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.     

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.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

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.     

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.     

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.     

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.     

Numerical analysis of doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic medium using the boundary integral equation method

In this paper, the boundary integral equation approaches are used to study the doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic plane medium. For the doubly periodic rigid-line inclusion problems, the special integral equation containing the axial and shear forces within the rigid-line inclusion is used. The doubly periodic crack problems are dealt with using the displacement discontinuous integral equation approach. Stress intensity factors, effective elastic properties for doubly periodic array of cracks/rigid-line inclusions are calculated and compared with the available numerical solutions.     

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.     

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.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Stress intensity factor for the cylindrical interface crack between nonhomogeneous coaxial finite elastic cylinders

The problem of determining the stress intensity factor for a cylindrical interface crack between two dissimilar nonhomogeneous coaxial finite elastic cylinders under axially symmetric longitudinal shear stress is considered. The mixed boundary conditions lead to a pair of dual series equations which are reduced to a Fredholm integral equation of the second kind and then finally to a system of algebraic equations. Numerical values of the stress intensity factor are presented graphically.     

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.     

On the numerical evaluation of stress intensity factors for an interface crack of a general shape

A numerical algorithm is presented for the problem of a crack along the interface of an elastic inclusion embedded in an elastic plane subjected to uniform stress at infinity. The algorithm is based on a Fredholm integral equation of the second kind and allows for fast and accurate solutions to geometries of great complexity. In an example crack opening displacement and stress intensity factors are computed for a crack in the interface of an inclusion with nineteen protruding arms. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings

Summary The solutions of an eccentric crack problem in a rectangular piezoelectric ceramic medium under combined anti-plane shear and in-plane electrical loadings are obtained by the continuous electric crack face condition. Fourier transforms and Fourier series are used to reduce the problem to two pairs of dual integral equations, which are then expressed by a Fredholm integral equation of the second kind. Numerical values of the stress intensity factor and the energy release rate are obtained to show the influence of the electric field.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

Single edge-cracked orthotropic strip under three-point load

The problem of an orthotropic strip having a crack of unit length normal to one edge and subjected to a bending moment resulting from three-point loading is solved using integral transform method. The mixed boundary conditions lead to dual integral equations which are ultimately reduced to a Fredholm integral equation of second kind. The integral equation thus obtained is solved by the method developed by Fox and Goodwin. Numerical solutions for a fibre-reinforced composite material have been carried out to determine the stress intensity factor of an orthotropic medium. The same has been compared with the isotropic case.     

Electromagnetic scattering from cylindrical objects above a conductive surface using a hybrid finite-element-surface integral equation method

This work presents a novel finite-element solution to the problem of scattering from a finite and an infinite array of cylindrical objects with arbitrary shapes and materials over perfectly conducting ground planes. The formulation is based on using the surface integral equation with Green's function of the first or second kind as a boundary constraint. The solution region is divided into interior regions containing the cylindrical objects and the region exterior to all the objects. The finite-element formulation is applied inside the interior regions to derive a linear system of equations associated with nodal field values. Using two-boundary formulation, the surface integral equation is then applied at the truncation boundary as a boundary constraint to connect nodes on the boundaries to interior nodes. The technique presented here is highly efficient in terms of computing resources, versatile, and accurate in comparison with previously published methods. The near and far fields are generated for a finite and an infinite array of objects. While the surface integral equation in combination with the finite-element method was applied before to the problem of scattering from objects in free space, the application of the method to the important problem of scattering from objects above infinite flat ground planes is presented here for the first time, to our knowledge.