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2022 | Book

# Weight Function Methods in Fracture Mechanics

## Theory and Applications

Authors: Prof. Xue-Ren Wu, Assoc. Prof. Wu Xu

Publisher: Springer Nature Singapore

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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.

##### Chapter 1. Standardized Analytical Weight Function Method Based on Crack Opening Displacements
Abstract
The stress intensity factor (SIF) is the foundation of fracture mechanics analysis for engineering structures and materials. Development of solution methods for SIFs of cracked bodies has been one of the central topics in fracture mechanics. The weight function method (WFM) is a powerful, accurate, efficient and easy-to-use method for computing SIFs due to arbitrary loadings. In this chapter, the Wu-Carlsson standardized analytical WFM based on crack opening displacement, and the generalized WFM for mixed boundary conditions are presented. Detailed derivation procedures for analytical WFs are described for 2D center and edge crack geometries, respectively. Closed-form SIF-expressions for various basic crack face loadings are derived. Verification of 2D WF-accuracy using Green’s function for point-by-point assessment is proposed. The influences of displacement boundary condition and the reference load case on analytical WFs are discussed. Analytical WFMs for crack analysis in mode II and in orthotropic composite material are briefly introduced.
Xue-Ren Wu, Wu Xu
##### Chapter 2. Analysis and Discussions on Weight Function Methods Based on Multiple Reference Load Cases
Abstract
This chapter presents two types of 2D weight function methods (WFMs) that are based on multiple reference states (MRS), instead of crack opening displacements. The two methods are: the direct adjustment method (DAM) of Fett and Munz, and the universal weigh function method (UWFM) of Glinka and Shen, with different series forms for the WF-expressions. Detailed discussions are made on the commonalities and differences between the two approaches. Various aspects affecting accuracy, stability and robustness of the two MRS WFMs are analyzed, including the appropriateness of assumed WF-expressions, use of geometric condition, influence of reference load cases, and problems for short cracks. The root cause of various problems associated with the two MRS WFMs is discussed from mathematical view point: the inherent ill-imposed Volterra integral equation of the first kind.
Xue-Ren Wu, Wu Xu
##### Chapter 3. Verification and Accuracy Evaluation of Various Weight Function Methods
Abstract
To ensure successful application of weight function methods (WFMs) to fracture mechanics analysis, and to achieve high accuracy of stress intensity factor (SIF) solutions for arbitrary load cases, weight functions must be thoroughly verified. The best way to reveal the intrinsic accuracy of WFs is to employ Green’s functions (GF) for point-by-point comparison, instead of the customarily adopted comparisons of SIFs for some load cases. In the present chapter, a highly accurate numerical WF, the weight function complex Taylor series expansion (WCTSE), is described and used as benchmark solutions for verification of different analytical WFMs. The evaluation results from several crack geometries show that the COD-based standardized analytical WFM is superior to the two multiple reference states (MRS) WFMs, especially in term of solution accuracy and reliability. Several important factors affecting the robustness MRS WFMs are discussed, and sources of sensitivity and instability are analyzed.
Xue-Ren Wu, Wu Xu
##### Chapter 4. Weight Functions for Center Crack Geometries
Abstract
In this chapter, standardized analytical weight functions (WFs) for several representative center crack geometries are derived. The derived WFs are verified based on the Green’s functions obtained by using the highly accurate numerical method of “weight function complex Taylor series expansion (WCTSE)”. Closed-form expressions of stress intensity factors (SIFs) for three basic crack line stresses, including point force, power stress and constant stress segment, are derived. Calculated non-dimensional SIFs and crack mouth opening displacements (CMODs) for power stresses are given in tables, allowing rapid determination of SIFs and CMODs for crack line polynomial stresses. Various application examples are presented. Comparisons are made to the available literature data wherever possible. Accurate SIF solutions for center crack geometries associated with various load cases are presented.
Xue-Ren Wu, Wu Xu
##### Chapter 5. Weight Functions for Edge Crack in Simply Connected Region
Abstract
In this chapter, standardized analytical weight functions (WFs) for a large variety of edge crack geometries in simply connected region are derived. The derived WFs are verified based on the Green’s functions obtained by using the highly accurate numerical method of “weight function complex Taylor series expansion (WCTSE)”. Closed-form expressions of stress intensity factors (SIFs) for three basic crack line stresses, including point force, power stress and constant stress segment, are derived. Calculated non-dimensional SIFs and crack mouth opening displacements (CMODs) for power stresses are given in tables, allowing rapid determination of SIFs and CMODs for crack line polynomial stresses. Many application examples are presented. Comparisons are made to the available literature data wherever possible. A large amount of accurate SIF solutions for edge crack geometries in simply connected region associated with various load cases are presented.
Xue-Ren Wu, Wu Xu
##### Chapter 6. Weight Functions for Edge Crack(s) in Multiply Connected Region
Abstract
In this chapter, standardized analytical weight functions (WFs) for various edge crack geometries in multiply connected region, including radial edge crack(s) emanating from a circular hole in infinite/finite plate, radial edge crack(s) in circular ring and circumferential crack in a pipe and cylindrical bar, are derived. The derived WFs are verified based on the Green’s functions obtained by using the highly accurate numerical method of “weight function complex Taylor series expansion (WCTSE)” or finite element. Closed-form expressions of stress intensity factors (SIFs) for several basic crack line stresses are derived. Calculated non-dimensional SIFs for power stresses are given in tables, allowing rapid determination of SIFs and CMODs for crack line polynomial stresses. Many practical examples are presented. Comparisons are made to the available literature data wherever possible. A large amount of accurate SIF solutions for edge crack geometries in multiply connected region associated with various load cases are presented.
Xue-Ren Wu, Wu Xu
##### Chapter 7. Weight Function Method for Crack in Orthotropic Materials
Abstract
Damage tolerance is increasingly important in the composite structural design community. However, there are very limited solutions of closed-form stress intensity factors (SIF), crack opening displacements (COD) and weight functions available for cracked composite plates. In this chapter, the Wu-Carlsson weight function method (WFM) for isotropic material is extended to crack analysis in orthotropic material. Explicit weight function for an edge crack in an orthotropic strip is provided. Through extensive verifications against results from finite element analyses (FEAs) and existing literature data, the present WFM is demonstrated to be highly accurate, efficient and versatile for calculating SIFs and CODs for edge cracks in orthotropic materials.
Xue-Ren Wu, Wu Xu
##### Chapter 8. Weight Function Method for Collinear Cracks and Its Application to Multiple Site Damage
Abstract
Structure with multiple site damage (MSD) is a serious threat to aircraft safety. The weight function method (WFM) for multiple collinear cracks is derived and exact weight functions (WF) for two and three collinear cracks are given this chapter. The WFM is then used to determine the stress intensity factors (SIFs), crack opening displacement (COD) and plastic zones of two and three collinear cracks in unstiffened and stiffened plates. These results are verified by finite element method (FEM) and existing results from literature. The WFM is used with CTOA criterion to predict the stable crack growths and residual strengths of sheets with MSD. The predicted crack behaviors and residual strengths agree well with the experimental results. Compared to FEM for predicting residual strength of plate with MSD, the WFM is much more efficient.
Xue-Ren Wu, Wu Xu
##### Chapter 9. Mode II Weight Functions and Mixed Mode Stress Intensity Factors
Abstract
Cracks in materials and structures are often subjected to complex mixed mode loadings. In this chapter, the crack opening displacement-based weight function method (WFM) for mode I crack is extended to derive the weight functions (WFs) for mode II edge and center cracks. Highly accurate mode II WFs for center cracked plate and Brazilian discs, edge cracked strip and Brazilian discs, and hole-edge crack(s) in an infinite sheet are provided. The mixed mode SIFs for these crack configurations are obtained by using the present mode II WF and the mode I WF presented in Chaps. 46. It is demonstrated that the WFM is very accurate and highly efficient for calculation of the mixed mode SIFs.
Xue-Ren Wu, Wu Xu
##### Chapter 10. Weight Function Methods for Three-Dimensional Crack Problems
Abstract
Three-dimensional (3D) crack problems are much more complicated and analytically intractable than 2D cases because the stress intensity factor (SIF) is a function of position along the crack front, the crack size and also several other geometric variables. The weight function methods (WFMs) provide efficient and reliable means for analysis of 3D crack problems, especially when multi-geometry parameters and complex load conditions are considered. In this chapter, several 3D WFMs are described. The slice synthesis weight function method (SSWFM) based on the combination of the slice-synthesis technique and the 2D analytical WFM is presented, and verified with various practical cases. Accurate SIFs for surface/corner cracks subjected to power stresses in various configurations are determined and tabulated. Various point load weight function methods (PWFMs) for 3D cracks in bi-variant stress fields are discussed. Simple engineering approaches for 3D SIFs at the deepest and surface points are briefly introduced.
Xue-Ren Wu, Wu Xu
##### Chapter 11. Analysis of Cracks in Thermal and Residual Stress Fields Using Weight Function Method
Abstract
In this chapter, the standardized analytical weight function method (WFM) is utilized to determine stress intensity factors (SIFs) for cracks in thermal and residual stress fields that are characterized by strong variations and steep gradients due to their self-equilibrating nature. The validity of superposition principle for SIFs of cracks subjected to thermal/residual stresses is proved, and several important aspects in crack problem analysis involving self-equilibrating internal stress fields are discussed. SIFs for typical 2D crack geometries with a variety of thermal/residual stresses profiles, including steady state and thermal shock stresses, residual stresses due to welding and plastic deformation etc., are calculated by using the analytical WFM. Solution accuracy and computational efficiency of the analytical WFM for crack analysis involving thermal and residual stresses are demonstrated.
Xue-Ren Wu, Wu Xu
##### Chapter 12. Computation of Crack Opening Displacements and Crack Opening Areas Using Analytical Weight Function Method
Abstract
Determination of CODs for arbitrary crack face loadings by using the standardized analytical weight function method (WFM) is described in this chapter. Analytical COD-expressions are derived for three types of basic crack line stresses: distributed stress over the entire crack, a segment of uniform pressure and a pair of point forces acting at arbitrary location of the crack. Calculation methods for crack opening displacement at crack mouth and at fictitious crack tip, CMOD and CTOD, by using the analytical WFM are presented. CMODs for center and edge cracks under polynomial stresses are easily determined. The methods are verified by comparisons to exact and highly accurate results obtained using other methods in the literature. By integration of the CODs from the analytical WFM, crack opening area is readily obtained.
Xue-Ren Wu, Wu Xu
##### Chapter 13. Weight Function Analyses of Crack Bridging, Cohesive Model and Crack Opening Stress
Abstract
In this chapter, the weight function method (WFM) and the weight functions (WFs) given in the previous chapters are applied to solve a variety of problems subjected to the crack surface loadings, for example, the crack bridging model, cohesive zone model and plasticity induced crack opening stress, etc. These models are widely used in different engineering fields. However, their mathematical essences are the same. It is demonstrated through several examples that the analytical WFM is the most powerful tool for solving these models involving various crack face loadings.
Xue-Ren Wu, Wu Xu
##### Chapter 14. Weight Functions and Stress Intensity Factors for Complex Crack Geometries
Abstract
Based on quantitative comparisons of weigh functions (WFs) for various crack geometries, this chapter discusses the effect of overall as well as local geometry of the cracked body on the WF; assesses the rationality of the “substitute geometry” concept by using simple model crack geometry for more complicated crack geometries; provides the way of using the “substitute geometry” concept to analyze engineering crack problems associated with more complicated real world crack geometries. A brief discussion is made on another approach for analyzing cracks in complex geometries, namely the composition of SIF weight functions.
Xue-Ren Wu, Wu Xu
##### Chapter 15. Determination of Crack-Line Stress by Using Inverse Weight Function Method
Abstract
In this chapter, an inverse weight function method (WFM) is developed for the determination of crack line stress distribution in un-cracked body. The method uses the analytical weight function (WF) and crack mouth opening displacement CMOD for the considered crack geometry as known inputs. The unknown crack line stress is assumed to be composed of a large number of constant segments. The integral equation that relates the CMOD, the WF and the crack line stress is solved. Stress distribution at the prospective crack line in the un-cracked body is determined segment by segment. The method is verified through comparisons to known stress distributions for a variety of 2D crack geometries and load cases. The inverse WFM will be useful for determination of residual stresses.
Xue-Ren Wu, Wu Xu
##### Backmatter
Title
Weight Function Methods in Fracture Mechanics
Authors
Prof. Xue-Ren Wu
Assoc. Prof. Wu Xu