Fracture Mechanics and Crack Growth / Edition 1

by Naman Recho

Overview

This book presents recent advances related to the following two topics:

• how mechanical fields close to material or geometrical singularities such as cracks can be determined;
• how failure criteria can be established according to the singularity degrees related to these discontinuities.

Concerning the determination of mechanical fields close to a crack tip, the first part of the book presents most of the traditional methods in order to classify them into two major categories. The first is based on the stress field, such as the Airy function, and the second resolves the problem from functions related to displacement fields. Following this, a new method based on the Hamiltonian system is presented in great detail. Local and energetic approaches to fracture are used in order to determine the fracture parameters such as stress intensity factor and energy release rate.

The second part of the book describes methodologies to establish the critical fracture loads and the crack growth criteria. Singular fields for homogeneous and non-homogeneous problems near crack tips, v-notches, interfaces, etc. associated with the crack initiation and propagation laws in elastic and elastic-plastic media, allow us to determine the basis of failure criteria.

Each phenomenon studied is dealt with according to its conceptual and theoretical modeling, to its use in the criteria of fracture resistance; and finally to its implementation in terms of feasibility and numerical application.

Contents

1. Introduction.
Part 1: Stress Field Analysis Close to the Crack Tip
2. Review of Continuum Mechanics and the Behavior Laws.
3. Overview of Fracture Mechanics.
4. Fracture Mechanics.
5. Introduction to the Finite Element Analysis of Cracked Structures.
Part 2: Crack Growth Criteria
6. Crack Propagation.
7. Crack Growth Prediction in Elements of Steel Structures Submitted to Fatigue.
8. Potential Use of Crack Propagation Laws in Fatigue Life Design.

Product Details

ISBN-13:	9781848213067
Publisher:	Wiley
Publication date:	01/12/2015
Series:	ISTE Series , #574
Pages:	480
Product dimensions:	6.20(w) x 9.30(h) x 1.40(d)

About the Author

Naman Recho is Professor at EPF-Ecole d'Ingénieurs, Sceaux, France and at Blaise Pascal University, Clermont Ferrand, France

Table of Contents

Preamble xiii

Preface xv

Notations xix

Chapter 1 1

Part 1 Stress Field Analysis Close to the Crack Tip 5

Chapter 2 Review of Continuum Mechanics and the Behavior Laws 7

2.1 Kinematic equations 9

2.2 Equilibrium equations in a volume element 16

2.3 Behavior laws 20

2.3.1 Modeling the linear elastic constitutive law 22

2.3.2 Definitions 24

2.3.3 Modeling of the elastic-plastic constitutive law 35

2.3.4 Modeling the law of perfect plastic behavior in plane stress medium 45

2.4 Energy formalism 50

2.4.1 Principle of virtual power 51

2.4.2 Potential energy and complementary energy 54

2.4.3 Stationary energy and duality 59

2.4.4 Virtual work principle - two-dimensional application 60

2.5 Solution of systems of equations of continuum mechanics and constitutive behavior law 63

2.5.1 Direct solution method 63

2.5.2 Solution methods using stationary energies 64

2.5.3 Solution with other formulation devices (Airy function) 68

2.6 Review of the finite element solution 72

2.6.1 The displacements 74

2.6.2 The strains 75

2.6.3 The stresses 76

2.6.4 Minimum potential energy principle 76

2.6.5 Assembly 78

Chapter 3 Overview of Fracture Mechanics 81

3.1 Fracture process 83

3.2 Basic modes of fracture 84

Chapter 4 Fracture Mechanics 87

4.1 Determination of stress, strain and displacement fields around a crack in a homogeneous, isotropic and linearly elastic medium 90

4.1.1 Westergaard Solution 90

4.1.2 William expansion solution 101

4.1.3 Solution via the Mushkelishvili analysis 106

4.1.4 Solution of a three-dimensional fracture problem in mode I 110

4.1.5 Solution using energy approaches 115

4.1.6 Plastic zone shape around a crack 137

4.2 Plastic analysis around a crack in an isotropic homogeneous medium 144

4.2.1 Irwin's approach 145

4.2.2 Dugdale's (COD) solution 146

4.2.3 Direct local approach of the stress state in a cracked elastic-plastic medium 151

4.2.4 Determination of the J-integral in an elastic-plastic medium 161

4.2.5 Asymptotic stress fields in an elastic-plastic medium: the Hutchinson, Rice and Rosengren solution 162

4.3 Case of a heterogeneous medium: elastic multimaterials 164

4.4 New modeling approach to singular fracture fields 165

4.4.1 The fracture Hamiltonian approach 165

4.4.2 Integral equations approach 174

4.4.3 Case of V-notches 179

Chapter 5 Introduction to the Finite Element Analysis of Cracked Structures 187

5.1 Modeling of a singular field close to the crack tip 188

5.1.1 Local method from a "core" element 192

5.1.2 Local methods from enhanced elements 198

5.2 Energetic methods 200

5.2.1 Finite variation methods 201

5.2.2 Contour integrals 203

5.2.3 Other integral/decoupling modes 205

5.3 Nonlinear behavior 208

5.3.1 Case of a power law 209

5.3.2 Case of a multilinear law 209

5.3.3 Relationship between COD and the J-integral 212

5.4 Specific finite elements for the calculation of cracked structures 213

5.4.1 Barsoum elements and P_u and Hussain 213

5.4.2 Verification of the strain field form 214

5.5 Study of a finite elements program in a 2D linear elastic medium 216

5.5.1 Definition and formulation of the conventional QUAD-12 element 217

5.5.2 Definition and formulation of the conventional TRI-9 element 220

5.5.3 Definition of the singular element or core around the crack front 221

5.5.4 Formulation and resolution by the core element method 222

5.5.5 The evaluation of stress intensity factor (K) as a function of the radius (r) 223

5.6 Application to the calculation of the J-integral in mixed mode 224

5.6.1 Partitioning of J in J_I and J_II 227

5.7 Different meshing fracture monitoring techniques by finite elements 229

5.7.1 The eXtended finite element modeling method 231

5.7.2 Crack box technique (CBT) 232

Part 2 Crack Growth Criteria 235

Chapter 6 Crack Propagation 237

6.1 Brittle fracture 239

6.1.1 Stress intensity factor criteria 240

6.1.2 Criterion of energy release rate, G 242

6.1.3 Crack opening displacement (COD) criterion 242

6.1.4 J-integral criterion 243

6.1.5 R-curve criterion 244

6.1.6 Feddersen's concept 246

6.1.7 Two criteria approach 248

6.1.8 Electro Power Research Institute Method 250

6.1.9 Leguillon's criterion 250

6.1.10 Tensile/shear transition criterion 255

6.2 Crack extension 265

6.2.1 Maximum circumferential stress criterion 266

6.2.2 Minimum local strain energy density criterion 268

6.2.3 Maximum energy release rate criterion 269

6.2.4 Discussion of criteria 271

6.3 Crack extension criterion in an elastic plastic medium 272

6.3.1 Crack extension criterion for tensile fractures 273

6.3.2 Crack-extension criterion for shear fracture 273

6.4 Crack-extension criterion from V-notches 275

6.5 Fracture following crack growth under high-cycle number fatigue 277

6.6 Crack propagation laws 279

6.6.1 Closure of the crack lips 284

6.6.2 Crack propagation laws in mixed mode 285

6.7 Approaches used for the calculation of fatigue lifetime 286

6.7.1 Standard approach by means of (S-N) curves 286

6.7.2 Approach by means of linear fracture mechanics 288

6.7.3 Quick calculation of the stress intensity factor in mode I 291

6.8 Case of the variable amplitude loading 296

6.8.1 Physical definitions of the damage law giving the fatigue resistance 296

6.8.2 Physical definitions of the cumulative damage law 298

6.8.3 Considered definitions of the damage and cumulative damage laws 298

6.8.4 Several types of associations of damage laws to cumulative damage laws 299

6.8.5 Fatigue dimensioning methodology of a mechanical component subjected to variable loading 301

6.8.6 Cycle-counting methods 302

6.8.7 Principle of the cumulative damage theories 305

6.8.8 Miner's rule 306

6.8.9 Drawbacks of Miner's rule 308

6.8.10 Mean lifetime 308

6.8.11 Other more complex theories 309

6.9 Crack retardation effect due to overloading 312

6.9.1 Phenomenon of crack closure 314

6.9.2 Cyclic strain hardening of the material at the crack tip 315

6.9.3 Phenomenon of residual compressive stresses at the crack tip 315

6.10 "Reliability-failure" in the presence of random variables 318

6.10.1 Reliability elements 320

6.10.2 Damage indicating integral 322

6.10.3 Case of random variable loading 324

6.10.4 Damaging cycles 325

6.10.5 Effect of the application sequence of solicitation 329

Chapter 7 Crack Growth Prediction in Elements of Steel Structures Submitted to Fatigue 331

7.1 Significance and analysis by calculation of stresses around the local effect 333

7.1.1 Tubular joints, geometry and position of the problem 335

7.1.2 First numerical local effect (the intersection of finite elements) 337

7.1.3 Second and third local effects: inertia of the weld bead and weld toe 338

7.1.4 Fourth local effect (defects at the weld toe) 342

7.2 Crack initiation under fatigue 343

7.2.1 Crack initiation fatigue 344

7.2.2 Initial crack size in angle welds 356

7.3 Localization and sensitivity to rupture of cracks 367

7.3.1 Definitions and position of the problem in cruciform welded joints 368

7.3.2 First approach 369

7.3.3 Count data and compare with the experimental results 371

7.3.4 Load-carrying cruciform welded joint submitted to bending 372

7.3.5 Conclusions relative to localization and sensitivity to rupture of cracks 375

7.4 Extension of the initiated crack under fatigue 375

7.4.1 Preliminary test campaign 376

7.4.2 Crack monitoring in an elastic-plastic medium 384

7.4.3 Simulation of crack propagation in mixed-mode test configurations 386

Chapter 8 Potential Use of Crack Propagation Laws in Fatigue Life Design 395

8.1 Calculation of the crack propagation fatigue life of a welded-joint 395

8.1.1 Case of a welded cruciform joint 396

8.2 Study of the influence of different parameters on fatigue life 402

8.3 Statistical characterization of the initial crack size according to the welding procedure 404

8.3.1 Crack propagation and a proposed relationship between n and C 406

8.3.2 Statistical approach and calculation of the initial crack depth, a₀ 408

8.4 Initiation/propagation coupled models: two phase models 410

8.4.1 Propagation period 411

8.4.2 Initiation period 414

8.4.3 S-N curve analysis from the coupled model 415

8.4.4 Coupled model application in the case of variable amplitude loading 417

8.5 Development of a damage model taking into account the crack growth phenomenon 419

8.5.1 Numerical determination of the number of cycles according to crack length or vice versa 422

8.6 Taking into account the presence of residual welding stresses on crack propagation 423

8.6.1 Distribution of residual stresses 423

8.6.2 Method for calculating the energy release rate, G 425

8.6.3 Numerical simulation 426

8.6.4 The influence of welded residual stresses on crack growth rate 428

8.7 Consideration of initial crack length under variable amplitude loading 430

8.7.1 Method description 431

8.8 Propagation of short cracks in the presence of a stress gradient 433

8.8.1 Parametric study of a sample in mode I opening of a notch 436

8.8.2 Application in the case of a welded joint 438

8.8.3 Conclusion and future extensions 439

8.9 Probabilistic approach to crack propagation fatigue life: reliability-failure 440

8.9.1 Modeling of crack retardation effect due to overloading 445

8.9.2 Evolution of the probability of failure 446

8.9.3 Study of sensitivity in terms of reliability 447

8.9.4 Inspection and reliability/failure 448

Conclusion 451

Bibliography 455

Index 477