Cracks emanating from a square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a square hole in a rectangular plate in tension is performed using a hybrid displacement discontinuity method (a boundary element method). Detailed solutions of the stress intensity factors (SIFs) of the plane elastic crack problem are given, which can reveal the effect of geometric parameters of the cracked body on the SIFs. By comparing the calculated SIFs of the plane elastic crack problem with those of the centre crack in a rectangular plate in tension, in addition, an amplifying effect of the square hole on the SIFs is found. The numerical results reported here also prove that the boundary element method is simple, yet accurate, for calculating the SIFs of complex crack problems in finite plate.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

Determination of crack surface displacements for cracks emanating from a circular hole using weight function method

Cracks emanating from a circular hole are of significant engineering importance, especially in aerospace industry. Accurate determination of key fracture mechanics parameters is essential for damage tolerance design and fatigue life predictions. The purpose of this paper is to provide an efficient and accurate closed‐form weight function approach to the calculation of crack surface displacements for radial crack(s) emanating from a circular hole in an infinite and finite‐width plate. Results were presented for two loading conditions: remote applied stress and uniform stress segment applied to crack surfaces, and extensively compared to recent studies using other methods in the literature. Both single and double radial cracks were considered, and also the effect of finite plate width on crack surface displacements has been investigated. A brief assessment was made on an engineering estimation of displacements based on a correction of stress intensity factor ratio. It has been demonstrated that the Wu‐Carlsson closed‐form weight functions are very efficient, accurate and easy‐to‐use for calculating crack surface displacements for arbitrary load conditions. The method will facilitate fatigue crack closure and other fracture mechanics analyses where accurate crack surface displacements are required.     

Stress intensity factors for cracks in thin plates

This paper presents stress intensity factor solutions for several crack configurations in plates. The loadings considered include internal pressure, and also combined bending and tension. The dual boundary element method is used to model the plate and mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. Several cases including centre crack, edge crack and cracks emanating from a hole in finite width plates are presented.     

Boundary element analysis of cracks at a fastener hole in an anisotropic sheet

Stress intensity factors for cracks emanating from a fastener hole are obtained, using the principle of superposition and a Green's function for a point force applied in an anisotropic sheet with an elliptical hole. Various loading cases, such as a point load, a uniform pressure applied on an arc, and a cosine distribution of pressure acting over half the circumference of the hole are considered to simulate the bolted-joint load. Both cases of a single crack and of two cracks of unequal lengths are studied. The accuracy and simplicity of the technique are demonstrated by comparing numerical results for various loading conditions and crack lengths with their known isotropic counterparts.     

Stress intensity factors for cracks emanating from a triangular or square hole in an infinite plate by boundary elements

This paper concerns stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure calculated by means of a boundary element method, which consists of constant displacement discontinuity element presented by Crouch and Starfied and crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors of plane elasticity crack problems. Specifically, the numerical results of stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure are given.     

Cracks emanating from a circular hole under biaxial load

The principle of linear superposition has been employed in studying cracks emanating from a circular hole in a finite sheet under biaxial loading. The series type analytical solution around the crack tip has been combined with numerical analysis for the purpose of this investigation. The method presented here makes it possible to demonstrate both analytically and numerically, the effects of applied load biaxiality on the stress intensity factor. The ratio of the hole-radius-to-crack-length (R/c) is shown to determine the effect of the applied load biaxiality on the crack emanating from a circular hole in a finite sheet.     

STRESS INTENSITY FACTORS FOR SURFACE CRACKS AT A HOLE BY A THREE-DIMENSIONAL WEIGHT FUNCTION METHOD WITH STRESSES FROM THE FINITE ELEMENT METHOD

Stress intensity factors for semielliptical surface cracks emanating from a circular hole are reported in this paper. The three-dimensional weight function method with three-dimensional finite element solutions for the uncracked stress distribution is used for the analysis. Two different loading conditions, i.e. remote tension and wedge loading, are considered for a wide range of geometrical parameters. Both single and double surface cracks are studied and compared with other solutions available in the literature. Typical crack opening displacements are also provided.     

Cracks emanating from circular hole or square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a circular hole (Fig. 1) or a square hole (Fig. 2) in rectangular plate in tension is performed by means of the displacement discontinuity method with crack-tip elements (a boundary element method) presented recently by the author. Detail solutions of the stress intensity factors (SIFs) of the two plane elastic crack problems are given, which can reveal the effect of geometric parameters of the cracked bodies on the SIFs. By comparing the SIFs of the two crack problems with those of the center crack in rectangular plate in tension (Fig. 3), in addition, an effect of the circular hole or the square hole on the SIFs of the center crack is discussed in detail. The numerical results reported here also illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

Numerical solution for the T-stress in branch crack problem with infinitesimal branch length

This paper investigates the T-stress in a branch crack problem with infinitesimal branch length. The branch crack is composed of a main crack and many branches. The ratio of the lengths for branch to main crack is very small. A singular integral equation method is suggested to solve the problem numerically and the stress intensity factor and T-stress can be evaluated immediately. Many computed results for T-stress under different conditions for branches are presented. It is found from the computed results that the interaction for T-stress among branches is very complicated.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

Weight function,stress intensity factor and crack opening displacement solutions to periodic collinear edge hole cracks

Periodic collinear edge hole cracks and arbitrary small cracks emanating from collinear holes, which are two typical multiple site damages occurred in the aircraft structures, are studied by using the weigh function method. An explicit closed form weight function for periodic edge hole cracks in an infinite sheet is obtained and further used to calculate the stress intensity factor and crack opening displacement for various loading cases. Compared to finite element method, the present weight function is accurate and highly efficient. The interactions of the holes and cracks on the stress intensity factor and crack opening displacement are quantitatively determined by using the present weight function. An approximate weight function method is also proposed for arbitrary small cracks emanating from multiple collinear holes. This method is very useful for calculating the stress intensity factor for arbitrary small cracks.     

Fracture of an Elastic rib Reinforced Plate Weakened by a Circular Cracked Hole

The present paper considers an important for engineering application problem of fracture of an elastic plate weakened by a circular hole with radius and emanating from its countur a crack of length. Refere the plate to the system of coorinates or in complex variables with an origin at the centre of the hole and axis directed along the crack, Fig.1.     

An analysis of cracks emanating from a circular hole in unidirectional fiber-reinforced composites

This paper presents an analytical study of cracks emanating from a circular hole in an off-axis unidirectional fiber-reinforced composite. A convenient and accurate method of analysis is formulated on the basis of conservation laws of elasticity and of fundamental relationships in anisotropic fracture mechanics. The problem is eventually reduced to a system of linear algebraic equations in mixedmode stress intensity factors. Superiority of the current analysis to other approaches in investigating the problem with very complicated crack geometry and material anisotropy is demonstrated when used in conjunction with any numerical method such as a finite element analysis. Mixed-mode stress intensity factors and the associated energy release rates in the crack problem are determined for the composites with various fiber orientations. Solutions for both single and double cracks emanating from the edge of a hole in the composites are presented also to illustrate the fundamental nature of the problem.     

Comments on “Approximate Green’s functions for singular and higher order terms of an edge crack in a finite plate” by Xiao and Karihaloo [Engng Fract Mech 2002;69:959-981]

T-stress dependence on the loading along the crack face is an important topic in fracture analysis. It is found that a statement for T-stress evaluation in Engng Fract Mech 69, 959-981 (2002) is incorrect. A rigorous derivation for T-stress for a semi-infinite crack is presented.     

Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids

Because the elastic T-stress and other coefficients of the higher-order terms play an important role in fracture mechanics such as the stability of crack kinking, crack path, and two-parameter characterization of elastic-plastic crack tip fields, determination of all the coefficients in the crack tip field expansion in an anisotropic linear elastic solid is presented in this paper. Utilizing conservation laws of elasticity and Betti's reciprocal theorem, together with selected auxiliary fields, T-stress and third-order stress coefficients near the crack tip are evaluated first from path-independent line integrals. To determine the T-stress terms using the J-integral and Betti's reciprocal work theorem, auxiliary fields under a concentrated force and moment acting at the crack tip are used respectively. Through the use of Stroh formalism in anisotropic elasticity, analytical expressions for all the coefficients including the stress intensity factors are derived in a compact form that has surprisingly simple structure in terms of one of the Barnett-Lothe tensors, L. The solution forms for degenerated materials, monoclinic, orthotropic, and isotropic materials are also presented.     

Analysis of the crack growth propagation process under mixed-mode loading

In the present paper, a computational model for crack growth analysis under Mode I/II conditions is formulated. The focus is on two issues – crack path simulation and fatigue life estimation. The finite element method is used together with the maximum principal stress criterion and the crack growth rate equation based on the equivalent stress intensity factor. To determine the mixed-mode stress intensity factors, quarter-point (Q-P) singular finite elements are employed. For verification purposes, a plate with crack emanating from the edge of a hole is examined. The crack path of the plate made of 2024 T3 Al Alloy is investigated experimentally and simulated by using the finite element method with the maximum tangential stress criterion. Then, the validation of the procedure is illustrated by applying the numerical evaluation of the curvilinear crack propagation in the polymethyl methacrylate (PMMA) beam and the Arcan specimen made of Al Alloy for which experimental results are available in the literature. In order to estimate fatigue life up to failure of the plate with crack emanating from the edge of a hole, the polynomial expression is evaluated for the equivalent stress intensity factor using values of stress intensity factors obtained from the finite element analysis. Additionally, the fatigue life up to failure of the Arcan specimen is analyzed for different loading angles and compared with experimental data. Excellent correlations between the computed and experimental results are obtained.     

Evaluation of the T-stress in branch crack problem

This paper investigates the T-stress in the branch crack problem. The problem is modeled by a continuous distribution of dislocation along branches, and the relevant singular integral equation is obtained accordingly. After discretization of the singular integral equation, the balance for the number of equations and unknowns is well designed. After the singular integral equation is solved, the equation for evaluating the T-stress is derived. The merit of present study is to provide necessary equation for evaluating T-stress, rather than to provide the integral equation. Many computed results for T-stress under different conditions for branch crack are presented. It is found from the computed results that the interaction for T-stress among branches is complicated.     

STRESS INTENSITY FACTORS FOR CORNER CRACKS AT A HOLE BY A 3-D WEIGHT FUNCTION METHOD WITH STRESSES FROM THE FINITE ELEMENT METHOD

Abstract— Stress intensity factors for quarter-elliptical corner cracks emanating from a circular hole are determined using a 3-D weight function method combined with a 3-D finite element method. The 3-D finite element method is used to analyze uncracked configurations and provide stress distributions in the region where a crack is likely to occur. Using this stress distribution as input, the 3-D weight function method is used to determine stress intensity factors. Three different loading conditions, i.e. remote tension, remote bending and wedge loading, are considered for a wide range of geometrical parameters. The significance of using 3-D uncracked stress distributions is studied. Comparisons are made with solutions available in the literature.     

An empirical formula for stress intensity factors of cracks emanating from a circular hole in a rectangular plate in tension

An empirical formula of the SIFs of cracks emanating from a circular hole in a rectangular plate in tension is presented and examined. It is found that the empirical formula is simple, yet accurate for evaluating the SIFs of the crack problem.