The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

Stress intensity factors at crack tips near boundaries or other geometrical discontinuities

A modification of the Lobatto-Chebyshev method for the numerical solution of Cauchy type singular integral equations appearing in plane or antiplane elasticity crack problems and the determination of stress intensity factors at crack tips is presented. This modification, based on a variable transformation, permits the selection of abscissas and collocation points used to be modified so as to obtain rapid convergence of the numerical results for the stress intensity factors to their correct values. The proposed technique is seen to be particularly effective for crack problems with crack tips near boundaries, interfaces or other geometrical discontinuities. Two applications of the method, to a periodic array of cracks in plane elasticity and to an antiplane shear crack near a boundary, show the effectiveness of the method.

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Résumé Une modification de la méthode Lobatto-Chebyshev pour la solution numérique des équations singulières intégrales de type Cauchy apparaissant en problèmes de fissures dans l'élasticité plane ou antiplane et la détermination des facteurs d'intensité des contraintes aux extrémités des fissures est présentée. Cette modification, basée sur une transformation de variable, permet de modifier la sélection des abscisses et des points de collocation utilisés pour obténir une convergence rapide des résultats numériques pour les facteurs d'intensité des contraintes à leurs valeurs correctes. La technique proposée est vue d'être particulièrement effective pour des problèmes des fissures avec des extrémités près de limites, d'interfaces ou d'autres discontinuités géometriques. Deux applications de la méthode, à une série périodique des fissures dans l'élasticité plane et à une fissure près d'une limite dans l'élasticité antiplane, montrent l'efficacité de la méthode.

     

Contact of the lips of an interphase crack in the field of concentrated mass forces

We consider a plane problem of the theory of elasticity for a crack located on the boundary of a fixed half plane subjected to the action of an arbitrary concentrated load applied to internal points of the domain. It is assumed that the crack lips contact without friction in the vicinity of the tip. By applying the Fourier integral transformation, we reduce the problem to a system of singular integral equations. In constructing the numerical solution of this system, we take into account the singular behavior of unknown functions near singular points. We establish the dependence of the size of the region of contact of the crack lips on the type of applied load. It is shown that, in this case, the combination of stress intensity factors obtained under the load applied at infinity remains quasiinvariant with respect to the length of the contact region. By using the elastic solution, we approximately determine the boundaries of the plastic domain near the crack tip and analyze their dependence on the intensity of the shearing field.Dnepropetrovsk State University, Dnepropetrovsk. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 2, pp. 35–42, March – April, 1995.     

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.     

The numerical solution of crack problems in plane elasticity in the case of loading discontinuities

The direct quadrature method of numerical solution of Cauchy type singular integral equations encountered in plane elasticity crack problems is applied to the case where the loading distribution along the crack edges presents jump discontinuities. This is made by using a well-known modification of the quadrature method which is free of undesirable errors due to the loading discontinuities. Hence, the method is ideal to treat the aforementioned class of crack problems and, particularly, crack problems where the Dugdale-Barenblatt elastic-perfectly plastic model is adopted. Finally, a numerical application of the method to the problem of a periodic array of cracks with a loading distribution presenting a jump discontinuity is made. The numerical results obtained in this problem compare favorably with the corresponding theoretical results available in this special problem.     

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.     

Three-dimensional nonplanar fracture model using the surface integral method

A computer code based on the use of the surface integral method, which represents a crack as a distribution of force dipoles, has been developed for modeling 3D nonplanar fractures. The nonplanar geometry was approximated as piecewise linear by subdividing the fracture surface into triangular elements that assume constant crack opening in the interior, and a p ^1/2 variation of opening along the crack front. The resulting singular integral equations were integrated using a combination of numerical and analytical techniques.Convergence studies using the surface integral formulation have yielded accurate stress intensity factors and crack opening displacements for both planar and nonplanar cracks under a variety of mixed mode loading conditions. Elliptical meshes were mapped on to cylindrical and spherical surfaces to model nonplanar fractures that could be compared to published results. Also, a high aspect ratio rectangular mesh was used to model a nonplanar kinked crack under plane strain conditions.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

Numerical solution of Cauchy type singular integral equations with logarithmic weight,based on arbitrary collocation points

Singular integral equations with a Cauchy type kernel and a logarithmic weight function can be solved numerically by integrating them by a Gauss-type quadrature rule and, further, by reducing the resulting equation to a linear system by applying this equation at an appropriate number of collocation points x _k. Until now these x _k have been chosen as roots of special functions. In this paper, an appropriate modification of the original method permits the arbitrary choice of x _k without any loss in the accuracy. The performance of the method is examined by applying it to a numerical example and a plane crack problem.     

On the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations

Complex boundary integral equations (Fredholm‐type regular or Cauchy‐type singular or even Hadamard–Mangler‐type hypersingular) have been used for the numerical solution of general plane isotropic elasticity problems. The related Muskhelishvili and, particularly, Lauricella–Sherman equations are famous in the literature, but several more extensions of the Lauricella–Sherman equations have also been proposed. In this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries (for anyone of the above equations: regular or singular or hypersingular, but with a prerequisite their actual numerical solution) through the appropriate use of the even more classical elementary Cauchy theorem in complex analysis. This approach has been already used for the accurate numerical computation of analytic functions and their derivatives by Ioakimidis, Papadakis and Perdios (BIT 1991; 31 : 276–285), without applications to elasticity problems, but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non‐analytic complex density function (such as an edge dislocation/loading distribution density) is numerically available on the boundary. The present results are also directly applicable to inclusion problems, anisotropic elasticity, antiplane elasticity and classical two‐dimensional fluid dynamics, but, unfortunately, not to crack problems in fracture mechanics. Brief numerical results (for the complex potentials), showing the dramatic increase of the computational accuracy, are also displayed and few generalizations proposed. Copyright © 2000 John Wiley & Sons, Ltd.     

On the penny-shaped crack in a non-homogeneous interlayer under torsion

This paper presents a theoretical treatment of a penny-shaped crack in an interfacial zone, along the thickness of which the elastic modulus is assumed as ₂(z) = ( +bz) ^k , wherek represents the distribution parameter independent of material properties and interlayer thicknessh. The theoretical formulations governing the torsion deformation behavior of the material are based on the use of a dislocation density function and integral transform technique. The stress intensity factor is obtained by solving a singular integral equation. Numerical examples are given to show the effects of material properties, interlayer thickness, and especially the distribution parameterk on the stress intensity factor. In the numerical procedure, modified Bessel functions are used, and the rate of convergence depends greatly on the ratio ofh/c, wherec is the crack radius.     

Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics

An exact expression is derived for the general finite-part integral over an inclined ellipticaldomain Ω. r denotes the distance of a point in Ω to the singular point $\left({x,y} \right).f = x_{^0 }^i y_0^j \sqrt {Z\left({x_{0,} y_0 }\right)}$ is a general function of the Cartesian co-ordinates x₀,y₀. The boundary of the region Ω represents the equation Z(x₀, y₀)=O. These integrals appear during the numerical solution of plane crack problems in three-dimensional elasticity where they are the dominant part of a hypersingular integral equation. The availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and, simultaneously, lead to quicker numerical results. The considered finite-part integral can be expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively. Formuias for special cases and some i, j values and their numerical verification are given in Appendices II and III.     

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.     

Moving crack at the interface between two dissimilar magnetoelectroelastic materials

Summary Analytical solutions for an anti-plane Griffith crack moving at the interface between two dissimilar magnetoelectroelastic media under the conditions of permeable crack faces are formulated using the integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic materials. Expressions for stresses, electric displacements, and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities, and it is found that the dynamic stress intensity factor (DSIF), the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF) are independent of the remote electromagnetic loads. The moving speed of the crack has influence on the DEDIF and the DMIIF. When the crack is moving at lower speeds 0 ≤ M≤ M_c1 or higher speeds M_c2 < M < 1, the crack will propagate along its original plane, while in the range of M_c1 < M < M_c2 , the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Contact problem for a half plane with cracks subjected to the action of a rigid punch on its boundary

We consider a contact problem of the theory of elasticity for an isotropic half plane with a system of curvilinear cracks and a rigid punch pressed into the half plane. We assume that the base of the punch has an arbitrary convex shape, that either the punch interacts with the half plane via friction forces or they are rigidly engaged, and that the crack lips are under the conditions of smooth contact. The problem is reduced to singular integral equations in unknown derivatives of singular displacements on the crack contours and contact forces under the punch. These equations are solved by the method of mechanical quadratures. We present the results of numerical analysis of the stressed state of the half plane with internal vertical crack under pressure of a flat punch without friction on the boundary of the half plane.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 6, pp. 7–16, November – December, 1995.     

A new boundary element method for mixed boundary value problems involving cracks and holes: Interactions between rigid inclusions and cracks

This paper derives a new boundary integral equation (BIE) formulation for plane elastic bodies containing cracks and holes and subjected to mixed displacement/ traction boundary conditions, and proposes a new boundary element method (BEM) based upon this formulation. The basic unknown in the formulation is a complex boundary function H(t), which is a linear combination of the boundary traction and boundary displacement density. The present BIE formulation can be related directly to Muskhelishvili's formalism. Singular interpolation functions of order r ^–1/2 (where r is the distance measured from the crack tip) are introduced such that singular integrand involved at the element level can be integrated analytically. By applying the BEM, the interaction between a rigid circular inclusion and a crack is investigated in details. Our results for the stress intensity factor are comparable with those given by Erdogan and Gupta (1975) and Gharpuray et al. (1990) for a crack emanating from a stiff inclusion, and with those by Erdogan et al. (1974) for a crack in the neighborhood of a stiff inclusion.     

Effects of T-stresses on fracture initiation for a closed crack in compression with frictional crack faces

This paper studies crack extension resulting from a closed crack in compression. The crack-tip field of such a crack contains a singular field relative to K _II and non-singular T-stresses T _x and T _y parallel and perpendicular to the crack plane, respectively. Using a modified maximum tensile stress criterion with the singular and non-singular terms, the kinking angle at the onset of crack growth is determined by a two parameter field involving the mode-II stress intensity factors and T-stresses, and at fracture initiation a wing crack may be created at an arbitrary angle from 0° to 90°. A compressive T _y increases the kinking angle and reinforces apparent mode-II fracture toughness, while a compressive T _x decreases the kinking angle and enhances apparent mode-II fracture toughness. The direction and resistance of fracture onset is strongly affected by T-stresses as well as frictional stress. The von Mises effective stress is determined for small-scale yielding near the crack tip. The effective stress contour shape exhibits a marked asymmetrical behavior unless 2T _x  = T _y  ≤ 0 for plane stress state. Coulomb friction between two crack faces generally increases the kinking angle, shrinks the size enclosed by the effective stress contour and enhances apparent fracture toughness. Field evidence and experimental observations of many phenomena involving the growth of closed cracks in compression agree well with theoretical predictions of the present model.     

Further studies on the characteristics of the T * ɛ integral: Plane stress stable crack propagation in ductile materials

Some Characteristic behavior of the T ^* _ɛ (Atluri, Nishioka and Nakagaki (1984)) is identified in this paper through an extensive numerical study. T ^* _ɛ is a near tip contour integral and has been known to measure the magnitude of singular deformation field at crack tip for arbitrary material models. In this paper, T ^* _ɛ is found to behave quite differently for different choices of near tip integral contours. If the integral contour moves with advancing crack tip (moving contour), then T ^* _ɛ measures primarily the energy release rate at the crack tip. It is very small for metallic materials, and tends to zero in the limit as Δa→0 for low hardening materials. Thus, T ^* _ɛ evaluated on a moving contour tends to zero as ɛ→0 and Δa→0, for low hardening materials. If the integral contour elongates as crack extends (elongating contour), then T ^* _ɛ measures total energy inside the volume enclosed by Γ_ɛ [i.e., the energy dissipated in the extending wake] plus the energy release at the crack tip. Furthermore, the difference in the behavior of CTOA and T ^* _ɛ, when the applied load is slightly perturbed, is identified. The CTOA is found to be quite insensitive to applied load change. T ^* _ɛ is found to be roughly proportional to the square of the applied load. The functional shape of T ^* _ɛ in terms of the size ɛ of integral contour (for the elongating contour case), is identified, using the crack tip asymptotic formula of Rice (1982). Also, the behaviors of CTOA and T ^* _ɛ are discussed from the view point of Rice's asymptotic solution. It is recommended that as a crack tip parameter for ductile materials, T ^* _ɛ with elongating path be used. CTOA is sometimes not very sensitive to the applied load change, therefore it may create some numerical problems in application phase crack propagation analysis.