Analysis of an edge crack in a semi-infinite composite with a long reinforced phase

For the purpose of clarifying the micro fracture of continuous fiber unidirectionally reinforced composite materials, the problem of an edge crack perpendicular to a long reinforced phase is considered on the basis of the plane strain theory of elasticity. The stress intensity factor at the tip of the crack, and the stresses on the interface between the matrix and the reinforced phase and in the reinforced phase are discussed. In the analysis, the method of continuous distributions of dislocations is used. Then, a singular integral equation is derived and is solved by the technique developed by Erdogan and Gupta. From the numerical results it was concluded that:

1. The stress intensity factor decreases monotonically as the crack tip approaches the reinforced phase. That is, the presence of the reinforced phase can result in crack arrest.
2. When the crack tip exists near the reinforced phase, the normal stress on the interface between the matrix and the reinforced phase has a maximum at the intersection of the extension of the edge crack and the reinforced phase, while the shear stress on the interface and the normal stress in the reinforced phase take, respectively, maxima at symmetric points with respect to the crack surface in the immediate vicinity of the intersection.
3. The maximum values of the stresses on the interface and in the reinforced phase increase monotonically as the crack tip approaches the reinforced phase.

     

Stresses in an orthotropic semi-infinite strip due to edge loads

Summary The stresses in an orthotropic elastic semi-infinite strip subject to plane strain are investigated. Symmetrical distributions of surface tractions are prescribed on the sides of the strip, while along the end the boundary conditions are arbitrary. By using an integral transform method the problem is reduced to a singular integral equation. The dependence of the stress singularity and the stress-intensity factors on the orthotropic properties of the strip is investigated. Stress distributions over the strip end are evaluated numerically.With 5 Figures     

The stress intensity for an edge crack in a semi-infinite orthotropic body

The problem of an edge crack in a semi-infinite plane of linear elastic orthotropic material is studied. The correction factor which relates the stress intensity factor for this problem to that for an isolated crack in an infinite body is evaluated for a range of orthotropic material properties. Calculations are restricted to mode I problems. The method requires the numerical solution of an integral equation, the integrands in which are derived from related complex variable solutions.

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Resumé On étudie le problème d'une fissure de bord dans un plan semi-infini en un matériau linéaire eastique et orthotrope. On évalue pour une gamme de propriétés orthotropes du matériau le facteur de correction qui permet de relier le facteur d'intensité de contraintes relatif à ce problème à celui relatif à une fissure isolée dans un corps infini. Ces calculs sont limités aux problèmes de rupture selon le mode. I. La méthode exige de trouver la solution numérique d'une intégrale dont les intégrants sont tirés de solutions associées à variables complexes.

     

The effective crack length of an edge slot in a semi-infinite sheet under tension

The rate of change of the elastic strain energy with respect to notch depth is considered for edge notches or slots with a finite constant radius, , at the tip. Numerical results are obtained for an edge slot in a semi-infinite sheet under tension. It is found that the slot can be considered as a crack of length L + 1.18 for this purpose. In finite sheet, this result is valid for small L/ ratios.

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Zusammenfassung Das Wechselverhältnis der elastischen Dehnungsenergie zur Kerbentiefe wird fuer Randkerben oder Schlitze mit endlichem konstanten Radius, , an der Spitze betrachtet. Numerische Ergebnisse werden für einen Randschlitz in einer halb-unendlichen Platte unter Zugspannung gefunden. Es wird gefunden, dass zu diesem Zwecke der Schlitz als Riss mit Länge L + 1.18 betrachtet werden kann. Für eine endliche Platte ist dieses Resultat für kleine L/ Verhältnisse gültig.

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Résumé L'evolution de l'énergie de deformation elastique en fonction de la profondear d'entaille est etudiée dans le cas d'entailles lateriales ou de saignées .....dont le rayen de courbure d'extrémité a une valeur finie et constante.Des données numeriques oat été oblénues dans le cas d'une saignée effectivée sur les bord d'un feuillard semi infini soumis à tension. On trouve que la saignee peut-être dans ce cas assimilee a une fissure qui aurait une longueur L + 1.18 . En ce qui concerne le cas de la tole finie, ce resultat n'est applicable que pout de faibles valeurs du rapport L/.

     

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.     

Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D combined mode loading

The dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body are analyzed. The crack is subjected to a traction distribution consisting of two pairs of suddenly-applied shear point loads, at a distance L away from the crack tip. The exact expression for the combined mode stress intensity factors as the function of time and position along the crack edge is obtained. The method of solution is based on the direct application of integral transforms together with the Wiener-Hopf technique and the Cagniard-de Hoop method, which were previously believed to be inappropriate. Some features of solutions are discussed and the results are displayed in several figures.     

Behaviour of an edge dislocation in a semi-infinite solid with surface energy effects

The method of continuously distributed dislocations and the method of discrete distribution of dislocations have been used to determine the effect of surface energy on the surface boundary conditions of a semi-infinite solid containing an edge dislocation. The surface dislocation model which incorporates two surface dislocation arrays, the primary and the secondary, in order of importance, is used to study the effect of surface energy. The surface dislocation model in conjunction with the method of continuously distributed dislocations enables the exact determination of the dislocation distribution function of the primary and secondary dislocation arrays and the effect of surface energy tends to lower both the total Burgers vector associated with the surface arrays and dislocations in evaluating the effect of surface energy is illustrated and is compared with the method of continuously distributed dislocations. It has been found that the surface energy tends to lower both the total Burgers vector associated with the surface arrays and the length of the region within which they are spread on the surface. Although the effect on the primary surface arrays is not very large, the secondary surface arrays are completely eliminated with normal values of surface energy encountered in real solids. Thus, the effect of surface energy is to bring non-vanishing stress components to the surface. The surface is also non-uniformly stressed. The superiority of the surface dislocation model over the other methods hitherto used in the literature is illustrated.     

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.     

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.     

A half plane and a strip with an arbitrarily located crack

The paper introduces a technique to deal with the problem of an elastic domain containing an arbitrarily oriented internal crack. The problem is formulated as a system of integral equations for a fictitious layer of body forces imbedded in the plane along a closed smooth curve encircling the original domain. The problems of a half plane with a crack in the neighborhood of its free boundary and of an infinite strip containing a symmetrically located internal crack with an arbitrary orientation are considered as examples. In each case the stress intensity factors are computed and are given as functions of the crack angle.     

Transient response of a crack in an anisotropic strip

Summary The problem of an anisotropic elastic strip containing a crack which is opened by stresses suddenly applied on the crack faces is considered here. The problem is reduced to a set of simultaneous Fredholm integral equations of the second kind which may be solved iteratively. Once the solutions of these integral equations are obtained, the dynamic stress intensity factors may be evaluated numerically. Numerical results are obtained for a particular transversely isotropic strip.With 1 Figure     

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.     

Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading

Summary The temperature distribution in structural elements in practical cases usually changes in two or three directions. Based on such facts, aiming at more effectiveness, a functionally graded material (FGM), whose properties change in two or three directions, is introduced, that investigated here called bi-directional FGM. The current study aims at the formulation, solution and investigation of a semiinfinite edge cracked FGM plate problem with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. The solution of the boundary value problem that one obtains from the mathematical formulation of the current crack problem under thermal loading reduces to an integral equation with a generalized Cauchy kernel. This integral equation contains many two-dimensional double strongly singular integrals, which can be solved numerically. In order to separate the singular terms and overcome the divergence of the integrals an asymptotic analysis for the singular parts in the obtained integral equation was carried out. Also, the exact solution for many singular integrals is obtained. The obtained numerical results are used in the representation of the thermal stress intensity factor versus the thermal/mechanical nonhomogeneous parameters. The numerical results show that it is possible to reduce and control the thermal stress intensity factor.     

Vertical crack inside a semi-infinite plane

In this paper we consider the problem of determining the stress-intensity factors and the crack energy in a semi-infinite plane containing an inside crack perpendicular to the straight boundary of the plane. By the use of Mellin transform, we reduce the problem to solving a single singular integral equation. Approximate solution of the integral equation is obtained as a series of Chebyshev polynomials of the first kind. The coefficients $\text{B}{}_{\mathrm{n}}$ of the series are determined from a system of linear algebraic equations. Expressions for the stress-intensity factors at the edges of the crack, the shape of the crack and the crack energy are derived in terms of the coefficients $\text{B}{}_{\mathrm{n}}$. The numerical values of these quantities have been displayed graphically for three particular cases.     

Partially stiffened elastic half-plane with an edge crack

A technique, using the Brazilian disk specimen, for measuring the fracture toughness of unidirectional fiber-reinforced composites, over the entire range of crack-tip mode mixities, was developed. The fracture toughness of a graphite/epoxy fiber-reinforced composite was measured, under both mode-I and mode-II loading conditions. We found that for certain material orientations the mode-II fracture toughness is substantially higher than the mode-I toughness. The complete dependence of the fracture toughness on the crack-tip mixity was determined for particular material orientations and the phenomenological fracture toughness curves were constructed. Using the Brazilian disk specimen, together with a hydraulic testing machine, the fracture toughness of the composite under moderate loading rates was measured. We observed that the mode-I fracture toughness was not sensitive to the loading rate at the crack tip, K, while the mode-II ‘dynamic’ fracture toughness increased approximately 50 percent over the quasi-static fracture toughness. A qualitative explanation of the dependency of fracture toughness on crack-tip loading rate is discussed. Finally, a mechanical fracture criterion, at the microscopic level, which governs the crack initiation under mixed-mode loading conditions is presented; these theoretical predictions closely follow the trend of experimental measurements. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Stressed state of a half plane with shallow edge crack under Hertzian loading (a survey)

We present a survey of works dealing with the stress intensity factors (SIF) for shallow edge cracks formed in elastic half planes subjected to the action of Hertzian loads moving along the boundaries of the planes. The numerical data on the SIF obtained by the method of singular integral equations are compared with the results of the other authors. The best agreement is observed with the results obtained by Keer and Bower by using the dislocation approach. The agreement with the results obtained by the finite-element method is satisfactory. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 1, pp. 23–34, January–February, 2008.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

Three-dimensional elastic stress analysis of a fracture specimen with an edge crack

A numerical stress analysis of an elastic three-dimensional specimen similar to the compact tension specimen used in fracture investigations is presented. The numerical results are achieved using singular integral equations which are analogous to Green's boundary formula in potential theory. The analysis yields details of the stresses near the crack tip and clearly shows their three-dimensional character. Some results are also given to indicate the influence of thickness and Poisson's ratio on the stresses.

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Zusammenfassung Es wird für eine dreidimensionale elastische Probe, ähnlich der kompakten Zugprobe für Bruchuntersuchungen, eine zahlenmässige Analyse der Spannungen dargelegt. Die numerischen Ergebnisse wurden über singulare Integralgleichungen analog den Greenschen Formeln in der Potentialtheorie ermittelt. Die Untersuchung ergibt Einzelheiten über die Spannungen in der Umgebung der Rißspitze und zeigt klar deren dreidimensionalen Charakter. Es werden auch einige Resultate mitgeteilt die den Einfluß der Dicke und des Poissonschen Moduls auf die Spannungen herausstellen.

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Résumé On présente une analyse numérique des contraintes dans une éprouvette à trois dimensions en sollicitation dans le domaine élastique, éprouvette similaire à l'éprouvette de traction utilisée dans les études de mécanique de rupture.Les résultats numériques sont obtenus en recourant à des équations à intégrales singulières, qui sont analogues à l'équation aux limites de Green dans la théorie du potentiel.L'analyse conduit à fournir la distribution complète des contraintes au voisinage de l'éxtremité de la fissure et démontre clairement le caractère tridimensionnel de celle-ci.L'influence de l'épaisseur et du module de Poisson sur les contraintes est également dégagée de certains résultats qui ont été obtenus.