Models and Phenomena in Fracture Mechanics

ISBN-13: 9783642078453 / Angielski / Miękka / 2010 / 576 str.

Leonid I. Slepyan

Opis

Presenting the most important results, methods, and open questions, this book describes and compares advanced models in fracture mechanics. The author introduces the required mathematical technique, mainly the theory of analytical functions, from scratch.

Spis treści

1 Fundamentals and Basic Relations.- 1.1 Energy Release and Energy Criterion.- 1.1.1 What is a Crack?.- 1.1.2 How Do Cracks Arise?.- 1.1.3 Energy Release.- 1.1.4 Energy Criterion.- 1.1.5 Surface Energy and the Failure Energy of a Sample.- 1.1.6 Strength and Weaknesses of the Energy Criterion.- 1.1.7 Surface Tension and Surface Energy.- 1.1.8 Nucleation of a Crack and Strength of a Material.- 1.2 Some Methods for Determination of Energy Release.- 1.2.1 Variational Approach.- 1.2.2 Convolution Formula.- 1.2.3 J-integral.- 1.2.4 J-integral for Steady-state Motion.- 1.3 Other Examples of the Energy Release Phenomenon.- 1.3.1 Shock Wave.- 1.3.2 Moving Ship.- 1.3.3 Vehicle Moving Along a Beam on an Elastic Foundation.- 1.4 Stress Intensity Criterion.- 1.4.1 Fracture Process Zone.- 1.4.2 Irwin Fracture Criterion.- 1.5 Some Fracture-Associated Phenomena.- 1.5.1 Size Effect.- 1.5.2 Difference Between Crack Initiation and Propagation Criteria.- 1.5.3 Instabilities in Crack Propagation.- 2 Fourier Transform and Related Topics.- 2.1 Continuous Fourier Transform.- 2.1.1 Definitions.- 2.1.2 The Inverse Fourier Transform.- 2.1.3 Cauchy-Type Integral Continuous Case.- 2.1.4 Laplace Transform.- 2.1.5 Fourier Transform of a Convolution.- 2.1.6 Some Asymptotes.- 2.2 Wiener-Hopf Technique.- 2.2.1 The Equation.- 2.2.2 Factorization.- 2.2.3 Solution in Terms of the Transform.- 2.2.4 Delta-function as a Generalized Limit.- 2.2.5 Solution.- 2.2.6 Considerations in Terms of Original Functions.- 2.3 Laplace and Fourier Transform.- 2.3.1 Straightforward Inversion Formula.- 2.3.2 Double Fourier Transform and Hankel Transform.- 2.4 Discrete Fourier Transform.- 2.4.1 Definition.- 2.4.2 Inverse Transform.- 2.4.3 Cauchy-Type Integral Discrete Case.- 2.4.4 Convolution.- 2.4.5 Some Asymptotes.- 2.4.6 Power Asymptotes and the Related Continuum.- 2.4.7 Wiener-Hopf Technique for the Discrete Transform.- 3 Waves.- 3.1 Waves of Sinusoidal and Exponential Types.- 3.1.1 Equations.- 3.1.2 Complex Wave and Dispersion Relations.- 3.1.3 What is a Uniform Waveguide?.- 3.1.4 Phase and Group Velocities.- 3.1.5 Energy Flux in a Wave.- 3.1.6 Additivity of Energy Fluxes in Different Waves.- 3.2 Waves in Periodic Structures.- 3.2.1 Discrete Chain.- 3.2.2 General System of Periodic Structure.- 3.3 Forced Waves.- 3.3.1 Complex Wave and Fourier Transform.- 3.3.2 Causality Principle for Steady-state Solutions.- 3.3.3 Pre-Limiting Location of a Zero Point and the Group Velocity.- 3.3.4 Contributions of Singular Points.- 3.3.5 Resonant Waves.- 3.4 Waves in Homogeneous Space and Half-Space.- 3.4.1 Linear Elastic Isotropic Space.- 3.4.2 Longitudinal and Shear Waves.- 3.4.3 Rayleigh Wave.- 3.5 Nonlinear Waves in a String.- 3.5.1 The Wavefront Conditions.- 3.5.2 Two-Step-Wave Configuration.- 3.5.3 Some Asymptotic Results.- 4 One-dimensional Models.- 4.1 String Model.- 4.1.1 String Attached to a Rigid Foundation.- 4.1.2 Cohesive Zone Model for a String.- 4.1.3 String on a Linear Elastic Foundation.- 4.1.4 Nonlinear Post-peak Softening Cohesive Forces.- 4.1.5 Discrete Bonds.- 4.1.6 Soundless Crack.- 4.1.7 Nonlinear String Model.- 4.1.8 Nonuniform Crack Propagation.- 4.1.9 Dynamic Fracture Under a Fracture Criterion.- 4.1.10 Tearing of a String from a Solid Under an Impact.- 4.1.11 Nonlinear Dynamic Problem.- 4.2 Bending Beam Model.- 4.2.1 Splitting of a Beam in Half.- 4.2.2 Size Effect.- 4.2.3 Steady-State Dynamic Problem.- 4.2.4 Thread Beam Problem.- 4.2.5 Wave Resistance in Crack Propagation.- 5 Static Cracks in a Linearly Elastic Body.- 5.1 Field Representations.- 5.2 Kolosov—Muskhelishvili Representation.- 5.2.1 Opening Mode.- 5.2.2 Shear Mode.- 5.2.3 Anti-plane Mode.- 5.2.4 Boundary Conditions, Harmonic Function and Integral Equation.- 5.3 Papkovich Representation.- 5.3.1 Opening Mode.- 5.3.2 Shear Mode.- 5.3.3 Opening Mode in Cylindrical Coordinates.- 5.3.4 Shear Mode in Cylindrical Coordinates.- 5.3.5 Axially Symmetric Case.- 5.4 Crack in an Unbounded Plane.- 5.4.1 Finite Plane Crack.- 5.4.2 Nonlinear Condition for Mode I.- 5.5 Asymptotes.- 5.5.1 Stress Intensity Factors.- 5.5.2 Crack Tip Singularity.- 5.5.3 Stresses in Polar Coordinates.- 5.5.4 Stress Intensity Factors and Energy Release.- 5.6 Homogeneous Solutions.- 5.6.1 Homogeneous Solution as a Limit.- 5.6.2 Other Homogeneous Solutions.- 5.7 Integral Equations for a General Crack System.- 5.7.1 Field Induced by a Dislocation.- 5.7.2 Superposition.- 5.8 Crack Interaction.- 5.8.1 Collinear Crack Array General Distribution.- 5.8.2 Periodic Collinear Crack Array.- 5.8.3 Parallel Cracks.- 5.8.4 Collinear Cracks Do Not Like to Meet Each Other.- 5.9 Energy Release Under Crack Kink.- 5.10 Cohesive Zone Model.- 5.10.1 Formulation and Solution.- 5.10.2 Energy Release Large and Small Cracks.- 5.11 Penny-Shaped Crack.- 5.11.1 Crack Under Normal Traction.- 5.11.2 Axially Symmetric Problems.- 5.11.3 Harmonic Green#x2019;s Function.- 5.12 Betti#x2019;s Theorem and the Weight Functions.- 5.12.1 Betti#x2019;s Reciprocity Theorem.- 5.12.2 Weight Function Method.- 6 Nonlinear Elastic Body.- 6.1 Some Data from Nonlinear Elasticity.- 6.1.1 Geometrical Relations.- 6.1.2 Physical Relations.- 6.2 Lagrangian and Eulerian Interpretation of Linear Elasticity.- 6.2.1 Boundary Conditions.- 6.2.2 Lagrangian Interpretation.- 6.2.3 Eulerian Interpretation.- 6.3 Strains in the Neighborhood of a Singular Point.- 6.3.1 Lagrange Variables.- 6.3.2 Euler Variables.- 6.3.3 Logarithmic Singularity is the Lower Bound.- 6.4 Exact Relationships for the Energy Release and Some Consequences.- 6.4.1 J-integral.- 6.4.2 Crack Opening and Stresses on the Crack Line.- 7 Viscoelastic Fracture.- 7.1 Some Data from Viscoelasticity.- 7.1.1 General Formulations.- 7.1.2 Standard Viscoelastic Material.- 7.1.3 Stability and Passivity.- 7.1.4 Correspondence Principle.- 7.1.5 Static Problems Time-dependent Boundary Regions.- 7.2 Stationary Crack and Collinear Crack System.- 7.3 Growing Crack.- 7.3.1 Steady-state Formulation.- 7.3.2 Energy Release and Crack Growth Paradox.- 7.4 Cohesive Zone for Viscoelastic Material.- 7.4.1 Elastic Cohesive Zone.- 7.4.2 Viscoelastic Cohesive Zone.- 7.4.3 Global-to-Local Energy Release Ratio.- 8 Elastic-Plastic Fracture.- 8.1 Elastic-Plastic Fields.- 8.1.1 Some Basic Relations.- 8.1.2 Stress Fields.- 8.1.3 Continuity Conditions.- 8.1.4 Strain Fields.- 8.1.5 Moving Strain Fields.- 8.1.6 Unloading Domain.- 8.2 Fixed Cracks.- 8.2.1 Proportional Loading.- 8.2.2 Mode III Crack.- 8.2.3 Crack Under Plane Strain.- 8.2.4 Barenblatt-Dugdale Model for Plane Stress Crack.- 8.3 Growing Cracks.- 8.3.1 Mode III Growing Crack.- 8.3.2 Mode I Growing Crack.- 8.3.3 Mode II Growing Crack.- 8.3.4 A Note on the Logarithmic Singularity.- 8.3.5 Modified Barenblatt-Dugdale Model for Crack Under Cyclic Loading.- 8.4 Elastic-Plastic Dynamic Fracture.- 8.4.1 Mode III Crack Propagation.- 8.4.2 Mode I Crack Propagation.- 9 Dynamic Fracture in a Homogeneous Elastic Medium.- 9.1 Some Basic Relations.- 9.1.1 Mode III and Hydrodynamic Analogue.- 9.1.2 Mode III Fundamental Solution.- 9.1.3 Plane Problem Fundamental Solutions.- 9.2 Crack Tip Asymptotes and the Energy Release.- 9.2.1 Validity of the Steady-State Formulation.- 9.2.2 Subsonic Crack.- 9.2.3 Intersonic Crack.- 9.3 Factorization of the Fundamental Solutions.- 9.3.1 Singular Points, Convolutions and Supports.- 9.3.2 Factorization for Transient Problems.- 9.3.3 Factorization for Uniform Crack Propagation.- 9.4 Uniform Crack Propagation.- 9.4.1 Steady-State and Static Solutions.- 9.4.2 Transient Problem with a Constant Crack Speed.- 9.5 Nonuniform Crack Speed Problem.- 9.5.1 Solution for a Free Sector.- 9.5.2 Mode III Explicit Solution.- 9.5.3 Crack Tip Asymptotes for Plane Problem.- 9.5.4 Energy Release Versus Current Crack Speed.- 9.5.5 Crack Speed Crosses the Critical Speed.- 9.6 Self-Similar Dynamic Problems.- 9.6.1 Formulation and Basic Relations.- 9.6.2 Homogeneous Solutions.- 9.6.3 Solution to the Problem.- 9.6.4 Stress Intensity Factors for Symmetric Case.- 9.7 Dynamic Crack in a Plate Under Bending.- 9.7.1 Formulation.- 9.7.2 Dynamic Fracture Equations.- 9.7.3 Bending Waves Under Plate-Fluid Interaction.- 9.7.4 Edge Bending Waves.- 9.7.5 Crack Tip Asymptotes and the Local Energy Release.- 9.8 Principle of Maximum Energy Dissipation Rate.- 9.8.1 Introductory Remarks.- 9.8.2 The Dynamic Fracture Criterion.- 10 Cracks in a Bending Plate.- 10.1 Asymptotic Solution for a Single Crack.- 10.1.1 Crack Closure Phenomenon.- 10.1.2 Plane-Bending Problem.- 10.1.3 Contact Problem.- 10.1.4 Energy Release.- 10.1.5 Limiting Cases and Asymptotes.- 10.1.6 Closure Force and Moment.- 10.1.7 Contact Stress Distribution.- 10.1.8 Asymptotic Closure Width.- 10.2 Radial Cracking with Closure.- 10.2.1 Few Versus Many Cracks.- 10.2.2 Formulation of the Coupled Problem.- 10.2.3 Crack Closure, Open Crack and Intact Regions.- 10.2.4 Solutions.- 10.2.5 Energy Release.- 10.3 Self-Similar Dynamic Problem.- 10.3.1 Formulation.- 10.3.2 General Solution.- 10.3.3 Energy Criterion and the Number of Cracks.- 10.3.4 Concluding Remarks.- 11 The Square-Cell Lattice.- 11.1 Preliminaries.- 11.2 Some Introductory Remarks.- 11.3 Elastic Lattice: Formulation and the Governing Equation.- 11.3.1 Formulation.- 11.3.2 Derivation of the Governing Equation.- 11.3.3 Zero Points of the Functions h(k) and r(k).- 11.4 Factorization.- 11.4.1 Direct Factorization.- 11.4.2 Other Type of Factorization.- 11.5 Solutions.- 11.5.1 General Homogeneous Solution.- 11.5.2 Macrolevel-Associated Solution.- 11.5.3 Layered and Homogeneous Material.- 11.5.4 Microlevel Solutions.- 11.5.5 Structure of Waves in the x y-Plane.- 11.5.6 Wave Amplitude in the x, y-Plane.- 11.5.7 Existence of Real Solutions.- 11.6 Viscoelastic Lattice.- 11.6.1 Introductory Remarks.- 11.6.2 Formulation and Basic Relations.- 11.6.3 Stress-Strain Relation in Terms of Fourier Transform.- 11.6.4 Local Energy Release.- 11.6.5 Unbounded Lattice.- 11.6.6 Lattice Strip.- 11.6.7 Quasi-static Crack Growth.- 12 Triangular-Cell Elastic Lattice.- 12.1 Introductory Remarks.- 12.2 General Properties of Fundamental Solutions.- 12.2.1 Lattice and Coordinates.- 12.2.2 Plan of the Solution.- 12.2.3 Some Properties of the Fundamental Solutions.- 12.3 Equations and General Solutions.- 12.3.1 Dynamic Equations.- 12.3.2 General Solution for the Intact Lattice.- 12.3.3 Symmetry and the Modes.- 12.3.4 Dynamic Equation for a Particle with n = 0.- 12.3.5 Green#x2019;s Function L(k) and Dispersion Relations.- 12.3.6 General Solution.- 12.4 Macrolevel-Associated Solution.- 12.4.1 Various Asymptotes.- 12.4.2 Asymptotes for L(k)...- 12.4.3 Asymptotes for L±-(k).- 12.4.4 Energy Release.- 12.4.5 Mode II Intersonic Crack Speed Inhomogeneous Problem.- 12.4.6 Dissipative Waves.- 12.5 Microlevel Solutions.- 12.5.1 General Characterization.- 12.5.2 Sub-Rayleigh Crack Speed.- 12.5.3 Super-Rayleigh Crack Speed.- 12.5.4 Intersonic Crack Speed.- 12.5.5 Supersonic Crack Speed.- 12.6 Concluding Remarks.- 13 Phase Transition Waves.- 13.1 Introductory Remarks.- 13.2 Macrolevel Solution.- 13.3 Discrete Chain.- 13.3.1 Formulation.- 13.3.2 Derivation of the Governing Equation.- 13.3.3 Factorization.- 13.3.4 General Homogeneous Solution.- 13.3.5 Macrolevel-Associated Solution.- 13.3.6 Chain-Based Macrolevel Solution.- 13.3.7 Dissipative Waves.- 13.3.8 Microlevel Solutions.- 13.4 Higher-Order-Derivative Model.- 13.4.1 Some General Considerations.- 13.4.2 Theorem on the Highest Modulus.- 13.4.3 Equation of the Fourth Order.- 13.4.4 Subsonic Speed.- 13.4.5 Intersonic Speed.- 13.4.6 Supersonic Speed.- 13.5 Concluding Remarks.- 14 Dynamic Amplification Factor in Fracture and Phase Transition.- 14.1 Introductory Remarks.- 14.2 Line of Viscoelastic Oscillators.- 14.3 DOR and SAR Domains for Viscoelastic Oscillator.- 14.4 Viscoelastic Square-Cell Lattice.- 14.4.1 Superposition.- 14.4.2 Derivation of a Governing Equation.- 14.4.3 Factorization.- 14.4.4 Division of the Right-Hand Side.- 14.4.5 Solution.- 14.5 Slow Phase Transition Wave in a Chain.- 14.5.1 Formulation.- 14.5.2 Superposition.- 14.5.3 Solution.- 14.5.4 Some Remarks.- 14.6 Triangular-Cell Lattice Irregularities in Fracture.- 14.6.1 Introductory Remarks.- 14.6.2 Superposition.- 14.6.3 Superposition Paradox.- 14.6.4 Transient Problem for an Intact Viscoelastic Lattice.- 14.6.5 Lattice with a Crack.- 14.6.6 Solution of the Auxiliary Problem.- 14.6.7 Solutions for Statics.- 14.6.8 Some Results of Numerical Simulations.- 14.6.9 Concluding Remarks.- References.

Blurb

Modern concepts of fracture mechanics are presented consecutively. Homogeneous and structured models, where microstructure plays an essential role, are considered for fracture and phase transition. Firstly, one-dimensional models are comprehensively studied allowing one to retrace the main phenomena without technical difficulties. More realistic models are then used as linear and nonlinear elastic mediums, such as elastic plates with crack closure, viscoelastic discrete lattices, chains and cohesive zone models. Also considered are, crack origination, equilibrium, slow and fast growth. Sub- and super critical crack speed regimes and transition from one regime to another are studied. Fourier transform and related topics, including a version of the Wiener-Hopf technique dealing with originals are presented, as well as required topics from wave theory. This book is targeted at researchers of materials and structures, also at lecturers and advanced students.