Signatures of the spatial extent of plastic events in the yielding transition in amorphous solids
SSpatial extent of plastic events influences the yielding transition in amorphous solids
Daniel Korchinski, C´eline Ruscher, ∗ and J¨org Rottler Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver BC V6T 1Z1, Canada Institut Charles Sadron, 23 rue du Loess, F-67034 Strasbourg, France (Dated: January 27, 2021)Amorphous solids are yield stress materials that flow when a sufficient load is applied. Theirflow consists of periods of elastic loading interrupted by rapid stress drops, or avalanches, comingfrom microscopic rearrangements known as shear transformations (STs). Avalanches of size S areexpected to be scale free in the thermodynamic limit following the distribution P ( S ) ∼ S − τ where τ is the avalanche exponent. However, for finite size system, they present an upper cutoff S c ∼ L d f where d f is the fractal dimension that characterises the geometry of the failure event. Here weshow that the spatial extent of avalanches in a steadily sheared amorphous solid has a profoundeffect on the distribution of local residual stresses x . We find that in this distribution, the mostunstable sites are located in a system size dependent plateau. While the entrance into the plateauis set by the lower cutoff of the mechanical noise produced by individual STs, the departure fromthe usually assumed power-law (pseudogap) form P ( x ) ∼ x θ comes from stress fluctuations inducedby collective avalanches. Interestingly, we observe that the average value of weakest sites (cid:104) x min (cid:105) is located in an intermediate power law regime between the pseudogap and the plateau regimes,whose exponent decreases with system size. Our findings imply a new scaling relation linking theexponents characterizing the avalanche size and residual stress distributions. Upon deformation, amorphous materials behave assolids when the applied shear stress is lower than theyield stress and start to flow when this threshold is ex-ceeded. In the limit of very slow shear rate and at lowtemperature, the stress response becomes very jerky asseen for instance in bulk metallic glasses [1, 2], foams [3],granular matter [4] or porous silica [5]. The sudden stressdrops, or avalanches, originate in localized microscopicrearrangements involving a small number of particles inshear transformation zones (STs) [6, 7], which have beenobserved both in atomistic simulations [8] and in colloidalglasses with confocal microscopy [9].Since the seminal work of Eshelby [10], we know thatSTs induce anisotropic elastic interactions in the sur-rounding medium that are proportional to the elasticpropagator G ( r ) ∼ cos(4 θ ) /r d , with d the dimensionalityof the system [11]. As these interactions are long-rangewith an alternating sign, they stabilize or destabilize dis-tant sites, and therefore act as a mechanical noise. Theproperties of this noise are of fundamental importancebecause they control the distribution P ( x ) of residualstresses in the material, i.e. the distance of a given siteto mechanical instability. This distribution has a pseu-dogap form, i.e P ( x ) ∼ x θ , in the limit where x → θ > et al. [13] proposed a mean-field descriptionof the yielding transition that makes specific predictionsfor the pseudogap exponent θ . Since failure is initiatedat the weakest sites in the material, it is the distribu-tion P ( x ) that determines the statistical properties of theavalanches and thus provides a fundamental link betweenlocal material properties and macroscopic behavior. ∗ [email protected] However, recent contributions raised the possibilitythat single rearrangements are not sufficient to describecompletely the phenomenology of the yielding transition.Upon approach to the yield point, plastic events becomecorrelated and their correlation length diverges with acritical exponent ν = 1 / ( d − d f ) [14]. The anisotropyof the Eshelby propagator promotes line-line events with d f < d reported by atomistic simulations [15–18]. Elasticinteractions from spatially extended objects is expectedto behave as G ( r ) ∼ /r s with s < d [19], and signaturesof such behavior have been observed both in atomisticand mesoscopic simulations of sheared amorphous solids[20, 21]. Oyama et al. showed by means of atomistic sim-ulations that depending on the size of avalanches consid-ered, the avalanche exponent τ changes significantly [22].These results hint at the importance of multiple scales inthe yielding transition.In this work, we investigate the role of the spatial ex-tent of avalanches in the yielding transition. We use atwo-dimensional elastoplastic mesoscopic model (EPM)of amorphous solids [23] in a finite element implementa-tion [24], for which the Eshelby stress propagator emergesnaturally. We implement a strain-controlled deforma-tion protocol using extremal dynamics [25], in whichavalanches are initiated by uniformly loading the systemin a simple shear configuration until the weakest site fails,and fixing the strain until the avalanche ends. In additionto the expected pseudogap regime in the distribution ofresidual stresses P ( x ), we find an intermediate power lawregime and in the very small x limit, a system size de-pendent plateau. In order to explain these observations,we extract the distribution of the underlying mechanicalnoise and show that while the plateau is related to thelower-cutoff of the mechanical noise produced by individ-ual STs, the intermediate power-law emerges as a con- a r X i v : . [ c ond - m a t . d i s - nn ] J a n sequence of a transition between individual and collec-tive plastic rearrangements occurring around a crossovervalue x c that we show to be related to the total stressfluctuations induced by large avalanches. For the systemsizes considered, (cid:104) x min (cid:105) , the average value of the weak-est spots is located in the intermediate power law regimeraising the question of the validity of the scaling relationlinking the flow through τ to the residual stress distribu-tion via θ . We show that this latter changes significantlydue to the influence of large correlated plastic events inour model. DISTRIBUTION OF RESIDUAL STRESSES
The probability distribution function P ( x ) of residualstresses is shown in Fig 1(a) for different system sizesand exhibits three distinct regimes. For larger values of x we observe that P ( x ) ∼ x θ with a pseudogap exponent θ ≈ . x c , P ( x ) departs from thispseudogap regime and finally saturates in a system sizedependent plateau value P for x →
0. Interestingly, inthe crossover region, P ( x ) ∼ x ˜ θ with ˜ θ < θ . Origin of the plateau
To gain more insight into what happens for x < x c ,we first investigate the origin of the plateau region. Asalready observed in ref. [28], the plateau depends on thesystem size as P ∼ L − p . We find that p ≈ .
60 as alsoreported in [21, 28]. In ref. [20] it was suggested that theemergence of the plateau is related to the discreteness ofthe underlying mechanical noise arising from the stressredistribution during avalanches. For a given site, onedefines the mechanical noise ξ i = x i ( n + 1) − x i ( n ) where n is the number of plastic events (single ST or collec-tive rearrangements[29]) in chronological order. Due tothe long-range nature of the elastic interaction describedby a stress propagator G ( r ) ∼ r − s , this noise is broadlydistributed [30] and can be expected to follow a L´evydistribution P ( ξ ) ∼ | ξ | − µ − [13]. The aforementionedexponent s = d/µ and when µ = 1, G ( r ) corresponds tothe Eshelby propagator for STs. Assuming the differentrearrangements are independent and correspond to sin-gle STs, one expects a lower cutoff ξ ST,lc ∼ L − d/µ for thenoise distribution. The definition of the lower cutoff isnot unique [31, 32] and depends on the nature of the plas-tic objects considered (single STs vs. large avalanches).This aspect will be discussed in more detail in next sec-tion. The noise distribution also presents an upper cutoff ξ uc ∼ ξ , also called ”kicks”, between avalanches measured inour EPM for different system sizes. These distributionscollapse when the stress change is rescaled by L − . and the distribution scales as P ( ξ ) ∼ ξ − , in agree-ment with µ = 1. In our system, the global mechan-ical noise exhibits the characteristics of the noise ξ ST generated by single STs. Indeed, in the inset we also ob-serve P ( ξ ST ) ∼ ξ − ST and ξ ST,lc ∼ L − as expected forEshelby interactions. We conclude that the mechanicalnoise associated with avalanches is dominated by point-like rearrangements. To highlight the role of the lowercutoff in the emergence of the plateau in P ( x ), we showa rescaled distribution P ( x ) L − p vs. x/ξ ST,lc in Fig 1(b).A good finite system size collapse is obtained for x be-low and in the vicinity of ξ ST,lc showing that this regionis dominated by the influence of the lower cutoff of themechanical noise.
Origin of the intermediate regime
However, this rescaling fails to collapse points at orabove (cid:104) x min (cid:105) as indicated by diamonds in Fig 1. We no-tice that values of (cid:104) x min (cid:105) are systematically located be-low the crossover x (cid:46) x c locating the departure from thepseudogap regime P ( x ) ∼ x θ and defined as the intersec-tion between the two power law regimes. This crossoverscales as x c ∼ L − c with c ≈ .
15 (see SI Fig. S1). InFig. 1(c), we show P ( x ) L . vs x/x c and find a good col-lapse of the upper power law regime in the region x > x c and of the plateau region. However, in the intermedi-ate region where ξ ST,lc < x < x c the collapse fails. Thephenomenology below and above x c is thus different assuggested by the different scaling with system size.To shed some light on the origin of this intermedi-ate power law regime, we focus our attention on (cid:104) x min (cid:105) ,the second smallest characteristic scale. It is intrinsi-cally related to the macroscopic flow, in particular theaverage value of stress drops (cid:104)| ∆ σ |(cid:105) ∼ (cid:104) x min (cid:105) ∼ L − α ,where α = 1 . P ( x )as the survival probability of a random walker perform-ing a L´evy flight near an absorbing boundary [13]. It isthus interesting to investigate P ( x ) subject to the con-dition of absorption or survival after the occurrence ofan avalanche. Results are shown in Fig 3(a), where wesee first that, as mentioned above, no site survives be-low ξ ST,lc and most of the sites in the plateau regionare likely to be absorbed after an avalanche. We thenobserve that P ( x ) conditioned on the surviving sitesdeparts from the pseudogap regime when x ≈ (cid:104) x min (cid:105) ,where both failing and surviving distributions are equal.Therefore, (cid:104) x min (cid:105) marks the onset of a transition; for x > (cid:104) x min (cid:105) sites are more likely to survive while below (cid:104) x min (cid:105) , absorption dominates. Moreover, the extremaldynamics protocol induces a global shift − x min of resid-ual stresses after the end of each avalanche. This invitesthe introduction of a drift velocity: v = N x min /(cid:96) overthe course of an avalanche with (cid:96) plastic events. Takenas an average over individual STs, the drift per ST is (cid:104) v (cid:105) = N (cid:104) x min /(cid:96) (cid:105) ST = N (cid:104) x min (cid:105) av / (cid:104) (cid:96) (cid:105) av , where the lastequality is derived in the SI. For a L´evy flight of index FIG. 1. (a) Probability distribution function of the residual stresses P ( x ) for different system sizes L . In panels (b) and (c) x has been rescaled by the lower cutoff ξ ST,lc ∼ L − . of the mechanical noise from single STs (see inset of Fig. 2(a)) and by x c ∼ L − . (see text and SI Fig. S1), respectively. Filled circles indicate the location of the lower cutoff of the mechanicalnoise ξ ST,lc and filled diamonds indicate the mean values of the weakest site (cid:104) x min (cid:105) . µ = 1 with a drift v , the persistence exponent, i.e. thepseudogap exponent in the context of sheared amorphoussolids, can be expressed as a function v and the ampli-tude of the mechanical noise A as θ = arctan( Aπ/v ) /π [13, 33]. Assuming A to be constant, one expects a de-crease of θ with increasing v that we compute in theregions x > (cid:104) x min (cid:105) and x < (cid:104) x min (cid:105) in Fig 3(b). Theinset reveals that, although the maximum drift scales as (cid:104) x min (cid:105) , the drift enhancement begins at x ˜ L − ≈ x c . Thisobservation is coherent with the increase in probabilityof failing shown above.The drift also reveals a change in the nature ofavalanches above and below (cid:104) x min (cid:105) . Indeed ˜ v > v im-plies that (cid:104) ˜ (cid:96) (cid:105) < (cid:104) (cid:96) (cid:105) . The typical avalanches are largerabove (cid:104) x min (cid:105) , which thus marks a transition between col-lective and individual rearrangements. To illustrate thisdifference, we probe after each avalanche for sites wellabove and well below (cid:104) x min (cid:105) the value of S they actuallyfelt. We then compute the conditional avalanche distri-butions P ( S | x (cid:28) (cid:104) x min (cid:105) ) and P ( S | x (cid:29) (cid:104) x min (cid:105) ) whichare shown in Fig 4(a). We emphasize that even if wefocus on x (cid:28) (cid:104) x min (cid:105) , we observe large values of S asthey can lead to values of x < (cid:104) x min (cid:105) . We notice thatthe avalanche exponent τ differs substantially. Indeed,for sites well above (cid:104) x min (cid:105) , τ ≈ .
37 corresponds to theavalanche exponent measured for the whole distributionwhile for x (cid:28) (cid:104) x min (cid:105) , τ ≈ . τ hasbeen measured for small avalanches in atomistic simula-tions [22]. Evolution of the exponents ˜ θ and p The decrease of ˜ θ with increasing system size L comesfrom the joint effect of the decrease in the density of sitesbelow (cid:104) x min (cid:105) with system size as the cumulative distri-bution function (cid:82) (cid:104) x min (cid:105) P ( x ) dx ∼ L − d [34], and of anincrease of the drift ˜ v with L as observed in Fig 3(b).Assuming that the average of failing sites per avalancheis (cid:104) ˜ (cid:96) (cid:105) ≈
1, the drift ˜ v ∼ (cid:104) S (cid:105) ∼ L d − α and therefore weimmediately see that the larger L the larger the drift inthe region where x < (cid:104) x min (cid:105) . More and more sites arebrought on the verge of instability and in the thermo-dynamic limit, when L → ∞ , one expects the drift tobecome infinite and the density of sites below (cid:104) x min (cid:105) tobe zero. This implies ˜ θ ( L ) → P ( x ) shouldplateau for x ≤ (cid:104) x min (cid:105) .From the discussion so far, we hypothesise that theprobability of residual stresses can be written as P ( x ; L ) = P ( L ) ∀ x ≤ ξ ST,lc c ( L ) x ˜ θ ( L ) ∀ ξ ST,lc ≤ x ≤ x c c x θ ∀ x ≥ x c (1)Continuity at x = ξ ST,lc implies c ( L ) ∼ L θ − p . In thethermodynamic limit, P ( x ) below x c reduces to P ( x ) =˜ P where ˜ P ∼ L − p . Moreover, the continuity at x = x c and the relation x c ∼ L − c allows ys to establish anexpression for the plateau exponent, p = cθ + ˜ θ ( L )(2 − c ) L →∞ = cθ (2)In the thermodynamic limit, when ˜ θ vanishes, with c ≈ .
15 and θ ≈ .
52, this relation predicts p ≈ .
60. Thiscorresponds to the value that we find when consideringonly a rescaling of the pseudogap region for which no
FIG. 2. Finite-size scaling for kick distributions experiencedby sites over the course of plastic rearrangements. (a) Kickdistribution, when considering kicks over the course of anavalanche. Inset: Kick distribution from single STs. (b) Dis-tribution of avalanche kicks, conditioned on avalanches withsize S ∈ ( S c / , S c ). Inset: Kick distribution conditioned onavalanches of size S ∈ ( (cid:104) S (cid:105) / , (cid:104) S (cid:105) ). size dependence is observed. When ˜ θ (cid:54) = 0, the value ofplateau is modified by the presence of the intermediatepower-law region and p increases as suggested by equa-tion (2). INFLUENCE OF THE SPATIAL EXTENT OFAVALANCHES
For now we have linked the emergence of the plateau in P ( x ) to the lower cutoff of mechanical noise and showedthat (cid:104) x min (cid:105) sets the scale below which the failure of weaksites is enhanced when using extremal dynamics. How-ever, the question of the origin of the crossover in P ( x )remains. What is the physical meaning of x c ? To tacklethis question we investigate the role of the spatial extentof avalanches on the distributions of residual stresses and FIG. 3. (a) Representation of the distribution P ( x ) rescaledby (cid:104) x min (cid:105) conditioned on survival and absorption in the nextavalanche. (b) Finite-size evolution of the drift v experiencedby sites with respect to the value of residual stresses. Inset: (cid:104) v (cid:105) ( x ), rescaled by x · L , showing that the onset of drift en-hancement occurs with a different scaling than the maximumdrift enhancement. mechanical noise. In practice, we adopt a top to bottomapproach by considering only specific sizes of avalanchesand by conditioning the probability distributions on thesespecific values.As a first step, we compute the unconditional distribu-tion of avalanches P ( S ) and find in Fig 4(b) an avalancheexponent τ ≈ .
37 and a fractal dimension d f ≈ .
95 con-sistent with results from atomistic simulations [16, 26]but slightly lower than results from other EPM imple-mentations [14, 21]. The average size of avalanches (cid:104) S (cid:105) plays a central role in the scaling relations for the yieldingtransition, as it represents the typical energy release oc-curring during the flow of amorphous solids.. When theresidual stress distribution is measured after avalanchesof size S ∼ (cid:104) S (cid:105) , as shown in Fig 5 (inset), we observea change with respect to the case where all events areconsidered. We still notice the presence of two power lawregimes and a plateau in the small x limit, but this timea good collapse of the plateau and intermediate regionsis obtained by rescaling x with L − . instead of L − . In-spection of the associated mechanical noise in Fig 2(b)reveals that the lower cutoff of the mechanical noise haschanged and is now scaling as L − . . This result can beexplained by the fact that, as mentioned above, (cid:104) S (cid:105) isrelated to the average distance to instability (cid:104) x min (cid:105) . Byfocusing on avalanches of the order of (cid:104) S (cid:105) , one cannotinvestigate the regime of residual stresses below (cid:104) x min (cid:105) FIG. 4. (a) Avalanche size distribution for L = 512 condi-tioned on x < (cid:104) x min (cid:105) and x > (cid:104) x min (cid:105) . (b) Unconditionedavalanche distributions P ( S, L ). Colors indicate system sizesas in Fig. 1. which sets the lower cutoff of the mechanical noise. More-over, the rescaling P ( x ) L p vs. x/ (cid:104) x min (cid:105) is not sufficientto obtain a good collapse in the pseudogap regime con-trary to what has been suggested recently in ref. [31].Another characteristic scale for avalanches is the up-per cutoff S c , which corresponds to system size spanningevents. When focusing on S ∼ S c , we see in Fig 5 that theintermediate power law regime disappears (˜ θ = 0) andthe pseudogap region immediately gives way to a plateau,whose scaling verifies equation (2). The crossover scal-ing of L − . is consistent with the apparent lower cutoffscale of the large avalanche kicks in Fig 2(b). By con-sidering only system size spanning events, we are screen-ing all the effects of small avalanches which are respon-sible for the appearance of an intermediate power-lawregime in P ( x ) at finite L . We therefore observe thatthe lower cutoff of the mechanical noise can change de-pending on the spatial extent of events. When the plasticactivity consists mainly of single STs rearranging inde-pendently, we expect a lower cutoff ξ lc ∼ ξ ST,lc ∼ L − for the corresponding mechanical kick distribution. In-deed, one would probe the typical far field response toan Eshelby inclusion with elastic interactions decaying FIG. 5. Residual stress distribution after avalanches with size S ∼ S c . Inset: Residual stress distribution after avalancheswith size S ∼ (cid:104) S (cid:105) . as G ( r ) ∼ r − d/µ ∼ r − . However, this description is nolonger appropriate for larger avalanches which result fromcollective instabilities; one unstable site triggers furtherrearrangements and so on. A distant site not only feelsone rearrangement but an apparent kick coming from theaccumulation of noise from consecutive single rearrange-ments. According to Fig. 2(b), the corresponding kicksdistribution exhibits an apparent value of µ = 1 . ξ S c ,lc ∼ L − . . The typical elas-tic interaction can thus be seen as even more long-ranged[32] with an effective kernel decaying as G ( r ) ∼ r − . . Wetherefore suggest that the departure from the pseudogapbehavior of P ( x ) occurs at x c ∼ ξ S c ,lc , in perfect agree-ment with the scaling of x c found by direct analysis of P ( x ) (Fig. S1). Origin of the crossover in P ( x ) The above analysis suggests that the crossover fromthe pseudogap regime is associated with the largestavalanches. In ref. [16], Salerno and Robbins showed thatthe fluctuations of the total stress δ Σ ∼ L − φ , and sug-gested that if the largest avalanches set the scale of thefluctuations of the total stress, i.e φ ≤ d − d f , a cor-relation length ξ Σ ∼ | Σ Y − Σ | − ν can be defined with ν = 1 / ( d − d f ). We fing that δ Σ ∼ L − . (Fig. S2), ver-ifying thus φ ≤ d − d f . The total stress is a sum over thelocal stresses σ i ; fluctuations scaling as δ Σ ∼ L − . im-ply that the σ i are correlated. It is thus reasonable expectthat the distribution of residual stresses P ( x ) reflects thiscorrelation. We therefore suggest that the crossover ex-ponent c = 1 /ν = d − d f and the crossover x c representsthe scale above which the residual stresses become corre-lated. Indeed the plastic strain released by one avalanchescales precisely as ∼ L − ( d − d f ) [28]. Scaling relation
An important consequence of our findings pertains tothe finite size scaling of the weakest sites in the system.Assuming that residual stresses can be seen as indepen-dent random variables, extreme value statistics dictatesthat P ( x ) ∼ x θ , (cid:104) x min (cid:105) ∼ L − α with α = d/ (1 + θ ). Set-ting (cid:104) x min (cid:105) ∼ (cid:104) S (cid:105) L d leads to an important scaling lawlinking the pseudogap exponent to the avalanche statis-tics, τ = 2 − θα/d f = 2 − θd/ ( d f ( θ + 1)) [14]. As shownin the SI, this relation continues to hold even in the pres-ence of an intermediate power law or plateau in P ( x ) aslong as the departure from the pseudogap regime scalesas x c ∼ (cid:104) x min (cid:105) [31]. However, the fluctuations of to-tal stress observed in our data indicate that correlationsamong the random variables play a significant role. Wealso show in the SI that when P ( x ) gives way to a plateaubelow x c ∼ ξ S c ,lc , α = d − p in the thermodynamic limit,and thus with c = d − d f τ = 2 − ( d − d f ) θd f . (3)With our measured value of p = 0 . θ = 0 .
5, and d f = 0 .
95 in the accessible range of system sizes, thisexpression is indeed in better agreement with our values α = 1 . τ = 1 .
37 than the previously derived scalinglaw (which would predict α = 1 .
33 and τ = 1 . CONCLUSION
In this work, we showed that in athermally shearedamorphous solids lim x → P ( x ) = P for finite L . Werelated the emergence of the plateau to the lower cut-off of mechanical noise as observed in the survival prob-ability of fractional Brownian motion [35, 36], and weshowed that the plateau appears whatever the charac-teristic size of plastic events. We observed a previouslyunnoticed intermediate power-law regime in the distri-bution P ( x ) characterized by a system-size dependentexponent ˜ θ ( L ). The average size (cid:104) S (cid:105) of the plastic eventsincreases with system size, implying an increase of thedrift and an enhanced reduction in the density of indi-vidual rearrangements, leading to the plateauing of theintermediate power-law in the thermodynamic limit.The departure from the pseudogap regime can beunderstood in analogy with depinning. The correla- tion length ξ Σ corresponds to the diameter above whichavalanches becomes unlikely [37]. The crossover inour system x c ∼ ξ − /ν Σ implying that below x c , largeavalanches become rare in agreement with our observa-tions. The crossover x c therefore marks a transition be-tween a collective and correlated dynamics to a dynamicsin which individual and independent rearrangements aredominant. Surprisingly, despite the role played by largeavalanches in the form of P ( x ), the unconditioned distri-bution of mechanical noise in our EPM implementationis still mostly dominated by small plastic events.As the average value of weakest site (cid:104) x min (cid:105) is locatedin the part of the residual stress distribution where x can be seen as independent random variables, any fi-nite size scaling based on (cid:104) x min (cid:105) would only reflect theindependence of events and the most likely point-likenature of the elastic interaction, which is reasonable ifthat small avalanches are predominant. As the inter-mediate power law flattens in the thermodynamic limitand the crossover is set by large avalanches, we expect (cid:104) x min (cid:105) to belong to an extended plateau region. Theratio x c / (cid:104) x min (cid:105) ∼ L α + d f − d → ∞ , and as a result, anon-zero fraction of sites always on the verge of stabilityare occupying this extended plateau region that could beconsidered as marginally stable [38].Our results offer possible new routes of interpretationfor the yielding transition. In particular, it would be in-teresting to see how the phenomenology observed herein two dimensions would manifest in three dimensions asthe geometry of avalanches encoded in d f would change.Moreover, one might wonder whether the picture of cor-related events inducing the departure of the pseudogapregime is still valid in the transient regime for which re-cent results from atomistic simulations reported the ap-pearance of a plateau in P ( x ) after only few percent ofdeformation [20] and an increase in the fractal dimension[22] with respect to the elastic regime [39]. ACKNOWLEDGMENTS
This research was undertaken thanks, in part, to fund-ing from the Canada First Research Excellence Fund,Quantum Materials and Future Technologies Program.High performance computing resources were provided byComputeCanada. C.R acknowledges financial supportfrom the ANR LatexDry project, grant ANR-18-CE06-0001 of the French Agence Nationale de la Recherche. [1] T. C. Hufnagel, C. A. Schuh, and M. L. Falk, ActaMaterialia , 375 (2016).[2] Y. Zhang, J. P. Liu, S. Y. Chen, X. Xie, P. K. Liaw,K. A. Dahmen, J. W. Qiao, and Y. L. Wang, Progressin Materials Science , 358 (2017).[3] I. Cantat and O. Pitois, Physics of Fluids , 083302(2006), https://doi.org/10.1063/1.2267062. [4] K. A. Dahmen, Y. Ben-Zion, and J. T. Uhl, NaturePhysics , 554 (2011).[5] J. Bar´o, A. Corral, X. Illa, A. Planes, E. K. H. Salje,W. Schranz, D. E. Soto-Parra, and E. Vives, Phys. Rev.Lett. , 088702 (2013).[6] A. Argon, Acta Metallurgica , 47 (1979).[7] F. Spaepen, Acta Metallurgica , 407 (1977). [8] M. L. Falk and J. S. Langer, Phys. Rev. E , 7192(1998).[9] P. Schall, D. A. Weitz, and F. Spaepen, Science ,1895 (2007).[10] J. D. Eshelby and R. E. Peierls, Proceedings of the RoyalSociety of London. Series A. Mathematical and PhysicalSciences , 376 (1957).[11] G. Picard, A. Ajdari, F. Lequeux, and L. Bocquet, TheEuropean Physical Journal E , 371 (2004).[12] J. Lin, A. Saade, E. Lerner, A. Rosso, and M. Wyart,EPL (Europhysics Letters) , 26003 (2014).[13] J. Lin and M. Wyart, Phys. Rev. X , 011005 (2016).[14] J. Lin, E. Lerner, A. Rosso, and M. Wyart, Proceedingsof the National Academy of Sciences , 14382 (2014).[15] N. P. Bailey, J. Schiøtz, A. Lemaˆıtre, and K. W. Jacob-sen, Phys. Rev. Lett. , 095501 (2007).[16] K. M. Salerno and M. O. Robbins, Phys. Rev. E ,062206 (2013).[17] D. Zhang, K. A. Dahmen, and M. Ostoja-Starzewski,Physical Review E , 032902 (2017).[18] M. Ozawa, L. Berthier, G. Biroli, A. Rosso, and G. Tar-jus, Proceedings of the National Academy of Sciences , 6656 (2018).[19] I. Fern´andez Aguirre and E. A. Jagla, Phys. Rev. E ,013002 (2018).[20] C. Ruscher and J. Rottler, Soft Matter , 8940 (2020).[21] E. E. Ferrero and E. A. Jagla, Soft Matter , 9041(2019).[22] N. Oyama, H. Mizuno, and A. Ikeda, “Unified viewof avalanche criticality in sheared glasses,” (2020),arXiv:2009.02635.[23] A. Nicolas, E. E. Ferrero, K. Martens, and J.-L. Barrat,Reviews of Modern Physics , 045006 (2018).[24] Z. Budrikis, D. F. Castellanos, S. Sandfeld, M. Zaiser,and S. Zapperi, Nature communications , 15928 (2017).[25] M. Talamali, V. Pet¨aj¨a, D. Vandembroucq, and S. Roux,Phys. Rev. E , 016115 (2011).[26] C. Liu, E. E. Ferrero, F. Puosi, J.-L. Barrat, andK. Martens, Phys. Rev. Lett. , 065501 (2016). [27] Z. Budrikis, D. F. Castellanos, S. Sandfeld, M. Zaiser,and S. Zapperi, Nature Communications , 15928 (2017).[28] B. Tyukodi, D. Vandembroucq, and C. E. Maloney,Phys. Rev. E , 043003 (2019).[29] In ref. [13], the elementary time unit is associated withsingle rearrangements. In this work, to emphasize the im-portance of avalanches we look either at individual rear-rangements or at the total stress drops that can containmultiple single rearrangements.[30] A. Lemaˆıtre and C. Caroli, “Plastic response of a 2damorphous solid to quasi-static shear ii - dynamicalnoise and avalanches in a mean field model,” (2007),arXiv:0705.3122.[31] E. E. Ferrero and E. A. Jagla, “Properties of the den-sity of shear transformations in driven amorphous solids,”(2020), arXiv:2009.08519.[32] J. Parley, S. Fielding, and P. Sollich, “Aging in a meanfield elastoplastic model of amorphous solids,” (2020),arXiv:2010.02593.[33] P. Le Doussal and K. J. Wiese, Phys. Rev. E , 051105(2009).[34] S. Karmakar, E. Lerner, I. Procaccia, and J. Zylberg,Phys. Rev. E , 031301 (2010).[35] A. Zoia, A. Rosso, and S. N. Majumdar, Phys. Rev. Lett. , 120602 (2009).[36] A. K. Hartmann, S. N. Majumdar, and A. Rosso, Phys.Rev. E , 022119 (2013).[37] D. S. Fisher, Physics Reports , 113 (1998).[38] M. M¨uller and M. Wyart, Annual Review of CondensedMatter Physics , 177 (2015).[39] B. Shang, P. Guan, and J.-L. Barrat, Proceedings of theNational Academy of Sciences , 86 (2020).[40] A. Logg and G. N. Wells, ACM Transactions on Mathe-matical Software (TOMS) , 1 (2010).[41] M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet,A. Logg, C. Richardson, J. Ring, M. E. Rognes, andG. N. Wells, Archive of Numerical Software (2015).[42] J. Lin, T. Gueudr´e, A. Rosso, and M. Wyart, Phys. Rev.Lett. , 168001 (2015). Supplementary Information
ELASTOPLASTIC MODEL IMPLEMENTATION
We implement an elastoplastic model (EPM) which contains N = L sites on a two-dimensional square lattice. Weuse a finite-element method on a triangular mesh (Fig. S3) to determine the stress propagation between sites and thestress field from applying displacements at the boundaries of the system.Each square site in the system is a plaquette consisting of four triangles. We use first-order Lagrange elements forthe displacements u , which can be understood as setting the displacement at each vertex of the mesh and assumingthat the displacement varies linearly within each cell (triangle). As the strain, γ = 12 (cid:0) ( ∇ u ) + ( ∇ u ) T (cid:1) , (1)involves the gradient of Lagrange-1 elements, the natural finite element space for the strain and stress are zeroth orderdiscontinuous Galerkin elements. These elements can be understood as setting a constant tensor within each cell. Wedivide the total strain, γ into elastic and plastic (stress-free) contributions, with γ = γ el + γ pl . We treat the elasticcontribution to strain with linear isotropic homogeneous elasticity, so σ = 2 µγ el + λ Tr( γ el ) , (2)where µ and λ are the Lam´e parameters. While irrelevant for critical dynamics, we use µ = 20 and λ = 10. Bycombining this with the equilibrium condition ∇ · σ = 0, fixing the displacements of the mesh at the boundaries, andsetting the plastic strain on each plaquette, the stress-field is completely determined.To each site we associate a local stress equal to the average of the stress field on the underlying plaquette. Sites yieldwhen the local stress exceeds their local yield threshold | σ xy | > σ Y . They reduce their local stress by accumulatingplastic strain. Each triangle within the plaquette has its local plastic strain incremented, with γ pl,xy → γ pl,xy + γ xy /µ .In this way, the local stress is reduced to zero. The yield threshold σ Y for that site is then redrawn from a Weibulldistribution with shape parameter k = 2, the same distribution that is used to initialize the yield thresholds.The system is driven to instability using a strain-controlled protocol. Unlike molecular dynamics simulations, whichoften load using a fixed-strain increment, we can precisely determine the strain necessary to destabilize a single site.Loading is done by specifying the displacement at the boundary vertices of the system. We use simple shear, with u = γy ˆ x , which generally induces a constant strain across the system. From a stable configuration, the system isloaded to until one site exactly meets its yield stress: | σ xy | = σ Y . The yielding site initiates an avalanche. After a siteyields, all sites are checked for stability. The most unstable site then yields next, until the system returns to stability.Throughout the avalanche, the displacements at the boundaries of the system remain fixed.The most expensive part of the simulation is in solving the stress field. We use Python to calculate the stress oneach plaquette, and to drive a parallelized FEM solver, FEniCS[40, 41]. CALCULATION OF THE DRIFT VELOCITY
The drift velocity is a concept originating in mean-field random-walker models [13]. To make contact with thatliterature, for instance, in the prediction for ˜ θ [33], we must define a drift velocity experienced after each elementarykick a site experiences. This is clearly a drift velocity on a per-ST basis. However, our loading protocol introduces adrift ( x min ) to initiate an avalanche, and so the drift velocity is the same for all the STs comprising that avalanche.For an avalanche (labelled i here) preceded by loading x i,min with (cid:96) i events, the drift velocity is therefore defined as N x i,min /(cid:96) i . Each of the (cid:96) i STs (labelled here ‘j’) comprising avalanche i experiences this drift. Hence, after measuring M avalanches, the average drift velocity on a per-ST basis can be written (summing first over avalanches, and theirover their constitutive STs): (cid:104) v (cid:105) ST ≡ N (cid:104) x min (cid:96) (cid:105) ST = N (cid:80) Mi =1 (cid:96) i M (cid:88) i =1 (cid:96) i (cid:88) j =1 x i,min /(cid:96) i = N (cid:104) (cid:96) (cid:105) av M M (cid:88) i =1 x i,min = N (cid:104) x min (cid:105) av (cid:104) (cid:96) (cid:105) av (3) SCALING RELATIONS FOR THE YIELDING TRANSITIONCrossover related to spanning plastic events
Our data suggests that the upper-transition between the θ and ˜ θ regimes is set by the lower cutoff scale of the kickdistribution for the largest avalanches, with s ≈ s c ∼ L . implying ξ lc ( s = s c ) ∼ L − d f ∼ L − . > L − . ∼ (cid:104) x min (cid:105) .That is, (cid:104) x min (cid:105) occurs in the ˜ θ regime. This impacts the finite size scaling. If we assume a residual stress distributionof the form P ( x ; L ) = p ( L ) ∀ x ≤ ξ lc c ( L ) x ˜ θ ( L ) ∀ ξ lc ≤ x ≤ x c c x θ ∀ x ≥ x c , (4)then the self-consistency equation for (cid:104) x min (cid:105) is given by: L − d = (cid:104) x min (cid:105) (cid:90) P ( x ; L )d x = p ξ lc + c ˜ θ + 1 ( (cid:104) x min (cid:105) θ − ξ θlc ) . (5)Using continuity of p ( x ; L ) at ξ lc , we obtain p ( L ) = c ( L ) ξ ˜ θlc , with which we can eliminate c in favour of p , toobtain after some algebra, (cid:104) x min (cid:105) θ = (cid:20) L − d p − ξ lc + ξ lc θ (cid:21) ξ θlc (1 + ˜ θ ) . (6)Now in the large L limit, the ˜ θ drop off, and the dominant term is L − d /p ∼ L p − d ∼ (cid:104) x min (cid:105) ∼ L − α , giving the scalingrelation α = d − p . As outlined in the main text, with p = cθ and c = d − d f one obtains α = d − θ ( d − d f ). Thisscaling relation can also be readily derived by considering the complementary integral, L − d = 1 − (cid:82) x max (cid:104) x min (cid:105) P ( x ; L )d x ,and applying continuity. Pseudogap or crossover at (cid:104) x min (cid:105) . Using the ”ideal” pseudogap form P ( x ) ∼ x θ with the self-consistency equation L − d = (cid:90) (cid:104) x min (cid:105) P ( x ) dx (7)yields immediately the scaling (cid:104) x min (cid:105) ∼ L − α with α = d/ (1 + θ ) [12]. Initial inspection would suggest that (cid:104) x min (cid:105) should depend on the form of P ( x ) below (cid:104) x min (cid:105) . As a consequence, one might imagine that the above scaling relationno longer holds. This is, however, not the case. Consider P ( x ) of the form: P ( x ; L ) = x > x max C ( L ) x θ x ≥ (cid:104) x min (cid:105) and x < x max f ( x ; L ) x < (cid:104) x min (cid:105) x < f ( x ), where (cid:104) x min (cid:105) ∼ L − α . Considering the complementary probability, equation 7, (cid:104) x min (cid:105) canbe written to depend only on the power-law above (cid:104) x min (cid:105) : L − d = 1 − (cid:90) x max (cid:104) x min (cid:105) Cx θ dx = 1 − C θ (cid:2) x max θ − (cid:104) x min (cid:105) θ (cid:3) . (9)The normalization factor, C , will necessarily exhibit finite-size scaling. C = (1 + θ )(1 − z d ) x max θ − (cid:104) x min (cid:105) θ (10)We can expand C as a Taylor series in z = L . Then to lowest order, C = (1 + θ ) (cid:20) x max θ + − d ! x max θ z d + ( α (1 + θ ))! x max θ z α (1+ θ ) (cid:21) (11)with higher-order terms omitted. Since, a priori α (1 + θ ) = t could be less than d (but still a positive integer, so asto ensure that all of terms in the Taylor series are well defined) we include both z d and z t terms. The requirementthat f is non-decreasing, along with continuity, ensures that L d = (cid:82) (cid:104) x min (cid:105) f ( x ) dx ≤ (cid:104) x min (cid:105) θ , implies that t ≥ d .Hence, the lowest order term in the Taylor expansion of C is O ( z d ). Thus, to lowest order, equation 7 gives (cid:104) x min (cid:105) θ ∼ L − d (12)which gives the scaling relation α (1 + θ ) = d . Summary of exponents and scaling laws