Rate dependent shear bands in a shear transformation zone model of amorphous solids

M. L. Manning,∗ E. G. Daub, J. S. Langer, and J. M. CarlsonDepartment of Physics, University of California, Santa Barbara, 93106

(Dated: October 1, 2008)

We use Shear Transformation Zone (STZ) theory to develop a deformation map for amorphoussolids as a function of the imposed shear rate and initial material preparation. The STZ formulationincorporates recent simulation results [Haxton and Liu, PRL 99 195701 (2007)] showing that thesteady state effective temperature is rate dependent. The resulting model predicts a wide range ofdeformation behavior, from homogeneous deformation to extremely thin shear bands and the onsetof material failure. In particular, the STZ model predicts homogeneous deformation for shorterquench times and lower strain rates, and inhomogeneous deformation for longer quench times andhigher strain rates. The location of the transition between homogeneous and inhomogeneous flowon the deformation map depends on the steady state effective temperature, which is likely material-dependent. This model also suggests that material failure occurs due to a runaway feedback betweenshear heating and the local disorder, and provides an explanation for the thickness of shear bandsnear the onset of material failure. We find that this model, which resolves dynamics within a shearedmaterial interface, predicts that the stress weakens with strain much more rapidly than a similarmodel which uses a single state variable to specify internal dynamics on the interface.

I. INTRODUCTION

Amorphous solids such as foams, dense colloids, bulkmetallic glasses, and granular fault gouge are ubiquitousin engineering applications and natural systems. Al-though these materials exhibit a yield stress on exper-imental time scales, they flow, deform, and fail in a man-ner which is different from crystalline solids or Newtonianfluids. Many of these materials undergo strain localiza-tion, where a small region deforms much more rapidlythan adjacent regions. For example, bulk metallic glassesdevelop very thin shear bands [1, 2], fault gouge in earth-quake faults develops a prominent fracture surface thataccommodates most of the slip [3], and colloidal systemsdevelop broad shear bands [4].

Surprisingly, the mechanisms that lead to this strainlocalization have remained elusive. An early theory ofshear banding [5] suggests that a small increase in thethermal temperature lowers the viscosity, resulting inmore rapid deformation and a local increase in temper-ature. However, Lewandowski and Greer showed thatshear bands in bulk metallic glasses can not be explainedby adiabatic thermal effects [6]. Although thermal heat-ing must play a role at high strain rates, it does notappear to govern the formation of shear bands in manymaterials.

In a recent paper [7], we found that at low strain ratesthe Shear Transformation Zone (STZ) theory for amor-phous solids predicts shear band formation. These bandsare generated by feedback between the local strain rateand the configurational disorder of an amorphous pack-ing. In a separate paper [8], we used STZ theory to fitdata over a wide range of strain rates from a simulation ofglassy disks by Haxton and Liu [9]. These data indicate

∗Electronic address: mmanning@physics.ucsb.edu

that dynamics of the glassy material change dramaticallyat large strain rates.

Experimental studies of bulk metallic glasses driven ata wide range of strain rates also show that deformation isstrain-rate dependent. Homogeneous deformation is seenat low strain rates and inhomogeneous flows dominate atlarge strain rates [10, 11].

In this paper, we study deformation in the STZ modelas a function of the initial material preparation and ex-ternally imposed strain rate, and include the rate depen-dence observed by Haxton and Liu in our model. ThisSTZ formulation is valid for a large range of strain ratesand predicts four different types of deformation behavior:homogeneous deformation, thick disorder limited shearbands, thin diffusion limited shear bands, and materialfailure. We describe the mechanism that generates shearbands and the processes that determine the thickness andlongevity of these inhomogeneous flows. We numericallyintegrate the STZ equations to produce a deformationmap for the model glass simulated by Haxton and Liuthat shows the type of deformation or failure predictedas a function of the initial conditions.

Shear bands in this STZ formulation occur due to afeedback between the effective temperature, which de-scribes the configurational disorder in a glassy or jammedmaterial, and the local strain rate. In a sheared, steadystate, non equilibrium amorphous material, the effectivetemperature can be calculated by measuring the fluctu-ations and linear response of an observable such as thepressure and applying the fluctuation-dissipation theo-rem (FDT) [9, 12, 13]. Ono, et al. have shown that inseveral simulated foams, measurements of different ob-servables yield a single, rate dependent steady state effec-tive temperature which is distinct from the thermal tem-perature [13]. In addition, these authors show that theFDT effective temperature is consistent with an entropicdefinition: the effective temperature is the derivative ofthe configurational entropy with respect to the (poten-

2

tial) energy. This definition suggests that slow steadyshearing causes the material to ergodically explore allpossible configurational packings, and therefore the sys-tem maximizes a configurational entropy.

The STZ model describes plastic deformation in amor-phous material as a series of localized plastic events thatoccur in susceptible regions, or zones [14–16]. Follow-ing Falk and Langer, we model STZs as bistable regionsthat are more likely than the surrounding material todeform under stress, and are created and annihilated asenergy is dissipated in the system [16]. This model hassuccessfully been used to describe bulk metallic glasses,thin films, and hard spheres in several different geome-tries [7, 17–20].

An important feature of all STZ formulations is thatthe zones are activated by an effective temperature orfree volume, and there is a feedback between packingstructure and deformation. In particular, we postulatethat an STZ is an unlikely, high energy configuration ofan amorphous packing. Because the effective temper-ature governs the statistics of configurational packings,the steady state density of STZs should correspond to aBoltzmann factor,

Λ = exp

[

−Ez

kB Teff

]

≡ exp

[

−1

χ

]

, (1)

where Λ is proportional to the steady state STZ density,Ez is an activation energy, kB is the Boltzmann constant,Teff is the effective temperature, and χ ≡ Ez/(kBTeff )is a dimensionless effective temperature [17].

Because plastic deformation occurs only at these STZs,the plastic strain rate in pure shear is proportional to thisdensity, and in many situations can be written as follows:

τ0γpl = 2 e−1/χf(s), (2)

where f(s) is a function of the deviatoric stress s andτ0 is an internal timescale such as the phonon frequency.For the remainder of this paper we will refer to the di-mensionless plastic strain rate, q = γplτ0.

For completeness, Appendix A reviews the STZ modeland defines the exact equations and parameters used inthis work. The model is vastly simplified by focusing onmaterials in a simple shear geometry at thermal temper-atures far below the glass transition temperature, andcan be summarized by two equations.

The first specifies that elastic deformation increasesthe shear stress, while plastic deformation decreases it:

ds

dγ= µ∗

(

1 −2

qf(s) Λ

)

, (3)

where µ∗ is the ratio of the elastic modulus to the yieldstress, and Λ is the spatial average of the STZ densityΛ = exp(−1/χ).

A second equation specifies that the effective temper-ature approaches its steady state value, χ(q) as plastic

work is dissipated, and it also diffuses:

dχ

dγ=

2 sχ

c0s0qf(s)e−1/χ(1 −

χ

χ(q)) + a2γpl

∂2χ

∂y2, (4)

where γ is strain, q = (V0/L) τ0 is the imposed averagestrain rate times the STZ time scale, c0 is a dimensionlessspecific heat, a is a diffusion length scale on the orderof an STZ radius, and s0 is the stress threshold for theonset of plastic deformation. For the remainder of thispaper, a symbol with an overline denotes a quantity thatis a spatial average or constant as a function of position.The function f(s) is detailed in the Appendix.

The remaining unspecified component of Eq. 4 is thesteady state effective temperature χ. This material-dependent parameter captures the physically intuitiveidea that there is a maximum possible disorder attainedin a sheared amorphous packing; above χ no heat canbe dissipated in the configurational degrees of freedom.Recent simulations by Haxton and Liu [9] suggest thatthe steady state effective temperature is dependent onthe plastic strain rate, denoted q. In steady state, theinverse strain rate 1/q can be viewed as a function of thesteady state effective temperature χ, much like the vis-cosity is a function of the thermal temperature T . Wehave shown [8] that the the glassy steady state effectivetemperature is well fit by the following functional form:

1

q(χ)=

1

q0exp

[

A

χ+ αeff (χ)

]

, (5)

A discussion of this effective temperature glass transitionis given in [8], we adopt the super-Arrhenius functionαeff identified in that paper:

αeff (χ) =χ1

χ − χ0exp[−b(χ − χ0)/(χA − χ0)], (6)

where χ0 is the thermal glass transition temperature, χA

is the temperature at which the system begins to exhibitArrhenius behavior, and b, χ1 are fit parameters. Thesuper-Arrhenius component (Eq. (6)) ensures that theeffective temperature approaches a constant, χ0, as thestrain rate approaches zero, which is seen in simulations.

As noted above, this paper focuses on deformation ingeometries where the equilibrium shear stress is constantacross the sample. Although there are many interestinggeometries with shear stress gradients (and STZ theorycan explain these shear bands [20]), stress effects gener-ally dominate the more subtle internal structural dynam-ics. In contrast, experiments in simple shear geometriesobserve strain localization even though the equilibriumstress is constant, indicating that some property of theinternal state governs shear band formation.

Shear banding is caused by the coupling in Eq. (2) be-tween a configurational or structural parameter and theplastic rate of deformation. Although the stress equili-brates quickly, the effective temperature dynamics con-tinue to evolve over much longer timescales. Materials

3

that develop long-lived shear bands exhibit a very differ-ent macroscopic rheology compared to those that deformhomogeneously [21]. The type of deformation has im-portant implications for the macroscopic material andfrictional properties — systems that localize also weakenrapidly.

The remainder of this paper is organized as follows.In section II we study the stability of the model givenby Eqs. (3) and (4) with respect to perturbations. Wenumerically integrate the STZ equations to validate ouranalytic stability results in Section III and study the dif-ferent types of deformation that persist for long times inSection IV. Section V concludes with a discussion of ourresults and open questions.

II. STABILITY ANALYSIS

We now study shear band formation for systems with arate dependent steady state effective temperature χ(q).The fact that shear bands persist in the STZ model issomewhat surprising, given that the only true stationarystate of Eqs. (3) and (4) is homogeneous deformation.In the following sections, we provide an explanation forshear band formation and evolution across a wide rangeof strain rates.

We emphasize that all of the shear bands discussed be-low are transient phenomena – at very large strains theshear bands diffuse across the entire width of the mate-rial and deformation once again becomes homogeneous.However, because the internal dynamics of the configu-rational degrees of freedom can be very slow comparedto the stress evolution timescale, shear bands persist forvery long times. For example, the shear stress in nu-merical solutions to the STZ equations generally appearsto reach a steady state within a few % strain, the effec-tive temperature profile often remains inhomogenous formore than 20% strain.

First, the evolution equations for χ and s can be writ-ten as follows:

s(s, χ) = µ∗

(

1 − 2f(s)

q

∫

dy e−1/χ

)

; (7)

χ(s, χ) = 2f(s)e−1/χ

(

sχ

qc0s0

(

1 −χ

ˆχ(q)

)

+ a2 ∂2χ

∂y2

)

,(8)

where the (·) operator indicates a derivative with re-spect to time in units of strain. The function χ(q) =χ(2f(s)e−1/χ) depends implicitly on the stress and the

effective temperature. Then the Jacobian J is given by:

J11 =ds

ds=

−2µ∗

qe−1/χf ′(s), (9)

J12 =ds

dχ=

−2µ∗

qe−1/χf(s)W (δχ)/χ2, (10)

J21 =dχ

ds=

2χe−1/χ

s0c0q×

[(

1 −χ

χ

)

(sf ′(s) + f(s)) +χsf

χ2

∂χ

∂s

]

, (11)

J22 =dχ

dχ=

2e−1/χf(s)s

s0c0q×

[(

1 −χ

χ

)(

1 +1

χ

)

+

(

−χ

χ+

χ2

χ2

∂χ

∂χ

)]

.(12)

The term W (δχ) is a spatial integral over one period ofthe perturbation function; it selects only the zero wavenumber component of the perturbing function becausethe other components must satisfy periodic boundaryconditions:

W (δχ) =

∞∑

k=−∞

1

2L

∫ +L

−L

dy δχk eikπy/L; (13)

=

1 for k = 0,

0 for k 6= 0.(14)

Details can be found in [7]. We will use the term “zero-mean perturbation” to refer to a perturbing function witha vanishing k = 0 component that does not change theaverage value of the underlying function across the widthof the sample.

For a specified externally imposed strain rate q =τ0V0/L, the steady state solution to Eqs. 7 and 8 isχ = χ(q), and sss is given by the solution to the alge-braic equation 1 = 2f(sss) exp[−1/χ(q)]/q. The Jaco-bian evaluated at this steady state is greatly simplified:

J11 =−2µ∗

qe−1/χ f ′(s), (15)

J12 =−µ∗ W (δχ)

χ2, (16)

J21 =s

c0s0

∂χ

∂s, (17)

J22 =s

c0s0

(

∂χ

∂χ− 1

)

. (18)

This system is linearly stable with respect to zero-mean perturbations from a homogeneous steady state if∂χ/∂χ < 1. This is because the upper right entry in theJacobian (J12) vanishes for zero-mean perturbations, theJacobian is upper triangular, and the J11 entry is strictlynegative.

The partial derivative ∂χ/∂χ can be written as follows:

∂χ

∂χ=

χ′(q)q

χ2, (19)

4

where χ′(q) is determined by the derivative of Eq. 5.This clarifies the physical meaning of the criterion

∂χ/∂χ < 1. At low strain rates in the glassy effectivetemperature regime, χ changes slowly with q so that thepartial derivative in Eq. 19 is significantly smaller thanunity. In [7] we used the approximation that χ(q) is con-stant at low strain rates; in this case the STZ equationsare always stable with respect to perturbations in steadystate.

At high strain rates, where αeff is zero and χ(q) =A/ ln(q0/q), it can be shown that ∂χ/∂χ = 1/A. In thisregime, the equations are stable with respect to pertur-bations in steady state if and only if the normalized ac-tivation energy A is greater than unity.

The material-dependent parameter A is greater thanunity in all materials where the steady state stress in-creases with increasing imposed strain rate. To see this,note that Eq. 5 can be rewritten as follows:

χ(q) = −A∗(q)

ln(q/q∗), (20)

where A∗ is an activation energy and q∗ is a dimensionlessconstant. In steady state, χ = χ everywhere, so insertingEq. 20 into Eq. 2 and taking the derivative with respectto q results in the following equation for s′(q), which isthe derivative of the steady state stress with respect tothe imposed driving rate:

s′(q) = [−1 + A∗(q) + q ln(q/q∗)A′∗(q))]

(

q

q∗

)−1/A∗(q)1

A∗(q)f ′(s)(21)

The function f increases monotonically with s. There-fore s′(q) is positive and the steady state stress increaseswith increasing imposed strain rate whenever the firstfactor in Eq. 21 is positive. In high strain regimes, theactivation energy is a constant, A∗(q) = A, and the ma-terial is steady state strengthening whenever A > 1.

In most simulations of disordered disks with hardcore repulsion (including simulation data by Haxton andLiu [9] and Shi et al [21]), the steady state stress in-creases with increasing strain rate, so that ∂χ/∂χ is lessthan unity and the system is stable with respect to zeromean perturbations. Materials with a steady state stressthat increases as a function of driving velocity are calledvelocity strengthening. In contrast, several experimentsthat study friction of granular fault gouge find velocity-weakening behavior; the steady state stress decreases asthe imposed strain rate increases. These materials mightbe susceptible to shear banding even in steady state. Inthis paper we focus on shear banding in materials thatare velocity strengthening, but velocity weakening mate-rials are an interesting avenue of further research.

STZ theory predicts that velocity strengthening mate-rials should continue to deform homogeneously once theyattain steady state. However, for certain initial condi-tions simulated velocity strengthening materials developshear bands before they reach this homogeneous steady

state. This leads us to study the stability of perturba-tions at each point in time along a time-varying trajec-tory, assuming that the system deforms homogeneouslyup to that point. This is a “frozen time” stability anal-ysis.

First note that due to the integral in Eq. (7), spa-tial perturbations to χ with zero mean do not changethe equation for s to first order in δχ : s(s, χ + δχ) =s(s, χ)+O(δχ2). In this case the linear stability is deter-mined solely by the equation for χ and the sign of J22.Above the yield stress, χ and f(s) are strictly greaterthan zero and therefore J22 is negative (and the trajec-tory is linearly stable) whenever the following criterion ismet:

χ > χcrit =1

4χ

(

−χ + χ2 + qχ′(q)+

√

8χ3 + (−χ + χ2 + qχ′(q))2)

. (22)

By inserting values for χ(q) into Eq. (22), we derive alinear stability prediction for the boundary between thesetwo regimes as a function of the average initial effectivetemperature χ and the applied strain rate q. We chooseχ(q) to fit the data from Haxton and Liu [8, 9].

−13 −12 −11 −10 −9 −8 −7

0.1

0.15

0.2

0.25

Ln(imposed strain rate)

Initi

al e

ffect

ive

tem

pera

ture

χcrit

χR=1

FIG. 1: (color online) Predicted deformation map based oninitial conditions only. The solid line marked with circles(black) represents the frozen time linear stability criterionpredicted by Eq. (22), which does not take into account fi-nite amplitude perturbations. R is a more accurate general-ized stability criterion that takes into account finite amplitudeperturbations. A line with constant localization ratio R = 1is marked with crosses(blue). Above R = 1 homogeneous de-formation is predicted and below R = 1 strain localizationis predicted. For reference, the upper dashed line shows theχ(q) fit to the data from Haxton and Liu [9].

Unlike linear stability analysis for steady states, frozentime stability analysis for time varying trajectories doesnot predict the final state of the system. It provides an

5

indication that a transient instability is possible, but itdoes not specify global stability. The frozen time analy-sis is accurate only when the perturbations grow rapidlycompared to the growth of the underlying trajectory.We therefore use the more general localization ratio Rto characterize the transient instability. First discussedin [7], this ratio compares the growth rate of pertur-bations (determined by frozen-time stability analysis) tothe growth of the underlying trajectory:

R = δχexp[J22(sm, χini)∆t/2]J22(sm, χ)

χ(sm, χ), (23)

where δχ is the magnitude of the perturbation, χini is theinitial effective temperature, sm is the approximate max-imum shear stress given by the solution of the equationq = 2f(sm) exp[−1/χini], and ∆t is the approximatetime in units of strain it takes to achieve the stress maxi-mum. Localization occurs when the rate at which heat isdissipated inside the band is larger than the rate outsidethe band; in this case R is greater than unity.

The localization ratio given by Eq. (23) depends on themagnitude of the perturbation δχ. In the low strain ratelimit, we have systematically studied the localization ra-tio R as a function of perturbation amplitudes and foundthat it accurately predicts that larger perturbations leadto enhanced localization [7]. For simplicity, we have cho-sen δχ to be 5 % of the average value of the effectivetemperature, which is consistent with perturbations tothe potential energy per atom for a Lennard-Jones glasscalculated by Shi, et al [21]. Systematically studying theeffects of perturbation magnitude as a function of strainrate is beyond the scope of this paper.

Figure 1 is a deformation map that predicts the typeof flow as a function of the initial conditions for the sim-ulated glassy material studied by Haxton and Liu [9].The bold line is the linear stability criterion defined byEq. (22). Because a frozen time analysis does not takeinto account finite amplitude perturbations or the growthrate of the underlying trajectory, we use the localiza-tion ratio, R, to predict localization. Using Eq. 23,we calculate R for each set of initial conditions withδχ = 0.05 × χini, and ∆t = 0.06= 6 % strain. Theline marked with crosses in Fig. 1 corresponds to a linewith constant R = 1. Localization is expected below thisline, where R > 1, and homogeneous flow above it.

III. NUMERICAL SOLUTIONS TO STZEQUATIONS

To check these analytic predictions, we numerically in-tegrate the STZ partial differential equations. The nu-merical solutions exhibit three broad categories of de-formation behavior: homogeneous deformation, shearbands, and melting or failure. This section discussesqualitative features of each kind of deformation, whileSection IV develops a deformation map using a quantita-

tive criterion for each category and discusses macroscopicimplications.

To resolve extreme localization, we use an irregularmesh and a combination of fixed-step and adaptive-stepfinite difference methods. For each pair of initial condi-tions, the average initial effective temperature χini andthe externally applied strain rate q = τ0(V0/L), we nu-merically integrate the STZ equations (Eqs. (3) and (4))from 0 to 20 % strain. The initial effective temperaturefunction χini(y) is a constant perturbed by a hyperbolicsecant function of height δχ and width L/10, normalizedso that its average is χini, and the initial shear stressis 0.0001. All stresses are in units of the yield stress sy

unless otherwise noted.

For comparison, we also numerically integrate a singledegree of freedom STZ model, where the effective tem-perature is constrained to be constant across the widthof the material, and no perturbations are permitted. Thesystem of ordinary differential equations given by Eqs. (3)and (4) (with no diffusion) is integrated numerically intime using the same average initial conditions as the STZPDE.

The simple ODE model cannot localize and has beenused to describe macroscopic frictional behavior forboundary lubrication in thin films [19] and on earthquakefaults [22]. The ODE model is an example of a “rate andstate” friction law. These laws are frequently used in geo-physical modeling of earthquake ruptures, and describethe response of a sheared frictional interface as a func-tion of the slip rate (or strain rate) and a single statevariable, often denoted θ. While the STZ PDE resolvesinternal dynamics of the effective temperature within theinterface, the STZ ODE is constant across the interface.Comparing the two models allows us to study the effect ofsmall scale dynamics such as strain localization on modelpredictions for macroscopic behavior.

In some simulations of the full PDE model, the steadystate effective temperature χ approaches infinity. Al-though the STZ model given by Eqs. (3) and (4) is stillwell-defined in this limit, the shear heating term becomesconsiderably amplified, indicating a situation where theamorphous packing becomes more and more disorderedinside the band. In every instance where χ → ∞, theshear band becomes so thin that the numerical integra-tion routine fails.

We suggest that this numerical failure corresponds tomaterial failure. The smallest length scale in the modelis a, the diffusion length scale which is on the order ofthe radius of an STZ. We do not expect the STZ modelto hold at length scales smaller than a, and because ournumerical mesh is fine enough to resolve a band ten timessmaller than a, numerical failure corresponds to a shearband that rapidly becomes so thin that the model itselfbreaks down.

Although the simple STZ model developed here doesnot specify the rheology at strain rates above this “melt-ing” point, it does suggest that the solid-like STZ theorymust be replaced by a liquid-like theory (such as mode

6

coupling or Bagnold scaling) inside these bands. There-fore, integration of the STZ model indicates that whenthe disorder temperature approaches infinity, the mate-rial can no longer support a static shear stress; it liquefiesand fails.

In simulations where the effective temperature remainsfinite, we numerically track the shear stress s and theeffective temperature field χ(y) as functions of time, orequivalently, strain. In each case, the stress first respondselastically, and then begins to deform plastically abovethe yield stress sy. As plastic deformation increases theeffective temperature, the material softens and the stressrelaxes to its flowing value, sf . The dashed blue line inFig. 2 is a plot of the stress vs. the strain for a numericalsolution to the STZ PDE model. Initial conditions aresuch that the material is highly unstable with respect toshear bands. For comparison, the dash-dotted (magenta)line in Fig. 2 shows the solution to the STZ ODE model,which is a rate and state law with a single internal statevariable. The STZ PDE solution develops a shear bandand weakens much more rapidly than a rate and statemodel with similar initial conditions.

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

strain

norm

alize

d she

ar st

ress

ODE

0.02390.02720.02810.02890.03220.04470.0781

PDE

FIG. 2: (color online) Shear stress s vs. strain calculatedby numerically integrating the STZ equations of motion withinitial conditions χini = 0.0674, and imposed strain rateq = 1.015 × 10−6. The dashed (blue) curve represents thesolution to the perturbed STZ PDE model, while the dash-dotted (magenta) curve represents a solution to the STZ ODEmodel with the same average initial conditions. The coloredsymbols correspond to the plots shown in Figs. 3 and 5. Atabout 2% strain, the perturbed system begins to localize andweakens much more rapidly than the magenta curve.

During this initial stress response, the effective temper-ature field also evolves in space and time. The effectivetemperature field is initially constant with a small, cen-tered perturbation, and the field remains static duringthe elastic response. At the onset of plastic deformation,the effective temperature begins to rise. In systems thatdeform homogeneously the average value of the effectivetemperature rises and the perturbation dissipates, whilein systems which develop shear bands the effective tem-

0 0.5 10.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

y position

Effe

ctive

tem

pera

ture

,

0.00000.02390.02720.02810.02890.03220.04470.07810.1031

0 0.050.06

0.1

0.14

0.18(a) (b)

FIG. 3: Diffusion-limited shear band(color online) (a)Time series of the effective temperature as a function of posi-tion for a material with initial conditions χini = 0.0674, andimposed strain rate q = 1.01510−6 . Each line represents theeffective temperature field as a function of position at a differ-ent time, as indicated by the legend (all times are in units ofstrain). The effective temperature field is initially a constantwith a small perturbation centered in the middle(blue). Thisperturbation grows rapidly (green) and forms a shear band,which then diffuses outward slowly (red). (b) Inset showsthe same data on a different scale. The strain associated witheach line is also indicated in Fig. 2; localization coincides withrapid dynamic weakening of the shear stress.

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

25

strain

Ave

rage

pla

stic

str

ain

rate

FIG. 4: Average plastic strain rate,R

dy γ(y) τ0/q as a func-tion of strain for the same integration data shown in Fig. 2.Initially the average plastic strain rate is zero during the ma-terial elastic response. The average plastic strain rate thenrises rapidly during the stress overshoot, when the system re-leases stored elastic energy. Finally, it relaxes back to unityin the flowing regime, when all the strain must be accommo-dated plastically.

perature rises rapidly inside the band and attains a slowlyevolving state where the shear bands diffuse outwards.

Figures 3(a) and (b) are a series of plots of the effec-

7

0 0.5 10

500

1000

1500

2000

2500

3000

y position

Norm

alize

d plas

tic st

rain

rate

0.00000.02390.02720.02810.02890.03220.04470.07810.1031

(a)

0 0.050

20406080

100120 (b)

FIG. 5: Diffusion-limited shear band(color online) (a)Time series of the normalized plastic strain rate γτ0/q as afunction of position. The plastic strain rate is initially zero(during elastic deformation) but rises at the onset of plasticdeformation and becomes very sharply peaked (green). Thisshear band is extremely narrow with a thickness of about0.015, which is approximately the same as the diffusion lengthscale a. As the stress relaxes the strain rate also relaxes,and the shear band becomes wider and less sharply peaked(red).(b)Inset shows magnified position and strain rate axes.Although the maximum strain rate in the band decays signif-icantly with time, it remains large (> 25 times the imposedstrain rate).

tive temperature as a function of position for a materialthat develops a thin shear band (later we will identify theinitial transient as diffusion limited). Each colored linerepresents a different time in units of strain. A small ini-tial perturbation to the effective temperature is driven bya dynamic instability to a much higher value, saturatingat χ ∼ 0.22, and the band then slowly diffuses outward.

The plastic strain rate also evolves during the initialtransient response. We first focus on the average plas-tic strain rate, shown in Fig. 4 as a function of time.At early times when the stress is below the yield stresssy, the system deforms elastically and the plastic strainrate is zero everywhere. At the onset of plastic deforma-tion the average plastic strain rate increases continuouslyfrom zero, attains a maximum, and then relaxes back tothe externally imposed strain rate (in the flowing regimeall the deformation is plastic.) While the plastic strainrate is greater than unity, stored elastic energy is beingdissipated.

Although the stress is constant across the width of thematerial, regions with a higher effective temperature de-form more rapidly. To effectively compare strain local-ization at various strain rates, we plot the dimension-less strain rate, which is the strain rate at each locationdivided by the externally imposed strain rate. In fig-ures 5(a) and (b), each colored line represents a differenttime in units of strain; plots show the plastic strain rateas a function of position. Initially the stress is below theyield stress and the plastic strain rate is zero (blue line).Localization of the effective temperature field results in a

very narrow peak in the strain rate field (green line). Thestrain rate in the center of the shear band is nearly 3000times larger than the externally imposed strain rate. Asthe stress continues to relax, the strain rate becomes lesssharply peaked (red line). The inset plots magnify theposition axis.

Comparing numerical results to analytic predictions re-quires a method for measuring the degree of localizationin a given numerical simulation. The degree of localiza-tion can be quantified using the Gini coefficient φ [23],defined as:

φ(t) =1

2n2Dpl

∑

i

∑

j

|Dpl(yi, t) −Dpl(yj , t)|, (24)

where yi is a uniform grid of n points in position space.The Gini coefficient is equal to zero if the material de-forms homogeneously and increases as the plastic strainrate field becomes more sharply peaked. A delta func-tion has a Gini coefficient of 1. During a given numericalsimulation, the Gini coefficient φ(t) starts out as a verysmall number and then increases rapidly as the shearband forms. Then, as the shear band diffuses the Ginicoefficient decreases. Because we are focusing on the ini-tial transient, we first study the maximum value of theGini coefficient attained during a given numerical simu-lation.

Figure 6(a) is an intensity plot of the maximum valueof the Gini coefficient as a function of the average initialeffective temperature, χini and the natural logarithm ofthe dimensionless imposed strain rate log(q). This defor-mation map indicates that material deformation grad-ually changes from homogeneous flow to shear bandingas a function of the initial conditions. In figure 6(a),black boxes indicate that χ approached infinity during aparticular numerical integration, and the STZ solid-likedescription breaks down.

While the Gini coefficient is a direct indicator of local-ization, it is perhaps a less familiar metric. For compari-son, Figure 6(b) shows that maximum plastic strain rateattained in the band as a function of the initial condi-tions. A larger plastic strain rate is attained in a thinner,more localized band, and therefore Fig. 6(b) is very sim-ilar to Fig 6(a). Again, black boxes correspond to shearbands where the plastic strain rate reaches the meltingpoint and the model breaks down.

IV. DEFORMATION MAP ANDMACROSCOPIC IMPLICATIONS

Numerical solutions presented in the previous sectionshow that transient dynamics can lead to inhomogeneousflows. We would like to understand how to characterizethese flows. What type of deformation occurs as a func-tion of the initial conditions? If shear bands form, whatsets their thickness? What are the implications of inho-mogeneous flows for macroscopic system response?

8

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FIG. 6: (color online) A diagram showing the degree of lo-calization found in by numerically integrating the STZ equa-tions. (a) The maximum Gini coefficient, Eq. (24), as a func-tion of the average initial effective temperature and the exter-nally imposed strain rate. A higher Gini coefficient indicatesmore localization. (b) The log of the maximum plastic strainrate attained in the band normalized by the externally im-posed strain rate. In both figures, black boxes correspondto numerical simulations where the magnitude of the strainrate was so large that χ → ∞, as discussed in the text. Thesolid white line corresponds to the predicted localization ratioR = 1: Localization is expected for initial conditions belowthis line. See, e.g. results for χini = 0.0674, q = 1.015× 10−6

in Fig.4.

A. Shear band thickness

As mentioned earlier, the stress appears to achievea steady state quickly – after less than 7 % strain allnumerical STZ solutions have acheived a steady stressthat changes by less than 5 % over the course of theremaining simulation (200 % strain). In comparison,the effective temperature field often remains highly local-ized for t > 20% strain, and broadens over much longertimescales than the stress.

The goal of this section is to calculate shear bandthickness as a function of initial conditions within the

STZ model, and determine what sets the thickness of theshear bands in this model. Because localized strain statesare transient – the bands diffuse outward over time – westudy the model predictions for how shear band thicknessevolves over large (20 %) strains. Importantly, many ini-tial conditions lead to numerical simulations and exper-imental materials that fail before reaching large strains.The following analysis identifies these events as well.

We calculate the shear band thickness for each numer-ical solution at two times: the time tqmax at which thestrain rate in the shear band attains its peak and theshear band thickness is minimized, and at a later timet = 20% strain where the stress appears to be in steadystate. For systematic study, we specify initial conditionsthat generate only a single shear band – multiple shearbands are often found experimentally at higher strainrates and will be a topic of future study.

We first study the shear bands at a time when the plas-tic strain rate is most highly localized. Let qmax(y) bethe normalized plastic strain rate γ(y, t)τ0 /q evaluatedat the time tqmax when the strain rate achieves its abso-lute maximum. sqmax is the shear stress at tqmax. Thethickness wqmax of the shear band in a numerical solu-tion is defined to be the fraction of the real line between−1 and 1 where the function qmax(y) is sufficiently large:

wqmax =

∫

I

dy. (25)

The region I is defined as follows:

I = y ∈ [−1, 1] | qmax(y) > 1 + h supy

qmax(y), (26)

where h is an arbitrary fraction. Although we chooseh = 1/10, in most cases the calculated thickness is insen-sitive to the value of h because the strain rate functionis sharply peaked. When the system deforms homoge-neously, the thickness wqmax is not well-defined. In thiscase Eq. (25) becomes extremely sensitive to the fractionh and is no longer accurate.

Figure 7 is a plot of the shear band thickness at thetime of maximum strain rate wqmax as a function of theinitial conditions. White boxes in Figure 7 correspondto solutions where the maximum Gini coefficient is lessthan 0.35. These homogeneously deforming solutions donot have a well-defined value wqmax.

The discussion in the previous paragraphs analyzesshear bands at their peak, when the plastic strain rateis maximized. This generally occurs at less than 7 %strain. Fig. 8 shows the shear band thickness at at 20 %strain. In each case the shear bands have become wider,as expected. (Note that the maximum thickness shownin this plot is 0.3, as compared to 0.2 in Fig. 7.) Al-though these systems do not acheive a stationary state,the slowly evolving shear band thickness is observable,and has been seen in molecular dynamics simulations [21]where periodic boundary conditions allow the system tobe studied at very large strains.

9

Shear band width at tqmax

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FIG. 7: (color online) Shear band thickness at the time ofmaximum strain rate wqmax, Eq. (29), for numerical STZ so-lutions as a function of the initial conditions χini and log(q).The black boxes correspond to initial conditions for whichχ → ∞ during an integration, while the white boxes corre-spond to initial conditions for which the flow is homogeneous(the maximum Gini coefficient, φmax < 0.35) The color scaleis set such that the maximum thickness is 0.2

Shear band width at 20 % strain

ln(imposed strain rate)

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FIG. 8: (color online) Shear band thickness at 20 % strainfor numerical STZ solutions as a function of the initial condi-tions χini and log(q). The black boxes correspond to initialconditions for which χ → ∞ during an integration, whilethe white boxes correspond to initial conditions for whichthe flow is homogeneous (the Gini coefficient at 20 % strain,φ(t = 0.2) < 0.35) Note that the scale for this plot is largerthan that in Figure 7 – the shear bands are significantly widerat 20 % strain than at tqmax

Perhaps the most interesting feature of Figs. 7 and 8is that the system exhibits no obviously preferred lengthscale – the shear band thickness varies continuously fromabout O(a) ≃ 0.015 to O(1) (homogeneous flow). More-over, the shear band thickness increases with time. Bothof these observations are a consequence of the fact thatlocalized states are transient solutions to the equationsof motion rather than steady state solutions.

Although there is no preferred shear band length scale,the shear band thickness is reproducible; the STZ modelgenerates shear bands of the same thickness given thesame average initial conditions, even if the perturbationsare random [7]. In addition, simulations of Lennard Jonesglasses generate reproducible shear band thicknesses asa function of time [21] and experiments on bulk metal-lic glasses find a characteristic shear band thickness [10].These results suggest that on a given observational timescale, the system does pick out a specific shear bandthickness. Because it is observable and reproducible, theshear band thickness must evolve very slowly comparedto the stress relaxation time scale. We exploit this fea-ture, showing that the STZ model singles out three dif-ferent deformation profiles that evolve slowly in time andwhich should therefore describe observable deformationmodes. In addition, we discuss another state - materialfailure – where a parameter in the STZ model divergesand the model fails.

We analyze Eq. 8 to determine what deformation pro-files generate the smallest change in the effective temper-ature. Because the stress is nearly stationary, flows withthe smallest average values for χ are the longest-livedtransients and are easily observable. We show deforma-tion profiles for these states and develop a deformationmap at tqmax and t = 20% strain. This provides an ex-planation for observed shear band thicknesses.

B. Relaxation towards homogeneous deformation

The first and simplest state minimizes χ everywhereand is “homogeneous deformation.” The effective tem-perature field is constant everywhere and equal to χ(q),where q is the externally imposed dimensionless strainrate. Since both the shear heating and diffusion termsare zero in Eq. (8), this is a true steady state that per-sists forever. The dashed blue line in Fig. 9 is a plot ofthe stress as a function of time for the full STZ modelwith a small initial perturbation to the effective tem-perature field, but the initial conditions are such thatthe perturbation relaxes and the deformation relaxes to-wards homogeneous flow. The simple ODE model stresssolution, shown in magenta, lies on top of the PDE stresssolution – the macroscopic stress response is the same forboth models. The colored symbols correspond to plotsin Figs. 10(a) and 10(b).

Homogeneous deformation is characterized by the dis-sipation of perturbations to the effective temperaturefield. The effective temperature as a function of positionis shown in Fig. 10(a), and each colored line representsthe state of the system a different time. A small ini-tial perturbation to the effective temperature dissipatesas a function of time, although the average value of theeffective temperature increases as plastic work is dissi-pated. The effective temperature never varies more than5% from its average value.

A similar plot for the plastic strain rate is shown in

10

Fig. 10(b). The plastic strain rate is zero during theelastic response, and although the perturbation to theeffective temperature generates a small perturbation tothe strain rate at the onset of plastic deformation, thestrain rate relaxes towards a homogeneous state. Themaximum plastic strain rate is remains within 20% of itsaverage value.

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ODEPDE0.01010.01350.01430.01510.01850.03100.06430.0893

FIG. 9: (color online) Deviatoric stress s vs. strain calculatedby numerically integrating the STZ equations of motion withinitial conditions χini = 0.0674, q = 1.015×10−6 . The dashed(blue) curve represents the solution to the perturbed system,while the dash-dotted (magenta) curve represents a homoge-neous solution where the effective temperature is constrainedto be constant inside the material. In this plot the two curvesare indistinguishable. The colored symbols correspond to theplots shown in Fig. 10. Because χini is large, the system be-gins with a large number of plasticity carriers and thereforethe stress peak is negligible. This system does not localize.

C. Diffusion limited shear bands and failure

A second slowly evolving state, called “diffusion lim-ited localization,” occurs when the shear heating and dif-fusion terms in Eq. (8) balance. In this case the effectivetemperature field is far from its steady state value χ atall points in space, so that the factor (1 − χ/χ) is closeto unity and the shear heating term sχ/(s0c0), which isof order one, balances the diffusion term a2. The balanceis not perfect, and χ is not exactly zero, so the bandcontinues to diffuse slowly outward.

This type of deformation is important because it setsthe minimum length scale for shear bands O(a) ≃ 0.015,in numerical STZ solutions. Although a subset of initialconditions generates shear bands that become thinnerthan this length scale, the local strain rate in these bandsbecomes so large that the steady state effective temper-ature χ approaches infinity. In other words, sχ/(s0c0)is too large inside the band, the diffusive flux can notbalance it, resulting in a runaway heating process. Thisfailure is not an artifact of our numerical methods; it sig-

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FIG. 10: (color online) Relaxation to homogeneous flow: (a)effective temperature and (b) normalized plastic strain rateγ(y)τ0/q as a function of position, for a material with initialconditions

R

dy χ(y, t = 0) = χini = 0.20, and imposed strainrate q = 1.015 × 10−6. Different (colored) lines represent dif-ferent times. A small initial perturbation to the effective tem-perature relaxes, although the average effective temperatureincreases. (The effective temperature scale is much smallerthan Fig. ??(a). At the onset of plastic deformation the plas-tic strain rate is also perturbed, but this perturbation alsodecays.

nifies a break down in the STZ model that occurs whenthe effective temperature increases without bound. Weassociate this runaway process with a third state, theonset of material failure, because the solid-like STZ de-scription fails as the material liquefies.

Although we can not track the thickness of the bandbelow the grid resolution during this runaway process, wedo observe that just prior to failure the effective temper-ature in these simulations is elevated significantly aboveits average in a region of thickness a. In other words,the diffusion length scale appears to be an upper boundon the size of the region where structural changes occurduring these shear failure events.

11

Excluding material failure, the thinnest shear bandspossible in this model are diffusion limited. These typesof shear band persist for long times in STZ simulationsfor earthquake faults, where the parameters are highlyvelocity weakening and chosen to match laboratory ex-periments for granular fault gouge [24]. Diffusion alsolimits the initial thickness of shear bands in several ofthe numerical simulations performed in this paper, al-though these shear bands continue to diffuse outward atlarger strains.

Diffusion limited shear bands can be identified by theirnarrow thickness, which is of order a, although the ex-act value varies with the stress overshoot and specificheat c0. In this 2D model, we use the term “thickness”to refer to the extent of the shear band in the direc-tion orthogonal to the slip plane, which is similar to themeaning of this term in three-dimensional systems [2].Although diffusion of potential energy (and presumablyeffective temperature) has been seen in simulations [21],the length scale a associated with this diffusion constantis relatively unconstrained by simulations or experiments.A reasonable postulate is that a is on the same order asthe radius of an STZ, or equivalently, a few particle radii.This suggests that diffusion limited shear bands are verynarrow.

The stress vs. strain curve for a material that developsa diffusion limited shear band is given by the dashed blueline in Fig. 2. The stress weakens very rapidly as thediffusion limited shear band forms. For comparison, thedash-dotted magenta curve in Fig. 2 is the stress responseof the ODE rate and state model with the same averageinitial conditions. This illustrates that thin shear bandsdrastically change the macroscopic system response, andthat this dynamic weakening is not captured by a singledegree of freedom rate and state model.

As discussed in Section III, Figures 3 and 5 show thetime evolution of a shear band which is initially diffusion-limited.

D. Disorder limited shear band

The fourth “disorder limited” localized state is less in-tuitively obvious, but occurs frequently in our numericalSTZ solutions. Neglecting the diffusion term, the righthand side of Eq. (8) is proportional to the product oftwo factors, exp(−1/χ) and (1 − χ/χ). The former isvery close to zero whenever χ is significantly less thanχ0, and the latter is zero when χ = χ. The disorder lim-ited state occurs exactly when the small-χ condition ismet outside the shear band and χ = χ inside the band,so that χ in Eq. (8) is always small. However, it is neverzero, so that the disorder limited shear bands are alsotransient solutions that diffuse outward with time. Thistype of shear band was first described in [7], and capturesfeatures of shear bands observed in simulations by Shi,et al. [21].

Figure 11 is a plot of the shear stress s vs. strain for a

system that develops a disorder-limited shear band. Thisplot is calculated by numerically integrating the STZequations of motion with initial conditions χini = 0.1042and q = 8.7 × 10−6. The blue curve represents the so-lution to the perturbed system, while the magenta curverepresents a homogeneous solution where the effectivetemperature is constant as a function of position insidethe material. The colored symbols correspond to theplots shown in Fig. 12. Although the localized systemweakens slightly faster than the homogeneous system,the effect is small and on this scale the two curves areindistinguishable.

We show the effective temperature and strain ratefields for a numerical solution that exhibits a disorderlimited shear band in Figs. 12(a) and 12(b). The per-turbation to the effective temperature field grows veryslowly at first, then more rapidly as χ → χ, and finallythe peak begins to diffuse slowly outward. Similarly, thenormalized plastic strain rate begins at zero (blue), thenrises quickly (green) and relaxes slightly (red).

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FIG. 11: (color online) Shear stress s vs. strain calculatedby numerically integrating the STZ equations. The dashed(blue) curve represents the solution to the STZ PDE, whilethe dash-dotted (magenta) curve represents a solution wherethe effective temperature is constant inside the material. Thecolored symbols correspond to the plots shown in Fig. 12.Although the localized system weakens slightly faster thanthe homogeneous system, the effect is small and on this scalethe two curves are indistinguishable.

The thickness of disorder limited bands is not set by asimple internal length scale such as a. Instead, the thick-ness is determined dynamically by the externally imposedstrain rate and the initial conditions.

Assume for the moment that a single shear band formsin the material. This is explicitly enforced for the numer-ical integration in this paper because the initial hyper-bolic secant perturbation at y = 0 leads to a single shearband at that position. In addition, a single shear band isobserved in simulations [21] and numerical integration ofthe STZ model with random perturbations to the initialeffective temperature [7] at low strain rates.

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FIG. 12: Disorder limited shear band(color online) (a)Normalized plastic strain rate γ(y)τ0 /q and (b) effective tem-perature as a function of position (y), for a material withinitial conditions χini = 0.1042, and imposed strain rateq = 8.7 × 10−6. Different (colored) lines represent differenttimes; cooler colors (blue) correspond to earlier times, whilewarmer colors (red) correspond to later times. The plasticstrain rate in the band increases significantly (about 800 %),although much less than in the diffusion limited shear band.The thickness of this band at its peak is about 0.2, muchlarger than the thickness of a diffusion limited shear band.

Under this assumption, almost all of the deformationis accommodated in a band of thickness w:

q = τ0V0/L ≃ τ0 (w/L) γband. (27)

Using Eqs. (2) and (27) we derive the following relation-ship between the stress s, the thickness of the shear bandw and the externally imposed strain rate q:

q 2L

w≃ 2f(s) exp

[

−1

χ(q2L/w)

]

. (28)

This is not a prediction for the thickness of the shearband, because the final stress, s is not specified. Unfor-tunately, we can not derive an approximate value for sbecause it depends on the entire history of deformation inthe material. In addition, the final value of s is generally

close to the yield stress, and f(s) is very sensitive to s inthis regime. However, in the next section we will checkto see if the shear bands in a given numerical simulationsatisfy the criterion given by Eq. (28).

E. Deformation map at the time of maximumdeformation rate

We now determine which of these states occur and per-sist as a function of the initial conditions in the numeri-cally integrated solutions. First, we use the following cat-egories to characterize the deformation at the time tmax:homogeneous deformation, diffusion limited shear band,disorder limited shear band, or material failure. Thiscategorization is somewhat arbitrary because none of thestates are stationary; all perturbed states will eventuallydecay towards homogeneous flow. However, by identi-fying these different regions in phase space we hope toidentify length scales and features that are observable inexperiments.

To determine if a shear band thickness is consistentwith disorder limited deformation, we rearrange Eq. (28),inserting wqmax and sqmax:

log

(

q 2L

2f(sqmax)wqmax

)

+1

χ(q2L/wqmax)= 0. (29)

A shear band in a numerical solution is said to be“disorder limited” if Eq. (29) is satisfied to within 8 %,(i.e., the left hand side equals 0 ± 0.08.) Similarly, ashear band is “diffusion limited” if its thickness is ap-proximately equal to the diffusion length scale a, (i.e.0 < wqmax < 0.03), and homogeneous if the maximumGini coefficient is less than 0.5. These cutoffs are chosento ensure that deformation regions are non-overlapping,which is a strong constraint. Finally, a material is saidto fail if χ approaches infinity during the course of inte-gration. Transition regions are expected when an inho-mogeneous flow does not fit into one of these categories.Figure 13 is a deformation map that indicates where eachof these criteria are satisfied.

F. Deformation map at 20 % strain

The same criteria for deformation categories that wereused at tqmax in Fig. 13 can also be used to categorizeshear bands at 20% strain. Inhomogeneous flows whichwere in a “transition regime” at tqmax should broadenquickly towards one of the slowly varying categories. Thisis seen in Fig. 14, which shows deformation categoriesat 20 % strain. The transition region between disorderlimited shear bands and failure has shrunk considerably.However, all of the diffusion limited flows have transi-tioned to disorder limited bands, and some disorder lim-ited shear bands have transitioned towards homogeneousflows. This highlights the fact that shear band thickness

13

Shear band type at tqmax

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FIG. 13: (color online) Deformation map that uses the thick-ness shown in Fig. 7 to determine if the deformation at timetqmax is diffusion or disorder limited localization. Diffusionlimited shear bands (very dark gray/dark red) and failure(black) occur where 0 ≤ w < 0.035, and disorder limited shearbands (medium gray/orange) occur where the left-hand sideof Eq. (29) is less than 0.7. The very light gray (light yellow)region indicates homogeneous flow. Because this is a snapshotof the the deformation types at t = tqmax, for some initialconditions the system is transitioning between two types offlows. The red region represents a transition regime betweendiffusion limited and disorder limited shear bands, while darkyellow represents a transition between disorder limited shearbands and homogeneous flow. Blue outline boxes indicateinitial conditions detailed in Figs. ??, 10, and 12.

and deformation type depend significantly on the amountof strain. Most experiments are limited to small strainsand therefore can not see this time evolution.

This analysis of the STZ model shows that for a largerange of initial conditions, shear bands are a robust fea-ture that persist for long times. STZ theory indicatesthat shear band thickness evolves slowly over very largestrains and suggests that the thickness is determined dy-namically by the initial and boundary conditions. STZtheory predicts that the minimum thickness of the bandsis set by an effective temperature diffusion parameter a,but a continuum of other thicknesses is also possible anddynamically determined.

While we predict that the types of deformationmapped in Figs. 13 and 14 will occur in a wide rangeof amorphous solids, the exact location of boundariesbetween types and the longevity of each type are likelymaterial-dependent. These deformation maps depend onthe definition of the steady state effective temperatureχ(q).

All deformation maps in this work apply specificallyto the model glass simulated by Haxton and Liu becausewe used a function χ(q) that fits their data for simulatedrepulsive disks; different materials may have slightly dif-

Shear band type at 20 % strain

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FIG. 14: (color online) Deformation map at 20 % strain,that uses the thicknesss shown in Fig. 8 to determine thetype of deformation. Failure (black) occurs where χ → ∞

during a numerical simulation, and disorder limited shearbands (medium gray/orange) occur where the left-hand sideof Eq. (29) is less than 0.7. The very light gray (light yel-low) region indicates homogeneous flow. At t = 0.2, thedark gray (red) transition region in Fig. 13 has disappeared– the shear bands have widened to become disorder limitedshear bands. In addition, the diffusion limited shear bandshave also widened to become disorder limited, and some ofthe shear bands which were disorder limited in Fig. 13 havetransitioned towards homogeneous flow (light gray/dark yel-low). Blue outline boxes indicate initial conditions detailedin Figs. ??, 10, and 12.

ferent steady state effective temperatures, although it isalso possible that χ(q) is universal. The steady state ef-fective temperature can be measured in simulations bycomparing the fluctuations and linear response of an ob-servable such as the pressure; this input is all that isneeded to generate a deformation map for a new mate-rial using STZ theory.

In the analysis, we have assumed a single shear band.Experiments and simulations of bulk metallic glassesshow that the material develops multiple shear bands athigher strain rates [10, 21]. Developing a model for thenumber and spacing between shear bands is beyond thescope of this paper, but the STZ model should providean excellent starting place for these analyses.

V. CONCLUSIONS

We have analyzed the stability of the STZ model witha strain-rate dependent effective temperature, and foundthat the details of the rate dependence specify the steadystate stability of homogeneous flows. Most simulatedglasses exhibit velocity strengthening, where the steadystate stress increases as a function of strain rate, and we

14

have shown that these materials are stable with respectto perturbations in steady state. In contrast, velocityweakening materials, such as granular fault gouge, areunstable with respect to shear bands in steady state.

Perhaps surprisingly, shear bands develop even in ve-locity strengthening materials. They result from an in-stability that develops during a transient stress response,when a material is driven from rest or driven at a newvelocity. Although the perturbations are unstable onlyfor small strains, the resulting inhomogeneous effectivetemperature profiles χ(y) are nearly stationary states ofthe model equations of motion and therefore these shearbands persist for long times.

By including information about the rate dependenceof the steady state effective temperature, χ(q), we showthat the STZ model generates a deformation map thatincludes homogeneous deformation, thick “disorder lim-ited” shear bands, thin “diffusion limited” shear bands,as well as the onset of material failure.

The shear bands that emerge spontaneously in the STZmodel capture several important features seen in simu-lations and experiments. First, the STZ model predictsthat shear band formation coincides with stress relax-ation after the initial stress overshoot in start-up flows.In cases where the material does not fail, the model pre-dicts that shear bands gradually broaden over large (> 20%) strains.

For a fixed initial effective temperature, the STZ modelpredicts that the shear bands become thinner and thattheir internal structure becomes more disordered as thestrain rate increases. This is similar to the “ductile tobrittle” transition seen in amorphous materials as a func-tion of the strain rate [10]. At lower strain rates the ma-terial deforms nearly homogeneously and appears ductile,but at higher strain rates all the deformation is localizedin a thin shear band or mode II crack.

At very high strain rates and low initial effective tem-peratures, the effective temperature approaches infinityat the center of the band during the transient responseand the system “melts”. The liquefied region is very thinand failure occurs near the maximum stress overshoot,which is consistent with material failure via shear band-ing seen at high strain rates in bulk metallic glasses.

The model predicts that for materials that fail viathis shear banding mechanism, the apparent shear bandthickness should be at most the diffusion length scale a.Although a has not been measured experimentally, a rea-sonable assumption is that it is on the order of an STZ ra-dius. In bulk metallic glasses this scale should be at most30 atomic radii, on the order of 10 nm [1], which is muchsmaller than the thermal diffusion length scale (100-240nm [6]), and this could explain the shear band thicknessmeasured in these materials. While the STZ radius hasnot been estimated in granular fault gouge, this mech-anism could provide an explanation for the scale of theprominent fracture surface, which is orders of magnitudesmaller than other length scales in earthquake faults.

We have also shown that these localization dynamics

can not be captured by a single degree of freedom rateand state friction law, and that analyzing steady statemodel dynamics can often be misleading. This is be-cause the structural degrees of freedom parametrized byχ(y) evolve much more slowly than the stress dynamics,so that the microstructure continues to evolve althoughthe stress appears to have reached a steady state. Thisinsight is particularly important for materials that de-velop highly localized shear bands, as the friction lawbased on homogeneous dynamics is vastly different fromone that accounts for transient shear band development.We suggest that localization may play a role in dynamicweakening seen at high shear speeds in granular mate-rials, and that the STZ PDE model generates a usefulfriction law in this case.

While this is an exciting starting point for studyingdeformation and failure for amorphous materials, manyfundamental questions remain. We discuss a few of thembelow.

What is χ(q) for various amorphous materials?

Throughout this paper, we have used a fit to data gen-erated by Haxton and Liu [9] as the definition for χ(q).Haxton and Liu simulate a 2D amorphous packing of har-monically repulsive discs at thermal temperatures aboveand below the glass transition temperature, and usedFDT to extract an effective temperature at each ther-mal temperature and strain rate. To our knowledge, thisis the only such data set. It would be very interestingto use FDT to extract effective temperatures from sim-ulations of other types of amorphous packings, such asthe Lennard Jones glass studied by Shi et al. [21], foams,amorphous silicon, or bulk metallic glasses. Is χ(q) sim-ilar for all of these materials? Is the effective glass tran-sition temperature, χ0, universal?

One possibility is that the transition from glassy be-havior to simply activated behavior should occur whenq = 1, (i.e., when the strain rate is the same as the inter-nal rate 1/τ0). However, it is also possible that in compli-cated materials like bulk metallic glasses, the transitionoccurs at slower rates than 1/τ0, since the STZs in thesesystems are large, multi-component regions that likelyevolve more slowly than the phonon frequency.

Are there other ways to measure χ(q), such aslooking at the behavior of a tracer harmonic oscillatorinside a simulation box? Is it possible to define theeffective temperature by quantifying the change inconfigurational entropy as a function of the potentialenergy? Numerical results from Ono, et al. suggestthat this type of calculation is possible, but they werenot able to sample enough low probability states tostate conclusively that the FDT and entropic definitionsgenerate the same effective temperature. These are im-portant questions because it is very difficult to measurefluctuations in position or stress precisely enough in ex-periments to extract an effective temperature using FDT.

What are the effects of geometry and boundary condi-

15

tions? We have so far restricted ourselves to the simplestpossible shear geometry and periodic boundary condi-tions on the effective temperature. The boundary con-ditions on the effective temperature help determine thelocation of shear bands within the material as well as thesteady states of the system. In many experiments andin some earthquake faults, shear bands tend to localizealong the boundary [25]. Why does this occur?

Different geometries can be modeled in STZ theoryby adjusting the boundary conditions on the effectivetemperature. For example, an amorphous material ad-jacent to a rigid solid surface could be modeled by noconduction boundary conditions on the effective temper-ature, because no disorder can propagate into the rigidsolid. Crystalline solids might impose a constant, moreordered boundary condition on the effective temperature.It would be very interesting to investigate the effects ofthese conditions on shear band evolution.

In addition, many engineering materials are testedunder tension and compression, or a “notch” is placedon the surface of the material. In these cases there is afree boundary which can deform, leading to a couplingbetween deformation and stress. The necking instabilityhas been investigated using earlier STZ models [26]– it would be interesting to repeat this analysis withour improved understanding of the coupling betweenstructure and deformation.

What is the connection between stick slip instabilitiesand shear banding?

In section II, we showed that the steady-state stabil-ity of homogeneous flow was dependent on whether thematerial was velocity strengthening or velocity weaken-ing – velocity weakening materials were unstable withrespect to shear bands. Interestingly, in rate-and-state(ODE) friction models, stick-slip instabilities can only oc-cur when the system is velocity weakening. In addition,formation of a shear band coincides with what looks likea slip event in the macroscopic stress-strain curve. Can ashear band be understood as the PDE analogue of a slipevent in a single degree of freedom ODE? Is is possiblefor the transient shear bands seen in ostensibly velocitystrengthening materials to generate stick-slip like behav-ior? Preliminary numerical solutions suggest that thetransient shear bands can not generate stick slip behav-ior, but more work is needed on this avenue of research.

Is the localized state weaker in absolute terms than thehomogeneous state? Does this matter? Although strainlocalization leads to a rapid decrease in the shear stress,the final stress state of the localized system is not neces-sarily lower than the final stress state of a homogeneoussystem. In the special case where the initial effective tem-perature perturbation is a step function, the final stressof the localized state is smaller than the homogeneousstress [24]. However, in most numerical solutions to theSTZ equations (e.g. Fig. 2), the final stress state in thelocalized system is equal to or higher than the stress inthe homogeneous system. In absolute terms, the local-

ized system is stronger (or at least no weaker) than thehomogeneous system, which is counter-intuitive.

There are several ways to reconcile this informationwith intuition. First, we note that the rate at whichthe localized system weakens is much more rapid thanthe homogeneous system. For dynamic phenomena, suchas stick-slip instabilities and stop-start experiments, theweakening rate and the total stress drop help determinethe dynamic response. Is the rapid weakening seen insystems that localize large enough to cause stick-slip?Another possibility is that many of these systems attainstrain rates at which the STZ solid-like description breaksdown. Although we do not explicitly model this here,it seems likely that the liquid-like material in the bandpossess a vastly reduced strength compared to the solidoutside the band.

Acknowledgments

This work was supported by the Southern CaliforniaEarthquake Center, the David and Lucile Packard Foun-dation, and NSF grant number DMR-0606092. M.L.M.acknowledges an NSF Graduate Research Fellowship.J.S.L. was supported by DOE grant number DE-FG03-99ER45762.

APPENDIX A: STZ MODEL DETAILS

A mean field theory for shear transformation zones hasbeen developed in a series of papers [16, 27–29], and weuse this theory as a general model for a wide range ofamorphous solids.

In analyzing the dynamics of shear transformationzones, we develop equations of motion for five internalvariables: the deviatoric stress s, the pressure p, the den-sity of STZs oriented parallel and perpendicular to theprincipal stress directions n±, and the effective tempera-ture, χ. In a simple shear geometry at low temperatures,the model can be further simplified so that the state ofthe system is entirely specified by s and χ alone. Thefollowing sections review the STZ equations and specifythe parameters and simplifications used in this paper.

1. Overview of equations of motion

In the slowly sheared materials we are modeling, thespeed of sound in the material is very fast compared tothe rate of plastic deformation. In this case the stressgradients equilibrate very quickly, and we take the zerodensity limit of the momentum conservation equations.This results in static elastic equations for the stress:

∂σij

∂xj= 0. (A1)

16

The rate of deformation tensor is the sum of elastic andplastic parts:

Dtotalij =

1

2

(

∂vi

∂xj+

∂vj

∂xi

)

=D

Dt

(

−p

2Kδij +

sy

2µsij

)

+ Dplastij , (A2)

where D/Dt is the material or co-rotational derivative.To simplify notation, the deviatoric stress has beennondimensionalized by an effective shear modulus sy thatspecifies the stiffness of the STZs. The stress scale sy alsocharacterizes the stress at which the material begins todeform plastically. This yield stress is distinct from themaximum stress attained, sm, and the steady state flowstress, sf , both of which are sometimes also referred toas the yield stress in the literature.

The plastic rate of deformation tensor can be writtenin terms of dynamical variables from STZ theory. Wepostulate that under shear stress, each STZ deforms toaccommodate a certain amount of shear strain, and can-not deform further in the same direction. This is modeledby requiring that each STZ be in one of two states: ori-ented along the principal stress axis in the direction ofapplied shear, which we will denote “+”, or in the per-pendicular direction, “−”.

Under applied strain, the STZ will flip in the directionof strain, from “−” to “+”. Under shear stress in theopposite direction, the STZs can revert to their originalconfigurations, which corresponds to a flip from “+” to“−”. We assume that the STZ density is small and eachSTZ interacts with other STZs through continuum fieldssuch as the stress. Therefore the rearrangements or flipsoccur at a rate R(s)/τ0, which depends on the stress anda characteristic attempt frequency 1/τ0.

Because each STZ can flip at most once in the directionof applied strain, STZs must be created and annihilatedto sustain plastic flow. Based on these considerations,the number density of STZs in each direction, n±, obeysthe following differential equation

τ0n± = R(±s)n∓ − R(∓s)n± + Γ(n∞

2e−1/χ − n±

)

,

(A3)where R(±s)/τ0 is the rate of switching per STZ as afunction of stress, Γ is the rate at which energy is dissi-pated per STZ, and n∞ e−1/χ is the steady state densityof STZs in equilibrium.

The plastic rate of deformation tensor is given by therate at which STZs flip:

Dpl =ǫ0

n∞τ0(R(s)n− − R(−s)n+) , (A4)

where ǫ0 is a strain increment of order unity and n∞ isa density roughly equal to the inverse of the volume perparticle.

The first two terms in Eq. (A3) correspond to STZsswitching from “+” to “−” states and vice-versa, whilethe last term enforces detailed balance: STZs are created

at a rate proportional to n∞e−1/χ and annihilated at arate proportional to their density. The creation rate isproportional to the probability of a configurational fluc-tuation that corresponds to an STZ. As discussed in theintroduction, this probability is exp[−1/χ], where χ isan internal state variable that characterizes the configu-rational disorder. To close the system of equations, themodel requires an equation of motion for χ.

Ono et al. [13] and Haxton and Liu [9] show that adriven amorphous system possesses a well-defined steadystate effective temperature, χ, at each value of the im-posed strain rate. In these simulations, a thermostatensures homogeneous deformation within the glass andthe particles are sheared for long periods of time beforethe steady state measurement is taken.

To model deformation in time varying systems, such asstart-up flows, we have to estimate how the effective tem-perature changes in time. As detailed in [17], we assumethat the heat content in the configurational degrees offreedom is driven by two independent sources, mechan-ical work and thermal fluctuations. Therefore we makethe simplest assumption: the mechanical heat drives theeffective temperature towards χ according to the con-ventional linear law of heating. The rate of heat per unitvolume that enters the configurational degrees of freedomis Qc = Teff (dSc/dt)mech.

In addition, we postulate that the heat produced bythermal fluctuations, QT = Teff (dSc/dt)therm, drivesthe effective temperature towards the thermal equilib-rium bath temperature θ according to the linear law ofcooling. The resulting equation of motion for χ is:

χ =1

Ceff Tz

[

Teff

(

dSc

dt

)

mech

[

1 −χ

χ

]

+ Teff

(

dSc

dt

)

therm

[

1 −χTz

T

]]

+ D∂2χ

∂y2,(A5)

where Ceff is a specific heat, Tz = Ez/kB is the STZformation energy in temperature units, and the last termrepresents diffusion of effective temperature.

Because the effective temperature governs the config-urational degrees of freedom, only configurational rear-rangements, i.e. plastic events, permit diffusion of theeffective temperature. This suggests that the diffusivityshould vary with strain rate, so that D = a2|γpl|, wherea is a length scale that corresponds to the radius of anSTZ.

2. Simplifying assumptions

Pechenik [30] generalized Eqs. (A3) and (A4) to thecase where the principal axes of the STZ orientation ten-sor nij are not aligned with principal axes of the stresstensor sij . These generalized equations can be written interms of two new variables, Λ and m, which appear often

17

in literature on STZs:

Λ ≡ ntot/n∞; (A6)

mij ≡ nij/n∞, (A7)

where ntot is the tensorial generalization of (n+ + n−)and nij is the tensorial generalization of (n+ −n−). Thescalar Λ is the total density of zones in a sample, whilethe tensor m corresponds to the STZ orientational bias.

In this paper we focus on materials in a 2D simpleshear geometry, so that the diagonal terms in the devi-atoric stress tensor (sxx, syy) and STZ orientational bias(mxx, myy) are significantly smaller than off-diagonalterms and can be neglected. Let s = sxy = syx andm = mxy = myx. In this geometry the pressure p doesnot change with time and Dp/Dt in Eq. (A2) is zero.

As noted in [29] the density of STZs, ǫ0Λ, is necessarilysmall. In a simple shear geometry, the equations of mo-tion for the stress s and the effective temperature χ eachcontain this factor in their numerators, and they equili-brate very slowly compared to m and Λ. Therefore wereplace Λ and m by their steady state values. CombiningEqs. (A3), (A6), and (A7), we find that the steady statevalue of Λ is exp[−1/χ], and that m exchanges betweentwo steady states (elastic vs. plastic deformation) nears = sy. Below the yield stress the deformation is almostentirely elastic because all the existing STZs are alreadyflipped in the direction of stress. Above the yield stressSTZs are continuously created and annihilated to sustainplastic flow. Details can be found elsewhere [17].

We also make the simplifying assumption that the ma-terial is below the thermal glass transition temperatureT0, so that particle rearrangements are not activated bythermal fluctuations. In this case the thermal entropycontribution to the effective temperature equation of mo-tion, (dSc/dt)therm in Eq. (A5), is zero. This is alwaystrue, for example, in granular materials. In addition, wemake the approximation that at very low temperaturesthe STZs do not flip in a direction opposite the directionof applied stress: R(−|s|) = 0.

The functional form of Γ, the energy dissipated perSTZ that appears in Eq. (A3), is considerably simplifiedfor T < T0. Under the assumption that an STZ doesnot flip in a direction opposite to the direction of appliedstress, no energy can be stored in the plastic degrees offreedom. This means that the plastic work is equal tothe energy dissipated:

Dplijsij = γs = Q. (A8)

Following Pechenik [30], we postulate that the total en-ergy dissipation rate, Q, is proportional to Γ:

Q = s0ǫ0τ0

Λ Γ, (A9)

where s0 is a stress scale we return to below.Combining Eqs. A4, A8, and A9 results in the following

expression for Γ:

Γ(s) =2

s0ǫ0s f(s), (A10)

where the function f(s), which also appears in Eq. (2),is defined as follows:

f(s) = ǫ0R(s)

2

[

1 − m(s)]

. (A11)

In simple shear below the thermal glass transition, thetwo steady states of the equation of motion for the STZbias are also very simple:

m(s) →

1 for s < sy

s0/ s for s > sy,(A12)

and s0 = sy is the yield stress – the stress at which thetwo stability branches intersect.

We now turn to the parameters in Eq. (A5). In [17] therate at which configurational entropy is being producedby mechanical deformation, (dSc/dt)mech, is assumed tobe proportional to the product of the STZ density andthe STZ creation rate:

(

dSc

dt

)

mech

=kBνZ

Ω

ǫ0τ0

ΛΓ, (A13)

where Ω is a volume per molecule and kBνZ is the entropyper STZ [17].

Putting everything together, we arrive at an equationof motion for the effective temperature:

dχ

dγ=

2 sχ

c0s0qf(s)e−1/χ(1 −

χ

χ(q)) + a2γpl

∂2χ

∂y2,(A14)

where γ is strain, q is the imposed strain rate times theSTZ time scale, (V0/L) τ0 and c0 = CeffΩ/ (kBνz). In-serting Eq. (2) into the equation for the rate of deforma-tion tensor, Eq. (A2), and integrating across the widthof the material (in the y-direction) results in a secondequation for the stress dynamics:

ds

dγ= µ∗

(

1 −2

qf(s) Λ

)

, (A15)

where µ∗ is the ratio of the elastic modulus to the yieldstress, and Λ is the spatial average of the STZ densityΛ = exp(−1/χ).

Determining an exact function R(s) from first prin-ciples is a difficult many body problem. However, wedo know how R(s) behaves in the limits of very smalland very large stresses, and we choose a function thatsmoothly interpolates between these two regimes. R(s)exhibits Eyring-like behavior far below the yield stress,and power law behavior above the yield stress:

R(s) = exp

[

−TE

Te−s/µ

]

[

1 +

(

s

s1

)2]n/2

. (A16)

The first factor on the right-hand side of Eq.(A16) is theEyring rate in a form similar to that used in [16], wherethe exponential function of s/µ causes the rate to sat-urate at large s. Here, TE is the height of the Eyring

18

activation barrier in units of temperature. The expo-nent n in Eq.(A16) specifies the large stress power lawbehavior; possible values are discussed in [8]. Analysisof simulation and experimental data suggest that n = 1

is valid for bulk metallic glasses [17] but in this paperwe use n = 1/2, which is relevant for purely repulsiveharmonic disks [8].

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