Set Of Singular Integral Equations Research Articles

On the plastic zone size and CTOD study for a Zener–Stroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener–Stroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green’s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan’s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson’s ratio, and loading condition are studied.

Crack tip opening displacement of a Dugdale crack in a three-phase cylindrical model composite material

The analytical investigation of the plastic zone size of a crack in three-phase cylindrical model composite material was carried out. The physical problem is simulated as a crack near a circular inclusion (a single fiber) in the composite matrix, while the three-phase cylindrical composite model is used to represent the composite matrix. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, a thin strip of yielded plastic zone is introduced at each crack tip. Using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function, the physical problem is formulated into a set of singular integral equations. By employing Erdogan and Gupta’s method, as well as iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements.

On the dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under electro-mechanical loads

The dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under in-plane steady-state electro-mechanical loads is considered. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which are solved using Chebyshev polynomial expansions. Numerical results are provided to show the variation of both the dynamic stress intensity factors and electric displacement intensity factor with frequencies of the applied electro-mechanical loads. A phenomenon similar to “resonance” is observed when the applied loads act in some specific ranges of frequencies, and both the dynamic stress intensity factors and electric displacement intensity factor may increase significantly, which will lead to the failure of piezoelectric material. The effects of applied electric fields, crack geometry and elastic layer thickness on the phenomenon are also discussed.

Analytical solution of the problem for a set of shear fractures with Coulomb friction

A simplified solution of a two-dimensional problem on a set of interaction shear fractures with Coulomb friction along their wings is proposed. In the classical statement of this problem, the equality between shear stresses and friction stresses must necessarily hold in a new equilibrium state at each point along the fracture plane. The solution of the problem is reduced to the solution of a set of singular integral equations with respect to unknown functions of a shear jump on fractures. In reality, our knowledge of fracture conditions is rather approximate. In addition, in the problems of seismology and tectonophysics it is sufficient to approximately estimate dynamic (taking into account the inertia forces) and static perturbations from a fracture in the form of the first terms of a true solution series. With this aim in view, point models of an earthquake source or continual representation of discontinuous deformations are used. Within the framework of the requirements of such approaches, it is assumed that the condition of equality between the sum of shear stresses and friction stresses on the shear fractures in a new equilibrium state is met. In addition, the complex potential function is calculated for each fracture based on the function of the jump of its wings obtained in the problem for a solitary fracture. Such statement of the problem of a set of neighboring and even intersecting fractures makes it possible to reduce its solution to a set of linear algebraic equations with respect to shear stresses relieved at each fracture and the average along its length, they being the unknown values.

Plastic zone size and crack tip opening displacement of a Dugdale crack interacting with a coated circular inclusion

An analytical investigation on the plastic zone size of a crack near a coated circular inclusion under three different loading conditions of uniaxial tension, uniform tension and pure shear was carried out. Both the crack and coated circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small-scale yielding [J. Mech. Phys. Solids 8 (1960) p. 100], two thin strips of yielded plastic zones are introduced at both crack tips. Using the solution for a coated circular inclusion interacting with a single dislocation as the Green's function, the physical problem is formulated into a set of singular integral equations. Using the method of Erdogan and Gupta [Q. J. Appl. Math. 29 (1972) p. 525] and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement.

Identification of weakenings in a rock block

The authors propose a process for identification of a weakening location in a rock block and the algorithm of its numerical realization based on the direct problem solutions and complementary data on boundary displacements. The inverse problems are solved by a set of singular integral equations of the boundary stresses and displacements.

On the plastic zone size and crack tip opening displacement of a Dugdale crack interacting with a circular inclusion

An analytical investigation on the plastic zone size (PZS) of a crack near a circular inclusion has been carried out. Both the crack and the circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small scale yielding, two stripe plastic zones at both crack tips are introduced. Using the solution of a circular inclusion interacting with a single dislocation as the Green’s function, the physical problem is formulated as a set of singular integral equations. With the aid of Erdogan and Gupta’s method and iterative numerical procedures, the singular integral equations are solved numerically for the PZS and the crack tip opening displacement. The results obtained in the current work can be reduced to those simpler cases of the Dugdale model.

Elastic wave scattering from a penny-shaped interface crack in coated materials

This paper is concerned with the elastic wave scattering induced by a penny-shaped interface crack in coated materials. Using the integral transform, the problem of wave scattering is reduced to a set of singular integral equations in matrix form. The singular integral equations are solved by the asymptotic analysis and contour integral technique, and the expressions for the stress and displacement as well as the dynamic stress intensity factors (SIFs) are obtained. Using numerical analysis, this approach is verified by the finite element method (FEM), and the numerical results agree well with the theoretical results. For various crack sizes and material combinations, the relations between the SIFs and the incident frequency are analyzed, and the amplitudes of the crack opening displacements (CODs) are plotted versus incident wavenumber. The investigation provides a theoretical basis for the dynamic failure analysis and nondestructive evaluation of coated materials.

Stress investigation on a Griffith crack initiated from an eccentric disclination in a cylinder

Disclinations are rotational line defects which may be introduced in metal wires during the manufacturing process. In this work, the relaxation of an eccentric (off-center) negative wedge disclination in a cylinder by nucleation of a Griffith crack is investigated. The nucleated crack is simulated with distributed edge dislocations. The stress intensity factors (SIFs) of the crack are evaluated by solving a set of singular integral equations. By enforcing the condition that the SIFs at the two crack tips should keep the same value in the nucleation process, the crack length growth on each side of the wedge disclination is determined. The critical disclination power and equilibrium crack lengths are then numerically determined. Some important characteristics of the Griffith crack nucleation are revealed. (1) The two tips of the nucleated Griffith crack grow asymmetrically when the disclination locates eccentrically. The tip closer to the cylinder edge travels a shorter length. This asymmetry is getting more severe as the normalized off-center distance increases. (2) The critical disclination power increases monotonically with the normalized off-center distance. (3) The normalized stable equilibrium crack length decreases as the normalized off-center distance increases while the normalized unstable equilibrium crack length shows an opposite dependence. The dependence of the critical disclination power and the equilibrium crack lengths on the disclination power and cylinder radius is also discussed in this work. It is believed that this work helps to predict the strength of disclinated metal wires at various length scales.

Subsonic slip waves along the interface between two piezoelectric solids in sliding contact with local separation

The Stroh formalism of piezoelectric crystals and singular integral equation technique are applied to study the propagation of possible slip waves in presence of local separation at the interface between two frictionless contact piezoelectric solids, which are pressed together by uniaxial pressure and laid in the electric field. The problem is cast into a set of singular integral equations of which the closed solutions are obtained. Discussion on the existence of such slip waves is presented. The results show that such slip waves, which have square-root singularities at both ends of the local separation zones, can propagate in some special material combinations. And the existence of such slip waves is related with the applied mechanical and electric fields.

Zener–Stroh crack at the interface of multi-layered structures

A Zener–Stroh (Z–S) crack can be nucleated on the interface of a multi-layered structure when a dislocation pileup is stopped by the interface which works as an obstacle. During the entire fracture of a crack, Z–S crack mechanism controls the initial stage, or the first phase of crack initiation and propagation. In our current research, investigation on a Z–S crack at the interface of a multi-layered structure is carried out. The problem is formulated into a set of singular integral equations by applying the distributed dislocation based fracture mechanics. The obtained integral equations are then solved with numerical method after the singularities at crack tips are carefully checked. In the solution procedure, the contact zone model is adopted to cease the oscillation behavior. The contact zone length, the stress field near the crack tips and the stress intensity factors (SIFs) of the crack are discussed based on the numerical results of two typical structures. It was found that the contact zone length could be very large and was determined by the properties of all the three materials and loading conditions. Our analysis also shows that the thickness of the middle thin layer plays a critical role for the fracture behavior of the crack when it is comparable to the crack length.

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.

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The Ji and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.

Thermo-elastic crack branching in general anisotropic media

Thermal fields may exist in addition to mechanical loading, for example, due to short term exposure to fire. In this paper, the branching of cracks in the presence of combined thermal and mechanical loads is investigated for general anisotropic media by employing the theory of Stroh’s dislocation formalism, extended to thermo-elasticity in matrix notation. A general solution to the thermo-elastic crack problem for an anisotropic material under arbitrary loading is obtained in a compact form. Green’s functions are also presented for a thermal dislocation (heat vortex) and a conventional dislocation (or, referred as mechanical dislocation), which are formulated considering the cuts located at an arbitrary angle with respect to the x 1 axis of the coordinate system ( x 1, x 2, x 3). Using the derived compact expressions, the interaction between the crack and the dislocation is studied and a closed form solution for this interaction is obtained. The branching portion of the thermo-elastic crack is modelled as a continuous distribution of dislocations. This problem is then converted into a set of singular integral equations. Numerical results are presented to illustrate the possible effects of thermal loading on the propagation of the branched crack.

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

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 Green’s 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.

Numerical analysis of waveguide bandpass filters using singular integral equations and construction of simple design formulas

AbstractA powerful numerical solution is presented for bandpass filters in rectangular waveguides. The filters are composed of cascading of perfectly conducting inductive obstacles with zero thickness: asymmetrical windows, symmetrical windows, or symmetrical strips. The problem is reduced to a set of singular integral equations, which is solved by an extension of Lewin's quasi‐static procedure. After the accuracy of the classical formulas for the equivalent reactance is investigated, the simple inverse formulas are constructed to yield proper window widths for desired equivalent parameters. Their effectiveness is demonstrated by designing practical filters having maximally flat or equal‐ripple characteristics. © 2004 Wiley Periodicals, Inc. Electr Eng Jpn, 148(4): 17–25, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10251

An inclined Zener–Stroh crack near a free surface

Machining of a material surface often produces large shear stress beneath the free surface, which causes dislocations piled up along a slip plane near the surface until those dislocations are stopped by an obstacle and nucleates a microcrack at the subsurface. The nucleated microcrack due to dislocation pile-up is called a Zener–Stroh crack which has many properties different to a conventional Griffith crack. In this paper, the physical mechanism of such microcrack initiation has been discussed. Stress investigation on the problem of a subsurface Zener–Stroh crack inclined to the free surface of a material has been carried out. By using the stress solution of a single dislocation in a half plane as the Green's function, the microcrack is simulated with distributed dislocations along the crack line. A set of singular integral equations are then formulated and solved with numerical method. Results show that the influence of the free surface on the stress intensity factors (SIFs) of the crack and the critical crack length greatly depends on the slant angle. The free surface also brings intrinsic coupling phenomenon for the Mode I and Mode II SIFs. As a result this coupling effect changes the fracture toughness of the material.

Transient anti-plane crack problem of a functionally graded piezoelectric strip bonded to elastic layers

This paper examined the dynamic electromechanical behavior of a crack in a functionally graded piezoelectric layer bonded between two elastic layers under the combined anti-plane mechanical shear and in-plane electric impacts. Fourier cosine transforms are used to reduce the problem to the solution of a set of singular integral equations. It is found that the impermeable crack condition is more reasonable than the permeable crack condition to analyze the influence of electric loading, and the energy density factor is more acceptable than the energy release rate to be used as the fracture criterion. In addition, numerical results are also presented to show the influences of the crack position, electromechanical combination factor and material gradient parameter on the fracture behavior.

Electro-elastic stress analysis for a Zener–Stroh crack at the metal/piezoelectric bi-material interface

A Zener–Stroh crack may nucleate at the interface of metal/piezoelectric bi-materials, when piled-up dislocations along a slip plane in metal material are stopped by the interface which works as an obstacle. As a kind of microcrack, Zener–Stroh crack controls the initial phase of crack nucleation. In the first step of our current research, the stress and electric displacement fields for a single interfacial dislocation in general piezoelectric/metal bimaterials are obtained in an explicit close form. With this solution as the Green function, the interfacial Zener–Stroh crack problem is formulated into a set of singular integral equations with distributed dislocation technique. These singular integral equations are then solved by numerical method based on the singularity nature at the crack tips. To deal with the oscillation behaviors near the interfacial crack tips, both open crack tip model and contact zone model are considered and compared each other in the paper. Solutions on the stress field, stress and electric displacement intensity factors and contact zone length are obtained. A numerical example of a Zener–Stroh crack at the Pt/PZT-5H bimaterial interface is given. Some useful electro-elastic characteristics related to the crack are found and discussed.

Frictional interface crack in anisotropic bimaterial under combined shear and compression

The problem of a frictionally sliding interface crack embedded in an anisotropic bimaterial is investigated. Under remote normal compressive and shear load the crack faces are treated as completely closed in the direction of the normal load while in other directions the crack faces are allowed to slide. The frictional coefficient over the sliding zone is assumed to be constant. A set of singular integral equations is formulated which is valid for general anisotropic bimaterial. The nature of singularities for frictionally sliding bimaterial is investigated. It is found that for general anisotropic bimaterial the problem may be treated as a homogeneous anisotropy as long as W ̃ = 0 and hence the stresses developed on the frictional surface would be uniform for such bimaterial. If W ̃ is not identically zero but with ( W ̃ ) 13=0 and with the surface being frictionless then the stresses over the crack faces are square root singular. The homogeneous anisotropy satisfying ( L ̃ ) 12=( L ̃ ) 32=0 would transmit the load freely across the crack faces without any interference. For monoclinic bimaterial, the orders of singularities are found to depend on the material constant Λ and the frictional coefficient f and are either of order δ or 1− δ depending on the direction of the frictional force. The stresses on the interface for monoclinic bimaterial are also given explicitly.