Cracked semi-infinite cylinder and finite cylinder problems

This work considers the analysis of a cracked semi-infinite cylinder and a finite cylinder. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to axial tension. Solution of this problem can be obtained by superposition of solutions for an infinite cylinder subjected to uniformly distributed tensile load at infinity (I) and an infinite cylinder having a penny-shaped rigid inclusion at z = 0 and two penny-shaped cracks at z = ±L (II). General expressions for the perturbation problem (II) are obtained by solving Navier equations with Fourier and Hankel transforms. When the radius of the inclusion approaches the radius of the cylinder, the end at z = 0 becomes fixed and when the radius of the cracks approach the radius of the cylinder, the ends at z = ±L become cut and subject to uniform tensile load. Formulation of the problem is reduced to a system of three singular integral equations. By using Gauss–Lobatto and Gauss–Jacobi integration formulas, these three singular integral equations are converted to a system of linear algebraic equations which is solved numerically.     

Extension of a finite strip bonded to a rigid support

This paper considers the elastostatic plane problem of a finite strip. One end of the strip is perfectly bonded to a rigid support while the other is under the action of a uniform tensile load. Solution for the finite strip is obtained by considering an infinite strip containing a transverse rigid inclusion at the middle and two symmetrically located transverse cracks. The distance between the two cracks is equal to twice the length of the finite strip. In the limiting case when the rigid inclusion and the cracks approach the sides of the infinite strip, the region between one crack and the rigid inclusion becomes equivalent to the finite strip. Formulation of the problem is reduced to a system of three singular integral equations using the Fourier transforms. Numerical results for stresses and stress intensity factors are given in graphical form.     

Interaction between cracks and rigid lines in an infinite plate

Summary Interaction between cracks and rigid lines in an infinite plate is investigated in this paper. The rigid lines are assumed in an equilibrium condition and may have some rotation in the deformation process of the adjacent material. After placing some distributed dislocations along the cracks and some distributed body forces along the rigid lines, a system of singular integral equations is obtained. The obtained system of the singular integral equations is reduced to a system of Fredholm integral equations by appropriate substitution of the unknown functions. The regularized integral equations are solved numerically. Stress intensity factors at the crack tips and stress singularity coefficients are investigated in the numerical examples.     

The elasticity problem for a thick-walled cylinder containing a circumferential crack

The elasticity problem for a long hollow circular cylinder containing an axisymmetric circumferential crack subjected to general nonaxisymmetric external loads is considered. The problem is formulated in terms of a system of singular integral equations with the Fourier coefficients of the derivative of the crack surface displacement as density functions. The stress intensity factors and the crack opening displacement are calculated for a cylinder under uniform tension, bending by end couples, and self-equilibrating residual stresses.     

Plane deformation of a body with rigid cylindrical inclusion

We study the problem of plane deformation of an infinite elastic body with thin rigid cylindrical inclusion with oval cross section. The body is loaded by biaxial uniform tensile forces at infinity. The solution of the problem is reduced to two singular integral equations with Cauchy kernels for the jumps of normal and tangential stresses on the surface of the inclusion. The solutions of these equations are obtained in the closed analytic form and, used to deduce the formulas for the concentration of stresses near the inclusion, for stresses inside the inclusion, and for the angle of rotation of the inclusion as a rigid body. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Ukrainian State University of Forestry Engineering, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 87–92, November–December, 1996.     

Asymmetrical Cracks Parallel to an Interface between Dissimilar Materials

This paper solves a plane strain problem for two bonded dissimilar planes containing a crack parallel to the interface in each layer. The bimaterial system is loaded by tractions distributed along the crack surfaces. Based on the Fourier transform, the problem is reduced to a system of Cauchy type singular integral equations which contain exact and explicit kernel functions. The solution of these equations is obtained easily by utilizing Gauss–Chebyshev integral formulae for various material combinations and geometrical parameters. Several numerical results of stress intensity factors, energy release rate and stress distribution along the interface are presented to exhibit the interaction among cracks and interface. This revised version was published online in July 2006 with corrections to the Cover Date.     

Axisymmetric crack problem of thick-walled cylinder with loadings on crack surfaces

This study is concerned with the fracture of an infinite thick-walled cylinder. The inner surface of the cylinder is stress free and the outer is rigidly fixed. The cylinder having a ring-shaped crack located at the symmetry plane is subjected to distributed compressive load on its surfaces. The Hankel and Fourier transform techniques are used for the solution of the field equations. By applying the boundary conditions, the singular integral equation in terms of crack surface displacement derivative is derived. By using an appropriate quadrature formula, the integral equation is reduced to a system of linear algebraic equations. Numerical results are obtained for the stress intensity factors at the edges of the crack, surfaces of which are subjected to uniform, linear and parabolic load distributions.     

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.     

The gauss-hermite numerical integration method for the solution of the plane elastic problem of semi-infinite periodic cracks

The problem of parallel semi-infinite periodic cracks subjected to a transversely directed load in an infinite isotropic and elastic medium under conditions of plane stress or plane strain can be reduced to the solution of a Cauchy-type singular integral equation along one of the cracks. This equation can be transformed into a system of linear equations by means of an approximation of the integrals through the Gauss-Hermite procedure and application of the equation to distinct points along the faces of the crack. Stress intensity factors thus determined for the crack tips under constant load along the cracks are in satisfactory agreement with corresponding values derived previously.     

Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

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.     

On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags

An analytical investigation on the plastic zone size (PZS) of a crack near a circular inclusion has been carried out. Both the crack and the circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, two stripe plastic zones at both crack tips are introduced. Using the solution of a circular inclusion interacting with a single dislocation as the Green’s function, the physical problem is formulated as a set of singular integral equations. With the aid of Erdogan and Gupta’s method and iterative numerical procedures, the singular integral equations are solved numerically for the PZS and the crack tip opening displacement. The results obtained in the current work can be reduced to those simpler cases of the Dugdale model.     

On the plastic zone size and CTOD study for a Zener�CStroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener?CStroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green??s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan??s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson??s ratio, and loading condition are studied.     

Longitudinal Shear of an Elastic Wedge with Cracks and Notches

We study the problem of longitudinal shear of an infinite wedge with cracks and notches. The integral representations of the complex stress potential are constructed in terms of the jumps of displacements and stresses on curvilinear contours identically satisfying the boundary conditions imposed on the faces of the wedge (stresses or displacements are equal to zero). By using these representations, we deduce singular integral equations of the analyzed problem for a wedge weakened by a system of cracks and holes of any shape. In some cases (a crack along the bisectrix of the wedge, a crack along a circular arc whose center is located at the edge of the wedge, and a circular notch near the edge of the wedge), we obtain exact closed solutions.     

Stress analysis of a kinked crack initiating from a rigid line inclusion. Part 1: Formulation

The problem of a kinked crack which has initiated from the tip of a rigid line inclusion is analyzed as a mixed boudary value problem. The stress distribution, stress intensity factors, singularity at the inclusion tip, and the resultant moment on the rigid line inclusion are investigated for various angles of the kinked crack and crack lengths. The rotation of the rigid line inclusion, when loaded by a uniform farfield stress, is calculated. The cases in which the inclusion is free to rotate or is fixed are separately considered.     

Indentation of a cantilever beam or plate

In this paper the general plane problem for a semi-infinite strip fixed at its short end, containing a crack perpendicular to its voundaries is considered. The strip is under the effect of a stamp. By extending the crack to the surfaces, one can reduce the problem to that of c cantilever beam or plate. Integral transform technique is used to provide an exact formulation of this problem, in terms of a system of four singular integral equations one of them being second kind. Stress singularities at the corners of the fixed-end, at the crack tips and at the end points of the contact region undermeath the stamp are obtained from the singular integral equations which are then solved numerically.     

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.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

On three-dimensional singular stress/residual stress fields at the front of a crack/anticrack in an orthotropic/orthorhombic plate under anti-plane shear loading

A novel eigenfunction expansion technique, based in part on separation of the thickness-variable, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness crack/anticrack weakening/reinforcing an infinite orthotropic/orthorhombic plate, of finite thickness and subjected to far-field anti-plane shear loading. Crack/anticrack-face boundary conditions and those that are prescribed on the top and bottom (free, fixed and lubricated) surfaces of the orthotropic plate are exactly satisfied. Five different through-thickness crack/anticrack-face boundary conditions are considered: (i) slit crack, (ii) anticrack or perfectly bonded rigid inclusion, (iii) transversely rigid inclusion (longitudinal slip permitted), (iv) rigid inclusion in part perfectly bonded, the remainder with slip, and (v) rigid inclusion located alongside a crack. Explicit expressions for the singular stress fields in the vicinity of the fronts of the through-thickness cracks, anticracks or mixed crack–anticrack type discontinuities, weakening/reinforcing orthotropic/orthorhombic plates, subjected to far-field anti-plane shear (mode III) loadings, are presented. In addition, singular residual stress fields in the vicinity of the fronts of these cracks, anticracks and similar discontinuities are also discussed.