Complex potentials and integral equations for curved crack and curved rigid line problems in plane elasticity

Summary In this paper, a simple layer potential and a double layer potential are suggested to solve the curved crack problem. The complex potentials in the simple layer case are formulated on the distributed dislocation along the curve. Meantime, the complex potentials in the double layer case are formulated on the crack displacement opening distribution. Behaviors of the complex potentials, for example the behaviors of increments of some physical quantities around a large circle, are analyzed in detail. Continuity and discontinuity of some physical quantities in the normal direction of the curve are analyzed, which are key points for formulating the integral equations of the problems. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. The relations between the kernels in different integral equations are addressed. Similarly, a simple layer potential and a double layer potential are suggested to solve the curved rigid line problem. The complex potentials in the simple layer case are formulated on the distributed forces along the curve. Meantime, the complex potentials in the double layer case are formulated on the resultant force function. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. When the resultant forces and moment are applied on the deformable line, the constraint equations are suggested. For more general cases, for example, in the case that the tractions applied on the two crack faces are not same in magnitude and opposite in direction, a singular integral equation is suggested. The equation is obtained by a superposition of two kinds of single layer potentials.     

Stress state of a half-plane with cracks under rigid punch action

A contact problem in elasticity theory for an isotropic half-plane with a set of curvilinear cracks, into which a rigid punch with the foundation of convex shape is indented, is considered. Coulomb friction is assumed to exist between the punch and the half-plane, while the crack faces are under conditions of either stick or smooth contact on contact parts. On the basis of integral representation for Kolosov-Muskhelishvili complex potentials by derivatives of displacement discontinuities along the crack contours and pressure under the punch, the problem is reduced to a system of complex Cauchy type singular integral equations of first and second kind. An algorithm is proposed to find solution of these equations by the method of mechanical quadratures using an iterative procedure. Two numerical examples are presented.     

An integral-equation solution for cracked half-planes bonded together and containing debondings along their interface

The general solution of an arbitrary system of microdefects (i.e. cracks and/or holes) in an isotropic elastic half-plane bonded partially, along an infinite number of straight line segments to another half-plane consisting of a different isotropic elastic material, is formulated in this paper using the complex variable technique. The solution in terms of complex potentials is given by integrals over the cracks and/or holes with integrands expressed in terms of Green's functions and an unknown complex density function. Finally, the problem is reduced to the solution of a singular integral equation for the complex density function only along the microdefects. The appropriate Green's functions are derived from the solution of the problem of a concentrated force or a dislocation existing in either of the two half planes. Numerical results are presented for the stress intensity factors in three different cases.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

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.     

Stress intensity factors for a surface-breaking crack in a half-plane subject to contact loading

Modes I and II stress intensity factors are derived for a crack breaking the surface of a half-plane which is subject to various forms of contact loading. The method used is that of replacing the crack by a continuous distribution of edge dislocations and assume the crack to be traction-free over its entire length. A traction free crack is achieved by cancelling the tractions along the crack site that would be present if the half-plane was uncracked. The stress distribution for an elastic uncracked half-plane subject to an indenter of arbitrary profile in the presence of friction is derived in terms of a single Muskhelishvili complex stress function from which the stresses and displacements in either the half-plane or indenter can be determined. The problem of a cracked half-plane reduces to the numerical solution of a singular integral equation for the determination of the dislocation density distribution from which the modes I and II stress intensity factors can be obtained. Although the method of representing a crack by a continuous distribution of edge dislocations is now a well established procedure, the application of this method to fracture mechanics problems involving contact loading is relatively new. This paper demonstrates that the method of distributed dislocations is well suited to surface-breaking cracks subject to contact loading and presents new stress intensity factor results for a variety of loading and crack configurations.     

USE OF THE DISTRIBUTED DISLOCATIONS METHOD TO DETERMINE THE T-STRESS

Abstract— This paper demonstrates a method to determine the elastic T -stress for a semi-infinite half-plane containing a surface-breaking crack which is loaded by an arbitrarily distributed far-field tension. The method consists of representing the crack by a continuous distribution of edge dislocations and forming singular integral equations to determine the equilibrium dislocation distributions. By numerically solving the integral equations, stress intensity factors and T -stresses are obtained for the example case of a crack which is normal and inclined to the free-surface of a half-plane and loaded by a uniform far-field tension.     

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.     

A thin cracked layer bonded to an elastic half-space under an antiplane concentrated load

The antiplane elasticity problem for a thin cracked layer bonded to an elastic half-space under an antiplane concentrated load is considered. The fundamental solution is obtained as a rapidly convergent series in terms of the complex potentials via iterations of Möbius transformation. The singular integral equation with a logarithmic singular kernel is derived to model a crack problem that can be solved numerically in a straightforward manner. The dimensionless mode-III stress intensity factors obtained for various crack inclinations and crack lengths are discussed in detail and provided in graphic form. A strip problem with an arbitrarily oriented crack is also considered.     

Evaluation of the T-stress in branch crack problem

This paper investigates the T-stress in the branch crack problem. The problem is modeled by a continuous distribution of dislocation along branches, and the relevant singular integral equation is obtained accordingly. After discretization of the singular integral equation, the balance for the number of equations and unknowns is well designed. After the singular integral equation is solved, the equation for evaluating the T-stress is derived. The merit of present study is to provide necessary equation for evaluating T-stress, rather than to provide the integral equation. Many computed results for T-stress under different conditions for branch crack are presented. It is found from the computed results that the interaction for T-stress among branches is complicated.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

Quasistatic edge crack growth with non-self-balancing stresses at the edges

Conclusions Singular integral equations have been applied to the quasistatic growth of an edge crack in an isotropic elastic half-plane when an arbitrary non-self-balancing load is applied to the edges, and in particular when one of the edges is subject to a localized force. The path has been constructed in a step fashion on the basis of the stress redistribution during crack growth. The singular integral equations are solved by mechanical quadrature.The paths and the stress intensity coefficients along them have been calculated for cases where one of the edges of a crack initially perpendicular to the boundary is, acted on by a constant normal pressure or normal localized force. The direction of the crack increment is derived from the criterion for maximal circumferential stresses.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 4, pp. 53–59, July–August, 1991.     

The gauss-hermite numerical integration method for the solution of the plane elastic problem of semi-infinite periodic cracks

The problem of parallel semi-infinite periodic cracks subjected to a transversely directed load in an infinite isotropic and elastic medium under conditions of plane stress or plane strain can be reduced to the solution of a Cauchy-type singular integral equation along one of the cracks. This equation can be transformed into a system of linear equations by means of an approximation of the integrals through the Gauss-Hermite procedure and application of the equation to distinct points along the faces of the crack. Stress intensity factors thus determined for the crack tips under constant load along the cracks are in satisfactory agreement with corresponding values derived previously.     

Dynamics response of complex defects near bimaterials interface by incident out-plane waves

Dynamics response of an elliptical cavity and a crack (on different sides) near bimaterials interface under incident out-plane waves is studied by applying the methods of complex variables and Green’s function. Firstly, based on “conjunction,” the analytical model is divided along the horizontal interface into an elastic half-plane possessing an elliptical cavity and a full elastic half-plane containing a crack. Using complex variables, the scattering displacement field of the half-plane containing an elliptical cavity under incident out-plane waves is then derived. According to the method of Green’s function, the corresponding Green’s functions of two half-planes impacted by an out-plane source load are further deduced. Combined with “crack division,” a crack at the full elastic the half-plane is created, and thus, expressions of displacement and stress are derived while the cavity coexists with the crack. Undetermined antiplane forces are loaded on the horizontal surfaces for conjunction of two sections and then solved by a series of Fredholm integral equations on account of continuity conditions of the interface. Finally, this paper focuses on the discussion of the influence law of different parameters on the dynamics response of complex defects near bimaterials interface by comprehensive numerical results.     

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.     

Effects of a viscoelastic substrate on a cracked body under in-plane concentrated loading

Summary The effect of a viscoelastic substrate on an anisotropic elastic cracked body under in-plane concentrated loading is studied in this paper. Based on the correspondence principle, the viscoelastic solution is directly obtained from the corresponding elastic one. The fundamental elastic solution is solved as three complex potentials via the property of analytical continuation to satisfy the continuity condition along the interface between dissimilar media. A singular integral technique in association with the dual coordinate transformation is applied to obtain the stress intensity factors for various crack orientations. Using the standard solid model to formulate the viscoelastic constitutive equation, some numerical examples are considered to demonstrate the use of the present approach.     

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.     

Singular stresses in an infinite solid with a flat annular crack around a spherical cavity under tension

The problem of singular stresses in an infinite elastic solid containing a spherical cavity and a flat annular crack subjected to axial tension is considered. By application of an integral transform method and the theory of triple integral equations the problem is reduced to that of solving a singular integral equation of the first kind. The singular integral equation is solved numerically, and the influence of the spherical cavity upon the stress intensity factor and the influence of the annular crack upon the maximum stress at the surface of the spherical cavity are shown graphically in detail.     

Two bonded half planes with a crack going through the interface

The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.     

Numerical solutions of a hypersingular integral equation for antiplane elastic curved crack problems of circular regions

Summary. In this paper, a hypersingular integral equation for the antiplane elasticity curved crack problems of circular regions is suggested. The original complex potential is formulated on a distribution of the density function along a curve, where the density function is the COD (crack opening displacement). The modified complex potential can also be established, provided the circular boundary is traction free or fixed. Using the proposed modified complex potential and the boundary condition, the hypersingular integral equation is obtained. The curve length method is suggested to solve the integral equation numerically. By using this method, the usual integration rule on the real axis can be used to the curved crack problems. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.