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.     

Plane strain stress intensity factors for branched cracks

A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.     

Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.     

Distributed cracks and kinked cracks in bonded dissimilar half planes with an interface crack

This paper is concerned with the interactions between an interface crack and other arbitrarily distributed cracks in two bonded dissimilar half planes. Special emphasis is placed on the cracks kinked at a tip of the interface crack, which remain unsolved as far as the authors are concerned. For the present, we pay attention to the stress intensity factors at the tips of the kinks or the distributed cracks, and not to those at the tips of the interface crack. The analysis is based on continuous distributions of the body forces along the cracks, and their densities are determined with a new procedure in order to get highly accurate results. The present analysis for distributed line cracks applies to kinked cracks, branched cracks and those piercing the interface just by joining some of the line cracks. Numerical calculations are performed for various important problems, and the effects of geometric and mechanical parameters on the stress intensity factors are examined.     

FRACTURE MECHANICS ASSESSMENT OF PLANAR BRANCHED CRACKS IN AN EQUIBIAXIAL STRESS FIELD

Abstract A simplified fracture mechanics assessment is presented of branched planar cracks in an equibiaxial stress state. In linear-elastic fracture mechanics the stress intensity factors which characterize the load at the crack tips depend, for a given external load, only on the crack geometry. The stress intensity factors of a large number of branched cracks were evaluated using the Boundary Element method, and correlations between the stress intensity factors and the crack geometry were investigated. Formulae are presented which assign an individual effective crack length to each crack tip of a branched crack and hence allow approximate stress intensity factors to be determined for very complicated crack geometries. An algorithm is used for the stochastic simulation of an irregular crack pattern formation in thermal fatigue.     

Stress intensity factors of parallel cracks in a finite width sheet

Two and three parallel cracks in a finite sheet subjected to remote tensile loading have been studied. This paper presents empirical stress intensity factor formulae for these crack configurations. The stress intensity factors used to develop these formulae were obtained from finite element analysis. For central cracks and edge cracks, the formulae were within 1 and 3% of the finite element results, respectively.     

Stress intensity factors for several groups of equal and parallel cracks in finite plates

Several groups of equal and parallel 2D cracks in finite width plates subjected to remote tensile loading have been studied. Formulae for calculating the stress intensity factors of these crack configurations have been proposed from the finite element analysis, and the difference between the formulae and the finite element results is smaller than 3%. On this basis, the influence of crack interactions on stress intensity factors (SIF) is discussed, and it can be seen that interaction between multiple cracks could produce either strong enhancement or shielding effects on the SIF depending on the crack positions and lengths.     

Crack retardation equations for the propagation of branched fatigue cracks

The stress intensity factors (SIF) associated with branched fatigue cracks can be considerably smaller than that of a straight crack with the same projected length, causing crack growth retardation or even arrest. This crack branching mechanism can quantitatively explain retardation effects even when plasticity induced crack closure cannot be applied, e.g. in high R-ratio or in some plane strain controlled fatigue crack growth problems. Analytical solutions have been obtained for the SIF of branched cracks, however, numerical methods such as Finite Elements (FE) or Boundary Elements (BE) are the only means to predict the subsequent curved propagation behavior. In this work, a FE program is developed to calculate the path and associated SIF of branched cracks, validated through experiments on 4340 steel ESE(T) specimens. From these results, semi-empirical crack retardation equations are proposed to model the retardation factor along the crack path. The model also considers the possible interaction between crack branching and other retardation mechanisms.     

Determination of the growth of branched cracks by numerical methods

In order to determine the growth of branched cracks in brittle materials, a static stress analysis for a branched crack model is performed by the finite element method. Assuming vanishing Mode II stress intensity factor as the governing criterion the growth is followed in several steps. Secondary branching is also analysed. In a quasi-dynamic analysis energy balance considerations are used to study growth of two branches of unequal lengths. It is shown that at low velocities the shorter branch will be rapidly arrested, while at very high velocities the two branches will continue to grow with nearly the same velocity.     

Transverse cracks in a strip with reinforced surfaces

The symmetrical problem of two transverse cracks in an elastic strip with reinforced surfaces is formulated in terms of a singular integral equation. The special cases of one central crack or two edge cracks are discussed. Numerical methods for solving the problems with internal cracks are outlined and stress intensity factors are presented for various geometries and degrees of surface reinforcement.     

Fully automated mixed mode crack propagation analyses based on tetrahedral finite element and VCCM (virtual crack closure-integral method)

The authors have been developing a crack propagation analysis system that can deal with arbitrary shaped cracks in three-dimensional solids. The system is consisting of mesh generation software, a large-scale finite element analysis program and a fracture mechanics module. To evaluate the stress intensity factors, virtual crack closure-integral method (VCCM) for the quadratic tetrahedral finite element is adopted and is included in the fracture mechanics module. The rate and direction of crack propagation are predicted by using appropriate formulae based on the stress intensity factors. In this paper, the crack propagation system is briefly described and some numerical results are presented.     

Arbitrary branched and intersecting cracks with the extended finite element method

Extensions of a new technique for the finite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended finite element method (X‐FEM) allows the representation of crack discontinuities and voids independently of the mesh. A standard displacement‐based approximation is enriched by incorporating discontinuous fields through a partition of unity method. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed. Computation of the stress intensity factors (SIF) in different examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Doubly-periodic array and zig-zag array of cracks in solids under uniaxial tension

In fatigue of materials, it is often observed that a number of cracks initiate from preexisting defects and propagate to form major cracks causing the final failure. In this paper, we consider a doubly-periodic array and a zig-zag array of cracks in a two-dimensional solid subjected to tension, as simplified models of randomly distributed cracks in materials. The analysis is based on the eigenfunction expansions of the complex stress potentials for properly chosen unit regions. Numerical results for the stress intensity factors, crack opening displacements and effects of cracks on the tensile stiffnesses of these solids are given for various combinations of the parameters. The results are then fitted to reliable polynomial formulae for convenience of engineering applications.     

Interaction of a wedge disclination dipole with interfacial cracks

The elastic interaction between a wedge disclination dipole and collinear interfacial cracks in bimaterials is investigated. The general solutions of complex potentials to this problem are presented by using complex potential theory. As illustrative examples, the closed-form solutions for a wedge disclination dipole interacting with a finite interfacial crack and a semi-infinite interfacial crack are obtained. The stress intensity factors at the tips of the crack and the force acting on the disclination dipole center are also given. The shield and anti-shield effect of the wedge disclination dipole upon the stress intensity factors is evaluated, and the equilibrium position of the disclination dipole is discussed for various crack geometries and material mismatch. The results indicate that the shielding or anti-shielding effect to the stress intensity factors increases acutely when the disclination dipole approaches the tip of the crack. If the center of the dipole is fixed, there always exists a critical value of angle of the dipole arm which the shielding or anti-shielding effect to the stress intensity factor is maximal. In addition, the length of the dipole arm and the material mismatch have significant influence on the stress intensity factors. The results also show that the interfacial crack always attracts the wedge disclination dipole and an equilibrium position of the disclination dipole may be available near the interface, which differs from the case of a perfect bonded interface, when the dipole approaches the surface of the crack from infinity. The present solutions contain a series of new and previously known results which can be shown to be special cases.     

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral–differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.     

Asymmetrical Cracks Parallel to an Interface between Dissimilar Materials

This paper solves a plane strain problem for two bonded dissimilar planes containing a crack parallel to the interface in each layer. The bimaterial system is loaded by tractions distributed along the crack surfaces. Based on the Fourier transform, the problem is reduced to a system of Cauchy type singular integral equations which contain exact and explicit kernel functions. The solution of these equations is obtained easily by utilizing Gauss–Chebyshev integral formulae for various material combinations and geometrical parameters. Several numerical results of stress intensity factors, energy release rate and stress distribution along the interface are presented to exhibit the interaction among cracks and interface. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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.     

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.