Debonding along the interface of an elliptic rigid inclusion

The two-dimensional problem of a crack lying along the interface of an elliptic rigid inclusion embedded in an infinite elastic matrix is theoretically studied. Based on the complex variable method of Muskhelishvili, closed form solutions of the stresses and displacements around the crack are obtained when both general biaxial loads at infinity and uniform normal internal pressure are applied. These solutions are then combined with the virtual work argument of Griffith to give a criterion of the crack extension, namely the growth of the debonding of the interface. The critical applied loads are expressed explicitly by a function of four parameters; the size, the ratio of the length of the minor axis to that of the major axis of the inclusion, the angle subtended by the crack arc and the polar angle of the middle point of the crack arc. It is shown that when the applied load is only a simple tension or only an internal pressure the critical load is inversely proportional to the square-root of the size of the inclusion. The variations of the critical load with the angle subtended by the crack arc and with the ratio of the length of the semi-axes are graphically shown and discussed.     

Elastic circular inclusion in an infinite plane containing two cracks

The elastic fields in an elastic circular inclusion and surrounding infinite matrix containing two cracks symmetrically situated, are determined when the matrix is subjected to loads at infinity. In this problem, the elastic properties of inclusion could differ from those of the matrix. The Muskhelishvili's technique is used. The solution depends upon two sets of suitable complex potentials Φ_m(z), Ψ_m(z), Φ_i(z), Ψ_i(z) for matrix and inclusion respectively, which solves the problem.     

Interaction between an elastic circular inclusion and two symmetrically placed collinear cracks

This paper investigates the fracture toughness of a flat large elastic cracked plate containing an elastic circular inclusion, using the plane stress crack-tip stress intensity factor as a criterion for fracture. In this plane stress geometry, each of the two collinear finite cracks is located on either side of the elastic circular inclusion and the geometry is subjected to uniform stresses at infinity. The analysis is based on the two-dimensional theory of clasticity by using the Muskhelishvili complex variable approach. Numerical calculations are reported for the case of simple tension normal to the crack direction, and show the variation of the stress intensity factor with the configuration and elastic properties of the plate and the inclusion.

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Résumé On étudie la ténacité à la rupture d'une grande plaque plane et élastique comportant des fissures et une inclusion élastique circulaire, en utilisant comme critère de rupture le facteur d'intensité des contraintes en état plan de tension. Dans ce type de géométrie, on aligne une paire de fissures linéaires et finies de part et d'autre de l'inclusion circulaire, et on soumet cette disposition à une contrainte uniforme à l'infini. L'analyse est basée sur la théorie de l'élasticité bidimensionnelle grâce à l'approche par variables complexes due à Muskhelishvili. Les calculs numériques sont effectués pour le cas de la traction simple normale à la direction des fissures et montrent la variation du facteur d'intensité des contraintes avec la configuration de la tôle et de l'inclusion ainsi qu'avec leurs propriétés élastiques.

     

An arbitrarily oriented,rigid, elliptic inclusion in a finite anisotropic medium

This analysis presents an effective method of addressing rigid inclusion problems in an anisotropic medium with finite boundaries. For an arbitrarily oriented, rigid, elliptic inclusion, a solution is obtained by using the modified mapping collocation method. Within the two-dimensional theory of elasticity, this method provides nearly exact solutions for the stress and displacement fields and for the rigid-body translation and rotation concerning rigid, elliptic inclusions in an anisotropic medium. The effectiveness of the method is demonstrated through several examples and their comparison with other solution methods. The validity and accuracy of the solutions are established by comparing the results from this analysis with those obtained analytically and by other numerical techniques.     

Dynamic stress intensity factors around two parallel cracks in an infinite elastic plate

Summary Dynamic stresses around two parallel cracks in an infinite elastic plate are obtained. An incoming shock stress wave impinges on the cracks at right angles to their faces. The Fourier-Laplace transform technique is utilized to reduce the problem to dual integral equations. To solve these equations, the differences in the crack surface displacements are expanded in a series of functions which are zero outside the cracks. The unknown coefficients occurring in those series are solved using the Schmidt method. The stress intensity factors defined in the Laplace transform domain are inverted numerically, in the physical space.     

On a rigid inclusion pressed between two elastic half spaces

A solution to the problem of a rigid cylindrical inclusion pressed between two elastic half spaces is obtained using the distributed dislocation technique. The solution is compared with previously published analytical and numerical results for a rigid cylindrical inclusion bounded by two parabolic arcs with rounded corners. A simplified solution to the problem based on the classical contact theory and well-known results for crack problems is also suggested and validated. The simplified solution agrees well with analytical results in the case when the length of the opening around inclusion is much larger than the length of the contact zone.     

Stress analysis of a debonding and a crack around a circular rigid inclusion

A model of a debonding and a crack occurring from a circular rigid inclusion in an infinite plate is analyzed as a mixed boundary value problem under uniform tension. A mapping function represented in the form of a sum of fractional expressions and complex stress functions are used. The stress distribution, stress intensity factors at the tip of a crack, and stress singular values at a debonded tip are presented. By using these stress singular values, the intensity of the debonded tip is also considered.

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Résumé On analyse un modèle de décollement et de fissuration au départ d'une inclusion rigide circulaire dans une tôle infinie, en le considérant comme un problème mixte de valeurs aux limites sous une tension uniforme. On utilise des fonctions complexes de la contrainte et une représentation sous forme d'une somme d'expressions fractionnelles. La distribution des contraintes, des facteurs d'intensité de contraintes à l'extrémité de la fissure, et les valeurs des contraintes singulières à l'extrémité de la zone décollée, sont présentées. En utilisant ces valeurs de contraintes singulières, on prend également en compte l'étendue de cette zone décollée.

     

Dynamic stress intensity factors around two coplanar Griffith cracks in an orthotropic layer sandwiched between two elastic half-planes

Dynamic stresses around two coplanar Griffith cracks in an orthotropic layer sandwiched between two elastic half-planes are determined. To the surfaces of the cracks, an internal pressure is applied suddenly. Application of the Fourier and Laplace transforms reduces the problem to the solution of a pair of dual integral equations in the Laplace transform plane. To solve these equations, the crack surface displacement is expanded in a series of functions which are zero outside of the cracks. The unknown coefficients accompanied in that series are solved with the aid of the Schmidt method. The stress intensity factors defined in the Laplace transform plane are inverted numerically in the physical plane. Numerical calculations are carried out for the case that the layer of carbon fiber is sandwiched by the two elastic half-planes of plastic.     

Internal fracture in an elastomer containing a rigid inclusion

Rubber blocks were prepared with thin glass rods in their centres, firmly bonded to the surrounding rubber. A tensile stress applied to the ends of a block in the direction of the rod axis induced the sudden formation of voids in the rubber near the flat ends of the rod. Approximate values of the local stresses have been calculated by FEM, assuming linear elastic behaviour. Voids were found to form when and where the local dilatant stress, –P (negative hydrostatic pressure), exceeded the magnitude of Young's modulus,E, for the rubber. A precursor void in a highly elastic solid would expand indefinitely under these circumstances, so that fracture seems to be the result of an elastic instability. The applied stress at which voids appear was of the same order asE for short rods, or for a butt joint between a rod and a rubber cylinder of the same diameter, but it became extremely small when the rod was thin compared to the block in which it was embedded, and relatively long. Under these circumstances the local dilatant stress is calculated to be à large multiple of the applied tensile stress.[/p]     

Elliptical hole enlarged by a rigid elliptical inclusion

Summary This note deals with the infinitesimal plane analysis of the displacement and stress field in an infinite block with an elliptical hole which is deformed by a rigid elliptical inclusion.     

Interaction between cracks and rigid lines in an infinite plate

Summary Interaction between cracks and rigid lines in an infinite plate is investigated in this paper. The rigid lines are assumed in an equilibrium condition and may have some rotation in the deformation process of the adjacent material. After placing some distributed dislocations along the cracks and some distributed body forces along the rigid lines, a system of singular integral equations is obtained. The obtained system of the singular integral equations is reduced to a system of Fredholm integral equations by appropriate substitution of the unknown functions. The regularized integral equations are solved numerically. Stress intensity factors at the crack tips and stress singularity coefficients are investigated in the numerical examples.     

Antiplane interaction of an anisotropic elliptic inclusion with an arbitrarily oriented crack

The antiplane interaction problem for an anisotropic elastic inclusion embedded in an anisotropic elastic matrix with an arbitrarily oriented crack, located either in the matrix or in the inclusion, is considered in this paper. The proposed analysis is based upon the use of conformal mapping, analytical continuation and Laurent series expansion of the corresponding complex potentials. By applying the existing solutions for dislocation functions, the integral equations for a line crack are formulated and the mode-III stress intensity factors are obtained numerically. Several numerical examples are given to demonstrate the effects of geometrical parameters and material property combinations on the strength of the antiplane stress singularity.     

Detachment of an elastic matrix from a rigid spherical inclusion

An approximate theoretical treatment is given for detachment of an elastomer from a rigid spherical inclusion by a tensile stress applied to the elastomeric matrix. The inclusion is assumed to have an initially-debonded patch on its surface and the conditions for growth of the patch are derived from fracture energy considerations. Catastrophic debonding is predicted to occur at a critical applied stress when the initial debond is small. The strain energy dissipated as a result of this detachment, and hence the mechanical hysteresis, are also evaluated. When a reasonable value is adopted for Young's modulus E of the elastomeric matrix, it is found that detachment from small inclusions, of less than about 0.1 mm in diameter, will not occur, even when the level of adhesion is relatively low. Instead, rupture of the matrix near the inclusion becomes the preferred mode of failure at an applied stress given approximately by E/2. For still smaller inclusions, of less than about 1 m in diameter, rupture of the matrix becomes increasingly difficult, due to the increasing importance of a surface energy term. These considerations account for the general features of reinforcement of elastomers. Small-particle fillers become effectively bonded to the matrix, whereas larger inclusions induce fracture near them, or become detached from the matrix, at applied stress that can be calculated from the particle diameter, the strength of adhesion, and the elasticity of the matrix material.     

Dynamic stresses around two cracks placed symmetrically to a large crack

Dynamic stresses around three cracks in an infinite elastic plate have been solved. Two cracks, which are small and equal, are situated ahead of a large crack so as to allow for geometrical symmetry. Time-harmonic normal traction acts on each surface of these cracks. To solve the problem, two solutions are combined. One of them is a solution for a crack in an infinite plate and another is that for two collinear cracks in an infinite plate. The Schmidt method is used to satisfy the boundary conditions on the cracks' surfaces with use of the combined solutions. Stress intensity factors are calculated numerically for some of these crack configurations.