On three coplanar cracks in an infinite transversely isotropic medium

This paper concerns the problem of determining the stress distribution in an infinite transversely-isotropic medium containing three coplanar cracks. The analysis is carried out by using a solution of the equilibrium equations expressed in terms of displacements under plane strain assumption. By the use of Fourier transforms, we reduce the problem to solving a set of four integral equations. An exact solution of these equations is obtained by using finite Hilbert transform and expressions for the quantities of physical interest are obtained in closed form.     

Thermal stresses in an elastic solid weakened by two coplanar circular cracks

Some linear thermoelastic problems are studied for the thermal stress and displacement fields in an infinite elastic medium weakened by cracks occupying the space interior to two coplanar circular regions with equal radii. The thermal stresses are caused by the uniform heating or heat flow disturbed by the presence of the coplanar cracks. The problem is reduced to the determination of the solution of infinite sets of Fredholm integral equations. Attention is given to the case when the plane occupying the space external to the cracks is insulated from uniform heat flow. The sets of integral equations are solved iteratively by assuming the spacing between the center of the cracks is large as compared to the radii. Physical quantity of interest such as crack-opening displacement is investigated.     

Diffraction of elastic waves by two coplanar Griffith cracks in an infinite transversely-isotropic medium

Summary The problem of diffraction of normally incident longitudinal waves by two parallel and coplanar Griffith cracks embedded in an infinite transversely-isotropic medium is considered. Approximate formulas are derived for stress intensity factors when the wave lengths are large compared, to the distance between the outer edges of the two cracks By taking appropriate limits we derive various interesting and new results.     

Circular inhomogeneity and two concentric symmetric circular arc cracks problem in an infinite isotropic elastic plate under tension

A homogeneous infinite isotropic elastic plate contains two symmetrical circular arc cracks of equal radii and a concentric inhomogeneity. The radius of the inhomogeneity is less than that of the circular arc cracks. The plate is subjected to a traction at infinity. The stresses are found within the circular region bounded by the circular arcs, including the inhomogeneity. The problem is solved as a two-dimensional using the complex variable technique. Some numerical results are given to show the effects of the inhomogeneity.     

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.     

On the opening of two coplanar griffith cracks

The problem of opening two coplanar Griffith cracks in an infinite elastic medium has been considered in this note. The application of Somigliana's method to this mixed boundary value problem leads to a Föppl integral equation, whose solution has been derived in closed form with the help of a convolution theorem connected with finite Hilbert transforms.

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Résumé On a considéré, dans ce mémoire, le problème de l'ouverture de 2 fissures coplanaires de Griffith dans un milieu élastique infini. L'application de la méthode de Somigliana au problème de la valeur de la frontière indéfinie conduit à une équation intégrale de Föppl dont la solution a été dérivée sous une forme fermée en utilisant un théorème de convolution connecté à une transformée finie de Hilbert.

     

Multiple coplanar embedded elliptical cracks in an infinite solid subject to arbitrary crack face tractions

This paper extends the earlier work by the present authors for a single embedded crack in an infinite solid and presents a solution to the problem of multiple coplanar cracks in an infinite medium. An alternating method in conjunction with an analytical solution for a single crack is used to determine the stress intensity factors for interacting multiple coplanar embedded cracks in an infinite body. The alternating method, as implemented here, leads to a highly accurate evaluation of the appropriate stress intensity factors.     

The longitudinal shear problem for an array of cracks at the edge of a circular hole in an infinite elastic solid

Summary A Mellin-type transform technique reduces the longitudinal shear problem for a set of cracks at the edge of a circular hole in an infinite elastic solid to that of solving a system of integral equations. The stress intensity factors and crack formation energy are calculated. Three special cases are considered in detail and graphical results given.     

Three collinear cracks in an infinite elastic medium

The plane strain problem of determining the distribution of stress in the vicinity of three cracks embedded in an infinite isotropic elastic medium is considered. The cracks are collinear, the two side cracks are equal in length and located symmetrically with respect to the middle crack. The surface tractions acting on the cracks are completely arbitrary. Some special cases of the loading are discussed in detail.     

Dynamic stress intensity factors around two parallel cracks in an infinite elastic plate

Summary Dynamic stresses around two parallel cracks in an infinite elastic plate are obtained. An incoming shock stress wave impinges on the cracks at right angles to their faces. The Fourier-Laplace transform technique is utilized to reduce the problem to dual integral equations. To solve these equations, the differences in the crack surface displacements are expanded in a series of functions which are zero outside the cracks. The unknown coefficients occurring in those series are solved using the Schmidt method. The stress intensity factors defined in the Laplace transform domain are inverted numerically, in the physical space.     

Array of periodic curvilinear cracks in an infinite isotropic medium

Summary The problem of an array of periodic curvilinear cracks in an infinite isotropic medium under conditions of generalized plane stress or plane strain is reduced, by using the method of complex potentials of Muskhelishvili [1], to a complex Cauchy-type singular integral equation on one of the cracks, which can be further numerically solved by reduction to a system of two real Cauchytype singular integral equations and application of the Lobatto-Chebyshev method of numerical solution of such equations. Applications to the cases of arrays of straight or arc-shaped cracks are also given.

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Anordnung von periodisch gekrümmten Rissen in einem unendlichen, isotropen Medium.

Zusammenfassung Das Problem der Anordnung periodisch gekrümmter Risse in einem unendlichen, isotropen Medium wird untersucht. Der ebene Spannungs-bzw. Dehnungszustand wird für einen der Risse mit Hilfe komplexer Potentiale des Muskhelishvilischen Verfahrens in eine komplexe Cauchysche singuläre Integralgleichung reduziert. Die numerische Lösung des Problems erfolgt durch eine weitere Reduktion in ein System mit zwei Cauchyschen singulären Integralgleichungen unter Anwendung des Lobatto-Chebyshevschen Verfahrens. Beispiele für den Fall der Anordnung gerader wie auch bogenförmiger Risse werden angegeben.

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With 9 Figures     

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.     

The elastic problem for an infinite solid containing a circular hole with a pair of radial edge cracks of different lengths

By using the principle of superposition and the results of an earlier paper the problem is reduced to that of solving a pair of singular integral equations. A number of special cases are considered and numerical results given.     

Dynamic stress intensity factors around two coplanar Griffith cracks in an orthotropic layer sandwiched between two elastic half-planes

Dynamic stresses around two coplanar Griffith cracks in an orthotropic layer sandwiched between two elastic half-planes are determined. To the surfaces of the cracks, an internal pressure is applied suddenly. Application of the Fourier and Laplace transforms reduces the problem to the solution of a pair of dual integral equations in the Laplace transform plane. To solve these equations, the crack surface displacement is expanded in a series of functions which are zero outside of the cracks. The unknown coefficients accompanied in that series are solved with the aid of the Schmidt method. The stress intensity factors defined in the Laplace transform plane are inverted numerically in the physical plane. Numerical calculations are carried out for the case that the layer of carbon fiber is sandwiched by the two elastic half-planes of plastic.     

Four co-planar Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to four co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement in an infinite isotropic elastic medium has also been considered. In both cases, employing the Fourier integral transform, the problems have been reduced to solving a set of five integral equations. These integral equations have been solved using the finite Hilbert transform technique to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.     

Two coplanar cracks in an infinitely long elastic strip bonded to semi-infinite elastic planes

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long elastic strip containing two coplanar Griffith cracks. We assume that the strip is bonded to semi-infinite elastic planes on either side and that the cracks are opened by constant internal pressure. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations withcosine kernel and a weight function. These equations are solved using Finite Hilbert transform techniques. Analytical expressions upto the order off δ^?10 where 2δ denotes the thickness of the strip and δ is much greater than 1 are derived for the stress intensity factors and the crack energy.     

A system of curvilinear cracks in an isotropic elastic half-plane

The plane elasticity problem of an arbitrary system of curvilinear cracks in an isotropic elastic half-plane bonded to another half-plane consisting of a different isotropic elastic material is formulated by using the complex variable technique and reduced to a complex Cauchy type singular integral equation along the cracks. The special cases of a half-plane with a stress-free boundary and of a periodic array of curvilinear cracks are also treated. The numerical techniques available for the solution of complex Cauchy type singular integral equations are presented and a discussion on them is made. Finally, three applications to some special cases of straight or circular-arc-shaped cracks are made.

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Résumé Le problème de l'élasticité plane d'un système arbitraire de fissures curvilignes dans un demi-plan élastique isotrope joint a un autre demi-plan constitué d'un autre matériel élastique isotrope est formulé en utilisant la technique des variables complexes et reduit a une équation intégrale singulière complexe de type Cauchy au long des fissures. Les cas particuliers d'un demi-plan avec une limite libre de contraintes et d'une série périodique de fissures curvilignes sont aussi étudiés. Les techniques numériques disponsibles pour la solution d'équations intégrales singulières complexes de type Cauchy sont présentées et une discussion sur elles est faite. Finalement, trois applications à quelques cas particuliers de fissures droites ou ayant la forme d'un arc de circle sont faites.

     

Three co-planar moving Griffith cracks in an infinite elastic medium

The dynamic in-plane problem of determining the stress and displacement due to three co-planar Griffith cracks moving steadily at a subsonic speed in a fixed direction in an infinite, isotropic, homogeneous medium under normal stress has been treated. The static problem of determining the stress and displacement around three co-planar Griffith cracks in an infinite isotropic elastic medium has also been considered. In both the cases, employing Fourier integral transform, the problems have been reduced to solving a set of four integral equations. These integral equations have been solved using finite Hilbert transform technique and Cook's result [16] to obtain the exact form of crack opening displacement and stress intensity factors which are presented in the form of graphs.