Antiplane shear crack terminating at and going through a bimaterial interface

The antiplane shear problem of two bonded elastic half planes containing a crack perpendicular to the interface is considered. The cases of a semi-infinite crack terminating at the interface, a finite crack away from and terminating at the interface, two cracks one on each side of the interface, and a finite crack crossing the interface are separately investigated. The nature of the stress singularity for the crack terminating at and going through the interface is studied, and it is shown that at the irregular point on the interface, for the former the power of singularity is not -1/2 and for the latter the stresses are bounded. For a material pair of aluminum-epoxy some numerical results giving the stress intensity factors, the density functions, and the crack opening displacements are presented.     

Plane deformation of a body with rigid cylindrical inclusion

We study the problem of plane deformation of an infinite elastic body with thin rigid cylindrical inclusion with oval cross section. The body is loaded by biaxial uniform tensile forces at infinity. The solution of the problem is reduced to two singular integral equations with Cauchy kernels for the jumps of normal and tangential stresses on the surface of the inclusion. The solutions of these equations are obtained in the closed analytic form and, used to deduce the formulas for the concentration of stresses near the inclusion, for stresses inside the inclusion, and for the angle of rotation of the inclusion as a rigid body. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Ukrainian State University of Forestry Engineering, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 87–92, November–December, 1996.     

Debonding of rigid curvilinear inclusions in longitudinal shear deformation

The problem of two symmetrically placed interface cracks at rigid curvilinear inclusions under longitudinal shear deformation is considered. A solution valid for arbitrary inclusion shapes is found. It depends on a parameter β describing the cracks. For $\text{β = e}{}^{\mathrm{i\alpha }}$ where α is an angle, the cracks lie in the interface. For β real and greater than unity, we have two radial cracks emanating from a curvilinear cavity. The solution for $\text{β = 1}$ corresponds to a completely debonded inclusion.Examples of elliptic, square with rounded corners, and rectangular inclusions are worked out in detail. It is shown that the crack tip stress intensity factor becomes infinite for interface cracks terminating at cusps and corners. This phenomenon is attributed to the change in the nature of the singularity as the crack tip approaches a cusp or corner. The singularity is three-quarter power at a cusp and two-thirds power at a corner of a rectangular inclusion. Finally, the application of the results to composite materials is indicated.     

Stress analysis near the tip of a curvilinear interfacial crack between a rigid spherical inclusion and a polymer matrix

Craze and shear band formation at poorly adhering glass spheres in matrices of glassy polymers are known to be preceded by the formation of a curvilinear interfacial crack between sphere and matrix. In this study the axisymmetric finite element method has been used to analyse the stress situation near the tip of a curvilinear interfacial crack formed between a rigid spherical inclusion and a polymer matrix upon an applied uniaxial tension. Important factors that determine the stress state near the crack tip were found to be the crack length, the orientation of the crack tip with regard to the tension direction and the extent of interfacial slip between the inclusion and matrix. The results of the analyses were compared with the physical reality of craze and shear band formation at poorly adhering glass spheres. Reasonable agreement was found with respect to both the maximum interfacial crack length that can be reached until a craze or shear band forms at the crack tip and the planar orientation of craze growth perpendicular to the direction of the major principal stress.     

Antiplane deformation of a cracked composite strip

We obtain the analytic solution of an antiplane problem of the theory of elasticity for a cracked layer made of a composite material whose cross section is formed by a periodic array of repeated rectangular elements. Each element, in turn, contains four rectangular cells of different types. A crack is located on one of the interfaces of materials of these cells. The distributions of displacements on the outer surfaces of the layer are regarded as given. The numerical analyses are performed for the stress intensity factors depending on the mechanical properties of materials and the sizes of the cells.     

A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 3. Antiplane deformation

Within the framework of the approach proposed by L. P. Mazurak, L. T. Berezhnyts'kyi, and P. S. Kachur [“Method for determination of elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 1. Mathematical foundations,”Fiz.-Khim. Mekh. Mater.,33, No. 6, 21–31 (1997)], we construct a new method for the determination of elastic equilibrium of cylindrical bodies with noncanonical curvilinear foreign elastic inclusions under conditions of longitudinal shear. Unlike the method of perturbation of the form of a boundary, this method imposes no restrictions on the form of inclusions. The method is based on a procedure of determination of contour integrals of the Cauchy type by using the Faber polynomials. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 2, pp. 21–26, March–April, 1999.     

Antiplane vibration of an elastic layer with a midplane crack

The oscillations of an elastic layer excited by antiplane tractions of uniform intensity along the faces of a midplane crack are investigated. A singular integral equation is derived and solved numerically by Gauss-Chebyshev integration technique. The stress intensity factor and crack surface response are computed as functions of frequency and layer thickness-to-crack length ratio, and the resonant frequencies of the cracked layer are determined.     

Antiplane problem of curvilinear cracks in bonded dissimilar materials

A general solution to the antiplane problem of curvilinear cracks in bonded dissimilar materials is provided. The analysis is based upon the Hilbert problem formulation and the technique of analytical continuation. To illustrate the use of the present approach, detailed results are given for a single circular-arc crack lying along the interface between dissimilar materials. The expressions of the complex potentials are derived explicitly in both the unit disk and the surrounding medium. Both the stress intensity factors and contact stress are provided in an explicit form and the former are verified by comparison with existing ones. The effect of material and geometrical parameters upon the contact stress and stress intensity factors has also been discussed and shown in graphic form.     

Antiplane shear interactions between a circular boundary and a radial crack

Using some simple coefficients in the alternating procedure, this paper solves the problem of antiplane shear interactions between a circular boundary and a radial crack systematically and succinctly. Exact formulae for the coefficients, which play key roles in the method of solution presented in this paper, are derived. Making use of these coefficients, any particular case of the interaction problem can be solved readily and accurately. A number of numerical results are given to demonstrate the accuracy and efficiency of the present variation of the alternating method used in fracture mechanics and other branches of physics and engineering.     

A crack along the interface of a rigid circular inclusion embedded in an elastic solid

The two-dimensional problem of a crack opened under uniform internal pressure and lying along the interface of a rigid circular inclusion embedded in an infinite elastic solid is examined. Based on the complex variable method of Musk helishvili closed form solutions of the stresses and displacements around the crack are obtained and these are then combined with the Griffith's virtual work argument to give a criterion of a crack extension, or decohesion of the interface. The critical pressure is expressed explicitly by a function of the radius of the inclusion and the central angle sublended by the half length of the crack; especially it is inversely proportional to the square root of the radius of the inclusion.

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Zusammenfassung Das zweidimensionale Problem eines unter gleichmässigem internen Druck geöffneten Risses der sich längs der Übergangsfläche von einer steifen kreisförmigen Inklusion befindet, die in einem unendlichen elastischen Körper liegt, wird untersucht.Man erhält Lösungen von geschlossener Form für die Spannungen und die Verschiebungen, die sich auf das Verfahren der komplexen Veränderlichen von Muskhelishvili gründet. Diese Resultate werden dann mit der Hypothese der virtuellen Arbeit von Griffith verbunden um ein Kriterium für eine Rißausbreitung, oder eine Auftrennung der Übergangsfläche zu erhalten.Der kritische Druck wird durch eine Funktion des Radius der Inklusion, und des Mittelpunktwinkels der die Hälfte der Rißlänge einschließt ausgedrückt. Besonders ist er umgekehrt proportional zur Quadratwurzel des Radius der Inklusion.

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Résumé On s'est penché sur le problème à deux dimensions d'une fissure s'ouvrant sous une pression interne uniforme, et située sur l'interface d'une inclusion circulaire rigide noyée dans un solide élastique infini.En se basant sur la méthode des variables complexes de Muskhelishvili, on a obtenu des solutions de forme fermée pour les contraintes et les déplacements au voisinage de la fissure. Ces solutions ont été combinées à l'hypothèse de travail virtuel de Griffith, pour conduire à un critère d'extension de la fissure ou de décohésion de l'interface.La pression critique s'exprime de façon explicite en fonction du rayon de l'inclusion et de l'angle central soustendu par la demi-longueur de la fissure. En particulier, elle est inversément proportionnelle à la racine carrée du rayon de l'inclusion.

     

Elastic equilibrium of an interface crack between rigid stamp and orthotropic strip

Rupture of the interface between an absolutely rigid stamp and an orthotropic infinite strip is investigated. A plane elasticity problem for an interface crack formally leads to oscillatory singularities at the crack tip. In order to overcome this nonphysical solution, a model of an interface crack with frictionless contact zones near the crack tips and the corners of the stamp is developed. By using the method of integral Fourier transforms the problem is reduced to a system of three singular integral equations. The system is solved by the method of collocations with the points of collocation chosen at zeros of the Chebyshev polynomials. The stress intensity factors at the crack tips and the stamp corner points are evaluated.     

Antiplane shear crack problem in bonded materials with a graded interfacial zone

In this paper the interface crack problem for two elastic half spaces bonded through a nonhomogeneous interfacial zone is considered. It is assumed that the medium is under antiplane shear loading. The problem is solved for two different interfacial zone models that may approximate the actual diffusion bonded materials or homogeneous solids bonded through a functionally gradient material. Extensive results are obtained by varying the stiffness and the interfacial zone thickness to crack length ratios. Also, for various limiting cases the behaviour of the stress intensity factors and the strain energy release rates are studied.