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E-raamat: Fracture and Size Effect in Concrete and Other Quasibrittle Materials

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Fracture and Size Effect in Concrete and Other Quasibrittle MaterialsSuurem pilt
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Focuses on the theory of scaling of the failure loads of structures, and particularly the size effect on the strength of structures. The discussion of quasibrittle materials centers on concrete, but also addresses rocks, toughened ceramics, ice, composites, and other materials. The authors consider the statistical and fractal aspects of size effect. The work is intended to serve as both a textbook for graduate level engineering courses and a reference volume for engineers and scientists. Annotation c. by Book News, Inc., Portland, Or.

Fracture and Size Effect in Concrete and Other Quasibrittle Materials is the first in-depth text on the application of fracture mechanics to the analysis of failure in concrete structures. The book synthesizes a vast number of recent research results in the literature to provide a comprehensive treatment of the topic that does not give merely the facts - it provides true understanding.

The many recent results on quasibrittle fracture and size effect, which were scattered throughout many periodicals, are compiled here in a single volume. This book presents a well-rounded discussion of the theory of size effect and scaling of failure loads in structures. The size effect, which is the most important practical manifestation of fracture behavior, has become a hot topic. It has gained prominence in current research on concrete and quasibrittle materials.

The treatment of every subject in Fracture and Size Effect in Concrete and Other Quasibrittle Materials proceeds from simple to complex, from specialized to general, and is as concise as possible using the simplest level of mathematics necessary to treat the subject clearly and accurately. Whether you are an engineering student or a practicing engineer, this book provides you with a clear presentation, including full derivations and examples, from which you can gain real understanding of fracture and size effect in concrete and other quasibrittle materials.
Preface v(2)
Vector and Tensor Notation vii
1 Why Fracture Mechanics?
1(22)
1.1 Historical Perspective
1(4)
1.1.1 Classical Linear Theory
1(2)
1.1.2 Classical Nonlinear Theories
3(1)
1.1.3 Continuum-Based Theories
4(1)
1.1.4 Trends in Fracture of Quasibrittle Materials
5(1)
1.2 Reasons for Fracture Mechanics Approach
5(4)
1.2.1 Energy Required for Crack Formation
5(1)
1.2.2 Objectivity of Analysis
5(2)
1.2.3 Lack of Yield Plateau
7(1)
1.2.4 Energy Absorption Capability and Ductility
7(1)
1.2.5 Size Effect
7(2)
1.3 Sources of Size Effect on Structural Strength
9(2)
1.4 Quantification of Fracture Mechanics Size Effect
11(5)
1.4.1 Nominal Stress and Nominal Strength
11(2)
1.4.2 Size Effect Equations
13(1)
1.4.3 Simple Explanation of Fracture Mechanics Size Effect
13(3)
1.5 Experimental Evidence for Size Effect
16(7)
1.5.1 Structures with Notches or Cracks
18(1)
1.5.2 Structures Without Notches or Cracks
19(4)
2 Essentials of LEFM
23(26)
2.1 Energy Release Rate and Fracture Energy
23(14)
2.1.1 The General Energy Balance
24(1)
2.1.2 Elastic Potentials and Energy Release Rate
25(3)
2.1.3 The Linear Elastic Case and the Compliance Variation
28(2)
2.1.4 Graphical Representation of Fracture Processes
30(1)
2.1.5 Rice's J-Integral
31(3)
2.1.6 Fracture Criterion and Fracture Energy
34(3)
2.2 LEFM and Stress Intensity Factor
37(5)
2.2.1 The Center Cracked Infinite Panel and the Near-Tip Fields
37(2)
2.2.2 The General Near-Tip Fields and Stress Intensity Factors
39(1)
2.2.3 Relationship Between K(I) and G
40(1)
2.2.4 Local Fracture Criterion for Mode I: K(Ic)
41(1)
2.3 Size Effect in Plasticity and in LEFM
42(7)
2.3.1 Size Effect for Failures Characterized by Plasticity, Strength, or Allowable Stress
43(1)
2.3.2 General Forms of the Expressions for K(I) and G
44(1)
2.3.3 Size Effect in LEFM
45(1)
2.3.4 Structures Failing at Very Small Cracks Whose Size is a Material Property
46(3)
3 Determination of LEFM Parameters
49(26)
3.1 Setting up Solutions from Closed-Form Expressions
49(6)
3.1.1 Closed-Form Solutions from Handbooks
49(2)
3.1.2 Superposition Methods
51(4)
3.2 Approximate Energy-Based Methods
55(5)
3.2.1 Examples Approximately Solvable by Bending Theory
55(1)
3.2.2 Approximation by Stress Relief Zone
56(2)
3.2.3 Herrmann's Approximate Method to Obtain G by Beam Theory
58(1)
3.2.4 Subsurface Cracking in Compression by Bucking
59(1)
3.3 Numerical and Experimental Procedures to Obtain K(I) and G
60(4)
3.3.1 Numerical Procedures
60(3)
3.3.2 Experimental Procedures
63(1)
3.4 Experimental determination of K(Ic) and G(f)
64(3)
3.5 Calculation of Displacements from K(I)-Expressions
67(8)
3.5.1 Calculation of the Displacement
67(1)
3.5.2 Compliances, Energy Release Rate, and Stress Intensity Factor for a System of Loads
68(1)
3.5.3 Calculation of the Crack Mouth Opening Displacement
69(2)
3.5.4 Calculation of the Volume of the Crack
71(1)
3.5.5 Calculation of the Crack Opening Profile
72(1)
3.5.6 Bueckner's Expression for the Weight Function
73(2)
4 Advanced Aspects of LEFM
75(26)
4.1 Complex Variable Formulation of Plane Elasticity Problems
75(5)
4.1.1 Navier's Equations for the Plane Elastic Problem
75(1)
4.1.2 Complex Functions
76(1)
4.1.3 Complex Form of Hooke's and Navier's Equations
77(1)
4.1.4 Integration of Navier's Equation: Complex Potentials
77(3)
4.2 Plane Crack Problems and Westergaard's Stress Function
80(3)
4.2.1 Westergaard Stress Function
80(1)
4.2.2 Westergaard's Solution of Center-Cracked Infinite Panel
80(2)
4.2.3 Near-Tip Expansion for the Center-Cracked Panel
82(1)
4.3 The General Near-Tip Fields
83(7)
4.3.1 In-Plane Near-Tip Asymptotic Series Expansion
83(2)
4.3.2 The Stress Intensity Factors
85(1)
4.3.3 Closer View of the Near-Tip Asymptotic Expansion for Mode I
86(1)
4.3.4 The Antiplane Shear Mode
87(1)
4.3.5 Antiplane Near-Tip Asymptotic Series Expansion
88(1)
4.3.6 Summary: The General Singular Near-Tip Fields
89(1)
4.4 Path-Independent Contour Integrals
90(4)
4.4.1 Path Independence of the J-Integral
90(1)
4.4.2 Further Contour Integral Expressions for G in LEFM
91(1)
4.4.3 Further Proof of the Irwin Relationship
92(1)
4.4.4 Other Path-Independent Integrals
93(1)
4.4.5 Exercises
94(1)
4.5 Mixed Mode Fracture Criteria
94(4)
4.5.1 Maximum Energy Release Rate Criterion
95(1)
4.5.2 Maximum Principal Stress Criterion
96(2)
Appendix: Strain Energy Density Criterion
98(3)
5 Equivalent Elastic Cracks and R-curves
101(34)
5.1 Variability of Apparent Fracture Toughness for Concrete
101(2)
5.2 Types of Fracture Behavior and Nonlinear Zone
103(5)
5.2.1 Brittle, Ductile, and Quasibrittle Behavior
104(1)
5.2.2 Irwin's Estimate of the Size of the Inelastic Zone
105(1)
5.2.3 Estimate of the Fracture Zone Size for quasibrittle Materials
106(2)
5.3 The Equivalent Elastic Crack Concept
108(4)
5.3.1 Estimate of the Equivalent LEFM Crack Extension
109(1)
5.3.2 Deviation from LEFM
109(1)
5.3.3 Intrinsic Size
110(1)
5.3.4 How Large the Size Must Be for LEFM to Apply?
111(1)
5.4 Fracture Toughness Determinations Based on Equivalent Crack Concepts
112(4)
5.4.1 Compliance Calibration of Equivalent Crack Length
112(1)
5.4.2 Modified Compliance Calibration Method
113(1)
5.4.3 Nallathambi-Karihaloo Method
114(2)
5.5 Two-Parameter Model of Jenq and Shah
116(5)
5.5.1 The Basic Equations of Jenq-Shah Model
117(2)
5.5.2 Experimental Determination of Jenq-Shah Parameters
119(2)
5.6 R-Curves
121(9)
5.6.1 Definition of an R-XXXa Curve
121(2)
5.6.2 Description of the Fracture Process
123(1)
5.6.3 The Peak Load Condition
124(2)
5.6.4 Positive and Negative Geometries
126(1)
5.6.5 R-Curve Determination from Tests
126(2)
5.6.6 R-CTOD Curves
128(2)
5.7 Stability Analysis in the R-Curve Approach
130(5)
5.7.1 Stability under Load-Control Conditions
130(1)
5.7.2 Stability under Displacement-Control Conditions
131(1)
5.7.3 Stability under Mixed-Control Conditions
131(4)
6 Determination of Fracture Properties From Size Effect
135(22)
6.1 Size Effect in Equivalent Elastic Crack Approximations
135(3)
6.1.1 Size Effect in the Large Size Range
135(1)
6.1.2 Size Effect in the Jenq-Shah Model
136(2)
6.2 Size Effect Law in Relation to Fracture Characteristics
138(2)
6.2.1 Defining Objective Fracture Properties
138(1)
6.2.2 Determination of Fracture Parameters from Size Effect
138(1)
6.2.3 Determination of Fracture Parameters from Size an Shape Effects and Zero Brittleness Method
139(1)
6.2.4 Intrinsic Representation of the Size Effect Law
139(1)
6.3 Size Effect Method: Detailed Experimental Procedures
140(10)
6.3.1 Outline of the Method
140(1)
6.3.2 Regression Relations
140(3)
6.3.3 RILEM Recommendation Using the Size Effect Method: Experimental Procedure
143(1)
6.3.4 RILEM Recommendation Using the Size Effect Method: Calculation Procedure
144(3)
6.3.5 Performance of the Size Effect Method
147(1)
6.3.6 Improved Regression Relations
147(3)
6.4 Determination of R-Curve from Size Effect
150(7)
6.4.1 Determination of R-Curve from Size Effect
150(2)
6.4.2 Determination of R-Curve from Bazant's Size Effect Law
152(2)
6.4.3 Determination of the Structural Response from the R-Curve
154(3)
7 Cohesive Crack Models
157(56)
7.1 Basic Concepts in Cohesive Crack Model
157(10)
7.1.1 Hillerborg's Approach: The Cohesive Crack as a Constitutive Relation
158(2)
7.1.2 Other Approaches to Cohesive Cracks
160(2)
7.1.3 Softening Curve, Fracture Energy, and Other Properties
162(2)
7.1.4 Extensions of the Cohesive Crack Model
164(1)
7.1.5 Cohesive Cracks with Tip Singularity
165(1)
7.1.6 Cohesive Cracks with Bulk Energy Dissipation
165(2)
7.2 Cohesive Crack Models Applied to Concrete
167(13)
7.2.1 Softening Curves for Concrete
167(3)
7.2.2 Experimental Aspects
170(2)
7.2.3 Computational Procedures for Cohesive Crack Analysis
172(3)
7.2.4 Size Effect Predictions
175(2)
7.2.5 Cohesive Crack Models in Relation to Effective Elastic Crack Models
177(1)
7.2.6 Correlation of Cohesive Crack with Bazant's and Jenq and Shah's Models
178(2)
7.3 Experimental Determination of Cohesive Crack Properties
180(10)
7.3.1 Determination of the Tensile Strength
181(1)
7.3.2 Determination of the Initial Part of the Softening Curve
182(2)
7.3.3 Determination of Fracture Energy G(F)
184(4)
7.3.4 Determination of a Bilinear Softening Curve
188(2)
7.4 Pseudo-Boundary-Integral Methods for Mode I Crack Growth
190(9)
7.4.1 The Underlying Problem
190(1)
7.4.2 Petersson's Influence Method
191(1)
7.4.3 Improved Solution Algorithm of Planas and Elices
192(1)
7.4.4 Smeared-Tip Method
193(2)
7.4.5 Scaling of the Influence Matrices
195(1)
7.4.6 Inclusion of Shrinkage or Thermal Stresses
196(1)
7.4.7 Inclusion of a Crack-Tip Singularity
197(1)
7.4.8 Computation of Other Variables
198(1)
7.4.9 Limitations of the Pseudo-Boundary Integral (PBI) Methods
199(1)
7.5 Boundary-Integral Methods for Mode I Crack Growth
199(14)
7.5.1 A Basic Boundary Integral Formulation
199(3)
7.5.2 Size-Dependence of the Equations
202(1)
7.5.3 The Dugdale and Rectangular Softening Cases
203(1)
7.5.4 Eigenvalue Analysis of the Size Effect
204(2)
7.5.5 Eigenvalue Analysis of Stability Limit and Ductility of Structure
206(1)
7.5.6 Smeared-Tip Superposition Method
207(2)
7.5.7 Asymptotic Analysis
209(4)
8 Crack Band Models and Smeared Cracking
213(48)
8.1 Strain Localization in the Series Coupling Model
213(4)
8.1.1 Series Coupling of Two Equal Strain Softening Elements: Imperfection Approach
214(1)
8.1.2 Series Coupling of Two Equal Strain Softening Elements: Thermodynamic Approach
215(1)
8.1.3 Mean Stress and Mean Strain
215(1)
8.1.4 Series Coupling of N Equal Strain Softening Elements
216(1)
8.2 Localization of Strain in a Softening Bar
217(3)
8.2.1 Localization and Mesh Objectivity
217(1)
8.2.2 Localization in an Elastic-Softening Bar
218(1)
8.2.3 Summary: Necessity of Localization Limiters
219(1)
8.3 Basic Concepts in Crack Band Models
220(8)
8.3.1 Elastic-Softening Crack Band Models
220(2)
8.3.2 Band Models with Bulk Dissipation
222(1)
8.3.3 Unloading and Reloading
223(1)
8.3.4 Fracture Energy for Crack Bands With Prepeak Energy Dissipation
224(1)
8.3.5 Simple Numerical Issues
225(1)
8.3.6 Crack Band Width
226(2)
8.4 Uniaxial Softening Models
228(6)
8.4.1 Elastic-Softening Model with Stiffness Degradation
228(1)
8.4.2 Elastic-Softening Model with Strength Degradation
229(1)
8.4.3 Elastic-Softening Model with Stiffness and Strength Degradation
229(1)
8.4.4 A Simple Continuum Damage Model
230(1)
8.4.5 Introducing Inelasticity Prior to the Peak
231(1)
8.4.6 Crack Closure in Reverse Loading and Compression
231(1)
8.4.7 Introducing Other Inelastic Effects
232(2)
8.5 Simple Triaxial Strain-Softening Models for Smeared Cracking
234(12)
8.5.1 Cracking of Single Fixed Orientation: Basic Concepts
234(1)
8.5.2 Secant Approach to Cracking of Fixed Orientation
235(2)
8.5.3 Scalar Damage Model for Cracking of Fixed Orientation
237(1)
8.5.4 Incremental Approach to Cracking of Fixed Orientation
238(1)
8.5.5 Multi-Directional Fixed Cracking
239(1)
8.5.6 Rotating Crack Model
240(2)
8.5.7 Generalized Constitutive Equations with Softening
242(1)
8.5.8 Mazars' Scalar Damage Model
243(1)
8.5.9 Rankine Plastic Model with Softening
243(1)
8.5.10 A Simple Model with Stiffness and Strength Degradation
244(2)
8.6 Crack Band Models and Smeared Cracking
246(9)
8.6.1 Stress-Strain Relations for Elements of Arbitrary Size
246(2)
8.6.2 Skew Meshes: Effective Width
248(2)
8.6.3 Stress Lock-In
250(1)
8.6.4 Use of Elements of Large Size
251(1)
8.6.5 Energy Criterion for Crack Bands with Sudden Cracking
252(3)
8.7 Comparison of Crack Band and Cohesive Crack Approaches
255(6)
8.7.1 Localized fracture: Moot Point Computationally
255(1)
8.7.2 Nonlocalized Fracture: Third Parameter
255(2)
8.7.3 Relation to Micromechanics of Fracture
257(1)
8.7.4 Fracture of Arbitrary Direction
258(3)
9 Advanced Size Effect Analysis
261(58)
9.1 Size Effect Law Refinements
261(10)
9.1.1 The Generalized Energy Balance Equation
261(2)
9.1.2 Asymptotic Analysis for Large Size
263(1)
9.1.3 Matching to the Effective Crack Model
263(1)
9.1.4 Asymptotic Formula for Small Sizes and Its Asymptotic Matching with Large Sizes
264(1)
9.1.5 Asymptotic Aspects of Bazant's Extended Size Effect Law
265(1)
9.1.6 Size Effect for Failures at Crack Initation from Smooth Surface
266(2)
9.1.7 Universal Size Effect Law for Cracked and Uncracked Structures
268(1)
9.1.8 Asymptotic Scaling Law for Many Loads
269(1)
9.1.9 Asymptotic Scaling Law for a Crack with Residual Bridging Stress
270(1)
9.2 Size Effect in Notched Structures Based on Cohesive Crack Models
271(9)
9.2.1 The General Size Effect Equation
271(2)
9.2.2 Asymptotic Analysis for Large Sizes
273(1)
9.2.3 Asymptotic Analysis for Small Sizes
274(1)
9.2.4 Interpolation Formula
275(2)
9.2.5 Application to Notched Beams with Linear Softening
277(1)
9.2.6 Application to Notched Beams with Bilinear Softening
277(2)
9.2.7 Experimental Evidence
279(1)
9.3 Size Effect on the Modulus of Rupture of Concrete
280(11)
9.3.1 Notation and Definition of the Rupture Modulus
280(1)
9.3.2 Modulus of Rupture Predicted by Cohesive Cracks
281(3)
9.3.3 Further Analysis of the Influence of the Initial Softening
284(1)
9.3.4 Modulus of Rupture According to Bazant and Li's Model, Bazant's Universal Size Effect Law, and Zero-Brittleness Method
284(3)
9.3.5 Modulus of Rupture Predicted by Jenq-Shah Model
287(1)
9.3.6 Carpinteri's Multifractal Scaling Law
288(1)
9.3.7 Comparison With Experiments and Final Remarks
289(2)
9.4 Compression Splitting Tests of Tensile Strength
291(6)
9.4.1 Cracking Process in Stable Splitting Tests
292(2)
9.4.2 Modified Bazant's Size Effect Law
294(1)
9.4.3 Size Effect Predicted by Jenq-Shah Model
295(1)
9.4.4 Size Effect Predicted by Cohesive Crack Models
296(1)
9.5 Compression Failure Due to Propagation of Splitting Crack Band
297(15)
9.5.1 Concepts and Mechanisms of Compression Fracture
297(3)
9.5.2 Energy Analysis of Compression Failure of Column
300(5)
9.5.3 Asymptotic Effect for Large Size
305(1)
9.5.4 Size Effect Law for Axial Compression of Stocky Column
305(2)
9.5.5 Effect of Buckling Due to Slenderness
307(1)
9.5.6 Comparison with Experimental Data
308(2)
9.5.7 The Question of Variation of Microcrack Spacing with Size D
310(1)
9.5.8 Special Case of Compression with Transverse Tension
310(1)
9.5.9 Distinction Between Axial Splitting and Failure Appearing as Shear
311(1)
9.6 Scaling of Fracture of Sea Ice
312(7)
9.6.1 Derivation of Size Effect for Thermal Bending Fracture of Ice Plate
314(2)
9.6.2 General Proof of 3/8-Power Scaling Law
316(3)
10 Brittleness and Size Effect in Structural Design
319(64)
10.1 General Aspects of Size Effect and Brittleness in Concrete Structures
319(7)
10.1.1 Conditions for Extending Bazant's Size Effect Law to Structures
320(1)
10.1.2 Brittleness Number
321(2)
10.1.3 Brittleness of High Strength Concrete
323(1)
10.1.4 Size Effect Correction to Ultimate Load Formulas in Codes
323(1)
10.1.5 Size Effect Correction to Strength-Based Formulas
324(1)
10.1.6 Effect of Reinforcement
325(1)
10.2 Diagonal Shear Failure of Beams
326(9)
10.2.1 Introduction
326(1)
10.2.2 Bazant-Kim-Sun Formulas
327(3)
10.2.3 Gustafsson-Hillerborg Analysis
330(1)
10.2.4 LEFM Analysis of Jenq and Shah and of Karihaloo
331(3)
10.2.5 Finite Element Solutions with Nonlocal Microplane Model
334(1)
10.2.6 Influence of Prestressing on Diagonal Shear Strength
334(1)
10.3 Fracturing Truss Model for Shear Failure of Beams
335(14)
10.3.1 Basic Hypotheses of Fracturing Truss Model
336(1)
10.3.2 Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams Without Stirrups
337(4)
10.3.3 Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups
341(3)
10.3.4 Analysis Based on Stress Redistribution and Complementary Energy
344(2)
10.3.5 Size Effect on Nominal Stress at Cracking Load
346(3)
10.3.6 Conclusions
349(1)
10.4 Reinforced Beams in Flexure and Minimum Reinforcement
349(16)
10.4.1 Lightly Reinforced Beams: Overview
349(1)
10.4.2 Models Based on LEFM
350(6)
10.4.3 Simplified Cohesive Crack Models
356(1)
10.4.4 Models Based on Cohesive Cracks
357(6)
10.4.5 Formulas for Minimum Reinforcement Based on Fracture Mechanics
363(2)
10.5 Other Structures
365(18)
10.5.1 Torsional Failure of Beams
365(1)
10.5.2 Punching Shear Failure of Slabs
366(1)
10.5.3 Anchor Pullout
367(1)
10.5.4 Bond and Slip of Reinforcing Bars
368(3)
10.5.5 Beam and Ring Failures of Pipes
371(1)
10.5.6 Concrete Dams
372(3)
10.5.7 Footings
375(1)
10.5.8 Crack Spacing and Width, with Application to Highway Pavements
376(1)
10.5.9 Keyed Joints
377(1)
10.5.10 Fracture in Joints
377(2)
10.5.11 Break-Out of Boreholes
379(1)
10.5.12 Hillerborg's Model for Compressive Failure in Concrete Beams
380(3)
11 Effect of Time, Environment, and Fatigue
383(54)
11.1 Phenomenology of Time-Dependent Fracture
384(6)
11.1.1 Types of Time-Dependent Fracture
384(1)
11.1.2 Influence of Loading Rate on Peak Load and on Size Effect
385(1)
11.1.3 Load Relaxation
386(2)
11.1.4 Creep Fracture Tests
388(1)
11.1.5 Sudden Change of Loading Rate
388(1)
11.1.6 Dynamic Fracture
389(1)
11.2 Activation Energy Theory and Rate Processes
390(8)
11.2.1 Elementary Rate Constants
391(1)
11.2.2 Physical Rate Constants
391(3)
11.2.3 Fracture as a Rate Process
394(1)
11.2.4 General Aspects of Isothermal Crack Growth Analysis
395(1)
11.2.5 Load-Controlled Processes for Power-Law Rate Equation
396(1)
11.2.6 Displacement-Controlled Processes for Power-Law Rate Equation
397(1)
11.3 Some Applications of the Rate Process Theory to Concrete Fracture
398(6)
11.3.1 Effect of Temperature on Fracture Energy of Concrete
398(1)
11.3.2 Effect of Humidity on the Fracture Energy of Concrete
399(2)
11.3.3 Time-Dependent Generalization of R-Curve Model
401(2)
11.3.4 Application of the Time-Dependent R-Curve Model to Limestone
403(1)
11.4 Linear Viscoelastic Fracture Mechanics
404(14)
11.4.1 Uniaxial Linear Viscoelasticity
404(3)
11.4.2 Compliance Functions for Concrete
407(1)
11.4.3 General Linear Viscoelastic Constitutive Equations
408(1)
11.4.4 The Correspondence Principle (Elastic-Viscoelastic Analogy)
408(1)
11.4.5 Near-Tip Stress and Displacement Fields for a Crack in a Viscoelastic Structure
409(3)
11.4.6 Crack Growth Resistance in a Viscoelastic Medium
412(1)
11.4.7 Steady Growth of a Cohesive Crack with Rectangular Softening in an Infinite Viscoelastic Plate
413(3)
11.4.8 Analysis of Crack Growth in a Viscoelastic Plate
416(1)
11.4.9 Crack Growth Analysis at Controlled Displacement
417(1)
11.5 Rate-Dependent R-Curve Model with Creep
418(4)
11.5.1 Basic Equations
418(1)
11.5.2 Approximate Solution for Small Crack Extensions
419(1)
11.5.3 Comparison with Tests
419(1)
11.5.4 Rate-Dependence of Process Zone Length
420(1)
11.5.5 Sudden Change of Loading Rate and Load Relaxation
420(2)
11.5.6 Summary
422(1)
11.6 Time Dependent Cohesive Crack and Crack Band Models
422(7)
11.6.1 Time-Independent Softening in a Viscoelastic Body
423(1)
11.6.2 Time-Dependent Softening in an Elastic Body
424(1)
11.6.3 Time-Dependent Cohesive Crack Model
425(1)
11.6.4 Analysis of Viscoelastic Structure with Rate-Dependent Cohesive Crack by Finite Elements
426(2)
11.6.5 Analysis of Viscoelastic Structure with Rate-Depedent Cohesive Crack by Compliance Functions
428(1)
11.7 Introduction to Fatigue Fracture and Its Size Dependence
429(8)
11.7.1 Fatigue Crack Growth in Metals
430(1)
11.7.2 Fatigue Crack Growth in Brittle Materials
431(1)
11.7.3 Size Effect in Fatigue Crack Growth in Concrete
432(2)
11.7.4 Fatigue Description by History-Dependent Cohesive Models
434(3)
12 Statistical Theory of Size Effect and Fracture Process
437(52)
12.1 Review of Classical Weibull Theory
439(10)
12.1.1 The Weakest-Link Discrete Model
439(1)
12.1.2 The Weakest-Link Model for Continuous Structures under Uniaxial Stress
440(1)
12.1.3 The Weibull Statistical Probability Distribution
441(2)
12.1.4 Structures with Nonhomogeneous Uniaxial Stress
443(2)
12.1.5 Generalization to Triaxial Stress States
445(1)
12.1.6 Independent Failure Mechanisms: Additivity of the Concentration Function
446(1)
12.1.7 Effective Uniaxial Stress
447(1)
12.1.8 Summary: Nonhomogeneous States of Stress
447(2)
12.2 Statistical Size Effect due to Random Strength
449(7)
12.2.1 General Strength Probability Distribution and Equivalent Uniaxial Volume
449(2)
12.2.2 Statistical Size Effect Laws
451(1)
12.2.3 Divergence of Weibull Failure Probability for Sharply Cracked Bodies
452(2)
12.2.4 The Effect of Surface Flaws
454(2)
12.3 Basic Criticisms of Classical Weibull-Type Approach
456(4)
12.3.1 Stress Redistribution
456(1)
12.3.2 Equivalence to Uniaxially Stressed Bar
457(1)
12.3.3 Differences between Two-and and Three-Dimensional Geometric Similarities
458(1)
12.3.4 Energy Release Due to Large Stable Crack Growth
459(1)
12.3.5 Spatial Correlation
460(1)
12.3.6 Summary of the Limitations
460(1)
12.4 Handling of Stress Singularity in Weibull-Type Approach
460(5)
12.4.1 A Simplified Approach to Crack Tip Statistics
461(1)
12.4.2 Generalization of the Thickness Dependence of the Crack Tip Statistics
462(1)
12.4.3 Asymptotic Size Effect
463(1)
12.4.4 Extending the Range: Bulk Plus Core Statistics
463(1)
12.4.5 More Fundamental Approach Based on Nonlocal Concept
464(1)
12.5 Approximate Equations for Statistical Size Effect
465(5)
12.5.1 Bazant-Xi Empirical Interpolation Between Asymptotic Size Effects
465(1)
12.5.2 Determination of Material Parameters
465(1)
12.5.3 The Question of Weibull Modulus m for the Fracture-Process Zone
466(1)
12.5.4 Comparison with Test Results
466(1)
12.5.5 Planas' Empirical Interpolation Between Asymptotic Size Effects
467(3)
12.5.6 Limitations of Generalized Weibull Theory
470(1)
12.6 Another View: Crack Growth in an Elastic Random Medium
470(9)
12.6.1 The Strongest Random Barrier Model
471(1)
12.6.2 The Statistical R-Curve
472(1)
12.6.3 Finite Bodies
472(2)
12.6.4 Frechet's Failure Probability Distribution
474(2)
12.6.5 Random R-curve
476(3)
12.6.6 Limitations of the Random Barrier Model
479(1)
12.7 Fractal Approach to Fracture and Size Effect
479(10)
12.7.1 Basic Concepts on Fractals
480(2)
12.7.2 Invasive Fractal and Multifractal Size Effect for G(F)
482(1)
12.7.3 Lacunar Fractal and Multifractal Size Effect for XXX N(u)
482(1)
12.7.4 Fracture Analysis of Fractal Crack Propagation
483(2)
12.7.5 Bazant's Analysis of Fractal Crack Initiation
485(1)
12.7.6 Is Fractality the Explanation of Size Effect?
486(3)
13 Nonlocal Continuum Modeling of Damage Localization
489(38)
13.1 Basic Concepts in Nonlocal Approaches
490(11)
13.1.1 The Early Approaches
490(1)
13.1.2 Models with Nonlocal Strain
491(1)
13.1.3 Gradient Models
492(1)
13.1.4 A Simple Family of Nonlocal Models
493(2)
13.1.5 A Second-Order Differential Model
495(1)
13.1.6 An Integral-Type Model of the First Kind
496(1)
13.1.7 An Integral-Type Model of the Second Kind
497(1)
13.1.8 Nonlocal Damage Model
498(3)
13.2 Triaxial Nonlocal Models and Applications
501(6)
13.2.1 Triaxial Nonlocal Smeared Cracking Models
502(1)
13.2.2 Triaxial Nonlocal Models with Yield Limit Degradation
502(4)
13.2.3 Nonlocal Microplane Model
506(1)
13.2.4 Determination of Characteristic Length
506(1)
13.3 Nonlocal Model Based on Micromechanics of Crack Interactions
507(20)
13.3.1 Nonlocality Caused by Interaction of Growing Microcracks
507(3)
13.3.2 Field Equation for Nonlocal Continuum
510(1)
13.3.3 Some Alternative Forms and Properties of the Nonlocal Model
511(2)
13.3.4 Admissibility of Uniform Inelastic Stress Fields
513(1)
13.3.5 Gauss-Seidel Iteration Applied to Nonlocal Averaging
514(1)
13.3.6 Statistical Determination of Crack Influence Function
515(2)
13.3.7 Crack Influence Function in Two Dimensions
517(3)
13.3.8 Crack Influence Function in Three Dimensions
520(2)
13.3.9 Cracks Near Boundary
522(1)
13.3.10 Long-Range Decay and Integrability
523(1)
13.3.11 General Formulation: Tensorial Crack Influence Function
523(1)
13.3.12 Constitutive Relation and Gradient Approximation
524(1)
13.3.13 Localization of Oriented Cracking into a Band
525(1)
13.3.14 Summary
525(2)
14 Material Models for Damage and Failure
527(38)
14.1 Microplane Model
528(15)
14.1.1 Macro-Micro Relations
529(3)
14.1.2 Volumetric-Deviatoric Split of the Microstrain and Microstress Vectors
532(1)
14.1.3 Elastic Response
533(2)
14.1.4 Nonlinear Microplane Behavior and the Concept of Stress-Strain Boundaries
535(2)
14.1.5 Numerical Aspects
537(1)
14.1.6 Constitutive Characterization of Material on Microplane Level
538(2)
14.1.7 Microplane Model for Finite Strain
540(2)
14.1.8 Summary of Main Points
542(1)
14.2 Calibration by Test Data, Verification and Properties of Microplane Model
543(5)
14.2.1 Procedure for Delocalization of Test Data and Material Identification
543(2)
14.2.2 Calibration of Microplane Model and Comparison with Test Data
545(1)
14.2.3 Vertex Effects
545(2)
14.2.4 Other Aspects
547(1)
14.3 Nonlocal Adaptation of Microplane Model or Other Constitutive Models
548(2)
14.4 Particle and Lattice Models
550(10)
14.4.1 Truss, Frame, and Lattice Models
552(2)
14.4.2 Directional Bias
554(1)
14.4.3 Examples of Results of Particle and Lattice Models
555(4)
14.4.4 Summary and Limitations
559(1)
14.5 Tangential Stiffness Tensor Via Solution of a Body with Many Growing Cracks
560(5)
References 565(34)
Reference Citation Index 599(8)
Index 607


Zdenek P. Bazant, Jaime Planas
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