Concrete Fracture Models and Applications

Shailendra Kumar · Sudhirkumar V. Barai

Concrete Fracture Modelsand Applications

123

Asst. Professor Shailendra KumarNational Institute of TechnologyCivil Engineering DepartmentJamshedpurIndiashailendrakmr@yahoo.co.in

Prof. Sudhirkumar V. BaraiIndian Institute of TechnologyCivil Engineering DepartmentKharagpurIndiaskbarai@civil.iitkgp.ernet.in

ISBN 978-3-642-16763-8 e-ISBN 978-3-642-16764-5DOI 10.1007/978-3-642-16764-5Springer Heidelberg Dordrecht London New York

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Dedicated to All Those who RelentlesslyEndeavour to Unite a Fractured World

Foreword

Fracture mechanics as a discipline of mechanics goes back to the early years of the20th century. It started with the description and explanation of the cracking behaviorand failure of glass which could not be explained by means of the strength of mate-rials approach. The material to which the new theory was applied had to be elasticand brittle. After glass, the failure of brittle types of metals was investigated. Later,linear elastic fracture mechanics was extended to elastic–plastic material behaviorwith well-established theories.

Although concrete exhibited brittleness in conventional force-controlled tensiletests, it was only in the early 1960s that fracture mechanics principles penetratedslowly into the field of concrete. First attempts were made to apply linear elasticfracture mechanics to concrete, but there was no great success. However, the ideato apply fracture mechanics to concrete and concrete structures was very impor-tant. Many researchers started to think of concrete and fracture mechanics. At thesame time, the testing facilities developed enormously. It became possible to per-form displacement-controlled tensile tests on concrete. One realized that concrete isnot perfectly brittle but strain softening, i.e., failure in tension, occurred only after aconsiderable nonelastic displacement in the post-peak region. The idea of cohesivestresses in a concrete crack emerged.

After recognizing the real behavior of concrete, nonlinear fracture mechanicstheories were developed. The present book presents these theories in detail. Thebook is a comprehensive treatment of the state of knowledge and adds some newfindings to the field. The authors succeeded to write the book in a way that some-times complicated theories can be followed with ease. I congratulate the authorsto this achievement and wish that not only many teachers will use the book in theirclasses but also code makers will use it as a compendium of the principles of fracturemechanics of concrete in order to introduce these principles finally into the designstandards.

Stuttgart, Germany Hans-Wolf ReinhardtJuly 10, 2010

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Preface

Concept of linear elastic fracture mechanics was first applied to pre-cracked con-crete elements in the early 1960s. Thereafter, extensive experimental and numericalresearch investigations proved that the classical form of linear elastic fracturemechanics cannot be applied to normal size concrete members. From the pastresearch and studies it also became clear that the modified form of linear elasticfracture mechanics or nonlinear fracture mechanics can be useful and powerful toolsfor the analysis of the growth of distributed cracking and its localization in concreteif the softening behavior of the material is taken into account. The nonlinear fracturemodels coupled and introduced the tension-softening constitutive law in the fracturemechanics concepts to study the crack initiation and its propagation in concrete andconcrete structures.

In due course of time, a number of nonlinear fracture models have been pro-posed and used to predict the nonlinear fracture behavior of cementitious materials.These are fictitious crack model or cohesive crack model, crack band model,two-parameter fracture model, size-effect model, effective crack model, KR-curvemethod based on cohesive force distribution in the fracture zone, double-K fracturemodel, and double-G fracture model. Fracture mechanics concept introduced energyapproach for crack development and its growth which can avoid the unobjectivity inthe results, predict the post-peak response with a less complexity and exhibit size-effect behavior. The brittleness of the material can quantitatively be defined andmore uniform safety of factors can be achieved in the structural design with the helpof fracture mechanics concept.

It is a well-known phenomenon that the fracture parameters of concrete dependon the softening function of concrete, concrete strength, specimen size, specimengeometry, geometrical factors like relative size of notch length and the loadingcondition. The literature reports extensive numerical and experimental investiga-tion on nonlinear fracture behavior of concrete. All the important nonlinear fracturemodels are widely applied to characterize the related fracture parameters and thesestudies are available in scattered literature. This book attempts to present the theo-retical development and applications of various nonlinear concrete fracture modelsin a unified manner using different fracture parameters. In this regard, the authors

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x Preface

investigated the behavior of fracture parameters of concrete at different phases ofcrack propagation phenomena of concrete.

There are six major chapters in the textbook which are mainly based on the recentresearch and studies carried out by the authors in the recent years. The detailedintroduction of the book is mentioned in the opening chapter. In the subsequentchapters, cohesive crack model for three-point bending test, four-point bending test,and compact tension specimens using important softening functions of concreteare developed. The numerical results are compared with the experimental resultsavailable in the literature. Further, a systematic study on the different cohesivecrack fracture parameters is carried out. Introduction of weight function method isexplained to determine the double-K fracture parameters and the KR-curve methodbased on cohesive stress distribution. Furthermore, attempts are made to put forwardsome new developments regarding behavior of different fracture parameters usingthe cohesive crack model as the reference. A comprehensive comparison betweenthe double-K and double-G fracture criteria is presented. Emphasis on the effect ofvarious parameters including specimen geometry, size effect, and loading conditionon the double-K and the KR-curve method is also focused. Finally, a comparativestudy among different fracture parameters obtained from important nonlinear mod-els is presented. Hence, the textbook presents results of a comprehensive study onthe crack initiation and its growth in concrete-like materials using various fracturemodels. At last, the flowcharts of various fracture models are presented in Appendix.

In this book, the authors have taken a small step to present a basic introduction onthe various nonlinear concrete fracture models considering the respective fractureparameters. It can be helpful to undergraduate and postgraduate students who arestudying this subject. An immense help to the beginners and researchers in the areaof fracture mechanics of concrete is expected from this book which will providea sound basis on the relevant subject to carry out further innovative research workin the future. Appendix can be of much use to the readers for computing differentfracture parameters using computer programs.

At last, the authors would be thankful to the readers and their invaluablesuggestions or comments for the further improvement of the book.

Jamshedpur, India Shailendra KumarKharagpur, India Sudhirkumar V. Barai

Acknowledgement

Experimental and numerical investigations showed that the classical form of linearelastic fracture mechanics (LEFM) cannot be applied to normal size concrete mem-bers because of the presence of large and variable size of fracture process zone.However, when the softening behavior of material was taken into account, nonlin-ear fracture mechanics models emerged as a powerful tool for analyzing growthof distributed cracking and its localization in concrete. There is a scattered litera-ture exploring nonlinear fracture behavior of concrete. It consists of numerical andexperimental applications of all important nonlinear fracture models to characterizerelated fracture parameters of concrete. This inspired us to attempt to present aunified view of the theoretical development and applications of these models onconcrete.

In the journey of writing this book we came across many legendary researcherswho have actively contributed directly or indirectly toward the content of this book.

First of all we would like to thank Prof. Dr.-Ing. Hans-Wolf Reinhardt, Universityof Stuttgart, Germany, who has been constant source of inspiration and support toour book and kindly agreed to write Foreword for this book.

We wish to extend our sincere thanks to Prof. Per-Erik Petersson (Sweden), Prof.Zdenek P. Bažant (USA), Prof. B.L. Karihaloo (U.K.), Prof. Jaime Planas (Spain),Prof. Alberto Carpinteri (Italy), Prof. S P Shah (USA), Prof. Shilang Xu (China),Prof. Grzegorz Glinka (Canada), Prof. John P. Dempsey (USA), Prof. GustavoV. Guinea (Spain), Prof. Goangseup Zi (South Korea), Prof. B.K. Raghuprasad(India), Prof. P.C. Pandey (India), Prof. T. Kant (India), Prof. B. Pradhan (India),Prof. S.K. Bhattacharyya (India), Prof. L.S. Ramachandra (India), Prof. D. Deb(India), Prof. N Dhang (India), Prof. Yanhua Zhao (China), Prof. Jeff Roesler(USA), Prof. Ravindra Gettu (India), and Dr. Xiufang Zhang (China) for enrichinginputs and feedbacks.

We are indeed grateful to Dr. Thomas Lehnert, senior editor engineeringSpringer-Verlag GmbH, Berlin, Germany, who helped us in publishing this bookwith Springer-Verlag.

Shailendra is thankful to National Institute of Technology, Jamshedpur, India,for providing a conducive environment for working on this book. Further, he wouldlike to express a word of appreciation to his lovely wife, Neelam Prabha, for her

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xii Acknowledgement

unprejudiced encouragement and his sons Somil and Shankhin for bearing with alldifficulties during this exciting journey of writing the book.

Sudhir is grateful to Indian Institute of Technology, Kharagpur, India, forproviding stimulating environment for working on this book. Behind every accom-plishment, the family plays an important role in sacrificing the precious time oftogetherness. He would like to thank his wife – Parama – and lovely daughters –Sristi and Shailey – for all their warmth and support and helping him to sail throughduring difficult times.

Above all, we would like to express profound gratitude and appreciation toAlmighty.

Jamshedpur, India Shailendra KumarKharagpur, India Sudhirkumar V. Barai10 July 2010

Contents

1 Introduction to Fracture Mechanics of Concrete . . . . . . . . . . 11.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Organization of the Book . . . . . . . . . . . . . . . . . . . . . 51.3 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Fracture Mechanics of Concrete – State-of-the-Art Review . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . 9

2.2.1 Significance of Stress Intensity Factor . . . . . . . . . . . 112.2.2 Concept of R Curve . . . . . . . . . . . . . . . . . . . . 11

2.3 Elastic–Plastic Fracture Mechanics . . . . . . . . . . . . . . . . 132.3.1 The CTOD Criterion . . . . . . . . . . . . . . . . . . . . 132.3.2 The J-Integral Approach . . . . . . . . . . . . . . . . . . 13

2.4 Early Research Using LEFM to Concrete . . . . . . . . . . . . . 142.5 Tensile Behavior of Concrete . . . . . . . . . . . . . . . . . . . 17

2.5.1 Strain Localization Effect . . . . . . . . . . . . . . . . . 172.5.2 Fracture Process Zone . . . . . . . . . . . . . . . . . . . 192.5.3 Nonlinear Behavior of Concrete . . . . . . . . . . . . . . 20

2.6 Specimen Geometry for Fracture Test of Concrete . . . . . . . . 212.6.1 Dimensions of Test Specimens . . . . . . . . . . . . . . 22

2.7 Nonlinear Fracture Mechanics for Concrete . . . . . . . . . . . . 242.7.1 Cohesive Crack Model (CCM) or Fictitious Crack

Model (FCM) . . . . . . . . . . . . . . . . . . . . . . . 252.7.2 Crack Band Model (CBM) . . . . . . . . . . . . . . . . . 322.7.3 Two-Parameter Fracture Model (TPFM) . . . . . . . . . 352.7.4 Size-Effect Model (SEM) . . . . . . . . . . . . . . . . . 362.7.5 Effective Crack Model (ECM) . . . . . . . . . . . . . . . 402.7.6 Double-K Fracture Model (DKFM) . . . . . . . . . . . . 422.7.7 The KR Curve Associated with Cohesive Stress

Distribution in the FPZ . . . . . . . . . . . . . . . . . . 452.7.8 Double-G Fracture Model (DGFM) . . . . . . . . . . . . 46

2.8 Comparative Study and Size-Effect Behavior . . . . . . . . . . . 47

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2.9 Weight Function Approach . . . . . . . . . . . . . . . . . . . . 492.9.1 Some Existing Weight Functions . . . . . . . . . . . . . 502.9.2 Universal Weight Function for Edge Cracks in

Finite Width Plate . . . . . . . . . . . . . . . . . . . . . 512.9.3 Computation of Stress Intensity Factor and Crack

Face Displacement . . . . . . . . . . . . . . . . . . . . . 512.10 Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . 522.11 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Fracture Behavior of Concrete using Cohesive Crack andSize-Effect Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Cohesive Crack Model for Three-Point Bending Test . . . . . . . 65

3.2.1 Formulation Based on Energy Principle . . . . . . . . . . 653.2.2 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 673.2.3 Finite Element Discretization . . . . . . . . . . . . . . . 683.2.4 Beam Deflection . . . . . . . . . . . . . . . . . . . . . . 713.2.5 Model Implementation . . . . . . . . . . . . . . . . . . . 73

3.3 Softening Function of Concrete . . . . . . . . . . . . . . . . . . 733.3.1 Linear Softening Function . . . . . . . . . . . . . . . . . 743.3.2 Bilinear Softening . . . . . . . . . . . . . . . . . . . . . 743.3.3 Exponential Softening . . . . . . . . . . . . . . . . . . . 763.3.4 Nonlinear Softening . . . . . . . . . . . . . . . . . . . . 763.3.5 Quasi-exponential Softening . . . . . . . . . . . . . . . . 77

3.4 Numerical Study Using TPBT Specimen . . . . . . . . . . . . . 773.4.1 Experimental Results and Numerical Computation . . . . 773.4.2 Comparison with Numerical Results Using Linear Softening 793.4.3 Influence of Softening Function on the Global

P-CMOD Response . . . . . . . . . . . . . . . . . . . . 803.4.4 Influence of Kink Point in the Bilinear Softening

on the Global P-CMOD Response . . . . . . . . . . . . . 803.4.5 Effect of Finite Element Mesh Size on Bearing

Capacity of the Beam . . . . . . . . . . . . . . . . . . . 833.4.6 Effect of Size Scale on the Type of Failure . . . . . . . . 853.4.7 Size-Scale Deviation From LEFM Concept . . . . . . . . 873.4.8 Influence of Softening Function on Size-Effect Curve . . 89

3.5 Numerical Study Using Compact Tension (CT) Specimen . . . . 903.5.1 Global P-COD Response Using Linear Softening Function 913.5.2 Influence of Softening Functions on the Global

P-COD Response . . . . . . . . . . . . . . . . . . . . . 943.5.3 Influence of Softening Functions on the

Size-Scale Transition Toward LEFM . . . . . . . . . . . 953.5.4 Influence of Softening Functions on the

Size-Effect Curve . . . . . . . . . . . . . . . . . . . . . 97

Contents xv

3.5.5 Evolution of Fracture Zone . . . . . . . . . . . . . . . . 973.5.6 Influence of Softening Functions on the

Development of Fracture Process Zone . . . . . . . . . . 1003.5.7 Influence of Softening Functions on the

Load-Carrying Capacity Corresponding to theFully Developed FPZ . . . . . . . . . . . . . . . . . . . 101

3.5.8 Influence of Softening Functions on the FictitiousCrack Length Corresponding to Peak Load . . . . . . . . 102

3.5.9 Influence of Softening Functions on the CTODc . . . . . 1033.5.10 Influence of Softening Functions on the Local

Stress Distribution at Notch Tip Corresponding toPeak Load . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5.11 Comparison with Experimental Result . . . . . . . . . . 1053.6 Size-Effect Study from Size-Effect Model . . . . . . . . . . . . 106

3.6.1 Size-Effect Law for Size-Effect Model . . . . . . . . . . 1073.6.2 Size Effects and Fracture Load from CCM and SEM . . . 107

3.7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 110References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4 Crack Propagation Study Using Double-K and Double-GFracture Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2 Weight Function Method . . . . . . . . . . . . . . . . . . . . . 1134.3 Determination of Universal Weight Function for Edge

Cracks in Finite Width Plate . . . . . . . . . . . . . . . . . . . . 1154.3.1 Four-Term Universal Weight Function . . . . . . . . . . 1174.3.2 Five-Term Universal Weight Function . . . . . . . . . . . 120

4.4 Linear Asymptotic Superposition Assumption . . . . . . . . . . 1224.5 Determination of Double-K Fracture Parameters . . . . . . . . . 125

4.5.1 Effective Crack Extension for the AnalyticalMethod and Weight Function Approach . . . . . . . . . . 125

4.5.2 Effective Crack Extension for Simplified Method . . . . . 1274.5.3 Calculation of Double-K Fracture Parameters . . . . . . . 1284.5.4 Determination of SIF due to Cohesive Stress

(KCI) in FPZ . . . . . . . . . . . . . . . . . . . . . . . . 130

4.6 Determination of Double-G Fracture Parameters . . . . . . . . . 1354.6.1 Unstable Fracture Energy Release . . . . . . . . . . . . . 1354.6.2 Determination of Critical Cohesive Breaking Energy . . . 1364.6.3 Determination of Initial Fracture Energy Release . . . . . 1394.6.4 Determination of Effective Double-K Fracture Parameters 139

4.7 Application of Weight Function Approach for Double-KFracture Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1394.7.1 Comparison with TPBT Geometry . . . . . . . . . . . . 1394.7.2 Comparison with CT and WST Geometries . . . . . . . . 145

4.8 Size-Effect Prediction from the Double-K Fracture Model for TPBT 154

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4.8.1 Size-Effect Law for Unstable Fracture Toughness KunIC . 155

4.8.2 Size-Effect Law for Initiation Fracture Toughness KiniIC . 157

4.8.3 Prediction of Size Effects from FCM and DKFM . . . . . 1584.9 Numerical Study on Double-K Fracture Parameters . . . . . . . 164

4.9.1 Material Properties . . . . . . . . . . . . . . . . . . . . . 1644.9.2 Comparison Between Methods of Determination

for DKFM . . . . . . . . . . . . . . . . . . . . . . . . . 1664.9.3 Influence of Specimen Geometry . . . . . . . . . . . . . 1704.9.4 Influence of Loading Condition . . . . . . . . . . . . . . 1734.9.5 Influence of Softening Function . . . . . . . . . . . . . . 176

4.10 Equivalence Between Double-K and Double-G Fracture Parameters 1774.10.1 Material Properties . . . . . . . . . . . . . . . . . . . . . 1774.10.2 Effect of Specimen Geometry on Double-K

Fracture Parameters . . . . . . . . . . . . . . . . . . . . 1794.10.3 Effect of Specimen Geometry on Double-G

Fracture Parameters . . . . . . . . . . . . . . . . . . . . 1814.10.4 Equivalence Between Double-K and Double-G

Fracture Parameters . . . . . . . . . . . . . . . . . . . . 1834.10.5 Influence of Softening Function on the Double-K

and Double-G Fracture Parameters . . . . . . . . . . . . 1854.11 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 188References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5 Fracture Properties of Concrete Based on the KR CurveAssociated with Cohesive Stress Distribution . . . . . . . . . . . . . 1935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.2 The KR-Curve Method . . . . . . . . . . . . . . . . . . . . . . . 1935.3 Analytical Method for Evaluation of the KR Curve . . . . . . . . 196

5.3.1 Case 1: When a = ao . . . . . . . . . . . . . . . . . . . . 1975.3.2 Case 2: When ao ≤ ao ≤ ac . . . . . . . . . . . . . . . . 1975.3.3 Case 3: When ac ≤ a ≤ awc . . . . . . . . . . . . . . . . 1985.3.4 Case 4: When a ≥ awc . . . . . . . . . . . . . . . . . . . 198

5.4 Weight Function Approach for Evaluation of the KR Curve . . . 1995.4.1 Derivation of Closed-Form Expression for

Cohesive Toughness . . . . . . . . . . . . . . . . . . . . 1995.4.2 Case 1: When a = ao . . . . . . . . . . . . . . . . . . . . 2015.4.3 Case 2: When ao ≤ a ≤ ac . . . . . . . . . . . . . . . . . 2025.4.4 Case 3: When ac ≤ a ≤ awc . . . . . . . . . . . . . . . . 2025.4.5 Case 4: When a ≥ awc . . . . . . . . . . . . . . . . . . . 202

5.5 Computation and Validation of the KR Curve . . . . . . . . . . . 2035.5.1 Details of Experimental Results . . . . . . . . . . . . . . 2035.5.2 Crack Extension Resistance Curves (KR Curves)

and Stability Criterion . . . . . . . . . . . . . . . . . . . 2055.5.3 Effect of Specimen Size on the KR Curves . . . . . . . . 2095.5.4 The P-CTOD Curves . . . . . . . . . . . . . . . . . . . . 210

Contents xvii

5.5.5 The Relationship Between CTOD and �a . . . . . . . . 2105.5.6 Analysis and Discussion . . . . . . . . . . . . . . . . . . 211

5.6 Numerical Study on the KR Curve . . . . . . . . . . . . . . . . . 2135.6.1 Material Properties and Numerical Computations . . . . . 2135.6.2 Crack Extension Resistance Curves (KR Curves)

and Stability Criterion . . . . . . . . . . . . . . . . . . . 2175.6.3 Effect of Specimen Geometry and Size Effect on

the KR Curves . . . . . . . . . . . . . . . . . . . . . . . 2265.6.4 Effect of Specimen Geometry and Size Effect on

the CTOD Curves . . . . . . . . . . . . . . . . . . . . . 2275.6.5 Effect of Specimen Geometry and Size Effect on

the Process Zone Length . . . . . . . . . . . . . . . . . . 2275.6.6 Effect of Loading Condition and Size Effect on

the KR Curves . . . . . . . . . . . . . . . . . . . . . . . 2295.6.7 Effect of Loading Condition and Size Effect on

the CTOD Curves . . . . . . . . . . . . . . . . . . . . . 2295.6.8 Effect of Loading Condition and Size Effect on

the Process Zone Length . . . . . . . . . . . . . . . . . . 2325.7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 232References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6 Comparison of Fracture Parameters of Concrete UsingNonlinear Fracture Models . . . . . . . . . . . . . . . . . . . . . . 2356.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.2 Material Properties and Determination of Fracture Parameters . . 2356.3 Size-Effect Analysis Using Various Fracture Models . . . . . . . 239

6.3.1 Size Effect of Critical Stress Intensity Factors . . . . . . 2396.3.2 Effect of Specimen Size on CTODcs and CTODc . . . . . 2426.3.3 Effect of Specimen Size on ae of ECM and ac of

DKFM or DGFM . . . . . . . . . . . . . . . . . . . . . 2446.3.4 Relation Between cf of SEM and acs∞ of TPFM . . . . . 245

6.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 246References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

List of Symbols and Abbreviations

List of Symbols

a crack lengthao initial crack lengthac effective crack length at peak (critical) loadae effective crack length at peak (critical) load from ECMB width of the test specimenbi boundary force distribution defined on AT[C] a vector of crack opening displacement at node i for unit

external loadc1, c2 material constants for nonlinear softening functioncf effective extension of crack length for infinitely large speci-

men{Cg} a vector of crack opening displacement at node i due to self-

weight of specimenCTODcs critical value of crack-tip opening displacement from TPFMcp length of fully developed fracture process zoneda maximum size of aggregatesD depth of the test specimen (characteristic dimension)Dg load point deflection due to self-weight of the specimenDL load point deflection when external load is unity{Dp} a vector of load point deflection when {p} = {1}E modulus of elasticity of concretef a functionF1, F2 geometric factors for different loading casesF(x/a, a/D) the standard Tada Green’s function for edge cracks subjected

to pair of forces normal to the crack facefc cylinder compressive strengthfck characteristics strength of concretefcm mean compressive strengthfcu cube compressive strength of concreteftm mean tensile strength

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xx List of Symbols and Abbreviations

Fi the body forceft uniaxial tensile strength of concreteGf initial fracture energy of concrete, defined as the area under

curve of the horizontal intercept of the initial tangent of thesoftening curve

GF, GFC fracture energy of concrete, defined as the area under curveof the post-peak stress and crack opening displacementrelation

gf(x) Hillerborg’s local cohesive breaking energy at any location xalong the FPZ

GFB fracture energy of concrete for infinitely large specimenGI–coh the average value of cohesive breaking energy per unit lengthGF

ini crack initiation fracture energy obtained using LEFM equa-tion from known value of KIC

ini

GFun unstable fracture energy obtained using LEFM equation from

known value of KICun

GICini the crack initiation fracture energy release

GICun the unstable fracture energy release

GICC the critical value of the cohesive breaking energy

Ho thickness of the clip gauge holder[K] a symmetric matrix and the value of Kij is the COD at node i

by a unit opening nodal force applied at node jKI stress intensity factor in mode I fractureKI

COH the cohesive toughness during crack extensionKC

t total crack extension resistanceKIC

ini the crack initiation toughnessKIC

un the unstable fracture toughnessKIC

C the cohesive toughness at critical conditionKIC

e critical value of SIF from ECMKIC

s critical value of SIF from TPFMKIC

b critical value of SIF from SEMKini

IC the effective crack initiation toughnessKun

IC the effective unstable fracture toughnessKINu stress intensity factor for Pu and aokP stress intensity factor at the process zone tip due to unit

applied loadKσ the stress intensity factor at the process zone tip due to

cohesive stressKIC, KC the critical stress intensity factor[KIC

C]NI KICC determined using numerical integration method

[KICini]NI KIC

ini determined using numerical integration method[KIC

C]WF4 KICC determined using weight function method with four

terms

List of Symbols and Abbreviations xxi

[KICini]WF4 KIC

ini determined using weight function method with fourterms

[KICC]WF5 KIC

C determined using weight function method with fiveterms

[KICini]WF5 KIC

ini obtained using weight function method with five termsKIC

un∗ KICun evaluated by Xu and Reinhardt (1999b, c) in Chap. 4

KICini∗ the KIC

ini evaluated by Xu and Reinhardt (1999b, c) inChap. 4

k(α) a geometric factorlch characteristic length of materialm(x,a) weight functionM1, M2, M3, M4 parameters of weight function{p} a vector of nodal forcesPini initial cracking loadPu maximum applied loadPu,exp maximum applied load obtained from experimentPu,CCM maximum applied load obtained using cohesive crack modelPu,LEFM maximum load obtained using LEFM conceptru undamaged length of the ligament during crack propagationS span of the test beamui an admissible displacement field for the systemW the strain energy density function defined in the bodyw crack opening displacement ahead of the crack tipw1 the horizontal intercept of the initial tangent of softeningwt crack opening displacement at the initial notch tipwc maximum COD when the cohesive stress becomes zerowg the self-weight per unit length of the beamws COD at slope change in the bilinear softeningα a/D ratioβB the brittleness numberδ load point deflection�ac effective crack extension at critical loadεij strain field the surface potential defined in the part of bound-

ary APυ Poisson’s ratio of concreteσ s(CTODc) cohesive stress at CTODcσ (wt) cohesive stress at the initial notch tipσ cohesive stress ahead of the crack tipσNu peak nominal stressσ s cohesive stress at slope change in the bilinear softeningσ tip stress value generated at the crack tip during loading the total potential

xxii List of Symbols and Abbreviations

List of Abbreviations

CBM crack band modelCCM cohesive crack modelCMOD crack mouth opening displacementCMODc critical value of crack mouth opening displacementCOD crack opening displacementCODc critical value of crack opening displacementCT compact tensionCTOD crack-tip opening displacementCTODc critical value of crack-tip opening displacementDGFM double-G fracture modelDKFM double-K fracture modelECM effective crack modelFCM fictitious crack modelFEM finite element methodFPBT four-point bending testFPZ fracture process zoneLEFM linear elastic fracture mechanicsLHS left-hand sideRHS right-hand sideSEM size-effect modelSIF stress intensity factorTPBT three-point bending testTPFM two-parameter fracture modelWST wedge splitting test