Fiber Bundle Model Research Articles

Mesoscopic modeling of Acoustic Emission through an energetic approach

The aim of the present work is to present a simple model for damage progression and Acoustic Emission that correctly accounts for energy dissipation due to the formation of micro-cracks and the creation of surfaces in a material undergoing external loading, and thus to derive the scaling behaviour observed in experiments. To do this, energy balance considerations are included in a Fibre Bundle Model approach. The model predictions are first illustrated in a uniaxial test under quasistatic loading conditions. Numerical results are then compared to experimental data relative to tests on masonry elements of various sizes subjected compression. The scaling properties of Acoustic Emission under the chosen energy balance assumptions is analyzed and compared to previous numerical and experimental results in the literature. Power-law scaling behaviour is found with respect to specimen dimensions.

Continuous damage fiber bundle model for strongly disordered materials

We present an extension of the continuous damage fiber bundle model to describe the gradual degradation of highly heterogeneous materials under an increasing external load. The breaking of a fiber in the model is preceded by a sequence of partial failure events occurring at random threshold values. In order to capture the subsequent propagation and arrest of cracks, furthermore, the disorder of the number of degradation steps of material constituents, the failure thresholds of single fibers, are sorted into ascending order and their total number is a Poissonian distributed random variable over the fibers. Analytical and numerical calculations showed that the failure process of the system is governed by extreme value statistics, which has a substantial effect on the macroscopic constitutive behavior and on the microscopic bursting activity as well.

Energy bursts in fiber bundle models of composite materials

A bundle of many fibers with stochastically distributed breaking thresholds for the individual fibers is considered as a model of composite materials. The bundle is loaded until complete failure, to capture the failure scenario of composite materials under external load. The fibers are assumed to share the load equally, and to obey Hookean elasticity right up to the breaking point. We determine the distribution of bursts in which an amount of energy E is released. The energy distribution follows asymptotically a universal power law E(-5/2) , for any statistical distribution of fiber strengths. A similar power law dependence is found in some experimental acoustic emission studies of loaded composite materials.

Critical ruptures in a bundle of slowly relaxing fibers

We study the damage enhanced creep rupture of disordered materials by means of a fiber bundle model. Broken fibers undergo a slow stress relaxation modeled by a Maxwell element whose stress exponent m can vary in a broad range. Under global load sharing we show that due to the strength disorder of fibers, the lifetime t(f) of the bundle has sample-to-sample fluctuations characterized by a log-normal distribution independent of the type of disorder. We determine the Monkman-Grant relation of the model and establish a relation between the rupture life t(f) and the characteristic time t(m) of the intermediate creep regime of the bundle where the minimum strain rate is reached, making possible reliable estimates of t(f) from short term measurements. Approaching macroscopic failure, the deformation rate has a finite time power law singularity whose exponent is a decreasing function of m. On the microlevel the distribution of waiting times is found to have a power law behavior with m-dependent exponents different below and above the critical load of the bundle. Approaching the critical load from above, the cutoff value of the distributions has a power law divergence whose exponent coincides with the stress exponent of Maxwell elements.

Scaling of the size and temporal occurrence of burst sequences in creep rupture of fiber bundles

We present a detailed statistical analysis of the size and temporal occurrence of burst sequences in the creep rupture of a proposed linear viscoelastic fiber bundle model. According to the model, the burst sequences of fiber breaks display a power law asymptotic behavior analogous to that of the static-fracture [Kloster et al., Phys. Rev. E 56, 2615, (1997)]. Moreover, power law asymptotics apply to inter-arrival times between successive bursts with a universal exponent close to unity.

Modelling Damage Progression by a Statistical Energy-Balance Algorithm

In this contribution some characteristics and predictive capabilities are discussed of a recently introduced model for damage progression and energy release, in view of modelling Acoustic Emission. The specimen is discretized in a network of connected springs, similar to a Fibre Bundle Model approach, with the spring intrinsic strengths statistically distributed according to a Weibull distribution. Rigorous energy balance considerations allow the determination of the dissipated energy due to crack surface formation and kinetic energy propagation. Based on results of simulations, the macroscopic behaviour emerging from different choices at “mesoscopic” level is discussed, in particular the relevance of model parameters such as the distribution of spring cross sections, Weibull modulus values, and discretization parameters in determining results like stressstrain curves and energy scaling versus time or specimen size.

FIBERS ON A GRAPH WITH LOCAL LOAD SHARING

We study a random fiber bundle model with tips of the fibers placed on a graph having co-ordination number 3. These fibers follow local load sharing with uniformly distributed threshold strengths of the fibers. We have studied the critical behavior of the model numerically using a finite size scaling method and the mean field critical behavior is established. The avalanche size distribution is also found to exhibit a mean field nature in the asymptotic limit.

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 thermodynamical fibre bundle model for the fracture of disordered materials

We investigate a disordered version of a thermodynamic fibre bundle model proposed bySelinger et al a few years ago. For simple forms of disorder, the model is analyticallytractable and displays some new features. At either constant stress or constant strain, thereis a non-monotonic increase of the fraction of broken fibres as a function of temperature.Moreover, the same values of some macroscopic quantities such as stress and strain maycorrespond to different microscopic configurations, which can be essential for determiningthe thermal activation time of the fracture. We argue that different microscopic states maybe characterized by an experimentally accessible analogue of the Edwards–Andersonparameter. At zero temperature, we recover the behaviour of the irreversible fibre bundlemodel.

Fatigue failure of disordered materials

We present an experimental and theoretical study of the fatigue failure of heterogeneousmaterials under cyclic compression considering asphalt as a specific example. Varying theload amplitude, experiments reveal a finite fatigue limit below which the specimen does notbreak, while approaching the tensile strength of the material a rapid failure occurs. In theintermediate load range, the lifetime decreases with the load as a power law. We introducetwo novel theoretical approaches, namely, a fibre bundle model and a fuse model, and showthat both capture the major microscopic mechanisms of the fatigue failure ofheterogeneous materials, providing an excellent agreement with the experimental findings.

Effect of discontinuity in the threshold distribution on the critical behavior of a random fiber bundle

The critical behavior of a random fiber bundle model with mixed uniform distribution of threshold strengths and global load sharing rule is studied with a special emphasis on the nature of distribution of avalanches for different parameters of the distribution. The discontinuity in the threshold strength distribution of fibers nontrivially modifies the critical stress as well as puts a restriction on the allowed values of parameters for which the recursive dynamics approach holds good. The discontinuity leads to a nonuniversal behavior in the avalanche size distribution for smaller values of avalanche size. We observe that apart from the mean field behavior for larger avalanches, a new behavior for smaller avalanche size is observed as a critical threshold distribution is approached. The phenomenological understanding of the above result is provided using the exact analytical result for the avalanche size distribution. Most interestingly, the prominence of nonuniversal behavior in avalanche size distribution depends on the system parameters.

Critical behavior of random fibers with mixed Weibull distribution

A random fiber bundle model with a mixed Weibull distribution is studied under the global load sharing scheme. The mixed model consists of two sets of fibers. The threshold strength of one set of fibers is randomly chosen from a Weibull distribution with a particular Weibull index, and another set of fibers with a different index. The mixing tunes the critical stress of the bundle and the variation of critical stress with the amount of mixing is determined using a probabilistic method where the external load is increased quasistatically. In a special case which we illustrate, the critical stress is found to vary linearly with the mixing parameter. The critical exponents and power-law behavior of burst avalanche size distribution is found to remain unaltered due to mixing.

Experimental Study on Tensile Behavior of Carbon Fiber and Carbon Fiber Reinforced Aluminum at Different Strain Rate

In this study, dynamic and quasi-static tensile behaviors of carbon fiber and unidirectional carbon fiber reinforced aluminum composite have been investigated. The complete stress–strain curves of fiber bundles and the composite at different strain rates were obtained. The experimental results show that carbon fiber is a strain rate insensitive material, but the tensile strength and critical strain of the Cf/Al composite increased with increasing of strain rate because of the strain rate strengthening effect of aluminum matrix. Based on experimental results, a fiber bundles model has been combined with Weibull strength distribution function to establish a one-dimensional damage constitutive equation for the Cf/Al composite.

ダメージ力学にもとづくレオロジー

The deformation of the brittle upper crust is primarily accompanied by displacements on faults. Nevertheless, it has often been found that continuum fluid models, usually based on a non-Newtonian viscosity, are applicable. I provide a new view of rheology for upper crustal deformation, using the fiber-bundle model, a model of damage mechanics. In my analyses, a yield stress σy was introduced. Below this stress, I assumed that no fiber failure occurs, and the crust behaves elastically without earthquakes. Above the yield stress, I hypothesized that the fibers begin to fail and a failed fiber is replaced by a new fiber. This replacement is analogous to an earthquake rupture. The deformation above the stress σy under a constant strain rate ε for a suffcient time can be modeled as an equation similar to that used for a non-Newtonian viscous flow. There is a power-law relation between mean stress in the crust σ and the strain rate ε with ε∝(σ — σy)n where n is constant. I derived the modified Omori’s law (Omori-Utsu’s law) for aftershock decay using a viscoelastic version of my model and got good agreement with observations taking n=5-13. I point out that further development of my model can contribute to fully understanding the upper crustal deformation.

Interface crack propagation in porous and time-dependent materials analyzed with discrete models

A model describing the crack propagation at the interface between a rigid substratum and a beam is considered. The interface is modeled using a fiber bundle model (i.e. using a discrete set of elements having a random strength). The distribution of avalanches, defined as the distance over which the crack is propagated under a fixed force, is studied in order to capture the effect of ageing and time-dependent response of the interface. The avalanches depend not only on the statistical distribution of strength but more importantly on time (or displacement) correlations. Namely, local fiber breakage kinetics is related to a correlation length, which sets the size of the fracture process zone which occurs ahead of the crack due to progressive failure. First, a variation of porosity of the interface is considered. It corresponds for instance to diffusion controlled dissolution processes. Interpreting the results in Delaplace et al. [Delaplace A, Roux S, Pijaudier-Cabot G (2001) J Eng Mech 127:646–652], it is shown that the size of the fracture process zone increases with increasing porosity in accordance with experimental observations [Haidar K, Pijaudier-Cabot G, Dubé J-F, Loukili A (2005) Mater Struct 38:201–210]. The creep–fracture interaction is analyzed in the second part of the paper. It is found based on a Maxwell model that the size of the process zone depends on the fracture propagating velocity and on the distribution of forces in the interface due to the interaction between the interface and the rest of the specimen. The observed decrease of the size of the process zone, in creep experiments, compared to the size of the process zone in a time-independent process, is justified by the proposed model for an interface that is less viscous than the rest of the material.

Extension of fibre bundle models for creep rupture and interface failure

We present two extensions of the classical fibre bundle model to study the creep rupture of heterogeneous materials and the shear failure of glued interfaces of solid blocks. To model creep rupture, we assume that the fibres of a parallel bundle present time dependent behaviour under an external load and fail when the deformation exceeds their local breaking threshold. Assuming global load sharing among fibres, analytical and numerical calculations showed that there exists a critical load below which only partial failure occurs while above which the system fails globally after a finite time. Approaching the critical point from both sides the system exhibits scaling behaviour which implies that creep rupture is analogous to continuous phase transitions. To describe interfacial failure, we model the interface as an array of elastic beams which experience stretching and bending under shear load and break if the two deformation modes exceed randomly distributed breaking thresholds. The two breaking modes can be independent or combined in the form of a von Mises type breaking criterion. In the framework of global load sharing, we obtain analytically the macroscopic constitutive behaviour of the system and describe the microscopic process of the progressive failure of the interface.

Cumulative Benioff strain-release, modified Omori's law and transient behaviour of rocks

Irreversible thermodynamic theories with internal state variables can be used to derive a general constitutive law for both transient and steady-state behaviours of rocks. This constitutive law can represent the concepts of damage and damage evolution in either the fibre-bundle model or continuum damage mechanics. We have previously proposed an empirically based constitutive law for both the transient and steady-state behaviours of rocks ultimately derived from laboratory experimental data. We show here that this law is concordant with the general constitutive law derived from irreversible thermodynamic theories, and that the relaxation modulus has a temporal power–law that depends on a structural fractal property of rocks. Our constitutive law predicts forms for the cumulative Benioff strain-release for precursory seismic activations and the modified Omori's laws of aftershocks, both aspects of the temporal fractal properties of seismicity. These seismic properties can also be derived by the fibre-bundle model or continuum damage mechanics. Our model suggests that these time-scale invariant processes of seismicity may be regulated by the fractal structures of crustal rocks.

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.

Failure process of a bundle of plastic fibers

We present an extension of fiber bundle models considering that failed fibers still carry a fraction 0 < or = alpha < or = 1 of their failure load. The value of alpha interpolates between the perfectly brittle failure (alpha = 0) and perfectly plastic behavior (alpha = 1) of fibers. We show that the finite load bearing capacity of broken fibers has a substantial effect on the failure process of the bundle. In the case of global load sharing it is found that for alpha --> 1 the macroscopic response of the bundle becomes perfectly plastic with a yield stress equal to the average fiber strength. On the microlevel, the size distribution of avalanches has a crossover from a power law of exponent approximately 2.5 to a faster exponential decay. For localized load sharing, computer simulations revealed a sharp transition at a well-defined value alpha(c) from a phase where macroscopic failure occurs due to localization as a consequence of local stress enhancements, to another one where the disordered fiber strength dominates the damage process. Analyzing the microstructure of damage, the transition proved to be analogous to percolation. At the critical point alpha(c), the spanning cluster of damage is found to be compact with a fractal boundary. The distribution of bursts of fiber breakings shows a power-law behavior with a universal exponent approximately 1.5 equal to the mean-field exponent of fiber bundles of critical strength distributions. The model can be relevant to understand the shear failure of glued interfaces where failed regions can still transmit load by remaining in contact.

Effect of loading frequency on fatigue life and dissipated energy of structural plywood under panel shear load

Wood-based panels are subjected to cyclic panel shear load caused by wind and seismic forces in such an application as the sheathing of bearing walls. The fatigue behavior of structural plywood under panel shear load with two different loading frequencies was examined. Pulsating panel shear load with a triangular waveform and loading frequency of 0.5 or 5 Hz was applied to the plywood specimens. Stress−strain hysteresis loops were measured throughout the fatigue tests. Fatigue life was highly dependent on loading frequency at more than 0.5 stress level. The deterioration of mechanical property and damage accumulation in plywood specimen was observed to be slower at higher loading frequency at more than 0.5 stress level. Analyses based on energy loss suggest that panel shear load with higher loading frequency causes less damage to the plywood specimen during one loading cycle at higher stress level, and that the fatigue damage accumulation causing failure might be dependent on stress level although it seems to be unaffected by loading frequency. Based on these results, a new fatigue failure model for plywood specimen was qualitatively developed by combining Weibull’s weakest link model and Daniels’ fiber bundle model.