Materials Transactions, Vol. 48, No. 11 (2007) pp. 2923 to 2927
#2007 The Japan Institute of Metals

Theory of Shear Banding in Metallic Glasses and Molecular
Dynamics Calculations
Futoshi Shimizu1;2 , Shigenobu Ogata3 and Ju Li2; *1Center for Computational Science & e-Systems, Japan Atomic Energy Agency, Tokyo 110-0015, Japan
Department of Materials Science and Engineering, Ohio State University, Columbus, OH 43210, USA
3
Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science,
Osaka University, Osaka 560-8531, Japan2

The aged-rejuvenation-glue-liquid (ARGL) shear band model has been proposed for metallic glasses (Acta Mater. 54 (2006) 4293), basedon small-scale molecular dynamics simulations up to 20,000 atoms and thermomechanical analysis. The model predicts the existence of a
critical lengthscale 10 nm, above which melting could occur in shear-alienated glass. Large-scale molecular dynamics simulations with up to 5
million atoms have directly veriﬁed this prediction. When the applied stress exceeds the glue traction (computed separately before in a shear
cohesive zone, or an amorphous-amorphous ‘‘generalized stacking fault energy’’ calculation), we indeed observe maturation of the shear band
embryo into bona ﬁde shear crack, accompanied by melting. In contrast, when the applied stress is below the glue traction, the shear band
embryo does not propagate, becomes diﬀuse, and eventually dies. Thus this all-important quantity, the glue traction which is a property of shearalienated glass, controls the macroscopic yield point of well-aged glass. We further suggest that the disruption of chemical short-range order(‘‘chemical softening’’) governs the glue traction microscopically. Catastrophic thermal softening occurs only after chemical alienation andsoftening in our simulation, after the shear band embryo has already run a critical length. [doi:10.2320/matertrans.MJ200769](Received December 1, 2006; Accepted August 17, 2007; Published October 18, 2007)
Keywords: shear cohesive zone, chemical softening, incubation lengthscale, thermal softening

1.

Introduction

Bulk metallic glasses (BMGs) are new materials1) withvery high strength, low internal friction and good corrosion
resistance. The uniaxial yield strain of most BMGs falls
within a small range around y ’ 2% at room temperature,1,2)
beyond which shear bands nucleate and propagate globally.3,4) We have previously proposed the aged-rejuvenationglue-liquid (ARGL) model of shear band in BMGs.5) The
model argues that the y condition corresponds to embryonicshear band (ESB) propagation, not its nucleation. To
propagate an embryonic shear band, the far-ﬁeld shear stress1  Ey =2 must exceed the quasi steady-state glue traction
glue of shear-alienated glass (shear cohesive zone) until theglass-transition temperature Tg is approached internally due
to frictional heating, at which point ESB matures as a
runaway shear crack, since the trailing T > Tg zone would
have much lower viscosity than the shear cohesive zone andcan be regarded as shear traction free. The magnitude of glue
is governed by competing alienation and recovery processes.
Alienation is the disruption of short-range order (including
chemical ordering6)) due to fast localized shearing. Recovery
is the counter-acting diﬀusionless downhill process thatrecovers part of that short-range order.
It was predicted that the alienation and recovery dynamics
fall into the accessible timescale of ordinary molecular
dynamics (MD) simulations. Thus MD calculations should be
able to capture the dynamics of ESB and hence the y value of
BMGs. In our previous paper,5) although we proposed the
ARGL model based on small-scale atomistic simulations
and theoretical analysis, we have not conﬁrmed it directly
using more realistic boundary condition and large-scale
MD simulation. In this paper we present a brief outline of
the ARGL model and the calculations of glue by small-scale
MD simulations of the shear cohesive zone (‘‘generalized*Corresponding author, E-mail: li.562@osu.edu

stacking fault energy’’5)), as well as large-scale MD simulations which directly verify the ARGL ‘‘march-to-melting’’
prediction, within a spatial-temporal domain of 200 nm and1000 ps.
2.

Shear Band Model

A mature shear band (MSB) in BMG is proposed to be
similar to a mode-II or III dynamical crack, driven by farﬁeld shear stress 1 .5) The tail of the band is liquid or nearliquid, due to friction-induced high temperature T  Tg (Tg is
the glass transition temperature), and is essentially tractionfree. The band tip is the rejuvenation zone, where the glass
undergoes transition from well-aged to rejuvenated state,7)
under a locally very high shear stress  > r , where r is the
rejuvenation stress. r is an intrinsic property of well-aged
glass, but its value is not relevant to the y value.5) The
intermediate region between the band tip and the liquid tail isthe ‘‘glue’’ zone, similar to the cohesive zone in fracture
mechanics8) in the shear sense: its temperature is below Tg
and consists of shear-rejuvenated and subsequently shearalienated glass. Alienation is a special form of rejuvenation7)in the limit of intense localized shearing at extremely fast
rates. Because of shearing, the original ﬁrst-nearest-neighbor
relations and chemical ordering6) are severely disrupted at theatomic level in the glue zone. The traction this shear cohesivezone or glue zone (T < Tg ) oﬀers, glue , is the main resistanceagainst shear band propagation. The shear cohesive law is theequivalent of generalized stacking fault energy in crystalline
materials,9) but for an amorphous-amorphous interface.5)
We deﬁne the embryonic shear band (ESB) as a shear band
embryo with long aspect ratio, but which has no liquid zone,
since the internal temperature has not risen high enough yet.
To propagate an ESB, the far-ﬁeld shear stress 1  Ey =2
must exceed the quasi steady-state glue traction glue of shearalienated glass in the shear cohesive zone until the glasstransition temperature Tg is approached internally due to2924

F. Shimizu, S. Ogata and J. Li

frictional heating, at which point ESB matures as a runaway
shear crack. At the pESB
ﬃﬃﬃﬃﬃﬃﬃﬃtip,
ﬃ the local shear stress is ampliﬁed
as  / ð1  glue Þ a=w according to the stress solution for
an elliptical hole, where a is the length of the ESB, and w is
its width as an approximant of the ESB tip curvature. Thismeans that as long as the embryo has a large aspect ratio a=w,
r can always be exceeded at the tip if 1 > glue . We further
assume there are plenty of pre-existing embryos in the BMG
sample. But if 1 < glue , even with plenty of embryos, they
still cannot propagate.
When 1 > glue , an embryo can run, but it cannot run tomaturity if the size of the 1 > glue region is smaller than an
incubation lengthscale linc , estimated to belinc 

c2v ðTg  Tenv Þ2
;
2 c
glue
s

ð1Þ

where  is the BMG’s thermal diﬀusivity, cv is its volumetricspeciﬁc
heat,
pﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ Tenv is initial temperature, glue  0:01E, andcs ¼ = is the shear wave speed. For Zr-based BMGs,
eq. (1) predicts linc  10 nm, below which mature shear
banding cannot happen.5)
3.

Molecular Dynamics Calculations
We model four metallic glasses with molecular dynamics:

a binary Lennard-Jones (LJ) system, two binary embedded
atom method (EAM) systems, and a quinternary EAM
system. The samples were prepared by the melt-quench
procedure, and deformed by supercell tilting (PBC) andmoving grip (non-PBC) methods. glue values were calculatedby small-scale MD simulations, and shear banding is directly
observed in large-scale MD simulations.
3.1 Calculation of glue traction
A binary LJ A80 B20 glass,10) and binary Cu40 Ag60 11) andCu46 Zr54 12) glasses by EAM potentials were well characterized in the original papers, and we simply followed thesame sample preparation scheme. For quinternary
Zr52:5 Cu17:9 Ni14:6 Al10 Ti5 system, we adopted the secondmoment potential (TB-SMA) of Cleri and Rosato13) and took
algebraic or geometric means for the cross-interaction
parameters. The glass structures were generated using themelt-quench procedure starting from an fcc structure with
ﬁve species of atoms randomly assigned (Fig. 1(a)). However, when quenched from above the melting temperature,
the original Cleri-Rosato potential resulted in glass-glass
phase separation, ZrTiAl with CuNi, shown in Fig. 1(b). To
ﬁx this problem, we tuned the potential by using newer
parameters for Ni and Ti by Lai et al.14) and enhancing an
attractive cross interaction uniformly by a single factor ,2()1=2 3
X 1X
X


4
5;
Ec ¼
A e p ðrij =r0 1Þ ð  Þ2 e2q ðrij =r0 1Þ
2 j6¼i
i
j6¼i

 ¼

1

for  ¼ for  6¼ ;

where  and  indicate atomic species, and rij is the distance
between atoms i and j. We found the system is fully miscible
when   1:3 (Fig. 1(c)). Table 1 shows the density (),shear modulus (), and bulk modulus (B) obtained at T ¼
300 K for diﬀerent ’s. We conﬁrm that any   1:3 can beused to demonstrate the shear band model. Here we chose ¼ 1:6 as a compromise between better ﬁtting of  and  tothe actual material.
We generated three quinternary BMG conﬁgurations offour thousand atoms each by the procedures illustrated inFig. 2(a). In Fig. 2(b), the responses of the glass to shear bysupercell tilting method at constant strain rate _ ¼ 109 s1
are shown. Here, the shear stress  is normalized by E=2,
where E ¼ 71 GPa is the numerically calculated Young’s
modulus and 2 is the ideal Schmid factor. The maximum ofthese curves gives the homogeneous nucleation or rejuvenation stress r , which is a property of well-aged glass. The
value depends sensitively on the conditions by which theglass was made; we read 2r =E to be 3.6%, 4.9% and 5.1%
for the systems of case I, II, and III, respectively. In addition,
r also depends on the rate of shearing16) as seen in Fig. 3(where 2r =E  4:6% for case III conﬁguration with ten
thousand atoms deformed at _ ¼ 5  109 s1 ). In contrast,
we can see the convergence of glue , which is the plateau
value of the stress response at large , for the quinternary
EAM system.

ð2Þ

Despite vast diﬀerences in the structures and interatomic
interactions, the glue values obtained from small-scale MD
calculations of the shear cohesive zone for the four metallic
glass systems give y predictions in the range of 2.1%–
2.9%,5) in fair agreement with experiments. Furthermore,these glue values are insensitive to the initial glass conﬁguration: even the inherent structure of a high-temperatureliquid,17) which is deﬁnitely not a glass, gives the same glue
value.
3.2 Observation of shear band
The incubation length linc  10 nm predicted from eq. (1)falls into the reach of direct MD simulations. We have
performed large-scale parallel MD simulations and visualization18) in a thin-slab geometry to verify the ARGL modeland the ‘‘march-to-melting’’ scenario of shear banding.5)
To quantify plastic deformation at the atomic level, we
introduce the atomic local shear strain Misesfor each atom i.
iCalculation of Mises
requirestwo
atomicconﬁgurations,oneicurrent, and one reference. First, we seek a local transformation matrix J i , that best maps
fd 0ji g ! fd ji g;

8j 2 Ni0 ;ð3Þwhere d’s are vector separations (row vectors) between atom
j and i (superscript 0 means the reference conﬁguration).
Here, j is one of atom i’s nearest neighbors, and Ni0 is the total

Theory of Shear Banding in Metallic Glasses and Molecular Dynamics Calculations

a

b

c

2925

a
Temperature
1500K ρ=6.70
quench under 200MPa(1200ps)

relax(1250ps)

freeze300K

ρ=7.00ρ=7.04case Iρ=7.01

ρ=7.03case IIcase III

freezeheat(1ps)heat(1ps)
relax(10ps)+
0K ρ=7.11
ρ=7.04 +relax(10ps)
Zr

Ti

Al

Cu

Time

Ni

b 0.06

Fig. 1 Atomic conﬁguration of Zr52:5 Cu17:9 Ni14:6 Al10 Ti5 model system;(a) before melting, (b) without cross-interaction enhancement ( ¼ 1), and(c) with  ¼ 1:6.

0.05
0.04

Properties of Zr52:5 Cu17:9 Ni14:6 Al10 Ti5 model glass (g/cm3 ) (GPa)B (GPa)

1.3

6.56

16

120

1.4

6.71

16

130

0.02

1.5
1.6

6.87
7.03

21
27

130
150

0.01

1.7

7.19

29

150

experimental values15Þ

6.73

32.3

114.1

0.03

0
0

number of nearest neighbors of atom i, at the reference
conﬁguration. J i is determined by minimizing19)
0
11 0
1
X
X
X
0A @
A
jd 0ji J i  d ji j2 ! J i ¼ @
d 0T
d 0T
ji d ji
ji d ji :
j2Ni0

j2Ni0

case II2τ / E

Table 1.
γ = 109 /s

case IIIcase I

0.05

0.1

γ

0.15

0.2

0.25

Fig. 2 (a) Preparation of three conﬁgurations for Zr52:5 Cu17:9 Ni14:6 Al10 Ti5
model glass. Numbers in the boxes are density (g/cm3 ). (b) Stress-strain
curves of four thousand atom systems (4:1 nm  4:1 nm  4:1 nm)
deformed by supercell tilting method with _ ¼ 109 s1 .

j2Ni0

ð4Þ
For each J i , the local Lagrangian strain matrix is computed
as

i ¼

1
ðJ i J Ti  IÞ:2

We can then compute atom i’s local shear invariant assﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð yy  zz Þ2 þ ð xx  zz Þ2 þ ð xx  yy Þ2Mises
2 þ 2 þ 2 þ
¼
:
i
yz
xz
xy
6
2 19) Mises
Like Falk and Langer’s Dmin,
is a good measure ofilocal inelastic deformation. This measure has been incorporated into our visualization program AtomEye.20)
In Fig. 3, atoms are colored by Mises
calculated withi
 ¼ 0:01 between the current and reference conﬁgurations.
Before the shear stress reaches maximum, no signiﬁcant local
deformation is seen (inset image a). At ¼ 0:1, small stressdrops can be seen in the stress-strain curve and correspondingly we see some shear transformation zones (STZs) insnapshot (b). We see STZs coalesce to become ESB when theshear stress drops in cascading fashion at ¼ 0:14 (c), andthen this ESB diﬀuses out at ¼ 0:18 (d). The same process
then recurs a few times with increasing .
Large-scale MD simulations of the binary LJ system with 5
million atoms were then performed. The system was prepared
identically as that of Varnik et al.,10) but with a much larger
dimension of 800b  400b  16b as illustrated in Fig. 4(a).
The thickness in z is intentionally chosen to be small so we

ð5Þ

ð6Þ

can reach the incubation(and exceeding the thermal
pﬃﬃﬃﬃsizescaleﬃ
diﬀusion lengthscale t) in x and y. Here, we use the mean
as length unit and b=cs as time unit,
atomic spacing b  1=3
N
where N is the number density of atoms. In Zr-based BMG,
they are estimated to be b  0:26 nm and b=cs  0:1 ps.
Mode-III (anti-plane shear) loading which was applied byconstant forcing on the grips to realize simple shear loading
condition. Note that mode-III generates only shear stresswhile mode-I and II will generate compressive or tensile
normal stress as well as shear stress. At each MD step, forcefz ¼ ðA=N Þgrip is applied to each atom in the upper/
lower grip region, where A is the area of grip and N is thenumber of atoms in the grip. No thermostat was applied after
equilibration. For each case of 2grip =E ¼ 0:021 and 0.027, a
shear band embryo was introduced at the beginning by the
instantaneous displacement of atoms according to thedisplacement ﬁeld solution of a smeared super screw
dislocation. In the former case of grip < glue , the introduced

2926

F. Shimizu, S. Ogata and J. Li
0.05.
γ = 5x108 /s

0.04

2τ / E

0.03

0.02

b

c
d

a
a

b

c

d

0.3

0.01

0
0

0.03
∆γ =0.01
0.1

0.2

γ

0.3

0.4

4.

Conclusion

0.5

Fig. 3 Stress-strain curves of ten thousand atom system (case III:
6:4 nm  10:8 nm  2:5 nm) deformed by supercell tilting method with_ ¼ 5  108 s1 . Inset images are snapshots of atomic conﬁguration at¼ 0:05 (a), 0.1 (b), 0.14 (c), and 0.18 (d). Atoms are colored according
to their atomic shear strain Mises.
iembryo (Fig. 4(b)) does not propagate, becomes diﬀuse, andeventually dies (Fig. 4(c)). In the latter case of grip > glue ,the introduced embryo started to develop into MSB. Near thetip of ESB, it is observed that STZs coalesce and the width of
the ESB has grown up to  75b (Fig. 4(d)). The local
temperature of the system is monitored simultaneously. It isseen that, at t ¼ 7800b=cs (Fig. 4(f)), the local temperature at
the center of the shear band has reached Tg ¼ 0:435,asignifying the ESB has indeed matured, becoming a bonaﬁde shear crack, since the newly formed T > Tg zone hasmuch lower viscosity and can be regarded as traction free.
Movies of the evolution of shear strain and temperature inthis simulation can be viewed at.18)
We also performed another control calculation, where
2grip =E ¼ 0:03 was applied but no embryo was introduced at
the beginning. In this case, nothing happened within the MD
simulation timeframe (t ¼ 4000b=cs ). This should be expected since 2grip =E < 2r =E ’ 0:05, the homogeneous
nucleation stress.

In the ARGL shear band model, it is argued that the farﬁeld shear stress, 1 , must exceed the quasi steady-state glue
traction, glue , for an embryonic shear band to propagate. glue
values were calculated by small-scale MD simulations of theshear cohesive zone and it is shown that glue values areinsensitive to the initial glass conﬁguration. The results oflarge-scale MD simulations are summarized as that shear
banding requires (a) an embryo with long aspect ratio, (b)grip > glue , and (c) the size of grip > glue region must
exceed linc  10 nm in eq. (1). When (a), (b), (c) are satisﬁed,
we will see mature shear bands form with eventual liquid ornear-liquid tails, dramatic softening and load shedding. If anyof (a), (b), (c) conditions are not satisﬁed, there will be no
sharp local temperature rise,21) and even if a shear band isintroduced it will smear out.
MD contributes to this theory of shear banding byb

fz+ = (A / N+) τgrip∆t =800 b/cs 0.04

0.4

0.2

0.5

800b × 400b × 16b(z-direction: PBC)ycefz− = − (A/N−) τgrip

z

x

d

f

Fig. 4 (a) Molecular dynamics simulations of a binary LJ system with 5 million atoms. Atomic conﬁguration snapshots for (b)2grip =E ¼ 0:021 at t ¼ 1000 b=cs and (c) 2500 b=cs ; (d) for 2grip =E ¼ 0:027 at t ¼ 2500 b=cs and (e) 4000 b=cs . Atoms in (b–e) arecolored according to their atomic shear strain Mises. (f) Temperature distribution (Tg ¼ 0:435) for 2grip =E ¼ 0:027 at t ¼ 7800 b=cs .
i
Melting, predicted in,5) indeed occurs within a simulation run length 100 nm.

Theory of Shear Banding in Metallic Glasses and Molecular Dynamics Calculations

providing decent values for glue and therefore y compared toexperiments, and elucidating its microscopic origin.5) It turnsout glue is not an intrinsic property of well-aged glass, butthat of a strongly driven, non-equilibrium material - the cold‘‘glue’’ state inside shear cohesive zone - which only exists
transiently, but nonetheless controls the macroscopic y of
BMG. glue reﬂects the eﬀect of alienation (disruption of
atomic short-range order, including chemical order), and its
steepest-descent-like recovery on the energy landscape,occurring at picoseconds timescales. The signiﬁcant ﬂattening of the amorphous-amorphous shear cohesive law (akageneralized stacking fault energy)5) can be well-explained bythis disruption of chemical order6) (‘‘chemical softening’’),without the need of thermal softening. Catastrophic thermal
softening does eventually arise, however, if an embryo runs
in a critical sized volume (l > linc ) with a critical stored
elastic strain-energy density ( > glue ), as shown in our
simulation.
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