Mean-Field Theory

To substantiate our proposal, we develop a simple analysis, which is inspired by the description of sheared materials in terms of elasto-plastic models (2). This widespread mesoscopic approach successfully reproduces the key phenomenology of deformation and flow in amorphous materials. Our main focus is on the role of the initial preparation, which has received much less attention (see, however, refs. 28⇓⇓–31).

In this approach, the system is decomposed in mesoscopic blocks i=1,⋯ ,Nb, in which elastic behavior is interrupted by sudden shear transformations. At each block is assigned a local stress, σi, drawn from an initial distribution Pini(σ), which encodes the degree of annealing. In the absence of plastic events, the response is purely elastic, and a small deformation increment, δγ, loads all of the blocks as σi→σi+2μ2δγ, with μ2 the shear modulus. However, when the local stress becomes larger than a threshold value σith (that for simplicity we consider uniform σith=σth), the block yields and the local stress drops by a random quantity x≥0 sampled from a given distribution g(x). After the drop, the stress is redistributed to the other sites as σj→σj+Gijx. The elastic kernel Gij is generally taken of the Eshelby form, which corresponds to the far-field solution of elasticity (it decays as 1/|i−j|d, where d is the spatial dimension, but changes sign and displays a quadrupolar symmetry) (32). There is no straightforward and generally accepted way to handle the nonlocal Eshelby interaction kernel at a mean-field level (33⇓–35). Here we consider a mean-field approximation that consists of replacing this nonlocal interaction by a fully connected kernel Gij=μ2Nb(μ1+μ2), with μ1>0. This description overlooks the effect of the anisotropic and nonpositive form of the Eshelby interaction kernel. Nonetheless, we expect that it provides a correct qualitative description of the yielding transition itself. [A similar behavior is indeed found by analyzing more involved mean-field models (36).] Below we discuss its limitations and how to go beyond them. Note that this model also has a natural interpretation as a mean-field model of depinning^† as well as earthquake statistics (38, 39).

The key quantity in this approach is the distribution Pγ(x) of the distances xi=σth−σi from the threshold stress. In the following, we study its macroscopic evolution with strain γ. As detailed in SI Appendix, it is governed by the equation∂Pγ(x)∂γ=2μ21−xcPγ(0)∂Pγ(x)∂x+Pγ(0)g(x),[1]where xc=(μ2/[μ1+μ2])x¯ and x¯=∫0∞dxxg(x) represent material-dependent parameters (here, we have xc<x¯<1 as we set σth=1 as the stress unit). The degree of annealing of the material is fully encoded in the initial distribution Pγ=0(x), which contains the same information as Pini(σ).

The properties of the macroscopic stress–strain curves can be obtained through Eq. 1 and the relation ⟨σ⟩=1−⟨x⟩, which is derived by taking the average of the equation defining xi. Our results, which hold for a generic g(x) (see SI Appendix), are shown in Fig. 1 for the explicit case g(x)=exp(−x/x¯)/x¯ and Pγ=0(x)=e−x/A−e−x/(1−A)/(2A−1), 1/2<A<1. With this choice, A is the unique parameter controlling the degree of annealing, with smaller values of A corresponding to better annealed samples.

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Fig. 1.

The different yielding regimes in the mean-field elasto-plastic model. Stress ⟨σ⟩ versus strain γ for increasing degree of annealing (decreasing A values) from bottom to top, using xc=0.9,x¯=0.92. The monotonic flow curve (black) transforms into a smooth stress overshoot (red), and above a critical point with infinite slope (blue) becomes a discontinuous transition (green) of increasing amplitude (dark blue). (Inset) Stress discontinuity Δ⟨σ⟩ versus the degree of annealing [here, the initial distribution Pγ=0(x) is parametrized by a single parameter A, which plays the same role as the preparation temperature in the simulations].

For a poor annealing, the stress–strain curve is monotonically increasing and yielding is a mere crossover. As one increases the degree of annealing, a stress overshoot first appears, but yielding remains a crossover, still not a bona fide phase transition. For the best annealing, the overshoot is followed by a spinodal and a sharp discontinous jump of the average stress. Mathematically, this occurs when, increasing γ, Pγ(0) reaches xc−1, thus inducing a singular behavior of Pγ(x) via Eq. 1. In this case, yielding takes place as a nonequilibrium first-order transition. Crucially, a critical point Ac separates the first-order regime from the smooth one. From Fig. 1, it is clear that an appropriate order parameter distinguishing the two regimes of yielding is the macroscopic stress drop Δ⟨σ⟩. As shown in Fig. 1, Inset, Δ⟨σ⟩ vanishes at large A, but it grows continuously by decreasing A below Ac. This critical point is therefore reached not only for a specific value of the strain and stress but also by tuning the degree of annealing of the material. The stress overshoot, frequently observed in colloidal materials (40, 41), yield stress fluids (42), and computer simulations (43), is simply a vestige of this critical point at larger disorder strength.

When a spinodal, followed by a discontinuity, is present, the stress displays a square-root singularity as the yield strain γY is approached from below, and the distribution of the avalanche size S becomes for large S⟨σ⟩−σsp∝(γY−γ)1/2,[2]P(S)∼S−3/2e−C(γY−γ)S,[3]where C>0 is a constant and γ→γY−. The discontinuous stress drop decreases as annealing becomes poorer, and it eventually vanishes with a square-root singularity at the critical point (γYc,σc), where a different behavior emerges:⟨σ⟩−σc∝sgn(γ−γYc)|γ−γYc|1/3,[4]P(S)∼S−3/2e−C′|γ−γYc|4/3S,[5]with C′>0 and γ→γYc±. All these scaling behaviors coincide with those found for the RFIM^‡ within the mean-field theory (45).

The presence of an annealing-controlled random critical point is the main finding of our mean-field approach. We stress that its presence, as well as that of the different regimes of yielding, does not require the introduction of any additional physical mechanism, such as dynamical weakening (37⇓–39, 46⇓–48). It only depends on the initial preparation of the amorphous material before shearing, in combination with the basic rules of elasto-plastic models. In finite dimensions, the above scaling behaviors will be modified. Whereas a spinodal instability can still be present in athermal conditions, it will likely not be associated to any critical behavior (49). On the other hand, the random critical point should always be in the universality class of the athermally driven RFIM, but this class is presumably distinct from that of the conventional model with only short-ranged ferromagnetic interactions.

This mean-field description is not meant to reproduce all aspects of the deformation-and-flow phenomenology. In particular, it does not allow criticality of the sheared system along the elastic and plastic branches (29, 50), nor can it describe spatial flow inhomogeneities, such as shear bands. Nonetheless, as we now show by computer simulations, the model correctly captures the preparation dependence of the yielding transition, the central question addressed by our work.

Atomistic Model and Numerical Procedures

We have numerically studied the yielding transition in a 3D atomistic glass model for different degrees of annealing, with our mean-field predictions as a guideline. We have used a size-polydisperse model with a soft repulsive potential (27). Glass samples have been prepared by first equilibrating liquid configurations at a finite temperature, Tini, and then performing a rapid quench to T=0, the temperature at which the samples are subsequently deformed. The preparation temperature Tini then uniquely controls the glass stability, and we consider a wide range of preparation temperatures, Tini=0.062−0.200. To obtain well-annealed systems, we have used the swap Monte Carlo (SWAP) algorithm, which allows equilibration at extremely low temperatures (27). The considered range of Tini describes very poorly annealed glasses (Tini≈0.2, corresponding to wet foam experiments), ordinary computer glasses (Tini≈0.12, corresponding to colloidal experiments), well-annealed glasses (Tini≈0.085−0.075, corresponding to metallic-glass experiments), as well as ultrastable glasses (Tini≈0.062, see ref. 51). No previous numerical work has ever accessed such a large range of glass stability.

We have performed strain-controlled athermal quasi-static shear (AQS) deformation using Lees–Edwards boundary conditions (18). Note that during the AQS deformation, the system is always located in a potential energy minimum, such that inertia and thermal fluctuations play no role. This method is considered as the zero-strain rate limit, γ̇→0, which bypasses the time scale gap between simulation and experiments (43). Thus, our simulation setting (SWAP/AQS) fully overcomes the time scale gap in both glass preparation and mechanical deformation between simulations and experiments. To study the finite-size effect, we have varied the number of particles N over a considerable range, N=1500−96000. More details are given in SI Appendix.

The Two Regimes of Yielding

In Fig. 2A, we show the evolution of typical stress–strain curves for large individual samples with N=96000. For a high Tini, the usual jerky succession of stress drops is found with no overshoot for Tini=0.200 [akin to wet foam experiments (52, 53)] and with a stress overshoot for Tini=0.120 [akin to colloids experiments (40, 41)]. Note that previous simulation studies about effects of annealing on yielding behavior would be restricted by this regime (43, 54, 55). Strikingly, the stress overshoot transforms into a sharp stress discontinuity for Tini≲0.1 near the yield strain γY≈0.12. This large stress drop is distinct from the smaller stress drops observed at other strain values. This is confirmed in Fig. 2D, where we plot for increasing values of N the averaged stress ⟨σ⟩ (obtained by averaging over many independent samples) for Tini=0.062. The stress discontinuity at yielding is the only one surviving after the average and it becomes sharper and better resolved as N increases. These data strongly suggest that, in the thermodynamic limit, the averaged stress–strain curve has a sharp discontinuity at yielding and is smooth everywhere else. This discontinuity is a signature of a nonequilibrium first-order transition, as confirmed by the growth of the associated susceptibilities, the so-called “connected” susceptibility χcon=−d⟨σ⟩dγ and “disconnected” susceptibility χdis=N(⟨σ2⟩−⟨σ⟩2). The peaks of the susceptibilities become sharper and their amplitude, χconpeak and χdispeak, increases with N with exponents expected for a first-order transition in the presence of quenched disorder, as discussed below. This is illustrated for χdis in Fig. 2E, and we find that χdispeak∼N (Inset) and χconpeak∼N (see SI Appendix) at large N.

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Fig. 2.

From ductile to brittle behavior. (A) The yielding regimes in the simulation of a sheared glass for different degrees of annealing. Shown is stress σ as a function of the strain γ for several preparation temperatures Tini. For each Tini, three independent samples are shown. (B and C) Snapshots of nonaffine displacements between γ=0 and yielding at γ=0.13 for Tini=0.120 (B) and at γ=0.119 for Tini=0.062 (C). (D and E) Evidence of a first-order yielding transition for well-annealed glasses. Shown is a system-size dependence of the averaged stress–strain curve for Tini=0.062, showing a sharper stress drop for larger N (D). The associated susceptibility, χdis=N(⟨σ2⟩−⟨σ⟩2), becomes sharper as N increases (E). (Inset) The divergence of the maximum of χdis is proportional to N shown with the straight line.

The similarity between the mean-field theory in Fig. 1 and the data in Fig. 2A is patent. In agreement with the mean-field theory, we indeed find two distinct types of yielding: a discontinuous one for well-annealed glasses, which is associated with a first-order transition that becomes weaker as the degree of annealing decreases, and a continuous one, corresponding to a smooth crossover, for poorly annealed materials. As discussed in the next section, we also find a critical point at Tini,c≈0.095 that marks the limit between the two regimes.

In addition, the simulations give direct real-space insight into the nature of yielding. We illustrate the prominent difference between the two yielding regimes in the snapshots of nonaffine displacements measured at yielding in Fig. 2 B and C (see SI Appendix for corresponding movies). For a smooth yielding, we find in Fig. 2B that the nonaffine displacements gradually fill the box as γ increases, and concomitantly the stress displays an overshoot, as recently explored (56, 57). For the discontinuous case, the sharp stress drop corresponds to the sudden emergence of a system spanning shear band. By contrast with earlier work on shear-banding materials (58, 59), the shear band in Fig. 2C appears suddenly in a single infinitesimal strain increment and does not result from the accumulation of many stress drops at large deformation. For an intermediate regime between the discontinuous and continuous yielding (Tini≈0.1), strong sample-to-sample fluctuations are observed. Some samples show a sharp discontinuous yielding with a conspicuous shear band (similar to Fig. 2C), whereas other samples show smooth yielding with rather homogeneous deformation (similar to Fig. 2B). Such large sample-to-sample fluctuations are typical for systems with random critical points.

The Random Critical Point

Having identified a regime where yielding takes place through a first-order discontinuity and a regime where it is a smooth crossover, we now provide quantitative support for the existence of a critical point separating them, as one would indeed expect on general grounds. The mean-field theory presented above supports this scenario and suggests that the critical point is in the universality class of an Ising model in a random field. This criticality should not be confused with the marginality predicted to be present in sheared amorphous solids irrespective of the degree of annealing and of the value of the strain (29, 50). This issue is discussed separately below and in SI Appendix.

As shown in Fig. 1, the order parameter distinguishing the two regimes of yielding is the macroscopic stress drop. In the simulations, we measure its evolution by recording for each sample the maximum stress drop Δσmax observed in the strain window γ∈[0,0.3]. We have measured the mean value ⟨Δσmax⟩ as a function of the preparation temperature Tini for several system sizes N. At the largest temperature, no macroscopic stress drop exists: ⟨Δσmax⟩ simply reflects stress drops along the plastic branch and vanishes as ⟨Δσmax⟩|Tini=0.2∼N−0.4, as shown in Fig. 3A, Inset. In the main panel of Fig. 3A, we subtract this trivial behavior from ⟨Δσmax⟩. We find that the maximum stress drop is zero above Tini≈0.1 and nonzero for lower temperatures. The system-size dependence confirms that this temperature evolution becomes crisp in the large-N limit, and we locate the critical point at Tini,c≈0.095. Complementary information is provided by studying the fluctuations of the maximum stress drop, which can be quantified through their variance N(⟨Δσmax2⟩−⟨Δσmax⟩2) [not to be confused with the disconnected susceptibility χdis=N(⟨σ2⟩−⟨σ⟩2)], shown in Fig. 3B. One finds that the variance goes through a maximum that increases with system size around Tini,c≈0.095.

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Fig. 3.

Evidence of a random critical point. Mean (A) and variance (B) of Δσmax versus the preparation temperature Tini for several system sizes N. In A, we plot ⟨Δσmax⟩*≡⟨Δσmax⟩−⟨Δσmax⟩|Tini=0.2 to subtract the trivial high-temperature dependence that vanishes at large N, as shown in Inset. The critical temperature Tini,c≈0.095 is determined from the vanishing of the order parameter ⟨Δσmax⟩* in A and the growth of the maximum of the susceptibility N(⟨Δσmax2⟩−⟨Δσmax⟩2) in B. (C) Parametric plot of the connected and disconnected susceptibilities for all system sizes and several preparation temperatures. The straight line corresponds to the scaling χdis∝χcon2, as found in the RFIM.

These results provide strong evidence of a critical point separating ductile from brittle behavior, with the mean stress drop ⟨Δσmax⟩ playing the role of an order parameter. Additional support comes from the study of the overlap function q introduced in ref. 24. We find that the finite-size analysis of q and of the overlap jump Δqmax at yielding follows the same pattern as σ and Δσmax qualitatively. This points toward a macroscopic discontinuity for well-annealed glasses and a mere crossover for poorly annealed cases (see SI Appendix). Contrary to what was found in ref. 60, we find that the first-order transition behavior terminates at a temperature Tini,c, above which a smooth stress overshoot is instead observed.

Our findings are also corroborated by the analysis of the criticality of the sheared glass. As previously shown (29, 61), an amorphous material quasi-statically sheared at zero temperature is marginal at all values of the strain. The physical reason is the presence of a pseudogap in the density of elementary excitation (50), which is characterized by a critical exponent θ>0. This criticality implies a scale-free distribution of avalanche sizes, and by carefully analyzing the stress-drop statistics, we have extracted the exponent θ as a function of γ and Tini. As shown in SI Appendix, we find that the discontinuous transition is associated with a discontinuous variation of θ and that large fluctuations of the stress drops associated with criticality generate a rapid change in θ versus γ with the presence of a large maximum for temperatures Tini close to the critical point.

Our data do not allow us to measure the critical exponents associated to the RFIM critical point in a robust way. Yet it is possible to obtain a strong indication that the critical point and the first-order transition are governed by the universality class of an Ising model in a random field. In this case indeed, the presence of quenched disorder leads to two distinct susceptibilities, χcon and χdis. A key signature of the presence of random-field disorder is that χdis∝χcon2. This scaling relation, which is exact in the mean-field limit, is valid in finite dimensions at the first-order transition and is also approximately verified by the conventional RFIM at the critical point (26). It indicates that disorder-induced sample-to-sample fluctuations provide the dominant source of fluctuations. By looking at the parametric plot of the maximum of χdis versus the maximum of χcon, which is shown in Fig. 3C for all system sizes and several preparation temperatures, one finds that the relation is indeed observed in our simulations, at least at and below a temperature Tini=0.100≳Tini,c and for large N.

Discussion and Conclusion

Our analysis shows that irrespective of the nature of the amorphous material, yielding can come with two qualitatively different types of behavior, corresponding either to a discontinuous transition or to a smooth crossover. The transition between these two regimes occurs at a random critical point related to the RFIM, which naturally explains the large sample-to-sample fluctuations observed in simulations. The type of yielding that a given material displays depends on its degree of annealing, a mechanism that differs dramatically from the processes at play in crystalline solids (62). Conceptually, increasing the annealing for a given particle interaction implies that the initial amorphous configuration is drawn from a deeper location of the glassy energy landscape, in which the local environments fluctuate less (lower disorder in the RFIM analogy). In practice, the degree of annealing can be tuned for some materials such as metallic and molecular glasses (48, 63⇓–65) but would be more difficult to vary for others like emulsions and wet foams. Our approach shows that given the particle size (for colloids), the preparation protocol (for emulsions), and the cooling rate (for metallic glasses), a given amorphous material must belong to either one of the two yielding regimes. We suggest that colloids with a well-chosen range of particle sizes could be used to experimentally probe the random critical point separating the two yielding regimes.

Our work is focused on the two possible yielding scenarios rather than on the stationary state reached at large deformation. In ductile glasses, one expects a stationary state independent of the initial condition as shear transformations are quickly healed so that plasticity can spread homogeneously. In the materials we dubbed brittle in the present work, large deformations would trigger cracks or shear bands that may remain well-localized in the sample (as we indeed find numerically). Our study does not allow us to study the propagation of the cracks themselves.

There are several directions worth further studies to extend our results. On the theoretical side it is important to introduce nonlocal elastic interactions mediated by an Eshelby-like kernel in the proposed framework of an effective random-field Ising theory, which could potentially yield anisotropic avalanches that are not described by the traditional RFIM. This is essential to describe the role of nonperturbative and non-mean field effects that have been argued to be important for the spinodal behavior of disordered finite-dimensional systems at zero temperature (49). These correspond physically to rare regions that are able to trigger the failure in the material and are related to the shear bands found in simulations. We present numerical evidence already supporting this scenario in SI Appendix (see also ref. 36). On the simulation side, it is interesting to study how the rheological setting affects the yielding scenario proposed in this work. Considering uniaxial tension or compression tests would be useful for a further detailed comparison between simulations and experiments. In addition, investigating the influence of a finite temperature and/or a finite strain rate on the simple situation studied here would also be a worthwhile extension. Finally, one would like to understand better how the evolution of ductility with the initial disorder impacts the deformation and failure of glasses at larger length scales and make a connection with studies of macroscopic fracture in glasses. Because controlling ductility in amorphous solids is desirable for practical applications (3, 66), our theoretical studies will hopefully lead to design-principle of more ductile glassy materials.