Penny-shaped crack in an infinite elastic cylinder bonded to an infinite elastic material surrounding it

This paper concerns with the state of stress in a long elastic cylinder, with a concentric penny-shaped crack, bonded to an infinite elastic medium. The crack is assumed to be opened by an internal pressure and that the plane of the crack is perpendicular to the axis of the cylinder. The elastic constants of the cylinder and the semi-infinite medium are assumed to be different. The problem is reduced to the solution of a Fredholm integral equation of the second kind. Closed form expressions are obtained for the stress-intensity factor and the crack energy. The integral equation is solved numerically and results are used to obtain the numerical values of the stress-intensity factor and the crack energy which are graphed.     

Axisymmetric solution for a semi-infinite elastic circular cylinder of one material indenting an elastic half-space of another material

The axisymmetric bonded contact problem of a semi-infinite right circular cylinder of one elastic material indenting a half-space of a different elastic material is reduced to a system of singular integral equations of the second kind. The kernels of the integral equations are found to contain Cauchy and generalized Cauchy-type singularities. The index of the singularity for various material parameters combinations is determined by solving a characteristic determinant, which is obtained by considering the dominant part of the kernels. Using a modification of the method employed in [1], the system of singular integral equations is reduced to a system of simultaneous algebraic equations. The latter may then be solved numerically as in [1].     

A problem of Reissner-Sagoci type for a semi-infinite composite elastic cylinder

Summary The problem considered is that of the torsion of a semi-infinite elastic cylinder which is embedded in a semi-infinite elastic cylindrical shell of different material. By the use of integral transforms and the theory of dual integral equations, the problem is reduced to the solution of a Fredholm integral equation of the second kind. Numerical solution of the integral equation is obtained and the numerical values of the torque are displayed graphically.     

Torsion of a non-homogeneous infinite elastic cylinder slackened by a circular cut

We consider the torsional deformation of a non-homogeneous infinite elastic cylinder slackened by an external circular cut. The shear modulus of the material of the cylinder is assumed to vary with the radial coordinate by a power law. It is assumed that the lateral surface of the cylinder as well as the surface of the cut are free of stress. The main object of this study is to establish the effect of the non-homogeneity on the stress intensity factor at the tip of the cut. The problem leads to a pair of dual series relations, the solution of which is governed by a Fredholm integral equation of the second kind with a symmetric kernel. This equation is solved numerically by reducing it to an algebraic system. It is concluded that for any degree of non-homogeneity and for D, the relative depth of the cut, greater than 0.6, the cylinder may be replaced by a half-space. However, as the non-homogeneity increases, D decreases.     

Torsional impact of a layer or a cylinder bonded to an elastic half-space

In this paper two torsional impact problems are considered. The first problem deals with the solution of a layer bonded to an elastic half-space when the layer is driven by the torsional impact over a bonded rigid circular disc. In the second problem sudden torsion by a rigid disc attached over the plane face of a circular cylinder is considered and the rest of the plane surface of the cylinder is stress free. The cylinder is bonded to the half-space, making use of Laplace and Hankel transforms the solution of each problem is reduced into Fredholm integral equations of the second kind. A numerical Laplace inversion technique is then used to recover the time depencence of the solution. The numerical values for the applied torque at the surface of rigid disc are calculated for each problem and then are displayed graphically.     

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.     

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.     

A partially debonded ellipsoidal inclusion in an elastic medium. Part I: Stress and displacement fields

The paper calculates the elastic field of an ellipsoidal inclusion which has undergone an internal deformation and has debonded over a part of its boundary from the surrounding medium. The problem is reduced to the solution of a (singular) integral equation for the displacement discontinuity across the debond. The essential steps of a method of solving this equation are outlined. The elastic field is used in the companion paper (Part II) to calculate the stress intensity factors along the edge of debond.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

Penny-shaped interface crack between dissimilar nonhomogeneous elastic layers under axially symmetric torsion

Summary The problem of axially symmetric torsion for dissimilar nonhomogeneous bonded elastic layers containing a penny-shaped interface crack is considered. The mixed boundary value problem is reduced to solving a Fredholm integral equation of the second kind. The Fredholm integral equation is solved numerically by reducing it to a system of simultaneous algebraic equations. Numerical results for the stress intensity factor are presented in the form of graphs.     

Interaction of shear waves with a griffith crack situated in an infinitely long elastic strip

This paper deals with the propagation of shear waves in a wave guide which is in the form of an infinite elastic strip with free lateral surfaces. This strip contains a Griffith crack. An integral transform method is used to find the solution of the equation of motion from the linear theory for a homogeneous, isotropic elastic material. This method reduces the problem into an integral equation. It has been observed that only shear waves with frequencies less than a parameter-value, depending on the width of the wave guide, can propagate. The integral equation is solved numerically for a range of values of wave frequency and the width of the strip. These solutions are used to calculate the dynamic stress intensity factor, displacement on the surface of the crack and crack energy. The results are shown graphically.     

Griffith cracks in an elastic medium in which body forces are acting

The plane strain problem of determining the distribution of stress in an infinite isotropic elastic medium containing Griffith cracks located on a single line is examined. The crack surfaces are assumed to be free from tractions, and the stress distribution in the medium is due to the action of body forces. Fourier transform methods are employed to reduce the problem to that of solving a singular integral equation of Cauchy type. The solution is completed in the case in which the medium contains a single crack. Particular distributions of concentrated loads are considered in detail, and the results are compared with those available in the literature.     

Plane strain problem of a punch indenting an initially stressed Neo-Hookean anisotropic elastic half plane

Summary The plane strain problem of an initially stressed neo-Hookean orthotropic incompressible elastic half plane indented by a smooth rigid punch is formulated through a generalised Fourier transform and is reduced to the solution of a well-known singular integral equation of first kind. The solution of the integral equation directly gives the surface pressure under the punch. The normal displacement outside the region of contact of the punch is also determined. Three different shapes of punches are considered. Results are deduced for an sotropic incompressible medium free of initial stress.     

Interaction of torsional waves with an annular crack in an infinitely long cylinder

The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infinite elastic cylinder, which is excited by normal torsional waves. The curved surface of the cylinder is assumed to be stress free. Solution of the problem is reduced to three simultaneous Fredholm integral equations. By finding the numerical solution of the simultaneous Fredholm integral equations the variations of the dynamic stress-intensity factors are obtained which are displayed graphically.     

Axisymmetrically loaded thin circular plate in adhesive contact with an elastic half-space

The interaction problem of adhesive contact between an axisymmetrically loaded thin circular plate and an isotropic or transversely isotropic elastic half-space is reduced to a problem of solving a pair of coupled integral equations for the unknown normal and shearing interfacial tractions. The system of integral equations is solved numerically and some results are presented.     

The axisymmetric double contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the smooth receding contact between an elastic layer and a half space when the layer is compressed by a frictionless semi-infinite elastic cylinder. Upon loading, the contact along the layer-subspace interface shrinks to a circular area, radius of which is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of contact and the stress intensity factor round the edge of the cylinder are calculated for various material combinations.     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

On crack problem in an elastic ponderable layer

The paper deals with the plane problem of stress distribution in an elastic ponderable layer with a stationary edge crack normal to the boundary plane. The layer is situated and fixed on a rigid foundation. The stresses are caused by action of body forces. By using the method of Fourier transforms the problem is reduced to a system of dual integral equations and next, to a Fredholm integral equation of the second kind. The numerical analysis of the Fredholm equation permitted to determine the stress intensity factor and the crack opening displacement. This revised version was published online in July 2006 with corrections to the Cover Date.     

Annular punch on an elastic layer overlying an elastic foundation

The problem solved here is the axisymmetric mixed boundary value problem of the isotropic homogeneous theory of elasticity, in which the normal displacement is specified inside an annular area $\text{a ≤ r ≤ b}$, the normal stress is zero in $\text{r < a, r \# b}$ and the shearing stress is zero on the whole face $\text{z = ?h}$, the upper face of the elastic layer; the continuity of the normal and radial displacements and the normal and shearing stresses is assumed at the interface $\text{z = 0}$ between the elastic layer and the elastic foundation having different elastic constants. The problem is reduced to the solution of a Fredholm integral equation of the first kind. The Fredholm integral equation is further put in terms of four simultaneous Fredholm integral equations of the second kind in four unknown functions. The iterative solution of these integral equations has been obtained for $\text{epsi = b/h ? 1}$, and $\text{λ = a/b ? 1}$ for the case of an annular cylindrical punch. The expressions for the normal stress $\text{σ}{}_{\mathrm{zz}}\text{(r, ?h)}$ for $\text{a ≤ r ≤ b}$ and the total load $\text{P}$ on the punch have been obtained.     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.