Abstract
The physical processes governing the onset of yield, where a material changes its shape permanently under external deformation, are not yet understood for amorphous solids that are intrinsically disordered. Here, using molecular dynamics simulations and meanfield theory, we show that at a critical strain amplitude the sizes of clusters of atoms undergoing cooperative rearrangements of displacements (avalanches) diverges. We compare this nonequilibrium critical behaviour to the prevailing concept of a ‘front depinning’ transition that has been used to describe steadystate avalanche behaviour in different materials. We explain why a depinninglike process can result in a transition from periodic to chaotic behaviour and why chaotic motion is not possible in pinned systems. These findings suggest that, at least for highly jammed amorphous systems, the irreversibility transition may be a side effect of depinning that occurs in systems where the disorder is not quenched.
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Introduction
Amorphous solids such as plastics, window glass and amorphous metals are an important and ubiquitous form of matter. Industrial processing of such materials commonly involves plastic deformation. An outstanding issue in the amorphous plasticity community is defining and understanding yield^{1}, the onset of irreversible behaviour, in terms of underlying universal processes. There is increasing evidence to suggest that the microscopic mechanism of plastic deformation is a local rearrangement of particles involving a change of nearest neighbours (so called shear transformation zones), which results in an Eshelbylike elastic field^{2,3,4,5}. However, it was recently shown that these rearrangements can be repetitive or irreversible depending on the strain amplitude^{6,7,8,9,10,11,12,13,14,15,16,17,18} (a similar phenomenon has been observed in the shearing of colloidal suspensions, granular systems, dislocations and superconducting vortices^{19,20,21,22,23,24,25,26,27,28} and in the compaction of granular matter^{29}).
We have previously studied highly condensed amorphous solids (well above the jamming transition) under applied oscillatory shear and showed that for small strain amplitudes these systems evolve into periodic limit cycles during which particles change their (mechanical) equilibrium positions but follow the same trajectories for consecutive cycles (Fig. 1; ref. 6). Above a critical strain amplitude, the system does not settle into a limit cycle and the motion is chaotic with a positive maximal Lyapunov exponent^{6}.
The transition from jammed to flowing behaviour in systems as diverse as earthquakes, charge density waves and disordered magnets is accompanied by the occurrence of avalanches of increasing sizes obeying powerlaw statistics, a signature of critical behaviour. In this work, we show using molecular dynamics simulations that in amorphous solids, at the same critical strain amplitude where irreversibility occurs, the system undergoes a nonequilibrium phase transition, which involves avalanches of diverging sizes. We analyse the avalanche statistics using a meanfield model for the depinning transition in plastic deformation that was used previously to describe plasticity in crystals and amorphous solids^{30,31}. We show that large avalanches exist even below the transition, as was also observed in ref. 32, so that the existence of avalanches alone is not sufficient to explain the irreversible behaviour (see Fig. 2 for an example of the displacement field generated by an avalanche that is repeated periodically under repeated cycles of subcritical strain). However, we show that the cause of irreversible behaviour for strain amplitudes that are larger than a critical value is rooted in the changes of the energy landscape topology at depinning, which suggests why depinning and irreversibility occur at the same point.
Results
Simulations
The data used in this work were generated using molecular dynamics simulations of systems of 1,024, 4,096 and 16,384 point particles in two dimensions interacting via an isotropic attractive–repulsive pairwise potential. In each case, we prepared an initial amorphous configuration and applied oscillatory athermal quasistatic shear while controlling the maximal strain amplitude in each run. The energy was kept at a minimum after each straining step using the Fast Inertial Relaxation Engine (FIRE) algorithm that reaches the steady state more quickly than standard overdamped dynamics algorithms (a more detailed discussion of the simulation methods can be found in the methods section and in refs 6, 33). In previous work^{6}, we have shown that when the strain amplitude is increased, the system undergoes a transition from periodic to chaotic dynamics. In Fig. 3a, we show the point of transition with respect to a stress–strain curve obtained by applying a constant positive strain rate. The point of transition is marked as a yellow line and the time to reach steady state as red points. In Fig. 3b, we show the potential energy per particle for three different strain amplitudes (top—low amplitude, middle—medium and bottom—large amplitude). We observe that as the amplitude becomes larger, it takes longer to reach a periodic limit cycle and at a strain amplitude comparable to the yield strain (whose system size dependence will be discussed below), the typical time to reach a limit cycle diverges as a function of the strain amplitude (Fig. 3a,d). This indicates a kind of dynamical transition, whose nature is yet to be clarified. Below we suggest that this behaviour is a result of a nonequilibrium phase transition that is related to the wellstudied front depinning universality class.
Theoretical background
To understand the role of avalanches in the transitiontochaos/yield, we studied the avalanche statistics for different maximal strain amplitudes and system sizes. Since the simulations were athermal, we identified drops in the potential energy (Fig. 3c) with plastic rearrangement events. For each simulation, we extracted all of the energy drops in the last shear cycle (to avoid transient effects), created a histogram of the energy drops and calculated the average energy drop for each maximal strain amplitude. We observe in Fig. 4 a cusp at the point at which the irreversibility transition occurs, followed by saturation to a value that depends on the system size, at large strain amplitudes. The cusp suggests that the irreversibility transition is related to a change in the avalanche dynamics, and the system size dependence of the saturation suggests that there is a saturating correlation length. To understand the avalanche statistics in this system, we invoke a simple model^{31} that belongs to the same universality class as the theory of front depinning (but with longrange interactions), which was originally developed to explain the motion of an interface in random media. This motion involves parts of an interface overcoming local energy barriers due to pinning sites and neighbouring locations in the interface ‘pulling’ the site back. The forward motion of the interface occurs in avalanches. In the case of longrange interactions, such as the ones that exist in elastoplastic systems, the notion of a ‘front’ becomes more abstract since sites that are far apart affect each other and the notion of locality becomes blurred (see Fig. 5 for illustration). This explains why the same equations can also describe avalanche behaviour associated with the plasticity of amorphous solids in which the dynamics involves overcoming random energy barriers and longrange interactions, even if an actual front may not exist. The equations of motion describing the time evolution of the plastic displacement field u(r,t) controlled by overdamped dynamics are^{31}:
where η is the viscosity, F is an externally applied force, r is a position of a deformable region (Shear transformation zone), J(r−r′) is the Green’s function for the elastic interaction between different ‘soft’ regions located at points r and r′ and f_{R}(u,r) is a random pinning potential representing the structural disorder inherent to such systems. This model assumes that the nature of the structure (the distribution of the random pinning forces f_{R}(u,r)) does not change as a function of time. In amorphous solids, the randomness is selfgenerated and can (and typically does) change under plastic deformation. However, when the system is at a steady state under linear or cyclic shearing, one can assume that the disorder is fixed. Also, the scaling behaviour of the model predictions do not change if the pinning stresses randomly change in time. This model shows a nonequilibrium phase transition between a pinned, static state and a flowing state as the stress is slowly increased past a critical force F_{c} (ref. 31). The transition is a critical point involving correlated displacement jumps. These correlations are described in terms of a scaling theory, which was derived from a meanfield (infinite interaction range) approximation and renormalization group theory^{30,31}. This theory was indeed shown to give a good description of the statistics of avalanches during plastic deformation in crystals^{34,35,36,37} and is now also being applied to amorphous solids^{38,39,40}. For an applied external force, at zero velocity (quasistatic limit), it was found that at a critical force F_{c} the avalanche size distribution scales as:
where S is the avalanche size and τ is a universal critical exponent. Below F_{c}, the distribution follows the same power law but with a maximal size (cutoff):
where σ is the cutoff exponent. Then, the distribution function takes the form:
where is a universal cutoff scaling function but the constants A and B are system specific^{30,31}. Below we will see how depinning theory can help explain the observed changes in the avalanche statistics as a function of the maximal strain amplitude.
Statistics under oscillatory shear
When applying the statistical theory of front depinning for amorphous solids under oscillatory shear, we have to modify the theory to take into account the different factors that were not included in the theory described above, which assumes a steady force. One issue is that the disorder in amorphous solids is not quenched, which can affect the statistics. For example, there could be weakening effects during an avalanche event, where the same site can be triggered more than once. This has been addressed by Dahmen et al.^{31} and was shown to affect the stress–strain curve but not the scaling exponents^{30}. The second effect of having dynamic disorder is that the distribution that describes the random variable f_{R}(u,r) can change during a cycle. We avoid this problem by performing statistics only for avalanches in ‘steadystate’ cycles, when the avalanche statistics is stable. It is known that the exact distribution of the disorder does not affect the avalanche statistics so even if the disorder is different in different cycles, that should not change the scaling functions. Another issue is that the forcing is a ‘sawtooth’, periodic strain profile. To take that into account, we have to rewrite equation (4) in terms of the strain and integrate over the different strain amplitudes. The relation between the stress and the strain shows hysteresis due to the nonlinear nature of plastic deformation (Supplementary Note 1; Supplementary Fig. 2). Since the forward and reverse straining branches of the hysteresis curve are statistically identical, we take into account only the forward direction. For the forward branch, we can model the relation between stress and strain using the scaling relation (Supplementary Note 1):
Where the critical strain Γ_{c} and critical stress Σ_{c} are related to the critical force F_{c}, the shear modulus μ and the system size L by:
where b is a systemdependent constant. This expression is explained and verified in Supplementary Note 2 and demonstrated in Supplementary Fig. 3. By fitting to stress–strain curves in the steady state, we find that δ∼1.25 (Supplementary Note 1; Supplementary Fig. 4).
We substitute (equation 5) into equation (4) and obtain a scaling relation for the avalanche size distribution as a function of the strain amplitude:
which would be expected to describe the avalanche statistics close to the critical strain amplitude. However, for oscillatory driving, the scaling function requires corrections since the avalanche size distribution measured is a result of integration over a varying amount of applied strain. Since the strain increases and decreases periodically, the system spends time both below and above the critical strain amplitude. Because we are averaging over cycles, we need to integrate over the different strain amplitudes. We also simplify the calculations by making some approximations (Supplementary Note 3). Below the transition we get the equation:
where is the distribution of avalanche sizes at maximal strain amplitude Γ, and ɛ is the instantaneous strain amplitude during a cycle ɛ∈[−Γ,Γ] (Supplementary Note 3 for a detailed derivation). If the maximal strain amplitude Γ is larger than the critical value, we have to average over the statistics both below and above the critical strain amplitude. Due to the quasistatic forcing (zero strain rate), for strains larger or equal to the critical strain amplitude, the system is expected to be exactly at criticality^{41}, and the avalanche statistics is expected to behave as a pure power law:
Substituting, we obtain:
where we have performed the integral over the last term. As explained in the Supplementary Note 3, by changing the variable of integration in equation (8) we can obtain a scaling function for the fluctuations below the critical point (see Supplementary Note 3 for the derivation details):
where λ=τ+σ/δ and χ=δ/σ. The scaling function is generally unknown. However, for mean field it was calculated to be , where γ(a,x) is the complementary gamma function and seems to agree with the data collapse (Figs 6 and 7; Supplementary Note 3). Avalanche sizes in plasticity are usually associated with the amount of slip, which is proportional to the stress drop. However, as was shown in refs 33, 42, in the steady state, the fluctuations of stress and potential energy drops are proportional due to a sum rule. We assume that to apply here as well (this is further explained and verified from the simulations data in the Supplementary Note 4 and Supplementary Fig. 5). Using equation (10), we find data collapses for five maximal strain amplitudes Γ=0.05, 0.07, 0.08, 0.085 and 0.093 at system size N=16,384 (Figs 6 and 7) from which we extract λ and χ (see Supplementary Note 5 for an explanation about the choice of Γ values). Since we have an estimate of δ, we can find the critical exponent values τ=1.04[0.26], σ=0.59[0.04]. The exponents deviate from the exponents found using meanfield theory, which are τ=1.5 and σ=0.5. However, a recent study by Salerno et al.^{42}, for simulations under direct shear (not alternating), obtained different critical exponents for overdamped, underdamped and damped dynamics and the value of τ ranged between τ=1 (critically damped), τ=1.25 (overdamped) and τ=1.5 (underdamped) depending on the dynamics. Since the FIRE algorithm uses an inertialike effect to minimize the energy, it is possible that inertial effects contribute to the deviation from meanfield theory, which was derived for overdamped dynamics (see Supplementary Note 6 and Supplementary Fig. 6 for a comparison of the algorithm relative to overdamped, underdamped and critically damped dynamics applied to a harmonic oscillator). Another possible reason for the deviation from meanfield exponents, which has been suggested recently, is the effect of anisotropic interactions^{43}, at least for simulations in steady state, under linearly increasing shear. Below we use the critical exponents τ and σ to explain the observed cusp in the energy fluctuations (Fig. 4).
Average fluctuations
From the relevant critical exponents, we can obtain the average avalanche size introduced in Fig. 4 using similar analysis as above (see Supplementary Note 7 for details):
where is a cutoff avalanche size that depends on the system size in an unknown way, and we have divided the integral by Γ to perform a cycle average. For the critical exponents τ and σ, we used the values τ=1.04 and σ=0.59 that were obtained from the data collapse shown in Figs 6 and 7. We also used Γ_{c}=0.135 for N=1,024, Γ_{c}=0.12 for N=4,096 and Γ_{c}=0.115 for N=16,384 that are the values found for the transition to chaos. The maximal cluster size was assumed to be proportional to a power law of the system size since at the steady state the correlations span the entire system (ξ∼L):
where and Δ are constants. We found the parameter values , A∼4.547, B∼30.51 and Δ∼0.482 by minimizing the normalized L_{2} norm of equation (11) with respect to the data from simulations:
the best fit resulted in L_{2}∼0.114. Note that the value of Δ∼0.482 is approximately consistent with avalanches concentrated along a shear band and thus proportional to the linear system size L∼N^{1/2}. Figure 8 shows the first moment of the potential energy fluctuations obtained from the simulations as a function of the maximal strain amplitude Γ, compared with equation (11) for three different system sizes. The most obvious features of as a function of the maximal strain amplitudes is the crossover (cusp) in behaviour at the critical point (Figs 8 and 9), which was mentioned above, and the system sizedependent saturation of for very large strain amplitudes. As one can see in the figures, both of these features are described by the theory. The saturation, and dependence on system size can be explained by noting that for very large maximal strain amplitudes Γ→∞, the normalized distribution function converges to the usual powerlaw statistics and, respectively, . One feature that we observed in the simulations that is not explained by the current theory is that Γ_{c} changes slightly with the strain amplitude due to structural rearrangements. In the theory (equation 1), structural rearrangements will amount to a change in the properties of the distribution of the random pinning f_{R}(u,t). However, this effect is small (changes in Γ_{c} are <5%) and we did not take that into account when fitting the data to the theory. By analysing the avalanche statistics using scaling forms predicted by depinning theory, we have shown that there is a critical point at a critical strain amplitude Γ=Γ_{c}, which is the same strain amplitude at which the system undergoes an irreversibility transition. However, this raises the question of why the two occur at the same point. Below we explain this intriguing concurrency.
Connection between dynamics and critical behaviour
Here we discuss the connection between depinning and the observed dynamics in the reversible and irreversible regime. The essence of this connection is that at depinning, the external force F suppresses all the energy barriers (Fig. 10a,b), which changes the topology of the energy landscape—instead of a set of disconnected energy minima, we have a fully connected set of energy minima in terms of strain. This affects the dynamics and reversibility of the system (a related explanation was suggested for the dynamics of supercooled liquids, see ref. 44).
Limit cycles
Since the system is dissipative, it will always flow to an attractor occupying a limited part of phase space (Fig. 10c; ref. 45). This attractor will be composed of a finite or infinite set of configurations of the system connected to each other by elastic or plastic displacement (Fig. 10d). For a system under linear shear, when the external forcing is below depinning, it is guaranteed that after some amount of strain the system will find a local minimum of the potential energy (will become pinned). For cyclic strain, if the maximal strain amplitude is below depinning, the system will find, after transient dynamics, a set of configurations all below the critical stress. Since the stress is lower than the critical depinning stress, this set of states is guaranteed to be linearly stable or nonlinearly stable. In the case that is nonlinearly stable, if the stress is increased, the system will overcome a closeby energy barrier but will ‘fall’ into an adjunct energy barrier (Fig. 10a), which means that the next configuration in the attractor is separated by a finite energy barrier. Therefore, in this case, the attractor is not chaotic and it must be a limit cycle (periodic). This situation is not so different to an absorbing phase transition, which was suggested as an explanation for similar phenomena^{14,21}, although we suggest that depinning provides greater physical insight into the reason for the system to reach an absorbing state.
Chaotic attractor
When the stress is close to depinnig values, a small increase in stress, due to a strain step will overcome all of the energy barriers (Fig. 10b). In this case, the system will be completely unstable, for a short time. In the quasistatic shearing scenario, the system will reach another minimum of the potential energy when the minimization algorithm or dissipation lowers the energy again but before that happens it will spend some time in boundless motion. Since there are effectively no energy barriers in this time, there are no retrieving forces and chaotic motion is possible (in some systems with quenched disorder and with a ‘no passing’ property fulfilled^{46}, such as charge density waves and certain random magnets, chaotic motion is not possible and there always is a limit cycle^{47}, but this is not the case in plasticity in amorphous solids in which disorder is not strictly quenched and for which the no passing rule is broken.).
Period doubling
When the system is close but still not exactly at criticality, there are less and less stable ‘pinned’ configurations. Therefore, the likelihood of the system being able to ‘construct’ a limit cycle that returns to the same point after one period is smaller and it may be required to have more than one cycle before the system can return to the initial configuration.
To summarize, if the strain amplitude is below depinning, the system can always self organize into cycles composed of states in which the stress fluctuations never reach depinning values. In that case the dynamics will always be bounded, either linearly or nonlinearly (overcoming one energy barrier). If the strain amplitude is large enough, there are always states in which the stress is very close to depinning. In that case small increase in the stress, due to straining, will generate stresses that are larger than the depinning value, and thus will cause unbounded motion that can lead to sensitivity to initial conditions and chaos.
Relaxation dynamics
Depinning meanfield theory predicts that close to a depinning transition, the system will ‘slide’ for a long time (displace) before it becomes pinned (this is a type of nonequilibrium critical slowing down). Therefore, the accumulated strain to reach a steady state (the number of cycles times 4Γ) is expected to diverge as a function of the applied force:
with meanfield depinning theory, which was derived for linear shear, predicting a value of zv=1 (ref. 31). Since the steady state is a limit cycle composed of a set of pinned states, we expect that also under oscillatory shear, the accumulated strain to reach a steady state will scale in the same way as the strain needed to pin one state. Since we control the maximal strain amplitude, we obtain on substituting:
In the simulations, we find powerlaw scaling with zv∼2.4 for a choice Γ_{c}=0.11 (Fig. 3d), and zv∼1.38 for a slightly smaller Γ_{c}=0.1 for the largest system that we studied (N=16,384). Note that previously^{6} we calculated the dynamical exponent by considering the number of cycles required to reach a periodic cycle, whereas here we consider the accumulated amount of strain required to reach the limit cycle, which is more compatible with the theory. The dynamical exponent zv=1 predicted by meanfield theory is in rough agreement with the scaling of the time to reach steady state measured in the experiments of Nagamanasa et al.^{14} on colloidal glasses which gave zv∼1.1/δ∼0.88.
From the arguments presented above, it is clear why the depinning and irreversibility transitions are inherently connected. Since depinning is a relatively wellstudied theory, this connection provides a basis for a deeper understanding of plasticity in amorphous solids.
Discussion
We have studied the avalanche statistics of amorphous solids under oscillatory shear and have shown that there is a critical maximal strain amplitude at which the correlation length diverges and the avalanche statistics follow theoretical meanfield scaling laws developed for the depinning transition. A key result is that the nonequilibrium critical point occurs at the same maximal strain amplitude as a dynamical reversibility–irreversibility transition, which was recently identified in the same system. We have explained why the depinning transition causes a topological change in the structure of the energy landscape, thereby facilitating a transition from periodic to chaotic dynamics. Furthermore, we have shown that the observed dynamical scaling close to the irreversibility transition is connected to the one obtained by a depinning meanfield theory^{31}.
Experiments on shearing of colloidal suspensions^{7,8,14} have found an irreversibility transition. Similarly, experiments on granular piles have also shown that the onset of irreversible behaviour is associated with the appearance of system spanning events^{24}, consistent with our findings. It would be interesting to see if the avalanche statistics that we reported here can also be observed in experiments such as the ones performed by Nagamanasa et al.^{14} and Keim et al.^{8} on jammed colloidal glasses.
An important issue that was not addressed in our current work is the effect of the dynamics on the structure of the material. For dilute colloidal suspensions^{13} and granular matter^{29}, it has been observed that the structure changes during the irreversibility transition. Structural changes in amorphous solids are very subtle and will be harder to observe^{48,49,50}. An implication of the current results is that the onset of plastic failure may be detected from powerlaw scaling of the power spectra or the slip size distributions of the slip avalanches.
Methods
Molecular dynamics simulations
The data used in this work were generated using molecular dynamics simulations of systems of 1,024, 4,096 and 16,384 point particles in two dimensions interacting via an isotropic attractive–repulsive pairwise potential (described in detail in refs 33, 6) where the effective radius of half the particles is 1.4 times larger than the other half. We use the mass m of the particles, the typical interaction distance σ and the typical interaction energy ɛ to define reduced units for the energy (E→E/ɛ), the stress Σ→Σ/(ɛσ^{−3}), the number density ρ→ρ/(σ^{−3}) and the time t→t/(ɛm^{−1}σ^{−2})^{1/2}. Positions of particles are given in terms of q→q/L, where L is the system size. The sample is kept at a constant volume and number of particles (NVT) with a number density ρ=0.75 in the reduced units that are significantly higher than the jamming transition. Initial frozen amorphous configurations are generated by equilibrating systems of particles at the fluid phase and then quenching them to zero temperature using a minimization algorithm^{51}. To verify that the structure is amorphous, we calculated the paircorrelation function. In Supplementary Fig. 1, we show the paircorrelation function for the small particles that shows the typical structure of a frozen liquid and no longrange order. The material is subject to small steps of shear strain (Δɛ=10^{−4}) using the Lees–Edwards boundary conditions^{52}. The dynamics under shear is quasistatic (after each shearing step the energy was minimized using the FIRE minimization algorithm^{51}). The strain is applied in a periodic manner: first, positive strain steps are applied. When a maximal predecided strain Γ is reached, the strain is reversed by applying strain steps in the opposite direction. This proceeds until the strain reaches the negative value of the maximal strain −Γ. At this point, the strain steps are reversed until the system returns to zero strain, completing the cycle. The cycle is then repeated.
Additional information
How to cite this article: Regev, I. et al. Reversibility and criticality in amorphous solids. Nat. Commun. 6:8805 doi: 10.1038/ncomms9805 (2015).
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Acknowledgements
We thank LANL institutional computing for resources. The work of T.L. ,C.R. and partly I.R. was carried out under the auspices of the US Department of Energy at Los Alamos National Laboratory under Contract No. DEAC5206NA25396. K.D. gratefully acknowledges grants (NSF) DMR 1005209 and DMS 1069224.
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I.R., K.A.D., C.R. and T.L. designed the research and wrote the paper. I.R. and J.W. performed the research.
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Regev, I., Weber, J., Reichhardt, C. et al. Reversibility and criticality in amorphous solids. Nat Commun 6, 8805 (2015). https://doi.org/10.1038/ncomms9805
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DOI: https://doi.org/10.1038/ncomms9805
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