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.     

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.     

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.     

Kinked cracks in anisotropic elastic materials

The problem of a kinked crack is analysed for the most general case of elastic anisotropy. The kinked crack is modelled by means of continuous distributions of dislocations which are assumed to be singular both at the crack tips and at the kink vertex. The resulting system of singular integral equations is solved numerically using Chebyshev polynomials and the reciprocal theorem. The stress intensity factors for modes I, II and III and the generalised stress intensity factor at the vertex are obtained directly from the dislocation densities. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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.     

Dislocation modeling of quasi-static crack propagation in an elasto-plastic medium

A dislocation model for simulating two-dimensional quasi-static crack propagation is presented. The crack and plastic flow along slip planes are described using dislocation dipoles. A stationary crack can be modeled as well as a propagating crack along a straight line inclined at an arbitrary angle to a free surface of a semi-infinite medium. Cracks are also allowed to kink. A superdipole algorithm is introduced to save simulation time without loosing important information and necessary geometric details. It reduces the number of dislocation dipoles on slip planes in the plastic wake. The paper gives results on crack shapes for stationary and advancing cracks as well as it describes how the size of the plastic zone depends on crack inclination angles. Results on stress intensity factors (SIF) are given using two different approaches as well as kinking cracks are introduced and SIF at kinked crack tips are calculated.     

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.     

Line inclusions at anisotropic bimaterial interface

A line inclusion at the interface of an anisotropic bimaterial is studied. The line inclusion is assumed to be inextensible but with negligible bending rigidity. Complete singular fields near tips of the line inclusion are derived. The near-tip stress field exhibits singularities of the types in general with r being the distance measured from the tips. The near-tip fields are similar to those for fully closed interface cracks. In analogy to the stress intensity factors defined for interface cracks, strain intensity factors are introduced to characterize the near-tip fields. It is shown that there are only two independent strain intensity factors and corresponding modes of deformation. Complete displacement and stress fields and the corresponding strain intensity factors as the line inclusion is under uniform remote loading are given. Strain intensity factors for a line inclusion very near some anisotropic bimaterial interface are also derived.     

A set of interface cracks with contact zones in a combined tension-shear field

Summary. A set of cracks lying along the interface of two dissimilar isotropic materials under a mixed-mode loading is considered. The interface cracks are assumed to be fully open, partially closed with frictionless contact zones and fully closed. The problem is reduced to a homogeneous combined Dirichlet-Riemann boundary value problem, which is solved in closed form. A set of transcendental equations for the determination of the contact zone lengths for an arbitrary number of cracks and the closed-form expressions for the stresses and the displacement jumps on the material interface are obtained. A single crack with one and two contact zones has been considered in details. An explicit set of two transcendental equations for the relative contact zone length and closed-form expressions for the stress intensity factors at the crack tips are obtained for both cases. The contact zone lengths and the stress intensity factors are investigated numerically for different material pairs under different values of the loading, and a comparison of the results for a crack with one and two contact zones is carried out.     

Interaction effect of two arbitrarily oriented cracks — Part I

The elastosatic problem solved in this paper is of an isotropic homogeneous infinite plate, with two arbitrarily oriented cracks of different lengths, subjected to uniform uniaxial tension at infinity. The problem is formulated in the complex plane using the Kolossoff-Muskhelishvili stress functions and further the Schwarz's alternating method is used to solve the problem of the doubly connected region. The mode I and mode II stress intensity factors at all the four crack tips for various crack length ratios, crack angles and crack spacings are found, and are in good agreement with those obtained by other research workers. The fracture angles at the four crack tips are evaluated using the strain energy density theory and maximum tangential stress theory. The minimum strain energy density factor is also found at all the tips.     

Exterior interface cracks

Continuing a recent investigation of interface cracks, attention is paid to the exterior crack. Two elastic solids bonded over a finite segment of their boundary and capable of transmitting shear and tensile tractions are considered. It is found that one of the edge cracks remains completely closed under shear alone, and opens gradually as the level of tension is increased. Both crack tips, however, must remain closed at least over a small interval. Stress intensity factors and bond and contact stresses are given for a specific example.     

Disc analogy for approaching circular arc cracks

The limiting situation of a pair of approaching circular arc crack tips in a homogeneous medium is examined to draw analogies with circular disc problems. As the crack tips approach each other, the narrow material ligament bridging the crack tips controls the stresses, stress intensity factors and energy release rates. In particular, this ligament sets up the length scale for singularity analysis for a given radius of the arc crack. Following a detailed analytical examination of approaching crack tips, a photoelastic visualisation of the stress field is presented. Experimental isochromatics are also compared with theoretical predictions for some specific cases. Finally, based on the ideas developed in this paper, existing notions on interacting cracks and cavities treated in the literature are reinterpreted.     

Kinked crack in a bending trapezoidal plate

The interaction problem of a kinked crack and the edges of a bending trapezoidal plate which takes the effects of transverse shear deformation into account is presented. The research method is based upon the complex potential technique of Muskhelishvili using conformal mapping. Furthermore, for the analysis of the moment intensities at the tips of the kinked crack, the concept of dislocation distribution is applied. The integral equations for the stress disturbance problem along the line that is the presumed location of the kinked crack are then obtained as a system of singular integral equations with simple Cauchy kernels. As a consequence, the variation of moment intensity factors at the crack-tips is also illustrated.     

Analysis of Asymmetric Kinked Cracks of Arbitrary Size, Location and Orientation – Part II. Remote Tension

In this paper, the stress intensity factors of interacting kinked cracks in a solid and the overall strains of the solid under uniaxial tension are determined numerically. The kinked cracks are in general asymmetric, unequal, and arbitrarily oriented and located in the solid. Each kinked crack, assumed to be traction free, consists of a main crack and kinks. The analysis makes use of the dislocation modeling of kinks, and the superposition of problems of straight cracks subjected to dislocation and traction loadings. The model is used to investigate the dependence of the stress intensity factors and the overall strains on crack geometry (straight, Z-shaped and U-shaped cracks) and crack configuration (collinear and stacked cracks, periodic and random crack arrays). This revised version was published online in August 2006 with corrections to the Cover Date.     

Stress intensity factors for cracks at the vertex of a rounded V-notch

The method of singular integral equations was applied to determine the stress intensity factors for a system of cracks emanating from the vertex of an infinite rounded V-notch subjected to symmetric loading. The numerical values were obtained for two cases—the case of a single crack and the case of a system of two cracks of equal length. The influence of the rounding radius of the vertex of the notch and its opening angle on the stress intensity factors at the crack tips was analyzed. The solution obtained as a result has a general nature—the stress intensity factors at the crack tip are expressed as a function of the V-notch stress intensity factor and, hence, this solution could be treated as an asymptotic relation for finite bodies with deep V-notches subjected to symmetric loads.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semi-infinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

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.     

Weight functions for interface cracks

Weight functions are developed for determining stress intensity factors of cracks along an interface between two linear, elastic materials. As a result of the interface, both mode I and II components will be present for all but very special loading cases. The weight functions are employed to produce exactly the known stress intensity factors of a crack along an interface loaded by tensile and shear point forces.Part of this work was carried out while the author was on sabbatical leave at the Materials Laboratory, Wright Patterson Air Force Base, Ohio, USA