Analysis of a kinked crack in anti-plane shear

A semi-infinite kinked crack in anti-plane shear is analyzed. The problem is formulated using the Mellin transform, and solved by the Wiener-Hopf technique. A closed form solution for displacement is obtained, from which the stress intensity factor is calculated. Particular emphasis is put on the stress intensity factor as the kinked length approaches zero, where two limit processes (both the distance from the crack tip and the kinked length approaching zero) are involved. It is found that the stress intensity factor depends on the order of performing the two limit processes. The results are compared with those by previous researchers. Also the energy release rate for this problem is computed.     

The effect of crack surface interaction on the stress intensity factor in Mode III crack growth in round shafts

Turbine-generator shafts are often subjected to a complex transient torsional loading. Such transient torques may initiate and propagate a circumferential crack in the shafts. Mode III crack growth in turbo-generator shafts often results in a fracture surface morphology resembling a factory roof. The interaction of the mutual fracture surfaces results in a pressure and a frictional stress field between fracture surfaces when the shaft is subjected to torsion. This interaction reduces the effective Mode III stress intensity factor.The effective stress intensity factor in circumferentially cracked round shafts is evaluated for a wide range of applied torsional loading by considering a pressure distribution between mating fracture surfaces. The pressure between fracture surfaces results from climbing of asperities respect to each other. The pressure profile not only depends on the fracture surface roughness (height and width (wavelength) of the peak and valleys), but also depends on the magnitude of the applied Mode III stress intensity factor. The results show that asperity interactions significantly reduce the effective Mode III stress intensity factor. However, the interactions diminish beyond a critical applied Mode III stress intensity factor. The critical stress intensity factor depends on the asperities height and wavelength. The results of these analyses are used to find the effective stress intensity factor in various Mode III fatigue crack growth experiments. The results show that Mode III crack growth rate is related to the effective stress intensity factor in a form of the Paris law.     

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.     

Hybrid stress finite element analysis of bending of a plate with a through flaw

A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated. The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.     

Kinked cracks in anisotropic elastic materials

The problem of a kinked crack is analysed for the most general case of elastic anisotropy. The kinked crack is modelled by means of continuous distributions of dislocations which are assumed to be singular both at the crack tips and at the kink vertex. The resulting system of singular integral equations is solved numerically using Chebyshev polynomials and the reciprocal theorem. The stress intensity factors for modes I, II and III and the generalised stress intensity factor at the vertex are obtained directly from the dislocation densities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Crack kinking under nonsymmetric loading

This paper considers an asymmetrically kinked, semi-infinite crack in a two-dimensional solid under mixed-mode loading and a stress acting parallel to the main crack, the latter providing the non-singular stress term, T, in the Irwin-Williams expansion of the crack tip field. The aim of the study is twofold: First, to obtain an almost closed-form solution for the stress intensity factors at the tip of the kink with a view to explaining a curious result observed by many authors that under predominantly Mode I loading the first order solution in small kink angle is accurate for considerably large kink angles and, second, to study the effect of the in-plane tensile stress on the stability of crack growth. Where possible, the results are compared with those available in the literature.     

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.     

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.     

Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads

In this paper, the brittle fracture initiation characteristics under general combination of the opening mode (Mode I), sliding mode (Mode II) and tearing mode (Mode III) were investigated both theoretically and experimentally.

First, the perfectly brittle fracture tests were conducted on specimens of PMMA (Polymethylmethacrylate) for all possible combinations of the fracture modes including respective pure modes. The experimental fracture strengths were compared with those predicted by the fracture criteria which are represented in terms of: (1) maximum tangential stress, [σ_gq]_max, extended to general combined modes, (2) maximum energy release rate at the propagation of a small kinked crack, [G^k(γ)]_max, and (3) newly derived maximum energy release rate at the initiation of a small kinked crack, [G(γ)]_max. It was found that the [G^k(γ)]_max or [G(γ)]_max criterion was very effective to predict both the direction of initial crack propagation and the fracture strength. These energy release rates are expressed in closed forms, and the interaction curves of the brittle fracture strength under arbitrary combinations of Modes I, II and III were derived.

Next, for fracture accompanied by plastic deformation, tests were carried out on specimens of mild steel (SM 41) imposing bi-axial tensile loads at various low temperatures. Then, brittle fracture with plastic deformation occurs under a combination of Modes I and II. In the case of brittle fracture with small scale yielding, the [G(γ)]_max criterion predicts well the direction of initial crack propagation but estimates only lower fracture strength than the experimental one. In the cases of brittle fracture with large scale yielding and under general yielding, it was found from the fracture tests that the direction of initial crack propagation was nearly normal to the resultant vector of the crack opening displacements in the opening and sliding modes at the notch tip. To this type of fracture, the modified COD criterion predicts well the direction of initial crack propagation, but lower fracture strength.

When brittle fracture occurs under the influence of plastic deformation, in such cases as the last three mentioned above, the actual fracture strength is higher than what the most reliable criterion predicts and it increases as deformation in Mode II becomes larger.     

Stress analysis of a kinked crack initiating from a rigid line inclusion. Part 1: Formulation

The problem of a kinked crack which has initiated from the tip of a rigid line inclusion is analyzed as a mixed boudary value problem. The stress distribution, stress intensity factors, singularity at the inclusion tip, and the resultant moment on the rigid line inclusion are investigated for various angles of the kinked crack and crack lengths. The rotation of the rigid line inclusion, when loaded by a uniform farfield stress, is calculated. The cases in which the inclusion is free to rotate or is fixed are separately considered.     

Crack face pressure loading of semielliptical cracks located along the bore of a hole

The finite element-alternating method is used to obtain Mode I stress intensity factors for semielliptical surface cracks centered along the bore of a hole in a large plate of finite thickness. The variation in stress intensity factor around the crack perimeter is provided for two loading configurations: remote uniaxial tension and a crack face pressure described by a third order polynomial. It is suggested that the tabulated solutions for the crack face pressure loading represent “general” results which can be used with linear superposition to compute stress intensity factors for many other practical loading configurations. Two example problems describe application of the superposition procedure with the crack face pressure results.     

均匀热流下导热裂纹Ⅱ型温度应力强度因子的解析解

研究含中心裂纹无限大板受远场均匀热流作用,热流密度方向与裂纹有一夹角的情况。当垂直于裂纹面方向有定量热流穿过裂纹时,采用复变函数理论,得出了温度、应力与位移场解析解。利用位移单值条件,确定出温度应力强度因子的解析表达式。针对铝合金LY12材料进行了数值计算,研究了裂纹导热情况与热流方向对温度场及温度应力强度因子的影响。研究表明:该文给定的温度边界条件下,只产生Ⅱ型温度应力强度因子,不产生Ⅰ型温度应力强度因子。热荷载可等效为一个远场均匀作用的剪应力。Ⅱ型温度应力场取决于热流密度沿垂直裂纹面方向的分量,平行于裂纹方向的热流分量对温度应力场没有影响。     

Prediction of fatigue crack growth path in mechanical joints using a weight function approach

Cracks often initiate from the mechanical joints which are widely used in structural components. It has been reported that cracks in mechanical joints are under mixed‐mode condition and there is a critical angle at which mode I stress intensity factor becomes maximum. The crack propagates in an arbitrary direction and the prediction of fatigue crack growth path is needed to provide against crack propagation and examine safety. In this study, mixed‐mode fatigue crack growth tests are performed for horizontal and critical inclined cracks in mechanical joints. Fatigue crack growth paths are predicted using a weight function approach and maximum tangential stress criterion.     

Fatigue crack growth of a railway wheel

Typically, fatigue crack propagation in railway wheels is initiated at some subsurface defect and occurs under mixed mode (I–II) conditions. For a Spanish AVE train wheel, fatigue crack growth characterization of the steel in mode I, mixed mode I–II, and evaluation of crack path starting from an assumed flaw are presented and discussed.Mode I fatigue crack growth rate measurement were performed in compact tension C(T) specimens according to the ASTM E647 standard. Three different load ratios were used, and fatigue crack growth thresholds were determined according to two different procedures. Load shedding and constant maximum stress intensity factor with increasing load ratio R were used for evaluation of fatigue crack growth threshold.To model a crack growth scenario in a railway wheel, mixed mode I–II fatigue crack growth tests were performed using CTS specimens. Fatigue crack growth rates and propagation direction of a crack subjected to mixed mode loading were measured. A finite element analysis was performed in order to obtain the K_I and K_II values for the tested loading angles. The crack propagation direction for the tested mixed mode loading conditions was experimentally measured and numerically calculated, and the obtained results were then compared in order to validate the used numerical techniques.The modelled crack growth, up to final fracture in the wheel, is consistent with the expectation for the type of initial damage considered.     

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ_z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ_z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.     

Analysis of Asymmetric Kinked Cracks of Arbitrary Size, Location and Orientation – Part I. Remote Compression

In this paper, the stress intensity factors of interacting kinked cracks in an elastic solid under remote compression and the overall strains of the solid are determined numerically. The kinked cracks are in general asymmetric, unequal, and arbitrarily oriented and located. Each kinked crack consists of a closed frictional main crack, and traction free kinks modeled by continuous dislocation distributions. The original problem is decomposed into straight crack problems such that the main cracks are subjected to dislocation and shear traction loadings. The model is used to investigate the dependence of the stress intensity factors and the overall strains on the crack configuration, i.e., a single fault of cracks, parallel faults, crossed faults, periodic and random crack arrays, and on the geometrical and physical parameters such as the fault angle and the lateral confinement. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analysis of Asymmetric Kinked Cracks of Arbitrary Size, Location and Orientation – Part II. Remote Tension

In this paper, the stress intensity factors of interacting kinked cracks in a solid and the overall strains of the solid under uniaxial tension are determined numerically. The kinked cracks are in general asymmetric, unequal, and arbitrarily oriented and located in the solid. Each kinked crack, assumed to be traction free, consists of a main crack and kinks. The analysis makes use of the dislocation modeling of kinks, and the superposition of problems of straight cracks subjected to dislocation and traction loadings. The model is used to investigate the dependence of the stress intensity factors and the overall strains on crack geometry (straight, Z-shaped and U-shaped cracks) and crack configuration (collinear and stacked cracks, periodic and random crack arrays). This revised version was published online in August 2006 with corrections to the Cover Date.     

A mode I crack parallel to the interfaces in a periodically layered medium

The plane problem of a single crack in a periodically layered bimaterial composite is considered. For the case of a long crack loaded by opening normal tractions, the universal relation obtained between the Mode I and Mode II stress intensity factors show that the most dangerous crack location lies in the midplane of the layer. This crack location of the Mode I finite length crack is examined in detail. A closed form expression of the Green's function for a single dislocation is derived and the problem is reduced to a singular integral equation of the first kind. The study of the dependence of the normalized stress intensity factor upon the crack length reveals a wavy nonmonotonic behavior. A simple analytic formula for the limiting case of a semi-infinite crack is derived. It is found to be valid for a broad range of parameters.     

Sickle-shaped crack in a round bar under complex Mode I loading

A sickle‐shaped surface crack in a round bar under complex Mode I loading is considered. First, the stress‐intensity factor (SIF) along the front of the flaw is numerically determined for five elementary Mode I stress distributions (constant, linear, quadratic, cubic and quartic) directly applied on the crack faces. The finite element method and linear elastic fracture mechanics concepts are employed. Then, a numerical procedure to calculate approximate values of SIF for a complex Mode I stress distribution on the crack faces is proposed based on both the power series expansion of the function describing such a stress distribution and the superposition principle. In order to validate the results obtained through the above procedure, a comparison with numerical data available in the literature is made.     

Examination of the Mode II fracture behaviour of wood with a short crack in an asymmetric four-point bending test

Asymmetric four-point bending tests of agathis specimens with a short crack along the neutral axis in a tangential–longitudinal system were conducted onto analyse the failure behaviour of wood with a short crack. The nominal shear strength and Mode II critical stress intensity factors of the specimens with various crack lengths were measured, and the influence of crack length on these properties was examined. The nominal shear strength of the cracked specimens was significantly lower than the strength of a crack-free specimen, even when the crack was extremely short. This finding suggests that the fracture mechanics theory is effective for analysing the failure behaviour of wood with a very short crack in this loading condition. However, the Mode II critical stress intensity factor still depends on the crack length. When the crack length was corrected with considering the formation of fracture process zone ahead of the crack tip, the critical intensity factor could be predicted effectively as well as the nominal shear strength.