2022 | Book

# Fiber Bundles

## Statistical Models and Applications

Authors: James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri

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​This book presents a critical overview of statistical fiber bundle models, including existing models and potential new ones. The authors focus on both the physical and statistical aspects of a specific load-sharing example: the breakdown for circuits of capacitors and related dielectrics. In addition, they investigate some areas of open research.

This book is designed for graduate students and researchers in statistics, materials science, engineering, physics, and related fields, as well as practitioners and technicians in materials science and mechanical engineering.

##### Chapter 1. Introduction and Preliminaries
Abstract
Over the last sixty years, fiber bundle models (FBMs) have played an indispensable role in “Modelling Critical and Catastrophic Phenomena.” The phrase in quotes is part of the title of a book on FBM (Bhattacharyya & Chakrabarti, 2006). This book consists of several tutorial introductory chapters, one of which is by Kun et al. (2006) entitled “Extensions of fibre bundle models,” where they state that “The fibre bundle model is one of the most important theoretical approaches to investigate the fracture and breakdown (BD) of disordered media extensively used both by the engineering and physics community.” The chapters after the introductory ones are specialized applications of the FBM in the geosciences.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri

#### Part I

##### Chapter 2. Electrical Circuits of Ordinary Capacitors
Abstract
Here, we give a brief review regarding traditional circuits of ordinary capacitors where current is described as a flow in the classical theory of circuits. We distinguish this from later chapters regarding thin dielectrics that are solid-state electronic devices. For such devices, the flow analogy is only approximate because, at the nanoscale, quantum effects have to be taken into consideration. This is discussed in Chap. 3. The discussion of capacitors, resistors, and classic electric circuits that follows is based on Jones (1971). Plate type capacitors are discussed in Sect. 2.1, while in Sect. 2.2, the electrical laws for parallel and series circuits of ordinary capacitors and the behavior of the charge distribution on a series circuit are given.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 3. Breakdown of Thin-Film Dielectrics
Abstract
We are considering the breakdown mechanisms of a thin-film dielectric. The earlier type of dielectric used in electronic circuits, silicon dioxide, is generated on a metal substrate by a chemical vapor deposition process (Chu, 2014). The deposition process is usually performed at relatively low temperatures to avoid defect formation, diffusion, and degradation of the metal layers.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 4. Cell Models for Dielectrics
Abstract
Other cell types of models have been considered earlier to analyze the BD of a dielectric, but these are defect-based rather than based on load-sharing. See Sections 3.2–3.4 where percolation and analytic cell models are discussed in detail.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri

#### Part II

##### Chapter 5. Electrical Breakdown and the Breakdown Formalism
Abstract
In the testing of capacitors and capacitor circuits, one is interested in their reliability. Thus, one studies various types of breakdowns under accelerated stress conditions: e.g., stressed under increasing voltage or current to determine voltage or current breakdown (VBD or CBD) and time to failure under static voltage or current load or cycles to failure. The BD formalism allows one to relate BD under different testing protocols and to project the reliability to normal operating conditions.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 6. Statistical Properties of a Load-Sharing Bundle
Abstract
As indicated in the introduction of Part II, some physical systems can be modeled using fiber bundles where a bundle is a load-sharing parallel reliability system of components. Here, a parallel system (bundle) fails when all the components in the system fail. Below we discuss load-sharing rules for a bundle and its consequences and then give the survival distribution and mixed distribution of the strength of a bundle under increasing load. After that, we give the joint distribution, the Gibbs measure, of the state (failed/working) of the components of a bundle and discuss the stochastic failure process for the bundle. Finally, we close the chapter with a discussion of size effects for the equal load-sharing rule and a local load-sharing rule.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 7. An Illustrative Application: Fibers and Fibrous Composites
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 8. Statistical Analysis of Time-to-Breakdown Data
Abstract
In this chapter, we consider the data from Figures 3, 6, and 14 in Kim and Lee (2004) as well as their Weibull analyses of these figures and analysis of their Figure 14. We use this data to see the role that the Weibull plays in their analysis, the physical interpretation of the Weibull shape and scale, and the BD formalism. In addition, other distributions are used to compare with the Weibull and to study the legitimacy of some of the BD formalism assumptions.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 9. Circuits of Ordinary Capacitors
Abstract
In this chapter, we illustrate how the BD distributional behavior of ordinary capacitors affects that of a series circuit of such capacitors. In Sect. 9.1, we investigate the VBD of the circuit when the VBD distribution is Weibull. The parameters are based on the earlier analysis of Kim and Lee’s Figure 6 data for illustrative purposes even though the data is not for ordinary capacitors but for thin dielectrics.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 10. Simulated Size Effects Relationships Motivated by the Load-Sharing Cell Model
Abstract
In this chapter, we discuss the load-sharing (LS) cell model and size effects raised regarding Figure 14 data. The data in Figure 14 is generated from static loading tests where three different protocols are considered. Here, the cycle time to failure is from an accelerated failure time (AFT) perspective where the load-sharing directly reduces the cycle lifetime. That is, the role of increased voltage to voltage BD is functionally replaced by increased time or the number of cycles to cycle breakdown.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Chapter 11. Concluding Comments and Future Research Directions
Abstract
Over the last 60 years, FBMs have played a fundamental role in the analysis of the reliability of certain types of complex load-sharing systems. It was first used in the study of the strength of yarns and threads followed by modeling the reliability of fibrous composites. This followed with their use in the investigation of the fracture and breakdown (BD) of disordered materials studied by material scientists.
James U. Gleaton, David Han, James D. Lynch, Hon Keung Tony Ng, Fabrizio Ruggeri
##### Backmatter
Title
Fiber Bundles
Authors
James U. Gleaton
David Han
James D. Lynch
Hon Keung Tony Ng
Fabrizio Ruggeri