Role of fragility of the glass formers in the yielding transition under oscillatory shear

Domain: physics
Published: 2026-03-27
Editorial outcome: major_revision

Abstract
Amorphous materials, especially metallic glasses, are known for their exceptional mechanical properties, such as high yield strength and large yield strain. Understanding the microscopic mechanisms behind their failure, particularly the yielding transition, remains an active area of research. Previous studies have shown that yielding behavior depends on the initial age of the sample. Through extensive computer simulations, we demonstrate that this age dependence varies across different materials and is influenced by the specific characteristics of the initial glass former, particularly its fragility. Both strong and fragile glass formers exhibit similar yielding behaviour in poorly annealed conditions with a critical yield strain, γ that does not depend on the initial conditions. However, below a critical degree of annealing, the yield point increases significantly with further annealing for fragile glasses, while it remains relatively constant for strong glasses. The results are found to be universal across a wide variety of model glassy systems with varying fragility, including metallic glasses, molecular glasses, model granular glasses, and network-forming glasses like Silica. We rationalise these findings by introducing a modified mean-field elastoplastic model that explicitly incorporates the crucial role of changing energy barrier with increasing annealing in the yielding process. This simple model reproduces all the simulation results and provides critical insights into how energy barriers influence the physics of the yielding transition including the critical yield strain under oscillatory shear deformation.

Introduction
Understanding the mechanical behaviour of amorphous solids is crucial due to their enormous applications ranging from bulk to nanoscales [1–4] in our day-to-day life. Specifically, the study of amorphous solids under external shear deformation has drawn significant interest in designing better materials for future applications in recent years. Among these, a large number of studies focus on comprehending the response of amorphous solids to uniform shear deformation [5–15]. At small amplitude deformations, the response closely resembles that of an elastic solids; however, under large deformation, numerous plastic deformations occur, eventually leading to yielding and the system starts to flow. Although the nature of the yielding transition is a subject of debate [16–19], energy, or stress shows a discontinuity when driven by oscillatory shear deformation at the yielding transition [20–24].

Recent investigations using cyclic shear predominantly under athermal quasistatic (AQS) [25] conditions indicate that the yielding transition depends on the initial sample’s degree of annealing or inherent structure (IS) energy. In poorly annealed systems, stroboscopic energy decreases with increasing amplitude of deformation (γ ) until a critical yield strain amplitude (γ ), after which it increases again. Near yielding, the system undergoes an absorbing to diffusing transition: for smaller amplitudes below yielding, the system reaches a steady state called the absorbing state with an invariant stroboscopic configuration; for amplitudes larger than yielding, γ > γ , the steady states are diffusing states where the configuration keeps changing with the number of cycles. This scenario changes significantly if the systems are sufficiently annealed so that the initial energy is below the threshold energy, shown to be the energy at the Mode Coupling Transition temperature T [26, 27]. The system’s energy remains constant until it reaches the yielding amplitude γ at those annealing conditions, where it undergoes a sudden increase and reaches the diffusive states. Note that γ ≥ γ in all systems that are studied. A similar discontinuous jump in energy is observed under uniform shear as well [27, 28] when a large system size is considered. This effect of annealing on yielding is robust across various model systems [28–34], different driving protocols [35, 36], finite temperature, finite shear rate [37, 38], and different dimensions [37, 39–41].

For poorly annealed glasses, the stress vs. strain curve exhibits a monotonic trend, and the transition to a flowing state is smooth or gradual, particularly under uniform shear conditions. Under cyclic shear, the yielding point is more precisely defined. However, in both cases, material failure resembles ductile behavior for poorly annealed glass. As the degree of annealing increases, a large discontinuous stress drop is observed at yielding in both uniform and cyclic shear, indicating an abrupt and catastrophic transition, characteristic of brittle behavior. Observing two distinct yielding behaviors, solely by adjusting the degree of annealing, which correlates with the temperature at which the liquid is prepared, raises questions about the existence of a potential phase transition. To answer these questions, various coarse-grained models have been introduced [28, 30, 42–48] which successfully reproduce the observed behavior in simulation and also attempt to answer the existence of a possible non-equilibrium phase transition belonging to the driven Random Field Ising Model (RFIM) universality class [47, 49]. Another recent work using random particle pinning showed strong evidence of such a non-equilibrium phase transition from brittle to ductile states at a finite critical disorder [49]. These results suggest a renewed interest in the community in understanding the nature of the yielding transition in amorphous solids. While many of these studies support the notion of a phase transition distinguishing between brittle and ductile yielding, the question of its persistence in the thermodynamic limit remains a topic of ongoing debate.

All these studies on yielding described above primarily focused on the initial energy of the system, providing limited insights into the nature of the initial glass former, notably its fragility, which characterises how rapidly the viscosity of the glass former increases as it approaches the putative glass transition point. Fragility is a crucial parameter for glass-forming liquids as it connects to various dynamical and structural properties such as dynamical heterogeneity, configurational entropy, violation of the Stokes-Einstein relation, glass-forming ability and so on [50–56]. In [27], the authors present the oscillatory shear responses of two model glasses: Silica, a well-known strong glass that forms networks, and the Kob-Andersen Model, which is recognised as a fragile glass former. One can clearly see that the oscillatory shear yielding diagrams for these two models are very different, notably for well-annealed states below the critical energy. The yield strain increases significantly with increasing annealing for the Kob-Andersen model (≈ 46.6%), whereas for silica, the increase is modest (≈4.3%). The results inspired us to study the fundamental role of fragility in the yielding transition under oscillatory shear.

In this article, we investigate whether yielding behavior is similar among different glass formers with varying degrees of fragility. Through extensive computer simulations, we demonstrate that the nature of the yielding transition can vary depending on the fragility of the initial glass former. We show that our observation is universal across a wide variety of glassy systems with a large difference in their fragilities. This includes metallic glass, molecular glass, model granular glass, and network-forming glass such as silica glass. Details of the models and simulation protocols are provided in the Methods section and the Supplementary Information (SI). Additionally, we introduce a mean-field elastoplastic model that incorporates the critical role of energy barrier in the yielding transition, successfully reproducing the simulation results. The analysis of this model allows us to rationalise the observed changes in yielding behaviour, attributing them to changes in energy barriers with the annealing history.

Results
First, we show results for the binary harmonic sphere model in 3D (referred to as 3D HP) with changing density, and then we discuss results from other models. In Fig.1A, we show the Angell plot to highlight the large variation of fragility for the 3D HP model with densities by plotting the relaxation time, τ against scaled temperature (T /T). We obtain the Angell plot by defining the calorimetric glass transition temperature T such that τ (T ) = 107 in appropriate reduced unit. Subsequently, we fit this relaxation time using the Vogel Fulcher Tamannam (VFT) formula:

τ =τ exp [ 1 / (K (T/T -1) ) ] (1)

where T denotes the temperature at which the relaxation time might diverge upon extrapolation and K represents the kinetic fragility, the rate at which the relaxation time diverges at T . In the inset of Fig.1A, we show K as a function of density. Our observations corroborate with the previous studies that fragility in this soft sphere glass model increases significantly with increasing density [55, 57, 58]. Specifically, the system demonstrates behavior reminiscent of a strong liquid at a density ρ = 0.681 with K ≃ 0.034. Note that the zero temperature jamming density for this model is ρ ≃ 0.654−0.674 depending on the protocol and initial configurations [59–61]. However, as the density increases, the system becomes more fragile. At the highest studied density (ρ=0.943), K ≃0.549. Thus, the change in fragility is around 16, corresponding to a density change by a factor of 1.38. We then perform cyclic shear on these four glasses at various degrees of annealing as represented by the parent temperatures (T ). Parent temperature refers to the temperature at which the liquid is equilibrated before quenching to zero temperature to prepare the amorphous solid samples. For each chosen cyclic shear amplitude (γ ), we perform a couple of hundred to a thousand cycles of shear deformation (depending on the fragility) until the energy reaches a steady state value E . Using these extensive cyclic shear simulations, we obtain the yielding diagrams of the systems by plotting E as a function of γ for all the four densities and the studied parent temperatures.

Variations of energy with cycles are shown in the SI. Fig.1(B), (C), (D), and (E) show the yielding diagrams for densities ρ = 0.681, 0.750, 0.855, and 0.943, respectively, for 3D HP model with large fragility change. In poorly annealed systems, all glass formers behave similarly: the steady-state energy decreases as γ increases until reaching the critical yielding point (γ ), after which E rises again. For all poorly annealed systems (T > T ), the systems yield at the same point. We call γ critical yield strain because this strain demarcates the difference between the poorly annealed and well-annealed samples. However, stark differences emerge among different glass formers in the well-annealed regime. Although E remains unchanged with increasing γ , the yielding point, γ , increases significantly with increasing annealing for fragile liquids, whereas it remains nearly the same for strong liquids. The yielding transition for strong liquid (Fig. 1(B)) occurs more or less at the same point irrespective of the degree of annealing, whereas for the most fragile liquid (Fig. 1(E)), the yielding point increases very rapidly with increasing annealing for temperatures T < T . To highlight the changes in the yield strain qualitatively, we estimated the percentage changes in the yield point γ with respect to the critical yield point γ , and for ρ = 0.681, the percentage change in yield point is around 15% whereas, for the fragile glass formers, the changes are as large as 35%. These results provide important insights into observations reported in previous studies, which we discuss below.

In Fig.1(F), we illustrate γ /γ as a function of T /T , where γ denotes the yielding point of the system at the corresponding T and γ denotes the yielding point of poorly annealed glasses for all T > T . The yielding point does not change for different glasses until T ; however, below T , it increases rapidly with increasing fragility, whereas for the strong glasses, it remains almost constant. The inset of the same figure panel shows that changes in stress drop, σmax, across the yielding transition normalised by the steady state stress, σmax. It is clear that for fragile glasses, the relative magnitude of stress drop at the yielding transition is much larger than for strong glass formers, and it becomes even larger with increasing annealing.

To further elucidate the yielding transition under cyclic shear, in Fig.1(G)-(J), we show the maximum steady state stress σmax as a function of γ for different parent temperatures. We first discuss ρ = 0.750,0.855 and 0.943 cases before discussing the ρ=0.681 case. We see very interesting behaviour for ρ = 0.681 which seems to be linked to an nearby unjamming transition at ρ ≃ 0.645. We discuss this case in detail in the subsequent paragraph. In the case of one of the strong glasses (Fig. 1(H)), we observe that for various parent temperatures, σmax increases as γ increases until it reaches the yield point and subsequently stress drops to a lower value. Interestingly, the yield points appear to be similar across different parent temperatures for strong glass. This stress-strain behaviour not only reaffirms that the yielding point does not change with the degree of annealing but also suggests that strong glasses remain ductile even in the most annealed samples.

However, the stress-strain behaviour changes significantly for fragile glass (Fig. 1(J)). With increasing annealing, the yielding point increases very rapidly and becomes brittle for well-annealed glasses with sharp stress drops at the yielding strain. Thus, this result, combined with the yielding diagram described above, reveals that the nature of yielding is highly dependent on the fragility of the glass formers. We have seen similar phenomena in our recent investigation, where we have tuned the fragility by pinning a fraction of particles chosen randomly in the 3D KA model [62]. As particle pinning can be challenging to implement in experiments, particularly with molecular glass formers, we have investigated a pinning-like effect using heavy mass particles as soft pinning sites. We found that increasing the mass of the soft-pinned particles leads to a decrease in fragility, similar to the effects observed with traditional particle pinning. By conducting finite-rate oscillatory shear simulations for this model, we demonstrated that γ increases more rapidly with respect to γ in fragile glass formers compared to strong ones. These findings suggest that this phenomenon is observed in different driving conditions and is very likely to be universal in nature.

In the 3D HP model, fragility increases with density or packing fraction. One could attempt to understand this increase by following the argument of [63], which suggests that the softness of the repulsive part of the interatomic potential dictates a system’s fragility. At higher densities, particles experience the steep repulsive part of the potential, leading to more fragile behavior. The jamming-unjamming transition, which this model exhibits at zero temperature, could provide an alternative explanation of changes in fragility. For the model system studied here the unjamming-to-jamming transition occurs a range of densities ( ρ ≃ 0.654−0.674) depending on the protocol and initial configurations [59–61]. One could argue that the origin of fragility is closely related to this underlying jamming transition, suggesting that the strong case at low density (ρ = 0.681) may be significantly influenced by this transition. Previous study [64] has shown that for a repulsive model system with a large polydispersity (around 23%), some low-density states near the jamming density can undergo an unjamming transition due to annealing under simple shear. It is important to note that with high polydispersity, the jamming density or packing fraction also increases to larger values.

In our binary model system, we are working at a density slightly greater than the largest jamming density for this model [61]. However, upon closer examination of the case where ρ = 0.681, we observe some anomalies. For instance, the curves of σmax versus γ for various parent temperatures do not resemble those observed at higher densities as shown in Fig.1(G). Stress for small amplitudes is significantly lower in well-annealed glasses compared to poorly annealed ones. This contrasts with higher densities, where the difference in stress for small amplitudes is negligible. Despite this, the shear modulus increases systematically with higher annealing temperatures or lower parent temperatures. Additionally, the oscillatory shear results clearly indicate that even after multiple cycles of annealing, the system does not undergo an unjamming transition at low temperatures, as evidenced by the non-zero inherent structure energy. The noticeable reduction in stress for a smaller amplitude may be related to the behavior associated with annealing-induced unjamming [64]. Future research exploring the impact of this unjamming transition on the oscillatory shear yielding diagram would be a fascinating direction to pursue.

Nonetheless, to understand whether our observation of a direct relation between fragility and the yielding transition is universal or not, we have studied four more different model glassy systems with varying fragility. These models are (1) the BKS (van Beest-Kramer-van Santen) Silica (SiO ) model [65, 66], (2) the ortho-terphenyl (OTP) model [67–70], (3) the 3D Kob-Andersen model (3D KA) [71, 72], and (4) the Copper-Zirconium (Cu-Zr) metallic glass model [73]. We have performed extensive molecular dynamics simulations of all four models (see the Methods section for the model details). In Fig.2(A), we show the Angell plot of all these models to highlight the significant changes in fragility across these models. Note that all these models are fundamentally different in their microscopic details; for example, BKS Silica is a network-forming glass, whereas OTP is a molecular glass made of three bonded monomers. The Kob-Andersen (KA) model is a model of metallic glass to model Ni P interacting via the simple Lenard-Jones potential. The Copper-Zirconium (Cu-Zr) binary metallic glass is a well-studied model of metallic glass former interacting via an Embedded Atom Model (EAM) potential [73]. The overall change in the fragility from Silica to Cu-Zr glass is around 5 times, which is similar to the shift in fragility between ρ = 0.750 and ρ = 0.943 in the 3D HP model. If the effect of fragility is universal in nature, then one would expect to see similar relative changes in the yield strain relative to their poorly annealed critical yield strain with increasing degree of annealing.

In Fig. 2(B), we show the yielding diagram for the steady state energy of the BKS Silica model. This is the strongest (lowest fragility) glass-former out of the four models. The critical energy scale for this model is T ≃ 3000K, which is close to the MCT transition temperature, and the shift in the yield strain below this temperature is very small. Note that the absolute value of the yield strain for Silica is γ ≃ 0.165, which is much larger than the other models studied. We reiterate that we are not making a direct comparison between the absolute value of the yield strain and the fragility parameter, but rather we are making a direct connection between the shift in yield strain from its critical value γ and fragility. Although, as suggested in [10, 15, 74], there is a possibility of a strong correlation between the yield strain and the fragility, which we have discussed later.

In Fig. 2(C-E), we show a similar yielding diagram for OTP (C), 3D KA model (D) and Cu-Zr metallic glass (E). One can clearly see that the yield strain increases significantly with increasing annealing for OTP, 3D KA and metallic glass models. Fig. 2(F) shows a normalised plot of changes in yield strain (γ /γ ) with respect to the scaled parent temperature (T /T ) for all four models. This figure clearly demonstrates that the largest fragile liquid (Cu-Zr glass in this case) shows the largest change in relative yield strain compared to the less fragile liquid (Silica glass) at a comparable degree of annealing. This result is in complete agreement with the results obtained for the 3D HP model. Thus, we can conclude that fragility solely controls the shift in the yield strain in well-annealed glass, irrespective of the origin of fragility in these model systems. This establishes beyond doubt that our results are universally valid across a wide range of glass-forming systems, irrespective of their microscopic details and the origin of their kinetic fragility.

In Fig. 2(G-J), we present the stress-strain curves (σ vs. γ ) for the Silica (G), OTP (H), 3D KA (I) models, and Cu-Zr metallic glass (J). It is evident that, within the studied temperature range, Silica glass exhibits less brittle behaviour than the Cu-Zr metallic glass model. In the OTP, 3D KA and Cu-Zr models, we observe a significantly larger stress drop, the magnitude of which increases with higher annealing, indicating brittle yielding via sharp shear band formation. The responses of these model systems under simple shear are detailed in the SI. In Fig. 3(A), we have presented the shift in γ for all the models studied in this work and additional results for randomly pinned systems as well as systems with soft pinning using heavy masses (from [62]).

To establish a quantitative correlation between shift in γ with fragility, we have computed the relative shift (γ − γ )/γ at a fixed degree of annealing by setting T = 0.75T for all the model systems and then plotted them against obtained kinetic fragility, K in Fig.3(B). We had to extrapolate the shift in yield strain for the most fragile case (ρ = 0.943 for 3D HP model and Cu-Zr model) to get the change at 0.75T . Across a wide range of model glassy systems, including granular glass, molecular glasses, network glass, and metallic glasses, we observe a strong correlation between (γ − γ )/γ and K . This clearly establishes the universal effect of fragility on the yielding transition in amorphous solids. Note that in some of the fragile cases, the relative change in the yield strain could be very large. In the inset of the same figure panel, we show the correlation between (γ − γ )/γ and K for T /T =1.07 such that we do not have to do any extrapolation. This also shows a strong correlation between the shift in the yield strain relative to the critical yield strain and increasing fragility.

Methods
Model and Simulation details: We have investigated a number of model systems which qualitatively represent granular, colloidal, molecular, metallic and network-forming glasses. We have given brief details of these models below. First model is a binary mixture (50 : 50) of particles that interact via a Harmonic potential given by [55, 91]:

V (r) = ϵ /2 * (1 - r/σ )^2, r ≤σ
= 0, r >σ

where α, β ∈ (A, B) indicates the type of particle. The two types of particles differ in their size, with σ = 1.4σ (and the diameters are additive), with the interaction strengths the same for all pairs (ϵ = ϵ = ϵ ). We have performed simulations at different packing fractions, ϕ, which are larger than ϕ as well as ϕ [92]. The relation between number density ρ and packing fraction ϕ for binary mixture is,

ϕ = ρ * π/2 * (c * σ^3 + c * σ^3) / Γ(1+ 3/2)

where ρ = N/V, N is the number of particles, V is the simulation box’s volume, and c = c = 1/2. We have performed NVT (constant particle number N, volume V, temperature T) Molecular Dynamics (MD) simulation using Nose-Hover thermostat in cubic box maintaining periodic boundary condition for N = 5000 particles at four packing fractions ϕ = [0.667,0.735,0.837,0.925] and the corresponding number densities (from Eq. 9) ρ=[0.681,0.750,0.855,0.943]. The time step of molecular dynamics is chosen dt=0.01.

Next model is the well-known van Beest-Kramer-van Santen (BKS) model [65] for Silica (SiO ) as implemented in [66]. We prepared the samples via NVT molecular dynamics at density ρ = 2.8 gm/cm3 with system size N = 4800 ions and integration time step dt = 1fs. The interaction potential is given by,

V(r )= q q / (4πϵ r ) + A exp(-B r ) + C r^-6 + ϕ(r )

where, ϕ(r )=4ϵ ( (σ /r )^30 - (σ /r )^6 ). The potential cut-off is at r = 10˚A and the Ewald parameter α=2.5˚A^-1. To study the shear band formation we simulated N =48000 at T =2700K.

Third model that we have simulated is the model of ortho-terphenyl (OTP) molecule [67–69] which is a well-known fragile glass-former. OTP is a rigid molecule with a three-site planar isosceles triangle structure, where each site represents a whole phenyl ring. The bond length of the two sides of the triangle is σ = 4.83˚A and the bond angle is 75◦. The interaction potential,

V(r)=4ϵ ( (σ/r)^12 - (σ/r)^6 ) + λ 1 + λ 2 r

where ϵ = 5.276 kJ/mol, λ = 0.461 kJ/mol, λ = -0.313kJ/(mol.nm), the interaction cutoff r = 12.616˚A as used by Mossa et al in [70]. In addition, in our model, the three sites within a single molecule interacts via a stiff harmonic potential,

V bond (r,θ)= 1/2 k(r -r )^2 + 1/2 k(θ -θ )^2

where for rigid bond, r = 4.83˚A, θ = 75◦ and k = 104 kcal.mol^-1.˚A^-2. We have prepared samples via NVT MD simulation with system size N = 4800 atoms, i.e. N =1600 molecules at density ρ=1.135 gm/cm3 with integration time step dt=5fs.

Fourth model is the well-known model of metallic glass Cu Zr interacting via embedded atom model (EAM) potential as in [73] via NPT (constant particle number N, pressure P, temperature T) MD simulation at zero pressure with system size N =5000 and integration time step dt = 1fs. We have also performed NVT MD for this model at density 7.40 gm/cm3. To study the shear band formation we also simulated N = 50000 at T =750K.

Finally, we have simulated the well-known Kob-Andersen binary mixture (80:20) in three dimensions (3D) [71, 72]. The interaction potential is given by

V (r)=4ϵ ( (σ /r)^12 - (σ /r)^6 ) + C + C * (r/σ)^2

where α,β ∈ A,B; interaction strength of pairs ϵ /ϵ = 1.5,ϵ /ϵ = 0.5; and diameters σ /σ = 0.8,σ /σ = 0.88. The interaction cut-off is at r = 2.5σ and number density ρ = 1.2. We have prepared samples via NVT MD simulation with system size N = 5000 and the integration time step dt=0.005.

Initial states: For each density, we equilibrated the liquids at a wide range of temperatures, including temperatures well below the T (Mode Coupling Temperature) to study the effect of annealing on yielding. Temperature range, in the liquid regime, is chosen such that 10^7 for all densities. τ is defined as the time when the two-point correlation function becomes q(t=τ )=1/e (the details are shown in the SI). We have used the conjugate gradient (CG) method on these equilibrated configurations to obtain the energy-minimized or inherent structure configurations (IS configurations).

Oscillatory Shear Protocol: We performed cyclic shear on IS configurations in xz direction at different strain amplitude γ via the athermal quasistatic (AQS) shear protocol where (i) each particle is displaced by affine transformation (r ′ → r + r * dγ, r y ′ → r y , r z ′ → r z ) and then (ii) energy minimization using the CG method. The strain γ varies as (0 → γ → -γ → 0) indicating a complete cycle, where elementary strain increment dγ = 2*10^-4. We prepared 12 independent samples at each temperature and density. See SI for further details.

Data availability: All data generated and analyzed for this study are included in the manuscript and its Supplementary Information. The source data underlying the figures are publicly available on Figshare at https://doi.org/10.6084/m9.figshare.31146514 [93].

Code availability: All simulations were performed using the open-source molecular dynamics package LAMMPS. Codes are publicly available on Figshare at https://doi.org/10.6084/m9.figshare.31146514 [93].

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "see attached report.", "reviewer_2": "I have read this manuscript with great interest. The authors present careful simulations studies of model glasses under shear deformation and highlight an important role played by fragility on the yielding behavior. The methods of investigation appear to be scientifically sound and technically valid and the results presented in the paper have the potential to be of great relevance for a broad scientific audience of physicists, materials scientists, engineers and chemists. Before making a final recommendation on this paper, which I very much enjoyed reading, I would like the authors to address the critical points listed below.\n- I think that the manuscript could be substantially improved and brought to the level required for Nature Communications, if the authors were able to obtain a relationship linking fragility to the yield strain. The authors mention a lot of papers on yielding and mesoscopic models, but ignore the papers which would be most useful to them in digging out this crucial relationship. For example, in recent work Nature Physics volume 19, pages694–699 (2023) it has been shown that a universal relation across a hundred different materials exists according to which the fragility m is proportional to the glass transition temperature Tg (divided by thermal expansion alpha). In turn, the Tg is proportional to the energy barrier for plasticity, and the yield strain decreases upon increasing the Tg, according to Eq. 4 in PHYSICAL REVIEW B 90, 140203(R) (2014). See also the caption of Fig. 5 in J. Phys.: Condens. Matter 32 395402 (2020) for a similar relation. Hence, the authors should try to put together the dependence of Tg on yield strain and the dependence of Tg on fragility in order to arrive at a relationship between yield strain and fragility, in the framework of the above cited model supported by their simulations data.\nIn this context, I find the mesoscopic elastoplastic model used by the authors much less revealing and less useful for achieving this important, crucial relationship.\n- in parts of the introduction the authors mention that increasing density leads to more fragile glasses, and fit their data with the old VFT model, which is just an empirical fitting. They also cite simulations work in the literature. They ignore that the microscopic mechanistic framework to explain that fragility increases with density is provided by the KSZ model, see PNAS 112 (45) 13762-13767 (2015), and Fig. 5 therein. The underlying rationale is that increasing the density leads to a steeper local energy landscape, whereby the particles are pushed more closely against each other, which, in turn leads to more fragile behaviour. Hence, these facts should be more mechanistically interpreted along this line.\n- not only the density or the pressure can change the fragility, but, as a matter of fact, also the shear deformation itself, as demonstrated in Nature Communications 6, Article number: 7179 (2015).\n- the authors mention \"metallic glasses\" prominently in the abstract, but never again in the manuscript. The reader thus gets the impression from the abstract that the focus of the paper is on metallic glasses, but actually all simulations refer to a coarse-grained model of soft repulsive jammed particles."}, "author_response": "Response to Referee Comments\nNCOMMS-25-08478-T\n“Role of Fragility of the Glass Formers in the\nYielding Transition under Oscillatory Shear”\nRoni Chatterjee, Monoj Adhikari, Smarajit Karmakar\nWe thank all the referees for their comments/questions on our manuscript. We have ad-\ndressed the referees’ questions below, and appropriate changes are also made in the revised\nmanuscript. We hope that the referees will now find the responses satisfactory and recommend\nour manuscript for publication.\nIn what follows, we have reproduced the referee comments (in full) in black/normal text\nand our responses blue/boldface text.\nI have read with interest the manuscript by Chatterjee et al. entitled ‘Role of Fragility of the\nGlass Formers in the Yielding Transition under Oscillatory Shear’. This work aims at relating\nmodelglass-formingsystems’fragility—namelythedegreetowhichglasses’parentsupercooled\nliquids feature Arrhenius or non-Arrhenius temperature dependence of relaxation times — to\nthose glasses’ elastoplastic mechanics under cyclic deformation. The main observations/results\nare:\n1) Some quantitative differences are seen between the ‘fragile’ and ‘strong’ models. In par-\nticular, the authors show that (i) ‘fragile’ system features a larger annealing-induced variability\nin the strain at which the steady-state energy jumps discontinuously between absorbing and\ndiffusing states, (ii) a larger annealing-induced variability in the stress drop seen for the max-\nimal stress, and (iii) a broader shear band (by 60%) between the ‘fragile’ and ‘strong’ models\nabove the yield point.\n2) An extension of the simple, non-interacting elastoplastic model by Sastry (PRL 2021) that\nincludes some modeling of fragility by introducing a function 〈∆E〉(E ) that is supposed\n0\nto capture the dependence of energy barriers ∆E on the minimum energies E . This extension\n0\ngives good agreement with the MD simulational results.\n3) A density-independent (but apparently non-universal, see Fig. 2f in [PRL 134, 018202\n(2025)]) relation between the timescale τ to approach steady state, and the number N\ntransient\nof plastic events before reaching the steady state.\nI find the manuscript to contain interesting observations, while I have reservations about\nits foundational scientific premise and associated degree of impact/interest. I describe my\nreservations below, followed by more minor comments that I hope will help the authors improve\nthe presentation.\nsented in this manuscript. We also appreciate the referee’s careful and critical\nassessment of our manuscript. The referee has raised some concerns and also made\ninsightful comments. We have addressed them satisfactorily, and we hope that\nnow the referee will not have any reservations about our work’s scientific premise.\nWe also hope that the referee will agree that this is probably the first of its kind\nwork, which shows a universal relation linking the dynamical and mechanical prop-\nerties of glass-forming liquids. We have now added results from 4 more different\nglass-forming models with significant variability in their respective fragility. These\nnew results confirm the universal relation between the fragility of glass-forming\nliquids and their mechanical responses under oscillatory shear. Thus, we believe\nthat we have shown that the fragility of a glass-forming liquid has a direct link to\nits mechanical response under oscillatory shear via the evolution of a characteristic\nenergy barrier with annealing.\nMy first concern has to do with the origin of fragility or lack thereof in the studied soft-\nsphere system and its relation to what is commonly understood as ‘strong’ behavior of the\ndynamics of (network) glass-forming systems. In essence, the authors study a model system\nwhose inherent states lie close to the ‘unjamming’ point at which solidity is lost. The authors\nreproduce previous results (Berthier & Witten 2009) showing that soft spheres near unjamming\nfeature ‘strong’ glassy dynamics, see Fig. 1a in the manuscript. More recent work [Nat. Comm.\n10, 5102 (2019)] (not cited) shows the following results:\nI note that while these data are generated for the WCA model, at low packing fractions\nthis model behaves just like harmonic soft spheres, i.e. exactly like the model studied in the\nconsidered manuscript. I wish to focus the readers’ attention to the open blue circles pointed at\nby the brown arrows; these are states that undergo a thermal-annealing-induced unjamming.\nThis means that these are equilibrium states that, when quenched to their underlying inherent\nstates, theirenergyisidenticallyzero(seecaption). Thesedataclearlyshowthat, inadditionto\nthegenericincreasingstabilityofglass-formerswiththermalannealing, softspheresthermalized\nat constant volume are also subject to a competing effect of annealing-induced softening, that\naccompanies unjamming. It seems obvious that this annealing-induced softening can have\ndramatic implications on the elastoplastic mechanics of underlying glassy states, and might\nexplain the observations reported in the present manuscript.\nagree with the referee that WCA model and Harmonic sphere model will have\nsimilar properties at low temperatures especially at AQS limit and higher density\nor packing fractions. But the figure highlighted by the referee is for a system\nwith very large degree of polydispersity ( ∼ 23%) and it is well-known that with\nincreasing polydispersity the jamming packing fraction ϕ will increase to much\nJ\nlarger value than the the jamming packing fraction for a binary system. For the\nbinary system (same as our model) a previous study (PNAS 117, 10203 (2020))\ndemonstrates that the system remains in a jammed state for densities exceeding\n0.674(ϕ = 0.661) even with cyclic shear deformation. The lowest density in our study\nis above this threshold. The referee also pointed out that at lower packing fraction\nthere is annealing induced softening and some of the inherent structure (IS) energy\nat lower packing fraction is identically zero indicating unjaming transition. We\nwant to point out that we are working at a density which is close to but above\nthe jamming density for binary soft sphere systems and when we obtain their IS\nenergy they never go to zero in fact even after the cyclic shear deformation their\nsteady state energy remain finite non-zero values, indicating that at the studied\ndensity the system is definitely not go through annealing induced unjamming or\nsoftening. Our data for the ρ = 0.681 seems to show hardening behaviour rather\nthan softening behaviour as obvious from the slope of the σxz vs γ plot shown\nmax max\nin the SI. However, we also acknowledge that the ρ = 0.681 data has some other\nanomalies, but our conclusion that changes in fragility have a strong bearing on\nthe oscillatory shear yielding diagram does not depend crucially on this data set.\nNote that we are not claiming anything about the relation between fragility and\nthe absolute value of the yield strain which might have no correlation with the\nchanges in fragility. We are claiming that changes in yield strain for well-annealed\nsamples are much larger for fragile glass-formers as compared to the strong glass-\nformers. This change is relative to the critical yield strain of the respective systems.\nMoreover, in the revised manuscript, we added a new result that showed that\ncritical yield strain is strongly correlated with the high-temperature energy barrier\nof the system and the dependence is found to be logarithmic in nature. All the\ndata from our simulations follow this relation, and it will be interesting in the near\nfuture to see whether experimental data also obey this relation. We are working in\nthis direction and hope to publish the results in a separate publication. We have\nadded clarifying discussions in the revised manuscript to remove any ambiguity.\nThe relation between the behavior of canonical ‘strong’ glass formers (such that the BKS\nmodel studied by Sastry in [PNAS 118 e2100227118 (2021)], ref. [25] in the manuscript)\nand the seemingly ‘strong’ behavior of soft spheres near unjamming, remains an interesting\nopen question, which, as far as I understand, has not yet been addressed in the glass physics\nliterature. What is clear is that the elastoplastic behavior of network forming ‘strong’ glasses\nand marginally-stable soft sphere packings near unjamming is very different, as evident e.g.\nby the very different yield strains seen between the two systems – in PNAS 2021 Sastry finds\nyield strains well above 20% for the BKS model, while soft spheres near unjamming feature\nyield strains of merely about 6.5% (see Fig. 1b). In addition, the trend observed is that\n3\nthe yield strain decreases with decreasing fragility in soft sphere packings, which is the\nexact opposite behavior of what is observed in [PNAS 118 e2100227118 (2021)], there the\nstronger system has a much larger yield strain. Considering all these different observations,\nit remains unclear whether there is any causal relation between the suppressed fragility and the nature\nof cyclic shear mechanics of soft spheres near unjamming. Moreover, it seems plausible that\nboth phenomena stem from the same thermal-annealing-induced unjamming discussed above. I\nbelieve the authors must absolutely clarify this point in a decisive manner IF they wish to argue\nfor a causal relation between fragility of parent liquids and cyclic shear elastoplastic mechanics.\nAs discussed in the previous section, we are claiming that the changes in the\nyield strain (γ ) with increasing annealing for well-annealed glasses with respect\nY\nto their poorly annealed universal value γ , i.e. ∆γ = γY−γc is larger for fragile\nc Y γc\nglass-formers as compared to strong glass-formers with similar degree of annealing.\nNote that the γ itself might not have any correlation with changes in fragility as\nc\nit depends on the value of the energy barrier near the yielding amplitude rather\nthan how the barrier itself changes with increasing annealing. For fragile glass-\nformers, it is expected that the energy barrier grows substantially with decreasing\ntemperature or increasing annealing, but its absolute value seems to depend on\nthe high-temperature energy barrier, ∆E , as shown in the revised manuscript.\n∞\nThe large value of yield strain in silica is indeed related to its large ∆E (see SM).\n∞\nThus, the large yield strain for BKS silica, as reported by Bhaumik et. al. in\nPNAS 118 e2100227118 (2021), and the smaller yield strain for soft sphere models\nin this work or in other works, do not contradict our results.\nAnnealing-induced unjamming and subsequent softening of the material do not\nseem to play any role here, as we observe that cyclic shear leads to an increase in\nshear modulus rather than a decrease in the modulus, so cyclic shear undoubtedly\nincreases the rigidity of the material universally, at least in the studied density\nwindow for the binary Harmonic sphere model. All our energy vs oscillatory shear\ncycle clearly shows that the steady state energy is not zero after the annealing.\nThus, we hope the referee will agree that annealing-induced unjamming or soft-\nening may not explain the results reported in this work. Moreover, we have now\nperformed oscillatory shear simulations of 4 different models with varying fragility\n(Fig 2 in the revised manuscript). These models are the BKS Silica model (a strong\nglass-former), the model OTP, the Kob-Andersen model (a fragile glass-former)\nand Cu-Zr Metallic glass (also a fragile glass-former within the temperature range\nstudied). The origin of fragility in these models is very different and does not have\nany connection with jamming physics. In these systems, we have shown in the\nrevised manuscript that ∆γ increases more rapidly with increasing annealing in\nY\nfragile glass-formers than in strong glass-formers. This confirms that the results\nobtained using the Harmonic sphere model are universally true for other generic\nmodels of glassy systems and, irrespective of the origin of fragility in these solids,\nthe increase in yield strain in well-annealed glasses with increasing degree of an-\nnealing is causally related to how their energy barrier grows with annealing and\nthereby their fragility.\nhowever the framing of the results in the context of the fragility of parent supercooled liquids is\nnot convincing. I would suggest instead that the authors interpret their observations in light of\nthe results of [Nat. Comm. 10,5102 (2019)], while highlighting the differences with respect to\nthe results of [PNAS 118 e2100227118 (2021)], and discussing at length the interesting observed\ncorrelations with the (suppression of) fragility of the parent supercooled liquids.\nWe thank the referee for acknowledging that the results reported in this\nmanuscript are interesting, and we hope that we have clarified the primary con-\ncerns of the referee related to the causal relation between the yielding diagram of\nan amorphous solid under oscillatory shear deformation with its fragility parame-\nter, which is a property that governs their relaxation process in the liquid regime.\nWe added a clarifying discussion in the revised manuscript along these lines of\nargument.\nMy second concern has to do with the impact of the results and insights they provide; in\nlight of the results of [Nat. Comm. 10, 5102 (2019)] the weak dependence of cyclic deformation\nelastoplastic mechanics on annealing in soft spheres near unjamming is not very surprising. In\naddition, the elasto-plastic model studied is an extension of a previously introduced one, where\nthe observed behavior is also not surprising given that the different dependencies of the barriers\non E0 are baked by hand into the extended model. Finally, the broadening of shear bands with\nproximity to the unjamming point is also expected, see e.g. [Phys. Rev. E 98, 060901 (2018)]\nwhere it is shown that the size of soft localized defects that carry plastic deformation increases\nwith proximity to unjamming. To summarize, putting aside that the authors overlooked the\npossible effects of thermal-annealing-induced unjamming discussed above, I do not find the\nresults presented to provide significant new insights into the yielding behavior of glassy solids\nunder cyclic deformation.\nWe hope that we have clarified all the doubts in the previous paragraphs al-\nready. However, we want to highlight that a direct relation with ∆γ with the\nY\nenergy barrier is the main result of this work, which is corroborated via a single-\nsite elastoplastic model, and we think this is significant discovery as such a strong\nconnection between liquid properties and mechanical properties have not been\ndemonstrated in the past although people conjectured about possible correlations\nbetween strong-fragile crossover with brittle-ductile transition without any micro-\nscopic understanding. This work, for the first time, demonstrates that one can\nstudy the oscillatory shear response of the solid and get a reliable estimate of how\nthe energy barrier changes with increasing annealing in these systems, which can\nthen be directly used to understand the Arrhenius or Super-Arrhenius dynami-\ncal behaviour of the system or vice versa. We have demonstrated in the revised\nmanuscript how, from the information of relaxation time vs temperature data,\none can predict the shift in yield strain under oscillatory shear for all the studied\n5\nmodels using a simple mean-field single-site elastoplastic model. Moreover, our\nelasotplastic model calculations pointed out a very interesting relation between\nthe high-temperature energy barrier and the critical yield strain. We found that\ncritical yield strain can be changed by changing the high-temperature energy bar-\nrier in the model, and it follows nearly linear dependence. We then plotted the\ncritical yield strain from the simulations and their respective high-temperature en-\nergy barrier for all the models (nearly 12 of them) with varying microscopic details,\nfragility etc and found that indeed there is a strong correlation between critical\nyield strain and the high-temperature energy barrier. The dependence is found\nto be logarithmic rather than linear as predicted by the elastoplastic model. This\ndifference is expected as the model is mean-field in nature. It will be exciting to\nsee the results when spatial information is added in the elastoplastic model, which\nwe are pursing greatly. Thus, we believe this is a very significant understanding\nin the field of glassy systems, especially in understanding their yielding transition,\nwhich plays an integral part in designing future smart materials for applications.\nWe hope that the referee will not have much objection now to recommending this\nmanuscript for publication in Nature Communications.\nBelow I list additional comments/questions for the authors to consider.\n1) It seems to me that the order in which results are presented is not optimal; I believe the\npresentation would benefit from concatenating the “Structure of Shear Bands” section and the\n“Microscopic Understanding of Diverging Timescales” section with the first part of the paper,\nand push the “Elastoplastic Mesomodel Results” section to the end of the manuscript (or to\nthe Supplemental Material). I find the current order confusing because the storyline jumps\nfrom the atomistic model to the elastoplastic model and back.\nWethanktherefereeforthissuggestion. Wehavenowreorganisedthemanuscript\nto take care of this point of the referee. We have added new results, both from\nsimulations and the elastoplastic model, and tried our best to make the storyline\nflow smoothly for ease of reading.\n2) For reasons not explained, the authors present results for ρ = 0.75 as a ‘strong’ repre-\nsentative in Figs. 1G,3,4, but Fig. 1A shows that their ‘strongest’ model is, in fact, ρ = 0.681.\nWhy do the authors not show data for ρ = 0.681 in Figs. 1G,3,4 as well?\nNow, we have show the results for ρ = 0.681 in the SI. At this density, which is\nclose to the jamming density, the 3dHP model shows non-standard results and we\nbelieve there might be an interesting link between jamming physics and oscillatory\nshear response of these solids near jamming density, as correctly guessed by the\n6\nreferee. We are taking up further studies to understand this physics better. These\nresults, however, are not crucial for the main conclusion of this manuscript. We\nhave now added these results to SI and also added a discussion in the main article.\n3) The authors take an exponent of 2/3 instead of 1/2 in Eq. (6); does this change the\nqualitative behavior of the elastoplastic model?\nWe have used an exponent of 2/3 instead of 1/2 because it is well-known in the\nliterature that energy barrier changes as ∆E ∼ δγ2/3 as predicted by a saddle-node\nbifurcation close to any plastic instability. While this changes the numerical value\nof the yield point slightly, it does not affect the underlying physics or qualitative\nbehaviour of the system at the mean-field level. We believe a fully developed elasto-\nplastic model with spatial information will have interesting implications of these\nchanges in the exponent. We are in the process of developing such an elastoplastic\nmodel where the energy barrier plays a crucial role, and we try to understand\nwhether the observation in our simulation can be better compared with the elasto-\nplastic model, including the shear band formation and its width.\n4) It would be useful if the authors could present uniform-shear stress-strain curves for the\nvarious packings in the Supplemental Information, in particular the low-density ones.\nWe have shown results related to uniform shear for all densities of the 3dHP\nmodel, as well as for all other models studied in the SI.\n5) It would be very interesting to see NpT dynamics instead of NVT dynamics of the soft\nspheres. Does the dynamics remain ‘strong’ if the NpT ensemble is employed?\nWe have now performed NPT dynamics of the soft sphere model and showed\nthat with increasing pressure, the fragility changes in the same manner as the\ndensity change in the NVT ensemble. We have also included oscillatory shear data\nfor Cu-Zr metallic glass, where the NPT ensemble is used for the simulation, along\nwith NVT for the same system, to understand the effect of different ensembles.\nThe results again reconfirm the universal relation between fragility and the shift\n7\nin yield strain with annealing.\n6) It would be useful if the authors reported on the elastic properties and packing fractions\nof their packings, then readers can assess how close they are to the unjamming point.\nWe have mentioned the packing fraction as well as the number density in the\nmethod sections of the main manuscript and in the SI.\n7) The authors write “Thus, this result, combined with the yielding diagram described\nabove, reveals that the nature of yielding is highly dependent on the fragility of the glass\nformers”. In addition they write “We found that fragile glass formers show a large increase in\nyielding strain with increasing annealing, whereas strong glass formers show almost no change in\ntheir yielding strain with better annealing.” These choices of phrasing do not indicate that\nthese differences in yielding between strong and fragile glass formers have already been shown\nby Sastry PNAS 2021.\nWe thank the referee for raising this point. It is true that Bhaumik et. al.\nin PNAS 2021, reported results for both Kob-Andersen model (fragile liquid) and\nBKS Silica model (strong liquid) but whether fragility has any role to play in the\nunderlying physics was completely missing and their elastoplastic modelling also\nsupport this argument as they have assumed that stability limit of any elastic\nbranch depends on absolute value of the energy E rather than energy barrier ∆E.\none can now rationalise why KA and Silica show different yielding responses, but it\nwas not known before. Thus, we believe that just the existence of raw data in the\nliterature does not make our results less novel, as the microscopic understanding\nin those previous studies is missing.\n8) Are the timescales plotted in Fig. 1a made dimensionless? If not, what is the meaning\nof comparing between timescale at different densities? How do the authors establish that the\ndegree of supercooling of the model at different densities is comparable?\nWe have now non-dimensionalized all our data to enable comparison across\ndifferent model systems and densities. In particular, we compare our results using\n8\nthe scaled temperature (T /T ).\nMCT P\n9) The authors write “Our findings are particularly noteworthy because they reveal how\nthe kinetic properties of liquids can strongly influence the mechanical properties of amorphous\nsolids, providing new insights into the behavior of these materials”, but no citations are\nprovided. Have no similar conclusions been made in the glass literature?\nAs far as we know, this is not reported in the literature. People often tried\nto make a correlation plot between fragility and various other dynamical proper-\nties like stretch exponential exponent, calorimetric glass transition temperature\nT . There are some studies where a correlation plot between fragility and brittle-\ng\nductile transition is made, but our results make a microscopic connection between\nfragility and changes in yield strain in well-annealed samples. We also made an\ninteresting observation of a strong correlation between the critical yield strain and\nthe high-temperature energy barrier of the respective systems. We don’t have\nany microscopic understanding apart from the elastoplastic model, also showing\nsimilar correlation at this point in time, but we believe this will have important\nimplications in the near future in completely understanding the yielding transition\nof amorphous solids. We have added discussion and necessary references in the\nrevised manuscript.\n10) The authors write “...we observe that strong glass formers take significantly more shear\ncycles to reach an absorbing state when the strain amplitude is below their yield strain, while\nfragile glass formers reach their absorbing state much faster at the same strain amplitude.” Is\nthe same true for BKS vs. LJ?\nWe have now added results from our new simulations for BKS and LJ models\nand showed that generically the stronger glass-formers take significantly larger\nnumber of cycles to reach steady state than the fragile glass-formers.\n11) The authors write “... a number of oscillatory cycles are needed to reach the absorbing\nsteady state, which universally depends on the number of plastic events the system has to\ngo through before reaching the absorbing state...”, but also that ”This observation is very\nsimilar to the observation made in [63] for Kob-Andersen glass former, although with a different\nexponent”; they might want to be more accurate in their phrasing.\nthe number of oscillatory cycles, τ, follows a power-law relation with the number of\nplastic events, but the exponent might be non-universal. Although with our new\nresults from nearly six different model systems, we see a power-law which is well\ndescribed by a universal power-law relation (with some scatter) with an exponent\nclose to 0.6. We have added these new results and added relevant discussion.\nI have read this manuscript with great interest. The authors present careful simulations studies\nof model glasses under shear deformation and highlight an important role played by fragility\non the yielding behavior. The methods of investigation appear to be scientifically sound and\ntechnicallyvalidandtheresultspresentedinthepaperhavethepotentialtobeofgreatrelevance\nfor a broad scientific audience of physicists, materials scientists, engineers and chemists. Before\nmaking a final recommendation on this paper, which I very much enjoyed reading, I would like\nthe authors to address the critical points listed below.\nWe thank the referee for appreciating the importance of this work and its great\nrelevance for a broad scientific audience across disciplines. We are also very pleased\nto know that the referee enjoyed reading the manuscript. We have addressed all\nthe critical comments of the referee carefully, and we have now added additional\nsimulation results from 4 more different models of glassy systems with varying\nfragility to demonstrate the universal nature of our findings. We believe the referee\nwill now enthusiastically recommend this manuscript for publication in Nature\nCommunications.\n1) I think that the manuscript could be substantially improved and brought to the level\nrequired for Nature Communications, if the authors were able to obtain a relationship linking\nfragility to the yield strain. The authors mention a lot of papers on yielding and mesoscopic\nmodels, but ignore the papers which would be most useful to them in digging out this crucial\nrelationship. For example, in recent work Nature Physics volume 19, pages694–699 (2023) it\nhas been shown that a universal relation across a hundred different materials exists according to\nwhich the fragility m is proportional to the glass transition temperature T (divided by thermal\ng\nexpansion alpha). In turn, the T is proportional to the energy barrier for plasticity, and the\ng\nyield strain decreases upon increasing the T , according to Eq. 4 in PHYSICAL REVIEW B\ng\n90, 140203(R) (2014). See also the caption of Fig. 5 in J. Phys.: Condens. Matter 32 395402 (2020)\nfor a similar relation. Hence, the authors should try to put together the dependence\nof T on yield strain and the dependence of T on fragility in order to arrive at a relationship\ng g\nbetween yield strain and fragility, in the framework of the above cited model supported by their\n10\nsimulations data. In this context, I find the mesoscopic elastoplastic model used by the authors\nmuch less revealing and less useful for achieving this important, crucial relationship.\nWe thank the referee for bringing these references to our notice. The results\nreported in these studies helped us answer some of the questions raised by the first\nreferee on how fragility relates to the yield strain. To highlight, our primary goal\nin this work is to show a direct connection between how the yield strain in the well-\nannealed sample changes with respect to its universal poorly annealed value with\nincreasing annealing, with the growth of the energy barrier. As the referee pointed\nout, the energy barrier is also directly related to the glass transition temperature\nT and the yield strain is directly related to the absolute energy barrier and thus\ng\nT . Also, the referee pointed out the reference which links T to fragility, thus\ng g\none might be able to make a direct link between the fragility parameter m and\nthe critical yield strain γ . We have added these new results to the main article,\nc\nalong with a discussion and citations of relevant studies. Although we do not\nfind a direct relation between the yielding point and kinetic fragility, possibly\ndue to the missing parameter, such as the coefficient of thermal expansion, we\ndo observe a connection between the energy barrier and the yielding point of\npoorly annealed glasses. In the revised version, we demonstrate this relation for\nboth the 3dHP model and the new models we study, and we find a logarithmic\ndependence between the two. We also found a power-law relation between ∆E\n∞\nand T as ∆E ∼ T0.65, leading to another power-law dependence between γ and\ng ∞ g c\nT as γ ∼ T0.35. Regarding the elastoplastic model, we have done the calculation\ng c g\nto show that just simple information of how the energy barrier changes with the\ndegree of annealing can directly predict how the yielding diagram of the same\nmaterial under oscillatory shear in its solid state will be. We have now added new\nresults that directly show that if one extracts the parent temperature dependence\nof the energy barrier from the relaxation time data and then uses this information\nin the elastoplastic model, then one can quantitatively find the shift in the yield\nstrain with respect to the critical yield strain. We showed this comparison for\nboth 3dKA and Silica models. The comparison is remarkably accurate. Thus, our\nstudy demonstrates that one needs to incorporate energy barrier information for\ndeveloping a good elastoplastic model for understanding plasticity at a large length\nscale using computer simulations. This has its own importance for future materials\nmodelling and studying their failure under external loading.\n2) in parts of the introduction the authors mention that increasing density leads to more\nfragile glasses, and fit their data with the old VFT model, which is just an empirical fitting.\nThey also cite simulations work in the literature. They ignore that the microscopic mechanistic\nframework to explain that fragility increases with density is provided by the KSZ model, see\nPNAS 112 (45) 13762-13767 (2015), and Fig. 5 therein. The underlying rationale is that\nincreasing the density leads to a steeper local energy landscape, whereby the particles are\npushed more closely against each other, which, in turn leads to more fragile behaviour. Hence,\nthese facts should be more mechanistically interpreted along this line.\nadded a discussion on the origin of fragility with changing density with citing these\nimportant work in the revised manuscript. This, in part, will also answer the\nquestions raised by the first referee, who argued that the origin of fragility in soft\nsphere models is ill-understood, and the jamming-to-unjamming transition has an\nimportant role to play in the changes in fragility with increasing density. Although\nour results do not directly depend on this matter, it is important to highlight the\npossible origin of fragility in model systems using existing theoretical frameworks\nlike KSZ theory, as suggested by the referee. We also reconfirmed our results on\nfour different model systems - BKS Silica, OTP, Cu-Zr metallic glass and Kob-\nAndersen model. Note that fragility in these models changes significantly in the\nstudied parameter ranges.\n3) not only the density or the pressure can change the fragility, but, as a matter of fact, also\nthe shear deformation itself, as demonstrated in Nature Communications 6, Article number:\n7179 (2015).\nWe have added a discussion to highlight these important facts in our revised\nmanuscript.\n4) the authors mention ”metallic glasses” prominently in the abstract, but never again in\nthe manuscript. The reader thus gets the impression from the abstract that the focus of the\npaper is on metallic glasses, but actually all simulations refer to a coarse-grained model of soft\nrepulsive jammed particles.\nWe apologize for this oversight in the first version of the manuscript. We now\nhave results from Cu-Zr Metallic glasses done using both NpT and NVT ensembles,\nand these new results are included in the revised manuscript. We also showed\nresults from 3D KA a model to describe metallic glass. We have also edited some\nof these texts to highlight the robustness of our observation across model systems.\n12"}, {"round": 2, "reviewer_comments": {"reviewer_1": "see attached report", "reviewer_2": "The authors have done a really serious work in revising the manuscript extensively to account\nfor all the queries brought forward by the reviewers. The manuscript has much improved\nand now meets the criteria for publication in Nature Comms. I am happy to recommend its\npublication as is."}, "author_response": "Response to Referee Comments\nNCOMMS-25-08478A\n“Role of Fragility of the Glass Formers in the\nYielding Transition under Oscillatory Shear”\nRoni Chatterjee, Monoj Adhikari, Smarajit Karmakar\nWe thank all the referees for their comments/questions on our manuscript. We have ad-\ndressed the referees’ questions below, and appropriate changes have also been made in the\nrevised manuscript. We hope that the referees will now find the responses satisfactory and\nrecommend our manuscript for publication.\nIn what follows, we have reproduced the referee comments (in full) in black/normal text\nand our responses blue/boldface text.\nI have read carefully the revised manuscript and the authors’ rebuttal. In my opinion, the\nrevised version still suffers from major problems. I append my comments below.\nWe now believe that we have addressed all the concerns of the referee very satis-\nfactorily and we hope that the referee will not have any hesitation in recommending\nthis manuscript for publication in Nature Communications.\n1)Theobservationthatthethevariabilityofyieldstraininoscillatoryshearchangesbetween\nstrong and fragile glass formers has been made by Sastry et al. in [PNAS 118 e2100227118\n(2021)] (ref. 27 in the revised manuscript). In their rebuttal, the authors assert that “... the\nexistence of raw data in the literature does not make our results less novel”. I disagree with\nthis assertion. Clearly, the choice of contrasting the KABLJ model and the BKS model in\nPNAS 2021 is not coincidental – this choice was made exactly because of the stark differences\nbetween these models, in particular in their fragility. I suggest that the authors acknowledge\nin their text that the effect they study has been clearly observed in previous work, and avoid\nstatements that could mislead readers about the novelty of their findings.\nOne of us (SK) had an in-depth discussion with Prof. Sastry about this matter\nprior to starting this work. Our lack of understanding of the potential role of\n1\nfragility in the yielding diagram motivated us to pursue this research. However,\nwe agree with the referee that the PNAS paper by Sastry et al. contains the\nrelevant data, and considering the significant difference in fragility between these\ntwo models, it is reasonable to expect that fragility plays an important role in this\ncontext.\nWe would like to emphasize to the referee that the observations in the afore-\nmentioned PNAS paper were instrumental to our work, and we have clearly stated\nthis. We have now revised our text to highlight that the differences in the yield di-\nagrams of silica and the 3DKA model have already been demonstrated in Sastry et\nal.’s PNAS paper. We have no intention of overstating the novelty of our findings,\nand we assure the referee that we are not making any exaggerated claims.\nAdditionally, we have revised the manuscript to remove the statement that\nsuggested uncertainty in concluding from Sastry’s PNAS paper that fragility plays\na crucial role in the yielding diagram, and that the observed differences cannot\nbe attributed solely to the varying microscopic details of the studied models. We\nhope that these revisions will address any concerns the referee may have and that\nthe manuscript will be recommended for publication in Nature Communications.\n2) I do not find the authors reply regarding the absence of annealing-induced unjamming\neffects to be compelling. In their rebuttal the authors write that “at the studied density the\nsystem is definitely not go through annealing induced unjamming or softening”. At the same\ntime, they report that their lowest density system “has some other anomalies”. Indeed, upon\nexamining Figs. S9, S12A, it is clear that annealing induced unjamming is present and has a\nmajor effect. How can the authors rule out that the same physics also shows up in the ρ = 0.750\nsystem? Additionally, the authors should absolutely not omit the σmax vs. γ data of the\nmax\nρ = 0.681 system from Fig. 1 just because it does not align with their narrative.\nWe would like to clarify our previous statement. By ”unjamming,” we intended\nto refer to the system reaching a state of zero energy, which is nearly within the\nbounds of machine precision. At the density we studied, the system does not\nachieve such a state. Regarding the issue of softening, a careful examination of\nthe data will reveal that the system’s shear modulus systematically increases at\nlower temperatures. Therefore, we can confidently assert that annealing-induced\nsoftening is not present in our findings.\nWe agree with the referee that the stress-strain curve at this density (ρ = 0.681)\nis indeed very different from those observed in higher density cases or other systems\nat elevated densities. We believe that as one approaches the jamming density from\nabove, there are other intriguing effects within the yielding diagram. We plan to\nexplore this observation further in our next project, and we thank the referee for\nthe suggestion.\nAdditionally, we strongly refute the referee’s claim that we chose not to include\nthis result simply because it does not align with our narrative. If that were the\ncase, we would not have mentioned it or included it in the supplementary materials.\nWe aimed to avoid excessive focus on this matter, as it slightly deviates from the\ncentral theme of the paper. We also want to emphasize that the deletion of this\n2\ndataset does not alter our main conclusions, which we have validated across a wide\nrange of model systems.\nNevertheless, we have included the data for this density in the main article as\nsuggested by the referee, and we have discussed the topics of annealing-induced\nunjamming and softening in the main text, along with the potential new effects\nthat may arise in the response of these solids as they approach the unjamming\ntransition under oscillatory shear deformation.\n3) I am confused about Fig. 3; the authors report (γ −γ )/γ of over 80% for ρ = 0.943.\nY c c\nBut from Fig. 1 and the accompanying text it is clear that the percentage is closer to 30%.\nWe thank the referee for pointing this out. As stated in the manuscript, “We\nhad to extrapolate the shift in yield strain for the most fragile case (ρ = 0.943\nfor 3dHP model and Cu-Zr model) to obtain the value at 0.75 T .” For these\nMCT\nsystems, the relaxation time becomes extremely large at 0.75 T , and it was not\nMCT\ncomputationally feasible to anneal the glasses to the required state point directly.\nTherefore, the value of (γ − γ )/γ reported in Fig. 3 for ρ = 0.943 was obtained\nY c c\nby a linear extrapolation of the trend from higher temperatures, rather than from\ndirect simulation data. We have now stressed this in the figure caption also. We\nalso made these extrapolated symbols different.\n4) Do the authors find that shear bands are more diffuse in the BKS system as well? They\nonly report shear banding measurements for the harmonic model. If the BKS model doesn’t\nshow more diffuse shear bands, it implies that the effect they see stems from the proximity to\nthe unjamming transition, and is not at all related to fragility. This point must be clarified, or\nthe entire section about shear banding should be removed.\nWe thank the referee for raising this point. We indeed observe that the BKS\nsilica system (a strong glass-former) develops a noticeably more diffuse shear band\ncompared to the metallic glass (a fragile glass-former). These results are shown in\nFig. 5 of the main text.\n5) Despite my comment from the first round, the authors do not refer to previous work\nshowing that the size of plastic defects grows near unjamming, which is consistent with their\nobservations of more diffuse shear bands in the lower density harmonic spheres system.\nthat, for the Harmonic sphere, this might be due to the growth in the size of plastic\ndefects as the system approaches the unjamming transition. On the other hand,\nwe now show results for Silica and Metallic glasses, and we see that for silica,\nthe shear band is more diffuse than for metallic glasses. Thus, we think it is related\nto the fragility, and that the microscopic mechanisms for diffuse shear bands in\nstrong glasses may differ from model to model. We have now added a discussion\non this aspect. Thank you once again for this comment and suggestion.\n6) I cannot understand Fig. 8 nor the conclusions deduced from it; how do the authors\nobtain/define T in Kelvin for the harmonic spheres or KABLJ computer models? What is the\ng\nmeaning of plotting a dimensionless γ vs. a non-dimensionless T ? The same question holds\nc g\nfor ∆E /k which is also non-dimensionless.\n∞ B\nWe thank the referee for this important clarification request. For the 3dKA\nmodel, the conversion from reduced Lennard-Jones units to physical units follows\nthe parametrization used for Ni P , as stated in the manuscript. We have cited\n80 20\nthe original source in the main text, where the mapping between real and reduced\nunits is described.\nTo ensure consistency, all quantities in Fig.8 have now been expressed in re\nduced, dimensionless units. Since γ is compared to the high-temperature activa\nc\ntion energy ∆E /k , we normalize ∆E /k by the onset temperature T . This\n∞ B ∞ B onset\nconversion allows a direct and meaningful comparison across different models.\n7) An entire discussion is held regarding the number of plastic drops that the system tra\nverses through to reach the absorbing state. The authors do not mention it, but they should\nclarify that this number must depend on the number of particles/molecules in their glass model,\nperhaps in a nontrivial way. It is actually unclear how to compare this number e.g. between\nthe OTP (molecular) model and harmonic spheres (indeed, in their data of Fig. 4, OTP has\nthe lowest number of events for most of the strains). The authors might want to discuss their\nfindings in the context of previous work [Soft Matter 10, 2728 (2014)] studying the size of elastic\nbranches near unjamming.\nWe thank the referee for this comment. Indeed, the number of plastic drops\nthat the system traverses through to reach an absorbing state will be a strong\nfunction of system size. In our previous paper (see M. Adhikari et. al. Phys. Rev.\nLett. 134, 018202 (2025)), we showed that the system obeys a universal relation\n4\nbetween the number of cycles needed to reach an absorbing state and the number\nof plastic drops traversed in this process. The relationship is a universal power-law\nrelation irrespective of system size. Thus, we believe that our comparison across\nmodels sytems with the same number of particles is meaningful and we could\neven put data for other system sizes as well without deviating from this universal\nbehaviour. We have now added a discussion in the revised manuscript and also\nadded the suggested reference.\n8) I suggest that the authors supplement their calculations with the Stillinger-Weber model,\nin par- ticular with large λ that sets the strength of the 3-body term. Does it align with the\nrest of their observations?\nWe appreciate the referee’s suggestion to supplement our calculations with the\nStillinger–Weber (SW) model at large values of the tetrahedrality parameter λ.\nHowever, we note that large λ values in the SW framework are known to intro\nduce several well-documented artefacts that make such comparisons potentially\nmisleading.\nFirst, increasing λ strongly over-stabilises the tetrahedral network, which drives\nthe melting temperature unrealistically high and leads to a liquid that is excessively\nordered and structurally rigid. Second, a large λ makes the tetrahedral crystalline\nphase overwhelmingly favoured.\nMoreover, with our current High Performance Computing capacity, it will take a\nsignificantly long time to explore these new systems. It already took us a significant\ntime to study shear band formation in silica with the equilibration done at the lower\ntemperatures and system size of N = 50,000 particles. Starting to do the Stillinger\nWeber model of Silicon would certainly take a few more months for sure. We hope\nthe referee will agree with us that additional results from the Stillinger-Weber\nmodel will probably not be very different, as we have already studied the BKS\nsilica network model and metallic glass, which interacts with a complicated multi\nbody EAM potential. Thus studying a specific model with 3-body potential would\nnot change the result qualitatively, otherwise it would have changed the results for\nmetallic glasses with EAM potential.\nTo summarize: the authors study an interesting correlation between fragility and the vari\nability of yield strain in oscillatory shear, which has been previously observed by others in\nPNAS 2021 and thus cannot be considered novel. This issue ought to be reflected in their text.\nInaddition, severalmajorissuesthatarediscussedinmycommentsabovemustbeconvincingly\naddressed for this work to be considered for publication.\ncorrelation between fragility and the nature of yielding, while still acknowledging\nthe contributions of Bhaumik et al. (PNAS 2021). As mentioned in our introduc\ntion, the PNAS 2021 study inspired our investigation into this phenomenon, and\nwe have cited it wherever relevant to our results. We have also cited PNAS 2021\nwherever it is relevant to our results. Following the referee’s suggestion, we have\nrevised the conclusion to make this point explicit.\nWe are disappointed to observe that the referee is attempting to diminish the\nsignificance and originality of our work by merely referencing the study published\nin PNAS in 2021. If the authors of that paper fully understood the role of fragility,\nwhy did they not address this aspect at all in their manuscript? A search for the\nterm ”fragile” reveals the phrase ”fragile to strong crossover” mentioned twice in\nthe main article and once in the title of a reference. However, the term ”fragility”\ndoes not appear anywhere in the text. Additionally, they did not modify their\nelastoplastic model to account for this concept. Regardless, I hold Srikanth in high\nregard as my close collaborator, and there is no reason not to acknowledge his\ncontributions in the field.\nWe are pleased to address all the suggestions made by the referee, and we\nhope they recognize the significance of this work. Referee B has acknowledged our\nthorough efforts in revising the manuscript and has recommended it for publication.\nWe believe we have sufficiently addressed all concerns raised by the referee and\ntrust that they will feel confident in recommending the manuscript for publication.\nThe authors have done a really serious work in revising the manuscript extensively to account\nfor all the queries brought forward by the reviewers. The manuscript has much improved\nand now meets the criteria for publication in Nature Comms. I am happy to recommend its\npublication as is.\nWe thank the referee for recommending the publication of our manuscript. We\nalso appreciate their feedback, which has helped us significantly improve the revised\nversion.\n6"}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have done a really serious work in revising the manuscript extensively to account for all the queries brought forward by the reviewers. The manuscript has much improved and now meets the criteria for publication in Nature Comms. I am happy to recommend its publication as is."}, "author_response": "Response to Referee Comments\n“Role of Fragility of the Glass Formers in the\nYielding Transition under Oscillatory Shear”\nRoni Chatterjee, Monoj Adhikari, Smarajit Karmakar\nWethanktherefereeforher/hiscomments/questionsonourmanuscript. Wehaveaddressed\nthe referee’s questions below and have made the appropriate changes in the revised manuscript.\nWe hope that the referees will now find the responses satisfactory, and there will not be any\nfurther delay in the publication of our manuscript in Nature Communications.\nIn what follows, we have reproduced the referee comments (in full) in black/normal text\nand our responses blue/boldface text.\nI have read carefully the 3rd revision manuscript and the authors’ rebuttal. My various points\nof criticism have been largely addressed (and see additional comments below). I am, however,\nstill not satisfied with the authors’ response to the novelty and credit issues. In the revised\nmanuscript the authors write: “In [27], the authors present the oscillatory shear responses of\ntwomodelglasses: Silica, awell-knownstrongglassthatformsnetworks, andtheKob-Andersen\nModel, which is recognised as a fragile glass former. The results reveal an interesting difference\nin the oscillatory shear yielding diagrams for these two model glasses”. I still find this disclosure\ninsufficient: theauthorsshoulddiscloseprecisely, indetail, whatwasthe‘interestingdifference’,\nand how those observations compare to their own findings.\nTheir misleading attempt to avoid full disclosure is unacceptable, as is the ‘disappointment’\nthe authors feel following my insistence on the integrity of their scientific conduct.\nWe have revised the main article to provide a detailed discussion of the key\nfindings from Sastry’s 2021 PNAS paper, explaining how their observations moti\nvated our study and how our results align with theirs. We believe this addresses\nthe referee’s concerns.\nRegarding the question of our scientific integrity, we wish to reiterate our ac\ncount of events as they occurred. Given the seriousness of this concern, we re\nspectfully invite the referee to verify our narrative directly with Prof. Sastry.\nIn Ref.[27], the authors investigated oscillatory shear responses in two distinct\nmodel glass formers—BKS silica and the Kob-Andersen Lennard-Jones mixture.\nThey showed qualitative differences in steady-state phase profiles (E vs. γ\nSS max\nacross different annealing conditions, also referred to here as ”yielding diagram”)\nunder oscillatory shear in these two models. However, no one-to-one comparison\n1\nwas performed, and no control parameter was identified that could unambiguously\nexplain the origin of this difference. In particular, although BKS silica is com\nmonly classified as a strong glass former and the Kob-Andersen model as a fragile\none, Ref.[27] did not establish fragility as the governing variable. Since the two\nsystems differ simultaneously in their interaction potentials, compositions, and mi\ncroscopic structures, the observed differences in steady-state phase profiles cannot\nbe uniquely attributed to fragility. The authors thus rightly refrained from draw\ning such a conclusion. In a one-to-one discussion between Prof. Sastry and Prof.\nKarmakar, Prof. Sastry emphasised that a detailed study is needed to determine\nwhether fragility alone plays a role or whether the microscopic differences between\nthe two models under study account for the observed differences in their yielding\ndiagrams. The current data in their PNAS 2021 paper is not enough to draw such\na conclusion.\nThe present work is designed specifically to address this unresolved issue. We\nhypothesise that the difference in steady-state phase profiles originates from differ\nences in glass fragility. From an energy-landscape perspective, the characteristic\nenergy barrier ∆E grows rapidly with annealing in fragile glass formers, whereas\nthis growth is much weaker in strong glass formers. Since yielding under oscillatory\nshear requires the system to overcome these barriers, a larger ∆E implies that a\nlarger strain amplitude is needed for yielding (∆E ∝ γ2).\nAccordingly, in the first part of this study, we employ a soft-sphere model\nin which the interaction potential and composition are held fixed, while fragility\nis tuned systematically via density. This enables a controlled comparison on an\nidentical microscopic footing. Within this framework, we demonstrate that changes\nin fragility lead to systematic variations in the steady-state yielding phase profile\nunder oscillatory shear.\nHaving established fragility as a relevant control parameter in a controlled model\nsystem, we then examine the generality of this connection by extending our analysis\nto four qualitatively different glass-forming systems, whose fragilities arise from\ndistinct physical mechanisms. We observe a consistent correlation between fragility\nand the steady-state yielding phase profile across a wide class of glass-forming\nsystems.\nOur results demonstrate a clear link between fragility and yielding behaviour\nunder oscillatory shear. In fragile glass formers, the rapid growth of energy barriers\nwith annealing leads to a pronounced increase in the yield strain. In contrast, for\nstrong glass formers the weak growth of energy barriers results in only a negligible\nchange in the yield strain. This distinction is accompanied by qualitative differ\nences in mechanical response: strong glass formers exhibit small stress drops at\nyielding, indicative of ductile behaviour with diffuse shear bands, whereas fragile\nglass formers display large stress drops, characteristic of brittle yielding with rela\ntively narrow shear bands. Our analysis further indicates that the relevant control\nparameter governing yielding is the energy barriers, rather than the mode-coupling\nthreshold energy alone.\nFurther comments:\n1) The authors aim to compare the shear band widths of silica and Cu-Zr. The compar\\isson in Fig.5(J) should be made with the z-axis made dimensionless (maybe it is – it is not\nmentioned).\nThe current data is in real units scaled by their respective particle size, but now\nwe have presented the data scaled by the simulation box.\n2) I did not understand the authors’ response to my comment re: the number of plastic drops that\nthe system traverses through to reach an absorbing state. The revised manuscript contains a confusing\nand vague statement: “Note that the number of plastic drops depends on the system sizes in a\npower-law manner, but it was shown in [74] that this power-law relationship is universally obeyed\nfor all system sizes as well”. The authors further state in their rebuttal that they “could even put\ndata for other system sizes as well without deviating from this universal behaviour” but this is not\nactually done.\nIn our previous work Ref.[74] (M Adhikari, R Sharma and S Karmakar, PRL\nPHYSICAL REVIEW LETTERS 134, 018202 (2025)\n134, 018202, 2025), we showed for 3dKA model itself that the universal relation\nPHYSICAL REVIEW LETTERS 134, 018202 (2025)\nbetween the time taken to reach an absorbing state under oscillatory shear and\nthe under of plastic drops encountered during the process hold equally well for all\nsystem sizes given that the number of plastic drops increases with system size in\na power-law manner. We have reattached the figure here in the rebuttal. As the\ndata for various system sizes are already presented in our previous PRL paper,\nwe did not replot them in this work; however, as the referee insists, we have now\nadded them to the revised figure. The referee will likely agree that our statement\nis correct.\nPHYSICAL REVIEW LETTERS 134, 018202 (2025)\nFFiIgGu.r2e. 1(:a)FThigeusyrsetempaisnteralisnetdawkiethnγfrom0M.06.AStdrohbiokscaorpiiceetn.eragly.,EPγRL01i3s4p,lo0tte1d8a2sa0f2u,n(c2ti0o2n5o)fNto sh.oTwhehstoeawdystate\nFIG.2. (a)Thesystemistrainedwithγ max ¼0.06.Stroboscopicenergy,Eðγ ¼0ÞisplottedasafunctionofNcyclces .Thesteadystate\nthfoerdnifufemrebntesrysotefmcsyizcelsesreanceheesdaetdanmtaioxnc ¼rreeaasicnhglyalasrtgeeranduymbsetraotfecy(cτl)e\nð\nsdw ¼eitpheÞ inncdresasoinngstyhsetemsyssizteesm. (b)siTzhceyecl(rceeNslax)at(iosneetimeto\np f r o e l a o r c t p t d t h eh e p r i a e f dn lt e f a o n h e e a c te r e r t su h g e e l s ya d n n - t t h A e t if a i e nu a v s s d ) n y s t e a h y t c s e r f f et t u a s o i s e so d nt a m r t a n y c e l tt a e t o i s s d r o t h i f i a y e z ns γ t e e l e s op a s it f n i a l 3 t s r oγ t i e d te t p oh i a tn K i e l e c n n o d h t t s h A a t e t a e b e e s c s d ae s y m a d at a c t e t s yl w af e o a ud a s n e a d ny t f nae c u e i s td n t n n e t l ia c E . o c f t r t n τ e o f i e i O t r f a o n i o at s s a f w r i n o o n n st o f wy b g d e s t s l d o a l y y t iN i e s d fn B m f l t i e e a e f d t d m f r rr r es g e s a o b ri n e h s z p e n y r t i ne s o z sp st n e w y . a s f s u sr y o W . m s at s W reb te b m e t o e e s m h o l r i y o s b c e s i b o s s z i f s e t f z ie n e t re est c r v u i s ,m v n y ,e a m e g c a s t l s t h o e s h b s sa f s ih a h e t z t t o w o h r e τ τ w w e i s ∼ ∼ t o n n E h a f γ N i N i n i n . n n d β β ( t t c d d , ,h h r r ) w ew e o e T γ a l h p l he he s m e eg s i ge r n re a e e e n g n v x n β u dβ . s d s m . i y . s i Δ b N s Δ s d e E t d e E i r , f i m ¼ f o f e ¼ f a f e r s E e d n r i E n e r z γ d o n t e γ − p t f s − o s P . fE r o i ( E a s f d b r it n p i f , ) d i f lt w f e i , T o e f l w h t f r h t e e e e e h C r n r d ee e t r n ar e E γ e s s t l m a γ h aE γ a x i x m o s fγ a . u a w t t i ( x n h i s c . o c e s ) t n ( th a i c Δ oe c) t E n i t a Δ m u co a i E e s tf l u t a i o s l\nenergsyysitnemthesizsetesaadtythsetastteeaindyasctyatcel.eTahnednEufmitbiseroobftadirnoepdsbalysopashraobwosliacpfiotwtinergloawft;hteheEsγl.o(pde)Ththeensuammbeefroroefvderroypγs.i(se)Plohtetetdotaaslanufmunbcetrioonf of\nsystedmrospisze(sNatrtanth)etostreeaadcyhtshtaetest.eTadhyesntuamtebiserpolofttderdopassaalfsuonschtioownsofaspyostwemerslaizwe,;Nthefosrlodpifefeirsetnhteγsam.e(f)foτriesvpelroyttγe.d(aes)aThfuentcotiaolnnoufmtobtealrof\ndropsnu(Nmbt\nd\nr\nr\nae\nonr\np\ndrtooprsetaocrheathchetshteeasdteyadsytastetatiesfpolroattleldγfaosradiffufenrcetnitoNn.oDfisfyfesrteenmtcsoilzoer,sNcorfroerspdoifrndetondtiγffmearxe.n(tfs)yτsteismpslioztetse.dWaessahfouwncdtaitoanfoorfatloltal\nnumbinedredpreonpdsetnotrseaamchpltehsetostedaedmyosntsattratefeorthaelluγnfiovrerdsiaflfenraetnutreNo.fDtihfefepreonwtecrollaowrsdceoprreensdpeonncde.Thdeifflienreenhtassyasteslmopseizeefs.aWrouensdho0w.5.dataforall\nindependent samples to demonstrate the universal nature of the power law dependence. The line has a slope of around 0.5.\nincreasing system size, the number of cycles required to a cycle. As shown in Fig. 2(c), a large system visits more\nFIG.2. (a)Thesystemistrainedwithγincrerae 0sa .i0 cnh 6g. t S hst eyross btteeo ams d c y ospis izt c aet e,e n ,tehτ reg , y inn, cuE rme γ abseers 0 aonfi d scp tyh lco elttef e isd narael sqe aun f eiurren gdc y ti ptooe n r ofNamciynciml.eT a.h tAh e ass n tseha ados ywm sna ta llitne syFsitgem. 2.(Ac)p,eartlaainrginegsqyusteesmtionviissitwshmatore\nmax cyclces\nfor different system sizes reaches at an inrcerae ¼ cahpsiantrhgtielcylseltaerdagedecyrrensautasmetesb.,eτW,oihfneccnyrcewlaes ð seewsp¼ ialtonhtd Þ itnhtchereeanfsuiinmnagblseeyrnsetoerfgmycsypiczelee3rss. (b)m hTa ihn pei p mreen alas txh iaf atit nohne ats siy mmset a e ltm los s y i s z t e em is . la A rg p e e e rt n a o i u n g in h g so qu th es a t t io it n h i a s s w to hat\nreachthesteadystateisplottedasafunctpioanrtoi r cfe l qseyusi dtree emd crestoiazser ese s.a .Wch Weot h hbe esnesrtvw eea e dthy palto sτt t a ∼ t t e hN, eβτ,, nww uhm eerb oee bβr seiosr f vdeicffy teh cra lee tnstitfordh v iaf i fp s e i pr t ee a nnt n sγ e\nm\ni a f\na\nr\nx\nl t. y h(e i c n )s f yΔ in sE i t t e e ims nu s m iz b e e i r s o l f ar m g i e ni e m n a o ; u c g a h n s i o ts t t h il a l t re it tr h a a ce s to\nplottedasafunctionofγinthesteadystaterefqoru gtiwr r e ood wdsitfo wfeirr tee hna ttc hsh eyss tty hesme te ssmitze sea isd z,e yasa s sstha aot p ewo ,nw τien, rtw lhaee wleo wgb iet snh edr a.v n Δe eEx th po at nEe iγ n t− t E fivt t,i hws e iht pea aret n hEea t γ o rilsy htahi v nei fan icn gtiut aaelc n l u os m ed be l r oo o p f ? minima;canit still retrace\nenergyinthesteadystateinacycleandE fgitrioswoβ sbatwasiinτ teh ∼dthbNye β sp, yawrsat i ebth molβisci cfz hietatin ang sgi a nog pfotshw yes eEt r e γ m l.a(a wdt)ic wTahi l t leh yna fur n ome mbxep 0ro .¼ o4 nf5 edn froot rpsistphleo I t n pte t a e dt r ha e s s t t oa in f g hu l an y vc , tiin t o h gn e oa n f u c m lo b s e e r d o l f oo pl p a ? stic drops in the steady\nsystemsizesatthesteadystate.Thenumb β er a o s fs τ mdr ∼ aolpl N sa βam , lsp w oli i st t hu h odw β eso c fa h γ a pmn oag wxi e n¼ r g la0 s w. y 0; s 2t t ,h e et m os a l0o t . i p9 c e0 al if l s y otrh f γe roms maaxm ¼0 e . f 4 0o 5 .r0e f 6 o vae r sryγ.( s e t ) a IT t n e thee a r l et s os o ttainl in gn c ul r ym e , a bt s eh e re s o w nfu it m h b s e y r st o e f m p s la iz s e tic as dr ∼ o N ps 0.6 in fo th r e st s ra te in ady\ndrops(Nt d r r a o n p )toreachthesteadystateisp s l m ot a te l s l dh a oa m swa p n l f i iu t n u nF d ct e iigo o .n f 2o( γ bf)s.yNsteomte 0 s . t 0 hiz 2 aet , ,t t Nh o ef 0 oe . sr 9 t 0 dimif f fa o et r reed γ ntyγie m l a d x .s 0 (trf . )a 0 i 6 τni a fso s prlotte s da ta ma t s e plai a tf l uu s d o nect i γi n omc narxe o a af s st e os s thalo w w it n h i s n y F st i e g m . 2( s d iz ). e T a h s e ∼ av N e 0 r . a 6 ge fo s r tr s a t i r n ain\nnumberdropstoreachthesteadystatefor\ns\na\nll\no\nγ\nt\nf\nh\no\nn\ni\nr\ns\ni\nd\nn\ns\ni\ny\nff\nF\ns\ne\nt\ni\nr\ne\ng\ne\nm\n. nt\n2\ni\nN\n(\n\ns\nb\n.\nγ\n)\nDY. i\nN\n≃m\nff\no\na\ne\n0x\nr\nt\n.\ne\ne\nn\nt\nt\nh\n.\nc\na\nT\no\nt\nh\nlo\nt\nu\nh\nr\ns\ns\ne\nc\nt\ne\no\nh\ns\nr\ni\nt\nr\ns\ni\ne\nm\no\nsp\nb\na\no\nt\ne\nn\ne\nr\nd\nd\nv\nt\na\ny\notmi\ni\nd\no\ne\na\ni\nl\nnx\nf\nd\nf\ni e¼m\ns\nr\nt\ne\np\nr\nn\na\nl\nt\ni\ni\ne\nn\ns\ns\ny\nf\ns\nt\no\nt\nh\ne\nr\na\nm\nt\nsiz\na\ne\nb\nm\ne\n.\nt\nW\np\nwl\ni\ne\ne\nt\nu\nsh\nn\nd\no\ne\nt\nw\nwo\nda\ns\nt\nu\na\nc\nf\na\no\nr\ns\na\ns\nl\nl\nv\no\ne\nw\np\nla\ni\ns\nn\ntic\nFi\nd\ngr\n.\nop\n2\ns\n(d\n).\ns\nT\nte\nh\na\ne\nav\nst\ne\nr\na\nt\na\ng\ne\ngo\ns\netr\ns\nain\nindependent samples to demonstrate the univ a er n sa in l f n in a i t t u e r l e y o la f rg th e e s p y o st w em er w la i w lln d e e v p e e r nd fi e n n d ce a . s T te h a e dy lin st e at h e a . s T a hi s s lopeaosf aΔroun∼dN0 m . − 5 2a . x=3inbothtwoandthree-dimensionalsystems\nthiss h y a s s te s m tro i n s g γ Yim≃pl 0 ic .0 at 7 io . n T s hu fo s r ,t e h n is co o d b i s n e g rv m a e ti m on or i y m i p n li t e h s e t r h m a o t - b f e o t r wh m ee o in le t c w u o lar su g c l c a e ss s - s f i o v r e m p e l r a s st [ i 5 c 5 d – r 6 o 4 p ] s . i I n f s w te e ad a y ss s u t m at e es th g e oes\nanin d f y in n i a te m ly ica la ll r y ge la s r y g s e te b m ulk wi s l y l s n te e m ve s r , f a i s nd th a e s t t i e m ad e y fo s r ta e te n . c T od h i i n s g assamΔeγsc∼alNin − g 2= b 3 eihnavbiootrhtfworotahnednthurmeeb-edrimofendsrioopnsalinsystehms\nh i\nincreasing system size, the number ohfascysdctirvloeensrggreesiqmauspiraleicpdaotwtiooenrslafwao.rcInyencSlcueop.dpAilnesgmsehmnotewamlnMoiranyteFirniiaglt.h[25e(2rcm]),,owa-elargfeaobrssyomsrtboeimlnegcvusiltsaairttesgmalasosarse-ffournmcteiorsn o[5f5s–y6s4te]m. Isfizweefoarsasugmiveenthe\nreach thesteady state,τ, increasesanddythneasmhfioincwaalltlehynatelaragrgsyiempbeiluralrkpsoymwseitner-milmaswa, atdhseaptnheneadtseimnmcaeellefsoxyrisstetsenmecvo.edAninfpgoerrtainssaitnmrgaeinqusacemastlpiilonitngudisbeewγhhavati,otrhefnorittihsenantuumrabletor eoxfpedcrtotphsatinthethe\nmax\nparticle decreases. When we plot thdeivneurpgamertsbiaealrsmaoefpmocworceylreelsanwco.dIinhnagSp.uppepnlsemifetnhtealsyMstaetmerisailz[e52is],lawrgee enaonbuusgmohrbbseiorngtohfasttpailttaeshtaaicss tdaorofpusncwtiiollngorfowsywstietmh ssyiszteemforsizaegaivsen\nrequired to reach the steady state, τs,hwowe toThboastdearevtseeirmmthiilnaaetrtiphtoewreearvs-oilsaniwtfoadrnetehpaeernilnydceirnneacfiseneietixneiτnstuwsmietbhveesrnysoftfoemrminims∼tarNa;ic2n=a3namwithpsiltciithluldirseetvrγea\nm\nrcy\na\ne\nx\nsitmheilnarittoisthneatoubrsaelrvtoatieoxnpewcet thhaavte.the\ngrowswiththesystemsizeasapowerplaawritisailzitemh,eawmneeolxroypookenneactnottdhiengn.tuhmebpearthoftoplahsatvicinegveantcslothseedsylostoepm? nOunmebmerighotfqpuleasstitoicntdhreovpaslidwitiyllogfrthoewscwaliitnhgasrygsutemmentsiaznedas\nβ as τ∼Nβ, with β changing systematicTaolelnydceoftreuornmmteir0ns.e4dt5uhrefionrgteashoenofsoIcnltelhareteosirntyicnrsgehlayes,aerthicneyτcnlwuesmi.thbItesryistotfwempelllastic∼thdNer2oe=xp3spwoinnheitcnhhte[i6ss5te]vaedarssythsiismisilatrruetofotrhethoebssteearvdaytisohneawr setahtea.ve.\nsmallamplitudeof γ 0.02, to0.9si0zefo,k rwn γ oewlnoothka 0 ta.t0 th 6the aes nnumumbsebtraetroefoafpllspaolsatisicnticcerveeeavnsetesnstiswnctihrteheassseyysssttweemmithsizeOHanosewm∼ evNige0rh,.6tfqifmuoeirlsatsristorsanciatnlhinegvawliitdhitaysolifgthhtelyscdaiflfienrgenatrgeuxmpoennetnatnd\nmax max\nshown in Fig. 2(b). Note ¼ that the estimenacteoi dunnc y rtiee e arlss d i¼nds gutrra siy inn sgte fo mthresizoesac[ mi5l4lpa] lt. iot U uryn d d esehr γeoasrcica llys act s oleh ryso.w sIh nteaiisr n dwF e i feg oll. r- 2(d)tha.leTsoehxehpoaolvndeesrnaftgoe[r6s5tth]reaaisnfitrhstispilsastrtiuceefoverntthestsatteisatdicyss[h5e4a]r. sItnate.\nmax\nmation, the stroboscopic configurations become the same Fig. 2(e), we observe that the total number of stress drops\nthissystemisγ ≃0.07.Thus,thisobskenrovwatnionthiamtpthlieesnthuamtber boeftwpleaesntictweovesnutcsceisnscirveeaspelasstwicitdhropsHinowsteevaedry,ssitmicalsargsoceaslingwithaslightlydifferentexponent\nY whenthesystemreachesasteadystate.However,between encountered during the approach to steady state also\naninfinitelylargesystemwillneverfinindcaresatesaindgyssytasttee.mThsiizse [54a]s. U Δ nγde∼r oNs−ci2l=l3ationrbyosthhetawrodaenfodr-three-adlsimoehnosilodnsaflosrysttheme sfirst plastic event statistics [54]. In\nthecycles,thesystemjumhpsfriomoneminimumtoanother, increaseswithsystemsize,againreinforcingthatencoding\nhas strong implications for encodingmmateimono,rytheinsttrhoebrmosoc-opicfcoornfmigoulreactuiolanrs gbleacsosm-foermtheerssa[m5e5–64]F.igIf. 2w(ee),awsseuombesetrhvee that the total number of stress drops\nandcorrespondingly,manyplasticdropscanbeobservedin memoryintolargesystemsismorechallenging.Moreover,\ndynamically large bulk systems, as thwehteinmtehefosrysetnecmodrienagchessaasmteeadsycasltiantge.Hbeohwavevioerr,fboertwtheeennumebnecrooufntedrreodpsdiunrinthgethe approach to steady state also\ndivergesasapowerlaw.InSupplementhtaelcMycalteesr,iathle[5s2]s,twemejumapbssforrobminognestmateiniamsuamfutoncatniootnheorf, systienmcresaiszeesfwoirthasgyisvteemn size,againreinforcingthatencoding\nshow that a similar power-law dependanendcceoerrxeissptsonedveinngfoormstrypaincamndplritpudle,γbothvnietd0i1n8202-3toexpcetthatthenumberofplasticdropswillgrowwithsystemsizeas\nmax\npartial memory encoding. number of plastic drops will grow with system size as\nTodeterminethereasonfortheincreaseinτwithsystem ∼N2=3 which is very similar to the observation we have.\n018202-3\nsize, we look at the number of plastic events the system Onemightquestionthevalidityofthescalingargumentand\nencounters during the oscillatory shear cycles. It is well the exponent [65] as this is true for the steady shear state.\nknown that the number of plastic events increases with However,similarscalingwithaslightlydifferentexponent\nincreasing system size [54]. Under oscillatory shear defor- also holds for the first plastic event statistics [54]. In\nmation, the stroboscopic configurations become the same Fig. 2(e), we observe that the total number of stress drops\nwhenthesystemreaches asteadystate.However, between encountered during the approach to steady state also\nthecycles,thesystemjumpsfromoneminimumtoanother, increaseswithsystemsize,againreinforcingthatencoding\nandcorrespondingly,manyplasticdropscanbeobservedin memoryintolargesystemsismorechallenging.Moreover,\n018202-3"}]

Decision letter