Rare events and disorder control the brittle yielding of amorphous solids

We use atomistic computer simulations to provide a microscopic description of the brittle failure of amorphous materials and assess the role of rare events and quenched disorder. We argue that brittle yielding originates at rare soft regions, similarly to Griﬃths eﬀects in disordered systems. We numerically demonstrate how localised plastic events in such soft regions trigger macroscopic failure via the propagation of a shear band. This physical picture, which no longer holds in poorly annealed ductile materials, allows us to discuss the role of ﬁnite-size eﬀects in brittle yielding and reinforces the similarities between yielding and other disorder-controlled nonequilibrium phase transitions such as depinning and hysteresis.

Yielding of slowly deformed amorphous solids can proceed in two qualitatively different ways. Whereas ductile materials reach a stationary plastic flow through a continuous evolution under applied deformation, brittle ones undergo a macroscopic failure at which the stress discontinuously drops via the formation of a system spanning shear band [1][2][3][4][5]. We recently showed that the two regimes can be observed in the same material if prepared over a wide enough range of annealing conditions and are representative of distinct phases of yielding, separated by a critical point in the universality class of the out-ofequilibrium random-field Ising model (RFIM) [6,7].
Here, we exploit this finding to provide a microscopic perspective on the brittle yielding of amorphous media, shedding light on two issues that are central to understand material failure. First, we focus on the microscopic origin of shear banding. This is a widely studied question in the context of metallic glasses where it is of great practical interest for improving their ductility [8]. Many studies on metallic glasses have in particular considered the differing characteristics of homogeneous and heterogeneous shear-band nucleation [9,10], the former being a property of an ideal bulk material whereas the latter is mainly due to extrinsic flaws such as impurities or surface imperfections. Second, we take a new angle to investigate how finite-size effects modify brittle yielding. This is again relevant for experiments on metallic glasses in relation with a possible evolution towards brittleness [11,12] and is also central for the theory of yielding. In fact, the existence of a critical point as a function of glass stability has recently been challenged [13], and attributed to putative finite-size effects beyond those already analysed in [6,7].
The crux of the present work stems from the analogy with the driven RFIM at zero temperature [14]. We demonstrate that similarly to the zero-temperature spinodal of the RFIM [15], brittle yielding in well-annealed amorphous solids is controlled by rare events and quenched disorder [15,16]. Disorder obviously refers to the amorphous structure of the material but it is worth discussing what are "rare events" in this context. It is well established [1,[17][18][19][20] that slow deformation of amorphous solids involves localized plastic events that are characterized by local-stress and nonaffine-displacement fields having a quadrupolar symmetry similar to that of Eshelby inclusions in continuum elasticity theory [21]. Furthermore, shear-band formation seems to be associated with a mechanical instability taking the form of a line (in 2d) or a plane (in 3d) of Eshelby quadrupoles [20,[22][23][24][25]. Although the structural origin of soft regions prone to plastic rearrangements is still vividly debated (see Ref. [26] for a critical review), we simply assume here that plastic events generically occur at "weak spots". Crucially, large weak spots become very rare in stable glassy materials. To understand this, recall that correlation lengths are rather modest in supercooled liquids near the glass transition temperature [27]. Let ξ be the typical glassy length and p the probability to find a soft poorly-annealed region of volume ξ d , where d is the spacial dimension, around a given point in a stable glass. The probability to find a much larger soft region of volume v is proportional to c is a constant of order unity. This exponential suppression, characteristic of Griffiths phases in disordered media [28], makes large soft regions statistically a rare event in well-annealed glasses.
Computer simulations are unable to directly probe such phenomenon, as only small soft regions are found even in the largest systems that can be studied numerically. As a result, the central role of rare events is completely missed. To get around this major flaw, we insert a soft region in an otherwise stable glass by fiat [6,16]. Knowing that shear bands extend preferentially along the direction of shear, we choose an elongated shape oriented arXiv:2102.05846v1 [cond-mat.soft] 11 Feb 2021 along the expected direction of shear banding. The key physical point is that in a well-annealed amorphous material such a rare soft region is more prone to rearrange under shear strain and provides a "seed" for shear bands. Under applied deformation this region yields early, relaxing stress before the bulk of the material. By doing so, it destabilizes surrounding particles, which as a consequence also yield before the bulk. This leads to an extension of the soft region along the principal direction of the seed and, for a certain value of the imposed shear strain, to a self-sustained process leading to macroscopic shear banding. Our numerical procedure mimicks the spontaneous nucleation of a shear-band embryo [12] at an intrinsic defect in a pristine macroscopic sample that we cannot simulate directly [29]. The strong finite-size effect due to the difficulty to find exponentially rare defects is thus fully circumvented.
For simplicity, we focus on the simple shear of a 2d glass since the brittle-to-ductile transition and the analogy with the RFIM hold in 2d [7]. In addition, to be able to make crisp statements about nonequilibrium phase transitions, instabilities and other singularities, we consider the limit of zero temperature and of a quasi-static applied strain. Our study is based on the simulation of a 2d glass composed of polydisperse soft disks [30]. We prepare equilibrium supercooled liquid configurations over a very wide range of temperatures T ini by means of optimized swap Monte-Carlo simulations [31]. A "seed" is then inserted as follows. We define an ellipsoidal region characterized by D a , the length of the major axis chosen along the direction of shear, and by D b , the length of the minor axis. In a 2d system of N atoms with N from 8000 to 64000, which corresponds to a linear size L from 89.4 to 253.0 (in units of the average atomic diameter), we fix D b = 8, which is of the order of the typical scale of an elementary rearranging region [32], and we vary D a up to D a = 50 which is of the order of half the smallest system size (see below for a discussion). We then perform additional Monte-Carlo simulations inside the seed region at a very high temperature T h = 10 (about 100 times larger than the mode-coupling crossover [30]). We then quench the obtained atomic configuration to zero temperature by using the conjugate gradient method. As a result, the glass sample contains a poorly-annealed elliptic seed inside an otherwise very stable material.
Finally, we deform the samples by athermal quasistatic shear simulations at zero temperature with Lees-Edwards periodic boundary conditions, which correspond to change the shear strain in a controlled way throughout the sample [19]. More details on the model and methods are given in the Supplementary Information (SI).
We start by establishing that rare seeds are essential in well-annealed glasses displaying brittle yielding. We first show stress σ vs. strain γ curves for stable glasses in Fig. 1(a). The curves with no seed show a large stress overshoot and a large abrupt stress drop, as found pre- viously [6,7]. When inserting a seed in the same initial configurations, the elastic-like regime of the σ vs γ curves seems barely modified. When the strain γ is further increased, the samples with a seed continue to yield abruptly, but they do so systematically earlier for longer seeds. This directly establishes that a single seed, which corresponds to a rare event in a pristine macroscopic glass, has a dramatic impact on bulk yielding. By contrast, the stress vs. strain curves for a poorlyannealed glass with T ini = 0.100 in Fig. 1(b) hardly show any change with the presence or the length of the seed, suggesting that ductile yielding is not affected by rare events. We quantitatively support this conclusion by locating the average yield strain γ Y for each degree of annealing (measured by T ini ) and each D a . We do so by using the maximum of the disconnected susceptibility introduced in [6] (see SI). The systematic evolution of γ Y is summarized in Fig. 1(c) for N = 64000. This plot is strongly reminiscent of the RFIM study presented in Fig.  1 of Ref. [15]. It shows that the effect of seed insertion on the yielding transition fades away, within numerical uncertainties, at the critical point that we have previously located through completely different means [7]. This result not only independently confirms the existence of a nonequilibrium phase transition separating brittle and ductile behaviors but also shows that the finite-size study performed without seeds [6,7], while being incomplete in the brittle regime where it misses the strong effect of very rare events, becomes trustworthy in ductile materials where rare events no longer play a role.
Having established in which regime rare events dominate yielding, we now focus on the microscopic behavior giving rise to the observed seed influence. The snapshots in Fig. 2 confirm that the tiny decrease of σ before yielding (γ = 0.03 < γ Y ) is due to local plasticity inside the seed (the corresponding stress drop is at most of order D a D b /L 2 1 is better seen in the SI). Locations with a large nonaffine displacement, D 2 min [18], correspond to the core of an Eshelby quadrupolar displacement field [33] taking place much before global failure. For small seeds, e.g., D a = 16, a single quadrupolar displacement is found, but as D a is increased, we see two (for D a = 32) or three (for D a = 50) quadrupolar relaxations aligned along the major axis of the seed.
The symmetry of the stress relaxation associated to these events implies that the redistributed stress is max-imal at the tip of the seed, which eventually leads to the sudden formation of a macroscopic shear band. To analyze the physical process behind the ensuing stress drop of order 1 associated with this brittle yielding, we perform gradient-descent dynamics starting from the configuration exactly on the verge of yielding. The time evolution of D 2 min during the stress drop is visualized in Fig. 3(a-f). Plastic events with large D 2 min first appear very near the tip of the seed, and then proliferate along the horizontal direction, ending up in a system-spanning shear band. For a homogeneous glass sample with no seed, the initial location of the shear band is random in the simulation box and its direction is either horizontal or vertical [34]. On the contrary, in the presence of a long enough seed (see the SI for a quantitative discussion), the location as well as the direction of the system-spanning shear band are fully determined by the seed.
We also show the time evolution of the stress during the gradient descent in Fig. 3(g). Considering the relatively smooth decrease of σ, one might naively conclude that shear-band formation is merely the continuous sliding of two rigid blocks. To characterize more precisely the dynamical deformation process of the shear band formation, we have measured the squared velocity v 2 (t) = 1 N i |v i (t)| 2 . This quantity is more sensitive to dynamical activity during the shear-band formation. The time dependence of v 2 (t) in Fig. 3(g) reveals that the deformation process is in fact highly intermittent. As shown in the SI, this intermittent behavior translates in real space into individual Eshelby-like displacements which repeatedly appear along the shear band. It is these multiple Eshelby relaxations that eventually lead to the stress drop of order 1. Since a single Eshelby event can only carry tiny displacements at its core, to have a macroscopic effect that is visible on the σ vs γ curve, Eshelby events have to appear repeatedly in the already formed shear band, as theoretically argued [20], and confirmed in our simulations.
We can now rationalize our numerical findings by the following physical picture which is based on the expectation that rare soft regions, and more specifically the largest of the softest ones appearing with a nonzero probability in the thermodynamic limit, play a crucial role in leading to shear banding for well-annealed glasses. By inserting by hand such regions, which in macroscopically large samples would be randomly located in space, we found that they rearrange much before the rest of the material. This local plastic activity leads to the formation and alignment of Eshelby-like quadrupoles inside the seeds, which corresponds to the most favorable configuration from an energetic point of view [20]. The key point to then explain shear banding is that the aligned Eshelby quadrupoles induce contributions to the local stress that add up constructively along the direction of the seed major axis, and only there. Due to the decay of the elastic interaction with distance (as 1/r d ), the local stress generated by Eshelby relaxations is maximum near the tips of the ellipsoidal seed. We have confirmed numerically (see the SI), that the stress increase near the tips is of order 1 during the initial shear-band propagation. This positive interference is at the root of shear banding. Indeed, provided the system is relatively homogeneous, which should be the case for well-annealed glasses in which the disorder strength is small, the tips should thus yield first upon further straining and then induce an even larger stress near the new tips. A self-sustained process and a system-spanning shear band naturally ensues.
The above picture corresponds to a "weak-disorder" regime in which fluctuations of the local yield stress are relatively small. Disorder is nonetheless crucial because it allows the spontaneous, albeit extremely rare, presence of a large soft region. Disorder also gives rise to the intermittent behavior during shear banding shown in Fig. 3. For larger disorder strength (less stable glasses) the shear-band propagation should take more complex paths to avoid strong pinning sites where the local yield stress may be high. The propagating shear band may then need to deform substantially to go through the softest regions which are no longer exactly at its tips [7]. Finally, when the disorder strength becomes really large we anticipate that the presence of too many strong pinning sites will eventually prevent the formation of a welldefined system-spanning shear band, resulting in ductile yielding where the physical picture based on rare defects breaks down. The role of disorder is largely neglected in arguments based on continuum elasticity [13,35].
We can also discuss the strong finite-size effects of the stress vs. strain curve in the brittle case and its proper thermodynamic limit. The limiting value of the shear stress at yielding, σ Y , is determined by the stress needed to nucleate a shear band from an arbitrary large seed embedded in a much larger, macroscopic, sample. To realize this scenario numerically we should in principle analyze the double limit of 1 D a L. This explains why we have limited the seed length D a so that the tips of the seed are not affected by their images in the periodically repeated simulation box. In practice, we have chosen the longest seed to be D a = 50, which is about half the smallest system size L considered (N = 8000 and L = 89.4), and we have studied the variation at fixed D a when increasing L: more details are given in the SI. For a large seed immersed in an even larger piece of material we expect that σ Y does not correspond to the steadystate regime obtained for large γ. The latter is the stress needed to induce plastic activity after the shear band has already formed and propagated through the entire system. A larger external stress instead is needed for the stress at the tip to reach its local yield stress value, as this still represents the yield stress of a yet unrelaxed stable glass region. We conclude that a nonzero stress drop of order 1 at yielding survives in the thermodynamic limit. A different conclusion, based on fracture mechanics, was reached in [16] [36].
Finally, recall that introducing large seeds by hand is a means to get around the strong finite-size effects associated with their spontaneous occurrence. The probability of a large soft region of volume v anywhere in a system of volume L d goes as (L d /v) exp[−(c/ξ d )v] and is of order 1 only for L exponentially large in v. As then clearly shown in Fig. 1, this on the one hand explains why the yield strain γ Y increases with decreasing system size in deformed sub-micron metallic-glass samples [10,11] and on the other hand provides evidence for the persistence of a critical brittle-to-ductile transition as a function of glass preparation in a bona fide thermodynamic limit, in contradiction to what was suggested in [13].
The propagation and growth of a shear band within a brittle amorphous solid present analogies with the depinning of interfaces in a random medium, yet with some crucial differences due to the anisotropic and long-range character of the elastic interactions and to the very nature and shape of the propagating object. Developing a theory for such a depinning process is a worthwhile challenge for the future which could deepen further the theoretical understanding of shear banding and yielding in amorphous materials.
We thank S. Fielding and M. Wyart for discussions. This work was supported by grants from the Simons Foundation (#454935 Ludovic Berthier and #454935 Giulio Biroli).

DETAILS ON THE SIMULATION METHODS
Model. The two-dimensional glass-forming model consists of particles with purely repulsive interactions and a continuous size polydispersity. Particle diameters, d i , are randomly drawn from a distribution of the form: where v 0 sets the unit of energy (and of temperature with the Boltzmann constant k B ≡ 1) and = 0.2 quantifies the degree of nonadditivity of particle diameters. We introduce > 0 in the model to suppress fractionation and thus enhance the glass-forming ability. The constants c 0 , c 1 and c 2 enforce a vanishing potential and continuity of its first-and second-order derivatives at the cut-off distance r cut = 1.25d ij . We simulate a system with N particles within a square cell of area V = L 2 where L is the linear box length, under periodic boundary conditions, at a number density ρ = N/V = 1. We study N from 8000 to 64000. We compute the shear stress σ through where v ij is the derivative of the potential.
Preparation of samples with a seed. Glass samples with a seed (soft region) have been prepared by first equilibrating liquid configurations at a finite temperature, T ini , which is sometimes referred to as the fictive temperature of the glass sample. We prepare equilibrium configurations for the polydisperse spheres using swap Monte-Carlo simulations [31], which allows us to access a wide range of T ini . With probability P swap = 0.2, we perform a swap move where we pick two particles at random and attempt to exchange their diameters, and with probability 1 − P swap = 0.8, we perform conventional Monte-Carlo translational moves.
In order to introduce a seed (soft region), we define an ellipsoidal region whose size is characterized by the length of the major axis D a and the length of the minor axis D b (see Fig. 2 of the main text). For most of our study D b is fixed to the value of 8, and we vary D a from 0 to 90.
The linear box length of the two-dimensional system for N = 8000 is L = 89.4 and for N = 64000 is L = 253. We then perform additional swap Monte-Carlo simulations only for the particles inside the ellipsoidal region defined above. The temperature of this additional Monte-Carlo simulations is T h = 10.0. The dynamical mode-coupling crossover [37] of the system is T mct ≈ 0.110. Thus T h is about 100 times higher than the mode-coupling crossover temperature of the system. We then quench the obtained configuration down to zero temperature by using the conjugate gradient method. Thus our glass samples contain a disordered, poorly-annealed region in the middle of the simulation box (see Fig. 2 of the main text).

Mechanical loading.
We have performed straincontrolled athermal quasi-static shear (AQS) deformation using Lees-Edwards boundary conditions [19]. The AQS shear method consists of a succession of tiny uniform shear deformation with ∆γ = 10 −4 , followed by energy minimization via the conjugate-gradient method. The AQS deformation is performed along the x-direction. Note that during the AQS deformation, the system is always located in a potential energy minimum (except of course during the transient conjugate-gradient minimization), i.e., it stays at T = 0.
Nonaffine displacement. We consider the local nonaffine displacement of a given particle relative to its nearest neighbor particles, D 2 min [18]. D 2 min (γ 0 , γ) is measured between the configuration at γ 0 and γ. We define nearest neighbors by using the cut-off radius of the interaction range, R cut = 3.0d. We determine the nearest neighbors of a particle from the configuration at γ 0 . For the gradient-descent dynamics, we replace γ 0 and γ by two times t 0 and t and consider D 2 min (t 0 , t).

DETERMINATION OF γY
We explain how to determine the yield strain γ Y for each T ini and D a . In Fig. 4, we show the mean stress σ and the corresponding variance, χ dis = N ( σ 2 − σ 2 ), averaged over 100 − 400 independent samples. For stable glasses, say T ini = 0.035, the macroscopic stress drop takes place earlier as one increases D a . Thus, the remarkable phenomenology shown in Fig. 1 of the main text for individual samples is also confirmed at the ensemble level. The corresponding variance, χ dis , shows a noticeable peak near the location of the macroscopic stress drop, and it shifts with increasing D a . We can therefore use the peak position of χ dis as an unambiguous way to determine the yield strain, γ Y . As T ini is increased (T ini = 0.060), the shift in both σ and χ dis plots becomes weaker. When T ini is further increased (T ini = 0.090 − 0.120), the D a -dependence is significantly suppressed. Above the critical point, T ini,c = 0.085, of the N = 64000 system, the D a -dependence essentially disappears, which means that the seed does not play any role in the ductile yielding regime.

LOCATION AND DIRECTION OF THE SHEAR BAND
Here we show that the presence of a soft ellipsoidal region in amorphous solids determines the location and direction of the shear band (hence the name of "seed"). For stable homogeneous samples without a seed, the shear band occurs in any place under the Lees-Edwards periodic boundary conditions. Moreover, the direction of the shear band takes either a horizontal or a vertical direction because of the isotropy of the material. Figure 5(a) shows the histogram of the location of the shear band for a the two-dimensional stable glass with T ini = 0.035 and N = 64000 (L = 253). The abscissa denotes the x- coordinate of the vertical shear band or the y-coordinate of the horizontal shear band. The number of independent realizations for this measurement is 100. The histogram for the homogenous samples is relatively uniform, confirming that the shear-band location is random in space. We also measure the fraction of the samples having the horizontal shear band, f H . We find f H = 0.45 for the homogenous case (no seed), which implies that the horizontal and vertical shear bands happen with essentially the same probability. We now examine the effects of the seed in terms of location and direction of the shear band separately. First, Fig. 5(c) shows the result for a small circular seed with diameter D = 8 (or D a = D b = 8). The histogram has a tiny peak at the center, which means that the probability of finding the shear band at the position of the seed is higher than in the homogeneous sample case. As the size of the circular seed D is increased, the peak is enhanced, and most of the samples have the shear band appearing at the center (see Fig. 5(d)). However, f H is still near 0.5 for both D = 8 and D = 16, as expected from the symmetry of the circular seed.
We now insert an elliptical seed. We vary D a while we fix D b = 8. We find that the peak at the center is defined more sharply with increasing D a , as shown in Figs. 5(e,f). Note that the area of the circular seed in Fig. 5(d) and of the elliptical seed in Fig. 5(f) are the same. Moreover, f H quickly departs from near ∼ 0.5 once the elliptical seed is introduced, which means that, now, the direction of the shear band is statistically controlled by the orientation of the seed. This is consistent with the observation that the shear band tends to take place where the Eshelby events are aligned. In Fig. 3 of the main text, we see the intermittent behavior in the mean squared velocity, v 2 , during the gradient-descent dynamics starting from the configuration right before the largest stress drop for a stable glass (N = 64000, T ini = 0.035, and D a = 32). To see this intermittent behavior in real space, we visualize the displacement vector fields between two successive time frames in Fig. 6. We observe that Eshelby-like displacements propagate along the shear band. Remarkably, the Eshelby vector fields propagate many times during the entire lapse of shear band formation. This observation appears reasonable because a single Eshelby displacement vector can carry only a tiny displacement at its core. Thus, to build a system-spanning macroscopic shear band, the Eshelby displacement vectors have to propagate many times, as theoretically argued [20].

STRESS INSIDE THE SEED
We monitor the stress evolution under external strain inside and outside the seed separately.
First, we introduce the stress inside and outside the seed. We rewrite Eq. (1) as where σ i is defined by We define the stress inside the seed, σ I , and outside the seed, σ O , by  Figure 7 shows the stress inside and outside the seed and for the entire sample as a function of the strain γ for a stable glass with a seed of size D a = 50. Three independent samples are shown. The stress inside the seed shows multiple drops much before the macroscopic yield strain, γ Y 0.06, and displays ductile-like yielding behavior. On the other hand, the stress outside the seed shows elastic response, having an essentially identical curve to that of the entire system. The number of particles inside the seed with D a = 50 is around N I ≈ 300, and hence c I ≈ 0.005. Thus, in terms of the stress value, the seed contributes only marginally to the entire sample. Yet, as demonstrated in Fig. 1 of the main text, the location of γ Y is significantly affected by the presence of the seed. This observation confirms that a rare soft region with a concentration that vanishes in the thermodynamic limit can impact the macroscopic yielding behavior. We investigate the local stresses near the seed at the initial stage of shear-band formation. We define the local stress σ local i by σ local where n i is the number of neighboring particles for i-th particle. The local stress is thus obtained by averaging over particles within a cut-off radius R. We set R = 5, following Ref.
[32]. Figure 8 shows the time evolution of the nonaffine displacement D 2 min (left panels) and of the local stress (right panels) during the gradient-descent dynamics for the largest stress drop. At t = 0, i.e., right before the macroscopic yielding, the entire sample is stressed rather homogeneously except near the seed (Fig. 8(b)). Because the seed region has already yielded along the quasi-elastic branch, it has a lower stress, which appears in color blue. This inhomogeneity of the local stress at the seed induces higher stresses near the tips of the seed, yet with a magnitude is order 1. The higher stress field near the tips then induces plastic activity at later time (Figs. 8(c,d)), eventually causing propagation of the shear band (Figs. 8(e,f)). We consider the variation of the yielding behavior with the system size L when the size of the seed is fixed. We have argued in the main text that the appropriate thermodynamic limit is to consider L → ∞ first, and only then D a → ∞. This avoids considering a seed that scales as the system size or, within a finite system with periodic boundary conditions, a seed that is influenced by its images. We illustrate this points by showing in Fig. 9 the finite-size effect on the stress vs. strain curves for fixed D a (and D b ) and for a well-annealed glass with T ini = 0.035. In Fig. 9(a), we consider a seed length D a = 50 which is about half of the linear box length of the smallest system under study, i.e., N = 8000 and we investigate how the curve evolves when increasing N up to 64000. All system sizes show a qualitatively similar behavior with a large stress overshoot. The slope of the drop after the overshoot increases with N , which is in agreement with the finite-size scaling of a discontinuous (brittle) yielding transition as already found in the absence of a seed.

FINITE-SIZE EFFECTS
Quite notably, the overshoot becomes more prominent as the system size increases, all in all confirming the brittle nature of the transition in the thermodynamic limit.
On the other hand, in Fig. 9(b), we set D a = 90, so that the seed now spans the entire sample for the smallest system of N = 8000. In this latter case, the stress overshoot is completely wiped out and a seemingly ductile behavior is observed. However, when increasing the system size the overshoot reappears and becomes more and more prominent with, as for D a = 50, the slope of the curve that becomes steeper as N increases. This clearly indicates a brittle yielding in the thermodynamic limit.