Analytical Fracture Mechanics

Valuable self-contained text, supplementing standard fracture mechanics texts by focusing on analytical methods for determining crack-tip stress and strain fields, presents the only proven solution technique for the second-order nonlinear partial differential equation governing a mode I elastoplastic crack problem. It also deals with plastic zone transitions, environmental cracking, and small-scale yielding versus exact linear elastic solutions. "Should prove to be a valuable resource to both the new student and the experienced researcher in fracture mechanics. It is recommended."-Applied Mechanics Review.

Preface

xi

Acknowledgments

xiii

Introduction

Equations of Continuum Mechanics

1

(4)

Equilibrium

1

(3)

Strain-Displacement

4

(1)

Change of Volume

4

(1)

Compatibility of Strains

4

(1)

Equations of Elasticity

5

(1)

Equations of Plasticity

6

(10)

Strain Hardening

9

(2)

Material Stability

11

(1)

Incremental Strain-Stress Relationships

12

(4)

Flow Theory versus Deformation Theory

16

(1)

Plane Problems of Elasticity Theory

16

(7)

Cartesian Coordinates

17

(2)

Polar Coordinates

19

(2)

Kolosov Equations

21

(2)

Boundary Conditions

23

(1)

Linear Elastic Fracture Mechanics

23

(11)

Mode I

25

(3)

Mode II

28

(1)

Mode III

29

(5)

Strip Models of Crack Tip Plasticity

34

(11)

Mode I Small-Scale Yielding Strip Model

39

(2)

Mode III Small-Scale Yielding Strip Model

41

(4)

Exact Elastoplastic Solutions for Mode III

45

(18)

Isotropic Hardening

52

(11)

Plane Strain Problems Involving Plastic Theory

63

(31)

Plane Strain

63

(12)

Prandtl-Hill Solution

75

(7)

Failure Criteria

82

(8)

Power Law Hardening Materials under Plane Strain

90

(4)

Plane Stress Problems Involving Plastic Material

94

(11)

Tresca Yield Condition

95

(4)

Mises Yield Condition

99

(6)

Power Law Hardening Materials under Plane Stress

105

(1)

Numerical Solutions of the Mode I Elastoplastic Problem

105

(9)

Numerical Solutions

108

(6)

Miscellaneous Mathematical Topics

114

(8)

Complete Solutions

114

(4)

Monge-Ampere Family of Partial Differential Equations

118

(2)

Order Symbols

120

(2)

On the Continuance of an Analytical Solution across the Elastic-Plastic Boundary of a Mode I Fracture Mechanics Problem

Elastoplastic Stress Analyses for Modes I and III

122

(21)

Mode III

122

(5)

Mode I

127

(9)

Parabolic-Hyperbolic Plastic Boundary

136

(1)

Uniqueness and Continuity of Stress

137

(5)

Stress Discontinuities

142

(1)

Developable Surfaces

143

(4)

Strain Rates for Plane Stress under the Tresca Yield Condition

147

(3)

Mode I Displacements

150

(13)

Speculations Concerning an Analytical Mode I Elastoplastic Solution

163

(9)

Plastic Zone Transitions

A Finite Width Dugdale Zone Model for Mode III

172

(14)

Elastic Solution

172

(6)

Plastic Solution

178

(5)

Recovery of Previous Solutions

183

(2)

Crack Tip Opening Displacement

185

(1)

Comments

186

(1)

An Energy-Dissipation Analysis for the Transition Model

186

(7)

Effective Crack Length for the Transition Model

193

(4)

Fracture Assessment Diagrams

197

(11)

Environmental Cracking

Hydrogen-Assisted Cracking

208

(25)

Kinetic Processes

210

(5)

Models

215

(1)

Growth of the Cohesive Zone

216

(4)

Crack Propagation

220

(11)

Overview of Some Previous Models of Environmental Cracking

231

(2)

Analysis for Impending Hydrogen-Assisted Crack Propagation

233

(13)

Crack Tip Opening Displacement

237

(3)

Function ψ (c,a,α)

240

(2)

Asymptotic Expansion

242

(3)

Discussion

245

(1)

A Modified Stefan Problem Related to Stress Corrosion Cracking

246

(15)

Analysis

250

(6)

Small and Large Values of v / V0

256

(2)

Discussion

258

(3)

Small-Scale Yielding versus Exact Linear Elastic Solutions

The Fundamental Modes of Fracture

261

(13)

Elastic-Plastic Loci as Predicted by Linear Elastic Fracture Mechanics

274

(6)

Inverse Cassinian Oval Coordinates for Mode III

280

(5)

Appendix

285

(14)

References

299

(8)

Index

307

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Analytical Fracture Mechanics

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Table of Contents

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