Inertia and universality of avalanche statistics: the case of slowly deformedamorphous solids

Kamran Karimi, Ezequiel E. Ferrero, and Jean-Louis BarratUniversite Grenoble Alpes, LIPHY, F-38000 Grenoble, France and

CNRS, LIPHY, F-38000 Grenoble, France

By means of a finite elements technique we solve numerically the dynamics of an amorphous solidunder deformation in the quasistatic driving limit. We study the noise statistics of the stress-strainsignal in the steady state plastic flow, focusing on systems with low internal dissipation. We analyzethe distributions of avalanche sizes and duration and the density of shear transformations whenvarying the damping strength. In contrast to avalanches in the overdamped case, dominated by theyielding point universal exponents, inertial avalanches are controlled by a non-universal dampingdependent feedback mechanism; eventually turning negligible the role of correlations. Still, somegeneral properties of avalanches persist and new scaling relations can be proposed.

I. INTRODUCTION

Avalanche behavior is ubiquitous in nature. Severalsystems respond to a slow constant driving with inter-mittent dynamics. Examples are: Barkhausen noise inferromagnets [1], stick-slip dynamics in the plasticity ofsolids [2], earthquakes [3], creep of magnetic domainwalls [4], serrated stress of driven foams [5] and crackpropagation [6]. The common phenomenon is that differ-ent regions of a heterogeneous system are loaded towardsinstability thresholds; a region that first gets unstableyields, destabilizes others, and this goes on producingan “avalanche”. Recently, such avalanche dynamics hasbeen evidenced in the time series of the stress tensor indeformation experiments of amorphous systems, such asgrains, foams or metallic glasses [7–9], and has attractedconsiderable theoretical interest [10–12].

Global quantities linked to such collective behavior areusually power law distributed and allow for the under-standing of the system parameter dependencies in termsof scaling functions. This is a gift from the self-organizedcriticality (SOC) paradigm, in which stationary stateswith “critical” behavior are attractors of the dynam-ics [13–15] and dominate the scenario as soon as a balancebetween branching and killing probabilities is fulfilled.Among the factors that may break-down this balance inSOC we count inertia [16–19].

Various works have addressed inertial-like effects inthe context of SOC, like sand-pile models with thresholdweakening [17, 19] or depinning-like models that includestress overshoots [18, 20] or softening [21, 22]. Moreover,inertia has been explicitly considered on the Burridge-Knopoff model [23–25] in the context of seismic faults.In all cases, avalanche size distributions were found todeviate from the critical power-law scaling. It should besaid, nevertheless, that none of this models was intendedto illustrate the effect of inertia in the deformation ofamorphous solids.

In this respect, recent studies on atomistic simulationsof glasses under deformation [26, 27] argue, on the con-trary, that inertial effects drive the system to a “newunderdamped universality class”, rather than taking it

away from criticality. On the other hand, the same kindof atomistic approach [28], as well as more coarse grainedmethod [29], have signaled a strong contrast of the finite-shear-rate rheology between the overdamped and the un-derdamped cases; with no signs of universal behavior inthe latter. Strongly inertial underdamped systems tendto produce, in particular, a non-monotonic flow curve andthe associated localization of the deformation [28, 29].

In this work, we study the influence of inertia in thestatistics of avalanches at the yielding point. By meansof a finite-elements based elasto-plastic model we ana-lyze the stress time series of a sheared system and relatethe stress-drops there observed with avalanches of severalsites yielding in a collective process. At high dampingwe recover the critical avalanche statistics found in mod-els of amorphous solids with overdamped dynamics [10–12, 30]. When damping is decreased, we observe that theavalanche statistics smoothly deviates from the criticaldistribution. Even when a power-law shape is observedfor a range of avalanche sizes, the distribution ceases tobe scale-free, developing a bump at large values set by thesystem size. Also the exponent of the power-law regimesystematically changes as damping is decreased, indicat-ing a departure from universal behavior.

We understand the avalanche statistics of systems withlow dissipation as a combination of two distinct kinds ofevents. On one hand, avalanches dominated by inertialeffects, typically large in size, frequently system-spanningand with a rather well defined size, that populates thebump or peak of the distributions. On the other hand,smaller and more localized avalanches, that mostly con-tribute to the power-law regimes of the distributions andremain controlled by the critical point attractor of theoverdamped dynamics, with little influence of inertia.

We analyze the emergence of this dichotomy by a care-ful study of the finite system-size scaling of the avalanchedistributions, their dependence on damping, and thechanges on avalanches geometry. Within this new sce-nario, we justify the observed alteration by inertia ofthe probability gap for the density of shear transforma-tions and propose a scaling relation that links it to theavalanches geometry.

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Case-Study: Steady Shear

1

L

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FIG. 1. Illustration of the simple-shear set-up: Periodicity isimposed along x and y. A typical size of the finite-elementsmesh is h (here L ≈ 20× h).

II. MODEL AND PROTOCOL

We perform a finite-element-based simulation of amor-phous systems in d = 2 as in [29]. Consider a continuousmedium with a displacement field u(r) from an equi-librium configuration with position r. The equation ofmotion for the displacement field u(r, t) in such a con-tinuum reads

ρu(r, t) = ∇.σ(r, t), (1)

where ρ is the uniform mass density and σ is the stress.Conventionally, the state of a stress tensor is expressedas σ = −pI + σdev with p = −tr(σ)/d the hydrostaticpressure, σdev the deviatoric stress, and I the identitytensor. We further define the effective shear stress asσ = ( 1

2σdev : σdev)12 . The contribution to the stress at

r is assumed to depend only on the gradients of u(r)and/or u(r), more accurately on their symmetric partsε and/or ε. Similarly, the deformation rate may be alsodecomposed as ε = εvI + εdev with εv = tr(ε) the vol-umetric strain rate and εdev the deviatioric strain rate.In this case, the effective shear rate can be expressed asε = ( 1

2 εdev : εdev)12 .

Each finite-size element represents microscopically re-arranging zones (as in real particulate packings) at acoarse-grained level. In their rigid phase, these zonesare modeled by a Kelvin-type solid, while past a yieldingthreshold σy(r, t), a simple Newtonian fluid is employed.Given this qualitative picture, the constitutive equationsare determined by

σdev(r, t) = µεdev(r, t)[1− n(r, t)] + ηεdev(r, t),

p(r, t) = Kεv(r, t), (2)

with K and µ the bulk and shear moduli, respectively.The first terms on the right-hand side of the equationsmimic the elastic contribution, while the second term,only present for the deviatoric piece, represents viscous

dissipation with the viscosity coefficient η. Here n(r, t)is equal to one during the fluid phase – whose life timeis limited by a threshold τp – and zero otherwise. Weset τp = τmin

v with τminv defined here later on. Fix-

ing this time gives a distribution of (plastic) strains inthe range %0.5− 1. Realistically, Eshelby zones undergostrong shear deformations, rather than dilation, to relaxthe stress, and are, therefore, nearly incompressible. In-compressibility is enforced in the elastic regime by settingK/µ ≈ 10. As in [29], yield stresses are drawn randomlyfrom an exponential distribution with a mean value σyand a lower cut-off σmin

y , reminiscent of structural disor-der in glassy dynamics. Also, we dynamically assign anew random threshold to each element after yielding.

Having defined a proper set of constitutive equations,an irregular set of grid-points and linear plane-strainelasto-plastic triangular elements is then employed in aL × L periodic cell with typical grid-size of h to dis-cretize Eq.(1) in space (see Fig. 1). An area-preserving(simple) shear is implemented to drive the system ina quasi-static manner; that is, an infinitesimal strain-step[31] ∆γ ≈ 10−5 is initially applied to the simulationbox each time, followed immediately by a stress quenchthanks to the dissipative terms present in both Kelvinand Newtonian descriptions. The quench runs until themaximum net force on any grid-point is less than 10−6

times the average force. The damping rate for the vis-cous term is τ−1d = ηq2/ρ, where the vibrational fre-

quency is τ−1v = csq with cs =√µ/ρ the transverse

wave speed. Here q is the wave number (spatial fre-quency). The largest q corresponds to the inverse ele-ment size h−1, from which the the shortest vibrationaltime-scale τmin

v = h/cs. The numerical time integrationduring quench is performed by means of the velocity Ver-let algorithm with ∆t � min(τd, τv). The dimensionlessquantity Γ = τ−1d /τ−1v , called damping ratio or dampingcoefficient hereafter, quantifies the relative impact of dis-sipation. We expect to capture an overdamped dynamicswith Γ � 1, while Γ ≤ 1 should in principle lead to anunderdamped regime.

III. RESULTS

As for MD simulations and scalar elastoplastic models,imposing a large enough shear deformation results, aftera transient regime, into a steady-state plastic flow. Thestandard observable is the mean stress σxy(γ) averagedacross the sample. The stress signal is characterized bya series of elastic loading periods, where the global stressgrows linearly with the applied deformation, interruptedby stress drops ∆σ, which are due to plastic activity andmark an abrupt release of stored elastic energy. Fig. 2-left illustrates this common scenario in both high andlow damping simulations. A clear qualitative differencebetween the high and low damping cases is evident fromthe data: while overdamped fluctuations tend to span abroad range of scales (absence of a characteristic size),

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0,1(σ

xy−<

σ xy>)

/σy

10-2

100

102

Γ -1

10-4

10-3

10-2

<∆σ>

FIG. 2. Stress time series and mean-stress drop varyingdamping. Left: Fluctuating stress (σxy − 〈σxy〉)/σy (nor-malized by mean yield stress) versus shear strain γ for twodifferent damping ratios Γ = 100 (blue full line) and 10−2

(dashed orange line) and a system linear size L = 40. Right:Mean stress drop 〈∆σ〉 plotted against inverse damping ratioΓ−1. At low Γ−1 (high damping), the average ∆σ is nearlyflat. It rises only after an inverse critical damping Γ−1

c ' 0.2,that depends on system size.

inertial signals seem to be better described by a typical,large, characteristic stress drop, appearing in a quasi-periodic fashion. The averaged stress drop value 〈∆σ〉 asa function of the inverse damping coefficient Γ−1 (biggerthe more inertial is the system) is shown in Fig. 2-right.At low values of Γ−1 the mean stress drop saturates,which can be used to define the overdamped limit.

A. Avalanche distributions

The magnitude of the stress drop reflects the number ofsites involved in a correlated sequence of yielding eventsor avalanche. Therefore, in agreement with earlier stud-ies [26, 27], we simply define an extensive avalanche sizeas S = 〈σxy〉∆σLd. By using the mean stress value 〈σxy〉as a multiplicative factor that depends on Γ, S acquires(within a constant factor) the dimensions of an energydrop, that better quantifies the collective behavior andallows for a comparison among systems with differentdamping.

Figure 3 shows distributions of avalanche sizes S fordifferent inverse damping coefficients at a fixed systemsize L = 40 built from a statistics of 10000 events. Wefirst notice that an overdamped system (Γ−1 = 0.2) dis-plays, after a characteristic lower cutoff, a power-law de-cay in S that is later on suppressed by an exponentialdecay. In fact, based on previous results, we expect toobserve in this limit a behavior PS ∼ S−τG(S/Sc) withG(·) a rapidly decaying function and the cutoff set by Sc.The inset of Fig. 3 shows curves for such a high dampingat different system sizes rescaled as PSS

τc vs S/Sc, where

we introduce the size dependent cutoff Sc(L) ∝ Ldf withdf the fractal dimension of the avalanches. A collapseonto a unique master curve is obtained for large S. Thepower-law fitting and collapse found are characterized

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P S [s

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S / Ld

f

100

102

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P S L

τdf

204080

Γ -1

S-1.27

L

FIG. 3. Avalanche size distributions varying damping. PSversus avalanche size S at different inverse dissipation coeffi-cients Γ−1. Data correspond to a linear system size L = 40(2046 blocks). Curves for different Γ are arbitrarily shiftedalong the vertical axis for clarity. A peak develops at large Swhen inertia becomes important. Also the exponent τ in thepower-law regime increases. Inset: PSS

τdf versus SL−df forthe highly overdamped case, collapsing different system sizes.τ ≈ 1.27 and df ≈ 0.95

by the values τ = 1.27 ± 0.05 and df = 0.95 ± 0.05, inagreement with previous results of various elastoplasticmodels [10, 12, 30] and molecular dynamics simulationsin the overdamped limit [12, 26, 27].

When the damping is decreased and inertia starts tobe relevant, two main new features become apparent: Onthe one hand the PS distributions deviate from a purepower-law shape; a shoulder develops at large values of Sand evolves into a clear local maximum at very low damp-ing. On the other hand, an apparently robust power-law regime survives at smaller values of S; neverthelessthe exponent τ characterizing the decay of PS systemati-cally deviates from its overdamped value as the dampingis decreased, reaching values of τ ' 1.5 for the lowerdamping displayed, with no signs of saturation. Further-more, a closer inspection of the distributions obtained atlow damping evidences the existence of an intermediateregime in which a probability depletion is produced; si-multaneously showing how the inertial peak at large Sdegrades the free-scale regime and suggesting a separa-tion between two different kind of events. Let us recallthat, indeed, a differentiation of two groups of events hasbeen proposed for the Burridge-Knopoff model [23–25].

In our case, this distinctive feature of the avalanchesize distribution can be better understood by analyzingthe distribution of avalanche durations, PT . Figure 4shows this distribution in units of τv, which is the rele-vant microscopic time for inertial systems [28]; similar toPS , the distribution shows at large damping a power-lawregime, exponentially suppressed for long durations. Asdamping is decreased, a peak at large values of T ap-pears. A vertical dashed line marks the size dependent

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0.21101001000

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T / TL

10-4

10-2

100

PT (

T/T

L)

204080

Γ−1

L

TL

FIG. 4. Avalanche duration distributions varying damping.PT versus avalanche duration T normalized by the system-size-dependent duration TL ≡ L/cs at different inverse dissi-pation coefficients Γ−1. Data correspond to a linear systemsize L = 40 (2046 blocks). When inertia becomes importantthe PT cutoff overcomes T/TL and a peak develops. Inset:PT versus T/TL for Γ−1 = 1000 and different system sizesL = 20, 40, 80.

value TL = L/cs.Figure 4 suggests a simple interpretation of the ap-

pearance of the inertial peak in the distributions. Atthe local level, the immediate consequence of the lackof dissipation is a stronger effect of a yield event in itssurrounding, that could be seen as an effective reductionin the local thresholds. Nevertheless, the most notori-ous inertial feature, as we understand now, comes froma long-range action. When energy is not rapidly dissi-pated it keeps traveling around the lattice in the formof shear elastic waves. Inertial effects become dominantwhen an avalanche duration is long enough such that theavalanche is able to reinforce itself through its own elasticwaves (traveling around the system or possibly being re-flected at the boundaries of a non-periodic system). Weexpect this time threshold to be proportional to TL, thetime needed by the elastic waves to propagate across oneof the system’s main axis. This is confirmed by the be-havior of PT at different damping coefficients. In thestill overdamped case (Γ−1 = 0.5, putting the limit atΓ = 1)), PT is already exponentially decaying by T ' TLmeaning that a big majority of avalanches are not af-fected by this finite-size dependent inertial effect. Onthe other hand, underdamped systems allow for largerdurations T > TL, meaning that a growing number ofavalanches are long-lived enough to sustain their activ-ity by using the energy stored in the elastic waves theyemitted.

Although the wave speed does not depend on Γ, an-other time scale relevant for the efficiency of this feed-back mechanism is the one governing the damping of theelastic waves emitted by one event, Tr. This time scalewill increase as damping decreases, and also be affectedby the density of active sites that may scatter the wave.

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Px

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101

Px

20 - 140 - 180 - 140 - 0.280 - 0.2

Γ -1x

0.52

L , Γ -1

FIG. 5. Distances to yielding distributions varying damping.Px versus x at different inverse dissipation coefficients Γ−1.Data correspond to a linear system size L = 40 (2046 blocks).In the overdamped limit Px ∼ xθ with θ ≈ 0.52. As inertiabecomes important there is a systematic change in Px. In-set: Distributions at different system sizes corresponding tothe overdamped case (Γ−1 = 0.2) and a slightly less dampedcase(Γ−1 = 1) illustrating the finite size effects.

The efficiency of the feedback effect will become strongerand create longer lasting avalanches as Tr increases com-pared to TL and the elastic waves are able to cross thesystem repeatedly. This is illustrated by the drift of thepeak in PT towards larger values of T as Γ decreases.

Finally, let us emphasize that the particular role playedby the system size in this discussion is intrinsically re-lated to the use of a quasistatic protocol. At most oneavalanche is taking place in the system at any given in-stant. On the other hand, for a system driven at a finitebut small strain rate, the size of the system should bereplaced by the strain rate dependent distance betweenavalanches.

A priori one would assume this relaxation time to beproportional increase with Γ−1. Therefore, the PT andPS transmutation onto distributions with characteristicpeaks and the emergence of a new kind of avalanchesfully controlled by inertia, are a result of the competitionbetween two characteristic time scales Tr (of the order ofa few τd) and TL.

B. Local distances to threshold

A key quantity that characterizes the specific avalanchebehavior in plastic solids is the distribution Px of localdistances from instability xi = σyi − σi, with i an indexrunning over the N blocks that compose the system. Ateach loading stage in the stress time series, the minimumx value in the lattice denoted by xmin determines theglobal instability threshold for the next event. The extentof an avalanche will be controlled by the full distributionPx and its evolution during the stress drop.

5

It has been shown, following stability arguments [32],that the functional form of Px near the yielding tran-sition should be a power-law Px ∼ xθ with θ > 0 forsmall x, a pseudo-gap. Moreover, assuming the indepen-dence of the xi, a simple scaling argument links the ex-ponent θ with the exponents τ and df controlling the dis-tribution of avalanche sizes, θ+1 = 1/ (1− (2− τ)df /d).This prediction was found to hold in numerical simula-tions of overdamped systems, both in quasi-static pro-tocols [10, 32] and approaching the limit of vanishingstrain-rate [12] from finite values, giving consistent setsof exponents {θ, τ, df} in two and three dimensions. Wenow report the influence of the damping on Px by analyz-ing configurations right after a stress drop has occurred.

Figure 5 shows Px for systems with different dampingΓ. Our estimation of a power law Px ∼ xθ in the over-damped case (Γ−1 = 0.2) agrees within error bars withprevious estimates of the exponent θ in 2D elastoplasticmodels [10, 12]. The estimate θ2D ≈ 0.52 [12] is displayedas a dash line in the inset of Fig. 5, which also illustratesthe size dependence of Px affecting the cutoff at smallvalues of x.

When lowering the dissipation rate Γ a systematicchange in the behavior of Px is observed, with the devel-opment of a steeper gap in Px at small values of x. Basi-cally, as inertial avalanches start to dominate, the prob-ability of surviving (not yielding) with a small value of x(i.e., very close to the local threshold) decreases. Smallbarriers will be overcome during a massive avalanche.An apparent tendency to preserve a behavior of the formPx ∼ xθ at low x is observed, with an exponent θ thatwould increase as damping lowers. This seems to con-trast with the behavior of PS and PT that show a clearcharacteristic peak appearing as inertia becomes rele-vant (making possible a distinction between two kindsof avalanches). However, the situation is a bit more com-plex. In fact, the steeper growth at the smallest valuesof x in each curve is indeed a trace of the presence of twokind of events. The fact that Px considers the full set oflocal values of a configuration in contrast to S or T thatonly provide a global avalanche property, just rendersmore difficult the possibility of visualizing the avalancheheterogeneity at this point.

If one insists on thinking the largest power-law growthfor each curve Fig.5, as a damping-modified marginal sta-bility scenario where only the exponent θ changes, de-pending on Γ, we can find no simple explanation for thatexponent θ(Γ). In fact, due to the emergence of the upperpeak in PS , the scaling relation linking τ , df and θ de-

rived in [10] for the overdamped case, i.e. τ = 2− θθ+1

ddf

,

is no longer expected to be valid in the inertial case. In-stead, we show below that the behavior displayed by Pxis the combined result of an alternation between eventsroughly classified in two different kinds, those similar tothe well-known overdamped avalanches and those domi-nated by inertial effects.

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min

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0.21101001000

1 10 100 1000Γ -1

0,001

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x cros

s

Γ−1

FIG. 6. Minimal distance to yielding distributions varyingdamping. Px(xmin) versus xmin at different inverse dissipa-tion coefficients Γ−1. Data correspond to a linear system sizeL = 40 (2046 blocks). In the overdamped limit Px(xmin) isflat at small xmin, in agreement with previous findings. As in-ertia becomes important Px(xmin) acquires a bimodal shape,clearly separating events that have left the system far awayfrom instability from those that are similar tose observed inthe overdamped case. Dashed gray lines mark the particularvalue of xmin at which we locate this crossover, xcrossmin In-set: xcrossmin as a function of Γ−1 represented by open circles.The full line is a fitting with a logarithmic function, yielding:xcrossmin ' 0.00064 + 0.000278 ln(Γ−1).

C. Identifying and splitting two types of avalanches

In order to better understand the response of PS , PTand Px to changes in the damping coefficient, we ana-lyze xmin ≡ min{xi}, the quantity that determines theloading needed to trigger the next stress drop after anavalanche ends.

Figure 6 shows the distributions of such xmin valuesduring the stationary plastic flow for systems with differ-ent damping Γ. In the overdamped limit Px(xmin) is flatat small xmin, in agreement with previous findings [32].As inertia becomes important Px(xmin) acquires a bi-modal shape; this is, two characteristic local maximaseparated by an intermediate local minimum. Denotingby xcrossmin the position of the minimum in Px(xmin) (even-tually a saddle point for Γ = 1), we find a logarithmicdependence of xcrossmin with Γ−1. xcrossmin roughly separatestwo kinds of events: On the one hand, those that havebeen presumably massive, affecting many sites, and haveleft the system far away from instability, accumulating ona peak of “large” xmin values. On the other hand, thosethat we can consider to be similar to the ones in theoverdamped case, more localized avalanches, involving asmall number of sites compared to the full system, andleaving back some relatively weak spots without yield-ing. A priori, we could link the characteristic “large”xmin values, with the typically big stress drops observedin Fig. 2-left for the underdamped systems.

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PS 10

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-510

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df’

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PS

204080

S -1.5

L

FIG. 7. Avalanche size distributions split in two contribu-tions. PS versus S shown to be composed by two contribu-tions, one coming form the overdamped-like events and otherone coming corm the “inertial” events. Data correspond toa linear system size L = 40 (2046 blocks) and Γ−1 = 1000.Lower Inset: The same PS shown for different system sizesL = 20, 40, 80. An alignment for the position of the right-

most peak in PS is obtained when rescaling S by Ld′f , with a

damping depending fractal dimension d′f (Γ−1 = 1000) ≈ 1.75.Upper Insets: Activity maps of selected avalanches. Red dotsrepresent points on the grid that were activated at least onceduring the avalanche. On the left, a typical avalanche corre-sponding to the overdamped sub-set. On the right, an inertialavalanche.

For the time being, let us take the proposed separationas an ansatz, and test its validity with a self-consistencycriterion. We then use xcrossmin to discriminate avalanchesaccording to the xmin value that they yield.

Figure 7 shows in the main plot the PS distribution forL = 40 and the lowest damping simulated Γ−1 = 1000.The same curve as in Fig.3 is shown by open triangles,and has been built with the contribution of all avalanchesin the set. Now we discriminate avalanches yielding smallvalues of xmin and avalanches yielding large values of xmin

and plot their distributions with filled light color circlesand filled dark color squares respectively. After rescalingthese contributions according to their weight in the fullPS , we see that the discrimination between two kinds ofavalanches, provided by xcrossmin is quite accurate. Othervalues of Γ (not shown) display the same beautiful split-ting of PS contributions, while, of course, the low-xmin

contribution earns more and more weight as the over-damped limit is approached.

Alike studies of avalanches in spin systems [33], ac-cepting the idea of a separation between different kindsof avalanches, opens the door to go further and try to an-alyze some scaling properties of their respective distribu-tions. In the following we focus on some scaling featuresof inertial avalanches that contrast with the overdampedcase.

D. Growing fractal dimension and incipient shearlocalization

Let us first consider the finite size scaling of the “new”cutoff of PS , this is, the position of the peak of iner-tial avalanches. When doing so, we find a dependence

Speak ∼ Ld′f , where d′f is a damping depending fractal

dimension for the avalanches. The lower inset of Fig.7shows the peaks alignment for three different system sizeswhere a value d′f = 1.75 is used to align curves corre-

sponding to Γ−1 = 1000.This suggest that the geometry should be very differ-

ent between overdamped and inertial avalanches. Indeed,this is clearly manifested in the spatial coverage of a typ-ical avalanche. As insets of Fig. 7 we show two imagesrepresenting a system sample, where we have depictedactivity maps of selected avalanches. Red dots repre-sent points on the grid that were activated at least onceduring the avalanche. On the left, a typical avalanchecorresponding to the overdamped sub-set shows a sparsequasi-1d arrangement of active sites, consistent with afractal dimension df ≈ 0.95 as the one obtained from thefinite size scaling of overdamped systems [12, 27]. On theright, an inertial avalanche belonging to the rightmostpeak of PS , shows a much more dense and broad struc-ture, compatible with a fractal dimension closer to the di-mension of the system (d = 2). Indeed, if we inspect thelargest avalanches for different damping coefficients, wefind that the generic form their pattern in space changessystematically, becoming denser as d′f evolves from 0.95to 1.75 in the studied regime.

Inertia, therefore, is found to modify a the geometryand fractal dimension of the avalanches at the yieldingpoint, creating much denser events. The geometry ofthese system spanning and broad avalanches are sugges-tive of incipient shear bands. In fact, very recent worksboth using finite elements methods [29] and moleculardynamics [28], have associated the absence of dissipa-tion with a non-monotonic flowcurve in the rheology ofthe system, and therefore mechanical instability and theemergence of strain localization (see also [22]). Althoughwe are working with a quasi-static protocol, the self-sustained effect of inertial waves generating the inertialavalanches may be the same effect that generates theshear bands in finite strain rate driving protocols.

E. Px contributions and stability-geometry scalingrelation

We now turn to the analysis of the different contribu-tions to Px. Again, using the splitting criterion betweenoverdamped-like avalanches, that leave a xmin < xcrossmin ,and inertial avalanches, that yield a xmin > xcrossmin , weplot the full-set distributions of distances to yielding Pxtogether with two sub-set distributions.

Figure 8 show such a superposition of curves for ourlowest damping system (Γ−1 = 1000). We first notice

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)1000

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01234567

θ’

100

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1

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x6.91

FIG. 8. Distances to yielding distributions varying dampingsplit in two contributions. Px versus x shown to be composedby two contributions, one coming form the overdamped-likeevents and other one coming corm the “inertial” events. Datacorrespond to a linear system size L = 40 (2046 blocks) andΓ−1 = 1000 An alternative exponent θ′(Γ−1) can be obtained

for the law Px ∼ xθ′

when considering only inertial events.The gray dashed line show this fit for Γ−1 = 1000, the dotdashed line replicates the same slope. Inset: θ′ and the cor-

responding d′f = d(

1− 11+θ′

)as a function of Γ−1.

that the distribution Px for the sub-set of configurationsafter an inertial avalanche has a much sharper gap thanthe global distribution. This is consistent, of course, withthe fact that these events typically have a larger xmin

value than the average xmin. In addition, a remarkablefinding is that the Px of inertial events display also apower-law growth at small x. In other words, despite themassive events that push all local sites to have a relativelylarge value of x after the avalanche, these values are stillarranged in a marginally stable fashion, as Px ∼ xθ

′,

but now with a steeper exponent θ′ > θoverdamped. Infact, assuming that the very general finite-size property

〈xmin〉 ∼ L−d

θ′+1 [32, 34] also holds for this sub-set, andasking 〈xmin〉 ' 〈∆σ〉 during stationary plastic flow, weexpect

L−d

θ′+1 ∼ Ld′f−d ⇒ θ′ =

1

1− (d′f/d)− 1 (3)

where d′f is the exponent ruling the finite-size dependency

of the peak position in PS (see Fig.7 inset), or equiva-lently, the scaling of the sub-set of inertial avalanches.

The above relation between θ′ and d′f holds well forall low-damping systems studied in this work, as long as〈∆σ〉 is controlled by the inertial peak. This can be seenby fitting θ′ in the inertial sub-set of the Px distribution.For example, for Γ−1 = 1000 we have θ′ ' 6.91 andd′f ' 1.75, for Γ−1 = 10, θ′ ' 3.12 and d′f ' 1.5.

Another interesting observation is that, even the subsetof configurations that are left behind by an overdamped-like avalanche show at the smaller values of x a growth

that tends to be compatible with the same exponentθ′ (see dotted line, parallel to the dashed fit in Fig.7).We interpret this feature as a fingerprint of inertialavalanches, with a depletion (θ′ > θ) in the amount ofsites close to yielding even after one or several smalleravalanches has taken place. This fingerprint of inertialavalanches could also explain the dependence on damp-ing of the exponent τ in the “overdamped-like” subsets ofavalanche size distributions. Even when short-durationavalanches do not directly feel the effect of inertia, theyhave to deploy correlations on a particularly heteroge-neous landscape left behind by inertial avalanches. Asa result, the separation in two classes appears as a con-venient classification, but does not fully account for thecomplexity introduced by inertial effects.

IV. SUMMARY

We have analyzed the noise statistics of stress signalsproduced by an amorphous solid under quasi-static defor-mation, through numerical simulations of a realistic con-tinuum model treated with classical finite-element tech-niques. In particular we have focused our analysis on thedependence of such statistics with the ability of the sys-tem to dissipate energy, spanning a wide range of damp-ing values.

We validate our model by comparing our results for thedistributions of different observables in the overdampedlimit with previously reported numerical results in a va-riety of different models and techniques.

Both the distributions of avalanche sizes PS and du-rations PT display a growing peak at high values of Sand T , respectively, when we lower the damping coef-ficient. We have associated these characteristic peakswith a “new” kind of avalanches, peculiar to inertialsystems, which are triggered and amplified by elasticwaves generated by other avalanches. Such events are,in nature, more related to effective thermal heating thanto the deployment of large spatial correlations, in linewith [28]. These system spanning inertial avalanches arealso characterized by a geometry that is reminiscent ofshear bands of strain localization, although our protocolis quasi-static.

The distribution of minimal distances to yieldingPx(xmin) allowed us to formulate an ad hoc criterion todiscriminate between these inertial avalanches and over-damped like events that are much more localized andnot affected by the elastic wave propagation. Using thiscriterion we have found an explanation for the behaviorat different damping of the full-set Px distribution, thatis considered as the steady state property that controlsthe stability of the system and indirectly all the prop-erties of its noise signal. In particular, following verygeneral arguments, we propose a scaling relation betweenthe distribution Px of inertial avalanches and their fractaldimension (Eq. 3).

8

V. OUTLOOK

About the upper size cutoff

As mentioned in the introduction, the irruption ofinertial effects in the –otherwise overdamped– drivendynamics has been found to break down the univer-sal avalanche statistics of several systems. It is worthstressing, though, an important difference between thesand-pile problem with local yield thresholds and invari-ably positive load redistributions (depinning models in-cluded), and the elastoplastic models that describe plas-tic flow in solids: Systems with a positive load redistri-bution produce avalanches that are compact objects inspace (df ≥ d). The system size L controls the cutoff forthe avalanche sizes. An overdamped system will explorethis upper boundary displaying some system-size span-ning avalanches, but there is no possibility to observebigger avalanches than Sc ∼ Ldf . When the overdampedcondition is released in such systems, the avalanche sizedistribution is modified [17]. The original power-law ofthe overdamped case deforms into a shorter scale-freeregime followed by a kink or peak that depends on thevalue of the damping. However, for all damping choices,the largest avalanche accessible for a given system sizeremains the same.

In the deformation of amorphous solids, instead, wehave avalanches with df < d due to the heterogeneous(Eshelby) redistribution of stresses. The cutoff of theavalanche distribution in overdamped systems is far frombeing a massive avalanche involving all sites of the sys-tem. In fact, both for 2D and 3D systems, df is foundto be close to one [12, 27]; meaning that even a system-spanning avalanche leaves most of the sites untouched.Therefore, when inertia comes into play, the system stillhas room to make avalanches grow further. This can beseen in Fig. 3. As we lower the damping starting in theoverdamped limit, first a plateau and then a peak devel-ops in PS ; all this happening to the right of the over-damped system-size cutoff. May this explain why, formoderate damping, inertial effects are not strongly evi-dent in yield stress systems and avalanche distributionsremain quite similar to the one of the overdamped case,with eventually the added features of a small plateau atlarge sizes and a weak change in the τ exponent. A pic-ture that makes it appealing to conclude about a stilluniversal scenario, slightly modified by inertia [26, 27].

However, going deeper in the inertial regime while un-derstanding the system size role, we realize that indeedcritical behavior breaks down while a damping-dependenttypical event size emerges in the form of a very clear peakin PS . Even though a scale-free power-law regime re-mains observable and is only weakly affected by inertia,the inertial peak increasingly dominates the statistics ofevents. Furthermore, since systems that dissipate energyat different rates show dissimilar avalanche distributions,even when characterized by some scaling exponents, wefind inaccurate to talk about universality.

Connections to other non universal statistics

We have seen that the characteristic avalanche dis-tribution of underdamped systems in our prescriptionis a superposition of two populations. Inertial andoverdamped-like events interleave in time producing aunique statistics that we cannot discriminate beforehand.Both kinds of events are results of the same dynamicalrules and boundary conditions. It simply happens thatfrom the competition of different time scales present inthe dynamics, both kinds emerge. Even more, sometimeswe cannot tell if one event is of one kind or the other.

Such a situation is not unique to the introduction ofinertial effects. It can also be observed in cases where asecond time scale is introduced by viscoelasticity [35] orretardation [22]. Indeed, a quasi-periodic oscillation ofthe stress field was argued in [35] to be a possible expla-nation for deviations from a pure power law (Gutenberg-Richter (GR) like) in the distribution of earthquake mag-nitudes. In fact, some years ago, a discussion arose inthe seismology community contrasting opposite modelsof earthquake statistics: On one hand the famous GRpower-law decay kind of distribution; on the other hand,the “characteristic earthquake” hypothesis predicting atime recurrence of typically big earthquakes of a charac-teristic magnitude on each individual fault (see [36, 37]and references therein). Although somehow balance infavor of a pure GR picture, the discussion remains some-what open [37–39], and specific models with such char-acteristics are developed for single faults [40]. Whetheror not our simple model of an amorphous material is rel-evant for the large scale and complex phenomenology ofearthquakes is of little importance and we don’t attemptto discuss it here. Nevertheless, in line with previoussuggestions [23–25], our model shows how a single set ofdynamical rules and boundary conditions can lead to theemergence of distinct kind of events, ones with no char-acteristic size and a Gutenberg-Richter distribution, andothers with a magnitude that fluctuates around a typicalvalue in a peaked fashion.

The particular role of inertia in our model can also beconnected to the idea of “dynamic triggering” of earth-quakes [41], which has attracted considerable attentionin the last ten years. In the present case we see that dy-namical amplification trough sound waves, rather thantriggering, appears as a dominant mechanism. Our largeavalanches could however be understood as consisting ofan initial event dynamically triggering a series of after-shocks, the total stress drop magnitude being the out-come of the whole series.

Finally we note that laboratory systems such as metal-lic glasses also display statistics that deviate from a purepower law with exponential cutoff, as can be seen fromthe inspection of cumulative distributions shown for ex-ample in reference [7]. Making similar studies in granularsuspensions, in which inertial effects can be controlled byusing solvents of various viscosities, would be of great in-terest.

9

Concluding remarks

Overall, inertia and amplification trough sound wavesappears as a possible mechanism for enriching the statis-tics of intermittent phenomena in deformed solids, withdeviations from universal power law statistics and largesystem spanning events that should be connected withthe possibility of shear band formation in these systems.We hope that this work may stimulate experimental stud-ies of intermittent behavior in systems that may displaystrong inertial effects and/or strain localization, and wealso consider generalizing this study to finite strain rates

in the future.

ACKNOWLEDGMENTS

The authors acknowledge financial support from ERCgrant ADG20110209. JLB is supported by IUF. Mostof the computations were performed using the Froggyplatform of the CIMENT infrastructure supported by theRhone-Alpes region (GRANT CPER07-13 CIRA) andthe Equip@Meso project (reference ANR-10-EQPX-29-01). Further we would like to thank Alexandre Nicolas,Alberto Rosso and Jerome Weiss for a careful readingand useful feedback.

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