Double Edge Cracks Research Articles

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.

Computation of stress intensity factors ( KI, KII) and T-stress for cracks reinforced by composite patching

To increase the operational life of defected structures, a repairing method using composite patches has been used to reinforce cracked components. Due to various advantages of composite materials, this method has received much attention from researchers and engineers. Considerable investigations have been performed to highlight the effect of bonded composite patches on the fracture parameters such as stress intensity factors (SIF) and J-integral. However the effect of composite patches on the T-stress, the constant stress term acting parallel to the crack, has not been investigated in the past. In this paper, the finite element method is carried out to analyze the effect of bonded composite patches for repairing cracks in pure mode I and also mixed mode I/II conditions, by computing the stress intensity factors and the T-stress, as functions of the crack length, the crack inclination angle and the type of composite material. In pure mode I condition, the finite element analysis is carried out for three different specimens: centre crack, double edge crack and single edge crack specimens. For mixed mode I/II condition the analysis is conducted on an inclined central crack of various slant angles. For both pure mode I and mixed mode I/II, the numerical results show that composite patching has considerable effect on the T-stress.

On a new method for calculating stress-intensity factors of multiple edge cracks

Stress-intensity factors for multiple edge cracks in bars and beams have been calculated using geometrical considerations to provide quick and easy-to-use estimates. The research concentrates on symmetric double and triple edge crack configurations. The results have been verified by FE calculations. Furthermore, the analytical estimate has been improved based on numerical results.

Stress intensity factors for radial cracks in annular and solid discs under constant angular velocity

The finite element method has been used to predict the stress intensity factors for single- and double-edge cracks in six annular and solid rotating discs under constant angular velocity. Linear elastic fracture mechanics finite element analyses have been performed and the results are presented in the form of crack configuration factors for a wide range of component and crack geometry parameters. These parameters are chosen to be representative of typical practical situations and have been determined from evidence presented in the open literature. The extensive range of crack configuration factors obtained from the analyses are then used to obtain equivalent prediction equations using a statistical multiple non-linear regression model. The accuracy of this model is measured using a multiple coefficient of determination, R2, where 0 ≤ R2 ≤ 1. This coefficient is found to be greater than or equal to 0.98 for all cases considered in this study, demonstrating the quality of the model fit to the data. Predictive equations for stress intensity factors enable designers to predict the fatigue life of these components easily. It is also suggested that one of the component configurations (i.e. the cracked slit rotating disc) can be selected as a suitable experimental sample to measure the real fracture toughness of rotating components, using the relevant predictive equation presented in this study. Finally a fracture criterion is also suggested graphically to determine the limit load value of angular velocity for similar rotating disc components.

Application of virtual crack closure integral method for interface cracks in low- k integrated circuit devices under thermal load

The virtual crack closure integral (VCCI) method is used to evaluate the stress intensity factor (SIF) and energy release rate (ERR) of an interface crack under thermal load. The VCCIs used in this work include the traditionally known “Mode I” and “Mode II” VCCIs and an additional coupling VCCI. The singularity element is used in the finite element method (FEM) implementation. The SIF and ERR calculated by the FEM are compared to the exact solution in the case of a joint dissimilar semi-infinite plates with double edge crack under thermal loading. The FEM result agrees well with the exact solution for relatively coarse meshes. The contribution of the mesh density and material mismatch to the FEM error is also explored. The VCCI method is used with the multi-scale FEM in a delamination risk assessment of a low- k integrated circuits device in flip-chip plastic ball grid array packages. The ERR is calculated for different package configurations and the prediction of the delamination risk is confirmed by reliability tests.

Crack initiation and direction for circumferential periodic cracks in pipe under tension and torsion

The S-theory is applied to determine crack initiation angle and critical load of circumferential periodic cracks in pipe. A technique for determining mode II stress intensity factor is proposed by using G ∗ path independent integral. The method applies also to single-edge crack, double-edge crack and center crack cylindrical panel configurations.

Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions

Finite element analyses have been conducted to calculate elastic T-stress solutions for cracked test specimens. The T-stress solutions are presented for single edge cracked plates, double edge crack plates and centre cracked plates. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were used to derive weight functions for T-stress for the corresponding specimens. The weight functions for T-stress are then verified against several linear and non-linear stress distributions. The derived weight functions are suitable for the T-stress calculation for cracked specimens under any given stress field.

Stress intensity factors and T-stress for single and double-edge-cracked circular disks under mixed boundary conditions

Stress intensity factors and T-stress solutions for components containing single and double edge cracks were computed predominantly by application of the boundary collocation method, the fracture mechanics weight function and a Green's function for T-stresses. The results are compiled in the form of tables, figures and approximate relations. Different mechanical boundary conditions are considered as pure traction conditions, mixed boundary conditions and pure displacement conditions.

Compendium of yield load solutions for strength mis-matched DE(T), SE(B) and C(T) specimens

In two previous papers, mis-match yield load solutions for middle crack tension and single edge cracked bars in pure bending were derived. These specimens contained cracks which were located in the centre of the weld metal as well as in the interface between weld metal and base plate, and in positions in between. The present paper investigates additional three specimen types, namely double edge crack tension, single edge cracked bend and compact specimens, with cracks in the centre of the weld metal and in the interface between weld metal and base plate. As in the previous investigations, the conditions of plane strain and plane stress are considered. The results are presented as closed form solutions and diagrams.

Stress intensity factors for single and double edge cracks in a simple beam subject to a moving load

A procedure is presented for determining stress intensity factors for single and double edge cracks in simply supported undamped Bernoulli–Euler beams under a moving load. The approach is based on using modal analysis to determine the equivalent load on the beam, then linear elastic fracture mechanics is used to calculate stress intensity factors (SIF). The results show that SIF is a function of time, speed of the moving load and crack size and location.

Single-Edge and Double-Edge Cracks in a Fully Anisotropic Strip

The mode I and II stress intensity factors in a fully anisotropic infinite strip with a single-edge or double-edge crack configuration are obtained from an approach based on the continuous dislocation technique. The elastic solution of a single dislocation in an anisotropic half plane is used in conjunction with an array of dislocations along the boundary of the infinite strip, which is supposed to be traction-free, to provide the solution of a single dislocation in an anisotropic infinite strip. The dislocation densities of the dislocation array are determined in such a way that the traction forces generated by the dislocation array cancel the residual tractions along the boundary due to the single dislocation in the half plane. The stress field of a single dislocation in the infinite strip is thus a superposition of that of the single dislocation and the dislocation array in the half plane. This solution is then applied to calculate the mixed mode I and II stress intensity factors for a single-edge and a double-edge crack in the anisotropic strip, by replacing the cracks with a series of dislocations and satisfying the crack surface traction-free conditions. To illustrate the results, typical material data for graphite/epoxy were used in a unidirectional construction with the fiber orientation, θ, measured from the load direction (perpendicular to the crack direction), varying between 0 and 90 degrees. It is found that the effect of anisotropy on the mode I stress intensity factor is significant between 30 and 60 degrees and depends strongly on the relative crack length, being larger for cracks of relative larger length. The mode mixity, defined such that it is zero for pure mode I and 90 degrees for pure mode II, is significant between 40 and 70 degrees, and is in general between zero and 20 degrees.

A CONTINUOUS CRACKED BEAM VIBRATION THEORY

A continuous cracked beam vibration theory is developed for the lateral vibration of cracked Euler–Bernoulli beams with single-edge or double-edge open cracks. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked beam as a one-dimensional continuum. The displacement field about the crack was used to modify the stress and displacement field throughout the bar. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. The results of two independent evaluations of the lowest natural frequency of lateral vibrations for beams with a single-edge crack are presented: the continuous cracked beam vibration theory developed here, and a lumped cracked beam vibration analysis. Experimental results from aluminum beams with fatigue cracks are very close to the values predicted. A steel beam with a double-edge crack was also investigated with the above mentioned methods, and results compared well with existing experimental data.

Stress intensity factors and T-stress inedge-cracked rectangular plates under mixed boundary conditions

Stress intensity factors and T-stress solutions are available in literature predominantly for cracked components under stress boundary conditions. In order to provide solutions for displacement boundary conditions, rectangular plates with various length-to-width ratios and different types of cracks (single and double-edge cracks) are studied by application of the boundary collocation method. the results are reported in the form of figures and tables.

Energy domain integral applied to solve center and double-edge crack problems in three-dimensions

A three-dimensional Boundary Element Method (BEM) implementation of the energy domain integral for the numerical computation of the crack energy release rate is presented in this paper. The domain expression of the energy release rate is naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using the appropriate boundary integral equations. The pointwise crack energy release rate is evaluated along the three-dimensional crack front over a cylindrical domains that surround a segment of the crack front. The accuracy of the implementation is demonstrated by solving several problems, which include geometries containing straight as well as curved crack fronts.

A Direct Caustics Simulation Method Based on FEM.

In this paper, for the sake of prediction of caustic experiments, explanation and verification of experimental results, a new numerical method based on the finite element method, which can directly simulate the caustic patterns, is presented. Based on this new idea, the program for directly determining the gradient solution from the 2 dimensional displacement solution is presented. It is also proven to be necessary for only 8 or 12 node isoparametric element to apply this new method to the 2 dimensional problem. For verifying the accuracy of this method, the simulations of 4 point bending problem and concentrated load problem are performed. The simulated results have good agreement with the theoretical values. As the applied examples, the simulation is carried out for the plate specimen with a single edge crack, or double edge crack, or a short single edge crack under mode I condition. The changes of caustic image under the influence of the interferometry between cracks and the influence of the boundary are given.

Stress Intensity Factors of Double and Multiple Edge Cracks

In this paper stress intensity factors of double and multiple edge cracks in a semi-infinite plate is considered. The problem is considered by solving a system of hypersingular integral equations based on the body force method. First, two edge cracks A and B having different crack lengths and inclination angles are analyzed and the effect of crack B upon crack A is investigated. Then, stress intensity factor(SIF) of crack A is found to be almost constant independent of the wide variation of inclination angle B if the crack tip B is fixed. Second, periodic edge cracks are systematically analyzed with varying the number, distance and angle of cracks. Then the interaction effect is found to appear mostly from the distances of crack tip independent of the angle. Numerical results are also shown when crack parameters are slightly changed from average values in almost equally spaced multiple edge cracks.

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.

Stress Intensity Factors of Double and Multiple Edge Cracks.

In this paper the interaction of multiple edge cracks in a semi-infinite plate is considered. The problem is formulated as a system of hypersingular integral equations using the stress field due to a force doublet as a fundamental solution. In the numerical calculations, unknown functions are approximated by fundamental density functions and Chebyshev polynomials. First, two edge cracks A and B having different crack lengths and inclination angles are analyzed and the effect of crack B upon crack A is investigated. The stress intensity factor (SIF) of crack A is found to be almost constant independent of inclination angle B if the tip of crack B is fixed. Second, periodic edge cracks are systematically analyzed varying the number, distance and angle of cracks. The interaction effect is found to occur mostly due to the distance independent of the angle. Analytical results are also shown when crack parameters are changed slightly from average values in almost equally spaced multiple edge cracks.

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIFs (stress intensity factors) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral term in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIFs using only one SIF solution, without any modification for the crack geometries symmetric in the two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential cracks or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straightforward with only one SIF solution. The necessary information in the analysis is two reference SIFs. The analysis results using several examples verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.

On the use of displacement extrapolation to obtain crack tip singular stresses and stress intensity factors

A displacement extrapolation method is developed which can be used to obtain crack tip singular stresses and stress intensity factors using only nodal displacements in the first layer of elements around the crack tip. A major benefit with this technique is that no analytical solutions are required. The triangular quarter-point elements are used around the crack tip to obtain nodal displacements with sufficient accuracy. Three geometries are considered: a single edge crack, a central crack and double edge crack. Two loading types are used: uniaxial tension and uniform pressure on the crack surface. Excellent results for singular stresses and displacements are obtained for Mode I crack problems.