Fracture and Size Effect in Concrete and Other Quasibrittle Materials

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ISBN: 9780849382840 | 084938284X
Cover: Hardcover
Copyright: 12/29/1997

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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

(2)

Vector and Tensor Notation

vii

1 Why Fracture Mechanics?

(22)

1.1 Historical Perspective

(4)

1.1.1 Classical Linear Theory

(2)

1.1.2 Classical Nonlinear Theories

(1)

1.1.3 Continuum-Based Theories

(1)

1.1.4 Trends in Fracture of Quasibrittle Materials

(1)

1.2 Reasons for Fracture Mechanics Approach

(4)

1.2.1 Energy Required for Crack Formation

(1)

1.2.2 Objectivity of Analysis

(2)

1.2.3 Lack of Yield Plateau

(1)

1.2.4 Energy Absorption Capability and Ductility

(1)

1.2.5 Size Effect

(2)

1.3 Sources of Size Effect on Structural Strength

(2)

1.4 Quantification of Fracture Mechanics Size Effect

(5)

1.4.1 Nominal Stress and Nominal Strength

(2)

1.4.2 Size Effect Equations

(1)

1.4.3 Simple Explanation of Fracture Mechanics Size Effect

(3)

1.5 Experimental Evidence for Size Effect

(7)

1.5.1 Structures with Notches or Cracks

(1)

1.5.2 Structures Without Notches or Cracks

(4)

2 Essentials of LEFM

(26)

2.1 Energy Release Rate and Fracture Energy

(14)

2.1.1 The General Energy Balance

(1)

2.1.2 Elastic Potentials and Energy Release Rate

(3)

2.1.3 The Linear Elastic Case and the Compliance Variation

(2)

2.1.4 Graphical Representation of Fracture Processes

(1)

2.1.5 Rice's J-Integral

(3)

2.1.6 Fracture Criterion and Fracture Energy

(3)

2.2 LEFM and Stress Intensity Factor

(5)

2.2.1 The Center Cracked Infinite Panel and the Near-Tip Fields

(2)

2.2.2 The General Near-Tip Fields and Stress Intensity Factors

(1)

2.2.3 Relationship Between K(I) and G

(1)

2.2.4 Local Fracture Criterion for Mode I: K(Ic)

(1)

2.3 Size Effect in Plasticity and in LEFM

(7)

2.3.1 Size Effect for Failures Characterized by Plasticity, Strength, or Allowable Stress

(1)

2.3.2 General Forms of the Expressions for K(I) and G

(1)

2.3.3 Size Effect in LEFM

(1)

2.3.4 Structures Failing at Very Small Cracks Whose Size is a Material Property

(3)

3 Determination of LEFM Parameters

(26)

3.1 Setting up Solutions from Closed-Form Expressions

(6)

3.1.1 Closed-Form Solutions from Handbooks

(2)

3.1.2 Superposition Methods

(4)

3.2 Approximate Energy-Based Methods

(5)

3.2.1 Examples Approximately Solvable by Bending Theory

(1)

3.2.2 Approximation by Stress Relief Zone

(2)

3.2.3 Herrmann's Approximate Method to Obtain G by Beam Theory

(1)

3.2.4 Subsurface Cracking in Compression by Bucking

(1)

3.3 Numerical and Experimental Procedures to Obtain K(I) and G

(4)

3.3.1 Numerical Procedures

(3)

3.3.2 Experimental Procedures

(1)

3.4 Experimental determination of K(Ic) and G(f)

(3)

3.5 Calculation of Displacements from K(I)-Expressions

(8)

3.5.1 Calculation of the Displacement

(1)

3.5.2 Compliances, Energy Release Rate, and Stress Intensity Factor for a System of Loads

(1)

3.5.3 Calculation of the Crack Mouth Opening Displacement

(2)

3.5.4 Calculation of the Volume of the Crack

(1)

3.5.5 Calculation of the Crack Opening Profile

(1)

3.5.6 Bueckner's Expression for the Weight Function

(2)

4 Advanced Aspects of LEFM

(26)

4.1 Complex Variable Formulation of Plane Elasticity Problems

(5)

4.1.1 Navier's Equations for the Plane Elastic Problem

(1)

4.1.2 Complex Functions

(1)

4.1.3 Complex Form of Hooke's and Navier's Equations

(1)

4.1.4 Integration of Navier's Equation: Complex Potentials

(3)

4.2 Plane Crack Problems and Westergaard's Stress Function

(3)

4.2.1 Westergaard Stress Function

(1)

4.2.2 Westergaard's Solution of Center-Cracked Infinite Panel

(2)

4.2.3 Near-Tip Expansion for the Center-Cracked Panel

(1)

4.3 The General Near-Tip Fields

(7)

4.3.1 In-Plane Near-Tip Asymptotic Series Expansion

(2)

4.3.2 The Stress Intensity Factors

(1)

4.3.3 Closer View of the Near-Tip Asymptotic Expansion for Mode I

(1)

4.3.4 The Antiplane Shear Mode

(1)

4.3.5 Antiplane Near-Tip Asymptotic Series Expansion

(1)

4.3.6 Summary: The General Singular Near-Tip Fields

(1)

4.4 Path-Independent Contour Integrals

(4)

4.4.1 Path Independence of the J-Integral

(1)

4.4.2 Further Contour Integral Expressions for G in LEFM

(1)

4.4.3 Further Proof of the Irwin Relationship

(1)

4.4.4 Other Path-Independent Integrals

(1)

4.4.5 Exercises

(1)

4.5 Mixed Mode Fracture Criteria

(4)

4.5.1 Maximum Energy Release Rate Criterion

(1)

4.5.2 Maximum Principal Stress Criterion

(2)

Appendix: Strain Energy Density Criterion

(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

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Fracture and Size Effect in Concrete and Other Quasibrittle Materials

Summary

Table of Contents

Supplemental Materials