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.     

Stress intensity solutions for the interaction between a hole edge crack and a line crack

The principle of superposition is used to solve the problem and the original problem is converted into two particular hole edge crack problems. The remote stresses are applied at infinity in the first problem. Meantime, a dislocation distribution is assumed outside the hole contour in the second problem. Singular integral equation is proposed for the solution of the second problem, in which the right hand side of the integral equation is obtained from the solution of the first problem. The first problem as well as the elementary solution of the second problem are solved by means of the rational mapping approach. Finally, numerical examples with the calculated results of stress intensity factors are presented.     

Stress Intensity Factors for an Interface Kinked Crack in a Bi-Material Plate Loaded Normal to the Interface

Stress intensity factors for a kinked crack originating at interface of two bonded dissimilar materials subjected to normal tension are found by the finite element method.     

Interaction of a penny-shaped crack with an elliptic crack under shear loading

The problem of interaction between equal coplanar elliptic cracks embedded in a homogeneous isotropic elastic medium and subjected to shear loading was solved analytically by Saha et al. (1999) International Journal of Solids and Structures 36, 619–637, using an integral equation method. In the present study the same integral equation method has been used to solve the title problem. Analytical expression for the two tangential crack opening displacement potentials have been obtained as series in terms of the crack separation parameter _i up to the order _i⁵,(i=1,2) for both the elliptic as well as penny-shaped crack. Expressions for modes II and III stress intensity factors have been given for both the cracks. The present solution may be treated as benchmark to solutions of similar problems obtained by various numerical methods developed recently. The analytical results may be used to obtain solutions for interaction between macro elliptic crack and micro penny-shaped crack or vice-versa when the cracks are subjected to shear loading and are not too close. Numerical results of the stress-intensity magnification factor has been illustrated graphically for different aspect ratios, crack sizes, crack separations, Poisson ratios and loading angles. Also the present results have been compared with the existing results of Kachanov and Laures (1989) International Journal of Fracture 41, 289–313, for equal penny-shaped cracks and illustrations have been given also for the special case of interaction between unequal penny-shaped cracks subjected to uniform shear loading.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

A mode I crack parallel to the interfaces in a periodically layered medium

The plane problem of a single crack in a periodically layered bimaterial composite is considered. For the case of a long crack loaded by opening normal tractions, the universal relation obtained between the Mode I and Mode II stress intensity factors show that the most dangerous crack location lies in the midplane of the layer. This crack location of the Mode I finite length crack is examined in detail. A closed form expression of the Green's function for a single dislocation is derived and the problem is reduced to a singular integral equation of the first kind. The study of the dependence of the normalized stress intensity factor upon the crack length reveals a wavy nonmonotonic behavior. A simple analytic formula for the limiting case of a semi-infinite crack is derived. It is found to be valid for a broad range of parameters.     

Numerical solution for multiple crack problem in an infinite plate under compression

This paper investigates a numerical solution for multiple crack problem in an infinite plate under remote compression. The influence of friction is taken into account. In the first step of the solution, we make a full contact assumption on the crack faces. The full contact assumption means that one component of the dislocation distribution vanishes, and the first mode stress intensity factors (K ₁) at the crack tips become zero. On the above-mentioned assumption, the problem can be solved by using integral equation method, and the second mode stress intensity factors (K ₂) at the crack tips can be evaluated. Meantime, after solving the integral equation the normal contact stress on the crack faces can be evaluated. The next step is to examine the full contact assumption. If the contact stresses on the crack faces are definitely negative, the solution is true. Otherwise, the obtained solution is not true. It is found from present study that in most cases the full contact condition is satisfied, and only in a few cases the full contact condition is violated. Numerical examples are given. It is found that the friction can lower the stress intensity factors at crack tips in general.     

Bilateral bending of a plate with nonsymmetric through-thickness arc crack with allowance for the contact of its edges

The problem of bending of plates with nonsymmetric through-thickness arc crack with distributed moments at infinity with allowance for the contact of the crack edges has been examined on the basis of the classical theory of plate bending. The conditions for the existence of the solution of the problem in this formulation have been established. A numerical analysis of intensity factors of forces and moments, contact force between the crack edges has been performed, the results of which have been graphically illustrated. __________ Translated from Problemy Prochnosti, No. 5, pp. 135–141, September–October, 2006. Report on International Conference “Dynamics, Strength, and Life of Machines and Structures” (1–4 November 2006, Kiev, Ukraine).     

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.     

Plane problem of macro-microcrack interaction taking account of crack closure

The fracture stability of a macrocrack under the tensile and shear loading in the presence of a system of microcracks is analysed. Interaction of cracks leads to full or partial closure of the crack edges. The boundary problem is formulated and a solution is obtained by the small parameter method. Domains where microcracks are closed, and regions where microcracks cause full or partial closure of the macrocrack are found. The influence of crack contact on the stress intensity coefficient is analysed under the friction free assumption.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

Analysis of a short interfacial crack from the corner of a rectangular inclusion

A useful method is proposed to analyze a short interfacial crack emanating from the corner of a rectangular inclusion. We first analyze the singular stress field (and the corresponding singularity intensity factor H) without the crack in an infinite medium having the rectangular inclusion. The singular stress field (and the corresponding stress intensity factor K) at the tip of the short interfacial crack lying in the interface of the rectangular inclusion is also analyzed, giving the relation between H and K. With this relation, the stress intensity factor K is easily obtained for the case of a short interfacial crack from the corner of a different rectangular inclusion with different external boundary. This method is based on the assumption that the singular K-field is embedded in another singular H-field, which is much smaller than the specimen geometry. To meet the assumption, it is found here that the eigenfunction corresponding to the next smallest eigenvalue of the singular H-field has to be considered. An example is presented to show the usefulness of the present method, where a short interfacial crack from the corner of a rectangular lead frame in epoxy compound used in electronic packaging is analyzed. It is found that the result of the present method is in good agreement with that of the well-known method.     

The impact response of a half plane crack in a transversely isotropic solid due to 3-D mixed mode loading

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to two opposed pairs of shear line loads on its faces. Transform methods are used to reduce the boundary value problem to a set of coupled integral equations that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. Exact expressions are derived for the mode II and III stress intensity factors as functions of time and position along the crack edge. Some features of the solutions are discussed through numerical results.     

Stress intensity factors for a mixed mode center crack in an anisotropic strip

An approach based on the continuous dislocation technique is formulated and used to obtain the Mode I and II stress intensity factors in a fully anisotropic infinite strip with a central crack. First, the elastic solution of a single dislocation in an anisotropic infinite strip is obtained from that of a dislocation in an anisotropic half plane, by applying an array of dislocations along the boundary of the infinite strip, which is supposed to be traction-free. 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. Subsequently, the elastic solution is applied to calculate the stress intensity factors for a center crack in an anisotropic strip. Crack length and material anisotropy effects are discussed in detail.     

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.     

Interaction between a cracked hole and a line crack under uniform heat flux

This article deals with the interaction between a cracked hole and a line crack under uniform heat flux. Using the principle of superposition, the original problem is converted into three particular cracked hole problems: the first one is the problem of the hole with an edge crack under uniform heat flux, the second and third ones are the problems of the hole under distributed temperature and edge dislocations, respectively, along the line crack surface. Singular integral equations satisfying adiabatic and traction free conditions on the crack surface are obtained for the solution of the second and third problems. The solution of the first problem, as well as the fundamental solutions of the second and third, is obtained by the complex variable method along with the rational mapping function approach. Stress intensity factors (SIFs) at all three crack tips are calculated. Interestingly, the results show that the interaction between the cracked hole and the line crack under uniform heat flux can lead to the vanishing of the SIFs at the hole edge crack tip. The fact has never been seen for the case of a cracked hole and a line crack under remote uniform tension.     

含不完美界面的磁电复合材料倾斜裂纹问题

研究了含非完美界面的双层压电/压磁复合材料中压电相存在一个倾斜于界面的Ⅲ型裂纹问题。采用弹簧型耦合界面模型模拟非完美界面,运用Fourier积分变换法将裂纹面条件转化为奇异积分方程,并使用Lobatto-Chebyshev方法数值求解了裂纹尖端应力强度因子（SIF）。详细地研究了裂纹尖端SIF与界面参数、压电/压磁材料参数和材料的层厚、裂纹的倾斜角、裂纹与界面的距离等几何参数的关系。结果表明：力学不完美性可以独立地增大SIF,而磁学、电学不完美性只有与力学不完美性耦合时才会减小SIF;力学-电学、力学-磁学不完美性的耦合会减小SIF,而磁学-电学不完美性的耦合不会影响SIF;磁场作用下,增大压磁层弹性模量会减小SIF,而增大压电层压电系数,减小压电层弹性模量和介电常数,均会减小SIF;界面不完美性会影响SIF随裂纹倾斜角度或裂纹与界面之间距离的变化规律;在一定范围内增加压电层或压磁层厚度可以减小SIF。     

Estimation of the effects of plasticity and resulting crack closure during small fatigue crack growth

On the basis of a plastic-strip model and the method of singular integral equations, a closed-form analytical solution of the problem of an elastic-plastic plate containing a rectilinear fatigue crack is considered. The solution is used for the prediction of fatigue growth of `mechanically-small' crack by accounting for reverse plastic yielding and plasticity-induced crack closure in the material. The main effects of these factors on the crack-growth rate are analyzed, and the predicted results are compared with experimental data on small fatigue-crack growth in a aluminum-lithium alloy 2091-T351 and Fe-3% Si alloy.     

The interaction between screw dislocations and an interfacial blunt crack and a sharp crack

The interaction between screw dislocations and an interfacial blunt crack and a sharp crack under loads at infinity is dealt with. Utilizing the Muskhelishvili complex variable method, the closed form solutions are derived. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The results show that the shielding effect increases with the increase of the shear modulus and the distance between the two cracks, but decreases with the increase of dislocation azimuth. The critical loads at infinity for dislocation emission increase with the increment of the emission angle and the distance.     

Study on crack propagation behavior in a quenched glass plate

Crack pattern transition and crack propagation behavior in a quenched glass plate are investigated. Theoretical analysis indicates that the distance between the crack tip and the cold front is closely related to the crack pattern transition. This theoretical result is examined experimentally using instantaneous phase-stepping photoelasticity. As expected theoretically, when the crack tip remains close enough to the cold front, crack propagation remains straight. When this distance reaches a given value, the crack oscillates. These experimental results are in good agreement with the theory of crack pattern transition. Therefore, present theoretical analysis is valid in predicting the instability of crack propagation. The crack tip stress field is also examined by the present experimental method. In particular, in the oscillating regime, the mode-I stress intensity factor frequently becomes larger than the fracture toughness, and the mode-II stress intensity factor has a nonzero value during propagation. For the former result, some reasons are discussed, but the cause of this problem is still unknown. However, the latter result can be explained by the theoretical analysis of an infinitesimal kinked edge crack just after crack initiation.