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.     

First and second order stress extrapolation for mode I and mode II edge cracks

The use of linear and second order stress extrapolation to obtain K_I and K_II in two-dimensional finite element models of a thick plate containing an edge crack was examined. Three loading cases were studied, including classical Mode I and Mode II problems and a problem of tribological contact. Linear extrapolation was observed to yield accurate predictions of K_I in cases of dominant Mode I loading. In Mode II situations, notably where the crack faces experienced compressive normal stresses, second order extrapolation was observed to improve estimates of K_II.     

Simplified model for the fatigue growth analysis of surface cracks in round bars under mode I

The fatigue growth of an edge flaw in a round bar under cyclic tension or bending loading is examined, using a two-parameter numerical model. First, it is shown that the crack front evolution is defined by a very small number of parameters, which varies during crack growth. Approximated solutions for both the crack propagation path and the stress intensity factor are derived, and the fatigue predictions using this simple analytical method are finally compared with the numerical results.     

Mode I and mode II stress intensities for plates with cracks of finite opening

For the finite opening crack, there are two eigenvectors which give stress singularities at the crack tip. The Reciprocal Work Contour Integral Method is extended so as to yield the coefficients associated with these two eigenvectors. Several examples are given. In one, results using the algorithm are compared to a known solution.

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Résumé Dans le cas d'une fissure finie en cours d'ouverture, deux eigenvecteurs déterminent les singularités de contraintes à l'extrémité de la fissure. On procède àune extension de la méthode d'intégration sur un contour du travail réciproque de manière à déterminer les coefficients associés à ces deux eigenvecteurs. Divers exemples sont fournis et, dans l'un, on compare les résultats utilisant l'algoritme à ceux d'une solution connue.

     

Computation of shear mode weight functions for circular and elliptical cracks

Three-dimensional shear mode fundamental fields in finite bodies with mixed boundary conditions are analyzed by a special finite element method for circular and elliptical cracks. A procedure for determining the Fourier coefficients of the stress intensity factor for circular cracks is presented. A special series is proposed to represent the computed crack face weight functions for elliptical cracks.     

Point weight function method application for semi-elliptical mode I cracks

The method of the approximate weight function construction for a semi-elliptical crack was suggested. The weight function sought was written as the sum of asymptotic (weight function for an elliptical crack in an infinite body) and correction components. To take into account the influence of a body free surface on the asymptotic component behavior, fictitious forces symmetric with respect to the body free surface were introduced.As an example of the efficiency of the proposed method semi-elliptical axial cracks in pressure vessels were considered. The results of the stress intensity factor prediction are in good agreement with the corresponding results obtained by Raju and Newman. The only exception are the results for the points located near the major ellipse axis. This may be explained by the shortcomings of the employed empirical weight function expression for an elliptical crack in an infinite body.     

Greens functions for boundary element analysis of anisotropic bimaterials

We present several Greens functions for anisotropic bimaterials for two-dimensional elasticity and steady-state heat transfer problems. The details of the various Greens functions for perfect, slipping, and cracked interfaces are given for mechanical loading conditions. Previously reported formulations for cubic materials are extended to materials with general anisotropy in which plane strain deformations can exist. We also give the steady-state Greens function for thermal loading of a bimaterial with a perfectly bonded interface. The Greens functions are incorporated in boundary integral formulations and method of fundamental solutions formulations for analysis of finite solids under general boundary conditions.     

Stress intensity factors for tunnelling corner cracks under mode I loading

Stress intensity factors (SIFs) presented in the literature for corner cracks are limited to ideal quarter-circular and quarter-elliptical crack shapes. This paper presents SIF solutions for corner cracks that exhibit tunnelling, extending the range of corner crack shapes illustrated in the literature. Solutions were developed in a parametric form, obtained by empirically fitting polynomials to numerical values of SIF obtained from the FEM. A parameter was defined to quantify the extent of tunnelling. It was observed that crack shape has a significant effect on the SIF, so the consideration of equivalent quarter-circular cracks can produce inaccurate results when significant tunnelling occurs. SIF solutions for quarter-circular cracks are also presented and compared with those quoted in the literature.     

Weight functions for circular cracks

The authors examined a problem for a linearly elastic solid containing internal or external circular normal separation crack. It is shown that the corresponding weight function is equal to the product of the axisymmetric weight function and Poisson's kernel. An approximate weight function for an internal circular crack in an unlimited cylinder is constructed as an example.Translated from Problemy Prochnosti, No. 12, pp. 25–28, December, 1990.     

Weight functions for interface cracks

Weight functions are developed for determining stress intensity factors of cracks along an interface between two linear, elastic materials. As a result of the interface, both mode I and II components will be present for all but very special loading cases. The weight functions are employed to produce exactly the known stress intensity factors of a crack along an interface loaded by tensile and shear point forces.Part of this work was carried out while the author was on sabbatical leave at the Materials Laboratory, Wright Patterson Air Force Base, Ohio, USA     

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.     

Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading

The elastic T-stress has been recognised as a measure of constraint around the tip of a crack in contained yielding problems. A review of the literature indicates that most methods for obtaining T are confined to simple geometry and loading configurations. This paper explores direct use of finite element analysis for calculating T. It is shown that for mode I more reliable results with less mesh refinement can be achieved if crack flank nodal displacements are used. Methods are also suggested for calculating T for any mixed mode I/II loading without having to calculate stress intensity factors. There is good agreement between the results from the proposed methods and analytical results. T-stress is determined for a test configuration designed to investigate brittle and ductile fracture in mixed mode loading. It is shown that in shear loading of a cracked specimen T vanishes only when a truly antisymmetric field of deformation is provided. However this rarely happens in practice and the presence of T in shear is often inevitable. It is shown that for some cases the magnitude of T in shear is much more than that for tension. The effect of crack length is also investigated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Computation of the stress intensity factors for repaired cracks with bonded composite patch in mode I and mixed mode

In this study, the finite element method is used to analyse the behaviour of repaired cracks with bonded composite patches in mode I and mixed mode by computing the stress intensity factors at the crack tip. The effects of the patch size and the adhesive properties on the stress intensity factors variation were highlighted. The plot of the stress intensity factors according to the crack length in mode I, shows that the stress intensity factor exhibits an asymptotic behaviour as the crack length increases. In mixed mode, the obtained results show that the Mode I stress intensity factor is more affected by the presence of the patch than that of mode II.     

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.     

Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

Threshold and growth mechanism of fatigue cracks under mode II and III loadings

ABSTRACT The fatigue crack growth behaviour of 0.47% carbon steel was studied under mode II and III loadings. Mode II fatigue crack growth tests were carried out using specially designed double cantilever (DC) type specimens in order to measure the mode II threshold stress intensity factor range, ΔK_IIth. The relationship ΔK_IIth > ΔK_Ith caused crack branching from mode II to I after a crack reached the mode II threshold. Torsion fatigue tests on circumferentially cracked specimens were carried out to study the mechanisms of both mode III crack growth and of the formation of the factory‐roof crack surface morphology. A change in microstructure occurred at a crack tip during crack growth in both mode II and mode III shear cracks. It is presumed that the crack growth mechanisms in mode II and in mode III are essentially the same. Detailed fractographic investigation showed that factory‐roofs were formed by crack branching into mode I. Crack branching started from small semi‐elliptical cracks nucleated by shear at the tip of the original circumferential crack.     

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.     

Analysis of temperature effects near mode I cracks in glassy polymers

A previous isothermal study (Estevez et al., Journal of Mechanics and Physics of Solids 48, 2585–2617, 2000) has shown that the toughness of glassy polymers is governed by the competition between shear yielding and crazing. The present work aims at investigating loading rates for which thermal effects need to be accounted for. The influence of the heat coming from the viscoplastic shear yielding and from crazing on their competition and on the toughness is examined. Crazing is shown to be the dominant heat source, and the dependence of the craze properties on temperature appears to be key in controlling the toughness of the material.*Author for correspondence (E-mail: rafael.estevez@insa-lyon.fr)     

Fracture toughness of polycrystalline tungsten under mode I and mixed mode I/II loading

The fracture toughness of swaged polycrystalline tungsten was tested parallel and perpendicular to the swaging direction and under mixed mode I/mode II loading. The fracture mode is dominated by the microstructure and changed from all-transgranular cleavage in mode I to almost all-intergranular fracture in mode II. The mixed mode results can be related to two common failure criteria, the maximum tensile stress criterion (Maximum σ) and the maximum energy release rate criterion (Maximum G), but the large scatter in the data prohibits a clear distinction between the two criteria. Tests at 77 K show that the polycrystal is significantly tougher than the single crystal at this temperature. This is a consequence of the deflection of the crack into the grain boundaries and the imperfect texture (as compared to a single crystal) of the polycrystalline material.     

A hybrid approach to determine fracture resistance for mode I and mixed‐mode I and II fracture specimens

This paper proposes a hybrid approach to determine the fracture resistance for mode I and mixed‐mode I and II fracture specimens, combining both numerically computed and experimentally measured load (P) versus load‐line displacement (LLD or Δ) relationships for metallic fracture specimens. The hybrid approach predicates on the same principle as the conventional, multiple‐specimen experimental method in determining the energy release rate. The hybrid method computes the P–Δ curves from multiple finite element (FE) models, each with a different crack depth. The experimental procedure measures the P–Δ curve from a standard fracture specimen with a growing crack. The intersections between the experimental P–Δ curve and the numerical P–Δ curves from multiple FE models dictate the LLD levels to compute the strain energy (U) using the area under the numerical P–Δ curves. This method provides accurate estimates of the J resistance data for both SE(B) specimen under mode I loading and single‐edge notched specimens under mixed‐mode I and II loading.