Load Sharing Fiber Bundle Model Research Articles

Criterion for Imminent Failure During Loading—Discrete Element Method Analysis

It has recently been reported that the equal load sharing fiber bundle model predicts the rate of change of the elastic energy stored in the bundle reaches its maximum before catastrophic failure occurs, making it a possible predictor for imminent collapse. The equal load sharing fiber bundle model does not contain central mechanisms that often play an important role in failure processes, such as localization. Thus, there is an obvious question whether a similar phenomenon is observed in more realistic systems. We address this question using the discrete element method to simulate breaking of a thin tissue subjected to a stretching load. Our simulations confirm that for a class of virtual materials which respond to stretching with a well-pronounced peak in force, its derivative and elastic energy we always observe an existence of the maximum of the elastic energy change rate prior to maximum loading force. Moreover, we find that the amount of energy released at failure is related to the maximum of the elastic energy absorption rate.

Size Distribution of Emitted Energies in Local Load Sharing Fiber Bundles

We study the local load sharing fiber bundle model and its energy burst statistics. While it is known that the avalanche size distribution of the model is exponential, we numerically show here that the avalanche size (s) and the corresponding average energy burst (〈E〉) in this version of the model have a non-linear relation (〈E〉 ~sγ). Numerical results indicate thatγ≈ 2.5 universally for different failure threshold distributions. With this numerical observation, it is then possible to show that the energy burst distribution is a power law, with a universal exponent value of −(γ+ 1).

Crack localization and the interplay between stress enhancement and thermal noise

We study the competition between thermal fluctuations and stress enhancement in the failure process of a disordered system by using a local load sharing fiber bundle model. The thermal noise is introduced by defining a failure probability that constitutes the temperature and elastic energy of the fibers. We observe that at a finite temperature and low disorder strength, the failure process, which nucleates in the absence of any thermal fluctuation, becomes spatially uncorrelated when the applied stress is sufficiently low. The dynamics of the model in this limit lies closely to the universality class of ordinary percolation. When applied stress is increased beyond a threshold value, localized fractures appear in the system which grow with time. We identify the boundary between the localized and random failure process in the space of temperature and applied stress, and find that the threshold of stress corresponding to the onset of localized crack growth increases with the increase of temperature.

Phase transitions and correlations in fracture processes where disorder and stress compete

We study the effect of the competition between disorder and stress enhancement in fracture processes using the local load sharing fiber bundle model, a model that hovers on the border between analytical tractability and numerical accessibility. We implement a disorder distribution with one adjustable parameter. The model undergoes a localization transition as a function of this parameter. We identify an order parameter for this transition and find that the system is in the localized phase over a finite range of values of the parameter bounded by a transition to the non-localized phase on both sides. The transition is first order at the lower transition and second order at the upper transition. The critical exponents characterizing the second order transition are close to those characterizing the percolation transition. We determine the spatiotemporal correlation function in the localized phase. It is characterized by two power laws as in invasion percolation. We find exponents that are consistent with the values found in that problem.

Scaling law in avalanche breaking of composite materials

PurposeThe fibers are loaded by the cosine component of the external load, when a fiber fails, and due to the local load-sharing nature, its force is shared by surviving neighboring fibers. The results show that the system presents a greater resistance and toughness toward the applied load compared to the classical one.Design/methodology/approachIn this paper, the authors adopt the dynamics of a local load-sharing fiber bundle model in two dimensions under an external load to study scaling law in failure process of composite materials with randomly oriented fibers. The model is based on the fiber bundle model where the fibers are randomly oriented. The system is different to the classical one where the fibers are arranged in parallel with the applied load direction.FindingsThe evolution time of the fraction of broken fiber is described by an exponential law with two characteristic times. The latter decrease linearly and exponentially respectively with both applied load and temperature.Originality/valueScaling behavior of the broken fiber numbers with the size system shows that the system exhibits a scaling law of Family–Vicsek model with universal exponents.

Burst Distribution by Asymptotic Expansion in the Equal Load Sharing Fiber Bundle Model

We derive an asymptotic series expansion for the burst size distribution in the equal load sharing fiber bundle model, a predominant model for breakdown in disordered media. Earlier calculations give expressions with correct asymptotic behavior for large bursts, but low accuracy for small bursts, up to an order of magnitude off. The approximations from the expansion we present here give relative errors of at most several percent when compared with exact results or simulation results for large systems. We also solve the burst size distribution exactly for the Weibull threshold distributions.

Can Local Stress Enhancement Induce Stability in Fracture Processes? Part II: The Shielding Effect

We use the local load sharing fiber bundle model to demonstrate a shielding effect where strong fibers protect weaker ones. This effect exists due to the local stress enhancement around broken fibers in the local load sharing model, and it is therefore not present in the equal load sharing model. The shielding effect is prominent only after the initial disorder-driven part of the fracture process has finished, and if the fiber bundle has not reached catastrophic failure by this point, then the shielding increases the critical damage of the system, compared to equal load sharing. In this sense, the local stress enhancement may make the fracture process more stable, but at the cost of reduced critical force.

Can Local Stress Enhancement Induce Stability in Fracture Processes? Part I: Apparent Stability

By comparing the evolution of the local and equal load sharing fiber bundle models, we point out the paradoxical result that stresses seem to make the local load sharing model stable when the equal load sharing model is not. We explain this behavior by demonstrating that it is only an apparent stability in the local load sharing model, which originates from a statistical effect due to sample averaging. Even though we use the fiber bundle model to demonstrate the apparent stability, we argue that it is a more general feature of fracture processes.

Time to failure of dynamic local load-sharing fiber bundle models in 1 to 3 dimensions

Extensive Monte Carlo simulations are carried out in one, two and three dimensions for dynamic local load-sharing fiber bundle models following a power-law breaking rule with exponent ρ. This exponent controls the degree of disorder of the bundle. The results are obtained using two methods of introducing disorder in the simulations. In the standard, or classical, Monte Carlo method the disorder is quenched; in the second, or radioactive method the disorder is annealed. Both methods give identical mean time-to-failure values for systems of the same size. However, the radioactive method proves to be more efficient due to the smaller standard deviation of the probability distribution function of the time-to-failure. We take advantage of this efficiency to compute the asymptotic mean time-to-failure of large systems as a function of the degree of disorder, as parameterized by ρ. Based on these extensive simulations, conclusions are drawn regarding the upper critical dimension of time-dependent local load-sharing fiber bundle models.

Mesoscopic description of the equal-load-sharing fiber bundle model

One aim of the equal load sharing fiber bundle model is to describe the critical behavior of failure events. One way of accomplishing this, is through a discrete recursive dynamics. We introduce a continuous mesoscopic equation catching the critical behavior found through recursive dynamics. It allows us to link the model with the unifying framework of absorbing phase transitions traditionally used in the study of non-equilibrium phase transitions. Moreover, it highlights the analogy between equal load sharing and spinodal nucleation. Consequently, this work is a first step towards the quest of a field theory for fiber bundle models.

A Renormalization Group Procedure for Fiber Bundle Models

We introduce two versions of a renormalization group scheme for the equal load sharing fiber bundle model. The renormalization group is based on formulating the fiber bundle model in the language of damage mechanics. A central concept is the work performed on the fiber bundle to produce a given damage. The renormalization group conserves this work. In the first version of the renormalization group, we take advantage of ordering the strength of the individual fibers. This procedure, which is the simpler one, cannot be generalized to other fiber bundle models such as the local load sharing one. The second renormalization group scheme based on the physical location of the individual fibers may be generalized to other fiber bundle models.

Crack diffusion in failure process of composite materials

ABSTRACTWe study the crack diffusion in composite materials subject to a local load-sharing fiber bundle model in two dimensions under an external load applied at a single point. By the use of the local load-sharing rule, the redistributed load remains localized along the boundary of the broken patch. We investigated the correlation function of the broken fibers. The results show that this magnitude decreases exponentially with time and it's characterized by a characteristic time which is inversely proportional to the diffusion coefficient of the localized crack. The results exhibit also that the crack correlation function in the failure process in composite materials decreases with both of applied load and temperature. We calculate then the diffusion coefficient of the localized crack. We have found that this diffusion coefficient increases by power law with the applied load and thermal noise.

Local load-sharing fiber bundle model in higher dimensions.

We consider the local load-sharing fiber bundle model in one to five dimensions. Depending on the breaking threshold distribution of the fibers, there is a transition where the fracture process becomes localized. In the localized phase, the model behaves as the invasion percolation model. The difference between the local load-sharing fiber bundle model and the equal load-sharing fiber bundle model vanishes with increasing dimensionality with the characteristics of a power law.

Cracks in random brittle solids:

Statistical models are essential to get a better understanding of the role of disorder in brittle disordered solids. Fiber bundle models play a special role as a paradigm, with a very good balance of simplicity and non-trivial effects. We introduce here a variant of the fiber bundle model where the load is transferred among the fibers through a very compliant membrane. This Soft Membrane fiber bundle mode reduces to the classical Local Load Sharing fiber bundle model in 1D. Highlighting the continuum limit of the model allows to compute an equivalent toughness for the fiber bundle and hence discuss nucleation of a critical defect. The computation of the toughness allows for drawing a simple connection with crack front propagation (depinning) models.

Self-organized dynamics in local load-sharing fiber bundle models

We study the dynamics of a local load-sharing fiber bundle model in two dimensions under an external load (which increases with time at a fixed slow rate) applied at a single point. Due to the local load-sharing nature, the redistributed load remains localized along the boundary of the broken patch. The system then goes to a self-organized state with a stationary average value of load per fiber along the (increasing) boundary of the broken patch (damaged region) and a scale-free distribution of avalanche sizes and other related quantities are observed. In particular, when the load redistribution is only among nearest surviving fiber(s), the numerical estimates of the exponent values are comparable with those of the Manna model. When the load redistribution is uniform along the patch boundary, the model shows a simple mean-field limit of this self-organizing critical behavior, for which we give analytical estimates of the saturation load per fiber values and avalanche size distribution exponent. These are in good agreement with numerical simulation results.

Crossover behaviors in one and two dimensional heterogeneous load sharing fiber bundle models

We study the effect of heterogeneous load sharing in the fiber bundle models of fracture. The system is divided into two groups of fibers (fraction p and 1 − p) in which one group follows the completely local load sharing mechanism and the other group follows global load sharing mechanism. Patches of local disorders (weakness) in the loading plate can cause such a situation in the system. We find that in 2d a finite crossover (between global and local load sharing behaviours) point comes up at a finite value of the disorder concentration (near p c ∼ 0.53), which is slightly below the site percolation threshold. We numerically determine the phase diagrams (in 1d and 2d) and identify the critical behavior below p c with the mean field behavior (completely global load sharing) for both dimensions. This crossover can occur due to geometrical percolation of disorders in the loading plate. We also show how the critical point depends on the loading history, which is identified as a special property of local load sharing.

Onset of localization in heterogeneous interfacial failure

We study numerically the failure of an interface joining two elastic materials under load using a fiber bundle model connected to an elastic half space. We find that the breakdown process follows the equal load sharing fiber bundle model without any detectable spatial correlations between the positions of the failing fibers until localization sets in. Depending on the elastic constant describing the elastic half space, localization sets in before or after the critical load causing the interface to fail completely is reached. There is a crossover between failure due to localization and failure without spatial correlations when tuning the elastic constant, not a phase transition as has been proposed earlier. Also, contrary to earlier claims based on models different from ours, we find that a finite fraction of fibers must fail before the critical load is attained, even in the extreme localization regime, i.e., for very small elastic constant. We furthermore find that the critical load remains finite for all values of the elastic constant in the limit of an infinitely large system.

Relaxation dynamics in strained fiber bundles

Under an applied external load the global load-sharing fiber bundle model, with individual fiber strength thresholds sampled randomly from a probability distribution, will relax to an equilibrium state, or to complete bundle breakdown. The relaxation can be viewed as taking place in a sequence of steps. In the first step all fibers weaker than the applied stress fail. As the total load is redistributed on the surviving fibers, a group of secondary fiber failures occur, etc. For a bundle with a finite number of fibers the process stops after a finite number of steps, tf. By simulation and theoretical estimates, it is determined how tf depends upon the stress, the initial load per fiber, both for subcritical and supercritical stress. The two-sided critical divergence is characterized by an exponent -1/2, independent of the probability distribution of the fiber thresholds.

A fiber bundle model of traffic jams

We apply the equal load-sharing fiber bundle model of fracture failure in composite materials to model the traffic failure in a system of parallel road network in a city. For some special distributions of traffic handling capacities (thresholds) of the roads, the critical behavior of the jamming transition can be studied analytically. This has been compared with that for the asymmetric simple exclusion process in a single channel or road.

Crack avalanches in the three-dimensional random fuse model

We analyze the scaling of avalanche precursors in the three-dimensional random fuse model by numerical simulations. We find that both the integrated and non-integrated avalanche size distributions are in good agreement with the results of the global load sharing fiber bundle model, which represents the mean-field limit of the model.