Analytical wide-range weight functions for various finite cracked bodies

Closed-form wide-range weight functions have been presented for various finite plane cracked bodies. A unified analytical procedure was used in the derivation. First, accurate crack face displacement expressions for center and edge cracks were determined for the polynomial type reference load case. These displacements were then used to derive analytical weight functions, whose accuracy was critically assessed using the related Green's functions. Stress intensity factors formulae for a number of basic load cases including concentrated forces, polynomial as well as a band of linearly varying stress, have been obtained. These basic solutions combined with superposition method enable stress intensity factors to be rapidly determined for complex loadings, as demonstrated by example engineering crack problems. Discussions were made on the reference load case dependence of the weight functions, and the significance of the number of terms contained in the crack face displacement representation on the solution accuracy at extended crack lengths. The analytical wide-range weight functions have been proved versatile, very cost-saving, easy-to-use, and accurate.     

Approximate weight function technique using single-layer potential for a cracked circular disc

An efficient weight function technique using the indirect boundary integral method was presented for cracked circular discs. The crack opening displacement field was presented by a single layer whose kernel was a modified form of the fundamental solution in elastostatics. The application of a single-layer potential to the weight function method leads to a unique closed-form SIF (stress intensity factor) solution. The solution can be applied to a cracked circular discs with or without an internal hole or opening. For these crack geometries over a wide range of crack ratios, the SIF solution can be applied without any modification.

The calculation procedure of SIFs for the various cracked circular discs using only one analytical solution is very simple and straightforward. The information necessary in the analysis includes only two or three reference load cases. In most cases the SIF solution using two reference SIFs gives reasonably accurate results while the SIF solution with three reference load cases may be used to improve the solution accuracy of the crack configurations, with an internal opening or hole, compared with the solutions of the available literature.     

Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet

The problem of two equal-length collinear cracks in an infinite sheet is treated using the weight function method. Exact weight functions for the inner and outer crack tips are derived based on the crack opening displacement solution for a reference load case. These weight functions are used to calculate stress intensity factors for different load cases, plastic zone sizes and crack tip opening displacements of the strip yield model. The approach is validated by the perfect agreement between the present strip yield model solutions and Collins and Cartwright’s analytical results based on the direct complex stress function formulation.     

WEIGHTED AVERAGED STRESS INTENSITY FACTORS OF CIRCULAR-FRONTED CRACKS IN CYLINDRICAL BARS

Stress intensity factors are calculated in weighted average at the surface and the deepest point of a circular-fronted surface crack in a cylindrical bar by use of the weight function method. A wide range of various crack shapes are studied, from a nearly straight-fronted edge crack to a semi-circular crack front. Use of the weight function method requires that the crack opening displacement field of a reference load has to be known. It was obtained by 3-D finite element analysis. Results are presented for the cracked cylinder subjected to a constant stress (tension) and a linear stress distribution acting perpendicular to the crack faces and they are compared with values found by other investigators.     

A wide range weight function for a single edge cracked geometry with clamped ends

A single edge cracked geometry with clamped ends is well suited for fracture toughness and fatigue crack growth testing of composites and thin materials. Stress intensity factors may be determined by the weight function method. A weight function for the single edge cracked geometry with clamped ends is developed and verified in this paper. It is based on analytical forms for the reference stress intensity factor and crack mouth opening displacement. The analytical forms are shown to be valid, by comparison with finite element results, over a wide range of crack depths and plate aspect ratios. Use of the analytical form enables the weight function to be calculated for any plate aspect ratio without the need for preliminary finite element analysis. Stress intensity factors and crack mouth opening displacements, predicted using this weight function, correlated well with finite element results for non-uniform crack surface stress distributions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite Element Weight Function Application for a Cracked Disk

A practical application of the weight function method is used in this paper in order to find values of the stress intensity factor for a cracked disk subjected to different loadings. The finite element method is used in order to obtain discrete values of the crack face displacement in a reference loading case, namely inertia forces due to uniform rotation. These values are interpolated and a general expression of the displacements is obtained, which is further used to determine the stress intensity factor in this case. With the weight function equation, stress intensity factors for other loadings are obtained and the results are compared with those reported by other authors. Very good agreement was obtained, showing thus the reliability of this approach.     

Some recent fatigue and fracture mechanics research in BIAM

The paper presents a brief review of some of the major research activities on fatigue and fracture mechanics in recent years at the Beijing Institute of Aeronautical Materials. Attention is mainly given to the studies on weight function methods for analyses of two- and three-dimensional crack problems, fatigue crack growth under variable amplitude loading, small crack effects and a fracture-mechanics-based total fatigue life prediction method.Abbreviations 2(3)D two- (three-) dimensional - BIAM Institute of Aeronautical Materials, Beijing - CCT center cracked tension - COD crack opening displacement - CTOD crack tip opening displacement - FEM finite element method - LEFM linear elastic fracture mechanics - SENT single edge notched tension - SIF stress intensity factor - WFM weight function method     

Fracture analysis of base-edge-cracked reverse-tapered plates

This paper presents a fracture mechanics analysis of the base-edge-cracked reverse-tapered (RT) fracture geometry. Motivation for this study was the use of this test geometry in Phase 1 of a recently completed joint-industry-agency project entitled Large-Scale Ice Fracture Experiments. Underlying the choice of the RT fracture geometry for Phase 1 was the desirability of achieving crack propagation in a controlled and stable manner; such control would allow a number of observations to be made on one testpiece. Reverse tapering greatly improves not only crack growth stability but also crack path stability. The weight function method was used to provide accurate wide-ranging stress intensity factor (SIF), crack face displacement (COD) and crack opening area (COA) expressions for the RT subject to any loading. The required weight function was obtained through a finite element analysis of this geometry subject to a reference load case which determined the associated stress intensity factor and crack opening displacements. The Wu and Carlsson procedure was followed. A key modification to the latter procedure facilitated the attainment of the reference CMOD for all crack lengths, including the zero ligament limit; this was achieved by considering an additional reference solution. This modification is general in nature and could be pursued whenever the reference CMOD is not known analytically. An analytical solution for the crack opening area (COA) was also achieved for the special case of concentrated loading at the crack mouth. This solution can be applied to any geometry where the reference CMOD expression is known.     

THE FRACTURE BEHAVIOUR OF A CENTRE CRACKED TENSILE SPECIMEN

Abstract— This paper reviews the stress intensity factor, limit load, compliance and J-integral functions for a centre cracked tensile (CCT) specimen available in the literature. Compliance and J-integral functions are derived from the optimum stress intensity factor and limit load solutions. The functions are compared with the results obtained from two-dimensional finite element analyses of the specimen.
The finite element results have confirmed the accuracy of the compliance and limit load functions available in the literature and suggest that the unloading compliance technique, based on crack mouth opening displacement, could be developed for a CCT specimen. Non-linear finite element analyses have shown that J can be estimated from the measured load versus load-point displacement behaviour providing a/W ≥ 0.5     

A new method of crack-tip opening displacement determined based on maximum crack opening displacement

In this paper a new method is presented to determine the crack-tip opening displacement (CTOD) for the center cracked plate with uniaxial uniform tension load. The maximum crack opening displacement (MCOD) is adopted to estimate CTOD. Based on the series of calculation results by elastic–plastic finite element simulation, an explicit function expression for the CTOD versus MCOD is determined, which enables to consider the influence effects of crack geometries, plate sizes, applied loads, plane state and material properties. Hence, the presented method of CTOD determined by MCOD is suitable to any center crack finite plate of any material under uniaxial tension.     

Numerical study on deformations in a cracked viscoelastic body with the extended finite element method

Deformations such as crack opening and sliding displacements in a cracked viscoelastic body are numerically investigated by the extended finite element method (XFEM). The solution is carried out directly in time domain with a mesh not conforming to the crack geometry. The generalized Heaviside function is used to reflect the displacement discontinuity across a crack surface while the basis functions extracted from the viscoelastic asymptotic fields are used to manifest the gradient singularity at a crack tip. With these treatments, the XFEM formulations are derived in an incremental form. In evaluating the stiffness matrix, a selective integration scheme is suggested for problems with high Poisson ratios often encountered in viscoelastic materials over different element types in the XFEM. Numerical examples show that the crack opening displacement and crack sliding displacement are satisfactory.     

Parametric study of oblique edge cracks under cyclic contact loading

The problem of a two-dimensional elastic body, carrying an inclined edge crack and loaded by a cylinder rolling on the surface, is solved by the weight function method. The load induced by the cylinder on the cracked body was represented by the Hertzian pressure distribution, and the nominal stress distribution in the uncracked body was numerically evaluated by the superposition principle. The crack opening displacement components were obtained by an analytical Green's function. The partial crack closure was considered and the influence of the mutual forces between the crack faces included in the analysis, by which the effective stress intensity factors K _I and K _II could be evaluated. By considering different friction conditions between the crack surfaces and several crack inclinations, the evolutions of the effective K _I and K _II during typical loading cycles were analysed.     

FEM for evaluation of weight functions for SIF, COD and higher-order coefficients with application to a typical wedge splitting specimen

In the evaluation of accurate weight functions for the coefficients of first few terms of the linear elastic crack tip fields and the crack opening displacement (COD) using the finite element method (FEM), singularities at the crack tip and the loading point need to be properly considered. The crack tip singularity can be well captured by a hybrid crack element (HCE), which directly predicts accurate coefficients of first few terms of the linear elastic crack tip fields. A penalty function technique is introduced to handle the point load. With the use of these methods numerical results of a typical wedge splitting (WS) specimen subjected to wedge forces at arbitrary locations on the crack faces are obtained. With the help of appropriate interpolation techniques, these results can be used as weight functions. The range of validity of the so-called Paris equation, which is widely used in the evaluation of the COD from the stress intensity factors (SIFs), is established.     

HIGH RESOLUTION COMPUTED TOMOGRAPHY AND IMPLICATIONS FOR FATIGUE CRACK CLOSURE MODELLING

Understanding the physical bases of a cracked sample’s macroscopic response to applied loading, e.g. fatigue crack closure, requires non-destructive, microscopic quantification of the crack face separations as a function of applied load. Ideally, these measurements should cover the entire crack face. Non-destructive sectioning with high resolution X-ray computed tomography has been used for in situ observations of the crack faces under applied load in samples of Al–Li 2090, and in this paper, the crack openings that were measured in the interior of the sample are related to crack face geometry and to changes in the slope of load–displacement curves. The implications of these results are discussed for physically based crack closure modelling.     

Reference Stress Approximations for J and COD of Circumferential Through-Wall Cracked Pipes

Reference stress approximations for the J integral and crack tip opening displacement (COD) for circumferential through-wall cracked pipes under tension and under bending are reported. The proposed J estimation equation is fully compatible with the existing reference stress based J estimation, currently embedded in the R6 assessment procedure, but involves a slightly different definition of the reference stress, using an optimised reference load instead of the limit load. This modification enhances the accuracy of the J estimation for circumferentially cracked pipes. Confidence in the proposed equation is gained from the significantly reduced hardening dependence of the plastic influence functions in the GE/EPRI method. The proposed COD estimation equation includes two further modifications. One is the use of a power-law fit to the plastic portion of the stress strain data, instead of the use of the actual stress strain data. In this context, a robust estimation equation for the strain hardening index is given. The other modification is to the plasticity correction term in contained yielding. A lower bound COD estimation equation is also given, similar to the R6 option 1 Jestimation curve, which is suitable when only limited tensile properties are available. The resulting estimation equations are simple to use. Comparisons with experimental pipe test data show that the proposed COD estimation equations provide overall good agreement, which gives confidence in applying them to Leak-before-Break (LBB) analyses.     

A new expression for crack opening stress determined based on maximum crack opening displacement under tension–compression cyclic loading

The maximum crack opening displacement is introduced to investigate the effect of compressive loads on crack opening stress in tension–compression loading cycles. Based on elastic–plastic finite element analysis of centre cracked finite plate and accounting for the effects of crack geometry size, Young's modulus, yield stress and strain hardening, the explicit expression of crack opening stress versus maximum crack opening displacement is presented. This model considers the effect of compressive loads on crack opening stress and avoids adopting fracture parameters around crack tip. Besides, it could be applied in a wide range of materials and load conditions. Further studies show that experimental results of da/dN ? ΔK curves with negative stress ratios could be condensed to a single curve using this crack opening stress model.     

Universal features of weight functions for cracks in mode I

An analysis of known analytical and numerical weight functions for cracks in mode I has revealed that they all have a similar singular term and that it is possible to approximate them with one universal expression with three unknown parameters. The unknown parameters can be determined directly from reference stress intensity factor expressions without using the crack opening displacement function. The universal weight function expression, with suitable reference stress intensity factors, was used to derive the weight functions for internal and external radial cracks in a thick cylinder. These weight functions were then further used to calculate the stress intensity factors for radial cracks in a cylinder subjected to several nonlinear stress fields and were compared to available numerical data.     

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.     

Crack growth in cement-based composites

A model that can be used to predict Mode I crack growth in cement-based composites is presented. The region ahead of a crack tip, where nonlinear deformations and aggregate interlock occur, is modeled as an extension of the actual stress-free crack subjected to a closing pressure that depends on the crack face displacements. In the case of concrete, crack propagation is assumed to occur when the crack opening displacement at the tip of the actual crack reaches a critical value. To predict results, the elastostatics problem of a layer containing a vertical edge crack was solved using a Green's function approach together with integral transform techniques. Stress intensity factors and crack opening displacements were obtained by numerically solving a singular integral equation. The closing pressure function and critical crack tip opening displacement were taken from experimental data for various materials, and the model was applied to the analysis of experiments performed on initially notched concrete and fiber-reinforced mortar beams.     

Determination of weight functions for elastic T-stress from reference T-stress solutions

ABSTRACT This paper presents the application of the weight function method for the calculation of elastic T -stress. First, the background of the weight function method for the calculation of T -stress is summarized. Then an analysis of known weight functions for T -stress revealed that it is possible to approximate them with one universal mathematical form with three unknown parameters with high accuracy. The existence of this weight function form significantly simplified the determination of weight functions for T -stress. For any particular crack geometry, the unknown parameters can be determined from reference T -stress solutions. The general weight function expression, with suitable reference T -stress solutions, was used to derive the weight functions for single edge cracked plate, double edge cracked plate and center cracked plate specimens. These weight functions were then further used to calculate the T -stress solutions for cracked specimens under several nonlinear stress fields and were compared to available numerical data.