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.     

Elasto-plastic analysis for a finite thickness rectangular plate containing a through-thickness central crack

The objective of this work is to present a three-dimensional analysis of the elasto-plastic response of a center-cracked rectangular panel by using an alternate method, the method of lines (MOL). The problem of a finite thickness rectangular plate containing a through-thickness central crack under uniaxial tension is studied in detail. An incremental procedure is used to seek the 3-D elasto-plastic stress and strain distribution. The Prandtl-Reuss equations and the associated von Mises flow rule are employed. The material is assumed to be elastic-plastic with nonlinear work-hardening. Effective stress and effective strain concepts are used. The MOL is used to reduce the governing equations on displacements to a coupled set of ordinary differential equations (ODE) along x, y, z lines. These are integrated by Peano-Baker method. Since the ODE are of constant coefficient the coefficient matrix is diagonalized and the matrizon is easily evaluated. Numerical results on stresses, displacements, the growth of plastic zone and the crack surface opening displacements are reported and their variation along the thickness are presented in graphs. The variation of the stress intensity factor K ₁, along the thickness direction is given. It is found that the computation time needed for determining the distribution of stresses and strains is much less than what has been reported by using other methods. A comparison of results on SIF with those obtained by other methods is also given.

---

Résumé L'objet du travail est de présenter une analyse tridimentionnelle de la réponse élastoplastique d'un panneau rectangulaire présentant une fissure centrale, et utilisant une méthode alternative, la méthode des lignes MOL. On étudie dans le détail le problème de la plaque rectangulaire d'épaisseur finie comportant une fissure centrale traversante et soumise à la contrainte uniaxiale. Une procédure incrémentale est utilisée en vue de rechercher la distribution des contraintes et des déformations élastoplastiques suivant les trois dimensions. On utilise les équations de Prandtl-Reuss et la règle d'écoulement de von Mises qui lui est associée. On suppose que le matériau est élastoplastique et qu'il présente un vieillissement non linéaire. On recourt au concept des contraintes effectives et des déformations effectives. Le MOL est utilisé en vue de ramener les équations gouvernant les déplacements à un ensemble d'équations différentielles ordinaires suivant les axes X, Y, Z; on fait appel à la méthode de Peano-Baker pour leur intégration. Comme les équations différentielles ont un coéfficient constant, la matrice de coéfficient peut être résolue par diagonales et le matriciel peut être aisément évalué. Les résultats numériques relatifs aux contraintes, aux déplacements, à la croissance de la zône plastique et aux déplacements d'ouverture de la surface de la fissure sont présentés, et leur variation sur l'épaisseur sont mises en graphiques. On fournit la variation du facteur d'intensité de contrainte K suivant la direction de l'épaisseur. On trouve que le temps de calcul nécessaire à la détermination de la distribution des contraintes et déformations est beaucoup moindre que celui nécessaire par l'utilisation d'autres méthodes. Une comparaison des résultats relatifs aux facteurs d'intensité de contraintes avec ceux obtenus par d'autres méthodes est également fournie.

     

Solution of a rectangular sheet containing a central crack propagating at constant velocity under antiplane shear

Making use of the Fourier transform and Fourier series, the dynamic stress intensity factor of a rectangular sheet with a central crack moving at a constant velocity under antiplane shear is obtained. It is also proved that the solutions of a strip with a central crack propagating at constant velocity for Mode III are the special cases of the solution in the present paper.     

Stress intensity factors of a central slant crack with frictional surfaces in plates with biaxial loading

The stress intensity factor is a traditional topic in mechanics and there have been many solutions for many different cases. The closed frictional crack problem has been modeled in the rock mechanics field where fractures are mostly under compression. Further, the effect of finite plate dimensions under biaxial loading has not been considered in the literature. The key contribution of the present paper is to evaluate the effect of the crack length to plate width ratio on the mode I and II stress intensity factors (SIF) of a central slant crack with frictional surfaces in plates with biaxial loading of different patterns, i.e. tension-tension, tension-compression, compression-tension or compression-compression. A plane strain elastic two-dimensional finite element analysis was adopted. Crack length to plate width ratios equal to 0.1, 0.3 and 0.5 with biaxial ratios from –1 to 1, crack angles from 0° to 90° and friction coefficients from 0 to 1 were considered. Contact regimes and the effect of the crack length to plate width ratio were found dependent on biaxial ratio and pattern, friction coefficient and crack angle.     

Photoelastic analysis of a thick square plate containing central crack and loaded by pure bending

Three-dimensional photoelastic analysis is employed to study the stresses in the neighborhood of the tip of a central crack in a square plate with finite thickness. The plate is being loaded by pure bending moment around its free edges. Both maximum shear stresses and maximum normal stresses were investigated in planes parallel to the crack and through the plate thickness. In addition, the variation of these stresses was investigated along various lines drawn from the crack tip and again in various planes of the plate. The locations of the largest shear stress and the largest normal stress were established.It was found that these variations of the stresses depend not upon the crack length, but upon the ratio of the plate thickness to crack length. Both the theoretical and the experimental analysis show that a hydrostatic state of stress exists at points along the longitudinal axis of the crack originating from the crack tip outward.The experimental photoelastic results are correlated with the existing three-dimensional theoretical solution.

---

Résué L'analyse photoélastique à trois dimensions est utilisée pour l'étude des contraintes au voisinage de la pointe d'une fissure centrale traversante dans une plaque carrée d'épaisseur finie. La plaque est sollicitée flexion pure autour de ses bords libres. On examine les tensions tangentielles maximum et les tensions normales maximum dans des tranches parallèles découpées sur l'épaisseur de la plaque. En outre, on étudie la variation de ces contraintes le long de différentes lignes tracées à partir de la pointe de la fissure, et ce dans les différents plans découpés. On situe l'emplacement où apparaissent la plus grande tension tangentielle et la plus grande tension normale. On trouve que ces variations des tensions ne dépendent pas de la longueur de la fissure mais bien du rapport de l'épaisseur de la plaque à la longueur de la fissure. L'analyse théorique et l'expérience montrent qu'il existe un état de tensions hydrostatiques le long de l'axe longitudinal de la fissure prenant naissance en la partie extérieure de l'extrémité de la fissure. Ces résultats expérimentaux obtenus par photoélasticité sont comparés à la solution théorique tridimensionnelle existante.

     

Some axially symmetric thermal stress distributions in an elastic solid containing an annular crack

The present paper seeks to solve the axisymmetric thermal stress distribution in an infinite, isotropic solid containing an annular crack. The problem is reduced to triple integral equations, which are ultimately reduced to the solution of an infinite set of simultaneous equations.     

Numerical solutions of a hypersingular integral equation for antiplane elastic curved crack problems of circular regions

Summary. In this paper, a hypersingular integral equation for the antiplane elasticity curved crack problems of circular regions is suggested. The original complex potential is formulated on a distribution of the density function along a curve, where the density function is the COD (crack opening displacement). The modified complex potential can also be established, provided the circular boundary is traction free or fixed. Using the proposed modified complex potential and the boundary condition, the hypersingular integral equation is obtained. The curve length method is suggested to solve the integral equation numerically. By using this method, the usual integration rule on the real axis can be used to the curved crack problems. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

A non-conventional approach for crack problems in piezoelectric media under electromechanical loading

In this paper the elastostatic solution of an inclined crack problem has been analyzed. The approach is analogous to the one proposed over the past years for orthotropic materials by Piva andViola (1988) (Eng Fract Mech, 29: 535–548), and it may be considered an alternative to the mathematical formalisms currently used. A general expression of the stress field is determined in terms of complex potentials. The stress distribution and the displacement field in the proximity of the crack tip are finally obtained. An expression of the energy release rate is provided in terms of the field intensity factors. Numerical results are presented and discussed under different remote electro-mechanical loading conditions and varying the crack inclination. Particular attention has been focused on the effect of the lateral load on the fracture quantities and on the direction of the incipient branching angle.     

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.     

Boundary integral equation method to solve embedded planar crack problems under shear loading

The solution of three-dimensional planar cracks under shear loading are investigated by the boundary integral equation method. A system of two hypersingular integral equations of a three-dimensional elastic solid with an embedded planar crack are given. The solution of the boundary integral equations is succeeded taking into consideration an appropriate Gauss quadrature rule for finite part integrals which is suitable for the numerical treatment of any plane crack without a polygonal contour shape and permit the fast convergence for the results. The stress intensity factors at the crack front are calculated in the case of a circular and an elliptic crack and are compared with the analytical solution.     

On the concentrated loading of certain elastic half-space problems and related external crack problems. A new approach

Using a new integral equation connecting different types of fundamental solutions, a variety of external crack problems are solved when the crack is situated in an infinite space and occupies the region exterior to a circle, with the loading on the crack faces being of a concentrated nature. As a consequence of a particular integral representation we are also able to discuss certain “punch” problems for an elastic half-space when the punch is rigid and frictionless, with a circular cross section, but with an arbitrary base profile     

Partial closure of a Griffith crack under a general loading

The plane problem of a Griffith crack, partially closed under the effect of a general loading, is considered. The three cases of: a) closure at one end under a nonsymmetrical loading; b) closure at the two ends under a symmetrical loading; and c) closure at the middle under a symmetrical loading, are considered separately. Examples are given from partial closures by concentrated forces of cracks opened by a uniform tension at infinity or a parabolic pressure on the crack surface. The effect of the concentrated forces in decreasing the stress intensity factors so as to prevent crack propagation is examined.     

The numerical solution of crack problems in plane elasticity in the case of loading discontinuities

The direct quadrature method of numerical solution of Cauchy type singular integral equations encountered in plane elasticity crack problems is applied to the case where the loading distribution along the crack edges presents jump discontinuities. This is made by using a well-known modification of the quadrature method which is free of undesirable errors due to the loading discontinuities. Hence, the method is ideal to treat the aforementioned class of crack problems and, particularly, crack problems where the Dugdale-Barenblatt elastic-perfectly plastic model is adopted. Finally, a numerical application of the method to the problem of a periodic array of cracks with a loading distribution presenting a jump discontinuity is made. The numerical results obtained in this problem compare favorably with the corresponding theoretical results available in this special problem.     

Twisting of an elastic plate containing a crack

The stress distribution caused by twisting an infinite plate containing a finite crack is analyzed in terms of Reissner's theory for the bending of thin plates. The singular character and the detailed structure of the stresses near the ends of the crack are determined in closed form. Numerical results are given for the magnitudes of the stress couples and stress resultants for a range of plate thicknesses.

---

Zusammenfassung Due Spannungsvertcilung, hervorgerufen durch die Torsion einer unendlichen Platte mit einem Ri\ begrenzter Länge, wird mit Hilfe der Reissner-Theorie für die Biegebeanspruchung dünner Platten untersucht. Der singulare Charakter und die genauc Verteilung der Spannungen in Nähe der Ri\enden werden bestimmt. Zahlenmä\ige Ergebenisse für die Gro\e der Spannungsparre und ihrer Resultanten werden für eine Reihe von Plattenstärken angegeben.

---

Résumé La distribution des contraintes dans une plague infinie comportant une fisure finic et soumise à torsion est analysée au moyen de la théorie de Reissner pour la flexion des tôles minces.Le caractère singulier, et la structure de détail des contraintes au voisinage des extrémités de la fisure sont explicités.Des résultats numériques sont fournis en ce qui regarde les grandeurs des couples de contraintes et de leurs résultantes, pour une certaine gamme d'éspaisseur de tôles.

     

Fatigue crack growth for a surface crack in a round bar under multi-axial loading condition

In this paper, the fatigue life, surface crack extension direction and crack growth rate in an elastic bar with a circular cross section are determined through experiments under cyclic torsion with axial static and cyclic tension/compression loading. The effects of the loading type, loading value and stress ratio on the crack growth behaviour are discussed. The results show that, under pure fatigue torsion loading, the crack extension direction is almost the same whatever the value of torsion loading. Under fatigue torsion with cyclic tension loading, it is found that the crack extension direction is mainly determined by the alternating parts of the stresses and is almost independent of the average parts of the stresses, whereas the fatigue life is obviously dependent on the average stress.