This book provides a systematic and standardized approach based on the authors’ over 30 years of research experience with weight function methods, as well as the relevant literature. Fracture mechanics has become an indispensable tool for the design and safe operation of damage-tolerant structures in many important technical areas. The stress intensity factor―the characterizing parameter of the crack tip field―is the foundation of fracture mechanics analysis. The weight function method is a powerful technique for determining stress intensity factors and crack opening displacements for complex load conditions, with remarkable computational efficiency and high accuracy.
The book presents the theoretical background of the weight function methods, together with a wealth of analytical weight functions and stress intensity factors for two- and three-dimensional crack geometries; many of these have been incorporated into national, international standards and industrial codes of practice. The accuracy of the results is rigorously verified, and various sample applications are provided. Accordingly, the book offers an ideal reference source for graduate students, researchers, and engineers whose work involves fracture and fatigue of materials and structures, who need not only stress intensity factors themselves but also efficient and reliable tools for obtaining them.
Author(s): Xue-Ren Wu, Wu Xu
Publisher: Springer
Year: 2022
Language: English
Pages: 664
City: Singapore
Foreword
Preface
Acknowledgments
Contents
Abbreviations and Symbols
Abbreviations
Symbols
1 Standardized Analytical Weight Function Method Based on Crack Opening Displacements
1.1 Introduction
1.2 The Generalized Weight Function for Mixed Load-Displacement Boundary Conditions
1.2.1 Weight Functions of Bueckner and Rice
1.2.2 Generalized Weight Function Method for Crack Problem with Mixed Boundary Conditions
1.2.3 Superposition Principle
1.2.4 Simple-Form Generalized Weigh Functions
1.3 Derivation and Verification of Standardized Analytical WFs
1.3.1 Standardized Analytical Representation of CODs
1.3.2 Analytical Integration of Function ϕ(α)
1.3.3 Measures for Improving Analytical COD Representations
1.3.4 Accuracy Assessment for Analytical COD-Expressions
1.4 Standardized Analytical WFs
1.4.1 Center Crack
1.4.2 Edge Crack
1.5 Accuracy Assessment and Verification of Standardized Analytical WFs
1.5.1 Weight Function and Green’s Function
1.5.2 Accuracy Verification of WFs by Using Green’s Functions
1.5.3 Effect of CMOD Condition on Improving WF Accuracy
1.6 Effect of Load–Displacement Boundary Condition and Crack Geometry on WFs
1.6.1 Importance of Load–displacement Boundary Condition
1.6.2 Effect of Crack Geometry
1.7 Effect of the Reference Load Case
1.8 Computation of SIFs Using Standardized Analytical WFs
1.8.1 Analytical SIF-Expressions for Basic Load Cases
1.8.2 Calculation of SIFs for Arbitrary Crack Line Stresses
1.8.3 Numerical Integration
1.9 Computation of CODs Using Standardized Analytical WFs
1.10 Standardized Analytical WF for Mode II Crack
1.10.1 Basic Equations for Mode II WF
1.10.2 Accuracy Assessment of Mode II WF
1.10.3 Computation of Mode II SIFs Using Standardized Analytical WFs
1.11 Analytical WFM for Orthotropic Composite Material
1.12 Procedures of Standardized Analytical WF Derivation-Verification-Application
1.13 Concluding Remarks
References
2 Analysis and Discussions on Weight Function Methods Based on Multiple Reference Load Cases
2.1 The Glinka-Shen Universal Weight Function Method
2.2 The Fett-Munz Direct Adjustment Method
2.3 Discussion of Some Aspects for the MRS Weight Function Methods
2.3.1 Discussions on the G-S UWFM
2.3.2 Discussions on the F-M DAM
2.4 Influence of Reference Load Cases Combination on MRS WFs
2.5 Discussions on the Inverse Solution of MRS WFs
References
3 Verification and Accuracy Evaluation of Various Weight Function Methods
3.1 Verification Criterion for WF-Accuracy—The Green’s Function
3.1.1 Shortcomings of Using SIFs for Weight Function Verification
3.1.2 Role of Green’s Function for WF-Accuracy Verification
3.2 Weight Function Complex Series Expansion-WCTSE
3.2.1 Development of WCTSE Methodology
3.2.2 Determination of WCTSE WFs
3.3 Verification of Accuracy of the WCTSE WFs
3.3.1 Load-Case Independence Examination of WCTSE WFs
3.3.2 Accuracy Verifications of WCTSE WFs for Typical Crack Geometries
3.4 Accuracy Assessment of Various WFMs
3.4.1 A Single Edge Crack in a Circular Disc
3.4.2 A Single Edge Crack in a Finite Rectangular Plate
3.5 Robustness Analysis for Various WFMs
3.5.1 Sensitivity of WFs to Reference Load Cases and Accuracy of Reference SIFs
3.5.2 Sensitivity of WFs to the Geometry Condition at Crack Mouth
3.5.3 Sensitivity of WFs to Small Changes of Reference SIFs
3.6 Concluding Remarks
References
4 Weight Functions for Center Crack Geometries
4.1 Derivation and Verification of WFs for Center Crack
4.1.1 Derivation of Center Crack Analytical WFs
4.1.2 Verification of Center Crack Analytical WFs
4.1.3 SIFs and CMODs for Basic Load Cases
4.2 A Periodic Array of Collinear Cracks in an Infinite Plate
4.2.1 Derivation and Accuracy Verification of Weight Function
4.2.2 SIFs and CMODs for Point Forces and Power Stresses
4.3 A Center Crack in a Finite Rectangular Plate
4.3.1 Derivation and Accuracy Verification of Weight Functions
4.3.2 SIFs and CMODs for Point Forces and Power Stresses
4.4 A Center Crack in a Circular Disc
4.4.1 Derivation and Accuracy Verification of Weight Function
4.4.2 SIFs and CMODs for Point Forces and Power Stresses
4.4.3 Application Examples
4.5 A Center Crack in a Diagonal Cube Specimen
4.5.1 Derivation and Accuracy Verification of Weight Function
4.5.2 SIFs and CMODs for Point Forces and Power Stresses
4.6 Summary and Concluding Remarks
References
5 Weight Functions for Edge Crack in Simply Connected Region
5.1 Derivation and Verification of WFs for Edge Crack Geometry
5.1.1 Derivation of Edge Crack Analytical WFs
5.1.2 Verification of Edge Crack Analytical WFs
5.1.3 SIFs and CMODs for Basic Load Cases
5.2 An Edge Crack in a Semi-Infinite Plate
5.2.1 Derivation and Accuracy Verification of Weight Function
5.2.2 SIFs and CMODs for Point Forces and Power Stresses
5.2.3 Application Examples
5.3 An Edge Crack in a Finite Plate
5.3.1 Derivation and Accuracy Verification of Weight Function
5.3.2 SIFs and CMODs for Point Forces and Power Stresses
5.3.3 Application Examples
5.4 Double Edge Cracks in a Finite Rectangular Plate
5.4.1 Derivation and Accuracy Verification of Weight Function
5.4.2 SIFs and CMODs for Point Forces and Power Stresses
5.5 Single Edge Cracked Finite Plate with Clamped Ends
5.5.1 Derivation and Accuracy Verification of Weight Function
5.5.2 SIFs and CMODs for Point Forces and Power Stresses
5.6 Radial Crack(s) in a Circular Disc
5.6.1 A Single Edge Crack in a Circular Disc
5.6.2 Double Edge Cracks in a Circular Disc
5.7 An Edge Crack in a Semi-Circular Disc
5.7.1 Derivation and Accuracy Verification of Weight Function
5.7.2 SIFs and CMODs for Point Forces and Power Stresses
5.7.3 Application Example
5.8 An Edge Crack in an Isosceles Right-Angled Triangle Plate
5.8.1 Derivation and Accuracy Verification of Weight Function
5.8.2 SIFs and CMODs for Point Forces and Power Stresses
5.9 Compact Tension Specimen
5.9.1 Derivation and Accuracy Verification of Weight Function
5.9.2 SIFs and CMODs for Point Forces and Power Stresses
5.10 A Radial Edge Crack in C-shaped Specimen
5.10.1 Derivation and Accuracy Verification of Weight Function
5.10.2 SIFs and CMODs for Point Forces and Power Stresses
5.10.3 Application Example
5.11 A Radial Edge Crack in Curved Beam
5.11.1 Derivation and Accuracy Verification of Weight Function
5.11.2 SIFs and CMODs for Point Forces and Power Stresses
5.12 An Edge Crack at a Semi-Circular Notch in a Finite Plate
5.12.1 Derivation and Accuracy Verification of Weight Function
5.12.2 SIFs and CMODs for Point Forces and Power Stresses
5.12.3 Application Examples
5.13 Wedge-Splitting Specimens
5.13.1 Derivation and Accuracy Verification of Weight Function
5.13.2 SIFs and CMODs for Point Forces and Power Stresses
5.14 Summary and Concluding Remarks
References
6 Weight Functions for Edge Crack(s) in Multiply Connected Region
6.1 Radial Edge Crack(s) Emanating from a Circular Hole
6.1.1 Derivation and Verification of WFs
6.1.2 Radial Edge Crack(s) Emanating from a Circular Hole in an Infinite Plate
6.1.3 Radial Edge Crack(s) Emanating from a Circular Hole in a Finite Plate
6.2 Radial Edge Cracks in a Circular Ring (Hollow Cylinder)
6.2.1 Derivation and Verification of WFs
6.2.2 Accuracy Verification of Analytical WFs
6.2.3 SIFs for Basic Load Cases
6.2.4 Application Examples
6.3 A Circumferential Crack in a Pipe
6.3.1 Derivation of Weight Functions
6.3.2 SIFs for Basic Load Cases
6.3.3 Comparisons and Application Examples
6.4 A Circumferential Crack in Cylindrical Bar
6.4.1 Derivation of Weight Function
6.4.2 SIFs for Basic Load Cases
6.5 Summary and Concluding Remarks
References
7 Weight Function Method for Crack in Orthotropic Materials
7.1 Introduction
7.2 Weight Function Method for Edge Crack in an Orthotropic Strip
7.3 Weight Function for an Orthotropic SENT
7.3.1 SIF and CMOD Solutions to an Orthotropic SENT
7.3.2 Weight Function for an Orthotropic SENT
7.3.3 Comparison of the Weight Functions for Isotropic and Orthotropic Materials
7.4 Applications of the Weight Function Method
7.4.1 Three-Point Bending
7.4.2 Four-Point Bending
7.4.3 Eccentrically Loaded Single-Edge-Notched Specimen
7.4.4 COD-Solutions for Uniform Segment Stress
7.5 Conclusions
References
8 Weight Function Method for Collinear Cracks and Its Application to Multiple Site Damage
8.1 Introduction
8.2 Weight Functions for Multiple Collinear Cracks
8.2.1 Construction of Cracked Body Subjected to Different Load Cases
8.2.2 Determination of Stress Intensity Factors Using Reciprocity Theorem
8.2.3 Basic Equations for Stress Intensity Factor Solutions
8.2.4 Basic Equations for Crack Opening Displacement Solutions
8.3 Weight Functions and Fracture Parameters Solutions to Two Unequal-Length Collinear Cracks
8.3.1 Reference Stress Intensity Factor and Crack Opening Displacement Solutions
8.3.2 Weight Functions for Two Collinear Cracks
8.3.3 Stress Intensity Factor Solutions
8.3.4 Crack Opening Displacement Solutions
8.4 Weight Functions and Fracture Parameter Solutions to Three Symmetrical Collinear Cracks
8.4.1 Reference Stress Intensity Factor and Crack Opening Displacement Solutions
8.4.2 Weight Functions for Three Symmetric Collinear Cracks
8.4.3 Stress Intensity Factor Solutions
8.4.4 Crack Opening Displacement Solutions
8.4.5 Strip-Yield Model Solutions
8.4.6 Approximate Method for a Lead Crack with Multiple Small Cracks
8.5 Stress Intensity Factor for Stiffened Panels with Multiple Collinear Cracks
8.5.1 Displacement Compatibility Equation
8.5.2 Rivet Forces and Stress Intensity Factors for Stiffened Panel with Two Collinear Cracks
8.5.3 Stress Intensity Factors for Stiffened Panel with Three Symmetrical Cracks
8.6 A Unified Method for Strip Yield Model of Collinear Cracks in Finite and Infinite Plate
8.6.1 The Concept of the Method
8.6.2 Three Symmetrical Cracks in an Infinite Plate
8.6.3 Two Symmetrical Cracks in a Finite Plate
8.7 Residual Strength Prediction and Experimental Verification of Finite Sheets with MSD
8.7.1 CTOA Criterion
8.7.2 Residual Strength Tests for Finite Sheets with MSD
8.7.3 Residual Strength Prediction and Verification
8.8 Summary
References
9 Mode II Weight Functions and Mixed Mode Stress Intensity Factors
9.1 Weight Function Methods for Mode II Cracks
9.1.1 Basic Equations for Mode II Center Crack
9.1.2 Basic Equations for Mode II Edge Crack
9.2 Mode II Weight Functions for Center Cracks and Stress Intensity Factor Solutions
9.2.1 Mode II Weight Function for a Center Crack in a Rectangular Plate
9.2.2 Weight Function for Center Cracked Brazilian Disc
9.2.3 Mixed Mode Stress Intensity Factors for Center Cracked Brazilian Disc
9.3 Mode II Weight Functions for Edge Cracks and Stress Intensity Factor Solutions
9.3.1 An Edge Crack in a Semi-Infinite Plate
9.3.2 An Edge Crack in a Rectangular Finite Plate
9.3.3 Edge Cracked Brazilian Disc
9.3.4 Hole-Edge Crack(s) in an Infinite Plate
References
10 Weight Function Methods for Three-Dimensional Crack Problems
10.1 The 3D Slice Synthesis Weight Function Method
10.1.1 Modelling and Analysis Procedure of Slice Synthesis Weight Function Method
10.1.2 Application of SSWFM to Various 3D Crack Problem Analyses
10.1.3 Verification and Applications of SSWFM
10.1.4 Discussions and Summary of the SSWFM
10.2 Point Weight Functions for Bi-Variant Stressing
10.2.1 The Orynyak and Wang-Glinka Point Weight Function Methods
10.2.2 Bi-Variant PWF Formulation of McClung and Lee
10.2.3 Wide-Range PWFM
10.2.4 Comments on PWFMs
10.3 Approximate Weigh Function Methods for Computing 3D SIFs at Two End Points
10.3.1 The Universal Approximate 3D WFM
10.3.2 The Root-Mean-Square (RMS) Averaged 3D WFM
10.4 Derivation of SIF- and COD-Equations Relating 2D-Slices and 3D-Crack
References
11 Analysis of Cracks in Thermal and Residual Stress Fields Using Weight Function Method
11.1 Characteristics of Crack Problems Involving Thermal/Residual Stresses
11.2 Validity of Superposition Principle for Crack Analysis with Thermal/Residual Stresses
11.3 Analysis of Crack Problems in Thermal/residual Stress with WFMs
11.4 Application Examples of WFMs for Crack Analysis with Thermal Stresses
11.4.1 Steady State Thermal Stress
11.4.2 Transient Thermal Stress—Thermal Shock
11.5 Application Examples of WFMs for Crack Analysis with Residual Stresses
11.5.1 Residual Stress Induced by Plastic Deformation
11.5.2 Laser Shock Peening Induced Residual Stress
11.5.3 Forging Residual Stress
11.5.4 Welding Residual Stress
11.5.5 Treatment of Residual Stresses in Fatigue Crack Growth and Fracture
References
12 Computation of Crack Opening Displacements and Crack Opening Areas Using Analytical Weight Function Method
12.1 Introduction
12.2 CODs for Distributed Pressure Over Entire Crack Length
12.2.1 CODs for Distributed Pressure by Integration
12.2.2 CODs by Direct Calculation for Polynomial Crack Line Stress
12.3 CODs for Segment Uniform Stress (Dugdale Stress) in Crack Tip Wake
12.3.1 COD Along the Entire Crack Length
12.3.2 COD at the End Point ξ = d, the CTOD
12.3.3 CMOD at ξ = 0
12.3.4 CODs for a Segment Uniform Stress at Arbitrary Part of the Crack
12.4 CODs for Concentrated Load at Arbitrary Point of the Crack
12.5 Rapid Determination of CMODs for Polynomial Crack Line Stresses
12.6 Crack Opening Area
References
13 Weight Function Analyses of Crack Bridging, Cohesive Model and Crack Opening Stress
13.1 Introduction
13.2 Crack Bridging and Cohesive Zone Model
13.2.1 Cohesive Zone Model
13.2.2 Fiber Bridging Model
13.3 Weight Function Method for Analyses of Cohesive and Bridging Models
13.3.1 Basic Equations for the Cohesive and Bridging Models
13.3.2 Determination of the Bridging Stress-Separation Law
13.3.3 Determination of the Cohesive Stress—Separation Law
13.4 Analysis of Dugdale Model, Cohesive SIFs and KR Curves with Weight Function Method
13.4.1 Dugdale Model Analysis
13.4.2 Analyses of Cohesive Toughness and KR Curves
13.5 Weight Function Analysis of Crack Opening Stress for the Newman Crack Closure Model
13.5.1 The Newman Crack Closure Model
13.5.2 Crack Opening Stress for an Edge Crack in a Semi-Infinite Plate
13.5.3 Crack Opening Stress for Edge Crack(s) at Stress Concentrations
13.6 Concluding Remarks
References
14 Weight Functions and Stress Intensity Factors for Complex Crack Geometries
14.1 Introduction
14.2 Effect of Overall Cracked Body Geometry on Weight Functions
14.3 The Substitute Geometry Approach for Crack Analysis
14.3.1 The Concept of Substitute Geometry
14.3.2 Rationale for the Substitute Geometry Weight Function
14.3.3 Use of Substitute Geometry Approach for Complex Crack Geometries
14.3.4 Application Example and Verification
14.4 Composition of SIF Weight Functions for Complex Geometries
14.5 Concluding Remarks
References
15 Determination of Crack-Line Stress by Using Inverse Weight Function Method
15.1 Introduction
15.2 Inverse Weight Function Method for Determination of Un-Cracked Stress
15.3 Demonstration and Verification of the Inverse Weight Function Method
15.3.1 A Periodic Array of Collinear Cracks Under Cosine Stress
15.3.2 An Edge Crack in Circular Disc Under Uniform Crack Face Pressure
15.3.3 An Edge Crack in a Finite Rectangular Plate Under Bending
15.3.4 Double Edge Cracks at Circular Hole in Infinite Plate Under Remote Tension
15.3.5 A Circular Disc with a Center/edge Crack Subjected to a Pair of Point Forces
15.4 Determination of Residual and Thermal Stresses Using Inverse WFM
15.4.1 Residual Stress in a Butt Welded Infinite Plate
15.4.2 Thermal Stress in a Circular Disc
15.5 Concluding Remarks
References
Appendix
βi (α)-Values for the Standardized Analytical Weight Functions of Various 2D Crack Geometries
Center Crack Geometry
Edge Crack Geometry
Mode II Center Crack
Mode II Edge Crack
