Fiber Bundles: Statistical Models and Applications

Preface
Acknowledgments
Contents
Acronyms
1 Introduction and Preliminaries
	1.1 Overall Introduction
		1.1.1 Early Origins of Fiber Bundles Model
		1.1.2 Organization of This Book
	1.2 Preliminaries
		1.2.1 Elements of Probability
			1.2.1.1 Sample Space and Events
			1.2.1.2 Axioms of Probability
			1.2.1.3 Conditional Probability and Independence
		1.2.2 Random Variables
			1.2.2.1 Expectation of a Random Variable
			1.2.2.2 Variance, Covariance, and Correlation of Random Variables
			1.2.2.3 Moments of a Random Variable
			1.2.2.4 Survival Function
			1.2.2.5 Hazard Function
			1.2.2.6 Quantile Function
			1.2.2.7 Distributions of Minimum and Maximum
		1.2.3 Some Commonly Used Discrete Distributions
			1.2.3.1 Binomial Distribution
			1.2.3.2 Poisson Distribution
		1.2.4 Some Commonly Used Continuous Distributions
			1.2.4.1 Uniform Distribution
			1.2.4.2 Normal Distribution
			1.2.4.3 Exponential Distribution
			1.2.4.4 Weibull Distribution
			1.2.4.5 Other Log-Location-Scale Distributions
			1.2.4.6 Other Lifetime Distributions
		1.2.5 Likelihood Inference
			1.2.5.1 Likelihood and Fisher Information Matrices
			1.2.5.2 General Maximum Likelihood Theory
		1.2.6 Statistical Inference
		1.2.7 Model Selection Criteria
		1.2.8 Regression
			1.2.8.1 Simple Regression Analysis
			1.2.8.2 Parametric Lifetime Regression Models (Weibull Regression, Exponential Regression)
			1.2.8.3 Semiparametric Regression Model (Cox Proportional Hazards Model)
		1.2.9 Censoring
		1.2.10 Kaplan–Meier Estimator of cdf
Part I Physical Aspects of Fiber Bundle Models
2 Electrical Circuits of Ordinary Capacitors
	2.1 Electrical Laws for Circuits of Capacitors
	2.2 Conservation Laws for Series and Parallel Circuits
		2.2.1 Conservation Laws for Series and Parallel Circuits
		2.2.2 Consequences of the Conservation Laws: The Capacitor Laws
		2.2.3 Parallel and Series Circuits of Capacitors with the Same Capacitance
		2.2.4 Behavior of the Charge and  Voltage Load Distributions for Series Circuits of Capacitors
3 Breakdown of Thin-Film Dielectrics
	3.1 Quantum Theory of Electron States in Solids
	3.2 The Two Dielectric Materials Being Examined
		3.2.1 Structure of Silicon Dioxide Thin Films
		3.2.2 Structure of Hafnium Oxide Thin Films
	3.3 Mechanisms of Conduction Through Dielectrics
		3.3.1 Electrode-Limited Conduction Mechanisms
		3.3.2 Bulk-Limited Conduction Mechanisms
	3.4 Breakdown in Silicon Dioxide Dielectrics
	3.5 Breakdown in Hafnium Oxide Dielectrics
4 Cell Models for Dielectrics
Part II Statistical Aspects of Fiber Bundle Models
5 Electrical Breakdown and the Breakdown Formalism
	5.1 The Breakdown Formalism
	5.2 Time-to-Breakdown (TBD) Formalism: Static Loads
		5.2.1 TBD Formalism: Dynamic Loads
6 Statistical Properties of a Load-Sharing Bundle
	6.1 Load-Sharing Rules
	6.2 The Bundle Strength Distribution as an Affine Mixture
	6.3 The Bundle Strength Density as a Gamma-Type of Mixed Distribution
	6.4 The Gibbs Representation of the Distribution of the States of a Bundle
	6.5 Examples of Size Effects
7 An Illustrative Application: Fibers and Fibrous Composites
	7.1 The Weibull Distribution and the Weakest Link Hypothesis
		7.1.1 The Bader–Priest Fiber Data
		7.1.2 The Bader–Priest Impregnated Tow Data
		7.1.3 Cumulative Damage Models
	7.2 Discussion of Rosen\'s Experiments
		7.2.1 Description of the Series A Experiments and the Analysis of the Specimen A-7 Photographs
		7.2.2 Discussion Regarding the Shape of the Bundle in the Chain-of-Bundles Model
8 Statistical Analysis of Time-to-Breakdown Data
	8.1 Fitting Breakdown Data with Different Statistical Distributions
	8.2 Breakdown-Time Regression Models
		8.2.1 Proportional Hazard Models for kimle2004\'s Figure 6 Data
		8.2.2 Fitting kimle2004\'s Figure 3 data with different parametric models and link functions
	8.3 Prediction of Hard Breakdown Based on Soft Breakdown Time
9 Circuits of Ordinary Capacitors
	9.1 Voltage Breakdown (VBD) of Series and Parallel Circuits Based on kimle2004\'s Figure 6 Data
	9.2 Parallel–Series Circuits Based on kimle2004\'s Figure 6 Data
	9.3 TBD and Cycle Times to Breakdown (CTBD) of Series Circuits
10 Simulated Size Effects Relationships Motivated by the Load-Sharing Cell Model
	10.1 Background
	10.2 Size Effect Simulations
11 Concluding Comments and Future Research Directions
	11.1 Book Summary
	11.2 Some Future Research Directions
		11.2.1 Curvature in Weibull Plots
		11.2.2 Modeling Roughness
		11.2.3 Degradation
		11.2.4 Nano-Sensors
A Appendices of Supplementary Topics
	A.1 Curvature in Weibull Plots and Its Implications
		A.1.1 Reliability Systems and Curvature in Related Weibull plots
		A.1.2 Curvature in Weibull Plots
		A.1.3 Size Effects and Mixed Hazards
		A.1.4 Weibull Plots of Mixed Weibull Hazards: Convex Curvature
		A.1.5 An Example of an Exact Weibull Plot with Concave Curvature
		A.1.6 The Weibull Chain-of-Links Hypothesis and Linearity in  Weibull Plots
	A.2 Load-Sharing Networks and Absorbing State Load-Sharing Rules
		A.2.1 Load-Sharing Networks
		A.2.2 Absorbing State Load-Sharing Rules
	A.3 Gibbs Measure Potentials and the Stresses and Potential Energies in Load-Sharing Bundles
References
Index