Renée El

Melhem Bât

Pascal Blaise

M Stéphane

ÉLECTROTECHNIQUE E E A Électronique

M Philippe Delachartre
email: philippe.delachartre@insa-lyon.fr

Bénédicte Lanza

Mme Sandrine
email: sandrine.charles@univ-lyon1.fr

Mme Sylvie

M Hamamache Kheddouci

M Stéphane Benayoun
email: stephane.benayoun@ec-lyon.fr

Stéphanie Cauvin

M Jocelyn Bonjour

M Bruno

Dr Guojian Lyu

Dr Hang Guo

Dr Pablo Griffiths

Dr Robins

Dr Xiaoting Zhang

Dr Rongrong

Meng Liang

Laurent Chazeau, for providing invaluable guidance throughout this research. Discussions with him have always been a learning experience

1.3 Ashby plot of the strength vs. the elastic limit of different classes of materials: amorphous metallic alloys combine the higher strength of crystalline metal alloys with the elasticity of polymers [Telford, 2004]. . . . . . . . . .

1.4 Deformation of Zr-based MG in experiment [START_REF] He | Effect of ta on glass formation, thermal stability and mechanical properties of a zr52. 25cu28. 5ni4. 75al9. 5ta5 bulk metallic glass[END_REF]). inger and Weber (SW) potential [START_REF] Stillinger | Computer simulation of local order in condensed phases of silicon[END_REF] from MD simulations and with Eshleby's inclusion model. The black curve is the result of the MD simulation. Red, dashed blue and orange curves are obtained with Eshelby's inclusion model [START_REF] Albaret | Mapping between atomistic simulations and eshelby inclusions in the shear deformation of an amorphous silicon model[END_REF]. . . . . . . . . . . . . 1.20 Fractions of the five dominant polyhedra types as a function of sample strain in the region that evolves into the localized band [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF].

1.21 The stress-strain curve (the red one), potential energy (the blue one), and unload-reload stress-strain curve (black) with strain [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF]. . .

2.1 2-dimensional schematic diagram of periodic boundary conditions [Stepanov, 2013]. MG with quenching rate 10 10 K/s before and after the formation of SB. . . 4.9 Fitting stress-strain curves and curves after correction with factor α of Cu 64 Zr

MGs with different quenching rates (10 10 K/s (a),10 11 K/s (b), 10 12 K/s (c), 10 13 K/s (d), and 10 14 K/s (e), respectively; left is the original curves and right is after correction by a factor α=1.01 for 10 10 K/s, α=1.03 for 10 11 K/s, α=1.03 for 10 12 K/s, α=1.09 for 10 13 K/s and α=1.1 for 10 14 K/s). . . . . . .

Abstract

Metallic glasses (MGs) are very promising new materials because of their excellent mechanical properties. They have many potential applications in micromechanics, microelectronics, jewelry, biomedical, aerospace and many other fields. However, their low plasticity at room temperature limits their wide diffusion in the industry for the moment.

Therefore, the improvement of their ductility has become a major issue in materials science. The objective of this thesis is to study the plasticity of metallic glasses prepared under different conditions, by molecular dynamics (MD) simulations and to characterize the plastic events occurring at the microscopic scale during their deformation in order to understand how the plasticity evolves, and finally to evaluate under which conditions the ductility can be improved. The system studied in this thesis is a binary Cu-Zr model. We studied the effect of quenching rate, composition, strain rate and temperature.

Our simulations show that during quasi-static shear deformation, the plasticity of Cu-Zr glasses is more homogeneous, i.e., the localization of deformation and the formation of shear bands are more limited, when these materials are prepared with higher quenching rates, when they contain less copper, and/or when they are subjected to higher deformation rates and/or temperatures. In order to understand the microscopic mechanisms involved in this plasticity, we have developed a method to identify and characterize individual plastic events. In this way, we can evaluate their number, plastic intensity, average size and orientation. Moreover, we are able to reconstruct the simulated mechanical behavior by considering the plastic events as Eshelby inclusions in an infinite elastic matrix.

Finally, we show that thermal cycling below the glass transition temperature can be, under certain conditions, an efficient way to improve the ductility of metallic glasses, via their rejuvenation.

1

Résumé

Les verres métalliques (MGs) sont de nouveaux matériaux très prometteurs du fait de

General Introduction

Metallic glasses (MGs) can be obtained when a liquid metal is cooled at a rapid quenching rate without crystallizing. The atomic structure of MG exhibits short-range order (SRO)

or medium-range order (MRO), but does not have a long-range order, like crystalline alloys. Therefore, their physical properties are isotropic. Moreoever, due to the absence of defects, such as grain boundaries and dislocations in crystals, MGs have high tensile strength, hardness, high elastic limit and can find applications in various fields. For example, in the field of electronics, due to their superior soft magnetic properties, they can be used in transformers, inductors, and sensors. In the medical field, due to their biocompatibility, MGs can be used to make medical and surgical devices and biological implant materials. In the aerospace field, their high elastic limit of can meet the performance requirements of spacecraft. In the military industry, their high strength can be used to make armor-piercing bullets. However, MGs have a low plasticity at room temperature, which dramatically limits their engineering applications. In order to improve this plasticity, many studies have been performed. Nevertheless, a thourough understanding of the origin of plasticity of MGs at the microscopic scale is still lacking.

Thus, the motivation of this thesis is to explore the parameters that promote the plastic deformation of MGs and to study the effect of these parameters on the plasticity at the microscopic scale. The system investigated in this work is a model Cu-Zr binary MG, and the method used is Molecular Dynamics (MD) simulations. Cu-Zr binary MGs have a simple chemical composition and excellent glass-forming ability (GFA). They have been the topic of many experiments and numerical simulations. MD simulations provide a method to study the thermodynamic, structural, and mechanical properties at the atomic scale. The parameters that will be studied are quenching rate, composition, strain rate, General Introduction and temperature. The originality of this thesis consists in characterizing and quantifying the plastic events occurring during deformation, in terms of their number, size, and plastic intensity, thus providing a better understanding of the microscopic plasticity of MGs.

The method we have used to identify these plastic events is based on the definition of continuous plastic energy field from which we can extract the positions of the centers of the plastic events and determine their size and plastic intensity. This technique is quite unique with respect to the way plasticity of MGs is studied in the literature, since it is a rare example allowing to address in a proper and systematic fashion the identification and the characterization of the shear transformations (STs), which are considered as the elementary plastic rearrangements in amorphous materials.

The manuscript is organized in five chapters.

Chapter 1 introduces the research background, including the MG material studied, structural characteristics, mechanical properties, and applications of MGs. In addition, the microstructural models and the deformation characteristics of MGs are reported from the literature. The explanations of shear band formation from STs are also presented. We then describe the Eshelby's inclusion model, which will be used in Chapter 4 to reconstruct the stress-strain curves from MD simulations. MD simulations of MGs are then introduced and their results regarding the obtained structure, the correlation between microstructure and plasticity, and the effects of quenching rate, temperature, and composition on the microstructure and the mechanical properties. Previous studies about strategies developed to improve the plasticity of MGs are also discussed. Chapter 4 then focuses on the origin of the macroscopic plasticity of MGs through the analysis of plastic events at the microscopic scale. In this chapter, the effect of parameters studied in Chapter 3, i.e., quenching rate and composition, on the plastic events, with respect to their average size, number, plastic intensity, and distribution of orientation is analysed. The stress-strain curves from MD simulations are also reconstructed using Eshelby's inclusion model.

The last Chapter 5 is devoted to the impact of thermal cycling of the materials, with the aim to tune the plastic behavior of the MG. The variation of the potential energy and nonaffine square displacement induced by these thermal cycling are investigated for MGs prepared with different quenching rates and compositions. The effect of thermal cycling on the mechanical properties as well as on the plastic events occurring during quasi-static shear deformation are also investigated.

The manuscript ends with a conclusion on the main results and on the research perspectives opened by this work.

Chapter 1

Introduction Contents

GENERAL FEATURES OF METALLIC GLASSES

General features of metallic glasses

Metallic glass (MG) is an amorphous alloy produced by cooling it from its liquid state with a rapid cooling rate [Greer, 1995]. As shown in Figure 1.1, when temperature decreases with an infinitely low cooling rate, the liquid crystallizes at the melting temperature (T m ). In contrast, with an extremely high cooling rate, larger than a critical value, it overcomes T m and keeps its liquid state, called supercooled liquid. When temperature further decreases, viscosity increases, and the liquid freezes and forms the glass at a temperature called glass transition temperature (T g ) [START_REF] Ediger | Supercooled liquids and glasses[END_REF]. T g is defined as the temperature where the viscosity takes the value of 10 12 Pa•s [Debenedetti andStillinger, 2001, Spaepen andTurnbull, 1984]. Unlike the liquid to crystal transition, a first-order phase transition, the glass transition is not a phase transition but a kinetic phenomenon because T g depends on the cooling rate. As shown in Figure 1.1, a glass obtained with a higher cooling rate has a higher T g . The cooling rate is a significant parameter for the formation of glasses. By cooling at high cooling rates, there is no chance for the atoms of MGs to form periodical lattices like crystalline alloy, as shown in Figure 1.2. Compared to crystalline alloy, the arrangement of atoms in MG is disordered at a long distance, Figure 1.1: Volume change as a function of temperature from liquid to solid state (the dashed red line represents the equilibrium liquid) [Ritter, 2012] whereas ordered at a short and intermediate distance, giving rise to respectively shortrange order (SRO) and medium-range order (MRO). Variation of cooling rates leads to 10 CHAPTER 1. INTRODUCTION different initial "frozen" microstructures of MGs and thus plays a vital role in the resulting mechanical properties of them [START_REF] Fu | The effect of cooling rate on microstructure and mechanical properties of zr-based bulk metallic glasses[END_REF], Li et al., 2012, Yue et al., 2018], as we will see in this thesis.

Moreover, the large size differences between constituent atoms are beneficial to forming denser atomic packing and hinder atomic diffusion, therefore forming more stable glasses after cooling [Inoue, 2000, Telford, 2004, Wang et al., 2004].

Mechanical properties of metallic glasses

Compared to crystalline counterparts, MGs present excellent mechanical properties, such as higher strength and hardness, higher elastic limit, superior corrosion resistance, etc [Inoue, 2000, Johnson, 1999, Trexler and Thadhani, 2010]. As shown in Figure 1.3, with respect to traditional materials, like crystalline steels, titanium alloys, wood etc., the strength of Mg-based and Ti-based MGs can reach twice and four times that of crystalline alloys [Inoue, 2000]. The elastic limit can be over 2 %, while for crystals, they are lower than 1 %. The fracture strength (tensile yield strength) of MGs is higher than 1 GPa, and elastic energy storage is eight times that of crystalline materials [Telford, 2004]. The hardness of Fe-B based MGs can reach 1250 HV, which is much higher than magnesium and titanium crystalline alloy [START_REF] Inoue | Formation and applications of bulk glassy alloys in late transition metal base system[END_REF].

Even though MGs own so many outstanding mechanical properties, they have some drawbacks, particularly low ductility at room temperature. When they reach their yield point under loading, shear bands form and the MG will crack in a narrow zone (10-20 1.3. APPLICATION OF METALLIC GLASSES nm width). Once the crack appears, it will expand rapidly along the stress direction, resulting in a brittle fracture. As shown in Figure 1.4, there is nearly no plastic deformation for Zr-based MG [START_REF] Inoue | Formation and applications of bulk glassy alloys in late transition metal base system[END_REF].

Figure 1.3: Ashby plot of the strength vs. the elastic limit of different classes of materials: amorphous metallic alloys combine the higher strength of crystalline metal alloys with the elasticity of polymers [Telford, 2004].

Application of metallic glasses

The high elastic limit makes MGs a material with excellent elastic storage properties, which can be applied to sports equipment atomic arrangements present some degree of short-range order (SRO) topologically and chemically [START_REF] Lamparter | Atomic structure of amorphous metallic ni81b19[END_REF]. RDF is shown in Figure 1.6, and it is proportional to the probability of finding an atom at a certain distance of a given central atom.

Figure 1.6: The illustration of radial distribution function [Weihua, 2013].

Many models of atomic arrangement in MGs have been proposed to facilitate the microstructural study of MGs. Bernal's dense random packing of hard spheres is one of them, which considers atoms as hard spheres, forming network structures randomly without cavities [Bernal, 1960a[START_REF] Bernal | The structure of liquids[END_REF]]. Cargill simulated Ni-P MG with this model and obtained the RDF in agreement with experimental results [Cargill III, 1975].

After that, many other structural models have been developed. Gaskell proposed the local coordination model, which assumes that local structure in MGs is similar to that in crystals with the same composition, i.e., the local structures both consist of polyhe-CHAPTER 1. INTRODUCTION dra with various shapes [Gaskell, 1978]). For metal-metalloid glasses, constituted by transition metals and non-metallic elements [Gaskell, 1978], the RDF calculated by this model corresponds to the result by neutron diffraction. According to Gaskell's local coordination model, metal-metalloid MGs have capped trigonal prism structural units with metalloid atoms as the cores and metal atoms distributed on six corners [Gaskell, 1979a[START_REF] Gaskell | On the structure of simple inorganic amorphous solids[END_REF]. Some structures of MGs can be explained by Gaskell's model, like Pd-Si [START_REF] Fukunaga | An experimental derivation of partial structure functions of amorphous pd 80 si 20 alloy using combination of x-ray, electron and neutron diffraction experiments[END_REF], Ni 82 B 18 [START_REF] Chadha | A diffraction study of ni82b18 and ni64b36 metallic glasses[END_REF], Fe-P, etc. However, for metal-metal glasses, this model cannot be applied [Chen, 2008]. Free volume model [Spaepen, 1977] is an alternative model, first issued by Cohen and Turnbull and then improved by Cohen and Spaepen, widely applied to the study of the structure and properties of MGs [START_REF] Cohen | Molecular transport in liquids and glasses[END_REF]. Free volume is thought to provide the physical space for the free motion of a single atom or atomic cluster without external energy. The free volume reduces when cooling the liquid until the system gets frozen with no free volume, thus forming the glass structure. However, the free volume model cannot describe the atomic structure associated with SRO. Therefore, to better understand the SRO of MGs, another model called polytetrahedral packing model was introduced. In this model, we identify basic structural units in the MG, called Vornoi polyhedral [START_REF] Nelson | Polytetrahedral order in condensed matter[END_REF], defined as the smallest closed convex polyhedra made up of perpendicular bisector line connecting a central atom with neighbor atoms [Finney, 1970]. Voronoi polyhedra are identified by Voronoi indexes <n 3 , n 4 , n 5 , n 6 ,. . . >, where n i indicates the number of surface with i edges. For example, n 3 means the number of triangles. As shown in Figure 1.7, the <0,0,12,0> icosahedron, if the polyhedron is unfolded, we can get zero triangles, zero quadrangles, twelve pentangles, zero hexagons, and zero polygons with more than six edges. <0,0,12,0> icosahedron is also called fullicosahedron(FI). There are various types of polyhedra in MGs, as shown in Figure 1.8.

With the study of supercooled liquid, Frank supposed that central atoms and their nearest neighbors form icosahedra structure [Frank, 1952]. These icosahedra structures present characteristic five-fold symmetry and are abundant in MGs, due to their dense packing and the fact that they are energetically more favorable [START_REF] Lee | Origin of the plasticity in bulk amorphous alloys[END_REF][START_REF] Cheng | Relationship between structure, dynamics, and mechanical properties in metallic glass-forming alloys[END_REF], Cheng et al., 2008a, Cheng et al., 2009a, Ding et al., 2014a], and are evidenced by both simulations [Cheng et al., 2009b, Cao et al., 2009[START_REF] Cheng | Relationship between structure, dynamics, and mechanical properties in metallic glass-forming alloys[END_REF]] and experi- ments [START_REF] Luo | Icosahedral short-range order in amorphous alloys[END_REF], Li et al., 2009].

Moreover, SRO clusters can be connect to each other to form larger structures. Miracle proposed connection rules of SRO clusters to form MRO [Miracle, 2006]. These To study the atomic-scale deformation of MGs, people proposed many models. The most mature and widely used one is the free volume model, already introduced in Section 1.4

to describe the atomic-level structure. It considers vacancies randomly distributed as defects [Spaepen, 1977]. The deformation of MGs is based on the "transition" of a single atom. As shown in Figure 1.10, if an atom has a large enough vacancy in its vicinity (approximately equal to the volume of a hard atomic sphere), and the activation energy of the atom is larger than transition energy, the atom will be squeezed into an adjacent vacancy, and free volume redistributes. The free volume model explains the shear localization of MGs at the atomic scale, but it cannot explain the atomic processes in the shear bands (SBs). To describe the deformation mechanism of MGs, the motion of atomic clusters, and not just a single atom, should be considered, as explained in the following subsection.

Atomic local shear transformation

Argon proposed to extend the theory of Spaepen by considering the transition of atomic clusters when deformation occurs [START_REF] Velasco | Plastic deformation induced anisotropy in metallic glasses: A molecular dynamics study[END_REF]. He introduced the concept of shear transformation (ST) as the deformation unit, which is a thermally activated local rearrangement of several up to a few hundred atoms under shear stress [Argon, 1979] around free volume abundant sites.

In

Formation of shear bands

The plastic deformation of MGs is very heterogeneous at room temperature, and it is likely to form localized shear bands (SBs) in extremely narrow region [START_REF] Bian | Microstructure and ductile-brittle transition of as-cast zr-based bulk glass alloys under compressive testing[END_REF], Chen et al., 2011, Jiang et al., 2008, Jiang and Dai, 2011, Pampillo, 1975[START_REF] He | Effect of ta on glass formation, thermal stability and mechanical properties of a zr52. 25cu28. 5ni4. 75al9. 5ta5 bulk metallic glass[END_REF]].

In the free volume theory, the transition of atoms or atomic clusters easily occurs where 1.5. ATOMIC-SCALE DEFORMATION OF METALLIC GLASSES free volume is large and bond energy is small. The plastic deformation of MGs can be divided into two steps: before the generation of SB, in the region where free volume is abundant, heterogeneous localized plastic deformation occurs; after the generation of SB, the expansion of SBs results in extended plastic deformation, accompanied by strain softening. If there are multiple SBs in various directions, a single SB will be hindered, yielding homogeneous deformation and better plasticity. Therefore, the research about the formation of SBs is essential for understanding the molecular mechanisms of plastic deformation and for improving the plasticity of MGs [START_REF] Chen | On the formation of shear bands in a metallic glass under tailored complex stress fields[END_REF], Liu et al., 2009, Sergueeva et al., 2005].

The formation of SBs is a multiscale behavior at the microscopic, mesoscopic, and macroscopic scales. Atomic rearrangements lead to plastic events, which are STZs, containing several tens of atoms. Percolation of STZs can facilitate the formation of SBs on the order of a tenth of a micrometer [START_REF] Schuh | Mechanical behavior of amorphous alloys[END_REF]. At a macroscopic scale, a single SB expands, or several SBs interact, which results in different plastic behaviors [START_REF] Su | Plane strain indentation of a zr-based metallic glass: experiments and numerical simulation[END_REF]].

Greer et al. proposed three possible explanations for the formation of SBs [START_REF] Greer | Shear bands in metallic glasses[END_REF]. In the first one, the formation of the SB is considered to be the percolation of homogeneously activated STZs. After the activation of STZs, there will be a large accumulation of them. When it exceeds the percolation limit, a band with concentrated shear strains will form along the path with maximum shear stress on the plane. The second one considers that an embryonic SB generates due to activating local STZs, and then it grows to a mature SB. With the help of computer simulation and a theoretical model, Shimizu et al. [START_REF] Shimizu | Yield point of metallic glass[END_REF] found that when the size of embryonic SB reaches around 100 nm, mature SB will form. In the second explanation, the SB is nucleated from local STZ resulting from local stress concentration. In this case, there is a temperature gradient along with SB. The SB can be divided in four zones from the front to the tail, as shown in Figure 1.12: aged glass, rejuvenated glass, glue, and liquid (ARGL), in agreement with the experimental results of Yang et al. [START_REF] Yang | Dynamic evolution of nanoscale shear bands in a bulkmetallic glass[END_REF], Yang et al., 2006a]. Therefore, the propagation of SB is heterogenous in this case. The third explanation involves two consecutive stages. The first stage is stress concentration (Figure 1.13 (a)), similar to the second explanation. The activation of STZs increases the structural disorder and results in structural rejuvenation. After that, the structural rejuvenation propagates heteroge- . This third explanation was verified by computer simulation of Cao et al. [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF], in which rejuvenation propagates with a disordered structure. In conclusion, the origin of the SB is attributed to the activation of STZs.

Generally, it is thought that the reason for forming localized SBs is the reduction of viscosity in some inhomogeneous regions in MGs. There are two factors for the reduction of viscosity: free volume softening [START_REF] Huang | Inhomogeneous deformation in metallic glasses[END_REF], Spaepen, 1977, Steif et al., 1982] and adiabatic softening [START_REF] Leamy | Plastic flow and fracture of metallic glass[END_REF], Liu et al., 1998]. In free volume softening the aggregation of localized free volume causes the localized SB of MGs without thermal effect, which has been proved by Li andGu [Gu et al., 2002, Li et al., 2002a]. As et al., 1992]. Based on free volume theory, Dai et al. revealed that shear band instability in MGs is the result of the synergistic effect of free volume and adiabatic softening [START_REF] Dai | Adiabatic shear banding instability in bulk metallic glasses[END_REF]. Among them, free volume plays the main controlling role, and heat plays the auxiliary role. Compared with the increase of adiabatic temperature, the free volume appears earlier and much faster [START_REF] Jiang | On the origin of shear banding instability in metallic glasses[END_REF], as shown in Figure 1.15.

Eshelby's inclusions

Several assumptions to analyze the plastic deformation of MGs have been introduced

previously in this chapter. One can make an analogy between local shear transformations (STs) and inclusions in an infinite elastic matrix based on the theory of Eshelby's [Eshelby, 1957,Picard et al., 2004,Rodney et al., 2011]. An Eshelby's inclusion can be described with the following three imaginary steps [Eshelby, 1957].

Figure 1.16: Scheme of inclusion's removal from the matrix and of its unconstrained transformation.

Step 1. In an infinite isotropic homogeneous elastic medium (D), we remove an inclusion

(Ω) with the surface area S 0 from it and make it unconstrained with stress-free transformation strain tensor ϵ T ij , as shown in Figure 1.16 (Ω→Ω ′ ). e T ij is called eigenstrain, i.e., the

ESHELBY'S INCLUSIONS

strain without undergoing stress. The elastic constant of inclusion Ω, C ijkl is identical to that of the matrix D. Thus, eigenstress σ T ij can be expressed as, σ T ij = C ijkl e T kl . The strain e ij , stress σ ij , and displacement u i of the inclusion after removing from the matrix are calculated as:

e ij = e T ij (1.1) σ ij = 0 (1.2) u i = e T ij x j (1.3)
where x j is the component of unit displacement vector in j direction.

Figure 1.17: Applying force on the inclusion and putting it back in the matrix.

Step 2. We apply a surface traction (P i ) on the inclusion to restore its original form and put it back in the hole in the matrix, as shown in Figure 1.17. The elastic strain should offset the eigenstrain, i.e., e el ij = -e T ij . Surface traction P i is related to the stress, P i =-σ T ij n j . n j is the component form of normal vector. The strain, stress, and displacement of the inclusion are:

e ij = e el ij + e T ij = 0 (1.4) CHAPTER 1. INTRODUCTION σ ij = C ijkl e el kl = -C ijkl e T kl = -σ T ij
(1.5)

u i = 0 (1.6)
Figure 1.18: Releasing the surface traction of the inclusion.

Step 3. We rejoin the material across the cut and remove the inclusion surface traction, which is identical to applying a body force, F i =-P i on the surface of the inclusion, as

shown in Figure 1.18. We define u i (x) as the displacement related to F i .

Based on the expression of Green's function on the elastic body [Challis andSheard, 2003, Kröner, 1990], G(x,x') is defined as the displacement at x due to the point force at x'.

u i (x) is calculated as:

u i = u i (x) = S 0 0 F i G ij (x, x ′ )dS(x ′ ) (1.7)
The strain and stress are calculated as :

e ij = e ij (x) = 1 2 (u ij (x) + u ji (x)) (1.8) σ ij = C ijkl (e ij (x) -e T ij ) (1.9) 1.6. ESHELBY'S INCLUSIONS
The strain inside inclusion e in ij can be obtained from the Eshelby's tensor S ijkl through the following relationship:

e in ij = S ijkl e T ij
(1.10)

The spherical Eshelby's tensor used in our work can be found in Toshio Mura's book [Mura, 2013]. It only depends on the Poisson's ratio, on the shear modulus and on the sphere radius. There are many studies on the plasticity of amorphous systems using Eshelby's inclusion model [START_REF] Bulatov | A stochastic model for continuum elasto-plastic behavior. i. numerical approach and strain localization[END_REF], Dasgupta et al., 2012, Homer et al., 2010, Jagla, 2007, Martens et al., 2011, Nicolas et al., 2015, Picard et al., 2004, Rodney et al., 2011, Talamali et al., 2011, Tanguy et al., 2006] due to its simplicity. The existence of an analytic solution to the inclusion problem for some particular cases, e.g., spherical or cylindrical inclusions, allows for instance to describe the formation of a shear band (SB)

as a linear arrangement of Eshelby's inclusions [START_REF] Dasgupta | Microscopic mechanism of shear bands in amorphous solids[END_REF]]. Eshelby's inclusion model is also used within mesoscopic approaches of plasticity [START_REF] Bulatov | A stochastic model for continuum elasto-plastic behavior. i. numerical approach and strain localization[END_REF], Picard et al., 2004, Rodney et al., 2011]. However, these approaches rely on empirical assumptions on the size and the strain of Eshelby's inclusions [START_REF] Dasgupta | Microscopic mechanism of shear bands in amorphous solids[END_REF], Fusco et al., 2014, Homer et al., 2010, Marmottant and Graner, 2007, Martens et al., 2011, Nicolas et al., 2015, Picard et al., 2004, Talamali et al., 2011, Tanguy et al., 2006],

and the mechanisms of nucleation of the inclusions are not well elucidated [START_REF] Rodney | Distribution of thermally activated plastic events in a flowing glass[END_REF]. Besides, there is still a debate whether stress-strain curves from atomistic simulation can be reproduced with Eshelby's inclusion model. In the recent work of Albaret et al, Eshelby's inclusion model has been successfully applied to study the atomistic plasticity of amorphous silicon for quasistatic shear deformation [START_REF] Albaret | Mapping between atomistic simulations and eshelby inclusions in the shear deformation of an amorphous silicon model[END_REF]. It was shown the atomistic stress-strain curves could be reproduced based on Eshelby's inclusion model by combining the information on the stress and displacement field generated by inclusions with atomic-scale displacement from atomistic simulations [START_REF] Albaret | Mapping between atomistic simulations and eshelby inclusions in the shear deformation of an amorphous silicon model[END_REF] (Figure 1.19). However, this approach has never been applied to the plasticity of Cu-Zr MGs. An atomic-scale description of plasticity of MG from which we can detect the individual plastic events and determine their numbers, distribution, and sizes, is still lacking.

Molecular Dynamics studies on metallic glasses

Experimentally it is difficult to probe the atomic response of MG during deformation. On the other hand, numerical simulation is very effective in helping us to better understand local dynamical movements at the atomic scale. Molecular dynamics (MD) simulation is a valuable technique to study the dynamics of MGs from the atomistic point of view [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF].

Studies about structure

SRO and MRO in MGs have been observed with MD simulation [Sha et al., 2011,Sun andShen, 2009]. Foroughi et al. studied the evolution of the local structure of Cu 50 Zr 50 MG during glass formation with the help of MD simulation [START_REF] Foroughi | Molecular dynamics study of structural formation in cu50-zr50 bulk metallic glass[END_REF]. In their research, full and defective full icosahedra (FI) with lower potential energy and mobility are observed from the formation of glass with temperature decrease and the conversion 1.7. MOLECULAR DYNAMICS STUDIES ON METALLIC GLASSES from other types of icosahedra, mainly <0,2,8,2> [START_REF] Foroughi | Molecular dynamics study of structural formation in cu50-zr50 bulk metallic glass[END_REF],Zemp et al., 2014].

With temperature decrease, these icosahedra link to each other, and the stability of these connections increases. Moreover, during the deformation of MGs, icosahedra are destructed and reconstructed. Thus, changes in atomic structure can also be observed with MD [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF]. It paves the way for studying the correlation between microstructure and plastic behavior of MGs.

Correlation between microstructure and plastic behavior

Many studies have proved that the microstructure of MGs, including morphology, fraction, and distribution, can significantly influence their mechanical properties [START_REF] Johnson | A universal criterion for plastic yielding of metallic glasses with a (t/t g) 2/3 temperature dependence[END_REF], Khonik et al., 2008, Lewandowski* et al., 2005, Park et al., 2007, Yang et al., 2006b]. Structural heterogeneities inherited from supercooled liquid [START_REF] Cheng | Atomic-level structure and structureproperty relationship in metallic glasses[END_REF] cause heterogeneities of mechanical properties. A great number of soft spots with low energy barriers have been identified in MGs [START_REF] Manning | Vibrational modes identify soft spots in a sheared disordered packing[END_REF] due to the rapid quenching process. As we mentioned before, the carriers of deformation are regarded as the shear transformation of atomic clusters in STZs [Argon, 1979, Falk and Langer, 1998, Velasco et al., 2017]. Soft spots can turn into potential sites for STZs, i.e., STZs commonly occur where soft spots are fertile [Ding et al., 2014b]. Even though the activation and evolution of STZs are difficult to explore, many relevant studies focus on them by establishing the correlation between microstructure and plastic behavior.

Upon deformation, the destruction of FI was identified as the microstructural indication of shear localization [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF]. According to the research of Cheng et al. [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF], the formation of a shear band (SB) is accompanied by the breakdown of FI and the localized shear strain squeezed into a narrow band.

Figure 1.20 plots the fraction variation of dominant types of Cu-centered polyhedra with strain. The fraction of FI shows the same trend as stress vs. strain. In detail, the fraction of FI changes slightly within the elastic region and plastic flow. However, there is a drop in its fraction when the drop of stress occurs. In general, FI is very stable and shearresistant [START_REF] Cheng | Relationship between structure, dynamics, and mechanical properties in metallic glass-forming alloys[END_REF], Sheng et al., 2006] and can be significantly destroyed with SB formation. The destruction of FI into fragments leads to structural softening, which CHAPTER 1. INTRODUCTION provides potential sites for SB. Therefore, the fraction of FI in the band is much lower than neighboring region.

Figure 1.20: Fractions of the five dominant polyhedra types as a function of sample strain in the region that evolves into the localized band [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF].

Figure 1.21: The stress-strain curve (the red one), potential energy (the blue one), and unload-reload stress-strain curve (black) with strain [START_REF] Cao | Structural processes that initiate shear localization in metallic glass[END_REF].

Meanwhile, it is observed that the temperature within the SB is higher than the neighboring region, i.e., in SB, there exists thermal softening. As shown in Figure 1.21, an abrupt drop of stress and potential energy is observed upon yielding. In detail, at the beginning of stress decrease after the yield point, potential energy still increases, which is associated with structural rejuvenation within the SB. The accumulated potential energy 1.7. MOLECULAR DYNAMICS STUDIES ON METALLIC GLASSES within the elastic period is released at the drop of stress. In conclusion, the initiation of unstable shear localization can be attributed to the combined effect of structural rejuvenation, structural softening, and following thermal softening.

Effect on microstructure and mechanical properties

As said previously, the plastic flow of MGs is localized into shear bands (SBs) with a few tens of nm thickness [START_REF] Greer | Shear bands in metallic glasses[END_REF] when applying loading on them. Therefore, multiple SBs can lead to more homogeneous plastic deformation; nevertheless, catastrophic fracture under a single narrow SB often occurs. The number and distribution of SBs, which can determine mechanical properties, are essential parameters when studying the plasticity of MGs [START_REF] Conner | Shear bands and cracking of metallic glass plates in bending[END_REF],Lewandowski* et al., 2005,Liu et al., 2005,Liu et al., 2007, Xing et al., 2001]. Structural characteristics, like icosahedra, free volume, chemical heterogeneity, etc., are related to the nucleation of SBs [Jiang et al., 2009, Lee et al., 2011, Liu et al., 2007, Park and Kim, 2010]. Many experimental researches have proved that the free volume content within SBs is higher than in the matrix, which makes the structural softening and leads to further strain localization or softening [START_REF] Flores | Characterization of free volume in a bulk metallic glass using positron annihilation spectroscopy[END_REF][START_REF] Li | Characterization of nanometer-scale defects in metallic glasses by quantitative high-resolution transmission electron microscopy[END_REF], Pan et al., 2011, Bei et al., 2006, Yoo et al., 2009].

Many factors can affect the plasticity of MGs, such as cooling rate, temperature, composition, etc. The cooling rate is a critical factor in the atomic arrangement, which plays a vital role in the fraction of FI and the distribution of free volume. Temperature mainly affects thermally atomic mobility. In addition, the fraction of FI is influenced by the atomic arrangement dependent on the difference in atomic sizes.

Cooing rate

Yue et al. used MD simulation to study the effect of cooling rate on the microstructure and mechanical properties of Cu 50 Zr 50 metallic glass (MG) [START_REF] Yue | Effect of cooling rate on structures and mechanical behavior of cu50zr50 metallic glass: a molecular-dynamics study[END_REF]. The result that MGs possess higher T g with a higher cooling rate is consistent with the experimental result. A lower cooling rate leads to more fraction of full icosahedra clusters [START_REF] Zemp | Icosahedral superclusters in cu 64 zr 36 metallic glass[END_REF] and lower free volume [START_REF] Wang | Effects of quenching rate on amorphous structures of cu46zr54 metallic glass[END_REF] rates. They found that more solid-like structures would be formed under lower quenching rates, which results in higher yield stress, flow stress, yield strain (strain where stress is maximum), and larger stress drop [START_REF] Fusco | Role of local order in the small-scale plasticity of model amorphous materials[END_REF]. Besides, they obtained detailed information on plastic events in the plastic deformation region under different quenching rates in a model system for amorphous silicon. According to the results, the total number of plastic events decreases when the quenching rate decreases. As for plastic energy, it increases with strain. They also found that plastic events are less and smaller for the amorphous system with a lower quenching rate. In contrast, the plastic energy is larger at the yield point, which means that at yield strain, a larger amount of plastic energy is released rapidly. Therefore, it is more likely to generate a shear band at the yield point for the amorphous system with lower quenching rates. However, similar studies about the detailed analysis of the local plastic events of MGs are lacking.

Temperature

At high temperature (2000 K), metallic glasses (MGs) are liquid-like structures as it can be seen from the radial distribution function (RDF) analysis [START_REF] Song | Atomic simulations of plastic deformation behavior of cu50zr50 metallic glass[END_REF], Wen et al., 2021]. When the temperature decreases to the glass transition temperature (T g ), the liquid is frozen to solid glass. Below the T g , the atomic packing density increases with the temperature decreasing while atomic mobility and diffusion decrease [START_REF] Song | Atomic simulations of plastic deformation behavior of cu50zr50 metallic glass[END_REF], Wen et al., 2021] because the atomic diffusion is thermally activated. Less shear transformation zones (STZs) are activated, attributed to the less atomic diffusion [START_REF] Sepulveda-Macias | Strain rate and temperature effect on zr50cu50 metallic glass under pure shear[END_REF]. More full icosahedra (FI) and less free volume are observed in MGs [START_REF] Yue | Deformation behavior of annealed cu64zr36 metallic glass via molecular dynamics simulations[END_REF]. Regarding mechanical properties, as a consequences, during deformation, the plastic flow is less homogeneous, and the plasticity is poorer [START_REF] Yue | Deformation behavior of annealed cu64zr36 metallic glass via molecular dynamics simulations[END_REF]. However, the yield strength, shear and bulk moduli increase when temperature decreases [START_REF] Sepulveda-Macias | Strain rate and temperature effect on zr50cu50 metallic glass under pure shear[END_REF], Yue et al., 2020].

IMPROVEMENT OF PLASTICITY

Composition

The effect of composition on the plasticity of MGs can be ascribed to its influence on the formation of icosahedra clusters with different central elements [Zhong et al., 2016].

In a binary metallic glass with Zr and Cu elements, the initial fraction of FI increases dramatically with the rise of copper composition. In a ternary metallic glass with Zr, Cu, and Al elements, when the fraction of Cu is fixed, the fraction of FI increases slightly with the fraction of Al [Cheng et al., 2008a]. Correspondingly, the mechanical behavior is strongly affected by the composition. For example, by increasing the Cu content, the overshot stress (τ over ) and flow stress (τ f low ) are larger. Moreover, , the difference between overshot stress and flow stress, ∆τ, measuring the propensity to strain localization,also increases with more Cu content, which can be correlated with a higher of FI. Likewise, , the fraction of FI and ∆τ both increase slightly with the Al fraction. Therefore, to sum up, Cu and Al content negatively affect the plasticity of MG due to the formation of more FI, which are resistant to shear transformation and reduce ductility [START_REF] Cheng | Relationship between structure, dynamics, and mechanical properties in metallic glass-forming alloys[END_REF],Park et al., 2007].

Improvement of plasticity

Many studies evaluate different methods to improve the plasticity of metallic glasses (MGs). On the experimental side, plasticity can be improved by shot peening (SP), laser-shot-peening (LSP), laser surface melting (LSM), etc. Both types of shot peening have a similar effect on MGs, i.e., introducing a soft layer and residual stress. This enhances shear band nucleation and structural dilation, further resulting in lower yield strength [START_REF] Wang | Effect of residual stresses on the onset of yielding in a zr-based metallic glass[END_REF] and plastic stability. Compared to SP, LSP can locate the target precisely via laser beam and cause a deeper residual stress layer [START_REF] King | Effects of fatigue and fretting on residual stresses introduced by laser shock peening[END_REF]. LSM applies laser on the surface of MGs to remelt and resolidify, introducing residual stress and improvement of strength [START_REF] Chen | Optimization of mechanical properties of bulk metallic glasses by residual stress adjustment using laser surface melting[END_REF]. In addition, after pre-deformation, like cold rolling, the entire material experiences structural dilation and softening [START_REF] Liu | Shear band evolution and hardness change in cold-rolled bulk metallic glasses[END_REF], which is beneficial to stable and homogeneous plastic deformation [Yokoyama, 2003, Yokoyama et al., 2002]. Apart from the surface modification, the addition of other CHAPTER 1. INTRODUCTION elements inducing a larger amount of excess free volume [START_REF] Chen | New class of plastic bulk metallic glass[END_REF] can also help to improve the plasticity of MGs [START_REF] Li | Minor cr alloyed fe-co-ni-p-b high entropy bulk metallic glass with excellent mechanical properties[END_REF]. Molecular Dynamics (MD) simulation is also widely applied to better understand the microstructural mechanisms allowing to improve plasticity of MGs. At the microscopic scale, plasticity can be improved by activating more STZs, generating multiple homogeneous shear bands (SBs) and avoiding the propagation of a single SB. In the following, studies about the enhancement of ductility of MGs via MD simulation will be introduced.

Composite metallic glasses

The introduction of a second phase is a practical approach to improving the ductility of MGs. The presence of a crystalline phase causes the existence of interfaces, which leads to the increase of free volume [START_REF] Shete | Effect of strain hardening and volume fraction of crystalline phase on strength and ductility of bulk metallic glass composites[END_REF]. In addition, STZ localization and embryo SBs are likely to generate near structural heterogeneities, such as interfaces. Simultaneously, interfaces hinder the catastrophic extension of a single SB, further promoting the formation of homogeneous distribution of local shear strain and improving the global plasticity of MGs [START_REF] Lee | Enhanced plasticity in a bulk amorphous matrix composite: Macroscopic and microscopic viewpoint studies[END_REF], Şopu et al., 2011, Zhou et al., 2013]. For instance, a copper crystallized phase was embedded in Cu 64 Zr 36 MG, and the generation and development of the SB were observed [START_REF] Albe | Enhancing the plasticity of metallic glasses: Shear band formation, nanocomposites and nanoglasses investigated by molecular dynamics simulations[END_REF]. Precipitates can also promote the generation of multiple SBs and block the propagation of a single SB. However, at high strains, SBs can still go through precipitates because the thickness of SBs can cover the diameter of precipitates.

The impact of the introduction of an amorphous second phase was also studied via MD simulation [START_REF] Song | Enhancing the plasticity of noncrystalline cuzr multilayer: Insights from molecular dynamics simulations[END_REF]. Song et al. introduced a soft phase with structural defects into the MG matrix. They observed that when the defective atomic fraction is up to 12.5 %, it can cause a free volume gradient at the interfaces. It results in a stress concentration, activating more STZs with large atomic shear strain. Interfaces can also be formed in nanoglasses, which are obtained by powder consolidation [START_REF] Jing | Modified atomic structure in a pd fe si nanoglass: A mössbauer study[END_REF]].

1.8. IMPROVEMENT OF PLASTICITY

Pre-deformation

The introduction of a notch can enhance the ductility of MGs, as it has been proved by experiments [START_REF] Guo | Notch toughness of fe-based bulk metallic glass and composites[END_REF], Pan et al., 2017, Qiao, 2013, Zhao et al., 2010a]. There is evidence of a transition from brittle to ductile fracture in the notched MG under deformation [Zhao et al., 2010b]. Zhao et al. found that two symmetrical notches can improve the plasticity by up to 10 % through the initiation of SBs and hinder the propagation of a single SB. They also found that the initiation of SBs can be attributed to the stress concentration around the notches, and the obstruction of propagation of a single SB was due to the intersection of multiple SBs. The notched samples will finally fracture via necking [START_REF] Guo | Tensile ductility and necking of metallic glass[END_REF].

In addition, computer simulations about notches in the MG were also performed, including finite element analysis [START_REF] Dutta | Plastic deformation and failure mechanisms in nano-scale notched metallic glass specimens under tensile loading[END_REF] and molecular dynamics (MD) [START_REF] Feng | Enhancing strength and plasticity by pre-introduced indent-notches in zr36cu64 metallic glass: A molecular dynamics simulation study[END_REF]. At the microscopic scale, it was demonstrated that the nucleation of SBs in the indent-notch affected region was hindered by the formation of densely-packed clusters.

Meanwhile, the stress concentration around notch roots is observed. Therefore, there will be a competition between the nucleation of SBs outside the indent-notch affected region and the localized deformation due to stress concentration around notch roots. The plastic behavior depends on the depth of notches. When the depth of notches is relatively larger, the nucleation of SBs outside the indent-notch affected region plays a dominant role, and there will be ductile behavior. it [START_REF] Feng | Rejuvenation by weakening the medium range order in zr46cu46al8 metallic glass with pressure preloading: A molecular dynamics simulation study[END_REF]. The stress-strain curves and distribution of atomic local shear strain obtained by MD simulation demonstrate that the preloading on MG leads to the increase of SRO and decrease of MRO, resulting in the structural rejuvenation and more stable plastic deformation.

Thermal pre-treatment

Thermal pre-treatment process is another useful method that can improve the plastic behavior of MGs without destroying the integrity of the material. For instance, an annealing treatment above T g brings in more flow units, STZs, and more free volume, which is CHAPTER 1. INTRODUCTION beneficial to homogeneous plastic deformation for MGs [START_REF] Feng | A molecular dynamics analysis of internal friction effects on the plasticity of zr65cu35 metallic glass[END_REF]. Through MD simulation, Wakeda et al. investigated the microstructure and mechanical properties of binary Zr-Cu MG under pre-thermal process, i.e., after a secondary heating and quenching process [START_REF] Wakeda | Controlled rejuvenation of amorphous metals with thermal processing[END_REF]. When the re-quenching rate is lower than or equal to the initial quenching rate, aging appears, no matter how high the annealing temperature. Rejuvenation occurs only when the re-quenching rate is higher than the initial quenching rate and the annealing temperature is higher than a specific value. Thus, it is concluded that the level of rejuvenation does not depend on the absolute value of the re-quenching rate, but rather on the ratio between the re-quenching rate and the initial quenching rate. Another commonly applied thermal treatment is the cryogenic thermal cycling (CTC), i.e.a thermal cycle from room temperature to a lower temperature repeatedly at a constant pressure [START_REF] Song | Improving plasticity of the zr46cu46al8 bulk metallic glass via thermal rejuvenation[END_REF]. It has been proved by Shang et al. that CTC can lead to a rejuvenation effect on MGs by inducing structural heterogeneity and triggering local shear transformations [START_REF] Shang | Role of thermal expansion heterogeneity in the cryogenic rejuvenation of metallic glasses[END_REF]. It is found that the number of cycles and amplitude of temperature are two significant factors for CTC. When the number of cycles increases, the density of plastic events grows gradually leading to a pronounced rejuvenation effect. Similarly, internal thermal stress increases correspondingly by increasing the amplitude of the cycle high temperature. Thus, an approximate increase of the number and amplitude (towards higher temperature) of thermal cycles has a positive effect on the improvement of plasticity. The influence of several relevant parameters on the rejuvenation of MGs was also studied by Shang et al. via MD simulation [Shang et al., 2021]. They concluded that a cycle with a lower initial quenching rate, less holding time at the lowest or highest temperatures, and higher re-quenching and re-heating rate is more efficient to rejuvenate MGs and therefore improve their ductility. Nikolai found the enhancement of CTC on the plasticity of MGs was supported by experimental studies. [Priezjev, 2019]. [START_REF] Guo | Rejuvenation and plasticization of metallic glass by deep cryogenic cycling treatment[END_REF] compared compressive stress-strain curves of MG sample before and after deep cryogenic treatment from 20 °C to -180 °C, repeated over 30 cycles (DCT30), and they observed that the MG sample can form more SBs during compressive deformation after cryogenic thermal cycling treatment. In conclusion, the improvement of the plasticity of MGs by various methods has attracted much attention, both in experiments and numerical simulations. However, the 1.9. OPEN QUESTIONS deep understanding of the impact of these methods on the development of plastic events during deformation is still missing. Thus, this thesis focuses on the detailed characterization of these plastic events.

Open questions

The literature devoted to the microstructure and mechanical properties of MGs is already very abundant. However, some open questions still remain, that we will address in this thesis.

How do we detect plastic events and extract the relevant information about them?

A well-established method to identify and characterize individual plastic events in MGs has not been fully developed yet. In this work, we will use an approach for their detection based on local energy variations, through a coarse-graining procedure [START_REF] Goldenberg | Small and large scale granular statics[END_REF]. In this way, the distribution, size, and intensity of plastic events will be obtained. This is mandatory to study the activation and propagation of plastic events with deformation.

What is the effect of quenching rates on the occurrence of plastic events of metallic glasses during deformation?

As already illustrated in Section 1.7.3.1, the effect of quenching rates on the microstructure of MGs has been studied a lot. However, only very few works address the effect of the quenching rate on the plastic events, which determine the initiation and development of shear transformation with deformation. This is crucial to understand the microscopic origin of the plastic behavior of MGs, and in particular the transition from localized to homogeneous plasticity when the quenching rate is increased.

What is the effect of cryogenic thermal cycling treatment (CTC) on the plastic events of metallic glasses occurring during deformation?

According to the previous studies, cryogenic thermal cycling (CTC) treatment promotes more homogeneous plastic deformation on metallic glasses, through structural rejuvenation, inducing extra free volume, higher energy, and enthalpy. The effect of CTC on the development of plastic events is however missing. We will establish the link between the local atomic dynamics and the observed mechanical properties by analyzing them.

MOLECULAR DYNAMICS SIMULATION

Molecular dynamics simulation

In Molecular Dynamics (MD) simulation, Newton's equations of motion determine the movement of atoms and molecules under the action of the force fields created by surrounding atoms. At time t, the force on each atom can be obtained from the interaction potential between atoms. After ∆t, the position and velocity of each atom i can be computed through the equation of motion. Moreover, thermodynamic, structural, and other macroscopic properties can be calculated using statistical methods.

Newton's equation and algorithms

The position and velocity of atoms are determined by Newton's equation in MD simulation, which is:

m i a i = f i = ∑ j̸ =i f ij (2.1)
where m i is the mass of atom i , a i is the acceleration of atom i and f i is the interaction force between atoms. Newton's equation needs to be solved by the numerical integration method to study the many-body problem. The numerical algorithms include Verlet algorithm [Verlet, 1967], Velocity-Verlet algorithm [START_REF] Swope | A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters[END_REF], Leap-frog algorithm [Hockney, 1970], Beeman algorithm [Beeman, 1976], Gear algorithm [Gear, 1971] and Rahman algorithm [Rahman, 1964], among which, Verlet algorithm and Velocity-Verlet algorithm are commonly used.

The principle of the Verlet algorithm is to compute the positions r(t+∆t) and r(t-∆t) by two Tertiary Taylor Polynomials about position r(t), respectively.

r(t + ∆t) = r(t) + v∆t + 1 2 a(t)∆t 2 + 1 6 b(t)∆t 3 + O(∆t 4 ) (2.2) r(t -∆t) = r(t) -v∆t + 1 2 a(t)∆t 2 - 1 6 b(t)∆t 3 + O(∆t 4 ) (2.3)
where r(t) is the position, v(t) is the velocity, a(t) is the acceleration, and b(t) is third derivative of the position with respect to the time. By combing these two equations, we CHAPTER 2. METHODOLOGY AND MODELING obtain:

r(t + ∆t) = 2r(t) -r(t -∆t) + a(t)∆t 2 + O(∆t 4 ) (2.4)
The truncation error is of order ∆t 4 . Verlet algorithm is accurate, stable, and easy to use, but it cannot compute velocity directly. Thus, the Velocity-Verlet algorithm, is used for this purpose. In this one, positions and velocities are updated every timestep. At first, the position is updated:

r(t + ∆t) = r(t) + v(t)∆t + 1 2 a(t)∆t 2 (2.5)
and then forces are obtained from potential. At last, velocities are updated:

r(t + ∆t) = r(t) + v(t + 1 2 ∆t)∆t (2.6) v(t + 1 2 ∆t) = v(t) + 1 2 a(t)∆t (2.7) a(t + ∆t) = - 1 m ∇V(r(t + ∆t)) (2.8) v(t + ∆t) = v(t + 1 2 ∆t) + 1 2 a(t + ∆t)∆t (2.9)
The accuracy of the Velocity-Verlet algorithm is controlled by the timestep, ∆t, i.e., smaller timestep means better accuracy, while larger timestep means faster performance. Therefore, it is crucial to choose a suitable ∆t. The optimal choice is 1⁄100 of the period of the fastest vibration of the system.

Interatomic potentials

Potentials describe interatomic potential energy based on atomic coordinates, which include pair potential and multibody potentials. The accuracy and relevancy of MD calculation depends on them. Pair potentials consider attractive and repulsive contribution between a pair of atoms. The most widely used one is Lennard-Jones (L-J) potential [Allen 2.1. MOLECULAR DYNAMICS SIMULATION and Tildesley, 2017, Aschcroft and Mermin, ], and it can be expressed as:

ϕ(r ij ) = ε[(σ/r ij ) 12 -(σ/r ij ) 6 ] (2.10)
ε is an energetic parameter; σ is a length parameter; r ij is the distance between atom i and atom j. The first term indicates the attractive long-range interaction, and the second indicates the short range repulsing interaction. Multibody potentials can describe the interaction of the N-body (N⩾3). Widely used multibody potentials are the embedded atomic method (EAM) [START_REF] Daw | Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals[END_REF] and Finnis-Sinclair (FS) potential [START_REF] Finnis | A simple empirical n-body potential for transition metals[END_REF]. According to the Hohenberg-Kohn theorem [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF], Stott and Zaremba supposed that the energy of an embedded atom at position x is a function of electron density ρ [START_REF] Stott | Quasiatoms: An approach to atoms in nonuniform electronic systems[END_REF]. The principle of EAM is to treat each atom in the system as an embedded atom in the matrix of other atoms [Daw andBaskes, 1984, Daw andBaskes, 1983]. The total potential energy is the sum of the pair potential between the atoms in the lattice and the embedded energy of the atoms embedded in the electron cloud. EAM potential can be expressed as:

E pot = ∑ i F i (ρ i ) + 1 2 ∑ ij ϕ ij (r ij ) (2.11)
where F(ρ i ) is embedding energy function, i.e., the energy of an atom surrounded by electron density ρ i , which is the sum of electron density of all atoms j surrounding atoms

of atom i (ρ i = ∑ j ρ(r ij )); ϕ ij (r ij
) is pair potential depending on the distance (r ij ) between atom i and j. Finnis-Sinclair (FS) potential is issued based on EAM potential. The only difference between them is the explanation of electron density, ρ i , and embedded potential energy [START_REF] Finnis | A simple empirical n-body potential for transition metals[END_REF].

Ensemble of molecular dynamics simulation for an equilibrium system

In statistical physics, an ensemble represents a large set of possible states of a system under certain conditions. In other words, an ensemble is a probability distribution of the states of a system. The commonly used ensembles in MD simulation are micro-canonical ensemble, canonical ensemble, isothermal isobaric ensemble and isothermal isenthalpic Temperature is controlled by methods such as velocity calibration method [Heerman, 1987], Berendsen method [START_REF] Berendsen | Molecular dynamics with coupling to an external bath[END_REF], Gaussian method [Hoover, 2012], and Nose-Hoover method [Hoover, 2012, Nosé, 1984].

In the velocity calibration method, due to the relationship between velocity and temperature of the system, the temperature can be adjusted by calibrating velocity. If the temperature of the system at time t is T(t), and velocity is calibrated with a factor λ, then the variation of temperature can be determined:

∆T = T exp -T(t) (2.12) ∆T = (λ 2 -1)T(t) (2.13) λ = T exp T(t) (2.14)
T exp is the expected value of temperature.

Berensden method is an another algorithm that assumes that the system couples to a heat bath, whose temperature is the target value and keeps the temperature of the system constant by heat exchange with the heat bath. The variation of temperature at each step is:

∆T = δt τ (T bath -T(t)) (2.15)
where δt is the time interval, T bath is the temperature of the heat bath, and τ is the time constant. Calibration factor of velocity, λ:

2.1. MOLECULAR DYNAMICS SIMULATION λ = 1 + δt τ ( T bath -T(t) T(t) ) (2.16)
The Gaussian method adds a frictional force, ξ G on each atom as follows:

m i d 2 r i dt 2 = F i -m i ξ G v i (2.17)
where,

m i ξ G v i is the term of friction force. ξ G = (∑ j v j F j )(∑ k m k v k • v k ) -1 is the damping factor.
If ξ G is positive, it means the temperature is higher than that of the heat bath, and the system should be cooled, otherwise, the system should be heated. The temperature control is based on the kinetic energy of the total system.

The Nose-Hoover method supposes that the system is coupled to a virtual heat bath, in which the system can release and receive heat freely from the heat bath. The equation of motion of this method is similar to that of the Gaussian method, and differs only by the expression of the factor of the friction force. In this method, a Hamiltonian H * (total energy of the system) with an extra degree of freedom for the heat bath, s, is defined:

H * = ∑ i p i 2 2ms 2 + 1 2 ∑ ij,i̸ =j U(r i -r j ) + p 2 s 2Q + 3Nk B T exp ln s (2.18)
where N is the number of particles, and k B = 1.38 × 10 -23 J/K is the Boltzmann's constant.

Q is an effective mass controlling the coupling of the system to the heat bath and p s is the conjugate momentum of s. p i and r i are are so-called "virtual coordinates", which are related to the real coordinates r i ′ and p i ′ as follows:

r i ′ = r i , p i = p i s (2.19)
where the coordinates with accents are the real ones. Hamiltian deduced the motion equation:

ṙ′ = ∂H ∂r i ′ = p i ′ m (2.20) η = p s Q (2.21) CHAPTER 2. METHODOLOGY AND MODELING ṗ′ i = - ∂H ∂p i ′ = F i -ηṙ ′ i (2.22) ri ′ = F i m -ηṙ ′ i (2.23) η = 1 τ 2 ( T T exp -1) (2.24) τ = Q 3Nk B T exp (2.25)
where

F i = ∂U ∂r i
′ is the force acting on atom i, η is the damping factor and τ is the relaxation time of heat bath. τ is the input data in the command file of LAMMPS. Typical values are used such as 1 ps. T exp is the target temperature imposed to the system.

The commonly used methods of controlling pressure are Berendsen [START_REF] Berendsen | Molecular dynamics with coupling to an external bath[END_REF] and Parrinello-Rahman [Parrinello andRahman, 1981,Parrinello andRahman, 1982].

In the Berendsen method (similar to that used for temperature control), it is assumed that the system is coupled to a virtual pressure bath. The calibration factor of volume, λ, is expressed as:

λ = 1 + k δt τ P [P(t) -P bath ] (2.26)
where k and τ P are coupling parameters, P bath is the expected pressure and P(t) is the instant pressure at time t. This method can adjust the pressure in three directions simultaneously. But it cannot realize the change of dimensions in three spatial directions of the system under shear stress. To realize the change of shape and volume simultaneously and obtain the equilibrium with external pressure, Parrinello-Rahman issued Parrinello-Rahman method of controlling pressure [START_REF] Parrinello | Polymorphic transitions in single crystals: A new molecular dynamics method[END_REF], which is widely used in MD simulation.

MOLECULAR DYNAMICS SIMULATION

Periodic boundary conditions

Due to the limitation of the number of particles that can be introduced in MD simulation and the necessity of a large system size to study the macroscopic properties of the material, periodic boundary conditions are implemented. The principle is to consider the system containing the particles as a cell and replicate it in three directions so that cells

Structural analysis

Radial distribution function

Several important parameters are used for the structural analysis of metallic glasses (MGs). First, the radial distribution function (RDF), g(r), is defined by the probability of finding another atom at a distance, r, from a given atom [Bernal, 1960a], which has been briefly introduced in Section 1.4. g(r) can be calculated as follows:

g(r) = L 3 N 2 [ N ∑ i=1 n(r)]/4πr 2 ∆r (2.27)
where L is the length of the simulation box, N is the number of atoms, and n(r) is the number of atoms within the radius of the spherical shell from r to r + ∆r. When more than one element is present in the modeling system, we can define partial RDFs, g αβ (r),

by calculating the number of β atoms within the radius of the spherical shell from r to r + ∆r, taking each α atom as the center atom of the sphere. Its expression is:

g αβ (r) = L 3 N α N β [ N α ∑ i=1 n iβ (r)]/4πr 2 ∆r (2.28)
L is still the length of the box, N α , N β is the number of α and β atoms, and N iβ is the number of β atoms within the radius of the spherical shell from r to r + ∆r when the α atom is the center atom of the sphere. According to the definition of partial RDFs, g αβ (r)=

g βα (r). There are N(N+1)/2 partial RDFs in an N-tuple system, and the total partial RDF 2.2. STRUCTURAL ANALYSIS is the weighted sum of N(N+1)/2 partial RDFs.

Voronoi polyhedra

Even Tool (OVITO) through the operation "Voronoi analysis". In our case, it will be used to calculate the fraction of <0, 0, 12, 0> icosahedra (full-icosahedra FI) at a given strain.

Local atomic shear strain

To visualize shear transformations (STs) and shear bands (SBs), and understand the atomic rearrangement during the deformation of MGs, the local atomic shear strain, η Mises j can also be calculated [START_REF] Shimizu | Theory of shear banding in metallic glasses and molecular dynamics calculations[END_REF]. The calculation needs two atomic configurations, the referenced one and the current one. First, a local transformation gradient tensor J i must be calculated. It conforms to the following relationship [START_REF] Shimizu | Theory of shear banding in metallic glasses and molecular dynamics calculations[END_REF]:

d 0 ji → d ji , ∀j ∈ N 0 i (2.29)
where d's are vector separations (row vectors) between atom j and i (superscript 0 means the reference configuration). Here, j is one of atom i's nearest neighbors, N 0 i is the total number of the nearest neighbor of atom i. J i can be obtained via minimizing the following expression:

∑ j∈N 0 i d 0 ji J i -d ji 2 → J i = ( ∑ j∈N 0 i d 0T ji d 0 ji ) -1 ( ∑ j∈N 0 i d 0T ji d ji ) (2.30)
And then Lagrangian strain matrix can be calculated by:

η i = 1 2 (J i J T i -1) (2.31) CHAPTER 2.

METHODOLOGY AND MODELING

The local shear invariant of atom i can be obtained by:

η Mises i = η 2 yz + η 2 xz + η 2 xy + (η yy -η zz ) 2 + (η xx -η zz ) 2 + (η yy -η xx ) 2 6 (2.32) η Mises j
is the parameter measuring local non-elastic deformation [START_REF] Falk | Dynamics of viscoplastic deformation in amorphous solids[END_REF]],

which can be calculated by analysis and visualization software OVITO. Meanwhile, the intensity of η Mises j can be visualized through color coding by OVITO [Stukowski, 2009].

Construction of model and deformation

The metallic glass (MG) model used in this thesis is Cu-Zr binary alloy, for which reliable interatomic potentials are available, and the atomic structure has been well investigated [START_REF] Cheng | Atomic-level structure and structureproperty relationship in metallic glasses[END_REF]. The potential we used is Sheng's EAM potential [START_REF] Cheng | Atomic-level structure and structureproperty relationship in metallic glasses[END_REF]. Most of the simulation performed in this thesis, where the material is sheared, include 32000 atoms. They are cubic lattices with lengths of approximately 8 nm. For tensile deformation simulation, parallelepiped boxes with 64000 atoms are created, i.e., the edge along the tensile direction is twice as large as that of the model for shear.

When studying the effect of composition, We selected three different compositions, i.e.,

Cu 64 Zr 36 , Cu 50 Zr 50 , and Cu 36 Zr 64 . The potentials used for all of the models are Sheng's potential [Cheng et al., 2009b]. First, we prepare the cubic or parallelepiped lattice with Cu atoms and then replace some of them randomly with Zr atoms to obtain the desired compositions. The MG system is obtained by melt quenching to obtain a random atomic arrangement. In our cases, the system is heated to 2000 K and maintained at this temperature for 2 ns to ensure the chemical homogeneity; it is then quenched down to 0.1 K, with different quenching rates, ranging from 10 10 K/s to 10 14 K/s, and kept at this low temperature for 2 ns likewise.

After the construction of the MG model, we apply quasi-static and dynamic shear and tensile deformation to the box.

For quasi-static deformation, strain reaches 30 %, and strain increment δγ=0.1 % at each step. After each strain step, the potential energy is minimized with the Fast Inertial Relaxation Engine (FIRE) algorithm at very low temperature, 0.1 K. FIRE algorithm can 2.4. ANALYSIS AND MAPPING OF PLASTIC EVENTS add velocity modifications and adaptive time steps according to the last iteration result, which is rather faster than the conjugate gradient (CG) algorithm [START_REF] Bitzek | Structural relaxation made simple[END_REF],Eidel et al., 2011]. The criterion of force after minimization should be magnitude of 10 -3 .

For dynamic deformation, different strain rates are tested: 10 7 s -1 , 10 8 s -1 and 10 9 s -1 .

Pressure is kept at 0 during preparation and deformation, and temperature is controlled at 0.1 K, or at higher temperatures if thermal effects are studied(cf. Chapter 3). The thermostat is Nose-Hoover thermostat.

Analysis and mapping of plastic events

Definition of plastic events

To analyze the quasi-static simulations, we treat plasticity as an inclusion problem. The local plastic transformation zones are considered as inclusions in an isotropic homogeneous elastic medium, as introduced in Section 1.6. The intensity of plastic energy is computed to identify the plastic events. Since the temperature is kept at 0.1 K all the time, thermal effect can be ignored. After application of quasi-static shear deformation on the system, we define {i} curr as the configuration of forward deformation with δγ at step i, and {i} rev as the configuration of reverse deformation with -δγ at step i. From step {i} curr to {i + 1} curr , the energy variation includes plastic and elastic contributions, as shown in Figure 2.3 . When applying reverse deformation on it, {i} rev is reached instead of {i} curr from {i + 1} curr , because the plastic contribution is irreversible. When, there is no plastic event occurring, the difference of energy between {i} rev and {i} curr ,

δE pl i→i+1 = E curr i -E rev
i is nearly zero within numerical accuracy. In contrast, if a plastic event exists, then the plastic energy between two successive steps can be computed. For each atom, it can be computed considering the interaction of the center atom with the CHAPTER 2. METHODOLOGY AND MODELING surrounding ones. The plastic energy on atom i a follows:

PE at (i a ) = ( ∑ j a V 2b i a ,j a ({r} curr ) -V 2b i a ,j a ({r} rev ) 2 + ∑ j a ∑ k a V mb i a ,j a ({r} curr ) -V mb i a ,j a ({r} rev ) 2 ) 1 2
(2.33)

where, {r} curr and {r} rev are the current configuration and reverse configuration, respec- tively. V 2b i a ,j a and V mb i a ,j a are respectively the two-body and multi-body part of the EAM potential introduced in Eq. 2.11.

In order to build a continuous plastic energy field PE(r) from the atomic plastic energy defined in Eq. 2.33, we use a coarse-graining procedure [START_REF] Goldenberg | Small and large scale granular statics[END_REF] by averaging the atomic plastic energy with a spatial Gaussian function as follows:

PE(r) = ∑ i a PE at (i a )( 1 (πω 2 ) 3 2 e -| r-r ia | 2 ω 2 ) (2.34)
where ω is the Gaussian width, which is taken of the order of the average first-neighbor distance. For the binary metallic glass system Cu 64 Zr 36 used, the value of ω is 2.8 Å. PE(r) can be plotted on a 3D grid ,where we can notice the presence of several peaks, as shown in Figure 2.4. We define these peaks as attractors, which are indicated in in Figure 2.5, and the whole region surrounding an attractor is defined as a basin of attraction, i.e., the region of the space traversed by all the gradient paths that terminate at this attractor.

ANALYSIS AND MAPPING OF PLASTIC EVENTS

Figure 2.5: Map of plastic energy, PE(r) surrounding a plastic event (the intensity of plastic energy from low to high) corresponding to the color from blue to red [START_REF] Fusco | Role of local order in the small-scale plasticity of model amorphous materials[END_REF].

The locations of plastic events are determined by the position of the attractors. The intensity of an individual plastic event is computed integrating the plastic energy density over its basin of attraction. In order to calculate the size of an attractor, we plot the radial distribution of the plastic energy around the attractors and fit it with an exponential function, f (r) = a + b * e -( r λ ) , as shown in Figure 2.6. The damped exponential parameter λ is defined as the size of the attractor and therefore used to estimate the size of the plastic events. Thus, it is not necessary to totally display the detailed characteristics of PE(r).

The input parameters used in this step are the number of atoms, the interatomic potential and the cut-off radius used to calculate the plastic intensity (PI) on a grid. The outputs are the results of plastic intensity and the position of each attractor at each step. At the same time, we can calculate the size of each attractor with the method mentioned before.

Selection of attractors

In fact, not all attractors are relevant. We set two criteria to select valid and important attractors. First, at low strain, due to numerical accuracy, there appears a large number of attractors with low intensities even after reversible deformation, as shown in Figure 2.4. In order to get rid of them, we order the n att attractors at a given strain as a function of their intensity so that the first in the list (n=1) has the highest intensity and the last one (n=n att ) the lowest one. If the plastic energy increment is more than 20 % between two successive attractors, n and n+1, i.e., PI n -PI n+1 PI n+1

> 20 %, then all attractors below n+1 are collected and the others are discarded. With this first criterion, we can select large plastic events at lower strain [START_REF] Fusco | Role of local order in the small-scale plasticity of model amorphous materials[END_REF]. On the other hand, at large strains, plastic events can bear larger PI even without large PI differences between events. In this case, the application of the first criterion would lead to discard significant plastic events. In order to avoid this, we set a threshold of minimum valid PI value. If the PI of attractors is larger than the threshold, they are selected. Here, this value is 1.5 eV to select 90 % valid plastic events. This arbitrary value of the energy threshold has been chosen in light of a parametric study, and leads to stable results with respect to the statistic of the plastic events, specifically yielding a good comparison between the behavior of the total plastic intensity integrated over all the basins and the total dissipated plastic energy calculated with the method explained in Section 2.4.1.

ANALYSIS AND MAPPING OF PLASTIC EVENTS

After having selected the valid plastic events with these two criteria, the cumulative number and cumulative energy of plastic events with strain can be computed.

At this stage, the main input parameters are two criteria for selecting valid attractors, and outputs are the selected attractors. Å for the Eshelby's inclusion radius corresponds to a "typical" size of plastic events, as calculated by the method illustrated in Section 2.4.1, and it has been chosen after testing several values of this parameter and searching for the one giving the best agreement between the simulated stress-strain curves and the ones obtained using the Eshelby's inclusion model (see Section 2.4.5). We notice that this value of 12 Å is larger than the typical size of the plastic events in amorphous Si, suggesting that the nature of the plastic events in these two amorphous systems is rather different. The input of this stage is the list of the selected relevant attractors obtained in the previous stage. All attractors at each step are ordered from the largest plastic energy intensity to the lowest. The output is the list of reorganized plastic events.

Rearrangement of plastic events

Atomic displacement fitting

At this stage, we fit the MD atomic displacements with the displacement field obtained from the Eshelby's inclusion model. For the sake of simplicity we consider that the inclusions are spherical. Additionally, homogeneous elastic constants are equal inside and outside the inclusions. Thus, a spherical inclusion with radius a, shear modulus G for CHAPTER 2. METHODOLOGY AND MODELING quasi-static shear deformation, and Poisson's ratio ν is embedded in an infinite homogeneous isotropic medium. The stress-free strain transformation tensor is ϵ T ij , and surface tension can be obtained by

p ij = 2G(ϵ T ij + νδ ij ϵ T kk 1-2ν
). The fitting is based on displacement fitting. Thus, we need to calculate the displacement inside and outside inclusions from MD simulation and Eshelby's inclusion. Displacement u i outside the inclusion and stress σ out ij follow [Bower, 2009].

u i = a 3 4(1 -ν)G ( 2p ik x k + p kk x i 15R 5 (3a 2 -5R 2 ) + p jk x j x k x i R 7 (R 2 -a 2 ) + 4(1 -ν)p ik x k 3R 3 ) (2.35)
where R is the distance from the center of the inclusion.

σ out ij = a 3 2(1 -ν)R 3 ( p ij 15 (10(1 -2ν) + 6 a 2 R 2 ) + p ik x k x j + p jk x k x i R 2 (2ν -2 a 2 R 2 ) + δ ij p kk 15 (3 a 2 R 2 -5(1 -2ν)) + δ ij p kl x k x l R 2 ((1 -2ν) - a 2 R 2 ) - x i x j p kl x k x l R 4 × (5 -7 a 2 R 2 ) + x i x j p kk R 2 (1 - a 2 R 2 ) ) (2.36)
Inside the inclusion, the strain ϵ in ij has the following relationship with stress-free transformation strain tensor ϵ in ij = S ijkl ϵ T kl , where S is the Eshelby tensor [Bower, 2009], and ϵ T kl is the transformation strain with the surrounding elastic matrix. Due to the presence of surface traction P ij , the stress inside the inclusion is

σ in ij = C ijkl ϵ in kl -P ij = C ijkl (ϵ in kl -ϵ T kl
). Since the displacement inside the inclusions is strongly dependent on the local atomic structure, only the displacements outside the inclusions are fitted using Eq. 2.35.

How does the fitting proceed? Input includes initial cell, initial configuration, the information of plastic events after selection (location, size, and the intensity of plastic energy), reverse configuration, numbers of atoms, Poisson's ratio, and Eshelby radius. From the atomistic simulation, the atomic displacement can be calculated by MD simulation. The atomic displacement d i,i a of atom i a at i direction is computed by d i,i a = r rev i,i ar curr i,i a .The total number of plastic events is defined as N sel (selected plastic events). The center of plastic events is identified by i c (e), the index of the closest atom to the center of the inclusion. If atom i a runs over the N out atoms outside the inclusions, i.e., r curr i a r curr i c (e) > a(e), where a(e) is the size of the plastic event, then we can set an objective function f to com-

ANALYSIS AND MAPPING OF PLASTIC EVENTS

= N out ∑ i a =1 ∑ i 1 2 d i,i a - N sel ∑ e=1 u i (r curr i a -r curr i c (e) , ϵ T (e)) 2 (2.37)
is minimized by damped MD, and then we can get the strain tensor, ϵ T (e). It can also be used to estimate the precision of fitting.

Fitting of the stress-strain curve

At this stage we compare the stress-strain curve from the model with the MD results.

From the previous results, we know the number and location of plastic events at each strain step. From Figure 2.3, we can see that the difference between {i} curr and {i -1} rev is the elastic part. So shear modulus at step i can be obtained by the ratio:

G(γ i ) = G i = (σ xy (γ i ) -σ rev xy (γ i-1 ))/δγ xy (2.38)
We decompose strain increment, δ i ϵ xy between step i and i+1, into an elastic and plastic part:

δ i ϵ xy = δ i ϵ el xy + δ i ϵ pl xy (2.39)
We can get therefore the shear stress variation: CHAPTER 2. METHODOLOGY AND MODELING According to the former Eq. 2.40, the stress can be considered to be based on the contribution of the stress field. Therefore, the plastic stress contribution δσ pl,esh xy (e) of a single inclusion can be deduced to be:

δ i σ xy = 2G(γ i )δ i ϵ el xy = 2G(γ i )(δ i ϵ xy -δ i ϵ pl xy ) = δ i σ *
δσ pl,esh xy (e) = - V esh (e) V cell σ in xy (e) = 2G(γ i )V esh (e) V cell (ϵ T xy (e) -ϵ in xy (e)) (2.41)
where σ in xy (e) is the shear stress inside inclusion, e. (σ out ij is the stress outside the inclusion, the angular sum over the terms of σ out ij vanishes). V esh (e) is the volume of the spherical inclusion e, and V cell is the volume of the simulation cell. The Eshelby theory assumes that Eshelby's inclusion is in an infinite medium. However, in our computation we use a fixed cell, since at each strain γ i the simulation box is first deformed in the forward direction and then comes back to its original shape after the reverse step. This condition implies a zero total strain on the cell. However, the total shear strain due to sum of the shear strains of all the inclusions in the cell is ϵ esh,tot xy

= ∑ N sel e=1 V esh (e) V cell ϵ in
xy (e). Thus, a homogeneous correction, -ϵ esh,tot xy , is required to preserve zero shear strain over the cell. Thus, when taking the boundary conditions into account, we can get:

δσ pl,esh+bc xy (e) = δσ pl,esh xy (e) + 2G(γ i )ϵ esh,tot xy = 2G(γ i ) V esh (e) V cell (ϵ T xy (e) -ϵ in xy (e) + ϵ in xy (e)) = 2G(γ i ) V esh (e) V cell ϵ T xy (e)
(2.42)

CONCLUSION

Finally, the shear stress from Eshelby's inclusion is

σ mod (γ i ) = i-1 ∑ j=0 δ j σ * xy - i-1 ∑ j=0 δ j σ pl,esh+bc xy (2.43)
The first term is the elastically extrapolated shear stress and the second term represents the sum of the plastic stress drops due to each Eshelby's inclusions (Eq. 2.42) and cumulated up to strain γ i .

Conclusion

This chapter describes the methods used in this thesis, the molecular dynamics (MD) simulation, and the way quasi-static and dynamic tests are implemented. It also introduces the calculation of the parameters used to analyse the evolution of the microstructure. In addition, we presented the method for defining and analyzing plastic events, and the reconstruction of the stress-strain curve from MD simulation from the use of the description of the plastic event as Eshelby's inclusions.

Introduction

As mentioned in Chapter 1 , the factors affecting the mechanical properties of metallic glasses (MGs) are various [START_REF] Mukai | Effect of strain rate on compressive behavior of a pd40ni40p20 bulk metallic glass[END_REF], He et al., 2003, Shen et al., 2007]. In this chapter we study the influence of quenching rate, composition, strain rate, and temperature on the microstructure and the mechanical properties of MGs. Through molecular dynamics (MD) simulation, we analyze the effect of these factors and establish the link between microstructure and mechanical properties of MGs.

Effect of quenching rate

Quenching rate is a vital factor affecting the microstructure and mechanical properties of MGs [START_REF] Wang | Effects of quenching rate on amorphous structures of cu46zr54 metallic glass[END_REF],Yan et al., 2007,Huang et al., 2014]. Here, we study the microstructural characteristics of a binary MG, Cu 64 Zr 36 , with 32000 atoms under quenching rates from 10 10 K/s to 10 14 K/s. The global characteristics at each time step, like volume and temperature, can be obtained through MD simulation [START_REF] Daw | The embedded-atom method: a review of theory and applications[END_REF]. for MGs with higher quenching rate at the same temperature, which shows that higher quenching rates can induce more free volume in MG system [START_REF] Cheng | Indicators of internal structural states for metallic glasses: Local order, free volume, and configurational potential energy[END_REF]. The CHAPTER 3. GENERAL ANALYSIS OF THE MECHANICAL BEHAVIOR OF METALLIC GLASS FROM MOLECULAR DYNAMICS SIMULATION inflection points correspond to the glass transition temperature T g , which can be obtained by extrapolation and intersection of the two straight lines tangent to the curves for temperatures below and above this point. To make it clear, the curves have been shifted up in Figure 3.1 (b), and the values of T g are shown in Table 3.1. We can conclude that T g depends on the quenching rate, and is higher for higher quenching rates. In fact, higher quenching rates provide less time for atoms to relax, and the materials "freezes" earlier, leading to higher T g [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF]. the typical features of an amorphous system, namely a highest sharp first peak, a splitting of the second peak, and a blunt third peak [START_REF] Yue | Effect of cooling rate on structures and mechanical behavior of cu50zr50 metallic glass: a molecular-dynamics study[END_REF]. A subpeak before the first peak is observed in Figure 3.2. The existence of subpeak is also evidenced by both experiments [START_REF] Mattern | Structural behavior of cuxzr100-x metallic glass (x= 35-70)[END_REF] and simulations [START_REF] Yue | Deformation behavior of annealed cu64zr36 metallic glass via molecular dynamics simulations[END_REF], Feng et al., 2018]. A zoom in of the peaks displayed in Figure 3.3, shows that this subpeak appears at around 2.5 Å , the highest one is at around 2.8 Å, and the second peak is located at around 4.75 Å.

On the other hand, by inspection of the Cu-Cu partial RDF in Figure 3.4 we see that the first peak of Cu-Cu partial RDF in Figure 3.4 (a), is located at around 2.5 Å, and the first peak on Cu-Zr partial RDF curve is at around 2.8 Å. Thus, it can be speculated that the first peak of Cu-Cu partial RDF leads to the first subpeak of the total RDF. To confirm it, we plotted the RDF and partial RDF curves for another composition, Cu 36 Zr 64 , shown in Figure 3.5. Indeed, it can be observed that there are two subpeaks around the first peak, the left one is at 2.5 Å, the right one is at 3.2 Å, and the main peak is at 2.8 Å, which respectively corresponds to the peak of Cu-Cu, Zr-Zr, and Cu-Zr partial RDF, respectively.

Dziegielewski et al. also made the same conclusion in Cu-Zr MGs [START_REF] Dziegielewski | High pressure atomic structure of zr-cu metallic glass via exafs spectroscopy and molecular dynamics simulations[END_REF].

As a general remark, the effect of the quenching rate on the RDF is very slight and the average interatomic distances are not affected, in agreement with previous results of the literature [START_REF] Wang | Effects of quenching rate on amorphous structures of cu46zr54 metallic glass[END_REF], Yue et al., 2018].

In order to study the effect of quenching rates on the mechanical behavior of MGs, we apply quasi-static shear deformation to MG system prepared with different quenching rates. The resulting shear stress-strain curves are shown in Figure 3.6. The shear modulus, measured as the slope of the linear part of the stress at very small strains, decreases when increasing the quenching rate, indicating that higher quenching rates result in easier deformation of MGs, as shown in Figure 3.7 (a). The yield stress, obtained as the

EFFECT OF QUENCHING RATE

maximum of shear stress and the corresponding yield strain also decrease by increasing the quenching rate, as shown in Figure 3. 7 (b). This means that the resistance to the initiation of flow decreases, and or in other words that it is easier to rejuvenate MG structure prepared with higher quenching rates [Cheng et al., 2008a, Shimizu et al., 2006]. We also calculate the flow stress as the average stress during the last 10 % strain.

CHAPTER 3. GENERAL ANALYSIS OF THE MECHANICAL BEHAVIOR OF METALLIC GLASS FROM MOLECULAR DYNAMICS SIMULATION

The stress-strain curves in Figure 3.6 show that the stress drop occurs between 12 % and 15 % strain. Figure 3.8 at strain 12.3 % shows the map of the atoms with large atomic shear strains (larger than 60 % of the maximum atomic shear strain) in Cu 64 Zr 36 MG prepared with a quenching rate 10 10 K/s. Several localized shear transformations (STs) are visible. At strain 12.4 %, some STs develop as a shear band (SB), while, some diffuse out. According to Shimizu et al. [START_REF] Shimizu | Theory of shear banding in metallic glasses and molecular dynamics calculations[END_REF], for Zr-based MGs, the length scale of SB should reach few ten nm. In our case, the length of the sample is 8 nm, thus, a band throughout the entire sample can be defined as an SB. Here we use a criterion to define the occurrence of an SB region: we consider all the atoms having an atomic shear strain larger than 60 % of the maximum local strain over the whole sample and we check whether this region percolates across the system. If it percolates we define it as an SB. In the case of Figure 3.8 we can identify 12.4 % as the strain where an SB forms.

Figure 3.8: Map of atomic shear strain in Cu 64 Zr 36 MG prepared with a quenching rate of 10 10 K/s, sheared at 12.3 % strain (a) and 12.4 % (b), showing the atoms with shear strain larger than 60 % of the maximum atomic shear strain over the whole sample.

Moreover, 20 % strain being in the flow stress region, we plot the distribution of atomic shear strain at 12.3 %, 12.4 %, 12.5 %, 15 %, and 20 %, respectively, as shown in Figure 3.9. We compare the distribution of atomic shear strain for these different configurations for different quenching rates. At 12.4 % strain we can observe a complete shear band (SB) for the Cu 64 Zr 36 MG with quenching rate of 10 10 K/s, while for the other quenching rates, although localized STs exist, they do not form a complete band. At 20 % strain, for quenching rate from 10 11 K/s to 10 13 K/s, multiple SBs with different orientations are formed (in this system, both horizontal and vertical SBs can be formed with equal 3.2. EFFECT OF QUENCHING RATE Figure 3.9: The atomic shear strain in Cu 64 Zr 36 MG prepared with different quenching rates at different shear strains, at 0.1 K.

probability, with a slight preference for the horizontal orientation since we shear in xy plane). For quenching rate 10 14 K/s, the distribution of atomic shear strain is gradually concentrated and localized with the strain increase. A clear SB is not visible even in the flow stress region. From these observations, it can be concluded that higher quenching rates lead to more homogeneous strain, in agreement with previous experimental [START_REF] Huang | The effect of cooling rate on the wear performance of a zrcualag bulk metallic glass[END_REF], Shen et al., 2007] and simulation results [START_REF] Yue | Effect of cooling rate on structures and mechanical behavior of cu50zr50 metallic glass: a molecular-dynamics study[END_REF], Zhong et al., 2016].

In order to relate the mechanical behavior to the MG microstructure, we plot in Figure 3.10 the fraction of FI (full icosahedra, FI), already introduced in Chapter 1, as a function of the shear strain of MG quenched under different quenching rates. It can be observed that a lower initial fraction of FI is obtained with higher quenching rates, which is consistent with previous results [START_REF] Zemp | Icosahedral superclusters in cu 64 zr 36 metallic glass[END_REF]. With increasing shear strain, the fraction of FI decreases, proving their destruction during the deformation. Especially in the 12 %-15 % strain domain, the decrease of FI is considerable for the lowest quenching rate.

Additionally, the fraction of FI in the SB region of Cu 64 Zr 36 with shear strain is plotted in

EFFECT OF COMPOSITION

Effect of composition

Many researchers have focused on the effect of composition on the mechanical properties of MGs [START_REF] Liu | Super plastic bulk metallic glasses at room temperature[END_REF], Chen et al., 2008, Kim et al., 2009]. In this section, to study this question, we selected three different compositions of binary Cu-Zr MG system, i.e.,

Cu 64 Zr 36 , Cu 50 Zr 50 , and Cu 36 Zr 64 , quenched at a quenching rate of 10 11 K/s. Figure 3.12 shows the volume change as a function of the temperature. The volume of the system increases with the ratio of the Zr element, which is expected since the equilibrium distance of Zr atom is larger than that of the Cu atom [START_REF] Wang | Effects of quenching rate on amorphous structures of cu46zr54 metallic glass[END_REF]. T g of the MG increases with the Cu content which agrees with the result of Mattern et al [START_REF] Mattern | Structural behavior of cuxzr100-x metallic glass (x= 35-70)[END_REF]. For the total RDF, the composition has a very small effect on the position of the main peak at 2.8 Å. This result is consistent with the work [START_REF] Wang | Structural properties of zrxcu90xal10 metallic glasses investigated by molecular dynamics simulations[END_REF] and Mattern et al [START_REF] Mattern | Structural behavior of cuxzr100-x metallic glass (x= 35-70)[END_REF]. two different characteristic interatomic distances typical of medium-range order (MRO), as suggested by a previous study [START_REF] Liang | Influence of icosahedral order on the second peak splitting of pair distribution function for mg70zn30 metallic glass[END_REF].

After the comparison of microstructure, we study the influence of the composition on the mechanical properties of the MGs by applying to them the same quasi-static shear deformation, as shown in Figure 3.16. The shear modulus, yield stress, yield strain and flow stress are reported in Figure 3.17. From Figure 3.17 (a), (b), and (d), we can conclude that the shear modulus and yield stress increase with the Cu ratio (in spite of a large error bar). The yield strain of Cu 64 Zr 36 , Cu 50 Zr 50 , and Cu 36 Zr 64 is 12.4 %, 10.6 % and 10.7 %, respectively. These results are in agreement with the simulation work of by Cheng et al [Cheng et al., 2008a], which also showed that the resistance to flow increases with the Cu ratio. Indeed, our results also suggest a slight increase of the flow stress with the Cu content. After normalizing the stress by the shear modulus (cf. Figure 3.18), it can be seen that the difference between the stress-strain curves of these three compositions is slight and that the yield stress is nearly proportional to the shear modulus, again in agreement with Cheng et al [Cheng et al., 2008a].

In order to compare the distribution of shear transformations (STs) of three compositions, we plot the map of atomic shear strain. The strains selected are the yield strains of the Additionally, it is found that the initial fraction of <0,0,12,0> icosahedra (full icosahedra, FI) is higher for the MG with a higher Cu fraction, as shown in Figure 3.21. The reason is that most FI are Cu-centered clusters. As shown in Figure 3.21 (a), e.g., for Cu 64 Zr 36 , the fraction of FI is around 12.5 %. As shown in Figure 3.22 (a), among these FI, the fraction of Cu-centered icosahedra is 12.4 %, and that of Zr-centered icosahedra is nearly zero indicating that the smaller size atom tends to be the central atom of the icosahedra structure [START_REF] Liang | Influence of icosahedral order on the second peak splitting of pair distribution function for mg70zn30 metallic glass[END_REF], Miracle et al., 2003]. In addition, the fraction of FI decreases with strain, especially during the stress drop on the stress-strain curves. We calculated the decrease of FI, △FI =

FI initial -FI f inal FI initial
, reported in Table 3.2. △FI is smaller for Cu 36 Zr 64 , which corresponds to a smaller drop on the stress-strain curve and less localized deformation.

EFFECT OF COMPOSITION

Figure 3.21: Comparison of fraction of FI with shear strain for the three compositions.

Effect of strain rate

The effect of strain rate on the mechanical properties of MGs has been widely studied [START_REF] Mukai | Effect of strain rate on compressive behavior of a pd40ni40p20 bulk metallic glass[END_REF], Zhang et al., 2007, Ma et al., 2009, Li et al., 2017]. Different effects of strain rate on the mechanical behavior were observed, depending on the material investigated, on the strain rate range and on the temperature. The deformations are implemented at 0.1 K, far from the glass transition temperature. On the same Cu 64 Zr 36 MG system, we apply dynamic shear deformation (cf. Chapter 2) with different strain rates, 10 7 s -1 , 10 8 s -1 , and 10 9 s -1 , to study the effect of strain rate on the mechanical properties of MGs. All of these systems possess the same initial microstructure and they have been obtained with a quenching rate of 10 11 K/s. the mechanical properties of MGs is small (although the shear modulus and yield stress are slightly increased when the strain rate of MGs is largely increased). It also showed that the effect of strain rate on MGs becomes more pronounced when the temperature is increased. Therefore, we also studied the stress-strain curves of Cu 64 Zr 36 MG at 50 K and 200 K after deformation at different strain rates (to do so, Cu 64 Zr 36 MG was quenched from the liquid state with a quenching rate of 10 11 K/s down to the temperature of the shear test, i.e. 50 K or 200 K).

Results are shown in Figure 3.26. Indeed, the influence of the strain rate on the yield stress slightly increases when the temperature increases. The yield strain is relatively Zr 36 MG at strain 30 % with strain rate of 10 7 s -1 , 10 8 s -1 and 10 9 s -1 when the temperature is 50 K and 200 K.

larger at higher strain rate. We have plotted in Figure 3.27, the atomic shear strain maps of Cu 64 Zr 36 MG when deformed at 30 % strain, at different strain rates for temperatures 50 K and 200 K. It is clear that at higher temperatures, the STs of Cu 64 Zr 36 MG are more homogeneous when the strain rate is higher, in agreement with the conclusion of Sepulveda-Macias and Albe et al [START_REF] Sepulveda-Macias | Strain rate and temperature effect on zr50cu50 metallic glass under pure shear[END_REF], Albe et al., 2013]. In conclusion, no significant effect of the strain rate has been found at very low temperatures. This could also be due to the high strain rate used in MD. Similar conclusions were obtained in previous studies [Zhong et al., 2016, Egami et al., 2013]. Preliminary results suggest that for temperatures closer to T g , the influence of the strain rate on the mechanical properties is more significant [START_REF] Fu | Temperature, strain rate and reinforcement volume fraction dependence of plastic deformation in metallic glass matrix composites[END_REF].

EFFECT OF TEMPERATURE

Effect of temperature

We have more deeply studied the influence of temperature. Cu 64 Zr 36 MG was quenched from the liquid state with a quenching rate of 10 11 K/s down to 0.1 K, 50 K, 200 K, or 500 K, and then sheared. Stress-strain curves obtained at strain rate 10 8 s -1 and different temperatures are shown in Figure 3.32. We have compared two types of deformation: shear and tensile (Figure 3.32 (a) and 3.32 (b) respectively). Deduced shear modulus and Young's modulus are plotted in Figure 3.33. They both strongly decrease with the increase in temperature.

In addition the yield stress, yield strain and flow stress decrease with the temperature, as classically observed in previous experimental or simulation study [START_REF] Wang | Effect of temperature on mechanical properties of ti-based metallic glass matrix composite[END_REF], Sepulveda-Macias et al., 2018]. We have plotted the distribution of atomic shear strain at 12 % strain (after the yield peak) in Figure 3.34. It shows that higher temperature leads to more homogeneous deformation, in agreement with the experimental results [START_REF] Yu | Low-temperature mechanical properties of ce68al10cu20co2 bulk metallic glass[END_REF].

Discussion and conclusion

By studying the effects of quenching rate, compositions, strain rate and temperature on the mechanical properties of Cu-Zr MGs, we obtain the following conclusions.

The glass transition temperature of MGs depends on the quenching rate, and for higher quenching rate, higher T g is obtained. However, the quenching rate has a slight effect on the nearest-neighbor distance of the atoms. Regarding its effect on the mechanical properties of MGs, we find that the shear modulus and yield stress decrease with increasing quenching rate and the plastic deformation is less localized. With lower quenching rates, more full icosahedra (FI) are formed, which is consistent with less free volume, and more FI are destroyed during the deformation.

Regarding the composition effect, T g increases with the increase in Cu ratio in Cu-Zr

MGs. In addition, since the central atoms of FI are mostly Cu atoms, when the Cu ratio increases, fraction of FI in the Cu-Zr MG increases substantially, and thus STs are more localized at the same strain.

Concerning the strain rate effect, it is not significant at very low temperature we studied, and needs temperature close to the glass transition to be significant. In that case, plastic deformation is found more uniform for MGs with higher strain rates.

Regarding the temperature influence, the atomic packing density of the MGs is higher and the SRO is more pronounced when temperature is decreased. When deformed, MGs have higher shear and Young's moduli and higher yield stresses at lower temperature, which is correlated to a larger fraction of FI and more localized plastic deformation.

In conclusion, a higher quenching rate, lower Cu ratio, and higher temperature all result in a more uniform deformation of Cu-Zr MGs. This is also the case for higher strain rates.

INTRODUCTION

Introduction

Chapter 3 described the effects of quenching rate, composition, strain rate and temperature on the mechanical properties of metallic glasses (MGs), through the analysis of stress-strain curves, atomic shear strain, etc. In order to understand the effect of these factors on the observed macroscopic plasticity, in this chapter, we characterize and discuss in detail the plasticity of the system by identifying the plastic events occurring at a given strain, in terms of number, plastic intensity and size. From this parametric study we can propose a strategy how to improve the ductility of MGs.

Effect of quenching rates and compositions on plastic events

This section takes focuses on the results of Section 3.1 and 3.2. From Figure 4.2, it is clear that there are three small jumps in the curves for the three quenching rates from 10 10 K/s to 10 12 K/s. The locations of these three small jumps, which indicate a sudden release of elastic energy, correspond to that of the yielding points in their stress-strain curves (orange ellipses in Figure 4.3). That means that there is a pronounced increase in the plastic intensity of the plastic events when the yielding stress is reached. At higher shear strains, in the flow stress region of the stress-strain curve, the increase in the cumulative plastic intensity is linear (cf. Figure 4.2).

Effect of quenching rates on plastic events

To go further, the number, the plastic intensity and the average size of the plastic events of Cu 64 Zr 36 MG with different quenching rates, are plotted in Figure 4.4 to 4.6. It can be observed that we can detect plastic events with appreciable sizes even at the beginning of the deformation since we use a quasi-static deformation procedure. In the quasistatic approach, we cannot track the time evolution of plastic events. The curves are scattered, indicating a large variation from one strain step to another. However, in spite of this, a curve trend can be guessed for all the data. To make the comparison evident, we smoothed the curves with Savitzky-Golay (SG) filter method (described in the appendix)

and selected two quenching rates, 10 10 K/s (the lowest one) and 10 14 K/s (the highest one).

EFFECT OF QUENCHING RATES AND COMPOSITIONS ON PLASTIC EVENTS

Even in the elastic region, plastic events can be observed, however their number, intensity and average size are small, especially for the lowest quenching rate. Said differently, more numerous and larger plastic events with more plastic intensity are activated at the beginning of the deformation with the highest quenching rate. With the shear strain increasing, all of the parameters increase up to the yield point, more rapidly for the slowest quenching rate. At higher strain, number, size and plastic intensity, still scattered, seem to stabilize around a value which is roughly the same for all the quenching rates, except maybe for the size parameter: this one is slightly larger for the slowest quenching rate.

This may be related to the easier formation of shear bands for this system (Figure 3.8).

Then we have calculated the orientation of the plastic events, defined as φ xy,max . It is deduced from the components of the strain tensor of the equivalent Eshelby's inclusion They increase as the quenching rate increases, indicating that for high quenching rates, the plastic event directions are more diverse. This may be related to plastic events more uniformly distributed. To check this assumption, Figure 4.8 reports the comparison between the orientation distributions of plastic events which occur before, and after a shear band (SB) formation for the lowest quenching rate. The very small difference between the two curves actually suggests that the plastic events orientation distribution cannot be easily correlated to SB formation. Further work would be necessary to clarify this point.

EFFECT OF QUENCHING RATES AND COMPOSITIONS ON PLASTIC EVENTS

Figure 4.9: Fitting stress-strain curves and curves after correction with factor α of Cu 64 Zr MGs with different quenching rates (10 10 K/s (a),10 11 K/s (b), 10 12 K/s (c), 10 13 K/s (d), and K/s (e), respectively; left is the original curves and right is after correction by a factor α=1.01 for 10 10 K/s, α=1.03 for 10 11 K/s, α=1.03 for 10 12 K/s, α=1.09 for 10 13 K/s and α=1.1 for 10 14 K/s).

CHAPTER 4. ANALYSIS OF THE PLASTIC EVENTS AT THE ORIGIN OF THE MACROSCOPIC PLASTICITY

Using the procedure based on the Eshelby's inclusions theory (described in Chapter 2), we have fitted the stress-strain curves, as shown in Figure 4.9. The fitting curve can perfectly reproduce the shape (fluctuations) of the stress-strain curve obtained from the MD simulation. For all quenching rates but the smallest one (10 10 K/s), the fitting curve (blue curve) is slightly above the MD stress-strain curve (black curve), and the discrepancy slowly increases with the strain. We speculate that it is due to the removal of the smallest plastic events in the procedure of selection of the relevant plastic events, thus underestimating the plastic stress. Moreover, due to the occurrence of many events for each configuration, the quality of the displacement's fitting decreases since the Eshelby's inclusions can no longer be considered as single isolated events. Conversely, for the slowest rate, the fitting curve overestimates the stress before the yield point, and underestimates it after the stress drop. In this case we see that the stress drop at 12.3 % strain is overestimated by the fitting curve of the model. This can be due to the use of the minimum image convention, illustrated in Chapter 2 . In fact, when the SB forms, many events participate in the atomic displacements, but only a limited fraction of them is taken into account in the minimum image convention due to the small size of the cell. These events induce large components of the Eshelby's strain tensor, thus leading to an overestimation of the plastic stress. In order to correct these discrepancies between the MD curves and the fitting stress-strain curves, we use a corrective factor α applied to the plastic stress, which ranges between 1.0 and 1.1 for the quenching rates different from 10 10 K/s.This factor compensates the underestimation of the plastic stress and increases with the quenching rate, which is expected since more plastic events are present for higher quenching rates and thereby we neglect a larger number of plastic events. In this way we can obtain a very good overlap of the MD curve and the fitting curve, except for the highest quenching rate. For the smallest quenching rate, 10 10 K/s we first apply a corrective factor of 0.7 on the stress drop at 12.3 % and then we apply a factor α=1.01 to the whole plastic stress.

Influence of the metallic glass composition

The cumulative number and the cumulative plastic intensity of plastic events with shear strain for three different MG compositions are reported in for the Cu 64 Zr 36 MG, i.e., with higher Cu ratio. This is expected since such composition leads to a higher shear modulus (Figure 3.17 The average size, the average number and the average plastic intensity, as a function of the shear strain are reported in Figure 4.12 to 4.14. The former increases with shear strain up to the yield point and then fluctuates (with a larger amplitude) around a stabilized value. There are no significant differences between the different formulations, except maybe larger fluctuations with the composition with less Cu. Regarding the curves about the number of plastic events, larger fluctuations are visible above the yield point for this same composition. Conversely, below the yield points the number of plastic events decreases with the increase in the Cu content in the composition. The curves of the intensity of the plastic events of the three compositions roughly superimpose before the yield point. They show large fluctuations above the yield point, which seem also to increase with the decrease of Cu in the composition.

To sum up, with the decrease of the Cu content, more plastic events occur below the yield point (their intensity and their size are similar for all the Cu content), and above the yield point, the plastic intensity and the average size seems to fluctuate more. As we have seen in the previous Section 4.1.1, larger and more numerous plastic events, as for instance in the case of high quenching rates, seem to be correlated to less pronounced strain localization. In fact, by inspecting Additionally, we have compared the distribution of the orientation of the plastic events.

The width of the distribution plotted in Figure 4.15 is slightly larger for the lowest Cu content, suggesting again more variability of the plastic events for this composition.

Prospect for tensile tests

In order to evaluate if a change in the soliciation type strongly modifies the previous conclusion, we also investigate the tensile properties of Cu 64 Zr 36 MG after different quenching rates. In this case, in order to avoid finite size effects, we take a simulation box twice larger in the X direction than the box used for shear since the plastic events tend to be stretched in the direction of the applied tensile strain, as schematically illustrated in Fig- ure 4.17. The total number of atoms in this case is 64000. The dimensions of the box are 160 Å in the x-direction and 80 Å in the y-and z-directions. The numerical sample is cooled down to 0.1 K from the liquid state through the same different quenching rates previously explored, 10 10 K/s to 10 14 K/s and relaxed at 0.1 K for 2 ns. The deformation is performed in the X direction by steps of 0.05 %. The dimensions on the Y and Z directions are adjusted to keep the global volume of the box constant. After each strain step, the potential energy of the system is minimized by FIRE minimize style (see Chapter 2).

The obtained tensile stress-strain curves are shown in Figure 4.18. The Young's modulus, yield stress and yield strain extracted from the stress-strain curves are reported in Figure 4.19. As expected, in agreement with our previous results from shear test, and with previous work [Mayr, 2005, Xiao et al., 2010], the Young's modulus and yielding stress increases with the decrease of the quenching rate, i.e., for more relaxed material.

As seen in Figure 4.20, at the same tensile strain, the cumulative number of plastic events and the cumulative plastic intensity are larger for MG with higher quenching rates. However, we do not observe small jumps on the cumulative plastic intensity curves like those for shear deformation. This is consistent with the absence of a large stress drop (such as the one observed in shear). The Eshelby's inclusion model that we have used for shear cannot be directly applied to tensile deformation in its present form, thus we do not show the fitting of the stressstrain curves in the tensile case. In fact, we find that the plastic stress is abnormally underestimated, yielding a bad fit especially at high strains. This issue seems to be related to the high number of large events formed at each configuration that cause a bad quality displacement's fitting. A possible solution to get rid of this problem would be to change the displacement's fitting routine by fitting the displacement field due to each inclusion one at a time and not as an ensemble for a given configuration. This strategy will be certainly explored in a future work.

Discussion and conclusion

Through the study of effect of the quenching rate and composition on the plasticity of Cu-Zr MGs at the microscopic scale and the comparison of the mechanical properties, the following conclusions are obtained. Before the yield point, more and larger plastic events with higher plastic intensity are observed for Cu-Zr MGs with higher quenching rate and less Cu ratio, which are beneficial to better plasticity. At the yield point, plastic intensity increases more sharply for systems at lower quenching rate and with more Cu content, which is consistent with a stronger localization, while the increase of plastic intensity, number and size of the plastic events is smoother when increasing the quenching rate and/or decreasing the Cu content, leading to more uniform plastic deformation at the macroscopic scale. The orientation of the plastic events is more scattered for high quenching rates, suggesting a more homogeneous distribution of the plastic events, while the formation of the shear band does not seem to produce a noticeable effect on the orientation.

99

Introduction

In this chapter, we investigate the effect of thermal cycling on the global plastic behavior of Cu-Zr metallic glasses (MGs) prepared with different quenching rates. Additionally, the effect of thermal cycling on their plastic events is analyzed at the microscopic scale, in terms of number, size, plastic intensity and orientation.

Thermal cycling treatment

Thermal cycling has been introduced in Chapter 1 as an effective thermal treatment method to improve the plasticity of metallic glasses (MGs). Moreover, there have been many studies on the effects of different parameters of thermal cycling on the properties of MGs, including experimental and simulation methods [START_REF] Ketov | Rejuvenation of metallic glasses by non-affine thermal strain[END_REF], Song et al., 2018, Grell et al., 2018, Priezjev, 2019, Shang et al., 2021, Shang et al., 2018]. Thermal cycling does not affect the glass forming ability (GFA) of MGs compared to other methods of improving plasticity, such as composition modification [START_REF] Guo | Rejuvenation and plasticization of metallic glass by deep cryogenic cycling treatment[END_REF]. In contrast to shot peening, it does not alter the surface of the MG [START_REF] Concustell | Structural relaxation and rejuvenation in a metallic glass induced by shotpeening[END_REF], Nieh et al., 2012], and it is easy to be applicable across the whole MG sample.

Figure 5.1 shows the variation of temperature with time during one thermal cycle we chose to apply to our MGs. The initial quenching rate down to 0.1 K of the simulated MG is varied from 10 10 K/s to 10 14 K/s. After this quenching, the system is heated up to 400 K and then cooled down to 0.1 K for 100 times. The heating and cooling rate are the same, 1.6 • 10 14 K/s, which corresponds to a nearly instantaneous process [START_REF] Shang | Atomistic modelling of thermal-cycling rejuvenation in metallic glasses[END_REF]. During this process, the pressure remains constant. The upper temperature, 400 K is almost half of the glass transition temperature, which is below the temperatures normally associated with α and β thermal relaxations. The upper temperature, 400 K is maintained for 200 ps, and the lower temperature, 0.1 K is maintained for 20 ps. We checked that this choice of the annealing times favors the rejuvenation of the MG.

Effect of thermal cycling on metallic glasses with different initial quenching rates

Figure 5.2 shows the variation of the potential energy (PE) with the number of cycles, Cu 64 Zr 36 MGs prepared with different initial quenching rates. For the lowest initial quenching rates, such as 10 10 K/s and 10 11 K/s, the PE increases with the number of cycles, meaning that the sample undergoes some rejuvenation. For higher quenching rates, such as 10 13 K/s and 10 14 K/s, PE decreases with the number of cycles, meaning, conversely, that the sample ages. The PE variation is slight for the quenching rate of 10 12 K/s, which can be defined as a critical quenching rate.

In terms of microstructure, we already know that the lower quenching rates, the more there are <0,0,12,0> (full icosahedra, FI) inside Cu 64 Zr 36 MG. As shown in Figure 5.3, after thermal cycling, the FI fraction decreases for the quenching rates 10 10 K/s and 10 11 K/s, suggesting an easier plasticity for these systems, while for Cu 64 Zr 36 MGs prepared

EFFECT OF THERMAL CYCLING ON METALLIC GLASSES WITH DIFFERENT INITIAL QUENCHING RATES

with the highest quenching rates (10 13 K/s and 10 14 K/s), the FI fraction (which is initially lower) . For the "critical" quenching rate (10 12 K/s), the FI fraction variation is very small.

In order to understand the microscopic evolution of the system after the thermal treat-104 CHAPTER 5. EFFECT OF THERMAL CYCLING ON METALLIC GLASSES ment we calculated the non-affine square displacement D 2 min [Hufnagel, 2015] (the deviation with respect to local linear deformation) at different numbers of cycles, using the previous cycle as reference, Their 3D mapping is reported in Figure 5. Conversely, it decreases with the two highest quenching rates, i.e., the less-relaxed Cu 64 Zr 36

MGs. Therefore, we can deduce that the change in energy state is caused by the accumulated nonaffine square displacement [Priezjev, 2019]. In fact, as we can see by comparing We now focus our attention on the influence of the thermal cycling on the mechanical properties of the same samples. the thermal cycling treatment and their decrease is larger when the quenching rate is lower. For the highest quenching rate (10 13 K/s and 10 14 K/s), thermal cycling leads to an increase in the shear modulus and yield stress, with a larger increase for higher quenching rate, while the flow stress does not vary significantly.

From Figure 5.7, where we compare the atomic shear strain, it can be found that thermal cycling decreases the localization of the shear transformations (STs) for well-relaxed MG and increases the localization of the STs for the less-relaxed Cu 64 Zr 36 MG, while for Cu 64 Zr 36 MG with a critical quenching rate (10 12 K/s), the variation is not significant.

To go further, we can compare some parameters of the plastic events occurring in the less and most relaxed MGs (i.e. prepared with the quenching rate 10 10 K/s and 10 14 K/s) submitted or not to thermal cycling treatment. .9 show the average size of the plastic events as a function of the shear strain. At small strains, for the system with quenching rate 10 10 K/s, it is larger when the system has been submitted to thermal cycling, while for the system prepared with quenching rate 10 14 K/s, the opposite is observed. After the yield strain, given the strong oscillations on the curves, the difference between them cannot be commented. of the shear strain for Cu 64 Zr 36 MG with initial quenching rates of 10 10 K/s and 10 14 K/s without and with thermal cycling. For the quenching rate of 10 10 K/s, curves are very similar at small strains (below the yield strain), whereas the plastic intensity is smaller after thermal cycling for sample prepared with the quenching rate of 10 14 K/s. for Cu 64 Zr 36 MG quenched at 10 14 K/s. To summarize these findings, thermal cycling has a rejuvenation effect for the lowest quenching rates (10 10 K/s and 10 11 K/s) and an aging effect for the highest ones (10 13 K/s and 10 14 K/s). The rejuvenation effect seen in the most relaxed system results in plastic events slightly more numerous, with a larger average size (especially at small strains), a slightly more uniform distribution of their orientation. These are all beneficial to a more uniform deformation of MGs, as shown by the atomic shear strain distribution in Figure 5.7. Conversely, the aging effect of thermal cycling seen in the less relaxed system leads to smaller average size, smaller number, smaller plastic intensity and slightly less uniform orientation distribution of plastic events, consistent with a more localized deformation at the macroscopic scale.

Effect of thermal cycling for different metallic glass composition

The quenching rate is a key parameter in thermal cycling treatment. However, there is a lack of research on the effect of cryothermal cycling on the mechanical properties of MGs with different compositions and its analysis at the microscopic scale. In this section, the quenching rate is fixed at 10 11 K/s, and the effects of thermal cycling on the mechanical properties of Cu-Zr MGs with different compositions, Cu 64 Zr 36 , Cu 50 Zr 50 and Cu 36 Zr 64 , are compared firstly and then analyzed at the microscopic scale.

The variation of the potential energy (PE), δE is defined as δE = E(i) -E(0), the difference between the current potential energy and the initial one. δE as a function of the number of thermal cycles is shown in Figure 5.15. For the three compositions, the PE increases after thermal cycling, indicating significant of a rejuvenation effect. The increase is larger for Cu 36 Zr 64 than for the other two, which show a very similar curve. The rejuvenation effect of the thermal cycling is therefore more important for the Cu 36 Zr 64 , which contains less Cu. Additionally, we visualize in 3D the atoms with non-affine square displacement larger than 100 Å 2 for the three compositions, submitted to different numbers of thermal cycling (Figure 5.16). This mapping shows that the increase in the number of The stress-strain curves obtained from quasi-static shear deformation at 0.1 K with samples previously submitted or not to a complete thermal cycling (100 cycles) are shown in Figure 5.17. Furthermore, the deduced shear modulus, yielding stress and flow stress are reported in Figure 5.18. All of them decrease after thermal cycling except for the flow stress for which the decreasing trend is not obvious (partly due to the large uncertainty for its value). Thus, thermal cycling of the different Cu-Zr MGs compositions is responsible for a softening effect that is helpful for their plastic deformation. ) at the 1st, 5th, 9th and 99th cycle, which the previous cycle is the reference for the three studied compositions.

Figure 5.17: Stress-strain curves in quasi static shear for the three compositions previously submitted or not to the complete thermal cycling. Moreover, we compared the effect of thermal cycling on the difference between yield and flow stresses, ∆τ = σ yσ f . As reported in Table 5.1, after thermal cycling, the stress drops compositions are smaller for the three tested composition. This corresponds to less localization of the STZs [START_REF] Feng | A molecular dynamics analysis of internal friction effects on the plasticity of zr65cu35 metallic glass[END_REF], as shown in Figure 5.19. We also compared the relative decrease in the shear modulus (γ G ), in the yielding strain ( γ σ y ) and in the difference between yielding and flow stress (γ ∆τ ) for the three compositions after thermal cycling. As seen in Table 5.2 all are larger for Cu 36 Zr 64 , which further confirms that the rejuvenation effect of thermal cycling is more important for this MG. The orientation distributions (more exactly their gaussian fit) of the plastic events oc-

EFFECT OF THERMAL CYCLING FOR DIFFERENT METALLIC GLASS COMPOSITION

curring over the whole shear test (strain up to 30 %) are reported for each composition (submitted or not to thermal treatment) on Figure 5.24 and the width of these distribution are reported in Table 5.3. The effect of thermal cycling on these distributions is very weak. The one for Cu 36 Zr 64 is very slightly more uniform after thermal cycling, maybe consistent with the less localized distribution of the plastic events after thermal cycling for this system.

Discussion and conclusion

In this chapter, we analyze the effect of thermal cycling treatment on different compositions of Cu-Zr metallic glasses (MGs) or on one composition initially prepared with different quenching rates. Thermal cycling increases the potential energy (PE) and atomic mobility of Cu 64 Zr 36 MG prepared with quenching rates of 10 10 K/s and 10 11 K/s and do the opposite for quenching rates of 10 13 K/s and 10 14 K/s. For Cu 64 Zr 36 MG quenched at 10 12 K/s, the difference is slight, and the thermal cycling has almost no consequences.

Thus, we can conclude that the thermal treatment studied has a rejuvenation effect on the most relaxed Cu 64 Zr 36 MGs and an aging effect on the less-relaxed ones. During quasistatic shear deformation, this results in a decrease in the shear modulus, yielding stress and stress drop for the most relaxed MGs, and the opposite for the less relaxed materials.

For the lowest quenching rate 10 10 K/s (for which the rejuvenation effect is the largest) the average size, number and plastic intensity of the plastic events occurring during shear (up to the yield strain) increase after thermal cycling, and they are more uniform. Thus, thermal cycling improves the deformation homogeneity. On the opposite, for the largest quenching rate (for which the aging effect is the largest) , the opposite is observed, and as a result the deformation is more localized. Regarding the effect of thermal cycling on different compositions of initially quenched at 10 11 K/s, we found that the increase of PE due to the thermal cycling is larger for the composition with the largest Cu content.

Conclusions and Perspectives Conclusion

In this thesis, we investigated the factors affecting the plasticity of Cu-Zr binary metallic glasses (MGs) through molecular dynamics (MD) simulations, explored the parameters which can improve the plasticity of MGs, and analyzed their effect at the microscopic scale to establish the link between microscopic plasticity and mechanical properties of MGs.

By studying the effects of quenching rate, compositions, strain rate, and temperature on the plasticity of MGs, we obtained the following conclusions: In terms of microstructure, higher quenching rates, less Cu content in Cu-Zr binary MGs and higher temperatures can reduce the atomic packing density and the initial fraction of <0,0,12,0> icosahedra (full-icosahedra, FI) of MGs. Concerning mechanical properties, Cu-Zr binary MGs with higher quenching rate, less Cu content, and higher temperature leads to relatively smaller shear modulus, yield stress, and stress drop after quasi-static shear deformation. Furthermore, their FI drops are less pronounced during the deformation. By comparing the local atomic shear strain maps, their deformation is less localized. Hence, a higher quenching rate, less Cu content, and higher temperature are beneficial for the plasticity of MGs. The strain rate effect on the microstructure and mechanical properties of MGs is slight at low temperatures, while its effect on the mechanical properties of MGs is more pronounced at temperatures close to the glass transition temperature (T g ). At higher strain rates, the deformation of MGs is slightly less localized.

We also analyzed the effects of quenching rate and composition on the number, intensity, and average size of plastic events for Cu-Zr binary MGs at the microscopic scale and ob-

Conclusions and Perspectives

tained the following conclusions:

More and larger plastic events with higher plastic intensity are activated before the yield point for MGs prepared with higher quenching rates. This also leads to smaller shear moduli. When the yield point is reached, there is a significant increase in the plastic intensity of plastic events for MGs prepared with a low quenching rate, while for higher quenching rates, the increase is less pronounced, corresponding to less localized deformation. In addition, the plastic events are distributed more uniformly for MGs with higher quenching rates.

Moreover, with the decrease of the Cu content, more plastic events occur below the yield point (their intensity and their size are similar for all the Cu content), and above the yield point, the plastic intensity and the average size seems to fluctuate more.

We also investigated the effect of thermal cycling below T g on MGs prepared with different quenching rates and compositions, in particular we analyzed the plastic events occurring during their deformation.

Thermal cycling has a rejuvenation effect for MGs with low quenching rates, and when the quenching rate is lower, the rejuvenation effect is more significant. This rejuvenation causes an increase in the number, plastic intensity, and average size of plastic events.

Conversely, for MGs prepared with higher quenching rates, there is an aging effect, and this one is more pronounced when the quenching rate is higher. Indeed, in this case, the number, average size and plastic intensity of the plastic events decrease.

Finally , the rejuvenation effect obtained for systems prepared with low quenching rate is more important for the lowest Cu content.

In terms of the relationship between the characteristics of microscopic plastic events and macroscopic plasticity, we can also conclude that when the number, plastic intensity, and average size of plastic events are larger and their distribution is more uniform, MGs experience less localization and more uniform plastic deformation at the macroscopic scale.

Perspectives

Our analysis of microscale plastic events is currently available only for quasi-static shear and tensile deformation. We have attempted a reconstruction of the stress-strain curve

Conclusions and Perspectives

for quasi-static tensile deformation using Eshelby's inclusion model. However, this reconstruction is not correct. All the atomic displacements of plastic events being fitted simultaneously at each strain step a large number of plastic events with very high plastic intensity are detected during tensile deformation, resulting in the interaction of plastic events and a poor quality of the atomic displacement fitting. We will try to optimize the fitting of the stress-strain curve by adjusting the relevant fitting parameters.

In addition, we have not yet analyzed the plastic events of MGs after dynamic deformation. This would enable a better understanding of the influence of the strain rate. Moreover, by dynamically deforming the system in the forward and in the backward directions it will be possible to separate the plastic component of the stress from the an elastic part (i.e., the reversible component of the stress) and estimate thus the contribution of these two components. Eventually, we could also compare our results on the characterization of the plastic events to analytical models of visco-elasto-plasticity in amorphous materials, such as the quasi-point defect model of Perez et al., and make some conclusions about the validity of this kind of modelling which is based on physical assumptions not easily ascertainable.

As for the cryogenic treatment to improve plasticity, it would be interesting to perform a more systematic study on the influence of the temperature and of the annealing time on the degree of rejuvenation of the system. Moreover, the link between the local thermal heterogeneities and the internal stresses causing irreversible local atomic rearrangements, and thus enhanced mobility, has to be further explored. 128 BIBLIOGRAPHY [START_REF] He | Effect of ta on glass formation, thermal stability and mechanical properties of a zr52. 25cu28. 5ni4. 75al9. 5ta5 bulk metallic glass[END_REF][START_REF] He | Effect of ta on glass formation, thermal stability and mechanical properties of a zr52. 25cu28. 5ni4. 75al9. 5ta5 bulk metallic glass[END_REF]. Difference in compressive and tensile fracture mechanisms of zr59cu20al10ni8ti3 bulk metallic glass. Acta Materialia, 51(4):1167-1179.

[ Zhao et al., 2010a] Zhao, J., Qu, R., Wu, F., Li, S., and Zhang, Z. (2010a)

  . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.8 Improvement of plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.8.1 Composite metallic glasses . . . . . . . . . . . . . . . . . . . . . . . 33 1.8.2 Pre-deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.8.3 Thermal pre-treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . 38 2.1.1 Newton's equation and algorithms . . . . . . . . . . . . . . . . . . . 38 2.1.2 Interatomic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.3 Ensemble of molecular dynamics simulation for an equilibrium system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.4 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . 44 2.2 Structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.1 Radial distribution function . . . . . . . . . . . . . . . . . . . . . . 45 2.2.2 Voronoi polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2.3 Local atomic shear strain . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Construction of model and deformation . . . . . . . . . . . . . . . . . . . . 47 2.4 Analysis and mapping of plastic events . . . . . . . . . . . . . . . . . . . . 48 2.4.1 Definition of plastic events . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.2 Selection of attractors . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4.3 Rearrangement of plastic events . . . . . . . . . . . . . . . . . . . . 52 2.4.4 Atomic displacement fitting . . . . . . . . . . . . . . . . . . . . . . . 52 2.4.5 Fitting of the stress-strain curve . . . . . . . . . . . . . . . . . . . . . 54 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Effect of quenching rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Effect of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ii List of Figures 1.1 Volume change as a function of temperature from liquid to solid state (the dashed red line represents the equilibrium liquid) [Ritter, 2012] . . . . . . 1.2 Microscope diagram of the atomic arrangement of crystalline and amorphous alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  (a) Compressive strain-stress curve (b) Compression fracture morphology characteristics (c) Tensile strain-stress curve (b) Tensile fracture morphology characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Application of MG. (a) Golf stick (b) Armor-piercing projectile (c) Amorphous transformer core (d) Amorphous gastroscope . . . . . . . . . . . . . 1.6 The illustration of radial distribution function [Weihua, 2013]. . . . . . . . 1.7 <0,0,12,0> icosahedron in Cu 64 Zr 36 metallic glass. . . . . . . . . . . . . . . 1.8 Representative atomic clusters for monatomic hafnium [Şengül and Çeltek, 2018]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 SRO clusters connected by (a) VS, (b) ES, (c) FS, and (d) IS in the Cu 50 Zr 50 metallic glass (Zr and Cu atoms are represented with blue and brown balls, respectively, and centered Zr and Cu atoms are represented using green and red balls, respectively) [Li et al., 2014]. . . . . . . . . . . . . . . . . . . 1.10 Free volume model in the flow of metallic glass [Spaepen, 1977]. (a) Single atomic transition (b) Free volume generation . . . . . . . . . . . . . . . . . iv LIST OF FIGURES 1.11 The schematic of the relationship between α relaxation and β relaxation on potential energy landscape (PEL) [Ritter, 2012]. . . . . . . . . . . . . . . . . 1.12 The schematic diagram of the temperature gradient of SB [Shimizu et al., 2006]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Diagram of formation of SB in the MG under compression loading(v p is propagation velocity of structural disorder and v s is shear velocity) [Greer et al., 2013]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Shear band of Zr-based metallic glass (a) High-resolution TEM image and (b) nanoscale voids within it [Li et al., 2002a]. . . . . . . . . . . . . . . . . . 1.15 Evolution of free volume and temperature during the shear localization process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Scheme of inclusion's removal from the matrix and of its unconstrained transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Applying force on the inclusion and putting it back in the matrix. . . . . . 1.18 Releasing the surface traction of the inclusion. . . . . . . . . . . . . . . . . 1.19 Comparison of shear stress-strain curves for amorphous silicon with Still-

  Filled circles to open circles mean the movement of atoms after displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Schematic of atomic displacement calculation with and without minimum image convention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The schematic diagram of forward deformation and reverse deformation. v LIST OF FIGURES 2.4 2D projection of density of PE of amorphous silicon over the basins on xyplane at shear strain 1.6 % (40 Å<z<50 Å)[START_REF] Fusco | Role of local order in the small-scale plasticity of model amorphous materials[END_REF]. . . . . . . . . 49 2.5 Map of plastic energy, PE(r) surrounding a plastic event (the intensity of plastic energy from low to high) corresponding to the color from blue to red[START_REF] Fusco | Role of local order in the small-scale plasticity of model amorphous materials[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6 Radial distribution of plastic energy PE(r) around the attractor and exponential fit curve [Albaret et al., 2016]. . . . . . . . . . . . . . . . . . . . . . . 51 2.7 The distribution of nonaffine atomic displacements between two successive steps from MD simulation (a) and with Eshelby's inclusion model of Cu 64 Zr 36 MG at strain 15 % during quasi-static shear deformation (b). Both images are replicated three times. . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1 Volume as a function of temperature (V-T) of Cu 64 Zr 36 MGs under different quenching rates (a) and the figure with shifted up curves (b). . . . . . 58 3.2 Radial distribution function (RDF) of Cu 64 Zr 36 MG at 0.1 K under different quenching rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 The first peak (a) and the second peak (b) on the RDF curves of Cu 64 Zr 36 MG at 0.1 K under different quenching rates. . . . . . . . . . . . . . . . . . 60 3.4 The first peaks on Cu-Cu partial RDF curves (a), the first peaks on Cu-Zr partial RDF curves (b), and the first peaks on Zr-Zr partial RDF curves (c) of Cu 64 Zr 36 MG with different quenching rates. . . . . . . . . . . . . . . . 61 3.5 The first peak on RDF and partial RDF of Cu 36 Zr 64 MG after quenching with a quenching rate of 10 10 K/s at 0.1 K. . . . . . . . . . . . . . . . . . . 61 3.6 Shear stress-strain curves of Cu 64 Zr 36 MG after quasi-static deformation with different quenching rates at 0.1 K. . . . . . . . . . . . . . . . . . . . . . 62 3.7 Shear modulus (a), yield stress and strain (b), and flow stress (c) for Cu 64 Zr 36 prepared with different quenching rates. . . . . . . . . . . . . . . . . . . . . 62 3.8 Map of atomic shear strain in Cu 64 Zr 36 MG prepared with a quenching rate of 10 10 K/s, sheared at 12.3 % strain (a) and 12.4 % (b), showing the atoms with shear strain larger than 60 % of the maximum atomic shear strain over the whole sample. . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vi LIST OF FIGURES 3.9 The atomic shear strain in Cu 64 Zr 36 MG prepared with different quenching rates at different shear strains, at 0.1 K. . . . . . . . . . . . . . . . . . . . . . 3.10 Fraction of FI of Cu 64 Zr 36 prepared with different quenching rates as a function of shear strain at 0.1 K. . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Fraction of FI in SB region of Cu 64 Zr 36 prepared with the quenching rate 10 10 K/s as a function of shear strain. . . . . . . . . . . . . . . . . . . . . . 3.12 Volume as a function of the temperature for the three different compositions quenched with a quenching rate of 10 11 K/s. Dotted lines are to determine the glass transition temperature. . . . . . . . . . . . . . . . . . . 3.13 The total radial distribution function (RDF) for three compositions of binary Cu-Zr MGs quenched at 0.1 K at a quenching rate of 10 11 K/s (a) and the magnification of the first peak (b). . . . . . . . . . . . . . . . . . . . . . 3.14 Cu-Cu (a) and Zr-Zr (b) partial radial distribution function (PRDF) for three compositions of binary Zr-Cu MGs quenched at 0.1 K at a quenching rate of 10 11 K/s. The insets in the Figures show a magnification of the first peaks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 The insets in the figures show a magnification of the first peaks. Magnification of the second peaks of the total RDF curves shown in Figure 3.13 (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Shear stress-strain curves of three compositions quenched at 0.1 K after quasi-static deformation at a quenching rate of 10 11 K/s. . . . . . . . . . . 3.17 Shear modulus (a), yield stress (b), yield strain (c) and flow stress (d) of three compositions quenched at 0.1 K at a quenching rate of 10 11 K/s. . . . 3.18 Normalized shear stress-strain curves by the shear modulus (a) and shear modulus vs. yield stress (b) of three compositions quenched at 0.1 K at a quenching rate of 10 11 K/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Map of atomic shear strain for the three compositions at strain 10.6 % (a), 10.7 % (b), 12.4 % (c), 15 % (d), and 20 % (e) at 0.1 K for a quenching rate of 10 11 K/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Comparison of difference between yield and flow stress of three compositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES 3.21 Comparison of fraction of FI with shear strain for the three compositions. 72 3.22 Map of central atoms of FI for the three compositions (brown balls are Cu atoms, and green balls are Zr atoms). . . . . . . . . . . . . . . . . . . . . . . 72 3.23 Shear stress-strain curves with different strain rates for of Cu 64 Zr 36 MG prepared with a 10 11 K/s quenching rate at 0.1 K. . . . . . . . . . . . . . . 73 3.24 Shear modulus (a) and yield and flow stress (b) as a function of the different strain rates for Cu 64 Zr 36 MG quenched at 0.1 K, at a quenching rate of 10 11 K/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.25 Distribution of atomic shear strain of Cu 64 Zr 36 MGs with different strain rates, at strain 10 %, 12.5 %, 15 %, and 20 %, at 0.1 K. . . . . . . . . . . . . . 74 3.26 Comparison of stress-strain curves of Cu 64 Zr 36 MGs sheared with different strain rates, at three different temperatures: (a) 0.1 K, (b) 50 K, (c) 200 K. . 75 3.27 Map of atomic shear strain of Cu 64 Zr 36 MG at strain 30 % with strain rate of 10 7 s -1 , 10 8 s -1 and 10 9 s -1 when the temperature is 50 K and 200 K. . 75 3.28 The total RDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c). . . . 76 3.29 The Cu-Cu PRDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c). . 76 3.30 The Cu-Zr PRDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c). . 77 3.31 The Zr-Zr PRDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c). . 77 3.32 Shear stress-strain curves (a) and tensile stress-strain curves (b) of Cu 64 Zr 36 MG at different temperatures and strain rate 10 8 s -1 . . . . . . . . . . . . . 77 3.33 Shear modulus (a) and Young's modulus (b) of Cu 64 Zr 36 MG deduced form shear tests and tensile tests at different temperatures and strain rate 10 8 s -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.34 Distribution of atomic shear strain at 20 % strain at different temperatures and strain rate 10 8 s -1 (scale doubled in the y-direction to better visualize). 78 3.35 Fraction of FI of Cu 64 Zr 36 MG quenched at different temperatures and sheared at the same temperature with a 10 8 s -1 shear strain rate. . . . . . . 78 viii LIST OF FIGURES 4.1 Cumulative numbers of plastic events with strain for Cu 64 Zr 36 MG with different quenching rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cumulative plastic intensity of plastic events for different quenching rates. 4.3 Shear stress and cumulative plastic intensity of plastic events for different quenching rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Numbers of plastic events as a function of the shear strain, for Cu 64 Zr 36 MG, for different quenching rates (a) and comparison between quenching rates 10 10 K/s and 10 14 K/s after smoothing with SG filter method (b). . . 4.5 Plastic intensity of plastic events for the different quenching rates (a) and comparison between quenching rates 10 10 K/s and 10 14 K/s after smoothing with SG filter method (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Comparison of the average size of plastic events with strain for different quenching rates (a) and comparison between quenching rates 10 10 K/s and 10 14 K/s after smoothing with SG filter method (b). . . . . . . . . . . . . . 4.7 Distribution of orientation of all the plastic events occurring during a shear test of Cu 64 Zr 36 MG quenched at different quenching rates. . . . . . . . . . 4.8 Comparing the distribution of orientation of plastic events for Cu 64 Zr 36

4. 10

 10 Figure 5.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.19 Map of the atomic shear strain in the three compositions previously submitted or not to thermal cycling, at 20 % shear strain. . . . . . . . . . . . . 114 5.20 Cumulative number of plastic events for each compositions submitted or not thermal cycling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.21 Cumulative plastic intensity for each composition submitted or not to thermal cycling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.22 Number of plastic events as a function of the shear strain, for each compositions submitted or not to thermal cycling (original curves (a)-(c) and smoothed curves (d)-(f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.23 Plastic intensity of plastic events as a function of the shear strain for each composition before and after thermal cycling (original curves (a)-(c) and smoothed ones (d)-(f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.24 Orientation distribution (gaussian fit) of the plastics events occurring over the whole shearing test for each composition before and after thermal cycling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 1 . 2 :

 12 Figure 1.2: Microscope diagram of the atomic arrangement of crystalline and amorphous alloy.

Figure 1 . 4 :

 14 Figure 1.4: Deformation of Zr-based MG in experiment [Zhang et al., 2003]). (a) Compressive strain-stress curve (b) Compression fracture morphology characteristics (c) Tensile strain-stress curve (b) Tensile fracture morphology characteristics

Figure 1 . 5 :

 15 Figure 1.5: Application of MG. (a) Golf stick (b) Armor-piercing projectile (c) Amorphous transformer core (d) Amorphous gastroscope

Figure 1 . 7 :

 17 Figure 1.7: <0,0,12,0> icosahedron in Cu 64 Zr 36 metallic glass.

Figure 1 . 8 :

 18 Figure 1.8: Representative atomic clusters for monatomic hafnium [ Şengül and Çeltek, 2018].

  SRO icosahedra structures as central clusters extend spatially in hexagonal close packing (hcp) or face-centered cubic (fcc) configurations forming MRO structures with vertexsharing (VS), edge-sharing (ES),face-sharing (FS), and intercross-shared (IS) connections, as shown in Figure 1.9 (SRO clusters are shown by dashed red circles in the figure). SRO clusters are remarked with dashed red circles. The connections between SRO clusters allow to form chains, loops, or even networks [Sheng et al., 2006].

Figure 1 . 9 :

 19 Figure 1.9: SRO clusters connected by (a) VS, (b) ES, (c) FS, and (d) IS in the Cu 50 Zr 50 metallic glass (Zr and Cu atoms are represented with blue and brown balls, respectively, and centered Zr and Cu atoms are represented using green and red balls, respectively) [Li et al., 2014].

  1988, Falk and Langer proposed the shear transformation zone (STZ) model of plasticity of MGs by extending the theory of Argon and Spaepen on the basis of Molecular Dynamics (MD) simulation of viscoplasticity of MGs [Falk and Langer, 1998]. How does individual STZ develop to macroscopic flow? Johnson and Samwer issued the cooperative shear model (CSM) for the plastic flow of MGs[START_REF] Johnson | A universal criterion for plastic yielding of metallic glasses with a (t/t g) 2/3 temperature dependence[END_REF]. They proposed that STZ and macroscopic flow are related respectively to β relaxation and α relaxation, as shown in Figure1.11[START_REF] Harmon | Anelastic to plastic transition in metallic glass-forming liquids[END_REF]. Yielding occurs when a critical fraction of STZs leads to global instability.

Figure 1 . 11 :

 111 Figure 1.11: The schematic of the relationship between α relaxation and β relaxation on potential energy landscape (PEL) [Ritter, 2012].

  Figure1.12: The schematic diagram of the temperature gradient of SB[START_REF] Shimizu | Yield point of metallic glass[END_REF].

Figure 1 . 13 :

 113 Figure 1.13: Diagram of formation of SB in the MG under compression loading(v p is propagation velocity of structural disorder and v s is shear velocity) [Greer et al., 2013].

  Figure 1.14: Shear band of Zr-based metallic glass (a) High-resolution TEM image and (b)nanoscale voids within it[Li et al., 2002a].

Figure 1 . 15 :

 115 Figure 1.15: Evolution of free volume and temperature during the shear localization process.

  Figure 1.19: Comparison of shear stress-strain curves for amorphous silicon with Stillinger andWeber (SW) potential[START_REF] Stillinger | Computer simulation of local order in condensed phases of silicon[END_REF] from MD simulations and with Eshleby's inclusion model. The black curve is the result of the MD simulation. Red, dashed blue and orange curves are obtained with Eshelby's inclusion model[START_REF] Albaret | Mapping between atomistic simulations and eshelby inclusions in the shear deformation of an amorphous silicon model[END_REF].

CHAPTER 2 .

 2 METHODOLOGY AND MODELING ensemble.Micro canonical ensemble, also called NVE ensemble, is an isolated, conservative statistical ensemble of systems. It keeps the total number of atoms N, volume V and energy E all constant. Canonical ensemble, also called NVT ensemble, keeps the number of atoms N, volume V and temperature T constant and total kinetic energy zero. Isothermal isobaric ensemble, which is NPT ensemble, keeps the number of atoms N, pressure P, and temperature T constant. The temperature and pressure are controlled as explained later.

  can extend periodically. Meanwhile, atoms around the left boundary and right boundary, atoms around the top and bottom boundary, and atoms around the front boundary and back boundary all have interaction, which can cause atoms to go through boundaries, as shown in Figure 2.1. In this thesis, we calculate the atomic displacement using minimum image convention, which is illustrated in Figure 2.2. When an atom moves from the position of light blue circle to the dark blue one, we can get two atomic displacement vectors due to periodic boundary condition: when we use minimum image convention, we just calculate the smallest one for the atomic displacement.

Figure 2 .Figure 2 . 2 :

 222 Figure 2.1: 2-dimensional schematic diagram of periodic boundary conditions[Stepanov, 2013].Filled circles to open circles mean the movement of atoms after displacement.

  though the RDF can give an idea of the distribution of atoms, it cannot describe the geometric diagram and characteristics of atomic clusters. Therefore, Voronoi polyhedra are used to describe the local structure of MGs. They have been already introduced In Section 1.4. Voronoi polyhedral volume can be used to study the local packing density, and the fraction variation of Voronoi polyhedron is a key parameter during MG's deformation. These two parameters can be calculated by the software Open Visualization

Figure 2 . 3 :

 23 Figure 2.3: The schematic diagram of forward deformation and reverse deformation.

Figure 2 . 4 :

 24 Figure 2.4: 2D projection of density of PE of amorphous silicon over the basins on xy plane at shear strain 1.6 % (40 Å<z<50 Å) [Fusco et al., 2010].

Figure 2 . 6 :

 26 Figure 2.6: Radial distribution of plastic energy PE(r) around the attractor and exponential fit curve [Albaret et al., 2016].

Figure 2 . 7 :

 27 Figure 2.7: The distribution of nonaffine atomic displacements between two successive steps from MD simulation (a) and with Eshelby's inclusion model of Cu 64 Zr 36 MG at strain 15 % during quasi-static shear deformation (b). Both images are replicated three times.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effect of quenching rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Effect of strain rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Effect of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3 . 1 :

 31 Figure 3.1: Volume as a function of temperature (V-T) of Cu 64 Zr 36 MGs under different quenching rates (a) and the figure with shifted up curves (b).

Figure 3 .

 3 Figure 3.1 shows the change of volume change of the sample as a function of temperature from the liquid state at 2000 K, to the solid glass state at very low temperatures. From Figure 3.1, it can be seen that below the inflection points, the volume per atom is larger

3. 2 .

 2 Figure 3.2: Radial distribution function (RDF) of Cu 64 Zr 36 MG at 0.1 K under different quenching rates.

Figure 3 . 3 :

 33 Figure 3.3: The first peak (a) and the second peak (b) on the RDF curves of Cu 64 Zr 36 MG at 0.1 K under different quenching rates.

Figure 3 . 4 :

 34 Figure 3.4: The first peaks on Cu-Cu partial RDF curves (a), the first peaks on Cu-Zr partial RDF curves (b), and the first peaks on Zr-Zr partial RDF curves (c) of Cu 64 Zr 36 MG with different quenching rates.

Figure 3 . 5 :

 35 Figure 3.5: The first peak on RDF and partial RDF of Cu 36 Zr 64 MG after quenching with a quenching rate of 10 10 K/s at 0.1 K.

Figure

  shows that except the quenching rate of 10 11 K/s, the flow stress slightly decreases with the quenching rate (even though the data are affected by large error bars due to the small signal to noise ratio, characteristic of the serrated flow in the stress-strain curves). This slight decreasing trend has been also observed with amorphous silicon by Fusco et al.[START_REF] Fusco | Role of local order in the small-scale plasticity of model amorphous materials[END_REF] and in Cu 46 Zr 47 Al 7 MG byCheng et al. (in this study, the flow stress is the average stress of the last 60 % strain)[Cheng et al., 2008a].

Figure 3 . 6 :

 36 Figure 3.6: Shear stress-strain curves of Cu 64 Zr 36 MG after quasi-static deformation with different quenching rates at 0.1 K.

Figure 3 . 7 :

 37 Figure 3.7: Shear modulus (a), yield stress and strain (b), and flow stress (c) for Cu 64 Zr 36prepared with different quenching rates.

Figure 3 .

 3 Figure3.11. From strain 12.3 % to 12.4 %, the fraction of FI in SB region decreases significantly, from 2.23 % to 1.43 % whereas the total fraction of FI in the system decreases only by 0.01 % (as shown in Figure3.10). In other words, the FI fraction decrease is mainly

Figure 3 . 10 :

 310 Figure 3.10: Fraction of FI of Cu 64 Zr 36 prepared with different quenching rates as a function of shear strain at 0.1 K.

Figure 3 .

 3 Figure 3.11: Fraction of FI in SB region of Cu 64 Zr 36 prepared with the quenching rate 10 10 K/s as a function of shear strain.

Figure 3 . 12 :

 312 Figure 3.12: Volume as a function of the temperature for the three different compositions quenched with a quenching rate of 10 11 K/s. Dotted lines are to determine the glass transition temperature.

  Figure 3.13: The total radial distribution function (RDF) for three compositions of binary Cu-Zr MGs quenched at 0.1 K at a quenching rate of 10 11 K/s (a) and the magnification of the first peak (b).

Figure 3 .

 3 Figure 3.14: Cu-Cu (a) and Zr-Zr (b) partial radial distribution function (PRDF) for three compositions of binary Zr-Cu MGs quenched at 0.1 K at a quenching rate of 10 11 K/s. The insets in the Figures show a magnification of the first peaks.

Figure 3 . 16 :

 316 Figure 3.16: Shear stress-strain curves of three compositions quenched at 0.1 K after quasi-static deformation at a quenching rate of 10 11 K/s.

Figure 3 .

 3 Figure 3.17: Shear modulus (a), yield stress (b), yield strain (c) and flow stress (d) of three compositions quenched at 0.1 K at a quenching rate of 10 11 K/s.

  Figure 3.18: Normalized shear stress-strain curves by the shear modulus (a) and shear modulus vs. yield stress (b) of three compositions quenched at 0.1 K at a quenching rate of 10 11 K/s.

Figure 3 . 19 :

 319 Figure 3.19: Map of atomic shear strain for the three compositions at strain 10.6 % (a), 10.7 % (b), 12.4 % (c), 15 % (d), and 20 % (e) at 0.1 K for a quenching rate of 10 11 K/s.

Figure 3 .

 3 Figure 3.20: Comparison of difference between yield and flow stress of three compositions.

Figure 3 . 22 :

 322 Figure 3.22: Map of central atoms of FI for the three compositions (brown balls are Cu atoms, and green balls are Zr atoms).

  Figure 3.23: Shear stress-strain curves with different strain rates for of Cu 64 Zr 36 MG prepared with a 10 11 K/s quenching rate at 0.1 K.

Figure 3 . 25 :

 325 Figure 3.25: Distribution of atomic shear strain of Cu 64 Zr 36 MGs with different strain rates, at strain 10 %, 12.5 %, 15 %, and 20 %, at 0.1 K.

CHAPTER 3 .

 3 Figure 3.26: Comparison of stress-strain curves of Cu 64 Zr 36 MGs sheared with different strain rates, at three different temperatures: (a) 0.1 K, (b) 50 K, (c) 200 K.

Figure 3 . 27 :

 327 Figure3.27: Map of atomic shear strain of Cu 64 Zr 36 MG at strain 30 % with strain rate of 10 7 s -1 , 10 8 s -1 and 10 9 s -1 when the temperature is 50 K and 200 K.

  Figure 3.28 to Figure 3.31 show a higher amplitude of the first peak and of the second peak of the total and partial radial distribution function (RDF) when the MG is quenched at a lower temperature [Wen et al., 2021], indicating a higher local density.

Figure 3 . 28 :

 328 Figure 3.28: The total RDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c).

Figure 3 .

 3 Figure 3.29: The Cu-Cu PRDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c).

Figure 3 .

 3 Figure 3.30: The Cu-Zr PRDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c).

Figure 3 .

 3 Figure 3.31: The Zr-Zr PRDF of Cu 64 Zr 36 MG at different temperatures for quenching rate 10 11 K/s (a), magnified first peak (b) and magnified second peak (c).

Figure 3 . 32 :

 332 Figure 3.32: Shear stress-strain curves (a) and tensile stress-strain curves (b) of Cu 64 Zr 36 MG at different temperatures and strain rate 10 8 s -1 .

  Figure 3.35 presents the evolution of the FI fraction as a function of the shear strain, for test at different temperatures. The initial value of the FI fraction decreases with the increase in the temperature, in a non linear way (the decrease accelerates). The final FI fraction, after shearing also 3.5. EFFECT OF TEMPERATURE decreases, though less strongly. Thus the decrease is sharper with the smallest temperature. This is consistent with a less homogeneous deformation, as shown by the map of atomic shear strain in Figure 3.34.

Figure 3 .

 3 Figure 3.33: Shear modulus (a) and Young's modulus (b) of Cu 64 Zr 36 MG deduced form shear tests and tensile tests at different temperatures and strain rate 10 8 s -1 .

Figure 3 . 34 :

 334 Figure 3.34: Distribution of atomic shear strain at 20 % strain at different temperatures andstrain rate 10 8 s -1 (scale doubled in the y-direction to better visualize).

Figure 3 .

 3 Figure 3.35: Fraction of FI of Cu 64 Zr 36 MG quenched at different temperatures and sheared at the same temperature with a 10 8 s -1 shear strain rate.

Figure 4 . 1 :

 41 Figure 4.1: Cumulative numbers of plastic events with strain for Cu 64 Zr 36 MG with different quenching rates.

Figure 4 . 2 :

 42 Figure 4.2: Cumulative plastic intensity of plastic events for different quenching rates.

4. 2 .

 2 Figure 4.3: Shear stress and cumulative plastic intensity of plastic events for different quenching rates.

Figure 4 . 4 :

 44 Figure 4.4: Numbers of plastic events as a function of the shear strain, for Cu 64 Zr 36 MG, for different quenching rates (a) and comparison between quenching rates 10 10 K/s and 10 14 K/s after smoothing with SG filter method (b).

Figure 4 . 5 :

 45 Figure 4.5: Plastic intensity of plastic events for the different quenching rates (a) and comparison between quenching rates 10 10 K/s and 10 14 K/s after smoothing with SG filter method (b).

Figure 4 . 6 :

 46 Figure 4.6: Comparison of the average size of plastic events with strain for different quenching rates (a) and comparison between quenching rates 10 10 K/s and 10 14 K/s after smoothing with SG filter method (b).

Figure 4 . 7 :

 47 Figure 4.7: Distribution of orientation of all the plastic events occurring during a shear test of Cu 64 Zr 36 MG quenched at different quenching rates.

Figure 4 . 8 :

 48 Figure 4.8: Comparing the distribution of orientation of plastic events for Cu 64 Zr 36 MG with quenching rate 10 10 K/s before and after the formation of SB.

  Figure 4.10: Cumulative number (a) and plastic intensity (b) of plastic events as a function of the shear strain for the three compositions.

  (a)). The jump on the cumulative plastic intensity curve of the Cu 64 Zr 36 seems to correspond to the stress drop on the stress-strain curve ( Figure 4.11).

  Figure 4.12: Average size of plastic events as a function of the shear strain for the three compositions (a) and the curves after smoothed with SG filter method (b).

Figure 4 . 13 :

 413 Figure 4.13: Number of plastic events as a function of the shear strain for the three compositions (original curves (a) and smoothed one with SG filter method (b)).

Figure 4 .

 4 Figure 4.14: Plastic intensity of the plastic events with a function of the shear strain for the three compositions (original curves (a) and smoothed one with SG filter method (b)).

CHAPTER 4 .

 4 ANALYSIS OF THE PLASTIC EVENTS AT THE ORIGIN OF THE MACROSCOPIC PLASTICITY followed when decreasing the Cu content, since the composition with less Cu displays a more uniform atomic strain distribution.

Figure 4 . 15 :

 415 Figure 4.15: Distribution of the orientation of the plastic events of MGs with three different compositions.

  Figure 4.16 (a)-(c) show the original fitting curves. The fitting quality decreases with the Cu content. As already explained in Section 4.1.1, this might come depend on the use of the minimum image convention. To 4.3. PROSPECT FOR TENSILE TESTS account for the underestimation of the stress we apply a corrective factor α to the plastic stress, to reduce the stress drop and improve the fitting of the stress-strain curves in Figure 4.16 (d)-(f): with α=0.93 on plastic stress from 12.5 % to 17.8 % for Cu 64 Zr 36 , α=0.97 for Cu 50 Zr 50 and α=0.88 for Cu 36 Zr 64 , the fitting is very satisfactory for all the compositions.

Figure 4 . 16 :

 416 Figure 4.16: The fitting stress-strain curves of three compositions after quenched with quenching rate 10 11 K/s before correction ((a), (b) and (c)) and after correction ((d), (e) and (f)).

Figure 4 .

 4 Figure 4.17: The schematic of plastic events during tensile deformation. Red ellipses represent the plastic events. Blue square represents the deformed MG system.

Figure 4 .

 4 Figure 4.18: Tensile stress-strain curves of Cu 64 Zr 36 MG with different quenching rates.

Figure 4 .

 4 Figure 4.19: Young's modulus (a), yield stress (b) and yield strain (c) of Cu 64 Zr 36 MG for different quenching rates deduced from the tensile stress-strain curve.

Figure 4 . 20 :

 420 Figure 4.20: Cumulative number (a) and cumulative plastic intensity (b) of plastic events of Cu 64 Zr 36 MG after tensile deformation with different quenching rates.

Figure 4 . 22 :

 422 Figure 4.22: Plastic intensity of plastic events as a function of the tensile strain for Cu 64 Zr 36 MG quenched at different quenching rates (a), smoothed curve for quenching rate 10 10 K/s and 10 14 K/s (b).

Figure 4 .

 4 Figure 4.23: Average size of plastic events as a function of the tensile strain for Cu 64 Zr 36 MG quenched at different quenching rates (a) and smoothed curves for 10 10 K/s and 10 14 K/s (b).

4. 3 .

 3 PROSPECT FOR TENSILE TESTS quenching rates, and therefore reach more rapidly a stabilised value which is roughly the same for the three quenching rates above the yield point. The data are also more scattered for the fastest quenching rate. Correspondingly, MGs for higher quenching rate have a more uniform deformation during tensile deformation, as shown in Figure4.24.

Figure 4 . 24 :

 424 Figure 4.24: Map of the atomic shear strain for tensile strain 12.5 %, 15 % and 20 % for Cu 64 Zr 36 MG quenched at different quenching rates.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2 Thermal cycling treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Effect of thermal cycling on metallic glasses with different initial quenching rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Effect of thermal cycling for different metallic glass composition . . . . 111 5.5 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 5 .

 5 Figure 5.1: Temperature as a function of time over one cycle.

Figure 5 . 2 :

 52 Figure 5.2: Potential energy (PE) as a function of the number of thermal cycles for Cu 64 Zr 36MGs with different quenching rates.

Figure 5 . 3 :

 53 Figure 5.3: The variation of FI fraction as a function of the number of thermal cycling, for Cu 64 Zr 36 MGs prepared with different quenching rates.

  4 for the different quenching rates. Only the atoms with D 2 min > 100Å 2 are visualized in the figure. Obviously, for Cu 64 Zr 36 MGs with the highest quenching rates, more atoms undergo a large nonaffine square displacement after experiencing the first thermal cycle. This means that the mobility of atoms is initially easier, which is consistent with the high initial PE of this less-relaxed MG. For the more relaxed Cu 64 Zr 36 MGs (i.e., prepared with the lowest quenching rates, 10 10 K/s, 10 11 K/s and 10 12 K/s ), the number of atoms showing large nonaffine square displacement increases when the number of cycles increases.

Figure 5 . 4 :

 54 Figure 5.4: Map of the distribution of the non-affine square displacement (D 2 min > 100Å 2 ) at the 1st, 5th, 9th and 99th cycle, calculated with the previous cycle as reference, for Cu 64 Zr 36 MGs prepared with different quenching rates.

Figure 5 .

 5 Figure 5.4 and 5.2, most of the variation of the nonaffine square displacement and of the PE occurs between the first cycle and the ninth cycle, while the PE stabilizes afterwards

Figure 5 . 5 :

 55 Figure 5.5: Stress-strain curves of Cu 64 Zr 36 MG quenched at different quenching rates (10 10 K/s (a), 10 11 K/s (b), 10 12 K/s (c), 10 13 K/s (d) and 10 14 K/s (e), respectively) and submitted or not to 100 thermal cycles.

  Figure 5.6: Shear modulus (a), yield stress and flow stress (b) of Cu 64 Zr 36 MG quenched at different initial quenching rates with and without thermal cycling.

Figure 5 . 7 :

 57 Figure 5.7: Comparison of atomic shear strain of Cu 64 Zr 36 MG quenched at different initial quenching rates with and without thermal cycling, at strain 12.7 %.

  Figure 5.8: Comparison of average size of plastic events of Cu 64 Zr 36 MG (initial quenching rate 10 10 K/s) without and with thermal cycling (a) and smoothed curves (b).

Figure 5 . 9 :

 59 Figure 5.9: Comparison of average size of plastic events of Cu 64 Zr 36 MG (initial quenching rate 10 14 K/s) without and with thermal cycling (a) and smoothed curves (b).

Figure 5 .

 5 Figure 5.8 and Figure 5.9 show the average size of the plastic events as a function of the

Figure 5 .

 5 Figure5.10 and Figure5.11 show the number of the plastic events as a function of the shear strain for Cu 64 Zr 36 MG initially prepared with quenching rates of 10 10 K/s and 10 14 K/s, and submitted or not to thermal cycling. For quenching rate 10 10 K/s, the curves are roughly the same except (in average) a slight increase of the number of plastic events with thermal treatment whereas for quenching rate 10 14 K/s, we can clearly see that at small strains (below the yield strain), less plastic events occur in the Cu 64 Zr 36 MG

Figure 5 .

 5 Figure 5.10: Number of plastic events of Cu 64 Zr 36 MG (initial quenching rate 10 10 K/s) without and with thermal cycling (a) and smoothed curves (b).

Figure 5 . 11 :

 511 Figure 5.11: Number of plastic events of Cu 64 Zr 36 MG (initial quenching rate 10 14 K/s) without and with thermal cycling (a) and smoothed curves (b).

Figure 5 .

 5 Figure 5.12 and Figure 5.13 show the plastic intensity of the plastic events as a function

Figure 5 . 12 :

 512 Figure 5.12: Plastic intensity of plastic events of Cu 64 Zr 36 MG (initial quenching rate 10 10 K/s) without and with thermal cycling (a) and smoothed curves (b).

Figure 5 . 13 :

 513 Figure 5.13: Plastic intensity of plastic events of Cu 64 Zr 36 MG (initial quenching rate 10 14 K/s) without and with thermal cycling (a) and smoothed curves (b).

Figure 5 . 14 :

 514 Figure 5.14: Distribution of orientation of plastic events of Cu 64 Zr 36 MG without and with thermal cycling fora quenching rate of 10 10 K/s (a) and a quenching rate of 10 14 K/s (b).

Figure 5 .

 5 Figure 5.14 shows a comparison of the orientation distribution of the plastic events for

Figure 5 . 15 :

 515 Figure 5.15: Variation of PE for the three compositions as a function of the number of thermal cycles.

Figure 5 .

 5 Figure 5.18: Shear modulus G (a), yield stress σ y (b) and flow stress σ f (c) deduced from Figure 5.17.

Figure 5 . 19 :

 519 Figure 5.19: Map of the atomic shear strain in the three compositions previously submitted or not to thermal cycling, at 20 % shear strain.

Figure 5 .

 5 Figure5.20 and 5.21 present the cumulative number and cumulative plastic intensity of plastic events occurring during shear strain, as a function of the shear strain for the three compositions, before and after thermal cycling. It shows that thermal cycling treatment promotes more plastic events for Cu 64 Zr 36 and maybe (it is a very slight effect) for Cu 36 Zr 64 . Conversely, for Cu 50 Zr 50 , cumulative number of plastic events is decreased after thermal cycling.The number and plastic intensity of plastic events at each strain for the three compositions before and after thermal cycling are presented in Figure5.22 and 5.23. The curves are smoothed for an easier comparison. In the small strain region (shown by an or-

Figure 5 . 21 :

 521 Figure 5.21: Cumulative plastic intensity for each composition submitted or not to thermal cycling.

Figure 5 . 22 :

 522 Figure 5.22: Number of plastic events as a function of the shear strain, for each compositions submitted or not to thermal cycling (original curves (a)-(c) and smoothed curves (d)-(f)).

Figure 5 . 23 :

 523 Figure 5.23: Plastic intensity of plastic events as a function of the shear strain for each composition before and after thermal cycling (original curves (a)-(c) and smoothed ones (d)-(f)).

Figure 5 . 24 :

 524 Figure 5.24: Orientation distribution (gaussian fit) of the plastics events occurring over the whole shearing test for each composition before and after thermal cycling.

Figure A. 2 :

 2 Figure A.2: Different points window of SG filter method when smoothing the curve of average size of plastic events with shear strain of Cu 64 Zr 36 MG with a quenching rate of 10 11 K/s.

Figure A. 3 :

 3 Figure A.3: Curves in Figure A.1 after smoothing with 50 points of window.

  Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	1.1 General features of metallic glasses . . . . . . . . . . . . . . . . . . . . .
	1.2 Mechanical properties of metallic glasses . . . . . . . . . . . . . . . . . .
	1.3 Application of metallic glasses . . . . . . . . . . . . . . . . . . . . . . . .
	1.4 Atomic-level structure of metallic glasses . . . . . . . . . . . . . . . . .
	1.5 Atomic-scale deformation of metallic glasses . . . . . . . . . . . . . . .
	1.5.1 Free volume model . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.5.2 Atomic local shear transformation . . . . . . . . . . . . . . . . .
	1.5.3 Formation of shear bands . . . . . . . . . . . . . . . . . . . . . .
	1.6 Eshelby's inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.7 Molecular Dynamics studies on metallic glasses . . . . . . . . . . . . . .
	1.7.1 Studies about structure . . . . . . . . . . . . . . . . . . . . . . .
	1.7.2 Correlation between microstructure and plastic behavior . . . .
	1.7.3 Effect on microstructure and mechanical properties . . . . . . .
	1.8 Improvement of plasticity . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.8.1 Composite metallic glasses . . . . . . . . . . . . . . . . . . . . .
	1.8.2 Pre-deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.8.3 Thermal pre-treatment . . . . . . . . . . . . . . . . . . . . . . . .
	1.9

  xy =δγ xy /2 and δ i σ xy is obtained from the atomic simulation. δ i σ * xy is the stress increment without plasticity, called elastically extrapolated stress. From configuration {i} curr to {i -1} rev through reverse deformation, in Figure 2.3, the difference between σ curr xy (γ i ) and σ rev xy (γ i-1 ) is the elastic component, thus, δ i σ * xy = σ curr xy (γ i )σ rev xy (γ i-1 ). Then, the total elastically extrapolated stress, σ * xy (γ i ) can be computed by σ * xy (γ i ) = ∑ i-1 j=0 δ j σ * xy . The total shear stress σ xy (γ i ) = ∑ i-1 j=0 δ j σ xy is obtained from MD simulation, and plastic stress σ pl xy (γ i ) = σ * xy (γ i )σ xy (γ i ) can be obtained and compared to the plastic shear stress computed with Eshelby's inclusion introduced in the following.

	pl xy	(2.40)
	where δ i ϵ

xyδ i σ

Table 3 .

 3 1: T g for Cu 64 Zr 36 MGs with different quenching rates.

	Quenching rate (K/s) 10 10 10 11 10 12 10 13 10 14
	T g (K)	690 730 783 821 897

Another indicator of the microstructure of the MG that we consider is the radial distribution function (RDF), already introduced in Chapter 2. The RDF for the final glass states after quenching with different quenching rates is presented in Figure

3

.2. It clearly shows

Table 3 . 2

 32 Composition Cu 64 Zr 36 Cu 50 Zr 50 Cu 36 Zr 64

	∆FI (%)	20.07	23.84	15.79

: Relative decrease of FI after a quasi-static shear deformation (up to flow) for the three compositions.

Table 4 .

 4 1: Sigma value of the distribution of orientation of shear transformation with different quenching rates.As expected, the distribution of φ xy,max is symmetric, as shown in Figure4.7, with a maximun at 0, which corresponds to the shearing direction. It can be fitted by a Gaussian law, plotted in Figure4.7. The sigma values used for fitting are reported in Table4.1.

	Quenching rate (K/s)	10 10	10 11	10 12	10 13	10 14
	Sigma	0.286265 0.322852 0.327813 0.352292 0.356218

Table 4 . 2 :

 42 Sigma values of the distribution of the orientation of the plastic events for the three compositions. Composition Cu 64 Zr 36 Cu 50 Zr 50 Cu 36 Zr 64

	Sigma	0.297643 0.318623 0.321432
	Finally, we have tried to reconstruct the MD stress-strain curves of the different compo-
	sitions with the Eshelby's inclusion model.

Table 5 .

 5 1: Yield and flow stress, in quasi static shear test at 0.1 K, for the three compositions submitted or not to thermal cycling.CompositionCu 64 Zr 36 Cu 50 Zr 50 Cu 36 Zr 64 Map of the distribution of the non-affine square displacement (D 2 min > 100Å 2

	∆τ(GPa)
	Without thermal cycling 0.88497	0.51827	0.57255
	With thermal cycling	0.61871	0.47711	0.26941
				112

Table 5 . 2 :

 52 Relative decrease in shear modulus, yield stress and in the difference between yield and flow stress for the three compositions after thermal cycling. Composition Cu 64 Zr 36 Cu 50 Zr 50 Cu 36 Zr 64

	γ G (%)	10.36	6.05	10.60
	γ σ y (%)	15.61	12.98	22.84
	γ ∆τ (%)	30.09	7.94	52.95

Table 5 . 3 :

 53 Width of the orientation distribution of the plastic events and relative variation deduced from Figure5.24.CompositionCu 64 Zr 36 Cu 50 Zr 50 Cu 36 Zr 64

	Width without thermal cycling	0.2976	0.3186	0.3214
	Width with thermal cycling	0.3095	0.3130	0.3405
	Increased percentage (%)	4.00	-1.76	5.94

  Moreover, after thermal cycling, the atomic mobility of Cu 36 Zr 64 , i.e., the sample with the lowest Cu content, increases significantly and the shear modulus, yield stress and stress drop calculated in quasi static shear test decrease the most. At last, thermal cycling increases the number and intensity of the plastic events occurring during shear for both compositions Cu 64 Zr 36 and Cu 36 Zr 64 , whereas it decreases them for Cu 50 Zr 50 .

  . Deformation behavior and enhanced plasticity of ti-based metallic glasses with notches. Philosophical Magazine, 90(29):3867-3877.[Zhao et al., 2010b] Zhao, J., Wu, F., Qu, R.,Li, S., and Zhang, Z. (2010b). Plastic deformability of metallic glass by artificial macroscopic notches. Acta Materialia, 58(16):5420-5432.[Zhong et al., 2016] Zhong, C.,Zhang, H., Cao, Q., Wang, X., Zhang, D., Ramamurty, U., and Jiang, J. (2016). Deformation behavior of metallic glasses with shear band like atomic structure: a molecular dynamics study. Scientific reports, 6(1):1-12.[Zhou et al., 2013] Zhou, H., Qu, S., and Yang, W. (2013). An atomistic investigation of structural evolution in metallic glass matrix composites. International Journal of Plasticity, 44:147-160. Les verres métalliques sont de nouveaux matériaux très prometteurs du fait de leurs excellentes propriétés mécaniques. Ils ont potentiellement de nombreuses applications en micromécanique, microélectronique, bijouterie, dans le domaine biomédical, l'aérospatial et bien d'autres domaines. Cependant, leur faible plasticité à température ambiante limite pour l'instant leur large diffusion dans l'industrie. Par conséquent, l'amélioration de leur ductilité est devenue un en enjeu majeur pour la science des matériaux. L'objectif de cette thèse est ainsi d'étudier la plasticité de verres métalliques préparés sous différentes conditions, par des simulations de dynamique moléculaire (MD) et de caractériser les événements plastiques survenant à l'échelle microscopique pendant leur déformation afin de comprendre comment la plasticité évolue, et finalement d'évaluer sous quelles conditions la ductilité peut être améliorée. Le système étudié dans cette thèse est un modèle binaire Cu-Zr. Nous avons étudié l'effet de la vitesse de trempe, de la composition, de la vitesse de déformation et de la température. Nos simulations montrent que lors de déformations en cisaillement quasi-statique, la plasticité des verres Cu-Zr est plus homogène, c'est-à-dire que la localisation des déformations et la formation des bandes de cisaillement sont plus limitées, lorsque ces matériaux sont préparés avec des vitesses de trempe plus élevées, lorsqu'ils contiennent moins de cuivre, et/ou lorsqu'ils sont soumis à des vitesse de déformation et/ou des températures plus élevés. Afin de comprendre les mécanismes microscopiques impliqués dans cette plasticité, nous avons développé une méthode pour identifier et caractériser les événements plastiques individuels. De cette façon, nous pouvons évaluer leur nombre, leur intensité plastique, leur taille moyenne et leur orientation. De plus, nous sommes capables de reconstruire le comportement mécanique simulé en considérant les événements plastiques comme des inclusions d'Eshelby dans une matrice élastique infinie. Enfin, nous montrons que le cyclage thermique sous la température de transition vitreuse peut être, dans certaines conditions, un moyen efficace d'améliorer la ductilité des verres métalliques, via leur rajeunissement. MOTS-CLÉS : Verres métalliques; Dynamique Moléculaire; plasticité ; propriétés mécaniques; cyclage thermique Laboratoire (s) de recherche : Laboratoire MATEIS -INSA Lyon

		FOLIO ADMINISTRATIF
	THESE DE L'UNIVERSITE DE LYON OPEREE AU SEIN DE L'INSA LYON
	NOM : LIANG			DATE de SOUTENANCE : le 9 décembre 2022
	(avec précision du nom de jeune fille, le cas échéant)
	Prénoms : Meng
	TITRE : Analysis of small-scale plasticity of bulk Cu-Zr metallic glasses under deformation by atomistic
	simulations
	NATURE : Doctorat			Numéro d'ordre : 2022ISAL0115
	Ecole doctorale : ED 34 Matériaux de Lyon
	Spécialité : Science des Matériaux
	RESUME : Directeur de thèse:	Laurent CHAZEAU INSA Lyon
		Claudio FUSCO	INSA Lyon
	Président de jury :	Anne TANGUY		INSA Lyon
	Composition du jury :	Rapporteur : Alain DEQUIDT, Université Clermont Auvergne
		Rapporteur : Jichao QIAO, Northwestern Polytechnical University
		Examinatrice : Kirsten MARTENS, Université Grenoble Alpes
		Invité:	Julien MORTHOMAS, INSA Lyon
		Invité:	Tristan ALBARET, Université Lyon 1
				1

Cette thèse est accessible à l'adresse : https://theses.insa-lyon.fr/publication/2022ISAL0115/these.pdf © [M. Liang],[2022], INSA Lyon, tous droits réservés

Acknowledgements

thank him for the suggestions and help he gave me, whether in the technical, scientific, professional and personal field. , insightful comments and present on my defense, Assistant professor Julien Morthomas and Tristan Albaret, gave me patient guidance and unselfish assistance during my research.

In addition, I wish to thank China Scholarship Council (CSC) for the financial support during my PhD period, providing precious opportunity to continue my academic study.

Bibliography

Author's publications

Smoothing method

Comparison of the effect of the different compositions or quenching rates on the evolution of the size, intensity and number of the plastic events with the strain is difficult due to large scattering of the data (as shown in Figure A.1). For this reason, we use a smoothing procedure. Origin software provides several methods to do it, including Savitzky-Golay filter method, adjacency averaging, percentile filtering, FFT filtering, LOWESS and LOESS smoothing methods, and binomial filter.

With the Savitzky-Golay (SG) filter method, each data point receives a new smoothed value after a local polynomial regression is conducted around it. Because it tends to preserve data properties like peak height and width that can be "washed out" by adjacency averaging, this approach is superior to adjacency averaging. Increasing the "window size" or the quantity of data points used in each local regression (using the "window points" option) will smooth out the results. The overall profile can still be preserved with Savitzky-Golay even for very large window sizes, where adjacency averaging may deviate too far from the input signal. Adjacency averaging, as the name suggests, effectively replaces each point in the data with the new average by averaging a user-specified number of data points around each point in the data. The 50 % percentile filtering (or median filtering) method might be most effective for signals with distributed grain noise when the noise displays local spikes. The median value of a group of nearby points is used instead of the signal value at each point in this method. The FFT filtering method can be applied if the signal contains high frequency noise. The genuine signal is left after the APPENDIX A. SMOOTHING METHOD filter eliminates all high frequency noise. When there are a lot of data points present, the LOWESS and LOESS smoothing methods are highly helpful for directly detecting in noisy data. A weighted moving average filter with weights derived from binomial coefficients is known as a binomial filter, a low-pass filter that can eliminate high frequency noise.

In our case, we wanted to preserve more features of the data, so we choose the SG filter method. Then, we have tested different sets of points window, including 5, 10, 20 and 50.

From