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.     

Two coplanar griffith cracks under shear loading in an infinitely long elastic layer

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 rigidly fixed. The cracks are located in the middle plane of the layer parallel to its faces. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with a cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions are derived for the stress intensity factors, shape of the deformed crack, and the crack energy. Solutions to some particular problems are derived as limiting cases. Numerical results are presented in the form of graphs.     

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.     

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.     

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.     

On the problem of two coplanar cracks inside an infinite isotropic elastic solid

A method is proposed for the approximate evaluation of normal displacements and normal stresses on the plane of two coplanar cracks located inside an infinite isotropic elastic solid and subjected to normal internal pressure. The formulation results in a single integral equation for the unknown normal stresses on the plane of the cracks. Numerical results are given for the stress intensity factor K_I of two coplanar circular cracks and two coplanar elliptical cracks opened up under a uniform internal pressure.     

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.     

Running penny-shaped crack in an infinite elastic solid under torsion

This paper is concerned with the problem of a running penny-shaped crack in an infinite elastic solid under torsion. A basic formulation for an arbitrary velocity crack is given. As an illustrative example, the penny-shaped crack is assummed to expand at a constant velocity. For a constant-speed crack, the crack shape is explicitly obtained in exact expression easily comparable to the associated static solution.

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Résumé L'étude est relative au problème de la propagation d'une fissure circulaire noyée dans un solide élastique infini soumis à torsion. On fournit une formulation de base, correspondant à une vitesse arbitraire de développement. A titre d'exemple, on suppose qu'une fissure circulaire s'étend suivant une vitesse constante. Dans ce cas, la forme de la fissure est obtenue selon une forme explicite, dont l'expression est aisément comparable à celle correspondant à une solution statique.

     

Problem of two coplanar Griffith cracks running steadily under three-dimensional loading

In this paper, the three-dimensional problem of two coplanar Griffith cracks propagating uniformly in an elastic medium has been considered. Equal and opposite tractions which are triaxial in nature are applied to the crack surfaces. The two-dimensional Fourier transforms have been used to reduce the mixed boundary value problem to the solution of triple integral equations. In order to solve the problem, the transformed surface displacement has been expanded in a series of Chebyshev polynomials which is automatically zero outside the cracks and also satisfies the edge conditions. Finally Schmidt method has been used to determine the unknown constants occurring in the series. Numerical calculations are carried out to obtain the crack opening displacement and also the stress intensity factors for different values of the parameters.     

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.     

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.     

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.     

Interaction of elastic waves by a Griffith crack in an infinite transversely-isotropic medium

The paper deals with the problem of finding the stress distribution near a Griffith crack located in an infinite transversely-isotropic medium. The crack is opened by the interaction of a plane harmonic elastic wave incident normally on the crack. A Fredholm integral equation is derived for the determination of diffracted field. From the integral equation asymptotic solution is obtained which is valid for wavelength long compared to the crack length. For wave lengths comparable with the size of the crack, the integral equation is solved numerically. The stress intensity factor and displacement field in the vicinity of crack are computed for a range of values of the frequency. The approximate solution is compared with exact solution.

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Résumé Le mémoire est relatif au problème de déterminer la distribution de contraintes au voisinage d'une fissure de Griffith localisée dans un milieu infini transversalement isotrope. La fissure est ouverte par l'interaction d'une onde élastique harmonique plane incidente sur une direction normale au plan de la fissure. On dérive une équation intégrale de Fredholm pour la détermination du champ de diffraction. Une solution asymptotique est obtenue à partir de l'équation intégrale, qui se révèle valide dans le cas de longueur d'onde grande par rapport à la longueur de la fissure. Lorsque les longueurs d'onde sont comparables à la taille de la fissure, l'équation intégrale peut être résolue de manière numérique. Le facteur d'intensité de contrainte et le champ des déplacements au voisinage de la fissure sont calculées pour une gamme large de valeur de la fréquence. La solution approximative obtenue est comparée avec la solution exacte.

     

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.     

Integral equations for axisymmetric torsion in an elastic body containing cracks

Conclusion Basic axisymmetric problems in the torsion of a body containing cracks and having any surfaces of rotation are considered. Integral representations are given for the displacment functions in terms of steps in the displacmeents and stresses at the axisymmetric discontinuity surfaces. These are used in boundary-value treatments for a space containing cracks at the edges of which one is given the stresses or displacements, which are reduced to integral equations of the first kind. In the case of closed cracks, integral equations are also derived for a finite body of rotation contining cavities and cracks. For an infinite body containing a disk-type crack whose edges are loaded by any non-self-balancing torsional forces, the solution to the integral equations can be found in quadratures.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 6, pp. 37–44, November–December, 1989.     

Dynamic stress intensity factors around four rectangular cracks in an infinite elastic medium under impact load

The 3-D dynamic problem is presented for an infinite elastic medium weakened by four plane rectangular cracks of equal size. The surfaces of the cracks are loaded by a uniform pressure with Heaviside-function time dependence. Fourier-Laplace transform technique is utilized to reduce the problem to a solution of two simultaneous integral equations which can be solved by using the series expansion method. The Laplace transformed stress intensity factors are defined and are inverted numerically in the physical space.     

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.