Problem of equilibrium of the timoshenko plate containing a crack on the boundary of an elastic inclusion with an infinite shear rigidity

A problem of equilibrium of a composite plate consisting of a matrix and an elastic inclusion with a through crack along the boundary of this inclusion is studied. The matrix deformation is described by the Timoshenko model, and the elastic inclusion deformation is described by the Kirchhoff-Love model. Conditions of mutual non-penetration of the crack edges are imposed on the curve that describes the crack. Unique solvability of the variational problem is proved. A system of boundary conditions on the curve bounding (in the mid-plane) the elastic inclusion is obtained. A differential formulation of the problem equivalent to the initial variational formulation is given.     

Problem of a crack on the boundary of a rigid inclusion in an elastic plate

The problem of equilibrium of a thin elastic plate containing a rigid inclusion is considered. On part of the interface between the elastic plate and the rigid inclusion, there is a vertical crack. It is assumed that, on both crack edges, the boundary conditions are given as inequalities describing the mutual impenetrability of the edges. The solvability of the problem is proven and the character of satisfaction of the boundary conditions is described. It is also shown that the problem is the limit problem for a family of other problems posed for a wider region and describing equilibrium of elastic plates with a vertical crack as the rigidity parameter tends to infinity.     

Interaction between crack and elastic inclusion

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Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Material under Triaxial Tension

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed     

Penny-shaped crack in a fiber-reinforced matrix

Using the slender inclusion model developed earlier the elastostatic interaction problem between a penny-shaped crack and elastic fibers in an elastic matrix is formulated. For a single set and for multiple sets of fibers oriented perpendicularly to the plane of the crack and distributed symmetrically on concentric circles the problem is reduced to a system of singular integral equations. Techniques for the regularization and for the numerical solution of the system are outlined. For various fiber geometries numerical examples are given and distribution of the stress intensity factor along the crack border is obtained. Sample results showing the distribution of the fiber stress and a measure of the fiber-matrix interface shear are also included.     

A penny-shaped crack in a filament-reinforced matrix—II. The crack problem

Using the filament model developed in the previous paper, the elastostatic interaction problem between a penny-shaped crack and a slender inclusion or filament in an elastic matrix is formulated. For a single filament as well as multiple identical filaments located symmetrically around the crack the problem is shown to reduce to a singular integral equation. The solution of the problem is obtained for various geometries and filament to-matrix stiffness ratios, and the results relating to the angular variation of the stress intensity factor and the maximum filament stress are presented.     

Interaction between curved crack and elastic inclusion in an infinite plate

Summary In this paper, the curved-crack problem for an infinite plate containing an elastic inclusion is considered. A fundamental solution is proposed, which corresponds to the stress field caused by a point dislocation in an infinite plate containing an elastic inclusion. By placing the distributed dislocation along the prospective site of the crack, and by using the resultant force function as the right-hand term in the equation, a weaker singular integral equation is obtainable. The equation is solved numerically, and the stress intensity factors at the crack tips are evaluated. Interaction between the curved crack and the elastic inclusion is analyzed. Received 8 October 1996; accepted for publication 27 March 1997     

Interaction between a compliant disk-shaped inclusion and a crack upon incidence of an elastic wave

The propagation of harmonic elastic wave in an infinite three-dimensional matrix containing an interacting low-rigidity disk-shaped inclusion and a crack. The problem is reduced to a system of boundary integral equations for functions that characterize jumps of displacements on the inclusion and crack. The unknown functions are determined by numerical solution of the system of boundary integral equations. For the symmetric problem, graphs are given of the dynamic stress intensity factors in the vicinity of the circular inclusion and the crack on the wavenumber for different distances between them and different compliance parameters of the inclusion.     

The effect of an elastic triangular inclusion on a crack

IntroductionWiththedevelopmentofparticleandfiberreinforcedcomposites,theinclusion_crackinteractionproblemisbecominganimportantfieldbeingstudied .Andasamodel,itisalsousedtostudytheeffectsofmaterialdefectsonthestrengthandfractureofengineeringstructure.TheinterationbetweencircularinclusionandcrackwasstudiedinRefs.[1 -6 ] ;InRefs.[7-1 2 ] ,theinterationbetweenlineinclusionandcrackswasdiscussed ;TheinterationbetweenellipticalinclusionandcrackwasstudiedinRefs.[1 3,1 4] .However,withthedevelopmento…     

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

The two-dimensional problem of an arc shaped crack lying along the interface of a circular elastic inclusion embedded in an infinite matrix with different elastic constants is considered. Based on the complex variable method of Muskhelishvili, closed-form solutions for the stresses and the displacements around the crack are obtained when general biaxial loads are applied at infinity. These solutions are then combined with A.A. Griffith's virtual work argument to give a criterion of crack extension, namely the de-bonding of the interface. The critical applied loads are expressed explicitly in terms of a function of the inclusion radius and the central angle subtended by the crack arc. In the case of simple tension the critical load is inversely proportional to the square-root of the inclusion radius. By analyzing the variation of the cleavage stress near the crack tip, the deviation of the crack into the matrix is discussed. The case of uniaxial tension is worked out in detail.     

Effect of a flat inclusion on stress intensity factor of a semi-infinite crack

The elastic plane interaction between an arbitrarily located and oriented flat inclusion and a semi-infinite crack subjected to a remote Mode I loading is considered. The method uses distributions of edge dislocations to formulate integral expressions of flat inclusion (including crack) tractions and is shown to be very accurate by a test problem. The stress intensity factors of the main crack tip are presented for a variety of crack inclusion geometries. It is seen that the flat inclusion could either yield a stress enhancement or stress shielding effect to the main crack tip depending upon the location, orientation and thickness of the flat inclusion, and depending upon the modulus ratios of the flat inclusion to matrix.     

Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions

We consider an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that nonpenetration conditions are imposed at the crack faces which do not allow the opposite crack faces to penetrate each other. We analyze the variational formulation of the problem and provide shape and topology sensitivity analysis of the solution in two and three spatial dimensions. The differentiability of the energy with respect to the crack length, for the crack located at the boundary of rigid inclusion, is established.     

A misfitting elastic inclusion in an infinite plane containing a crack

The problem discussed in this paper is that of a misfitting circular inclusion in an infinite elastic medium which contains a straight crack. The crack is stress free. The stresses develop in the elastic medium because of the misfit. The point force method is used to solve the problem. The problem reduces to finding two sets of complex potential functions: {(z), (z)}: One for the infinite medium and the other for the misfitting inclusion. The solution has been obtained in closed form. Graphs are drawn for stress intensity at the crack tip and also for normal, shear and hoop stresses at the common interface of medium and misfitting inclusion.     

Internally indented penny-shaped cracks: A comparison of analytical and boundary integral equation estimates

The problem of the axisymmetric internal indentation of a penny-shaped crack by a rigid circular inclusion is discussed. The paper presents a comparison of analytical and boundary integral equation results for the stress intensity factor at the boundary of the penny-shaped crack indented by a smooth inclusion. Numerical results presented in the paper examines the influence of features such as adhesion at the inclusion-elastic medium interface and finite geometry of the elastic solid containing the penny shaped crack.     

Interaction between rigid-disc inclusion and penny-shaped crack under elastic time-harmonic wave incidence

Influence of a rigid-disc massive inclusion on a neighboring penny-shaped crack induced by the time-harmonic wave propagation in an infinite elastic matrix is investigated by the numerical solution of associated 3D elastodynamic problem. No restrictions on the mutual orientation of interacting objects and direction of wave incidence are assumed. The inclusion is perfectly bonded with a matrix and supposes the translations and rotations, the crack faces are load-free. Frequency-domain problem is reduced to a system of boundary integral equations (BIEs) relative to the interfacial stress jumps (ISJs) on the inclusion and the crack opening displacements (CODs). The subtraction technique in conjunction with mapping technique, under taking into account the structure of solution at the fronts of inclusion and crack, is applied for regularization of BIEs obtained. A discrete analogue of equations is constructed by using the collocation scheme. Numerical calculations are carried out for the grazing incidence of a plane P-wave on the crack, where the interacting inclusion is coplanar and perpendicular to the crack, and has the same radius. The shielding and amplification effects of inclusion are assessed by the analysis of mode-I stress intensity factor (SIF) in the crack vicinity depending on the wave number, incident wave direction, position of the crack front point, inclusion mass, crack-inclusion orientation and distance.     

Thermal stresses in a two-dimensional infinite medium containing a rigid inclusion embedded in a line crack

This paper examines the problem of finding thermal stresses, caused by a symmetric indentation of a line crack by an inclusion in an infinite isotropic elastic heat conducting solid. The thermal and elastic problems are reduced to a system of triple integral equations. In each case the solution of the triple integral equations is obtained in a closed form. The expressions for the stress intensity factor at the edge of the line crack, the strain energy density function and the resultant pressure applied to the inclusion are obtained. The expression for the displacement component is also obtained. Finally the results for the physical quantities are displayed graphically.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

反平面剪切裂纹和刚性线问题和带裂纹圆轴扭转问题中的新型积分方程

如果把通常裂纹问题中奇异积分方程中的右端项由应力改为合力,此时积分方程的核也要由奇异核改为对数型奇异核。文中对于反乎面剪切裂纹和刚性线问题和带裂纹圆轴扭转问题,推导出了这种带对数核的积分方程。     

Analysis of the Singularity for the Vertical Contact Problem of Line Crack-Inclusion

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Interaction between a circular inclusion and a symmetrically branched crack

The interaction problem between a circular inclusion and a symmetrically branched crack embedded in an infinite elastic medium is solved. The branched crack is modeled as three straight cracks which intersect at a common point and each crack is treated as a continuous contribution of edge dislocations. Green's functions are used to reduce the problem into a system of singular equations consisting of the distributions of Burger's dislocation vectors as unknown functions through the superposition technique. The resulting integral equations are solved numerically by the method of Gauss-Chebychev quadrature. The proposed integral equation approach is first verified for two limiting cases against the literature. More effort is paid on the effect of inclusion on both the Mode I and Mode lI stress intensity factors at the branch tips. The effect of inclusion on the branching path is also investigated.