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.     

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.     

Frictionless indentation by a rigid wedge: The effect of tangential displacements in the contact zone

Recent dynamic analyses of frictionless rigid indentation on linearly elastic half-planes assume that the tangential displacement of half-plane points in contact with the indentor is negligible. This assumption is used as a justification for denning the contact zone size, although not known in advance, on the undeformed half-plane, and for uncoupling the tangential and normal displacements in the contact zone. This article examines the validity of this assumption for the symmetric wedge with a constant indentation speed and a contact zone which, as denned on the undeformed half-plane, extends at a constant subcritical rate. Both the coupled and uncoupled solutions are obtained analytically and compared. The difference between the half-plane size defined on the undeformed half-plane and its size on the deformed half-plane may actually be greater for the uncoupled solution. Moreover, the coupled solution does not exhibit the wedge apex stress singularities found for the uncoupled solution.     

Various integral equations for a single crack problem of elastic half-plane

Two kinds of the complex potentials used for the crack problem of the elastic half-plane are suggested. First one is based on the distribution of dislocation along a curve, and second one is based on the distribution of crack opening displacement along a curve. Depending on the use of the complex potentials and the right hand term in the integral equation, two types of the singular integral equation for a single crack problem of elastic half-plane are derived. Regularization of the suggested singular integral equations gives three types of the Fredholm integral equation for the relevant problem. The weaker singular integral equation and the hypersingular integral equation are also introduced. Seven types of the integral equation are finally obtainable. The relation between the kernels of the various integral equations is also discussed.     

Structure of an edge-dislocation wave originating in plane-wave diffraction by a half-plane

A new treatment of the well-known Sommerfeld solution of the problem of plane-wave diffraction from a perfectly conducting half-plane is reported. We show, in both theory and experiment, that the diffraction field (E-polarization) can be represented as a superposition of real physically existing waves, in contrast to geometrical and boundary waves postulated in Sommerfeld's representation. Our representation includes two pairs of wave components: one pair propagates along the direction of the incident wave, and the other in a mirror-reflected direction. Each wave pair consists of a plane-wave component with an amplitude half that of the incident wave and a nearly plane-wave component with an infinitely extended edge dislocation. On the basis of the proposed interpretation, all features of the half-plane diffraction are explained.     

A note on plastic deformation at the tip of an edge crack

Summary An integral transform solution is given for the problem of an edgecrack forming at the free boundary of a half-plane. The plastic zone is taken in precisely the form as it appears experimentally in such materials as low-C steels. The method used is a further extension of the work of Sneddon and Das [1]. Using Dugdales hypothesis, the length of plastic zone is obtained. When the plastic zone tends of zero length, the solution of the stretching of an elastic half-plane with an edge crack is obtained.With 3 Figures     

Edge-crack growth

Conclusion Singular integral equations have been applied to the quasistatic growth of an edge crack in an isotropic elastic half-plane when a self-balancing load is given at the crack edges. The path has been calculated by a step method with allowance for the stress redistribution during the crack growth, and the growth direction has been determined from the criterion. The resulting singular integral equation is solved numerically by mechanical quadratures.Paths have been constructed and stress intensity coefficients have been determined for them for uniaxial stretching of a half-plane at infinity and also for normal pressure and localized forces applied to the edges of an initially rectilinear but arbitrarily oriented crack. Some trends in edge break-away are identified.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 5, pp. 42–48, September–October, 1991.     

Tangential-displacement effects in the wedge indentation of an elastic half-space – an integral-equation approach

The idea of considering tangential-displacement effects in a classical elastostatic contact problem is explored in this paper. The problem involves the static frictionless indentation of a linearly elastic half-plane by a rigid wedge, and its present formulation implies that the tangential surface displacements are not negligible and should thus be coupled with the normal surface displacements in imposing the contact zone boundary conditions. L.M. Brock introduced this idea some years ago in treating self-similar elastodynamic contact problems, and his studies indicated that such a formulation strongly influences the contact-stress behavior at half-plane points making contact with geometrical discontinuities of the indentor. The present work again demonstrates, by studying an even more classical problem, that the aforementioned considerations eliminate contact-stress singularities and therefore yield a more natural solution behavior. In particular, the familiar wedge-apex logarithmic stress-singularity encountered within the standard formulation of the problem (i.e. by avoiding the tangential displacement in the contact boundary condition) disappears within the proposed formulation. The contact stress beneath the wedge apex takes now a finite value depending on the wedge inclination angle and the material constants. By utilizing pertinent integral relations for the displacement/stress field in the half-plane, an unusual mixed boundary-value problem results whose solution is obtained through integral equations.