Interaction effect of stress intensity factors for any number of collinear interface cracks

In this study, the stress intensity factors for any number of interface cracks are calculated for various spacings, elastic constants and number of cracks and the interaction effect of interface cracks is discussed. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, the unknown functions of the body force densities which satisfy the boundary conditions are expressed by the products of fundamental density functions and power series. Here, the fundamental density functions are chosen to express the stress field due to a single interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers and examining the compliance with boundary conditions. The calculation shows that the present method gives rapidly converging numerical results for those problems as well as ordinary crack problems in homogeneous materials. The interaction effect of interface crack appears in a similar way to ordinary collinear cracks having the same geometrical condition and the maximum stress intensity factor is shown to be linearly related to the reciprocal of number of interface cracks. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

Effect of curvature at the crack tip on the stress intensity factor for curved cracks

In this paper, the numerical solution of the hypersingular integral equation using the body force method in curved crack problems is presented. In the body force method, the stress fields induced by two kinds of standard set of force doublets are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form r ^–2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density functions and power series. The calculation shows that the present method gives rapidly converging numerical results for curved cracks under various geometrical conditions. In addition, a method of evaluation of the stress intensity factors for arbitrary shaped curved cracks is proposed using the approximate replacement to a simple straight crack.     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. 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 mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

The effect of an elliptical inclusion on a crack

The interaction between an elliptical inclusion and a crack is analyzed by body force method. The investigated stress field is simulated by superposing the fundamental solutions for a point force applied at a point in an infinite plate containing an elliptical inclusion. Based on numerical results, effects of the inclusion shape on the crack tip stress intensity factor are discussed. It is found that for small cracks emanating from a stress-higher point on the inclusion interface the stress intensity factors are mainly determined by the stresses, occurring at the crack starting point before the crack initiation, and the inclusion root radius, besides the crack length. However, for the cracks occurring in a stress-lower region around the inclusion, it is difficult to characterize the effect of the inclusion geometry on the stress intensity factors of small cracks by the inclusion root radius alone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I cracks propagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-I crack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface. The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamic fracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite stable. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found to depend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.     

Tension of a cylindrical bar having an infinite row of circumferential cracks

In this paper, the crack problems in the case of a cylindrical bar having a circumferential crack and a cylindrical bar having an infinite row of circumferential cracks under tension are analyzed by the body force method. The stress field for a periodic array of ring forces in an infinite body is used to solve the problems. The solution is obtained by superposing the stress fields of ring forces in order to satisfy a given boundary condition. The stress intensity factors are calculated for various geometrical conditions. The obtained values of stress intensity factor of a single circumferential crack are considered to be more reliable than the results of other paper's. As the crack becomes very shallow, the stress intensity factor of a row of circumferential cracks approaches the value corresponding to that of a row of edge cracks in a semi-infinite plate under tension. As the crack becomes very deep, it approaches the values corresponding to that of a single deep circumferential crack.     

Boundary element analysis of thermal stress intensity factors for interface griffith and cusp cracks

The thermal stress intensity factors for interface cracks of Griffith and symmetric lip cusp types under vertical uniform heat flow in a finite body are calculated by the boundary element method. The boundary conditions on the crack surfaces are insulated or fixed to constant temperature. The relationship between the stress intensity factors and the displacements on the nodal point of a crack-tip element is derived. The numerical values of the thermal stress intensity factors for an interface Griffith crack in an infinite body are compared with the previous solutions. The thermal stress intensity factors for a symmetric lip cusp interface crack in a finite body are calculated with respect to various effective crack lengths, configuration parameters, material property ratios and the thermal boundary conditions on the crack surfaces. Under the same outer boundary conditions, there are no appreciable differences in the distribution of thermal stress intensity factors with respect to each material property. However, the effect of crack surface thermal boundary conditions on the thermal stress intensity factors is considerable.     

Application of an orthotropy rescaling method to edge cracks and kinked cracks in orthotropic two-dimensional solids

Oblique edge cracks and kinked cracks in orthotropic materials with inclined principal material directions under inplane loadings are investigated. The Stroh formalism is modified by introducing new complex functions, which recovers a classical solution for a degenerate orthotropic material with multiple characteristic roots. An orthotropy rescaling technique is presented based on the modified Stroh formalism. Stress intensity factors for edge cracks as well as kinked cracks are obtained in terms of solutions for a material with cubic symmetry by applying the orthotropy rescaling method. Explicit expressions of the stress intensity factors for a degenerate orthotropic material are obtained in terms of solutions for an isotropic material. The effects of orthotropic parameter, material orientation, and crack angle on the stress intensity factors for the degenerate orthotropic material are discussed. The stress intensity factors for cubic symmetry materials are calculated from finite element analyses, which can be used to evaluate the stress intensity factors for orthotropic materials. The energy release rate for the kinked crack in an orthotropic material is also obtained.     

Strong interactions of morphologically complex cracks

Previous works on crack morphology have focused on such cracks as a kinked crack, a branched crack, and an inclined array of identical branched cracks. In this paper, the strong interactions between two cracks in two-dimensional solids under remote tension are investigated. Three morphological types are considered: kinks, branches and zigzags. The method of analysis follows the singular integral equation approach in which the deviations from the main cracks are modeled by distributions of dislocations. Investigations are made on the dependence of the stress intensity factors on the asymmetry of the crack configuration, the crack separation, and the shape of the cracks. The results show that (i) strong interactions can have significant effects on the mode mixity of the stress intensity factors, (ii) a small asymmetry of the crack configuration can cause significant changes to the stress intensity factors, and (iii) zigzag cracks with rectangular steps reduce the stress intensity factors more efficiently than those with triangular or trapezoidal steps.     

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.     

Analysis of stress intensity factors for three-dimensional cruciform cracks

The stress intensity factors for three-dimensional cruciform surface cracks in a semi-infinite body are numerically calculated by the body force method. Mindlin's point force solution is used for the derivation of basic equations to express the influence coefficient of triangular elements, into which the crack is divided. The interactions between crossed crack planes as well as contact between crack surfaces are considered in the iterative manner. Stress intensity factors for a cruciform median crack and a cruciform semicircular crack under a point force on the surface of a semi-infinite solid are analyzed. The possibility of growth of a median crack toward the free surface of the semi-infinite solid is discussed. A cruciform semicircular surface crack under remote uniaxial tension, or under combined tension and compression is also analyzed. The effect of contact of crack surfaces on stress intensity factors is discussed.     

Interacting dissimilar semi-elliptical surface flaws under tension and bending

Stress intensity factors for two dissimilar interacting semi-elliptical coplanar surface flaws (cracks) in a semi-infinite elastic body are obtained under overall tension and bending. First the basic equations for a general planar crack normal to the free surface are established, using the method of equivalent eigen- or transformation strains (the body force method). Then the results are specialized for application to elliptical cracks. Numerical values are obtained for various configurations and crack shapes. Results are compared with those of two-dimensional collinear cracks. Finally, an approximate procedure for estimating the stress intensity factors for a general three-dimensional crack is suggested.     

Stress intensity factors at tips of branched cracks under various loadings

Several papers have been published on branched cracks by using various analytical methods, but most of them are concerned with special crack geometries or special loading conditions, and often give unreliable values for cracks with short branches or with small branching angles. The purpose of this paper is to give reliable formulae and new results of the stress intensity factors of various branched cracks in a wide plate. The analysis is based on the body force method combined with a perturbation procedure, and the stress intensity factors at the tips of all the branches and the main crack are given by power series formulae. Numerical results for typical branched cracks are discussed.     

各向异性平面内双边半无限裂纹问题的基本解

应用复变函数的方法，对于含双边半无限裂纹的各向异性平面，给出了其在任意集中力作用下的复应力函数基本解与应力强度因子基本解；结果表明：当外载作用在裂纹表面上时，其应力强度因子与相应各向同性的情形相同     

Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

A direct determination of weight functions for arbitrary three-dimensional cracks – a combined analytical and numerical approach

A combined analytical and numerical method is proposed for the determination of the weight functions of stress intensity factors of cracks in an arbitrary three-dimensional elastic body. Having defined the weight functions for a given geometry of a structure, the stress intensity factors for arbitrary loading conditions can be obtained by a simple inner product of the weight function and a traction vector. Traditionally weight functions are defined in the two ways; the one is defined by the hyper-singular term of the eigen-function expansion of the displacement field of a cracked body, and the other is defined by the variation of displacement field with respect to a virtual extension of a crack. In the present paper, the weight functions for stress intensity factors are defined by applying the Maxwell-Betti's reciprocal theorem to an original problem and the auxiliary problems subjected to three kinds of force-couples acting on the crack surfaces near the limiting periphery of an arbitrary three-dimensional crack. In the present formulation, weight functions can be calculated by using a general-purpose finite element code combined with analytical expressions near the condensation point, where hyper-singularities exist. The validity of the method is confirmed by two- and three-dimensional illustrative examples.     

Variation of stress intensity factor and crack opening displacement of 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 surface crack. Stress field induced by body force doublet in a semi infinite body is used as a fundamental solution. Then the problem is formulated as an integral equation with a singularity of the form of r ^-3. In solving the integral equations, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function; that is, the exact density distribution to make an elliptical crack in an infinite body. The calculation shows that the present method gives the smooth variation of stress intensity factors along the crack front and crack opening displacement along the crack surface for various aspect ratios and Poisson's ratio. The present method gives rapidly converging numerical results and highly satisfactory boundary conditions throughout the crack boundary.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r ^–3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary.