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.

     

Interaction of longitudinal wave with a penny-shaped crack at the interface of two bonded dissimilar elastic solids-II

The paper deals with the problem of finding the stress distribution near a penny-shaped crack situated at the interface of two bonded dissimilar elastic solids. The crack is opened by the interaction of plane harmonic longitudinal elastic wave, incident normally on the crack. The problem is first reduced to a set of simultaneous dual integral equations which are further transformed to a set of simultaneous singular integral equations. These are solved numerically by reducing them to a set of algebraic equations. The solution is used to calculate the stress-intensity factors and the size of the overlapping zones at the edge of the crack.

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Résumé Ce mémoire est relatif au problème de trouver la distribution des contraintes au voisinage d'une fissure circulaire à l'interface de deux solides élastiques dissemblables et collés. La fissure est ouverte sous l'effet d'une onde élastique longitudinale plane et harmonique, dont l'incidence est normale au plan de la fissure. Le problème est en premier lieu ramené à un système d'équations intégrales doubles, qui sont transformées ensuite en un système d'intégrales singulières. On résoud ces dernières par voie numérique en les réduisant à un système d'équations algébriques.La solution est appliquée au calcul des facteurs d'intensité de contraintes, et la dimension des zones de recouvrement aux bords de la fissure.

     

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar elastic half-spaces

The problem considered is that of determining the displacement field in the vicinity of a penny-shaped crack situated at the interface of two half-spaces of different elastic materials bonded together along their common plane boundary. The deformation in the two half-spaces is a result of the application of a symmetrically-distributed pressure to the faces of the crack. The representation of the displacement field in the form of Hankel transforms leads to a set of simultaneous dual integral equations for two unknown functions. These equations are transformed by the use of Abel operators to a similar set involving both the Fourier cosine and the Fourier sine transforms of each of these two unknown functions.These equations are transformed in turn to a singular integral equation for a complex-valued function in terms of whose real and imaginary parts it is possible to completely specify the stress and displacement fields. An explicit formula is obtained for the crack energy in the case in which the applied pressure is constant; it is also indicated how simple integral expressions may he obtained for the components of stress along the interface in this case.     

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half-spaces

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half spaces is studied. The problem is first reduced to a system of dual integral equations which are further reduced to the solution of Riemann-Hilbert problem. Further stresses at the rims of cracks and in the vicinity have been evaluated.     

Influence of initial stress on fracture of a halfspace containing a penny-shaped crack under radial shear

In this study, an axisymmetrical problem for a penny-shaped crack under radial shear is considered. The crack is located parallel to the surface of a halfspace, which is subjected to initial stress parallel to the crack plane. An approach proposed by Guz (1983) in the framework of the three-dimensional linearised solid mechanics is used. Analysis involves reducing the problem to a system of Fredholm integral equations of the second kind, where the solutions are identified with harmonic potential functions. The representations of the stress intensity factors K _I and K _II near the crack edges are obtained. These stress intensity factors are both influenced by the initial stress.     

Biaxial load effects on a crack between dissimilar media

An analysis of the in-plane biaxial loading of two dissimilar materials with a crack along their common interface is considered. The elastic solution is obtained by the application of the complex variable technique coupled with the principle of superposition. The stress and displacement fields near the tip of the crack are described and the influence of the load parallel to the crack direction is shown.Then a failure criterion is formulated to discuss the debonding along the interface or the brittle fracture of the elastic system. In particular the effect of the non-singular terms on the critical applied tensile stress and on the deviation of the interfacial crack into one of the adjacent media is analyzed.     

Thermal stresses in an infinite non-isotropic plate having a penny-shaped crack

Of concern in the paper is the distribution of thermal stresses in the vicinity of a penny-shaped crack in a thick elastic plate made of a non-isotropic material. The problem pertains to the situation where the crack is opened by a prescribed normal pressure and a prescribed heat-flux or a prescribed temperature.     

Plastic deformation in a thick transversely isotropic layer containing a penny-shaped crack

Summary The width of a thin plastic annular zone formed during the deformation of a pennyshaped crack in a transversely isotropic layer of an ideal elasto-plastic material is determined. Considered are the cases where the penny-shaped crack is extended by normal stresses and by torsional stresses. The faces of the layer are shear free and deformation of the plastic zone around the penny-shaped crack occurs according to the Dugdale hypothesis. For each case, the solution of the problem is reduced to a Fredholm integral equation of the second kind. Iterative solutions are obtained for small values of the parameters and numerical results for the width of the plastic zone are determined. Graphical results showing the effect of transverse isotropy upon the width of the plastic zone are also presented.With 6 Figures     

A fiber-reinforced matrix containing a penny-shaped crack under mode III loading condition

One of the fundamental problems related to the fracture of composite materials, that is, a penny-shaped crack in a fiber-reinforced matrix is solved under the Mode III loading condition, where the fibers are perpendicular to the crack plane and located along the crack border. An elastic fiber model is developed to the above torsional problem, yielding a Fredholm-type integral equation of the second kind for a set of fibers distributed symmetrically on a circle concentric with the crack. The integral equation is numerically evaluated, and the stress intensity factors are presented with the parameter of the fiber to matrix Young's modulus ratio for various geometries.     

Experimental solution of the problem of a curvilinear crack in bonded dissimilar materials

An experimental technique based on the optical method of caustics for the solution of the two-dimensional problem of an arc-shaped crack lying along the interface of a circular inclusion embedded in a matrix and subjected to a uniaxial tension at infinity was developed. The study was directed towards the determination of the complex stress intensity factorK ^*characterizing the near to the crack tip stress field. A detailed analysis of the caustics obtained by illuminating the near to the crack tip region by a coherent light beam was undertaken and nomograms for the direct experimental determination ofK ^*in the Dundurs parallelogram, incorporating all the physically relevant material combinations of the two bodies, were given. Extensive experimental evidence for the new technique was given.

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Résumé Une technique expérimentale basée sur une méthode optique par caustiques a été développée pour établir la solution d'un probléme bidimensionnel d'une fissure en forme d'arc située le long de l'interface d'une inclusion circulaire noyée dans une matrice et soumise à une tension uniaxiale à l'infini. L'étude a été dirigée en vue de la détermination du facteur d'intensité des contraintes complexesK ^*caractérisant le champ de contrainte situé au voisinage de l'extrémité de la fissure. Une analyse détaillée des caustiques obtenues en illuminant la région située au voisinage de la pointe de l'entaille à l'aide d'un rayon lumineux cohérent a été effectuée, et des nomogrammes ont été obtenus pour une détermination expérimentale directe deK ^*dans un parallélogramme de Dundurs, en incorporant toutes les combinaisons physiquement possibles des matériaux constituant les deux corps. Plusieurs vérifications expérimentales de la nouvelle technique sont fournies.

     

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...     

The elastostatic axisymmetric problem of a sphere containing a penny-shaped crack in a nonequatorial plane

The axisymmetric problem of a sphere containing a penny-shaped crack in a nonequitorial plane is solved with the use of Bousinesq stress functions. Two coordinate systems—oblate spheroidal for representing the crack surface and spherical polars for the spherical surface, translated along the z-axis with respect to each other—are used to satisfy boundary conditions. Integral representations and transformations of harmonic functions are used to relate stress functions in the two coordinate systems. This procedure-leads to a system of algebraic equations which is solved, for axisymmetric tractions on both the surfaces. Graphical results are presented for a specific loading case.     

A permeable interface crack between dissimilar thermopiezoelectric media

Summary This paper presents an explicit treatment of the generalized 2D thermopiezoelectric problem of an interfacial crack between two dissimilar thermopiezoelectric media by means of the extend Stroh formalism. In comparison with the other relevant studies, the present work has two features: one is that the crack is assumed to be a permeable slit across which the normal electric displacement and the tangential electric field are continuous. The other is that the heat loading is applied at infinity, rather than on the crack faces. As a result, the field intensity factors and the electric field inside the crack are obtained in explicit closed-forms, respectively. As examples, the solutions of several particular cases, including that of an impermeable crack and that of a homogeneous material with a crack are also presented. It is shown that the electric field inside a crack may be singular and oscillatory for the case of an interfacial crack, while for the case of a crack in a homogeneous medium it is linearly variable. Moreover, it is also found that for a homogeneous medium with a crack the stress intensity factors based on the impermeable model and permeable model are same, but the intensity factor of the electric displacement is not.     

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.     

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.     

Penny-shaped crack in an elastic layer bonded to dissimilar half spaces

Two axially symmetric mixed boundary value problems in an elastic dissimilar layered medium are considered. It is assumed that an elastic layer is bonded to two semi-infinite half spaces along its plane surfaces, and contains a penny-shaped crack parallel to the interfaces. In the first problem the two half spaces are assumed to have the same elastic properties and the crack is located in the mid-plane of the layer. In the second problem we consider the case of three different materials and arbitrary crack location in the layer. The numerical examples are given for a constant pressure on the crack surface. The stress intensity factors are evaluated and are plotted as functions of the layer thickness-to-crack radius ratio or the relative distance of the crack from an interface.     

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.     

Diffraction of torsional waves by a penny-shaped crack in an infinitely long cylinder bonded to an infinite medium

This paper contains an analysis of the interaction of torsional waves with penny-shaped crack located in an infinitely long cylinder which is bonded to an infinite medium. Both the cylinder and infinite medium are of homogeneous and elastic but dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack.     

Mixed mode fracture criterion of interface crack between dissimilar materials

This study presents an application of fracture mechanics to the interface crack between dissimilar materials. In this study, a concept of the stress intensity factors of an interface crack is discussed, and various types of specimens are tested experimentally for investigating the mixed mode fracture toughness criterion of an interface crack. The fracture toughness based on the stress intensity factors of an interface crack is decided by the fracture test and the boundary element analysis using the contour integral method. The mixed mode fracture toughness criterion is successfully characterized by the stress intensity factors of an interface crack.     

Steady state propagation of a mode III interface crack between dissimilar viscoelastic media

The steady state propagation of a semi-infinite crack between two dissimilar viscoelastic solids is considered. By means of the Wiener-Hopf technique, the stress intensity factor is found as a function of the crack tip velocity and the material parameters. Results for an interface crack between an elastic and a viscoelastic medium are obtained as a special case. Various limiting cases are examined as a check on the accuracy of the results. Finally, graphs are presented which examine the salient features of the stress intensity factor.