P d 77 Si 23 glass [START_REF] Volkert | Eect of sample size on deformation in amorphous metals[END_REF]. Below this size, in some systems, the plastic deformation is found to be homogeneous and no initiation of shear bands is observed [START_REF] Yi | Sample size and preparation eects on the tensile ductility of Pd-based metallic glass nanowires[END_REF][START_REF] Volkert | Eect of sample size on deformation in amorphous metals[END_REF][START_REF] Jang | Transition from a strong-yet-brittle to a stronger-and-ductile state by size reduction of metallic glasses[END_REF]. In other systems, this brittle to ductile transition when reducing the system size is associated to a transition from an unstable Figure 1.6 Metallic glass strength dependency on the system size. Smaller systems show a higher yield strength. From [START_REF] Wang | Sample size matters for Al88Fe7Gd5 metallic glass: Smaller is stronger[END_REF].

shear banding regime to a stable one [START_REF] Yang | Characteristic length scales governing plasticity/brittleness of bulk metallic glasses at ambient temperature[END_REF]. In the stable shear banding regime, the material still exhibits a shear band upon loading but the material does not fail immediately and shows a ductile behavior. Finally, similar results are obtained for indentation experiments in metallic glasses: below a critical size of the indenter, a change the plastic behavior is measured [START_REF] Wang | Softeninginduced plastic ow instability and indentation size eect in metallic glass[END_REF].

For metallic glass ribbons or plates on the other hand, when the thickness becomes smaller, the bending ductility increases and the spacing between shear bands decreases [START_REF] Conner | Shear bands and cracking of metallic glass plates in bending[END_REF]. Indeed, the shear band spacing are found to be proportional to the ribbons thickness [START_REF] Conner | Shear band spacing under bending of Zr-based metallic glass plates[END_REF]. Thus, for ribbons with the same length but dierent thicknesses, the smaller ribbons have more shear bands in which the deformation concentrates. Therefore, a bigger applied strain is needed before one of the band spawn the entire system size [START_REF] Ma | Controlling plastic instability[END_REF]. As a consequence, the thicker plates are more brittle [START_REF] Schuh | Mechanical behavior of amorphous alloys[END_REF].

The metallic glass preparation, or more precisely here its quench protocol, changes the plastic response of the material. Indeed, for metallic glasses with the same composition, those quenched with a high cooling rate will be more ductile compared to those obtained with a slower cooling rate [START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF][START_REF] Li | Eect of cooling rate on plastic deformation of Zr-based bulk metallic glasses[END_REF].

On gure 1.7 from [START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF], we can see the bending strain to failure as a function of the normalized ctive temperature T f /T g with T g the glass transition temperature for dierent metallic glasses. The lower the ctive temperature, the slower the cooling rate. We can observe in this gure that the higher the cooling rate, the higher the strain to failure. The ductility thus increase with the cooling rate. A brittle to ductile transition is also observed at low enough parent temperature [START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF].

Systems obtained with a slow cooling have a higher yield strength and elastic modulus [START_REF] Yue | Eect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: A molecular-dynamics study[END_REF]. They also have less free volume and more structural order [START_REF] Yue | Eect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: A molecular-dynamics study[END_REF][START_REF] Huang | The eect of cooling rate on the wear performance of a ZrCuAlAg bulk metallic glass[END_REF]. Furthermore, the higher the cooling rate, the more shear bands tend to nucleate in the system [START_REF] Shen | Plasticity of a TiCu-based bulk metallic glass: Eect of cooling rate[END_REF][START_REF] Li | Eect of cooling rate on plastic deformation of Zr-based bulk metallic glasses[END_REF][START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF].

Similarly with the metallic ribbon with a low thickness, a metallic glass obtained with a high cooling rate shows many shear bands in which the deformation concentrates. Therefore, the applied strain at which it fails tend to be higher compared to a system obtained with a lower cooling rate.

Plasticity dependence on mechanical history : the Bauschinger eect

In the previous part, we saw how the plastic behavior of metallic glasses depends on their thermal history and on their size. In this part, we will see how the plastic behavior depends on the mechanical history of the system by focusing on a particular example, the Bauschinger eect.

The Bauschinger eect [START_REF] Bauschinger | Über die Veränderung der Elastizitätsgrenze und der Festigkeit des Eisens und Stahls durch Strecken und Quetschen, durch Erwärmen und Abkühlen und durch oftmals wiederholte Beanspruchung[END_REF] is the remarkably common property that after experiencing plastic strain, materials generally exhibit a softer stress response under reverse loading as compared with reloading [START_REF] Patinet | Origin of the Bauschinger Eect in Amorphous Solids[END_REF]. While this phenomenon was rst observed in metal crystals [START_REF] Asaro | Mechanics of Solids and Materials[END_REF], it was later observed in a lot of materials such as polymers [START_REF] Senden | Strain hardening and its relation to Bauschinger eects in oriented polymers[END_REF] or amorphous solids, including metallic glasses [START_REF] Deng | Simulation of Plastic Deformation in a Two-Dimensional Atomic Glass by Molecular Dynamics IV[END_REF][START_REF] Karmakar | Plasticityinduced anisotropy in amorphous solids: The Bauschinger eect[END_REF][START_REF] Frahsa | On the Bauschinger eect in supercooled melts under shear: Results from mode coupling theory and molecular dynamics simulations[END_REF][START_REF] Hao Sun | Pure shear stress reversal on a Cu-based bulk metallic glass reveals a Bauschinger-type eect[END_REF]. It is observed in almost all material classes.

In the specic case of amorphous solids, the Bauschinger eect is observed experimentally [START_REF] Hao Sun | Pure shear stress reversal on a Cu-based bulk metallic glass reveals a Bauschinger-type eect[END_REF] and also reproduced through atomistic simulations [START_REF] Karmakar | Plasticityinduced anisotropy in amorphous solids: The Bauschinger eect[END_REF] as seen on gure 1.8. In this gure, we can observe on the left panel that as the directions from the as-quenched state, the mechanical response in the stressstrain curve remains the same. In the right handside, the same deformation protocol is performed, but starting from a system already loaded in the positive direction and then unloaded until zero stress at γ = γ 0 . An anisotropy can be observed as the material clearly exhibit softer stress response in the negative direction than the positive one. From [START_REF] Karmakar | Plasticityinduced anisotropy in amorphous solids: The Bauschinger eect[END_REF].

material is sheared on two opposite directions (the so-called positive and negative one) from the as-quenched state, the sample-averaged stress-strain curve, remains the same.

However, upon loading the samples in the positive direction, then unloading it down to zero-stress at γ = γ 0 , an anisotropy is observed in the mechanical response between the positive and the negative loading. Indeed, the material clearly exhibit softer stress response in the negative direction than the positive one. This is a direct manifestation of the Bauschinger eect.

While the Bauschinger eect can be observed and reproduced in many amorphous systems, the origin of this phenomena remains unclear. In a later chapter, we study the Bauschinger eect in numerical simulations of metallic glasses through the measure of their local structural properties.

Phenomenology at microscopic scale

In the previous section, the plastic behavior on metallic glasses was detailed.

This section will now focus on the phenomenology of plasticity in amorphous solids at the scales of the particles.

In crystals, the atoms are organized on a lattice and have a well dened structure. In these materials, regions more susceptible to deform plastically corresponds to defects in the structure called dislocations. As the material is loaded, these dislocations move along well-dened glide planes, thus causing an irreversible deformation as seen gure 1.9.

Figure 1.9 Schematic example of plasticity in crystals. The material under stress deforms plastically from the migration of a dislocation along specic planes. From [64].

In amorphous solids such as metallic glasses, the materials do not possess a crystalline order from which defects can be dened [START_REF] Greer | Shear bands in metallic glasses[END_REF]. Yet these materials can ow plastically [START_REF] Zhang | Stable owing of localized shear bands in soft bulk metallic glasses[END_REF]. To dene plasticity in amorphous structures, Argon [START_REF] Argon | Plastic deformation in metallic glasses[END_REF] proposed the concept of Shear Transformations (STs) which corresponds to local irreversible rearrangement of atoms (see gure 1.10). This elementary process is localized both in space (few dozens of atoms) and in time (a few picoseconds) [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. The STs are now recognized as the main elementary mechanism of plasticity and, as a such, are widely employed in for modeling amorphous plasticity. [START_REF] Greer | Shear bands in metallic glasses[END_REF]. Unlike crystals where the plasticity originates from defects in the structure, understanding the location of a ST in an amorphous material is much more challenging. In a parallel to these localized weaker regions, M.J. Falk proposed the principle of Shear Transformation Zones (STZ) [START_REF] Falk | Dynamics of viscoplastic deformation in amorphous solids[END_REF][START_REF] Michael | Deformation and Failure of Amorphous, Solidlike Materials[END_REF]. These regions, composed of ve to ten atoms, are more susceptible to deform inelastically when submitted to shear stress in a given direction. When the material is loaded in another direction, the STZ susceptible to be activated may change [START_REF] Falk | Dynamics of viscoplastic deformation in amorphous solids[END_REF].

The atoms within an ST undergo important displacement as can be seen in gure 1.10. As a consequence, the shape of this group of atoms changes. The incompatibility of the associated plastic strain induces stress in the surrounding material. To understand how this local reorganization of atoms inuence the surrounding system, Argon proposed to model it as a plastic inclusion within an elastic matrix [START_REF] Argon | Plastic deformation in metallic glasses[END_REF] based on a theory develloped by Eshelby [START_REF] Douglas | The determination of the elastic eld of an ellipsoidal inclusion, and related problems[END_REF]. By solving this model system, one can obtain the strain eld caused by the plastic inclusion on the surrounding system, as seen gure 1.11. This quadrupolar shape, associated with the Eshelby inclusion, obtained here analytically can be measured experimentally in colloidal glasses [START_REF] Hufnagel | Deformation of metallic glasses: Recent developments in theory, simulations, and experiments[END_REF] and directly observed in atomistic simulations of metallic glasses by calculating the displacement eld caused by a ST, as seen gure 1.12. Indeed, the atoms surrounding the location of the plastic event are still elastic and are displaced quasi-linearly by the stress eld induced by the plastic rearrangement.

This stress eld is now commonly used to describe the inuence of the ST in mesoscopic models [START_REF] Tyukodi | Finite-size eects in a model for plasticity of amorphous composites[END_REF][START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF].

Simulation of the mechanical response and plasticity of amorphous solids

The previous sections presented the plastic behavior metallic glasses and the phenomenology of plasticity at the scale of the particles. In the current section, we will review the dierent approaches performed to study plasticity and the mechanical response of amorphous solids through numerical simulations.

The numerical approach to study amorphous solids is realized at dierent time scales and length scales [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. These length scales are shown on gure 1.13. . From these models one can extract constitutive laws to perform nite element simulation. It is important to note that the images serve as illustrations and correspond to simulations performed independently. From [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF].

These studies often consist of simulating a liquid, composed of atoms with pre-dened interactions, through molecular dynamics before quenching it to obtain an amorphous solid. The interactions can be simple model interaction to reproduce the phenomenology of amorphous plasticity and to retain the essential features the system while keeping the numerical cost moderate [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. These interactions can also be much more sophisticated and use the details of the quantum chemistry between the atoms [START_REF] Cheng | Atomic-level structure and structureproperty relationship in metallic glasses[END_REF]. These more complex interactions are often used to simulate specic systems [START_REF] Ganesh | Signature of nearly icosahedral structures in liquid and supercooled liquid copper[END_REF][START_REF] Gu | Mechanical properties, glass transition temperature, and bond enthalpy trends of high metalloid Fe-based bulk metallic glasses[END_REF].

These systems can then be deformed either quasi-statically, which in the low temperature limit corresponds to shearing at an innitely low shear rate [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF][START_REF] Singh | Brittle yielding of amorphous solids at nite shear rates[END_REF]. They can also be performed at nite shear rate and temperature through molecular dynamics. These atomistic simulations are widely used, either in two dimensions [START_REF] Shi | Strain Localization and Percolation of Stable Structure in Amorphous Solids[END_REF][START_REF] Regev | Onset of irreversibility and chaos in amorphous solids under periodic shear[END_REF][START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF][START_REF] Jana | Correlations of non-ane displacements in metallic glasses through the yield transition[END_REF] or in three dimensions [START_REF] Toxvaerd | Stability of supercooled binary liquid mixtures[END_REF][START_REF] Trond | Crystallization Instability in Glass-Forming Mixtures[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF][START_REF] Gan | Elucidating how correlated operation of shear transformation zones leads to shear localization and fracture in metallic glasses: Tensile tests on Cu Zr based metallic-glass microwires, molecular dynamics simulations, and modelling[END_REF], to study mechanical properties and plasticity in amorphous solids and link it with the local structure.

Yet, these simulations require quite heavy computations and are at very limited length scale and time scale (few nanometers and few tens of nanoseconds typically [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]) which makes dicult the direct comparison with experiments.

To perform simulation at larger length scales while taking into account the heterogeneity of the local properties of the material, mesoscopic approaches are developed (shown on gure 1.13 (b)). As explained in the previous section, plasticity in glasses takes the form of local atomistic rearrangements at the scale of few dozens of atoms called Shear Transformation (ST). The mesoscopic models average the atomistic eects over a length scale of the size of a ST to take into account only the dynamics of STs [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF].

As detailed in [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF], mesoscopic models require in general four elementary ingredients: (i) a local yield criterion to determine when a local plastic event occurs, (ii) the description of how the elastic matrix is aected by a plastic rearrangement, (iii) the details of how the local yield criterion is aected by a plastic event, as a plastic event alters locally the structure which could lead to local softening or hardening. Finally, (iv) a dynamical rule to associate a time scale to the elementary process.

Then, by changing the ingredients, dierent eects can be probed through mesoscopic simulations such as the inuence on the mechanical behavior or plasticity of inertia [START_REF] Karimi | Role of inertia in the rheology of amorphous systems: A nite-element-based elastoplastic model[END_REF], softening [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF][START_REF] Tüzes | Disorder is good for you: the inuence of local disorder on strain localization and ductility of strain softening materials[END_REF][START_REF] Popovi¢ | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF] or times scales [START_REF] Martens | Spontaneous formation of permanent shear bands in a mesoscopic model of owing disordered matter[END_REF]. Mesoscopic simulations are able to reproduce many plastic behaviors observed experimentally and in atomistic simulations such as the shear banding (localization of the plastic deformation on a shear band) [START_REF] Martens | Spontaneous formation of permanent shear bands in a mesoscopic model of owing disordered matter[END_REF]. However, the comparison between mesoscopic and atomistic remains more qualitative. Indeed, the ingredients implemented in the mesoscopic models still rarely take into account the results obtained at the atomistic scale and are performed independently. Yet, as mentioned in recent reviews [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF], the transfer of information from one scale to another becomes more and more possible.

Finally, simulations at the macroscopic scale mostly uses the Finite Element Method [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. They thus require a constitutive law that relates the plastic strain rate to the state of stress and the history of deformation of the glass. This approach is mostly phenomenological [START_REF] Gao | An implicit nite element method for simulating inhomogeneous deformation and shear bands of amorphous alloys based on the free-volume model[END_REF][START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF].

Linking structural properties and plasticity

In the previous section, we saw that atomistic numerical simulations are used to study plasticity at the microscopic scale in amorphous solids. In this section, we will do a brief overview of the dierent types of existing local structural indicators used in atomistic simulations of amorphous solids to link plasticity and the local structural properties. The specicities of the indicator used in this thesis, the local yield stress [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF], will also be presented.

The question whether the location of plasticity of amorphous solids can be understood and predicted by the knowledge of the material local structure is still not fully answered. Indeed, some theories such as the Shear transformation Zones (STZ) [START_REF] Falk | Dynamics of viscoplastic deformation in amorphous solids[END_REF] previously introduced, or the Soft Glassy Rheology [START_REF] Sollich | Rheology of Soft Glassy Materials[END_REF][START_REF] Hébraud | Mode-Coupling Theory for the Pasty Rheology of Soft Glassy Materials[END_REF] are based on the idea that plasticity originates from local defect in the structure.

The search of the existence of structural defects was greatly helped by the development of atomistic numerical simulations of amorphous solids. Indeed, by giving a direct access to the material structure and by the knowledge of the potential linking the atoms, they allow the creation of wide variety of structural indicators [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF].

Among these indicators, some of the rst and most commonly used are the local density and the local free volume [START_REF] Starr | What Do We Learn from the Local Geometry of Glass-Forming Liquids[END_REF][START_REF] Steif | Strain localization in amorphous metals[END_REF]. Indeed, an increase of local free volume was measured experimentally after plasticity which led many theories of deformation in amorphous solids [START_REF] Spaepen | A microscopic mechanism for steady state inhomogeneous ow in metallic glasses[END_REF] to be based on these two conjugate quantities.

In a recent publication comparing systematically the propensity of many structural indicators to predict the location of the plastic events, these indicators were classied in ve main categories [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF].

The rst category (i) consists of the purely structural indicators, which are only based on the position of the particles and do not take into account the force eld between the atoms. Among them can be found the free volume, the local density (which both showed a poor correlation with plasticity [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF]) [START_REF] Starr | What Do We Learn from the Local Geometry of Glass-Forming Liquids[END_REF][START_REF] Steif | Strain localization in amorphous metals[END_REF] or also the bond orientation order [START_REF] Tanaka | Revealing key structural features hidden in liquids and glasses[END_REF][START_REF] Tong | Structural order as a genuine control parameter of dynamics in simple glass formers[END_REF].

Then (ii), very recently developed indicators are based on machine learning [START_REF] Bapst | Unveiling the predictive power of static structure in glassy systems[END_REF][START_REF] Cubuk | Identifying Structural Flow Defects in Disordered Solids Using Machine Learning Methods[END_REF][START_REF] Schoenholz | A structural approach to relaxation in glassy liquids[END_REF] which, after dierent types of training, are able to extract a local indicator from the sole knowledge of the material structure.

A third category (iii) is based on the local linear response of the material, without considering the strain which will be applied to the system. Among them one can nd the soft modes based on the computation of the low fequency vibrationnal modes of the structure [START_REF] Widmer-Cooper | Irreversible reorganization in a supercooled liquid originates from localized soft modes[END_REF].

Another class of indicators (iv), other indicators also considered the linear response of the material coupled with the applied strain, as the local elastic moduli tensor [START_REF] Tsamados | Local elasticity map and plasticity in a model Lennard-Jones glass[END_REF].

Finally (v), the last category of indicators corresponds to those considering the local non-linear response of the material. The saddle point sampling [START_REF] Ding | Universal structural parameter to quantitatively predict metallic glass properties[END_REF][START_REF] Yue Fan | How thermally activated deformation starts in metallic glass[END_REF] falls into this last category.

In this manuscript, we are considering an indicator called the local yield stress [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF]. This indicator is non-linear and allows us to compute at which at which stress value a local region will plastify when shear in a given direction. In [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF], this indicator was shown to have a good capacity to predict the location of plastic events. The measure of this indicator is based on a non-perturbative approach and gives access to the distribution of stress thresholds and orientations. This indicator will be extensively used in this manuscript.

One can note that the local yield stress is a commonly used parameter in mesoscale models [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF]. This indicator constitutes the rst direct measurement of this quantity at the atomistic scale [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF].

Shear banding

In the previous section, we reviewed the phenomenology of plasticity in amorphous solids and the numerical methods used to study it. This section will concentrate on shear banding in amorphous solids.

In the rst part of this section, a description of shear bands will be given. The second part will then present a theoretical study of the condition of existence of shear bands from the mechanical point of view. The third part will give a brief summary about the local properties of shear bands in metallic glasses measured in experiments. After that, the fourth part will relate the shear bands width evolution and will present a simple model to link shear band width with the applied strain. Finally, the fth part will describe the dierent origins of shear banding in amorphous solids.

Denition

Shear banding is a form of mechanical instability that correspond to the localization of large plastic strain in relatively thin and persistent bands when a material is deformed [START_REF] Greer | Shear bands in metallic glasses[END_REF]. This phenomenon, that seems to have been rst identied at the end of the nineteenth century [START_REF] Divoux | Shear Banding of Complex Fluids[END_REF], is present in a wide variety of materials.

Indeed, this phenomena have been observed in crystalline [START_REF] Morii | Development of shear bands in f.c.c. single crystals[END_REF] and amorphous [START_REF] Zhang | Stable owing of localized shear bands in soft bulk metallic glasses[END_REF] metals, in granular solids [START_REF] Nguyen | Experimental study of shear band formation: Bifurcation and localization[END_REF][START_REF] Fortin | Acoustic Emissions Monitoring during Inelastic Deformation of Porous Sandstone: Comparison of Three Modes of Deformation[END_REF][START_REF] Alshibli | Shear Band Formation in Plane Strain Experiments of Sand[END_REF] or in polymers, among many other materials [START_REF] Greer | Shear bands in metallic glasses[END_REF][START_REF] Schall | Shear Bands in Matter with Granularity[END_REF].

Furthermore, shear banding can be even observed in complex uids under shear [START_REF] Divoux | Shear Banding of Complex Fluids[END_REF][START_REF] Olmsted | Coexistence and phase separation in sheared complex uids[END_REF][START_REF] Höhler | Rheology of liquid foam[END_REF]. In this uid-like context, shear banding corresponds to the localization of high shear rate in a small region. In this manuscript, we will focus on shear bands in amorphous solids, corresponding to the concentration of the plastic deformation into persistent bands.

Shear bands local properties in experimental metallic glasses

In previous parts, we saw that shear bands can be observed in a wide variety of dierent systems. This section will concentrate on the shear bands local properties observed in experimental studies of metallic glasses.

The rst observation of shear bands in metallic glasses was realized by Masumoto and Maddin in 1971 [START_REF] Masumoto | The mechanical properties of palladium 20 a/o silicon alloy quenched from the liquid state[END_REF]. Their estimate of the shear band thickness was around 20nm. In agreement, more recent studies found the shear bands thickness to be of the order of tens of nanometer [START_REF] Zhang | Thickness of shear bands in metallic glasses[END_REF], mostly between 10 nm and 20 nm, depending on the metallic glass composition [START_REF] Donovan | The structure of shear bands in metallic glasses[END_REF][START_REF] Li | Characterization of nanometerscale defects in metallic glasses by quantitative high-resolution transmission electron microscopy[END_REF].

Recent studies were able to probe the inner properties of the shear bands.

When deforming a metallic glass, a temperature rise is measured within the shear band [START_REF] Lewandowski | Temperature rise at shear bands in metallic glasses[END_REF][START_REF] Georgarakis | Shear band melting and serrated ow in metallic glasses[END_REF], as we can see on picture on gure 1.17 showing that the coating of the metallic glass has melted after the shear band formation.

Figure 1.17 Picture of metallic glass coating melted at the shear bands locations due to local heating. From [START_REF] Lewandowski | Temperature rise at shear bands in metallic glasses[END_REF].

The question of whether this increase of temperature is a cause or a consequence of shear banding remains in debate [START_REF] Li | Temperature rises during strain-rate dependent avalanches in bulk metallic glasses[END_REF][START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF].

From a more structural perspective, an increase of the free volume of the order of few percent is measured after the shear band nucleation in the system [START_REF] Flores | Flow and Fracture of Bulk Metallic Glass Alloys and Their Composites[END_REF] and within the shear bands [START_REF] Pan | Softening and dilatation in a single shear band[END_REF].

Furthermore, indentation studies were able to measure the local hardness within shear bands in metallic glasses and showed that the shear bands are softer than the rest of the material [START_REF] Pan | Softening and dilatation in a single shear band[END_REF][START_REF] Maaÿ | A single shear band in a metallic glass: Local core and wide soft zone[END_REF][START_REF] Liu | Elastic Fluctuations and Structural Heterogeneities in Metallic Glasses[END_REF]. For instance, the gure 1.18 represents the local hardness prole centered on the shear band of a metallic glass. From this gure, a hardness contrast of 25% between inside and outside the shear band is found. Plasticity thus softens the material. From these studies, we can retain that the plastic deformation of metallic glasses occurs through the formation of shear bands of the order of few dozens of nanometers. The formation of these shear bands is associated with a local increase of temperature. These shear bands are also associated with a local increase of the free volume, a local softening but also with lower local relaxation time in its vicinity.

Shear bands width evolution

In the previous part, we saw the local properties of the material within the shear band. This section will relate the evolution of the shear bands width as the material is further deformed.

An increase of the shear band width as the strain applied to the system increases has been observed experimentally in metallic glasses [START_REF] Maaÿ | A single shear band in a metallic glass: Local core and wide soft zone[END_REF][START_REF] Pan | Softening and dilatation in a single shear band[END_REF][START_REF] Liu | Correlating defects density in metallic glasses with the distribution of inherent structures in potential energy landscape[END_REF].

However, in these systems, the experimental study of shear bands is limited by the small length and short time scale over which the shear band forms [121].

After the shear band formation, the material tend to quickly fail where the shear band is located [START_REF] Lewandowski | Temperature rise at shear bands in metallic glasses[END_REF][START_REF] Pampillo | Comprehensive plastic deformation of a bulk metallic glass[END_REF][START_REF] Greer | Shear bands in metallic glasses[END_REF].

Molecular dynamics simulation allows to study the evolution of shear bands in amorphous systems from the very beginning of plastic deformation [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF].

Indeed, by imposing periodic boundary conditions on a deformed system, it become possible to remove stress concentration and thus to keep the system integrity as it is deformed. It allows the observation of shear bands evolution [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF][START_REF] Yao | Evolution of shear banding ows in metallic glasses characterized by molecular dynamics[END_REF].

In [125], Jagla proposed a simple geometric argument relating the strain rate imposed to the system with the strain rate within the shear band to extract an equation giving the evolution of the shear band width w with the applied strain . It is dened as follows:

Let γ be the strain rate imposed to the system, L the system linear size, w the shear band width. If we consider that the strain rate is concentrated within the shear band, then the strain rate within the shear band γSB can be expressed as:

γSB = L w γ (1.1)
Moreover, a dimensional argument allows us to suppose the proportionality of dw/dt and γSB . Indeed, no other dependence can be constructed in the absence of additional parameters. We thus get

dw dt = A L w γ (1.2)
with A a proportionality constant. By integrating the equation 1.2, we obtain:

w = w 2 0 + 2AL γ(t -t 0 ), (1.3)
with w 0 the shear band width right after its nucleation. We dene t 0 to be the time at which the shear band nucleates. As γ = γt, we thus have that the relation between the shear band width w and the applied strain γ can be expressed as:

w = w 2 0 + 2AL(γ -γ 0 ), (1.4)
with γ 0 the applied strain at the shear band nucleation.

This relation between the shear band width and the applied strain is consistent with the shear band width evolution measured through molecular dynamics in [START_REF] Yao | Evolution of shear banding ows in metallic glasses characterized by molecular dynamics[END_REF]. This relation will be compared with our measurement of the shear band width evolution in the studied numerical system.

In [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF], another relation based on a model taking into account shear transformation zones is also derived. However, this is developed for glasses at nite temperature. As our study of shear banding will be done on systems at zero temperature, this will not be detailed here.

Shear band criterion: continuum mechanics point of view

In the previous section, we saw a model describing the widening of the shear bands. In this section, we will present a continuum mechanics approach, pre-sented in [START_REF] Trung | Modelisation multi-echelle des materiaux granulaires frottant-cohesifs[END_REF], for the condition of existence of a shear band in a solid.

This approach, developped by Rice [START_REF] James R Rice | The localization of plastic deformation[END_REF], considers a homogeneous and homogeneously deformed solid subjected quasi-statically to increments of deformation. These increments of deformation could give rise to homogeneous strain and stress rate eld: ˙ 0 and σ0 as shown on gure 1.19 (a). (a) homogeneous eld, (b) localized eld. From [START_REF] Trung | Modelisation multi-echelle des materiaux granulaires frottant-cohesifs[END_REF].

The following calculation aims to determine if a bifurcation within a localized band of orientation n is possible. In the considered localization mode, from the kinematical conditions we have that the strain rate within the band ˙ 1 should be of the form [START_REF] James R Rice | The localization of plastic deformation[END_REF]:

˙ 1 kl = ˙ 0 kl + g k • n l , (1.5)
with g k the velocity gradient in the direction n. Thus, the deformation within the band is the combination of the homogeneous eld ˙ 0 kl with an additional eld: g k • n l , as shown on gure 1.19 (b).

The continuity of the normal stress σ • n between the inside and the outside of the band gives:

σ1 ij • n j = σ0 ij • n j (1.6)
Furthermore, for the sake of simplicity, the incremental response of the stress for a strain increment will be considered as linear. It can thus be written as:

σij = L ijkl ˙ kl (1.7)
Then, the equations 1.6 and 1.7 gives:

L ijkl ˙ 1 kl • n j = L ijkl ˙ 0 kl • n j (1.8)
Combining the equation 1.8 and 1.5 nally gives:

(n j • L ijkl • n l ) • g k = 0 (1.9)
This equation gives the condition of existence of a band in the considered system. The trivial solution is g k = 0 corresponding to a homogeneous system without a band with dierent strain rate. On the other hand, localization appears if the following condition is respected:

det(nLn) = 0, (1.10)
where the tensor nLn is called the acoustic tensor.

From similar approach, it can be shown that strain-softening, i.e. det(nLn) ≤ 0, is a condition for the existence of a shear band in a solid [START_REF] Forest | Local Approach to Fracture[END_REF].

Causes of shear banding

In the previous parts, we reviewed the shear bands properties and evolution, as well as the criteria of existence. In this section, we will describe the supposed causes of shear banding in amorphous materials. This section is inspired by [START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF].

Many elasto-plastic models were able to reproduce the formation of persistent shear bands in mesoscopic simulations [START_REF] Martens | Connecting Diusion and Dynamical Heterogeneities in Actively Deformed Amorphous Systems[END_REF][START_REF] Karimi | Role of inertia in the rheology of amorphous systems: A nite-element-based elastoplastic model[END_REF][START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF][START_REF] Tüzes | Disorder is good for you: the inuence of local disorder on strain localization and ductility of strain softening materials[END_REF][START_REF] Popovi¢ | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF]. In these models, the main cause of localization of the deformation is the insucient healing of the regions in which a plastic event just occured [START_REF] Nicolas | Rheology of athermal amorphous solids: Revisiting simplied scenarios and the concept of mechanical noise temperature[END_REF]. The origin of this insucient healing can be categorized in three main causes which will be developed in the next three sections: (i) the inuence of the time scales in the system, (ii) the local inertia and (iii) the shear rejuvenation upon plasticity. In this manuscript, we focus on shear rejuvenation. Indeed, our model system does not take into account time scale or inertial eects as we will see in chapter 2 dealing with the methods and systems.

Time scale

When an amorphous solid is deformed, its stress-strain curve shows a succession of stress increases and stress drops. The stress drops correspond to the activation of plastic events [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF].

Based on a mean eld model developped by Coussot and Ovarlez [START_REF] Coussot | Physical origin of shear-banding in jammed systems[END_REF], K.

Martens et al. [START_REF] Martens | Connecting Diusion and Dynamical Heterogeneities in Actively Deformed Amorphous Systems[END_REF] proposed a mesoscopic model for the owing of the disordered matters. In this model, they dene the parameter τ el which corresponds to the characteristic time scale of the stress drop, thus corresponding to the plastic rearrangement healing time.

From this model, they were able to observe the formation of persistent shear bands for large enough τ el or, equivalently, when the strain rate γ is small enough. For instance, the gure 1.20, representing instantaneous plastic activity maps for dierent strain intervals, show that reducing the strain rate from 1 to 10 -2 change the plastic behavior from a homogeneous one to the formation of a persistent shear band.

For large enough τ el or a low enough γ, the local stress generated by a plastic event will have enough time to diuse to the surrounding regions. This added local stress can trigger new plastic events and can lead to collective eects. These collective events can in their turn induce the formation of shear bands. Figure 1.20 Instantaneous plastic activity maps for dierent strain intervals after a rapide reduction of the shear rate from 1 to 10 -2 . As the strain rate is reduced, the plastic activity changes from a homogeneous behavior to the formation of persistent shear bands. [START_REF] Martens | Spontaneous formation of permanent shear bands in a mesoscopic model of owing disordered matter[END_REF] 1.

Inertia

The inuence of inertia in the formation of shear bands in amorphous systems is observed in molecular dynamics simulations [START_REF] Nicolas | Eects of Inertia on the Steady-Shear Rheology of Disordered Solids[END_REF] and in nite element based elasto-plastic model [START_REF] Karimi | Role of inertia in the rheology of amorphous systems: A nite-element-based elastoplastic model[END_REF].

In the molecular dynamics simulation [START_REF] Nicolas | Eects of Inertia on the Steady-Shear Rheology of Disordered Solids[END_REF], Nicolas et al. added a damping term in the equations of motion of the particles. By tuning this damping term for a system under a nite strain rate, they could reach an underdamped regime. This underdamped regime causes inertial oscillations of the particles after a localized plastic rearrangement. As a consequence, the kinetic energy generated by a plastic event slowly dissipates to the rest system due to these inertial eects. This kinetic energy remains concentrated in a localized region can and help to trigger more easily new plastic events in this region. This leads to collective eects which in their turn induce the formation of shear bands.

Similar observations were done in nite element based elasto-plastic model [START_REF] Karimi | Role of inertia in the rheology of amorphous systems: A nite-element-based elastoplastic model[END_REF] in which, here again, by controlling the damping of the system and putting in an underdamped regime, the formation shear bands is observed for large enough systems. For instance, in gure 1.21 (a) is represented the velocity proles taken at dierent times for the nite element simulations. In this gure, we can observe non-linearities in the velocity prole due to the presence of a shear band. The location of the shear band is indicated by a symbol. As it can be seen in the gure 1.21 (b) representing the position of the shear band as a function of the time, the shear band location is not x in this simulation but changes with time.

Rejuvenation

In [START_REF] James R Rice | The localization of plastic deformation[END_REF], Rice obtained theoretically that the localization of the plastic deformation into a shear band in elasto-plastic systems requires softening upon plasticity, i.e. shear rejuvenation.

Moreover, mesoscopic models conrmed that implementing rejuvenation in their models by making the system locally softer after the occurrence of a plastic event leads to the formation of persistent shear bands [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF][START_REF] Tüzes | Disorder is good for you: the inuence of local disorder on strain localization and ductility of strain softening materials[END_REF][START_REF] Popovi¢ | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF]. They observed that the more the softening between the initial state of the system and the rejuvenated one is important, the more the deformation is localized.

For a large enough softening, the rst region to yield is rejuvenated in soft state more likely to plastify again. As a consequence, the plasticity accumulates and the system get trapped in a banded structure. This can be seen on gure 1.22 taken from [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF], representing maps of plastic strain for increasing strain (from left to right). In the rst row, without shear softening, no localization of the deformation in a shear band can be observed. Then, when softening is implemented in the model, formation of shear band can be observed as shown in the two remaining rows. Furthermore, in the row having the higher shear softening (see lower row) a more important localization of the plastic deformation can be observed. Figure 1.22 Maps of plastic strain for increasing strain (left to right) for mesocopic system without shear softening (upper row) and with increasing shear softening (from the middle to the lower row). From [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF].

Conclusion

In this chapter, we introduced the main elements needed to read this manuscript.

We have seen that metallic glasses are amorphous solids characterized by a high yield strength, but also often exhibit a brittle behavior due to the formation of shear bands. Their plastic behavior were found to depend on their thermal and mechanical history, and also on their size. To understand plasticity and the formation of shear bands, numerical simulations are performed in dierent scales: atomistic, mesoscopic and macroscopic. In atomistic simulations, to link plasticity and structure in atomistic simulations, a wide variety of local structural indicators were developed.

In this context, we perform numerical atomistic simulations of model metallic glasses. We then use a novel local indicator called the local yield stress to study plasticity and shear banding in our simulated system. As metallic glasses share many common properties with other amorphous solids, the results obtained in this thesis are thus not exclusive to metallic glasses and can be expected to apply to other amorphous solids. The next chapter will be dedicated to the presentation of the system and the methods we are using to obtain and study numerically plasticity at the atomic scale.

Chapter 2

Numerical System and methods

In the previous chapter, we introduced the principal elements needed to read this manuscript. We detailed the main phenomenology of plasticity in amorphous solids and the methods applied to study numerically plasticity. The present chapter will be dedicated to the presentation of the methods and the systems used in this manuscript to study plasticity in amorphous solids.

In a rst section, we will detail the system under study, a two-dimensional binary Lennard-Jones glass. The second section will explain the methods used to synthesize model glasses through molecular dynamics. In this section, we will also introduce the athermal quasi static protocol used to deform our glassy systems. Finally, the third section will describe the approaches used to calculate the local properties of our systems in order to link plasticity with the local structure.

2D binary Lennard Jones Glass: a reasonable system for metallic glasses and colloidal suspensions

In this manuscript, we chose to focus on a model system commonly used to study plasticity in amorphous solids in two dimensions: the 2D binary Lennard-Jones glass. This system constitutes a reasonable model for metallic and colloidal glasses [START_REF] Guzmán | An eective-colloid pair potential for Lennard-Jones colloidpolymer mixtures[END_REF][START_REF] Shi | Structural transformation and localization during simulated nanoindentation of a noncrystalline metal lm[END_REF].

The Lennard-Jones potential is a simple pair potential [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF] which has the advantage of requiring a low computational cost compared with more complex potentials. This potential does not aim to reproduce a specic metallic glass composition. Indeed, this potential can be applied to describe dierent amorphous solids such as metallic or colloidal glasses. By using this potential, previous atomistic simulation studies were able to reproduce most of the plastic phenomenology of amorphous solids such as the shear band formation [START_REF] Shi | Strain Localization and Percolation of Stable Structure in Amorphous Solids[END_REF].

Using a two-dimensional system also helps to reduce the computational cost.

It also allows the simple visualization of the whole system and of its dierent local elds [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. The 2D binary Lennard-Jones glass is a commonly used model for metallic glass simulations [START_REF] Kob | Scaling Behavior in the β -Relaxation Regime of a Supercooled Lennard-Jones Mixture[END_REF][START_REF] Shi | Strain Localization and Percolation of Stable Structure in Amorphous Solids[END_REF][START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF]. Using this model also allows to compare the obtained results with a vast literature.

In this thesis, we use a similar model glass as in [START_REF] Shi | Strain Localization and Percolation of Stable Structure in Amorphous Solids[END_REF] which was shown to feature shear bands for stable enough systems. This model glasses is considered as a good glass former as no crystallization is observed during the glass formation.

In gure 2.1, we show an example of the systems we study in this manuscript.

We consider two-dimensional square system of various sizes and at xed volume.

A constant density of ρ N ≈ 1.02 is used. Also, as in [START_REF] Shi | Structural transformation and localization during simulated nanoindentation of a noncrystalline metal lm[END_REF], we use two sizes of particles with a ratio between large (L) and small (S) particles number equal to These particles interact with a Lennard-Jones inter-atomic potential. However, in contrast with [START_REF] Shi | Structural transformation and localization during simulated nanoindentation of a noncrystalline metal lm[END_REF], the potential is slightly modied by adding a smoothing function at a cut-o distance R cutof f . Indeed, to measure the force applied on a particle, we consider the inuence of all the particles of the system.

N L /N S = (1 + √ 5)/4.
This procedure requires a high computational cost. Yet, the Lennard-Jones potential used in [START_REF] Shi | Structural transformation and localization during simulated nanoindentation of a noncrystalline metal lm[END_REF] converges towards zero following a law of r -6 , with r the distance from the particle. The contribution of far particles is then very limited.

By adding a cut-o distance on the potential, we neglect the particles which are too far to have signicant contribution on the force applied on particles. The use of a smoothing function ensure the derivability of the potential.

In practice, for inter-particles distances greater than R in , the Lennard-Jones expression is replaced by a smooth quartic function vanishing at a cut-o distance R cutof f . For two particles i and j separated by r ij , their interaction is dened as:

U (r ij ) =                  4 σ rij 12 -σ rij 6 + A, for r ij < R in 4 k=0 C k (r ij -R in ) k , for R in < r ij < R cutof f 0, for r ij > R cutof f (2.1) with A = C 0 -4 σ R in 12 - σ R in 6 C 0 = -(R cutof f -R in ) [3C 1 + C 2 (R cutof f -R in )] /6 C 1 = 24 σ 6 (R 6 in -2σ 6 )/R 13 in C 2 = 12 σ 6 (26σ 6 -7R 6 in )/R 14 in C 3 = -[3C 1 + 4C 2 (R cutof f -R in )] / 3(R cutof f -R in ) 2 C 4 = [C 1 + C 2 (R cutof f -R in )] / 2(R cutof f -R in ) 3
with and σ the parameters describing the energy and the length scale of interparticle interactions, respectively. Under this formalism, the cut-o distances are dened as: R in = 2σ and R cutof f = 2.5σ.

The interaction energy for the SS (small-small) and the LL (large-large)

bonds are half the one for the SL (small-large) bonds as: SS = LL = 1/2 SL . Furthermore, we have σ SS = 2σ SL sin(π/10) and σ LL = 2σ SL sin(π/5).

We choose in this manuscript SL and σ SL as the reference for the energy scale and the length scale, respectively. Note that in this system, all the particles have the same mass m.

In this manuscript, the reference time scale is then dened as: t ref = σ SL m/ SL . Also, the temperature is referred in units of σ SL /k with k the Boltzmann factor. In the following, all the quantities will be in Lennard-Jones units

System production and deformation

In the previous section, we dened the potential and the parameters of the model glasses used in this manuscript.

In this section, we will rst describe the molecular dynamics method we use to simulate an equilibrated liquid at nite temperature. Then, we will detail the dierent protocols we apply to prepare glasses from equilibrated liquids.

Finally, we will explain the method we use to deform the glasses.

Note that the preparation and the deformation of the glasses are done using the LAMMPS open software [START_REF] Plimpton | Fast Parallel Algorithms for Short Range Molecular Dynamics[END_REF].

Molecular dynamics

Molecular dynamics (MD) is a method used to simulate the time evolution of interacting particles. It basically consists of solving numerically the Newtons's equation of motion [START_REF] Becquart | Dynamique moléculaire appliquée aux matériaux[END_REF].

Through MD simulations, we can obtain equilibrated liquids or supercooled liquids at a given temperature. These equilibrated liquids are then quenched to obtain a glass. The dierent quench methods used in this manuscript will be detailed in the next section.

We consider in this manuscript MD simulations performed in the canonical ensemble (NVT) in which a system of constant volume V with a xed number of N particles is in interaction with a heat bath of temperature T. The system can exchange energy with the heat bath and this heat bath will maintain the temperature of the system around T.

In order to simulate a liquid in the canonical ensemble, we apply a thermostat to our MD simulations. More specically, we use the Nose-Hoover thermostat, a commonly used thermostat [START_REF] Nosé | A unied formulation of the constant temperature molecular dynamics methods[END_REF][START_REF] William | Canonical dynamics: Equilibrium phase-space distributions[END_REF]. The advantage of this thermostat is that the obtained temperature uctuations are consistent with a real canonical ensemble [START_REF] Hünenberger | Thermostat algorithms for molecular dynamics simulations[END_REF]. This thermostat is represented by a virtual particle coupled to the other particles in the system. This virtual particle corresponds to the heat bath of the system at xed temperature T. This virtual particle slows down or accelerates the other particles and maintain the temperature of the system around T.

The Hamiltonian of the system corresponds to the Hamiltonian of the physical system extended with terms corresponding to the virtual particle [START_REF] Kleinerman | Implementations of NoséHoover and NoséPoincaré thermostats in mesoscopic dynamic simulations with the united-residue model of a polypeptide chain[END_REF]. It can be written as:

H N H = E k (p v ) + V e (r) + p ξ 2Q + (g + 1)kT η (2.3)
with p v the physical particle's momentum, E k the kinetic energy of the system, V e represents the potential energy of the system as a function of particles coordinates r.

Here V e corresponds to the Lennard-Jones potential introduced in section 2.1.

Moreover, p ξ is the momenta of the virtual particle representing the thermostat. We also have we have η = ln(s), where s is the virtual particle position. Q is the articial mass of the thermostat. Then g is the number of degree of freedom in the system, k is the Boltzmann constant and T is the desired temperature.

In this manuscript, we use the Nose-Hoover thermostat implemented in LAMMPS. The equations of motion used are taken from [START_REF] Shinoda | Rapid estimation of elastic constants by molecular dynamics simulation under constant stress[END_REF]. For a system with a conservation of the volume, it will give [START_REF] Martyna | Constant pressure molecular dynamics algorithms[END_REF][START_REF] Kleinerman | Implementations of NoséHoover and NoséPoincaré thermostats in mesoscopic dynamic simulations with the united-residue model of a polypeptide chain[END_REF]:

ṙi = p v i m , ṗv i = - ∂ ∂r i V e (r) -p v i p ξ Q (2.4)
with m the particles mass. For the virtual particles, we have

η = p ξ Q , ṗξ = 2E k (p v ) -gkT (2.5)
Finally, these equations of motion are time-integrated following the protocol described in [START_REF] Tuckerman | A Liouville-operator derived measurepreserving integrator for molecular dynamics simulations in the isothermalisobaric ensemble[END_REF].

It is important to note that the time scales accessible with molecular dynamics are very limited (few tens of nanoseconds typically [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]) which makes dicult the direct comparison with experiments. The number of particles we can simulate through molecular dynamics is also limited. Indeed, the larger the system, the longer the simulation time. For instance, when equilibrating with MD our larger system (composed of 10000 atoms) at the lower considered temperature, we need few months of computation.

In a system with xed boundary conditions, the particles near the boundaries have a dierent behavior than those in the middle of the system as the number of neighbors diers. As the accessible system sizes through MD simulations are limited, the xed boundaries inuence is expected to an important inuence on the particle's behavior in our simulations. To remove the xed boundary eect, we use in our simulations periodic boundaries conditions.

Preparing a glass

In the previous section, we introduced the MD simulation we use to simulate a liquid at a given temperature T. In this section, we will detail the protocols we follow to prepare glasses at dierent degrees of relaxation.

To prepare a glass, we rst simulate liquids equilibrated at a given temperature T through MD simulations. We perform two verications in order to know if our liquid is equilibrated.

We begin (i) with estimating its relaxation time τ α by calculating its selfintermediate scattering function F SI . To do so, we follow the protocol described in the annex 7.3. The value of τ α is calculated recursively with a xed time interval as the liquid is being equilibrated. Indeed, τ α varies when the liquid is not at thermodynamical equilibrium. When the value of the calculated τ α has stabilized, we equilibrate our system until the function F SI converges towards its equilibrium form and becomes invariant of the starting time from which we calculate it.

To conrm if the system is indeed equilibrated, we also (ii) measure the potential energy of the system. We consider the liquid to be equilibrated at the temperature T if its potential energy has converged as shown in the annex 7.3.

In practice, we observe that the liquids are equilibrated after an equilibration time of the order of t ≈ 400τ α From this equilibrated liquid, we use two dierent quench protocols to obtain a glass. In gure 2.2 representing the temperature dependence of a liquid's enthalpy at constant pressure, a schematic representation of the quench protocols is shown. In this gure, T m corresponds to the melting temperature. A liquid equilibrated below T m thus corresponds to a supercooled liquid in a metastable state.

The rst (i) quench protocol consists of an instantaneous quench of the equilibrated liquid at T down to T = 0. In practice, we change the velocity of all the particles to zero. Then, we minimize the system energy with a conjugate gradient algorithm described in [START_REF] Polak | Note sur la convergence de méthodes de directions conjuguées[END_REF]. From this energy minimization, the system falls in a local minimum of the potential energy landscape (PEL) of the liquid at parent temperature T p . The obtained system after energy minimization corresponds by denition to an inherent state (IS) of the liquid at temperature T [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF][START_REF] Widmer-Cooper | Irreversible reorganization in a supercooled liquid originates from localized soft modes[END_REF].

The instantaneous quench protocol is represented in this gure by the black and blue arrows. The black arrow correspond to an instantaneous quench from a high temperature liquid (T > T m ), while the blue arrow corresponds to an instantaneous quench from a supercooled liquid (T < T m ).

For the second protocol (ii), in order to obtain more relaxed glasses, we apply a gradual quench. The principle is to take an equilibrated liquid at nite temperature T and then reduce gradually its temperature until it is close to T = 0. At this point, we minimize the system potential energy with a static relaxation. In practice, in this manuscript, all the glasses obtained through gradual quench are following the exact same protocol. It is dened as follows: we consider a supercooled liquid equilibrated at T = 0.351. We then reduce its temperature down to T = 0.030 over a period of 10 6 t 0 using a Nose-Hoover thermostat. Finally, we quench the system instantaneously down to zero temperature. In gure 2.2, this quench protocol is represented by the green slope.

Note that due to the limited time scales of molecular dynamics, the cooling rate of a gradual quench through MD is always much higher than experimental cooling rates [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF].

As a glass is in an out-of-equilibrium state, its properties depend on its thermal history. As a consequence, the properties of a gradually quenched system depends on its cooling rate. A common way to describe the state of a glass is to use its ctive temperature T f [146].

In gure 2.2, we can see that the variation of the enthalpy for a liquid (for T > T gb ) is dierent than for a glass. The ctive temperature is dened as the intersection of the line of the enthalpy for the liquid state with the line for the glassy state. This is represented by T ga and T gb in the gure 2.2. The ctive temperature is thus dependent on the cooling rate. The lower the cooling rate, the lower the ctive temperature. For instance, as T ga < T gb , the glass (a) in gure 2.2 corresponds to a glass obtain with a lower cooling rate than the glass (b). As a consequence, the lower the ctive temperature, the more relaxed a system is.

During an instantaneous quench from an equilibrated liquid at T = T l , the temperature is instantaneously reduced down to T = 0. As a consequence, the ctive temperature T f of the inherent state of a liquid at T l corresponds to T f = T l .

To be able to compare the systems obtained with a gradual quench with those obtained by an instantaneous quench, we want to measure the ctive temperature of the gradually quenched system.

To do so, we measure the potential energy per atom of the inherent states obtained by an instantaneous quench from dierent temperatures. We can see on gure 2.3 the potential energy as a function of the temperature from which the glass was instantaneously quenched. The results are reported in the gure as gray circles. As expected, the lower the temperature from which the instantaneous quench was performed, the lower the system potential energy. The ctive temperature T f is estimated as that where the liquid IS energy extrapolates to that of the gradually quenched system (see blue dotted lines).

To estimate the ctive temperature of the gradually quenched system, we rst t the potential energy of the inherent states as a function of their parent temperatures with the equation A/T B + C (green lines). From this, we measure A ≈ -0.08, B ≈ 1.12 and C ≈ -2.08. We then measure the potential energy per atom for the gradually quenched system. From this value, we use the tted function to estimate the corresponding ctive temperature T f . For the gradually quenched systems, we compute T f = 0.291 ± 0.007.

Note that in the model glass under study, the mode-coupling temperature T M CT introduced in 1.1.3 is estimated as T M CT = 0.357. The details of its estimation can be found in the annex 7.1.

Athermal Quasi Static Deformation

In the previous section, we explained the quench protocols we follow to obtain model glasses at zero temperature with dierent degree of relaxation. In this section, we will now detail the method we use to deform these glasses.

To deform our model glasses, we use an Athermal Quasi-Static (AQS) shear deformation. This deformation protocol is commonly used to deform numerical amorphous systems. Indeed, by deforming the material with MD, the accessible shear rate would be of the order of 10 8 s -1 which is much higher than the experimental shear rates which are of the order of 10 -3 s -1 [START_REF] Rodney | Modeling the mechanics of amorphous solids at dierent length scale and time scale[END_REF]. This is due to the fact that the time scales are very limited in MD simulations compared to experimental studies (few tens of nanoseconds). As a consequence, if we want to obtain a strain of the order of 1 for a deformation under MD simulation, a very high shear rate would be needed.

A way to avoid this diculty is to use AQS deformation. It consists of shearing the material with small strain increments. After each strain increment, system energy is minimized [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF][START_REF] Singh | Brittle yielding of amorphous solids at nite shear rates[END_REF]. As the energy minimization removes all thermal uctuations, the AQS protocol corresponds to the limit T → 0. Also, as the system can fully relax to a new energy minimum before the following strain increment, this protocol also corresponds to the limit γ → 0.

In a system deformed by an AQS protocol, all the plastic events will then be mechanically activated as no thermal uctuations are present. This protocol also rules out time scales and inertia eects for the formation of persistent shear bands, leaving shear rejuvenation as the most likely cause of shear banding in the studied systems.

In practice, the AQS deformation consists in the iteration of two steps: (i) we move anely the particles position with a small strain increment ∆γ. Then, (ii) we minimize the energy of the system by using here again the conjugate gradient algorithm. During this minimization, the particles relaxes and instabilities, i.e. plastic events, can be triggered. Indeed, the strain increments leads the material to cross locally an energy barrier [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF]. Then the minimization of the energy causes the system to fall in a new energy minima. This crossing of the local energy barrier leads to the occurrence of plastic events in the system [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF][START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF]. These instabilities cause a drop in the stress-strain curve. In gure 2.4, we show an example of a stress-strain curve obtained with AQS deformation. We can see on the curve stress drops corresponding to the occurrence of plastic events. The inset shows the shape of a single stress drop. We can remark on this curve the presence at higher applied strain (i.e. γ xy > 0.06) of larger stress drops. These large stress drops correspond to avalanches, i.e. the activation of many energy barriers during a single minimization.

We want to determine the strain increment ∆γ we use for AQS deformation.

To do so, we deform the same system with dierent strain increments. We then take the increment value below which we obtain the same stress strain curve up to the machine precision for an applied strain below γ xy = 0.06. Indeed, above this applied strain, a strain step can cause the activation of many energy barriers, i.e. an avalanche, which can provoke a chaotic motion of the particles before reaching a new minimum of the potential energy as shown in [START_REF] Regev | Reversibility and criticality in amorphous solids[END_REF]. As a consequence, at higher strain, no matter how small ∆γ is, we would observe dierent stress-strain responses for dierent strain steps at large strain (see the annex 7.4 for more details). Note that the larger the system, the lower the strain The inset shows a zoom of a stress drop corresponding to one plastic event.

increment to get a convergent trajectory. The value of ∆γ is thus determined for our larger system (i.e. with 10000 atoms). In this manuscript, we use a strain increment of ∆γ = 10 -4 .

When we minimize the system energy, its relaxation depends on the boundary conditions. As we explained in the section 2.2.1, in a system with xed boundaries conditions, the particles near the boundaries have a dierent behavior than those in the center of the system. For this reason, we apply periodic boundary conditions.

In this manuscript, to deform our model glasses, we apply a simple shear.

To be able to maintain periodic boundaries conditions during the simple shear, we use the Lees and Edwards periodic boundary conditions [START_REF] Lees | The computer study of transport processes under extreme conditions[END_REF]. Indeed, the Lees-Edwards periodic boundary conditions is a method developed to combine shear ow with periodic boundaries. In this way, we can deform our system while avoiding the xed boundary eects.

Probing local properties

In the last section, we detailed the methods we use to prepare our model glasses.

We also explained the protocol we follow to apply a deformation on our systems.

We saw that as the system is loaded, plastic events are activated leading to drops in the stress strain curve. In this section, we will explain the methods we use to study locally our glasses. A particular attention will be paid to the local yield stress [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF], a novel indicator we apply to probe the materials local structural properties.

In a rst part, we will explain the method we use to estimate the local strain and the local stress of the system. In a second part, we will detail the local yield stress probing method.

Local strain and local stress 2.3.1.1 Local strain

In this manuscript, the local strain is measured following the same method as in [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF]. It is based on the computation of the coarse-grained local deformation gradient tensor F.

In continuum mechanics, F relates the reference and the current (deformed) conguration through the Cauchy-Born rule. In the case of an atomic system under ane deformation, this tensor links the vectors X αβ and x αβ , where X αβ and x αβ are the vectors connecting the atoms α and β in the reference and current congurations, respectively.

The relation between these two vectors and the local deformation gradient tensor F can be written as:

x αβ ij = F ij X αβ ij (2.6)
with i and j the coordinates components. However, in glassy systems, the deformation is non-ane [START_REF] Saw | Nonane displacements and the nonlinear response of a strained amorphous solid[END_REF], thus the equation 2.6 is not valid in our systems.

We use here a protocol proposed by Zimmerman et al. [START_REF] Zimmerman | Deformation gradients for continuum mechanical analysis of atomistic simulations[END_REF] to measure local deformation gradient tensor F. We consider two congurations separated by a small macroscopic strain interval ∆γ. We dene the atomic level deformation tensor for the atom α: F α ij . The aim of this protocol is to nd F α ij corresponding to the best t, in the sense of the least square method, of the equation 2.6. To do so we minimize the equation:

B α = n β=1 2 i=1 g(r αβ 0 )(x αβ i -F α ij X αβ j ) 2 , (2.7)
where r αβ 0 corresponds to the distance between the atoms α and β. g(r αβ 0 ) is a smooth weighting function that only depends on r αβ 0 [151] and which contin- uously vanishes at a coarse-graining length scale R CG . g(r αβ 0 ) is dened as in [START_REF] Lemaître | Structural Relaxation is a Scale-Free Process[END_REF]:

g(r) = 15 8πR 2 CG (1 -2( r R CG ) 4 + ( r R CG ) 8 ), for r < R CG 0, otherwise.
(2.8) Through this denition, the quantity n introduced in equation 2.7 corresponds to the number of neighbors of α, dened here as the atoms β, for which r αβ 0 < R CG . Note the coarse-graining length scale was chosen to be R CG = 5. This value corresponds to the minimum scale from which applying laws of continuum mechanics on discrete systems begins to be valid [START_REF] Goldenberg | Particle displacements in the elastic deformation of amorphous materials: Local uctuations vs. non-ane eld[END_REF][START_REF] Tsamados | Local elasticity map and plasticity in a model Lennard-Jones glass[END_REF].

When minimizing equation 2.7 with respect to F α , we obtain:

F α ij = 2 k=1 Y ik Z -1 jk , (2.9)
with

Y ik = n β=1 (x αβ i X αβ k )g(r αβ 0 ) and Z ik = n β=1 (X αβ i X αβ k )g(r αβ 0 ).
(2.10)

We then apply these equations to compute the deformation gradient tensor F for each atom for congurations separated by a strain interval of ∆γ = 0.01.

For a larger strain interval, we calculate F as

F = F n F n-1 • • • F 2 F 1 where
F i is the deformation gradient tensors between states i and i + 1 separated by ∆γ = 0.01. By multiplying the deformation gradient tensors F i over stress intervals of ∆γ = 0.01, we can measure the local strain of highly deformed systems.

From the deformation gradient tensor F obtained by multiplying F i for small strain increments, we measure the logarithmic Hencky strain tensor E. We use the logarithmic Hencky strain tensor E to measure the local strain as it was shown to be an appropriate measure of the local strain for large simple shear deformation [START_REF] Onaka | Appropriateness of the Hencky Equivalent Strain as the Quantity to Represent the Degree of Severe Plastic Deformation[END_REF]. This strain tensor is expresses as:

E = ln(U) = xx xy yx yy , (2.11)
with U the right stretch tensor, i.e. the unique symmetric and positive denite tensor such that:

F = RU, (2.12)
with R a rotation. As U is symmetric, the logarithmic Hencky strain tensor E is also symmetric. We then have xy = yx .

Since we have F T F = U 2 , we can also write:

E = 1 2 ln(U 2 ) = 1 2 ln(F T F) (2.13)
We write λ + and λ -the two eigenvalues of U 2 , with λ + > λ -. From these eigenvalues, we dene local strain scalar quantity as the square root of the second tensor invariant of E as:

= 1 4 | ln(λ + /λ -) |= xx -yy 2 2 + 2 xy (2.14)
In the following manuscript, the local strain elds will be used following this method.

Local stress

In the previous section, we detailed the method we use to measure the local strain. In this section, we will explain how the local stress and the macroscopic stress are measured in this manuscript. From the measure of the macroscopic stress, we will be able to calculate the stress-strain curves of our systems.

To estimate the stress of the system, we rst measure the stress per atom.

In our systems, the atoms are linked with the Lennard-Jones potential. As the Lennard-Jones potential is a pair interaction potential, the stress tensor for the atom α can be written as:

σ α ij = - 1 2 N β=1 (r α i -r β i )f α,β j (2.15)
where i and j take on values x, y. N corresponds to the number of neighbors of the atom α. Moreover, f α,β corresponds to the force between the atoms α and β and r α , r β correspond to the positions of the atoms α and β, respectively.

In order to measure the stress of the whole system, or of a specic region of our system, we apply the following formula:

Σ ij = 1 Ω k∈Ω σ k ij (2.16)
with Ω the volume of the considered system. For instance, to compute the stress strain curve of the whole system deformed with a simple shear, we take Ω to be the volume of the whole system and measure Σ xy as a function of the imposed strain. Note that in our simulations, we use the measure of the stress per atom implemented in LAMMPS.

Local yield stress probing : a mechanical mesoscope

In order to probe the local structural properties of the material, we use the same method as in [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF]. This method allows us to measure locally the stress at which the material yields. This method is similar to those used in [START_REF] Mizuno | Measuring spatial distribution of the local elastic modulus in glasses[END_REF] to estimate the local elastic moduli, and in [START_REF] Puosi | Probing relevant ingredients in mean-eld approaches for the athermal rheology of yield stress materials[END_REF] to probe the yield stresses in a single direction for model glasses.

The method is illustrated in gure 2.5. We rst dene a grid in our system in which each sites are separated with a mesh size R sampling . Let us concentrate on one of those sites. At the center of this site, we dene a patch. As shown in gure 2.5 (a), this patch is composed of two regions. The rst region (region I), in green, is a disk of radius R f ree . The second region (region II), in blue, is a ring surrounding region I. The width of this ring is 2R cut .

In this system, otherwise mentioned, we use R f ree = 5 consistently with the coarse-graining scale R CG = 5 used to measure the local strain. The choice of the length scale and its consequences is discussed in depth in section 3.3.1.

Also, the value of R sampling is taken as: R sampling ≈ 2.5 = R f ree /2 = R cut . This value of R sampling ensures that all the regions of the system are included in at least one patch.

The patches are then deformed by using an AQS pure shear following the loading direction α (see the arrows on 2.5 (a)) until a plastic event occurs within the region I. As explained in the section 2.2.3, the AQS shear protocol consists of alternating two steps: First (i), we apply a small linear displacement of the atoms following the wanted shear protocol. Then (ii), we relax the atoms by minimizing the potential energy of the system. Here, we apply ane displacement increments to the whole patch, but we only relax the region I. In that way, the plastic rearrangements are only allowed to occur within region I. The atoms within region II thus only deform anely.

To detect if a plastic event occurred within region I, we measure its stress at each AQS step, following the procedure explained in section 2.3.1.2. We then plot its stress-strain curve as shown in gure 2.5 (b). This gure represents the stress-strain curves of a given patch for four dierent shear directions: α = 0°, 45°, 90°and -45°. We can see from these curves that at this length scale the stress response depends on the loading direction. We name τ 0 (α) the local stress in the direction α within the patch prior to loading. We can observe in gure 2.5 (b) that τ 0 (α) is non zero and depends on the loading direction. We can also observe that the mechanical responses, including the shear modulus, change depending on the loading direction. The reason is that while for a scale of R f ree = 5, the solid is still anisotropic and heterogeneous, as shown in [START_REF] Goldenberg | Particle displacements in the elastic deformation of amorphous materials: Local uctuations vs. non-ane eld[END_REF][START_REF] Tsamados | Local elasticity map and plasticity in a model Lennard-Jones glass[END_REF].

In these stress-strain curves, the plastic events correspond to the occurrence of stress drops. These stress drops can be seen on the curves shown in gure 2.5 (b). In practice, we measure in each loading direction the stress strain curve until the end of the stress drop, i.e. until the stress begins to increase again.

We can observe in gure 2.5 (b) that the shape of the stress drop can be very dierent: it can be a very abrupt drop followed instantaneously by a new stress increase as in the curve representing α = 45°. But this stress drop can be much smoother as we can see in the curve representing α = 90°. We can also remark that the stress drops occur at dierent stress and strain values depending on the loading direction α.

For a given patch, the stress-strain curves are then measured until the stress drops along dierent loading directions. By repeating this process over the different grid sites, we can probe locally our system at dierent locations. From the measure of the local mechanical response of material along dierent directions, we construct scalar indicators to probe the material local properties. We focus in this section on the three local indicators used in this manuscript.

The rst one (i) is the local threshold along the direction α: τ c (α). As we can see on gure 2.5 (b), τ c is the stress value at which the patch plasties when loaded in the direction α. It is independent by construction on the elastic local stress τ 0 (α) within the system prior to loading and so gives directly informations about the local structural properties of the material, whether it is locally hard or soft.

In order to measure the propensity of a site to plastify along a given direction α for a given state, we also construct two other indicators. Among them, we dene (ii) the distance to threshold along the direction α: ∆τ c (α) = τ c (α)τ 0 (α) with τ 0 (α) the initial stress within the patch before deforming it. This indicator corresponds to the stress increment needed to trigger a plastic event in the direction α.

In the thesis of M. Lerbinger [START_REF] Lerbinger | Local shear rearrangements in glassy systems: From micromechanics to structural relaxation in supercooled liquids[END_REF], it is shown that each patches possess a discrete number of weak slip planes. When loading the system along a given direction α l , the triggered plastic event corresponds to the closest weak slip plane in the direction α min .

However, in the patches we use to measure the local yield stress, the atoms within region I, in which the plastic rearrangements can occur, are highly constrained by the surrounding region II. Indeed, the atoms within region II are xed and act as xed boundaries conditions to the atoms of region I. As a consequence, the atoms of region I close to the boundaries with region II have a dierent behavior than those in the center of region I. This high constrain might bias the activated weak plane α min when loading the patch in a given direction α l .

We consider the slip direction α min triggered when loading the patch in the direction α l in the absence of constrains. α min is dened as the angle minimizing the projection of ∆τ c (α) along α l , as:

min α ∆τ c (α) cos(2[α -α l ])
.

(2.17)

For a given loading direction α l , it gives that if the corresponding activated slip direction α min is too dierent from α l , the slip direction α min might not be activated due to the constrains on the patch.

Based on this idea, we develop a third indicator (iii) ∆τ y (α l ) corresponding to the minimum distance to threshold ∆τ c (α) projected along the loading direction α l . This indicator corresponds roughly to the local distance to threshold we should have measured in the absence of constrains. This indicator is expressed as:

∆τ y = min α ∆τ c (α) cos(2[α -α l ]) with |α -α l | < 45 • .
(2.18)

In the absence of constrains within the patch, we thus expect ∆τ y (α l ) = ∆τ c (α l ). The comparison between ∆τ y and ∆τ c in their ability to predict the location of future plastic events will be shown in chapter 3. This indicator is useful to predict and understand the location of plastic events from the sole knowledge of the ∆τ y (α l ) map of the system prior to any deformation. This will be studied in chapter 3.

Contrarily with ∆τ y , ∆τ c (α) is not biased by a projection along the loading direction. It is then a more precise quantity to study anisotropy induced by plastic deformation. It will be used on chapter 5 to study the Bauschinger eect in our systems.

In addition to the measurement of the thresholds and the distance to thresholds, the local yield stress method allows us to estimate the stress relaxation upon plasticity. More precisely, we can measure the amplitude of the stress drop ∆τ r corresponding to a plastic event. ∆τ r is represented in gure 2.5 (b).

From this measure, we can study the link between the local yield stress and the following stress drop.

When considering the quantity ∆τ y (α l ), the activated plastic rearrangement occurs along the weakest direction α min . Thus, the stress relaxation of the plastic event observed in the direction α l corresponds to the stress relaxation of the event occurring along the direction α min and projected along the direction α l . The stress relaxation associated with ∆τ y (α l ) can then be expressed as:

∆τ p = ∆τ r (α min ) cos(2[α min -α l ]) (2.19)
with α min the activated weak plane.

It is important to note that the local yield stress method we employ has some limitations. We already pointed out that the constrains applied on region I of the patches by the surrounding xed boundary, region II, can hinder certain degrees of freedom and relaxations. The measured thresholds are then likely to be slightly overestimated. These constraints are also expected to impose an important bias and minor the amplitude of the stress relaxation. As the patches are overlapping to include all the atoms within the system, some weak planes might be activated by many patches. This will change the local yield stress statistics.

The inuence of the patches size and of their overlapping will be studied in the chapter 3. Nevertheless, this technique allows us to eciently sample the local plastic rearrangements thresholds, in a non-perturbative manner, for dierent loading directions and on well controlled length scales. These advantages will be exploited in the next chapters.

Conclusion

In this chapter, we introduced the model system we will be studying in this manuscript: the 2D binary Lennard-Jones glass. We saw that this simple model system can be used to describe metallic glasses but also other amorphous solids such as colloidal glasses. By choosing to use a simple model, we seek to observe and understand generic plastic behaviors which can relate to many dierent amorphous systems.

We also presented how our glasses are obtained by using molecular dynamics and energy minimization. By performing an instantaneous quench from a supercooled liquid equilibrated at a given temperature T , we obtain an amorphous solid that corresponds to an inherent state (IS) of this supercooled liquid at T . The lower this temperature, the lower the potential energy of the inherent state and the more relaxed the system will be. To describe the relaxation state of a glass, we will refer in this manuscript to its parent temperature T from which it was equilibrated and instantaneously quenched.

By deforming the glasses with Athermal Quasi Static (AQS) simple shear, we can perform very low shear rate simulations, unreachable through molecular dynamics. This protocol corresponds to the limit T → 0 and γ → 0.

Finally, we presented the methods used to compute the local strain and the local yield stress. We also introduced the dierent local indicators which can be extracted from the local yield stress method. These indicators will be extensively used in this manuscript to study the link between structure and plastic behavior of our glassy systems.

In the next chapter, we will study the inuence of the degree of relaxation on the local yield stress statistics. We will also investigate the slip threshold susceptibility regarding the patch size and the loading orientation.

Chapter 3

Probing the inuence of thermal history on glass structure through the mechanical mesoscope

In the last chapter, we presented the model system we study in this manuscript: the two-dimensional Lennard-Jones glass. We detailed the protocols we apply to prepare the glasses and deform them. We nally explained the methods we use to probe the local properties of our systems. A particular attention was given to the measure of the local yield stress, a mechanical mesoscope we apply extensively in this manuscript.

In this chapter, we will use the local yield stress to study the inuence of the thermal history on the material local properties. the link between the local properties of the systems and plasticity will also be investigated. A discussion of the inuence of the patch size on the measure of the local yield stress will nally be given.

The rst section will be dedicated to the local yield stress statistics in our systems. In a second section, we will examine the inuence of the loading direction on plasticity and on the local yield stress. Finally, in a third section, we will address the eect of the length scale over which the local yield stress eld is computed. This chapter refers to the results published in [START_REF] Barbot | Local yield stress statistics in model amorphous solids[END_REF].

Local yield stress statistics

In this section, we will study the local yield stress statistics: their link with the thermal history and plastic events. We will in a rst part measure the local yield stress distribution for the dierent quench protocols. Then, we will calculate the correlation between the plastic activity and the local soft regions. The last part of this section will focus on the variation of the amplitude of the stress relaxation upon plasticity with the local yield stress.

Distribution

In this part, we study the inuence of the thermal history on the local yield stress statistics. To do so, we consider glasses at three dierent degrees of relaxation.

We rst prepare well relaxed glasses obtained with a gradual quench (GQ) from an equilibrated supercooled liquid. The two other systems are obtained by an instantaneous quench from an equilibrated supercooled liquids (ESL) and from a high-temperature liquid (HTL). The corresponding parent temperature, or ctive temperature, for these three systems are T p =0.29, 0.351 and 2.98 for GQ, ESL and HTL, respectively. The lower the parent temperature, the more the system is relaxed. More details about the glass preparation are given in the section 2.2.2.

In gure 3.1, we represent the average stress-strain curves for GQ (in blue); ESL (in green) and HTL (in red). We can see in this gure that HTL shows a softer mechanical response compared to ESL and HTL. This gure shows that the more relaxed a system is, the harder it is. We can also remark in gure 3.1 the occurrence of stress drops (as shown in the insert). These stress drops correspond to the occurrence of plastic events. We can then see that the degree of relaxation of the system changes its mechanical properties. We want to study the inuence of the degree of relaxation on the local properties of our system. To do so, apply the local yield stress method on our three systems: GQ, ESL and HTL.

The local yield stress method consists of identifying the rst plastic event undergone by atoms inside a small circular patch (of radius R f ree = 5) by imposing on this patch a pure strain in the loading direction α l . From this method, we measure in this chapter the local distance to threshold projected along the loading direction α l : ∆τ y (α l ). This indicator gives the stress increment in the direction α l needed to trigger locally a plastic event. More details about this indicator are given in section 2.3.2.

From this indicator, we rst measure the local yield stress maps of our glasses, as shown in gure 3.2. In this gure, we represent the local yield stress maps computed on a regular grid for the three dierent quench protocols: (a)

HTL, (b) ESL, and (c) GQ. From these maps, we can remark the abundance of soft regions (in red) in the HTL system, while the GQ system is mostly composed of hard regions (in blue) with few soft zones embedded in this hard skeleton [START_REF] Shi | Strain Localization and Percolation of Stable Structure in Amorphous Solids[END_REF][START_REF] Shi | Does metallic glass have a backbone? The role of percolating short range order in strength and failure[END_REF]. For ESL, the degree of relaxation is comprised between HTL and GQ, thus its local yield stress landscape presents an intermediate situation. and GQ (in blue). This result shows that the more a system is relaxed, the more its probability density is shifted towards high values. This result is consistent with the observation done on the local yield stress maps gure 3.2.

By using this method, we are able to study the statistics of the sites at the onset of plasticity, i.e. very close to plastify. These sites correspond to the low ∆τ y tail of the distribution shown in gure 3.3. These low ∆τ y are important as they control the early plastic response of the system. Indeed, they are the regions which are the most likely to plastify as the system is loaded.

Previous studies based on mean eld approaches [START_REF] Lin | Mean-Field Description of Plastic Flow in Amorphous Solids[END_REF], as well as atomistic [START_REF] Karmakar | Statistical physics of the yielding transition in amorphous solids[END_REF][START_REF] Hentschel | Stochastic approach to plasticity and yield in amorphous solids[END_REF] and mesoscopic simulation [START_REF] Lin | On the density of shear transformations in amorphous solids[END_REF][START_REF] Lin | Criticality in the Approach to Failure in Amorphous Solids[END_REF] predicted that the statistics of the sites at the onset of plasticity follow a power law with a universal exponent θ, independent of the degree of relaxation of the considered system. More precisely, they proposed a scaling for these soft regions such as lim ∆τy→0 P (∆τ y ) ∼ ∆τ θ y where θ is a non-trivial exponent. This exponent θ was estimated to be θ ≈ 0.6 [START_REF] Karmakar | Statistical physics of the yielding transition in amorphous solids[END_REF][START_REF] Hentschel | Stochastic approach to plasticity and yield in amorphous solids[END_REF].

To see if we observe a similar evolution of the low ∆τ y tails, we measure the tail of the ∆τ y Cumulative Distribution Function (CDF) as shown in the inset of gure 3.3. If the low ∆τ y population follow a power-law with an exponent θ, From this gure, we observe that the tail of the CDF is dependent on the quench protocol. Indeed, the tail for ESL follows a power-law with an exponent of ≈ 1.6. The yellow curve represents a power-law with an exponent of exactly 1.6. When comparing it with ESL we indeed observe a similar trend. On the other hand, HTL seems to have a higher exponent while GQ seems to have a lower one.

From these observations, the behavior of P (∆τ y ) in the limit of low ∆τ y seems to depend on the degree of relaxation of the system. There may be many explanations for this disagreement between our observations and the studies observing an exponent θ independent on the degree of relaxation. For instance, the lack of statistics for small ∆τ y values or the strain increments ∆γ applied on the patches could explain this dierence. Also, the xed boundary conditions used to compute the local yield stresses may prevent some relaxation of the system.

To study the onset of plasticity, the previous approaches have considered the distribution of the critical strains applied to the whole system. The critical strains corresponding to the applied strain at which the rst plastic event is triggered in the system. Our approach would be strictly equivalent to this one if our systems were elastically homogeneous. But assuming that the systems are elastically homogeneous is a strong assumption at this length scale [START_REF] Tsamados | Local elasticity map and plasticity in a model Lennard-Jones glass[END_REF]. The answer to this question deserves more investigation, which is outside the scope of the present study.

Correlation with plastic activity

In the previous section, we saw that the lower the degree of relaxation, the harder the system will be. In this section, we will link the measure of the local yield stress with the plastic activity in the model glasses.

As we explained in the previous section, we can see in the inset of gure 3.1 a stress drop corresponding to the occurrence of a plastic event. More specically, the inset corresponds to the manifestation of a single plastic event. On gure 3.4, we show the atom displacement eld (arrows) and the local strain map (colormap) measured between the onset of instability and just after the event, i.e. just before the stress drop and just after the stress drop, respectively. In this gure, we can see that the atom displacement eld takes the shape of a quadrupole similar to an Eschelby inclusion. In the center of this quadrupole, we can remark an increase of the local strain. This increase of local strain corresponds to the location of the activated plastic event. Based on this gure, we dene the location of the plastic event as the maximum of the local strain map measured between the onset of instability and just after the event.

By using this method, we measure the location of the plastic events occurring as we load our system. In the gure 3. To quantify this correlation, we apply the same method as [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF]. We use a correlation coecient to relate the correlation between the plastic events location as they appear and a local scalar eld, here the local yield stress. The aim is to compute if our indicator is able to predict for the location of successive plastic rearrangements from the sole knowledge of the as-quenched state of the system.

This correlation coecient is determined as follows: (i) we rst calculate the cumulative distribution function of our structural indicator, here ∆τ y . Then (ii) as we deform our system, we measure for each plastic event its location on the grid (i max ). (iii) For each of these plastic events, we note the applied strain γ xy at which they were triggered.

After that, (iv) for a plastic event triggered in the site i max at an applied strain of γ xy , we measure the corresponding local yield stress ∆τ y (i max , γ xy ) in the as-quenched state. We next (v) consider the value of the cumulative distribution function (CDF) corresponding to ∆τ y (i max , γ xy ): CDF [∆τ y (i max , γ xy )].

If ∆τ y (i max , γ xy ) is soft compared to the rest of the system, then CDF [∆τ y (i max , γ xy )] would be close to zero. On the opposite, if it is harder than the rest of the system, then the corresponding CDF would be close to 1.

From this, our correlation coecient is dened as:

C ∆τy = 1 -2CDF [∆τ y (i max , γ xy )] (3.1)
where CDF is the disorder average of the cumulative distribution function.

As it is dened, C ∆τy ∼ 1 corresponds to a perfect correlation between the location of the plastic rearrangements and the location of the lowest local yield stress grid point. Also, C ∆τy ∼ 0 corresponds to an absence of correlation.

Note that in the Eq. 3.1, the stress redistribution caused by successive rearrangements is neglected. Furthermore, only the position in the grid of the maximum of the local strain i max is considered. As a consequence, the possibility that a plastic event is composed of several rearrangements is not taken into account. Also, we do not take into account the renewal process (i.e. the change of local yield stress after a plastic event) and we neglect the elastic heterogeneities.

In gure 3.5 (a) we show the variation of C ∆τy with the applied deformation γ xy for our three quench protocols: HTL (black symbols), ESL (red symbols) and GQ (blue symbols). In this gure, in agreement with [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF], we observe an excellent correlation between plasticity and the initially soft regions.

Moreover, this correlation shows a slow decrease with the applied strain γ xy . This decrease is mainly due to the elastic noise caused by each activated rearrangements. Indeed, the stress released by a plastic event modies the local stress of its surrounding regions. This change of local stress thus changes the local distance to threshold ∆τ y . Still, we can see from this gure that this noise is weak compared to the values of ∆τ y . This leads to the slow decorrelation.

When comparing the dierent quench protocols, we measure that the rst ten plastic event rearrangements occur in regions belonging to the softest 23, 13 and 8.5% sites for the HTL, ESL and GQ systems, respectively. This shows a more robust correlation for well relaxed systems.

For our most relaxed system, GQ, it is interesting to note that the correlation decreases sharply for a deformation corresponding to the formation of a shear band in which the strain will be localized. It occurs for an applied strain around γ xy ∼ 0.06.

Moreover, we can understand the better correlation for the more relaxed systems directly from their local yield stress distribution. Indeed, the more relaxed systems have a much lower population of low yield threshold zones. As a consequence, the plastic events are concentrated in the few existing soft regions. This will increase the correlation between the softest zones and plasticity. . From this, we obtain a characteristic decorrelation strain γ d of 0.034, 0.044 and 0.059 for the HTL, ESL and GQ systems respectively. This measure shows that the more stable the system, the slower its decorrelation strain. Two main reasons can be found to explain this observation.

First (i), as we saw on gure 3.3 representing the ∆τ y distribution for the three quench protocols, a more relaxed system has on average a higher local yield stress. The more stable systems are then less perturbed by the mechanical noise caused by the plastic events.

The second reason (ii) is that as we saw on gure 3.1 representing the stress strain curves for the three quench protocols, the poorly relaxed systems are more ductile and have a very small initial elastic branch. As a consequence, at a given strain γ xy , more plastic events have been activated for the less relaxed system. This leads to a longer characteristic decorrelation strain for the more stable systems. We can also remark that for GQ, the decorrelation strain and the peak stress, corresponding roughly to the end of its elastic branch, are both around γ xy ≈ 0.06.

As we previously expressed, this method used to correlate the local yield stresses in the as-quenched systems with plastic activity assumes that each plastic events correspond to an individual and well localized local plastic rearrangement. However, as shown in [START_REF] Maloney | Subextensive Scaling in the Athermal, Quasistatic Limit of Amorphous Matter in Plastic Shear Flow[END_REF][START_REF] Ratul Dasgupta | Microscopic Mechanism of Shear Bands in Amorphous Solids[END_REF], while this hypothesis is relatively well satised for small deformations, it does not hold anymore at higher deformation. Indeed, at higher deformation we can observe occurrence of avalanches in the form of system spanning events.

To circumvent this problem, we deal directly with the correlation between the entire local yield stress eld of the as-quenched state ∆τ y and the cumulative deformation eld (γ xy ) for an applied strain of γ xy . The cumulative deformation eld (γ xy ) corresponds to the measure of the cumulated local strain eld at γ xy measure following the protocol described in 2.3.1.1.

To meausure the correlation between the ∆τ y eld and the (γ xy ) eld, we use a similar method as [START_REF] Smessaert | Structural relaxation in glassy polymers predicted by soft modes: a quantitative analysis[END_REF]. In practice, we calculate the cross-correlation, or

Pearson's correlation, as a function of the applied strain γ xy as

ρ P e ∆τy, (γ xy ) = - N i=1 (∆τ i y -∆τ y )( i -) N σ ∆τy σ (3.2)
with N the number of points on the grid on which the thresholds are calculated. σ ∆τy and σ correspond to the standard deviation of ∆τ y and respectively.

The minus sign is added here to obtain a positive value since large are expected for small ∆τ y regions (i.e., anticorrelation). ∆τ y and correspond to the ensemble average of corresponding quantity. Note that explicit dependence on γ xy of is omitted in the right-hand side for the sake of simplicity.

The evolution of ρ ∆τy, as a function of the applied strain is shown on gure 3.1 (b). As for C ∆τy , we observe a better correlation between plasticity and local yield stress for the well relaxed system. However, the trend of these two quantities are dierent. Indeed, for ρ P e ∆τy, , the correlation begins by increasing before decreasing. We understand this increase as being caused by the accumulation of plastic rearrangements on weak sites.

We also observe that the decay of the correlation for the GQ system is accelerated around γ xy ∼ 0.06. This applied strain corresponds to the typical strain at which a shear band occurs in this system. Within the shear band, the plastic events is concentrated and very few plastic events occur outside of it. As a consequence, the initially soft regions outside of the shear are ignored as the plasticity accumulates in the band. This lead to the rapid decorrelation after γ xy ∼ 0.06.

In the systems HTL and ESL we can see their correlation coecient is decaying more slowly compared to GQ. We can also remark that in the HTL and ESL systems, no persistent shear band formation is observed. In the absence of persistent shear band the plastic events are then not restricted to a localized region. We can imagine that this lead to the observed slower decorrelation.

While the correlations appears smaller than what we could have expected from the computation based on local rearrangements in Eq. 3.1, we can remark that this calculation was based on simple assumptions. For example, we have not dissociated the elastic part from the plastic part when calculating .

Still, we obtain a good result for the correlation between plasticity and ∆τ y (Eq. 3.1) and for the cross-correlation based on the cumulated strain eld (Eq. 3.2). These correlations also are signicantly better and more persistent with deformation than those obtained for the local classical structural indicators reviewed in [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF].

Local relaxation vs local yield stress

In the previous section, we measured the correlation between the plastic activity and the local yield stress eld. In this section, we will study the link between the amplitude of the local plastic relaxation that follows plastic rearrangements with the corresponding local yield stress.

By construction, the quantity ∆τ y (α l ) corresponds to the lowest distance to threshold ∆τ c in the direction α min projected in the direction α l . The aim of this indicator is to measure the propensity of a local region to plastify as the whole system is loaded in a given direction α l .

To be consistent with ∆τ y we have to consider the plastic relaxation amplitude ∆τ r (α min ) in the direction α min . From this, we measure ∆τ p (α l ) corresponding to the projection of ∆τ r (α min ) in the direction α l . More details about the measure of these quantities are given in the section 2.3.2.

Note that the estimator of the stress relaxation ∆τ p slightly underestimates the plastic relaxations because of the projection. Nevertheless, it gives access to a suciently simple scalar indicator. We veried that the absence of projection does not qualitatively change our results.

We report in gure 3.6 (a) the distributions of ∆τ p for the three quench protocols in a lin-log scale. From this gure, we can rst observe that the distributions of the three protocols show an exponential decay. Also, by measuring the mean plastic relaxation amplitudes with an exponential regression, we obtain: ∆τ p =0.164, 0.269 and 0.337 for HTL, ESL and GQ systems, respectively. This result shows that on average, the amplitude of the stress drop increases with the degree of relaxation of the system. We saw that ∆τ p gives an estimation of the amplitude of the stress relaxation upon plasticity. From this quantity, we can estimate the amplitude of the plastic deformation γ p upon a rearrangement, or slip increment. This can be measured as γ p ∼ ∆τ p /µ where µ is the average shear modulus of the glass.

In the inset of gure 3.6 (a), we show the γ p distributions for the three quench protocols in a lin-log scale. We can see on this gure that the three distributions collapse on a single curve with a mean of γ p =0.00887 independently of the quench protocol. These results are in agreement with previous atomistic computations based on dierent methods such as the mapping between elastic eld and Eshelby inclusion model [START_REF] Albaret | Mapping between atomistic simulations and Eshelby inclusions in the shear deformation of an amorphous silicon model[END_REF][START_REF] Boioli | Shear transformation distribution and activation in glasses at the atomic scale[END_REF] and automatic saddle point search techniques [START_REF] Yue Fan | Energy landscape-driven non-equilibrium evolution of inherent structure in disordered material[END_REF]. On gure 3.6 (b) we show the relation between the stress relaxation amplitude ∆τ p and the local yield stress for as-quenched systems ∆τ y for the three quench protocols. The rst striking result is that ∆τ p increases on average with ∆τ y . But remarkably, this dependence seems independent of the quench protocol.

The fact that the relaxation amplitude increases with the local yield seems reasonable. Indeed, a site with a high local yield stress has likely stored more elastic energy at the onset of plasticity. We can then expect it to release more stress upon yielding. However, we have no explanation to derive this relationship at the moment. By tting empirically with an exponential, we obtain ∆τ p ∼ Ae B∆τy with A ≈ 0.054 and B ≈ 0.976.

Though we are condent in the capacity of this local method to quantify the plastic thresholds, the measurement of the relaxation amplitude is much more subject to caution. Indeed, the frozen boundary conditions prevent some relaxations and might bias the measured relaxation amplitude.

The picture that emerges from gure 3.6 is nevertheless interesting and sheds new light on the plastic deformation as it greatly simplies representation of relaxation in glassy systems. Indeed, we were able to measure a characteristic relaxation which does not seem to depend on the degree of relaxation of the system.

Orientational inuence

So far, we have only considered the local yield stress in a single direction α l = 0°.

Indeed, this angle corresponds to the loading direction applied on the whole system, as described in the section 2.2.3.

Most of the proposed local indicators are scalar quantities and do not take into account dierent loading directions [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF], with the exception of indicators based on soft modes by taking advantage of the vectorial aspect of vibrational eigenmodes [START_REF] Rottler | Predicting plasticity with soft vibrational modes: From dislocations to glasses[END_REF][START_REF] Smessaert | Structural relaxation in glassy polymers predicted by soft modes: a quantitative analysis[END_REF].

Here, we take advantage of one of the strength of the local yield stress probing method: its capacity to be calculated for any loading directions. This orientational aspect is important because loading a system in dierent directions does not necessarily activate the same plastic rearrangements [START_REF] Gendelman | Shear Transformation Zones: State determined or protocol dependent? EPL[END_REF].

In this section we will study how the local yield stress varies as the loading direction is changed and investigate the uctuations in the orientation of plastic rearrangements around the loading direction α l .

Loading direction

We want to test how the activated plastic events and the local yield stress depends on the directions of the mechanical loading. To do so, we deform our glasses with the same AQS protocol but for dierent orientations.

Until now, we only applied on our systems a simple shear in the positive direction as described section 2.2.3 and illustrated by the red arrows on gure 3.7 (a). This deformation corresponds in the innitesimal strain limit to shearing along α l = 0°.

Here we also deform our systems with a pure shear by applying on our systems strain increments ∆γ/2 = -∆ xx = ∆ yy . This corresponds to the red arrows gure 3.7 (b). Finally, we also load our system with a simple shear in the negative direction by applying deformation increments ∆γ = -∆γ xy . It is illustrated with the red arrows gure 3.7 (c). These remote loadings corresponds in the innitesimal strain limit to shearing along α l = 45°and α l = 90°d irections, respectively.

With this denition, the pure shear produces a diagonally oriented shear.

On the other hand, the negative simple shear corresponds to a laterally oriented shear. This laterally oriented shear is in the opposite direction with respect to the positive simple shear remote loading employed so far.

In gure 3.7, we represent the local yield stress maps measured for dierent loading directions (top row). These maps are obtained from the same GQ glass.

We consider in this gure three loading directions: α l = 0°(a), α l = 45°(b) and α l = 90°(c). We can observe dierent local yield stress maps for the dierent loading directions. The location of the softest regions also seem to be inuenced by the loading direction. In the gure 3.7, the open black symbols correspond to the location of the ten rst plastic events activated when loading the systems in the same directions as the patches. We can see from the gures 3.7 that most of the plastic events seem to occur in dierent regions for the dierent loading protocols. Only few of them occasionally appear in the same location for dierent loading directions.

Thus, the location of plastic rearrangements show a strong dependence on the loading protocol. This result is in agreement with [START_REF] Gendelman | Shear Transformation Zones: State determined or protocol dependent? EPL[END_REF].

We can understand these dierences by looking at the associated local yield stress eld. Indeed, we observe a strong dependence on the loading directions for the ∆τ y maps. At the same time, we know that the plastic events tend to occur in the soft zones. The rotation of the shear orientation then results in the appearance or disappearance of soft and hard regions. As a consequence, the regions in which the plastic events are the more likely to occur are shown to change with the loading direction.

For example, the areas close to the rst and third plastic rearrangements for positive simple shear in gure 3.7 (a) disappear in the case of negative sim-ple shear in gure 3.7 (c). Conversely, soft areas appear close to the fth and seventh rearrangements in Fig. 3.7 (c). The quantication of correlations for α l = 45°and α l = 90°through Eq. 3.1 and 3.2, as described in section 3.1.2, is quantitatively similar.

To highlight the discrete aspect of the variation of the local yield stress eld, we compute the threshold contrast (TC) existing between two loading directions.

We dene this contrast as the ratio between their dierences and their averages as

T C(α 1 l , α 2 l ) = |∆τ y (α 1 l ) -∆τ y (α 2 l )| (∆τ y (α 1 l ) + ∆τ y (α 2 l ))/2 (3.3)
where α 1 l and α 2 l are two dierent remote loading directions.

The contrast maps are shown in the bottom row of gure 3.7 (d)-(f ). On these maps, we can observe regions with a high contrast corresponding to an important change of the threshold values for dierent loading orientations. These maps conrm that changing the loading direction of the patches provokes the appearance or the disappearance of the soft zones.

When comparing the dierent contrast maps, we remark that a greater difference between the loading direction angles ∆α l = |α 1 l -α 2 l | leads to more regions with a high contrast. Indeed, the contrast map of gure 3.7 (f ), which corresponds to ∆α l = 90°, has a higher number of high contrast zones than the maps of gure 3.7 (d) and (e) for which ∆α l = 45°.

To quantify this trend, we compute the cross-correlation of the local yield stress eld as a function of the loading angle dierence ∆α l . We show the result for the three quench protocols on gure 3.8. We observe from this gure that the correlation of the local yield stress eld decreases quickly with the loading angle for all the quench protocols.

Moreover, we also notice that this correlation is never completely zero even in the opposite loading direction, i.e. ∆α l = 90°. We attribute this eect to the small correlation existing between stable (unstable) zones and their tendency to have large (small) slip barriers as seen in [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF][START_REF] Rodney | Yield stress in metallic glasses: The jamming-unjamming transition studied through Monte Carlo simulations based on the activation-relaxation technique[END_REF]. The measure of the local stability is independent on the direction [START_REF] Rodney | Yield stress in metallic glasses: The jamming-unjamming transition studied through Monte Carlo simulations based on the activation-relaxation technique[END_REF]. As a consequence, a stable region have a lower probability to have a soft orientation. Still, the decorrelation due to the directional aspect is clearly the dominant eect.

This result shows that local stress thresholds are a very sensitive probe of the loading protocol insofar as it is possible to accurately predict the plastic activity as a function of the orientation of the load.

Unlike [START_REF] Gendelman | Shear Transformation Zones: State determined or protocol dependent? EPL[END_REF], we thus believe that these results are consistent with a plasticitybased view of shear transformation zones. Indeed, from our point of view, the dependence of the plastic activity upon the loading protocol does not rule out the existence of plastic deformation via discrete units encoded in the structure and that these discrete units clearly preexist within the material prior to loading.

Our results show rather that the plastic deformations of an amorphous solid, at least for the transient regime at small deformations, can be seen as a sequence of activation of discrete shear transformation zones having weak slip orientations.

Fluctuations around the loading direction

In the previous section, we studied the inuence of the loading direction in the measured local yield stress eld. In this section, we want to study the uctuations of the plastic rearrangement's orientation around the remote loading direction α l . More precisely, we want to study the uctuations of α min , the weakest slip direction triggered for a given loading direction α l .

We rst conrmed that the local yield stress distribution for as-quenched glasses does not depend on the loading orientation, the as-quenched amorphous solids being on average isotropic. This result is discussed more in depth in chapter 5 where we analyze the mechanical anisotropy induced by severe plastic deformation.

In the gure 3.9, we show the distributions of the weakest local slip direction α min around loading direction α l for the three quench protocols HTL (black symbols), ESL (red symbols) and GQ (blue symbols). We can see from these distributions that the uctuations of α min around α l seems to depend on the quench protocol.

To conrm this trend, we t these distributions with a Gaussian function (lines) and measure the corresponding standard deviation σ αmin . The evolution of the measured standard deviation with the quench protocol is shown in the insert of gure 3.9. In this gure, we can see that the standard deviation decreases slightly as the system is more relaxed.

Inuence of the orientation mismatch on the correlation

In the last section, we measured the uctuations of α min , the weakest slip direction triggered for a given loading direction α l . In this section, we will study the inuence of the orientation mismatch on the correlation.

More particularly, we now want to conrm the pertinence of using the projected distance to threshold ∆τ y (α l ) to predict the location of plasticity in our study. Indeed, this indicator is based on the hypothesis that when we load the patch along the direction α l , the corresponding weakest slip direction α min might not be activated due to the presence of constrains within the patch due to the xed boundaries. More details are given in the section 2.3.2.

To conrm the pertinence of ∆τ y , we compare three dierent indicators based on the local yield stress probing method: (i) the minimum ∆τ c (α) over all directions, (ii) the distance to threshold in the loading direction ∆τ c (α = α l ) and (iii) the distance to threshold projected along the loading direction ∆τ y (α = α l ). Note that in the absence of constrain, we should measure ∆τ c = ∆τ y .

We measure in our GQ glasses the correlation between plasticity and our three indicators as a function of the applied strain by using the Eq. 3.1. The results are plotted on gure 3.10.

First of all, we observe from this gure that the minimum of ∆τ c (α) over all direction (i), i.e. min α ∆τ c (α) gives the best correlation for the very rst plastic event. The correlation for this quantity then quickly decreases with deformation. This indicator gives the propensity of a region to plastify when subjected to an isotropic excitation, such as uctuation on thermal energy. It is therefore sensitive to small barriers, thus explaining its good correlation at low strain.

Conversely, the correlation for ∆τ c (α = α l ) (ii) is the lowest at small deformation meaning that it misses the low thresholds which are slightly disoriented with respect to the loading direction of the system. But its correlation becomes better for higher deformation comparatively with min α ∆τ c (α).

Finally, ∆τ y (iii) shows the best correlation for both small and large defor- mations. Indeed, due to the projection, it is sensitive to the small thresholds responsible of plasticity at small deformation. It is also sensitive to the macroscopic loading direction which have a bigger inuence on plasticity at higher strain as shown by the behavior of the correlation for ∆τ c (α = α l ).

We can see here the importance of having access to a directional quantity.

The local yield stresses dened in this chapter is therefore a good compromise between simplicity (purely local and scalar) and performance to understand the onset of plasticity from as-quenched glass. This result also conrms the inuence on the measured threshold by the constrains imposed on the particles within the patches by the xed boundaries conditions. This will be also discussed in the next section.

Patch size inuence

In the previous section, we studied the inuence of the loading direction on the measure of the local yield stress. In this section, we will focus on the inuence of the size of the patches over which the local yield stress is measured.

Optimal size

In this section, we aim to understand the eect of changing the patch size R f ree over which the local yield stress is computed. Indeed, through the local yield stress method, we aim to have a better understanding of the discrete origin of the mechanical properties of our systems.

To do so, we use laws of continuous mechanics to measure the stress strain curve when we deform the patches. In this way, we can probe the occurrence of local discrete atomic rearrangements, corresponding to a stress drop in the stress-strain curve.

The patches in the local yield stress probing method are in the limit between the material's discrete and continuous properties. Indeed, we want the patches to be as small as possible to increase the spatial resolution. However, if the patches are too small, the laws of continuum mechanics would not be relevant anymore [START_REF] Goldenberg | Particle displacements in the elastic deformation of amorphous materials: Local uctuations vs. non-ane eld[END_REF][START_REF] Tsamados | Local elasticity map and plasticity in a model Lennard-Jones glass[END_REF].

Also, if the patches are too small, the atoms within the patches are overconstrained. This over-constrain bias the activated plastic events. This would lead to a biased measure of the yield thresholds.

To look for a good compromise for the patch size, we use the correlation of the threshold with plastic activity as a function of the applied strain C ∆τy (γ)

as dened by the equation 3.1. We are interested in the inuence of the patch size R f ree on this correlation.

Indeed, if the spatial resolution is poor, the plasticity and the local yield stress eld are likely to decorrelate quickly as the system is further deformed.

On the other hand, if R f ree is too small, we can expect that the measured local yield stress is too biased, which will reduce the correlation for the rst plastic events (i.e. C ∆τy ((γ → 0)).

To probe which patch size is a good compromise between consistency and spatial resolution, we rst consider many values of R f ree , going from 2.5 to 15σ. The size of the grid R sampling over which ∆τ y (R f ree ) is probed is kept constant and equal to L/39 ≈ R cut .

To quantify the degree of correlation for the dierent R f ree , we construct three quantities based on the correlation function C ∆τy (γ) dened in Eq. 3.1.

The rst (i) one is the correlation of the local yield stress eld with the rst plastic rearrangements C ∆τy (R f ree , γ xy → 0 + ).

The In this gure, we can see that the correlation with the rst plastic event C ∆τy (R f ree , γ xy → 0 + ) (top) increases with R f ree for all quench protocols. Also, for R f ree < 5 we observe a clear drop of the correlation. To understand the reduction of the correlation when we decrease R f ree , we identify three main causes.

The rst one (i) is that the relevance of using laws of continuum mechanics to measure the mechanical response within the patches reduces as R f ree decreases.

Indeed, in [START_REF] Goldenberg | Particle displacements in the elastic deformation of amorphous materials: Local uctuations vs. non-ane eld[END_REF], Goldenberg et al. found that the continuum description should apply for a scale of the order of R ≈ 5. This result is consistent with the correlation drop for R f ree < 5.

Also (ii), at small scale, the measured local elastic moduli is strongly heterogeneous with soft and sti regions [START_REF] Tsamados | Local elasticity map and plasticity in a model Lennard-Jones glass[END_REF]. Therefore, for big enough patches, the loading heterogeneities within the patches is more consistent with the heterogeneities within the whole system. Subsequently, when increasing the radius R f ree , the loading felt within it corresponds more and more to the loading this same region would have felt when the whole system is deformed.

Finally, (iii) the smaller the patch size, the more the frozen boundary conditions over-constrain the measurement of the local shear stress thresholds. This over-constrain bias the measure of the local yield stress. The inuence of this eect was observed in the section 3.2.3.

From (ii) and (iii), when increasing R f ree , the plastic event activated within the patches are more likely consistent with the one triggered as the whole system is deformed. This leads to a better correlation between the rst plastic events with the regions with the lowest local yield stress, i.e. the most unstable regions.

In gure 3.11, when looking at the characteristic decorrelation strain γ d (middle), we can see that this quantity reduces for all quench protocols as the size of the probing zone R f ree increases. The main eect of the evolution of γ d with R f ree we identify is the following:

When deforming a patch in a given direction to measure its local yield stress, only its weakest local plastic rearrangements are observed. All the other potential plastic rearrangements present within the patch are then ignored as the amount of stresses they require to be activated are higher.

With too large probing zones R f ree only the weakest sites are detected, all the slightly harder sites which would have been activated at larger strain are ignored. As a consequence, as R f ree increases, the slightly harder sites are less and less detected. By loosing information about the hardest sites, the weakest sites are over-represented. This reduces spatial resolution. This leads to the reduction of the correlation for larger applied strain and so to a lower characteristic decorrelation strain γ d .

Our last correlation is the average of C ∆τy computed as:

C ∆τy (R f ree ) = (1/γ * ) γ * 0 C ∆τy (R f ree , γ xy )dγ xy (3.4)
The upper bound of the interval of integration γ * is chosen equal to the largest decorrelation strain γ * = γ d (R f ree = 2.5), i.e. computed for the smallest R f ree . This indicator of correlation is more global than the two previous one (C ∆τy (R f ree , γ xy → 0 + ) and γ d ) as it contains informations about the degree of correlation at low strain, corresponding to the rst plastic events, but also during the deformation, corresponding to when the glass looses its memory.

In gure 3.11 (bottom), we can see that the correlation average decreases as R f ree increases, except when R f ree goes from 2.5 to 5. In this case, C ∆τy (R f ree ) is almost unchanged. We even observe that the maximum of correlation is attained for R f ree = 5 for GQ and HTL. Noticeably, while the trends of the correlation indicators as a function of R f ree are independent on the quench protocols, we observe a better correlation for more relaxed systems. This observation is consistent with the results from section 3.1.2.

From these results, we can see empirically that R f ree = 5 is a good compromise to calculate the correlation between the plastic activity and the local yield stress. Indeed, it is large enough to predict the location of the rst plastic events, while preserving a good spatial resolution.

Statistical size eect

In the previous section, we measured the inuence of the size of the probing zone R f ree on the correlation between the local yield stress and plasticity. In this section, we will study how R f ree aects the slip barrier statistics.

In the previous section, when looking for the optimal patch size, we observed that changing the patch had two main eects. We can classify them as a mechanical and a statistical one.

The mechanical size eect corresponds here to the elastic heterogeneities and the inuence of the frozen boundary conditions. Indeed, increasing R f ree will make the loading heterogeneities within the patch more consistent with those in the whole system. Also, the frozen boundary condition constrains the atoms in its vicinity. The frozen boundary being anely deformed, the atoms within the patch have a more ane displacement due to the constrains. A larger patch size has a lower proportion of atoms aected by the frozen boundaries. As a consequence, its mechanical response is closer to the one of a non-constrained system.

The statistical size eect, corresponds to the bias towards small local stress thresholds when increasing the patch size. Indeed, the local yield stress is primarily controlled by the weakest zones in the patch since its amplitude is given by the smallest threshold within it. As a consequence, a larger patch is more likely include a soft region thus reducing the measured threshold.

To illustrate the statistical size eect, we show on gure 3.12 the maps of ∆τ y computed for R f ree going from 5 to 15 (top row). We can clearly observe that increasing the patch size decreases the proportion of hard zones. To see the inuence on the thresholds statistics, we plot on gure 3.13 the ∆τ y distributions obtained for dierent patch sizes and for dierent quench protocols. From this gure, we can see that increasing R f ree causes a signicant shift of the distributions towards the smallest values of ∆τ y . This result conrms the over-representation of the weakest sites as R f ree increases. When observing the size evolution of the softest regions on gure 3.12 (top row), we remark that these regions seems circular with a radius comparable to R f ree . This can be understood by the fact that the local yield of a patch corresponds to the threshold of the weakest region it contains. As a consequence, a very unstable plastic rearrangement, i.e. with a very low local yield stress, would be the one activated by all the patches surrounding it within a radius of R f ree .

To conrm this, we consider the following: We take as a reference the ∆τ y map for R f ree =5. Then, we consider that for the larger patch sizes, all the thresholds ∆τ y (R f ree ) of the grid points will take the value of the smallest local minima ∆τ y (R f ree = 5) located inside a disk of radius R f ree .

The maps obtained following this protocol are shown on gure 3.12 (bottom row). This purely geometric approach shows a remarkable agreement compared to the local yield stress maps calculated by actually varying R f ree .

From this approach, we also deduce the evolution of the thresholds distributions for R f ree > 5 based on the distribution obtained for R f ree = 5. The obtained distributions are shown on gure 3.13, represented by continuous lines.

We obtain a good agreement between the measured distributions with dierent R f ree and those deduced from R f ree = 5.

From these results, we can deduce that the variation of the distribution of ∆τ y is dominated by statistical eects. Indeed, the increase of R f ree plays the role of a low-pass lter for the thresholds, shifting their distributions toward the smaller yield stress values. However, we observe that the agreement between the measured distributions and those obtained by this geometric approach is lower for less relaxed systems and for large values of R f ree .

We attribute this discrepancy to two main reasons: (i) the less relaxed systems have a higher elastic disorder [START_REF] Feng | The role of congurational disorder on plastic and dynamic deformation in Cu 64 Zr 36 metallic glasses: A molecular dynamics analysis[END_REF]. Therefore, the elastic properties within the patch might depend more of the patch size for HTL compared to GQ. As a consequence, the activated rearrangements might be biased by this elastic disorder.

Moreover, (ii) as we can see on gure 4.9, representing the local yield stress distribution for the three quench protocols in their as-quenched states, HTL has a narrower ∆τ y distribution compared to the other quench protocols. The average ∆τ y of HTL is also much lower. Because of that, the contrast between the softest zones and the ∆τ y landscape is much lower for HTL than for GQ.

As the contrast between the soft and hard region is lower, the mechanical size eects become non-negligible. Indeed, when the softest region is near the xed boundary of the patch, the constraints hinder the activation of the rearrangement. If the contrast within the patch is too small, the constraints might cause the activation of another rearrangement. As a consequence, the sensitivity of the patches to the soft regions might be reduced for less relaxed systems.

Conclusion

In this chapter, we applied the local yield stress probing method on as-quenched glasses obtained from three dierent quench protocols. From this method, we measured ∆τ y (α l ), an indicator designed to measure the propensity of a local region in our glasses to plastify when loaded in the direction α l .

We measure that on average, the more a glass is relaxed, the higher its distance to thresholds ∆τ y . As a consequence, the more a glass is relaxed, the harder it is. Also, a good correlation was found between the plastic activity and the low local yield stress regions. This correlation is better for more stable glasses.

By using the local yield stress method, we were able to probe the amplitude of the stress relaxation after a plastic event ∆τ p . Interestingly, the relaxation amplitude increases on average with the local yield stress and follows an exponential distribution.

We then observed that the local yield stress ∆τ y (α l ) depends on the loading direction α l . Consequently, the locations of the soft and hard regions in our systems are also dependent on the loading orientation. We also showed that the variation of the plastic activity with the loading direction is well captured by the variation of the local yield stress eld calculated using our method.

Finally, we studied how changing the patches size over which the local yield stress is measured will inuence the correlation between the plastic activity and the local thresholds. Based on these results, we found that a radius of R f ree = 5 is a good compromise between the mechanical size eects when the patch is too small and the statistical size eects when it is too large. Moreover, we were able to reproduce the statistical size eects with simple geometric arguments.

Overall, this study showed the impact of the quench protocol on the local structural properties. As the system is deformed, we observed a decorrelation between the plasticity and our structural indicator ∆τ y meaning that the material looses its memory as it plasties. In the next chapter, we will study the inuence of the plastic events on the glass structure by using the local yield stress method. Chapter 4 Rejuvenation and aging: how plasticity changes the local structure

In the previous chapter, we showed that the measure of the local yield stress has a good ability to predict the locations of rst plastic events occurring in the system upon loading. As more plastic events are triggered by mechanical loading, the correlation between the plastic events locations and the local yield stress decreases. This decorrelation is caused by the change of local structural properties after the occurrence of plastic rearrangements. The plastic events are erasing the memory of the initial state of the material [START_REF] Sun | Thermomechanical processing of metallic glasses: extending the range of the glassy state[END_REF][START_REF] Fiocco | Encoding of Memory in Sheared Amorphous Solids[END_REF][START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Fiocco | Oscillatory athermal quasistatic deformation of a model glass[END_REF][START_REF] Nicolas | Deformation and ow of amorphous solids: Insights from elastoplastic models[END_REF]. In this chapter, we will study how plasticity changes the local structural properties of atomistic model glasses.

To do so, we will in a rst section measure the evolution of the local yield stress with plasticity in a deeply quenched glass and see how it captures the changes of structural properties after plastic events. We will in particular study the properties of this post-yield state, dubbed renewed state. The rapidity of this renewal process will also be investigated. Finally, in the last section, the inuence of the thermal history on the post-yield state will be analyzed.

Rejuvenation in a deeply quenched system

In this section we will study the impact of the shear-induced plasticity on the structural properties for our deeply quenched glass GQ introduced in Chapter 3. The interest here would be to study the characteristics of the renewed state and understand how the material evolves as plasticity accumulates.

One of the characteristics of the GQ systems is the formation of a shear band (around γ xy ∼ 0.06) in which the strain is localized. As a consequence, the system will be divided in a yielded region (within the band) and a unyielded one (outside of the band) thus allowing us to study the evolution of the local properties for both regions as the system is deformed.

In a rst part, we will study the evolution of the yield threshold with the local plastic strain. We will then study in a second part how the local yield stress depends on pressure and free volume and see that their impact is on second order compared with rejuvenation. Furthermore, we will measure in a third part the local yield stress distribution in the renewed state, i.e. upon plasticity.

We will then compare the renewed state with the as quenched state. Based on the renewed state, we will try to understand steady ow state properties.

Finally, in a fourth part, we will study if the local yield stress remains isotropic in the renewed and steady state.

Local yield stress vs local plastic strain

To trigger plastic events, the model glasses are deformed up to linear shear strain γ xy = 5 with an athermal quasi-static simple shear deformation in the positive direction (see section 2.2.3 for more details). This loading corresponds, in the innitesimal strain limit, to shearing along α l = 0°. As the glasses are deformed, we measure their local strain as the second invariant of the logarithmic Hencky strain. We use the Hencky strain as it was shown to be a pertinent method to measure the local strain at high deformation [START_REF] Onaka | Appropriateness of the Hencky Equivalent Strain as the Quantity to Represent the Degree of Severe Plastic Deformation[END_REF] (see section 2.3.1.1 for more details). To be able to compare the applied strain and the local strain quantitatively, the applied global strain is expressed following the Hencky formalism, as: E=ln(γ/2 + 1 + γ 2 /4) (see 7.2 for more details).

In gure 4.1, we represent the sample-averaged stress-strain curve for the GQ system. The evolution of the pressure as a function of the strain is also shown (in blue). This curve displays a clear stress overshoot followed by a softening.

We can observe a quick increase of the pressure until the peak stress. After this peak stress, the pressure continues to grow more slowly. This pressure increase in a constant volume simulation is the signature that plastic activity creates atomic congurations which are less well packed compared with the initial state. This phenomenon is often used to justify the freevolume theory [START_REF] Falk | Simulating the mechanical response of amorphous solids using atomistic methods[END_REF]. We will see in a later section that the free volume has a minor contribution on strain softening. As our GQ system is deformed, we systematically observe the formation of a shear band in which the plastic activity concentrates as illustrated in gure 4.2 (a). This gure represents the local strain eld map for a system deformed until E=0.05. We can clearly observe that the plastic deformation is concentrated in a band (in violet/black) while the rest of the system is almost undeformed.

Note that as our glass is deformed using an AQS protocol, this shear band can only arise from structural causes.

We now want to probe the local structural properties of our systems to link it with plasticity. To do so, we are here again using the local yield stress method.

In the present chapter, we aim to understand how the local structural properties change upon plasticity. As a consequence, we need to consider a dierent local indicator than the one used until now.

The indicator ∆τ y (α l ) used in the previous chapter was designed to provide information about the propensity of a local region to plastify as the whole system is loaded in a given direction α l . While ∆τ y is a pertinent indicator to predict the location of plastic events, it directly depends on the local stress, i.e. the local elastic loading. To probe the material structural properties, we need a quantity independent on the system internal elastic stress.

In this chapter, the local structural properties will be estimated through the local yield threshold τ c (α) as dened in section 2.3.2. This indicator corresponds to the stress value at which a plastic event is triggered in a local inclusion following the direction α. This quantity has the advantage of being by construction independent of the local stress initially present in the inclusion. Also, this indicator is able to probe the local structural properties for dierent orientations.

To see how the yield thresholds evolve upon plasticity in a system with shear banding, we plot in gure 4. We now want to have a more quantitative measurement of the softening within the shear band and its evolution. To do so, we rst focus on samples where a single horizontal band nucleates (53 out of 100 samples). Then, for each sample we identify the position of the band center y SB .

To identify y SB , we average the strain elds for a macroscopic strain of E = 0.1 over the x-axis and dene the band center y SB as the position for which the x-averaged local strain reaches its maximum. From this, we plot the The strain prole is represented in gure 4.2 (c). In this gure, we can clearly observe that the shear band grows over time as observed in [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF]. This trend would be inaccessible in experiments as the material would quickly fail after the formation of the shear band. But due to the applied periodic boundary conditions, the system keeps its integrity. The characteristics and evolutions of these shear bands will be studied with further details in section 6.3 but is out of the scope of the present chapter.

From the local yield stress proles, we can see that the material is clearly softer where the glass is locally highly deformed, i.e. within the shear band.

Based on this observation, we can see that plasticity renews the glass locally and that this renewed state is softer than the initial state. Surprisingly, at high deformation, when comparing E = 0.48 and E = 1.44, we can see a hardening in the band as the system is further deformed. One can also note that the glass is measurably harder outside of the band with respect to the quench state at the same location, another unexpected phenomenon.

To have a better understanding of the observed hardening at high deformation and also of how fast the system is locally rejuvenated, we plot on gure 4.3 (a) the average local yield stress as a function of the local strain for dierent applied mechanical strains E. Note that here the average is performed over all the 100 samples, independently on the shear band orientation.

To measure after how many successive plastic rearrangements the system is locally rejuvenated, we need to know the typical strain increment caused by a single plastic event. To do so, we plot on gure 4.3 (b) the distribution of the local strain for the same value of E as in 4.3 (a). This way, we can measure the evolution of the strain heterogeneity as the applied strain increases. We observe from these distributions the formation of a two-peak structure as the shear band nucleates and grows. The low strain peak corresponds to the unyielded regions while the high strain peak corresponds to the regions which have plastied. From these distributions, we introduce the characteristic strain separating these two peaks: * . It corresponds roughly to the scale of the local strain change caused by a plastic rearrangement. We measured that * ∼ 2 c with c ∼ 0.054 corresponding to the typical strain needed to trigger a plastic event withing the patches.

On gure 4.3 (a), we observe that τ c xy systematically decays with the local strain . It indicates a softening caused by plasticity. Remarkably, we measure that the characteristic strain of this decay is of the order of * . This means that the system locally looses the memory of its initial state after essentially a single plastic event.

When looking more closely at the curves of τ c xy as a function of , we notice some noteworthy features. Indeed, the unplastied regions corresponding to a very low strain ( / * 0.1) are signicantly harder than the as-quenched state. Also, the curves show that the variation of τ c xy with the local strain depends on the global strain E. Indeed, for a given , the local yield threshold increases with E. This corresponds to a hardening eect.

Upon yielding, we observe a local softening, followed by a hardening. To understand these puzzling phenomena, we will in the next section study the inuence of the local density and pressure on the local yield stresses.

Evolution of the local yield stress with the local density and pressure

As we saw on gure 4.1, the pressure P increases with E. As the volume is kept constant, the increase of pressure is associated with the decrease of the local density within the plastied regions. In this section, we will study if the reduction of local density and the increase of pressure upon plasticity is related with the observed softening.

We rst focus on E = 0.05 and separate the yielded regions with the unyielded ones. To do so, we dene the regions which have undergone a plastic rearrangement in the following way: if the local strain veries < * /8 we consider that this site is unyielded. Then, if > 2 * , we consider that this local region has yielded. We use here such an important gap in the value of between the yielded and unyielded regions as we want to remove any ambiguity about whether a region has yielded or not.

From this, we plot on gure 4. We also notice that in a same state, the local yield stress slightly increases with the local density. However, this contribution is negligible in the observed softening. Indeed, the dierence of local yield stress between the yielded and the unyielded states is higher than between the low and high density regions.

It is then obvious that the leading cause of softening is the production by plasticity of packing in a dierent state, i.e. presenting dierent τ c xy vs ρ l relation than the initial material. Also, these results clearly show that the local density, while being commonly used in the literature, is not a relevant indicator to understand the mechanical properties of metallic glasses.

In addition to the yielded state, we dene the renewed regions for a given value of the applied strain E. These renewed state at E corresponds to the regions that have yielded between the state E and E + ∆E, with ∆E = 0.05. In this case, to know if a region has yielded, we look at the local strain increments ∆ measured between E and E+∆E. The regions for which ∆ veries ∆ > 2 * are considered as yielded.

On gure 4.4 we show the local yield stress τ c xy as a function of the local density ρ for the renewed regions from E = 0.2, E = 0.88 and from the system in the steady state. We can observe that the relation between τ c xy and ρ l for the renewed state does not show a clear dependence on E. Indeed, we obtain very similar curves for the regions renewed from the as-quenched state and renewed from the steady state.

The model glasses are deformed with a simple shear at constant volume.

This constant volume might hinder the decrease of local density upon yielding.

In this case, we should measure an increase of the local pressure upon yielding.

For this reason, we show in gure 4.5 the variation of the local yield stress with the local pressure. states and the unyielded one. We observe once more that for a given state, the variation of τ c xy with the pressure is lower than the variation measured between the yielded and the unyielded state. As a consequence, the xed volume does not explain why the local density can not capture these two coexisting populations: the plastied one and the non-plastied one.

Interestingly, when looking at the variation of the local yield stress τ c xy with the local pressure P l , we observe that τ c xy increases linearly with P l as: τ c xy = 0.25P +Cst. Also, we measure that the relations between τ c xy and local pressure P l for the dierent renewed states does not collapse on a single master curve.

Although the variations observed are minor, it implies a specic relation for each renewed ensemble.

Invariance of the local yield stress renewed distribution

As we just saw in the previous parts, the material locally rejuvenated and falls into a new structural state upon yielding. We also saw that for a well relaxed system (GQ) this new state is on average softer than the as-quenched state.

The system is locally rejuvenated very quickly: after essentially one plastic event. However, we observed that upon deformation, the unyielded regions are on average harder that the average as-quenched state. We also observed that as we further deform the system, the yielded regions become harder and harder until reaching a steady state. Also, by looking at the variation of the pressure with the local yield stress, we measured slightly dierent curves for the renewed states for dierent applied strain E (see gure 4.5).

In this section, we aim to understand through local yield stress distributions in the dierent states why (i) the unyielded regions are harder than the as-quenched state. Also, (ii) how to explain the measured hardening between the yielded sites at E = 0.05 and the steady state? And nally (iii) we want to see if the renewed state depends on the applied strain.

We show on gure 4.6 (a) the local yield stress distributions for the asquenched state (in black), the steady state (in blue), the unyielded site at E = 0.05 (in red) and nally the yielded site at E = 0.05 (in green).

We rst focus on the distributions of the as-quenched state and of the unyielded regions at E = 0.05. We can clearly see that the unyielded sites at E = 0.05 are harder than the as-quenched state. To understand this hardening, we pick out the locations of the unyielded sites at E = 0.05. We then look at their values in the as-quenched state and plot the corresponding distribution on gure 4.6 (b) (orange diamonds). By comparing it with the as-quenched distribution (in black) we observe that the unyielded sites at E = 0.05 are on average harder than the whole initial system. This shows that this hardening is mainly caused by the quick exhaustion of the weakest sites as the system is deformed until E = 0.05. Indeed, the weakest sites are the rst one to plastify and as a consequence, the remaining unyielded sites at E = 0.05 are composed of the initial hardest sites.

We now want to understand the slight dierence between the distribution of the unyielded sites at E = 0.05 with the distribution of these same sites in their as-quenched state. We notice that between the initial state and the state at E = 0.05, the pressure level is not the same. We then correct this dierence by applying the relation linking τ c xy and the pressure P l obtained from gure 4.5: τ c xy = 0.25P l + Cst.

As we can see on gure 4.6 (b), the pressure-corrected distribution of the unyielded sites at E = 0.05 collapses with the distribution of these same sites in their as-quenched state. From this, we can understand the dierence between the as-quenched state and the unyielded regions at E = 0.05 as the superposition of two distinct phenomena.

The rst (i) is the quick exhaustion of the weakest sites which quickly plastify as the material is deformed. As a consequence, only the initial hardest sites remain unyielded as the system is deformed. The second reason of this dierence (ii) is the increase of the pressure as the material is loaded. This increase of pressure increases the local yield stress following the relation τ c xy = 0.25P l +Cst. By looking on gure 4.6 (a) we can see that the steady state is on average harder than the renewed state at E = 0.05. To understand this evolution, we rst want to see if the renewed distribution depends on the applied strain E.

To do so, we show on 4.6 (c) the local yield stress distributions of the renewed states measured from E = 0.2, E = 0.88 and measured from the steady state. We also compare them with the distributions for the yielded sites at E = 0.05 and the steady state.

Surprisingly, we measure a unique local yield stress distribution in the renewed state, independent on the applied strain E. Indeed, as the local yield stress increases linearly with pressure, we could have expected a shift between the renewed distribution for dierent E. As our renewed ensembles show mean pressure dierences up to 0.25, a yield stress dierence of ∼ 0.0625 between the renewed regions at E = 0.05 and the regions renewed from the steady state could have been anticipated.

We can then argue that the renewal process does not produce a unique structural state but rather a unique local yield stress distribution under dierent conditions. While this idea was already proposed in the construction of meaneld or mesoscopic models [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF][START_REF] Tyukodi | Finite-size eects in a model for plasticity of amorphous composites[END_REF][START_REF] Popovi¢ | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF], it was never directly measured.

The hardening observed for yielded sites as the macroscopic strain E increases seems to be due to a statistical hardening. Indeed, we just saw that the renewal process is independent on pressure. Among the freshly renewed sites, the weakest ones tend to plastify more quickly while the hardest resists longer.

This phenomenon will create a bias towards the highest values of τ c xy in the distribution of the yielded regions. This bias increases with E towards a steady state. As a consequence, we obtain a steady state harder than the renewed state due to statistical hardening.

Local yield stress in other directions

We saw in the previous parts that upon plasticity, the material locally changes from an initial state to a renewed state with a unique local yield stress distribution. As the material is further deformed, the yielded sites reach a harder steady state due to statistical hardening.

Previously, we only considered the yield thresholds in the forward direction, i.e. α = 0 and did not verify if the system remains isotropic upon plasticity. In gure 4.7, we show the local yield stress distributions for the forward (α = 0) and the backward (α = π/2) global shear direction. More precisely, we show the distributions for the as-quenched state (black), the renewed state (green), the steady state (blue) and for the unyielded sites at E = 0.05 (red). From the gure 4.7, we can rst observe that upon yielding, the renewed state is softer in all directions. The steady state that follows the renewed state is also clearly harder in all direction compared with the renewed state, and softer compared with the as-quenched state. We can also remark that the unyielded sites at E = 0.05 are harder in all directions.

By looking at the dierence between the forward and the backward directions, we can rst see that the local yield stress distribution in the as-quenched state is isotropic. When we look now at the steady state distribution, we measure a clear anisotropy between the forward and the backward barriers. When looking at the renewed distribution, which corresponds here to the distribution of the yielded sites at E = 0.05, we again observe an anisotropy between the forward and the backward barriers. This shows that after the occurrence of plastic events, the model glass is locally anisotropic.

When we look at the local yield stress distribution of the unyielded sites at E = 0.05, we also observe a slight anisotropy. Indeed, we can see that in the backward direction, the unyielded sites are on average slightly harder than the as-quenched state. But this hardening is much lower compared to the forward direction.

We can understand this result by remembering that in the as-quenched state, the correlation between local yield stresses in the forward direction and the backward direction is not completely zero (see the gure 3.8 in chapter 3 showing the cross-correlation if the local yield stress as a function of the loading direction shift). Therefore, there is a slight correlation between the local yield stresses of two opposite directions in the same region.

Due to this correlation, a hard site in the forward direction tends to be associated with a harder backward yield threshold. As in the forward direction, the unyielded sites at E = 0.05 are composed of the initial hardest thresholds, the backward yield stress is on average slightly harder than the as-quenched state but also slightly softer than in the forward direction. It explains this anisotropy.

Upon heavy plasticity, the deeply quenched GQ system is in an anisotropic steady state. In the next section, we will look at the steady state for three dierent quenched protocols. We will also look at how fast we attain this steady depending on the degree of relaxation of the system.

Rejuvenation dependency on quench protocols

In section 4.1, we studied shear rejuvenation by focusing on the most relaxed system: GQ. It has the advantage of having a clear separation between the plastied regions (inside the shear band) and the unyielded ones (outside of the shear band). In this system, we measured a unique renewal distribution. We also observed that after further loading, the yielded sites in which plasticity has accumulated enter in a steady state. This steady state is on average harder than the renewed state. We then found this hardening to be caused by statistical hardening: the weakest sites are more renewed than the harder ones. The question remains of whether this steady state and the renewed distribution are independent of the quench protocol.

To answer this question, we consider the three systems dened in chapter 3 and ordered by the descending level of relaxation: GQ, ESL and HTL. These systems are then deformed up to γ = 5, corresponding to E ≈ 1.64.

We show on gure 4.8 the sample-averaged stress-strain response obtained during a constant volume AQS simple shear. We observe again a stress overshoot followed by a softening towards a plateau for GQ. For ESL, we seem to observe a small stress overshoot after which the system reaches the same plateau as GQ but more quickly. Finally, HTL, our less stable glass, does not show any stress overshoot but shows a mechanical hardening as the system is loaded. The average stress of HTL then converges very slowly towards the same steady state as GQ and ESL. One can notice that HTL does not seem to have completely converged at E = 1.64. From these systems, we calculate the local yield stress distributions for the two opposite directions 2α = 0 and 2α = π in the as-quenched state but also for the steady state. To measure the distribution in the steady state, we consider our systems deformed at E = 1.64 and limit ourselves to the most plastied sites, i.e. the sites for which > 2.

In gure 4.9 (a), we show the distributions of the as-quenched states for the three quench protocols. We can rst see that the as-quenched distributions are isotropic. Also, we measure that the more a system is relaxed, the harder it is.

We now look at the steady state distributions on gure 4.9 (b). Interestingly, we observe that the distributions for the forward and backward direction collapse for the three protocols. From this result, we obtain that the accumulation of plasticity makes the material lose its memory and converge towards a unique steady state, independent on the initial state.

We saw in the previous sections that local yield stress distribution in the steady state can be explained by the statistical hardening of the unique re- newed distribution measured in GQ. We can then argue that at least at high deformation, we have a unique renewed distribution, independent on the thermal preparation of the system. Finally, we remark that the steady state is harder than the as-quenched state for the HTL system. This shows that in HTL, the material is locally harder upon plasticity. This corresponds to an aging process. To know which systems are hardened or softened upon plasticity, we plot on gure 4.10 the average τ c xy computed for dierent glasses obtained by an instantaneous quench from equilibrated liquids at dierent parent temperatures T p . Note that the x-axis representing the parent temperature is normalized with T M CT , the mode-coupling temperature, equal to T M CT ≈ 0.373 in our system (see the annex 7.1 for more details).

From this plot, we recover that a more relaxed system (with a lower parent temperature) is on average harder. Remarkably, we measure for the renewed state an average yield threshold of τ c xy ∼ 1.32, a comparable value with the average local yield stress of a supercooled liquid equilibrated close to T p = T M CT . We also measured in the steady state a mean yield threshold similar to the inherent states of a supercooled liquid close to T p = 0.9T M CT . This is consistent with the statistical hardening observed in the section 4.1.3.

It is important to notice that the renewed states and the inherent states cannot be strictly compared due to the anisotropy in the yielded state (see chapter 5) and the slight increase of pressure in the ow state. We consider only their averaged values.

As we saw in chapter 1, T M CT corresponds to the temperature below which the dynamics of the supercooled liquids start to be dominated by the thermally activated events. It then corresponds to the upper layer of the potential energy landscape (PEL) [START_REF] David R Reichman | Mode-coupling theory[END_REF][START_REF] Bouchaud | Out of equilibrium dynamics in spin-glasses and other glassy systems[END_REF]. From our results, we can imagine that the critical parent temperature separating the brittle and ductile behavior is in the vicinity of T M CT [START_REF] Ozawa | Random critical point separates brittle and ductile yielding transitions in amorphous materials[END_REF][START_REF] Ozawa | The role of uctuations on the yielding transition of two-dimensional glasses[END_REF]. Indeed, at a lower parent temperature, plasticity causes a local softening. This softening upon plasticity leads to the formation of shear bands as we saw in chapter 1. This seems to be conrmed by the observation in gure 4.11 of the cumulated local strain maps at E=0. In these maps, we can see that for a parent temperature much below T M CT , the formation of a persistent shear band is observed. When the parent temperature in the vicinity of T M CT , the strain eld is heterogeneous but we do not see the formation of a persistent shear band. Finally, when the parent temperature is higher than T M CT , the strain eld is almost homogeneous. The link between shear banding and the thermal history will be studied with further details in the chapter 6.

The observation of the transition between plastic hardening to softening near T M CT is consistent with recent ndings obtained on oscillatory shear simulations [START_REF] Yeh | Glass stability changes the nature of yielding under oscillatory shear[END_REF][START_REF] Bhaumik | The role of annealing in determining the yielding behavior of glasses under cyclic shear deformation[END_REF]. These results show that the transition from annealed to yielded behavior is found for parent temperatures in the vicinity of T M CT .

Conclusion

The idea of local yield thresholds in amorphous solids which, after a local plastic event, are renewed from a post-yield distribution is used in many mesoscopic models [START_REF] Vandembroucq | Mechanical noise dependent aging and shear banding behavior of a mesoscopic model of amorphous plasticity[END_REF][START_REF] Tyukodi | Finite-size eects in a model for plasticity of amorphous composites[END_REF][START_REF] Popovi¢ | Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading[END_REF]]. Yet, the details of implementation if this principle differs from a model to another. Indeed, no direct observation of the local yield thresholds and their evolution were performed.

With the local yield stress method, we were able for the rst time to measure directly the local yield thresholds in atomistic simulations of amorphous solids.

Because of the limitations of the method, already discussed in chapter 3, we do not pretend to provide a recipe for the construction of mesoscopic models.

Nevertheless, we are able to study the rejuvenation process in model glasses at the atomic scale.

By applying the local yield stress method on a well relaxed system, we measured that upon plasticity, the local stress thresholds is drawn from a welldened post-yield threshold distribution. This renewed distribution is insensitive of the initial state of the material before the rearrangement. In the case of a well relaxed system, we showed that this renewed state is on average softer than the as-quenched state. Also, we measured that this renewal is rapid process. Indeed, only a single plastic event is required to bring a local region to a unique renewed state, independently of its initial state. This rapid renewal is often used in mesoscale models, but it is here directly measured for the rst time.

By looking at the local yield stress as a function of the local density for the as-quenched state and the renewed state, we were able to show that the local density can not capture the local dierence between the yielded state and the unyielded state. We thus observed that the local density does not seem to be a pertinent indicator to understand the local mechanical properties of metallic glasses despite being commonly used.

Thanks to the tensorial aspect of the local yield stress method, we were able to measure the local stress thresholds along dierent directions. From this, we rst observed that for a well relaxed system, the material becomes locally softer along all directions. We also showed that the post-yield stress thresholds distribution is anisotropic.

In a later part, we measured that model glasses with very dierent thermal histories and dierent mechanical behavior reach the same steady state as they are suciently deformed. This steady state was found to be slightly harder than the post-yield state due to statistical hardening. The stress thresholds of the steady state and of the post-yield state were found to be comparable to the inherent states of a supercooled liquid around the mode-coupling temperature T M CT .

Finally, we noticed that when the parent temperature of a model glass is lower than T M CT , the deformation localizes in persistent shear bands. On the other hand, if the parent temperature is higher than T M CT , a more homogeneous plastic behavior was observed.

We will study in the next chapter how the anisotropy observed in the stress thresholds upon plasticity inuences the mechanical response of the system. We will investigate then in another chapter the inuence of thermal history on the plastic behavior and the formation of persistent shear bands.

Chapter 5

Bauschinger eect in amorphous solids

In the previous chapter, we showed that upon plasticity, the studied model glass is locally renewed into a unique state independently of the initial state of the material. We also found that as the glasses are highly deformed, they will reach a steady state independently of the initial degree of relaxation. Finally, we found the local yield stress in the renewed state and the steady state to be anisotropic.

In metallic glasses, the mechanical history inuences the mechanical response. One example of this eect is the Bauschinger eect [START_REF] Bauschinger | Über die Veränderung der Elastizitätsgrenze und der Festigkeit des Eisens und Stahls durch Strecken und Quetschen, durch Erwärmen und Abkühlen und durch oftmals wiederholte Beanspruchung[END_REF]. The Bauschinger eect is a property shared by many dierent materials. It consists of after being plastically deformed and unloaded to zero stress, these materials show a softer stress-strain response in the reverse loading compared to reloading (see section 1.2.1.2 for more details). This phenomenon has rst been observed in metallic crystals [START_REF] Buckley | The bauschinger eect in super-pure aluminum single crystals and polycrystals[END_REF][START_REF] Asaro | Elastic-plastic memory and kinematic-type hardening[END_REF] or polymers [START_REF] Senden | Strain hardening and its relation to Bauschinger eects in oriented polymers[END_REF], but it has been observed more recently in amorphous solids such as metallic glasses, either experimentally [START_REF] Hao Sun | Pure shear stress reversal on a Cu-based bulk metallic glass reveals a Bauschinger-type eect[END_REF] and numerically [START_REF] Karmakar | Plasticityinduced anisotropy in amorphous solids: The Bauschinger eect[END_REF]. Still, its origin remains unclear across the concerned disciplines and is still an active research subject.

In parallel, when a mechanical loading is applied on metallic glasses, a polarization of the structure is observed experimentally [START_REF] Dmowski | Observation of structural anisotropy in metallic glasses induced by mechanical deformation[END_REF] and numerically [START_REF] Egami | Mechanical Properties of Metallic Glasses[END_REF]. The link between the structural anisotropy induced by plasticity and the Bauschinger eect in metallic glasses is speculated [START_REF] Hao Sun | Pure shear stress reversal on a Cu-based bulk metallic glass reveals a Bauschinger-type eect[END_REF]. However, no methods allowed to directly link these two phenomena.

In this chapter, through the measure of the local yield stress, we will study the link between the Bauschinger eect and the structural anisotropy induced by plasticity. We will use here the tensorial aspect of the local yield stress method to probe the structural anisotropy. From this method, we will try to understand the polarization of the mechanical response.

In a rst section, we will study the evolution of the anisotropy induced by mechanical loading through the measure of the local yield stress. We will also observe how this anisotropy inuences the early plastic response when loading in dierent direction. In a second section, we will study an elementary model to understand and reproduce the Bauschinger eect through the knowledge of the local yield stress distribution of the system.

Plastic response anisotropy and local yield stress

In this rst section, we will use the local yield stress method to investigate the mechanical and local anisotropy in model glasses.

In a rst part we will study the evolution of the anisotropy through the measure of the local yield stress. Then in a second part, we will investigate the link between the anisotropy and the local stress of the system. Finally, we will in a third part observe the relation between the early plastic response and the polarization of the low-yield barriers.

Evolution of the anisotropy with loading

In this chapter, we want to understand the Bauschinger eect [START_REF] Bauschinger | Über die Veränderung der Elastizitätsgrenze und der Festigkeit des Eisens und Stahls durch Strecken und Quetschen, durch Erwärmen und Abkühlen und durch oftmals wiederholte Beanspruchung[END_REF] in model glasses through the measure of the local yield stress. In gure 5.1, we show a schematic representation of the Bauschinger eect in model glasses. The system is rst loaded in the forward direction (in blue) beyond its yield point, up to the steady ow state. Then, the system is unloaded down to zero stress (in black) in the backward direction. From this unloaded state, the glass is either reverse loaded (in red) in the backward direction (note that the reverse loading corresponds to the continuation of the unloading beyond zero stress), or the glass is reloaded (in green) in the forward direction. The Bauschinger eect consists of observing a more ductile response during the reverse loading compared to the reloading.

To study the Bauschinger eect, we start from three very dierent glasses at pronounced dierent degree of relaxations. The rst one is prepared by an instantaneous quench from a high temperature liquid (HTL). The second one is obtained by an instantaneous quench from an equilibrated supercooled liquid (ESL). And the last one is relaxed by a slow gradual quench (GQ). All these three systems are the same as those studied in the previous chapters (these three quench protocols are introduced with more details in chapter 3.

When the shear is maintained beyond the initial transient response, all the glasses are driven towards a unique ensemble, independent of the initial degree of relaxation. This state corresponds to what we called steady state in the previous chapter. This steady state is normally inaccessible in experiments on hard glass due to the formation of shear bands leading to the fracture of the material. In atomistic simulation, this steady state can be easily attained by imposing periodic boundary conditions which prevents the fracture of the material.

In this chapter, we want to evidence the Bauschinger eect independently of the degree of relaxation of the system. Indeed, the preparation of the glasses induces a hardening or a softening in the transient mechanical response of the glass. This is why we consider the Bauschinger eect from the steady state of our system. It corresponds to a state in which the glasses have lost their memory of their initial state, thus simplifying the analysis.

To observe the Bauschinger eect, we rst impose a simple shear in the positive direction until the glasses reach the steady state. In gure 5.2, we show the results of the loading tests applied to evidence the Bauschinger eect it the model glasses. The system is rst loaded until the steady state (not shown here).

Then, from the steady state, we unload the material by imposing a simple shear in the negative direction until the stress reaches zero (black symbols). From this unloaded state, we either reload the system (loading back the system in the positive direction, in green) or impose a reverse loading (loading the system in the negative direction, in red).

From this curve, we can clearly observe that the reverse loading is much more ductile than the reloading. This eect corresponds to the Bauschinger eect. When comparing the unloading (in black) and the reloading (in green), a small hysteresis can be seen. This hysteresis indicates that the unloading entails a small amount on plasticity and is not perfectly elastic.

To understand the origin of this phenomenon, we want to analyze the structural properties of the steady state, and in particular its structural anisotropy.

To do so, we use the local yield stress method. As introduced in the chapter 2, this method consists of identifying the rst plastic event undergone by atoms inside a small circular patch (of radius R f ree = 5) by imposing on this patch a pure strain in the loading direction α.

As the stress and the strain tensor are π-periodic (see the annex 7.5), we deform the patches over the 2α ∈ [0, 2π]. Indeed, 2α = 0 corresponds to loading the patch in the same direction as simple shear imposed on the glass and 2α = π corresponds to reverse loading. The patches are sampled on a grid whose sites are separated by R sampling ≈ 2.5. For each patch, the local yield stress is measured by considering regular π/9 intervals for 2α. From the local yield stress method, we then construct the indicator ∆τ c (2α) which corresponds to the stress increment within the patch that leads to the rst plastic event.

More precisely, we dene ∆τ c (2α) = τ c (2α) -τ (2α), where τ (2α) is local elastic stress present within the patch prior to the loading. τ c (2α) is the stress value at which the rst plastic event is triggered when loading the system in the direction 2α. This indicator thus corresponds to the local residual strength of the material, or its local barrier.

Based on this indicator, we measure the local residual strength ∆τ c in the model glasses along dierent directions. From this, we can estimate the probability distribution function P (∆τ c ; 2α) of the local barriers ∆τ c in the direction 2α, 2α ∈ [0, 2π] plane. To represent it, we show in the left panels of gure 5.3 the polar maps of the function P (∆τ c ; 2α) in the (∆τ c , 2α). The right panels show cuts of the polar plots along the x-axis, i.e., plots of P vs ∆τ c xy = ∆τ c for 2α = 0 (forward), and vs ∆τ c xy = -∆τ c for 2α = π (backward).

In gure 5.3 (a), we represent the polar plot for the ESL system in the asquenched state, i.e., prior to any deformation. We can see in this plot that the as-quench system is isotropic. In gure 5.3 (d), we show the cuts of the asquenched polar plots for GQ, ESL and HTL. From this gure, we can recover that the more relaxed the system, the higher its local residual strength. In the three cases, we observe that the distribution of the local residual strength is symmetric. This agrees with the results obtained in the previous chapters. We now represent in gure 5.3 (b) the polar plot in the steady state. From this steady state, we can clearly observe that P (∆τ c ; 2α) is anisotropic. To observe more clearly this anisotropy, we show in this polar plot the mean local residual strength in each direction: ∆τ c (2α) in white. We remark that this mean local residual strength keeps a circular shape despite the anisotropy (a curious feature we cannot explain so far). We can also see that this circle is shifted horizontally by χ = ( ∆τ c (0) -∆τ c (π))/2.

(5.1)

We call χ the mean barrier polarization. This indicator χ is a measure of the degree of anisotropy in the system.

For the steady state in gure 5.3 (b), we measure χ -0.31 < 0. This means that the stress barriers are on average closer in the forward direction (2α = 0). This is expected since the steady ow state is under a positive average stress τ f low xy = 0.53. We now concentrate on the unloaded state in which τ xy = 0. The corresponding polar plot is shown in gure 5.3 (c). In this gure, we can see that the mean local residual strength ∆τ c (2α) (in white) is still circular but shifted horizontally towards the right. We observe here an inversion of the mean barrier polarization with χ 0.14 > 0.

When observing the polar plots of P (∆τ c ; 2α) and ∆τ c (2α) for many strain levels, we systematically nd ∆τ c (2α) to be hardly distinguishable from a circle. The center of these circles corresponds to the mean barrier polarization χ. From this gure, we observe that the barrier distributions develop a forwardbackward asymmetry due to the mechanical history of the material. During the unloading, we can see an inversion of the asymmetry. This observed asymmetry in the unloaded state raises the question of the link between the mechanical polarization and the Bauschinger eect. As the glass is reverse loaded (loaded in the backward direction), we remark that the mean barrier polarization χ overshoots the steady state value (red dotted line). As slight overshoot of the mean barrier polarization χ can also be seen in the forward direction (green dotted line). This overshoot seems to indicate that χ is not sucient to characterize the material state. If χ is sucient to characterize the material state, we should recover a symmetric barrier distribution when χ = 0. We then show in gure 5.5 the residual strength ∆τ c xy distribution along the forward/backward loading direction for χ ∼ 0. It corresponds to an applied strain of γ xy = 0.015. In this gure, we can directly remark that the barriers distribution is still asymmetric. Indeed, the forward barrier distribution is narrower and shows a higher peak than the backward direction. We then can directly see that the mean barrier polarization χ allows to observe the evolution of the anisotropy but is not sucient to characterize the material state. This result is consistent with Karmakar et al. [START_REF] Karmakar | Plasticityinduced anisotropy in amorphous solids: The Bauschinger eect[END_REF]. In this study, the authors also observed that a scalar quantity is insucient to characterize the material state. This observed anisotropy of the material structural properties are often presume to arise from the local stress asymmetry [START_REF] Asaro | Elastic-plastic memory and kinematic-type hardening[END_REF][START_REF] Rice | On the Structure of Stress-Strain Relations for Time-Dependent Plastic Deformation in Metals[END_REF][START_REF] Mughrabi | Dislocation clustering and long-range internal stresses in monotonically and cyclically deformed metal crystals[END_REF]. In the next section, we will study the evolution of the local stress to see if it can explain the asymmetry in the barrier distribution.

Anisotropy and local stress

In the previous section, we observed through the measure of the local residual strength that the system is anisotropic in the steady state and in the unloaded state. Indeed, in the steady state we measured a polarization of the barriers distribution and found that the barriers are on average smaller in the forward direction (χ -0.31 < 0). When unloading the material down to zero stress, we measured an inversion of the polarization and observed that the barriers are on average smaller in the backward direction.

The Bauschinger eect is often supposed to be caused by an asymmetry of the local stress [START_REF] Asaro | Elastic-plastic memory and kinematic-type hardening[END_REF][START_REF] Mughrabi | Dislocation clustering and long-range internal stresses in monotonically and cyclically deformed metal crystals[END_REF][START_REF] Rice | On the Structure of Stress-Strain Relations for Time-Dependent Plastic Deformation in Metals[END_REF]. To verify this assumption, we plot on gure 5.6 the distribution of the local stress τ 0 xy (i.e. the initial local stress within the patches) in the steady ow state (in black) and after unloading down to zero stress (in red). states; the latter distribution is also plotted after the x → -x transformation (green) to show that it is nearly symmetric. The steady state distribution shifted by the ow stress is also very similar (blue).

The rst observation when looking at the distribution in the steady state is that this distribution seems symmetric and centered around the average stress in this state, i.e. τ f low xy = 0.53. As we unload, we obtain a very similar symmetric shape for the distribution but centered around zero. This seems to indicate an absence of anisotropy of the local stress in the unloaded state.

To verify that the local stress distribution is indeed symmetric in the unloaded state, we ip it by imposing a x → -x transformation. The ipped distribution is shown in gure 5.6 (in green). We can see from this curve that the ipped distribution collapses very well with the unloaded distribution (the un-ipped one) which indicates that the distribution is indeed symmetric.

Then, to see if the local stress distribution changes its shape during the unloading, we compare the local stress distribution in the steady state with the distribution in the unloaded state. To do so, we shift the steady state distribution by the mean stress τ f low xy (in blue). We can see that the shifted distribution collapses pretty well with the unloaded distribution. This indicates that the local stress distribution in the unloaded state basically corresponds to the elastic shift by the mean stress τ f low xy of its counterpart in the steady ow state.

These results clearly show that in our systems, the anisotropy observed in the unloaded state does not arise from an anisotropy in the local stress. In the next section, we will study the link between the anisotropy of the local residual strength distributions and the early plastic response.

Linking the early plastic response with the tail of the local residual strength distribution

We saw in the last part that the local residual strength anisotropy observed in the unloaded state does not arise from an anisotropy of the local stress. In this part, we will study how the forward-backward asymmetry of the barrier distribution may be linked with the Bauschinger eect.

We rst need to recall that during an AQS loading, the plastic events occur when atomic packings are mechanically brought beyond the local yield barriers [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF][START_REF] Karmakar | Predicting Plastic Flow Events in Athermal Shear-Strained Amorphous Solids[END_REF]. We can then expect that during the Bauschinger tests (i.e. loading in the forward and backward direction from the unloaded state), the early plastic response is controlled by the sites characterized by a low local distance to threshold ∆τ c . These small barriers correspond to the tail of the local residual strength distribution P (∆τ c ; 2α).

To study the evolution of the small barriers tails between the steady state and the unloaded state, we show in gure 5.3 (e) the cuts of the polar plots in the steady state (in blue) and the unloaded state (in red). In the gure 5.3

(f ), we focus more precisely of the small barrier tails of the distribution in the forward and backward directions.

From these distributions, we can observe that the steady state has a higher population of small barriers in the forward direction compared to the backward direction. Conversely, we observe in the unloaded state a higher population of low barriers in the backward direction compared to the forward directions.

This indicate that the forward-backward asymmetry of the small barriers tails of P (∆τ c ; 2α) is inverted during the unloading. We now want to illustrate the importance of these small barriers in the early plastic response. To do so, we take an arbitrary system in the unloaded state (i.e. in the zero stress conguration) and show in gure 5.7 its local residual strength maps in the backward (a) and forward (b) direction. We can observe that the backward direction has a higher population of low barriers (in red) than the forward direction. This is consistent with the observation done on gures 5.3 (e) and (f ) in which the small barriers tails of P (∆τ c ; 2α) show the small barrier population is higher in the backward direction compared to the forward direction. These maps clearly show that the forward direction is harder than the backward direction.

Then, in the same maps, we show the location of the plastic rearrangements triggered during the rst 2% of strain in the corresponding loading direction (black symbols). We observe that more plastic events are activated in the rst 2% of strain in the backward direction (14 events) compared to the forward direction (4 events). This corroborates the idea that a higher population of small barriers leads to the activation of more plastic events. Also, this supports the idea that the asymmetry of the local residual strength distribution in the unloaded state is responsible for the more ductile response during the reverse loading (backward loading) compared to the reloading (forward loading), i.e.

the Bauschinger eect.

To comfort this idea, we propose in the next section an elementary model to connect the early plastic response with the local residual strength distribution in the unloaded state.

Elementary model from local residual strength distributions

In the previous section, we observed an anisotropy for the local residual strength distribution in the unloaded state. We saw that the backward local residual strength distribution has a higher small barrier population compared to the forward direction. We supposed that this dierence might be responsible for the more ductile response when loading in the backward direction from the unloaded state compared to loading in the forward direction. To test this idea, we present in this section an elementary model to relate the early plastic response with the barrier distribution in the initial state from which the system is loaded. From this model, we hope to be able to reproduce and understand the Bauschinger eect in the studied model glasses from the local residual strength distributions in the unloaded state.

We will in a rst section give a description of our model and its main hypotheses. Then in a second section we will explain how we estimate the dierent parameters of the model. And nally we will in a third part apply the model to our system and compare the predicted result with the one we measured.

Description of the model

Our aim is to reproduce and understand the stress-train curve for the three considered loadings of the Bauschinger tests. For that reason, we only consider three types of loading in this model: the unloading from the steady state, the reloading and the reverse loading.

We name in the model δγ xy the macroscopic strain increment. Its reference value corresponds to the initial state from which the system is loaded. As the system is loaded, the elastic response increases the stress τ xy of the system, while the plastic events lead to stress drops and so reduce the applied stress.

From this, we write the strain-induced macroscopic stress change δτ xy induced by δγ xy as:

δτ xy = µδγ xy -δτ pl xy (5.2)
with µ the shear modulus and δτ pl xy the stress released by plastic events.

In this model, we assume that no avalanches are activated during the three considered loadings: the unloading, the reverse loading and the reloading. To test this hypothesis, we looked at the plastic drops during the rst few percents of deformation for the considered loading, and compared it with the plastic drops observed when we continue to deform the system from the steady state in the forward direction. From this observation, we neglect in our model the occurrence of avalanches for the considered applied loadings. We also assume that the mechanical noise and the elastic heterogeneities can be neglected. Moreover, we do not consider rejuvenation upon plasticity. Indeed, we want to understand only the beginning of the stress response. This means that we consider that a region that has yielded will not be triggered again during the considered loading. From this last hypothesis, we thus assume that the early plasticity is controlled by the weak regions, i.e. the low barriers ∆τ c xy , preexisting in the initial state.

To model the evolution of the macroscopic stress change δτ xy with the in- crement strain δγ xy , we need to know the evolution of δτ pl xy , the stress released by plastic events. For a given value of the stress δτ xy , this quantity can be expressed as the product of the typical stress drop amplitude by their number:

δτ pl xy (δτ xy ) = ∆τ event xy • N • r yielded (δτ xy ), (5.3)
where ∆τ event xy corresponds to the typical stress released by a single plastic event,

N corresponds to the number of yield barriers in the system susceptible to be triggered. Finally, r yielded (δτ xy ) corresponds to the proportion of barriers that have yielded at δτ xy .

We rst look for the expression of r yielded (δτ xy ). In the absence of the elastic heterogeneities, the preyield local stress τ xy (δγ xy ) of a given yield barrier can be written as: where ∆τ c xy (0) is the distance to threshold in the initial state. The yielding of this barrier occurs when ∆τ c xy (δγ xy ) vanishes. This equation shows that in the model, the barriers are mechanically shifted by the macroscopic stress δτ xy up to instabilities. This also mean that the barriers in the opposite direction of the macroscopic stress are shifted towards higher values. This constitutes the core assumption of this model.

From this assumption, the proportion of sites that has yielded r yielded for a stress level of δτ xy is:

r yielded = δτ xy 0 p(∆τ c xy )d∆τ c xy , (5.7)
with p the distribution of the local distance to threshold in the initial state.

Then, we consider the number of yield barriers in the system. It can be expressed as:

N = ρ • V, (5.8)
with ρ the spatial density of yield barriers in the system and V the volume of the system. Here we have V = L 2 , with L the size of the system (as we consider a 2D square system as in the simulations). We thus have:

N = ρ • L 2
(5.9)

Finally, we now look for the expression of ∆τ event xy , the typical stress released by a single plastic event on the system. It can be written as:

∆τ event xy = 2µ • ∆ L , (5.10)
with µ the shear modulus of the system, and ∆ L the typical average strain increase for the whole system due to a single plastic event. Note that the factor 2 comes from the expression of ∆ L in Hencky formalism in the limit of small strain (see the annex 7.2 for more details).

The plastic events we are considering in this model are isolated and induce a local strain increase. This is why we measure in the atomistic simulations the local strain from a coarse-graining procedure. We name ∆ 0 the typical strain released measured over a coarse-graining size a. The surface of the coarsegraining patch is then of the order of a 2 . From this, we have the relation:

∆ L = a 2 L 2 ∆ 0 , (5.11)
which gives:

∆τ event xy = 2µ • a 2 L 2 ∆ 0 (5.12)
Finally, by assembling the equations 5.2, 5.3, 5.7, 5.9 and 5.12 altogether, we obtain: and the Bauschinger eect. To do so, we need an estimation of the dierent parameters present in this equation, i.e. µ, ρ, a and ∆ 0 .

Estimation of the model parameters

In the previous part, we obtained through a model a relation between the mechanical response and the local residual strength distribution. To compare this model with the measured mechanical response in our model glasses, we rst need to estimate from our numerical simulations the dierent unknown parameters present in the relation. These parameters are the shear modulus µ, the density of yield barrier ρ, the typical strain increase after a plastic event ∆ 0 measured over a typical region of size a.

The shear modulus is easily obtained by looking at the beginning of the stress-strain curve for the unloading in the innitesimal strain limit. From this, we estimate µ ∼ 15.7 which is by principle equal in both forward and reverse direction.

When looking at the equation 5.13, we remark that it depends on a single unknown parameter equal to ρa 2 ∆ 0 . To obtain an estimation of this quantity, we t the beginning of the unloaded curve in gure 5.9 with the relation from our model (solid black line). To do so, we use for p the backward barrier distribution of the steady state. From this, we obtain ρa 2 ∆ 0 ∼ 0.25. To conrm the consistency of this value, we also estimate separately a 2 ∆ 0 and ρ. First, we estimate the typical released strain ∆ 0 . We consider only single isolated plastic rearrangements. We then measure the coarse-grained local strain map for single plastic rearrangements over a coarse-graining size of a = 5

(following the method explained in section 2.3.1.1). The obtained strain maps show a local strain increase in an isolated region, i.e. the region that has yielded.

These maps are similar with the one showed in the chapter 3 in gure 3.4. From this analysis, we obtain for ∆ 0 a value in the interval 0.016 -0.028, which corresponds to a value of a 2 ∆ 0 in the range 0.4 -0.7.

We now want to estimate the density of yield barriers ρ. To do so, we rst consider a conguration in the steady state, just after the end of a stress drop caused by a plastic event. Previous studies [START_REF] Craig | Universal Breakdown of Elasticity at the Onset of Material Failure[END_REF][START_REF] Karmakar | Plasticityinduced anisotropy in amorphous solids: The Bauschinger eect[END_REF] have shown that in such a state, the length of the following elastic branch is determined by the next region which plasties when deformed by external loading.

Based on this principle, we estimate ρ independently from the steady state. To do so, we consider a 2D system of size L and volume V = L 2 in the steady state. The next plastic event happens in one of the N y = ρL 2 barriers. We assume these barriers to be independent and note p their distribution. In prac-tice, we dene p equal to the forward local residual strength distribution in the steady state, as shown in gure 5.3 (e) and (f ). In the elastic branch, the local residual strength varies as:

∆τ c xy (δγ xy ) = ∆τ c xy (0) -µδγ xy , (5.14)
up to the next instability.

We place ourselves at an applied strain value of γ . Under these two assumptions, we have δγ pl xy > γ , with δγ pl xy the strain interval at which the next plastic event occurs, if all of the N y barriers verify ∆τ c xy (0) > µγ . This happen with the probability:

P (δγ pl xy > γ ) = ∞ µγ p(δτ )dδτ Ny (5.15)
We remark that the value of the cumulative distribution function of δγ pl xy associated with γ is equal to F δγ pl xy (γ ) = 1 -P (δγ pl xy > γ ) by denition. Thus, the density of probability of δγ pl xy is the derivative of F δγ pl xy : -P (δγ pl xy ).

From this, we have that the average strain interval can be calculated as:

δγ pl xy = - ∞ 0 xP (x)dx = ∞ 0 P (x)dx (5.16)
We measure from our data p and δγ pl xy independently. Then, from the knowledge of these two quantities, we nd the value of N y by tting the equations 5.16 and 5.16. In our model glasses, this gives a number of yield barriers equal to N y 3800 which corresponds to ρ ∼ 0.39. From this, we obtain a mean distance between the yield barriers of ξ 1.6.

Based on the values of a 2 ∆ 0 and ρ measured independently, we estimate ρa 2 ∆ 0 to be in the range 0.16-0.27 which is consistent with the value obtained by tting the beginning of the unloading curve, i.e. ρa 2 ∆ 0 ∼ 0.25.

Application of the model to understand the Bauschinger eect

In this part, we will now apply the equation 5.13 obtain by the model we developed by using the parameters estimated in the last part. This equation will be used to predict the evolution of the loadings performed for the Bauschinger tests.

We rst focus on reloading (forward loading) and reverse loading (backward loading) from the unloaded state at zero stress. To predict the reloading, we consider for the equation 5.13 the local residual strength distribution for the unloaded state in the forward direction P (∆τ c ; 2α = 0), shown in gure 5.3 (e) (in red). For the reverse loading, we use the local residual strength distribution in the backward direction, i.e. P (∆τ c ; 2α = π).

From these distributions and the equation 5.13, we are now able to predict the evolution of the function δτ xy (δγ xy ) for the two loadings as a function of the applied strain. In gure 5.9, we show the obtained evolution of the stress strain curve for the reloading (solid green line) and reverse loading (solid red line). We observe that we recover a more ductile behavior in the reverse loading due to the higher population of small barriers.

In this gure, we can see that the predictions match strikingly well the measured stress-strain curves up to at least 5% of applied strain. This agreement establishes that in our model glass, the Bauschinger eect does result from the forward-backward asymmetry of the small barrier tails in the unloaded state.

The small barrier tails refer here to ∆τ c xy 0.5 which corresponds to the macroscopic stress change in the tted strain range.

To predict the evolution of the unloading, we consider the local residual strength distribution in the steady state for the backward barriers P (∆τ c ; 2α = π) (see gure 5.3 (e) (in blue)). From this distribution, we can again estimate the evolution of the strain as a function of the stress change. This is shown in gure 5.9 (solid black line). The obtained stress strain curve matches also very well the one we measure by unloading the system.

The remarkable ability of the model to reproduce the three mechanical tests of the Bauschinger eect supports that its core assumption is reasonable up to the rst few percent of strain. This assumption is that the barriers are mechanically shifted by the mechanical stress up to instabilities. We can then use this model to understand how unloading to zero stress lead to the small barriers asymmetry, which we have just shown to be responsible of the Bauschinger eect.

To do so, we consider the local residual strength distribution in the steady state, shown in gure 5. The obtained distributions P (m) predicted by the model for the unloaded state are shown in gure 5.3 (e) and 5.3 (f ) (black dashed lines). We name P (u) the measured local residual strength distribution in the unloaded state (in red).

Note that P (m) is multiplied by an arbitrary factor to better observe how it departs from P (u) .

We observe from these gures that for a local residual strength of |∆τ c xy | 1, P (m) and P (u) fall right atop. We conclude from this that the assumed elastic shift of the distribution is a very reasonable assumption during unloading, in which very few plastic events are triggered.

When we now look in these gures at the small barrier tails of P (m) , we can see that the model remarkably predicts the inversion observed during the unloading. Yet, we can remark two discrepancies compared to the measured distribution P (u) . First (i), we can see that the model overestimates the small barrier population in the backward direction. Indeed, we observe that P (u) essentially vanishes at ∆τ xy = 0 which is not the case for the distribution P (m) predicted by the model. We expect this dierence to arise from the neglected mechanical noise due to the stress released after each plastic events. Indeed, the mechanical noise facilitates the activation of small barriers. This mechanical noise requires P (u) to essentially vanish at ∆τ xy = 0 [START_REF] Lemaître | Plastic response of a twodimensional amorphous solid to quasistatic shear: Transverse particle diusion and phenomenology of dissipative events[END_REF][START_REF] Lemaître | Rate-Dependent Avalanche Size in Athermally Sheared Amorphous Solids[END_REF].

Then (ii), we can remark that the model predicts for the forward direction the appearance of a gap with no barriers for ∆τ xy < τ f low xy . In this range, we remarkably observe a pseudo-gap in P (u) . This pseudo-gap corresponds to a weak initial rise of the barrier distribution for ∆τ xy < τ f low xy compared with its rise beyond τ f low xy . Yet, the barrier distribution for P (u) does not strictly vanish for ∆τ xy < τ f low xy , unlike P (m) . We presume that this dierence is due to the rejuvenation and/or the noise associated with the small plastic activity during unloading.

This analysis suggests that the elastic shift of the barriers up to instabilities is the main reason behind the inversion of the small barrier populations during unloading. It also suggests that the eects neglected in the model, i.e. the rejuvenation and the noise, which arise from unloading-induced plasticity are only mitigating factors.

To test this interpretation, we compute again the reloading and the reverse loading from the unloaded state by using this time the model-predicted local residual strength distribution in the unloaded state P (m) instead of the measured one P (u) . From this, we want to see if we recover again the Bauschinger eect.

In gure 5.9, we show the predicted response for the reloading (green dashed lines) and for the reverse loading (red dashed lines). We can see from this gure that the reloading is purely elastic. This is due to the gap in the local residual strength distribution. We also observe that the reverse loading is softer and corresponds to the continuation of the unloading predicted by the model (solid black line). These curves clearly exhibit a Bauschinger eect of very reasonable amplitude, although slightly overestimated.

This result conrms that the discrepancies observed between P (u) and P (m) only reects compensation mechanisms like mechanical noise or rejuvenation.

It also conrms that the elastic shift of the barriers up to instability, which is the main hypothesis of the model, captures the core mechanism responsible for the small barrier inversion leading to the Bauschinger eect.

Additionally, the model predicts that (i) for a partial unloading (i.e. the unloading is stopped before the stress reaches zero), we should still observe a gap in the forward barrier distribution, yet smaller. This gap leads to an elastic response during the reloading. Also, (ii) the model predicts that from its onset, the unloading initiates reverse plasticity and softening. Indeed, the small barrier populations in the backward direction of the steady state is non-zero.

We can then expect that when we do a reverse loading from a nite unloading (which corresponds to continue to unload the system), we should observe a softer response compared to reloading (which is expected to be elastic). We should thus observe a Bauschinger eect for partial unloading.

In gure 5.10, we show the reloading (solid lines) and reverse loading (dashed In all cases, strain is measured with reference to the zero-stress state.

lines) curves for dierent level of partial unloading and for a full unloading. We can see in this gure that for all the levels of partial unloading, we measure a more ductile response in the reverse-loading compared with the reloading.

We can also see that the higher the amplitude of the unloading, the softer the reverse loading response. This unambiguously conrms the prediction of the model.

Conclusion

In this chapter, we studied the origin of the Bauschinger eect in model glasses at the atomistic scale. By measuring the local residual strength of the system, we were able to observe the evolution of the local residual strength anisotropy when unloading the glass from the steady state. We also measured a more ductile response when reverse loading from the unloaded state compared with reloading.

By using a simple model, we found the Bauschinger eect in the studied glasses to be caused by the inversion of the small barrier populations anisotropy between the forward and the backward direction during the unloading. This model also showed that this inversion of small barrier populations is essentially caused by the elastic shift of the barriers during the unloading. Although we used here a 2D model, we expect that our conclusions are still valid in 3D since our main results do not depend on the system dimension.

In the next chapter, we will study the heterogeneity and the persistence of the plastic deformation observed in the cumulated strain maps. The shear bands nucleation and broadening will also be investigated.

Chapter 6

Heterogeneity of plastic deformation: persistence and shear band broadening

In the previous chapter, we studied the Bauschinger eect in model glasses with the local yield stress method. By developing an elementary model, we could show that the Bauschinger eect in model glasses arises from the polarization of the small yield barriers population as the system is loaded along dierent directions.

In this chapter, we will study the inuence of the thermal history and of the system size on the plastic behavior. By comparing this result with the local yield stress, we seek to have a better understanding of the formation of persistent shear bands. The inuence of parent temperatures near the mode-coupling cross over T M CT will notably be discussed. After studying the conditions to have a persistent shear band, we will examine if we can understand the location of the shear band nucleation from the knowledge of the initial local yield stress eld.

Finally, the shear band width evolution will be discussed.

In a rst section, the inuence of the thermal history and of the system size on the heterogeneity and the persistence of plasticity will be discussed. In this section, we will rst study under which conditions we observe an heterogeneous local strain eld corresponding to the presence of persistent localization of the deformation. Then we will study how the plasticity behaves and persists in the ow state, i.e. after the elastic branch. In a second section, we will see if the location of the shear bands can be understood from the local yield stress eld prior to any deformation. Finally, in a third section, we will examine the evolution of the shear band width.

Size and thermal history dependence of the plastic behavior

In this section, we will study model glasses at dierent sizes and degrees of relaxation, i.e. at dierent parent temperatures. From this, we will study the inuence of the system size and of the degree of relaxation on the plastic behavior of the systems.

To do so, we will in a rst part study the heterogeneity of the cumulated plastic strain for systems at dierent sizes and parent temperatures. Then in a second part, we will observe the plastic events individually and investigate how the system parameters inuence the frequency and the size of the plastic events.

Finally, in a third part, we will study the persistence of the individual plastic events, i.e. if they tend to be located in the same regions or if they are spatially uncorrelated.

6.1.1 Heterogeneity of the cumulated plastic strain 6.1.1.1 First estimate of the heterogeneity through the measure of the variance

Changing the parent temperature of a glass, i.e. the equilibrium temperature from which a system was instantaneously quenched, inuences its plastic behavior. Indeed, we observed in chapter 4 the localization of the plastic deformation into a persistent shear band for a low parent temperature compared to the modecoupling temperature T M CT . When the parent temperature is high compared to T M CT , no persistent localization of the deformation was observed.

In addition, in experimental studies, the size of the system was found to inuence the plastic behavior. It was measured that the smaller the system, the more ductile its mechanical response.

To study these phenomena, we perform numerical atomistic simulation of 2D square model glasses (introduced in the chapter 2). We simulate square glasses at dierent size and dierent parent temperatures T p . Note that in this section, the parent temperature will be referred by T p /T M CT , with T M CT = 0.384 (see the annex 7.1).

To refer to the system size, we will use either the number of atoms N contained in the square box, of the length L of one of the square side. Since we use a density close to unity in these model glasses, we have that L ≈ √ N .

We rst want to have a visual observation of whether the cumulated local strain eld is localized or not depending on the parent temperature and of the system size. To do so, we plot on gure 6.1 the cumulated strain maps obtained for an applied strain of γ xy = 0.3. We consider systems at three dierent sizes: N=10000, 1000 and 300 (upper to lower row) and at two dierent parent temperatures: T p /T M CT ≈ 0.84 (left column) and 1.3 (right column).

We can see from these maps that for well relaxed systems (i.e. with a low parent temperature) in the left column, the local deformation is localized in persistent shear bands. For a system at higher parent temperature (right column), no persistent localization of the deformation is observed and the cumulated strain maps is much more homogeneous. We can also see that for the small systems (N=300), a persistent shear band is observed at low parent temperature.

No clear variation of the shear band size with the system size can be observed from these maps.

We want to estimate the persistent localization of the plastic deformation in these cumulated strain maps for dierent system sizes and parent temperatures. To do so, we measure their variance. Indeed, the variance measures the degree of heterogeneity of a system. The more persistent and localized the plastic deformation, the more heterogeneous the cumulated local strain eld.

As a consequence, we expect to have a high variance for systems in which the plasticity is localized and persistent. Conversely, the absence of persistent localization should lead to a more heterogeneous cumulated strain map as visible on gure 6.1 (right column).

In practice, we consider dierent samples for many system sizes going from N = 100 to N = 10000 and many parent temperatures going from T p /T M CT ≈ 0.84 to T p /T M CT ≈ 1.3. For a given parent temperature and system size, we simulate between 10 and 100 samples. Then, we measure the variance of the cumulated strain maps at γ xy = 0.3 for these dierent systems. At each system size and parent temperature, we measure the sample-averaged variance. From this, we are able to obtain the variance phase map as a function of the system size and of the parent temperature shown in gure 6.2. More details about how the phase map is obtained are given in the annex 7.6. From this phase map of the variance, we can see that the lower the parent temperature is (in the left of the phase map), the higher the variance. This is consistent with the observations of gure 6.1, i.e., that the plastic deformation is localized and persistent at low parent temperature. We also see that at low T p , the smaller the system is, the lower the variance. To study more precisely the inuence of the system size and of the parent temperature, we show on gure 6.3 the variance as a function of the parent temperature of the system T p /T M CT for dierent system sizes N .

We can see from these curves that the variance begins to increase when the parent temperature becomes smaller than the mode coupling temperature T M CT (black vertical line). We also observe that at low parent temperature, an important size eect is observed on the variance. At high parent temperature, we observe that the variance seems to converge towards a nite value as the size of the system increases.

In the case of N=100, corresponding to L ≈ 10, we do not observe a variation of the variance with the parent temperature. This length L = 10 then seems to correspond to the characteristic size of the plastic rearrangements. Indeed, if the system size is of the order of the size of a plastic rearrangement, no evolution of the variance with the parent temperature should be seen as no localization of the plasticity can be measured. We can also note this size is similar to the optimal size for the local yield stress patches estimated in chapter 3.

To verify if the evolution of the variance with T p for L = 10 is not due to the coarse-graining size R CG = 5, we perform the same measure of the variance with a coarse graining size of R CG = 2.5. Here again, no clear evolution of the variance with the parent temperature can be seen (see annex 7.7). To understand the evolution of the variance with the system size for the dierent parent temperatures, we propose in the following simple models of the strain eld. We rst begin with models above for high parent temperatures that do not show signicant temperature dependence above T M CT .

Toy models for the high parent temperature regime

In this part, we will present two toy models to describe plasticity at high parent temperature. From these models, we seek to have a better understanding of the plastic behavior at high parent temperature from the evolution of the variance with system size.

We rst name = the mean local strain of the system. In the limit of low strain, we have that ≈ γ xy /2 as shown in the annex 7.2. The variance shown until now was measured for a local strain eld at γ xy = 0.3. We thus have ≈ 0.15.

We consider that = el + pl with el the contribution of the elastic deformation and pl the contribution of the plastic deformation. In the following models, the elastic deformation is considered to be homogeneous along all the strain eld. As a consequence, it does not contribute to the variance as shown in the annex 7.8.

In practice, we estimated in our simulations the variance due to the elastic deformation to be of the order of 3 • 10 -5 independently of the system size for an applied strain of γ xy = 0.3 (not shown here). The variance due to the elastic strain is then several orders of magnitude lower than the total variance of the system at γ xy = 0.3. The hypothesis of neglecting the inuence of elasticity for the variance in the model is then reasonable.

For a system at high parent temperature, we rst assume that the local plastic strain eld results from an accumulation of independent elementary plastic events. We suppose that the plastic event have a length l and thus a size equal to l 2 (as the system is in two dimensions). We now consider a square system of size L containing independent square sites of length l in which plastic events can randomly occur. Since we consider all sites to be independent, the probability of a plastic event to occur is the same for all sites. A schematic representation of the system is shown in gure 6.4. Figure 6.4 Schematic representation of the local strain map of a system of size L in which the plastic deformation takes the shape of independents square events of size l. After each plastic event, the local strain of increases in the plastic event location by inc .

In this system, the number of independent sites n S is then:

n S = L 2 l 2 .
(6.1)

In the following, otherwise mentioned, we only consider the plastic strain eld in the system. We note pl the mean plastic strain within the system.

When a plastic event occurs in a site, it will increase its local strain by inc . As all the sites have the same probability to plastify, the mean local deformation within a site will be equal to the mean plastic strain pl . As a consequence, the mean local plastic strain within a site is expressed as:

= pl = inc N 1site , (6.2)
with N 1site the mean number of plastic events activated per sites. The expression of N 1site is then:

N 1site = pl inc . (6.3)
From this, the total number N pl of plastic events activated for a mean plastic strain of pl is expressed as:

N pl = N 1site • n S . (6.4)
We now look at the variance for a single plastic event. We remember that a single plastic event will locally increase the strain by c in one of the n inde- pendent sites.

We note p 1site the probability of a site to activate a plastic event. It is equal to:

p 1site = 1 n S . (6.5)
Thus, the probability for a given site to not be activated is then 1 -

p 1site = 1 -1 n S
. From this, we have that the mean local plastic strain of the system for one plastic event is:

1event = 1 n S • inc + (1 - 1 n S ) • 0 = inc n S . (6.6)
Similarly, we have:

2 1event = 2 inc n S . (6.7)
From this, the variance of the system after a single plastic event VAR 1event is:

VAR 1event = 2 1event -2 1event = 2 inc n S (1 - 1 n S ). (6.8)
As the plastic events are independent, then the variance for N plastic events VAR N events is:

VAR N events = N pl • VAR 1event .

(6.9)

We consider that N pl is the number of plastic events needed to have a mean plastic strain of pl . Thus, we have VAR N events = VAR( pl ).

As we considered that the elastic eld is homogeneous, it does not contribute to the variance. As a consequence, we have:

VAR( ) = VAR( el + pl )
= VAR( pl ).

(6.10)

The expression of VAR( ) is then:

VAR( ) = N pl • 2 inc n S (1 - 1 n S
). From this, we nally obtain:

VAR( ) = pl inc (1 - 1 n S ) = ( -el ) inc (1 - l 2 L 2 ).
(6.12)

We obtain a variance depending on 1 L 2 with the length L of the system.

In a second approach, we consider again that for a system at high temperature the local plastic strain eld is obtained by an accumulation of independent plastic events. Dierently with the previous model, we consider now that the plastic events correspond to avalanches with a size of l • L (in two dimensions).

After a plastic event, the local strain within the avalanche increases by inc . We also note pl the mean plastic strain of the system. A schematic representation of this system is shown in gure 6.5. To simplify, we illustrate in gure 6.5 the plastic events with bands of width l and of length L. We can note that here we supposed the avalanches to have a size l • L, we do not suppose any specic shape. This model corresponds to measuring the variance of a system with n S independent sites of size l • L in which independent plastic events will randomly happen. This system is then strictly equivalent as the one studied just before, with the only dierence that the number of independent sites is:

n S = L 2 lL = L l (6.13)
From this, we obtain that the variance of the system for an applied strain of 0 is:

VAR( ) = pl inc (1 - 1 n S ) = ( -el ) inc (1 - l L ). (6.14)
In this case, we obtain a variance depending on 1

L with the length L of the system.

In gure 6.6, we show the evolution of the variance at high parent temperature of the local strain eld for an applied strain of γ xy = 0.3 as a function of

1

L with the length L of the system (in blue). We also show in this gure the t obtained by the equation 6.12 (for independent plastic events of size l 2 , in orange) and 6.14 (for independent avalanches of size l • L, in green). Note that we did not considered L = 100 in the t as its meaning is questionable since it has the same size than the coarse graining of the system. In this gure, we can see that the variance seems to depend linearly on 1 L .

This result tends to show that the plastic events at high temperature corresponds to independent avalanches of size l • L. We can also remark that the t with the equation 6.14 (for independent avalanches of size l • L, in green) is working better than with the equation 6.12 (for independent plastic events of size l 2 , in orange). However, more statistics will be needed to really conrm this trend.

From the t with the equation VAR( ) = (el ) inc (1 -l L ), we obtain an estimation of l ≈ 5.1 and (el ) inc ≈ 0.0085. We can remark that the length scale l of avalanches is of the order of the size of the coarse graining used to measure the local strain R CG = 5.

From the stress strain curves of the systems for dierent sizes, we estimated the plastic strain to be of the order of (el ) ≈ 0.129 (see the annex 7.9 for more details). From this, we obtain a characteristic plastic strain of inc ≈ 0.066. This result is of the same order of the characteristic strain for plastic events measured in chapter 4 in which we estimated * ≈ 0.1.

From this model, we obtain that the plastic events seem to take the shape of avalanches whose size increase linearly with the length L of the system. This result is consistent with previous study on avalanche's statistics in steady state of glasses deformed with athermal quasi static shear [START_REF] Maloney | Subextensive Scaling in the Athermal, Quasistatic Limit of Amorphous Matter in Plastic Shear Flow[END_REF][START_REF] Salerno | Avalanches in Strained Amorphous Solids: Does Inertia Destroy Critical Behavior?[END_REF][START_REF] Ferrero | Criticality in elastoplastic models of amorphous solids with stress-dependent yielding rates[END_REF]. Here again, we name = the mean local strain of the system. We consider that = el + pl with el the contribution of the elastic deformation and pl the contribution of the plastic deformation. We will consider in the following model that the elastic deformation is homogeneously distributed in the system. As a consequence, it does not contribute to the variance as shown in the annex 7.8.

We assume that at low parent temperature, the plastic deformation consists of a shear band of length L and of width l. A schematic representation of the model is shown in gure 6.7. Figure 6.7 Schematic representation of the local strain map of a system of size L in which the elastic deformation el is homogeneous while the plastic deformation is concentrated on a shear band of width l. We note el + SB the local strain within the shear band.

We rst only measure the variance of the system by focusing only on the plastic contribution of the strain eld. We do not consider the elastic contribution yet. We call pl the mean plastic deformation of this system. We also consider SB , the mean local plastic strain within the band. The relation between SB and pl is SB = L l pl (6.15)

In this system, the mean strain writes:

= pl = l L SB (6.16) and 2 = l L 2 SB (6.17)
From this, the variance of the system for a mean strain plastic of pl is

VAR( pl ) = 2 -2 (6.18)
which corresponds to

VAR( pl ) = 2 SB l L (1 - l L ) = L 2 l 2 2 pl l L (1 - l L ). (6.19)
Finally we have: From this result, we expect to nd at low parent temperature a linear evolution of the variance with the system size and with a slope of (el ) 2 l . We show on gure 6.8 the evolution of the variance at T /T M CT ≈ 0.84 of the local strain eld at γ xy = 0.3 as a function of the length L of the system (in blue). When applying directly the equation 6.22 to t the curve of the variance as a function of the length L of the system, we do not obtain a satisfactory result. The hypotheses in this model are likely to be too simplistic. In gure 6.8 we t the measure with the equation VAR( ) = (el ) 2 ( L l -1) + C with C a constant to see the evolution of (el ) 2 L l (green curve).

VAR( pl ) = 2 pl ( L l -1).
We measure in our simulations that at low parent temperature, (el ) ≈ 0.133 for γ xy = 0.3. From the slope of the t, we estimate l ≈ 42. This result is clearly higher than the actual band width as we can see on gure 6.1.

We next assume that instead of having the plastic deformation localized in a uniform shear band, the shear band prole is now characterized by a Gauss function along the x-axis and uniform along the y-axis. From this denition, the x-averaged local strain prole x for the plasticity is written as:

x (y) = SB exp( -y 2 2σ 2 ), (6.23) with SB the strain amplitude of the shear band and σ the half of the shear band width l such as l = 2σ. As we will see in the section 6.3, it is reasonable to assume a Gaussian prole for the local strain prole. We consider the shear band width l to be small compared to the linear size L of the system. We then have that the mean plastic strain is:

= pl = 1 L ∞ -∞ SB exp( -y 2 2σ 2 )dy. (6.24)
This gives:

pl = √ 2πσ SB L .
(6.25)

Similarly, we have:

2 = 1 L ∞ -∞ 2 SB exp( -2y 2 2σ
2 )dy, (6.26) which gives

2 = √ πσ 2 SB L (6.27)
We can now compute the variance as:

VAR( pl ) = 2 -2 , (6.28)
which corresponds to

VAR( pl ) = √ π 2 SB σ L (1 -2 √ π σ L ) = 2 pl ( L 2 √ πσ -1). (6.29)
As previously, since the elastic eld does not change the variance, we also have:

VAR( ) = VAR( el + pl )

= VAR( pl ), (6.30) and as σ = l 2 , we nally obtain:

VAR( ) = (el ) 2 ( L √ πl -1). (6.31) This model also gives a linear evolution between the variance and the linear system size L for large enough systems. For a gaussian shear band, the slope between the variance and L is ( -el ) 2 √ πl

. For a uniform shear band, we obtained a slope of (el ) 2 l

. From this, if we note l uni the width estimated from the t for a uniform shear band and l gauss the width estimated for the Gaussian shear band, we obtain:

l gauss = l uni √ π ≈ 23.
(6.32)

The obtained estimation of the shear band width is closer to the observation but is still grossly overestimated. Indeed, by estimating the shear band width for systems at N=10000 and T /T M CT ≈ 0.84 with a t from a Gauss function, we obtain l ≈ 11. See the section 6.3.1.1 for more details about the shear band width estimation method.

several reasons can be invoked for this discrepancy. Among them, (i) we did not take into account in our model the possibility of a vertical shear band [START_REF] Golkia | Flow heterogeneities in supercooled liquids and glasses under shear[END_REF].

In our systems, we do see at low parent temperature the occurrence of vertical shear bands in some systems. As systems with a vertical shear band are further loaded, a horizontal shear band will also appear at a higher applied strain. A study with more samples would be needed to verify this hypothesis.

Also, (ii) with molecular dynamics simulations, it is not possible to have very stable systems. We can thus imagine that we have in this case a combination of persistent plastic event with some non persistent plastic events. Indeed, in gure 6.1 (a), we can see some plasticity outside of the shear band. In this local strain map, all the local strain is not concentrated within the shear band. Using the novel method of Monte-Carlo Swap [START_REF] Berthier | Ecient swap algorithms for molecular dynamics simulations of equilibrium supercooled liquids[END_REF] to obtain systems at a very low parent temperature would be a way to verify this hypothesis.

Finally, (iii) when looking at gure 6.1 (a), we can see that the shear band is not uniform along the x-axis. The uctuations of the shear band are not taken into account in the model and can lead to a reduction of the variance.

A more precise model taking into account these uctuations could improve the understanding of the variance for the systems at low parent temperature.

In the next parts, we will study the evolution of plasticity. We aim here to have a better understanding of the persistence of the plastic deformation into shear band at low parent temperature. In addition, we want to understand the homogeneous local strain eld obtained at high parent temperature. In the following parts of this section, we will not study the nucleation of shear bands, but their persistence. We will place ourselves at high enough applied strain to ensure that the shear bands have already nucleated for well relaxed systems. In practice, we consider γ xy > 0.1.

Frequency of plastic events

The persistent localization of plasticity is caused by the accumulation of plastic events in the same regions. At high parent temperature, we obtained through a model that plastic events consist of avalanches whose size depends linearly with the length of the system. We want here to study individually these dierent plastic events. To do so, instead of considering the accumulated strain maps as we did until now, we will measure the strain maps taken from strain windows of ∆γ xy . More precisely, we place ourselves at a given applied strain γ xy and we compute the local strain map between the states γ xy and γ xy + ∆γ xy with ∆γ xy = 0.01. This strain windows map will show the shape of the plastic events occurring in this interval.

We show on gure 6.9 a comparison between the cumulated strain maps (upper row) and the strain maps measured on windows of ∆γ xy = 0.01 (lower row) for a system at N = 10000 and T p /T M CT ≈ 0.84 for an applied strain going from γ xy = 0.1 to 0.14. The cumulated strain maps correspond to the accumulation of the local strain measured in the strain windows.

We plot in gures 6.10, 6.11 and 6.12 the same comparison for systems We observe from these gures that for the systems at T p /T M CT ≈ 0.84 (gures 6.9 and 6.11), the plastic event seems to occur in the same location in the dierent strain windows. This leads to the persistence of a shear band as we can see on the upper row. Also, we remark that most of the plastic events at low parent temperature have the shape of a band.

In systems at T p /T M CT ≈ 1.3 (gures 6.10 and 6.12), the plastic events in strain windows do not occur in the same regions anymore. The plastic events now take dierent shapes and are localized in dierent regions from a strain Figure 6.11 Same as gure 6.9 for a system of 300 atoms at T p /T M CT ≈ 0.84 for an applied strain γ xy in the range 0.1-0.14. Figure 6.12 Same as gure 6.9 for a system of 300 atoms at T p /T M CT ≈ 1.3 for an applied strain γ xy in the range 0.1-0.14. window to another. This leads to more homogeneous cumulated local strain maps as seen in the upper row. We can also see that plastic events with the shape of a band can be observed at high parent temperature.

When we now focus on the eect of the system size, the most striking result is that in small systems, some strain windows do not seem to show any plastic events. To determine if a strain window displays a plastic event, we use the following criteria:

In each strain windows of ∆γ xy = 0.01, the average local strain is, in the Hencky formalism, equal to ≈ 0.005. Then we consider that if the maximum on the local strain M ax( ) in a strain window is lower than three times the mean strain , then no plastic events occurred in this window. More precisely, we consider that a strain window contains a plastic event if it veries:

M ax( ) > 0.015.

We want to quantify how often plastic events occurs for a given system size and parent temperature. To do so, we calculate for dierent samples the local strain windows for an applied strain between γ xy = 0.1 and 0.3. For each of these strain windows, we verify if M ax( ) > 0.015, corresponding to the presence of a plastic event. We nally count the number of windows containing a plastic event. We then dene the Plasticity rate Γ pl as the ratio between the windows containing a plastic event and the total number of windows (see the annex 7.11

for more details).

In gure 6.13, we show the phase map of the plasticity rate for dierent system sizes and parent temperatures. From this phase map, we observe that the plasticity rate increases with the system size. No clear dependency on the parent temperature can be observed. Figure 6.13 Phase map of the variation with the system size and the normalized parent temperature T p /T M CT of the plasticity rate measured in windows of 1% of deformation for an applied strain γ xy in the range 0.1-0.3. The higher the plasticity rate, the more plastic events take place in strain windows of 1%.

To conrm this observation, we plot on gure 6.14 the variation of the plasticity rate as a function of the parent temperature T p /T M CT for dierent system size N . In this gure, we observe that the plasticity rate seems to be independent on the parent temperature. The plateau equal to 1 shows that for a large enough system, we observe a plastic event in all the strain windows of ∆γ xy = 0.01. For smaller system, the plasticity rate depends almost linearly on L.

To understand this linear evolution, we develop here again an elementary toy model. In the previous section, we found through a toy model that the plastic events at high parent temperature take the shape of avalanches whose size is of the order of lL with L the length of the system and l length of the avalanches width. As the plasticity rate Γ pl is independent of the parent temperature, we suppose that independently on the parent temperature, the size of the avalanches is of the order of S ≈ lL.

We consider inc the mean local strain increment after a local plastic event.

If the plastic events are organized in avalanches whose average size is lL, then the mean local strain increase after an avalanche av would be

av = = lL L 2 • inc = l L • inc (6.33)
We consider av the mean strain we need to apply to trigger an avalanche. If the mean local strain increase av is higher than the strain window ∆γ xy = 0.01 which corresponds in Hencky formalism to ∆ = 0.005, then we will not observe a plastic event in all strain windows. The plasticity rate would then be equal to:

Γ pl = ∆ av , (6.34)
which gives a plasticity rate of:

Γ pl = ∆ l inc • L. (6.35)
Conversely, if we have av < ∆ , then we should obtain an avalanche in all strain windows and measure Γ pl = 1.

We apply this obtained relation on gure 6.15 (orange curve). We can see that the relation obtained by the model is quite consistent with the measure plasticity rate (in blue). For av > ∆ , we measured with the t a slope of ∆ l inc ≈ 0.035. If we consider l = 2R CG = 10, we obtain inc ≈ 0.015. We obtain a value of the mean local strain increment inc four times smaller than the strain increment obtained in section 6.1.1.2. If we consider l = 5 as we obtained in section 6.1.1.2, we nd inc ≈ 0.03 which is two times smaller than in the previous section. This model is thus too simplistic to describe the evolution of the plasticity rate with the linear system size. One reason might be that we only considered averaged quantities. For instance, as the system size increase, we can imagine that the probability to nd a weak local yield threshold is higher. This weak region might trigger more easily a plastic event. Also, the criteria used to describe the plasticity rate is also very simple. The evolution of the plasticity rate thus requires a more precise study.

Another model we develop to understand the plasticity rate is the following: we rst consider a system of linear size L containing n S sites of linear size l. The number of sites n S in this system is then:

n S = L 2 l 2 .
(6.36)

We consider that if no plastic event is triggered in a strain window ∆γ xy , then this strain window is part of an elastic branch in the stress-strain curve. If a plastic event is observed in the strain window, it means that a plastic event has been triggered in at least one of the n S sites. We consider one of the n S sites i and dene ∆τ c xy,i , its local residual strength taken at the beginning of the strain window ∆γ xy . Let us call ∆τ xy = µ∆γ xy the elastic stress associated to the strain increment ∆γ xy . No plastic events has been triggered in the system for a stress increment of ∆τ xy if ∀i, ∆τ c xy,i > ∆τ xy .

The sites susceptible to plastify corresponds to the lowest yield barriers. In [START_REF] Tyukodi | Avalanches, thresholds, and diusion in mesoscale amorphous plasticity[END_REF][START_REF] Ferrero | Criticality in elastoplastic models of amorphous solids with stress-dependent yielding rates[END_REF], dierent authors obtained a distribution for the low yield barriers of:

p(∆τ c xy → 0 + ) ∼ L -θ , (6.37)
with θ ∼ 2/3.

From this equation, the probability that a plastic event has been triggered in a site with a local residual strength of ∆τ c xy for a stress increment of ∆τ xy is:

P (∆τ c xy < ∆τ xy ) ∆τxy 0 AL -θ d∆τ A∆τ xy L -θ , (6.38)
with A a constant.

Then, the probability that no plastic event has been triggered in a site for a stress increment of ∆τ xy is:

P (∆τ c xy > ∆τ xy ) 1 -A∆τ xy L -θ . (6.39)
As we have n S = L 2 l 2 sites in the system, the probability P no that no plastic event has been triggered in the whole system for a stress increment of ∆τ xy is then:

P no = 1 -A∆τ xy L -θ L 2 l 2 , (6.40)
which gives in the limit of A∆τ xy L -θ small:

P no exp -A∆τxy L 2-θ l 2 . (6.41)
Finally, the probability that a plastic event has been triggered in the whole system for a strain increment of ∆γ xy = ∆τxy µ is:

P pl 1 -exp -Aµ∆γxy L 2-θ l 2
.

(6.42)

The probability P pl corresponds by denition to the plasticity rate Γ pl In gure 6.16, we show the t of the plasticity rate Γ pl with the equation 6.42 by considering θ = 2/3 as in [START_REF] Tyukodi | Avalanches, thresholds, and diusion in mesoscale amorphous plasticity[END_REF]. We can see in this gure a good correspondence between the t and the measured plasticity rate. From this t, we estimate Aµ∆γxy l 2 ∼ 0.0198. In this section, we can nevertheless retain that the frequency of plasticity in our model glasses is independent on the degree of relaxation of the system.

The evolution of the plasticity rate with the size of the system would require a deeper study.

Persistence of the plastic deformation

In the last section, we saw that in the ow state, the frequency of plastic events is independent of the degree of relaxation of the system. We also observed through local strain maps taken for small strain windows ∆γ xy that at low parent temperature, the avalanches mostly take the shape of a band. Furthermore, at high parent temperature, avalanches with a band shape can also occur.

In this part, we will be looking at the persistence of the individual plastic events. We do not consider here the onset of plasticity. To do so, we consider the local strain maps for a strain window of ∆γ xy = 0.01 for an applied strain γ xy in the range 0.1-0.3. In the Hencky formalism, the mean local strain of each window is ¯ = 0.005. Then, out of these strain windows, we only consider those featuring plastic events. As in the previous part, we consider a strain window to feature a plastic event if its maximum value of local strain max( ) corresponds to max( ) > 3 • ¯ = 0.015.

We then consider a sample for a given system size N and parent temperature T p /T M CT . In this sample, we have N w strain windows with plasticity. The local strain eld of the N w strain windows is written as: { i } i=1..Nw .

We then consider two of the local strain elds : i and j and measure their Pearson correlation coecient ρ i, j as:

ρ P e i, j = ( i -i )( j -j ) σ i σ j , (6.43)
with i the average of the local strain eld i and σ i the standard deviation of i .

After that, we measure the Pearson correlation coecient between all the strain windows and dene the P lasticity P ersistence as:

P lasticity P ersistence = 1 N w (N w -1)/2 Nw i=1 Nw j>i ρ P e i, j , (6.44)
which corresponds to the average Pearson correlation coecient between all the N w strain windows (see Annex 7.12 for more details). From that, we measure for each system size and parent temperature the sample-averaged plasticity persistence.

We show on gure 6.17 the phase map of the plasticity persistence for dierent system size and parent temperature. In this phase map, we can see that the plasticity persistence increases as the parent temperature decreases, i.e. as the system is more relaxed. We also observe a slight dependence with the system size.

We show on gure 6.18 the evolution of the plasticity persistence with the parent temperature T p /T M CT for dierent system sizes N . In this gure, we can Figure 6.17 Phase map of the plasticity persistence with the system size and the normalized parent temperature T p /T M CT measured for an applied strain γ xy in the range 0.1-0.3. The higher the plasticity persistence, the more plastic events tend to occur in the same location.

observe that as we saw on the phase map, the plasticity persistence increases as the parent temperature decreases. Remarkably, we see that this increase seems to begin around T p ≈ T M CT . We also see no clear increase of the persistence for N = 100. This can be expected as for N = 100, the system size is similar to the coarse graining size. At this size, we observe very noisy local strain eld for strain windows with plasticity (not shown here). We also remark a slight dependence of the plasticity persistence with the system size. More statistics would be needed to clearly conrm and study this trend. Yet, this size dependence is on second order compared to the parent temperature dependence.

We saw in the chapter 4 that the local yield thresholds τ c xy decreases with the parent temperature T p ≈ T M CT . On gure 6.19, we show the variation of the mean local yield threshold τ c xy as a function of the parent temperature T p /T M CT for dierent system size N . We can observe as in the chapter 4 that τ c

xy increases as the parent temperature decreases. Also, we see that the mean local yield threshold does not show any clear dependency on the system size.

In chapter 4, we saw that for a system size of N = 10000, the mean local yield threshold of yielded sites is in the vicinity of T M CT . Then for T T M CT we expect a softening upon plasticity and for T T M CT we expect a hardening upon plasticity.

As the local threshold τ c xy does not seem to depend on the system size as we see on gure 6.19, we expect the separation between hardening and softening to be in the vicinity of T M CT independently of N . Additionally, we can see from gure 6.19 that for T T M CT , as the parent temperature decreases, the amplitude of the softening (i.e. τ c xy (T p ) -τ c xy (T M CT )) increases.

From this, we can see on gure 6.18 that the plasticity persistence increases with the amplitude of softening caused by plasticity. Based on this result, we can assume that the persistence of plasticity is mainly due to softening. Indeed, as a plastic event occurs, the system will be on average locally softer. Then, this softer region will be more likely to plastify again. On the opposite, at high parent temperature, after a plastic event occurs, the system will be on average locally harder. As a consequence, plastic events will tend to occur in other regions which are on average softer. This will reduce the persistence of plasticity in the systems at high parent temperature.

As we saw in the gure 6.9 and 6.11 representing strain maps for a system at low parent temperature for an applied strain in the range 0.1-0.14, the plastic events tend to take the shape of a band. These gures show that these bands occur at the same location. This observation is consistent with the result obtained with the measure of the persistence of plasticity. At high parent temperature, as shown in the gures 6.10 and 6.12, the shape of the plastic events is more heterogeneous but we can still observe the occurrence of plastic events with the shape of a band. Also, consistently with the measure of the plastic persistence, we observe no persistence of the plastic deformation at high parent temperature.

We can link these results with the experimental observations of [START_REF] Shen | Plasticity of a TiCu-based bulk metallic glass: Eect of cooling rate[END_REF][START_REF] Li | Eect of cooling rate on plastic deformation of Zr-based bulk metallic glasses[END_REF][START_REF] Kumar | Critical ctive temperature for plasticity in metallic glasses[END_REF].

Indeed, in these experiments, the authors observed that for metallic glasses obtained with a low cooling rate, i.e. a glass with a high degree of relaxation, the deformation will concentrate in few persistent shear bands. Then, as the degree of relaxation of metallic glasses decreases, the number of shear bands in which the plasticity concentrate increases and the material will show a more ductile response. Based on our observations, we can imagine that as the metallic glass is less relaxed, the persistence of its plasticity will reduce which will lead to the formation of multiple shear bands.

6.2 Linking the shear band location and the local yield stress in the as-quenched state

In the previous section, we saw that in the ow state, the frequency of plastic events is independent of the parent temperature and increase linearly with the length of the system. We also saw that the persistence of plasticity depends mostly on the parent temperature. More precisely, the plastic deformation persists in systems featuring shear-softening. This will lead to the formation of persistent shear bands for systems at low enough parent temperature (i.e. for T p T M CT ).

In this section, we will study the correlation between the location of shear bands and the initial local yield threshold τ c xy eld of the systems. To do so, we concentrate on the gradually quenched system (GQ) presented in chapter 4.

In the gure 6.20, already shown in chapter 4, we represent the x-averaged (a)

local strain and (b) local yield stress as a function as a function of the vertical position y centered on the shear band location y SB for several macroscopic strains E. Note that these proles were obtained by focusing on samples with a horizontal shear band.

On the gure 6.20 (b), we can see that the local yield stress prole for the as-quenched state (in black) is locally softer at the shear band location. As in the as-quenched state, no deformation was applied, it means that in our system, the shear bands tends to nucleate in initially softer regions.

Based on this observation, we will now measure the correlation between the local yield stress prole in the as-quenched state and the location of the shear As shown in gure 6.21 (a), we rst (i) consider the x-averaged local strain prole of a given sample for a macroscopic strain of E = 0.1. We dene the band center y SB as the position for which the local strain prole reaches its maximum (in red).

Then, (ii) we look at the local yield stress prole in the as-quenched state.

From this, we measure the local yield stress corresponding to the shear band location y SB as shown in gure 6.21 (b) (the green arrow).

Finally (iii), we measure the cumulative distribution function of the local yield stress prole CDF ( τ c xy ) as shown in gure 6.21 (c). We look at the value of CDF ( τ c xy (y SB )) corresponding to the value of the as-quenched local yield stress at the shear band location τ c xy (y SB ).

Based on this quantity, we measure the sample-averaged correlation C SB between the shear band location and the as-quenched local yield stress prole as:

C SB = 1 -2CDF [ τ c xy (y SB )] (6.45)
As it is dened, C SB ∼ 1 corresponds to a perfect correlation between the location of the shear band and the location of the softest region in the local yield stress prole. Also, C ∆τy ∼ 0 would correspond to an absence of correlation. In the gradually quenched system, we obtain a correlation of C SB ≈ 0.64

It means that on average, the bands tend to nucleate in the softest 18% strips of the initial system. We can thus conclude that their is indeed a correlation between the initially softer bands and the location of the shear band after its nucleation. This result shows that the shear bands tend to nucleate in initially weak planes of the material structure. The results has been conrmed in ultra stable glasses [START_REF] Richard | Predicting plasticity in disordered solids from structural indicators[END_REF].

Shear band width evolution

In the last section, we saw that in a gradually quenched glass (GQ) in which a persistent shear band nucleates upon deformation, the shear band tends to nucleate in an initially softer region.

In this section, we will study the evolution of the shear band width in the same gradually quenched system GQ. We will only focus on the horizontal shear bands. On gure 6.20 (b) we show the local strain prole centered on the shear band location for dierent values of applied strain E. We can see that as the macroscopic strain E increases, the shear band width increases. Based on the local strain proles for dierent values of E, we will estimate the shear band width and look at its evolution with E.

In a rst part, we will detail the three dierent methods used to estimate the shear band width. Then in a second part, we will study the evolution of the shear band width with applied strain and compare it with a model and previous studies on similar systems.

Estimation of the shear band width

In this part we present the three methods used to estimate the shear band width.

In a rst part, we will explain the methods developed by Parmar et al. [START_REF] Anshul | Strain Localization Above the Yielding Point in Cyclically Deformed Glasses[END_REF] in which we t the x-averaged local strain with a gaussian function. Then in a second part, we will detail the method based on the work of Alix-Williams and Falk [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF] which consists on a binarization of the local strain map with a pre-dened cuto. Finally, we will dene a method based on measuring the participation ratio of the x-averaged local strain proles.

Gaussian method

We present here the method based on Parmar et al. [START_REF] Anshul | Strain Localization Above the Yielding Point in Cyclically Deformed Glasses[END_REF] to estimate the shear band width. This method is based on using a Gauss function to t the proles of the Mean Square Displacement of the atoms at dierent applied strain. In this section, we apply a similar method but based on the local strain prole.

In our method, we rst isolate the sample featuring a horizontal shear band.

Then, for each sample, we measure the applied strain γ 0 corresponding to the nucleation of the shear band (see annex 7.13 for more details). In practice, we measure that the mean strain at nucleation is γ 0 ≈ 0.086.

Then we calculate the local strain proles x centered on the shear band location y SB for dierent samples. For each sample, we start to calculate the proles from an applied strain of γ 0 and up to γ xy = 5.0. From this, we measure the sample-averaged local strain prole starting from γ 0 . Note that the value of γ 0 is sample dependent. We consider then γ 0 as the strain reference. From this, we perform a sample average of the local strain proles with γ 0 as the initial strain for each samples.

An example of the obtained prole is shown in gure 6.22 (black disks) and corresponds to the local strain prole measured for an applied strain of γ xy = 1.0 (γ xy -γ 0 =≈ 0.91). y SB (black dots). Gaussian t of the local strain prole (blue line). From this t, we extract and estimation of the shear band width w gauss as w gauss = 2σ.

After that, we t the local strain proles with the following gaussian function:

x (y -y SB ) = x (0) exp(

-(y -y SB ) 2 2σ
2 ) (6.46)

The t corresponds to the blue line in gure 6.22. Based on this t, we dene the shear band width w gauss = 2σ (purple arrow).

From this method, we measure that the shear band width at its nucleation is on average w gauss,0 ≈ 11.8. This result is of the order of the diameter of the coarse graining used to measure the local strain, which is: D CG = 2R CG = 10.

Binary mask method

The second method we use to estimate the shear band width is based on Alix-Williams and Falk [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF]. In gure 6.23 we show a schematic representation of the method.

We rst consider the samples with a horizontal shear band and measure their corresponding applied strain at the shear band nucleation γ 0 . Then, starting from γ 0 , we measure for each sample their local strain elds on a regular lattice (gure 6.23 (a)). After that, as shown in gure 6.23 (b), we construct a binary mask from this local strain eld by using the cuto value cut = 0.125. When the local strain is lower than cut , the binary mask will be equal to 0. When > cut , the binary mask is equal to 1. Note that the cuto value cut = 0.125 corresponds to the one used in [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF] after conversion to the Hencky formalism. ), with i > cut → 1 and i < cut → 0. We nally consider width of the largest continuous line as the shear band width w line (c). This gure was modied from [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF].

Examples of binary masks are shown in gure 6.24 for two dierent samples (rst and second column) for an applied strain of γ xy = 0.2 (rst row) and of γ xy = 1.0 (second row).

Finally, we dene the shear band width w line as the largest continuous line in the binary mask as shown in gure 6.23 (c). We employ this method for a growing applied strain and for dierent samples to have the evolution of the shear band with the applied strain starting from γ 0 .

From this method, we measure that the shear band width at its nucleation is on average w line,0 ≈ 2.3. We obtain with this method a lower shear band width at the nucleation compared with w Gauss,0 measured by tting the strain prole with a Gauss function. We understand this dierence by the fact that the binary mask is applied with xed cuto, independently on the value of the applied strain. As a consequence, at low strain only the center of the shear band will be detected in the binary mask.

Participation Ratio method

We present in this part the method we constructed to estimate the shear band width based on the participation ratio of the x-averaged shear band prole. The principle of this method is the following: let us consider a glass of size L with a horizontal shear band of width w. We call SB the deformation within the In gure 6.26, we show a schematic representation of the x-averaged local strain prole

x of this model system as a function of the vertical y position.

In this prole, we see that all the deformation is located in a shear band of width w.

If we measure the participation ratio of the local strain prole P R( x ), we obtain:

P R( x ) = 1 L • ( y=0..L ( x (y) 2 )) 2 y=0..L ( x (y)) 4 =
w L (6.47) Figure 6.26 Schematic representation of the local strain averaged along the x axis as a function of the vertical position y for a system of size L in which all the deformation is localized in a horizontal band of width w. We note SB the deformation within the shear band.

Based on this elementary model, we measure the x-averaged local strain proles for the dierent samples and starting from an applied strain γ 0 (the strain at which the shear band nucleates). We then calculate the participation ratio of the local strain proles P R( x ) and dene the shear band width as:

w P R = P R( x ) • L, (6.48)
with L the size of the system.

From this method, we measure that the shear band width at its nucleation is w P R,0 ≈ 15.5. This result is consistent with w Gauss,0 measured by tting the strain prole with a Gauss function and is also of the order of the coarse graining diameter D CG = 2R CG = 10.

Shear band width evolution and comparison with model

In the last part, we presented the three methods used to estimate the shear band width in our gradually quenched system GQ. We obtain the three estimations of the shear band width: w Gauss , w line and w P R . In this part, we will show the evolution of these three quantities for an applied strain up to γ xy = 5.0.

In [125], Jagla proposed a simple geometric argument relating the strain rate imposed to the system with the strain rate within the shear band to extract an equation giving the evolution of the shear band width w with the applied strain γ xy . He obtains:

w = w 2 0 + 2AL(γ xy -γ 0 ), (6.49)
with A a proportionality constant, w 0 the shear band width at its nucleation, γ 0 the applied strain at which the shear band nucleates and L the size of the system. (See the section 1. The orange curve corresponds to a power-law t starting from γ xy = 1.0.

From the t, we nd that the exponent c of the power-law is: 0.37, 0.55 and 0.38 for w Gauss , w line and w P R , respectively. We recover the sub-diusive regime of the shear band width evolution for the width measured with a Gauss t. We also recover this sub-diusive regime for a width measured with the participation ratio. For the shear band width measured from a binary mask, we obtain a shear band width evolution closer to a diusive regime.

From this result, we can see that the power-law exponent of the shear band width evolution depends on the applied strain taken as a reference and on the method used to estimate the shear band width. By taking as a reference the strain at the shear band nucleation, we could observe that the model by Jagla

[125] which predicts a diusive evolution of shear band is reasonable for our deeply quenched model glasses.

Conclusion

In this chapter, we rst saw that for large systems and low parent temperatures, the plastic deformation is localized. This leads to the heterogeneity of the cumulative local strain elds. The lower the system size, the closer it is to the size of a plastic rearrangement. This causes a reduction of the strain heterogeneity at low parent temperature.

From the measure of the variance at high parent temperature, we obtained through an elementary model that the size of the plastic events seems to increase linearly with the linear length of the system. We also observed that the frequency of plastic events in the ow state is independent on the degree of relaxation of the system and only depends on the system size. We showed in a later section that plasticity persists along shear bands in relaxed systems featuring strain-softening upon plasticity. In practice, this corresponds to a system whose parent temperature is lower than the mode-coupling temperature T M CT .

By comparing the local yield stress eld and the shear bands location, we showed that in well relaxed systems where shear bands persist, the shear bands tend to nucleate in initially softer bands. Finally, we measured how the width of the shear bands increases as the system is loaded. The increase of the shear band width was found to be consistent with the diusive behavior proposed by a phase diagram characterizing the spatial extent and persistence of plastic events as a function of the system size and its initial stability; the diusive broadening of shear bands upon deformation in the athermal and quasi-static limit.

In this thesis, we performed numerical simulations of model glasses at the atomistic scale to study plasticity and shear banding in amorphous solids. By using a novel structural indicator, the measure of the local yield stress [START_REF] Patinet | Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids[END_REF], we could characterize the dependence of the local properties of glasses depending on the degree of relaxation. We measured that the more a glass is relaxed, the harder it is.

By comparing the location of plastic events with the local yield stress eld, we obtained a good correlation between the initial softest regions and the location of the rst plastic events. The locations of the initial soft regions were found to depend on the orientation of the loading applied to the system. Moreover, we also discussed the inuence of the patch size used to measure the local yield stress. The most relevant results were obtained for a typical patch containing about 80 atoms.

After the occurrence of a plastic event, we measured that the glass is locally renewed in a post-yield state with a unique local yield stress distribution, independently of the initial state of the system. As the glasses are further loaded, we found that the renewed state and the steady have a mean local yield stress similar to the inherent state of supercooled liquids close to the mode-coupling temperature. This raises the question of the role of the mode-coupling temperature as a separation between the shear softening and the shear hardening regime.

Furthermore, by comparing the local yield stress with the local density, we showed that the local density has a second order contribution of free volume compared to structural rejuvenation to explain the changes of softness upon plastic deformation. We thus observed that the local free volume does not seem to be a pertinent indicator to understand the local mechanical properties of model glasses despite being commonly used.

By using the tensorial aspect of the local yield stress, we measured that the glasses are locally anisotropic upon plasticity. We then studied the evolution of this anisotropy as the system is loaded in many directions.

After unloading down to zero stress model glasses in the steady ow state, we observed an asymmetry of the mechanical response. A further unloading shows a more ductile response compared to reloading. This phenomenon is called the Bauschinger eect. By the mean of an elementary model, we found that the origin of the Bauschinger eect in the amorphous solids arises from the inversion of the low yield barriers population anisotropy during the unloading.

We next investigated the plastic behavior in model glasses at dierent sizes and degrees of relaxation. From these simulations, we obtained through simple arguments that the plastic events seem to correspond in average to avalanches whose size increases linearly with the length of the system. Furthermore, we found that the frequency of plastic events is independent on the initial degree of relaxation of the system. We also showed that the persistence of plasticity is principally inuenced by the degree of relaxation of the system. By using the parent temperature of the system as an indicator of its degree of relaxation, we found that the persistence of plasticity arises for parent temperatures lower than the mode-coupling temperature.

The mode coupling temperature being the separation between the hardening and the softening regime, the persistence of plasticity seems to be mainly due to softening upon plasticity. As the avalanches persists at low parent temperature, we observe the formation of persistent shear bands.

Finally, we focused on the properties of persistent shear bands in well relaxed glasses. We measured a correlation between the shear band location and the initial softest bands in the system. The location of the shear bands seems then to be governed by the initial weakest regions in the glasses. We also studied the evolution of the shear band width with the applied deformation on the system. We observed a diusive evolution of the shear band consistently with the model developed by Jagla in [125].

imposing exible boundary conditions [START_REF] Sinclair | Flexible boundary conditions and nonlinear geometric eects in atomic dislocation modeling[END_REF]. This could allow a more sensitive indicator.

Finally, the local yield stress is a quantity often used for mesoscopic simulations of amorphous solids. It can thus be interesting to implement the local yield stress statistics measured in this manuscript in mesoscopic models to observe if we can reproduce the phenomenology of plasticity observed at the atomistic scale.

Hencky equivalent global shear-strain

Our model glasses are deformed using Athermal Quasi-Static (AQS) simple shear. Some systems, particularly in chapter 3 and 4, are loaded up to linear strains γ xy = 500%. As we detailed in section 2.3.1.1, we dene our local strain by using the Hencky's logarithmic strain denition [START_REF] Ne | The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky[END_REF].

To be able to compare and characterize macroscopic and microscopic strains using comparable quantities, the macroscopic strain is expressed in this manuscript using the Hencky strain formalism. Following the calculation detailed in [START_REF] Onaka | Hencky Strain as a Measure to Evaluate Large Deformations Caused by Giant Straining Processes[END_REF],

we will express in this annex the relation between the linear strain γ and the equivalent Hencky strain E.

Similarly with section 2.3.1.1, we consider F the deformation gradient. The logarithmic (Hencky) strain is then dened as E = ln(U), where U is the right stretch tensor, i.e. the unique symmetric and positive-denite tensor such that F = RU, with R a rotation. The right Cauchy-Green strain can be expressed as:

F T F = U 2 , (7.1)
We have that E = 1 2 ln(U 2 ). Also, as U 2 is symmetric and positive-denite, it can be diagonalized, i.e. written as U 2 = P -1 DP with D diagonal and P the associated change of basis matrix.

From this, we obtain that the invariants of E = P -1 ln(D)P are those of ln(D). For a 2D simple shear deformation, the deformation gradient is:

F = 1 γ 0 1 , (7.2)
with γ the linear strain. From this, we obtain that the right Cauchy-Green strain is

U 2 = 1 γ γ 1 + γ 2 (7.3)
The eigenvalues of U are λ ± = γ 2 ± 1 + γ 2 4 2

. Since λ + λ -= 1, we have that det(U 2 ) = 1. From this, we obtained that the square-root of the second tensor invariant of E is:

E ≡ 1 4 | ln(λ + /λ -)| = ln γ 2 + 1 + γ 2 4 (7.4)
This is the quantity we refer to in the manuscript as the macroscopic Hencky strain.

We can also remark that at small strain, γ ≈ 2E.

Checking the equilibration of supercooled liquids

In order to verify if our liquids are equilibrated, we perform two verications.

We rst (i) measure their self intermediate scattering F S function starting from dierent equilibration times. On gure 7.2, we show the sample-averaged self intermediate scattering function for a supercooled liquid composed of 10000 atoms. In this example, we want to equilibrate our system at T = 0.325. We can see that as we equilibrate the systems, the scattering functions converge. From this converged function, we are then able to measure the relaxation time τ α following the protocol described in the annex 7.1 and we obtain τ α = 848 ± 16. In practice, we observe that the system seems equilibrated after an equilibration time of the order of few hundred times τ α .

Then (ii), the second verication consists of measuring the potential energy of the system as a function of the equilibration time and verify if it has converged. We nally apply this two-steps protocol for our systems with dierent numbers of atoms and temperatures until they are equilibrated before quenching them up to T = 0 as explained in section 2.2.2.

Finding a reasonable strain increment for AQS

To determine the strain steps used to load our systems through AQS loading, we do the following protocol:

We deform our systems up to γ xy = 0.06, which is close to the peak stress for well relaxed systems (γ peakstress xy ≈ 0.07). The system is deformed with dierent values of strain steps. In gure 7.4 an example of the stress-strain curves obtained in a system of 10000 atoms for three dierent strain steps: ∆γ = 10 -3 (green), ∆γ = 10 -4 (red) and ∆γ = 10 -5 (blue). We can see on this gure that for a strain step of ∆γ = 10 -3 , we obtain a dierent mechanical response starting from γ xy = 0.05 compared to the two other strain steps. From this gure, we can see that ∆γ = 10 -4 seems to be a reasonable strain step as it gives the same mechanical response than the smaller strain steps. This value of the strain step value is consistent with Maloney and Falk in [START_REF] Craig | Amorphous systems in athermal, quasistatic shear[END_REF] which found that ∆γ = 10 -4 is a reasonable strain step such that all the loading curves have well resolved elastic segments for similar systems of size up to N=40000.

Stress and strain tensor π-periodicity

In this annex, we show the π-periodicity of the 2D stress and the strain tensor. This π-periodicity arises from the symmetry of the stress and strain tensor. In the following demonstration, we will consider the stress tensor, however this demonstration also applies for the strain tensor.

Let us consider the 2D stress tensor Σ. As the stress tensor is symmetric, it can be written as: By multiplying the matrices, we obtain: In this annex, we show that adding a homogeneous elastic strain eld to a plastic local strain eld does not change its variance.

To do so, we consider a local plastic strain eld composed of N sites. We name the mean local strain of the system. Also, for a given site i, the corresponding local strain is written i . The variance V AR of this system can then be written as:

V AR = 1 N N i=1 ( i -) 2 (7.10)
If we add to the system an homogeneous strain of el , then for each sites i, we the new local strain i is i = i + el (7.11)

Measure of the plasticity rate

We present here the protocol we follow the measure the plasticity rate in our model glasses. For a given system size N (number of atoms) and parent temperature T p /T M CT , we measure the local strain elds for strain windows of ∆γ xy = 0.01. We do so for an applied strain γ xy in the range 0.1-0.3 as shown in gure 7.16. This gure shows more precisely the local strain windows of ∆γ xy = 0.01 for an applied strain γ xy in the range 0.11-0.3, for a system of size N = 300 at a parent temperature of T p /T M CT ≈ 0.84. To know if a plastic event has occurred in these strain windows of ∆γ xy = 0.01 we do as follows:

In the Hencky formalism, the mean local strain of each window is ¯ = 0.005.

We consider a strain window to feature a plastic event if its maximum value of local strain max( ) corresponds to max( ) > 3 • ¯ = 0.015. The local strain maps without plastic event are shown with a red cross in gure 7.16.

From this, we dene the plasticity rate Γ pl as the ratio between the number of strain windows with a plastic event (those without a red cross) and the total number of windows:

Γ pl = N umber of windows with a plastic event T otal number of windows (7.16) In the example of gure 7.16, out of 20 strain windows, 7 of them show a plastic event. This corresponds in this case to a plasticity rate equal to Γ pl = 7 20 = 0.35.

Measure of the plasticity persistence

We present in this annex the protocol we follow the measure the plasticity persistence in our model glasses.

For a given system size N (number of atoms) and parent temperature T p /T M CT , we measure the local strain elds for strain windows of ∆γ xy = 0.01. We do so for an applied strain γ xy in the range 0.1-0.3 as shown in gure 7.17. As in the previous annex, this gure shows more precisely the local strain windows of ∆γ xy = 0.01 for an applied strain γ xy in the range 0.11-0.3, for a system of size N = 300 at a parent temperature of T p /T M CT ≈ 0.84. We then only consider the strain elds calculated showing a plastic event. We consider that a strain eld calculated on a window of ∆γ xy = 0.01 shows a plastic event if the max of its local strain max( ) respects the condition max( ) > 3 with the mean local strain.

For a window of ∆γ xy = 0.01, the mean local strain is = 0.05 in the Hencky formalism. The local strain maps without plastic events are shown with a red cross in the gure 7.17. We obtain in this example N w = 7 local strain elds with a plastic event.

From this, we measure the Person correlation coecient between these N w = 7 strain windows with a plastic event. To measure the Pearson correlation coecient ρ i, j between two elds i and j , we apply the following equation:

ρ P e i, j = ( ii )( jj ) σ i σ j (7.17) with i the average of the local strain eld i and σ i the standard deviation of i .

Based on this equation, we measure the Pearson correlation between all the N w = 7 local strain elds. This is illustrated in gure 7.18. In this gure, the black arrows links the elds between which the Pearson correlation coecient will be measured. This corresponds to N w (N w -1)/2 dierent measure correlation coecients. From the obtained Pearson correlations between the train elds, we dene the Plasticity persistence indicator of our system as:

P lasticity P ersistence = 1 N w (N w -1)/2 Nw i=1 Nw j=i+1 ρ P e i, j (7.18) This indicator corresponds to the average Pearson correlation coecient between all the N w strain windows. This protocol is reproduced for all the samples and for the dierent considered system size and parent temperature.

Determination of the strain corresponding to the shear band nucleation

In this section, we will explain the method we use to measure the strain at which a horizontal shear band nucleates γ 0 in gradually quenched glasses. This method is based on the measure of the participation ratio of the local strain proles. We use here again the gures of section 6.3.1.3 to illustrate a system with a horizontal shear band and the corresponding local strain proles.

We consider in gure 7.19 a model system of size L in which the deformation is concentrated in a band of size w. We note SB the deformation within the shear band. x of this model system as a function of the vertical y position.

In this prole, we see that all the deformation is located in a shear band of width w. If the system shows a vertical shear band, then we have P Rx P Ry = L w . Also, if the deformation is localized in a square of side w, then the local strain proles in the x and y direction will both be similar to the prole in gure 7.20. This will then give P Rx P Ry = 1.

Based on these results, we consider that the local strain eld of the glass contains an horizontal shear band if its horizontal and vertical participation ratio, P R x and P R y , respect the following condition:

P R x P R y < 0.4 (7.21) This condition corresponds to the presence of a horizontal shear band in the system with a width smaller than half of the system size, or more precisely here 0.4L with L the width of the system.

To estimate the broadening of the shear bands, we only consider samples with a horizontal shear band. In each of these samples, we deform them and look at the rst value of the applied strain at which the condition of 7.21 is respected. This applied strain is then dened as γ 0 . Après avoir déchargé jusqu'à une contrainte nulle des verres depuis l'état stationnaire, nous observons une anisotropie de la réponse mécanique. En eet, nous mesurons une dureté plus importante dans la direction de la déformation initale par rapport à la direction inverse. Ce phénomène correspond à l'eet Bauschinger. Nous montrons que l'eet Bauschinger est provoqué par l'inversion de l'anisotropie de la distribution des seuils les plus faibles durant la décharge.

Figure 1 . 1

 11 Figure 1.1 Overview of amorphous solids. From left to right, top row: examples of metallic glasses, colloidal gels, emulsion, foam and granular solids. just below, a chart of dierent amorphous materials classied by the size and the

Figure 1 . 2

 12 Figure 1.2 Evolution of the critical casting thickness of metallic glasses since 1960. From [10].

Figure 1 . 3

 13 Figure 1.3 Chart representing the fracture thoughness as a function of the yield strength. Metallic glasses (black crosses) show a higher yield strength compared to their crystalline counterpart (gray regions). The silicate glasses show a high yield strength and a very low fracture toughness. From [3].

Figure 1 . 4

 14 Figure 1.4 Temperature dependence of a liquid's volume or enthalpy at constant pressure. Depending on the cooling rate, the liquid will either crystalize at

Figure 1 . 5

 15 Figure 1.5 Schematic representation of the evolution of the normalized intermediate scattering function as a function of time as predicted by the Mode Coupling Theory. The curve A represents the evolution for system above the melting temperature, exhibiting an exponential decay due to Brownian-motion. The curve B represents the evolution in a supercooled liquid above the Mode Coupling Temperature T M CT , exhibiting a non monotonic decay caused by the surrounding structure around the atoms. Finally, the curve C represents the evolution for a supercooled liquid below T M CT , showing an innite plateau.

Figure 1 . 7

 17 Figure 1.7 Bending strain to failure as a function of the normalized ctive temperature T f /T g with T g the glass transition temperature for dierent metallic glasses. The lower the ctive temperature, the slower the cooling rate. We can see that the strain to failure increases with the ctive temperature. From [5].

Figure 1 . 8

 18 Figure 1.8 Observation of the Bauschinger Eect in atomistic simulation of metallic glass. In the left handside, as the material is sheared in two opposite

Figure 1 .

 1 Figure 1.10 Schematic example of plasticity in amorphous materials. The material under stress deforms plastically from local irreversible rearrangements of atoms. From [17].

Figure 1 .

 1 Figure 1.11 Strain eld around a plastic inclusion in an elastic matrix submitted to a pure shear biaxial loading Γ = Γ yy = -Γ xx . We can recognize the quadrupolar symmetry associated with the Eshelby inclusion. From [67].

Figure 1 .

 1 Figure 1.12 Displacement eld of atoms due to a plastic event obtained from a binary Lennard Jones atomistic simulation. We can observe a quadrupolar shape similar to an Eshelby inclusion.

  At the smallest scale, i.e. the nanometer scale, shown on gure 1.13 (a), the simulations are performed at the scale of the atoms. The simulated systems are then deformed to study their mechanical properties while having access to the local structure and dynamics, which is very dicult experimentally.

Figure 1 .

 1 Figure 1.13 Illustration of multiscale approach for modelling glasses. Starting from atomistic simulation (a) in which measured quantities related to plasticity can be used in mesoscopic models (b). From these models one can extract

Figure 1 .

 1 Figure 1.14 Experimental correlation map of the scattered light from the surface layer of a granular material under compression. We can observe that the decorrelation, corresponding to local deformation, is concentrated in a shear band. From [102].

Figure 1 .

 1 Figure 1.15 Shear band in a metallic glass under compression (a). Under further loading, the material breaks at the shear band location (b). From [65].

Figure 1 .

 1 Figure 1.16 Shear band example in a nite element simulation of sand compression. From [108].

Figure 1 .

 1 Figure 1.18 Local hardness prole within a shear band after 6% of deformation(green) and after annealing upon the deformation (red). We can observe a softening in the shear band location. From[START_REF] Pan | Softening and dilatation in a single shear band[END_REF]

Figure 1 .

 1 Figure 1.19 Schematic representation of the localization within a shear band:

Figure 1 .

 1 Figure 1.21 Velocity prole (a) of nite element simulations studying the inuence of inertia on shear banding. The velocity proles taken at dierent times show shear bands induced by inertia whose location change with time. This is explicited in (b) where the shear band location is showed as a function of time. From [82].

Figure 2 . 1

 21 Figure 2.1 Snapshot of a 2D binary glass of 1000 atoms.

Figure 2 . 2

 22 Figure 2.2 Temperature dependence of a liquid's volume or enthalpy at constant pressure. The black and the blue arrows represent the instantaneous quench to zero temperature from an equilibrated liquid at high temperature and in a supercooled state, respectively. The green curve represents the gradual quench to obtain well relaxed glasses. Note that in our system, the quench is performed at constant volume. Modied from [4].

Figure 2 . 3

 23 Figure 2.3 Inherent-state (IS) potential energy per atom as a function of temperature in equilibrated liquids (gray circles) and in our gradually quenched system (blue diamond). The green line is a t of the equilibrated liquid data.

Figure 2 . 4

 24 Figure 2.4 Typical stress-strain curves for a system deformed with AQS shear.

Figure 2 .

 2 Figure 2.5 (a) Schematic drawing of the local yield stress computation on a regular square grid of mesh size R sampling . Region I (of radius R f ree ) is fully relaxed while region II (of width 2R cut ) is forced to deform following an ane pure shear deformation in the α direction. (b) Typical local stress-strain curves of the Region I for dierent loading directions a = 0, 45, 90, and -45°. The measurement of the threshold τ c , the distance to threshold ∆τ c and relaxation ∆τ r are represented for α = 90°.

Figure 3 . 1

 31 Figure 3.1 Typical stress-strain curves for the three quench protocols: instantaneous quench from a high-temperature liquid (HTL, continuous line), instantaneous quench from an equilibrated supercooled liquid (ESL, dashed line), and a gradual quench (GQ, dash-dotted line). The inset shows a zoom of a stress drop corresponding to one plastic event.

Figure 3 . 2

 32 Figure 3.2 Local yield stress maps computed on a regular grid for the three dierent quench protocols: (a) HTL, (b) ESL, and (c) GQ. The rst ten plastic event locations are shown as open black symbols numbered by order of appearance during remote shear loading. Plastic events clearly tend to occur in regions characterized by low-yield stresses.

Figure 3 . 3

 33 Figure 3.3 Probability distribution function of the local yield stresses for the three dierent quench protocols. The corresponding cumulative distribution functions are represented in log-log scale in the inset. The straight line is a power law of exponent 1 + θ = 1.6, i.e., the expected scaling of the integral of the probability distribution function as ∆τ y approaches zero [159, 160].

Figure 3 . 4

 34 Figure 3.4 Plastic event computed between the onset of instability and just after the event: the arrows and the color scale are the displacement u and local shear strain elds, respectively. For the sake of clarity, the arrows are magnied by a factor 200 and deleted in the core region. The atom with the maximum shear strain gives the location of the plastic rearrangement.

  2, representing the local yield stress maps for HTL (a), ESL (b) and GQ (c), we show with open black symbols the location of these ten rst plastic events triggered during remote loading. In this gure, we can see that plastic events tend to occur in regions characterized by low local yield stresses, i.e. in initially soft regions.

Figure 3 . 5

 35 Figure 3.5

Figure 3 .

 3 Figure 3.6 (a) Probability distribution function of the stress drops for the three dierent quench protocols in lin-log scale. The same quantities rescaled by the shear moduli, i.e., the slip increments, are represented in the inset. The lines are exponential ts. (b) Average stress drop as a function of the average local yield stress for the free dierent quench protocols. The line is an exponential t.

Figure 3 . 7 1 l , α 2 l

 3712 Figure 3.7 Top row: local yield stress maps computed on a regular grid for dierent loading directions for the GQ protocol: (a) simple shear α l = 0°, (b) pure shear α l = 45°, and (c) negative simple shear α l = 90°. The red arrows correspond to the applied strains. The rst ten plastic event locations are shown as open black symbols numbered by order of appearance during remote shear loading. Bottom row: local yield stress contrasts TC(α 1 l , α 2 l ) dened in Eq. (8) between the above loading directions: (d) α 1 l = 0°, α 2 l = 45°, (e) α 1 l = 45°, α 2 l = 90°, and (f ) α 1 l = 0°, α 2 l = 90°.

Figure 3 . 8

 38 Figure 3.8 Cross-correlation of the local yield stress eld as a function of the loading direction shift. Error bars are smaller than symbols.

Figure 3 . 9

 39 Figure 3.9 Probability distribution functions of angles for the three quench protocols for which the projected local yield stress is minimal α min . The lines correspond to Gaussian ts which standard deviations are reported in the inset.

Figure 3 .

 3 Figure 3.10 Correlation between the local yield stresses computed in the quenched state and the locations of the plastic rearrangement as a function of the applied strain for the GQ protocol for dierent local yield stress elds: its minimum over all orientations min α ∆τ c (α), its value along the loading direction ∆τ c (α = α l ) and its minimum once projected along the loading direction min α ∆τ c (α)/cos[2(α -α l )].

  second one (ii) is the characteristic value of the deformation γ d over which the correlation decreases with the imposed deformation. This quantity is obtained by tting the curves C ∆τy (R f ree , γ xy ) with the expression: A + Be -(γxy/γ d ) 2 as shown in gure 3.5 (a). Finally, our last indicator (iii) is the average of the correlation for the strain interval [0, γ d (R f ree = 2.5)]: C ∆τy (R f ree ) . The variation of these three indicators for the dierent quench protocols is shown in gure 3.11. The upper bound of the strain window: γ d (R f ree = 2.5) corresponds to the largest decorrelation strain.In gure 3.11, we represent the evolution the three correlation indicators as a function of the size of the probing zone R f ree .

Figure 3 .

 3 Figure 3.11 Correlation indicators computed as a function of the size of the probing zone R f ree . Top: Correlation with the rst plastic rearrangement locations. Middle: decorrelation strain. Bottom: Average correlation.

Figure 3 .

 3 Figure 3.12 Top row: Local yield stress maps of a GQ glass computed for dierent inclusion sizes R f ree : (a) 5, (b) 7.5, (c) 10, and (d) 15. Bottom row: Corresponding local yield stress maps deduced from local minima obtained from the map computed for R f ree = 5 shown in panel (a).

Figure 3 .

 3 Figure 3.13 Probability distribution function of the local yield stress for the three dierent quench protocols as a function of the inclusion size R f ree : (a) HTL, (b) ESL, and (c) GQ. The lines correspond to the zoom-out process exemplied in the bottom row of gure 3.12, where the local yield stresses are deduced from maps computed with R f ree = 5.

Figure 4 . 1

 41 Figure 4.1 Mean shear stress τ xy (black) and pressure P (blue) as a function of Hencky's equivalent shear strain E during simple shear AQS loading. The color symbols on the stress curve correspond to particular states where the shear band prole is analyzed.

  2 (b) the local yield stress map measured for the same loading orientation applied on the whole system: τ c (α = 0) = τ c xy . From this map, we can clearly observe that the local yield stress is lower where the shear band is located. It indicates a softening upon plasticity.

Figure 4 . 2

 42 Figure 4.2 Left handside: Local maps at E = 0.05 of: (a) the accumulated local strain and (b) the local yield stress τ c xy . Right handside: Local averages computed along the shear direction x for dierent strain levels E as a function of the vertical position y with respect to the shear band position y SB for: (c) the local strain

Figure 4 . 3

 43 Figure 4.3 Top: Average yield stresses τ c xy as a function of the normalized local strain / * for various macroscopic strain E. The continuous, dashed, and dash-dotted horizontal lines correspond to τ c xy in the as-quenched, steady- state, and renewed ensembles, respectively. Bottom: Distribution of log 10 ( / * ).

  4 the local yield stress as a function of the local density ρ l for the yielded (in green) and unyielded (in red) regions. We can observe that for a same value of the local density ρ l , the yielded and unyielded regions have a very dierent value of local yield stress τ c xy . This shows that in our system, local packings of atoms can have a similar density but drastically dierent stability and local properties.

Figure 4 . 4

 44 Figure 4.4 Average local yield stress τ c xy as conditioned by local density in unyielded (red) and yielded (green) states at E = 0.05 and in the renewed ensembles (shades of brown).

Figure 4 . 5

 45 Figure 4.5 Local yield stress τ c xy as conditioned by local pressure P in yielded (green) and unyielded (red) regions at E = 0.05 and for renewed sites (shades of brown) at dierent macroscopic strains.

Figure 4 .

 4 Figure 4.6 (a) Distribution of local yield stresses τ c xy in the as-quenched state (black), the steady ow state (blue), at E = 0.05 in the unyielded (red) and yielded (green) areas and the renewed state (shades of brown). [(b) and (c)] The eect of pressure and statistical hardening (see text for details) for the as-quenched state and unyielded sites (b); the steady ow state and yielded sites (c).

Figure 4 . 7

 47 Figure 4.7 Distribution of local yield stresses τ c xy in the loading (continuous lines) and reverse directions (dashed lines) in the as-quenched state (black), the steady ow state (blue), at E = 0.05 in the unyielded (red) and yielded (green) areas Note that here, τ c xy = τ c for 2α = 0 and τ c xy = -τ c for 2α = π so the forward barriers correspond to the positive abscissa and the backward barriers correspond to the negative abscissa.

Figure 4 . 8

 48 Figure 4.8 Mean shear stress τ xy as a function of Hencky's equivalent shear strain E during simple shear AQS loading for three dierent quench protocols : GQ, ESL and HTL.

Figure 4 . 9

 49 Figure 4.9 Local yield stress distribution for the quench protocols GQ, ESL and HTL in the as quenched state (a) and steady state (b).

Figure 4 .

 4 Figure 4.10 Average local yield stresses τ c xy of inherent states as a function

  1 of three dierent quench protocols GQ (a), ESL (b) and HTL (c) (corresponding respectively to (T p /T M CT ≈ 0.78, 0.94 and 8).

Figure 4 .

 4 Figure 4.11 Local strain maps at E = 0.1 for GQ (a), ESL (b) and HTL (c).

Figure 5 . 1

 51 Figure 5.1 Schematic representation of the stress strain curves to investigatethe Baushinger eect. The system is rst loaded in the forward direction up to the steady state (in blue). Then, the system is unloaded (in the backward direction) down to zero stress (black). From this unloaded state, the glass is either reloaded (green) or reverse loaded (red). The Bauschinger eect in our model glasses consists of a more ductile mechanical response during the reverse loading compared to the reloading.

Figure 5 . 2

 52 Figure 5.2 Mean stress vs strain (both in absolute values) during three tests: unloading from steady ow (black), backward (red) and forward (green) loading from fully unloaded (zero stress) congurations. In all three cases, strain is measured with reference to the zero-stress state.

Figure 5 . 3

 53 Figure 5.3 Left frames: the polar function P (∆τ c , 2α), with the mean local residual strength ∆τ c (2α) in white. (a) As-quenched isotropic (ESL) state, (b) steady state, and (c) unloaded steady state congurations. Right frames: cuts of P (∆τ c , 2α) along the forward (2α = 0) and backward (2α = π) directions (d) in our three as-quenched glasses of dierent degrees of relaxation (HTL, ESL, and GQ). (e),(f ) In steady ow (blue) and unloaded states (red), and (up to a scaling factor) model prediction for unloading (see text, dashed black).

Figure 5 . 4

 54 Figure 5.4 Evolution of mean barrier polarizations with the strain during three tests: unloading from steady ow (black), backward (red) and forward (green) loading from fully unloaded (zero stress) congurations. The mean barrier polarizations asymptotic values are represented with the dotted lines.

  xy |=0.015; χ~0

Figure 5 . 5

 55 Figure 5.5 Residual strength ∆τ c xy distribution along the forward/backward loading direction in the unloading branch at γ xy = 0.015 that corresponds to mean barrier polarizations χ ∼ 0.

Figure 5 . 6

 56 Figure 5.6 The distribution of local stresses in ow (black) and unloaded (red)

Figure 5 . 7

 57 Figure 5.7 Maps of the local residual strength in the unloaded state for shearing (a) in the backward (2α = π) and (b) forward (2α = 0) directions. Symbols show the loci of plastic events in the 2% of strain in the corresponding direction.

Figure 5 . 8

 58 Figure 5.8 Example of displacement elds for plastic drops under dierent types of loading conditions. (a) in steady state, an event is typically an avalanche; in contrast, during unloading (b), reverse loading (c), or reloading (d), typical events are well separated Eshelby-like single zone ips.

τ

  xy (δγ xy ) = τ xy (0) + δτ xy , (5.4) with τ xy (0) the stress level in the initial state. We consider τ c xy (δγ xy ) the local threshold of the barrier. The corresponding local residual strength is by denition: ∆τ c xy (δγ xy ) = τ c xy (δγ xy ) -τ xy (δγ xy ), xy ) = ∆τ c xy (0) -δτ xy , (5.6)

  µδγ xy -δτ xy = δτ pl xy = 2µρa 2 ∆ 0 denes the evolution of δτ xy (δγ xy ). From it, we can test the relation between the small barrier tails in the local residual strength distribution

Figure 5 . 9

 59 Figure 5.9 Mean stress vs strain (both in absolute values) during three tests: unloading from steady ow (black), backward (red) and forward (green) loading from fully unloaded (zero stress) congurations. In all three cases, strain is measured with reference to the zero-stress state. Solid black line: t of the unloading curve using Eq. 5.13 and ρa 2 ∆ 0 = 0.25. Solid green and red lines: consequent predictions for the Bauschinger tests. Dashed lines: predicted reloading and backward loading curves, when the model is used starting from the steady ow barrier distribution (see text), i.e., while taking into account its prediction for the unloading-induced asymmetry.

  3 (e) (in blue). In the steady state, the mean stress is equal to τ f low xy = 0.53 in the forward direction. During the unloading, we induce a macroscopic stress change of τ f low xy in the backward direction to have a zero mean stress. Thus, the macroscopic stress imposed on the forward barriers is reduced by the unloading. At the same time, the macroscopic stress increases for the backward barriers. Following the model assumptions, this corresponds to a shift of the forward barriers of τ f low xy towards higher values. For the backward direction, the macroscopic stress shifts the barriers by τ f low xy towards smaller values. When a barrier in the backward direction becomes negative, it becomes considered as yielded and is not taken into account anymore.

Figure 5 .

 5 Figure 5.10 Mean stress vs strain for dierent responses: full unloading from steady ow (black symbols); after full or partial unloading (by γ u ) from steady state, during reloading (continuous lines), and backward loading (dashed lines).

Figure 6 . 1

 61 Figure 6.1 Cumulated strain maps at an applied strain of γ xy = 0.3. The rst row ((a), (b)) corresponds to systems of 10 000 atoms, the second row corresponds to systems of 1000 atoms and the third row corresponds to systems of 300 atoms. Moreover, the rst column ((a), (c), (e)) corresponds to systems with a parent temperature of T p = 0.315 (i.e. T p /T M CT ≈ 0.84), the second column ((b), (d), (f )) corresponds to T p = 0.488 (i.e. T p /T M CT ≈ 1.3). A shear band can be observed for the most relaxed systems (i.e. obtained by an instantaneous quench from T p = 0.315), while the distribution of local strain seems much more homogeneous for the less relaxed system.

Figure 6 . 2

 62 Figure 6.2 Phase map of the variation with the system size and normalized parent temperature T p /T M CT of the strain variance within the cumulated local strain maps for an applied strain of γ xy = 0.3. An increase of the variance, representing the degree of heterogeneity of the local strain, can be observed for more relaxed systems and for bigger systems.

Figure 6 . 3

 63 Figure 6.3 Evolution of the variance of the local strain eld for an applied strain of γ xy = 0.3 as a function of the normalized parent temperature T p /T M CT for dierent numbers of atoms N . The error bars correspond to the standard error of the sample average.

(6. 11 )

 11 Moreover, we know thatN pl = N 1site • n, with N 1site = pl c and n S = L 2 l 2 .

Figure 6 . 5

 65 Figure 6.5 Schematic representation of the local strain map of a system of size L in which the plastic deformation corresponds of an accumulation of independent avalanches of size l • L. After each a plastic event, the local strain of increases in the avalanche by inc . The avalanches are represented by bandsof length L and width l. However, the variance of this system is independent of the shape of the avalanche.

Figure 6 . 6

 66 Figure 6.6 Evolution of the variance at high parent temperature of the local strain eld at γ xy = 0.3 as a function of 1 L with the length L of the system (in blue). The variance here corresponds to a variance averaged for parent temperature T p /T M CT in the range 1.1-1.3. The orange curve corresponds to the equation 6.12 obtained from a model of independent plastic events of size l 2 . The green curve corresponds to the equation 6.14 obtained from a model of independent avalanches of size l • L.

116 6. 1 . 1 . 3

 116113 Toy model for the low parent temperature regimeWe now want to understand the variance at low parent temperature. While the variance at high parent temperature seemed to converge toward a nite value for increasing L, we do not observe the convergence of the variance with L at low parent temperature.

(6. 20 )

 20 As previously, since the elastic eld does not change the variance, we also have:VAR( ) = VAR( el + pl ) = VAR( pl ),(6.21) so we nally obtain VAR( ) = (el ) 2 ( L l -1).

Figure 6 . 8

 68 Figure 6.8 Evolution of the variance at T /T M CT ≈ 0.84 of the local strain eld at γ xy = 0.3 as a function of the length L of the system (in blue). The green curve corresponds to t for the equation 6.22 obtained from a model with a persistent shear band of width l.

  with [N = 10000 and T p /T M CT ≈ 1.3], [N = 300 and T p /T M CT ≈ 0.84] and [N = 300 and T p /T M CT ≈ 1.3], respectively.

Figure 6 . 9

 69 Figure 6.9 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 10000 atoms at T p /T M CT ≈ 0.84 for an applied strain γ xy in the range 0.1-0.14.

Figure 6 .

 6 Figure 6.10 Same as gure 6.9 for a system of 10000 atoms at T p /T M CT ≈ 1.3 for an applied strain γ xy in the range 0.1-0.14.

Figure 6 .

 6 Figure 6.14 Evolution of the plasticity rate with the normalized parent temperatures T p /T M CT for dierent numbers of atoms N in the system. The error bars correspond to the standard error of the sample average.

Figure 6 .

 6 Figure 6.15 Evolution of the plasticity rate with the length of the side of the system L averaged over the dierent the parent temperatures. We can observe an increase of the plasticity rate with L before it saturates. We show in orange the t obtained from the equation 6.35. The error bars are smaller than the symbols

Figure 6 .

 6 Figure 6.16 Evolution of the plasticity rate with the length of the side of the system L averaged over the dierent the parent temperatures. We can observe an increase of the plasticity rate with L before it saturates. We show in orange the t obtained from the equation 6.42. The error bars are smaller than the symbols To estimate the values of A and l, a more thorough study of the plasticity rate with this model would be needed. It would require the study of the evolution of the plasticity rate with the size of the strain window ∆γ xy for dierent

Figure 6 .

 6 Figure 6.18 Evolution of the plasticity persistence as a function of the normalized parent temperature T p /T M CT for dierent the number of atoms N . We can see the plasticity persistence increases as the parent temperature decreases. The error bars correspond to the standard error of the sample average.

Figure 6 .

 6 Figure 6.19 Evolution of the local yield stress as a function of the normalized parent temperature T p /T M CT for dierent the number of atoms N . We can see the local yield stress increases as the parent temperature increases. We see no inuence of the system size in the local yield stress. The error bars correspond to the standard error of the sample average.

Figure 6 .

 6 Figure 6.20 Local averages computed along the shear direction x for dierent strain levels E as a function of the vertical position y with respect to the shear band position y SB : a) strain x and b) yield stress τ c xy x .

Figure 6 .

 6 Figure 6.21 Illustration on a single sample showing an horizontal shear band of the method used to estimate the correlation between the x-averaged local yield stress in the as-quenched state and the shear band location y SB . (a) xaveraged local strain prole as a function of the vertical position y at E = 0.1. We use the maximum in the prole (red line) as the shear band position y SB . (b) x-averaged local yield stress prole τ c xy as a function of the vertical position y in the as-quenched state. The red line corresponds to the shear band position at E = 0.1. The green arrow indicates the value of τ c xy at the shear band location. (c) Cumulative Distribution Function (CDF) of τ c xy . The green line

Figure 6 .

 6 Figure 6.22 Local strain computed along the shear direction x for γ xy = 1.0 as a function of the vertical position y with respect to the shear band position

Figure 6 .

 6 Figure 6.23 Schematic representation of the method used to estimate the shear band width. The local strain eld { i } is calculated on a grid (a). We then construct a binary mask (b) with the cuto cut = 0.125 (which is the conversion in Hencky formalism of the cuto of γ cut = 0.25 used in [123]), with i > cut → 1 and i < cut → 0. We nally consider width of the largest continuous line as the shear band width w line (c). This gure was modied

Figure 6 .

 6 Figure 6.24 Examples of binary map obtain with a cuto of cut = 0.125 for two dierent samples (rst and second column) with a horizontal shear band for an applied strain of γ xy = 0.2 (rst row) and of γ xy = 1.0 (second row).

Figure 6 .

 6 Figure 6.25 Schematic representation of a system of size L in which all the deformation is concentrated on a horizontal shear band of width w. We note SB the deformation within the shear band.

3 . 3

 33 in chapter 1 for more details) This model predicts a linear relation between w 2 -w 2 0 and γ xy -γ 0 . To test this model, we plot on gure 6.27 the evolution of w 2 -w 2 0 with γ xy -γ 0 for To test whether we obtain a similar trend, we show in gure 6.28 the evolution of the shear band width w as a function of γ xy -γ 0 for our three shear band width indicators: w Gauss (a), w line (b) and w P R (c) (blue curve). We can see in these gures that w Gauss and w P R show a slower increase compared to w line . Then, for γ xy > 1.0 we t the width evolution with the power-law function w = (B 1 + B 2 (γ xy -γ 0 )) c (orange curve).

Figure 6 .

 6 Figure 6.28 Evolution of w the shear band width as a function of the applied strain shifted by the nucleation strain: γ xy -γ 0 (blue curve). In this system, γ 0 ≈ 0.086. We consider in this gure three shear band width indicators: w Gauss (a), w line (b) and w P R (c) obtained by tting the local strain proles with a Gauss function, applying a binary mask on the strain eld with a pre-dened cuto and by measuring the participation ratio of the local strain proles, respectively.

  Jagla in [125]. Conclusion Throughout this manuscript, we highlighted seven main results about plasticity and shear banding in model glasses at the atomistic scale: the identication of shear transformation weak planes encoded in the structure; the measurement of a unique post-yield shear threshold distribution, independent of the initial state; the second order contribution of free volume compared to structural rejuvenation to explain the softening upon plastic deformation; the determination of a simple criterion to separate brittle and ductile glasses based on their initial degree of relaxation; the identication of the origin of the Bauschinger eect in amorphous solids;

Figure 7 . 1

 71 Figure 7.1 Average local yield stresses τ c xy of inherent states as a function

Figure 7 . 2

 72 Figure 7.2 Sample-averaged self-intermediate scattering function, starting from dierent equilibration times, for supercooled liquids of 10000 atoms that we want to equilibrate at T = 0.325. After they converge, we measure τ α = 848±16 using the method detailed in annex 7.1.

Figure 7 . 3

 73 Figure 7.3 Sample-averaged potential energy as a function of the equilibration time normalized by the relaxation time τ α for supercooled liquids of 10000 atoms that we want to equilibrate at T = 0.325.

Figure 7 . 4

 74 Figure 7.4 Example of stress response for a system with 10000 atoms loaded up to the peak stress (i.e. γ xy ≈ 0.07 with AQS simple shear with three dierent shear steps: 10 -3 , 10 -4 , 10 -5 .

Σ = σ xx σ xy σ xy σ yy ( 7 . 5 )( 7 . 6 )σ

 7576 Let us now consider the rotation matrix for an angle (θ): R(θ), which can be written as:R = cos(θ) sin(θ) -sin(θ) cos(θ)The stress tensor rotated by θ: Σ corresponds to:Σ = RΣR T ,xx σ xy σ xy σ yy = cos(θ) sin(θ) -sin(θ) cos(θ) σ xx σ xy σ xy σ yy cos(θ) -sin(θ) sin(θ) cos(θ)

Figure 7 . 5

 75 Figure 7.5 Scatter plot of the variation with the system size and normalized parent temperature T p /T M CT of the strain variance within the cumulated local strain maps for an applied strain of γ xy = 0.3. The variance is divided by the mean local strain, which for γ xy = 0.3 would be ≈ 0.15. An increase of the variance, representing the degree of heterogeneity of the local strain, can be observed for more relaxed systems and for bigger systems.

Figure 7 . 6

 76 Figure 7.6 Phase map of the variation with the system size and normalized parent temperature T p /T M CT of the strain variance within the cumulated local strain maps for an applied strain of γ xy = 0.3. The variance is divided by the mean local strain, which for γ xy = 0.3 would be ≈ 0.15. An increase of the variance, representing the degree of heterogeneity of the local strain, can be observed for more relaxed systems and for bigger systems.

7. 7

 7 Cumulated local strain variance with a smaller coarse-graining sizeIn this annex, we show the evolution of the variance of the cumulated local strain elds, measured with a coarse-graining size of R CG = 2.5, with the parent temperature T p /T M CT for dierent numbers of atoms N . We can observe on this gure that the variance for N = 100 shows no clear evolution with the parent temperature.

Figure 7 . 7

 77 Figure 7.7 Evolution of the variance of the local strain eld measured with a coarse graining size R CG = 2.5 for an applied strain of γ xy = 0.3 as a function of the normalized parent temperature T p /T M CT for dierent numbers of atoms N . The error bars correspond to the standard error of the sample average.

7. 8

 8 Variance of a strain eld with an homogeneous elastic eld

Figure 7 . 8

 78 Figure 7.8 Sample averages stress strain curve at T p /T M CT ≈ 1.3 and for N=10000 (blue curve). The green arrow represents the unloading obtained from the knowledge of the shear modulus of the system in the elastic branch.

Figure 7 . 9

 79 Figure 7.9 Sample averages stress strain curve at T p /T M CT ≈ 0.84 and for N=10000 (blue curve). The green arrow represents the unloading obtained from the knowledge of the shear modulus of the system in the elastic branch.

7. 10

 10 Cumulated and window strain mapsIn this annex, we show the cumulated local strain maps (upper row) and local strain maps measured in strain windows of ∆γ xy = 0.01 (lower row) for an applied strain γ xy in the range 0.01 -0.3.The gures 7.10 and 7.11 corresponds to a system of 10000 atoms at T p /T M CT ≈ 0.84 and T p /T M CT ≈ 1.3, respectively.

T

  p /T M CT ≈ 0.84 and T p /T M CT ≈ 1.3, respectively.Finally, the gures 7.14 and 7.15 corresponds to a system of 300 atoms at T p /T M CT ≈ 0.84 and T p /T M CT ≈ 1.3, respectively.

Figure 7 .

 7 Figure 7.10 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 10000 atoms at T p /T M CT ≈ 0.84 for an applied strain γ xy in the range 0.01 -0.3.

Figure 7 .

 7 Figure 7.11 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 10000 atoms at T p /T M CT ≈ 1.3 for an applied strain γ xy in the range 0.01 -0.3.

Figure 7 .

 7 Figure 7.12 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 1000 atoms at T p /T M CT ≈ 0.84 for an applied strain γ xy in the range 0.01 -0.3.

Figure 7 .

 7 Figure 7.13 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 1000 atoms at T p /T M CT ≈ 1.3 for an applied strain γ xy in the range 0.01 -0.3.

Figure 7 .

 7 Figure 7.14 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 300 atoms at T p /T M CT ≈ 0.84 for an applied strain γ xy in the range 0.01 -0.3.

Figure 7 .

 7 Figure 7.15 Cumulated local strain maps (rst row) and local strain maps calculated in strain windows of ∆γ xy = 0.01 (second row) for a system of 300 atoms at T p /T M CT ≈ 1.3 for an applied strain γ xy in the range 0.01 -0.3.

Figure 7 .

 7 Figure 7.16 Local strain maps calculated on windows of ∆γ xy = 0.01 for an applied strain γ xy in the range 0.11-0.3 for a system of size N = 300 at a parent temperature of T p /T M CT ≈ 0.84. The local strain maps showing no plastic event are shown with a red cross.

Figure 7 .

 7 Figure 7.17 Local strain maps calculated on windows of ∆γ xy = 0.01 for an applied strain γ xy in the range 0.11-0.3 for a system of size N = 300 at a parent temperature of T p /T M CT ≈ 0.84. The local strain maps showing no plastic event are shown with a red cross.

Figure 7 .

 7 Figure 7.18 Local strain maps calculated on windows of ∆γ xy = 0.01 showing plasticity for an applied strain γ xy in the range 0.11-0.3, for a system of size N = 300 at a parent temperature of T p /T M CT ≈ 0.84. This gure illustrates with black arrows between which strain eld the Pearson correlation coecient is measured. We measure in practice the Pearson correlation coecient between all the local strain maps showing plastic event for an applied strain γ xy in the range 0.1-0.3.

Figure 7 .

 7 Figure 7.19 Schematic representation of a system of size L in which all the deformation is concentrated on a horizontal shear band of width w. We note SB the deformation within the shear band.

Figure 7 .( 7 . 19 )

 7719 Figure 7.20 Schematic representation of the local strain averaged along the x axis as a function of the vertical position y for a system of size L in which all the deformation is localized in a horizontal band of width w. We note SB the deformation within the shear band.

Figure 7 . 1 ( 7 . 20 )

 71720 Figure 7.21 Schematic representation of the local strain averaged along the y axis as a function of the horizontal position x for a system of size L in which all the deformation is localized in a horizontal band of width w. We note SB the deformation within the shear band.

θξpR

  Abbreviation AQS Athermal Quasi-Static CDF Cumulative Distribution Function CG Coarse-Graining ESL Equilibrated Supercooled Liquid GQ Gradual Quench HTL High Temperature Liquid IS Inherent state LJ Lennard-Jones LL Large-Large MD Molecular Dynamics PEL Potential Energy Landscape PR Participation Ratio SB Shear Band SL Small-Large SS Small-Small Symbols α

  10 2 10 1 10 0 10 1 10 2 10 3 10 4

		1.0
		0.8
	F s	0.0 0.2 0.4 0.6	t start / =0 t start / =108 t start / =217 t start / =326 t start / =435 t start / =543 t start / =652	t

  In gure 7.3, we represent the sample-averaged potential energy as a function of the equilibration time normalized by the relaxation time τ α for the systems considered in gure 7.2. We can see in this example that the potential energy seems to converge after a waiting time of the order of t ∼ 500τ α .

	2.060 2.055 2.050 2.045 2.040 2.035 2.030 2.025 2.020	0	200	400	t/	600	800 1000

Remerciements

0 as a function of γ xy -γ 0 with w the shear band width for an applied strain of γ xy (blue curve). w 0 corresponds to the shear band width at its nucleation and γ 0 corresponds to the strain at which the shear band nucleates. In this system, γ 0 ≈ 0.086. We consider in this gure three shear band width indicators: w Gauss (a), w line (b) and w P R (c) obtained by tting the local strain proles with a Gauss function, applying a binary mask on the strain eld with a pre-dened cuto and by measuring the participation ratio of the local strain proles, respectively. We name w Gauss,0 , w line,0 and w P R,0 the shear band width at the nucleation for the three indicators. The orange curve corresponds to a linear t of the shear band width evolution on the model proposed in [125].

We also perform a linear t with the function w 2 -w 2 0 = C(γ xy -γ 0 ) with C a proportionality constant (orange curve) equal to C = 2ALw 2 0 . We can see on these gures that a linear relation between w 2 -w 2 0 and γ xy -γ 0 is a good approximation. We measure A=0.0081, 0.666 and 0.0075 with the quantities w Gauss , w line and w P R , respectively. We observe that the model constitutes a good approximation of the shear band width evolution in our gradually quenched model glass.

In [START_REF] Golkia | Flow heterogeneities in supercooled liquids and glasses under shear[END_REF], in accordance with [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF], a sub-diusive evolution of the shear band was observed for an applied strain of γ xy > 1.0. Indeed, they observed an evolution of the shear band width following a power-law function with an exponent of 0.32. To obtain the shear band width, they used a method similar to [START_REF] Anshul | Strain Localization Above the Yielding Point in Cyclically Deformed Glasses[END_REF] which consists of tting the mean-square displacement prole with a Gauss function. Note that the model glasses studied in [START_REF] Golkia | Flow heterogeneities in supercooled liquids and glasses under shear[END_REF] is not strictly equivalent to ours. The studied model glass is a three-dimensional binary Lennard-Jones glass at low temperature. Furthermore, in [START_REF] Alix-Williams | Shear band broadening in simulated glasses[END_REF], the studied model glass is a two-dimensional binary Lennard-Jones glass at low temperature.

Perspectives

In the continuity of this thesis, many improvements can be thought. They can be classied in two categories: the continuations of the work described in this manuscript and an evolution towards more realistic models.

When studying the averaged local yield stress proles evolution with the applied strain, we observe the formation of a region within the shear band with a constant local yield stress. This region widens as the system is loaded.

Studying the shape of the local yield stress prole and its evolution could allow a better understanding of the inuence of shear bands in the local structural properties of model glasses.

In the simulations performed for this thesis, we only considered systems at zero temperature and deformed with an AQS shear protocol, that corresponds to the low temperature limit and an innitely low shear rate. Observing how the local yield stress statistics and the shear band properties depend on the temperature and nite shear rate could allow the observation of dierent phenomenology.

In chapter 6, we have shown that the plastic events frequency is independent on the degree of relaxation of the model glasses. To understand its evolution with the system size, we proposed two elementary models. However, these models are not able yet to fully reproduce and explain the evolution of the plastic events frequency. An improvement of the models is thus needed to fully understand this behavior.

By quenching a glass using molecular dynamics, the degree of relaxation of the obtained system is limited. It would be interesting to observe the inuence of the degree of relaxation of glass by also using ultra-stable glasses obtained with the Monte-Carlo swapping method [START_REF] Berthier | Ecient swap algorithms for molecular dynamics simulations of equilibrium supercooled liquids[END_REF]. Using these ultra-stable glasses could for instance allow a better understanding of the heterogeneities of the cumulated local strain maps observed for well relaxed system in chapter 6. The Mode-Coupling Temperature T MCT is determined from a functional t of the relaxation time τ α in the dynamical regime for dierent temperatures T [START_REF] Schrøder | Crossover to potential energy landscape dominated dynamics in a model glass-forming liquid[END_REF][START_REF] Cavagna | Supercooled liquids for pedestrians[END_REF].

We rst compute the self-intermediate scattering function F S (q, t) = cos(q• (r j (t) -r j (0))) , where the subscript L refers to the large particles, r j (t) is the position of the j th particle at time t and . . . denotes the average over j and the time origin. F S (q, t) is averaged over 32 samples, containing 1024 atoms each, for every temperature.

We dene the relaxation time τ α as F S (q SF = τ α ) = 1 e with |q SF | = 6.07

corresponding to the primary peak of the static structure factor. As reported in Fig. 7.1, τ α , as computed for temperatures larger than 0.4, is well tted by the power law relation τ α ∝ (T -T MCT ) γ . We obtain γ = -0.962 ± 0.054 and T MCT = 0.384 ± 0.005.

From these relations, we can observe that all the components of the rotated stress-tensor Σ : σ xx , σ xy and σ yy are π-periodic. From this, we obtain the π-periodicity of the stress tensor Σ.

Creation of phase maps

In this annex, we present the method we use to obtain phase maps as a function of the number of atoms N and of the normalized parent temperature T p /T M CT .

To illustrate the method, we use the variance of the cumulated local strain elds measured for an applied strain of γ xy = 0.3.

We rst select dierent system sizes (number of atoms) and parent temperatures. For a given combination of system sizes and parent temperatures, we simulate many independent samples of model glasses. In practice, the number of samples is in the range 10-100. These samples are then deformed up to γ xy = 0.3.

After that, for each of these samples, we measure the variance of the cumulated local strain eld at γ xy = 0.3. From the obtained result, we do a scatter plot, as shown in gure 7.5, of the sample-averaged variance as a function of the system sizes and of the parent temperatures. The color of the disks in the scatter plot corresponds to the value of the variance.

Based on this scatter plot, we perform a 2D linear interpolation with the package Matplotlib [START_REF] Hunter | Matplotlib: A 2D graphics environment[END_REF] in Python3 [START_REF] Van Rossum | Python 3 Reference Manual[END_REF]. From this linear interpolation, we are able to obtain the phase maps as shown in gure 7.6.

Similarly, the new mean local strain is:

Finally, the variance V AR of this system is:

which gives:

We nally obtain that adding an homogeneous elastic strain to the system does not change the variance.

Estimation of the mean plastic deformation

To estimate the mean plastic deformation, we rst consider the sample-averaged stress strain curves for dierent system size N and at dierent parent temperature T p /T M CT . In gure 7.8 and 7.9 we show the sample-averaged stress strain curve for N=10000 at T p /T M CT ≈ 1.3 and T p /T M CT ≈ 0.84, respectively (blue curve).

From this stress-strain curves, we measure the shear modulus in the elastic branch µ. Then we dene the plastic deformation γ pl for an applied strain of γ xy as:

with Σ xy (γ xy ) the mean stress at γ xy .

For a given temperature, we estimate the plastic strain for the dierent system size. As we do not observe an inuence of the system size on the value γ pl , we use for a given temperature the size-averaged γ pl .

Note that in this manuscript, we use the plastic strain in the Hencky for- In this thesis, we perform atomistic simulations with a simple two-dimensional binary Lennard-Jones model glass. This model allows us to reproduce most of the plastic behaviors such as the formation of shear bands, and is generic enough to describe dierent kinds of amorphous solids such as metallic or colloidal glasses. To link plasticity and the material structure, we use a novel structural indicator, the local yield stress, which shows a good ability to predict the locations of plastic events based on the initial structure of the system.

Using the local yield stress method, the inuence of the thermal history of metallic glasses on their local properties in investigated. An increase of the material average local yield stress is found as the degree of relaxation increases.

The existence of initial soft regions in which plastic events are more likely to be triggered is shown. These soft regions are also found to depend on the orientation of the loading applied on the material.

By looking at the inuence of plasticity on the local yield stress, the existence of a unique post-yield shear threshold distribution, independent on the initial state of the material, is shown. Interestingly, the yielded state is found to share a similar local yield stress mean value with glasses whose parent temperature is of the order of the mode-coupling temperature. This mode-coupling temperature was found to constitute a separation between the shear softening and shear hardening regime in the model glass.

After unloading down to zero stress model glasses in the steady ow state, an asymmetry of the mechanical response is observed. A further unloading shows a more ductile response compared to reloading. This phenomenon is called the Bauschinger eect. By the mean of an elementary model, the origin of the Bauschinger eect in the amorphous solids is found to arise from the inversion of the low yield barriers population anisotropy during the unloading.

By considering systems of many dierent sizes and degrees of relaxation, plastic events are found to take the shape of avalanches whose size seems to increase linearly with the length of the system independently of degree of relaxation. On the other hand, the persistence of plasticity is shown to mostly depend on the degree of relaxation of the system.

Finally, in well relaxed glasses, a correlation between the location of the shear band and the initial soft regions is shown. As further loading is applied on the material, a diusive broadening of the shear band is observed.

Cette interprétation est soutenue par un modèle reproduisant quantitativement l'anisotropie plastique induite dans les premiers stades de la déformation.

En étudiant des systèmes de diérentes tailles et à diérents niveaux de relaxation, il apparaît que les événements plastiques s'organisent sous la forme d'avalanches dont la taille semble varier linéairement avec la longueur du système, et ce indépendamment de la préparation du système. De plus, nous constatons que la persistance de la plasticité dépend principalement de l'état de relaxation initial du système.

Enn, en considérant des verres stables, nous montrons l'existence d'une corrélation entre la position des bandes de cisaillement et les régions initialement plus molles. En continuant à déformer ces systèmes, nous mettons en évidence que les bandes de cisaillement s'élargissent en suivant un comportement diusif.