Disordered materials, when subjected to external stress, go through local damages that eventually emerge as a catastrophic fracture when the load exceeds a critical value. These breaking events, called avalanches, are usually detected as acoustic emissions [1]. Systematic statistical analysis of the time series of these events has led to a considerable insight into the failure dynamics of disordered samples across scales. In particular, the scale-free nature of the size distribution of these avalanches has motivated a “critical phenomenal” description of fracture processes. From laboratory-scale experiments to earthquake statistics, the Guttenberg–Richter-like law is widely and accurately verified [2, 3]. The statistics of damage avalanches and the corresponding emitted energies are studied also in other systems such as dislocation dynamics and magnetic systems. both theoretically and experimentally [4–7].

The advantage of having a critical phenomenal description is that the dynamics are expected to be independent of the microscopic details of the individual systems studied and will depend on a few parameters such as the dimensionality of the system, the interaction range, and order parameter definition (see, for e.g., [8]). This means simplified models with very few parameters should be able to reproduce the “critical description” of these avalanching phenomena. Indeed, there have been many such attempts. Some well-explored models in this regard include the fiber bundle model [9], the random fuse model [10], and Burridge–Knopoff model for earthquakes [11]. There are also multiple efforts in molecular dynamics simulations to obtain scale-free avalanche statistics (see, for e.g., [12] for a review).

The two quantities, avalanche size and the corresponding energy bursts, are generally distinct. While the former gives a measure of the area opened up during a fracture event (or the number of spins changing orientation in a disordered magnetic system under an external field), the latter gives a measure of the (strain) energy released due to such events. Depending on how the avalanche size is defined in a system, a generally nonlinear relationship can be obtained between avalanche size and energy burst. This relationship has been studied analytically using mean-field models, numerically using lattice models or discrete elements models, and also experimentally in rupture cascades in laboratories and earthquakes for crackling noise in magnetic domains [7]. While the average values of energy bursts (⟨E⟩) and avalanche sizes or damage (⟨s⟩) are known in those systems, in this work, we focus on how does the individual correspondence between the avalanche size and energy of a particular event vary with localization of load redistribution in the context of the fiber bundle model in one and two dimensions. It was noted earlier [13] that in the extreme limit of the nearest neighbor load sharing, those avalanches of large sizes can have small energy release and vice versa. In terms of the size distribution functions, the avalanche size distribution is exponential, but the energy distribution is a power law. It was also shown recently in a rock deformation experiment through a combination of acoustic energy measurement and an in situ x-ray imaging of the microstructure that local strain and the corresponding acoustic emission amplitudes can be uncorrelated [14]. This can be attributed to changes in the local stiffness in a damaged region in the experiment. As for the theoretical modeling is concerned, tuning the range of load redistribution in the fiber bundle model can shed light on this phenomenon. Therefore, in this work, we systematically explore the intermediate ranges between the nearest neighbor and the equal load-sharing limits and study the correspondence of avalanche size and energy bursts and their size distributions.

In the fiber bundle model, the mode of failure depends mainly on two factors: the strength of disorder in the individual failure threshold of the fibers and the range of stress release [15–18] when one such fiber is broken. Such interplay between disorder and damage nucleation not only affects the failure mode but also influences the spatial correlation during the failure process [17–20]. The range of stress release has two extreme limits, viz. the equal and nearest neighborhood load sharing. However, in most disordered materials, the effective mechanisms are in between these two extreme cases. Therefore, it is crucial to know to what extent one can infer the magnitude of one of these quantities when the other quantity is known. In this work, we study the fiber bundle model in one and two dimensions with variable ranges of load sharing. We look at the distributions of the avalanche size and energy. We also look at the correlation measures for the time series of avalanche sizes and the corresponding energy bursts as a function of the range of stress release. We show that as soon as localization in the load redistribution is introduced, the one-to-one linear correspondence between the avalanche size and the corresponding energy bursts decreases. The nature of decreases and the size distribution of the two quantities depend strongly on the dimensionality of the model.

The fiber bundle model has been proven to be a very useful yet very simple model to study failure processes in disordered systems. It has gained a lot of attention among engineers, material scientists, and physicists after its introduction by Pierce in 1926 [9]. A conventional fiber bundle model of size L consists of L parallel Hookean fibers for 1d and L × L fibers in 2d, placed vertically between two supporting clamps (1d) or plates (2d). The clamps/plates are pulled apart by a force F, creating stress f = F/N per fiber. Here, N is the total number of fibers. For 1d, N = L, and for 2d, N = L ². The disorder is introduced within the model as fluctuation of failure strength values in the individual fibers. In the present work, such strength values (h) are chosen from a uniform distribution between (0, 1). A certain fiber i will break irreversibly if the applied stress σ(i) on that fiber crosses its threshold value h(i). The load carried by the broken fibers is then redistributed to the other surviving fibers. As mentioned earlier, the two extreme limits of such redistribution schemes are the global or equal load sharing (all surviving fibers get an equal fraction of the redistributed load [9, 21]) and the nearest neighbor load sharing (only the nearest neighbors share the redistributed load) [22–28].

Here, we have adopted an intermediate load-sharing scheme that interpolates continuously between these two extreme limits [17, 29]. If σ _i is the stress of the ith broken fiber, then for a certain fiber j, the local stress profile after redistribution will obey the following rule: σ j → σ j + r i j − γ Z σ i (1) where r _ij is the distance between the fibers i and j. Z is the normalization factor given by Z = ∑ k r i k − γ (2) where k runs over all surviving fibers. With the present rule, a high γ stands for a low-stress release range and hence closer to the LLS scheme. On the other hand, for low γ, the model enters the ELS or mean-field limit. Earlier studies pointed out a critical γ _c for both one- [17] and two-dimensional [29] fiber bundle model around which this transition from LLS to ELS limit takes place. The values of γ _c for one and two dimensions are 4/3 and 2.15 ± 0.05, respectively [17]. After such redistribution, other fibers can break due to the elevated local stress profile, and the stress is again redistributed. This starts an avalanche at the same external stress. The size s of an avalanche is the number of fibers broken until the system reaches the next stable configuration (loads on all surviving fibers are lower than their respective failure thresholds). The energy emitted during an avalanche of size s is represented as follows: E s = ∑ 1 s 1 2 h i 2 (3) where h _i ’s are the threshold values of the s broken fibers during the avalanche. An average energy ⟨E⟩ can be calculated then by averaging overall E values corresponding to a certain avalanche size s. The total bundle might break through a single avalanche, or it can come to a stable state after an avalanche. In the latter case, we increase the applied stress just to break the next weakest fiber and start a new avalanche. This process goes on until all fibers are broken.

Numerically, we have studied both one- and two-dimensional fiber bundle models with a variable stress release range, characterized by γ (Eq. 1). The system sizes vary from 10³ to 10⁵ for one dimension and from 50 × 50 to 300 × 300 for two dimensions. The failure threshold distribution for the fibers is kept uniform in (0, 1). The value of γ was increased from 0 to 8, which allows us to explore the whole region between ELS to LLS, during the failure process.

We start by observing the relation between s and ⟨E⟩ with variation in γ. This will give us an idea about the correlation that the avalanche size and the energy has in between. Such correlation is then explored explicitly through a correlation function C(γ) and compared with s vs. ⟨E⟩ behavior. Finally, the distribution of avalanches and emitted energies are shown, and a connection between them is established through the correlation function.

In driven disordered systems in general and fracture propagation in particular, an avalanche of size s corresponds to the opening up of a fractured surface (or reorientation of magnetic spins in a disordered magnetic system), while the corresponding energy burst E is the strain energy released, usually in the form of acoustic emission. In experiments and theory, the usually nonlinear relationship between these two quantities has been extensively studied (see, for e.g., [4, 7]). In this work, we have studied the changes in this relationship with the variation in the range of stress release following a local failure, using the fiber bundle model as a prototypical example. In particular, we assess the correlation between the individual avalanches and the corresponding energy bursts, as a function of this stress release range (parameterized by γ in Eq. 1). We show that the correlation decreases from a value close to 1.0 (fully correlated) in the mean-field limit (γ → 0) to a significantly lower value in the nearest neighbor load-sharing limit, for both one and two dimensions. This is in line with recent experimental observations [14] in rock deformations, where an in situ measurement of a large change in the local strains (through x-ray imaging) did not always correspond with a large acoustic emission and vice versa.

In the fiber bundle model, the failure mode depends on the range of stress release, once the disorder distribution is fixed in a moderate range (uniform in (0, 1) here). The reason for the different modes of failure is the competition of damage progression due to stress localization near an already damaged region and vulnerable spots at some distance from the damaged region. While the first mechanism promotes a nucleation-driven failure, the latter leads toward percolative damage. The critical range that separates these two limits has been studied elsewhere [17, 18]. The underlying mechanism is not only valid for the fiber bundle model [17–20] but also seen in the random fuse model [32, 33] and in molecular dynamics simulations [34]. The present study is a manifestation of these two competing effects. Indeed, for a wide range of stress release, load increases on almost all elements, and hence the weaker among those elements break first. Given that there is no spatial correlation between the failure thresholds of the fibers, the damage is percolative. When the load redistribution range is more localized (for higher γ values), the correspondence between an avalanche size and the corresponding energy burst decreases. Indeed, we see that beyond a critical value γ _c , which depends on the dimension of the model, the correlation between avalanche and energy drops drastically. Incidentally, this is also where the size distribution of avalanches becomes exponential from the power-law behavior for γ < γ _c .

Specifically, we have plotted s vs. E in Figure 1, where a clear linearity is observed for a wide range of stress redistribution. But for localized ranges, the linearity is not present. The correlation between the two quantities drops from one as the stress release is localized. However, a positive correlation always remains, which is anyway expected. But, there could be individual events that are of larger sizes which still can emit relatively less energy. In the distribution functions, the differences are more apparent. For very high values of γ, the avalanche size distribution always remains exponential, but the energy distribution always shows a power-law behavior with a higher value of the exponent (see Figure 4) in both one and two dimensions. This highlights the fact that in the localized load sharing, one can see power-law distributions in energy, which is what is usually measured in the experiments, while the size distributions can already shift to exponential.

In conclusion, the interchangeability of the avalanche size and energy in the fiber bundle model is only valid when the stress release range is very wide. Specifically, this equivalence is not valid in the nucleation failure mode. However, not only that the correlations between the two quantities are less, but also the size distribution of the energy still shows a scale-free behavior where the avalanche size distribution may not, in both one and dimensions. Given that in real materials that the stress release range is always somewhat local, the observations here can explain the simultaneous existence of damage localization and scale-free distributions of emitted energies.