Weight functions and stress intensity factor solutions pdf

Weight functions and stress intensity factor solutions pdf
As an example, a weight function for the SEN strip is obtained in this manner. Moreover, closed form infinite body weight functions are also developed and used to derive some well-known stress intensity factor …
stress intensity factors [9-13].One of these methods is a numerical method like Green’s function, weight functions, boundary collocation, alternating method, integral transforms,
Since the stress intensity factor (K) is the foundation of fracture mechanics of aircraft structuresand damage tolerance , a analysis significant focus of development efforts in the past years has been geared towards fifteen enhancing legacy solutions and developing new and effK icient numerical K solutions that can handle the complicated stress gradients nalysts using detailed finite computed
The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of a planer surface crack and an embedded crack subjected to fluctuating non-linear stress.
Evaluating Stress Intensity Factors due to Weld Residual Stresses by the Weight Function and Finite Element Methods Rui Bao1, Xiang Zhang2*, Norvahida Ahmad Yahaya2 1Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China 2Department of Aerospace Engineering, School of Engineering, Cranfield University, Bedford, MK43 0AL, U.K
This page provides stress intensity factor solutions for common cases. Contents
A weight function to evaluate the stress intensity factor (SIF) of a circumferential crack, subjected to arbitrarily distributed stress on the crack surfaces, in a finite length thin-walled cylinder was derived based on the closed form SIF
An efficient boundary weight function method for the determination of mode I stress intensity factors in a three-dimensional cracked body with arbitrary shape and subjected to …
5.1 Stress intensity factor force Q, the closed form solution for the weight functions are, Figure 5.3 Infinitely large plate with a central crack of length 2a. SIF A Q1 Qax gx,a πa a-x = = + (5.3a) SIF B Q1 gx,a πa ax= = + Qa-x (5.3b) When it is not possible to obtain such functions analytically, numerical methods based on finite elements are often used. An example of determining SIF
Weight function and stress intensity factor for a semi-elliptical surface saddle crack in a tubular welded joint E Chang* and W D Dover Department of Mechanical Engineering, University College London, London, UK
The general 2-D weight function accounting for the free boundary effect was found to be: Weight Functions and Stress Intensity Factors NUMERICAL TECHNIQUE The stress intensity factor due to a continuous stress field applied to the crack surface is calculated by integrating the product (5) of the weight function and the stress field over the entire crack area. In the case of numerical
A weight function technique is used to obtain mode I stress intensity factor solutions for radially cracked rings loaded with arbitrary crack face pressure. When the crack face pressure is defined as the hoop stress occurring in an
useful and versatile method of calculating stress intensity factors for cracks subjected to non-uniform stress fields, such as residual stress or thermal loading. Using the weight function …
Stress intensity factor solutions for the ring-shaped crack in an infinite body loaded by constant stress normal to the crack plane (Fig. 2.1b) were compiled by Rosenfelder [1]. For an arbitrarily given stress distribution σ( x ) in the uncracked body normal to the crack
Stress Intensity Factors of a Semi-Elliptical Crack in a
https://www.youtube.com/embed/H5yo2dvaalU

Stress Intensity Factor Evaluation of a Circumferential
In this paper, we use the weight function for an elliptical crack embedded in an infinite elastic media in conjunction with the alternating method to derive the exact analytical solution for the stress intensity factor for a semi-elliptical surface crack subjected to an arbitrary mode I loading.
The weight functions h and t can be interpreted as the stress intensity factor and as the T-term for a pair of single forces P acting at the crack face at the location x 0 (Fig.3.2), i.e. the weight functions (h, t) are Green’s functions for K I and T.
11/8/00 ME111 Lecture 16 10 • The units of the stress intensity factor are, for examplK e, MPa m, or • The stress intensity factor describes the state of stress near a crack tip. K • It is found experimentally that existing cracks will propagate (I.e. grow) when the stress intensity factor reaches a critical valu e called the fracture toughness. K • The fracture toughness is
3 a v K G h a x 1 1 1 1 8, (2) The Green’s function is an earlier approach that is similar to the weight function method [13].

are presented in terms of weight functions, allowing stress intensity factors to be evaluated for arbitrary stress fields. Smith [A.13] has compared R6 K-solutions for cylinders with those of other procedures; consequently
The derived weight functions are then validated against stress intensity factor and T-stress solutions for several linear and nonlinear two-dimensional stress distributions. These derived weight functions are particularly useful for the development of two-parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from
Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .
using the J-integrals, dynamic stress intensity factors are calculated. Numerical results including the values of Numerical results including the values of dynamic stress intensity factors for a crack in an infinite domain subjected to P and SV waves are presented.
Evaluation of stress intensity factors and higher order asymptotic terms of displacement and stress elds represents a crucial issue for perturbative analysis of many interfacial crack problems (Bercial-Velez et al., 2005; Piccolroaz et al., 2010).
Fracture Mechanics Lecture notes – course 4A780 Concept version Dr.ir. P.J.G. Schreurs Eindhoven University of Technology Department of Mechanical Engineering
Universal Weight Function Method and Polynomial Stress Distribution Method The existing stress intensity factor (K) solutions for surface cracks in pipe typically require a polynomial stress distribution through the pipe wall thickness. (e.g., in API RP 579, the through thickness stress distribution can be represented as a 4th order polynomial fit) However, if the through thickness stress
Stress intensity factors in crack closure problems ENESCU IOAN Department of Mechanical Engineering Transylvania University of Brasov 500036 Bvd.

11/12/2017 · Watch Weight Functions and Stress Intensity Factor Solutions – connerspencer994 on dailymotion
Approximate weight functions derived from stress intensity factor solutions; Weight functions based on BCM computations; Round-CT-specimen Chapter 10 Cracks in front of internal notches
The weight function method is one of the most reliable, versatile, and cost-effective methods of evaluating the stress intensity factors and crack opening displacements. This book provides a valuable account of the author’s research in these fields. It has two aims: firstly to provide a theoretical background to the weight function method in fracture mechanics for accurate analysis of two
Weight Function for Stress Intensity Factors in Rotating Thick-Walled Cylinder 31 form of polynomial expression, which is fit for calculation in weight function method.
one reference stress intensity factor solution. This method was used to derive the weight functions for This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body.
The stress intensity factors are calculated from equation (3.4) by integration of the product of the weight function (3.2) and the stress distribution given by equation (3.22)

Catalog Record Weight functions and stress intensity
Weight functions and stress intensity factors 103 intensity factors using the weight functions derived in this approach was of the same order as the reference stress intensity factor solutions.
Approximate weight functions for a quarter-elliptical crack in a fastener hole were derived from a general weight function form and two reference stress intensity factors.
Stress intensity solutions for semi-infinite, part-throughwall flaws in thin and thick walled cylindrical vessels can be obtained by various methods, including weight functions and finite-element analysis.Analytical Solutions for Stress Intensity Factor, T-Stress and Weight Function for the Edge-Cracked Half-Space
Wang, X., Bell, R. Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates subject to non-uniform stress distributions Engng Fracture Mechanics, 2004, 71, 1477 – 1496.
the crack-face weight function, the efficient calculation of the stress intensity factor is illustrated. The size of the The size of the Dugdale plastic zone ahead of the crack tip for a finite plate is estimated from the available weight functions.
stress eld is by using the weight function for a given cracked body. The weight function The weight function represents the SIF induced by a unit concentrated load.
Weight functions allow for the determination of stress intensity factors K and T-stresses T under various crack surface tractions. Whereas the mode-I stress intensity factor in most cases is determined by the normal tractions only and the mode-II stress intensity factor …
Calculating Stress Intensity Factor (Mode I) for Composite
The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting
Stress intensity factor solutions are used in the assessment of crack-like flaws (see Section 9). C.1.1.2 A summary of the stress intensity factor solutions is contained in Table C.1.
is consistent with the stress intensity factor derived for the non-symmetric distribution of forces obtained by Piccolroaz et al. (2009). Finally, in Appendix A, the evaluated skew-symmetric weight function is compared to those
Glinka weight function solutions to calculate the Mode I crack tip stress intensity factor, KI, in complex ship structure details. The project was divided into the following 6 tasks:
stress intensity factors and weight functions – technical background 5 3. universal weight functions for one-dimensional cracks 9 4. universal weight functions for two-dimensional part- through surface and corner cracks 11 5. sequence of steps for calculating stress intensity factors using weight functions 13 6. determination of weight functions 13 7. numerical integration of the weight
National Conference on Innovative Paradigms in Engineering

(PDF) Weight functions and stress intensity factors for
https://www.youtube.com/embed/PhBnDOA8WsA
a two-dimensional weight function approach is used to determine stress intensity factors for cracks in either tensile or compressive stress fields, due to one of three mechanisms: remote tension overload, remote compression overload or hole cold expansion.
The weight function may be thought of as Green’s function for the stress intensity factor of cracked bodies. Once the weight function for a cracked body is determined, the stress intensity factor for any arbitrary loading can be simply and efficiently evaluated through the integration of the product of the loading and weight function. A numerical method for the determination of weight
This paper presents the weight functions for the determination of the stress intensity factor and T -stress solutions for edge-cracked plates with built-in ends under complex stress distributions. First, a compliance analysis approach is used to calculate stress intensity factor and T -stress for edge cracks in finite width plates with built-in ends with uniform or linear stress distributions
3-D Weight Function Method with Stresses from Finite Element Method W. Zhao stress intensity factors for semi-elliptical surface cracks emanating from a circular hole are determined. The 3-D weight function method with the 3-D finite element solutions for the uncracked stress distribution as in Part-I is used for the analysis. Two different loading conditions, i.e. remote tension and wedge
dure for calculating stress intensity factors and crack opening dis-placements of cracks subjected to nonlinear stress distributions such as those in autofrettaged cylinders or near notches is dis-cussed below. Stress Intensity Factors and Weight Functions Most of the existing methods of calculating stress intensity fac-tors require separate analysis of each load and geometry configu-ration
A stress-intensity-factor weight function for the edge-cracked rectangular plate has been determined. • A crack-opening-displacement Green’s function for the …
and [[ weight functions were derived from corresponding reference displacement fields and stress intensity factors calculated by finite element method. Normalized mode ~ and mode ]] stress intensity factors…
Stress intensity factors and weight functions in

PDF Fracture Mechanics Materials Technology
stress intensity factor solutions were given, methods for the determination of weight functions were reported and numerical results for a number of crack geometries were compiled. In the meantime, further crack problems have been evaluated which will be addressed in this
In contrast with BS 7910, where solutions are presented in terms of bending and membrane stress only, many of the R6 solutions are presented in terms of weight functions, allowing stress intensity factors to be evaluated for arbitrary stress fields. R6 provides valuable critical comment on the accuracy of solutions, sometimes citing (but not necessarily incorporating) solutions from other
Geometric functions of stress intensity factor solutions for spot welds in U-shape specimens P.-C. Lina,*, D.-A. Wangb a Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi …
Weight function method, suggested by Bueckner [4] and Rice [5], is one of the most effective tools of stress intensity factor determination in cases of continuous load symmetrically distributed along both sides of the crack.
The stress intensity factors obtained by the present numerical approach are compared with analytical solutions. The errors in the stress intensity factors for opening fracture mode I are less than 1% although the model mesh is relatively coarse. Key words: Element free Galerkin method, two dimensional elasticity problems, Fracture mechanic, Stress intensity factors. 1. Introduction The
the calculation of K1c (Chap. 3) stress intensity factor solutions for these speci­ mens are necessary. Also knowledge of weight functions is of interest, e.g. for the
The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress
determine the stress intensity factor for the other load system Q2′ Of course, the 1 and 2 systems may represent any arbitrarily chosen load systems and thus it is being shown that if a solution for the displacement field and stress intensity factor is known for any particular
COMPUTATION OF THE WEIGHT FUNCTION FROM A STRESS INTENSITY FACTOR” . by H. J. Petroski and J. D. Achenbach 4 SI’ One obstacle to the direct use of Eq. 1 is that the solutions for the stress intensity factor that are available in the literature to serve as reference data are often not accompanied by data for crack-face displacements. To overcome this obstacle, a simple method has …
Weight functions and stress intensity factors for embedded
A STRESS 3 INTENSITY FACTOR SOLUTION INSPIRED BY SOAP BUBBLES FRAMEWORK Fig. 2. Discretized values seen as a mesh. 2.2 1D shape functions The 2D shape functions …
In this investigation, the weight function method was employed to calculate stress intensity factors for semi-elliptical surface crack in a hollow cylinder. The weight functions at …
Weight functions and stress intensity factor solutions. Xue-Ren Wu, Janne Carlsson. Pergamon Press, 1991 – Science – 513 pages. 0 Reviews. From inside the book . What people are saying – Write a review. We haven’t found any reviews in the usual places. Contents. A Center Crack in a Finite Rectangular Plate . 39: A Center Crack in a Circular Disc . 63: Periodic Array of Collinear Cracks in …
Hardbound. An important element of work in fracture mechanics is the stress intensity factor – the characterizing parameter for the crack tip field in a linear elastic material; something reflected in its intense research over the last 30 years.
Weight functions and stress intensity factor solutions / Xue-Ren Wu and A. Janne Carlsson.
Bibliography Includes bibliographical references (p. [347]-354) and index. Contents. Chapter 1 – Stress intensity factors and weight functions Stress intensity factors– Mode-1 weight functions for one-dimensional cracks Chapter 2 – The extended Petroski-Achenbach method (PAM) Approximate weight functions for a component with an external crack zig ziglar books pdf free download

https://www.youtube.com/embed/nsIhQmi-nSg
(PDF) Weight Functions and Stress Intensity Factors for

WEIGHT FUNCTION FOR STRESS INTENSITY FACTORS IN
Stress intensity factors in crack closure problems iaras.org
Stress Intensity Factor Solutions MechaniCalc

A STRESS INTENSITY FACTOR SOLUTION INSPIRED BY SOAP

Unitary weight functions for semi-infinite ABSTRACT

Weight functions and stress intensity factors for embedded
Weight functions for the determination of stress intensity

The general 2-D weight function accounting for the free boundary effect was found to be: Weight Functions and Stress Intensity Factors NUMERICAL TECHNIQUE The stress intensity factor due to a continuous stress field applied to the crack surface is calculated by integrating the product (5) of the weight function and the stress field over the entire crack area. In the case of numerical
dure for calculating stress intensity factors and crack opening dis-placements of cracks subjected to nonlinear stress distributions such as those in autofrettaged cylinders or near notches is dis-cussed below. Stress Intensity Factors and Weight Functions Most of the existing methods of calculating stress intensity fac-tors require separate analysis of each load and geometry configu-ration
A STRESS 3 INTENSITY FACTOR SOLUTION INSPIRED BY SOAP BUBBLES FRAMEWORK Fig. 2. Discretized values seen as a mesh. 2.2 1D shape functions The 2D shape functions …
The weight functions h and t can be interpreted as the stress intensity factor and as the T-term for a pair of single forces P acting at the crack face at the location x 0 (Fig.3.2), i.e. the weight functions (h, t) are Green’s functions for K I and T.
This page provides stress intensity factor solutions for common cases. Contents
Stress intensity factor solutions for the ring-shaped crack in an infinite body loaded by constant stress normal to the crack plane (Fig. 2.1b) were compiled by Rosenfelder [1]. For an arbitrarily given stress distribution σ( x ) in the uncracked body normal to the crack
determine the stress intensity factor for the other load system Q2′ Of course, the 1 and 2 systems may represent any arbitrarily chosen load systems and thus it is being shown that if a solution for the displacement field and stress intensity factor is known for any particular

CALCULATION OF DYNAMIC STRESS INTENSITY FACTORS IN
Weight function method lmafsrv1.epfl.ch

the calculation of K1c (Chap. 3) stress intensity factor solutions for these speci­ mens are necessary. Also knowledge of weight functions is of interest, e.g. for the
In contrast with BS 7910, where solutions are presented in terms of bending and membrane stress only, many of the R6 solutions are presented in terms of weight functions, allowing stress intensity factors to be evaluated for arbitrary stress fields. R6 provides valuable critical comment on the accuracy of solutions, sometimes citing (but not necessarily incorporating) solutions from other
In this paper, we use the weight function for an elliptical crack embedded in an infinite elastic media in conjunction with the alternating method to derive the exact analytical solution for the stress intensity factor for a semi-elliptical surface crack subjected to an arbitrary mode I loading.
Stress intensity factor solutions are used in the assessment of crack-like flaws (see Section 9). C.1.1.2 A summary of the stress intensity factor solutions is contained in Table C.1.
A weight function to evaluate the stress intensity factor (SIF) of a circumferential crack, subjected to arbitrarily distributed stress on the crack surfaces, in a finite length thin-walled cylinder was derived based on the closed form SIF
This page provides stress intensity factor solutions for common cases. Contents
Stress intensity factor solutions for the ring-shaped crack in an infinite body loaded by constant stress normal to the crack plane (Fig. 2.1b) were compiled by Rosenfelder [1]. For an arbitrarily given stress distribution σ( x ) in the uncracked body normal to the crack

Weight functions and stress intensity factors for embedded
Weight function method lmafsrv1.epfl.ch

stress intensity factor solutions were given, methods for the determination of weight functions were reported and numerical results for a number of crack geometries were compiled. In the meantime, further crack problems have been evaluated which will be addressed in this
Weight functions and stress intensity factor solutions. Xue-Ren Wu, Janne Carlsson. Pergamon Press, 1991 – Science – 513 pages. 0 Reviews. From inside the book . What people are saying – Write a review. We haven’t found any reviews in the usual places. Contents. A Center Crack in a Finite Rectangular Plate . 39: A Center Crack in a Circular Disc . 63: Periodic Array of Collinear Cracks in …
Weight functions allow for the determination of stress intensity factors K and T-stresses T under various crack surface tractions. Whereas the mode-I stress intensity factor in most cases is determined by the normal tractions only and the mode-II stress intensity factor …
Weight function and stress intensity factor for a semi-elliptical surface saddle crack in a tubular welded joint E Chang* and W D Dover Department of Mechanical Engineering, University College London, London, UK
A stress-intensity-factor weight function for the edge-cracked rectangular plate has been determined. • A crack-opening-displacement Green’s function for the …
is consistent with the stress intensity factor derived for the non-symmetric distribution of forces obtained by Piccolroaz et al. (2009). Finally, in Appendix A, the evaluated skew-symmetric weight function is compared to those
This page provides stress intensity factor solutions for common cases. Contents

SOLID MECHANICS Българска академия на
Weight functions for Tstress for semi-elliptical surface

stress intensity factors [9-13].One of these methods is a numerical method like Green’s function, weight functions, boundary collocation, alternating method, integral transforms,
Geometric functions of stress intensity factor solutions for spot welds in U-shape specimens P.-C. Lina,*, D.-A. Wangb a Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi …
is consistent with the stress intensity factor derived for the non-symmetric distribution of forces obtained by Piccolroaz et al. (2009). Finally, in Appendix A, the evaluated skew-symmetric weight function is compared to those
Stress intensity factor solutions for the ring-shaped crack in an infinite body loaded by constant stress normal to the crack plane (Fig. 2.1b) were compiled by Rosenfelder [1]. For an arbitrarily given stress distribution σ( x ) in the uncracked body normal to the crack
Evaluating Stress Intensity Factors due to Weld Residual Stresses by the Weight Function and Finite Element Methods Rui Bao1, Xiang Zhang2*, Norvahida Ahmad Yahaya2 1Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China 2Department of Aerospace Engineering, School of Engineering, Cranfield University, Bedford, MK43 0AL, U.K
This paper presents the weight functions for the determination of the stress intensity factor and T -stress solutions for edge-cracked plates with built-in ends under complex stress distributions. First, a compliance analysis approach is used to calculate stress intensity factor and T -stress for edge cracks in finite width plates with built-in ends with uniform or linear stress distributions
Weight Function for Stress Intensity Factors in Rotating Thick-Walled Cylinder 31 form of polynomial expression, which is fit for calculation in weight function method.
The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress
a two-dimensional weight function approach is used to determine stress intensity factors for cracks in either tensile or compressive stress fields, due to one of three mechanisms: remote tension overload, remote compression overload or hole cold expansion.
and [[ weight functions were derived from corresponding reference displacement fields and stress intensity factors calculated by finite element method. Normalized mode ~ and mode ]] stress intensity factors…
In this investigation, the weight function method was employed to calculate stress intensity factors for semi-elliptical surface crack in a hollow cylinder. The weight functions at …
The weight functions h and t can be interpreted as the stress intensity factor and as the T-term for a pair of single forces P acting at the crack face at the location x 0 (Fig.3.2), i.e. the weight functions (h, t) are Green’s functions for K I and T.
A weight function to evaluate the stress intensity factor (SIF) of a circumferential crack, subjected to arbitrarily distributed stress on the crack surfaces, in a finite length thin-walled cylinder was derived based on the closed form SIF
The stress intensity factors are calculated from equation (3.4) by integration of the product of the weight function (3.2) and the stress distribution given by equation (3.22)

Stress Intensity Factor Solutions for Part-Throughwall
(1) (2) arXiv

3 a v K G h a x 1 1 1 1 8, (2) The Green’s function is an earlier approach that is similar to the weight function method [13].
An efficient boundary weight function method for the determination of mode I stress intensity factors in a three-dimensional cracked body with arbitrary shape and subjected to …
Weight functions and stress intensity factors 103 intensity factors using the weight functions derived in this approach was of the same order as the reference stress intensity factor solutions.
A STRESS 3 INTENSITY FACTOR SOLUTION INSPIRED BY SOAP BUBBLES FRAMEWORK Fig. 2. Discretized values seen as a mesh. 2.2 1D shape functions The 2D shape functions …
Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .
The derived weight functions are then validated against stress intensity factor and T-stress solutions for several linear and nonlinear two-dimensional stress distributions. These derived weight functions are particularly useful for the development of two-parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from
stress eld is by using the weight function for a given cracked body. The weight function The weight function represents the SIF induced by a unit concentrated load.
This paper presents the weight functions for the determination of the stress intensity factor and T -stress solutions for edge-cracked plates with built-in ends under complex stress distributions. First, a compliance analysis approach is used to calculate stress intensity factor and T -stress for edge cracks in finite width plates with built-in ends with uniform or linear stress distributions
Weight function method, suggested by Bueckner [4] and Rice [5], is one of the most effective tools of stress intensity factor determination in cases of continuous load symmetrically distributed along both sides of the crack.
stress intensity factors and weight functions – technical background 5 3. universal weight functions for one-dimensional cracks 9 4. universal weight functions for two-dimensional part- through surface and corner cracks 11 5. sequence of steps for calculating stress intensity factors using weight functions 13 6. determination of weight functions 13 7. numerical integration of the weight
Wang, X., Bell, R. Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates subject to non-uniform stress distributions Engng Fracture Mechanics, 2004, 71, 1477 – 1496.

Calculating Stress Intensity Factor (Mode I) for Composite
Weight Functions for Stress Intensity Factors and T-Stress

stress intensity factors [9-13].One of these methods is a numerical method like Green’s function, weight functions, boundary collocation, alternating method, integral transforms,
Evaluating Stress Intensity Factors due to Weld Residual Stresses by the Weight Function and Finite Element Methods Rui Bao1, Xiang Zhang2*, Norvahida Ahmad Yahaya2 1Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China 2Department of Aerospace Engineering, School of Engineering, Cranfield University, Bedford, MK43 0AL, U.K
a two-dimensional weight function approach is used to determine stress intensity factors for cracks in either tensile or compressive stress fields, due to one of three mechanisms: remote tension overload, remote compression overload or hole cold expansion.
The stress intensity factors are calculated from equation (3.4) by integration of the product of the weight function (3.2) and the stress distribution given by equation (3.22)
A stress-intensity-factor weight function for the edge-cracked rectangular plate has been determined. • A crack-opening-displacement Green’s function for the …
Approximate weight functions for a quarter-elliptical crack in a fastener hole were derived from a general weight function form and two reference stress intensity factors.
Weight functions and stress intensity factor solutions / Xue-Ren Wu and A. Janne Carlsson.
determine the stress intensity factor for the other load system Q2′ Of course, the 1 and 2 systems may represent any arbitrarily chosen load systems and thus it is being shown that if a solution for the displacement field and stress intensity factor is known for any particular
Stress intensity factor solutions are used in the assessment of crack-like flaws (see Section 9). C.1.1.2 A summary of the stress intensity factor solutions is contained in Table C.1.

Stress intensity factors for part-elliptical cracks
Determination of approximate point load weight functions

stress intensity factor solutions were given, methods for the determination of weight functions were reported and numerical results for a number of crack geometries were compiled. In the meantime, further crack problems have been evaluated which will be addressed in this
Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .
Stress intensity factor solutions are used in the assessment of crack-like flaws (see Section 9). C.1.1.2 A summary of the stress intensity factor solutions is contained in Table C.1.
In this investigation, the weight function method was employed to calculate stress intensity factors for semi-elliptical surface crack in a hollow cylinder. The weight functions at …
stress eld is by using the weight function for a given cracked body. The weight function The weight function represents the SIF induced by a unit concentrated load.
Fracture Mechanics Lecture notes – course 4A780 Concept version Dr.ir. P.J.G. Schreurs Eindhoven University of Technology Department of Mechanical Engineering
Stress intensity factors in crack closure problems ENESCU IOAN Department of Mechanical Engineering Transylvania University of Brasov 500036 Bvd.
Geometric functions of stress intensity factor solutions for spot welds in U-shape specimens P.-C. Lina,*, D.-A. Wangb a Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi …
Approximate weight functions derived from stress intensity factor solutions; Weight functions based on BCM computations; Round-CT-specimen Chapter 10 Cracks in front of internal notches
Universal Weight Function Method and Polynomial Stress Distribution Method The existing stress intensity factor (K) solutions for surface cracks in pipe typically require a polynomial stress distribution through the pipe wall thickness. (e.g., in API RP 579, the through thickness stress distribution can be represented as a 4th order polynomial fit) However, if the through thickness stress
The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting
Weight Function for Stress Intensity Factors in Rotating Thick-Walled Cylinder 31 form of polynomial expression, which is fit for calculation in weight function method.
This paper presents the weight functions for the determination of the stress intensity factor and T -stress solutions for edge-cracked plates with built-in ends under complex stress distributions. First, a compliance analysis approach is used to calculate stress intensity factor and T -stress for edge cracks in finite width plates with built-in ends with uniform or linear stress distributions

PDF Fracture Mechanics Materials Technology
Effect of Weld Residual Stress Fitting on Stress Intensity

Weight functions and stress intensity factor solutions. Xue-Ren Wu, Janne Carlsson. Pergamon Press, 1991 – Science – 513 pages. 0 Reviews. From inside the book . What people are saying – Write a review. We haven’t found any reviews in the usual places. Contents. A Center Crack in a Finite Rectangular Plate . 39: A Center Crack in a Circular Disc . 63: Periodic Array of Collinear Cracks in …
The weight function may be thought of as Green’s function for the stress intensity factor of cracked bodies. Once the weight function for a cracked body is determined, the stress intensity factor for any arbitrary loading can be simply and efficiently evaluated through the integration of the product of the loading and weight function. A numerical method for the determination of weight
The weight function method is one of the most reliable, versatile, and cost-effective methods of evaluating the stress intensity factors and crack opening displacements. This book provides a valuable account of the author’s research in these fields. It has two aims: firstly to provide a theoretical background to the weight function method in fracture mechanics for accurate analysis of two
the calculation of K1c (Chap. 3) stress intensity factor solutions for these speci­ mens are necessary. Also knowledge of weight functions is of interest, e.g. for the

Determination of approximate point load weight functions
Weight Function Method With Segment-Wise Polynomial

The derived weight functions are then validated against stress intensity factor and T-stress solutions for several linear and nonlinear two-dimensional stress distributions. These derived weight functions are particularly useful for the development of two-parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from
The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress
Stress intensity factor solutions for the ring-shaped crack in an infinite body loaded by constant stress normal to the crack plane (Fig. 2.1b) were compiled by Rosenfelder [1]. For an arbitrarily given stress distribution σ( x ) in the uncracked body normal to the crack
stress intensity factors [9-13].One of these methods is a numerical method like Green’s function, weight functions, boundary collocation, alternating method, integral transforms,
Fracture Mechanics Lecture notes – course 4A780 Concept version Dr.ir. P.J.G. Schreurs Eindhoven University of Technology Department of Mechanical Engineering

Progress Report No afgrow.net
Calculation of Stress Intensity Factors and Crack Opening

the crack-face weight function, the efficient calculation of the stress intensity factor is illustrated. The size of the The size of the Dugdale plastic zone ahead of the crack tip for a finite plate is estimated from the available weight functions.
stress intensity factors [9-13].One of these methods is a numerical method like Green’s function, weight functions, boundary collocation, alternating method, integral transforms,
The weight function may be thought of as Green’s function for the stress intensity factor of cracked bodies. Once the weight function for a cracked body is determined, the stress intensity factor for any arbitrary loading can be simply and efficiently evaluated through the integration of the product of the loading and weight function. A numerical method for the determination of weight
Approximate weight functions for a quarter-elliptical crack in a fastener hole were derived from a general weight function form and two reference stress intensity factors.
5.1 Stress intensity factor force Q, the closed form solution for the weight functions are, Figure 5.3 Infinitely large plate with a central crack of length 2a. SIF A Q1 Qax gx,a πa a-x = = (5.3a) SIF B Q1 gx,a πa ax= = Qa-x (5.3b) When it is not possible to obtain such functions analytically, numerical methods based on finite elements are often used. An example of determining SIF
Weight function method, suggested by Bueckner [4] and Rice [5], is one of the most effective tools of stress intensity factor determination in cases of continuous load symmetrically distributed along both sides of the crack.
dure for calculating stress intensity factors and crack opening dis-placements of cracks subjected to nonlinear stress distributions such as those in autofrettaged cylinders or near notches is dis-cussed below. Stress Intensity Factors and Weight Functions Most of the existing methods of calculating stress intensity fac-tors require separate analysis of each load and geometry configu-ration
useful and versatile method of calculating stress intensity factors for cracks subjected to non-uniform stress fields, such as residual stress or thermal loading. Using the weight function …
Fracture Mechanics Lecture notes – course 4A780 Concept version Dr.ir. P.J.G. Schreurs Eindhoven University of Technology Department of Mechanical Engineering
Stress intensity solutions for semi-infinite, part-throughwall flaws in thin and thick walled cylindrical vessels can be obtained by various methods, including weight functions and finite-element analysis.
The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting
one reference stress intensity factor solution. This method was used to derive the weight functions for This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body.
and [[ weight functions were derived from corresponding reference displacement fields and stress intensity factors calculated by finite element method. Normalized mode ~ and mode ]] stress intensity factors…
determine the stress intensity factor for the other load system Q2′ Of course, the 1 and 2 systems may represent any arbitrarily chosen load systems and thus it is being shown that if a solution for the displacement field and stress intensity factor is known for any particular
3-D Weight Function Method with Stresses from Finite Element Method W. Zhao stress intensity factors for semi-elliptical surface cracks emanating from a circular hole are determined. The 3-D weight function method with the 3-D finite element solutions for the uncracked stress distribution as in Part-I is used for the analysis. Two different loading conditions, i.e. remote tension and wedge

Weight functions for the determination of stress intensity
Stress intensity factors and weight functions in

Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .
Stress intensity factors in crack closure problems ENESCU IOAN Department of Mechanical Engineering Transylvania University of Brasov 500036 Bvd.
Weight function and stress intensity factor for a semi-elliptical surface saddle crack in a tubular welded joint E Chang* and W D Dover Department of Mechanical Engineering, University College London, London, UK
A STRESS 3 INTENSITY FACTOR SOLUTION INSPIRED BY SOAP BUBBLES FRAMEWORK Fig. 2. Discretized values seen as a mesh. 2.2 1D shape functions The 2D shape functions …
the calculation of K1c (Chap. 3) stress intensity factor solutions for these speci­ mens are necessary. Also knowledge of weight functions is of interest, e.g. for the
The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of a planer surface crack and an embedded crack subjected to fluctuating non-linear stress.

oTWO DIMENSIONAL STRESS INTENSITY FACTOR SOLUTIONS FOR
Weight Functions and Stress Intensity Factors for Axial

3-D Weight Function Method with Stresses from Finite Element Method W. Zhao stress intensity factors for semi-elliptical surface cracks emanating from a circular hole are determined. The 3-D weight function method with the 3-D finite element solutions for the uncracked stress distribution as in Part-I is used for the analysis. Two different loading conditions, i.e. remote tension and wedge
Weight function and stress intensity factor for a semi-elliptical surface saddle crack in a tubular welded joint E Chang* and W D Dover Department of Mechanical Engineering, University College London, London, UK
The weight functions h and t can be interpreted as the stress intensity factor and as the T-term for a pair of single forces P acting at the crack face at the location x 0 (Fig.3.2), i.e. the weight functions (h, t) are Green’s functions for K I and T.
Approximate weight functions derived from stress intensity factor solutions; Weight functions based on BCM computations; Round-CT-specimen Chapter 10 Cracks in front of internal notches
Evaluating Stress Intensity Factors due to Weld Residual Stresses by the Weight Function and Finite Element Methods Rui Bao1, Xiang Zhang2*, Norvahida Ahmad Yahaya2 1Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China 2Department of Aerospace Engineering, School of Engineering, Cranfield University, Bedford, MK43 0AL, U.K
Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .

Weight functions for the determination of stress intensity
Unitary weight functions for semi-infinite ABSTRACT

Approximate weight functions derived from stress intensity factor solutions; Weight functions based on BCM computations; Round-CT-specimen Chapter 10 Cracks in front of internal notches
Stress intensity solutions for semi-infinite, part-throughwall flaws in thin and thick walled cylindrical vessels can be obtained by various methods, including weight functions and finite-element analysis.
stress eld is by using the weight function for a given cracked body. The weight function The weight function represents the SIF induced by a unit concentrated load.
The stress intensity factors are calculated from equation (3.4) by integration of the product of the weight function (3.2) and the stress distribution given by equation (3.22)
useful and versatile method of calculating stress intensity factors for cracks subjected to non-uniform stress fields, such as residual stress or thermal loading. Using the weight function …
This page provides stress intensity factor solutions for common cases. Contents
the calculation of K1c (Chap. 3) stress intensity factor solutions for these speci­ mens are necessary. Also knowledge of weight functions is of interest, e.g. for the
A weight function technique is used to obtain mode I stress intensity factor solutions for radially cracked rings loaded with arbitrary crack face pressure. When the crack face pressure is defined as the hoop stress occurring in an
3 a v K G h a x 1 1 1 1 8, (2) The Green’s function is an earlier approach that is similar to the weight function method [13].
The stress intensity factors obtained by the present numerical approach are compared with analytical solutions. The errors in the stress intensity factors for opening fracture mode I are less than 1% although the model mesh is relatively coarse. Key words: Element free Galerkin method, two dimensional elasticity problems, Fracture mechanic, Stress intensity factors. 1. Introduction The
Since the stress intensity factor (K) is the foundation of fracture mechanics of aircraft structuresand damage tolerance , a analysis significant focus of development efforts in the past years has been geared towards fifteen enhancing legacy solutions and developing new and effK icient numerical K solutions that can handle the complicated stress gradients nalysts using detailed finite computed
In contrast with BS 7910, where solutions are presented in terms of bending and membrane stress only, many of the R6 solutions are presented in terms of weight functions, allowing stress intensity factors to be evaluated for arbitrary stress fields. R6 provides valuable critical comment on the accuracy of solutions, sometimes citing (but not necessarily incorporating) solutions from other

lecture 16 Stanford University
Weight functions for Tstress for semi-elliptical surface

Wang, X., Bell, R. Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates subject to non-uniform stress distributions Engng Fracture Mechanics, 2004, 71, 1477 – 1496.
determine the stress intensity factor for the other load system Q2′ Of course, the 1 and 2 systems may represent any arbitrarily chosen load systems and thus it is being shown that if a solution for the displacement field and stress intensity factor is known for any particular
11/8/00 ME111 Lecture 16 10 • The units of the stress intensity factor are, for examplK e, MPa m, or • The stress intensity factor describes the state of stress near a crack tip. K • It is found experimentally that existing cracks will propagate (I.e. grow) when the stress intensity factor reaches a critical valu e called the fracture toughness. K • The fracture toughness is
the calculation of K1c (Chap. 3) stress intensity factor solutions for these speci­ mens are necessary. Also knowledge of weight functions is of interest, e.g. for the
stress intensity factors [9-13].One of these methods is a numerical method like Green’s function, weight functions, boundary collocation, alternating method, integral transforms,
Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .
A STRESS 3 INTENSITY FACTOR SOLUTION INSPIRED BY SOAP BUBBLES FRAMEWORK Fig. 2. Discretized values seen as a mesh. 2.2 1D shape functions The 2D shape functions …
Evaluating Stress Intensity Factors due to Weld Residual Stresses by the Weight Function and Finite Element Methods Rui Bao1, Xiang Zhang2*, Norvahida Ahmad Yahaya2 1Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China 2Department of Aerospace Engineering, School of Engineering, Cranfield University, Bedford, MK43 0AL, U.K
using the J-integrals, dynamic stress intensity factors are calculated. Numerical results including the values of Numerical results including the values of dynamic stress intensity factors for a crack in an infinite domain subjected to P and SV waves are presented.
Stress intensity solutions for semi-infinite, part-throughwall flaws in thin and thick walled cylindrical vessels can be obtained by various methods, including weight functions and finite-element analysis.
3 a v K G h a x 1 1 1 1 8, (2) The Green’s function is an earlier approach that is similar to the weight function method [13].
5.1 Stress intensity factor force Q, the closed form solution for the weight functions are, Figure 5.3 Infinitely large plate with a central crack of length 2a. SIF A Q1 Qax gx,a πa a-x = = (5.3a) SIF B Q1 gx,a πa ax= = Qa-x (5.3b) When it is not possible to obtain such functions analytically, numerical methods based on finite elements are often used. An example of determining SIF
The weight function method is one of the most reliable, versatile, and cost-effective methods of evaluating the stress intensity factors and crack opening displacements. This book provides a valuable account of the author’s research in these fields. It has two aims: firstly to provide a theoretical background to the weight function method in fracture mechanics for accurate analysis of two
As an example, a weight function for the SEN strip is obtained in this manner. Moreover, closed form infinite body weight functions are also developed and used to derive some well-known stress intensity factor …
a two-dimensional weight function approach is used to determine stress intensity factors for cracks in either tensile or compressive stress fields, due to one of three mechanisms: remote tension overload, remote compression overload or hole cold expansion.

CALCULATION OF DYNAMIC STRESS INTENSITY FACTORS IN
National Conference on Innovative Paradigms in Engineering

COMPUTATION OF THE WEIGHT FUNCTION FROM A STRESS INTENSITY FACTOR” . by H. J. Petroski and J. D. Achenbach 4 SI’ One obstacle to the direct use of Eq. 1 is that the solutions for the stress intensity factor that are available in the literature to serve as reference data are often not accompanied by data for crack-face displacements. To overcome this obstacle, a simple method has …
one reference stress intensity factor solution. This method was used to derive the weight functions for This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body.
The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of a planer surface crack and an embedded crack subjected to fluctuating non-linear stress.
Analytical Solutions for Stress Intensity Factor, T-Stress and Weight Function for the Edge-Cracked Half-Space
Stress intensity factors in crack closure problems ENESCU IOAN Department of Mechanical Engineering Transylvania University of Brasov 500036 Bvd.
Weight functions and stress intensity factor solutions / Xue-Ren Wu and A. Janne Carlsson.
useful and versatile method of calculating stress intensity factors for cracks subjected to non-uniform stress fields, such as residual stress or thermal loading. Using the weight function …
Evaluating Stress Intensity Factors due to Weld Residual Stresses by the Weight Function and Finite Element Methods Rui Bao1, Xiang Zhang2*, Norvahida Ahmad Yahaya2 1Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China 2Department of Aerospace Engineering, School of Engineering, Cranfield University, Bedford, MK43 0AL, U.K
Weight Function for Stress Intensity Factors in Rotating Thick-Walled Cylinder 31 form of polynomial expression, which is fit for calculation in weight function method.
and [[ weight functions were derived from corresponding reference displacement fields and stress intensity factors calculated by finite element method. Normalized mode ~ and mode ]] stress intensity factors…

Boundary Weight Functions for Cracks in Three-Dimensional
Determination of approximate point load weight functions

Weight functions allow for the determination of stress intensity factors K and T-stresses T under various crack surface tractions. Whereas the mode-I stress intensity factor in most cases is determined by the normal tractions only and the mode-II stress intensity factor …
Geometric functions of stress intensity factor solutions for spot welds in U-shape specimens P.-C. Lina,*, D.-A. Wangb a Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi …
Stress intensity solutions for semi-infinite, part-throughwall flaws in thin and thick walled cylindrical vessels can be obtained by various methods, including weight functions and finite-element analysis.
and [[ weight functions were derived from corresponding reference displacement fields and stress intensity factors calculated by finite element method. Normalized mode ~ and mode ]] stress intensity factors…

A STRESS INTENSITY FACTOR SOLUTION INSPIRED BY SOAP
Weight functions for cracks in finite rectangular plates

are presented in terms of weight functions, allowing stress intensity factors to be evaluated for arbitrary stress fields. Smith [A.13] has compared R6 K-solutions for cylinders with those of other procedures; consequently
Wang, X., Bell, R. Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates subject to non-uniform stress distributions Engng Fracture Mechanics, 2004, 71, 1477 – 1496.
The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress
is consistent with the stress intensity factor derived for the non-symmetric distribution of forces obtained by Piccolroaz et al. (2009). Finally, in Appendix A, the evaluated skew-symmetric weight function is compared to those

Weight Function Approach to Study a Crack Propagating
Weight Functions and Stress Intensity Factors for Axial

using the J-integrals, dynamic stress intensity factors are calculated. Numerical results including the values of Numerical results including the values of dynamic stress intensity factors for a crack in an infinite domain subjected to P and SV waves are presented.
Weight functions allow for the determination of stress intensity factors K and T-stresses T under various crack surface tractions. Whereas the mode-I stress intensity factor in most cases is determined by the normal tractions only and the mode-II stress intensity factor …
Weight functions and stress intensity factor solutions / Xue-Ren Wu and A. Janne Carlsson.
The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of a planer surface crack and an embedded crack subjected to fluctuating non-linear stress.
The general 2-D weight function accounting for the free boundary effect was found to be: Weight Functions and Stress Intensity Factors NUMERICAL TECHNIQUE The stress intensity factor due to a continuous stress field applied to the crack surface is calculated by integrating the product (5) of the weight function and the stress field over the entire crack area. In the case of numerical

WEIGHT FUNCTION FOR STRESS INTENSITY FACTORS IN
TWI Compilation of stress intensity factor and load

Stress intensity factor solutions for the ring-shaped crack in an infinite body loaded by constant stress normal to the crack plane (Fig. 2.1b) were compiled by Rosenfelder [1]. For an arbitrarily given stress distribution σ( x ) in the uncracked body normal to the crack
A STRESS 3 INTENSITY FACTOR SOLUTION INSPIRED BY SOAP BUBBLES FRAMEWORK Fig. 2. Discretized values seen as a mesh. 2.2 1D shape functions The 2D shape functions …
The general 2-D weight function accounting for the free boundary effect was found to be: Weight Functions and Stress Intensity Factors NUMERICAL TECHNIQUE The stress intensity factor due to a continuous stress field applied to the crack surface is calculated by integrating the product (5) of the weight function and the stress field over the entire crack area. In the case of numerical
Weight function method, suggested by Bueckner [4] and Rice [5], is one of the most effective tools of stress intensity factor determination in cases of continuous load symmetrically distributed along both sides of the crack.
Since the stress intensity factor (K) is the foundation of fracture mechanics of aircraft structuresand damage tolerance , a analysis significant focus of development efforts in the past years has been geared towards fifteen enhancing legacy solutions and developing new and effK icient numerical K solutions that can handle the complicated stress gradients nalysts using detailed finite computed
5.1 Stress intensity factor force Q, the closed form solution for the weight functions are, Figure 5.3 Infinitely large plate with a central crack of length 2a. SIF A Q1 Qax gx,a πa a-x = = (5.3a) SIF B Q1 gx,a πa ax= = Qa-x (5.3b) When it is not possible to obtain such functions analytically, numerical methods based on finite elements are often used. An example of determining SIF
Stress intensity factors along corner crack fronts at the rivet-filled dimpled hole are systematically studied for different crack length a , elliptical shape factor t ,and far-end stress S .
In this investigation, the weight function method was employed to calculate stress intensity factors for semi-elliptical surface crack in a hollow cylinder. The weight functions at …