Shear banding instabilities in amorphous solids: Predicting the yield strainRatul Dasgupta, H. George E. Hentschel, Ashwin Joy, and Itamar Procaccia Citation: AIP Conf. Proc. 1518, 162 (2013); doi: 10.1063/1.4794563 View online: http://dx.doi.org/10.1063/1.4794563 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1518&Issue=1 Published by the American Institute of Physics. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Shear Banding Instabilities in Amorphous Solids: Predictingthe Yield Strain

Ratul Dasgupta, H. George E. Hentschel, Ashwin Joy and Itamar Procaccia

Department of Chemical Physics, The Weizmann Institute of Science

Abstract. We present a short review of the theory of shear localization which results in shear bands in amorphous solids.As this is the main mechanism for the failure of metallic glasses, understanding the instability is invaluable in finding how tostabilize such materials against the tendency to shear localize. We explain the mechanism for shear localization under externalshear-strain, which in 2-dimensions is the appearance of highly correlated lines of Eshelby-like quadrupolar singularitieswhich organize the non-affine plastic flow of the amorphous solid into a shear band. We prove analytically that such highlycorrelated solutions in which N equi-distant quadrupoles are aligned with equal orientations are minimum energy stateswhen the strain is high enough. The line lies at 45 degrees to the compressive stress. We use the theory to first predict the yieldstrain at zero temperature and quasi-static conditions, but later generalize to the case of finite temperature and finite shear

rates, deriving the Johnson-Samwer T 2/3 law.

Keywords: Amorphous Solids, Plastic Yield, Mechanical Propoeties, Shear LocalizationPACS: 61.43.Dq,81.40.Np,62.20.M-

FIGURE 1. A typical example of the failure of a metallicglass sample when subjected to compressive stress. The mate-rial localizes the stress in a plane that is at 45 degrees to thecompressive stress axis, and then breaks along this plane [1].

INTRODUCTION

In Fig.1 we show the typical failure of a sample of metal-lic glass when subjected to compressive stress. When thestress exceeds some critical level (known as the yieldstress), the sample, rather than flowing homogeneouslyin a plastic flow, localizes all the shear in a plane thatis at 45 degrees to the compressive stress axis, and thenbreaks along this plane [1]. Despite considerable amountof research [2, 3, 4, 5], the fundamental instability thatgives rise to this spectacular phenomenon remains elu-sive. In this short review we summarize the work donerecently to understand this phenomenon [6, 7, 8].

Amorphous solids are obtained when a glass-formeris cooled below the glass transition [1, 9, 10] to a statewhich on the one hand is amorphous, exhibiting liquidlike organization of the constituents (atoms, moleculesor polymers), and on the other hand is a solid, react-ing elastically (reversibly) to small strains. There is a

large variety of experimental examples of such glassysystems, and theoretically there are many well studiedmodels [11, 12, 13] based on point particles with a vari-ety of inter-particle potentials that exhibit stable super-cooled liquids phases which then solidify to an amor-phous solids when cooled below the glass transition.Typically all these materials, both in the lab and on thecomputer, begin to have plastic (irreversible) responseswhen the external strains increases beyond some limit.All these systems also exhibit a so-called yield-stressabove which the material fails to a plastic flow, eitherhomogeneously or by shear localization as seen in Fig.1.

PLASTICITY IN AMORPHOUS SOLIDS

To understand this phenomenon one needs to briefly re-view recent progress in understanding plasticity in amor-phous solids [12, 13, 14, 15, 16]. Below we deal with2-dimensional systems composed of N point particles inan area A, characterized by a total energy U(rrr1,rrr2, · · ·rrrn)where rrri is the position of the i’th particle. General-ization to 3-dimensional systems is straightforward ifsomewhat technical. The fundamental plastic instabilityis most cleanly described in athermal (T = 0) and quasi-static (AQS) conditions when an amorphous solid is sub-jected to quasi-static strain, allowing to system to regainmechanical equilibrium after every differential stress in-crease. Higher temperatures and finite strain rates in-troduce fluctuations and lack of mechanical equilibriumwhich cloud the fundamental physics of plastic instabili-

4th International Symposium on Slow Dynamics in Complex SystemsAIP Conf. Proc. 1518, 162-169 (2013); doi: 10.1063/1.4794563

© 2013 American Institute of Physics 978-0-7354-1141-8/$30.00

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ties with unnecessary details [14]. We choose to developthe theory for the case of external shear since then thestrain tensor is traceless, simplifying some of the theoret-ical expressions. Applying an external shear, one discov-ers that the response of an amorphous solids to a smallincrease in the external shear strain δγ (we drop tensorialindices for simplicity) is composed of two contributions.The first is the affine response which simply follows theimposed shear, such that the particles positions rrri = xi,yichange via

xi → xi +δγ yi ≡ x′i

yi → yi ≡ y′i. (1)

This affine response results in nonzero forces betweenthe particles (in an amorphous solid) and these are re-laxed by the non-affine response uuui which returns the sys-tem to mechanical equilibrium. Thus in total rrri → rrr′

i +uuui.The nonaffine response uuui solves an exact (and model in-dependent) differential equation of the form [12, 17]

duuui

dγ= −H−1

i j Ξ j (2)

where Hi j ≡ ∂ 2U(rrr1,rrr2,···rrrn)∂ rrri∂ rrr j

is the so-called Hessian ma-

trix and Ξi ≡ ∂ 2U(rrr1,rrr2,···rrrn)∂γ∂ rrri

is known as the non-affine

force. The inverse of the Hessian matrix is evaluated af-ter the removal of any Goldstone modes (if they exist). Aplastic event occurs when a nonzero eigenvalue λP of HHHtends to zero at some strain value γP. It was proven thatthis occurs universally via a saddle node bifurcation suchthat λP tends to zero like λP ∼ √

γP − γ [16]. For valuesof the stress which are below the yield stress the plasticinstability is seen [12] as a localization of the eigenfunc-tion of HHH denoted as ΨP which is associated with theeigenvalue λP, (see Fig. 2 left panel). While at γ = 0 allthe eigenfunctions associated with low-lying eigenvaluesare delocalized, ΨP localizes as γ → γP (when λP → 0)on a quadrupolar structure as seen in Fig. 2 for the non-affine displacement field. when the plastic instability isapproached. These simple plastic instabilities involve themotion of a relatively small number of particles but thestress field that is released has a long tail.

THE SHEAR BANDING INSTABILITY

The main analytic calculation that is reported in [7]shows that when the stress built in the system is suffi-ciently large, the nature of the plastic instabilities canchange in a fundamental way. Instead of the eigenfunc-tion localizing on a single quadrupolar structure, it cannow localize on a series of N such structures, whichare organized on a line that is at 45 degrees to the

principal stress axis, with the quadrupolar structureshaving a fixed orientation relative to the applied shear[5]. In Fig. 3 we show a typical stress vs. strain curve ofa glassy material (the details of the simulations are pre-sented in [7]), and show the non-affine displacement fieldassociated with the plastic instability after exceeding theyield stress as indicated by the ellipse on the curve. Fig.4 shows the non-affine field that is identical to the eigen-function which is associated with this instability, clearlydemonstrating the series of quadrupolar structures thatare now organizing the flow such as to localize the shearin a narrow strip around them. This is the fundamentalshear banding instability. Note that this instability is rem-iniscent of some chainlike structure seen in liquid crys-tals, arising from the orientational elastic energy of theanisotropic host fluid [18], and ferromagnetic chains ofparticles in strong magnetic fields [19].

To explain why this mechanism for shear banding canappear only at values of the stress that exceed the yieldstress, we turn now to analysis. As a first step [12] wemodel the quadrupolar stress field which is associatedwith the simple plastic instability as a circular Eshelbyinclusion of radius a and a traceless eigenstrain ε∗

αβ =ε∗(2n̂α n̂β − δαβ ) where n̂nn is a unit vector along theprincipal direction of the eigen strain tensor [20]. Theinclusion is inserted in a homogeneous elastic mediumwith Young modulus E and Poisson ratio ν which issubjected to a homogeneous shear strain ε∞

αβ . It was

shown by Eshelby that inside the inclusion we have,due to the effect of the constraining elastic medium, aconstant strain εc

αβ = Sαβγδ ε∗γδ where Sαβγδ is a constant

tensor for any elliptical inclusion. For a circular inclusionthe Eshelby tensor reads

Sαβγδ =4ν −1

8(1− ν)δαβ δγδ +

3−4ν8(1− ν)

(δαδ δβγ +δβδ δαγ).

(3)Computing accordingly we find that the constrainedstrained is proportional to the eigenstrain, i.e. εc

αβ =3−4ν4(1−ν)ε∗

αβ . Inside the inclusion, using the fact that a

traceless strain field εεε induces a stress field σσσ = E1+ν εεε

and displacement field uα = εαβ Xβ (with XXX denoting anarbitrary cartesian point in the material) we find in theinclusion

σαβ = σ cαβ − σ∗

αβ +σ∞αβ =

−E4(1− ν2)

ε∗αβ +σ∞

αβ ,(4)

uα = [3−4ν4(1− ν)

ε∗αβ + ε∞

αβ ]Xβ . (5)

Now outside of the inclusion the displacement fieldcan be written as uα(XXX) = uc

α(xxx)+ ε∞αβ Xβ where uc

α(xxx)solves the bi-Laplacian equation

∇2∇2ucα(XXX) = 0 , (6)

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FIGURE 2. (Color Online). Left panel: the localization of the non-affine displacement onto a quadrupolar structure which ismodeled by an Eshelby inclusion, see right panel. Right panel: the displacement field associated with a single Eshelby circularinclusion of radius a, see text. The best fit parameters are a ≈ 3.4 and ε∗ ≈ 0.09 with a Poisson ration ν = 0.363. To remove theeffect of boundary conditions, the best fit is generated on a smaller box of size (x,y) ∈ [25.30,75.92]. We find the parameters a andε∗ to be very weakly dependent on the external strain for a given quench rate.

0 0.1 0.2γ

-20300

-20200

-20100

-20000

-19900

-19800

-19700

-19600

U

0 0.005 0.01 0.015

-20290

-20280

-20270

0.05 0.06 0.07-20300

-20200

-20100

-20000

a) b)

FIGURE 3. (Color online). The energy vs. strain in a typi-cal numerical simulation in our system. Shown are the pointson the curve for a regular plastic event involving a singlequadrupolar structure, (marked in a red circle on the main graphand in insert (a) which is a blow up of the same graph), andalso the point on the curve that results in a plastic instabilityleading to a shear band, yellow ellipse on the main graph andthe blow up in insert (b). The regular plastic event is not evenseen without a blow up.

subject to continuity on the surface of the inclusion andzero at infinity. Remembering the radial solutions of thebi-Laplacian in 2-dimensions (i.e. 1, lnr,r2,r2 lnr), wecan write the most general displacement field that islinear in the traceless eigenstrain and that tends to zero atinfinity as

ucα(XXX) = Aε∗

αβ∂ lnr∂Xβ

+Bε∗βγ

∂ 3 lnr∂Xα ∂Xβ ∂Xγ

+ Cε∗βγ

∂ 3(r2 lnr)∂Xα ∂Xβ ∂Xγ

, (7)

FIGURE 4. (Color Online). Left panel: The nonaffine dis-placement field associated with a plastic instability that resultsin a shear band. Right panel: the displacement field associatedwith 7 Eshelby inclusions on a line with equal orientation. Notethat in the left panel the quadrupoles are not precisely on a lineas a result of the finite boundary conditions and the random-ness. In the right panel the series of 7 Eshelby inclusions, eachgiven by Eq. 8 and separated by a distance of 13.158, usingthe best fit parameters of Fig. 2, have been superimposed togenerate the displacement field shown.

since the third derivative of r2 vanishes identically. Wedetermine the coefficients A,B,C as usual by fitting theboundary conditions. The calculation is lengthy but stan-dard (see [7]) with the final result

ucα(XXX) = (8)

ε∗

4(1− ν)

(ar

)2 [2(1−2ν)+

(ar

)2 ][2n̂αnnn · XXX −Xα

]

+ε∗

2(1− ν)

(ar

)2 [1−

(ar

)2 ][2(nnn ·XXX)2

r2−1

]Xα .

Having the displacement field associated with each Es-helby inclusion at hand, we can now turn to the main cal-culation of the energy of N such inclusions arranged atrandom positions in the material and with a random ori-entation of their quadrupole. Denoting the N inclusions

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with an index i = 1,2, · · ·N and the externally inducedstress field by σ∞

αβ we can write the total energy of the

material as E = E∞ +Einc +Eesh +Emat. The first is dueto the externally induced shear interacting with the strainfield of the inclusions, the second is the interaction ofthe inclusions themselves (i.e. the stress of one with thestrain of the other). The third is the self energy of theinclusions and the fourth the self energy of the strainedmaterial without inclusions. For the purpose of this cal-culation we need only the first two:

E∞ = −πa2

2σ∞

αβ

n

∑i=1

ε(∗,i)αβ , (9)

Einc = −πa2

2∑<i j>

[ε(∗,i)

βα σ (c, j)αβ (Ri j)+ ε(∗, j)

βα σ (c,i)αβ (Ri j)

].

Here Ri j is the distance between two inclusions and the

stress field σ (c,i)αβ (Ri j) is that induced by inclusion i at a

distance Ri j away. This stress can be readily computedfrom Eq. 8. The computation of Einc is very lengthy, andis reproduced in the supplementary material in full. Thefinal result is

Einc=−πa2(ε∗)2E8(1− ν2) ∑

<i j>(

aRi j

)2

×{

−8[(1−2ν)+(a

Ri j)2][4n̂nn(i) · n̂nn( j)n̂nn(i) · r̂rri jn̂nn( j) · r̂rri j

−2(n̂nn(i) · r̂rri j)2 −2(n̂nn( j) · r̂rri j)2 +1]

+4[2(1−2ν)+(a

Ri j)2][2(n̂nn(i) · n̂nn( j))2 −1]

− 8[1−2(a

Ri j)2][2(n̂nn(i) · r̂rri j)2 −1][2(n̂nn( j) · r̂rri j)2 −1]

+32[1− (a

Ri j)2][n̂nn(i) · r̂rri jn̂nn( j) · r̂rri jn̂nn(i) · n̂nn( j)

− (n̂nn(i) · r̂rri j)2(n̂nn( j) · r̂rri j)2]}. (10)

On the other hand we find

E∞ = −πa2Eγε∗

1+ν

n

∑i

n(i)x n(i)

y . (11)

Our task is now to find the configuration of n Eshelbysthat minimize the total energy. Obviously, if the externalstrain γ is sufficiently large, we need to minimize E∞

separately, since it is proportional to γ . The minimumof (11) is obtained for

n(i)x = n(i)

y =1√2. (12)

Substituting this result in Eq. (10) simplifies it consider-ably. We find

Einc = −πa2(ε∗)2E

× 8(1− ν2) ∑<i j>

(a

Ri j)2

{−8[(1−2ν)+(

aRi j

)2]

+ 4[2(1−2ν)+(a

Ri j)2]

−8[1−2(a

Ri j)2][2(n̂nn · r̂rr)2 −1]2

+32[1− (a

Ri j)2][(n̂nn · r̂rr)2 − (n̂nn · r̂rr)4]

}. (13)

We can find the minimum energy very easily. Denotex ≡ (n̂nn · r̂rr)2, and minimize the expression A[2x − 1]2 −B[x − x2]. The minimum is obtained at x = 1/2, or

cosφ =√

1/2. We thus conclude that when the line ofcorrelated quadrupole forms under shear, this line is in45 degrees to the compressive axis, as is indeed seen inexperiments. Note that for other external strains whichare not consistent with a traceless strain tensor (or in 3-dimensions) this conclusion may change.

The physical meaning of this analytic result is that itis cheaper (in energy) for the material to organize Nquadrupolar structures on a line of 45 degrees with thecompressive stress, all having the same orientation, thanany other arrangement of these N quadrupoles, includ-ing any random distribution. This explains why sucha highly correlated distribution appears in the strainedamorphous solid, and why it can only appear when theexternal strain (or the built-up stress) are high enough.This fact, in addition to the observation that such an ar-rangement of Eshelby quadrupoles organize the displace-ment field into a shear band, explains the origin of thisfundamental instability.

It should be noted that in the present calculation wedid not predict the number of quadrupoles that appearat the instability. To achieve this we must consider theother terms in the total energy (i.e. Eesh and Emat above);this had been done in [7] and the result is quoted andemployed in the next sections.

TOWARDS RHEOLOGY

A satisfactory derivation of the rheology of amorphoussolids like metallic glasses under various mechanical andmagnetic external strains is still far from being accom-plished. Examples of universal phenomena and universalrelations are rare and far between [16]. One of these rareexamples is the Johnson-Samwer T 2/3 law [21] whichpertains to the temperature dependence of the yield-strain (the value of the strain where the material fails via

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FIGURE 5. (Color Online). The total plastic energy Eq. (16)for the creation of an array of quadrupoles with density ρfor three values of γ: γ = γ

Y− 0.1 (upper curve), γ = γ

Y−

0.05 (middle curve), and γ = γY

(lower curve). In the presentcase γ

Y= 0.07. To generate this picture we use the measured

constants E ≈ 37.2, ν ≈ 0.31, ε∗ ≈ 0.082 and a = 1.83. FinallyUp ≈ 0.22.

plastic instabilities). Examining a large group of metallicglasses these authors proposed the law

γY(T, γ̇) = γ

Y(T = 0, γ̇ = 0)

{1− [AT ln(ω0/Cγ̇)]2/3

},

(14)where γ

Y(T = 0, γ̇ = 0) is the yield-stress at athermal

quasi-static conditions, γ̇ is the external strain rate, A andC are material constants and ω0 is a microscopic inversetime scale. Johnson and Samwer offered a derivation ofthis law [21] based on Frenkel’s theory [22] which is ap-propriate for an infinite crystals having a periodic elasticenergy density, leaving a derivation which is proper foran amorphous solid for the future. This task was pickedup in Ref. [23]. These authors recognized that in amor-phous solids plastic yielding follows a saddle-node bi-furcation where an eigenvalue λP of the system’s Hessianhits zero at a value of the strain γ = γP. Such a bifurcationleads to a local rearrangement which is characterized byan energy barrier δE which scales like δE ∼ (γP −γ)3/2.Assuming that this energy barrier is much larger than kT(k being Boltzmann’s constant), Ref. [23] showed thata temperature induced barrier crossing would result in alaw of the form of Eq. 14. In our work [8] we arguedthat the energy barrier that needs to be crossed to achieveshear localization is sub-extensive; on the other hand theenergy barrier associated with a single localized plasticinstability is minute, and in the thermodynamic limit itscales like 1/N where N is the number of particles [15].Therefore the conditions at which the system yields via acoherent shear localized band need to be reconsidered.

YIELDING VIA SHEAR LOCALIZATION

We explained above that when the external strain ap-proaches the shear-localization yield-strain the nature ofplastic instabilities can change qualitatively [6, 7]. At lowexternal strains a plastic instability results in a local rear-rangement such that the non-affine displacement of parti-cles can be very well modeled by the displacement fieldassociated with a single Eshelby inclusion in an elasticmatrix [20]. At larger values of the strain the instabil-ity results in 2D in a highly correlated array of Eshelbyquadrupoles that are aligned at 45 degrees to the prin-cipal stress axis. All the quadrupoles are in phase andin total they result in shear localization in a narrow strip.One could show that this highly correlated array is a min-imum energy state which depends on the density ρ ofthe quadrupoles. For γ < γ

Ythe only solution is ρ = 0,

i.e. isolated qudrupoles, but at γ = γYa bifurcation opens

up a new solution with a finite density, cf. Fig. 5. Underconditions of athermal quasistatic straining (AQS), it wasshown that the yield strain γ

Yis given by the expression

[7]

γY ≡ γY (T = 0, γ̇ = 0) ≡ ε∗

2(1− ν). (15)

where ε∗ is the eigenstrain induced by single plastic in-stability in the background elastic matrix whose Pois-son’s ratio is ν . Each quadrupole can be very well mod-eled as an Eshelby inclusion with eigenstrain ε∗ and coresize a [20]. The energy density cost of creating a lineararray of N quadrupoles all with the same orientation,separated by distance R = L/N (in a 2-dimensional sys-tem of size L2) was computed analytically [7] in the form

E(ρ,γ)La

=Up

[(1− γ

γY

)aρ −B(aρ)3+C(aρ)5

](16)

whereUp = [E π(ε∗)2]/[4(1−ν2)] with E being Young’smodulus; while B = 4ζ (2) and C = 6ζ (4), where theRiemann zeta functions are respectively ζ (2) = π2/6and ζ (4)= π4/90. This energy is shown in Fig. 5. Exam-ining this energy we realize that in AQS conditions yieldvia shear localization can occur only at γ = γ

Y. With fi-

nite temperature we can have thermally-assisted transi-tions which we consider next.

Thermally assisted plastic yield: Considering theyield stress at any finite temperature T = 0; we real-ize that thermal fluctuations can always (if given enoughtime) cross any energy barrier and therefore under con-ditions of quasistatic straining the yield stress should al-ways vanish

γY (T, γ̇ = 0) = 0. (17)

Accordingly, for strictly quasistatic straining γ̇ = 0 thelimit T → 0 yield-stress is different from the T = 0 yield-stress. This jump in the limit was observed in [21] and

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was interpreted as a “quantum effect". There is no needto invoke quantum mechanics to understand this simpleissue.

In reality one always strains at some finite value of γ̇ .Then thermal fluctuations can lead to barrier crossing ifthe timescale for such crossing τ is smaller or similar tothe straining time scale γ̇−1, leading to the condition foryielding due to thermal fluctuations given by

τγ̇ = O(1). (18)

If we had a process in which every particle in the systemacted independently, it would be enough to compute theenergy barrier per particles, say Ub, and compare it to thethermal energy, to be used in the Arrhenius form

τ = τ0 exp [Ub/kT ], for independent particle motion(19)

where τ0 is the inverse of the attempt frequency for bar-rier crossing. In fact, the situation here is more complex.The way that shear localization occurs in practice is thatthere exists a mode that gets localized on N∗ � N parti-cles which are involved in the first quadrupole that formsand then the rest of the linear array of qudrupoles formsinstantly, as seen in Fig. 6. Thus the rate determining stepis the creation of the first quadrupole, and we estimate theappropriate times scale as

τ = τ0 exp [N∗Ub/kT ], for concerted particle motion(20)

Combing these results gives the expression for yieldingas

τ0γ̇ exp [N∗Ub/kT ] = O(1). (21)

Note that N∗ remains independent of N in the thermo-dynamic limit, representing the number of particles in-volved in the concerted barrier crossing of the creationof one quadrupolar non-affine displacement field. We ex-pect N∗ to be of the order of 100, give or take a factor of2.

We now use these result to estimate how finite temper-atures and finite strain rates change the value of the AQSyield strain due to shear localization.

Calculating the energy barrier: Examining Eq. (16)we see that the plastic energy density due to shear local-ization at strain γ can be written in terms of the dimen-sionless variable x = ρa. The height of the barrier perparticle Ub is determined by the energy density E∗/Laat x∗ = ρ∗a which can be found from ∂ (E/La)/∂x = 0with the result that the barrier occurs at

x∗ = ρ∗a =√

[1− γ/γY ]/2/π, (22)

and its strip energy density is given by

E∗/La =

√2Up

3π[1− γ/γY ]

(3/2). (23)

Eq. (23) implies that the barrier energy per particle in thestrip is

Ub = (E∗/La)/n = [√

2Up/(3πn)][1− γ/γY ](3/2), (24)

where n is the number density of particles in the strip ofdimension La.

Finally we need to estimate τ0, the inverse attempt fre-quency. There are two candidates. The first is the eigen-value of the Hessian matrix λp which is associated withthe eigenfunction that gets localized. The square root ofλp is an inverse time scale. This time scale however di-verges to infinity when γ is close to γ

Y, and the purely

thermal time scale becomes shorter and more relevant.We thus argue here that the relevant time scale can beestimated from the thermal fluctuations of the individualparticles in the solid which determines the time scale ofa concerted motion of N∗ particles. For a single particlethe thermal time scale is l/v where l ≈ 1/

√n is the typ-

ical distance between particles and the typical velocity vcan be found from the equipartition theorem m〈v2〉 = kT .Thus for N∗ particles moving together we estimate

τ0 = N∗√m/(nkT ). (25)

Now combining Eq. (21) for the yield strain with Eq. (24)for the barrier height and Eq. 25 for the bare time scalewe finally find

γY (T, γ̇)/γY = 1−[

3πnkT√2N∗Up

]2/3

log2/3

[√nkT/mN∗γ̇

].

(26)We note that at this point only N∗ is not known withcertainty, only in order of magnitude. The expressionhas the T (2/3) temperature reduction in the yield strainfound experimentally by Johnson and Samwer, cf. Eq.(14), including the logarithmic correction term.

Comparison with numerical simulations: We haveperformed 2D Molecular Dynamics simulations on a bi-nary system which is an excellent glass former and isknown to have a quasi-crystalline ground state. Eachatom in the system is labeled as either “small”(S) or“large”(L) and all the particles interact via LennardJones (LJ) potential. All distances are normalized byσSL, the distance at which the LJ potential between thetwo species becomes zero and the energy is normal-ized by εSL which is the interaction energy between twospecies. For detailed information on the model poten-tial and its properties, we refer the reader to Ref [? ].The number of particles taken in all our simulations is100489 at a number density n = 0.985 with a particleratio NL/Ns = (1+

√5)/4. The mode coupling temper-

ature TMCT for this system is known to reside close to0.325εSL/kB. All particles have identical mass m0 and

hence the time is normalized to t0 =√

εSLσSL2/m0. For

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FIGURE 6. (Color Online). Visualization of the process of shear localization in the AQS simulations whose full details can befound in Ref. [7]. Even with the quasistatic protocol with arbitrary long waiting times if necessary one cannot resolve the formationof the full structure of shear localized band of quadrupoles (right panel) from the creation of a single quadrupole (left panel). Theright panel appears instantly after the left panel.

the sake of computational efficiency, the interaction po-tential is smoothly truncated to zero along with its firsttwo derivatives at a cut-off distance rc = 2.5σSL. To pre-pare the glasses, we start from a well equilibrated liq-uids at a high temperature of 1.2εSL/kB which are su-percooled to 0.35εSL/kB at a reduced quenching rate of3.4×10−3t−1

0 . We then equilibrate these supercooled liq-uids for times greater than 10τrel, where τrel is the timetaken for the self intermediate scattering function to be-come 1% of its initial value. Following this equilibration,we quench these supercooled liquids deep into the glassyregime at a temperature of 0.03εSL/kB at a much slowerquenching rate of 3.2×10−6t−1

0 .We perform simple shear loading experiments by inte-

grating the SLLOD system of equations [? ] along withLees-Edwards periodic boundary conditions [? ]. As canbe expected, work done on the system will lead to heatdissipation and hence we keep our system connected tothe thermostat in order to keep the temperature constantthroughout the mechanical deformation. We employ astrain rate γ̇ = 10−5 in all our loading experiments. Atypical stress vs. strain curve is shown in Fig. 7. The dataof stress vs strain were fitted to a cubic in γ and the yieldstrain was estimated from the maximum of the curve.

To compare with the theory above we need also γY atzero temperature and the value of Up. We obtained theseby minimizing the energy of one of the glasses generatedas explained above to T = 0. Performing athermal quasi-static simulations on this sample we determined all theparameters appearing in Eq.(26), with the result γY ≈0.06 and Up = 0.22. Armed with these and all the otherknown numbers we compare the data for the temperaturedependent yield strains to the prediction of Eq. (26) usingN∗ to get a best fit, which is obtained for N∗ ≈ 250.The fit is shown in Fig. 8. Having in mind the inherentunknown factors hidden in the estimates Eqs. (18),(19)and (25), we find the fit very satisfactory.

In summary, we have used the analytically computedenergy associated with a correlated series of quadrupolarstructures that add up to a shear localizing instability

0.02 0.04 0.06 0.08 0.10γ

A = -1663.81

B = 23.83

C = 13.36

D = 0.01

γY (γ̇,T)

MD data

Aγ3 +Bγ2 +Cγ+D

FIGURE 7. (Color Online). Typical stress vs. strain curveobtained at finite temperature and finite strain rate. The yieldstress was estimated by fitting a cubic to the curve (see inset)and finding the maximum of the curve.

0.05 0.10 0.15 0.20 0.25 0.30

T

Up =0.2214γY =0.0600

N ∗ =250

γY [1−{

3πρT√

2N ∗Up

log(√ρT

N ∗ γ̇)}2/3

]

MD

FIGURE 8. (Color Online). Comparison of the simulationaldata of γ(T, γ̇) to the prediction Eq. (26). The dotted line wasobtained by finding the best fit for N∗. The error bars indicateaveraging over three independent realizations.

168

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to study the thermally assisted yield strain associatedwith plastic failure in two dimensions. This energy issub-extensive and its strip-density remains unchanged inthe thermodynamic limit. We could derive the Johnson-Samwer T 2/3 law essentially without a free parameterexcept for an uncertainty regarding the value of N∗. Thefitted value of this number appears to be in the right orderof magnitude, lending strong support to the approachdetailed above.

ACKNOWLEDGMENTS

This work had been supported in part by the Israel Sci-ence Foundation, the German-Israeli Foundation and bythe European Research Council under an “ideas" grant"STANPAS".

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