Sudden twisting of an external circular crack in an infinite medium with a cylindrical inclusion

In this paper the torsional impact response of an external circular crack in an infinite medium bonded to a cylindrical inclusion has been investigated. The infinite medium and cylindrical inclusion are assumed to be of different homogeneous isotropie elastic materials. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the results are expressed in terms of a Fredhol integral equation of the second kind. By solving Fredholm integral equation of the second kind the numerical results for the dynamic stress-intensity factor are obtained which measure the load transmission on the crack.     

An axisymmetric external crack problem for an infinite medium with a cylindrical inclusion

Summary This paper deals with the determination of stresses in an infinite medium containing an external crack surrounding a cylindrical inclusion. The two media are assumed to be homogeneous, isotropic and elastic but with different elastic constants. The continuity of stresses and displacements is assumed at the common cylindrical surface due to perfect bonding. The problem is reduced to the solution of a Fredholm integral equation of the second kind. A closed-form expression is obtained for the stress-intensity factor. The integral equation is solved numerically and the results are used to obtain the numerical values of the stress-intensity factor which are displayed graphically.The authors thank the National Research Council of Canada for supporting this research through NRC Grant No. A-4177.     

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.     

Diffraction of torsional waves by a penny-shaped crack in an infinitely long cylinder bonded to an infinite medium

This paper contains an analysis of the interaction of torsional waves with penny-shaped crack located in an infinitely long cylinder which is bonded to an infinite medium. Both the cylinder and infinite medium are of homogeneous and elastic but dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack.     

Electroelastic response of a flat annular crack in a piezoelectric fiber surrounded by an elastic medium

Following the theory of linear piezoelectricity, the electroelastic problem of a flat annular crack in a piezoelectric fiber embedded in an elastic medium is considered. Fourier and Hankel transform techniques are employed to formulate the mixed-boundary-value problem as a singular integral equation. The stress-intensity factor, energy-release rate and energy-density factor are computed for some piezoelectric composites, and the influence of applied electric fields on the normalized values is displayed graphically.     

Penny-shaped crack in a sphere embedded in an infinite medium

We consider the problem of determining the stress intensity factor and the crack energy in an Isotropie, homogeneous elastic sphere embedded in an infinite Isotropie, homogeneous elastic medium when there is a diametrical crack in the sphere. We assume that the crack is opened by an internal pressure and the sphere is bonded to the surrounding material. The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function φ($\text{t}$). Expressions for the stress intensity factor and the crack energy are obtained in terms of φ($\text{t}$). The integral equation is solved numerically and the numerical values of the stress intensity factor and the crack energy are graphed.     

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.     

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.     

Vibrations of elastic infinite and semi-infinite media containing cracks

The problems of stress distribution in an infinite medium and in an elastic half-plane containing line cracks, when the pressure which opens the crack is periodic in time, are considered. These are (1) a cruciform crack in an elastic infinite medium, (2) an edge crack perpendicular to the surface of an elastic half-plane, and (3) their corresponding “exterior” problems. The integral equations corresponding to these problems are obtained. Expressions for the stress intensity factor and the crack energy are derived and numerical results are presented. The equivalence of the stress intensity factor and the crack energy for “exterior” and “interior” problems as established by Stallybrass for the static case is obtained from the dynamic results by letting the frequency tend to zero.     

Theoretical stress analysis for a non-isotropic body of cylindrical configuration containing a row of cracks

A problem of stress analysis for a long circular cylinder has been dealt with in this paper by analytical methods. The cylinder is assumed to be made of an elastic material which is not isotropic but the elastic properties are considered to be similar in directions perpendicular to the axis of the cylinder. The body under consideration is supposed to contain an infinite row of penny-shaped cracks which are parallel to each other and located periodically along the cylinder-axis. All the cracks are assumed to be opened by the same distribution of internal pressure on their surfaces. By choosing appropriate potential functions the problem is treated mathematically through the use of integral equation approach. Numerical results for the stress-intensity factor, the strain energy and the critical pressure, obtained on the basis of the analysis are also given.     

Axisymmetric crack problem of a thick-walled Cylinder with cladding

The paper considers the elastostatic axisymmetric problem for a long thick-walled cylinder containing an axisymmetric circumferential internal or edge crack with cladding. The cladding is assumed to be bonded to inner wall of the hollow cylinder. Using the standard transform technique, the problem is formulated in terms of an integral equation of the first kind which has a generalized Cauchy kernel as the dominant part. The integral equation is solved numerically by using appropriate quadrature formulas. The related stress-intensity factors are calculated for the hollow cylinder with cladding under axial load. The influence of the geometrical configuration and the cladding on the stress-intensity factors is shown graphically.     

Stress-intensity factors for a cruciform crack with equal arm in an infinite elastic strip of finite thickness

The stress-intensity factors and the crack energy for a cruciform crack with equal arm, in an infinite elastic strip of finite thickness have been determined by solving the corresponding boundary value problem by using the Fourier Transform technique. The dual integral equations generated by using the necessary boundary conditions have been reduced to solving a pair of integral equations of Fredholm type involving two unknown functions. The integral equations have been solved by a numerical technique and the formulae for determining the quantities of physical interest have been expressed in terms of the solutions of the integral equations. Numerical results for these physical quantities have been furnished at the end for different values of the thickness of the strip.     

High-frequency torsional oscillations—I: Penny-shaped crack in an inhomogeneous medium

The problem of an inhomogeneous medium, whose shear modulus and density vary exponentially with radius, containing a penny-shaped crack undergoing high-frequency torsional oscillations is reduced asymptotically to Wiener-Hopf integral equation and solved by Carleman's method. Uniformly valid asymptotic results are obtained. Explicit expressions are derived for the normal displacement gradient outside the crack region, the stress-intensity factor and the energy of the crack.     

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.     

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.     

Crack interaction in an infinite elastic strip

The problem considers an arbitrary number of colinear and unequal size Griffith cracks opened by a non-uniform internal pressure in an infinite elastic strip. The cracks are located halfway between and parallel to the surfaces of the 2-dimensional medium. By appropriate integral transformations the mixed boundary value problem is reduced to singular integral equations. The stress intensity factors, crack openings and crack energies are then determined for many different cases.     

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.     

The stress intensity factors of a star-shaped array of cracks in an infinite elastic solid

The problem of determining the stress intensity factors and crack formation energy of a radial system of line cracks in an infinite elastic solid is reduced to the solution of a singular integral equation. The equation is solved numerically for the special case in which the cracks are opened by a constant pressure.     

Two coplanar Griffith cracks in an infinite elastic layer under torsion

Summary We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are assumed to be stress free. The cracks are located in the middle plane of the layer parallel to its faces. By using Fourier transforms, we reduce the problem to the solution of a set of triple integral equations with a cosine kernel and a weight function. These equations are solved exactly by using finite Hilbert transform techniques. Finally we derive the closed form expressions for the stress intensity factors and the crack energy. Solutions to the following problems are derived as particular cases: (i) a single crack in an infinite layer under torsion, (ii) two coplanar cracks in an infinite space under torsion, (iii) a single crack in an infinite space under torsion.     

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.