Dugdale plastic zone of a penny-shaped crack in a piezoelectric material under axisymmetric loading

The Dugdale plastic zone ahead of a penny-shaped crack in a piezoelectric material, subjected to electric and axisymmetric mechanical loadings, is evaluated analytically. Hankel transform is employed to reduce the mixed boundary-value problem of the penny-shaped crack to dual integral equations, which are solved exactly under the assumption of electrically permeable crack face conditions. A closed-form solution to the mixed boundary-value problem is obtained to predict the relationship between the length of the plastic zone and the applied loading. The stress distribution in and outside of the yield zone has been derived analytically, and the crack opening displacement has been investigated. The electric displacement has a constant value in the strip yield zone. The current Dugdale crack model leads to non-singular stress and electric fields near the crack front, and it is observed that the material properties affect the crack opening displacement.     

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.     

Dynamic penny-shaped crack in a transversely isotropic material

The scattering of a harmonic longitudinal wave by a penny-shaped crack in a transversely isotropic material is investigated using the techniques of Hankel transform. The wave impinges normally on the crack surfaces. A complete contour integration is employed to simplify the expressions of the results. An exact expression of the dynamic stress-intensity factor is obtained as a function of the frequency factor and the anisotropic material constants. The normalized dynamic stress-intensity factor is shown to have different maximum values at different wave frequencies for the sample composite and metallic materials. The distortion of the dynamic crack shape and the displacement at the crack center are also shown to be dependent of the wave frequency and the anisotropy of the material.     

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.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Lack-of-contact conditions for a penny-shaped crack under a polynomial normal loading

Summary The classical problem of a penny-shaped crack inside an infinite three-dimensional isotropic elastic medium and under a polynomial normal loading (with axial symmetry) on both crack faces is reconsidered. By using elementary results from computational quantifier elimination techniques in computer algebra and applied logic, such as cylindrical algebraic decomposition and Sturm (or Sturm-Habicht) sequences, it is possible to satisfy the funcdamental inequality constraint about the positivity of the crack opening displacement inside the whole crack. This constraint assures us about the lack of contact of the crack faces, due to the loading of the crack, and the derived quantifier-free formula constitutes the related necessary and sufficient condition involving the loading parameters, that is the coefficients of the loading polynomial. Several such low-degree polynomial loadings are considered in detail (with the help of elementary and well-known solutin techniques for the present penny-shaped crack problem) as an application of the approach. Further possibilities for generalizations are also discussed in brief.     

A hollow oblate spheroid subjected to internal or external pressure and an oblate spheroid with a penny-shaped crack

This paper contains a three-dimensional elastic solution for a region bounded by two oblate spheroidal surfaces, subjected to axisymmetric internal or external pressure. In the analysis, Boussinesq's two harmonic stress functions approach is employed and the solution is given in the form of a series involving Legendre functions expressed by oblate spheroidal coordinates.Numerical calculations are carried out when the shape ratio of the inner surface varies, keeping the thickness of the hollow spheroid in thexy-plane constant. Also when the thickness in thexy-plane varies, the shape ratio of the inner surface is kept constant. Lastly, the results are compared with those obtained by the membrane theory.In addition, the stress intensity factors are calculated for an oblate spheroid containing a penny-shaped crack under constant internal pressure.

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Résumé Le mémoire comporte une solution élastique tridimensionnelle pour une région limitée par deux surfaces sphéroïdales concentriques sujettes à une pression interne ou externe axisymétrique. Dans l'analyse, l'approche des deux fonctions de contrainte harmonique de Bossineq est utilisée et la solution est fournie sous la forme d'une série comportant des fonctions de Legendre exprimée en coordonnées sphéroïdales.Les calculs numériques sont effectués lorsque varie le rapport de forme de la surface interne tout en maintenant l'épaisseur du sphéroöde creux constante dans le plan xy. On considère également le cas où l'épaisseur dans le planxy varie et où le rapport de forme de la surface interne est maintenu constant. Enfin, les résultats sont comparés avec ceux obtenus par la théorie des membranes.En outre, on calcule les facteurs d'intensité de contrainte pour un sphéroïde comportant une fissure en forme d'ongle soumise à une pression interne constante.

     

A transversely isotropic composite with a penny-shaped crack

Summary We consider the problem of determining the stress intensity factor and the crack energy in a transversely isotropic composite medium, containing a penny-shaped crack. We assume that the crack surface is perpendicular to the bond face and the crack is opened by constant internal pressure. By use of integral transform, we reduce the problem to solving a Fredholm integral equation of the second kind. Numerical results are given for the combination of some practical materials such as magnesium and cadmium. The effect of transverse isotropy upon the stress intensity factor, the crack energy and the deformation on the crack surface is discussed.

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Ein transversal, isotropes, komposites Medium mit einem münzenförmigen Riß

Zusammenfassung Das Problem der Bestimmung des Spannungsintensitätsfaktors und der Rißenergie, in einem transversalen, isotropen, kompositen Medium mit einem münzenförmigen Riß, wird betrachtet. Es wird vorausgesetzt, daß die Rißoberfläche normal zur Verbundfläche liegt, und der Riß sich durch konstanten inneren Druck öffnet. Durch Anwendung einer Integraltransformation, wird das Problem auf die Lösung einer Fredholmschen Integralgleichung zweiter Art reduziert. Numerische Ergebnisse werden für die Kombination einiger Materialien, wie Magnesium und Cadmium angegeben. Der Einfluß der transversalen Isotropie auf den Spannungskonzentrationsfaktor, die Rißenergie und die Deformation an der Rißoberfläche werden diskutiert.

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With 3 Figures

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This work is supported by the Board of Scientific and Industrial Research, Orissa (India).     

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

In this paper we deal with finding the stress intensity factors under the transient thermal loading in a circular cylinder with infinite length containing a penny-shaped crack. Variations of the stress intensity factors with time, which are closely related with the crack propagations, are obtained and illustrated in figures. From these figures we can obtain useful suggestions respecting crack propagation.

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Résumé On traite, dans cette étude, de la recherche des facteurs d'intensité de contraintes dans un cylindre circulaire de longueur infinie comportant une fissure en forme d'angle et soumise à sollicitation thermique en régime transitoire.On obtient, et on donne des valeurs à titre d'exemples pour les variations de facteur d'intensité des contraintes en fonction du temps, qui sont en relation étroite avec la propagation de la fissure. Ces valeurs conduisent à des suggestions utiles en ce qui regarde la propagation d'une fissure.

     

Fracture from a straight crack subjected to mixed mode loading

Fracture from a straight crack under mixed mode loading conditions and small scale of yielding is studied. It is assumed that crack growth occurs in either mode I or mode II. Comparison of theoretically obtained values of the mode I stress intensity factor at incipient kinking with experimental results indicates that mode I is preferred to mode II when the loading is such that the crack surfaces are traction-free, i.e. in the absence of a confining pressure. Mode II would be preferred only if the ratio K _{II c}/K _{I c} between the stress intensity factors is very low, between 0.38–0.81, depending on the load situation. Since there are reasons to believe that most materials are characterized by higher values of K _{II c}/K _{I c}, the conclusion is that mode II hardly occurs in the absence of a high confining pressure.

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Résumé On etudie la rupture entraînée par une fissure droite sujette à des sollicitations selon un mode mixte et comportant une plastification sur une petite échelle. On suppose que la croissance de la fissure se produit sous un mode I ou un mode II.En comparant les valeurs théoriques du facteur d'intensité de contraintes selon le mode I correspondant à l'évasement initial et les résultats expérimentaux, il s'indique que le mode I est privilégié par rapport au mode II lorsque la sollicitation est telle que les surfaces de la fissure sont libres de toutes contraintes, c'est-à-dire en l'absence de toute pression de confinement.Le mode II ne serait privilégié que si le rapport K _{II c}/K _{I c} des facteurs d'intensité de contraintes est très faible, à savoir entre 0,38 et 0,81, selon les conditions de sollicitation.Comme il y a de bonnes raisons de croire que la plupart des matériaux sont caractérisés par des rapports K _{II c}/K _{I c} plus élevés, on en conclut que, en l'absence de hautes pressions de confinement, le mode II a peu de chances de se manifester.

     

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.     

Fracture analysis of a penny-shaped magnetically dielectric crack in a magnetoelectroelastic material

In this paper we investigate the magnetoelectroelastic behavior induced by a penny-shaped crack in a magnetoelectroelastic material. The crack is assumed to be magnetically dielectric. A closed-form solution is derived by virtue of Hankel transform technique with the introduction of certain auxiliary functions. Field intensity factors are obtained and analyzed. The results indicate that the stress intensity factor depends only on the mechanical loads. However, all the other field intensity factors depend directly on both the magnetic and dielectric permeabilities inside the crack as well as on the applied magnetoelectromechanical loads and the material properties of the magnetoelectroelastic material. Several special cases are further discussed, with the reduced results being in agreement with those from literature. Finally, according to the maximum crack opening displacement (COD) criterion, the effects of the magnetoelectromechanical loads and the crack surface conditions on the crack propagation and growth are evaluated.     

Transient response of a finite crack subjected to dynamic anti-plane loading

In this study, the transient response of a finite crack in an elastic solid subjected to dynamic antiplane loading is investigated. Two specific loading situations, a body force near the finite crack and a concentrated point loading applied on the crack face, are analyzed in detail. In analyzing this problem, an infinite number of diffracted waves generated by two crack tips must be taken into account which will make the analysis extremely difficult. The solutions are determined by superposition of proposed fundamental solutions in the Laplace transform domain.The fundamental solutions to be used are the problems for applying exponentially distributed traction and screw dislocation to the crack faces and along the crack-tip line respectively. Exact transient closed-form solutions for the dynamic stress intensity factor are obtained and expressed in very simple and compact formulations. The solutions are valid for an infinite length of time and have accounted for the contributions of an infinite number of diffracted waves. Numerical calculations for the two problems are evaluated and results indicate that the dynamic stress intensity factors will oscillate near the corresponding static values after the first three waves have passed through the specified crack tip.     

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.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

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...     

Fracture of a finite piezoelectric layer with a penny-shaped crack

This paper studies a penny-shaped crack in a finite thickness piezoelectric material layer. The piezoelectric medium is subjected to a thermal flux on its top and bottom surfaces. Both thermally insulated crack and heated crack are considered. Numerical solution for the finite layer thickness is obtained through the solution of a pair of dual integral equations. The result reduces to the closed form solution when the thickness of the piezoelectric layer becomes infinite. Exact expressions for the stress and electric displacement at the crack border are given as a function of the stress intensity factor, which is determined by the applied thermal flux. This paper is useful for the reliability design of piezoelectric materials in thermal environments.     

Fatigue crack propagation in a piezoelectric ceramic strip subjected to mode III loading

Summary Following the theory of linear piezoelectricity, a forth-power stress intensity factor crack growth equation in an orthotropic piezoelectric ceramic strip is developed under mode III loading. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Dugdale's assumption regarding the plastic zone in metals is applied to estimate the effects of yield around the crack tips. Fourier transforms are used to reduce the electroelastic problem to one involving the numerical solution of a Fredholm integral equation of the second kind. A direct approach based on the accumulated plastic displacement criterion for crack propagation is used to develop the equation to predict the fatigue crack growth. Graphical results showing the effect of electroelastic interactions on the fatigue crack growth rate are presented.