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.     

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.     

A general weight function for inclined kinked edge cracks in a semi-plane

A general method for evaluating the Stress Intensity Factors (SIFs) of an inclined kinked edge crack in a semi-plane is presented. An analytical Weight Function (WF) with a matrix structure was derived by extending a method developed for an inclined edge crack. The effects of the principal geometrical parameters governing the problem were studied through a parametric Finite Element (FE) analysis, carried out for different reference loading conditions. The WF can be used to produce efficient and accurate evaluations of the SIFs for cracks with initial inclination angle in the range −60° to +60° and kinked angle in the range from −90° to +90°. The agreement between the results with those obtained by accurate FE solutions suggests that the proposed WF can be used as a general tool for evaluating the fracture mechanics parameters of an inclined kinked crack.     

A mode II weight function for subsurface cracks in a two-dimensional half-space

The general properties of a mode II Weight Function for a subsurface crack in a two‐dimensional half‐space are discussed. A general form for the WF is proposed, and its analytical expression is deduced from the asymptotic properties of the displacements field near the crack tips and from some reference cases obtained by finite elements models. Although the WF has general validity, the main interest is on its application to the study of rolling contact fatigue: its properties are explored for a crack depth range within which the most common failure phenomena in rolling contact are experimentally observed, and for a crack length range within the field of short cracks. The accuracy is estimated by comparison with several results obtained by FEM models, and its validity in the crack depth range explored is shown.     

Wide-range weight function for center cracks

A closed-form wide-range weight function for center cracks in various finite bodies was presented based on a more accurate crack surface displacement representation. This weight function covers a large relative crack length a/w up to 0.85. High accuracy level was established through careful comparisons with several existing solutions. Closed-form stress intensity factor expressions were developed for a number of basic load cases, which makes the analysis possible for center cracks subjected to arbitrary crack face loading. Of particular interest is the segment uniform pressure loading on the crack surface which can be efficiently used for the Dugdale Model based fracture and fatigue analyses.     

The evolution of ‘cone’ cracks under axi-symmetric loading conditions

Cone fractures have been produced in epoxy resin samples in four different test configurations using a cylindrical indenter. The shapes of the cracks were determined using optical microscope focusing methods and it was found that they were strongly dependent on test geometry. Large changes in cone angle with crack length were observed. The evolution of the cracks was determined by mapping the fine river line markings on the fracture surface. In all cases crack nucleation occurred at point sources close to the edge of the indenter and then the crack grew in the material round the base of the indenter before expanding outwards. Numerous crack arrest markings were also mapped. It was found that the number and distribution of crack arrests was dependent on the test configuration and re-nucleation occurred after each arrest. The results are interpreted in terms of the no-twist growth constraint, which applies to crack growth in brittle solids. This leads to characteristic patterns on the fracture surface. It is shown that point nucleation results in a strongly non-symmetric fracture path and that the requirement that nucleation occurs at a point has a significant effect on the interpretation of fracture data in other test configurations.Emeritus Goldsmiths' Professor of Metallurgy, University of Cambridge, U.K.     

Probabilistic failure analysis for real glass components under general loading conditions

In this paper, the generalized local model (GLM) is applied to derive the primary failure cumulative distribution function (PFCDF) of annealed glass in order to achieve the failure prediction of structural glass. The uniqueness of the glass characterization is demonstrated irrespective of the test, specimen size, and geometry used. Consequently, the strength of glass is unequivocally derived in a probabilistic way as a material property, so that the definition of normalized testing specimens in international standards might be put under question. Furthermore, the application of the GLM to the results assessment allows to ensure a correct transferability of the laboratory data from simple specimens to the practical design of real glass components and vice versa. The feasibility of the GLM to characterize the strength of annealed glass from different test types is illustrated by means of an extensive experimental program.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

Accurate loading analyses of curved cracks under mixed-mode conditions applying the J-integral

In linear elastic fracture mechanics the path-independent J-integral is a loading quantity equivalent to stress intensity factors (SIF) or the energy release rate. Concerning plane crack problems, $J_k$ J k is a 2-dimensional vector with its components $J_1$ J 1 and $J_2$ J 2 . These two parameters can be related to the mode-I and mode-II SIFs $K_{\mathrm{I}}$ K I and $K_{\mathrm{II}}$ K II . To guarantee path-independence for curved crack geometries, an integration path along the crack faces must be considered. This paper deals with problems occurring at the numerical calculation of the J-integral in connection with the FE-method. Two new methods for accurately calculating values of $J_2$ J 2 for arbitrary cracks are presented.     

Evolution of subsurface radial cracks in bi-material structures undergoing indentation loading

Mathematical modelling is used to study the evolution of damage caused by indentation loading on curved bilayers consisting of brittle shells filled with polymer support material. Such loads are pertinent to all-ceramic crown structures on tooth dentin in occlusal function. The aim is to develop tools to assist in the design of such structures to ensure both high damage resistance and high damage tolerance. Specifically, the initiation and propagation of a radial crack emanating from the interface is studied using the boundary element method (BEM) in three dimensions. The system that is analysed consists of a spherical indenter and both flat and convex bi-material samples. A semi-circular intrinsic flaw/crack is assumed to lie on the axis of indentation at the interface of the two materials, in the coating. Upon application of an indentation load, the mode I stress intensity factor distribution along the crack front is determined and the crack front is propagated using a small increment. By repeating this process, the critical load for propagation of the crack is obtained as a function of crack size. The results compare well with experimental crack propagation studies in bi-materials, as well as observed damage in porcelain crowns that have been used to repair teeth. The convex models show that radial cracks can exist in the brittle coating, without leading to catastrophic failure, up to a critical crack length. An increase in the applied load, causing the crack to grow beyond this length, causes the coating to fail in an unstable way. The results show that there is an optimum combination of design parameters for maximising the damage resistance. It is shown that larger convex radii of curvature lead to higher damage tolerance.     

A weight function technique for partially closed inclined edge cracks analysis

The Green Functions, giving the Crack Opening Displacement components of an inclined edge crack under general loading conditions, were obtained starting from the matrix-like structure of the Weight Functions developed for determining the Stress Intensity Factors. The mathematical formulation of the problem is presented and the computational efficiency of the method is demonstrated by solving and discussing the non linear problem of a partially closed inclined edge crack under bending. By means of an iterative procedure, the closed portion of the crack is determined and the effects of normal and friction contact forces on the Crack Opening Displacement components and on the Stress Intensity Factors are discussed. The efficiency and the accuracy of this approach are assessed by comparison with Finite Element solutions.     

A strain-based Dugdale model for cracks under generally yielding conditions

To develop an analytical method for quantifying the growth behaviour of short cracks embedded in notch plastic zones, a critical assessment of the Dugdale model is first made by comparison against finite element analysis for an edge-cracked plate subjected to an applied strain varying linearly along the crack path. It is shown that the conventional stress-based Dugdale model provides accurate estimates for the crack-tip opening displacement and the plastic zone size provided that the applied strain does not exceed one third of the yield strain. These estimates become significantly inaccurate at higher strain levels. To overcome this limitation of the conventional model, a strain-based implementation of the Dugdale model is proposed in which the conventional equilibrium equation is replaced by strain compatibility. Comparison with finite element results shows that this strain-based model provides accurate values for both the crack-tip-opening displacement and the plastic zone size for applied strains up to four times the yield strain and with no evidence of decreasing accuracy with increasing strain. Furthermore, it is shown that the relevant plastic constraint factor to be used for plane strain is that appropriate for the notch plastic zone in the absence of a crack, rather than the more usual choice which is appropriate only for small-scale yielding conditions. This provides a practical and physically plausible approach for extending the scope of current predictive software for fatigue crack growth based on the Dugdale model to include conditions of large-scale yielding.     

A closed-form method for integrating weight functions for part-through cracks subject to Mode I loading

Numerical integration of weight functions tends to be computationally inefficient because of the singularity in a typical weight function expression. An alternative technique has been developed for surface and corner cracks, which greatly improves both efficiency and accuracy of K_I estimates. Exact analytical solutions for the weight function integral are obtained over discrete intervals, and then summed to obtain the stress intensity factor. The only numerical approximation in this approach is the way in which the variation in stress between discrete known values is treated. Closed-form weight function integration methods are presented for three approximations of the stress distribution: (1) constant stress over each integration interval, (2) a piecewise linear representation, and (3) a piecewise quadratic fit. A series of benchmark analyses were performed to validate the approach and to infer convergence rates. The quadratic method is the most computationally efficient, and converges with a small number of integration increments. The piecewise linear method gives good results with a modest number of stress data points on the crack plane. The constant-stress approximation is the least accurate of the three methods, but gives acceptable results if there are sufficient stress data points.     

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.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

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.     

Constructing the weight function for flat solids with cracks

An approximated method is proposed for constructing the weight function for flat solids with a crack based on the available exact solutions for an infinite plane and the available values of the stress intensity factor at the crack tip in the examined solid for any distribution of the load at its edges. Examples are given of determining the weight function for edge and central cracks in right-angled plates and disks.Translated from Problemy Prochnosti, No. 8, pp. 10–14, August, 1990.     

Approximation of curved cracks under mixed-mode loading

The calculation of stress intensity factors or mechanical energy release rate for non-straight cracks can be complicated. Approximation to equivalent crack shapes can simplify calculations considerably, but this requires an understanding of the influence of key shape parameters on crack-tip stresses. A simple analytical model has been developed, based on the concept of a relaxed volume, to predict mechanical energy release rate and deflection angle for a range of crack shapes under mixed-mode loading. Results from this model compared well with those obtained from finite element (FE) simulations, and with predictions from previous analytical models. It was found that the crack length and orientation of the crack-tip with respect to loading direction are the key influences on fracture parameters, whilst curvature near the crack-tip can also affect results.     

A 3D full‐field study of cracks in a nuclear graphite under mode I and mode II cyclic dwell loading conditions

Three‐dimensional (3D) full‐field deformation around crack tips in a nuclear graphite has been studied under mode I and mode II cyclic dwell loading conditions using digital volume correlation (DVC) and integrated finite element (FE) analysis. A cracked Brazilian disk specimen of Gilsocarbon graphite was tested at selected loading angles to achieve mode I and mode II cyclic dwell loading conditions. Integrated FE analysis was carried out with the 3D displacement fields measured by DVC injected into the FE model, from which the crack driving force J‐integral was obtained using a damaged plasticity material model. The evolution of near‐tip strains and the J‐integral during the cyclic dwell loading was examined. Under cyclic dwell, residual strain accumulation was observed for the first time. The results shed some light on the effect of dwell time on the 3D crack deformation and crack driving force in Gilsocarbon under cyclic mode I and II loading conditions.