Atomistic simulation and modeling of localized shear deformation in metallic glasses

Progress in Materials Science 56 (2011) 785–816

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Progress in Materials Science

journa l homepage : www.e lsev ie r . com/ loca te /pmatsc i

Atomistic simulation and modeling of localized sheardeformation in metallic glasses

Shin Takeuchi a,⇑, Keiichi Edagawa b

a Tokyo University of Science, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japanb Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Available online 31 January 2011

0079-6425/$ - see front matter � 2011 Elsevier Ltdoi:10.1016/j.pmatsci.2011.01.007

⇑ Corresponding author at: Tokyo University of3260 4271x1516; fax: +81 3 5228 8116.

E-mail address: [email protected] (S. T

Since the end of 1980s, bulk metallic glasses became available forvarious multi-component alloys. Because bulk metallic glassesare applicable to structural materials, their mechanical propertieshave become a matter of great interest in these decades. A charac-teristic feature of plastic deformation of metallic glasses at theambient temperature is the localized shear deformation. Sincewe have no appropriate experimental technique, unlike crystallinematter, to approach microscopic deformation process in amor-phous materials, we have to rely on computer simulation studiesby use of atomistic models to reveal the microscopic deformationprocesses. In this article, we review atomistic simulation studiesof deformation processes in metallic glasses, i.e., local shear trans-formation (LST), structural characterization of the local sheartransformation zones (STZs), deformation-induced softening, shearband formation and its development, by use of elemental andmetal–metal alloy models. We also review representative micro-scopic models so far proposed for the deformation mechanism:early dislocation model, Spaepen’s free-volume model, Argons’sSTZ model and recent two-state STZ models by Langer et al.

� 2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7862. Atomistic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

2.1. Ingredients of atomistic simulation of deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

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akeuchi).

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786 S. Takeuchi, K. Edagawa / Progress in Materials Science 56 (2011) 785–816

2.1.1. Model construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7892.1.2. Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7892.1.3. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790

2.2. Deformation simulation in the first era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7902.3. Deformation simulation in the second era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

2.3.1. 2-D simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7932.3.2. Macroscopic behavior in 3-D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7942.3.3. Shear band evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7972.3.4. Atomic structure of localized shear deformation site . . . . . . . . . . . . . . . . . . . . . . . . . . 7992.3.5. Potential energy landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

2.4. Summary of simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802
3. Microscopic deformation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804
3.1. Dislocation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8053.2. Free-volume model by Spaepen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8073.3. Shear transformation zone model by Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8093.4. Two-state shear transformation zone model by Langer et al. . . . . . . . . . . . . . . . . . . . . . . . . . . 811

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814

1. Introduction

The metallic glass here simply means metallic alloy having an amorphous structure produced bycooling process from the melt. Researches on metallic glasses became popular worldwide since1970s due to their specific properties and their possible applications, particularly as soft magneticmaterials. Until the end of 1980s, however, metallic glasses had been produced only by rapid quench-ing from the melt, so that metallic glasses had been available mostly as thin ribbon samples, whichhad restricted the application field. As a result, the activity of research on metallic glasses was decreas-ing in the late 1980s. However, from the end of 1980s, it has been demonstrated that bulk metallicglasses with a thickness larger than a few mm can be produced for multi-component alloys with avariety of compositions even by the usual casting process, see, e.g. [1]. The advent of bulk metallicglasses has opened new application fields as structural materials. Thus, the mechanical propertiesof metallic glasses became again a matter of great interest. In this paper, the history of the metallicglass researches will be divided into the first era until 1989 and the second era after 1990.

Mechanical properties of metallic glasses exhibit various features different from crystalline alloys,ceramic crystals and other kinds of glass, as mentioned below. For recent reviews of mechanicalbehavior of metallic glasses, see [2–4].

(1) Localized shear deformation: Other kinds of inorganic glasses are quite brittle at room tempera-ture, exhibiting a smooth fracture surface as a result of mode I brittle fracture. Meanwhile,metallic glasses generally deform by localized shear deformation (LSD) at room temperature,and are ductile for bending and rolling deformation. Under uni-axial tension or compression,metallic glass specimens undergo shear fracture immediately after the yielding without show-ing appreciable macroscopic plastic strain, but the fracture occurs not in a brittle manner micro-scopically but as a result of extremely localized shear deformation, exhibiting a vein-fracturesurface. Thus, metallic glasses are generally not brittle in nature in the microscopic sense, butin the macroscopic sense, they are not ductile both for tension and compression. A lot of effortshave been devoted to increase the macroscopic ductility, see, e.g. [3].

Fig. 1 shows an example of the temperature dependence over a wide temperature range of the yieldstress under the uni-axial tension of ribbon samples of a Cu–Zr binary metallic glass [5]. Above a crit-ical temperature near the glass transition temperature, specimens exhibit homogeneous viscous flowwith the yield stress being quite strain-rate sensitive, and hence the critical temperature is strain-rate

Fig. 1. Temperature dependence of critical resolved shear stress of ribbon samples of Cu57Zr43 metallic glass [5].

S. Takeuchi, K. Edagawa / Progress in Materials Science 56 (2011) 785–816 787

dependent. Such a viscous flow at high temperatures due to diffusional motion of atoms is commonlyobserved in any kind of glass. Below the critical temperature, metallic glasses generally deform plas-tically by LSD, as mentioned above. The yield stresses in this temperature region are considerably scat-tered in comparison with crystalline matter, with no appreciable strain-rate dependence. Thetemperature dependence of the yield stress by LSD is weak but still slightly larger than the tempera-ture dependence of the elastic modulus.

As the origin of the localization of deformation, two mechanisms have been proposed and longbeen discussed: the deformation-induced softening mechanism and the softening due to adiabaticheating by work-done in the shear band (SB). A considerable temperature rise was actually observedby various experimental techniques, and the vein-fracture surface, which is known to be the charac-teristic fracture surface of viscous materials, is indicative of the temperature rise in the SB up to abovethe glass transition temperature. The problem is whether or not the temperature rise is a cause of theshear localization or a result of it. Recent ingenious experiments [6] showed that the extent of thetemperature rise is far wider than the SB width, which strongly suggests that the temperature riseis not the main cause but a result of the localized deformation caused by deformation-induced disor-dering, e.g. [7].

(2) Homologous nature: Fig. 2 shows the correlation between the Vickers hardness value HV at roomtemperature in units of GPa and Young’s modulus E at room temperature for a variety of metal-lic glasses: ribbon and bulk samples; metal–metal and metal–metalloid systems; and binaryand multi-component alloys. Both HV and E vary over an order of magnitude, but there existsa strong correlation between them, satisfying

HV ; 0:06E: ð1Þ

Such a correlation was first pointed out by Whang et al. [8]. Correspondingly, there is a correlationbetween the yield stress ry and Young’s modulus E as

ry ; 0:02E; ð2Þ

meaning that the yield strain is almost the same in any metallic glass and 2% of the elastic limit ishigher than those of any crystalline materials. The relation HV ffi 3ry from Eqs. (1) and (2) indicates

Fig. 2. Relation between Vickers hardness and Young’s modulus of various metallic glasses.


that metallic glasses can be regarded as almost ideal plastic solid having a constant flow stress. Sincethe temperature dependence of ry is quite weak, almost the same correlation holds even at 0 K. This isin sharp contrast to the intrinsic strength or the Peierls stress sP in crystals, where sP/G (G: shear mod-ulus) value varies over four orders of magnitude depending mainly on the crystal structure, e.g. [9].The homologous nature expressed by Eqs. (1) and (2) of the strength of metallic glasses suggests thatthe mechanism of SB deformation is largely common to any kind of metallic glass, in spite of a varietyof short range order structure in different kinds of metallic glasses.

The mechanism of plastic deformation by slip process in crystalline materials is well established bythe theory of dislocations and through the direct observation of dislocation processes by electronmicroscopy. In contrast, because of the lack of structural periodicity, we have no appropriate experi-mental technique to detect localized stress distributions and localized structural changes in amor-phous structure. There remain a lot of questions to be answered: (1) Is the yield stress by the LSDinitiation-controlled or propagation-controlled? (2) Does the propagation of the SB occur purelymechanically (athermally) or accompanied by a thermal activation (thermally)? (3) Why the widthof SBs are kept so narrow (10–20 nm [10]) even after a severe deformation? (4) Whether or not theSB propagation can be described as the dislocation propagation? (5) Why the yield strain is insensitiveto alloy systems? (6) Why SB’s are often wavy in the perpendicular direction to the shear?

In these circumstances, atomistic simulation studies of the SB formation is of great significancewith multiple reasons: (1) the atomistic simulation is the only available method to approach theatomistic process of the SB formation; (2) the mechanism of the SB formation seems to be commonto any metallic glass, meaning that the phenomenon is not sensitive to the details of interatomicpotentials; (3) the thickness of the SB is as narrow as 10–20 nm, the size of which can be accessibleby the present large scale computers.

In this review article, we focus on the formation mechanism of the shear bands. The macroscopicplastic strain or the macroscopic ductility of metallic glasses is a matter of great interest from thepractical point of view as mentioned above, but will not be treated in this article, because the


macroscopic ductility is quite structure sensitive. We will treat the SB formation as a common genericnature of metallic glasses. We will not review the high temperature viscous flow, the fracture phe-nomenon nor the structural relaxation. The present review article is organized as follows. In Section 2,we review the atomistic simulations of deformation in model metallic glasses. Simulations have beenmade both for 2-dimensional and 3-dimensional model metallic glasses. Although 2-D simulationscan serve to visualize characteristics of amorphous structure and its deformation processes, thereare limitations to argue the mechanism of the real 3-D metallic glasses, particularly on quantitativeargument. So, we focus mainly on 3-D simulations. At first, we review earlier simulations made inthe first era by use of models of a size smaller than the SB width. Then, we review simulations per-formed in the second era for the models with larger sizes. In Section 3, we review main microscopicdeformation models proposed so far for the SB deformation in metallic glasses. We review only rep-resentative models such as the dislocation model, Spaepen’s free-volume model, Argon’s thermallyactivated local shear transformation (LST) model in the first era, and the mean-field theory of thetwo-state shear transformation zone (STZ) models developed in the second era. Finally in Section 4,we conclude by trying to link the simulation, modeling and experiment.

2. Atomistic simulation

2.1. Ingredients of atomistic simulation of deformation

An atomistic simulation study consists of (1) the construction of a model structure, (2) the defor-mation simulation, (3) analyses of the structure and properties of deformation sites and their changeswith plastic deformation and (4) extraction of some physical concepts concerning the deformationmechanism.

2.1.1. Model construction2.1.1.1. Interatomic potentials. For our purpose, we have to treat a number of atoms that are arrangedin a non-periodic way, and hence ab initio simulation is practically impossible. Various levels of two-body potentials have been used; empirical potential, the Morse potential, the Lennard–Jones (L–J)potential, the potentials determined based on the pseudo-potential theory and also a semi-empiricaltight-binding potential. The original 6–12 L–J potential applicable to inert gas elements is too hard formetallic material, and hence sometimes modified as a 4–8 L–J potential. Rather recently, as a many-body potential, the embedded atom method (EAM) [11] or the effective medium theory (EMT) [12] hasalso been employed to construct a more realistic model. As far as the deformation simulation is con-cerned, the reality of the potential does not seem to be a crucial problem, because, as mentioned in theprevious section, experiments indicate that the deformation mechanism is common to any kind ofmetallic glass, meaning that it is insensitive to the details of interatomic potentials.

2.1.1.2. Model structure construction. In an early stage where computer performance is still low, a ran-dom positioning of atoms followed by static relaxation procedure was used to produce an amorphousstructure. In recent years, the following procedure has become a standard method to produce a modelmetallic glass, i.e., by use of the molecular dynamics (MD) method, a molten structure at high temper-ature is cooled down to far below the glass transition temperature, followed by annealing at an ele-vated temperature. The model size used has increased naturally as the progress of computerperformance. Early models consisted of 103–4 atoms and recent models 105–6 atoms for 3-Dsimulations.

2.1.2. DeformationIn early simulations, deformation of a model was made by applying a homogeneous strain to the

model structure stepwise and then relaxing the structure quasi-statically to attain equilibrium statefor each strain step under a fixed boundary condition on some surface layers and the periodic bound-ary condition on other surfaces in most cases. In recent simulations, the MD method has been usedwidely for the relaxation. In many MD simulations, NPT constant (N: number of atoms, P: pressure,


T: temperature) condition has been applied, and for the constant temperature simulation, Nosé-Hoover thermostat [13] has often been employed. The deformation mode was either uni-axial tensionor compression, or pure shear deformation. The indentation simulation was also conducted. The defor-mation of the model has been done by controlling either strain or stress with the fixed boundary con-dition on some surfaces and the periodic or free boundary condition on other surfaces. The strain-rateimposed on the model has been 107–10 s�1, which is more than ten orders of magnitude higher thanexperiments. To attain excited states as well as the ground state of the model, Monte Carlo simulationhas also been employed.

2.1.3. AnalysisThe distribution of the strain in the model has been analyzed by the atomic displacement field.

Localized inhomogeneous plastic deformation has been visualized by a map of the deviatory atomicdisplacements from the affine transformation.

Local atomic packing characteristics have often been analyzed in terms of the Voronoi polyhedronanalysis, where the way of arrangement of nearest neighbor atoms around each atom is representedby a polyhedron consisting of faces bisecting all the nearest neighbor bonds. Voronoi index (F3, F4, F5,F6, . . .) indicates that the faces of the Voronoi polyhedron consist of F3 triangular face, F4 square face, F5

pentagonal face, F6 hexagonal face, etc. (0, 0, 12, 0) Voronoi index indicates the 13 atom icosahedralpacking, which is known to be a typical close-packed cluster unit present abundantly in metallicglasses. Nc = RFi corresponds to the coordination number. Voronoi volume provides us with the infor-mation of the free volume.

The concept of ‘‘the atomic stress’’ has been introduced by Egami et al. [14]. Atomic stresses aredefined by the derivative of the atomic energy with respect to a uniform deformation of the structure,and are given for a central force interatomic potential /(r) by

rabi ¼

12Xi

Xj

1rij

@/ðrijÞ@rij

raijr

bij; ð3Þ

where rij = |rij| (rij : the vector between atoms i and j), and raij and rb

ij are the a and b components of thevector rij, respectively. The atomic modulus is also defined by the second derivative of the atomicenergy with respect to a uniform strain. As atomic level defects other than geometrical ones, p(positive atomic pressure) and n (negative atomic pressure) defects, and s (atomic shear stress) defecthave been introduced and used for the atomic level analysis [15].

In recent years, to obtain information of the potential energy landscape, simulations have beentried to detect activation energy barrier by giving an artificial disturbance to the model.

2.2. Deformation simulation in the first era

Before 3-D simulations, pioneering 2-D simulations were performed for a computer model [16] andfor a bubble raft model [17]. Based on the latter model, Argon introduced the concept of the local sheartransformation (LST), which is now recognized widely as an elementary process of plastic deformationin metallic glasses. Maeda and Takeuchi proposed a chain-reacting propagation of LSTs that occur atlarge free area sites to develop into a SB [16]. Argon et al. further investigated the behavior of the de-formed bubble raft model and 2-D atomistic model to reveal the nature of the formation process ofLSTs and their development to a SB [18,19]. They found that there exist two types of LST, the sharpshear transformation and the diffuse shear transformation, and that the distribution of calculated freeenergy barrier for LST has a negative correlation to the free area of the region. These observations on2-D models provided us with an insight into the deformation mechanism of metallic glasses.

Weaire et al. [20] first performed a deformation simulation for a realistic 3-D model to reveal elas-tic properties of amorphous metals. They showed that internal displacements of atoms cause a soft-ening of elastic moduli of amorphous metals in comparison with the crystalline state. The first 3-Dsimulation of macroscopic plastic deformation of metallic glass was performed by Yamamoto et al.by stretching a cylindrical monatomic model of 2 nm diameter [21]. They observed plastic shear defor-mation after about 6% of elastic strain. In 3-D simulations that followed, atomic scale inhomogeneities

Fig. 3. (a) Map of internal atomic displacement magnified by 5 times in a slice of model Cu57Zr43 metallic glass after 2% plasticshear strain. (b) The initial distribution of atomic shear stress rxy in the same slice [22].


of the amorphous structure were analyzed. Kobayashi and Takeuchi [22] simulated plastic deforma-tion of a cubic model binary metallic glass of a size of about 10 atomic distances in one side. Sheardeformation was applied by displacing the top and bottom layers. The changes with plastic strainof the internal energy, the internal displacement of atoms, the atomic pressure distribution and theatomic shear stress distribution were analyzed. The model underwent plastic deformation by diffuseLSTs in an area of 4–5 atomic diameters at a stress of G/15 (G: shear modulus of the model). Fig. 3ashows an example of the internal displacement map in a slice of the model, showing diffuse LSTs indi-cated by shaded arrows. The region of a large LST was characterized as the region where atoms arecollectively subjected initially to an internal shear stress acting in the same direction as the appliedstress, as shown in Fig. 3b. A similar modeling was performed for an elemental model metallic glassby Maeda and Takeuchi [23]. Deformation sites were analyzed in terms of atomic stresses, atomicshear modulus and Voronoi polyhedron. LSTs occur in regions of 3–4 atomic diameters, slightly smal-ler than the binary case. LST sites were characterized by a pressure distribution that assists the sheardeformation and also by a region of low atomic shear modulus which was related to Voronoi polyhe-dron type. Maeda and Takeuchi further correlated deformation sites with various atomic parameters:atomic pressure P, Voronoi volume V, atomic shear stress rxy and atomic shear modulus Cxyxy [24].Fig. 4 shows the spatial distribution of internal atomic displacements (a), the vacant space definedas the open space that can accommodate a sphere of diameter of 0.3 atomic size (b), the Voronoi poly-hedron type (c), the deviatory shear modulus (d) and the atomic shear stress (e). Note that the struc-ture is periodic in x-direction. Vacant spaces exist around the deformation sites but not necessarilyexist in the site. Most of atoms in the deformation sites have miscellaneous Voronoi polyhedron typeother than icosahedron (0, 0, 12, 0) and its derivatives with large F5 value, and possess low modulusand subject to positive shear stress, as seen in (c), (d) and (e), respectively. Table 1 lists the correlationcoefficients among magnitudes of atomic displacement d, atomic pressure P, Voronoi volume V,atomic shear modulus Cxyxy and atomic shear stress rxy. It is seen that d has a considerable correlationwith V, Cxyxy and rxy assisting the deformation, but not with P, though P and V have a strongcorrelation.

Then, Srolovitz et al. [25] also performed a shear deformation simulation up to a much higher strainfor the same elemental amorphous model as used by Maeda and Takeuchi. They observed a stress–strain curve showing a yield drop with the upper yield stress of 0.14G. They observed various kindsof localized deformations involving several to a hundred atoms. They showed that localized deforma-tion regions have little correlation with local density fluctuations expressed by n- and p-defects,apparently in disagreement with the free-volume model for the localized deformation, but correlatedwith high s-defect regions. Fig. 5 is an example showing that s-defect centers are present close to thedeformation sites. It was also shown that some of s-defects are preserved and its magnitude isincreased with increasing plastic deformation, while n- and p-defects initially present are annihilatedby mechanical annealing after deformation.

Fig. 4. (a) map of atomic displacement between 6% and 8% shear strain in a slice of a monatomic metallic glass model. (b–e)show distributions of vacant space, Voronoi polyhedron type, deviatory atomic shear modulus and atomic shear stress,respectively, in the state of 6% strain [24].

Table 1Correlation coefficient among atomic parameters in an elemental model structure [24]. d: atomic displacement, P: atomic pressure,V: Voronoi volume, Cxyxy: atomic shear modulus component, rxy: atomic shear stress component. Considerable correlations areshown in bold.

d P V Cxyxy rxy

d 1.0 0.033 0.178 �0.140 0.172P 1.0 0.482 �0.271 0.053V 1.0 �0.256 0.040Cxyxy 1.0 �0.003rxy 1.0


A characteristic feature of these early simulations was to identify and characterize the shear trans-formation zones (STZs). A fundamental concept that was established is that the elementary deforma-tion process is localized internal atomic rearrangements in an area of several atomic diameters, whichcorresponds to the LST of diffuse type proposed by Argon. A great effort was devoted to identify thedeformation sites, but no clear-cut characterization as expressed by a single atomic parameter wasmade. A simplest expression of the heterogeneity in metallic glasses may be to divide the structureinto well-packed (or closely packed) region and ill-packed (or loosely packed) region, where the for-mer region is stable while the latter region is less stable for the shear stress. From the above analyses,the following picture of the localized deformation may be drawn. The well-packed region is composedmainly of typical Voronoi polyhedra of (0, 0, 12, 0) and its derivatives with smaller Voronoi volumeand with lower atomic stresses, while the ill-packed region which is susceptible to localized sheardeformation is characterized geometrically by non-typical Voronoi polyhedron with larger Voronoivolume, and physically by large s-defect and low atomic modulus, but is not much concerned withn- and p-defects. Localized deformations occur preferentially at ill-packed regions, and they reproducenew ill-packed regions in the vicinity and thus localized deformation continues to result in intense SB.

Fig. 5. (a) Map of internal atomic displacement between 2 and 4% shear strain. (b) That between 14% and 16%. The longestarrow corresponds to a displacement of 0.3 Å. The solid square marks indicate the centers of s-defects at 2% strain in (a) and 14%strain in (b). Reproduced with permission from [25].


2.3. Deformation simulation in the second era

After passing an inactive period from the late 1980s to 1990s, quite active research has startedagain from the end of 1990s and continued up to the present, in accordance with the experimentalactivities on bulk metallic glasses. Some features of simulations in the second era different from thoseof the first era are: (1) due to the development of the computer performance, the model size isincreased as mentioned in the previous section, (2) temperature is controlled in many cases due tothe MD method and (3) information concerning the potential energy landscape becomes the targetof research. In this section, instead of reviewing paper by paper chronologically, we overview eachaspect of phenomena concerning the shear deformation in metallic glasses.

We review the following aspects of the deformation simulations: macroscopic properties, deforma-tion structures, including the process of SB formation, characterization of deformation sites, changesof the structure and temperature with plastic deformation, including the origin of deformation soften-ing and activation energy barrier calculation of LST.

2.3.1. 2-D simulationEven in the second era of history of deformation simulation of metallic glasses, 2-D model struc-

tures have also been employed as an analog system of metallic glass [26–35]. Before going into de-tailed review of 3-D simulations, we briefly review 2-D simulations in the second era. As alreadyseen in the pioneering works in the first era, 2-D simulations have contributed to modeling the plasticdeformation process of metallic glass. The plastic behavior of 2-D model metallic glasses is largelysimilar qulitatively to that of 3-D model as mentioned in the following sub-section: (1) The modelexhibits yielding with a large work softening for slowly cooled samples [27,31], followed by steady-state flow [26,28,30]. (2) The elementary plastic deformation event is a STZ formation [26,27,29],which is associated with a quadrupolar energy fluctuation [29,31] and a large excess free area[30,31]. (3) The cascade of STZs precedes a SB formation [31,28,30,32]. The difference of the SB forma-tion in 2-D models from that in 3-D models is that SBs are rather diffuse and densely formed in a sam-ple, and the strain does not localize in one SB. Some noticeable findings are that after plasticdeformation in one direction the specimen exhibits a significant Bauschinger effect, suggesting thatsome of the STZs are reversibly deformable units [26], which are actually observed in a 2-D simulation[33]. Based on the reversible nature of the STZ, a dynamic deformation model of the two-state STZ wasproposed by Falk and Langer [26] and further developed [36–40].

Recently, by use of the Monte Carlo analysis of the activation relaxation technique (ART), Rodneyand Schuh investigated the activation energy spectrum of plastic events and showed that the plastic

Fig. 6. Distributions of activation energies of plastic events before deformation and after deformation obtained by Monte Carlosimulation of ART for a 2-D model metallic glass. Reproduced with permission from [35]. Copyright (2009) by the AmericanPhysical Society.


deformation remarkably changes the spectrum from the undeformed state to the flow state [34,35].Fig. 6 shows the changes of the activation energy spectrum from the undeformed state to the flowstate [35]. Note that these spectra are obtained in the unloaded state. As seen in the figure, the distri-bution of the activation energy is quite wide unlike crystalline cases. A high density of low energy bar-riers is a characteristic of the flow state, indicating that the plastic deformation brings the energylandscape of the glassy state into a higher potential energy state with a high density of low energybarriers. It was also shown that after unloading a negative strain is produced, indicating that the defor-mation induces a microstructural polarization.

It may be an interesting problem to compare the deformation processes of 2-D model metallicglasses and those of 3-D model metallic glasses, but this is out of the scope of the present review.

2.3.2. Macroscopic behavior in 3-D modelIn Table 2, we list 3-D atomistic simulations of deformation of model metallic glasses performed in

the first and the second eras. We review below each aspect of deformation of model metallic glassesrevealed in the simulations performed in the second era.

2.3.2.1. Stress–strain curve. Under uni-axial tension or compression, or pure shear deformation of mod-el metallic glasses, stress vs. strain relations have been obtained for various models. Some commonfeatures are as follows: (1) Stress vs. strain curves show more or less serrated feature with the ampli-tude being larger for smaller model size. This indicates that the plastic deformation takes place discon-tinuously as successive localized deformation events. Malandro and Lacks [41] investigated thesystem size effect (in the range 102 < N < 2 � 103 which is below the size of SB formation) of sheardeformation in a monatomic model metallic glass and showed that the number of atoms participatingin each localized deformation event increases with the system size and levels off at 50 ± 30, and theaverage shear flow stress decreases rapidly with increasing model size to the leveling-off stress of0.04G. They described the deformation behavior based on a phenomenological model assuming a sta-bility range function of a group of atoms. (2) In well-aged or slowly cooled model metallic glasses, thestress vs. strain relation shows a yield drop followed by a steady flow of nearly constant flow stress[42–51], whereas in quenched or rapidly cooled samples, yielding occurs without upper yielding[42,43] and their steady-state flow stress is nearly the same as the steady-state flow stress of the agedsample of the same model [42]. The well-aged sample which has been pre-deformed to the steadyflow stage exhibits no more upper yielding [51]. Fig. 7 shows schematically the stress vs. strain behav-ior of model metallic glasses with three different sample histories, together with structure energyvariations.

It should be noted at this stage that the cooling rates of samples in the simulations are severalorders of magnitude higher than those of experiments. In the MD simulations, the cooling rates of

Table 2List of 3-D atomistic simulations of model metallic glasses. Meanings of abbreviations are as follows. Potential column: W–S:Weber–Stillinger, L–J: Lennard–Jones, EAM: Embedded Atom Method, EMT: Effective Medium Theory, TB: Tight–Binding, F–S:Finnis–Sinclair; H–H: Hausleitner–Hafner. Simulation column: MC: Monte Carlo, MD: Molecular Dynamics, SR: Static Relaxation.Analysis column: PEL: Potential Energy Landscape, SRO: Short Range Order, AE: Activation Energy, FV: Free Volume, SB: ShearBand, STZ: Shear Transformation Zone, VA: Voronoi Analysis, Cijkl: atomic modulus, X: atomic volume, NAD: Non-AffineDisplacement.

System Size Potential Simulation Deform. mode Analyses Ref. No.

Fe N = 567 Johnson SR Tension Atomic flow, E–e [21]Cu57Zr43 N = 1.5 � 103 4–8 L–J SR Shear E–e, STZ, p, r [22]Fe N = 2 � 103 Johnson SR Shear E�e, p, r, Cijkl, VA [23]Fe N = 2 � 103 Johnson SR Shear r�e, r, p, n, s [25]Monatomic N = 2 � 103 W–S MD Shear PEL [41]L–J mixture N = 4 � 103 L–J MC Plane strain (E, r, p)–e [42]Ni50Zr50 N = 5 � 103 Teichler MD Shear at high T (r, E)–e [52]Cu50Zr50 N = 2.8 � 103 4–8 L–J SR Tension-compr. r�e [55]L–J N = 500 L–J MD Cyclic shear E–e [61]Ni40Zr60 N = 2.4 � 106 EAM MD Tension r�e, notch effect [58]Binary 70 � 100 � 15a3 L–J, etc. MD Compression r–e, SRO [43]Cu50Ti50 N = 4.5 � 103 EAM MD Const. AE barrier [71]Mg85Cu15 28 � 28 � 14 nm3 EMT MD Tension r–e, SB [44]CuTi N = 3.5 � 104 EAM MD Shear Spatial corr. [72]Ni67Al33 N = 1.1 � 105 EAM MD Tension VA, Cijkl [45]L–J system – L–J – Cyclic loading FV [70]Cu57Zr43 N = 5 � 105 4–8 L–J MD Shear r–e, SB [47]Cu64Zr36 N = 2.9 � 105 TB MD Shear r–e, SB, X [53]CuxZr1�x N = 3 � 104 4–8 L–J MD Compression VA, PD vs. plasticity [68]CuxZr1�x – 4–8 L–J MD Shear r–e, VA [48]Ni50Zr50 N = 6.5 � 104 TB MD Shear STZ [73]Cu65Zr35 N = 2.5 � 104 F–S MD Biaxial loading r–e, X [49]Ni50Zr50 10 nm/ sphere TB MD Compression r–e, PE of STZ, VA [62]Composite N = 2.5 � 105 TB MD Shear Mobile atom, SB [66]CuZr N = 5 � 105 4–8 L–J MD Compression r–e, SB, T–e [50]L–J N = 2 � 104 L–J MD Compression E, B, m, ry [59]L–J N = 5.6 � 106 L–J MD Mode II def. SB, T [65]Cu50Ti50 N = 3.5 � 104 EAM MD Shear r–e, (s, p, E, X)–NAD [69]Cu64Zr36 N = 5 � 105 EAM MD Compression r–e, e–T, SB [51]Ni50Zr50 / = 1–6 nm T.B. MD Compression r�e, X, STZ [57]Cu–Zr, Mg–Cu 53 � 12 � 2 nm3 EAM MD Shear, indent SB [63]Ni33Y67 N = 1 � 104 H–H MD Shear Dislocation, SB [64]


the samples are 109–12 K/s [41,44,47,51], while the fastest quenching rate in experiments is known tobe 106 K/s. In the Monte Carlo simulation by Utz et al. [42], the mean square displacement of atomsafter the fastest cooling process is reported to be only less than half of the nearest neighbor distanceand hence the structure is not much changed from the molten state. Thus, the structure produced bythe fast quenching rate used in the simulations can never been realized in experiments.

The temperature and strain-rate effects have been studied. High temperature simulations showedthat the flow stress decreases monotonically with increasing temperature [52,53]. Activation volumesof deformation at high temperatures have been estimated to be 70 Å3 [52] and 25 Å3 [54] from thestrain-rate dependence of the shear stress. Upper yielding is more pronounced for faster strain-rate[44,49,50], and the steady flow stress also increases with increasing strain-rate of the order of 107–10

s�1 [44,50]. It should be noticed, however, the strain-rates in the simulations are almost ten orders ofmagnitude higher than that of usual experiment, and the simulation results cannot be compared withthe strain-rate effect in experiments. For example, strain-rate of 1010 s�1 means that the sample de-forms 1% in ps, which is probably shorter time than the propagation time of a shear band acrossthe sample, indicating an extremely special situation.

A tension–compression anomaly of the flow stress was also observed by Lund and Schuh [55], indi-cating that the normal stress affects the shear flow stress. They showed that the calculated yield stress

Fig. 7. Schematic diagrams of stress vs. strain curves and structure energy vs. strain curves for model metallic glasses producedby three different quenching rates. After Utz et al. [42].


obeys the Mohr–Coulomb criterion expressed by sy = s � arn (sy: effective yield stress, s: yield stressin pure shear, rn: normal stress and a: effective friction coefficient), rather than the Tresca yield cri-terion or maximum shear yield criterion. a value was estimated to be �0.12, which is slightly largerthan the experimentally estimated value of 0.07–0.10 [56]. The Mohr–Coulomb type yield criterionwas also suggested by Ogata et al. [47] in their shear deformation simulation.

Flow stresses observed in smaller size simulations in the first era are commonly much higher thanthe flow stresses observed in larger size simulation in the second era, indicating a sample size effect ofthe flow stress. The sample diameter dependence of the stress vs. strain relation was investigated [57],which showed that the steady flow stress increases and the serration becomes pronounced as thesample diameter decreases, as shown in Fig. 8. These facts are considered to reflect inhomogeneityof deformation, including SB formation. Notch effects on the tensile deformation of a wire samplewere also investigated, which showed that the flow stress decreases down to half and the elongationdecreases from 80% to 20% by the notch effect [58].

For the deformation simulation where the SB formation occurs, the steady flow stress ss normal-ized by the shear modulus G of the model is listed in Table 3. It is to be noted that ss/G values ofthe models at 0 K are in the range of 0.03–0.04. This range agrees with the experimental yield stressesof metallic glasses at helium temperatures, which are around 0.035 [59].

Gilabert and Castellanos [60] succeeded in producing L–J amorphous models with different atomicsizes but with almost identical elastic and plastic properties by scaling the potential parameters. Sucha methodology may be useful to reduce the numerical density of atoms in describing particularmechanical properties.

2.3.2.2. Rejuvenation. Rejuvenation means that the amorphous state becomes excited into higherstructural energy state, or high temperature state. The flow stress behavior shown in Fig. 7 indicatesthat the deformed structure of aged samples becomes similar to that of the quenched state of a highstructural energy. Computation of structural energy actually increases after a large plastic deformation[42,52,61]. When only a small strain is applied, the structural energy can be decreased due to amechanical annealing effect [33,22,23,25]. As schematically shown in the upper figure of Fig. 7 forthree different samples, the sample cooled with a fast rate, that cooled with an intermediate rate

Fig. 8. Sample radius dependence of steady flow stress rs and amplitude of serration Dr. Reproduced with permission from[57]. Copyright (2009) by the American Physical Society.

Table 3Steady state flow stress normalized by the shear modulus of the model for various model metallic glasses.

System Temperature (K) ss/G Ref. No.

Mg85Cu15 100 0.03 [44]Cu64Zr36 100 0.022 [53]

300 0.017600 0.011

Cu30Zr70 0 0.038 [48]Cu50Zr50 0 0.040 [48]Cu80Zr20 0 0.043 [48]Cu50Ti50 10 0.033 [68]


and that cooled with a slow rate, the initial structural energy is lower in the above order but after thedeformation to the steady state, the energy becomes the same for the three samples [42]. By compar-ing the energy of a structure that experienced some treatment history with the energies of the struc-tures in the thermal-equilibrium state at various temperatures, one can assign the former state as Teff

[61]. After plastic deformation of a sample aged at a low temperature initially having a low Teff, the Teff

increases, i.e., the structure is rejuvenated, and Teff almost levels off in the steady deformation state.Brinkmann and Teichler showed that after plastic deformation at 810 K and 1000 K of a binary metal-lic glass, the structure changes to the state equivalent to the thermal-equilibrium state at a temper-ature about 200 K higher in both cases, i.e., Teff increases by 200 K after deformation [52]. However,even Teff’s for the samples with different histories are the same, it was shown by Lacks and Osborne[61] that the states are not identical with each other; they showed by calculating the normal modefrequencies at energy minima that the energy landscapes are different for the two samples, meaningthat the structures are not identical. This is expected because the structure of thermally equilibratedstate is isotropic while the structures of plastically deformed samples must be more or less polarized,as shown by an analysis of bond orientation anisotropy by Tomida and Egami for deformed modelmetallic glasses [54].

2.3.3. Shear band evolutionNo SB formation was realized in simulations of small model size, indicating that the nucleation of

shear band requires collective and cooperative operations of STZs over several nm region. The shearband formation depends not only on the model size but also on the sample history and on the

Fig. 9. The effect of the cooling rate on the shear band formation for different three models, (a–c), represented by deviatoryshear strain after 10% compression. Cooling rate is slower in the right direction. The ratio of the cooling time of the right endsample to that of the left end sample is 102 in (a), 4 � 102 in (b) and 2 � 102 in (c). Reproduced with permission from [43].


deformation process of the model including the boundary condition. Ogata et al. [47] were probablythe first who reported the development of shear bands in a 3-D model metallic glass by shear defor-mation. Since then, the SB formation has been observed in a number of 3-D simulations[43,44,53,48,62,50,51,63,64]. Shi and Falk [43] showed in their 3-D simulations for three differentmodel systems that rapidly quenched samples commonly exhibit homogeneous deformation withoutshowing upper yielding and as the quenching rate is decreased the SB formation becomes clearer, andthe upper yielding becomes pronounced in the stress vs. strain relation. Fig. 9 shows the effect of thecooling rate of the samples on the shear band formation for three different models, (a)–(c), repre-sented by deviatory shear strain after 10% compression. The cooling rates of the samples in (a)–(c)are slower in the right direction. Bailey et al. [44] performed detailed investigations on the shear bandformation. They showed that the width of the shear band produced is about 10 nm, SBs develop acrossthe sample within a few tens of ps, the local temperature rises by 50–100 K and the density decreasesby 1% within the SB. The width of the SB formed was reported to be 10 nm [53], 3 nm [51], 5 and10 nm [63], 2 nm [64]. These shear band widths in simulations are considerably narrower than theexperimental values of 10–20 nm [10]. In the mode II deformation simulation of a notched sampleshowed that the temperature rise near the notch reaches higher than 1000 K as plastic strain increases[65]. In shear deformation, the temperature difference between the shear band and the matrixbecomes larger than 100 K [46]. After compressing a sample up to e = 0.2, the average temperaturegoes up above the melting point of the model [50]. Such a large temperature rise, however, is related


to the extremely high strain-rate in the simulation, and cannot be compared with experiments. Wak-eda et al. [50] investigated effects of the strain-rate (5 � 108 to 5 � 1010 s�1) on the SB formation andshowed that the shear band density increases and the upper yielding becomes pronounced as thestrain-rate increases. We should note that for the strain-rate of 1010 s�1 the compression speed isalmost the same as the sound velocity and can be higher than the SB propagation, meaning that thedeformation condition is quite special.

Delogu simulated the deformation of metallic glass composites and showed that the shear bandformed is deflected at the phase boundary and the deflection angle depends on the boundary orienta-tion and increases with increasing strain [66].

The propagation speed of SB has been shown to be quite fast and is almost the same as the soundvelocity [51,64]. This fact indicates that the propagation of the SB front in model metallic glassesoccurs purely mechanically or athermally without accompanying any thermally activated event.Fig. 10a shows propagation of a shear band, nucleated on the surface, in a compressed sample repre-sented by the atomic shear strain map [51], and Fig. 10b shows propagation of a shear band nucleatedat an introduced dislocation site, represented by the strain contour map [64]. An interesting observa-tions concerning Fig. 10a is that once a shear band is formed, further plastic strain inside the band doesnot proceed by a propagation mode but by simultaneous deformation in the band [51]. This is inagreement with the recently reported experimental observation [67].

2.3.4. Atomic structure of localized shear deformation siteLocalized deformations take place as a result of the inherent heterogeneity of metallic glass struc-

ture. As mentioned in a previous section, many trials have been reported for characterizing the defor-mation sites, which are regarded as ill-packed region or loosely packed region, by use of atomisticstructure parameters of the free volume and the Voronoi index, and atomistic physical parametersof the atomic energy, atomic stresses and atomic moduli.

Shi and Falk [43] analyzed the packing structure in terms of the coordination shell consisting ofnear neighbor atoms around a central atom, and searched for the shell consisting only of triangularcoordination of the surface atoms, which is named as the triangular coordination shell (TCS). Theyshowed that the SB is formed in a region of low density of TCS. They suggested that the SB formationis related to the percolation of TCS-SRO (short range order). Lee et al. [68] and Wakeda et al. [48] cor-related the composition dependence of plastic behavior of a series of CuxZr1�x model metallic glassesand showed that the lower the packing density of the structure and the lower the density of

Fig. 10. Propagation of a SB nucleated at the surface of a compressed sample represented by atomic shear strain map,reproduced with permission from [51] (a), and that in a sheared sample nucleated at a dislocation site [64] (b). Time differencebetween A and G in (a) is 60 ps.

Fig. 11. Illustration showing arrangement of seven atoms around F5 face of Voronoi polydedron.

Fig. 12. (a) shows density distribution of F5 and (b) distribution of non-affine displacement after 3% shear deformation in a Cu–Zr model metallic glass. Arrows indicate shear directions. Reproduced with permission from [48].


pentagonal face (F5) in Voronoi polyhedra, the higher is the plasticity. We note that the atomic clusteraround F5 face consists of seven atoms constituting five tetrahedral cells, as illustrated in Fig. 11. Theyalso suggested that the localized deformation occurs preferentially in the region of non-pentagonal site.Fig. 12 shows an example of the correlation between the density distribution of F5 and the non-affinedisplacement of atoms in a slice of a model Cu–Zr metallic glass [48]. It is seen that dark areas in (a)with low density of pentagons correspond to lighter areas in (b) with large atomic displacements.Neudeker and Mayr [69] investigated the correlation of the non-affine displacements of atoms forthe shear deformation eyz with the atomic stresses (ryz, p), atomic energy (E) and atomic volume (X,and showed that there is a clear anti-correlation with Dryz, but little correlation with p, E and X.

The structural changes that occur within the developed SB in the deformation simulations can besummarized as follows: (1) The atomic volume is increased [53,65,49,57]. The amount of the increasewas reported to be as high as 3% [57]. (2) Voronoi analyses showed that the decrease of (0, 0, 12, 0)cluster [61,48,51] and the decrease in number of F5 [66], which is equivalently expressed as theincrease in non-pentagonal sites [48]. (3) The structural energy in the SB is increased, as a result ofchange of the structure to a looser packed one as expressed by (1) and (2) above. (4) The number


of unstable atoms with negative atomic modulus increases [45]. These facts clearly indicate that thestructure inside the SB changes to a looser packing state, which enhances the plasticity in the band.

A cyclic deformation simulation for a L–J model [70] showed that the regions with the excess freevolume deform preferentially and the increase of the free volume is accelerated under a cyclic loadingcondition. It was also reported that when the load is reversed, deformation events happen in differentlocations, which is not consistent with the two-state STZ model.

2.3.5. Potential energy landscapeIn discussing structural changes of metallic glass, potential energy landscape (PEL) picture has be-

come popular in recent years. The well-aged structure of an amorphous structure is in a stable state ata potential energy minimum, and the structural modifications from the stable state changes the struc-tural energy as schematically shown in Fig. 13 as a potential energy vs. configuration coordinate dia-gram, i.e., PEL. Under a stress higher than r, the state 0 will be thermally activated to the state A withthe activation energy of EþA but after unloading, the state will be recovered with a smaller activationenergy of E�A . Depending on the scale of the structural modification, the PEL changes, and in any casethe energy fluctuates to have many minimal points in the PEL.

3-D simulations have recently been performed to investigate the activation energy barrier for thetransition of local structures. Mayr [71] tried to determine the activation energy of local deformationby giving a Gaussian noise in a fixed area in a model. All the atoms in an excited region (ER) within theradius R from an arbitrary chosen point in a model CuTi metallic glass at 10 K were exposed to theGaussian white noise with an energy EE under a compressive biaxial stress, and after the life timeof the relaxation, the same procedure was repeated. For different ERs and EE’s, the same procedurewas repeated. Fig. 14a shows plastic strain produced by repeated Gaussian excitations in randomlychosen spherical region of R = 0.8 nm (containing �140 atoms) at various excitation energies EE undera fixed stress of 0.5 GPa. It is seen that in this spherical region, plastic relaxations continuously occurabove a critical excitation energy, or activation energy EA, of 0.35 eV. Fig. 14b shows the ER (repre-sented by number of atoms N in the abscissa scale) dependence of the activation energy. A strikingfeature is that EA decreases rapidly with increasing ER toward minimum at N � 150 and then graduallyincreases. Thus, in this model a typical thermally activated LST takes place in a region of R = 0.8–1.0 nm with the activation energy of �0.35 eV. Zink et al. [72] investigated for the same CuTi modelthe spatial correlation of non-affine displacements of atoms with the plastic deformation at 10 Kand 500 K. They showed that cooperative motions of �120 atoms occur for cluster diameter of1.5 nm or below insensitively to the strain, strain-rate and temperature.

More recently, Delogu investigated STZ by artificially disturbing the structure of Ni50Zr50 modelmetallic glass constructed by a semi-empirical tight-binding potential [73]. He studied local atomicrearrangements by displacing an arbitrary chosen atom to a point within its Voronoi cell in randomdirections, relaxing the system at 300 K, and examining irreversible rearrangement (IR) of atoms. Withseveral percentages of probability, IR takes place by displacements of atoms by 0.5 ± 0.5rnn (rnn: near-est neighbor distance). The number of atoms that induce IR in the sample increases with increasingquenching rate of the sample. Effects of application of stress was studied and the results showed thatIR is facilitated by increasing strain (equivalent to stress in the elastic region), as a matter of course.The activation energy EA deduced from the observed energy change profile at e = 0 is 27 kJ/mol and

Fig. 13. Schematic illustration of potential energy landscape.

Fig. 14. (a) Cumulative plastic strain in model CuTi metallic glass at 10 K under a biaxial stress of 0.5 GPa with successiveGaussian excitations with various energies given in the figure in regions of radius R = 8 Å (139 atoms). (b) Size (represented bynumber of atoms) dependence of activation energy necessary for continuous flow at a biaxial stress of 0.5 GPa. Data arereproduce with permission from [71].


decreases linearly with increasing elastic strain, giving the activation volume of 7 Å3. Delogu pointedout that the activation energy and activation volume of IRs are close to those for low-energy diffusionmode in the same alloy. It is, however, more interesting to see that the presently obtained EA of 27 kJ/mol (= 0.28 eV) is similar to the EA obtained for different model and different method by Mayr, 0.35 eV.

Delogu further investigated, by use of the same model of Ni50Zr50, the potential energy barrier bytracing potential energy fluctuations with time of individual mobile atoms and searched for collectiverearrangements of a group of atoms [57]. He observed two groups of potential energy distribution asgiven in Fig. 15a and interpreted the lower EA group as the surface rearrangements and the higher EA

group, which includes those observed in his previous artificial disturbance method, as the bulk rear-rangements. Lower value of EA of STZ at the surface is due to smaller coordination number of surfaceatoms and to easier accommodation of strain at the surface. Higher value of EA of STZ in the bulk is dueto its opposite situation. In the simulation, Delogu pointed out the important role of more mobile spe-cies of atom Ni in the STZ activity. In Fig. 15b, the statistical distribution of number of atoms partic-ipating in the bulk STZs, which indicates the average diameter of about 1 nm. He also analyzed thecorrelation between successive STZs in the bulk. Fig. 16 shows the distribution of distances betweensuccessive STZs (a) and that of the time intervals between them (b). It is found that the average dis-tance of successive STZs is 1.2 nm and the velocity of transfer of STZs is order of 102 m/s.

2.4. Summary of simulation

The following facts have been deduced from the simulation for elemental and metal–metal alloymodels.

(1) The elementary deformation process is a localized shear transformation (LST) in an area of1–2 nm diameter region with 20–200 atoms.

(2) The successive LST events occur with a separation of 1–2 nm.(3) The LSTs occur preferentially in ill-packed regions characterized by a low density of tetrahedral

packing units, having a low local atomic density, a low local modulus and a local internal stressassisting the LST.

Fig. 15. Statistical distribution of potential energy barrier for STZ’s in a model Ni50Zr50 model metallic glass (a), and that ofnumber of atoms in bulk STZ’s, after Delogu. Reproduce with permission from [57]. Copyright (2009) by the American PhysicalSociety.

Fig. 16. Statistical distribution of the distance rij between the successive ith and jth STZ (a) and that of the time interval Dtij

between the onsets of two successive STZ’s (b), after Delogu. Reproduce with permission from [57]. Copyright (2009) by theAmerican Physical Society.


(4) The yield stress decreases with increasing model size and approaches a bulk value above adiameter of 10 nm. This fact indicates that the yielding occurs through a cooperative activationof LSTs, which is restricted in small model size.

(5) Slowly cooled samples having a lower structure energy exhibit an upper yielding phenomenonfollowed by a steady flow stress, whereas rapidly cooled samples with higher structure energyundergo a smooth yielding. The flow stresses and the structure energies in the steady flow statein different samples of the same model are the same irrespective of the sample history. Thesefacts indicate that the initial density of LST sites in slowly cooled sample is low giving rise toa high yield stress due to weak interaction between LST sites, and the density is increased byplastic deformation leading to a flow stress softening (yield drop) and then it is saturated inthe steady flow region by the balance of the consumption rate of the LST sites and the produc-tion rate of new LST sites under plastic deformation. The upper yielding phenomenon may beunderstood along a similar line as the Johnston-Gilman type yielding of covalent crystals. Wenote again here that the fast cooling rates in simulations can never been realized in experi-ments, and hence the above cooling effects occur only in the simulation world.

(6) The shear band formation occurs in slowly cooled samples of a size larger than 10 nm, but not inrapidly cooled samples. The steady shear flow stress by the shear band formation at 0 K is in therange 0.03–0.04G, which is consistent with experiments. The widths of the produced shearbands are considerably smaller than those experimentally observed, probably because the sizeof the model is still too small. The essential conditions for a localized deformation to continue ina shear band are a work softening as manifested in the yield drop phenomenon and the follow-ing steady flow without work hardening. Under these conditions, once a shear band is formedthe local flow stress in the shear band never exceeds the local yield stress of the surroundingundeformed matrix. For a localized deformation region to be sufficiently work-softened, thelocalized region must undergo plastic shear strain of about 0.1 (see, Fig. 7), which is difficultto be realized in a small model due to the boundary constraint.

Fig. 17. Fundamental unit processes assumed in Spaepen’s model (a) and in Argon’s model (b).


(7) A large plastic deformation is accompanied by a large temperature rise due to the work-done.(8) Thermal activation energy barriers for LST have been calculated. The activation energies distrib-

ute around 0.3 eV. These activation energies do not seem to be evidenced by experiments, butthe calculated activation volumes are almost consistent with experiments

3. Microscopic deformation models

In early years, dislocations and their roles in plastic deformation of metallic glasses were arguedcontroversially. The concept of a dislocation was originally introduced by Volterra and others (see,e.g. [74]) as a line defect in continuum elastic media, which does not require any periodic lattices. Thisfact may indicate that dislocations can exist stably in metallic glasses. In 1970s, Gilman [75,76] and Li[77] tried to describe the slip process in metallic glasses in terms of glide of dislocations. However,dislocations are accompanied inevitably by an intense stress field at their center, which should be re-laxed rapidly through configurational rearrangements possible in amorphous structures due to thelack of translational symmetry, as Spaepen [78] pointed out. Some of the early studies [79,80] andour recent study [64] on the stability of dislocations in metallic glasses are briefly reviewed inSection 3.1.

Two pioneering deformation models [7,81] were proposed in the late 1970s, which have survivedtill today; they provide basic and important concepts in deformation mechanism of metallic glasses.One of the models is the free-volume model by Spaepen [7], and the other is the shear transformationzone (STZ) model by Argon [81]. These two models are based on the assumption that the fundamentalunit process of deformation is a local rearrangement of atoms which accommodates the local shearstrain. These unit processes are schematically depicted in Fig. 17a and b, respectively. Here, the groupof atoms change its configuration under a shear stress from one relatively low energy configuration toa next such configuration by thermal activation.

In these models, we may write the macroscopic shear strain-rate _cðs; TÞ under an applied stress sand a temperature T as

_cðs; TÞ ¼ V0 � c0 � n � f ðs; TÞ; ð4Þ

where V0 is the volume of the deformation unit, c0 is the increment of the local shear strain by a unitprocess, n is the number density of the deformation unit, and f(s, T) is the net frequency of the unitprocess. f(s, T) is generally given by

Fig. 18relaxatindicatof the x


f ðs; TÞ ¼ f0 exp �DGþðsÞkT

� �� exp �DG�ðsÞ

kT

� �� ; ð5Þ

where f0 is the frequency factor, DG+(s) and DG�(s) denote the activation free energies for the forwardand backward processes, respectively. Under the conditions of low temperature and high stress,exp � DGþ

kT

� �� exp � DG�

kT

� , and hence, we can rewrite Eq. (5) as

f ðs; TÞ ¼ f0 exp �DGþðsÞkT

� �: ð6Þ

As the deformation proceeds, a strain-softening may occur, which could be due to the increase in nand/or the increase in f(s, T) through the decrease of DG+. After a certain elapse of time, the systemmay reach a steady-state with:

_csðs; TÞ ¼ V0 � c0 � nsðs; TÞ � fsðs; TÞ; ð7Þ

where _csðs; TÞ, ns(s, T) and fs(s, T) are the steady-state values. Here, we have assumed that ns dependson s and T.

In both models by Spaepen and Argon, the explicit forms of the above equations were derived.Strain-softening mechanism was discussed and formulated. Shear-localization resulting from thestrain-softening was simulated. These results are reviewed in Sections 3.2 and 3.3 for Spaepen’sand Argon’s models, respectively. Recently, Langer et al. have developed a ‘‘two-state STZ model,’’which is based on the previous two models [26,36–40,82]. This model is reviewed in Section 3.4. Sev-eral other deformation models have so far been proposed, some of which are reviewed in previous re-view articles [2–4].

3.1. Dislocation model

In early years, stability of dislocations were investigated for model metallic glasses [79,80], but theresults were rather contradictory. Chanudhari et al. [79] investigated the stability of edge and screwdislocations in an amorphous Lennard–Jones solid. After the introduction of a dislocation, the struc-ture was relaxed, and then the remaining stress field was evaluated. The results showed that the screwdislocation can exist stably but the edge dislocation cannot. Shi [80] studied the stability of

. The displacement field after the initial introduction of an edge dislocation (a) and the displacement field afterion (b). The displacement of each atom with respect to the atom position before the introduction of the dislocation ised by an arrow, where the length of the arrow is three times larger than the actual atomic displacement. The magnitude-component of the displacement is shown in the color scale.


dislocations in essentially the same model amorphous structure as used by Chaudhari et al. [79], andreported that both edge and screw dislocations are stable.

Recently, we have studied the stability of edge and screw dislocations in a model amorphous struc-ture of a Ni33Y67 alloy [64], which was constructed according to the standard MD method using two-body interatomic potentials [83]. The size of the model is approximately x � y � z = 10 � 10 � 5 nm3.An edge or a screw dislocation with the Burgers vector of b = 0.5 nm was introduced parallel to z-axisat the center in the x–y plane by a cut-and-shift procedure, where the initial displacement field wasgiven according to the elasticity theory. Then, the structure of the dislocated model was staticallyrelaxed under the periodic boundary condition in z-axis and the free boundary condition in x- andy-axes. The displacement field and the stress field were calculated to examine the stability of thedislocations in the model metallic glass.

Fig. 18a shows the atomic displacement field after the initial introduction of an edge dislocationand (b) shows the displacement field after relaxation. Obviously, the initial displacement field is partlyrelaxed but a large fraction of it remains. In Fig. 19a and b, we have evaluated the stress component rxx

on the plane x = 0 as a function of y for the structures before and after the relaxation, respectively.Here, the solid circles indicate the rxx values calculated for the structures, and the solid lines indicatethe function rxx(y) according to the elasticity theory. Though the calculated stress values are consid-erably scattered, it is clear that the stress near the edge dislocation core, at least in the range y < 1 nm,is almost completely relaxed. It should be noted that this result is not inconsistent with the result thata large fraction of the initial displacement field remains after the relaxation; to relax the stress field,the structure does not need to go back to the initial structure without the dislocation. In the picture ofthe potential energy landscape, there should be many local potential minima near the point corre-sponding to the structure before relaxation in the configurational space. The stress relaxation canbe achieved by reaching one of these nearby minima.

The result for the screw dislocation was qualitatively the same as that for the edge dislocation;while the displacement field of the screw dislocation largely remained after the relaxation, the stressfield vanished almost completely. Fig. 20 shows the stress component rhz evaluated as a function ofthe distance r from the dislocation center for the structures before and after relaxation. Here, rhz val-ues were evaluated by averaging over many h values to cancel out the effect of the structural inhomo-geneity. As a result, the scatter of the data is largely suppressed, compared with that for the edgedislocation shown in Fig. 19b and also with those for screw dislocations in the previous works byChaudhari et al. [79] and Shi [80]. The solid line shows the function rhz(r) according to the elasticitytheory. As seen in the figure, the stress field around the dislocation in the model before the relaxation

Fig. 19. The stress component rxx on the plane x = 0 as a function of y for the structures before relaxation (a) and afterrelaxation (b) for an edge dislocation. Here, the solid circles indicate the rxx values calculated for the structures, and the solidlines indicate the function rxx(y) according to the elasticity theory.

Fig. 20. The stress component rhz evaluated as a function of the distance r from the dislocation center for the structures beforeand after relaxation for a screw dislocation.


agrees very well with the elasticity theory whereas after the relaxation the stress field has beenrelaxed almost completely.

Our calculations have indicated that both edge and screw dislocations cannot exist stably in metal-lic glasses without a significant core-blunting, i.e., a large stress-relaxation around the dislocationcore. However, a nascent shear band should be formed by propagation of a shear front. The line ofthe shear front represents the boundary between a region which has been sheared and a region whichhas not, and therefore it is nothing else but a dislocation. Here, it is reasonable to assume that thestress concentration at the shear front itself makes the shear front move forward; the stress concen-tration should activate the deformation units such as STZs ahead of the shear front, as pointed out bySchuh et al. [2]. As shown in Section 2.3.3, the propagation speed of shear front is quite high, which isconsistent with the above assumption, because the shear front could keep the stress concentrationunder such a high-speed propagation.

3.2. Free-volume model by Spaepen

The unit process of deformation in this model is schematically depicted in Fig. 17a, where an atomjumps into an adjacent open space. Spaepen [7] derived the number density n of the deformation unitand the net frequency f of the unit process, as follows. We define a free volume vf of a given atom as

v f ¼ v � v0; ð8Þ

where v denotes the volume of the space that the atom occupies, and v0 is the volume of the atom.Here, vf v0 is generally satisfied in metallic glasses. vf values of the atoms are distributed statisticallyin a metallic glass structure, and only the atoms with sufficiently large vf can make a jump. Accordingto the free volume theory of Cohen and Turnbull [85], the number density of such atoms has theform:

n ¼ 1v0

exp � avvave

f

!; ð9Þ

where v⁄ is a critical free volume, vavef is the average free volume over all the atoms, and a is a geo-

metrical factor between 1 and 1/2. The net frequency f of the process is given in the same form asEq. (5), i.e.,

Fig. 21. Schematic illustrations of the effective free energy curves without applied stress (a) and with an applied stress (b).


f ¼ f0 exp �DGþ

kT

� �� exp �DG�

kT

� �� : ð10Þ

Spaepen [7] gave DG± as

DG� ¼ DG0 � sv0

2; ð11Þ

where DG0 is the activation free energy without applied stress. The above equation is based on theassumption that the dependence of the effective free energy h(cloc) on the local shear strain cloc underapplied stress s is given by

hðclocÞ ¼ gðclocÞ � sV0cloc; ð12Þ

as shown in Fig. 21, where g(cloc) is the potential curve without applied stress, V0 is the volume of thedeformation unit. Here, the term �sV0cloc corresponds to the work-done. In the Spaepen’s model,V0 � v0 and c0 � 1 (c0: the increment of shear strain by a unit process). Then, DG± can be approximatedby Eq. (11), when s is small. For the region with large stresses close to the athermal shear resistence sthat satisfies DGþðsÞ ¼ 0, Johnson and Samwer [59] have derived the functional form of DG+(s), whichwill be reviewed in Section 3.3. Inserting Eqs. (9)–(11) into (4), we obtain the strain-rate _c as

_c ¼ exp � avvave

f

!� 2f 0 � sinh

sv0

2kT

� �� exp �DG0

kT

� �; ð13Þ

where we used the relations V0 � v0 and c0 � 1.Spaepen [7] discussed the mechanism of strain-softening within the framework of his free-volume

model. Here, the plastic strain is assumed to increase the average free volume vavef , resulting in the

increase in the density n of the deformation unit, leading to the increase in the macroscopic strain-rate_c. Microscopically, the increase in the free volume results from an atomic jump into a relatively smallspace: the atom is squeezed into the space to create a certain amount of free volume. The rate þ _vave

f ofthe increase in the average free volume vave

f was derived to be

þ _vavef ¼ av

vavef

� 3l� 1� m1þ m

� kT coshsv0

2kT

� �� 1

n o� Nf0 exp �DG0

kT

� �� exp � av

vavef

!; ð14Þ

Fig. 22. The geometry of the structure for the calculation of shear localization [84].


where l is the shear modulus, m is Poisson’s ratio, and N is the total number of atoms. The annihilationrate � _vave

f of the free volume vavef by atomic diffusion was also estimated:

� _vavef ¼ v

nDNf0 exp �DG0

kT

� �� exp � av

vavef

!; ð15Þ

where nD is the number of diffusive jumps necessary to annihilate a free volume equal to v⁄, which is anumber between 1 and 10. The net rate _vave

f is given as _vavef ¼ þ _vave

f � � _vavef . In steady-state, _vave

f ¼ 0,leading to:

smavef ¼ aND �

3l� 1� m1þ m

� kT coshsv0

2kT

� �� 1

n o; ð16Þ

where smavef denotes the average free volume in steady-state. Substituting this smave

f for vavef in Eq. (13),

we can obtain the steady-state macroscopic strain-rate _cs.Steif et al. [84] simulated the shear localization based on the free-volume model. The geometry of

the structure used for the calculation is shown in Fig. 22. Here, the structure consists of the matrix anda thin band of volume fraction q, which is typically of the order of 10�6. Both the matrix and the bandobey the same type of constitutive equations:

_slþ _cb ¼ _ctot

b

_slþ _cm ¼ _ctot

m

(; ð17Þ

q _ctotb þ ð1� qÞ _ctot

m ¼ _ca;

where the first term _s=l denotes the elastic strain-rate, and _cb and _cm are the inelastic strain-rate forthe band and matrix, respectively, which are given by Eq. (13). In the initial condition, vave

f for the bandis slightly larger than that for the matrix, i.e., the band is slightly softer than the matrix. _ca is the rate ofthe strain given to the sample, and is assumed to be constant. An example of the shear strain in theband cb calculated as a function of the applied strain ca is shown in Fig. 23. Here, we can see that acatastrophic strain-localization occurs once a critical point has been reached.

3.3. Shear transformation zone model by Argon

The unit process of deformation in this model is schematically depicted in Fig. 17b, where a localatomic cluster called a ‘‘shear transformation zone (STZ)’’ undergoes an inelastic shear deformationunder an applied stress. In the high stress range, Eqs. (4) and (6) give the shear strain-rate:

_c ¼ V0 � c0 � n � f0 exp �DGþ

kT

� �: ð18Þ


In analogy with the nucleation of a dislocation loop, Argon derived DG+ as:

F

DGþ ¼ C � s � V0 1� ss

� �2; ð19Þ

where C � 4.6 is a constant and s is the athermal shear resistence that satisfies DGþðsÞ ¼ 0. Eqs. (18)and (19) lead to

s ¼ s� s 1B

ln_cA

� �12

ðkTÞ12 ðA ¼ V0 � c0 � n � f0; B ¼ C � s � V0Þ: ð20Þ

This gives the dependence of the critical shear stress s on the strain-rate _c and the temperature T,which can be compared with experimental results. Here, we notice a small strain-rate sensitivity ofthe form ðln _cÞ

12 and also a relatively small temperature dependence of the form T

12 .

Johnson and Samwer [59] have derived the functional form of DG+ in a high stress region on thebasis of the model depicted in Fig. 21 in the preceding section. At a stress close to the athermal shearresistance s, the resistance stress vs. strain relation can be approximated by s ¼ s� C1ðDcÞ2 (Dc: acti-vation strain, C1: a constant). Then, the activation free energy can be written as

DGþðsÞ ¼ C2 � s � 1� ss

� �32: ð21Þ

Inserting this equation in Eq. (18) leads to

s ¼ s� s � C3 � T23: ð22Þ

This is similar to Eq. (20) in the Argon’s model except that the exponent of T is 2/3. Johnson andSamwer [59] have collected the experimental data of the temperature dependence of the yield stressfor various metallic glasses reported in literature, and have shown that they satisfy well the functionalform of Eq. (22). Fig. 24 reproduces the data, which are fitted to the relation of Eq. (22). However,Schuh et al. [2] have pointed out that the functional form of Eq. (20) can also be fitted to the same dataas well.

The shear strain-rate in the low-stress range with s s was derived as

_c ¼ V0 � c0 � n � 2f 0 � sinhs � c0 � V0

2kT

� �� exp �DG0

kT

� �; ð23Þ

which is essentially the same form as Eq. (13) in the Spaepen’s model. For s 2kTc0 �V0

, Eq. (23) can berewritten as

s ¼ kTA � V0 � c0

� _c � expDG0

kT

� �: ð24Þ

ig. 23. An example of the shear strain in the band cb calculated as a function of the applied shear strain ca [84].

Fig. 24. The yield stress normalized by shear modulus plotted against temperature normalized by glass transition temperaturefor various metallic glasses. These data were collected by Johnson and Samwer [59]. The solid line indicates the fitting curve ofthe form in Eq. (22).


In contrast to Eq. (20), Eq. (24) exhibits a strong strain-rate sensitivity of the _c-linear form and a strongtemperature dependence of the exponential form of exp DG0

kT

� . This shows a Newtonian viscous flow,

which is observed experimentally at high temperatures.Argon [81,86] discussed the mechanism of strain-softening in view of the increase in the free vol-

ume during deformation, as Spaepen did. However, Argon attributed the strain-softening not directlyto the increase in the number density n of the deformation unit but rather to the decrease in the ather-mal shear resistance s, i.e., the decrease in the activation free energy DG0. He also analyzed shearlocalization in a model similar to that shown in Fig. 22.

3.4. Two-state shear transformation zone model by Langer et al.

The basic assumption in this model is similar to the Spaepen’s and Argon’s models: it is assumedthat the fundamental unit process of deformation is a local rearrangement of atoms, which is essen-tially the same as an STZ operation. The major revision in this model for the earlier models lies in theassumption that the STZs are intrinsically two-state systems. In response to an applied stress, theymay transform from one orientation to another, but they cannot transform further in the same direc-tion. Here, they may transform back to the original orientation when a stress is applied in the reversedirection. This is the first fundamental assumption in this model. Such a property of STZs has beenobserved in 2-D model simulations [26,27,33]. Below, formulation of deformation behaviors in thismodel is reviewed, mainly for a low temperature region where thermal effects are negligible.

Under the assumption of the two states, the macroscopic shear strain-rate can be written as

_c ¼ V0 � c0fRðsÞ � n� � Rð�sÞ � nþg; ð25Þ

instead of Eq. (4). Here, ± denote the two states, and n± are the number densities of the STZs in thesetwo states. We assume that an applied stress s (>0) promotes the transition of the STZs from ‘‘�’’ to‘‘+’’. R(s) and R(�s) denote the transition rates in the forward (�? +) and the backward (+ ? �) direc-tions, respectively. The time derivatives of n± are given as

_nþ ¼ RðsÞ � n� � Rð�sÞ � nþ þ A� B � nþ_n� ¼ Rð�sÞ � nþ � RðsÞ � n� þ A� B � n�

�; ð26Þ

where the first two terms in the right-hand side represent the transitions between the two states, andthe last two terms correspond to the creation and annihilation rates of STZs. Here, both A and B areassumed to be proportional to the energy dissipation rate per STZ, i.e., s� _c

nþþn�, which is the second major

revision to the earlier models. We write


A ¼ B2 ne

B ¼ s� _cV0 �c0 �sc �ðnþþn�Þ

(; ð27Þ

where ne is the steady-state value of the total number density n+ + n� of STZs, which is deduced from thesum of the two equations of Eq. (26). sc, which has the dimension of stress, is defined here to give the pro-portionality factor, which will be shown later to be a critical stress in the ‘‘jamming’’–‘‘unjamming’’transition.

The third fundamental assumption in this model is that ne is given as

ne ¼ n1 exp � EkTeff

� �; ð28Þ

where E is a typical formation energy of a STZ, Teff is an effective temperature, and n1 is a number den-sity of the order of V�1

0 . Teff is different from the ordinary thermal temperature; it is an indicater of thedegree of configurational disorder in the system that is not in thermal equilibrium. Here, Teff plays asimilar role to the average free volume vave

f in the Spaepen’s model, where the number density ofthe active sites is assumed to be proportional to expð�const:=vave

f Þ (see Eq. (9)). Inserting Eqs. (27)and (28) into (26) yields

_nþ ¼ RðsÞ � n� � Rð�sÞ � nþ þ s� _cV0 �c0 �sc �ðnþþn�Þ

n12 expð�1=vÞ � nþ�

_n� ¼ Rð�sÞ � nþ � RðsÞ � n� þ s� _cV0 �c0 �sc �ðnþþn�Þ

n12 expð�1=vÞ � n��

8<: ; ð29Þ

where v kTeff/E is a reduced effective temperature. We define the following quantities:

e0 V0 � c0 � n1; ð30Þ

K nþ þ n�n1

; ð31Þ

m nþ � n�nþ þ n�

; ð32Þ

CðsÞ 12fRðsÞ þ Rð�sÞg; ð33Þ

and

TðsÞ RðsÞ � Rð�sÞRðsÞ þ Rð�sÞ : ð34Þ

Here, m is an order parameter of the orientation of STZs: m = ±1 when all the STZs are in the ± orien-tations, respectively, and m varies from �1 to 1 as the number ratio between the two orientations var-ies. Using Eqs. (30)–(34), Eqs. (25) and (29) can be rewritten as

_c ¼ e0 �K � CðsÞ � fTðsÞ �mg; ð35Þ

_m ¼ 2 � CðsÞ � fTðsÞ �mg 1�m � ssc� expð�1=vÞ

K

� �; ð36Þ

and

_K ¼ B � fexpð�1=vÞ �Kg: ð37Þ

Eq. (37) indicates K = exp (�1/v) in the steady-state, which is consistent with Eq. (28). Eq. (36) hastwo steady-state solutions: m = T(s) and m = sc/s, giving _c ¼ 0 and _c – 0, respectively. These two stea-dy-states are called ‘‘jammed’’ and ‘‘unjammed’’ states in the language of granular matters. Now, weconsider the case of s > 0. At low temperatures, R(s)� R(�s), indicating T(s) � 1. Then, in the jammedstate, m � 1, i.e., all the STZs are in the + direction. We can see from Eq. (36) that this jammed state isdynamically stable for s < sc. Here, s is so small that the creation of the ‘‘�’’ STZs cannot supply their


consumption, and finally the ‘‘�’’ STZs become exhausted to stop further deformation. On the otherhand, the unjammed state with m = sc/s and _c – 0 is dynamically stable for s > sc. Here, new ‘‘�’’ STZsare created as fast as existing ‘‘�’’ STZs zones are consumed.

In view of an energy-balance, Langer [38] and Bouchbinder et al. [82] wrote the differential equa-tion for v as

c0 � _v ¼ e0 �K � B � ðv1 � vÞ; ð38Þ

at low temperatures. Here, c0 is a factor related to the specific heat of the configurational degrees offreedom, and v1 is the state-state value of v. Eqs. (35)–(38) give a complete set of equations of motionin this model. To calculate explicitly the deformation behaviors, we need explicit forms of C(s) andT(s), or R(s). Different forms of these functions have been given using different approximations[26,36,38,82]. Thermal effects and deformation behaviors at high temperatures have been discussedand formulated in [37,40]. Shear localization has been analyzed in [87,88].

4. Conclusion

In this decade, a considerable progress has been made and still being made both in computer sim-ulation studies of deformation. The atomistic process of deformation in metallic glasses has been elu-cidated to a great extent, reproducing various experimental features such as the shear band formationand the homologous flow stress. However, detailed atomistic process of realization of the steady flowstate and the detailed process of the shear band nucleation and its propagation have not been fullyclarified yet. Namely, the cascade process of the LSTs, and the generation and annihilation processof STZs have not been fully understood. The widths of the produced shear bands are still narrowerthan those of experiments and so it seems necessary to perform simulations for larger size models.

So far, only elemental and metal–metal alloy systems have been employed for simulations, becauseinteratomic potentials for these systems are simpler than those for metal–metalloid systems.Although no distinctly different features are recognized experimentally for the deformation behaviorof the two kinds of systems, it seems necessary to perform also for metal–metalloid systems to eluci-date why the mechanical behavior is common in spite of the presumably different packing units be-tween the two kinds of metallic glasses.

In the shear band deformation, it is necessary to make clear which process is controlling the flowstress, nucleation or propagation, and in which process a thermal activation is involved. Consideringthe quite fast shear propagation observed in experiments, it is highly probable that a thermal activa-tion is involved in the nucleation process and not in the propagation process. It seems necessary tosimulate a thermally activated nucleation of a shear band.

In relation to the free-volume model of plastic deformation, the simulation results indicate that thelarger is the average free volume in the region, the lower is the deformation stress in the region, whichis apparently consistent with the free-volume model. It seems, however, that the low deformationstress is related directly to the ill-packingness of the region and the large average free volume ofthe region is a result of the ill-packingness. As mentioned in a previous section, the simulation byDelogu [73] showed that an artificial displacement of one atom can trigger LST with a probability ofseveral percentages. It seems important to characterize the environments of those atoms that triggerLSTs to see whether or not they are consistent with the Spaepens’s model as illustrated in Fig. 17a.

The two-state STZ model is attractive in its comprehensive scheme. However, the fundamentalassumption of the two-state nature of the STZs seems too simple and idealized: two-state nature doesnot seem to be evidenced by 3-D simulations nor by experiments. The theoretical models of Spaepenand Argon and the two-state STZ model are all essentially the mean-field theory. In these models, thedeformation localization is described by a bifurcation process as a result of the spatial structure fluc-tuation. The nucleation and propagation process of a shear band observed in experiments and simu-lations have not been incorporated satisfactorily in those models, although the effective temperaturetheory in the two-state STZ model has been shown to contain an instability against shear banding [38],and further developments along this line is expected [39].


As for the modeling of localized shear deformation, it is clear that the deformation mechanism bydislocation glide in an atomic plane in metallic glasses is not a realistic model, but the zonal disloca-tion concept, which was developed for the non-basal slip in hcp metals and the twinning deformation,may be worth to apply to the shear band propagation.

For the full understanding of the microscopic deformation mechanism of metallic glasses, three dif-ferent approaches, experiments, computer simulations and theoretical modeling, should be consistentwith each other. At the present stage, among three consistencies, the consistency between simulationand experiment is coming to a considerably level, but that between simulations and modeling is stillin a low level. Any theoretical model involves some parameters that must be determined by experi-ment or computer simulation. Particularly, parameters describing microscopic processes must bedetermined with the help of computer simulations. In this respect, it seems essential to collaboratebetween the groups of the computer simulation and the theoretical modeling to construct a reason-able and realistic model.

Acknowledgments

The authors are grateful to Professor J. Th. De Hosson for providing us with the opportunity of writ-ing this review article as one of the chapters in a special issue of Festschrift for Vasek Vitek. They areindebted to Dr. Y. Kamimura of Institute of Industrial Science, The University of Tokyo for his help inpreparing the manuscript.

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Atomistic simulation and modeling of localized shear deformation in metallic glasses