Cauchy Type Kernel Research Articles

Fracture of a burning solid propellant attenuated by a crack-type cavity

A plane steady problem of fracture mechanics for a burning deformable solid propellant attenuated by a crack-type cavity with a burning surface is considered. The crack-type cavity is assumed to have tip zones with bonds between the faces, and mixed boundary conditions are imposed on the propellant charge boundary. The problem of equilibrium of a propellant charge containing a crack-type cavity reduces to solving a system of nonlinear singular integrodifferential equations with a Cauchy-type kernel. Based on the solution obtained, the normal and tangential forces in the tip zones of the crack-type cavity are found. Local conditions determining stability (safety) of the solid propellant burning regime are found.

Modeling of crack nucleation in a variable-thickness band under nonuniform heating

A mathematical model of crack nucleation in a variable-thickness band (rod) under nonuniform heating is constructed. As the band (rod) is thermally loaded, pre-fracture zones are assumed to appear; these zones are modeled as regions with attenuated bonds between material particles. The presence of bonds between the pre-fracture zone faces is modeled by adhesion forces applied to the pre-fracture zone surface. Solving the problem of equilibrium of an isotropic variable-thickness band with a prefracture zone is reduced to solving a nonlinear singular integrodifferential equation with a Cauchy-type kernel, which establishes the relation between the adhesion forces in the crack-nucleation zone and the distance between the crack faces. The condition of crack nucleation in a variable-thickness band is formulated with allowance for the criterion of ultimate tension of bonds in the material.

Adapted quadratic approximation for singular integrals

The goal of this work is to present an adapted modification to the parabolic approxi- mation of the density function for singular integrals of Cauchy type. This approximation serves to eliminate the singularity of the integral and gives the help to obtain the numerical solution of singular integral equations with Cauchy type kernel on an oriented smooth contour.

Convergence of the collocation methods for singular integro-differential equations in Lebesgue spaces

In this article, the numerical schemes of collocation methods for an approximate solution of singular integro- differential equations with kernels of Cauchy type are explained. The equations are defined on arbitrary smooth closed contours. The theoretical background of collocations methods in Lebesgue spaces is obtained.

Nucleation of cracks in a perforated fuel cell

A mathematical model is constructed for crack nucleation in an isotropic fuel cell (heat-releasing solid material) attenuated by a biperiodic system of cooling cylindrical channels with a circular cross section. Cracks are assumed to appear with increasing heat-release intensity in the bulk of the material. The solution of the problem on equilibrium of an isotropic perforated fuel cell with crack nuclei reduces to the solution of a nonlinear singular integral equation with a Cauchy-type kernel. The solution of the latter equation yields the forces in the band of crack nucleation. The condition of crack nucleation is formulated with allowance for the criterion of ultimate extension of bonds in the material.

Generalized models and local invariants of Kohn–Nirenberg domains

The main obstruction for constructing holomorphic reproducing kernels of Cauchy type on weakly pseudoconvex domains is the Kohn–Nirenberg phenomenon, i.e. nonexistence of supporting functions and local nonconvexifiability. In this paper we give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models.

Crack initiation in a stiffened plate

The fracture mechanics problem of crack initiation in a stiffened plate is considered. The crack nucleus is modeled by a prefracture zone with bonds between the crack faces, which is treated as a region of weakened interparticle bonds of the material. The boundary-value problem of the equilibrium of a stiffened plate with a crack nucleus reduces to a nonlinear singular integrodifferential equation with a Cauchy type kernel. The strains in the crack initiation zone are found by solving this equation. The case of the stress state of the plate with a periodic system of prefracture zones is considered.

Approximate solution of singular integro- differential equations in generalized Hölder spaces

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejer points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized Holder spaces.

The solution of a problem in contact fracture mechanics on the nucleation and development of a bridged crack in the hub of a friction pair

The contact deformation of the hub of a plunger pair is considered. It is assumed that, during the repeated reciprocating motion of the plunger, a crack is initiated and fracture of the materials of the elements of the contact pair occurs. The problem of the equilibrium of the hub of a friction pair with a crack nucleus reduces to solving a system of non-linear singular integrodifferential equations with a Cauchy-type kernel. The normal and shear forces in the zone where the crack originates are found from the solution of this system of equations. The condition for the appearance of a crack is formulated, taking account of the criterion of the limit traction of the bonds in the material. A problem for the plunger of a friction pair as applied to a borehole sucker rod pump is considered as an example. In conclusion, the case when there are several arbitrarily distributed rectilinear bridged cracks, with bonds between the crack faces in the end zone, close to the contact surface of the hub is investigated.

Cracks with bonding between the lips in bushings of friction couples

We study the problem of contact deformation of a bushing of friction couple. The friction surface of the bushing contains rectilinear cracks with end zones characterized by the presence of interaction between the crack lips. In the case of repeated reciprocal motion of a plunger, the material of the bushing finally fails as a result of contact interaction. The boundary-value problem of equilibrium of the bushing of friction couple with cracks whose lips are interacting is reduced to the solution of a system of nonlinear singular integro-differential equations with Cauchy-type kernel. The solution of this system is used to find the normal and tangential forces in the bonds. The condition of limiting equilibrium of a crack with end zone is formulated with regard for the criterion of ultimate length of the bonds in the material.

Thermoelastic Cracked Plates Carrying Nonstationary Electrical Current

ABSTRACT Problems related to the distribution of electromagnetic field and of the Joule heating along/around the cracks in a thermo- and electroconductive plate are addressed. It is assumed that the plate carries a nonstationary electrical current and contains two finite collinear cracks that are located perpendicular to the current direction. The cracks are free of mechanical loads. The formulated problem is reduced to a system of singular integral equations with Cauchy-type singular kernels and solved numerically. The influence of interaction of the cracks as well as non-stationarity of the current on the distribution of current and Joule heating are clarified. The results are instrumental toward arrest and de-acceleration of crack propagation in electro-conductive material structures.

Torsional impact response of a cylindrical interface crack between a functionally graded interlayer and a homogeneous cylinder

The torsional impact problem of a cylindrical interface crack between a functionally graded material (FGM) interlayer and its external homogeneous cylinder is investigated in this paper. The shear modulus and mass density of the FGM interlayer are assumed to vary continuously between those of the two coaxial dissimilar homogeneous elastic cylinders. The mixed boundary value problem is first reduced to a singular integral equation with a Cauchy type kernel in the Laplace domain by applying Laplace and Fourier integral transforms and introducing dislocation density function. The singular integral equation is then solved numerically and the dynamic stress intensity factor (DSIF) in the physical domain is also obtained by a numerical Laplace inversion technique. It is found that the DSIF rises rapidly to a peak and then oscillates with a decreasing magnification and tends to the corresponding static value, and that both crack geometry configuration and material gradient of the FGM interlayer have all important influences on the DSIF.

Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading

In this paper the problem of a cylindrical crack located in a functionally graded material (FGM) interlayer between two coaxial elastic dissimilar homogeneous cylinders and subjected to a torsional impact loading is considered. The shear modulus and the mass density of the FGM interlayer are assumed to vary continuously between those of the two coaxial cylinders. This mixed boundary value problem is first reduced to a singular integral equation with a Cauchy type kernel in the Laplace domain by applying Laplace and Fourier integral transforms. The singular integral equation is then solved numerically and the dynamic stress intensity factor (DSIF) is also obtained by a numerical Laplace inversion technique. The DSIF is found to rise rapidly to a peak and then reduce and tend to the static value almost without oscillation. The influences of the crack location, the FGM interlayer thickness and the relative magnitudes of the adjoining material properties are examined. It is found among others that, by increasing the FGM gradient, the DSIF can be greatly reduced.

Dynamic stress intensity factor of a cylindrical interface crack with a functionally graded interlayer

This paper presents the dynamic stress intensity factor of a cylindrical interface crack located between two coaxial dissimilar homogeneous cylinders that are bonded with a functionally graded interlayer and subjected to a torsional impact loading. The shear modulus and mass density of the functionally graded interlayer are taken to vary continuously between those of the two coaxial cylinders. The mixed boundary value problem involved is reduced to a singular integral equation with a Cauchy-type kernel in the Laplace domain by applying Laplace and Fourier integral transforms. The expressions for the dynamic stress intensity factor at the crack tips are derived in detail and the results are displayed after solving the singular integral equation numerically and performing numerical Laplace inversions. It is found that the relative magnitudes of the adjoining material properties, the graded interlayer thickness, and the crack length, all have significant effects on the dynamic stress intensity factor.

Contact Problem for Porous Elastic Half-Plane

The paper is concerned with a static contact problem about a rigid punch on the free surface of a linear porous elastic half-plane. With the use of the Fourier transform the problem is reduced to a singular integral equation holding over the contact zone. This integral representation permits consideration of the Flamant problem (a line load on the half-plane) to be explicitly reduced to some quadratures. It is shown that in the classical linear elasticity limit the main integral equation has a Cauchy-type kernel, so distribution of the contact pressure is like in the Sadowsky punch-problem. For arbitrary porosity a numerical co-location technique is applied that allows one to analyze in detail the distribution of the contact pressure versus porosity. Both in the Flamant and Sadowsky problems we demonstrate a higher compliance of the porous foundation, with respect to the classical linear elastic results.

Local Buckling of a Circular Interface Delamination Between a Layer and a Substrate With Finite Thickness

An analytical study investigating the local buckling response of a circular delamination along the interface of an elastic layer and a dissimilar substrate with finite thickness is presented. The solution method utilizes the stability equations of linear theory of elasticity under axisymmetry conditions. In-plane loading and the presence of mixed boundary conditions on the bond-plane result in a homogeneous system of coupled singular integral equations of the second kind with Cauchy-type kernels. Numerical solution of these integral equations leads to the determination of local buckling stress and its sensitivity to geometric parameters and material properties. [S0021-8936(00)01503-8]

Singular integral equations of the second kind with generalized Cauchy-type kernels and variable coefficients

A numerical solution method is presented for singular integral equations of the second kind with a generalized Cauchy kernel and variable coefficients. The solution is constructed in the form of a product of regular and weight functions. The weight function possesses complex singularities at the ends of the interval. The parameters defining the power of these singularities are obtained by solving for the characteristic equations. A Gauss–Chebychev quadrature formula is utilized in the numerical solution of the integral equations. Benchmark examples are considered in order to illustrate the validity of the solution method. Copyright © 1999 John Wiley & Sons, Ltd.

Composition Profile for Improving the Brittle Fracture Characteristics in Semi-Infinite Functionally Graded Materials.

In this study, we analyze the composition profile of semi-infinite functionally graded materials(FGMs) with a view to improving the brittle fracture characteristics of these materials. First, a method is developed to calculate the stress intensity factor of an edge crack in semi-infinite homogeneous media with distributed eigenstrain under a far-field uniform applied load. The crack is represented by the distribution of edge dislocations, and a singular integral equation with a Cauchy-type kernel is obtained using the complex potential functions of the edge dislocations. Second, a useful approximation method is established to simulate the nonhomogeneity of semi-infinite FGMs by an equivalent eigenstrain. The stress intensity factor is calculated for this equivalent eigenstrain. Finally, the stress intensity factor of the original semi-infinite FGMs is obtained by principle of superposition. By equating the stress intensity factor to the intrinsic fracture toughness of the FGMs, the apparent fracture toughness is calculated for prescribed composition profiles of the semi-infinite FGMs. Conversely, the composition profiles of prescribed apparent fracture toughness are also determined.

Application of Cauchy integrals and singular integral equations in scattered data problems

A new approach is suggested for constructing “smooth” surfaces which contain discontinuities in functional values or derivatives at prescribed locations. The approach is based on solving singular integral equations with Cauchy-type kernels. It is applicable in the interpolation or approximation of scattered data. We investigate, in particular, several two-dimensional biharmonic and triharmonic problems which have smooth solutions except on given line segments, across which different types of discontinuities occur. We also discuss some issues concerning the application of the approach in practical problems.

Ancillary solutions for curved crack stress analysis

A method to generate a weight function for use in the stress analysis of arbitrarily shaped cracks with arbitrary boundary conditions and mixed mode loading is presented. The method only involves the numerical resolution of a singular integral equation with a Cauchy–type kernel whose singularity is straight forward to remove. The generation of the weight function is done through the superposition of a fundamental solution and a complementary solution. The fundamental solution obeys the complex Bueckner formula asymptotically and has a finite traction potential along the curved crack. The complementary solution is made up of a distributed load plus a simple crack solution such that the weight function is stress free along the crack and has the right boundary conditions. Only the simple crack solution needs to be solved using the numerical resolution of a singular integral equation with a Cauchy–type kernel. A numerical example is added to illustrate the method.