Crack–inclusion problem for a long rectangular slab of superconductor under an electromagnetic force

The interaction between a crack and an inclusion in a type-II superconductor is investigated in this paper. Using the finite element method, the crack–inclusion problem can be solved. Numerical results are presented to illustrate fracture behavior of superconductor under electromagnetic force. The magnetic behavior of the superconductor is described by the critical-state Bean model. The stress intensity factors at the crack tip are obtained and discussed for decreasing field after zero-field cooling. Numerical results show that the stress intensity factors at crack tip are always larger with an elastic inclusion than for a rigid inclusion. Because of the barrier effect of the rigid inclusion, the values of the stress intensity factors decrease when the crack approaches the inclusion. Relative to rigid inclusion and no inclusion cases, elastic inclusion leads to the largest value of the stress intensity factor at crack tip. Thus, the crack propagation is easier near an elastic inclusion and the rigid inclusion is helpful for crack arrest.     

On the axisymmetric loading of an annular crack by a disk inclusion

This paper examines the axisymmetric elastostatic problem related to the loading of an annular crack by a rigid disk–shaped inclusion subjected to a central force. The integral equations associated with the resulting mixed–boundary–value problem are solved numerically to determine the load–displacement result for the rigid inclusion and the Mode II stress–intensity factors at the boundaries of the annular crack. The results presented are applicable to a wide range of Poisson's ratios ranging from zero to one half.     

Stress analysis of a kinked crack initiating from a rigid line inclusion. Part 1: Formulation

The problem of a kinked crack which has initiated from the tip of a rigid line inclusion is analyzed as a mixed boudary value problem. The stress distribution, stress intensity factors, singularity at the inclusion tip, and the resultant moment on the rigid line inclusion are investigated for various angles of the kinked crack and crack lengths. The rotation of the rigid line inclusion, when loaded by a uniform farfield stress, is calculated. The cases in which the inclusion is free to rotate or is fixed are separately considered.     

Axisymmetric finite cylinder with one end clamped and the other under uniform tension containing a penny-shaped crack

This study considers the axisymmetric analysis of a finite cylinder containing a penny-shaped transverse crack. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to uniform axial tension. Solution is obtained by superposing the solutions for an infinite cylinder loaded at infinity and an infinite cylinder containing four cracks and a rigid inclusion loaded along the cracks and the inclusion. When the radius of the inclusion approaches the radius of the cylinder, its mid-plane becomes fixed and when the radius of the distant cracks approach the radius of the cylinder, the ends become cut and subject to uniform tensile loads. General expressions for the perturbation problem are obtained by solving Navier equations with Fourier and Hankel transforms. Formulation of the problem is reduced to a system of five singular integral equations. By using Gauss-Lobatto and Gauss-Jacobi integration formulas, these five singular integral equations are converted to a system of linear algebraic equations which is solved numerically. Stress distributions along the rigid support, stress intensity factors at the edges of the rigid support and the crack are calculated.     

Stress intensity factors for curvilinear cracks loaded under anti-plane strain (mode III) conditions

A general solution to the elastic and thermoelastic problems with a rigid circular-arc inclusion is presented. The proposed analysis is based upon the complex variable theory dealing with sectionally holomorphic functions which is reduced to the solution of the Hilbert problem. It is indicated that both the stress and thermal stress fields near the inclusion tip possess a square-root singularity similar to that for the corresponding crack problem. In analogy to the stress intensity factors defined for crack problem, stress singularity coefficients are introduced in this paper to characterize the near tip fields. Complete stress fields and the corresponding stress singularity coefficients as the circular-arc inclusion are under uniform remote load, concentrated force and uniform heat flux are given. Failure initiation of an infinite plate embedded with a rigid arc inclusion under different loading conditions is also discussed.     

The interaction of a curved crack with a circular elastic inclusion

The solution to a curved matrix crack interacting with a circular elastic inclusion is presented. The problem is formulated using the Kolosov–Muskhelishvili complex stress potential technique where the crack is represented by an unknown distribution of dislocations. After an appropriate parameterization, the resulting singular integral equations are solved with the Lobatto-Chebyshev quadrature technique. The accuracy of the current solution is shown by comparing these results to previously published results. A preliminary investigation is conducted to study the effects of crack curvature and inclusion stiffness on the stress intensity factors and it is shown that in certain instances, the effect of the crack curvature and the inclusion stiffness are competing influences.     

On the expansion of a penny-shaped crack by a rigid circular disc inclusion

This paper examines the problem of the symmetric indentation of a penny-shaped crack by a smoothly embedded rigid circular thin disc inclusion. The analysis of the problem yields a system of triple integral equations which are solved in an approximate manner. An expression for the stress intensity factor at the boundary of the penny-shaped crack is evaluated in the form of a series which involves the ratio of the radius of the rigid circular inclusion to the radius of the penny-shaped crack.

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Résumé Dans le mémoire, on examine le problème du marquage symétrique d'une fissure en angle noyée par une inclusion mince et lisse, en forme de disque circulaire rigide. L'analyse du problème conduit à un système d'équations intégrales triples, que l'on résoud par approximations. On obtient une expression du facteur d'intensité de contrainte aux frontières de la fissure sous la forme d'une série comportant le rapport du rayon de l'inclusion rigide circulaire au rayon de la fissure.

     

Computation of sensitivity of periodically excited dynamical systems

An analytical treatment is presented for bonded contact of a rigid disk inclusion embedded in a penny-shaped crack in a transversely isotropic full-space. Theoretical analysis is carried out using a complete potential function method, and with the aid of Hankel transforms. Boundary conditions propel the problem toward a set of triple integral equations, which are solved analytically and then reduced to a pair of Fredholm integral equations of the second kind. Furthermore, the results are evaluated and presented graphically using numerical schemes, and comparison is made with well-known classical solutions in transversely isotropic and isotropic media. This can be obtained as special cases for the problem to reveal the efficacy of the proposed method. Eventually, it can be seen that not only the presence of the crack around the disk inclusion decreases the axial stiffness, but also the extension of its length also reduces the fracture parameter, stress intensity factor, for different degrees of material anisotropy.     

Scattering of elastic waves by a crack with spring-mass contact

The scattering problem of elastic waves by a crack with spring-mass contact is investigated. Such a crack may be regarded as a simplified model of a thin elastic inclusion. Boundary integral equations are formulated for both displacement and traction on crack faces and are solved numerically. Numerical results are presented for stress intensity factors, crack-opening displacements and scattering cross-sections. Our results are in good agreement with other published solutions. It is also found that the effect of a mass can not be neglected in evaluation of scattering cross-sections, even if the mass is small.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Extension of a finite strip bonded to a rigid support

This paper considers the elastostatic plane problem of a finite strip. One end of the strip is perfectly bonded to a rigid support while the other is under the action of a uniform tensile load. Solution for the finite strip is obtained by considering an infinite strip containing a transverse rigid inclusion at the middle and two symmetrically located transverse cracks. The distance between the two cracks is equal to twice the length of the finite strip. In the limiting case when the rigid inclusion and the cracks approach the sides of the infinite strip, the region between one crack and the rigid inclusion becomes equivalent to the finite strip. Formulation of the problem is reduced to a system of three singular integral equations using the Fourier transforms. Numerical results for stresses and stress intensity factors are given in graphical form.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semi-infinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Elastic–plastic fracture behavior of a radial matrix crack interacting with a circle inclusion with generalized Irwin corrections

A generalized Irwin plastic zone model is proposed to investigate the interaction problem for a circular inclusion embedded in an elastic–plastic matrix that contains a radial crack, oriented at an arbitrary angle from a remote load. The distributed dislocation technology is applied to formulate the current problem. The effective stress intensity factors, the plastic zone size, and the crack tip opening displacement are evaluated by solving the formulated singular integral equations. In the numerical examples, the inclusion is taken as a void and a rigid body. The effects of the crack angle and the inclusion–crack distance (the distance from the inclusion center to the crack center) on the effective stress intensity factors, the plastic zone size, and the crack tip opening displacement are discussed in detail. Numerical results show that if the crack angle is not large, the values of the plastic zone size and the crack tip opening displacement are less than the corresponding values in the homogenous case when the inclusion is a rigid body; when the inclusion is a void, these values are larger than the corresponding values in the homogenous case.     

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.     

The indentation of a precompressed penny-shaped crack

The paper examines the axisymmetric problem related to the indentation of the plane surface of a penny-shaped crack by a smooth rigid disc inclusion. The crack is also subjected to a far-field compressive stress field which induces closure over a part of the crack. The paper presents the Hankel integral transform development of the governing mixed boundary value problem and its reduction to a single Fredholm integral equation of the second kind and an appropriate consistency condition which considers the stress state at the boundary of the crack closure zone. A numerical solution of this integral equation is used to develop results for the axial stiffness of the inclusion and for the stress intensity factors at the tip of the penny-shaped crack.     

Stress intensity factor for a Griffith crack interacting with a coated inclusion

Stress investigation for the interaction problem between a coated circular inclusion and a near-by line crack has been carried out. The crack and the coated inclusion (a coated fiber) are embedded in an infinitely extended isotropic matrix, with the crack being along the radial direction of the inclusion. Two loading conditions, namely, the tensile and shear loading ones are considered. During the solution procedure, the crack is treated as a continuous distribution of edge dislocations. By using the solution of an edge dislocation near a coated fiber as the Green's function, the problem is formulated into a set of singular integral equations which are solved by Erdogan and Gupta (1972) method. The expressions for the stress intensity factors of the crack are then obtained in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Several numerical examples are given for various material and geometric parameters. The solutions obtained from the integral equations have been checked and confirmed by the finite element analysis results.     

Thermal analysis of carbon-nanotube composites using a rigid-line inclusion model by the boundary integral equation method

The boundary integral equation (BIE) method is applied for the thermal analysis of fiber-reinforced composites, particularly the carbon-nanotube (CNT) composites, based on a rigid-line inclusion model. The steady state heat conduction equation is solved using the BIE in a two-dimensional infinite domain containing line inclusions which are assumed to have a much higher thermal conductivity (like CNTs) than that of the host medium. Thus the temperature along the length of a line inclusion can be assumed constant. In this way, each inclusion can be regarded as a rigid line (the opposite of a crack) in the medium. It is shown that, like the crack case, the hypersingular (derivative) BIE can be applied to model these rigid lines. The boundary element method (BEM), accelerated with the fast multipole method, is used to solve the established hypersingular BIE. Numerical examples with up to 10,000 rigid lines (with 1,000,000 equations), are successfully solved by the BEM code on a laptop computer. Effective thermal conductivity of fiber-reinforced composites are evaluated using the computed temperature and heat flux fields. These numerical results are compared with the analytical solution for a single inclusion case and with the experimental one reported in the literature for carbon-nanotube composites for multiple inclusion cases. Good agreements are observed in both situations, which clearly demonstrates the potential of the developed approach in large-scale modeling of fiber-reinforced composites, particularly that of the emerging carbon-nanotube composites.     

Bending of a thin plate containing a rigid inclusion and a crack

The bending of a thin infinite plate with a line crack and an arbitrarily shaped rigid inclusion is analyzed. The superposition principle is used to reduce the original formulation to two subsidiary problems. A distribution of dislocation is assumed along the crack line. The solution is obtained in an integral form by using the Green function of a point dislocation. The stress functions for both subsidiary problems are obtained by employing the rational mapping function technique. The stress intensity factors are obtained in terms of the dislocation density function. Numerical results are demonstrated for the plate containing a square rigid inclusion and a line crack.     

Electroelastic analysis for a Griffith crack interacting with a coated inclusion in piezoelectric solid

A Mode III Griffith crack interacting with a coated inclusion in piezoelectric media is investigated. The crack, the coated inclusion are embedded in an infinitely extended piezoelectric matrix media, with the crack being along the radial direction of the inclusion. In the study, three different piezoelectric material phases are involved: the inclusion, the coating layer, and the matrix. A far-field loading condition is considered. During the solution procedure, the crack is simulated as a continuous distribution of screw dislocations. By using the solution of a screw dislocation near a coated inclusion in piezoelectric media as the Green function, the problem is formulated into a set of singular integral equations, which are solved by numerical method. The stress and electric displacement intensity factors are derived in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Numerical examples are given for various material constants combinations and geometric parameters.     

Electro-elastic stress analysis for a Zener-Stroh crack interacting with a coated inclusion in a piezoelectric solid

Summary. The problem of a Zener-Stroh crack initiated near a coated circular inclusion in a piezoelectric medium is investigated in this paper. By using the solution of a single piezoelectric screw dislocation near a coating inclusion as the Greens function, a Mode III displacement loaded crack is investigated. The proposed problem is formulated as a set of singular integral equations which are solved by numerical techniques. The influence of various parameters, such as the material constants of the inclusion, the coating, the matrix, the coating layer thickness, etc., on the crack behavior is studied. The stress and electric displacement intensity factors of the crack are derived. Several numerical examples are given and the results obtained are discussed in detail.