Simultaneous Singular Integral Equations Research Articles

Thermomechanical Effects due to Hot Rolling Contact on the Energy Release Rates of Multiple Interface Cracks in Layered Media.

This paper deals with the mutual interference of multiple two-dimensional interface cracks in a surface coating layered material under thermal stresses due to rolling/sliding contact with heat input. Contact loading is simulated as a contact pressure load with both the normal and shear components having parabolic distribution. Attention here is focused on the energy release rate at the crack tips which provide a measure for quantifying the magnitude of interface crack growth. In the present crack analysis, the interface cracks are replaced by the distributed edge dislocations, and the crack face friction is negrected. The problem is reduced to simultaneous singular integral equations for dislocation densities. The integral equations can be solved numerically by considering the nature of the singularities at the crack tips. The numerical results of the energy release rate showing the effects of the mutual interference of a pair of interface cracks are given for some tribological material coatings on a steel substrate. The effects of the frictional coefficient, the heat input strength and the coating thickness upon the magnitude of the energy release rate and it's mutual interference are considered numerically.

Stress intensity factors for moving interfacial crack between bonded dissimilar fixed orthotropic layers

The plane strain problem of determining the dynamic stress intensity factors for a moving Griffith crack situated at the interface of two dissimilar orthotropic fixed layers is considered. The mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which have finally been solved by using Jacobi’s polynomials. Numerical results for some particular cases of the problem are also presented graphically.

Stress intensity factors for an interfacial crack between an orthotropic half-plane bonded to a dissimilar orthotropic layer with a punch

The plane strain problem of determining Stress Intensity Factors (SIF) for a moving interfacial Griffith crack between an elastic orthotropic half-plane and a dissimilar orthotropic layer with a moving punch situated along the boundary of the layer have been considered. The problem is reduced to the solution of three simultaneous singular integral equations with Cauchy-type singularities. Expressions for SIF for the case of a general loading are obtained. Numerical results for some particular cases are also presented graphically.

The steady linear thermoelastic problem for a circular hole with two radial edge cracks of unequal length

The problem discussed in this paper is that of determining the thermal stresses in a linear thermoelastic solid when a uniform heat flow is disturbed by the presence of a circular hole with two radial edge cracks of unequal length. Using a Mellin-type transform and superposition, the problem is reduced to a pair of simultaneous singular integral equations. The integral equations are solved numerically and results for the stress intensity factors and crack formation energy are given in graphical form.

Moving Griffith crack at the interface of two dissimilar orthotropic half planes

The plane strain problem for determining the dynamic stress intensity factor in orthotropic medium when a Griffith crack is situated at the interface of two dissimilar half spaces is considered. By use of the Fourier transform technique, the mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which finally have been solved by using Jacobi polynomials. Numerical results for the dynamic stress intensity factors are obtained for some particular cases and the results are presented graphically.

A griffith crack located asymmetrically in an infinitely long elastic strip

Of concern in this paper is the distribution of stress in the vicinity of a Griffith crack located in a strip of finite width parallel to the edges. The edges of the strip are free from tractions and the crack is opened out by the application of a uniform internal pressure on its faces. The crack is not necessarily situated on the centreline of the strip, and so the mode II stress intensity factor is not necessarily zero even though the loading is purely of mode I type. By means of Fourier transforms the problem is reduced to that of solving a pair of simultaneous singular integral equations of Cauchy type. The mode I and mode II stress intensity factors are computed and presented in tables and figures for various combinations of h/δ and °/a, where 2δ is the strip width, 2a is the crack length and h is the distance of the crack from the centreline of the strip.For large δ/a one can derive approximate expressions for stress intensity factors by retaining terms up to a certain order of a/δ, say, (a/°)6. A comparison between the numerical results obtained from the approximate expressions and those obtained from the direct numerical solution of singular integral equations shows that the approximate expressions are good only for h = 0 and perhaps for small values of h/δ. It is observed that the convergence of the numerical results increases with the increase in the value of (δ-h)/a. This suggests that one should obtain approximate formulae for the stress intensity factors by retaining terms of a certain order of a/(δ-h). It is obviously done for the case when h = 0. However, for nonzero values of h/δ, the problem does not seem to be an easy one.A limiting case of the strip problem described above is a half plane problem. The latter problem of determining the distribution of stress in the neighbourhood of a crack, subjected to a uniform internal pressure and situated in the half plane parallel to its boundary, has been studied in the literature independently. We deduce the formulae for the half plane problem from the formulae for the strip problem and compare our results with those in the literature.

Theoretical Analysis of Elastic Singular Stress Field by use of Singular Integral Equation Method.

Simultaneous pairs of dual integral equations with trigonometric kernels encountered in the analysis of mixed boundary value problems by Fourier transforms are solved by reduction to a coupled system of singular integral equations. The method of reduction indicated here is based on the relationship between the Fourier transform and the Hilbert transform. The numerical solution of the simultaneous singular integral equations is obtained by using two types of integration formulae. The accuracy and the convergency of the results are examined by examples of crack problems.

Dynamic Response of an Orthotropic Half-Space With a Subsurface Crack: In-Plane Case

Scattering of elastic waves by a subsurface crack in an orthotropic half-space subjected to a surface line load of arbitrary angle of inclination is studied. Green’s functions are developed and used along with the representation theorem to reduce the problem to a set of simultaneous singular integral equations in the Fourier transformed domain. Solution to these equations is then obtained by expanding the unknown crack opening displacement (COD) in terms of Chebyshev polynomials. Numerical results are given for specific examples involving orthotropic materials.

Static stresses in a periodically layered anisotropic elastic composite containing a periodic array of planar cracks

The problem of calculating the static elastic stresses in a periodically layered anisotropic composite containing a periodic array of planar cracks is considered. We formulate the problem in terms of a system of simultaneous finite-part singular integral equations which can be solved numerically using collocation techniques. The solution of the integral equations enables relevant quantities such as the stress intensity factors to be computed. Numerical results are obtained for specific cases of the problem.

Coplanar cracks at the interface of bonded dissimilar orthotropic elastic half-planes

This paper deals with the problem of determining the stress intensity factors when two coplanar Griffith cracks are located at the interface of two bonded dissimilar orthotropic elastic half-planes. Fourier transforms are employed to reduce the mixed boundary value problem to that of solving a set of simultaneous singular Fredholm integral equations of the second kind. These equations are further reduced, by using Jacobi polynomials, to a system of simultaneous algebraic equations. The stress intensity factors are calculated by using numerical methods.

An arbitrarily oriented inclusion in a half-space subjected to an inclined surface load

The plane elastostatic problem for a half-space which contains an arbitrarily oriented flat inclusion is considered. The medium is loaded from its surface by an inclined concentrated force. It is assumed that the inclusion may be rigid or elastic with negligible bending rigidity. Fourier integral transforms are applied and the problem is formulated in terms of the unknown interface stresses using concentrated force solutions as Green's functions. The unknowns are then solved from a system of simultaneous singular integral equations. Stress intensity factors are presented for various inclusion and load geometries.

A griffith crack at the interface of two bonded dissimilar orthotropic elastic half-planes

The problem considered is that of determining the stress intensity factors when a Griffith crack is located at the interface of two bonded dissimilar orthotropic elastic half-planes. A Fourier transform technique is employed to reduce the mixed boundary value problem to solving a set of simultaneous singular integral equations of the second kind. By the use of Jacobi polynomials, these equations are further reduced to a system of simultaneous algebraic equations. It is shown that both the stress intensity factors K 1 and K 2 occur and also that they depend on elastic properties of the materials when the crack is opened by an applied normal stress, which is contrary to the results obtained by Zhang.

Application of a Singular Integral Equation Method to a Crack Branching at an Interface

The plane elastostatic problem of a crack branching at a fiber/matrix interface in carbon fiber reinforced composites is considered using a micromechanical model. The model is a crack which originates in one half-plane and branches at a bimaterial interface of a bonded plane consisting of an isotropic and an orthotropic half-planes. The method of continuous distribution of dislocations reduces the problem to a system of a simultaneous singular integral equations of the second kind, which is solved nemerically by use of the Lobatto-Jacobi integration rule. The singular stress field at the crack tips is investigated in terms of stress intensity factors and crack opening displacements.

Application of a singular integral equation method to a crack branching at an interface.

The model is a crack which originates in one half-plane and branches at a bimaterial interface of a bonded plane consisting of an isotropic and an orthotropic half-planes. The method of continuous distribution of dislocations reduces the problem to a system of a simultaneous singular integral equations of the second kind, which is solved nemerically by use of the Lobatto-Jacobi integration rule

Plane elastostatic analysis of a crack passing through a bimaterial interface.

The plane elastostatic problem for a bonded plane consisting of an isotropic and an anisotropic half planed is considered when a straight crack originating in one half-plane is passing through a bimaterial interface into an other. It is assumed that the crack is normal to the interface and that the bonded plane is subjected to a constant strain away from, and perpendicular to, the crack. A fundamental stress field due to two colinear edge dislocations existing in each half plane is firstly analyzed, and then the method of continuous distribution of dislocations reduces the problem to a system of simultaneous singular integral equations for the dislocation density. The singular behaviour of stresses at the point of intersection of the crack and the interface as well as at the crack-tips in investigated in detail.

Interaction of longitudinal wave with a penny-shaped crack at the interface of two bonded dissimilar elastic solids-II

The paper deals with the problem of finding the stress distribution near a penny-shaped crack situated at the interface of two bonded dissimilar elastic solids. The crack is opened by the interaction of plane harmonic longitudinal elastic wave, incident normally on the crack. The problem is first reduced to a set of simultaneous dual integral equations which are further transformed to a set of simultaneous singular integral equations. These are solved numerically by reducing them to a set of algebraic equations. The solution is used to calculate the stress-intensity factors and the size of the overlapping zones at the edge of the crack.

Interaction of elastic waves in two bonded dissimilar elastic half-planes having Griffith crack at interface—I

The paper deals with the problem of finding the stress distribution near a Griffith crack located at the interface of two bonded dissimilar elastic half-spaces. The crack is opened by the interaction of a plane harmonic elastic wave, incident normally on the crack. The problem is first reduced to a set of simultaneous dual integral equations which are further transformed to a set of simultaneous singular integral equations. These are solved numerically by reducing them to a set of algebraic equations. The solution is used to calculate the stress intensity factors.

Influence of an insulated circular hole on thermal stress singularities at tips of a crack

On the basis of the steady-state two-dimensional theory of thermoelasticity, the influence of an insulated circular hole on the thermal stress singularities at the tips of a line crack, whose surface is cooled, is discussed. The crack is located arbitrarily in an infinite matrix. The method of continuous distribution of dislocations is extended by newly introducing a continuous distribution of quasi-Volterra dislocations corresponding to line heat sources. Replacing the crack by the continuous distributions of quasi-Volterra dislocations and edge dislocations, we obtain a set of simultaneous singular integral equations for dislocation density functions. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. The variations of the stress intensity factors and the initial direction of crack extension with the distance of the crack from the insulated circular hole and the angle of inclination of the crack are shown graphically.

Thermal stresses near tips of an insulated line crack in a semi-infinite medium under uniform heat flow

On the basis of the two-dimensional theory of thermoelasticity, the thermal stress field near the tips of a thermally insulated line crack in a semi-infinite medium under steady-state uniform heat flow is discussed. The crack is replaced by continuous distributions of sources of temperature discontinuity and edge dislocations. Then we obtain a set of simultaneous singular integral equations for density functions. The solution is given in the form of the product of the series of Tchebycheff polynomials of the first kind and their weight function. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. The effects of the distance from the crack tip to the bounding plane surface of the semi-infinite medium and the angle of inclination of the line crack on the stress intensity factors are shown graphically.

An Elastic Half Plane Weakened by a Rectangular Trench

The problem of an elastic half plane loaded in tension and weakened by a rectangular trench is set up in a manner leading to a system of simultaneous singular integral equations. The method is to consider the superposition of solutions for two cracks perpendicular and one parallel to the surface of the half plane. The limiting case when the three cracks intersect appropriately leads to the geometry of a rectangular trench. A collocation scheme developed by Erdogan and Gupta leads to numerical solution of the equations to determine the stress intensity factors at the corners and the horizontal displacement of the trench wall.