A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIFs (stress intensity factors) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral term in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIFs using only one SIF solution, without any modification for the crack geometries symmetric in the two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential cracks or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straightforward with only one SIF solution. The necessary information in the analysis is two reference SIFs. The analysis results using several examples verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.     

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.     

Stress intensity factors for cracks in bolted joints

The mixed-mode stress intensity factors (SIFs) of the bolted joint with single and double cracks were examined. Changes in friction, clearance, applied force and crack angle were included in the nonlinear contact finite element analysis. A fine mesh was made between the contact surface and the crack tip in order to obtain an accurate solution. The least-squares method was used to determine the mixed-mode SIFs. Finite element results indicate that reasonable changes in the applied force, frictional coefficient and the clearance will not make significant changes in the normalized SIFs. The pure opening mode for cracked bolted joints does not occur at the horizontal crack but occurs at the crack with the crack angle between 0° and 22.5°. Nevertheless, using the SIF for a horizontal crack as the maximum opening-crack mode is sufficiently reliable. The maximum mode-II crack is approximately at a crack angle of 45° for both isotropic and orthotropic materials; however, at that angle the maximum mode-II SIF is only about one half of the mode-I SIF. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Stress intensity factors of square crack inclined to interface of transversely isotropic bi-material

This paper analyzes a square crack in a transversely isotropic bi-material solid by using dual boundary element method. The square crack is inclined to the interface of the bi-material. The fundamental solution for the bi-material solid occupying an infinite region is incorporated into the dual boundary integral equations. The square crack can have an arbitrary angle with respect to the plane of isotropy of the bi-material occupying either finite or infinite regions. The stress intensity factor (SIF) values of the modes I, II, and III associated with the square crack are calculated from the crack opening displacements. Numerical results show that the properties of the anisotropic bi-material have evident influences on the values of the three SIFs. The values of the three SIFs are further examined by taking into account the effect of the external boundary of the internally cracked bi-material.     

Three-dimensional stress intensity factors of a central square crack in a transversely isotropic cuboid with arbitrary material orientations

In this paper, we present the dual boundary element method (dual-BEM) or single-domain BEM to analyze the mixed three-dimensional (3D) stress intensity factors (SIFs) in a finite and transversely isotropic solid containing an internal square crack. The planes of both the transverse isotropy and square crack can be oriented arbitrarily with respect to a fixed global coordinate system. A set of four special nine-node quadrilateral elements are utilized to approximate the crack front as well as the outer boundary, and the mixed 3D SIFs are evaluated using the asymptotic relation between the SIFs and the relative crack opening displacements (COD) via the Barnett–Lothe tensor.Numerical examples are presented for a cracked cuboid which is transversely isotropic with any given orientation and is under a uniform vertical traction on its top and bottom surfaces. The square crack is located in the center of the cuboid but is oriented arbitrarily. Our results show that among the selected material and crack orientations, the mode-I SIF reaches the largest possible value when the material inclined angle ψ₁=45° and dig angle β₁=45°, and the crack inclined angle ψ₂=0° and dig angle β₂=0°. It is further observed that when the crack is oriented vertically or nearly vertically, the mode-I SIF becomes negative, indicating that the crack closes due to an overall compressive loading normal to the crack surface. Variation of the SIFs for modes II and III along the crack fronts also shows some interesting features for different combinations of the material and crack orientations.     

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.     

Fracture analysis of semi-circular and semi-circular-bend geometries

A fracture mechanics analysis of the semi-circular (SC) and semi-circular-bend (SCB) fracture geometries is presented. The weight function method is implemented to obtain wide ranging stress intensity factor (SIF) and crack opening displacement (COD) expressions. This study has as its basis a finite element analysis of the semi-circular disk (SC) subjected to a reference loading case. The latter is required to determine both the associated reference stress intensity factor and the weight function for the base-edge-cracked semi-circular geometry. With this information, SIF and COD expressions for the full range of crack lengths are obtained. The special cases of the SC subject to a concentrated crack mouth loading and the SCB are analyzed in detail. The weight function for the SCB is fully developed, with an accurate expression for the SIF and and a numerical result for the crack mouth opening displacement (CMOD). The latter wide ranging expressions can, in turn, be applied as a reference solution. From this weight function approach, SIF's and COD's for the SC and SCB subject to any other loading can be obtained.     

Effect of Reference Loading and Crack Configuration on the Stress Intensity Factors In Weight Function Method

In calculating the stress intensity factors (SIFs) using the weight function method, the accuracy of the result depends upon proper selection of the set of reference loadings. The objective of the present work is to find a unique set of reference loadings to calculate the SIF at the tip of a crack of any configuration. Additionally, the universality of these reference loadings with respect to calculating the SIF for various crack configurations is examined. Two sets of crack configurations were considered, with and without a pre-existing crack. In each set, a horizontal crack and a slanted crack were analyzed.     

A numerical procedure for the approximate WF solution of mode I SIFs in 3D geometries under arbitrary load

A numerical procedure was developed for the approximate weigth function (AWF) evaluation of reliable stress intensity factor (SIF) for part-through Mode I cracks for general load. Different from other WF procedures which require closed form reference SIFs, this procedure requires only limited number of discrete SIF solutions directly obtained from other numerical methods as reference SIFs to compute continuous SIFs as function of both the crack size and the location along the crack front. As an implement to the general numerical methods in the Damage and Safe Life analysis, this procedure substantially increases the value of numerical SIF results. The present procedure is relative simple, with most of basic relations being analytically soved, and therefore efficient in use. Several examples were presented to demonstrate the accuracy of this procedure.     

External surface cracks in shells under cyclic internal pressure

The stress analysis and fatigue crack growth behaviour of a part‐through‐cracked double‐curvature thin‐walled shell is examined. An external surface crack is assumed to lie in one of the principal curvature planes of the shell, and to present a semi‐elliptical shape. The stress intensity factors (SIFs) along the crack front for different elementary opening stresses acting on the crack faces are determined through a three‐dimensional finite element analysis. Then approximate values of SIF in the case of a cracked pressure vessel are computed by employing the above results together with the superposition principle and the power series expansion of the actual opening stress. Finally, a numerical simulation procedure is carried out to predict the crack growth under cyclic internal pressure. Some results are compared with those of other authors.     

Interaction between a cracked hole and a line crack under uniform heat flux

This article deals with the interaction between a cracked hole and a line crack under uniform heat flux. Using the principle of superposition, the original problem is converted into three particular cracked hole problems: the first one is the problem of the hole with an edge crack under uniform heat flux, the second and third ones are the problems of the hole under distributed temperature and edge dislocations, respectively, along the line crack surface. Singular integral equations satisfying adiabatic and traction free conditions on the crack surface are obtained for the solution of the second and third problems. The solution of the first problem, as well as the fundamental solutions of the second and third, is obtained by the complex variable method along with the rational mapping function approach. Stress intensity factors (SIFs) at all three crack tips are calculated. Interestingly, the results show that the interaction between the cracked hole and the line crack under uniform heat flux can lead to the vanishing of the SIFs at the hole edge crack tip. The fact has never been seen for the case of a cracked hole and a line crack under remote uniform tension.     

On the influence of reference load case on the crack face weight functions

The influence of reference load case on the crack face weight functions for different finite cracked bodies was investigated by examining the Green's functions which are related to the crack face point force loading. The weight functions were analytically determined based on a general crack opening displacement equation. It was found that the degree of reference-load-case-dependence is not only a function of the reference stress itself, but also the crack length, overall cracked configuration and the number of terms contained in the crack opening displacement equation. Some useful guidelines for the choice of reference load cases in generating the weight functions have been introduced, and the way to improve the weight function accuracy at extended crack lengths recommended.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

Fracture Analysis of Double-Side Adhesively Bonded Composite Repairs to Cracked Aluminium Plate Using Line Spring Model

A line spring model is developed for analyzing the fracture problem of cracked metallic plate repaired with the double-sided adhesively bonded composite patch. The restraining action of the bonded patch is modeled as continuous distributed linear springs bridging the crack faces provided that the cracked plate is subjected to extensional load. The effective spring constant is determined from 1-D bonded joint theory. The hyper-singular integral equation (HSIE), which can be solved using the second kind Chebyshev polynomial expansion method, is applied to determine the crack opening displacements (COD) and the crack tip stress intensity factors (SIF) of the repaired cracked plate. The numerical result of SIF for the crack-tip correlates very well with the finite element (FE) computations based on the virtual crack closure technique (VCCT). The present analysis approaches and mathematical techniques are critical to the successful design, analysis and implementation of crack patching.     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.     

Determination of crack-tip stress intensity factors in complex geometries by the composition of constituent weight function solutions

Linear elastic fracture mechanics (LEFM) is the science frequently used to understand the stable and progressive fatigue crack growth that often occurs in engineering components under varying applied stress. The stress intensity factor (SIF) is its basis and describes the stress state at the crack tip. This can be used with the appropriate material properties to calculate the rate at which the crack will propagate in a linear elastic manner. Unfortunately, the SIF is difficult to compute or measure, particularly if the crack is situated in a complex three‐dimensional geometry or subjected to a non‐simple stress state. This is because the SIF is not only a function of the crack and component geometry but is also dependent on the applied stress field. In the last 20 years, the SIF weight function has gained prominence as a method for calculating and presenting SIFs independent of applied stress. This paper demonstrates that the real promise of the SIF weight Function lies in its use to rapidly generate SIF solutions for cracks in complex geometries by simple composition of geometric influences from reference constituent solutions.     

Approximate weight function methods using a new condition

A method is presented to derive the approximate weight functions, by using a new condition on the crack mouth, for edge cracks in the semi-infinite or finite plate under Mode I loadings. This method requires only one information, say, the reference stress intensity factor (SIF), K_r. Numerical examples show that the present method is efficient in evaluation of the SIFs for edge cracks subjected to the polynomial loadings on the crack faces.     

DBEM and FEM analysis on non-linear multiple crack propagation in an aeronautic doubler-skin assembly

The performance of a riveted patch repair, applied on a cracked panel, is simulated by using both a commercially available boundary element code (BEASY) and a finite element code (ANSYS). A two-dimensional stress analysis on a single-sided repaired configuration is performed by both methodologies; consequently, the occurrence of out-of-plane bending and its effect on the through-thickness stress intensity factor (SIF) variation is neglected. The connection between the two layers (patch and panel) is realised by 32 rivets, with through-cracks initiated on the most loaded holes. Special elements are used to model the crack: discontinuous elements in the dual boundary element method (DBEM) approach or quarter point elements in the finite element method (FEM) approach. Different loading configurations are considered depending on the presence of a biaxial or uniaxial remote load and the non-linear hole/rivet contact is simulated by gap elements. The most stressed skin holes are highlighted, and the effect of a through crack from such holes is analysed in terms of SIFs and stress redistribution. The accuracy in SIFs assessment by DBEM and FEM and the respective computational and pre-processing efforts are determined. Such a two-dimensional analysis allows us a straightforward pre-processing phase, and very short run times are needed. A peculiar arrangement of the pin configuration in the DBEM analysis allows us to take into account the real in-plane plate stiffness and the transversal pin stiffness, even in a 2D analysis (this is straightforward by using FEM).     

基于断裂力学的裂隙岩体强度分析初探

基于断裂力学讨论裂隙岩体的强度一直是断裂力学与岩石力学等学科的重要课题。通过分析单个线性裂纹在压缩载荷作用下的剪切断裂条件,明确了压缩状态下张破裂剪切断裂韧性的物理意义及求解办法。使用复变函数和边界配置法求解了裂尖应力强度因子无量纲系数,进而对张破裂压剪准则进行了改进,使之更加便捷地进行有限裂纹体断裂的预测。基于修正的压剪判据推导了裂隙岩体试件的抗压强度求解公式,算例表明该文方法所建立的强度公式是准确、可靠的。最后,详细讨论了摩擦系数、裂隙倾角、裂隙长度、黏聚力以及围压对裂隙岩体试件抗压强度的影响机理。