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.     

Stress intensity factors in two bonded elastic layers containing cracks perpendicular to and on the interface—I. Analysis

In this paper the basic crack problem which is essential for the study of subcritical crack propagation and fracture of layered structural materials is considered. Because of the apparent analytical difficulties, the problem is idealized as one of plane strain or plane stress. An additional simplifying assumption is made by restricting the formulation of the problem to crack geometries and loading conditions which have a plane of symmetry perpendicular to the interface. The general problem is formulated in terms of a coupled system of four integral equations. For each relevant crack configuration of practical interest the singular behavior of the solution near and at the ends and points of intersection of the cracks is investigated and the related characteristic equations are obtained. The edge crack terminating at and crossing the interface, the T-shaped crack consisting of a broken layer and a delamination crack, the cross-shaped crack which consists of delamination crack intersecting a crack which is perpendicular to the interface and a delamination crack initiating from a stress-free boundary of the bonded layers are some of the practical crack geometries considered as examples. The formulation of the problem is given in Part I of the paper. Part II deals with the solution of the integral equations and presentation of the results.     

A griffith crack problem for an inhomogeneous elastic material

Summary This paper examines the problem of a Mode I crack in a nonhomogeneous elastic medium. It is assumed that the shear modulus varies exponentially with the coordinate perpendicular to the plane of the crack. The problem is reduced to a Fredholm integral equation and in terms of its solution the normal components of stress and displacement are described. Expressions are also derived for the stress intensity factor and the crack energy. The effect of the inhomogeneity is examined and comparisons made with the corresponding results for the homogeneous material.     

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.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials

A generalised plane strain crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material are considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

A semi-analytical solution method for problems of cohesive fracture and some of its applications

Cohesive zone models are extensively used for the failure load estimates for structure elements with cracks. This paper focuses on some features of the models associated with the failure load and size of the cohesive zone predictions. For simplicity, considered is a mode I crack in an infinite plane under symmetrical tensile stresses. A traction–separation law is prescribed in the crack process zone. It is assumed by the problem statement that the crack faces close smoothly. This requirement is satisfied numerically by a formulation of the modified boundary conditions. The critical state of a plate with a cohesive crack is analyzed using singular integral equations. A numerical procedure is proposed to solve the obtained systems of integral equations and inequalities. The presented solution is in agreement with other published results for some limiting cases. Thus, an effective methodology is devised to solve crack mechanics problems within the framework of a cohesive zone model. Using this methodology, some problems are solved to illustrate the (i) influence of shape parameters of traction–separation law on the failure load, (ii) ability to account for contact stress for contacting crack faces, (iii) influence of getting rid of stress finiteness condition in the problem statement.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

A general method for multiple crack problems in a finite plate

A novel method for the multiple crack problems in a finite plate is proposed in this paper. The basic stress functions of the solution consist of two parts. One is the Fredholm integral equation solution for the crack problem in an infinite plate, and the other is that of the weighted residual method for general plane problems. The combined stress functions are used in the analysis and the boundary conditions on the crack surfaces and the boundary are considered. After the coefficients of the functions have been determined, the stress intensity factors (SIF) at the crack tips can be calculated. Some numerical examples are given and it was observed that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the influence of the boundary can be considered. This method can be used for arbitrary multiple crack problems in a finite plate.     

General solution of a certain mixed boundary value crack problem

An exact analytical solution is derived for a certain class of mixed boundary value crack problems which occur for example when analysing crack closure effects during fatigue crack growth. A general analysis is presented which is derived from the solution of a singular integral equation. The analysis is checked by comparing the general results to a known special solution (the triple crack problem). A technique is described for calculating the stress intensities for a central crack opened by two wedges of general shape which preserve symmetry about the crack plane and its perpendicular bisector. The stress distribution acting on the wedges is also calculated.     

The plastic stress ahead of a mixed mode I and II plane stress crack

The small scale yielding for mixed mode I and II plane stress crack problems in elastic perfectly-plastic solids is analysed by considering the stress field near the crack line. By expanding the stresses near the crack line and matching the stress field in the plastic zone with the elastic dominant field for a blunt crack near the crack line at the elastic-plastic boundary, the problem is reduced to solving a system of nonlinear algebraic equations. The relationship between the near-field mixity parameter M^p and the far-field mixity parameter M^e is detennined by solving the system of equations numerically. Analogous to Shih's calculation by the finite element method for the small scale yielding of mixed mode plane strain crack problems, the numerical results indicate that the shift from a mixed mode to a pure mode may not be a smooth one.     

Bonded Cracked Half Planes with an Interface and an Interesting Crack

The solution of the plane elasticity problem of two bonded isotropic linearly elastic half-planes of different elastic properties having a crack L along the interface as well as a crack S in one of the half planes which intersects the interface crack, is given by using the Muskhelishvili's complex variable method with sectionally holomorphic functions. The initial problem is reduced to a Hilbert problem, the solution of which in the case of a dislocation existing in either half-planes constitutes the Green's functions of the problem. Finally, a singular integral equation is derived for the problem only along the crack S. The singular integral equation is solved numerically and results are presented for the stress intensity factors. This revised version was published online in July 2006 with corrections to the Cover Date.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.     

Cracked Elastic Bi-Material Layer Under Compressive Loads

The boundary value problem of an elastic bi-material layer containing a finite length crack under compressive mechanical loadings has been studied. The crack is located on the bi-material interface and the contact between crack surfaces is frictionless. Based on Fourier integral transformation techniques the solution of the formulated problem is reduced to the solution of singular integral equation, then, with Chebyshev`s orthogonal polynomials, to infinite system of linear algebraic equations. The expressions for contact stresses in the elastic compound layer are presented. Based on the analytical solution it is found that in the case of frictionless contact the shear and normal stresses have inverse square root singularities at the crack tips. Numerical solutions have been obtained for a series of examples. The results of these examples are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mixture of mode I and mode II type singularities. The numerical solutions show that an interfacial crack under compressive forces can become open in certain parts of the contacting crack surfaces, depending on the applied forces, material properties and geometry of the layers.     

The mode‐III fracture analysis of an inclined crack embedded in thin layer systems by analytical alternating methods

Abstract

This paper analyzes the mode‐III stress intensity factor of an inclined crack, embedded in a thin layer, bonded to a half plane, subjected to arbitrary distributed anti‐plane loads. Special alternating procedures are presented to evaluate the mode‐ III S.I.F. and the numerical results confirm the validity of the proposed alternating procedure. The solution of a bi‐material problem in an infinite plane with an inclined crack and the analytical solution of a thin layer, without crack, bonded to a half plane, subjected to an anti‐plane point force applied on the boundary are referred to as fundamental solutions. By using these fundamental solutions and alternating procedures, the stress intensity factors of a crack in a thin layer bonded to a half plane are evaluated. The numerical results of some reduced problems are computed and excellent agreements with existing solutions are obtained.     

A Method to Extract Interface Stress Intensity Factors for Bimaterial Problems using Interlayers

A numerical method is presented here to determine stress intensity factors for interface cracks in plane, isotropic, elastic bimaterial fracture problems. The method relies on considering a companion problem wherein a very thin elastic interlayer with a crack, is artificially inserted between the two material regions of the original bimaterial problem. Modes I and II stress intensity factors are obtained for the companion problem using the modified virtual crack closure method. These stress intensity factors for the companion problem are then converted to the stress intensity factors for the original interface crack problem with the help of a universal relation. This universal relation between the stress intensity factors of the two problems is established by considering an asymptotic problem where the thickness of the interlayer is small compared with all other length scales. Two benchmark problems are considered to demonstrate the effectiveness of the interlayer approach for determining interface stress intensity factors.     

An integral equation method for dynamic crack growth problems

Within the assumptions of linear elastic fracture mechanics, dynamic stresses generated by a crack growth event are examined for the case of an infinite body in the state of plane strain subjected to mode I loading.The method of analysis developed in this paper is based on an integral equation in one spatial coordinate and in time. The kernel of this equation, i.e., the influence or Green's function, is the response of an elastic half-space to a concentrated unit impulse acting on its edge. The unknown function is the normal stress distribution in the plane of the crack, while the free term represents the effect of external loading.The solution for the stresses is obtained with the assumption that its spatial distribution contains a square root singularity near the tip of the crack, while its intensity is an unknown function of time. Thus, the orginal integral equation in space and time reduces to Volterra's integral equation of the first kind in time. The equation is singular, with the singularity of the kernel being a combined effect of the singularity of the influence function and the singularity of the dynamic stresses at the tip of the crack. Its solution is obtained numerically with the aid of a combination of quadrature and product integration methods. The case of a semi-infinite crack moving with a prescribed velocity is examined in detail.The method can be readily extended to problems involving mode II and mixed mode crack propagation as well as to problems of dynamic external loadings.     

Elastic equilibrium of an interface crack between rigid stamp and orthotropic strip

Rupture of the interface between an absolutely rigid stamp and an orthotropic infinite strip is investigated. A plane elasticity problem for an interface crack formally leads to oscillatory singularities at the crack tip. In order to overcome this nonphysical solution, a model of an interface crack with frictionless contact zones near the crack tips and the corners of the stamp is developed. By using the method of integral Fourier transforms the problem is reduced to a system of three singular integral equations. The system is solved by the method of collocations with the points of collocation chosen at zeros of the Chebyshev polynomials. The stress intensity factors at the crack tips and the stamp corner points are evaluated.     

An infinitely large plate weakened by a circular-arc crack subjected to partially distributed loads

On the basis of the complex-variable approach for the first boundary condition problems, a mapping function is proposed to transform the contour surface of a circular arc crack into a unit circle. By this mapping, direct stress integration along the contour surface can be performed for the case when uniform tractions are applied on part of the crack edge. General complex stress functions are obtained by evaluating the Cauchy integral for the governing boundary equation. After the obtained stress functions are differentiated with respect to a reference angle in the mapped plane, the general complex stress functions for the circular-arc crack problem, when concentrated loads are applied on the crack surface, can be obtained. The importance of this solution lies in its general applicability to crack problems with arbitrary loading.     

The plane elastostatic solution for a symmetrically loaded crack in a strip composite

The plane elastostatic problem for a crack in a strip composite loaded with normal or shearing traction is reduced to a single integral equation. The dependence of the solution on the material parameters is exhibited explicitly in the integral equation through two composite parameters. The integral equation is solved numerically and the dependence of the stress intensity factors on the material parameters is displayed graphically for all physically relevant composites for each of several chosen values of the crack length to strip width ratio.