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Fracture and Size Effect in Concrete and Other Quasibrittle Materials
By: Jaime Planas, Zdenek P. Bazant
Hardcover | 29 December 1997 | Edition Number 1
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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.
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 | p. v |
| Vector and Tensor Notation | p. vii |
| Why Fracture Mechanics? | p. 1 |
| Historical Perspective | p. 1 |
| Classical Linear Theory | p. 1 |
| Classical Nonlinear Theories | p. 3 |
| Continuum-Based Theories | p. 4 |
| Trends in Fracture of Quasibrittle Materials | p. 5 |
| Reasons for Fracture Mechanics Approach | p. 5 |
| Energy Required for Crack Formation | p. 5 |
| Objectivity of Analysis | p. 5 |
| Lack of Yield Plateau | p. 7 |
| Energy Absorption Capability and Ductility | p. 7 |
| Size Effect | p. 7 |
| Sources of Size Effect on Structural Strength | p. 9 |
| Quantification of Fracture Mechanics Size Effect | p. 11 |
| Nominal Stress and Nominal Strength | p. 11 |
| Size Effect Equations | p. 13 |
| Simple Explanation of Fracture Mechanics Size Effect | p. 13 |
| Experimental Evidence for Size Effect | p. 16 |
| Structures with Notches or Cracks | p. 18 |
| Structures Without Notches or Cracks | p. 19 |
| Essentials of LEFM | p. 23 |
| Energy Release Rate and Fracture Energy | p. 23 |
| The General Energy Balance | p. 24 |
| Elastic Potentials and Energy Release Rate | p. 25 |
| The Linear Elastic Case and the Compliance Variation | p. 28 |
| Graphical Representation of Fracture Processes | p. 30 |
| Rice's J-Integral | p. 31 |
| Fracture Criterion and Fracture Energy | p. 34 |
| LEFM and Stress Intensity Factor | p. 37 |
| The Center Cracked Infinite Panel and the Near-Tip Fields | p. 37 |
| The General Near-Tip Fields and Stress Intensity Factors | p. 39 |
| Relationship Between K[subscript I] and G | p. 40 |
| Local Fracture Criterion for Mode I: K[subscript Ic] | p. 41 |
| Size Effect in Plasticity and in LEFM | p. 42 |
| Size Effect for Failures Characterized by Plasticity, Strength, or Allowable Stress | p. 43 |
| General Forms of the Expressions for K[subscript I] and G | p. 44 |
| Size Effect in LEFM | p. 45 |
| Structures Failing at Very Small Cracks Whose Size is a Material Property | p. 46 |
| Determination of LEFM Parameters | p. 49 |
| Setting up Solutions from Closed-Form Expressions | p. 49 |
| Closed-Form Solutions from Handbooks | p. 49 |
| Superposition Methods | p. 51 |
| Approximate Energy-Based Methods | p. 55 |
| Examples Approximately Solvable by Bending Theory | p. 55 |
| Approximation by Stress Relief Zone | p. 56 |
| Herrmann's Approximate Method to Obtain G by Beam Theory | p. 58 |
| Subsurface Cracking in Compression by Buckling | p. 59 |
| Numerical and Experimental Procedures to Obtain K[subscript I] and G | p. 60 |
| Numerical Procedures | p. 60 |
| Experimental Procedures | p. 63 |
| Experimental determination of K[subscript Ic] and G[subscript f] | p. 64 |
| Calculation of Displacements from K[subscript I]-Expressions | p. 67 |
| Calculation of the Displacement | p. 67 |
| Compliances, Energy Release Rate, and Stress Intensity Factor for a System of Loads | p. 68 |
| Calculation of the Crack Mouth Opening Displacement | p. 69 |
| Calculation of the Volume of the Crack | p. 71 |
| Calculation of the Crack Opening Profile | p. 72 |
| Bueckner's Expression for the Weight Function | p. 73 |
| Advanced Aspects of LEFM | p. 75 |
| Complex Variable Formulation of Plane Elasticity Problems | p. 75 |
| Navier's Equations for the Plane Elastic Problem | p. 75 |
| Complex Functions | p. 76 |
| Complex Form of Hooke's and Navier's Equations | p. 77 |
| Integration of Navier's Equation: Complex Potentials | p. 77 |
| Plane Crack Problems and Westergaard's Stress Function | p. 80 |
| Westergaard Stress Function | p. 80 |
| Westergaard's Solution of Center-Cracked Infinite Panel | p. 80 |
| Near-Tip Expansion for the Center-Cracked Panel | p. 82 |
| The General Near-Tip Fields | p. 83 |
| In-Plane Near-Tip Asymptotic Series Expansion | p. 83 |
| The Stress Intensity Factors | p. 85 |
| Closer View of the Near-Tip Asymptotic Expansion for Mode I | p. 86 |
| The Antiplane Shear Mode | p. 87 |
| Antiplane Near-Tip Asymptotic Series Expansion | p. 88 |
| Summary: The General Singular Near-Tip Fields | p. 89 |
| Path-Independent Contour Integrals | p. 90 |
| Path Independence of the J-Integral | p. 90 |
| Further Contour Integral Expressions for G in LEFM | p. 91 |
| Further Proof of the Irwin Relationship | p. 92 |
| Other Path-Independent Integrals | p. 93 |
| Exercises | p. 94 |
| Mixed Mode Fracture Criteria | p. 94 |
| Maximum Energy Release Rate Criterion | p. 95 |
| Maximum Principal Stress Criterion | p. 96 |
| Strain Energy Density Criterion | p. 98 |
| Equivalent Elastic Cracks and R-curves | p. 101 |
| Variability of Apparent Fracture Toughness for Concrete | p. 101 |
| Types of Fracture Behavior and Nonlinear Zone | p. 103 |
| Brittle, Ductile, and Quasibrittle Behavior | p. 104 |
| Irwin's Estimate of the Size of the Inelastic Zone | p. 105 |
| Estimate of the Fracture Zone Size for quasibrittle Materials | p. 106 |
| The Equivalent Elastic Crack Concept | p. 108 |
| Estimate of the Equivalent LEFM Crack Extension | p. 109 |
| Deviation from LEFM | p. 109 |
| Intrinsic Size | p. 110 |
| How Large the Size Must Be for LEFM to Apply? | p. 111 |
| Fracture Toughness Determinations Based on Equivalent Crack Concepts | p. 112 |
| Compliance Calibration of Equivalent Crack Length | p. 112 |
| Modified Compliance Calibration Method | p. 113 |
| Nallathambi-Karihaloo Method | p. 114 |
| Two-Parameter Model of Jenq and Shah | p. 116 |
| The Basic Equations of Jenq-Shah Model | p. 117 |
| Experimental Determination of Jenq-Shah Parameters | p. 119 |
| R-Curves | p. 121 |
| Definition of an R-[Delta]a Curve | p. 121 |
| Description of the Fracture Process | p. 123 |
| The Peak Load Condition | p. 124 |
| Positive and Negative Geometries | p. 126 |
| R-Curve Determination from Tests | p. 126 |
| R-CTOD Curves | p. 128 |
| Stability Analysis in the R-Curve Approach | p. 130 |
| Stability under Load-Control Conditions | p. 130 |
| Stability under Displacement-Control Conditions | p. 131 |
| Stability under Mixed-Control Conditions | p. 131 |
| Determination of Fracture Properties From Size Effect | p. 135 |
| Size Effect in Equivalent Elastic Crack Approximations | p. 135 |
| Size Effect in the Large Size Range | p. 135 |
| Size Effect in the Jenq-Shah Model | p. 136 |
| Size Effect Law in Relation to Fracture Characteristics | p. 138 |
| Defining Objective Fracture Properties | p. 138 |
| Determination of Fracture Parameters from Size Effect | p. 138 |
| Determination of Fracture Parameters from Size and Shape Effects and Zero Brittleness Method | p. 139 |
| Intrinsic Representation of the Size Effect Law | p. 139 |
| Size Effect Method: Detailed Experimental Procedures | p. 140 |
| Outline of the Method | p. 140 |
| Regression Relations | p. 140 |
| RILEM Recommendation Using the Size Effect Method: Experimental Procedure | p. 143 |
| RILEM Recommendation Using the Size Effect Method: Calculation Procedure | p. 144 |
| Performance of the Size Effect Method | p. 147 |
| Improved Regression Relations | p. 147 |
| Determination of R-Curve from Size Effect | p. 150 |
| Determination of R-Curve from Size Effect | p. 150 |
| Determination of R-Curve from Bazant's Size Effect Law | p. 152 |
| Determination of the Structural Response from the R-Curve | p. 154 |
| Cohesive Crack Models | p. 157 |
| Basic Concepts in Cohesive Crack Model | p. 157 |
| Hillerborg's Approach: The Cohesive Crack as a Constitutive Relation | p. 158 |
| Other Approaches to Cohesive Cracks | p. 160 |
| Softening Curve, Fracture Energy, and Other Properties | p. 162 |
| Extensions of the Cohesive Crack Model | p. 164 |
| Cohesive Cracks with Tip Singularity | p. 165 |
| Cohesive Cracks with Bulk Energy Dissipation | p. 165 |
| Cohesive Crack Models Applied to Concrete | p. 167 |
| Softening Curves for Concrete | p. 167 |
| Experimental Aspects | p. 170 |
| Computational Procedures for Cohesive Crack Analysis | p. 172 |
| Size Effect Predictions | p. 175 |
| Cohesive Crack Models in Relation to Effective Elastic Crack Models | p. 177 |
| Correlation of Cohesive Crack with Bazant's and Jenq and Shah's Models | p. 178 |
| Experimental Determination of Cohesive Crack Properties | p. 180 |
| Determination of the Tensile Strength | p. 181 |
| Determination of the Initial Part of the Softening Curve | p. 182 |
| Determination of Fracture Energy G[subscript F] | p. 184 |
| Determination of a Bilinear Softening Curve | p. 188 |
| Pseudo-Boundary-Integral Methods for Mode I Crack Growth | p. 190 |
| The Underlying Problem | p. 190 |
| Petersson's Influence Method | p. 191 |
| Improved Solution Algorithm of Planas and Elices | p. 192 |
| Smeared-Tip Method | p. 193 |
| Scaling of the Influence Matrices | p. 195 |
| Inclusion of Shrinkage or Thermal Stresses | p. 196 |
| Inclusion of a Crack-Tip Singularity | p. 197 |
| Computation of Other Variables | p. 198 |
| Limitations of the Pseudo-Boundary Integral (PBI) Methods | p. 199 |
| Boundary-Integral Methods for Mode I Crack Growth | p. 199 |
| A Basic Boundary Integral Formulation | p. 199 |
| Size-Dependence of the Equations | p. 202 |
| The Dugdale and Rectangular Softening Cases | p. 203 |
| Eigenvalue Analysis of the Size Effect | p. 204 |
| Eigenvalue Analysis of Stability Limit and Ductility of Structure | p. 206 |
| Smeared-Tip Superposition Method | p. 207 |
| Asymptotic Analysis | p. 209 |
| Crack Band Models and Smeared Cracking | p. 213 |
| Strain Localization in the Series Coupling Model | p. 213 |
| Series Coupling of Two Equal Strain Softening Elements: Imperfection Approach | p. 214 |
| Series Coupling of Two Equal Strain Softening Elements: Thermodynamic Approach | p. 215 |
| Mean Stress and Mean Strain | p. 215 |
| Series Coupling of N Equal Strain Softening Elements | p. 216 |
| Localization of Strain in a Softening Bar | p. 217 |
| Localization and Mesh Objectivity | p. 217 |
| Localization in an Elastic-Softening Bar | p. 218 |
| Summary: Necessity of Localization Limiters | p. 219 |
| Basic Concepts in Crack Band Models | p. 220 |
| Elastic-Softening Crack Band Models | p. 220 |
| Band Models with Bulk Dissipation | p. 222 |
| Unloading and Reloading | p. 223 |
| Fracture Energy for Crack Bands With Prepeak Energy Dissipation | p. 224 |
| Simple Numerical Issues | p. 225 |
| Crack Band Width | p. 226 |
| Uniaxial Softening Models | p. 228 |
| Elastic-Softening Model with Stiffness Degradation | p. 228 |
| Elastic-Softening Model with Strength Degradation | p. 229 |
| Elastic-Softening Model with Stiffness and Strength Degradation | p. 229 |
| A Simple Continuum Damage Model | p. 230 |
| Introducing Inelasticity Prior to the Peak | p. 231 |
| Crack Closure in Reverse Loading and Compression | p. 231 |
| Introducing Other Inelastic Effects | p. 232 |
| Simple Triaxial Strain-Softening Models for Smeared Cracking | p. 234 |
| Cracking of Single Fixed Orientation: Basic Concepts | p. 234 |
| Secant Approach to Cracking of Fixed Orientation | p. 235 |
| Scalar Damage Model for Cracking of Fixed Orientation | p. 237 |
| Incremental Approach to Cracking of Fixed Orientation | p. 238 |
| Multi-Directional Fixed Cracking | p. 239 |
| Rotating Crack Model | p. 240 |
| Generalized Constitutive Equations with Softening | p. 242 |
| Mazars' Scalar Damage Model | p. 243 |
| Rankine Plastic Model with Softening | p. 243 |
| A Simple Model with Stiffness and Strength Degradation | p. 244 |
| Crack Band Models and Smeared Cracking | p. 246 |
| Stress-Strain Relations for Elements of Arbitrary Size | p. 246 |
| Skew Meshes: Effective Width | p. 248 |
| Stress Lock-In | p. 250 |
| Use of Elements of Large Size | p. 251 |
| Energy Criterion for Crack Bands with Sudden Cracking | p. 252 |
| Comparison of Crack Band and Cohesive Crack Approaches | p. 255 |
| Localized fracture: Moot Point Computationally | p. 255 |
| Nonlocalized Fracture: Third Parameter | p. 255 |
| Relation to Micromechanics of Fracture | p. 257 |
| Fracture of Arbitrary Direction | p. 258 |
| Advanced Size Effect Analysis | p. 261 |
| Size Effect Law Refinements | p. 261 |
| The Generalized Energy Balance Equation | p. 261 |
| Asymptotic Analysis for Large Sizes | p. 263 |
| Matching to the Effective Crack Model | p. 263 |
| Asymptotic Formula for Small Sizes and Its Asymptotic Matching with Large Sizes | p. 264 |
| Asymptotic Aspects of Bazant's Extended Size Effect Law | p. 265 |
| Size Effect for Failures at Crack Initiation from Smooth Surface | p. 266 |
| Universal Size Effect Law for Cracked and Uncracked Structures | p. 268 |
| Asymptotic Scaling Law for Many Loads | p. 269 |
| Asymptotic Scaling Law for a Crack with Residual Bridging Stress | p. 270 |
| Size Effect in Notched Structures Based on Cohesive Crack Models | p. 271 |
| The General Size Effect Equation | p. 271 |
| Asymptotic Analysis for Large Sizes | p. 273 |
| Asymptotic Analysis for Small Sizes | p. 274 |
| Interpolation Formula | p. 275 |
| Application to Notched Beams with Linear Softening | p. 277 |
| Application to Notched Beams with Bilinear Softening | p. 277 |
| Experimental Evidence | p. 279 |
| Size Effect on the Modulus of Rupture of Concrete | p. 280 |
| Notation and Definition of the Rupture Modulus | p. 280 |
| Modulus of Rupture Predicted by Cohesive Cracks | p. 281 |
| Further Analysis of the Influence of the Initial Softening | p. 284 |
| Modulus of Rupture According to Bazant and Li's Model, Bazant's Universal Size Effect Law, and Zero-Brittleness Method | p. 284 |
| Modulus of Rupture Predicted by Jenq-Shah Model | p. 287 |
| Carpinteri's Multifractal Scaling Law | p. 288 |
| Comparison With Experiments and Final Remarks | p. 289 |
| Compression Splitting Tests of Tensile Strength | p. 291 |
| Cracking Process in Stable Splitting Tests | p. 292 |
| Modified Bazant's Size Effect Law | p. 294 |
| Size Effect Predicted by Jenq-Shah Model | p. 295 |
| Size Effect Predicted by Cohesive Crack Models | p. 296 |
| Compression Failure Due to Propagation of Splitting Crack Band | p. 297 |
| Concepts and Mechanisms of Compression Fracture | p. 297 |
| Energy Analysis of Compression Failure of Column | p. 300 |
| Asymptotic Effect for Large Size | p. 305 |
| Size Effect Law for Axial Compression of Stocky Column | p. 305 |
| Effect of Buckling Due to Slenderness | p. 307 |
| Comparison with Experimental Data | p. 308 |
| The Question of Variation of Microcrack Spacing with Size D | p. 310 |
| Special Case of Compression with Transverse Tension | p. 310 |
| Distinction Between Axial Splitting and Failure Appearing as Shear | p. 311 |
| Scaling of Fracture of Sea Ice | p. 312 |
| Derivation of Size Effect for Thermal Bending Fracture of Ice Plate | p. 314 |
| General Proof of 3/8-Power Scaling Law | p. 316 |
| Brittleness and Size Effect in Structural Design | p. 319 |
| General Aspects of Size Effect and Brittleness in Concrete Structures | p. 319 |
| Conditions for Extending Bazant's Size Effect Law to Structures | p. 320 |
| Brittleness Number | p. 321 |
| Brittleness of High Strength Concrete | p. 323 |
| Size Effect Correction to Ultimate Load Formulas in Codes | p. 323 |
| Size Effect Correction to Strength-Based Formulas | p. 324 |
| Effect of Reinforcement | p. 325 |
| Diagonal Shear Failure of Beams | p. 326 |
| Introduction | p. 326 |
| Bazant-Kim-Sun Formulas | p. 327 |
| Gustafsson-Hillerborg Analysis | p. 330 |
| LEFM Analyses of Jenq and Shah and of Karihaloo | p. 331 |
| Finite Element Solutions with Nonlocal Microplane Model | p. 334 |
| Influence of Prestressing on Diagonal Shear Strength | p. 334 |
| Fracturing Truss Model for Shear Failure of Beams | p. 335 |
| Basic Hypotheses of Fracturing Truss Model | p. 336 |
| Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams Without Stirrups | p. 337 |
| Analysis Based on Stress Relief Zone and Strain Energy for Longitudinally Reinforced Concrete Beams With Stirrups | p. 341 |
| Analysis Based on Stress Redistribution and Complementary Energy | p. 344 |
| Size Effect on Nominal Stress at Cracking Load | p. 346 |
| Conclusions | p. 349 |
| Reinforced Beams in Flexure and Minimum Reinforcement | p. 349 |
| Lightly Reinforced Beams: Overview | p. 349 |
| Models Based on LEFM | p. 350 |
| Simplified Cohesive Crack Models | p. 356 |
| Models Based on Cohesive Cracks | p. 357 |
| Formulas for Minimum Reinforcement Based on Fracture Mechanics | p. 363 |
| Other Structures | p. 365 |
| Torsional Failure of Beams | p. 365 |
| Punching Shear Failure of Slabs | p. 366 |
| Anchor Pullout | p. 367 |
| Bond and Slip of Reinforcing Bars | p. 368 |
| Beam and Ring Failures of Pipes | p. 371 |
| Concrete Dams | p. 372 |
| Footings | p. 375 |
| Crack Spacing and Width, with Application to Highway Pavements | p. 376 |
| Keyed Joints | p. 377 |
| Fracture in Joints | p. 377 |
| Break-Out of Boreholes | p. 379 |
| Hillerborg's Model for Compressive Failure in Concrete Beams | p. 380 |
| Effect of Time, Environment, and Fatigue | p. 383 |
| Phenomenology of Time-Dependent Fracture | p. 384 |
| Types of Time-Dependent Fracture | p. 384 |
| Influence of Loading Rate on Peak Load and on Size Effect | p. 385 |
| Load Relaxation | p. 386 |
| Creep Fracture Tests | p. 388 |
| Sudden Change of Loading Rate | p. 388 |
| Dynamic Fracture | p. 389 |
| Activation Energy Theory and Rate Processes | p. 390 |
| Elementary Rate Constants | p. 391 |
| Physical Rate Constants | p. 391 |
| Fracture as a Rate Process | p. 394 |
| General Aspects of Isothermal Crack Growth Analysis | p. 395 |
| Load-Controlled Processes for Power-Law Rate Equation | p. 396 |
| Displacement-Controlled Processes for Power-Law Rate Equation | p. 397 |
| Some Applications of the Rate Process Theory to Concrete Fracture | p. 398 |
| Effect of Temperature on Fracture Energy of Concrete | p. 398 |
| Effect of Humidity on the Fracture Energy of Concrete | p. 399 |
| Time-Dependent Generalization of R-Curve Model | p. 401 |
| Application of the Time-Dependent R-Curve Model to Limestone | p. 403 |
| Linear Viscoelastic Fracture Mechanics | p. 404 |
| Uniaxial Linear Viscoelasticity | p. 404 |
| Compliance Functions for Concrete | p. 407 |
| General Linear Viscoelastic Constitutive Equations | p. 408 |
| The Correspondence Principle (Elastic-Viscoelastic Analogy) | p. 408 |
| Near-Tip Stress and Displacement Fields for a Crack in a Viscoelastic Structure | p. 409 |
| Crack Growth Resistance in a Viscoelastic Medium | p. 412 |
| Steady Growth of a Cohesive Crack with Rectangular Softening in an Infinite Viscoelastic Plate | p. 413 |
| Analysis of Crack Growth in a Viscoelastic Plate | p. 416 |
| Crack Growth Analysis at Controlled Displacement | p. 417 |
| Rate-Dependent R-Curve Model with Creep | p. 418 |
| Basic Equations | p. 418 |
| Approximate Solution for Small Crack Extensions | p. 419 |
| Comparison with Tests | p. 419 |
| Rate-Dependence of Process Zone Length | p. 420 |
| Sudden Change of Loading Rate and Load Relaxation | p. 420 |
| Summary | p. 422 |
| Time Dependent Cohesive Crack and Crack Band Models | p. 422 |
| Time-Independent Softening in a Viscoelastic Body | p. 423 |
| Time-Dependent Softening in an Elastic Body | p. 424 |
| Time-Dependent Cohesive Crack Model | p. 425 |
| Analysis of Viscoelastic Structure with Rate-Dependent Cohesive Crack by Finite Elements | p. 426 |
| Analysis of Viscoelastic Structure with Rate-Dependent Cohesive Crack by Compliance Functions | p. 428 |
| Introduction to Fatigue Fracture and Its Size Dependence | p. 429 |
| Fatigue Crack Growth in Metals | p. 430 |
| Fatigue Crack Growth in Brittle Materials | p. 431 |
| Size Effect in Fatigue Crack Growth in Concrete | p. 432 |
| Fatigue Description by History-Dependent Cohesive Models | p. 434 |
| Statistical Theory of Size Effect and Fracture Process | p. 437 |
| Review of Classical Weibull Theory | p. 439 |
| The Weakest-Link Discrete Model | p. 439 |
| The Weakest-Link Model for Continuous Structures under Uniaxial Stress | p. 440 |
| The Weibull Statistical Probability Distribution | p. 441 |
| Structures with Nonhomogeneous Uniaxial Stress | p. 443 |
| Generalization to Triaxial Stress States | p. 445 |
| Independent Failure Mechanisms: Additivity of the Concentration Function | p. 446 |
| Effective Uniaxial Stress | p. 447 |
| Summary: Nonhomogeneous States of Stress | p. 447 |
| Statistical Size Effect due to Random Strength | p. 449 |
| General Strength Probability Distribution and Equivalent Uniaxial Volume | p. 449 |
| Statistical Size Effect Laws | p. 451 |
| Divergence of Weibull Failure Probability for Sharply Cracked Bodies | p. 452 |
| The Effect of Surface Flaws | p. 454 |
| Basic Criticisms of Classical Weibull-Type Approach | p. 456 |
| Stress Redistribution | p. 456 |
| Equivalence to Uniaxially Stressed Bar | p. 457 |
| Differences between Two- and Three-Dimensional Geometric Similarities | p. 458 |
| Energy Release Due to Large Stable Crack Growth | p. 459 |
| Spatial Correlation | p. 460 |
| Summary of the Limitations | p. 460 |
| Handling of Stress Singularity in Weibull-Type Approach | p. 460 |
| A Simplified Approach to Crack Tip Statistics | p. 461 |
| Generalization of the Thickness Dependence of the Crack Tip Statistics | p. 462 |
| Asymptotic Size Effect | p. 463 |
| Extending the Range: Bulk Plus Core Statistics | p. 463 |
| More Fundamental Approach Based on Nonlocal Concept | p. 464 |
| Approximate Equations for Statistical Size Effect | p. 465 |
| Bazant-Xi Empirical Interpolation Between Asymptotic Size Effects | p. 465 |
| Determination of Material Parameters | p. 465 |
| The Question of Weibull Modulus m for the Fracture-Process Zone | p. 466 |
| Comparison with Test Results | p. 466 |
| Planas' Empirical Interpolation Between Asymptotic Size Effects | p. 467 |
| Limitations of Generalized Weibull Theory | p. 470 |
| Another View: Crack Growth in an Elastic Random Medium | p. 470 |
| The Strongest Random Barrier Model | p. 471 |
| The Statistical R-Curve | p. 472 |
| Finite Bodies | p. 472 |
| Frechet's Failure Probability Distribution | p. 474 |
| Random R-curve | p. 476 |
| Limitations of the Random Barrier Model | p. 479 |
| Fractal Approach to Fracture and Size Effect | p. 479 |
| Basic Concepts on Fractals | p. 480 |
| Invasive Fractal and Multifractal Size Effect for G[subscript F] | p. 482 |
| Lacunar Fractal and Multifractal Size Effect for [sigma] N[subscript u] | p. 482 |
| Fracture Analysis of Fractal Crack Propagation | p. 483 |
| Bazant's Analysis of Fractal Crack Initiation | p. 485 |
| Is Fractality the Explanation of Size Effect? | p. 486 |
| Nonlocal Continuum Modeling of Damage Localization | p. 489 |
| Basic Concepts in Nonlocal Approaches | p. 490 |
| The Early Approaches | p. 490 |
| Models with Nonlocal Strain | p. 491 |
| Gradient Models | p. 492 |
| A Simple Family of Nonlocal Models | p. 493 |
| A Second-Order Differential Model | p. 495 |
| An Integral-Type Model of the First Kind | p. 496 |
| An Integral-Type Model of the Second Kind | p. 497 |
| Nonlocal Damage Model | p. 498 |
| Triaxial Nonlocal Models and Applications | p. 501 |
| Triaxial Nonlocal Smeared Cracking Models | p. 502 |
| Triaxial Nonlocal Models with Yield Limit Degradation | p. 502 |
| Nonlocal Microplane Model | p. 506 |
| Determination of Characteristic Length | p. 506 |
| Nonlocal Model Based on Micromechanics of Crack Interactions | p. 507 |
| Nonlocality Caused by Interaction of Growing Microcracks | p. 507 |
| Field Equation for Nonlocal Continuum | p. 510 |
| Some Alternative Forms and Properties of the Nonlocal Model | p. 511 |
| Admissibility of Uniform Inelastic Stress Fields | p. 513 |
| Gauss-Seidel Iteration Applied to Nonlocal Averaging | p. 514 |
| Statistical Determination of Crack Influence Function | p. 515 |
| Crack Influence Function in Two Dimensions | p. 517 |
| Crack Influence Function in Three Dimensions | p. 520 |
| Cracks Near Boundary | p. 522 |
| Long-Range Decay and Integrability | p. 523 |
| General Formulation: Tensorial Crack Influence Function | p. 523 |
| Constitutive Relation and Gradient Approximation | p. 524 |
| Localization of Oriented Cracking into a Band | p. 525 |
| Summary | p. 525 |
| Material Models for Damage and Failure | p. 527 |
| Microplane Model | p. 528 |
| Macro-Micro Relations | p. 529 |
| Volumetric-Deviatoric Split of the Microstrain and Microstress Vectors | p. 532 |
| Elastic Response | p. 533 |
| Nonlinear Microplane Behavior and the Concept of Stress-Strain Boundaries | p. 535 |
| Numerical Aspects | p. 537 |
| Constitutive Characterization of Material on Microplane Level | p. 538 |
| Microplane Model for Finite Strain | p. 540 |
| Summary of Main Points | p. 542 |
| Calibration by Test Data, Verification and Properties of Microplane Model | p. 543 |
| Procedure for Delocalization of Test Data and Material Identification | p. 543 |
| Calibration of Microplane Model and Comparison with Test Data | p. 545 |
| Vertex Effects | p. 545 |
| Other Aspects | p. 547 |
| Nonlocal Adaptation of Microplane Model or Other Constitutive Models | p. 548 |
| Particle and Lattice Models | p. 550 |
| Truss, Frame, and Lattice Models | p. 552 |
| Directional Bias | p. 554 |
| Examples of Results of Particle and Lattice Models | p. 555 |
| Summary and Limitations | p. 559 |
| Tangential Stiffness Tensor Via Solution of a Body with Many Growing Cracks | p. 560 |
| References | p. 565 |
| Reference Citation Index | p. 599 |
| Index | p. 607 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780849382840
ISBN-10: 084938284X
Series: New Directions in Civil Engineering
Published: 29th December 1997
Format: Hardcover
Language: English
Number of Pages: 640
Audience: Professional and Scholarly
Publisher: Taylor & Francis Ltd
Country of Publication: GB
Edition Number: 1
Dimensions (cm): 18.9 x 26.4 x 4.2
Weight (kg): 1.36
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