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.     

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.     

Cracks emanating from a rhombus hole in infinite and finite plate subjected to internal pressure

This note deals with the stress intensity factors (SIFs) of cracks emanating from a rhombus hole in a rectangular plate subjected to internal pressure by means of the displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author. Moreover, an empirical formula of the SIFs of the crack problem is presented and examined. It is found that the empirical formula is very accurate for evaluating the SIFs of the crack problem.     

Rectangular Tensile Sheet with Double Edge Notch Cracks

This paper deals with the rectangular tensile sheet with symmetric double edge notch cracks. Such a crack problem is called an edge notch crack problem for short. By using a hybrid displacement discontinuity method (a boundary element method), two edge notch models are analyzed in detail. By changing the geometrical forms and parameters of the edge notch, and by comparing the stress intensity factors (SIFs) of the edge notch crack problem with those of the double edge cracked plate tension specimen (DECT), which is a model frequently used in fracture mechanics, the effect of the geometrical forms and parameters of the edge notch on the SIFs of the DECT specimen, is revealed. Some geometric characterestic parameters are introduced here, which are used to formulate the notch length and the branch crack length, which are to be determined in mechanical machining of the DECT specimen So we can say that the geometric characterestic parameters and the formulae used to determine the notch length and the branch crack length presented in this paper perhaps have some guidance role for mechanical machining of the DECT specimen.     

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.     

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).     

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.     

Evaluating stress intensity factors of a surface crack in lubricated rolling contacts

The three-dimensional finite element method and the least-squares method were used to find the stress intensity factors (SIFs) of a surface crack in a lubricated roller. A steel roller on a rigid plane was modeled, in which a semi-elliptical surface crack is inclined at an angle ψ to the vertical axis. A distance c is set between the crack base and the roller edge. The results indicate that the mode-I SIF reaches the maximum value when the angle θ is equal to 0° (on the roller surface), and the mode-II SIF reaches the absolute maximum value when the angle θ is near or equal to 90° (inside the roller), where θ is the angle of the semi-ellipse from 0° to 180°. The influence of mode-III SIFs in this model is minor since they are much smaller than the mode-I and mode-II SIFs. The SIFs increase greatly when the crack location approaches the uncrowned edge. At this time, a crowned profile can be used to significantly reduce the SIFs near the roller edge. This revised version was published online in July 2006 with corrections to the Cover Date.     

Comparison of crack growth simulation by DBEM and FEM for SEN-specimens undergoing torsion or bending loading

The 3D fatigue crack growth behaviour in 3PB-specimens or in cantilever beam specimens under, respectively, bending or torsion loading, with inclined planes for the initial crack, is investigated by the dual boundary element method (DBEM). Mixed mode conditions along the crack edge are characterized: stress intensity factors (SIFs) are determined using the J-integral method, and the crack growth direction is computed by the minimum strain energy density criterion. The evolving crack shape (spatially twisted or twisted and warped crack faces), computed by the DBEM model, is successfully compared with experimental findings and FEM results.     

压电弹性体中光滑顶点的正三角形孔边裂纹的反平面问题分析

通过构造新的保角映射,利用复变函数的方法,研究了含光滑顶点的正三角形孔边裂纹的横观各向同性的压电弹性体的反平面问题。在电可穿透和电不可穿透裂纹、孔周及裂纹面为自由表面的假设下,充分利用Cauchy积分公式和复变函数方法,得到了裂纹尖端的场强度因子和能量释放率的表达式。数值算例显示了在不同边界条件下裂纹的几何尺寸、机电载荷对能量释放率和机械应变能释放率的影响规律。结果表明:在电可通和电不可通边界条件下,裂纹长度和三角形边长的增加会导致能量释放率增加,机械载荷则总是促进裂纹的扩展。在电不可通边界条件下电位移可以促进或抑制裂纹的扩展,而在电可通边界条件下电位移对裂纹扩展没有影响。     

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.     

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.     

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.     

An improved technique for the solution of edge crack problem for finite plate

This paper provides a detailed examination for the edge crack problem of finite plate. The Williams expansion for the crack problem is used first. Secondly, the complex potentials for the central crack problem are used in the present study, which is called the improved technique hereafter. In both techniques, the eigenfunction expansion variational method (EEVM) is used for evaluating the undetermined coefficients in the expansion form. The ratio of height versus width of plate (h/w) is varying from 1.5, 1.0, 0.75, 0.5, 0.4, 0.3 to 0.25. The ratio of edge crack length versus width of plate (a/w) takes two sets: (1) a/w = 0.1, 0.2, … to 0.9, (2) a/w = 0.01, 0.02, … to 0.09. The detailed computation proves that for moderate cases of the a/w ratio, for example, 0.2 < a/w < 08, the deviations for SIFs and T-stress from two techniques are minor. However, for the case of short edge crack length, for example, a/w < 0.05, the deviations for SIFs and T-stress from two techniques are significant. It is found that the Williams expansion may not be suitable for the short edge crack, for example, a/w < 0.05.     

SGBEM analysis of crack-particle(s) interactions due to elastic constants mismatch

The effects of elastic constants mismatch on the interaction between a propagating crack and single or multiple inclusions in brittle matrix materials are investigated using numerical simulations. The simulations employ a quasi-static crack-growth prediction tool based upon the symmetric-Galerkin boundary element method (SGBEM) for multiregions, a modified quarter-point crack-tip element, the displacement correlation technique for evaluating stress intensity factors (SIFs), and the maximum principal stress criterion for crack-growth direction. It is shown that, even with this simple method for calculating SIF, the crack-growth prediction tool is both highly accurate and computationally effective. This is evidenced by results for the case of a single inclusion in an infinite plate, where the SGBEM results for the SIFs show excellent agreement with known analytical solutions. The simulation results for crack growth and stress intensity behaviors in particulate media are very stable. The crack-tip shielding and amplification behaviors, as seen in similar studies using other numerical approaches, can be clearly observed.     

A brief note on elastic T-stress for centred crack in anisotropic plate

The stress intensity factors (SIFs) and the T-stress for a planar crack with anisotropic materials are evaluated by the fractal finite element method (FFEM). The FFEM combines an exterior finite element model and a localized inner model near the crack tip. The mesh geometry of the latter is self-similar in radial layers around the tip. A higher order displacement series derived from Laurent series and Goursat functions is used to condense the large numbers of nodal displacements at the inner model near the crack tip into a small set of unknown coefficients. In this study, the variations of the SIFs and the T-stress with material properties and orientations of a crack are presented. The separation of the analytical displacement series into four fundamental cases has shown to be necessary in order to cover all the material variations and the orientations of a crack in the plate with general rectilinear anisotropic materials.     

Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field

An improvement to the extended finite element method (XFEM) and generalised finite element method (GFEM) is introduced. It enriches the finite element approximation of the crack tip node as well as its surrounding nodes with not only the first term but also the higher order terms of the linear elastic crack tip asymptotic field using a partition of unity method (PUM). Numerical results show that together with a reduced quadrature rule to the enriched elements, this approach predicts accurate stress intensity factors (SIFs) directly (i.e. without extra post‐processing) after constraining the enriched nodes properly. However, it does not predict accurately the coefficients of the higher order terms. For that a hybrid crack element (HCE) is introduced which is powerful and convenient not only for directly determining the SIF but also the coefficients of higher order terms in the plane linear elastic crack tip asymptotic field. Finally, the general expressions for the coefficients of the second to fifth terms of the linear elastic crack tip asymptotic field of three‐point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are extended to very deep cracks with the use of the HCE.     

Fatigue crack growth simulation in heterogeneous material using s-version FEM

A fully automatic fatigue crack growth simulation system is developed using the s-version Finite Element Method (s-FEM). This system is extended to fractures in heterogeneous materials. In a heterogeneous material, the crack tip stress field has a mixed-mode condition, and the crack growth path is affected by inhomogeneous materials and mixed-mode conditions. Stress intensity factors (SIFs) in the mixed-mode condition are evaluated using the virtual crack closure method (VCCM). The criteria for the crack growth amount and crack growth path are based on these SIFs, and the growing crack configurations are obtained. At first, the basic problem is solved, and the results are compared with some results available in the literature. It is shown that this system gives an adequate accurate estimation of the SIFs. Then, two-dimensional fatigue crack growth problems are simulated using this system. The first example is a plate with an interface between hard and soft materials. The cracks tend to grow in soft materials through the interface. A second example is a plate with distributed hard inclusions. The crack takes a zig-zag path by propagating around the hard inclusions. In each case, the crack growth path changes in a complicated manner. Changes of the SIFs values are also shown and discussed.     

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.     

Fatigue growth prediction of center slant crack in rectangular plate

This paper presents a numerical prediction model of mixed‐mode crack fatigue growth in a plane elastic plate. It involves a formulations of fatigue growth of multiple crack tips under mixed‐mode loading and a displacement discontinuity method with crack‐tip elements (a boundary element method) proposed recently by Yan is extended to analyse the fatigue growth process of multiple crack tips. Due to an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single‐region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modelled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the present numerical approach is used to analyse the fatigue growth of a centre slant crack in a rectangular plate. The numerical results illustrate the validation of the numerical prediction model and can reveal the effect of the geometry of the cracked plate on the fatigue growth.