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engineering - fracture mechanics

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Engineering - Fracture Mechanics

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engineering - fracture mechanics

  1. 1. Draft Lecture Notes in: FRACTURE MECHANICS CVEN-6831 c VICTOR E. SAOUMA, Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428 May 17, 2000
  2. 2. Draftii Victor Saouma Fracture Mechanics
  3. 3. Draft Contents I PREAMBULE 3 1 FINITE ELEMENT MODELS FOR for PROGRESSIVE FAILURES 1 1.1 Classification of Failure Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Continuum Mechanics Based Description of Failure; Plasticity 1 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1.1 Uniaxial Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1.2 Idealized Stress-Strain Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Hardening Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Review of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1.1 Hydrostatic and Deviatoric Stress Tensors . . . . . . . . . . . . 5 2.2.1.2 Geometric Representation of Stress States . . . . . . . . . . . . . 6 2.2.1.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2.1 Hydrostatic and Deviatoric Strain Tensors . . . . . . . . . . . . 8 2.2.2.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Yield Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Rate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 J2 Plasticity/von Mises Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5.1 Isotropic Hardening/Softening(J2− plasticity) . . . . . . . . . . . . . . . . 14 2.5.2 Kinematic Hardening/Softening(J2− plasticity) . . . . . . . . . . . . . . . 15 3 Continuum Mechanics Based Description of Failure; Damage Mechanics 1 II FRACTURE MECHANICS 3 4 INTRODUCTION 1 4.1 Modes of Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4.2 Examples of Structural Failures Caused by Fracture . . . . . . . . . . . . . . . . 2 4.3 Fracture Mechanics vs Strength of Materials . . . . . . . . . . . . . . . . . . . . . 3 4.4 Major Historical Developments in Fracture Mechanics . . . . . . . . . . . . . . . 6 4.5 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
  4. 4. Draftii CONTENTS 5 PRELIMINARY CONSIDERATIONS 1 5.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5.1.1 Indicial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5.1.2 Tensor Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5.1.3 Rotation of Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.1.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.1.5 Inverse Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1.6 Principal Values and Directions of Symmetric Second Order Tensors . . . 6 5.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2.1 Force, Traction and Stress Vectors . . . . . . . . . . . . . . . . . . . . . . 7 5.2.2 Traction on an Arbitrary Plane; Cauchy’s Stress Tensor . . . . . . . . . . 9 E 5-1 Stress Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.2.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2.4 Spherical and Deviatoric Stress Tensors . . . . . . . . . . . . . . . . . . . 11 5.2.5 Stress Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.3 Kinematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3.1 Strain Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3.2 Compatibility Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.4 Fundamental Laws of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . 15 5.4.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4.2 Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4.3 Conservation of Mass; Continuity Equation . . . . . . . . . . . . . . . . . 17 5.4.4 Linear Momentum Principle; Equation of Motion . . . . . . . . . . . . . . 17 5.4.5 Moment of Momentum Principle . . . . . . . . . . . . . . . . . . . . . . . 18 5.4.6 Conservation of Energy; First Principle of Thermodynamics . . . . . . . . 19 5.5 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.5.1 Transversly Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.5.2 Special 2D Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.5.2.1 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.5.2.2 Axisymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.5.2.3 Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.6 Airy Stress Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.7 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.7.1 Complex Airy Stress Functions . . . . . . . . . . . . . . . . . . . . . . . . 24 5.8 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.9 Basic Equations of Anisotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . 26 5.9.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.9.2 Plane Stress-Strain Compliance Transformation . . . . . . . . . . . . . . . 29 5.9.3 Stress Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.9.4 Stresses and Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 III LINEAR ELASTIC FRACTURE MECHANICS 33 6 ELASTICITY BASED SOLUTIONS FOR CRACK PROBLEMS 1 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Victor Saouma Fracture Mechanics
  5. 5. DraftCONTENTS iii 6.2 Circular Hole, (Kirsch, 1898) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.3 Elliptical hole in a Uniformly Stressed Plate (Inglis, 1913) . . . . . . . . . . . . . 4 6.4 Crack, (Westergaard, 1939) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.4.1 Stress Intensity Factors (Irwin) . . . . . . . . . . . . . . . . . . . . . . . . 10 6.4.2 Near Crack Tip Stresses and Displacements in Isotropic Cracked Solids . 12 6.5 V Notch, (Williams, 1952) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.6 Crack at an Interface between Two Dissimilar Materials (Williams, 1959) . . . . 17 6.6.1 General Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.6.3 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.6.4 Solve for λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.6.5 Near Crack Tip Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.7 Homogeneous Anisotropic Material (Sih and Paris) . . . . . . . . . . . . . . . . . 24 6.8 Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 LEFM DESIGN EXAMPLES 1 7.1 Design Philosophy Based on Linear Elastic Fracture Mechanics . . . . . . . . . . 1 7.2 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7.3 Fracture Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 7.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 7.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7.5 Additional Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.5.1 Leak Before Fail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.5.2 Damage Tolerance Assessment . . . . . . . . . . . . . . . . . . . . . . . . 15 8 THEORETICAL STRENGTH of SOLIDS; (Griffith I) 1 8.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8.1.1 Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8.1.1.1 Ideal Strength in Terms of Physical Parameters . . . . . . . . . 1 8.1.1.2 Ideal Strength in Terms of Engineering Parameter . . . . . . . . 4 8.1.2 Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8.2 Griffith Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9 ENERGY TRANSFER in CRACK GROWTH; (Griffith II) 1 9.1 Thermodynamics of Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9.1.1 Brittle Material, Griffith’s Model . . . . . . . . . . . . . . . . . . . . . . . 2 9.1.2 Critical Energy Release Rate Determination . . . . . . . . . . . . . . . . . 5 9.1.2.1 From Load-Displacement . . . . . . . . . . . . . . . . . . . . . . 5 9.1.2.2 From Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.1.3 Quasi Brittle material, Irwin and Orrowan Model . . . . . . . . . . . . . . 8 9.2 Energy Release Rate; Equivalence with Stress Intensity Factor . . . . . . . . . . 8 9.3 Crack Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 9.3.1 Effect of Geometry; Π Curve . . . . . . . . . . . . . . . . . . . . . . . . . 10 9.3.2 Effect of Material; R Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 11 9.3.2.1 Theoretical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 9.3.2.2 R vs KIc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Victor Saouma Fracture Mechanics
  6. 6. Draftiv CONTENTS 9.3.2.3 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 9.3.2.4 Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 10 MIXED MODE CRACK PROPAGATION 1 10.1 Analytical Models for Isotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . 2 10.1.1 Maximum Circumferential Tensile Stress. . . . . . . . . . . . . . . . . . . 2 10.1.2 Maximum Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . 3 10.1.3 Minimum Strain Energy Density Criteria. . . . . . . . . . . . . . . . . . . 4 10.1.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 10.2 Emperical Models for Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10.3 Extensions to Anisotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.4 Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 10.4.1 Crack Tip Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 10.4.2 Dimensions of Bimaterial Stress Intensity Factors . . . . . . . . . . . . . . 16 10.4.3 Interface Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . 17 10.4.3.1 Interface Fracture Toughness when β = 0 . . . . . . . . . . . . . 19 10.4.3.2 Interface Fracture Toughness when β = 0 . . . . . . . . . . . . . 19 10.4.4 Crack Kinking Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 10.4.4.1 Numerical Results from He and Hutchinson . . . . . . . . . . . 20 10.4.4.2 Numerical Results Using Merlin . . . . . . . . . . . . . . . . . . 22 10.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 IV ELASTO PLASTIC FRACTURE MECHANICS 29 11 PLASTIC ZONE SIZES 1 11.1 Uniaxial Stress Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 11.1.1 First-Order Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.1.2 Second-Order Approximation (Irwin) . . . . . . . . . . . . . . . . . . . . . 2 11.1.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.1.3 Dugdale’s Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.2 Multiaxial Yield Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 11.3 Plane Strain vs. Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 12 CRACK TIP OPENING DISPLACEMENTS 1 12.1 Derivation of CTOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12.1.1 Irwin’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 12.1.2 Dugdale’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 12.2 G-CTOD Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 13 J INTEGRAL 1 13.1 Genesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 13.2 Path Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 13.3 Nonlinear Elastic Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . 3 13.3.1 Virtual Crack Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 13.3.2 †Generalized Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 13.4 Nonlinear Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 13.5 J Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Victor Saouma Fracture Mechanics
  7. 7. DraftCONTENTS v 13.6 Crack Growth Resistance Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 13.7 † Load Control versus Displacement Control . . . . . . . . . . . . . . . . . . . . . 11 13.8 Plastic Crack Tip Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 13.9 Engineering Approach to Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 16 13.9.1 Compilation of Fully Plastic Solutions . . . . . . . . . . . . . . . . . . . . 18 13.9.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 13.10J1 and J2 Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 13.11Dynamic Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 13.12Effect of Other Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 13.12.1 Surface Tractions on Crack Surfaces . . . . . . . . . . . . . . . . . . . . . 32 13.12.2 Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 13.12.3 Initial Strains Corresponding to Thermal Loading . . . . . . . . . . . . . 33 13.12.4 Initial Stresses Corresponding to Pore Pressures . . . . . . . . . . . . . . 35 13.12.5 Combined Thermal Strains and Pore Pressures . . . . . . . . . . . . . . . 37 13.13Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 V SUBCRITICAL CRACK GROWTH 39 14 FATIGUE CRACK PROPAGATION 1 14.1 Experimental Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 14.2 Fatigue Laws Under Constant Amplitude Loading . . . . . . . . . . . . . . . . . 2 14.2.1 Paris Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 14.2.2 Foreman’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 14.2.2.1 Modified Walker’s Model . . . . . . . . . . . . . . . . . . . . . . 3 14.2.3 Table Look-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 14.2.4 Effective Stress Intensity Factor Range . . . . . . . . . . . . . . . . . . . 4 14.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 14.2.5.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 14.2.5.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14.2.5.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14.3 Variable Amplitude Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14.3.1 No Load Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 14.3.2 Load Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 14.3.2.1 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 14.3.2.2 Retardation Models . . . . . . . . . . . . . . . . . . . . . . . . . 6 14.3.2.2.1 Wheeler’s Model . . . . . . . . . . . . . . . . . . . . . . 7 14.3.2.2.2 Generalized Willenborg’s Model . . . . . . . . . . . . . 8 VI FRACTURE MECHANICS OF CONCRETE 11 15 FRACTURE DETERIORATION ANALYSIS OF CONCRETE 1 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 15.2 Phenomenological Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 15.2.1 Load, Displacement, and Strain Control Tests . . . . . . . . . . . . . . . . 2 15.2.2 Pre/Post-Peak Material Response of Steel and Concrete . . . . . . . . . . 3 15.3 Localisation of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Victor Saouma Fracture Mechanics
  8. 8. Draftvi CONTENTS 15.3.1 Experimental Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 15.3.1.1 σ-COD Diagram, Hillerborg’s Model . . . . . . . . . . . . . . . . 4 15.3.2 Theoretical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 15.3.2.1 Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 15.3.2.2 Dynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 10 15.3.2.2.1 Loss of Hyperbolicity . . . . . . . . . . . . . . . . . . . 11 15.3.2.2.2 Wave Equation for Softening Maerials . . . . . . . . . . 11 15.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 15.4 Griffith Criterion and FPZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 16 FRACTURE MECHANICS of CONCRETE 1 16.1 Linear Elastic Fracture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 16.1.1 Finite Element Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 2 16.1.2 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.1.3 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.2 Nonlinear Fracture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 16.2.1 Hillerborg Characteristic Length . . . . . . . . . . . . . . . . . . . . . . . 6 16.2.1.1 MIHASHI data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 16.2.2 Jenq and Shah Two Parameters Model . . . . . . . . . . . . . . . . . . . . 6 16.2.3 Size Effect Law, Baˇzant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 16.2.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 16.2.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 16.2.4 Carpinteri Brittleness Number . . . . . . . . . . . . . . . . . . . . . . . . 11 16.3 Comparison of the Fracture Models . . . . . . . . . . . . . . . . . . . . . . . . . . 12 16.3.1 Hillerborg Characteristic Length, lch . . . . . . . . . . . . . . . . . . . . . 12 16.3.2 Baˇzant Brittleness Number, β . . . . . . . . . . . . . . . . . . . . . . . . . 13 16.3.3 Carpinteri Brittleness Number, s . . . . . . . . . . . . . . . . . . . . . . . 14 16.3.4 Jenq and Shah’s Critical Material Length, Q . . . . . . . . . . . . . . . . 14 16.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 16.4 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 17 FRACTURE MECHANICS PROPERTIES OF CONCRETE 1 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 17.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 17.2.1 Concrete Mix Design and Specimen Preparation . . . . . . . . . . . . . . 2 17.2.2 Loading Fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 17.2.3 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 17.2.4 Acoustic Emissions Monitoring . . . . . . . . . . . . . . . . . . . . . . . . 5 17.2.5 Evaluation of Fracture Toughness by the Compliance Method . . . . . . . 6 17.3 Fracture Toughness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 17.4 Specific Fracture Energy Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 17.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 17.6 Size Effect Law Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 17.7 Notation and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Victor Saouma Fracture Mechanics
  9. 9. DraftCONTENTS vii 18 FRACTALS, FRACTURES and SIZE EFFECTS 1 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 18.1.1 Fracture of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 18.1.2 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 18.1.3 Numerical Determination of Fractal Dimension . . . . . . . . . . . . . . . 3 18.1.4 Correlation of Fractal Dimensions With Fracture Properties . . . . . . . . 4 18.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 18.2.1 Fracture Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 18.2.2 Profile Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 18.2.3 Computation of Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . 8 18.3 Fractals and Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 18.3.1 Spatial Variation of the Fractal Dimension . . . . . . . . . . . . . . . . . . 10 18.3.2 Correlation Between Fracture Toughness and Fractal Dimensions . . . . . 14 18.3.3 Macro-Scale Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . 14 18.4 Fractals and Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 18.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 VII FINITE ELEMENT TECHNIQUES IN FRACTURE MECHAN- ICS 23 19 SINGULAR ELEMENT 1 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 19.2 Displacement Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 19.3 Quarter Point Singular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 19.4 Review of Isoparametric Finite Elements . . . . . . . . . . . . . . . . . . . . . . . 3 19.5 How to Distort the Element to Model the Singularity . . . . . . . . . . . . . . . . 5 19.6 Order of Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 19.7 Stress Intensity Factors Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 7 19.7.1 Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 19.7.2 Anisotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 19.8 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 19.9 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 19.10Other Singular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 20 ENERGY RELEASE BASED METHODS 1 20.1 Mode I Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 20.1.1 Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 20.1.2 Virtual Crack Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 20.2 Mixed Mode Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 20.2.1 Two Virtual Crack Extensions. . . . . . . . . . . . . . . . . . . . . . . . . 3 20.2.2 Single Virtual Crack Extension, Displacement Decomposition . . . . . . . 4 21 J INTEGRAL BASED METHODS 1 21.1 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 21.2 Mixed Mode SIF Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 21.3 Equivalent Domain Integral (EDI) Method . . . . . . . . . . . . . . . . . . . . . 5 21.3.1 Energy Release Rate J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Victor Saouma Fracture Mechanics
  10. 10. Draftviii CONTENTS 21.3.1.1 2D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 21.3.1.2 3D Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 8 21.3.2 Extraction of SIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 21.3.2.1 J Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 21.3.2.2 σ and u Decomposition . . . . . . . . . . . . . . . . . . . . . . . 10 22 RECIPROCAL WORK INTEGRALS 1 22.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 22.2 Volume Form of the Reciprocal Work Integral . . . . . . . . . . . . . . . . . . . . 5 22.3 Surface Tractions on Crack Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 7 22.4 Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 22.5 Initial Strains Corresponding to Thermal Loading . . . . . . . . . . . . . . . . . 8 22.6 Initial Stresses Corresponding to Pore Pressures . . . . . . . . . . . . . . . . . . . 10 22.7 Combined Thermal Strains and Pore Pressures . . . . . . . . . . . . . . . . . . . 10 22.8 Field Equations for Thermo- and Poro-Elasticity . . . . . . . . . . . . . . . . . . 11 23 FICTITIOUS CRACK MODEL 1 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 23.2 Computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 23.2.1 Weak Form of Governing Equations . . . . . . . . . . . . . . . . . . . . . 2 23.2.2 Discretization of Governing Equations . . . . . . . . . . . . . . . . . . . . 4 23.2.3 Penalty Method Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 23.2.4 Incremental-Iterative Solution Strategy . . . . . . . . . . . . . . . . . . . 8 23.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 23.3.1 Load-CMOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 23.3.2 Real, Fictitious, and Effective Crack Lengths . . . . . . . . . . . . . . . . 11 23.3.3 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 23.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 23.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 24 INTERFACE CRACK MODEL 1 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 24.2 Interface Crack Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 24.2.1 Relation to fictitious crack model. . . . . . . . . . . . . . . . . . . . . . . 8 24.3 Finite Element Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 24.3.1 Interface element formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 9 24.3.2 Constitutive driver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 24.3.3 Non-linear solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 24.3.4 Secant-Newton method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 24.3.5 Element secant stiffness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 24.3.6 Line search method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 24.4 Mixed Mode Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 24.4.1 Griffith criterion and ICM. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 24.4.2 Criterion for crack propagation. . . . . . . . . . . . . . . . . . . . . . . . . 23 24.5 Examples and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 24.5.1 Direct shear test of mortar joints. . . . . . . . . . . . . . . . . . . . . . . . 24 24.5.2 Biaxial interface test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 24.5.3 Modified Iosipescu’s beam. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Victor Saouma Fracture Mechanics
  11. 11. DraftCONTENTS ix 24.5.4 Anchor bolt pull-out test. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 24.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Victor Saouma Fracture Mechanics
  12. 12. Draftx CONTENTS Victor Saouma Fracture Mechanics
  13. 13. Draft List of Figures 1.1 Kinematics of Continuous and Discontinuous Failure Processes . . . . . . . . . . 1 1.2 Discrete-Smeared Crack Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Typical Stress-Strain Curve of an Elastoplastic Bar . . . . . . . . . . . . . . . . . 1 2.2 Elastoplastic Rheological Model for Overstress Formulation . . . . . . . . . . . . 2 2.3 Bauschinger Effect on Reversed Loading . . . . . . . . . . . . . . . . . . . . . . . 2 2.4 Stress and Strain Increments in Elasto-Plastic Materials . . . . . . . . . . . . . . 3 2.5 Stress-Strain diagram for Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . 4 2.6 Haigh-Westergaard Stress Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.7 General Yield Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.8 Isotropic Hardening/Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.9 Kinematic Hardening/Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1 Cracked Cantilevered Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.2 Failure Envelope for a Cracked Cantilevered Beam . . . . . . . . . . . . . . . . . 4 4.3 Generalized Failure Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.4 Column Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.1 Stress Components on an Infinitesimal Element . . . . . . . . . . . . . . . . . . . 8 5.2 Stresses as Tensor Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.3 Cauchy’s Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.4 Flux Through Area dS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.5 Equilibrium of Stresses, Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 18 5.6 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.7 Transversly Isotropic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.8 Coordinate Systems for Stress Transformations . . . . . . . . . . . . . . . . . . . 29 6.1 Circular Hole in an Infinite Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2 Elliptical Hole in an Infinite Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.3 Crack in an Infinite Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6.4 Independent Modes of Crack Displacements . . . . . . . . . . . . . . . . . . . . . 11 6.5 Plate with Angular Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.6 Plate with Angular Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.1 Middle Tension Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7.2 Single Edge Notch Tension Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7.3 Double Edge Notch Tension Panel . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7.4 Three Point Bend Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7.5 Compact Tension Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
  14. 14. Draftii LIST OF FIGURES 7.6 Approximate Solutions for Two Opposite Short Cracks Radiating from a Circular Hole in an Infinite Plate under Tension . . . . . . . . . . . . . . . . . . . . . . . . 5 7.7 Approximate Solutions for Long Cracks Radiating from a Circular Hole in an Infinite Plate under Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7.8 Radiating Cracks from a Circular Hole in an Infinite Plate under Biaxial Stress . 6 7.9 Pressurized Hole with Radiating Cracks . . . . . . . . . . . . . . . . . . . . . . . 7 7.10 Two Opposite Point Loads acting on the Surface of an Embedded Crack . . . . . 8 7.11 Two Opposite Point Loads acting on the Surface of an Edge Crack . . . . . . . . 8 7.12 Embedded, Corner, and Surface Cracks . . . . . . . . . . . . . . . . . . . . . . . 9 7.13 Elliptical Crack, and Newman’s Solution . . . . . . . . . . . . . . . . . . . . . . . 11 7.14 Growth of Semielliptical surface Flaw into Semicircular Configuration . . . . . . 14 8.1 Uniformly Stressed Layer of Atoms Separated by a0 . . . . . . . . . . . . . . . . 2 8.2 Energy and Force Binding Two Adjacent Atoms . . . . . . . . . . . . . . . . . . 2 8.3 Stress Strain Relation at the Atomic Level . . . . . . . . . . . . . . . . . . . . . . 3 8.4 Influence of Atomic Misfit on Ideal Shear Strength . . . . . . . . . . . . . . . . . 5 9.1 Energy Transfer in a Cracked Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9.2 Determination of Gc From Load Displacement Curves . . . . . . . . . . . . . . . 5 9.3 Experimental Determination of KI from Compliance Curve . . . . . . . . . . . . 6 9.4 KI for DCB using the Compliance Method . . . . . . . . . . . . . . . . . . . . . 6 9.5 Variable Depth Double Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . 8 9.6 Graphical Representation of the Energy Release Rate G . . . . . . . . . . . . . . 9 9.7 Effect of Geometry and Load on Crack Stability, (Gtoudos 1993) . . . . . . . . . 11 9.8 R Curve for Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 9.9 R Curve for Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 9.10 Plastic Zone Ahead of a Crack Tip Through the Thickness . . . . . . . . . . . . 16 10.1 Mixed Mode Crack Propagation and Biaxial Failure Modes . . . . . . . . . . . . 2 10.2 Crack with an Infinitesimal “kink” at Angle θ . . . . . . . . . . . . . . . . . . . . 4 10.3 Sθ Distribution ahead of a Crack Tip . . . . . . . . . . . . . . . . . . . . . . . . . 6 10.4 Angle of Crack Propagation Under Mixed Mode Loading . . . . . . . . . . . . . . 7 10.5 Locus of Fracture Diagram Under Mixed Mode Loading . . . . . . . . . . . . . . 7 10.6 Fracture Toughnesses for Homogeneous Anisotropic Solids . . . . . . . . . . . . . 9 10.7 Angles of Crack Propagation in Anisotropic Solids . . . . . . . . . . . . . . . . . 12 10.8 Failure Surfaces for Cracked Anisotropic Solids . . . . . . . . . . . . . . . . . . . 14 10.9 Geometry and conventions of an interface crack, (Hutchinson and Suo 1992) . . . 15 10.10Geometry of kinked Crack, (Hutchinson and Suo 1992) . . . . . . . . . . . . . . . 17 10.11Schematic variation of energy release rate with length of kinked segment of crack for β = 0, (Hutchinson and Suo 1992) . . . . . . . . . . . . . . . . . . . . . . . . 18 10.12Conventions for a Crack Kinking out of an Interface, (Hutchinson and Suo 1992) 21 10.13Geometry and Boundary Conditions of the Plate Analyzed . . . . . . . . . . . . 23 10.14Finite Element Mesh of the Plate Analyzed . . . . . . . . . . . . . . . . . . . . . 23 10.15Variation of G/Go with Kink Angle ω . . . . . . . . . . . . . . . . . . . . . . . . 25 11.1 First-Order Approximation of the Plastic Zone . . . . . . . . . . . . . . . . . . . 2 11.2 Second-Order Approximation of the Plastic Zone . . . . . . . . . . . . . . . . . . 3 11.3 Dugdale’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.4 Point Load on a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Victor Saouma Fracture Mechanics
  15. 15. DraftLIST OF FIGURES iii 11.5 Effect of Plastic Zone Size on Dugdale’s Model . . . . . . . . . . . . . . . . . . . 6 11.6 Barenblatt’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 11.7 Normalized Mode I Plastic Zone (Von-Myses) . . . . . . . . . . . . . . . . . . . . 8 11.8 Plastic Zone Size Across Plate Thickness . . . . . . . . . . . . . . . . . . . . . . . 9 11.9 Plastic Zone Size in Comparison with Plate Thickness; Plane Stress and Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 11.10Plate Thickness Effect on Fracture Toughness . . . . . . . . . . . . . . . . . . . . 11 12.1 Crack Tip Opening Displacement, (Anderson 1995) . . . . . . . . . . . . . . . . . 1 12.2 Estimate of the Crack Tip Opening Displacement, (Anderson 1995) . . . . . . . . 2 13.1 J Integral Definition Around a Crack . . . . . . . . . . . . . . . . . . . . . . . . . 1 13.2 Closed Contour for Proof of J Path Independence . . . . . . . . . . . . . . . . . 3 13.3 Virtual Crack Extension Definition of J . . . . . . . . . . . . . . . . . . . . . . . 4 13.4 Arbitrary Solid with Internal Inclusion . . . . . . . . . . . . . . . . . . . . . . . . 6 13.5 Elastic-Plastic versus Nonlinear Elastic Materials . . . . . . . . . . . . . . . . . . 8 13.6 Nonlinear Energy Release Rate, (Anderson 1995) . . . . . . . . . . . . . . . . . . 8 13.7 Experimental Derivation of J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 13.8 J Resistance Curve for Ductile Material, (Anderson 1995) . . . . . . . . . . . . . 10 13.9 J, JR versus Crack Length, (Anderson 1995) . . . . . . . . . . . . . . . . . . . . 12 13.10J, Around a Circular Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13.11Normalize Ramberg-Osgood Stress-Strain Relation (α = .01) . . . . . . . . . . . 13 13.12HRR Singularity, (Anderson 1995) . . . . . . . . . . . . . . . . . . . . . . . . . . 15 13.13Effect of Plasticity on the Crack Tip Stress Fields, (Anderson 1995) . . . . . . . 16 13.14Compact tension Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 13.15Center Cracked Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 13.16Single Edge Notched Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 13.17Double Edge Notched Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 13.18Axially Cracked Pressurized Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 23 13.19Circumferentially Cracked Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 26 13.20Dynamic Crack Propagation in a Plane Body, (Kanninen 1984) . . . . . . . . . . 30 14.1 S-N Curve and Endurance Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 14.2 Repeated Load on a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 14.3 Stages of Fatigue Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 14.4 Forman’s Fatigue Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 14.5 Retardation Effects on Fatigue Life . . . . . . . . . . . . . . . . . . . . . . . . . . 7 14.6 Cause of Retardation in Fatigue Crack Grwoth . . . . . . . . . . . . . . . . . . . 7 14.7 Yield Zone Due to Overload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 15.1 Test Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 15.2 Stress-Strain Curves of Metals and Concrete . . . . . . . . . . . . . . . . . . . . . 3 15.3 Caputring Experimentally Localization in Uniaxially Loaded Concrete Specimens 4 15.4 Hillerborg’s Fictitious Crack Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5 15.5 Concrete Strain Softening Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 15.6 Strain-Softening Bar Subjected to Uniaxial Load . . . . . . . . . . . . . . . . . . 8 15.7 Load Displacement Curve in terms of Element Size . . . . . . . . . . . . . . . . . 10 15.8 Localization of Tensile Strain in Concrete . . . . . . . . . . . . . . . . . . . . . . 13 15.9 Griffith criterion in NLFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Victor Saouma Fracture Mechanics
  16. 16. Draftiv LIST OF FIGURES 16.1 Servo-Controlled Test Setup for Concrete KIc and GF . . . . . . . . . . . . . . . 3 16.2 *Typical Load-CMOD Curve from a Concrete Fracture Test . . . . . . . . . . . . 4 16.3 Effective Crack Length Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 16.4 Size Effect Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 16.5 Inelastic Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 17.1 *Wedge-splitting specimen geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 3 17.2 *Wedge fixture and line support. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 17.3 *Block diagram of the experimental system. . . . . . . . . . . . . . . . . . . . . . 5 17.4 *Typical PSP vs. CMOD curve for a “Large” specimen. . . . . . . . . . . . . . . 5 17.5 *Typical AE record for a “Large” WS specimen test. . . . . . . . . . . . . . . . . 6 17.6 *The three stages of the fracture toughness vs. effective crack length curve. . . . 7 17.7 *Mean fracture toughness values obtained from the rounded MSA WS specimen tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 17.8 *Mean specific fracture energy values obtained from the rounded MSA WS spec- imen tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 17.9 Size effect for WS specimens for da=38 mm (1.5 in) (Br¨uhwiler, E., Broz, J.J., and Saouma, V.E., 1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 18.1 (A) Straight line initiator, fractal generator, and triadic Koch curve; (B) Quadratic Koch curve; (C) Modified Koch curve. . . . . . . . . . . . . . . . . . . . . . . . . 2 18.2 Frontal view of wedge-splitting-test specimen showing forces applied to specimen by lateral wedge loading (FS) between two circular pins located near top of specimen on either side of the vertical starting notch. Crack Mouth Opening Displacement (CMOD) gage straddles the initial notch. . . . . . . . . . . . . . . 6 18.3 Orientations of measured profiles over the fractured surface, horizontally, verti- cally, and diagonally. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 18.4 Typical grid overlying an object. Dashed lines indicate adjustable sidFixed grid boundaries; B, Flexible grid boundaries. . . . . . . . . . . . . . . . . . . . . . . . 9 18.5 Plot of box counting method applied to the profile of a typical fractured concrete specimen. Number of occupied boxes (N) is plotted versus box size. Slope of line fit to data is the fractal dimension (D). . . . . . . . . . . . . . . . . . . . . . 10 18.6 A-B) GF and KIc versus D; C) GF versus D for concrete (this study), ceramics, and alumina (Mecholsky and Freiman, 1991); D) KIc versus D for concrete (this study); ceramics, and alumina (Mecholsky and Frieman, 1991); Flint (Mecholsky and Mackin, 1988); polystyrene (Chen and Runt, 1989); and silicon (Tsai and Mecholsky (1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 18.7 GF versus D for concrete (this study), ceramics, and alumina (Mecholsky and Freiman, 1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 18.8 KIc versus D for concrete (this study); ceramics, and alumina (Mecholsky and Frieman, 1991); Flint (Mecholsky and Mackin, 1988); polystyrene (Chen and Runt, 1989); and silicon (Tsai and Mecholsky (1991) . . . . . . . . . . . . . . . . 18 18.9 Variation of L(S) in terms of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19.1 Stress Intensity Factor Using Extrapolation Technique . . . . . . . . . . . . . . . 2 19.2 Isoparametric Quadratic Finite Element: Global and Parent Element . . . . . . . 3 19.3 Singular Element (Quarter-Point Quadratic Isoparametric Element) . . . . . . . 6 19.4 Finite Element Discretization of the Crack Tip Using Singular Elements . . . . . 7 Victor Saouma Fracture Mechanics
  17. 17. DraftLIST OF FIGURES v 19.5 Displacement Correlation Method to Extract SIF from Quarter Point Singular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 19.6 Nodal Definition for FE 3D SIF Determination . . . . . . . . . . . . . . . . . . . 10 20.1 Crack Extension ∆a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 20.2 Displacement Decomposition for SIF Determination . . . . . . . . . . . . . . . . 4 21.1 Numerical Extraction of the J Integral (Owen and Fawkes 1983) . . . . . . . . . 2 21.2 Simply connected Region A Enclosed by Contours Γ1, Γ0, Γ+, and Γ−, (Anderson 1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 21.3 Surface Enclosing a Tube along a Three Dimensional Crack Front, (Anderson 1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 21.4 Interpretation of q in terms of a Virtual Crack Advance along ∆L, (Anderson 1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 21.5 Inner and Outer Surfaces Enclosing a Tube along a Three Dimensional Crack Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 22.1 Contour integral paths around crack tip for recipcoal work integral . . . . . . . . 2 23.1 Body Consisting of Two Sub-domains . . . . . . . . . . . . . . . . . . . . . . . . 3 23.2 Wedge Splitting Test, and FE Discretization . . . . . . . . . . . . . . . . . . . . . 11 23.3 Numerical Predictions vs Experimental Results for Wedge Splitting Tests . . . . 12 23.4 Real, Fictitious, and Effective Crack Lengths for Wedge Splitting Tests . . . . . . 13 23.5 Effect of GF on 50 ft Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 23.6 Effect of wc on 50 ft Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 23.7 Effect of s1 on 3 ft Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 24.1 Mixed mode crack propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 24.2 Wedge splitting tests for different materials, (Saouma V.E., and ˇCervenka, J. and Slowik, V. and Chandra Kishen, J.M. 1994) . . . . . . . . . . . . . . . . . . 3 24.3 Interface idealization and notations. . . . . . . . . . . . . . . . . . . . . . . . . . 4 24.4 Interface fracture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 24.5 Failure function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 24.6 Bi-linear softening laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 24.7 Stiffness degradation in the equivalent uniaxial case. . . . . . . . . . . . . . . . . 8 24.8 Interface element numbering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 24.9 Local coordinate system of the interface element. . . . . . . . . . . . . . . . . . . 11 24.10Algorithm for interface constitutive model. . . . . . . . . . . . . . . . . . . . . . 12 24.11Definition of inelastic return direction. . . . . . . . . . . . . . . . . . . . . . . . . 14 24.12Influence of increment size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 24.13Shear-tension example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 24.14Secant relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 24.15Line search method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 24.16Griffith criterion in NLFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 24.17Mixed mode crack propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 24.18Schematics of the direct shear test setup. . . . . . . . . . . . . . . . . . . . . . . 25 24.19Direct shear test on mortar joint. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 24.20Experimental set-up for the large scale mixed mode test. . . . . . . . . . . . . . . 27 24.21Nonlinear analysis of the mixed mode test. . . . . . . . . . . . . . . . . . . . . . 29 Victor Saouma Fracture Mechanics
  18. 18. Draftvi LIST OF FIGURES 24.22Crack propagation in Iosipescu’s beam, (Steps 1 & 3). . . . . . . . . . . . . . . . 31 24.23Crack propagation in Iosipescu’s beam, (Increment 11 & 39 in Step 6). . . . . . . 32 24.24Multiple crack propagation in Iosipescu’s beam (Steps 3,4). . . . . . . . . . . . . 33 24.25Multiple crack propagation in Iosipescu’s beam (Step 5). . . . . . . . . . . . . . . 34 24.26Meshes for crack propagation in Iosipescu’s beam (Steps 1,3,4,5). . . . . . . . . . 35 24.27Iosipescu’s beam with ICM model. . . . . . . . . . . . . . . . . . . . . . . . . . . 36 24.28Crack paths for Iosipescu’s beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 24.29Large Iosipescu’s beam, h = 50 x 100 mm. . . . . . . . . . . . . . . . . . . . . . . 37 24.30Crack propagation for anchor bolt pull out test I. . . . . . . . . . . . . . . . . . . 38 24.31Crack propagation for anchor bolt pull out test II. . . . . . . . . . . . . . . . . . 39 24.32Crack patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 24.33Load displacement curve for test I. . . . . . . . . . . . . . . . . . . . . . . . . . . 40 24.34Load displacement curve for test II. . . . . . . . . . . . . . . . . . . . . . . . . . 41 Victor Saouma Fracture Mechanics
  19. 19. Draft List of Tables 4.1 Column Instability Versus Fracture Instability . . . . . . . . . . . . . . . . . . . . 5 5.1 Number of Elastic Constants for Different Materials . . . . . . . . . . . . . . . . 27 6.1 Summary of Elasticity Based Problems Analysed . . . . . . . . . . . . . . . . . . 1 7.1 Newman’s Solution for Circular Hole in an Infinite Plate subjected to Biaxial Loading, and Internal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7.2 C Factors for Point Load on Edge Crack . . . . . . . . . . . . . . . . . . . . . . . 9 7.3 Approximate Fracture Toughness of Common Engineering Materials . . . . . . . 12 7.4 Fracture Toughness vs Yield Stress for .45C − Ni − Cr − Mo Steel . . . . . . . . 12 10.1 Material Properties and Loads for Different Cases . . . . . . . . . . . . . . . . . . 23 10.2 Analytical and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 10.3 Numerical Results using S-integral without the bimaterial model . . . . . . . . . 26 12.1 Comparison of Various Models in LEFM and EPFM . . . . . . . . . . . . . . . . 2 13.1 Effect of Plasticity on the Crack Tip Stress Field, (Anderson 1995) . . . . . . . . 15 13.2 h-Functions for Standard ASTM Compact Tension Specimen, (Kumar, German and Shih 1981) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 13.3 Plane Stress h-Functions for a Center-Cracked Panel, (Kumar et al. 1981) . . . . 21 13.4 h-Functions for Single Edge Notched Specimen, (Kumar et al. 1981) . . . . . . . 22 13.5 h-Functions for Double Edge Notched Specimen, (Kumar et al. 1981) . . . . . . . 24 13.6 h-Functions for an Internally Pressurized, Axially Cracked Cylinder, (Kumar et al. 1981) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 13.7 F and V1 for Internally Pressurized, Axially Cracked Cylinder, (Kumar et al. 1981) 26 13.8 h-Functions for a Circumferentially Cracked Cylinder in Tension, (Kumar et al. 1981) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 13.9 F, V1, and V2 for a Circumferentially Cracked Cylinder in Tension, (Kumar et al. 1981) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 15.1 Strain Energy versus Fracture Energy for uniaxial Concrete Specimen . . . . . . 10 16.1 Size Effect Law vs Column Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 16.2 Summary relations for the concrete fracture models. . . . . . . . . . . . . . . . . 15 16.3 When to Use LEFM or NLFM Fracture Models . . . . . . . . . . . . . . . . . . . 15 17.1 Concrete mix design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 17.2 Experimentally obtained material properties of the concrete mixes used. . . . . . 3
  20. 20. Draftii LIST OF TABLES 17.3 Wedge-splitting specimen dimensions. . . . . . . . . . . . . . . . . . . . . . . . . 3 17.4 Test matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 17.5 Summary of fracture toughness data obtained from the WS tests. . . . . . . . . . 8 17.6 Fracture toughness values obtained from the CJ-WS specimens. . . . . . . . . . . 8 17.7 Summary of specific fracture energy values obtained from the WS tests. . . . . . 10 17.8 Fracture energy values obtained from the CJ-WS specimens. . . . . . . . . . . . . 10 17.9 Size Effect Law model assessment from the WS test program (average val- ues)(Br¨uhwiler, E., Broz, J.J., and Saouma, V.E., 1991) . . . . . . . . . . . . . . 12 18.1 Fractal dimension definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 18.2 Concrete mix design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 18.3 Range and resolution of the profilometer (inches) . . . . . . . . . . . . . . . . . . 7 18.4 CHECK Mapped profile spacing, orientation, and resolution for the two specimen sizes investigated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 18.5 Computed fractal dimensions of a straight line with various inclinations . . . . . 9 18.6 Computed fractal dimensions for various synthetic curves . . . . . . . . . . . . . 9 18.7 Fractal dimension D versus profile orientations . . . . . . . . . . . . . . . . . . . 11 18.8 Fractal dimension for various profile segments and distances from centerline in specimen S33A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 18.9 Comparison between D, KIc, and GF for all specimens (MSA, Maximum size aggregate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 18.10Linear regression coefficients between GF and KIc with D . . . . . . . . . . . . . 16 18.11Amplification factors for fractal surface areas with D = 1.1 . . . . . . . . . . . . 20 18.12Comparison between “corrected” G∗ F and Gc values based on Swartz Tests (1992). 21 24.1 Material properties for direct shear test. . . . . . . . . . . . . . . . . . . . . . . . 27 24.2 Material properties for direct shear test. . . . . . . . . . . . . . . . . . . . . . . . 28 24.3 Material properties for ICM for Iosipescu’s test. . . . . . . . . . . . . . . . . . . . 29 Victor Saouma Fracture Mechanics
  21. 21. DraftLIST OF TABLES 1 COVERAGE Mon. Day COVERAGE Jan. 14 Intro, Coverage 1 19 Overview, Elasticity 21 Elasticity, Kirch, Hole 2 26 Crack, Griffith 29 Notch, Williams 3 Feb. 2 LEFM 4 LEFM, Examples 4 9 Bi-Material, Merlin 11 Theoretical Strength 5 16 Theoretical Strength 18 Energy 6 23 Energy 26 MERLIN 7 Mar. 2 Mixed Mode 4 Plastic Zone Size 8 9 CTOD, J 11 J 9 16 Fatigue 18 Fatigue Fatigue 10 30 Concrete Apr. 1 Fatigue 11 6 Concrete 8 Concrete 12 13 EXAM 15 Concrete 13 20 Num. Methods 22 Num. Methods 14 27 Experiment 29 Anisotropic Victor Saouma Fracture Mechanics
  22. 22. Draft2 LIST OF TABLES Victor Saouma Fracture Mechanics
  23. 23. Draft Part I PREAMBULE
  24. 24. Draft
  25. 25. Draft Chapter 1 FINITE ELEMENT MODELS FOR for PROGRESSIVE FAILURES 1.1 Classification of Failure Kinematics 1 We can distinguish three separate modes of failure according to the degree of C-(dis)continuity in failure processes, Fig. 1.1: (?). 000000000000000000 000000000000000000000000000000000000 000000000000000000000000000 000000000000000000000000000 111111111111111111 111111111111111111111111111111111111 111111111111111111111111111 111111111111111111111111111 u = u. . . .+ - + ε = - ε = = 000000000000 000000000000000000000000 000000000000000000 000000000000000000 111111111111 111111111111111111111111 111111111111111111 111111111111111111 000000000000 000000000000000000000000 111111111111 111111111111111111111111 000000000000000000 000000000000000000000000000000000000 000000000000000000000000000000000000 111111111111111111 111111111111111111111111111111111111 111111111111111111111111111111111111 + - - - + + 1 0 -1 C Continuity C Continuity C Continuity u = u. . . .+ - + ε - ε u = u . . . .+ - + ε - ε Figure 1.1: Kinematics of Continuous and Discontinuous Failure Processes Diffuse failure: corresponds to C1-continuity of motion. We do have a continuity of both displacement (increments) and strains across a potential failure surface. Localized failure: corresponds to a C0 continuity of motion. Whereas continuity of displace- ments is maintained, we do have a discontinuity in strains. Discrete failure: corresponds to C−1-continuity of motion, where we have a discontinuity of both strain and displacements.
  26. 26. Draft2 FINITE ELEMENT MODELS FOR for PROGRESSIVE FAILURES 2 The numerical simulation of progressive failure in solids has been historically addressed within the frameworks of either computational plasticity (and more recently damage mechanics), or within the framework of fracture mechanics. 3 With reference to Fig. 1.1, solutions rooted in continuum mechanics (plasticity and damage mechanics) have historically addressed problems with C1 and C0 continuity, whereas fracture mechanics has focused on problems with C−1 continuity. 4 In the context of finite element analysis of failure processes different strategies can be adopted, Fig. 1.2: 00001111 00001111 0011 0011 00001111 001101010011 0011 01 01 0011 0000111100110011 01 01 0011 0000111100110011 0011 00001111 0000111100001111 0011 01 001100001111 00001111 0011 0011 0011 0011001100001111 0011 01 01 0011 000011110011001100001111 00001111 0011 0011 00001111 0011010100110011 0011 0011 00001111 0011 00001111 0011 00001111 0000111100001111 00001111 00001111 0011 0011 Constant Number of Elements/Nodes Increase No. of Elements Decrease No. of Elements Continuum Mechanics (Plasticity, Damage) Fracture Mechanics Smeared Embedded Discrete Lattice Figure 1.2: Discrete-Smeared Crack Models Smeared Crack Embedded Crack Discrete Crack Lattice Model 5 Broadly speaking, we have two class of solutions Continuum mechanics based solutions, in which the solid is assumed to be inherently con- tinuous. This includes Plasticity theory Damage mechanics Victor Saouma Fracture Mechanics
  27. 27. Draft1.1 Classification of Failure Kinematics 3 Fracture mechanics based solutions, where a physical discontinuity is assumed to exist from the onset. 6 Smeared and embedded cracks formulations are both based on continuum mechanics formu- lations, whereas discrete cracks are more suitably analysed within the framework of fracture mechanics. INSERT FMC1 here Victor Saouma Fracture Mechanics
  28. 28. Draft4 FINITE ELEMENT MODELS FOR for PROGRESSIVE FAILURES Victor Saouma Fracture Mechanics
  29. 29. Draft Chapter 2 Continuum Mechanics Based Description of Failure; Plasticity 2.1 Introduction 2.1.1 Uniaxial Behavior 7 The typical stress-strain behavior of most metals in simple tension is shown in Fig. 2.1 σy2 O1 ε1 e σ ε σ σy0 y1 O 1 O2 0A A A2 ε1 p ε ε ep 2 2 Figure 2.1: Typical Stress-Strain Curve of an Elastoplastic Bar 8 Up to the A0, the response is linearly elastic, and unloading follows the initial loading path. O − A represents the elastic range where the behavior is load path independent. 9 At point A0, the material has reached its elastic limit, from A0 to C the material becomes plastic and behaves irreversibly. In this plastic range, the stiffness decreases progressively, and eventually fails at C.
  30. 30. Draft2 Continuum Mechanics Based Description of Failure; Plasticity 10 Unloading from any point between A0 and C results in a proportionally decreasing stress and strain along A1O1 parallel to the initial elastic loading O0A0. A complete unloading would leave a permanent strain or a plastic strain εp 1. Thus only part of the total strain ε1at A1 is recovered upon unloading, that is the elastic strain εe 1. 11 The Rheological model for elastoplasticity, Fig. 2.2, has only one spring element and a friction element. σy 1 + ε E σεp ( ) Figure 2.2: Elastoplastic Rheological Model for Overstress Formulation 12 In case of reversed loading, Fig. 2.3, when the material is loaded in compression after it has been loaded in tension, the stress-strain curve will be different from the one obtained from pure tension or compression. The new yield point in compression at B corresponds to stress σy0 σy0 2σy0 2σy0 σ O 0A ε B C σyB σyA A Figure 2.3: Bauschinger Effect on Reversed Loading σB which is smaller than σy0 and is much smaller than the subsequent yield stress at A. This phenomena is called Bauschinger effect. Victor Saouma Fracture Mechanics
  31. 31. Draft2.1 Introduction 3 13 It is thus apparent that the stress-strain behavior in the plastic range is path dependent. In general the strain will not depend on the the current stress state, but also on the entire loading history, i.e. stress history and deformation history. 14 Since there is not a one to one relationship between stress and strain in the plastic state, it is not possible to express the stress-strain relationship in terms of total strain or stress. Thus for elastic-plastic materials only an incremental relationship between stress and strain increments can be written, Fig. 2.4. σ ε 1 E 1 Et ε ε dσ p e εd d d B A Figure 2.4: Stress and Strain Increments in Elasto-Plastic Materials 15 There are two theories corresponding with elastoplasticity. One is the Total Formulation of Plasticity (Deformation Theory) and the other is the Rate Formulation of Plasticity (Rate Theory). 16 The deformation theory is a total secant-type formulation of plasticity that is based on the additive decomposition of total strain into elastic and plastic components = e + p (2.1) while the rate theory is defined by ˙ = ˙e + ˙p (2.2) if σ ≤ σy(elasticity), then ˙ = ˙e = ˙σ E (2.3) if σ > σy(plasticity), then, Fig. 2.5. ˙ = ˙e + ˙p (2.4) ˙ = ˙σ E + ˙σ Ep = ˙σ ET (2.5) where ET = EEp E + Ep (2.6) Victor Saouma Fracture Mechanics
  32. 32. Draft4 Continuum Mechanics Based Description of Failure; Plasticity 17 Note that ET =    > 0, Hardening = 0, Perfectly Plastic < 0, Softening σy εe εp σ ε E ET Figure 2.5: Stress-Strain diagram for Elastoplasticity 2.1.2 Idealized Stress-Strain Relationships 18 There are many stress-strain models for the elastic-plastic behavior under monotonic loading: Elastic-Perfectly Plastic where hardening is neglected, and plastic flows begins when the yield stress is reached ε = σ E for σ < σy0 (2.7-a) ε = σ E + λ for σ = σY 0 (2.7-b) Elastic-Linearly Hardening model, where the tangential modulus is assumed to be constant ε = σ E for σ < σy0 (2.8-a) ε = σ E + 1 Et (σ − σy0 for σ = σY 0 (2.8-b) Elastic-Exponential Hardening where a power law is assumed for the plastic region ε = σ E for σ < σy0 (2.9-a) ε = kεn for σ = σY 0 (2.9-b) Victor Saouma Fracture Mechanics
  33. 33. Draft2.2 Review of Continuum Mechanics 5 Ramberg-Osgood which is a nonlinear smooth single expression ε = σ E + a σ b n (2.10) 2.1.3 Hardening Rules 19 A hardening rule describes a specific relationship between the subsequent yield stress σy of a material and the plastic deformation accumulated during prior loadings. 20 First we define a hardening parameter or plastic internal variable, which is often denoted by κ. κ = εp = √ dεpdεp Equivalent Plastic Strain (2.11) κ = Wp = σdεp Plastic Work (2.12) κ = εp = dεp Plastic Strain (2.13) 21 a hardening rule expresses the relationship of the subsequent yield stress σy, tangent modulus Et and plastic modulus Ep with the hardening parameter κ. The most commonly hardening rules are: Isotropic hardening rule states that the progressively increasing yield stresses under both tension and compression loadings are always the same. |σ| = |σ(κ)| (2.14) Kinematic hardening rule states that the difference between the yield stresses under tension loading and under compression loading remains constant. If we denote by σt y and σc y the yield stress under tension and compression respectively, then σt y(κ) − σc y(κ) = 2σy0 (2.15) or alternatively |σ − c(κ)| = σy0 (2.16) where c(κ) represents the track of the elastic center and satisfies c(0) = 0. 2.2 Review of Continuum Mechanics 2.2.1 Stress 2.2.1.1 Hydrostatic and Deviatoric Stress Tensors 22 If we let σ denote the mean normal stress p σ = −p = 1 3 (σ11 + σ22 + σ33) = 1 3 σii = 1 3 tr σ (2.17) then the stress tensor can be written as the sum of two tensors: Hydrostatic stress in which each normal stress is equal to −p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. σhyd = −pI =    −p 0 0 0 −p 0 0 0 −p    (2.18) Victor Saouma Fracture Mechanics
  34. 34. Draft6 Continuum Mechanics Based Description of Failure; Plasticity Deviatoric Stress: which causes the change in shape. s =    s11 − σ s12 s13 s21 s22 − σ s23 s31 s32 s33 − σ    (2.19) 2.2.1.2 Geometric Representation of Stress States 23 Using the three principal stresses σ1, σ2, and σ3, as the coordinates, a three-dimensional stress space can be constructed. This stress representaion is known as the Haigh-Westergaard stress space, FIg. 2.6. σ1 σ2 σ3 σ1 σ2 σ3 (s ,s ,s ) 1 2 3 N(p,p,p) ρ ξ P( , , ) deviatoric plane HydrostaticaxisO π/2 π/2 Figure 2.6: Haigh-Westergaard Stress Space 24 The decomposition of a stress state into a hydrostatic, pδij and deviatoric sij stress com- ponents can be geometrically represented in this space. Considering an arbitrary stress state OP starting from O(0, 0, 0) and ending at P(σ1, σ2, σ3), the vector OP can be decomposed into two components ON and NP. The former is along the direction of the unit vector (1 √ 3, 1/ √ 3, a/ √ 3), and NP⊥ON. 25 Vector ON represents the hydrostatic component of the stress state, and axis Oξ is called the hydrostatic axis ξ, and every point on this axis has σ1 = σ2 = σ3 = p. 26 Vector NP represents the deviatoric component of the stress state (s1, s2, s3) and is perpen- dicular to the ξ axis. Any plane perpendicular to the hydrostatic axis is called the deviatoric plane and is expressed as σ1 + σ2 + σ3 √ 3 = ξ (2.20) Victor Saouma Fracture Mechanics
  35. 35. Draft2.2 Review of Continuum Mechanics 7 and the particular plane which passes through the origin is called the π plane and is represented by ξ = 0. Any plane containing the hydrostatic axis is called a meridian plane. The vector NP lies in a meridian plane and has ρ = s2 1 + s2 2 + s2 3 = 2J2 (2.21) 27 The projection of NP and the coordinate axis σi on a deviatoric plane are shown in Fig. COMPLETE HERE AND CHECK 2.2.1.3 Invariants 28 The principal stresses are physical quantities, whose values do not depend on the coordinate system in which the components of the stress were initially given. They are therefore invariants of the stress state. 29 If we examine the deviatoric stress invariants, s11 − λ s12 s13 s21 s22 − λ s23 s31 s32 s33 − λ = 0 (2.22-a) |σrs − λδrs| = 0 (2.22-b) |σ − λI| = 0 (2.22-c) When the determinant in the characteristic Eq. 2.22-c is expanded, the cubic equation takes the form λ3 − J1λ2 − J2λ − J3 = 0 (2.23) where the symbols J1, J2 and J3 denote the following scalar expressions in the stress compo- nents: J1 = σ11 + σ22 + σ33 = σii = tr σ (2.24) J2 = −(σ11σ22 + σ22σ33 + σ33σ11) + σ2 23 + σ2 31 + σ2 12 (2.25) = 1 2 (σijσij − σiiσjj) = 1 2 σijσij − 1 2 I2 σ (2.26) = 1 2 (σ : σ − I2 σ) (2.27) J3 = detσ = 1 6 eijkepqrσipσjqσkr (2.28) 30 In terms of the principal stresses, those invariants can be simplified into J1 = σ(1) + σ(2) + σ(3) (2.29) J2 = −(σ(1)σ(2) + σ(2)σ(3) + σ(3)σ(1)) (2.30) J3 = σ(1)σ(2)σ(3) (2.31) Victor Saouma Fracture Mechanics
  36. 36. Draft8 Continuum Mechanics Based Description of Failure; Plasticity 2.2.2 Strains 2.2.2.1 Hydrostatic and Deviatoric Strain Tensors 31 The lagrangian and Eulerian linear strain tensors can each be split into spherical and deviator tensor as was the case for the stresses. Hence, if we define 1 3 e = 1 3 tr E (2.32) then the components of the strain deviator E are given by Eij = Eij − 1 3 eδij or E = E − 1 3 e1 (2.33) We note that E measures the change in shape of an element, while the spherical or hydro- static strain 1 3 e1 represents the volume change. 32 If we let σ denote the mean normal strain p σ = −p = 1 3 (σ11 + σ22 + σ33) = 1 3 σii = 1 3 tr σ (2.34) then the stress tensor can be written as the sum of two tensors: Hydrostatic stress in which each normal stress is equal to −p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. σhyd = −pI =    −p 0 0 0 −p 0 0 0 −p    (2.35) Deviatoric Stress: which causes the change in shape. s =    s11 − σ s12 s13 s21 s22 − σ s23 s31 s32 s33 − σ    (2.36) 33 Similarly, the lagrangian and Eulerian linear strain tensors can each be split into spherical and deviator tensor as was the case for the stresses. Hence, if we define 1 3 e = 1 3 tr E (2.37) then the components of the strain deviator E are given by Eij = Eij − 1 3 eδij or E = E − 1 3 e1 (2.38) We note that E measures the change in shape of an element, while the spherical or hydro- static strain 1 3 e1 represents the volume change. Victor Saouma Fracture Mechanics
  37. 37. Draft2.3 Yield Criteria 9 2.2.2.2 Invariants 34 Determination of the principal strains (E(3) < E(2) < E(1), λ3 J1λ2 − J2λ − J3 = 0 (2.39) where the symbols J1, J2 and J3 denote the following scalar expressions in the strain compo- nents: J1 = e11 + e22 + e33 = eii = tr E (2.40) J2 = −(e11e22 + e22e33 + e33e11) + e2 23 + e2 31 + e2 12 (2.41) = 1 2 (eijeij − eiiejj) = 1 2 eijeij − 1 2 J 2 1 (2.42) = 1 2 (E : E − I2 E) (2.43) J3 = detE = 1 6 eijkepqrEipEjqEkr (2.44) 35 In terms of the principal strains, those invariants can be simplified into J1 = e(1) + e(2) + e(3) (2.45) J2 = −(e(1)e(2) + e(2)e(3) + e(3)e(1)) (2.46) J3 = e(1)e(2)e(3) (2.47) 2.3 Yield Criteria 36 In uniaxial stress states, the elastic limit is obtained by a well-defined yield stress point σ0. In biaxial or triaxial state of stresses, the elastic limit is defined mathematically by a certain yield criterion which is a function of the stress state σij expressed as f(σij) = 0 (2.48) For isotropic materials, the stress state can be uniquely defined by either one of the following set of variables f(σ1, σ2, σ3) = 0 (2.49-a) f(I1, J2, J3) = 0 (2.49-b) f(ξ, ρ, θ) = 0 (2.49-c) those equations represent a surface in the principal stress space, this surface is called the yield surface. Within it, the material behaves elastically, on it it begins to yield. The elastic-plastic behavior of most metals is essentially hydrostatic pressure insensitive, thus the yield criteria will not depend on I1, and the yield surface can generally be expressed by any one of the following equations. f(J2, J3) = 0 (2.50-a) f(ρ, θ) = 0 (2.50-b) Victor Saouma Fracture Mechanics
  38. 38. Draft10 Continuum Mechanics Based Description of Failure; Plasticity 2.4 Rate Theory 37 Using Flow Theory of Plasticity we decompose the strain rate ˙, into elastic and plastic components as ˙ = ˙e + ˙p. (2.51) 38 The total strain rate ˙ is defined kinematically in terms of the strain-displacement matrix B by redefining the rate of change of the displacement field u as a function of time (velocity vector ˙u) as ˙ = B ˙ue. (2.52) 39 In order to define the plastic strain rate ˙p, we need to first determine if plasticity has occurred. This is done by introducing a Yield Function F in terms of stresses σ and internal variables q. 40 Internal variables define hardening and softening by describing motions and deformations of the yield surface. 41 Assuming the Yield Function F = 0 is satisfied then the rate of change distinguished between plastic loading and elastic unloading is ˙F(σ, q)    < 0, elastic = 0, plastic > 0, not permitted . (2.53) If the yield function is less than zero then the state of stress and internal variables is in the elastic range, equal to zero it has yielded entering the plastic range, and if the yield function is greater than zero the stresses are no longer following the defined yield function and this state is not permitted. This is further illustrated in Fig. 2.7. σ F > 0F < 0 F = 0 Figure 2.7: General Yield Surface. 42 Back to Eq. 2.51 and focusing our attention to the plastic strain rate ˙p, we define the Plastic Flow Rule as ˙p = ˙λ ∂Qp ∂σ m = ˙λm, (2.54) Victor Saouma Fracture Mechanics
  39. 39. Draft2.4 Rate Theory 11 where ˙λ is the plastic multiplier and Qp is the plastic flow potential. 43 When associated flow is used, Qp = F and when non-associated flow is used, Qp = F. 44 The associated flow is referred to as a normality of plastic behavior. In the case of plastic loading, the following Kuhn-Tucker condition is always true ˙λ ˙F = 0. (2.55) Therefore,the plastic multiplier must be equal to zero (˙λ = 0) in order for the rate of change Yield Function to be less than zero (F < 0). 45 After establishing the Plastic Flow Rule, a Consistency Condition is defined to enforce that the rate of change of stress and internal variables satisfy the yield condition having the plastic loading ( ˙F = 0). As a result, we differentiate F(σ, q) with respect to time so ˙F = ∂F ∂t = 0, (2.56-a) ˙F = ∂F ∂σ : ˙σ + ∂F ∂q : ˙q = 0, (2.56-b) or ˙F = ∂F ∂σ : ˙σ + ∂F ∂q : ∂q ∂λ ˙λ = 0. (2.57) Introducing the normal n and the hardening parameter Hp n = ∂F ∂σ (2.58-a) Hp = − ∂F ∂q : ∂q ∂λ (2.58-b) we can rewrite Eq. ?? as ˙F = n : ˙σ − Hp : ˙λ = 0 (2.59) then by substitutions ˙F = n : E : [˙ − ˙p] − Hp ˙λ = 0 (2.60-a) ˙F = n : E : [˙ − ˙λm] − Hp ˙λ = 0 (2.60-b) ˙F = n : E : ˙ − ˙λn : E : m − Hp ˙λ = 0 (2.60-c) 46 Eq. 2.60-c enables use to define the plastic multiplier as ˙λ = n : E : ˙ Hp + n : E : m (2.61) 47 Finally, we use the tangential stress-strain relation as follows: ˙σ = E : ˙ = E : [˙ − ˙p] = E : [˙ − ˙λm] (2.62) 48 substituting Eq. 2.61), the plastic multiplier, we get ˙σ = E : [˙ − m( n : E : ˙ Hp + n : E : m )] (2.63) Victor Saouma Fracture Mechanics
  40. 40. Draft12 Continuum Mechanics Based Description of Failure; Plasticity 49 Rearranging we find ˙σ = [E − E : m ⊗ n : E Hp + n : E : m ] : ˙ (2.64) or ˙σ = Eep : ˙ (2.65) where Eep is the elastoplastic material operator (modulus of elastoplasticity) defined by Eep = E − E : m ⊗ n : E Hp + n : E : m (2.66) 50 Note that for associated flow, n = m and for non-associated flow, n = m. 2.5 J2 Plasticity/von Mises Plasticity 51 For J2 plasticity or von Mises plasticity, our stress function is perfectly plastic. Recall per- fectly plastic materials have a total modulus of elasticity (ET ) which is equivalent to zero. We will deal now with deviatoric stress and strain for the J2 plasticity stress function. 1. Yield function: F(s) = 1 2 s : s − 1 3 σ2 y = 0 (2.67) 2. Flow rate (associated): ˙ep = ˙λ ∂Qp ∂s = ˙λ ∂F ∂s = ˙λs (2.68) 3. Consistency condition ( ˙F = 0): ˙F = ∂F ∂s : ˙s + ∂F ∂q : ˙q = 0 (2.69) since ˙q = 0 in perfect plasticity, the second term drops out and ˙F becomes ˙F = s : ˙s = 0 (2.70) Recall that ˙s = 2G : ˙ee = 2G : [˙e − ˙ep] (2.71) finally substituting ˙ep in ˙s = 2G : [˙e − ˙λs] (2.72) substituting ˙s back into (2.70) ˙F = 2Gs : [˙e − ˙λm] = 0 (2.73) and solving for ˙λ ˙λ = s : ˙ s : s (2.74) Victor Saouma Fracture Mechanics
  41. 41. Draft2.5 J2 Plasticity/von Mises Plasticity 13 4. Tangential stress-strain relation(deviatoric): ˙s = 2G : [˙e − s : ˙e s : s s] (2.75) then by factoring ˙e out ˙s = 2G : [I4 − s ⊗ s s : s ] : ˙e (2.76) Now we have the simplified expression ˙s = Gep : ˙e (2.77) where Gep = 2G : [I4 − s ⊗ s s : s ] (2.78) is the 4th order elastoplastic shear modulus tensor which relates deviatoric stress rate to deviatoric strain rate. 5. Solving for Eep in order to relate regular stress and strain rates: Volumetric response in purely elastic tr ( ˙σ) = 3Ktr (˙) (2.79) altogether ˙σ = 1 3 tr ( ˙σ) : I2 + ˙s (2.80) ˙σ = Ktr (˙) : I2 + Gep : ˙e (2.81) ˙σ = Ktr (˙) : I2 + Gep : [˙ − 1 3 tr (˙) : I2] (2.82) ˙σ = Ktr (˙) : I2 + Gep : ˙ − 1 3 tr (˙)Gep : I2 (2.83) ˙σ = Ktr (˙) : I2 − 2 3 Gtr (˙)I2 + Gep : ˙ (2.84) ˙σ = KI2 ⊗ I2 : ˙ − 2 3 GI2 ⊗ I2 : ˙ + Gep : ˙ (2.85) and finally we have recovered the stress-strain relationship ˙σ = [[K − 2 3 G]I2 ⊗ I2 + Gep] : ˙ (2.86) where the elastoplastic material tensor is E Ô = [[K − 2 3 G]I2 ⊗ I2 + Gep] (2.87) Victor Saouma Fracture Mechanics
  42. 42. Draft14 Continuum Mechanics Based Description of Failure; Plasticity s1 s2 s3 n+1 n+1 (Hardening) F = 0 for p nF = 0 for (Perfectly Plastic) H = 0 (Softening) F = 0 for H < 0p p H > 0 Figure 2.8: Isotropic Hardening/Softening 2.5.1 Isotropic Hardening/Softening(J2− plasticity) 52 In isotropic hardening/softening the yield surface may shrink (softening) or expand (harden- ing) uniformly (see figure 2.8). 1. Yield function for linear strain hardening/softening: F(s, p eff ) = 1 2 s : s − 1 3 (σo y + Ep p eff )2 = 0 (2.88) 2. Consistency condition: ˙F = ∂F ∂s : ˙s + ∂F ∂q : ∂ ˙q ∂ ˙λ ˙λ = 0 (2.89) from which we solve the plastic multiplier ˙λ = 2Gs : ˙e 2Gs : s + 2Ep 3 (σo y + Ep p eff ) 2 3 s : s (2.90) 3. Tangential stress-strain relation(deviatoric): ˙s = Gep : ˙e (2.91) where Gep = 2G[I4 − 2Gs ⊗ s Gs : s + 2Ep 3 (σo y + Ep p eff ) 2 3s : s ] (2.92) 53 Note that isotropic hardening/softening is a poor representation of plastic behavior under cyclic loading because of the Bauschinger effect. Victor Saouma Fracture Mechanics
  43. 43. Draft2.5 J2 Plasticity/von Mises Plasticity 15 s2 s 3 s1 Fn+2 = 0 Fn+1 = 0 Fn = 0 O R P S Q Figure 2.9: Kinematic Hardening/Softening 2.5.2 Kinematic Hardening/Softening(J2− plasticity) 54 Kinematic hardening/softening, developed by Prager [1956], involves a shift of the origin of the yield surface (see figure 2.9). Here, kinematic hardening/softening captures the Bauschinger effect in a more realistic manner than the isotropic hardening/ softening. 1. Yield function: F(s, α) = 1 2 (s − α) : (s − α) − 1 3 σ2 y = 0 (2.93) 2. Consistency condition (plastic multiplier): ˙λ = 2G(s − α) : ˙e (s − α) : (s − α)[2G + C] (2.94) where C = Ep (2.95) and C is related to α, the backstress, by ˙α = C ˙e = ˙λC(s − α) (2.96) For perfectly plastic behavior C = 0 and α = 0. 3. Tangential stress-strain relation (deviatoric): ˙s = Gep : ˙e (2.97) where Gep = 2G[I4 − 2G(s − α) ⊗ (s − α) (s − α) : (s − α)[2G + C] ] (2.98) Victor Saouma Fracture Mechanics
  44. 44. Draft16 Continuum Mechanics Based Description of Failure; Plasticity Victor Saouma Fracture Mechanics
  45. 45. Draft Chapter 3 Continuum Mechanics Based Description of Failure; Damage Mechanics
  46. 46. Draft2 Continuum Mechanics Based Description of Failure; Damage Mechanics Victor Saouma Fracture Mechanics
  47. 47. Draft Part II FRACTURE MECHANICS
  48. 48. Draft
  49. 49. Draft Chapter 4 INTRODUCTION In this introductory chapter, we shall start by reviewing the various modes of structural failure and highlight the importance of fracture induced failure and contrast it with the limited coverage given to fracture mechanics in Engineering Education. In the next section we will discuss some examples of well known failures/accidents attributed to cracking. Then, using a simple example we shall compare the failure load predicted from linear elastic fracture mechanics with the one predicted by “classical” strength of materials. The next section will provide a brief panoramic overview of the major developments in fracture mechanics. Finally, the chapter will conclude with an outline of the lecture notes. 4.1 Modes of Failures The fundamental requirement of any structure is that it should be designed to resist mechanical failure through any (or a combination of) the following modes: 1. elastic instability (buckling) 2. large elastic deformation (jamming) 3. gross plastic deformation (yielding) 4. tensile instability (necking) 5. fracture Most of these failure modes are relatively well understood, and proper design procedures have been developed to resist them. However, fractures occurring after earthquakes constitute the major source of structural damage (Duga, Fisher, Buxbam, Rosenfield, Buhr, Honton and McMillan 1983), and are the least well understood. In fact, fracture often has been overlooked as a potential mode of failure at the expense of an overemphasis on strength. Such a simplification is not new, and finds a very similar analogy in the critical load of a column. If column strength is based entirely on a strength criterion, an unsafe design may result as instability (or buckling) is overlooked for slender members. Thus failure curves for columns show a smooth transition in the failure mode from columns based on gross section yielding to columns based on instability. By analogy, a cracked structure can be designed on the sole basis of strength as long as the crack size does not exceed a critical value. Should the crack size exceed this critical value, then
  50. 50. Draft2 INTRODUCTION a fracture-based failure results. Again, on the basis of those two theories (strength of materials and fracture mechanics), one could draw a failure curve that exhibits a smooth transition between those two modes.1 4.2 Examples of Structural Failures Caused by Fracture Some well-known, and classical, examples of fracture failures include: • Mechanical, aeronautical, or marine 1. fracture of train wheels, axles, and rails 2. fracture of the Liberty ships during and after World War II 3. fracture of airplanes, such as the Comet airliners, which exploded in mid-air during the fifties, or more recently fatigue fracture of bulkhead in a Japan Air Line Boeing 747 4. fatigue fractures found in the Grumman buses in New York City, which resulted in the recall of 637 of them 5. fracture of the Glomar Java sea boat in 1984 6. fatigue crack that triggered the sudden loss of the upper cockpit in the Air Aloha plane in Hawaii in 1988 • Civil engineering 1. fractures of bridge girders (Silver bridge in Ohio) 2. fracture of Statfjord A platform concrete off-shore structure 3. cracks in nuclear reactor piping systems 4. fractures found in dams (usually unpublicized) Despite the usually well-known detrimental effects of fractures, in many cases fractures are man-made and induced for beneficial purposes Examples include: 1. rock cutting in mining 2. hydrau-fracturing for oil, gas, and geothermal energy recovery 3. “Biting” of candies (!) Costs associated with fracture in general are so exorbitant, that a recent NBS report (Duga et al. 1983) stated: [The] cost of material fracture to the US [is] $ 119 billion per year, about 4 percent of the gross national product. The costs could be reduced by an estimated missing 35 billion per year if technology transfer were employed to assure the use of best practice. Costs could be further reduced by as much as $ 28 billion per year through fracture-related research. In light of the variety, and complexity of problems associated with fracture mechanics, it has become a field of research interest to mathematicians, scientists, and engineers (metallurgical, mechanical, aerospace, and civil). 1 When high strength rolled sections were first introduced, there was a rush to use them. However, after some spectacular bridge girder failures, it was found that strength was achieved at the expense of toughness (which is the material ability to resist crack growth). Victor Saouma Fracture Mechanics
  51. 51. Draft4.3 Fracture Mechanics vs Strength of Materials 3 4.3 Fracture Mechanics vs Strength of Materials In order to highlight the fundamental differences between strength of materials and fracture mechanics approaches, we consider a simple problem, a cantilevered beam of length L, width B, height H, and subjected to a point load P at its free end, Fig. 4.1 Maximum flexural stress Figure 4.1: Cracked Cantilevered Beam is given by σmax = 6PL BH2 (4.1) We will seek to determine its safe load-carrying capacity using the two approaches2. 1. Based on classical strength of materials the maximum flexural stress should not exceed the yield stress σy, or σmax ≤ σy (4.2) Thus, based on this first approach, the maximum load which can be safely carried is: PSOM max = BH2 6L σy (4.3) 2. In applying a different approach, one based on fracture mechanics, the structure cannot be assumed to be defect free. Rather, an initial crack must be assumed. Eq. 4.2 governed failure; for the strength of materials approach in the linear elastic fracture mechanics approach (as discussed in the next chapter), failure is governed by: KI ≤ KIc (4.4) where KI is a measure of the stress singularity at the tip of the crack and KIc is the critical value of KI. KI is related to σmax through: KI = 1.12σmax √ πa (4.5) where a is the crack length. KI is a structural parameter (analogous to σmax), and KIc, is a material parameter (analogous to σy). Fracture toughness is a measure of the material ability to resist crack growth (not to be confused with its tensile strength, which is associated with crack nucleation or formation). Thus, the maximum load that can be carried is given by: PF M max = BH2 6L KIc 1.12 √ πa (4.6) 2 This example is adapted from (Kanninen and Popelar 1985). Victor Saouma Fracture Mechanics
  52. 52. Draft4 INTRODUCTION The two equations, Eq. 4.3 and 4.6 governing the load capacity of the beam according to two different approaches, call for the following remarks: 1. Both equations are in terms of BH2 6L 2. The strength of materials approach equation is a function of a material property that is not size dependent. 3. The fracture mechanics approach is not only a function of an intrinsic material property,3 but also of crack size a. On the basis of the above, we can schematically represent the failure envelope of this beam in Fig. 4.2, where failure stress is clearly a function of the crack length. Figure 4.2: Failure Envelope for a Cracked Cantilevered Beam On the basis of this simple example, we can generalize our preliminary finding by the curve shown in Fig. 4.3. We thus identify four corners: on the lower left we have our usual engineering design zone, where factors of safety are relatively high; on the bottom right we have failure governed by yielding, or plasticity; on the upper left failure is governed by linear elastic fracture mechanics; and on the upper right failure is triggered by a combination of fracture mechanics and plasticity. This last zone has been called elasto-plastic in metals, and nonlinear fracture in concrete.4 Finally, we should emphasize that size effect is not unique to fractures but also has been encountered by most engineers in the design of columns. In fact, depending upon its slenderness ratio, a column failure load is governed by either the Euler equation for long columns, or the strength of materials for short columns. Column formulas have been developed, as seen in Fig. 4.4, which is similar to Fig. 4.2. Also note that column instability is caused by a not perfectly straight element, whereas fracture failure is caused by the presence of a crack. In all other cases, a perfect material is assumed, as shown in Table 4.1. As will be shown later, similar transition curves have also been developed by Baˇzant (Baˇzant, Z.P. 1984) for the failure of small or large cracked structures on the basis of either strength of materials or linear elastic fracture mechanics. 3 We will see later that KIc is often a function of crack length. Similarly compressive strength of concrete is known to be slightly affected by the cylinder size. 4 This curve will be subsequently developed for concrete materials according to Baˇzant’s size effect law. Victor Saouma Fracture Mechanics
  53. 53. Draft4.3 Fracture Mechanics vs Strength of Materials 5 Figure 4.3: Generalized Failure Envelope Figure 4.4: Column Curve Approach Governing Eq. Theory Imperfection Strength of Material σ = P A Plasticity σy Dislocation Column Instability σ = π2E (KL r )2 Euler KL r Not Perfectly straight Fracture σ = Kc√ πa Griffith KIc Micro-defects Table 4.1: Column Instability Versus Fracture Instability Victor Saouma Fracture Mechanics
  54. 54. Draft6 INTRODUCTION 4.4 Major Historical Developments in Fracture Mechanics As with any engineering discipline approached for the first time, it is helpful to put fracture mechanics into perspective by first listing its major developments: 1. In 1898, a German Engineer by the name of Kirsch showed that a stress concentration factor of 3 was found to exist around a circular hole in an infinite plate subjected to uniform tensile stresses (Timoshenko and Goodier 1970). 2. While investigating the unexpected failure of naval ships in 1913, Inglis (Inglis 1913) extended the solution for stresses around a circular hole in an infinite plate to the more general case of an ellipse. It should be noted that this problem was solved 3 years earlier by Kolosoff (who was the mentor of Muschelisvili) in St Petersbourg, however history remembers only Inglis who showed that a stress concentration factor of S.C.F. = 1 + 2 a ρ 1/2 (4.7) prevails around the ellipse (where a is the half length of the major axis, and ρ is the radius of curvature)5. 3. Inglis’s early work was followed by the classical studies of Griffith, who was not orginally interested in the strength of cracked structures (fracture mechanics was not yet a disci- pline), but rather in the tensile strength of crystalline solids and its relation to the theory based on their lattice properties, which is approximately equal to E/10 where E is the Young’s Modulus (Kelly 1974). With his assistant Lockspeiser, Griffith was then working at the Royal Aircraft Estab- lishment (RAE) at Farnborough, England (which had a tradition of tolerance for original and eccentric young researchers), and was testing the strength of glass rods of different diameters at different temperatures (Gordon 1988). They found that the strength in- creased rapidly as the size decreased. Asymptotic values of 1,600 and 25 Ksi were found for infinitesimally small and bulk size specimens, respectively. On the basis of those two observations, Griffith’s first major contribution to fracture mechanics was to suggest that internal minute flaws acted as stress raisers in solids, thus strongly affecting their tensile strengths. Thus, in reviewing Inglis’s early work, Griffith determined that the presence of minute elliptical flaws were responsible in dramatically reducing the glass strength from the theoretical value to the actually measured value. 4. The second major contribution made by Griffith was in deriving a thermodynamical cri- terion for fracture by considering the total change in energy taking place during cracking. During crack extension, potential energy (both external work and internal strain energy) is released and “transferred” to form surface energy. Unfortunately, one night Lockspeiser forgot to turn off the gas torch used for glass melting, resulting in a fire. Following an investigation, (RAE) decided that Griffith should stop wasting his time, and he was transferred to the engine department. 5. After Griffith’s work, the subject of fracture mechanics was relatively dormant for about 20 years until 1939 when Westergaard (Westergaard 1939a) derived an expression for the stress field near a sharp crack tip. 5 Note that for a circle, a stress concentration of 3 is recovered. Victor Saouma Fracture Mechanics
  55. 55. Draft4.4 Major Historical Developments in Fracture Mechanics 7 6. Up to this point, fracture mechanics was still a relatively obscure and esoteric science. However, more than any other single factor, the large number of sudden and catastrophic fractures that occurred in ships during and following World War II gave the impetus for the development of fracture mechanics. Of approximately 5,000 welded ships constructed during the war, over 1,000 suffered structural damage, with 150 of these being seriously damaged, and 10 fractured into two parts. After the war, George Irwin, who was at the U.S. Naval Research Laboratory, made use of Griffith’s idea, and thus set the foundations of fracture mechanics. He made three major contributions: (a) He (and independently Orowan) extended the Griffith’s original theory to metals by accounting for yielding at the crack tip. This resulted in what is sometimes called the modified Griffith’s theory. (b) He altered Westergaard’s general solution by introducing the concept of the stress intensity factor (SIF). (c) He introduced the concept of energy release rate G 7. Subcritical crack growth was subsequently studied. This form of crack propagation is driven by either applying repeated loading (fatigue) to a crack, or surround it by a cor- rosive environment. In either case the original crack length, and loading condition, taken separately, are below their critical value. Paris in 1961 proposed the first empirical equa- tion relating the range of the stress intensity factor to the rate of crack growth. 8. Non-linear considerations were further addressed by Wells, who around 1963 utilized the crack opening displacement (COD) as the parameter to characterize the strength of a crack in an elasto-plastic solid, and by Rice, who introduce his J integral in 1968 in probably the second most referenced paper in the field (after Griffith); it introduced a path independent contour line integral that is the rate of change of the potential energy for an elastic non-linear solid during a unit crack extension. 9. Another major contribution was made by Erdogan and Sih in the mid ’60s when they introduced the first model for mixed-mode crack propagation. 10. Other major advances have been made subsequently in a number of subdisciplines of fracture mechanics: (i) dynamic crack growth; (ii) fracture of laminates and composites; (iii) numerical techniques; (iv) design philosophies; and others. 11. In 1976, Hillerborg (Hillerborg, A. and Mod´eer, M. and Petersson, P.E. 1976) introduced the fictitious crack model in which residual tensile stresses can be transmitted across a portion of the crack. Thus a new meaning was given to cracks in cementitious materials. 12. In 1979 Baˇzant and Cedolin (Baˇzant, Z.P. and Cedolin, L. 1979) showed that for the objective analysis of cracked concrete structure, fracture mechanics concepts must be used, and that classical strength of materials models would yield results that are mesh sensitive. This brief overview is designed to make detailed coverage of subsequent topics better un- derstood when put into global perspective. Victor Saouma Fracture Mechanics
  56. 56. Draft8 INTRODUCTION 4.5 Coverage Following this brief overview, chapter two will provide the reader with a review of elasticity. In particular we shall revisit the major equations needed to analytically solve simple problems involving elliptical holes or sharp cracks. Those solutions will be presented in detail in chapter three. This chapter, mathematically the most challenging, is an important one to understand the mathematical complexity of solu- tions of simple crack problem, and to appreciate the value of numerical based solutions which will be discussed later. First Inglis solution of a circular and elliptical hole will be presented, then the problem of a sharp crack in an infinite plate will be solved using the two classical methods. The first one is based on Westergaard’s solution, and the second on Williams’s clas- sical paper. Through Westergaard’s solution, we shall introduce the concept of stress intensity factors, and William’s solution will be extended to cracks along an interface between two dis- similar materials. Also covered in this chapter will be the solutions of a crack in a homogeneous anisotropic solid based on the solution of Sih and Paris. With the rigorous derivation of the stress field ahead of a crack tip performed, Chapter four will formalize the Linear Elastic Fracture Mechanics approach, and show how it can be used in some practical design cases. A complementary approach to the stress based one, will be presented in chapter five which discusses Energy Methods in linear elastic fracture mechanics. First, we shall thoroughly ex- amine the theoretical strength of crystalline materials and contrast it with the actual one, then we will define the energy release rate G, and discuss the duality between the stress based and the energy based approaches. Chapter six will extend the simple mode I crack propagation criteria to mixed modes (where a crack is simultaneously subjected to opening and sliding) by discussing some of the major criterions. Subcritical crack growth, and more specifically fatigue crack growth will be covered in chapter seven. Elasto-Plastic fracture mechanics, and derivation of the J integral will then be covered in chapter eight. First we will derive expressions for the size of the plastic zone ahead of the crack tip, then criteria for crack growth presented. In chapter nine, we shall examine some of the fracture testing techniques, with emphasize on both metallic and cementitious materials. Fracture of cementitious material, such as concrete and rock, will be studied in chapter ten. In this extensive chapter, we shall review some of the major models currently investigated, and examine some applications. Numerical techniques will then be discussed in chapter eleven. First techniques of modelling the stress singularity at the crack tip will be examined, followed by methods to extract the stress intensity factors from a finite element analysis and evaluation of J integral will be presented. The last chapter, twelve, will focus on numerical techniques for cementitious materials. For more detailed coverage, the reader is referred to the numerous excellent books available, such as Broek (Broek 1986, Broek 1989), Cherepanov (Cherepanov 1979), Kanninen (Kanninen and Popelar 1985), Knott (Knott 1976), Barsom and Rolfe (Barsom and Rolfe 1987), and Anderson’s (Anderson 1995). Finally, a recent book by Baˇzant (Baˇzant, Z.P. and Cedolin, L. 1991) covers (among other things) some of the issues related to fracture of concrete. Victor Saouma Fracture Mechanics

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