Dislocation Based Fracture Mechanics

The following sections are included:

• PREFACE

• TABLE OF CONTENTS

• SELECTED SYMBOLS LIST

The stress is very large at the tip of a sharp crack in a lightly stressed solid. Why is the stress there so big when the applied stress is so small? Saying a crack is a ‘stress concentrator’ doesn’t explain it. That just ‘explains’ it by giving the phenomenon a name. A mathematical solution that takes into account the equations of elasticity, the equilibrium equations and the boundary conditions, and does not ignore the strain compatibility equations, leads to a nice, exact description of the high magnitude crack tip stress field. But such a solution does not answer the question: Why is the stress at the tip so big? The physical answer: The stress is large because it is the sum of stress fields of a huge number of like sign dislocations concentrated near the crack tip. Dislocations are piled-up and concentrated at crack tips because a small applied stress, depending upon the length of the crack and the applied stress level, can create at the crack plane the continuum equivalent of tens of thousands or even millions of discrete crystal lattice dislocations and push these dislocations towards the crack tips. From this physical answer we learn at once that fracture mechanics cannot be understood at a deep level unless its study includes dislocations. The goal of this book is to provide for a student an account of the dislocation foundation of fracture mechanics. (Various fracture mechanics and dislocation texts are listed at the end of this book.)…

The crack problems in this book are solved using the stress fields of discrete dislocations of Burgers vector b. These stress fields are the Green’s functions for crack problems. The first topic of this chapter is the stress field of dislocations of Burgers vector b…

The following sections are included:

• HILBERT TRANSFORMS

• MUSKHELISHVILI EQUATIONS
  • Derivation of Muskhelishvile Equations for a Single Zone

• DETERMINATION OF THE DISTANCE C TO THE OUTER CRACK BOUNDARY

• SUMMARY OF EQUATIONS FOR A SINGLE DISLOCATION ZONE

• SOLUTION WHEN THE STRESS IS INFINITE AT THE CRACK TIPS

• NON-CENTERED COORDINATE SYSTEM

• MULTIPLE DISLOCATION ZONES
  • Double Zone
  • Infinite crack tip stress for symmetric double zone
  • Symmetric triple zone

• PLANE STRESS

• RESOLUTION OF A HILBERT TRANSFORM PARADOX WITH GHOST DISLOCATION DISTRIBUTIONS

• Table 3.1 Values of Terms on Segments of Contour Integrals ‡

• HOME WORK PROBLEMS FOR CHAPTER 3

The following sections are included:

• BILBY-COTTRELL-SWINDEN-DUGDALE CRACK
  • Griffith-Inglis Crack Limit
  • Crack Extension force of BCSD Crack
  • Crack Tip Shielding of BCSD Crack
  • Mixed Mode I & II BCSD Crack

• BCSD TYPE ZENER-STROH-KOEHLER CRACK
  • Classical Limit of Zener Stroh-Koehler Crack
  • Stressed Zener-Stroh-Koehler Crack

• STRESS FIELDS OF BCSD CRACKS
  • Mode III BCSD Crack
  • Mode II BCSD Crack
  • Mode I BCSD Crack

• ASYMMETRIC CRACK SOLUTIONS

• DOUBLE SLIP PLANE CRACK MODEL
  • Stationary Crack
  • Crack Tip Stress Intensity Factor
  • Growing Crack

• HOME WORK PROBLEMS FOR CHAPTER 4

This chapter considers the interaction of a dislocation with a crack. Once this interaction is understood it is possible to proceed (in later chapters) to the solution of plastic zones at the tips of cracks in elastic-plastic solids. The solution of the problem of the interaction of a dislocation with a crack also is needed to understand crack tip blunting by dislocation emission, a topic treated later in this chapter…

This chapter presents solutions of the problem of the formation of a plastic zone at a crack tip of a mode III crack in an elastic-plastic solid. The solutions are found by recognizing that the formation of the plastic zone requires the injection of screw dislocations from the crack tip, or from the crack plane near the crack tip, into the plastic zone. The screw dislocations move along stress trajectories. The stress trajectories are the trajectories of the plane on which the shear stress attains its maximum value.

In the previous chapter are found the solution for a mode III crack in an elastic perfectly plastic solid in small scale yielding and for a mode III crack in a linear work hardening solid in general yielding. In this chapter the dislocation method used to find those mode III crack solutions is extended to the mode II crack in such solids under plane strain conditions.

In this chapter the methods used in the two previous chapters to find solutions for the problems of mode II and mode III cracks are employed to find solutions for the mode I crack in elastic-plastic solids.

The problem of the fast moving propagating crack is a difficult one because the motion is non-uniform and the crack radiates elastic energy. A simpler moving crack problem was proposed by Orowan and was solved by Yoffe (1951) for the mode I crack. In this crack problem it is imagined that the crack length a remains constant but the crack is placed in uniform motion (see Fig. 9.1). That is, both crack tips move in the same direction. As one crack tip opens the solid, the other crack tip seals the solid. Because the motion is uniform, no energy is radiated away from the crack. It is assumed that the stress field in the vicinity of a crack tip approximates, and gives insight into, the stress field of a more realistic moving crack tip.

The following sections are included:

• CREVASSES AND DIKES
  • Crevasses
  • Propagation of Magma Filled Cracks
  • Dikes

• UNSTABLE SLIPPAGE

• EDGE CRACKS IN ELASTIC SOLIDS

• INTERFACE CRACKS

• FATIGUE CRACK PROPAGATION
  • Fatigue Crack Growth Controlled by Crack Tip Blunting
  • Paris Law

• REDUNDANT AND NON-REDUNDANT DISLOCATIONS
  • Irwin-Orowan Fracture Equation

• CHAPTER 10 APPENDIX STRESS FIELD OF A MODE I AND OF A MODE II INTERFACE CRACK
  • Mode I Interface Crack
  • Mode II Interface Crack

• HOME WORK PROBLEMS FOR CHAPTER 10

The following sections are included:

• APPENDIX A SHORT TABLE OF HILBERT TRANSFORMS

• APPENDIX B TABLE OF USEFUL INTEGRALS FOR CRACK PROBLEMS

• APPENDIX C STRESS FIELDS OF DISLOCATIONS NEAR OR ON AN INTERFACE
  • SCREW DISLOCATION
  • GLIDE EDGE DISLOCATION
  • CLIMB EDGE DISLOCATION
  • DISLOCATION AT THE INTERFACE
  • COMNINOU-DUNDURS EQUATIONS
  • INVERSE COMNINOU-DUNDURS EQUATIONS

• APPENDIX D TABLE OF USEFUL EQUATIONS
  • STRESS AND STRAIN COMPONENTS
  • ELASTIC CONSTANTS
  • STRESS AND STRAIN COMPONENTS IN ROTATED COORDINATE SYSTEM
  • CONDITIONS FOR ANTIPLANE STRAIN, PLANE STRAIN AND PLANE STRESS
  • EQUATIONS OF DYNAMIC EQUILIBRIUM
  • NON-REDUNDANT DISLOCATION DENSITY FIELD
  • STRESS-STRAIN-ROTATION-DISPLACEMENT FIELDS IN PLANE STRAIN FROM STRESS AND STRAIN FUNCTIONS
  • THE DEL DIVERGENCE, CURL AND GRADIENT OPERATORS (WHEN _z COMPONENT OF A VECTOR = 0 AND ∂{}/∂z = 0)
  • EQUATIONS OF STATIC EQUILIBRIUM IN SHIFTING CENTER CYLINDRICAL COORDINATE SYSTEM
  • TRIGONOMETRIC RELATIONSHIPS
  • DIRAC DELTA FUNCTIONS
  • CURVES
  • SURFACE DISLOCATION DENSITY TENSOR
  • SINGLE STRESS COMPONENT - DISLOCATION DENSITY RELATIONSHIPS

• APPENDIX E DERIVATION OF THE DEL GRADIENT OPERATOR ∇_grad, THE DEL DIVERGENCE OPERATOR ∇_div AND THE DEL CURL OPERATOR ∇_curl FOR THE SHIFTING CENTER CYLINDRICAL COORDINATE SYSTEM
  • DERIVATION OF EQUATIONS OF STATIC EQUILIBRIUM IN SHIFTING CENTER CYLINDRICAL COORDINATE SYSTEM

• APPENDIX F ORTHOGONAL CURVILINEAR COORDINATES
  • CURVILINEAR COORDINATES
  • STATIC EQUILIBRIUM EQUATIONS
  • GRADIENT ∇_grad, DIVERGENCE ∇_div, AND CURL ∇_curl OPERATORS

• APPENDIX G DISLOCATIONS IN STRESS SPACE
  • CRACKS IN ELASTIC STRESS SPACE
  • PLANE STRAIN STRESS SPACE

• REFERENCES

• INDEX