Srutarshi Pradhan

Failure processes in elastic fiber bundles

The fiber bundle model describes a collection of elastic fibers under load. The fibers fail successively and, for each failure, the load distribution among the surviving fibers changes. Even though very simple, this model captures the essentials of failure processes in a large number of materials and settings. A review of the fiber bundle model is presented with different load redistribution mechanisms from the point of view of statistics and statistical physics rather than materials science, with a focus on concepts such as criticality, universality, and fluctuations. The fiber bundle model is discussed as a tool for understanding phenomena such as creep and fatigue and how it is used to describe the behavior of fiber-reinforced composites as well as modeling, e.g., network failure, traffic jams, and earthquake dynamics.

Crossover Behavior in Burst Avalanches: Signature of Imminent Failure

The statistics of damage avalanches during a failure process typically follows a power law. When these avalanches are recorded only near the point at which the system fails catastrophically, one finds that the power law has an exponent which is different from that one finds if the recording of events starts away from the vicinity of catastrophic failure. We demonstrate this analytically for bundles of many fibers, with statistically distributed breakdown thresholds for the individual fibers and where the load is uniformly distributed among the surviving fibers. In this case the distribution D(Delta) of the avalanches (Delta) follows the power law Delta-xi with xi=3/2 near catastrophic failure and xi=5/2 away from it. We also study numerically square networks of electrical fuses and find xi=2.0 near catastrophic failure and xi=3.0 away from it. We propose that this crossover in xi may be used as a signal of imminent failure.

The fiber bundle model : modeling failure in materials

Description: The book contains different applications of the fiber bundle model. As example, the authors have included fatigue and creep in their discussion. The understanding of materials and their behavior under different conditions is undergoing a revolution due to the coming–of–age of computational atomistic modeling. Even though this technique allows for the precise knowledge of what every atom is doing in such materials, it is still necessary to understand the results in terms of models such as the fiber bundle model. From the contents: The Fiber Bundle Model Average Properties Fluctuation Effects Local and Intermediate Load Sharing Recursive Breaking Dynamics Predicting Failure Fiber Bundle Model in Material Science Snow Avalanches and Landslides

Crossover behavior in failure avalanches.

Composite materials, with statistically distributed thresholds for breakdown of individual elements, are considered. During the failure process of such materials under external stress (load or voltage), avalanches consisting of simultaneous rupture of several elements occur, with a distribution D(Delta) of the magnitude Delta of such avalanches. The distribution is typically a power law D(Delta) proportional to Delta (-xi). For the systems we study here, a crossover behavior is seen between two power laws, with a small exponent xi in the vicinity of complete breakdown and a larger exponent xi for failures away from the breakdown point. We demonstrate this analytically for bundles of many fibers where the load is uniformly distributed among the surviving fibers. In this case xi=3/2 near the breakdown point and xi=5/2 away from it. The latter is known to be the generic behavior. This crossover is a signal of imminent catastrophic failure of the material. Near the breakdown point, avalanche statistics show nontrivial finite size scaling. We observe similar crossover behavior in a network of electric fuses, and find xi=2 near the catastrophic failure and xi=3 away from it. For this fuse model power dissipation avalanches show a similar crossover near breakdown.

Rupture Processes in Fibre Bundle Models

Fibre bundles with statistically distributed thresholds for breakdown of individual fibres are interesting models of the statics and dynamics of failures in materials under stress. They can be analyzed to an extent that is not possible for more complex materials. During the rupture process in a fibre bundle avalanches, in which several fibres fail simultaneously, occur. We study by analytic and numerical methods the statistics of such avalanches, and the breakdown process for several models of fibre bundles. The models differ primarily in the way the extra stress caused by a fibre failure is redistributed among the surviving fibres. When a rupture occurs somewhere in an elastic medium, the stress elsewhere is increased. This may in turn trigger further ruptures, which can cascade to a final complete breakdown of the material. To describe or model such breakdown processes in detail for a real material is difficult, due to the complex interplay of failures and stress redistributions. Few analytic results are available, so computer simulations is the main tool (see [1, 2, 3] for reviews). Fibre bundle models, on the other hand, are characterized by simple geometry and clear-cut rules for how the stress caused by a failed element is redistributed on the intact fibres. The attraction and interest of these models lies in the possibility of obtaining exact results, thereby providing inspiration and reference systems for studies of more complicated materials. In this review we survey theoretical and numerical results for several models of bundles of N elastic and parallel fibres, clamped at both ends, with statistically distributed thresholds for breakdown of individual fibres (Fig. 1). The individual thresholds xi are assumed to be independent random variables with the same cumulative distribution function P (x) and a corresponding density function p(x):

Stress-Induced Fracturing of Reservoir Rocks: Acoustic Monitoring and μCT Image Analysis

Stress-induced fracturing in reservoir rocks is an important issue for the petroleum industry. While productivity can be enhanced by a controlled fracturing operation, it can trigger borehole instability problems by reactivating existing fractures/faults in a reservoir. However, safe fracturing can improve the quality of operations during CO2 storage, geothermal installation and gas production at and from the reservoir rocks. Therefore, understanding the fracturing behavior of different types of reservoir rocks is a basic need for planning field operations toward these activities. In our study, stress-induced fracturing of rock samples has been monitored by acoustic emission (AE) and post-experiment computer tomography (CT) scans. We have used hollow cylinder cores of sandstones and chalks, which are representatives of reservoir rocks. The fracture-triggering stress has been measured for different rocks and compared with theoretical estimates. The population of AE events shows the location of main fracture arms which is in a good agreement with post-test CT image analysis, and the fracture patterns inside the samples are visualized through 3D image reconstructions. The amplitudes and energies of acoustic events clearly indicate initiation and propagation of the main fractures. Time evolution of the radial strain measured in the fracturing tests will later be compared to model predictions of fracture size.

Crossover behavior in a mixed-mode fiber bundle model

We introduce a mixed-mode load sharing scheme in a fiber bundle model. This model reduces exactly to equal-load-sharing (ELS) and local-load-sharing (LLS) models at the two extreme limits of a single-load-sharing parameter. We identify two distinct regimes: (a) the mean-field regime where the ELS mode dominates and (b) the short-range regime dominated by the LLS mode. The crossover behavior is explored through a numerical study of the strength variation, the avalanche statistics, susceptibility and relaxation time variations, the correlations among the broken fibers, and their cluster analysis. Analyzing the moments of the cluster size distributions we locate the crossover point of these regimes. We thus conclude that even in one dimension, the fiber bundle model shows crossover behavior from mean-field to short-range interactions.

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.

Breaking-rate minimum predicts the collapse point of overloaded materials.

As a model of composite materials, we choose a bundle of fibers with stochastically distributed breaking thresholds for the individual fibers. The fibers are assumed to share the load equally, and to obey Hookean elasticity right up to the breaking point. We study the evolution of the fiber breaking rate at a constant load in excess of the critical load. The analysis shows that the breaking rate reaches a minimum when the system is half-way from its complete collapse.

Failure due to fatigue in fiber bundles and solids.

We consider first a homogeneous fiber bundle model where all the fibers have got the same stress threshold (sigma(c)) beyond which all fail simultaneously in absence of noise. At finite noise, the bundle acquires a fatigue behavior due to the noise-induced failure probability at any stress sigma. We solve this dynamics of failure analytically and show that the average failure time tau of the bundle decreases exponentially as sigma-->sigma(c) from below and tau=0 for sigma>or=sigma(c). We also determine the avalanche size distribution during such failure and find a power law decay. We compare this fatigue behavior with that obtained phenomenologically for the nucleation of the Griffith cracks. Next we study numerically the fatigue behavior of random fiber bundles having simple distributions of individual fiber strengths, at stress sigma less than the bundles strength sigma(c); (beyond which it fails instantly). The average failure time tau is again seen to decrease exponentially as sigma-->sigma(c); from below and the avalanche size distribution shows similar power law decay. These results are also in broad agreement with experimental observations on fatigue in solids. We believe, these observations regarding the failure time are useful for quantum breakdown phenomena in disordered systems.