Interaction of axially symmetric waves with an annular crack in an infinitely long hollow cylinder

In this paper an analysis of the interaction of longitudinal waves with an annular crack in an infinitely long hollow cylinder is presented. Using Fourier sine and cosine as well as Hankel integral transforms, formal complete solutions to the governing equations are given. By means of Abel integral transform, the problem is reduced to the solution of a Fredholm integral equation of the second kind which is, then, solved numerically for a range of values of the frequencies of the incident waves. The numerical values of the dynamic stress intensity factor at the rim of the crack have been calculated.     

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.     

Plane steady problem of heat-conduction theory for a hyperbolic cylinder with boundary conditions of the third kind

Fredholm integral equations of the second kind are obtained for the temperature distribution at the surface of a hyperbolic cylinder which is cooled in accordance with Newton's law. The integral equations permit solution by the successive-approximation method at small values of the Biot number.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 38, No. 1, pp. 150–153, January, 1980.     

Transient response of a crack in an anisotropic strip

Summary The problem of an anisotropic elastic strip containing a crack which is opened by stresses suddenly applied on the crack faces is considered here. The problem is reduced to a set of simultaneous Fredholm integral equations of the second kind which may be solved iteratively. Once the solutions of these integral equations are obtained, the dynamic stress intensity factors may be evaluated numerically. Numerical results are obtained for a particular transversely isotropic strip.With 1 Figure     

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.     

Forced torsional oscillations of multilayered solids

A new approach is proposed for obtaining the dynamic elastic response of a multilayered elastic solid caused by a forced torsional oscillation inside the solid. The elastodynamic Green’s function of the center of rotation and a point load method are used to solve the problem. The solution of the center of rotation for multilayered solids is obtained by solving a set of simultaneous linear algebraic equations using the boundary conditions for the singularity and for the layer interfaces. The solution of the forced torsional oscillation is formulated by integrating the Green’s function over the contact area with unknown surface traction. The dual integral equations of the unknown surface traction are established by considering the boundary conditions on the contact surface of the multilayered solid, which can be converted into a Fredholm integral equation of the second kind and solved numerically.     

Stress analysis of an elastic layer attached to an elastic half space of the same material

The problem of an elastic layer attached to a half space of the same material by a spot weld is formulated for the case when the loading is tension shear. The problem is reduced to the solution of three simultaneous Fredholm integral equations of the second kind. Solution is obtained to integral equations for several values of the ratio of radius of weld to thickness of layer and quantities of physical interest are calculated.     

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.     

Dynamic stress intensity factor of a cracked dielectric medium in a uniform electric field

Summary We consider the scattering of normally incident longitudinal waves by a finite crack in an infinite isotropic dielectric body under a uniform electric field. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor versus frequency is computed, and the influence of the electric field on the normalized values is displayed graphically.     

Impact response of a finite crack in an orthotropic piezoelectric ceramic

Summary The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.     

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.     

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.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

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.     

Diffraction of SH waves by a moving crack

Summary The problem of diffraction of anti-plane shear waves by a running crack of finite length is investigated analytically. Fourier transform method is used to solve the mixed boundary value problem which reduces to two pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The iterative solution of the integral equations has been obtained for small wave number. The solution is used to calculate the dynamic stress intensity factor at the edge of the crack.With 2 Figures     

Diffraction of torsional waves by a flat annular crack at the interface of two bonded dissimilar elastic solids

The paper deals with the problem of finding the stress distribution near an annular crack located at the interface of two bonded dissimilar elastic solids. The crack is opened by the interaction of a torsional wave incident normally on the annular crack. The problem is reduced to the solution of three simultaneous Fredholm integral equations. The numerical solution of these simultaneous integral equations has been obtained. The solution is used to calculate the stress-intensity factors at the tips of the crack.     

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.     

Acoustic diffraction by a rigid annular disk

Summary The subject of this paper is the problem of acoustic diffraction by a perfectly rigid annular disk. The method of solution rests on formulating the problem in terms of an integral equation which embodies the steady state wave equation as well as the boundary conditions. This Fredholm integral equation of the first kind is converted into four simultaneous integral equations of the second kind by using Williams' integral equation technique. These four integral equations are subsequently solved by the standard iterative procedure when the frequency of the incident wave is low and the inner radius of the annulus is small.     

An arbitrarily-oriented plane crack in an anisotropic elastic slab

The problem of an arbitrarily-oriented plane crack in an anisotropic elastic slab is considered. Through the use of a Fourier transform technique, the problem is reduced to a system of simultaneous Fredholm integral equations of the second kind. Once these integral equations are solved, relevant quantities such as the crack energy can be readily computed. Numerical results pertaining to the stability of a plane crack in a particular elastic slab are given.