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 factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

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.     

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.     

Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials

Explicit analytical expressions are obtained for the longitudinal and transverse effective spring stiffnesses of a planar periodic array of collinear cracks at the interface between two dissimilar isotropic materials; they are shown to be identical in a general case of elastic dissimilarity (the well-known open interface crack model is employed for the solution). Since the interfacial spring stiffness can be experimentally determined from ultrasound reflection and transmission analysis, the proposed expressions can be useful in estimating the percentage of disbond area between two dissimilar materials, which is directly related to the residual strength of the interface. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness are clearly represented in the solution. It is shown that in general the crack interaction weakly depends on material dissimilarity and, for most practical cases, the crack interaction is nearly the same as that for crack arrays between identical solids. This allows approximate factorization of the effective spring stiffness for an array of cracks between dissimilar materials in terms of an elastic dissimilarity factor and two factors obtained for cracks in a homogeneous material: the effective spring stiffness for non-interacting (independent) cracks and the crack interaction factor. In order to avoid the effect of the crack surface interpenetration zones on the effective spring stiffness, the range of the tensile to transverse load ratios is obtained under the assumption of small-scale contact conditions. Since real cracks are often slightly open (due to prior loading history and plastic deformation), it is demonstrated that for ultrasound applications the results obtained are valid for most practical cases of small interfacial cracks as long as the mid-crack opening normalized by the crack length is at least in the order of 10⁻⁵.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.     

A finite element modelling for directly determining intensity factors of piezoelectric materials with cracks

Recently, authors(Cao et al., Acta Aeronautica et Astronautica Sinica 25(5): 470–472, 2004) extended the singular crack element originally introduced by Wang et al. (Eng Fract Mech 37(6):1195–1201, 1990) for evaluating the stress intensity factors (SIFs). Extensive studies have proved the versatility and accuracy of the element. This study is to show the versatility of the element for piezoelectric materials. In this paper, electric potential and displacement fields near a crack tip of piezoelectric materials are first used to construct a finite element version for directly determining intensity factors of piezoelectric materials with cracks. A singular finite element is constituted and a new method to calculate intensity factors of piezoelectric materials with cracks is obtained without any post-processing procedures. Detailed derivations are given and the results obtained with present method are good agreement with those of theoretical results, the FEM data by ANSYS and singular electromechanical crack tip elements. The results to the different selections of the structural dimensions are carried out. Numerical examples demonstrate the accuracy and validity of the novel element of present method.     

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.     

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.     

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.     

Stress intensity factors for surface cracks at countersunk holes

Fatigue crack growth from countersunk fastener holes loaded in remote tensile loading was studied using the transparent polymer PMMA. A single edge corner crack at the bottom of the plate and a single internal surface crack at the sharp intersection between the bore and the countersink were induced in the PMMA specimens by pre-cracking. The specimens were then fatigue tested under constant amplitude remote tensile loading and the ‘back-calculation’ method was used to determine stress intensity factors at several crack front locations. When variations in fatigue crack closure were taken into account, the experimental stress intensity factors agreed well with the computational results at selected crack fronts.     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

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.     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

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 analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

Parametric equations for T-butt weld toe stress intensity factors

This paper describes the generation of parametric equations for weld toe stress intensity factors. The methodology employed used a two-dimensional finite element analysis to evaluate the ‘crack opening’ stress distribution in the uncracked plane of T-butt geometries. This was then used as input into a dedicated weight function solution for the determination of stress intensity factors. The final parametric equations describe the stress intensity factor distributions for tension and bending as a function of plate thickness, weld attachment width, weld angle, weld root radius, crack length and crack shape. The equations are compared and validated against a wide spectrum of published values and appear by comparison accurate and wide ranging. The validation exercise uncovered situations where present design guidance is unconservative.     

Stress intensity factors for cracks in bolted joints

The mixed-mode stress intensity factors (SIFs) of the bolted joint with single and double cracks were examined. Changes in friction, clearance, applied force and crack angle were included in the nonlinear contact finite element analysis. A fine mesh was made between the contact surface and the crack tip in order to obtain an accurate solution. The least-squares method was used to determine the mixed-mode SIFs. Finite element results indicate that reasonable changes in the applied force, frictional coefficient and the clearance will not make significant changes in the normalized SIFs. The pure opening mode for cracked bolted joints does not occur at the horizontal crack but occurs at the crack with the crack angle between 0° and 22.5°. Nevertheless, using the SIF for a horizontal crack as the maximum opening-crack mode is sufficiently reliable. The maximum mode-II crack is approximately at a crack angle of 45° for both isotropic and orthotropic materials; however, at that angle the maximum mode-II SIF is only about one half of the mode-I SIF. This revised version was published online in August 2006 with corrections to the Cover Date.     

Variation of mixed modes stress intensity factors of an inclined semi-elliptical surface crack

In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D inclined semi-elliptical surface crack in a semi-infinite body under tension. The stress field induced by displacement discontinuities in a semi-infinite body is used as the fundamental solution. Then, the problem is formulated as a system of integral equations with singularities of the form r ^–3. In the numerical calculation, the unknown body force doublets are approximated by the product of fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately for various geometrical conditions. The effects of inclination angle, elliptical shape, and Poisson's ratio are considered in the analysis. Crack mouth opening displacements are shown in figures to predict the crack depth and inclination angle. When the inclination angle is 60 degree, the mode I stress intensity factor F _I has negative value in the limited region near free surface. Therefore, the actual crack surface seems to contact each other near the surface.     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.