A general weight function for inclined cracks at sharp V-notches

A general method for evaluating the Stress Intensity Factors of an inclined edge crack originated at the tip of a sharp V-notch in a semiplane is presented. An analytical Weight Function with a matrix structure was derived by extending a method developed for an inclined edge crack in an unnotched semi-plane. The effects of the principal geometrical parameters governing the problem were studied through a parametric finite element analysis, carried out for different reference loading conditions. The Weight Function can be used to produce efficient and accurate evaluations of the stress intensity factors for cracks with inclination angle in the range −72°, +72° emanating from V-notches with opening angle in the range from 18° to 144°. For a crack length up to the 10% of the characteristic notch dimension, the maximum estimated error of the Stress Intensity Factor is lower than 2% (typical errors less than 1%) in the whole ranges of the angular parameters. The capability of the proposed method to analyse cracked notches in finite-size bodies was also considered. The agreement between the results with those obtained by accurate Finite Element solutions suggests that the proposed Weight Function can be used as a general tool for evaluating the Fracture Mechanics parameters of a short crack at any V-notch tip.     

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.     

A weight function for 2D subsurface cracks under general loading conditions

A procedure for evaluating the fracture mechanics parameters of a subsurface two-dimensional crack parallel to the boundary in an elastic half plane is presented. A Weight Function (WF) with a matrix structure is proposed, to account for the coupling effects between modes I and II, typical in non-symmetrical problems. In order to face any loading condition, the WF was formulated by symmetric and anti-symmetric components and the ‘multiple reference loading’ approach was used to derive their analytical expressions. To this purpose, a parametric Finite Element (FE) analysis was set up and the Stress Intensity Factors (SIFs) were determined for several independent loading conditions. The analysis was carried out for different ratios between crack length and in-depth position and, consequently, the dependence of the WF on this parameter was studied. The WF accuracy was assessed by considering different loading and the method applied for evaluating the SIFs produced by a point-like load travelling on the semi-plane surface. The results indicated that the correct fracture mechanics analysis requires crack closure (either complete or partial) to be taken into account. Consequently, the crack opening displacement (COD) components under general loading conditions have to be evaluated. On the basis of the WF, the related Green Function (GF) was also derived by which the COD components can be efficiently evaluated for any applied load including the contact due to crack closure.     

Weight Function for an Inclined Edge Crack in a Semiplane

The matrix-like structure of the Weight Function (WF) for determining the Stress Intensity Factors (SIFs) in a nonsymmetric plane body is obtained from the general properties of the elastic field. General asymptotic and symmetric properties of the WF are discussed. By extending a previously proposed methodology, an analytical approximate WF is determined for an edge crack in a semiplane within the range of inclination (-75^–75^). Finite Element evaluations considering the minimum number of loading conditions were performed to this purpose. The accuracy of the SIFs obtained by the WF is found in the order of a few tenths of percent. The solution of a typical problem illustrates the practical usefulness of the WF. This revised version was published online in July 2006 with corrections to the Cover Date.     

Green’s functions for the T-stress of small kink and fork cracks

Green’s functions for T-stresses used so far are restricted to symmetric crack problems, for instance, edge cracks normal to the free surface of a rectangular plate. In the case of more complicated cracks, component geometries, and loading cases, a more general formulation of the Green’s function has to be used. Such a relation was recently proposed by the authors. As an example, Green’s functions for T-stresses acting at the tip of a kinked crack were determined by extending the Cotterell-Rice procedure. Application to the edge-cracked plate revealed an excellent agreement of local T-stress with results obtained directly from FE computations. A simple approximate relation with a high accuracy for kink angles of up to 30° is proposed that allows an estimate of the T-stress from the stress field of the unkinked crack. It is shown that the T-stress of a kinked crack under pure mode-I loading can be computed roughly as the difference of the radial and tangential stresses of the unkinked crack at the tip of the kink.     

A new weight function for one‐dimensional subsurface cracks under general loading

In the present study, weight functions (WFs) of a subsurface crack were derived by proposing a new general form for approximate one‐dimensional WF. The WFs coefficients were considered as a function of crack length to depth ratio and were obtained using reference stress intensity factors (SIFs) of 16 cracks under uniform, linear, and parabolic normal and shearing loadings. The verification was performed by comparison of the straight and coupled SIFs calculated by WF and finite element modelling under some complicated loadings. In conclusion, the WFs can be simply and effectively employed for evaluating the cracks under any complex stress distributions.     

Fatigue cracks emanating from sharp notches in high-strength aluminium alloys: The effect of loading direction, kinking, notch geometry and microstructure

The effect of notch geometry on the propagation of fatigue cracks emanating from sharp V-shaped notches was investigated. To this purpose, an experimental campaign has been conducted on Al-7075–T651 specimens carrying notches with aperture angles of 45°, 90°, and 135°. In order to investigate the role of microstructure texture, specimens were extracted from the plates with the main axis either in the longitudinal rolling direction (L-samples) or in the transversal direction (T-samples), or 45° inclined with respect to both directions (LT-samples). The effect of stress amplitude was investigated by performing tests at two load levels. Three loading directions θ = 0°, 45° and 90° were considered. Some specimens experienced pure Mode I loading condition, whereas the remaining ones were subjected to combined Mode I and Mode II loading condition. The crack deflection induced by the variation in loading direction was determined by measuring the kinking angle. A linear elastic fracture mechanics approach was adopted for the analysis of experimental results. Stress intensity factors (SIF) of straight cracks were calculated using an appropriate weight function set up for studying inclined edge cracks emanating from sharp V-notches. On the contrary, a finite element model has been built up to derive the SIFs at the tip of the kinked cracks. The influence of K_II on the crack propagation was discussed on the basis of theoretical and semi-empirical models. It has been found that (i) the crack initiation at the notch root occurred in mixed mode conditions, (ii) a decreasing Mode II component with growing crack length was observed under initial loading direction θ₀ = 45° and θ₀ = 90°, (iii) a crack deflection was observed after 45° rotation of the initial loading direction; a good prediction of the kinking angle was obtained using the maximum tangential stress criterion, and (iv) a fairly good rationalization of all the collected crack growth rate data is obtained if the driving force for crack propagation is expressed in terms of K_I.     

Rectangular tensile sheet with single edge crack or edge half-circular-hole crack

By using the displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author, this note presents the stress intensity factors (SIFs) of a rectangular tensile plate with single edge crack. Further this note studies the SIFs of crack emanating from an edge half-circular hole. By comparing the calculated SIFs of the single edge half-circular-hole crack with those of the single edge crack, a shielding effect of the half-circular hole on the SIFs of the single edge crack is discussed. It is found that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

Application of an orthotropy rescaling method to edge cracks and kinked cracks in orthotropic two-dimensional solids

Oblique edge cracks and kinked cracks in orthotropic materials with inclined principal material directions under inplane loadings are investigated. The Stroh formalism is modified by introducing new complex functions, which recovers a classical solution for a degenerate orthotropic material with multiple characteristic roots. An orthotropy rescaling technique is presented based on the modified Stroh formalism. Stress intensity factors for edge cracks as well as kinked cracks are obtained in terms of solutions for a material with cubic symmetry by applying the orthotropy rescaling method. Explicit expressions of the stress intensity factors for a degenerate orthotropic material are obtained in terms of solutions for an isotropic material. The effects of orthotropic parameter, material orientation, and crack angle on the stress intensity factors for the degenerate orthotropic material are discussed. The stress intensity factors for cubic symmetry materials are calculated from finite element analyses, which can be used to evaluate the stress intensity factors for orthotropic materials. The energy release rate for the kinked crack in an orthotropic material is also obtained.     

Mechanics of two-stage crack growth in fretting fatigue

Motivated by experimental observations, we carry out a numerical analysis of the two-stage crack growth under fretting fatigue by using an efficient and accurate boundary element method. To start with, the variation of stress field during a loading cycle is analyzed. Various values of friction coefficient in the contact zone are considered, which is shown to considerably affect the stress field. Then, by assuming crack initiation to occur in the shear mode, a surface-breaking crack is introduced to the specimen at the location of highest shear-stress amplitude. The crack-tip stress intensity factors (SIFs) are calculated for various crack lengths and at various crack angles ranging from 25° to 45° about the contact surface. It is shown that, for a loading ratio of 0.5, the cyclic mode-II SIF amplitude decreases with increasing crack length, whilst its mean value increases. It suggests that the (first-stage) shear crack would sooner or later become dormant, or switch to another mode that can provide continuous support of growth. Then, the first-stage shear crack is manually kinked into a second-stage opening crack, and the follow-on driving force is analyzed. It is shown that the kinking event is only favored after the first-stage crack has grown to a certain length. The present study thus provides insights in the mechanics of two-stage crack growth that has been frequently observed in a typical dovetail joint under fretting fatigue. It also suggests an improved experimental setup to quantitatively investigate the fretting fatigue in dovetail joints.     

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.     

Crack kinking of a delamination at an inclined core junction interface in a sandwich beam

The paper considers a general interface delamination and crack kinking from an inclined core junction in a sandwich beam. This particular problem is relevant for a newly developed peel-stopper component for sandwich structures.A finite element model (FEM) was developed and calibrated against a known model by He and Hutchinson. The numerically and analytically determined solution coefficients were in a perfect agreement with each other, so the necessary generalisation of results can be obtained through the application of FEM-analyses. The FE-model was used to determine solution coefficients for a number of interface compositions of practical interest. As expected some of the coefficients were quite sensitive to the specific material combination, which confirms that accurate solution strategies are important.The solution coefficients obtained were further applied to the analysis of the crack propagation and kinking process in three different sandwich beam configurations, each of which contained an inclined junction of 20°, 30°, or 40°. The objective was to examine how the core junction angle and the fracture mechanical properties of the sandwich components influenced the crack kinking tendency. The latter is vital for the design and functionality of a newly developed peel-stopper. It was shown that smaller core junction angles will lead to longer crack propagation (delamination) along the core-core interface prior to a possible kinking. The physical insight obtained is essential for optimal design of peel-stoppers.     

A through interface crack between a transversely isotropic pair of materials (+30°/−60°, −30°/+60°)

This study focuses on a delamination between two layers of a fiber-reinforced composite material oriented in the directions θ/(θ − 90°). Two specific interfaces are examined: the +30°/−60° interface and −30°/+60° interface. The delamination in these cases is treated effectively as a crack between two monoclinic materials. The behavior of the stress and displacement fields near the crack tip is studied. The first term of the asymptotic expansion for the stress and displacement fields are found by means of the Stroh and Lekhnitskii formalisms. A general solution is obtained for an interface crack in the x₂ = 0 plane. The crack is between two monoclinic materials with x₂ = 0 a symmetry plane.In order to calculate the stress intensity factors, a three-dimensional interaction energy or conservative M-integral is extended and implemented in conjunction with the finite element method. For the M-integral, the auxiliary fields used are particular cases of the stress and displacement fields obtained earlier. The displacement extrapolation method is also extended for this case. The crack surface displacements obtained from a finite element analysis are employed. The methods are independent of each other; hence, they may be used for validation of the results determined.Three test cases are analyzed to examine the accuracy of the results obtained by means of the M-integral method. In addition, two problems of a central crack in a symmetric composite under different loadings are solved. Those loadings are tension and in-plane shear. Stress intensity factors and the interface energy release rate are obtained along the crack front for all cases.     

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.     

表面裂纹应力强度因子计算及扩展跟踪的权函数法

 介绍了一种显式的权函数法,并将这种方法用于圆柱形容器接管外拐角表面裂纹的应力强度因子计算和扩展跟踪上.结果表明,权函数法可以用于分析各种载荷下不同形状的裂纹.就一般的工程问题而言,权函数法不失为一种与有限元法互补的方便有效的分析方法.     

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.     

Statistical scatter of fracture toughness in the ductile-brittle transition of a structural steel

The statistical scatter of fracture toughness in the ductile-brittle transition temperature range was experimentally examined on a 500 MPa class low carbon steel. Fracture toughness tests were replicatedly performed at −60 °C, −20 °C and −10 °C. The tests at −60 °C resulted in a single modal Weibull distribution with a shape parameter of 4 for the critical stress intensity factor converted from J-integral, whereas the Weibull distributions of the critical stress intensity factor at −20 °C and −10 °C showed a bilinear pattern with an elbow point, which caused a wider scatter than that at −60 °C. Such scatter transition behavior was discussed with reference to stable crack initiation. A model of the statistical scatter transition has been proposed in this work and the model reasonably explains the experimental results.     

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 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.