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.     

Mode I stress intensity factor estimate for a half-penny shaped crack in a trilayer structure by reduction to a bilayer model

The stress intensity factor (SIF) of a half-penny shaped crack normal to the interface in the top layer of a three-layer bonded structure is obtained by the finite element method for a wide range of parameters. To obtain a simple estimate of the SIF, the method of reduction of an idealized cracked trilayer domain to that of a corresponding bilayer domain has been introduced based on the notion of an equivalent homogeneous material substitution for the two bottom layers. The results obtained are utilized in estimating the SIF of a small crack at the interface in a trilayer structure subjected to an indentation load based on the stress calculations in a corresponding uncracked structure. The simplification method may be useful in predicting brittle failure initiating from interfacial flaws in layered structural components with complex geometries that would normally require extensive computational modeling.     

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.     

Stress intensity factor solutions for cracks in finite-width three layer laminates with and without residual stress effects

Approximate stress intensity factor solutions for cracks in finite-width three layer laminates, with the crack located in the middle layer, were derived on the basis of force-balance between the applied stress and the modified Westergaard form of normal stress distribution ahead of the crack tip. This yielded a simple and closed form equation for the stress intensity factor that included the effects of the ratio of the moduli of the layers and the relative layer thicknesses. A comparison of the stress intensity factor values from this equation and with finite element data indicated that the difference between these two data sets was small for most of the crack lengths and the modulus ratio of the layers. The maximum difference occurred at crack lengths approaching the interface and at high moduli ratios, but was less than 10%, in general. The equations were also modified to incorporate the effects of residual stresses that arise during cooling after laminate processing, on the stress intensity factor. A comparison of the analytical data with the finite element data obtained by imposing thermal and mechanical boundary loads on the laminate specimens indicated a good agreement. The present closed form approximate solutions may be useful in fracture analyses of finite-width laminates containing cracks.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

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.     

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.     

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 finite element based interior collocation method for the computation of stress intensity factors and T-stresses

A new, simple and efficient method for simultaneous estimation of the mixed-mode stress intensity factors (SIFs) and T-stresses using finite element computations is proposed in this paper. The current work is based on the formation of overdetermined system of equations using the displacement components near the crack tip. The proposed method can be easily implemented in the existing finite element codes. The results obtained from the present investigation for plane stress problems are validated by comparing with the published results and found to be in very good agreement with them.     

Stress intensity factors of a central interface crack in a bonded finite plate and periodic interface cracks under arbitrary material combinations

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary material combinations. This paper deals with a central interface crack in a bonded finite plate and periodic interface cracks. Then, the effects of material combination and relative crack length on the stress intensity factors are discussed. A useful method to calculate the stress intensity factor of interface crack is presented with focusing on the stress at the crack tip calculated by the finite element method.     

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.     

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.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.     

Numerical methods for determination of stress intensity factors of singular stress field

The asymptotic solution of the singular stress field near a singular point is generally comprised of one or more singular terms in the form of Kr^λ-1f_ij(θ). Based on the asymptotic solution of the singular stress field and the common numerical solution (stresses or displacements) obtained by an ordinary tool such as the finite element method or boundary element method, a simple and effective numerical method is developed to calculate stress intensity factors for one and two singularities. Three examples show that the stress intensity factors evaluated using the method proposed in this paper are very accurate.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Application of the substructured finite element/extended finite element method (S-FE/XFE) to the analysis of cracks in aircraft thin walled structures

The substructured finite element/extended finite element (S-FE/XFE) approach is used to compute stress intensity factors in large aircraft thin walled structures containing cracks. The structure is decomposed into a ‘safe’ domain modeled with classical shell elements and a ‘cracked’ domain modeled using three-dimensional extended finite elements. Two applications are presented and discussed, supported by validation test cases. First a section of stiffened panel containing a through-thickness crack is investigated. Second, small surface cracks are simulated in the case of a generic ‘pressure membrane’ with realistic crack configurations. These two semi-industrial benchmarks demonstrate the accuracy, robustness and computational efficiency of the substructured finite element/extended finite element approach to address complex three-dimensional crack problems within thin walled structures.     

Stress intensity factor solutions for cracks in railway axles

The aim of this paper is a collection of stress intensity factor solutions for cracks in railway axle geometries which the authors of the present special issue developed and/or used for damage tolerance analyses. These solutions comprise closed form analytical as well as tabled geometry functions and they refer to solid as well as hollow axles and various crack sites such as the T- and V-notch and the axle body.     

Stress intensity factor computation using the method of fundamental solutions

In this paper, we study the application of the method of fundamental solutions to the computation of stress intensity factors in linear elastic fracture mechanics. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity and the leading terms for the displacement near the crack tip. The applicability of two formulations of the method is demonstrated on two mode I crack problems, where it is shown that accurate approximations for the stress intensity factors can be obtained with relatively few degrees of freedom. Parts of this work were undertaken while the first author was a Visiting Professor in the Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A.     

Piezoelectric sensor for stress intensity factor measurement of two dimensional cracks

A piezoelectric sensor for the measurement of stress intensity factors (SIFs) of two dimensional cracks induced in a structure is developed. Two small pieces of piezoelectric elements are adhered near the crack tip so that the piezoelectric elements are placed close to each other and the crack tip’s position is between them. The electric currents from the piezoelectric elements are integrated by integration circuits and the output voltages which are proportional to the electric charge induced in the piezoelectric elements are measured. The SIFs of Mode I (K_I) as well as of Mode II (K_II) based on the piezoelectric constitutive law and fracture mechanics are calculated.     

A direct method for calculating the stress intensity factor in BEM

The proposed algorithm employs singular crack tip elements in which the stress intensity factor appears as a degree of freedom. The additional degrees of freedom are compensated by constraint conditions which originate from imposing continuity across elements and a contour integration formula. The two benchmark problems indicate the proposed algorithm can accurately predict the stress intensity factor and the distribution of the primary and secondary variables in fracture problems.