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.     

Evaluation of T‐stresses in multiple crack problems of finite plate

This paper provides a solution for T‐stresses for multiple cracks in a finite plate. The results for stress intensity factors (SIFs) are also presented. The case of two cracks in a rectangular plate is taken as an example. In the problem, the crack faces are applied by some loadings, and tractions are free along edges of a rectangular plate. The whole stress field is considered as a superposition of three particular stress fields. The first and second stress fields are initiated by loadings on the first and second crack faces in an infinite plate. The third field is chosen in a polynomial form of complex potentials. After discretization, the loadings on two cracks and the undetermined coefficients in the complex potentials become the unknowns. The relevant algebraic equations are formulated. The solution of algebraic equations will lead to the results of SIFs and T‐stresses at the crack tips. Several numerical examples are presented, which were not reported previously.     

On the computation of the pure Neumann problem in 2-dimensional elasticity

The accurate computation of stress intensity factors (SIFs) plays a decisive role in the determination of crack paths. The calculation of SIFs with the help of singular weight functions leads to pure Neumann problem for anisotropic elasticity in a plane domain with a crack. Here a method is presented to overcome the specific numerical difficulties which arises while calculating these solutions with Finite Element methods. The accuracy and advantage of this method are shown by a numerical example, the calculation of SIFs of a compact tension specimen.      

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Stress intensity factors for cracks emanating from a circular hole in an infinite quasi‐orthotropic plane

An infinite quasi‐orthotropic plane with a cracked circular hole under tensile loading at infinity is studied analytically. To this end, complex variable theory of Muskhelishvili is used. In addition, to obtain analytical functions, a new conformal mapping is proposed and expanded to series expressions. Stress intensity factors (SIFs) for two unequal cracks emanating from a circular hole are obtained. To validate the analytical SIFs in a quasi‐orthotropic plane, the results are compared with FEM and the results of isotropic plane. The SIFs for small cracks in a quasi‐orthotropic and an isotropic plane are different, because of difference between stress concentrations in points which cracks emanate from the hole. However, the results of quasi‐orthotropic plane converge to isotropic plane for the large cracks. Therefore, the SIFs of the large cracks in a quasi‐orthotropic plane can be replaced by the results of the center crack with equivalent length in an isotropic plane.     

Stress intensity factors for interacting cracks and complex crack configurations in linear elastic media

In this paper, an effective numerical method for analyzing interacting multiple cracks and complex crack configurations in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple-crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by taking into account the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are included to illustrate that the method is very simple and effective for analyzing interacting multiple cracks and complex crack configurations in an infinite linear elastic media under remote uniform stresses. Specifically, analysis of perpendicular cracks under general in-plane loading is performed using the numerical approach and many numerical results are given in the form of tables.     

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.     

A rectangular crack in an infinite solid,a semi-infinite solid and a finite-thickness plate subjected to tension

This paper is concerned with the tension problems of a semi-infinite solid with an embedded rectangular crack and a rectangular surface crack as well as an infinite solid with a rectangular crack as a special case. The analyses are based on the body force method, and are performed by generalizing the previous analyses of an embedded elliptical crack and a semi-elliptical surface crack. Furthermore, approximate results are given for a finite-thickness plate with an embedded rectangular crack, by superposing the effects of the free surfaces obtained in the above analysis for a semi-infinite solid.     

Complex fracture parameters for an interface crack between twohardening materials: a photoelastic study

In this paper, complex stress intensity factors (SIFs) at an interface crack are determined for a range of applied loads, crack lengths and remote mode mixes using automated photoelasticity. The specimen geometries comprise epoxy resin and aluminium alloy halves bonded together, and are loaded in either compact tension in mixed‐mode conditions or in three‐point bend under mode I conditions. In the experiments, full‐field isochromatic data were obtained from the epoxy half using an established phase‐stepping technique. A reworked approach to the determination of the SIFs was developed by combining a least‐squares over‐deterministic method for fitting crack‐tip stress equations to the data and a weighting factor that ensures that only data in the singularity zone are used. For comparison, some of the specimens were tested using a linear‐elastic finite element (FE) analysis and/or by experiment using homogeneous test specimens. Excellent agreement between the experimental and numerical SIF moduli was achieved for remote mode I loadings. However, for good agreement to be made between the phase angle results requires an additional phase term to be added to the FE solution at each load to account for the development of a crack‐tip plastic zone. Further, results for the SIFs from remote mixed‐mode loadings of the compact tension specimen only have a meaningful interpretation in light of small‐scale yielding conditions. It is shown, qualitatively, that the experiments verify some of the predictions made in the literature of asymptotic behaviour at interface crack tips from results of elasto‐plastic FE analyses.     

Computation of the stress intensity factors of sharp notched plates by the fractal‐like finite element method

The fractal‐like finite element method (FFEM) is an accurate and efficient method to compute the stress intensity factors (SIFs) of different crack configurations. In the FFEM, the cracked/notched body is divided into singular and regular regions; both regions are modelled using conventional finite elements. A self‐similar fractal mesh of an ‘infinite’ number of conventional finite elements is used to model the singular region. The corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co‐ordinates after performing a global transformation by using global interpolation functions. In this paper, we extend this method to analyse the singularity problems of sharp notched plates. The exact stress and displacement fields of a plate with a notch of general angle are derived for plane‐stress/strain conditions. These exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the SIFs. The use of the global interpolation functions reduces the computational cost significantly and neither post‐processing technique to extract SIFs nor special singular elements to model the singular region are needed. The numerical examples demonstrate the accuracy and efficiency of the FFEM for sharp notched problems. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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 unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

Closure of circular arc cracks under general loading: effects on stress intensity factors

The two-dimensional circular arc crack solution of Muskhelishvili (Some basic problems of the mathematical theory of elasticity, P. Noordhoff Ltd, Groningen, Holland, 1953) has been used widely to study curved crack behavior in an infinite, homogeneous and isotropic elastic material. However, for certain orientations and magnitudes of the remotely applied loads, portions of the crack will close. Since the analytical solution is incorrect once the crack walls come into contact, the displacement discontinuity method is combined with a complementarity algorithm to solve this problem. This study uses stress intensity factors (SIFs) and displacement discontinuities along the crack to define when the analytical solution is not applicable and to better understand the mechanism that causes partial closure under various loading conditions, including uniaxial tension and pure shear. Closure is mainly due to material from the concave side of the crack moving toward the outer crack surface. Solutions that allow interpenetration of the crack tips yield non-zero mode I SIFs, while crack tip closure under proper contact boundary conditions produce mode I SIFs that are identically zero. Partial closure of a circular arc crack will alter both mode I and II SIFs at the crack tips, regardless of the positioning or length of the closed section along the crack. Friction on the crack surfaces in contact changes the total length and positioning of closure, as well as generally decreases the magnitude of opening along the portions of the crack that are not closed.     

Two-dimensional finite element method for stress intensity factor using adaptive mesh strategy

Finite element analysis (FEA) combined with the concepts of Linear Elastic fracture mechanics (LEFM) provides a practical and convenient means to study the fracture and crack growth of materials. A numerical analysis (FEM) of cracks was developed to derive the SIF for two different geometries, i.e., a rectangular plate with half circle-hole and central edge crack plate in tension loading conditions. The onset criterion of crack propagation is based on the stress intensity factor, which is the most important parameter that must be accurately estimated and facilitated by the singular element. Displacement extrapolation technique (DET) is employed, to obtain the stress intensity factors (SIFs) at crack tip. The fracture is modeled by the splitting node approach and the trajectory follows the successive linear extensions of each crack increment. These comprehensive tests are evaluated and compared with other relevant numerical and analytical results obtained by other researchers.     

Thermal striping fatigue damage in multiple edge-cracked geometries

Thermal fatigue striping damage may be caused when incompletely mixed hot and cold fluid streams pass over the surface of a component or structure containing a defect. Stress intensity factor (SIF) fluctuations are developed in response to the surface temperature fluctuations. An existing methodology for the analysis of striping damage in geometries containing a single edge‐crack geometry is extended to such an analysis of multiple edge cracks. SIFs are calculated as functions of crack depth, when an edge‐cracked plate and semi‐infinite solid, each containing multiple cracks, are subjected to thermal striping. The effect of various restraint conditions and striping frequencies on the SIF values for a stainless steel plate is examined. The degree of conservatism is shown when an assessment of thermal fatigue striping damage is based on a single, rather than multiple, crack analysis. Accurate curve fits are developed resulting in practical weight functions for an edge‐cracked plate and semi‐infinite solid.     

无限大平面中斜裂纹的压剪断裂分析

根据Muskhelishvili的复势理论,结合裂面边界条件和位移单值条件,将无限大平面受压应力作用的裂纹问题转化为对应的Hilbert问题,并运用复变函数法分别给出了在伪集中力作用下,不同裂面形态的基本解。对不同裂面形态的摩擦力大小和分布进行了详细分析,建立了新的摩擦力计算模型。采用“伪力法”和叠加原理,结合所求的基本解,给出了含中心斜裂纹的岩石类材料在压缩荷载作用下的应力强度因子(SIF)的解法。研究表明:裂面状态对KⅠ的大小没有影响,而对KⅡ的影响却很大,相同应力条件下,裂面状态会影响裂纹的开裂角和开裂方式。     

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.     

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.