Probabilistic Mechanics of Quasibrittle Structures: Strength, Lifetime, and Size Effect

Contents
Foreword
Preface
1 Introduction
	1.1 The Problem of Tail of Probability Distribution
	1.2 History in Brief
		1.2.1 Classical History
		1.2.2 Recent Developments
	1.3 Safety Specifications in Concrete Design Codes and Embedded Obstacles to Probabilistic Analysis
	1.4 Importance of Size Effect for Strength Statistics
	1.5 Power-Law Scaling in the Absence of Characteristic Length
		1.5.1 Nominal Strength of Structure and Size Effect
	1.6 Statistical and Deterministic Size Effects
	1.7 Simple Models for Deterministic Size Effects
		1.7.2 Type 2 Size Effect for Structures with Deep Cracks or Notches
		1.7.1 Type 1 Size Effect for Failures at Crack Initiation
	1.8 Probability Distributions of Strength of Ductile and Brittle Structures
2 Review of Classical Statistical Theory of Structural Strength and Structural Safety, and of Statistics Fundamentals
	2.1 Weakest-Link Model
	2.2 Weibull Theory
	2.3 Scaling of Weibull Theory and Pure Statistical Size Effect
	2.4 Equivalent Number of Elements
	2.5 Stability Postulate of Extreme Value Statistics
	2.6 Distributions Ensuing from Stability Postulate
	2.7 Central Limit Theorem and Strength Distribution of Ductile Structures
	2.8 Failure Probability When Both the Strength and Load Are Random, and Freudenthal Integral
3 Review of Fracture Mechanics and Deterministic Size Effect in Quasibrittle Structures
	3.1 Linear Elastic Fracture Mechanics
	3.2 Cohesive Crack Model
	3.3 Crack Band Model
	3.4 Nonlocal Damage Models and Lattice-Particle Model
	3.5 Overcoming Instability of Tests of Post-Peak Softening of Fiber–Polymer Composites
	3.6 Dimensional Analysis of Asymptotic Size Effects
	3.7 Second-Order Asymptotic Properties of Cohesive Crack or Crack Band Models
	3.8 Types of Size Effect Distinguished by Asymptotic Properties
	3.9 Derivation of Quasibrittle Deterministic Size Effect from Equivalent LEFM
		3.9.1 Type 2 Size Effect
		3.9.2 Type 1 Size Effect
	3.10 Nonlocal Weibull Theory for Mean Response
	3.11 Combined Energetic-Statistical Size Effect Law and Bridging of Type 1 and 2 Size Effects
4 Failure Statistics of Nanoscale Structures
	4.1 Background of Modeling of Nanoscale Fracture
	4.2 Stress-Driven Fracture of Nanoscale Structures
	4.3 Probability Distribution of Fatigue Strength at Nanoscale
	4.4 Random Walk Aspect of Failure of Nanoscale Structures
5 Nano–Macroscale Bridging of Probability Distributions of Static and Fatigue Strengths
	5.1 Chain Model
	5.2 Fiber-Bundle Model for Static Strength
		5.2.1 Brittle Bundle
		5.2.2 Plastic Bundle
		5.2.3 Softening Bundle with Linear Softening Behavior
		5.2.4 Bundle with General Softening Behavior and Nonlocal Interaction
	5.3 Fiber-Bundle Model for Fatigue Strength
	5.4 Hierarchical Model for Static Strength
	5.5 Hierarchical Model for Fatigue Strength
6 Multiscale Modeling of Fracture Kinetics and Size Effect under Static and Cyclic Fatigue
	6.1 Previous Studies of Fracture Kinetics
	6.2 Fracture Kinetics at Nanoscale
	6.3 Multiscale Transition of Fracture Kinetics for Static Fatigue
	6.4 Size Effect on Fracture Kinetics under Static Fatigue
	6.5 Multiscale Transition of Fracture Kinetics under Cyclic Fatigue
	6.6 Size Effect on Fatigue Crack Growth Rate and Experimental Evidence
	6.7 Microplane Model for Size Effect on Fatigue Kinetics under General Loading
7 Size Effect on Probability Distributions of Strength and Lifetime of Quasibrittle Structures
	7.1 Probability Distribution of Structural Strength
	7.2 Probability Distribution of Structural Lifetime
		7.2.1 Creep Lifetime
		7.2.2 Fatigue Lifetime
	7.3 Size Effect on Mean Structural Strength
	7.4 Size Effects on Mean Structural Lifetimes and Stress-Life Curves
	7.5 Effect of Temperature on Strength and Lifetime Distributions
8 Computation of Probability Distributions of Structural Strength and Lifetime
	8.1 Nonlocal Boundary Layer Model for Strength and Lifetime Distributions
	8.2 Computation by Pseudo-random Placing of RVEs
	8.3 Approximate Closed-Form Expression for Strength and Lifetime Distributions
	8.4 Analysis of Strength Statistics of Beams under Flexural Loading
	8.5 Optimum Fits of Strength and Lifetime Histograms
		8.5.1 Optimum Fits of Strength Histograms
		8.5.2 Optimum Fits of Histograms of Creep Lifetime
		8.5.3 Optimum Fits of Histograms of Fatigue Lifetime
9 Indirect Determination of Strength Statistics of Quasibrittle Structures
	9.1 Relation between Mean Size Effect Curve and Probability Distribution of RVE Strength
	9.2 Experimental Verification
		9.2.1 Description of Experiments
		9.2.2 Analysis of Test Results
	9.3 Determination of Large-Size Asymptotic Properties of the Size Effect Curve
	9.4 Comparison with the Histogram Testing Method
	9.5 Problems with the Three-Parameter Weibull Distribution of Strength
		9.5.1 Theoretical Argument
		9.5.2 Evidence from Histogram Testing
		9.5.3 Mean Size Effect Analysis
	9.6 Alternative Proof of Strength Distribution of an RVE Based on Stability Postulate and Atomistic Analysis
10 Statistical Distribution and Size Effect on Residual Strength after Sustained Load
	10.1 Nanomechanics Based Relation between Monotonic Strength and Residual Strength of One RVE
	10.2 Analysis of Residual Strength Degradation for One RVE
	10.3 Probability Distribution of Residual Strength
		10.3.1 Formulation of Statistics of Residual Strength for One RVE
		10.3.2 Formulation of Residual Strength cdf of Geometrically Similar Structures of Different Sizes
	10.4 Comparison among Strength, Residual Strength, and Lifetime Distributions
	10.5 Experimental Validation
		10.5.1 Optimum Fits of Strength and Residual Strength Histograms of Borosilicate Glass
		10.5.2 Optimum Fits of Strength Histograms and Prediction of Lifetime and Mean Residual Strength for Unidirectional Glass/Epoxy Composites
		10.5.3 Prediction of Strength Degradation Curve for Soda-Lime Silicate Glasses
	10.6 Comparison of Size Effects on Mean Strength, Residual Strength, and Lifetime
11 Size Effect on Reliability Indices and Safety Factors
	11.1 Size Effect on the Cornell Reliability Index
	11.2 Size Effect on the Hasofer–Lind Reliability Index
	11.3 Approximate Equation for Scaling of Safety Factors
	11.4 Analysis of Failure Statistics of the Malpasset Arch Dam
		11.4.1 Model Description
		11.4.2 Discussion of Cornell and Hasofer–Lind Indices
		11.4.3 Discussion of Central and Nominal Safety Factors
12 Crack Length Effect on Scaling of Structural Strength and Type 1 to
2 Transition
	12.1 Type 1 Size Effect in Terms of Boundary Strain Gradient
	12.2 Universal Size Effect Law
	12.3 Verification of the Universal Size Effect Law by Comprehensive Fracture Tests
13 Effect of Stress Singularities on Scaling of Structural Strength
	13.1 Strength Scaling of Structures with a V-Notch under Mode I Loading
		13.1.1 Energetic Scaling of Strength of Structures with Strong Stress Singularities
		13.1.2 Generalized Finite Weakest-Link Model
	13.2 Numerical Simulation of Mode I Fracture of Beams with a V-Notch
		13.2.1 Model Description
		13.2.2 Results and Discussion
	13.3 Scaling of Fracture of Bimaterial Hybrid Structures
		13.3.1 Energetic Scaling with Superposed Multiple Stress Singularities
		13.3.2 Finite Weakest-Link Model for Failure of Bimaterial Interface
	13.4 Numerical Analysis of Bimaterial Fracture
		13.4.1 Description of Analysis
		13.4.2 Results and Discussion
14 Lifetime of High-k Gate Dielectrics and Analogy with Failure Statistics of Quasibrittle Structures
	14.1 Deviation of Lifetime Histograms of High-k Dielectrics from the Weibull Distribution
	14.2 Breakdown Probability
		14.2.1 Analogy with Strength of Quasibrittle Structures
		14.2.2 Application to Dielectric Breakdown
		14.2.3 Microscopic Statistical Models
		14.2.4 Breakdown Voltage Distribution
	14.3 Breakdown Lifetime under Constant Voltage
		14.3.1 Relation between Lifetime and Breakdown Voltage
		14.3.2 Microscopic Physics
		14.3.3 Probability Distribution of Breakdown Lifetime
	14.4 Breakdown Lifetime under Unipolar AC Voltage
	14.5 Experimental Validation
		14.5.1 Breakdown under Constant Gate Voltage Stress
		14.5.2 Breakdown under Unipolar AC Voltage Stress
	14.6 Size Effect on Mean Breakdown Lifetime
Appendix A: Power-Law Scaling of Boundary Value Problems
Appendix B: Proof of Transitional Size Effects of Types 1 and 2 by Dimensional Analysis and Asymptotic Matching up to Second Order
Appendix C: Proof of Small-Size Asymptotics of Cohesive Crack Model up to Second Order
References
Author Index
Subject Index