Error estimation of stress intensity factors for mixed‐mode cracks

This paper presents an a posteriori error estimator for mixed‐mode stress intensity factors in plane linear elasticity. A surface integral over an arbitrary crown is used for the separate calculation of the combined mode's stress intensity factors. The error in the quantity of interest is based on goal‐oriented error measures and estimated through an error in the constitutive relation. Copyright © 2006 John Wiley & Sons, Ltd.     

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

This study presents a novel development of a new semi‐analytical method with diagonal coefficient matrices to model crack issues. Accurate stress intensity factors based on linear elastic fracture mechanics are extracted directly from the semi‐analytical method. In this method, only the boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw–Curtis numerical integration result in diagonal Euler's differential equations. Consequently, when the local coordinates origin is located at the crack tip, the stress intensity factors can be determined directly without further processing. In order to present infinite stress at the crack tip, a new form of nodal force function is proposed. Validity and accuracy of the proposed method is fully demonstrated through four benchmark problems, which are successfully modeled using a few numbers of degrees of freedom. The numerical results agree very well with the analytical solution, experimental outcomes and the results from existing numerical methods available in the literature.     

On the I–II mixed mode fracture of granite using four‐point bend specimen

Four‐point bend experiments on black granite are conducted. The fracture behaviours of granite under pure mode I, pure mode II and I–II mixed mode are investigated, and the corresponding stress intensity factors K_I , K_II and the non‐singular term T‐stress are obtained through numerical–experimental method. The results are compared with the theoretical predictions of generalized maximum tangential stress criterion and other conventional criteria. It shows that generalized maximum tangential stress criterion fits the experimental results better for considering the effect of T‐stress. Contrasting with other loading configurations, the values of T‐stress for asymmetric four‐point bend specimens are much smaller, especially for pure mode II specimens, which provide an asymmetric deformation field where the T‐stress is approaching zero.     

Three‐dimensional mixed‐mode (I and II) crack‐front fields in ductile thin plates — effects of T‐stress

It has been well‐established that the non‐singular T‐stress provides a first‐order estimate of geometry and loading mode (e.g. tension versus bending) effects on elastic–plastic crack‐front field under mode I loading conditions. The objective of this paper is to exam the T‐stress effect on three‐dimensional (3D) crack‐front fields under mixed‐mode (modes I and II) loading. To this end, detailed 3D small strain, elastic–plastic simulations are carried out using a 3D boundary layer (small‐scale yielding) formulation. Characteristics of near crack‐front fields are investigated for a wide range of T‐stresses (T/σ₀ = ?0.8, ?0.4, 0.0, 0.4, 0.8). The plastic zones and thickness and angular and radial variations of the stresses are studied, corresponding to two values of the remote elastic mixity parameters M_e = 0.3 and 0.7, under both low and high levels of applied loads. It is found that different T‐stresses have a significant effect on the plastic zones size and shapes, regardless of the mode mixity and load level. The thickness, angular and radial distributions of stresses are also affected markedly by T‐stress. It is important to include these effects when investigating the mixed‐mode ductile fracture failure process in thin‐walled structural components.     

On the use of an anti‐symmetric four‐point bend specimen for mode II fracture experiments

The edge‐cracked beam specimen subjected to anti‐symmetric four‐point bend (ASFPB) loading has been conventionally used in the past for investigating the pure mode II fracture experiments in many engineering materials. However, it is shown through finite element analysis that the ASFPB specimen sometimes fails to produce pure mode II conditions. For anti‐symmetric loads applied close to the crack line, there are considerable effects from K_I and T‐stress in the ASFPB specimen. Pure mode II is provided only when the applied loads are sufficiently far from the crack plane.     

Computation of the stress intensity factors of sharp notched plates by the fractal‐like finite element method

The fractal‐like finite element method (FFEM) is an accurate and efficient method to compute the stress intensity factors (SIFs) of different crack configurations. In the FFEM, the cracked/notched body is divided into singular and regular regions; both regions are modelled using conventional finite elements. A self‐similar fractal mesh of an ‘infinite’ number of conventional finite elements is used to model the singular region. The corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co‐ordinates after performing a global transformation by using global interpolation functions. In this paper, we extend this method to analyse the singularity problems of sharp notched plates. The exact stress and displacement fields of a plate with a notch of general angle are derived for plane‐stress/strain conditions. These exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the SIFs. The use of the global interpolation functions reduces the computational cost significantly and neither post‐processing technique to extract SIFs nor special singular elements to model the singular region are needed. The numerical examples demonstrate the accuracy and efficiency of the FFEM for sharp notched problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical and experimental investigation of mixed‐mode fracture parameters on silicon nitride using the Brazilian disc test

Engineering applications of ceramics can often involve mixed‐mode conditions involving both tensile and shear loading. Mixed‐mode fracture toughness parameters are evaluated for applicability to ceramics using the Brazilian disc test on silicon nitride. Semi‐elliptical centrally located surface flaws are induced on the disc specimens using Vickers indentation and compression loaded to fracture with varying levels of mode mixity. The disc specimens are modelled via 3D finite element analysis and all three modes of stress intensity factors computed along the crack front, at failure load. We present a numerical and experimental investigation of four widely used mixed‐mode fracture criteria and conclude that the critical strain energy release rate criterion is simple to implement and effective for silicon nitride under mixed‐mode conditions.     

A High‐order extended finite element method for extraction of mixed‐mode strain energy release rates in arbitrary crack settings based on Irwin's integral

An analytical formulation based on Irwin's integral and combined with the extended finite element method is proposed to extract mixed‐mode components of strain energy release rates in linear elastic fracture mechanics. The proposed formulation extends our previous work to cracks in arbitrary orientations and is therefore suited for crack propagation problems. In essence, the approach employs high‐order enrichment functions and evaluates Irwin's integral in closed form, once the linear system is solved and the algebraic degrees of freedom are determined. Several benchmark examples are investigated including off‐center cracks, inclined cracks, and two crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, because of its analytical nature, no special post‐processing is required. Thus, we conclude that this method may provide a good and simple alternative to the popular J‐integral method. In addition, it may circumvent some of the limitations of the J‐integral in 3D modeling and in problems involving branching and coalescence of cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

An accurate scheme for mixed‐mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models

The interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed‐mode stress intensity factors (SIFs). The present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions (e.g. exponentially graded materials) or micromechanics models (e.g. self‐consistent, Mori–Tanaka, or three‐phase model). In the latter case, there is no closed‐form expression for the material‐property variation, and thus several quantities, such as the explicit derivative of the strain energy density, need to be evaluated numerically (this leads to several implications in the numerical implementation). The SIFs are determined using conservation integrals involving known auxiliary solutions. The choice of such auxiliary fields and their implications on the solution procedure are discussed in detail. The computational implementation is done using the finite element method and thus the interaction energy contour integral is converted to an equivalent domain integral over a finite region surrounding the crack tip. Several examples are given which show that the proposed method is convenient, accurate, and computationally efficient. Copyright © 2003 John Wiley & Sons, Ltd.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

Calculation of mixed mode (I/II) stress intensities at sharp V‐notches using finite element notch opening and sliding displacements

In this paper, a simple, robust, and an efficient technique has been proposed for accurate estimation of mixed mode (I/II) notch stress intensity factors (NSIFs) of sharp V‐notched configurations using finite element notch opening and sliding displacements at the selected number of nodes along the notch flanks. Unlike the crack problems, displacement field is rarely employed in the notch problems due to complexities introduced by the presence of rigid body displacements. One of the main emphasis of the present work is to neatly bypass these rigid body displacements and develop a simple approach for accurate computation of the NSIFs so that it can be easily incorporated in the existing code. Several benchmark problems have been analyzed. The results obtained using the present method show excellent agreement with the solutions available in the literature. Some new results have also been reported in the present work.     

Weight functions for the determination of stress intensity factor and T‐stress for semi‐elliptical cracks in finite thickness plate

This paper presents the application of weight function method for the calculation of stress intensity factors (K) and T‐stress for surface semi‐elliptical crack in finite thickness plates subjected to arbitrary two‐dimensional stress fields. New general mathematical forms of point load weight functions for K and T have been formulated by taking advantage of the knowledge of a few specific weight functions for two‐dimensional planar cracks available in the literature and certain properties of weight function in general. The existence of the generalised forms of the weight functions simplifies the determination of specific weight functions for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalised weight function expression. These unknown parameters can be determined from reference stress intensity factor and T‐stress solutions. This method is used to derive the weight functions for both K and T for semi‐elliptical surface cracks in finite thickness plates, covering a wide range of crack aspect ratio (a/c) and relative depth (a/t) at any point along the crack front. The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.     

Extraction of the generalized stress intensity factor in gross sliding complete contacts using a path‐independent integral*

A new procedure to extract the generalized stress intensity factor (GSIF) in full sliding complete contact problems is proposed. It is based on a path‐independent integral derived from the reciprocal theorem of Maxwell‐Betti and on a specially conceived extraction field. This extraction field avoids the computation of path integrals along the contact plane that start from the singular point. The methodology presented here is also applicable to configurations with large relative displacements of the indenter (gross sliding). The procedure overcomes the problems that arise when extracting the GSIF via the stress extrapolation technique, because it does not need highly refined meshes, and it can be applied far from the singularity‐dominated zone and with much less user intervention. The robustness of the method has been verified with numerical examples, studying the effect of large displacements of the indenter and different discretizations.     

Experimental and numerical investigation of the effect of temperature on mixed‐mode fracture behaviour of AM60 Mg alloy

The aim of present paper is to experimentally investigate mixed‐mode fracture behaviour of AM60 Mg alloy at low and elevated temperatures. For this purpose, mode I, 45° mixed‐mode, and mode II tests were conducted using a modified version of Arcan device at three different temperatures. An elastic‐plastic finite element model was used to extract necessary geometric parameters. Crack resistance curves (J‐R) and critical J‐integral of the material were extracted. The results indicated that, for all loading modes, maximum critical J‐integral value was observed at ambient temperature and decreased by either increasing or decreasing the temperature. It was observed that effect of temperature on fracture behaviour is much larger at temperatures above 0°C rather than sub‐zero temperatures. By changing the loading angle to go from mode I to mode II, a decreasing trend was observed in the values of critical fracture parameters at all temperatures. Finally, the surfaces were examined using scanning electron microscopy (SEM).     

An investigation on the effects of equal channel angular pressing on the mixed‐mode fracture toughness and mechanical properties of 6063 aluminium alloy

In this study, effects of equal channel angular pressing (ECAP) on the mixed‐mode fracture toughness of Al‐6063 were investigated. The ECAP process continued up to 5 passes without failure. Grain refinement was obvious after 5 passes of the ECAP process. The average grain size reduced from 45μm to less than 1μm, and textural studies shows aligning the grains in known directions. After 4 passes, yield and ultimate strengths increase respectively from 100 and 209 MPa to 300 and 375 MPa and reduction in elongation was also observed. The microhardness improved after the process. The fracture toughness for different orientations was measured. For pure mode I (opening mode), its value decreased after the first pass from 18.4 to 15.71  ; however, it increased to about 18.8  after the fifth pass. For mixed‐mode loading condition, different orientations were investigated. The results revealed different fracture toughness reductions after the first passes of the process for specimens with different orientations. The fracture surfaces were studied by using scanning electron microscope, and refined equiaxed dimples were observed after the ECAP process.     

Non‐planar mixed‐mode growth of initially straight‐fronted surface cracks,in cylindrical bars under tension,torsion and bending,using the symmetric Galerkin boundary element method‐finite element method alternating method

In this paper, the stress intensity factor (SIF) variations along an arbitrarily developing crack front, the non‐planar fatigue‐crack growth patterns, and the fatigue life of a round bar with an initially straight‐fronted surface crack, are studied by employing the 3D symmetric Galerkin boundary element method‐finite element method (SGBEM‐FEM) alternating method. Different loading cases, involving tension, bending and torsion of the bar, with different initial crack depths and different stress ratios in fatigue, are considered. By using the SGBEM‐FEM alternating method, the SIF variations along the evolving crack front are computed; the fatigue growth rates and directions of the non‐planar growths of the crack surface are predicted; the evolving fatigue‐crack growth patterns are simulated, and thus, the fatigue life estimations of the cracked round bar are made. The accuracy and reliability of the SGBEM‐FEM alternating method are verified by comparing the presently computed results to the empirical solutions of SIFs, as well as experimental data of fatigue crack growth, available in the open literature. It is shown that the current approach gives very accurate solutions of SIFs and simulations of fatigue crack growth during the entire crack propagation, with very little computational burden and human–labour cost. The characteristics of fatigue growth patterns of initially simple‐shaped cracks in the cylindrical bar under different Modes I, III and mixed‐mode types of loads are also discussed in detail.     

Classical and Bayesian inference on progressive‐stress accelerated life testing for the extension of the exponential distribution under progressive type‐II censoring

In this paper, a progressive‐stress accelerated life test under progressive type‐II censoring is considered. The cumulative exposure model is assumed when the lifetime of test units follows an extension of the exponential distribution. The maximum likelihood and Bayes estimates of the model parameters are obtained. The approximate and credible confidence intervals of the estimators are derived. Furthermore, a real lifetime data set is analyzed to illustrate the proposed procedures. Finally, the simulation studies are used to compare between 2 different designs of the progressive‐stress test (simple and multiple ramp‐stress tests).     

Calculation of the crack tip parameters in the holed‐cracked flattened Brazilian disk (HCFBD) specimens under wide range of mixed mode I/II loading

In this paper, the crack tip parameters including the stress intensity factors (K_I and K_II), T‐stress and the third terms of the stress field (A₃ and B₃) are determined comprehensively for a disk‐type sample named holed‐cracked flattened Brazilian disk (HCFBD) under various combinations of mode I and mode II loading. The HCFBD specimen is a circular disk containing a central hole in which the initial cracks are created radially from the hole circumference. Moreover, the ends of HCFBD are flattened for the sake of convenient loading. Performing enormous finite element analyses and calculating the stress intensity factors K_I and K_II, the states of pure mode II are determined for different configurations of HCFBD. Furthermore, the sign and magnitude of parameter A₃ which plays an important role to justify the geometry and size effects on the fracture toughness of quasi‐brittle materials are also determined for HCFBD with different geometrical ratios.