On solutions of crack surface opening displacement of a penny-shaped crack in an elastic cylinder subject to tensile loading

A simple analytical expression for the surface displacement of a penny-shaped crack in an elastic cylinder subject to remote tensile loading is proposed based on a modified shear-lag model. The results are then compared with the dilute solution [1] and those of finite element calculation. It is found that the present work gives much better result than the dilute model.     

Exact stress distribution,crack shape and energy for a running penny-shaped crack in an infinite elastic solid

The integral solutions for an axisymmetrical crack propagating at arbitrary speed in an infinite elastic solid are obtained as sums of associated static solutions and stress-waves integrals. For a circular crack running at a constant speed, exact dynamic solutions for crack shape and stress distribution with singularities in the crack plane are obtained in closed forms easily comparable to associated static solutions. The dynamic solution reduces to the static solution at zero crack speed and deviates at speed other than zero. Deviation between dynamic and static solutions is governed by dynamic correction factors which are nondimensional functions of Poisson's ratio and the ratio between crack speed and shear-wave speed. Values of these dynamic factors are obtained for large range of crack speed and deviation can clearly be determined from the results obtained. Exact expressions for dynamic stress-intensity factor and energy functions are also obtained in terms of crack speed.

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Zusammenfassung Die Integrallösungen für einen assymetrischen Riß, der sich mit einer beliebigen Geschwindigkeit in einem unendlichen elastischen Körper ausbreitet ergeben sich als die Summe von zugehörigen statischen Lösungen und Integrale von Spannungswellen. Die exakten dynamische Lösungen für Rißform und Spannungsverteilung mit Besonderheiten in der Rißebene, im Falle eines kreisförmigen Rissesder sich mit gleicher Geschwindigkeit ausbreitet ergeben sich in geschlossener Form, leicht vergleichbar mit der zugehörigen statischen Lösung.Die dynamische Lösung stimmt mit der statischen überein wenn die Ausbreitungsgeschwindígkeit null ist und weicht davon ab wenn die Ausbreitungsgeschwindigkeit größer als null ist. Der Unterschied zwischen den dynamischen und den statische Lösungen wird durch Korrektionsfaktoren, die undimensionale Funktionen dem Poissonverhältnis und dem Verhältnis zwischen der Rißverbreitungsgeschwindigkeit und der Geschwindigkeit der Querwellen. Die Werte dieser dynamischen Faktoren können für einen großen Bereich von Rißgeschwindigkeiten aufgestellt werden und der Unterschied kan einwandfrei von diesen Resultaten bestimmt werden.Exakte Formeln für Spannungsintensitätsfaktoren und Energieformeln werden in Form von Rißgeschwindigkeit aufgestellt.

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Résumé Les solutions intégrales pour une fissure axisymétrique progressant à une vitesse quelconque dans un solide élastique infini se présentent sous la forme de sommations d'intégrales associées de solutions statiques et d'ondes de contraintes.Dans le cas d'une fissure circulaire se développant à vitesse constante, les solutions dynamiques exactes pour la forme de la fissure, la distribution des contraintes et leurs singularités dans le plan de la fissure, sont obtenues sous des formes fermées faciles à comparer aux solutions statiques correspondantes.La solution dynamique se ramène à la solution élastique lorsque la vitesse de fissuration est égale à zéro. Elle s'en distingue pour les vitesses supérieures à zéro. La divergence entre la solution statique et la solution dynamique est régie par des facteurs de correction dynamique, fonctions non dimensionnelles du rapport de Poisson et du quotient de la vitesse de la fissure par la vitesse des ondes transversales.Des valeurs de ces facteurs dynamiques ont été obtenues pour une gamme étendue de vitesses de fissuration; la divergence peut être clairement exprimée à partir des résultats obtenus.Des expressions exactes du facteur dynamique d'intensité des contraintes et des fonctions d'énergie ont également été trouvées en fonction de la vitesse de fissuration.

     

Note on the growth of a penny-shaped crack in a general uniform applied stress field

Energetic arguments are used to discuss the growth of a penny-shaped crack situated within an infinite solid which is subject to tensile and shear stresses that are respectively normal and parallel to the crack plane. The most favourable growth mode is that for which the circular periphery becomes an ellipse, such that there is no growth perpendicular to the direction of application of the. shear stress; the appropriate growth condition is derived and compared with that obtained by assuming the circular crack to expand uniformly.     

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.     

The stability of a wedge crack under a uniform applied stress

The usual approach in finding the conditions for crack extension is to determine the stationary values of the total energy of the system with respect to crack length; this necessitates the calculation of all the terms which combine to form the total energy. However, a very simple alternative approach is described; this allows the crack extension condition to be expressed in terms of the local dislocation distribution functions which represent the displacement of the crack surface near a tip. Such functions can quite often be obtained in simple form, and the method is used to discuss the stability of a wedge shaped crack subject to a general uniform applied stress system. It is shown how microcrack models, discussed by other workers, can be treated as special cases.

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Zusammenfassung Um die Bedingungen zu bestimmen, unter denen sich ein Riss verlägert, stellt man normalerweise die gleichbleibenden Werte der gesamten Energie des Systems in Bezug auf die Rissläge fest; zu diesem Zwecke müssen alle Glieder, die gemeinsam die Gesamtenergie ergeben, errechnet werden. Hier wird jedoch eine andere Methode beschrieben, die sehr einfach ist. Dabei kann die Bedingungen für die Rissverlängerung durch die Funktionen der lokalen Verlagerungsverteilung ausgedrückt werden, die der Verschiebung der Rissoberfläche im Bereiche eines Rissendes entsprechen. Solche Funktionen lassen sich recht oft in einfacher Form gewinnen, und die Methode wird angewandt, um die Stabilität eines keilförmigen Risses, auf den ein allgemeines, gleichmässiges Spannungssystem einwirkt, zu untersuchen. Es wird gezeigt, wie man Mikrorissmodelle, die von anderen Forschem erörtert werden, als Sonderfalle behandeln kann.

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Résumé L'approche habituelle en matiée de recherche des conditions d'extension d'une craque est de déterminer les valeurs stationnaires de l'énergie totale du système, eu égard à la longueur de la crique; ceci implique le calcul de tons les termes qui se combinent pour former l'énergie totale. Toutefois, une approche alternative très simple est décrite; elle permet d'exprimer la condition d'extension de la crique suivant les fonctions de répartition de la dislocation locale, lesquelles représentent le déplacement de la surface de la crique dans le voisinage d' une extríté. De relies fonctions peuvent très souvent être obtenues sous une forme simple, et la méthode est utilisé pour discuter la stabilité d' une crique en forme de coin soumise à un système d' effort général uniformént appliqué. On montre comment des modèles de micro-criques, discutés par d' autres chercheurs, peuvent être traités à part.

     

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.

     

The penny-shaped crack problem for a finitely deformed incompressible elastic solid

In this paper the theory of small deformations superposed on large is used to examine the axisymmetric problem of a penny-shaped crack located in an incompressible elastic infinite solid which is subjected to a uniform finite radial stretch. The small axisymmetric deformations are due to a uniform stress applied in the axial direction. Formal integral expressions are derived for the displacements and stresses in the elastic solid. An exact expression is developed for critical stress necessary for the propagation of a penny-shaped crack in a finitely deformed elastic solid.

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Résumé Dans le mémoire, on utilise la théorie des petites déformations superposées à de larges déformations pour examiner le problètrique d'une fissure en disque noyée dans un solide élastique infini incompressible soumis à un étirement uniforme fini radial. Les déformations axisymétriques de faible amplitude sont dues à une contrainte uniforme appliquée suivant la direction axiale. Des expressions intégrales formelles sont déduites des déplacements et des contraintes dans le solide élastique. Une expression exacte relative à la contrainte critique nécessaire pour la propagation d'une fissure en forme de disque est développée dans le cas d'un solide élastique déformé de manière finie.

     

Interaction of a penny-shaped crack with an elliptic crack

Three-dimensional problem of crack-microcrack interaction is solved. Both the crack and microcrack are embedded in an infinite isotropic elastic medium which is subjected to constant normal tension at infinity. One of the cracks is circular while the other is elliptic and they are coplanar and are positioned in such a way that the axis of the elliptic crack passes through the centre of the circular crack. A recently developed integral equation method has been used to solve the corresponding two dimensional simultaneous dual integral equation involving the displacement discontinuity across the crack faces that arises in such an interaction problem. A series of transformations first reduce them to a quadruplet infinite system of equations. A series solution is finally obtained in terms of crack separation parameter which depends on the separation of the crack microcrack centre. Analytical expression for the stress intensity factors have been obtained up to the order ⁶. Numerical values of the interaction effect have been computed for and results show that interaction effects fluctuate from shielding to amplification depending on the location of each crack with respect to the other and crack tip spacing as well as the aspect ratio of the elliptic crack. The short range interaction can play a dominant role in the prediction of crack microcrack propagation.     

Thermal stress in an elastic layer containing a penny-shaped crack and bonded to dissimilar half-spaces

This paper gives an analysis of the distribution of thermal stress in an elastic layer bonded to two half-spaces along its plane surfaces and contains a penny-shaped crack parallel to the interfaces. The crack is situated in the mid-plane of the layer. The thermal and elastic properties of the layer and of the half-spaces are assumed to be different. The problem is first reduced to dual integral equations. These equations are further reduced to Fredholm integral equations of the second kind which are solved iteratively. Expressions for quantities of physical interest are derived.

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Résumé Le mémoire fournit une analyse de la distribution des contraintes thermiques dans une couche élastique solidaire de deux demi-espaces situés le long de ses surfaces planes et comportant une fissure en forme de disque parallèle à ses interfaces. La fissure est située dans le plan moyen de la couche élastique. Les propriétés thermiques et élastiques de cette couche ainsi que celles des demi-espaces sont supposées différentes. Le problème est en premier lieu ramené à des équations intégrales. Ces équations sont ensuite ramenées à des équations intégrales de Fredholm du second genre qui sont résolues par itération. Des expressions pour les quantités présentant un intérêt physique sont déduites de ce travail.

     

The interaction between a penny-shaped crack and a spherical inhomogeneity in an infinite solid under uniaxial tension

Summary This paper presents an approximate three-dimensional analytical solution to the elastic stress field of a penny-shaped crack and a spherical inhomogeneity embedded in an infinite and isotropic matrix. The body is subjected to an uniaxial tension applied at infinity. The inhomogeneity is also isotropic but has different elastic moduli from the matrix. The interaction between the crack and the inhomogeneity is treated by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. The stress intensity factor at the boundary of the penny-shaped crack and the stress field inside the inhomogeneity are evaluated in the form of a series which involves the ratio of the radii of the spherical inhomogeneity and the crack to the distance between the centers of inhomogeneity and crack. Numerical calculations are carried out and show the variation of the stress intensity factor with the configuration and the elastic properties of the matrix and the inhomogeneity.     

On a transient thermal stress problem in an infinite circular cylinder with a penny-shaped crack

In the present paper, we deal with finding the stress intensity factors under transient thermal loading in an infinitely long circular cylinder containing a penny-shaped crack. Variations of the stress intensity factors with time, which will be closely related with crack extension, are illustrated in the figures.     

The indentation of a precompressed penny-shaped crack

The paper examines the axisymmetric problem related to the indentation of the plane surface of a penny-shaped crack by a smooth rigid disc inclusion. The crack is also subjected to a far-field compressive stress field which induces closure over a part of the crack. The paper presents the Hankel integral transform development of the governing mixed boundary value problem and its reduction to a single Fredholm integral equation of the second kind and an appropriate consistency condition which considers the stress state at the boundary of the crack closure zone. A numerical solution of this integral equation is used to develop results for the axial stiffness of the inclusion and for the stress intensity factors at the tip of the penny-shaped crack.     

Stress analysis of a penny-shaped crack locted between two oblate spheroidal cavities in an infinite solid

The problem of a penny-shaped crack located between two oblate spheroidal cavities in an infinite solid subjected to uniaxial loads is considered. Using transformations between harmonic functions in cylindrical coordinates and those in oblate spheroidal ones, the problem is reduced to non-homogeneous linear equations. The obtained equations are solved numerically and the stress intensity factors at the penny-shaped crack tip under the influence of the two oblate spheroidal cavities are shown graphically.     

The effect of couple stresses on the stress field around a penny-shaped crack

The present paper is concerned with the influence of couple stresses on the stress concentration around a penny-shaped crack in an infinite body within the framework of both the micropolar and couple stress (indeterminate) theory. Taking into account the symmetry of the boundary value problem considered this one is converted into solving dual integral equations, and finally into a Fredholm integral equation of the second kind. Approximating the kernel the latter is solved analytically.

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Résumé Dans le cadre de la théorie des contraintes micropolaires et des couples, on étudie l'influence d'un couple et d'une concentration de contraintes sur une fissure en ongle dans un corps infini. En prenant en compte la symétrie qui caractérise les contours du problème considéré, celui-ci est ramené `a la solution d'une double équation intégrale, et ensuite à une équation intégrale de Fredholm du second genre. Celle-ci est résolue par voie analytique en faisant une approximation sur le noyau.

     

Interaction of a penny-shaped crack with an elliptic crack under shear loading

The problem of interaction between equal coplanar elliptic cracks embedded in a homogeneous isotropic elastic medium and subjected to shear loading was solved analytically by Saha et al. (1999) International Journal of Solids and Structures 36, 619–637, using an integral equation method. In the present study the same integral equation method has been used to solve the title problem. Analytical expression for the two tangential crack opening displacement potentials have been obtained as series in terms of the crack separation parameter _i up to the order _i⁵,(i=1,2) for both the elliptic as well as penny-shaped crack. Expressions for modes II and III stress intensity factors have been given for both the cracks. The present solution may be treated as benchmark to solutions of similar problems obtained by various numerical methods developed recently. The analytical results may be used to obtain solutions for interaction between macro elliptic crack and micro penny-shaped crack or vice-versa when the cracks are subjected to shear loading and are not too close. Numerical results of the stress-intensity magnification factor has been illustrated graphically for different aspect ratios, crack sizes, crack separations, Poisson ratios and loading angles. Also the present results have been compared with the existing results of Kachanov and Laures (1989) International Journal of Fracture 41, 289–313, for equal penny-shaped cracks and illustrations have been given also for the special case of interaction between unequal penny-shaped cracks subjected to uniform shear loading.     

Axisymmetric problems for an externally cracked elastic solid. I. Effect of a penny-shaped crack

The present paper examines the problem of a penny-shaped flaw which is located in the plane of an external crack in an isotropic elastic solid. The penny-shaped flaw is subjected to uniform internal pressure. The paper develops power series representations for the stress intensity factors at the boundary of the penny-shaped flaw and at the perimeter of the externally cracked region. These series representations are in terms of a non-dimensional parameter which is the ratio of the radius of the penny-shaped flaw to the radius of the externally cracked region.     

The stress intensity factor for a penny-shaped crack in an elastic body under the action of symmetric body forces

A formula is derived for the stress intensity factor at the rim of a penny-shaped crack in an infinite solid in which there is an axisymmetric distributing of body forces acting in a direction normal to the original crack surfaces. An expression for the surface displacement of the crack is also given. The use of these formulae is illustrated by a consideration of the special case in which the solid is deformed by the action of two point forces situated symmetrically with respect to the crack.

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Zusammenfassung Eine Formel für den Spannungsintensitätfaktor am Rande eines pfenninggeformten Risses in einem unendlichen Festkörper ist gewonnen. Ein achsensymmetrisches Verteilen der Körperkräite fand statt, welches in einer Richtung, normal zu der originalen Rissoberfläche wirkt. Es ist auch Ausdruck für den Oberflächenverschiebung des Risses gegeben. Die Benutzung dieser Gleichungen wird verdeutlicht durch die Betrachtung eines Spezialfalles bei dem der Festkörper durch die Wirkung zweier Punktkräfte deformiert wird, die symmetrisch zum Riss angebracht sind.

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Résumé On a établi une formule donnant le facteur de concentration de tension aux extrémités d'une fissure ferrnée disposée dans un solide infini au sein duquel une distribution de forces internes á symétrie axiale agit dans une direction normale par rapport aus surfaces de la fissure. On fournit également une expression du déplacement de ces surfaces. L'utilisation de ces formules est appliquée, à titre d' exemple, au cas spécial d'un solide soumis à l'action de deux forces concentrées symétriques par rapport à la fissure.

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This paper was prepared as a part of the work of the Applied Mathematics Research Group at North Carolina State University through the Grant AF-AFOSR-444-66 and is under the joint sponsorship of AFOSR, ARO, and ONR through the Joint Services Advisory Group.     

The distribution of stress around a flat parabolic crack in an elastic solid

A simple limiting procedure is revealed whereby the three-dimensional state of stress and deformation around parabolic cracks or flaws embedded in elastic solids may be obtained. Several results are derived from available solutions concerning elliptical cracks or thin-sheet rigid inclusions. In particular, stress-intensity factors used in the Griffith-Irwin theory of fracture are evaluated and shown in curves. The findings of this investigation may also be exploited to determine the stress-strain field in thin sheets containing parabolic boles (and hence semi-infinite cracks) from existing solutions concerning similar bodies with elliptic holes.     

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.     

Plastic energy dissipation due to a penny-shaped crack

International Journal of Fracture - A penny-shaped crack in a material which is ideally elastic-plastic has been envisaged with the assumption that the plastic zone forms a very thin layer...