[Paper Review] Rupture processes in fiber bundle models

P. C. Hemmer, Alex Hansen+1 more|arXiv (Cornell University)|Feb 16, 2006

Optical Network Technologies5 references4 citations

TL;DR

This paper investigates rupture processes in fiber bundle models with different stress redistribution mechanisms, focusing on avalanche size statistics. Using analytic and numerical methods, it shows that equal-load-sharing models exhibit a power-law avalanche distribution with exponent ξ = 5/2 in the thermodynamic limit, while local-load-sharing models show crossover behavior and emergent universality classes, including a power-law cluster size distribution with exponent −1.6 at criticality.

ABSTRACT

Fiber bundles with statistically distributed thresholds for breakdown of individual fibers are interesting models of the static 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 fiber bundle avalanches, in which several fibers fail simultaneously, occur. We study by analytic and numerical methods the statistics of such avalanches, and the breakdown process for several models of fiber bundles. The models differ primarily in the way the extra stress caused by a fiber failure is redistributed among the surviving fibers.

Motivation & Objective

• To understand the statistical properties of failure avalanches in fiber bundle models under varying stress redistribution rules.
• To determine whether avalanche size distributions follow universal power laws across different load-sharing mechanisms.
• To investigate the critical behavior near complete breakdown, particularly the emergence of self-organized criticality.
• To compare equal-load-sharing (global) and local-load-sharing models in terms of avalanche statistics and failure dynamics.
• To explore the role of elastic coupling and spatial correlations in triggering catastrophic failure.

Proposed method

• Uses equal-load-sharing (global) and local-load-sharing models to simulate fiber bundle rupture under increasing load.
• Employs a recursive fixed-point algorithm to compute the average strength of the bundle at each load step.
• Applies a preconditioned iterative solver based on the Taylor expansion of the inverse matrix (I + K G)⁻¹ for efficient computation.
• Uses fast Fourier transforms (FFTs) to accelerate matrix multiplications involving the Green's function G.
• Implements a Hoshen-Kopelman algorithm to identify and analyze clusters of failed bonds at the onset of localization.
• Controls the system via either applied force F or displacement D to study different failure instabilities.

Experimental results

Research questions

• RQ1 Does the avalanche size distribution in equal-load-sharing fiber bundles follow a power law in the thermodynamic limit?
• RQ2 How does the avalanche exponent ξ change when the system is near criticality or under local load redistribution?
• RQ3 What is the nature of the failure cluster size distribution at the onset of catastrophic failure in local-load-sharing models?
• RQ4 Is the failure process in local-load-sharing systems self-organized, and does it belong to a new universality class?
• RQ5 How do elastic constants and spatial correlations influence the transition to catastrophic failure?

Key findings

• In the equal-load-sharing model, the avalanche size distribution follows a power law with exponent ξ = 5/2 in the thermodynamic limit.
• When the system is near criticality, the avalanche exponent crosses over to ξ = 3/2, indicating a change in scaling behavior.
• For local-load-sharing models with finite elastic constants, the avalanche distribution still follows a power law with ξ ≈ 2.6, consistent with the global model's ξ = 2.5.
• At the onset of localization under displacement control, the cluster size distribution follows a power law with exponent −1.6, indicating a new universality class distinct from percolation (−2.05).
• Catastrophic failure occurs due to spatial correlations and localization when the elastic constant e is small; this instability is absent under force control for large e.
• The preconditioned iterative solver based on the Taylor expansion of (I + K G)⁻¹ enables stable and efficient simulation, scaling as nL²ln(L), where n is the number of terms in the expansion.