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.

     

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.

     

Axisymmetric contact and crack problems for an initially stressed neo-Hookean elastic layer

A solution of the contact problem for an initially stressed neo-Hookean infinite layer is obtained. The layer is assumed to be bonded to a rigid foundation. The punch is taken to be axisymmetric and in particular, the conical and cylindrical shapes of the punch are considered in detail. The expression for the total load applied to the punch to maintain a given displacement is obtained. We have also considered the penny-shaped crack in the layer under an initial stress. The crack is taken to lie in the central plane of the layer. The surfaces of the layer are considered to be either stress free or fixed. Expressions for the stress intensity factors are obtained. Numerical values of the physical quantities are exhibited graphically.     

Axisymmetric graph model of an elastic solid

The method of a deformable solid analysis based on graph theory principles is used in mechanics to construct discrete models that allow a numerical simulation of the solid displacement, strain, and stress fields. It provides the alternative opportunity to derive solid equations by transformation of the generalized coordinate systems introduced by the graph structural objects (paths, cut-sets, cycles, chords).Published in Problemy Prochnosti, No. 6, pp. 83–103, November–December, 1996.     

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.     

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.     

Torsion of an elastic medium containing a nanosized penny-shaped crack with surface effects

International Journal of Fracture - Theoretical and experimental studies have revealed that at small length scales, surface effects contribute significantly to overall elastic deformation. In this...     

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.     

Axisymmetric dynamic response of a penny-shaped crack to a pair of transient concentrated forces

The three-dimensional axisymmetric elastodynamic response of a penny-shaped crack embedded in an infinite elastic solid subjected to a pair of transient concentrated forces is investigated. The forces are applied on the symmetry axis perpendicular and symmetric to the crack surfaces, including the special case when the forces act precisely on the crack surfaces. A time-domain boundary integral equation method is applied for computing the crack-opening displacement and subsequently the time dependence of the dynamic stress intensity factors. Numerical calculations are carried out for various geometry parameters and the results are discussed. It is found that the location of the applied concentrated forces inducing the highest dynamic stress intensity factors differs from that producing the highest static values.     

Effect of elastic coating on fracture behaviour of piezoelectric fibre with a penny-shaped crack

In this paper, the problem of a penny-shaped crack in a piezoelectric fibre with an elastic coating is investigated. By using the potential function method and Hankel transform, this problem is formulated as the solution of a system of dual integral equations which are reduced to a Fredholm integral equation of the second kind. Numerical studies are conducted to investigate the effect of the thickness and the elastic material properties of the coating on the fracture behavior of piezoelectric fibre composites.     

Edge crack in a strip of an elastic solid

The problem of determining the distribution of stress and the deformation of a long strip of an elastic material, damaged by a crack normal to an edge of the strip, is investigated. The strip is deformed by pressure applied to the faces of the crack. The stress intensity factor is calculated and its variation with the depth of the crack, relative to the width of the strip, in the special case of uniform pressure, is illustrated.     

Contact and crack problems for an elastic wedge

In this paper the contact and the crack problems for an elastic wedge of arbitrary angle are considered. The problem is reduced to a singular integral equation which, in the general case, may have a generalized Cauchy kernel. The singularities under the stamp as well as at the wedge apex are studied and the relevant stress intensity factors are defined. The problem is solved for various wedge geometries and loading conditions. The results may be applicable to certain foundation problems and to crack problems in symmetrically loaded wedges in which cracks may initiate from the apex.     

The stability of a penny-shaped crack in a solid subject to an applied tensile stress

The stability of a penny-shaped crack in a solid subject to an applied tensile stress is investigated using a model in which plastic deformation is assumed to be confined to the plane of the crack, the growth criterion being that a constant crack tip opening angle is maintained at the crack tip during growth. The results are compared with those for the analogous two-dimensional crack problem. It is shown that a penny-shaped crack is appreciably more stable than a two-dimensional crack, when the applied stress levels are high relative to the yield stress.

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Résumé On étudie la stabilité d'une fissure circulaire noyée dans un solide soumis à une contrainte de traction en utilisant un modèle dans lequel on suppose que la déformation plastique est limitée au plan de la fissure, le critère de croissance étant que se maintient à l'extrémité de la fissure un angle d'ouverture de fissure constant au cours de cette croissance. On compare les résultats avec ceux obtenus dans un problème analogue d'une fissure à deux dimensions. On montre qu'une fissure circulaire noyée présente une stabilité plus grande qu'une fissure à deux dimensions, dès lors que les niveaux de contrainte appliquée sont élevés en regard de la limite d'élasticité.

     

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.     

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.

     

Axisymmetric problem for a spherical crack on the interface of elastic media

A problem concerning a spherical interfacial crack is solved by the eigenfunction method. The problem is reduced to a coupled system of dual-series equations in terms of Legendre functions and then to a system of singular integral equations for two unknown functions. The behaviour of the solution near the edge of the spherical crack, and the stress-intensity factors and crack-opening displacements are studied. The case when the crack surfaces are under normal internal pressure of constant intensity is examined.     

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.