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Acta Materialia 55 (2007) 4317–4324

Stress-induced structural transformation and shear bandingduring simulated nanoindentation of a metallic glass

Yunfeng Shi, Michael L. Falk *

Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109-2136, USA

Received 13 December 2006; received in revised form 23 March 2007; accepted 24 March 2007Available online 25 May 2007

Abstract

Simulated nanoindentation tests on a three-dimensional model of a binary metallic glass-forming alloy reveal how the stress field andmaterial structure interact to control deformation beneath the indenter. Initially, the stress field follows the Hertzian solution with fluc-tuations due to heterogeneities. Homogeneous or localized plastic deformation arises, depending on the processing history of the mate-rial. In the case of localized deformation, the first shear band initiates sub-surface, relaxing the local shear stress. The subsequent shearband morphology is observed to be rate-dependent. Deformation induced changes in material structure are characterized in terms ofshort-range ordering and free volume, with the former providing significantly greater signal-to-noise.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Molecular dynamics; Nanoindentation; Metallic glasses; Shear bands; Incipient plasticity

1. Introduction

Metallic glass (MG) is of great technological interestbecause of its excellent mechanical properties and its poten-tial for wider use in structural applications and emergingnanoelectromechanical systems [1–4]. Like many nanocrys-talline materials [5], however, when MG materials aremechanically loaded in unconfined testing geometries theplastic deformation localizes into shear bands while the restof the material deforms elastically, resulting in little overallductility. Strain localization in MG appears to arise fromstructural rather than thermal softening [6,7]. The temper-ature rise for final fracture of MG samples can be wellabove the melting point [7], but estimates of the tempera-ture rise inside a shear band before failure have rangedfrom less than a few degrees Kelvin to thousands of degrees[6,8]. Recent reports indicate that the temperature rise isdelayed with respect to the plastic deformation [9] andthe width of the heated zone is much larger than the shear

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.03.029

* Corresponding author. Tel./fax: +1 734 615 8086.E-mail address: mfalk@umich.edu (M.L. Falk).

band width [7]. Furthermore, suppression of localizeddeformation by high strain rate has been demonstratedboth experimentally [10] and numerically [11]. These resultsall provide evidence that the instability does not arise fromthermal softening.

The most often considered atomistic structural softeningmechanism, local free volume creation [12–15], has beenmeasured experimentally [16], but the difficulty of measur-ing free volume in situ with sufficient temporal and spatialresolution has prevented its role in localized deformationfrom being directly confirmed experimentally. In simula-tions this is not an issue, but in recent studies the signal-to-noise ratio of such a signature in the density wasobserved to be low [9]. Although density or free volumeare promising candidates, there are other more subtlechanges in structure that can be quantified and related toliquid structure and glass formability [17,43,44]. For exam-ple, we have shown in previous simulations of a particulartwo-dimensional glass-forming system that there is deple-tion of short-range order (SRO) within the shear bandwhich results in a structural softening mechanism [18]. Sim-ilar observations have been made in several three-dimen-sional systems as well [19]. Recently, work-hardening

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4318 Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324

accompanied by exceptional ductility has been observed ina CuZrAl alloy, which is attributed to the existence ofnanoscale structural inhomogeneities [20]. However, theprecise nature of these inhomogeneities and the means bywhich they control the mechanical response remain to bedetermined. A coherent understanding of the underlyingphysics of deformation capable of explaining both work-softening and work-hardening in MG and its compositesis needed.

Nanoindentation has become the preferred testingmethod to measure the mechanical properties of bulkmaterials as well as nanometer scale structures [21–24]because it provides the means to apply load to a confinedsample region. However, the resulting plastic deformationoccurs primarily sub-surface. As a result, only the conse-quences of the deformation in forming external surfacesteps and in influencing the load–displacement curves areimmediately accessible. Moreover, it has been observedthat both the contact area and stress distribution withinthe contact deviate from continuum theory due toatomic-scale roughness making the load displacementcurves difficult to analyze [25]. It is not yet clear whethercontact theory holds for the stress distribution inside thebulk of the loaded sample on the nanometer scale.

Here, we present simulations of a three-dimensional bin-ary Lennard–Jones model glass mechanically tested in ananoindentation geometry. The incipient plasticity is char-acterized by the deformation morphology and stress fieldpattern underneath the surface. Furthermore, we quantifythe structural difference between the material inside theshear bands and that outside for the sample that exhibitsthe most localized deformation. It is found that the degreeof ductility of metallic glass is controlled by structuraltransformations on the atomic scale induced by deforma-tion. These transformations are quantified in terms of freevolume and SRO.

2. Simulation setup

We have performed a molecular dynamics simulation ona three-dimensional binary alloy interacting with a Wahn-strom potential [26]. The alloy consists of two species,which will be referred to as S and L for small and large,interacting via a Lennard–Jones potential of the form

/ij ¼ 4errij

� �12

� rrij

� �6" #

ð1Þ

Table 1Cooling schedule for three samples

Tstart (K) Tend (K) Pstart (GPa) Pend (GPa)

Step I 2100 1290 16.23 9.73Step IIa 1290 790 9.73 5.77Step III 790 60 5.77 0.00

a This temperature range crosses the glass transition point where most of thedepends most sensitively on the quenching rate of step II.

where e represents the bonding energy and r provides alength scale, the distance at which the interaction energyis zero. The SS and LL bond energies are equal to thatof the SL bond energy so that eSS = eSL = eLL. The SSand LL length scales are related to the SL length scale by

rSS ¼5

6rLL; rSL ¼

11

12rLL ð2Þ

The reference length scale is chosen to be rLL and the ref-erence energy scale to be eLL. The two types of atoms havedifferent masses such that mL = 2m0,mS = m0. The refer-ence time scale is t0 ¼ rLL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffim0=eLL

p. TMCT, the mode cou-

pling temperature, was measured to be 0.57eLL/k [27],where k is the Boltzmann factor. The closest binary metal-lic glass, in terms of composition and atom radius ratio, isperhaps Ni50Nb50, with Nb being the larger species. Bothquantities match the current model glass exactly. However,a less important quantity, the mass ratio for Ni50Nb50 is 1.6instead of 2.0. Therefore the reduced units for this Wahn-strom system can be matched to physical units:t0 � 0.5 ps, rLL � 2.7 A, m0 � 46 amu and TMCT � 1000 K[28,29].

The glassy samples were created by starting from super-cooled liquids equilibrated at 2100 K. Subsequent to equil-ibration, the temperature of the liquid was reduced to 6%of TMCT or 60 K. We employed a three-step coolingscheme to carefully control the rate of the quench at tem-peratures near the glass transition (Table 1). The tempera-ture and pressure of the system were continuously reducedduring each cooling step. There were three samples pro-duced by this procedure which are referred to as samplesI–III, with cooling rates increased 50-fold progressively instep II (Table 1). All three samples were then tiled eightacross by three down to create a single slab of 600,000atoms (50% Nb atoms and 50% Ni atoms) in a box of size�100 · 35 · 2.5 nm3 for the three samples. Periodic bound-ary conditions were imposed along the X- and Z-directions.This thin-slab geometry was selected to maximize the in-plane spatial dimension and to mimic the plane straincondition.

During nanoindentation tests, a frictionless cylindricalindenter was lowered into the glassy thin films as illustratedin Fig. 1. The indenter was modeled by a purely repulsivepotential with the form:

/Ii ¼ eLL

rIi � RI

0:6rLL

� ��12

ð3Þ

tcool (ps), sample I tcool (ps), sample II tcool (ps), sample III

50 50 5025,000 500 10

50 50 50

TCS SRO will be formed. Therefore the structure of the final glassy sample

Fig. 1. Illustration of the nanoindentation testing geometry. Periodicboundary conditions (PBC) apply on the X–Y and Y–Z planes.

0 1 2 3 4 5Displacement (nm)

0

0.2

0.4

0.6

Loa

d (μ

N)

Sample I

Sample II

Sample III

Fig. 2. The load displacement curves from nanoindentation simulationsperformed on samples I (red), II (green) and III (blue). The dotted lineshows the elastic prediction for indentation in a material for sample I. (Forinterpretation of the references to colour in the figure legend, the reader isreferred to the web version of this article.)

0.5

Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324 4319

where1 rIi is the distance between atom i and the center ofthe cylindrical indenter projected to the X–Y plane. RI, theradius of the indenter, was chosen to be 40 nm. The inden-ter acted as a material of infinite stiffness with no surfacefriction. The lower edge of the slab was held fixed as ifthe amorphous film were deposited upon a substrate of infi-nite stiffness. The indenter was lowered into the substrateunder displacement control at a velocity of 0.54 m s�1 toa maximum displacement of 5.0 nm. Although these inden-tation rates are high compared to typical experimentalrates, they are still well below the rates at which shockwaves or other high-impact phenomena become important.This can be quantified by noting that the sound speed ofthe glassy sample is �2 km s�1 for sample I. Therefore,the indenter speed was more than three orders of magni-tude smaller than the sound speed. A sound wave needsroughly 50 ps to travel along the longest spatial dimensionof the specimen, the time it takes to depress our indenter by�0.3 A. During indentation, the system was maintained ata constant temperature of 6% TMCT, or 60 K, using aNose–Hoover thermostat [30,31].

0 1 2 3 4Y/a

-1

-0.5

0

Stre

ss/p

0

τ

σx

σy

Fig. 3. Local in-plane stresses along the center line of the sample directlyunderneath the indenter just before the incipient of plasticity for sample I.The indenter is held at a nominal indentation depth of 1.22 nm for 100 psin order to reduce temporal local stress fluctuations. Y is the depth awayfrom the contact surface in the Y-direction; a, the half projected contactlength, is measured to be 7.2 nm and p0, the maximum pressure, ismeasured to be 6.6 GPa. The solid lines are Hertzian solutions for stress inthe X-direction, the Y-direction and maximum shear stress.

3. Simulation results

3.1. Hertzian contact

The vertical force on the indenter rises elastically fromzero according to Hertz theory before the inception of plas-tic deformation. The load follows [32]:

P ¼ pa2E�

4RLz ð4Þ

where P is load, R is the reduced elastic constant and a ishalf of the projected contacted length. Atoms are consid-ered in contact with the indenter if they are within0.22 nm of the surface of the indenter. Lz is the length ofthe simulation box in the Z-direction. E* is the reducedYoung’s modulus in plain strain conditions:

1

E�¼ 1� v2

s

Es

þ 1� v2i

Ei

ð5Þ

where Es and Ei are the elastic modulus of the sample andthe indenter and v is the Poisson’s ratio. For sample I, Es ismeasured to be 70 GPa, and vs is 0.364. Ei is infinite for therigid indenter. Fig. 2 shows the simulated load–displace-ment curves as well as the Hertzian solution for sample I.Although the overall load on the indenter follows Eq. (4),it is also important to determine whether the spatial distri-bution of the stress underneath the indenter obeys contin-uum elasticity. This is done by calculating the stresstensor for each volume element with dimensions1.0 · 1.0 · 2.5 nm3, over the entire system. The stress pro-file for rxx, ryy and maximum shear directly beneath the in-denter for sample I just before the incipient plasticity isshown in Fig. 3. The result agrees well with the Hertziansolution.

4320 Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324

3.2. Incipient plasticity

As later demonstrated, sample I exhibits the most local-ized deformation underneath the indenter among the threesamples. Therefore, we focus on the onset of plasticity insample I. The first shear band nucleates at a displacementbetween �1.24 nm and 1.30 nm. The plastic deformationcauses a discontinuity on the load–displacement curve, asshown in Fig. 4a. The local atomic rearrangements canbe characterized by the deviatoric shear strain using thenumerical method introduced in Ref. [33]. Fig. 4b showsthe plastic deformation of the system evolving from a nom-inal displacement of 1.22–1.27 nm, the midpoint of the

1 1.2Displace

0.15

0.2

0.25

Loa

d (μ

N)

(

(b)

(d)

a

b

d

Fig. 4. (a) The load displacement curve for sample I during the formation onucleation of the first shear band of sample I. The local shear strain is calcu1.22 nm to those at a depth of 1.27 nm (b) and 1.22 nm to 1.32 nm (c). Black delocal shear stress distribution before the nucleation of the first shear band of sa100 ps in order to reduce temporal local stress fluctuations; (e) local shear streindenter is held at a nominal indentation depth of 1.32 nm for 100 ps in order todenotes the magnitude and direction of the maximum shear for each volumedenotes the maximum shear stress (1.99 GPa) predicted by elasticity.

nucleation process. Fig. 4c shows the plastic deformationfrom a nominal displacement of 1.22–1.32 nm, the end ofthe first nucleation of a band. The initial shear bandnucleus runs from the upper-left to the lower-right follow-ing the a-lines in the slip-line field. This initial shear bandnucleus is apparently off-center and away from the locusof maximum shear. There is also very little shear bandgrowth along this direction. Instead, the shear band prop-agates along the b-lines as shown in Fig. 4c. The centerpoint of the shear band lies �5.4 nm below the surfaceand is �45� away from the vertical direction. At a nominaldisplacement of 1.22 nm where elasticity still holds, Hertz-ian theory predicts that the maximum shear will be located

1.4 1.6ment (nm)

c)

(e)

c

e

f the first shear band; (b and c) local shear strain distribution during thelated by comparing the configurations at a nominal indentation depth ofnotes 10% strain while white denotes 0% strain. The scale bar is 2.7 nm; (d)mple I. The indenter is held at a nominal indentation depth of 1.22 nm forss distribution after the nucleation of the first shear band of sample I. The

reduce temporal local stress fluctuations. The size and orientation of crosselement with dimensions of 0.54 · 0.54 · 2.5 nm3. The cross on the right

Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324 4321

at 0.78 a or 5.37 nm beneath the indenter, with a 45� orien-tation away from the vertical direction, both in excellentagreement with simulation results.

We also calculates the stress field before and after thefirst shear band nucleated to see whether there is any signa-ture in the stress field that would predict the resulting shearband morphology. The stress field at a nominal indentationdisplacement of 1.22 nm is shown in Fig. 4d. The stressfield fluctuates spatially due to structural heterogeneities,but there was no recognizable high shear stress region thatcorrelated to the shear band later observed. The orienta-tion of the shear band, however, does follow the local max-imum shear direction. As shown in Fig. 4e, the stress fieldin the area where the shear band emerges diminished signif-icantly, resulting in a considerably more inhomogeneousstress field. The resulting stress field is crucial for determin-ing the morphology of further shear band development.

3.3. The effect of quenching rates

As shown in Fig. 2, sample I, produced at the slowestquench rate, exhibits the highest hardness, while sampleIII, produced at the fastest quench rate, exhibits the lowest.After yield, serrated flow is evident in the load displace-ment data for sample I while the load displacement curvesare progressively smoother for the other two samples. The

Fig. 5. Visualization of the magnitude of the net local deviatoric shear strain usample II; (IIIa) sample III. Yellow denotes regions of high local strain saturasamples at the maximum indentation depth are characterized using TCS SROrespectively. Both methods employ a coarse-grain averaging over volume elemrepresents the mean value of the structural measurements, 15.0% for TCS SROstructural measurements, 5.0% for TCS SRO and 39.5% for free volume.

discontinuities in the load–displacement curves corresponddirectly to shear band activity [18,34].

Fig. 5Ia, IIa and IIIa show images of the regions of highlocal strain in all three samples at the maximum indenta-tion depth. The deformation morphology follows the pre-diction of the slip-line theory [35]. A few shear bandscarry the majority of the plastic flow in sample I. Two ofthose grow to the surface causing surface steps and pile-ups. Sample III exhibits the majority of deformation in aregion immediately beneath the indenter and the plasticflow is more evenly distributed with smaller spacingbetween plastic events. Sample II exhibits an intermediatebehavior; the deformation is much less localized than thatof the sample I, yet one shear band still propagates to thesurface. The reason for this relationship between quenchrate and plastic flow will be discussed further in the contextof the structural analysis to follow.

3.4. The effect of strain rates

To study the strain rate effects, we have indented allthree samples using two additional indenter rates, 2.7 and13.5 m s�1, which are five and twenty-five times the originalrate, respectively. For each sample, the load–displacementcurves with different indenter rates are rather similar, withan overlapping elastic regime and only showing higher load

nder the indenter at maximum depth of three samples: (Ia) sample I; (IIa)ting at 40%. Red regions denote zero strain. The structures of the glassy(Ib, IIb and IIIb) and by free volume (Ic, IIc and IIIc) for samples I–III,ents of 2.0 · 2.0 · 2.8 nm3. To maximize the contrast for sample I, blackand 38.2% for free volume; white represents the disordered extrema of the

Fig. 6. Visualization of the magnitude of the net local deviatoric shear strain under the indenter at maximum depth of three samples: sample I, sample IIand sample III (top to bottom) under three indenter rates: 0.54, 2.7 and 13.5 m s�1 (left to right). Here, only a slice of the simulation system on the X–Y

plane is shown, while atoms with large z-values are hidden behind. However, the morphology of deformation does not vary as a function of the Z-positionof the slice. The color coding is the same as in Fig. 5.

4322 Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324

for higher indenter rate at the maximum indentation depth.However, the deformation morphology is quite sensitive tothe indenter rate. As shown in Fig. 6, two types of shearband arise: (1) wing-like shear bands propagating to thesurface, such as the left wing of sample I for the lowestindentation rate and (2) wedge-like shear bands penetratinginto the film, such as the one from upper-right to lower-leftof sample I at the lowest indentation rate. For sample I,two wing-like shear bands emerge on both the left andthe right for the lowest indentation rate. At the intermedi-ate indentation rate, the right wing-like shear banddisappears. At the highest indentation rate, there are nowing-like shear bands in sample I. The same holds for sam-ple II. Indentation rate does not seem to affect the homoge-neous deformation morphology of sample III.

The formation of wing-like shear bands usually leads tolarger discontinuities in the load–displacement curves [36].Therefore, it is conceivable that higher strain rate sup-presses wing-like shear bands, leading to fewer discontinu-ities in the load–displacement curves as seen in experiments[21].

4. Structural analysis

In order to reveal the structural difference between theshear band region and the undeformed region, we usedan analysis based on the idea of efficient local packingfor hard spheres by quantifying the triangulated coordina-tion shell (TCS) SRO [19] as well as the free volume. TCSSRO is defined as the fraction of atoms in configurationsthat maximize the tetrahedral packing as determined by ahard-sphere approximation. A criterion derived from

Euler’s theorem can be written for the coordination poly-hedron [37]:X

q

ð6� qÞvq ¼ 12 ð6Þ

Here, the surface coordination number, q, is defined foreach atom in the coordination shell as the number of neigh-boring atoms also residing in the coordination shell; vq isthe number of atoms in the coordination shell with a sur-face coordination number q. The near-neighbor cutoffsare approximately 0.39, 0.36 and 0.33 nm for the AA, ABand BB bonds, respectively, as determined by the averageof the first and second peak positions in the radial distribu-tion function. We refer to atoms with a coordination shellsatisfying Eq. (6) as SRO atoms. By constraining the sur-face coordination number to be only 5 or 6, Frank andKasper [37] found four of such triangulated coordinationshells with coordination numbers of 12 (icosahedron), 14,15 and 16. Such restriction of the surface coordinationnumber is appropriate only for nearly identical atoms.Here, we do not limit the range of q so that such a criterioncan be applied to general binary systems. However, for theWahnstrom system, q ranges from 3 to 7.

Free volume is defined by Cohen and Turnbull [38] asfollows:

vf ¼ �v� v0 ð7Þwhere �v is the average volume per molecule and v0 is thevan der Waals volume of the molecule. We have found thatour ability to resolve variations in free volume dependscritically on how we define the van der Waals volume. Infact, the most straightforward definition, i.e., that the

0

100

200

300

0

100

200

300

His

togr

am

0 0.1 0.2 0.3Relative TCS SRO

0

100

200

300

Relative Free Volume0.37 0.38 0.39 0.4

Before indentationMax indentationHeavily deformed

Sample I

Sample II

Sample III

Sample I

Sample II

Sample III

Fig. 7. The distribution of TCS SRO (left panel) and free volume (rightpanel) in each sample before indentation testing (blue) and at themaximum load (red) for samples I–III. The TCS SRO and free volume arenormalized by the total number of atoms and the volume of each volumeelement, respectively. The green bars represent the distributions consid-ering only the heavily deformed material. The black arrows (left panel)indicate the shift in the distribution due to plastic deformation. (Forinterpretation of the references to colour in the figure legend, the reader isreferred to the web version of this article.)

Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324 4323

hard-sphere radius of the small atoms is simply rS/2 andthe large atoms are rL/2, results in any free volume con-trast becoming entirely lost in the noise. To obtain anappreciable signal we find that we need to define the vander Waals volume of each atom to be a spherical volumeassociated with the largest radius the atom could adoptwithout overlapping other atoms similarly approximatedas spheres. Formally, we express the radius of this sphere as

ri ¼ minj2N

ri

ri þ rjdij

� �ð8Þ

where atom j is one of the N near neighbors of atom i, anddij is the distance between atom i and atom j. If the neigh-boring atom, j, is of the same species as atom i, then themaximum effective van der Waals radius of i is half ofthe distance between these two atoms. If j is a different spe-cies, then the upper bound on the van der Waals radius of i

due to atom j is calculated by multiplying the interatomicseparation between i and j by the ratio of the hard-sphereradius of i to the sum of the hard-sphere radii of i and j.For the potential used here, ri takes on the value rLL forL atoms (Nb) and rSS for S atoms (Ni). The smallest valueof all the candidate radii for a given atom becomes its vander Waals radius. During coarse-graining, the system is di-vided into a number of cells. The boundaries of each cellmay cut through the spherical shells of many atoms. In thatcase, the van der Waals volume of each atom is partitionedamong the cells accordingly. The average free volume ineach cell is then defined as:

�vf ¼ v�X

i

P i �4

3p � r3

i ð9Þ

where v is the volume of each cell, ri is the hard-sphere ra-dius for each atom and Pi is the partitioning factor of eachatom with respect to this specific cell. It is important tonote that in this free volume characterization method wefollow Cohen and Turnbull by considering all the space be-tween the atoms. This method is different than that pro-posed by Sietsma and Thijsse [39], which is closer to theTCS SRO method since in that method spherical voidsare constructed to exist at tetrahedral centers.

The coarse-grained distributions of the TCS SRO areshown in Fig. 5Ib, IIb and IIIb and those of the free vol-ume are shown in Fig. 5Ic, IIc and IIIc, for all three sam-ples at the maximum indentation depth. For sample I, theTCS SRO maps clearly resemble the plastic deformationbehavior, while the free volume maps only exhibit marginalfeatures with significantly higher noise.

To evaluate the correlation of the structural signature tothe deformation behavior quantitatively, we calculated thesignal-to-noise ratios for both structural analysis methods.If a particular coarse-grained volume element contains amajority of atoms with a local strain larger than 10%, thenthe element is designated as heavily deformed. We esti-mated the noise to be the standard deviation of the TCSSRO or free volume in regions that are not heavily

deformed, and the signal to be the absolute difference ofthe average SRO or free volume between the heavilydeformed and not heavily deformed regions. The signal-to-noise ratio of the TCS SRO map is �50% higher thanthat of the free volume. It is not surprising that the free vol-ume maps correlate to some degree with the TCS SROmeasurements, since the latter is based on efficient packing,but the degree to which the TCS SRO more clearly delin-eates the region of deformation is notable. Unlike sampleI, sample III appears to be structurally homogenous andthere is no structural signature resembling the deformationmorphology in Fig. 5IIIa. The structural feature in sampleII is marginal.

The structural evolution of the material during deforma-tion is depicted in Fig. 7. Although similar transformationis evident in both sets of histograms, the changes in the dis-tribution of the TCS SRO is clearer. For this reason thefollowing discussions only refer to the TCS SRO. Fig. 7shows that due to the interplay between mechanical disor-dering and thermal reordering upon deformation, thestructure converges toward a common flow state. Depend-ing on the initial state, each sample fares differently: sampleI undergoes significant disordering; sample II undergoesmoderate disordering; and sample III undergoes reorderingfacilitated by plastic deformation. Since the yield stress ofthe sample is a positive function of the degree of TCSSRO [19], sample I can be structurally softened due to itshighly ordered initial state. Consequently, sample I exhibitsthe most dramatic strain localization. Sample III hardensdue to the deformation induced enhancement of its SRO.Therefore, plastic deformation in sample III is nearly hom-ogenously distributed.

4324 Y. Shi, M.L. Falk / Acta Materialia 55 (2007) 4317–4324

5. Conclusions

We present a systematic study of a model three-dimen-sional binary glass-forming system subjected to simulatednanoindentation. The elastic and later plastic response ofthe specimen is determined by the combined effects of theexternal mechanical force as well as the structure of theglassy material itself. The external mechanical force varieswith the indenter shape, indentation rate and depths. Thestructure of the specimen varies sensitively with its thermalhistory. In the elastic regime, continuum elasticity theoryholds down to a spatial scale of �5–10 atomic spacing,below which the stress field fluctuates due to structural het-erogeneities. There is no recognizable region with highshear stress that leads to the nucleation of the first shearband, which agrees with the picture that incipient plasticitycomes from the instability due to structural transforma-tions instead of stress concentration. The first shear bandemerges off-center in a location that does not coincide withthe location of maximum shear predicted by elasticity.However, the subsequent shear band that eventually prop-agates occurs at the location of maximum shear stress pre-dicted by elasticity theories. The formation of the shearband relaxes the local shear stress field significantly, lead-ing to a non-uniform stress field.

Quenching rate determines the structure of the glassymaterial, which in turn controls the qualitative nature ofthe plastic deformation in the material. Graduallyquenched samples exhibited a higher initial degree ofSRO and a consequently increased degree of strain locali-zation. The conversion of material from ordered to disor-dered plays a crucial role in the softening process thatleads to the shear banding instability. Our observationsindicate that the local SRO provides a more precise charac-terization of the relevant local structural degrees of free-dom than the free volume. This is indicated by the highersignal-to-noise ratio in the structural signature of the defor-mation. For this reason, TCS SRO appears to be a morerobust measure of the structural changes that accompanythe plasticity of metallic glasses. These degrees of freedommay more closely correspond to the parameters tradition-ally associated with free volume in constitutive models[12,15] and couple to the thermodynamic changes mea-sured in glasses subjected to deformation [40–42]. Encour-agingly, such structural signatures could, in principle, beobtained experimentally [17,43,44].

Acknowledgements

The authors acknowledge the support of the NSF undergrant DMR-0135009 and in part under Grant No. PHY99-07949. The authors thank Drs. Naida Lacevic, SharonGlotzer, Michael Atzmon and John Kieffer for stimulatingdiscussions. We are also grateful for the support of the

Center for Advanced Computing (CAC) at the Universityof Michigan.

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