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Draft Lecture Notes in: FRACTURE MECHANICS Victor E. Saouma Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO 80309-0428
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Draft Contents 1 INTRODUCTION 1 1.1 Modes of Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Examples of Structural Failures Caused by Fracture . . . . . . . . . . . . . . . . 2 1.3 Fracture Mechanics vs Strength of Materials . . . . . . . . . . . . . . . . . . . . . 3 1.4 Major Historical Developments in Fracture Mechanics . . . . . . . . . . . . . . . 6 1.5 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 PRELIMINARY CONSIDERATIONS 1 2.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1.1 Indicial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.2 Tensor Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.3 Rotation of Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.5 Inverse Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.6 Principal Values and Directions of Symmetric Second Order Tensors . . . 6 2.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Force, Traction and Stress Vectors . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Traction on an Arbitrary Plane; Cauchy’s Stress Tensor . . . . . . . . . . 8 E 2-1 Stress Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.4 Spherical and Deviatoric Stress Tensors . . . . . . . . . . . . . . . . . . . 10 2.2.5 Stress Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Kinematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Strain Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 Compatibility Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Fundamental Laws of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . 14 2.4.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.2 Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.3 Conservation of Mass; Continuity Equation . . . . . . . . . . . . . . . . . 16 2.4.4 Linear Momentum Principle; Equation of Motion . . . . . . . . . . . . . . 16 2.4.5 Moment of Momentum Principle . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.6 Conservation of Energy; First Principle of Thermodynamics . . . . . . . . 18 2.5 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.1 Transversly Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2 Special 2D Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2.1 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
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Draft CONTENTS iii 5.1.1.2 Ideal Strength in Terms of Engineering Parameter . . . . . . . . 4 5.1.2 Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.2 Griffith Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 ENERGY TRANSFER in CRACK GROWTH; (Griffith II) 1 6.1 Thermodynamics of Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.1.1 General Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6.1.2 Brittle Material, Griffith’s Model . . . . . . . . . . . . . . . . . . . . . . . 2 6.2 Energy Release Rate; Global . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2.1 From Load-Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6.2.2 From Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.3 Energy Release Rate; Local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6.4 Crack Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.4.1 Effect of Geometry; Π Curve . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.4.2 Effect of Material; R Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.4.2.1 Theoretical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.4.2.2 R vs K Ic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.4.2.3 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.4.2.4 Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 MIXED MODE CRACK PROPAGATION 1 7.1 Analytical Models for Isotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . 2 7.1.1 Maximum Circumferential Tensile Stress. . . . . . . . . . . . . . . . . . . 2 7.1.2 Maximum Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . 3 7.1.3 Minimum Strain Energy Density Criteria. . . . . . . . . . . . . . . . . . . 4 7.1.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7.2 Empirical Models for Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.3 Extensions to Anisotropic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7.4 Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7.4.1 Crack Tip Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7.4.2 Dimensions of Bimaterial Stress Intensity Factors . . . . . . . . . . . . . . 16 7.4.3 Interface Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . 17 7.4.3.1 Interface Fracture Toughness when β = 0 . . . . . . . . . . . . . 19 7.4.3.2 Interface Fracture Toughness when β 6 = 0 . . . . . . . . . . . . . 20 7.4.4 Crack Kinking Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7.4.4.1 Numerical Results from He and Hutchinson . . . . . . . . . . . 21 7.4.4.2 Numerical Results Using Merlin . . . . . . . . . . . . . . . . . . 22 7.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 II ELASTO PLASTIC FRACTURE MECHANICS 29 8 PLASTIC ZONE SIZES 1 8.1 Uniaxial Stress Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8.1.1 First-Order Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8.1.2 Second-Order Approximation (Irwin) . . . . . . . . . . . . . . . . . . . . . 2 8.1.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Victor Saouma Fracture Mechanics
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Draft CONTENTS v 11.11Dynamic Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 11.12Effect of Other Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 11.12.1Surface Tractions on Crack Surfaces . . . . . . . . . . . . . . . . . . . . . 33 11.12.2Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 11.12.3Initial Strains Corresponding to Thermal Loading . . . . . . . . . . . . . 34 11.12.4Initial Stresses Corresponding to Pore Pressures . . . . . . . . . . . . . . 35 11.12.5Combined Thermal Strains and Pore Pressures . . . . . . . . . . . . . . . 37 11.13Epilogue
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