Calculation of stress intensity factors by the force method

The force method is a simple and accurate technique for calculating stress intensity factors (SIFs) from finite element (FE) models, but it has been scarcely used. This paper shows three important advantages of the force method, which make it particularly attractive for designers and researchers. First, it can be employed without special singular quadratic finite elements at the crack tip. Actually, linear reduced integration elements may be used. Second, the force method can be applied to highly anisotropic materials without requiring knowledge of complicated elasticity relations for the stress field around the crack tip. Third, it can handle mixed-mode fracture problems.     

Method for efficient computation of stress intensity factors from weight functions by singular point elimination

Mode I stress intensity factor K_I can be computed by integration of a function representing a stress profile (e.g., variation of stress with depth), modified by an appropriate weight function. Usually, numerical integration is required. However, widely used weight functions cause the end (s) of integration intervals to be singular points, complicating numerical integration. Approaches for computing K_I that deal with singularities by approximating stress profiles by a linear function near a singular point, or transforming a weight function to a form that enables Gauss–Chebyshev integration, are reviewed. As an alternative to those approaches, this study presents a different method for numerical integration involving weight functions. First, a general, variable transformation method to eliminate singularities is introduced. Elimination of singular point enables elementary integration approaches such as Simpson’s rule, as well more involved methods, such as adaptive-Lobatto integration, to be applied. Benchmark tests using a variety of numerical integration formulas show the singular point elimination method to provide accurate, robust and computationally efficient integrations.     

Dislocation-based semi-analytical method for calculating stress intensity factors of cracks: Two-dimensional cases

Determination of the stress intensity factors of cracks is a fundamental issue for assessing the performance safety and predicting the service lifetime of engineering structures. In the present paper, a dislocation-based semi-analytical method is presented by integrating the continuous dislocation model with the finite element method together. Using the superposition principle, a two-dimensional crack problem in a finite elastic body is reduced to the solution of a set of coupled singular integral equations and the calculation of the stress fields of a body which has the same shape as the original one but has no crack. It can easily solve crack problems of structures with arbitrary shape, and the calculated stress intensity factors show almost no dependence upon the finite element mesh. Some representative examples are given to illustrate the efficacy and accuracy of this novel numerical method. Only two-dimensional cases are addressed here, but this method can be extended to three-dimensional problems.     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

Multiple penny-shaped cracks interaction in a finite body and their effect on stress intensity factor

In this paper we investigate the stress intensity factors (SIFs) of multiple penny-shaped cracks in an elastic solid cylinder under mode I (axial tension) loading. The cracks are located symmetrically and in parallel to one another in the isotropic cylinder. The fractal-like finite element method (FFEM) is employed to study the interaction of multiple cracks and to demonstrate the efficiency of the FFEM for multiple crack problems. The results show that the SIF values of the inner cracks, which are denoted as crack number 1,2,3,…,(n+1)/2 of a stack of n parallel cracks, are lower than the SIF values of a single crack by between 16% and 48%. Also, the outermost crack, that is the crack closest to the boundaries of a multiple cracked body, has the highest SIF values and is, therefore, likely to fail first.     

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.     

Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients

This paper proposed numerical methods to determine the multiple stress singularities (including the oscillatory stress singularities) and the related stress intensity coefficients, by the use of common numerical solutions (stresses or displacements) obtained by an ordinary numerical tool such as finite element method (FEM) or boundary element method (BEM). To verify the efficiency of the present methods, two models of bonded dissimilar materials under the plane strain state are analyzed by BEM, and the orders of the stress singularities and the related stress intensity coefficients are examined numerically. The results show that all the orders of the stress singularities at an interface edge can be determined precisely by the present method, and the related stress intensity coefficients can be determined by the extrapolation method with a very good linearity. It is found that the methods presented in this paper are very simple and efficient. Moreover, they can be easily extended to any singular problem.     

Stress intensity factors in spot welds

This paper looks at stress intensity factors of cracks in resistance spot welded joints. Stress intensity factors have been used in the past to predict fatigue crack propagation life of resistance spot welds. However, the stress intensity factors from all previous work was based on assumed initial notch cracks at the nugget, parallel to the sheets. Physical evidence shows, however, that fatigue cracks in spot welds propagate through the thickness of the sheets rather than through the nugget. In this work, stress intensity factors of assumed notch cracks and through thickness cracks in tensile shear (TS) and modified coach peel (MCP) specimens were determined by the finite element method. The finite element results from the assumed notch cracks were compared with the results in the literature and were found to be in agreement with the results from Zhang’s equations [Int. J. Fract. 88 (1997) 167]. The stress intensity factors of assumed notch cracks were found to be different from those of through thickness cracks. To date, no analytic equations for stress intensity factors of through thickness cracks in spot welds have been published. In the current work, simple equations are proposed to estimate the K_I and K_II values of through thickness cracks in TS and MCP specimens.     

Evaluating stress intensity factors due to weld residual stresses by the weight function and finite element methods

This paper presents a study on the application of the weight function and finite element methods to evaluate residual stress intensity factors in welded test samples. Three specimen geometries and various residual stress profiles were studied. Comparisons of the two different methods were made in terms of the accuracy, easiness to use, conditions and limitations. Calculated residual stress intensity factors by the two different methods are in general in good agreement for all the configurations studied. Computational issues involved in executing these methods are discussed. Some practical issues are also addressed, e.g. treatment of incomplete or limited residual stress measurements, influence of transverse residual stresses, and modelling residual stress in short-length specimens. The finite element method is validated by well-established weight functions and thus can be applied to complex geometries following the procedures recommended in this paper.     

Stress intensity factors for functionally graded solid cylinders

This paper presents the mode I stress intensity factors for functionally graded solid cylinders with an embedded penny-shaped crack or an external circumferential crack. The solid cylinders are assumed under remote uniform tension. The multiple isoparametric finite element method is used. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors. The influence for embedded cracks is quite different from that for external cracks.     

Radial locations of strain gages for accurate measurement of mode I stress intensity factor

Strain gage methods are popular in experimental determination of stress intensity factors (SIFs). Radial location of gages with respect to the crack tip plays an important role in accuracy of strain measurements and thus accurate determination of SIFs. The present work proposes a finite element based simple, accurate and consistent method for determination of the limiting value of the radial distance (r_max) of a strain gage. This parameter is in turn useful in deciding the valid strain gage location for accurate measurement of opening mode SIF. The results obtained from the present investigation agree well with the theoretical predictions and could be used for experimental determination of SIFs for both single ended and double ended cracked specimens. The r_max values of center cracked and edge cracked plates with different crack length to width ratio are estimated. The results of the present investigation show that the relative size of the crack length and net ligament length strongly influences the r_max value and the effect of Poisson’s ratio is marginal on the r_max value.     

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.     

Optimum strain gage location for evaluating stress intensity factors in single and double ended cracked configurations

A general methodology has been proposed for calculation of optimum radial strain gage locations for measurement of the stress intensity factors using strain gage technique. The upper bound (r_max) of strain gage locations for complex single ended and double ended cracked configurations has been determined using the proposed method. Further, dependency of the r_max on the crack length to width ratio and on the state of stress is investigated. Numerical results obtained from the present investigation are observed to be in accordance with the theoretical predictions. Using the proposed approach the correctness of strain gage locations used by the earlier researchers is also verified.     

Noise-robust stress intensity factor determination from kinematic field measurements

Stress intensity factors are often estimated numerically from a given displacement field through an interaction integral formalism. The latter method makes use of a weight, the virtual crack extension field, which is under-constrained by first principles. Requiring a least noise sensitivity allows one to compute the optimal virtual crack extension. Mode I and mode II specialized fields are obtained and particularized for a given displacement functional basis. The method is applied to an experimental case study of a crack in a silicon carbide sample, whose displacement field is obtained by a digital image correlation technique. The optimization leads to a very significant uncertainty reduction up to a factor 100 of the non-optimized formulation. The proposed scheme reveals additional performances with respect to the integral domain choice and assumed crack tip geometry, which are shown to have a reduced influence.     

Numerical analysis of the stress intensity factors for repaired cracks from a notch with bonded composite semicircular patch

In this paper, we investigated the crack growth behaviour of cracked thin aluminium plate repaired with bonded composite patch. The finite element method is used to study the performance of the bonded composite reinforcement or repair for reducing the stress concentration at a semicircular lateral notch and repairing cracks emanating from this kind of notch. The effects of the adhesive properties and the patch size on the stress intensity factor variation at the crack tip in mode I were highlighted. The obtained results show that the stress concentration factor at the semicircular notch root and the stress intensity factor of a crack emanating from notch are reduced with the increase of the diameter and the number of the semicircular patch. The maximal reduction of stress intensity factor is about 42% and 54%, respectively, for single and double patch. However, the gain in the patch thickness increases with the increase of the crack length and it decreases when the patch thickness increases. The adhesive properties must be optimised in order to increase the performance of the patch repair or reinforcement.     

Static finite element stress intensity factors for annular cracks

Results of finite element static stress intensity factor calculations for an annular crack around a spherical inclusion (void) are presented and compared with those from approximate analytical methods.     

An interaction integral method for computing mixed mode stress intensity factors for curved bimaterial interface cracks in non-uniform temperature fields

In finite element analysis the interaction integral has been a useful tool for computing the stress intensity factors for fracture analysis. This work extends the interaction integral to account for non-uniform temperatures in the calculation of stress intensity factors for three dimensional curvilinear cracks either in a homogeneous body or on a bimaterial interface. First, the derivation of the computational algorithm, which includes the additional terms developed by the non-zero gradient of the temperature field, is presented in detail. The algorithm is then implemented in conjunction with commercial finite element software to calculate the stress intensity factors of a crack undergoing non-uniform temperatures on both a homogeneous and a bimaterial interface. The numerical results displayed path independence and showed excellent agreement with available analytical solutions.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Determining stress intensity factors of composites using crack opening displacement

The main purpose of this paper is to find the mixed-mode stress intensity factors of composite materials using the crack opening displacement (COD). First, a series solution of the composite material with a crack was used to evaluate COD values. Then, the least-squares method was used to calculate mixed-mode stress intensity factors. This algorithm can be applied to any method that generates or measures COD values. The major advantage of this method is that COD values very near the crack tip are not necessary. Both finite element simulations and laboratory experiments were applied to validate this least-squares method with acceptable accuracy if the even terms of the series solution are removed.     

Experimental calculation of mixed-mode notch stress intensity factors for anisotropic materials

This study developed a least-squares method to find the notch stress intensity factors (SIFs) of anisotropic materials using image-correlation experiments without the requirement of smooth procedures. Complex displacement functions are deduced into a least-squares form, and then displacements from the image-correlation experiments are substituted into the least-squares equation to evaluate the notch SIFs for anisotropic materials. Experimental results compared with the H-integral and finite element analyses show that the SIFs evaluated from the current method are acceptably accurate if more than five displacement terms are included. Moreover, the least-squares SIFs are not sensitive to the maximum or minimum radius of the area from which data is included, so experimental data very near the notch tip is not necessary.