Fiber Bundle Model Research Articles

Discussion of the statistical representativeness of the results in: Lomov, Breite, Melnikov, Mesquita, Swolfs and Abaimov [Int. J. Solids Struct 225 (2021) 111061

The variability of the statistical physics descriptors of failure in an impregnated fibre bundle model is investigated. Using the same bundle and fibre parameters as in (Lomov et al., 2021), 50 random realisations were built for VF50 and for VF60 models. The same descriptors of the bundle damage, clusters, and avalanches were calculated, and their statistical scatter characteristics were analysed. The conclusions of (Lomov et al., 2021) are confirmed to be statistically representative for an ensemble of random fibre placement realisations.

From Nucleation to Percolation: The Effect of System Size when Disorder and Stress Localization Compete

A phase diagram for a one-dimensional fiber bundle model is constructed with a continuous variation in two parameters guiding the dynamics of the model: strength of disorder and range of stress relaxation. When the range of stress relaxation is very low, the stress concentration plays a prominent role and the failure process is nucleating where a single crack propagates from a particular nucleus with a very high spatial correlation unless the disorder strength is high. On the other hand, a high range of stress relaxation represents the mean-field limit of the model where the failure events are random in space. At an intermediate disorder strength and stress release range, when these two parameters compete, the failure process shows avalanches and precursor activities. As the size of the bundle is increased, it favors a nucleating failure. In the thermodynamic limit, we only observe a nucleating failure unless either the disorder strength is extremely high or the stress release range is high enough so that the model is in the mean-field limit. A complex phase diagram on the plane of disorder strength, stress release range, and system size is presented showing different failure modes - 1) nucleation 2) avalanche, and 3) percolation, depending on the spatial correlation observed during the failure process.

Social inequality analysis of fiber bundle model statistics and prediction of materials failure.

Inequalities are abundant in a society with a number of agents competing for a limited amount of resources. Statistics on such social inequalities are usually represented by the Lorenz function L(p), where p fraction of the population possesses L(p) fraction of the total wealth, when the population is arranged in ascending order of their wealth. Similarly, in scientometrics, such inequalities can be represented by a plot of the citation count versus the respective number of papers by a scientist, again arranged in ascending order of their citation counts. Quantitatively, these inequalities are captured by the corresponding inequality indices, namely, the Kolkata k and the Hirsch h indices, given by the fixed points of these nonlinear (Lorenz and citation) functions. In statistical physics of criticality, the fixed points of the renormalization group generator functions are studied in their self-similar limit, where their (fractal) structure converges to a unique form (macroscopic in size and lone). The statistical indices in social science, however, correspond to the fixed points where the values of the generator function (wealth or citation sizes) are commensurately abundant in fractions or numbers (of persons or papers). It has already been shown that under extreme competitions in markets or at universities, the k index approaches a universal limiting value, as the dynamics of competition progresses. We introduce and study these indices for the inequalities of (prefailure) avalanches, given by their nonlinear size distributions in fiber bundle models of nonbrittle materials. We show how prior knowledge of the terminal and (almost) universal value of the k index for a wide range of disorder parameters can help in predicting an imminent catastrophic breakdown in the model. This observation is also complemented by noting a similar (but not identical) behavior of the Hirsch index (h), redefined for such avalanche statistics.

A minimal model to understand the discrepancies between acoustic and electromagnetic emissions in fracturing processes

Despite their common origin, experiments show that acoustic and electromagnetic emissions evolve in different ways during the fracture processes of brittle materials. It is common to find peaks of acoustic activity without the presence of electromagnetic emissions and vice versa, it is possible to find high electromagnetic radiation without significant acoustic activity. As far as we know, there are currently no models that explain these discrepancies in a simple way. In this work we propose a minimal model that is able to generate the frequently observed discrepancies between acoustic and electromagnetic emissions during fracture processes of brittle materials. The model is an electrical version of the fiber bundle model where the fibers are replaced by fuses in a resistor. The resistors are placed on a Wheatstone bridge subjected to a gradually increasing voltage. A capacitor is added to generate charge and discharge processes that will generate the electromagnetic emissions, while the breaking of the fuses will generate the acoustic emissions. The model allows to follow the evolution of acoustic and electromagnetic activities and shows that, despite their common origin, the two phenomena evolve differently.

Rheology of Immiscible Two-phase Flow in Mixed Wet Porous Media: Dynamic Pore Network Model and Capillary Fiber Bundle Model Results

Immiscible two-phase flow in porous media with mixed wet conditions was examined using a capillary fiber bundle model, which is analytically solvable, and a dynamic pore network model. The mixed wettability was implemented in the models by allowing each tube or link to have a different wetting angle chosen randomly from a given distribution. Both models showed that mixed wettability can have significant influence on the rheology in terms of the dependence of the global volumetric flow rate on the global pressure drop. In the capillary fiber bundle model, for small pressure drops when only a small fraction of the tubes were open, it was found that the volumetric flow rate depended on the excess pressure drop as a power law with an exponent equal to 3/2 or 2 depending on the minimum pressure drop necessary for flow. When all the tubes were open due to a high pressure drop, the volumetric flow rate depended linearly on the pressure drop, independent of the wettability. In the transition region in between where most of the tubes opened, the volumetric flow depended more sensitively on the wetting angle distribution function and was in general not a simple power law. The dynamic pore network model results also showed a linear dependence of the flow rate on the pressure drop when the pressure drop is large. However, out of this limit the dynamic pore network model demonstrated a more complicated behavior that depended on the mixed wettability condition and the saturation. In particular, the exponent relating volumetric flow rate to the excess pressure drop could take on values anywhere between 1.0 and 1.8. The values of the exponent were highest for saturations approaching 0.5, also, the exponent generally increased when the difference in wettability of the two fluids were larger and when this difference was present for a larger fraction of the porous network.

A Coupled Mechanobiological Model of Muscle Regeneration In Cerebral Palsy

Cerebral palsy is a neuromusculoskeletal disorder associated with muscle weakness, altered muscle architecture, and progressive musculoskeletal symptoms that worsen with age. Pathological changes at the level of the whole muscle have been shown; however, it is unclear why this progression of muscle impairment occurs at the cellular level. The process of muscle regeneration is complex, and the interactions between cells in the muscle milieu should be considered in the context of cerebral palsy. In this work, we built a coupled mechanobiological model of muscle damage and regeneration to explore the process of muscle regeneration in typical and cerebral palsy conditions, and whether a reduced number of satellite cells in the cerebral palsy muscle environment could cause the muscle regeneration cycle to lead to progressive degeneration of muscle. The coupled model consisted of a finite element model of a muscle fiber bundle undergoing eccentric contraction, and an agent-based model of muscle regeneration incorporating satellite cells, inflammatory cells, muscle fibers, extracellular matrix, fibroblasts, and secreted cytokines. Our coupled model simulated damage from eccentric contraction followed by 28 days of regeneration within the muscle. We simulated cyclic damage and regeneration for both cerebral palsy and typically developing muscle milieus. Here we show the nonlinear effects of altered satellite cell numbers on muscle regeneration, where muscle repair is relatively insensitive to satellite cell concentration above a threshold, but relatively sensitive below that threshold. With the coupled model, we show that the fiber bundle geometry undergoes atrophy and fibrosis with too few satellite cells and excess extracellular matrix, representative of the progression of cerebral palsy in muscle. This work uses in silico modeling to demonstrate how muscle degeneration in cerebral palsy may arise from the process of cellular regeneration and a reduced number of satellite cells.

A generic form of fibre bundle models for root reinforcement of soil

PurposeThe mechanical contribution of plant roots to the soil shear strength is commonly modelled using fibre bundle models (FBM), accounting for sequential breakage of roots. This study provides a generic framework, able to includes the many different existing approaches, to quantify the effect of various model assumptions.MethodsThe framework uses (1) a single model parameter determining how load is shared between all roots, (2) a continuous power-law distribution of root area ratio over a range of root diameters, and (3) power-law relationships between root diameters and biomechanical properties. A new load sharing parameter, closely resembling how roots mobilise strength under landslide conditions, is proposed. Exact analytical solutions were found for the peak root reinforcement, thus eliminating the current need for iterative algorithms. Model assumptions and results were validated against existing biomechanical and root reinforcement data.ResultsRoot reinforcements proved very sensitive to the user-defined load sharing parameter. It is shown that the current method of discretising all roots in discrete diameter classes prior to reinforcement calculations leads to significant overestimations of reinforcement. Addition of a probabilistic distribution of root failure by means of Weibull survival functions, thus adding a second source of sequential mobilisation, further reduced predicted reinforcements, but only when the reduction due to load sharing was limited.ConclusionThe presented solutions greatly simplify root reinforcement calculations while maintaining analytical exactness as well as clarity in the assumptions made. The proposed standardisation of fibre bundle-type models will greatly aid comparison and exchange of data.

Exploring the variability in elastic properties of roots in Alpine tree species

Quantifying the soil reinforcement provided by roots is essential for assessing the contribution of forests to reducing shallow landslide susceptibility. Many soil-root models were developed in the literature: from standard single root model to fibre bundle model. The input parameters of all models are the geometry of roots (diameter and length) and the biomechanical properties (maximum tensile force and elastic modulus). This study aims to investigate the elastic properties estimated by the stress-strain curves measured during tensile tests. A standard procedure detected two different moduli of elasticity: one due to the root tortuosity, and the other due to the woody fibres of roots. Based on a large dataset of tensile tests on different Alpine tree species, the relationships between elastic modulus and root diameter was estimated for each series. Further, the interspecific and intraspecific variability in such relationships was investigated by a statistical analysis. The results showed more intraspecific differences in the elastic modulus vs. root diameter relationships compared to the interspecific ones. This outcome could be an important criterion of discrimination to explain the variability of the elastic properties and to provide representative biomechanical properties for specific environmental conditions.

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.

Clusters and avalanches of fibre breaks in a model of an impregnated unidirectional fibre bundle under tension

Development of clusters and avalanches of damage events is a key characteristic in statistical physics of structures susceptible to a catastrophic failure. This paper applies the concepts of such descriptions to the process of fibre breakage under tension in an impregnated fibre bundle model (IFBM). The following parameters are analysed: (1) susceptibility (damage caused by change in loading); (2) spatial clustering of breaks and its interdependency with avalanches; (3) the size distribution of fibre break clusters and avalanches; (4) fractal dimensions; and (5) correlation lengths of breaks. A systematic workflow and algorithms for this analysis are presented and applied for two realisations of a random fibre placement of carbon fibre/epoxy bundles with fibre volume fractions of 50% and 60%. For these two realisations: (1) the susceptibility divergence near the failure point is affected by the finite-size effect, being nearly constant for the last ~1/20th of strain life; this behaviour can be considered as a failure predictor; (2) the morphology of the break system is a spatially distributed multi-defect occurrence with the dimensionality of 3 for all breaks together and weak planarity demonstrated only by the largest clusters near failure; (3) the exponents of the cluster-size and avalanche-size power law distributions reached 2.5–3 for both statistics near the bundle failure, which may constitute an imminent failure indicator. Damage evolution of the carbon/epoxy IFBM up to the final failure is found to be short-correlated (with correlation length of the order of one fibre break stress redistribution zone) in the absence of collective long-range defect interactions. During the catastrophic avalanche, the spatially distributed morphology of breaks is gradually reorganized into a translaminar fracture mode.

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).

Stick-Slip Dynamics in Fiber Bundle Models with Variable Stiffness and Slip Number

We present an extension of fiber bundle models to describe the mechanical response of systems which undergo a sequence of stick-slip cycles taking into account the changing stiffness and the fluctuating number of slip events of local material elements. After completing all stick-slip cycles allowed, fibers can either ultimately break or can keep their final stiffness leading to softening or hardening of the bundle, respectively. Under the assumption of global load sharing we derive analytic expressions for the constitutive response of the bundle with both quenched and annealed disorder of the failure thresholds where consecutive slips occur. Our calculations revealed that on the macro-scale the bundle exhibits a plastic behavior, which gets more pronounced when fibers undergo a higher number of stick-slip cycles with a gradually degrading stiffness. Releasing the load a permanent deformation remains, which increases monotonically for hardening bundles with the maximum deformation reached before unloading starts, however, in the softening case a non-monotonous behavior is obtained. We found that the macroscopic response of hardening bundles is more sensitive to fluctuations of the number of stick-slip cycles allowed than of the softening ones. The quenched and annealed disorder of failure thresholds gives rise to the same qualitative macro-scale behavior, however, the plastic response is found to be stronger in the annealed case.

Transverse Tensile Deformation and Failure of Three-dimensional Five-directional Braided Carbon Fiber Composites

Accurate characterisation of transverse tensile deformation and damage evolution is of importance for evaluating the failure behaviors of three-dimensional (3D) braided composites. In the present study, a finite element method (FEM) and several non-destructive testing methods including acoustic emission, digital image correlation, and infrared thermography are developed to investigate the transverse tensile deformation and damage evolution of 3D five-directional braided composites. In the finite element approach, a matrix-impregnated fiber bundles (MIFB) model and a representative volume cell (RVC) model, which take into account the fiber bundles and matrix, are respectively established to predict the effective mechanical properties of fiber bundles and simulate the deformation and progressive damage of such composites. The damaged locations and the failure modes including matrix crack, fiber debonding and shear fracture of fiber are predicted and verified by experimental tests. The non-destructive tests show that the transverse tensile fracture process can be divided into four stages which correspond to acoustic emission signals severally. The combination of the FEM based numerical modeling and multiple non-destructive characterisation methods can accurately monitor the deformation and damage behaviors of 3D braided composites under transverse tensile loads and thus provide a reference for structural health monitoring of composites in practical application.

Cooperative Dynamics in the Fiber Bundle Model

We discuss the cooperative failure dynamics in the fiber bundle model where the individual elements or fibers are Hookean springs that have identical spring constants but different breaking strengths. When the bundle is stressed or strained, especially in the equal-load-sharing scheme, the load supported by the failed fiber gets shared equally by the rest of the surviving fibers. This mean-field-type statistical feature (absence of fluctuations) in the load-sharing mechanism helped major analytical developments in the study of breaking dynamics in the model and precise comparisons with simulation results. We intend to present a brief review on these developments.

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.

Characteristic load-elongation behavior of weak electrospun fiber texture

The main purpose of this research is to establish the characteristic load-displacement behavior of weak, planar, randomly oriented fiber bundles. In this paper unidirectional strain-controlled force measurements have been performed on brittle, randomly oriented fibrous scaffold, prepared by electrospinning from polysuccinimide (PSI) based polymers. We observed that the mechanical behavior of these weak 2D networks deviated significantly from that of traditional materials treated usually within the framework of continuum mechanics. These loading curves show a maximum type dependence due to damage formation that results in stiffness reduction at elongation. We are able to monitor the rupture formation by sensitive force measurements and study how evolution of the ruptures develops during elongation. On the basis of the Fiber Bundle Model, the loading force can be directly related to the cumulative probability distribution function of failures, appearing on the loading curve as abrupt force drops during extension. We are able to determine the sequence of rupture force during extension as well as the magnitude of force drops. Large number of failures observed on each loading curve, made possible to analyze the statistical properties of damage formation. We estimated the cumulative empirical distribution function of rupture force and analyzed them on the basis of Weibull distribution. The shape parameter of distribution proves that, the rupture force of polysuccinimide electrospun fiber bundles follows the exponential distribution function. This experimental technique can provide the initial stiffness, as well as to explain load bearing capacity of several synthetic and biological textures that are composed of fibers. It also suggests improved probabilistic approaches to the development of more sophisticated statistical models.

Influence of cluster shaped defects on fracture process of fiber bundle

Defects that exist inside composites have an important effect on the tensile fracture properties of composites. The fiber bundle model is a theoretical model commonly used to study the tensile fracture properties of disorder materials. Existing work on fiber bundle models with single fiber defects shows that after single fiber defects are introduced into the fiber bundle model, the defects have a significant effect on the tensile fracture properties of the model. Since there are more complex microscopic defect structures in actual materials, such as voids, gaps, impurities, dislocations, micro-cracks, etc, it is necessary to build a multi-size defect model. In order to study the defects of different sizes and damage degrees existing in actual materials, the spatial size of the defect, the degree of defect and the distribution of fiber damage levels within the defect and other influencing factors are introduced to construct an extended fiber bundle model with cluster shaped defects. For the model, it is first assumed that the degree of defect of the fiber inside each cluster decays linearly from the center to the outside in two spatial attenuation forms: exponential decay and constant degree of defect. In the fiber bundle model of this cluster-shaped defect, the two most important factors are the number of defects <i>α</i> and the upper limit of defect size <i>β</i>. The numerical simulation method is used to analyze the influence of the number of defects, the upper limit of defect size, and spatial distribution of degree of defective fibers inside defect on the macroscopic mechanical properties and statistical properties of fracture when the model is subjected to quasi-static load borne under the nearest neighbor stress redistribution. Through the simulation analysis, it is found that owing to the overlapping competition mechanism of the defect spatial distribution, when the upper limit <i>β</i> of the defect size is large, the influence of the number of defects on the system load capacity trends to saturation. Since the defect degree of the defect center fiber is proportional to the defect size, with the upper limit <i>β</i> of the defect size increasing, its influence on the load capacity of the model becomes more and more significant. When large size defects exist, even if the number of defects is small, the load bearing performance of the material will be significantly reduced. The spatial distribution function of the damage degree of fiber inside the defect has no substantial influence on the above rules, and only changes the specific value of each fracture property. The simulation analysis results in this paper have certain theoretical significance in improving the mechanical properties of composite materials.

INFLUENCE OF ROOT SYSTEMS OF DECIDUOUS TREES ON SOIL REINFORCEMENT - A CASE STUDY FROM THE CARPATHIANS, POLAND

Forests play an significant role in the protection of slopes against shallow landslides, and the presence of the tree roots in the soil is considered the most relevant factor increasing its shear strength. Most research on root reinforcement carried out in the European mountains refer to coniferous tree species but there is still not enough information about the root systems of decidous trees and their role in enhancing the slope stability. The aim of this study was to determine root reinforcement of four typical tree species (European hornbeam, common birch, black locust and small-leaved linden) growing in the forests of the Polish Carpathians. The measurements of root systems were performed using the trench wall method and the root reinforcement values were calculated using two fiber-bundle models -RBMw and RBMe, which represented different approaches to tensile force distribution within the root bundle. Studies have shown that the hornbeam roots have the highest, and the black locust roots the lowest values of the tensile strength. Calculations showed that the values of root reinforcement obtained using the RBMw model are 17% higher than the ones calculated using the RBMe model. Root reinforcement values for common birch, black locust and small-leaved linden did not exceed 10 kPa, and in case of hornbeam they were about 20 kPa max. As part of the work, the parameters of the Root Distribution Model were estimated, which after integration with the bundle models allowed to determine the relation between the root reinforcement, the distance from the tree and its diameter.

Hooked on mushrooms: Preparation and mechanics of a bioinspired soft probabilistic fastener.

Probabilistic fasteners are known to provide strong attachment onto their respective surfaces. Examples are Velcro® and the "3M dual lock" system. However, these systems typically only function using specific counter surfaces and are often destructive to other surfaces such as fabrics. Moreover, the design parameters to optimize their functionality are not obvious. Here, we present a surface patterned with soft micrometric features inspired by the mushroom shape showing a nondestructive mechanical interlocking and thus attachment to fabrics. We provide a scalable experimental approach to prepare these surfaces and quantify the attachment strength with rheometric and video-based analysis. In these "probabilistic fasteners," we find that higher feature densities result in higher attachment force; however, the individual feature strength is higher on a low feature density surface. We interpret our results via a load-sharing principle common in fiber bundle models. Our work provides new handles for tuning the mechanical attachment properties of soft patterned surfaces that can be used in various applications including soft robotics.

Scaling the Roots Mechanical Reinforcement in Plantation of Cunninghamia R. Br in Southwest China

The degree of mechanical reinforcement provided by plants depends upon its roots distribution in the soil and mechanical properties of the roots. The mechanical properties and distribution of root traits (root diameter and number) in the soil of the standing forest depends on the tree stem diameter. This variation of root traits with tree stem diameter is rarely investigated. Therefore, this research presents the effect of tree stem diameter on the distribution of roots within the standing forest of Cunninghamia in the Longchi forest area, Sichuan province, China. In this area, shallow landslides take place frequently. We investigated the root traits distribution for trees with different stem diameters, i.e., 220 mm, 320 mm, 450 mm, and 468 mm, to show the variation of roots distribution in the soil with stem diameter. The root architecture of the selected trees was studied by step excavation method of the root zone accompanied by measurement of roots physical parameters (roots number and roots diameter) and indices (roots area ratio (RAR), roots biomass (RB), and roots distribution (RD)). We measured the root’s maximum tensile strength by performing root tensile tests in the laboratory. The field and laboratory-measured data were used to estimate the root cohesion by both the commonly used model Wu and Waldron Model (WWM) and Fiber Bundle Model (FBM). The results indicate that the tree stem diameter correlates with both the root distribution and the tensile strength. The roots indices and root cohesion increase with an increase in the diameter of the tree. Further, RAR decreases with depth and lateral distance from the tree stem, while the maximum values were observed in 10 cm depth. The relationship between roots diameter and roots tensile strength is established through power function. The average root cohesion estimated for a tree with stem diameter 220 mm is 23 kPa, 29 kPa for 320 mm, 54 kPa for 450 mm, and 63 kPa for 460 mm. This effect of stem diameter on the increase of soil shear resistance should be considered while evaluating the stability of slopes in standing forests. The comparison between WWM and FBM for investigated species suggests that WWM estimates the cohesion values greater than FBM by 65%.