Cauchy Type Kernel Research Articles

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

LONGITUDINAL SHIFT OF THE BINDER AND INCLUSIONS OF A COMPOSITE WEAKENED BY TWO-PERIODIC RIGHT LINEAR CRACKS

In this paper, we consider antiplane deformation for an isotropic elastic material consisting of an infinite system of parallel identical circular cylindrical fibers covered with a uniform cylindrical film uniformly covering the surface of each fiber and a bonding medium weakened by a doubly periodic system of rectilinear cracks. Each washer has a centrally located crack that is less than the diameter of the washer. The presented stresses and their displacements are expressed in terms of an analytical function. For the solution, the well-known position is used that the displacement in the case of an antiplane shear is a harmonic function. A known representation of the solution in each area is applied through the corresponding complex analytical function. Three analytic functions are represented by Laurent series. Satisfying the boundary condition on the contours of holes and crack faces, the problem is reduced to two infinite algebraic systems with respect to the sought coefficients and to two singular integral equations with a Cauchy-type kernel. Then the singular integral equation is reduced to a finite algebraic system of equations by the Multopp – Kalandia method. The procedure for calculating the stress intensity coefficients is given. The numerical implementation of the described method is given at IBM. The results of calculations of the critical load depending on the crack length and elastic geometric parameters of the perforated medium are presented. Keywords: isotropically elastic material, doubly periodic lattice, rectilinear cracks, stress intensity factor, mean stresses, critical load, circular hole, longitudinal shear.

Global Plemelj Formula of Slice Dirac Operator in Octonions with Complex Spine

Slice analysis is generalized to the setting of octonions with new book structure whose spine consists of a complex plane instead of the real axis. The theory leads to the study of the slice Dirac operator. The associated Cauchy type kernel turns out to be operator-valued rather than function-valued, in contrast to the associative case. The global Plemelj jump formula is established on the boundary of any axially symmetric domain of octonions.

A study of fracture of a fibrous composite

We develop design model within which nucleation and propagation of crack in a fibrous composite is described. It is assumed that under loading, crack initiation and fracture of material happens in the composite. The problem of equilibrium of a composite with embryonic crack is reduced to the solution of the system of nonlinear singular integral equations with the Cauchy type kernel. Normal and tangential forces in the crack nucleation zone are determined from the solution of this system of equations. The crack appearance conditions in the composite are formed with regard to criterion of ultimate stretching of the material\'s bonds. We study the case when near the fiber, the binder has several arbitrary arranged rectilinear prefracture zones and a crack with interfacial bonds. The proposed computational model allows one to obtain the size and location of the zones of damages (prefracture zones) depending on geometric and mechanical characteristics of the fibrous composite and applied external load. Based on the suggested design model that takes into account the existence of damages (the zones of weakened interparticle bonds of the material) and cracks with end zones in the composite, we worked out a method for calculating the parameters of the composite, at which crack nucleation and crack growth occurs.

Effect of induced temperature field on development of curvilinear crack with bonds between the faces in end zones

Temperature changes near the end of a curvilinear cohesive crack and their influence on crack growth are investigated. The problem of local temperature changes consists in a delay or retardation of the cohesive crack growth. The bonds between the curvilinear crack faces in the end zones are modeled by application to the crack surface cohesive forces caused by the presence of bonds. The boundary value problem of equilibrium of the curvilinear crack with interfacial bonds in the end zones under action of external tensile loads, induced temperature field and tractions in the bonds preventing to its opening, is reduced to a system of singular integral equations with a Cauchy-type kernel. From the solution of this equation system, normal and tangential tractions in the bonds are found. Analysis of the limit equilibrium of the crack using the end zone model is performed on the basis of a criterion of bonds limiting stretching and includes: 1) establishment of tractions depending on opening of the crack faces; 2) evaluation of the stress state near the curvilinear crack with taking into account tensile loads, induced temperature field, tractions in the bonds; 3) determination of the critical external tensile loads.

On some subclasses of hypergeometric functions with Djrbashian Cauchy type kernel

On some subclasses of hypergeometric functions with Djrbashian Cauchy type kernel

Torsion analysis of a hollow cylinder with an orthotropic coating weakened by multiple cracks

This paper presents an analytical solution for an infinite hollow cylinder with an orthotropic coating containing multiple cracks under Saint-Venant torsion by means of the distribution dislocation technique. First, the solution of a hollow cylinder with an orthotropic coating weakened by a Volterra-type screw dislocation is achieved with the aid of the finite Fourier sine transform. Next, the problem is reduced to a set of singular integral equations with a Cauchy type kernel in the hollow cylinder by using distribution dislocation technique. The singular integral equations are then solved numerically and the stress intensity factors of the crack tips and torsional rigidity in the whole domain are also obtained. Finally, several examples are presented to show the accuracy and efficiency of the dislocation technique in Saint-Venant torsion problems.

Mode III SIFs for interface cracks in an FGM coating-substrate system

In this study, interaction of several interface cracks located between a functionally graded material (FGM) layer and an elastic layer under anti-plane deformation based on the distributed dislocation technique (DDT) is analyzed. The variation of the shear modulus of the functionally graded coating is modeled by an exponential and linear function along the thickness of the layer. The complex Fourier transform is applied to governing equation to derive a system of singular integral equations with Cauchy type kernel. These equations are solved by a numerical method to obtain the stress intensity factors (SIFs) at the crack tips. The effects of non-homogeneity parameters for exponentially and linearly form of shear modulus, the thickness of the layers and the length of crack on the SIFs for several interface cracks are investigated. The results reveal that the magnitude of SIFs decrease with increasing of FG parameter and thickness of FGM layer. The values of SIFs for FGM layer with exponential form is less than the linear form.

Inhibition of a curvilinear bridged crack by induced thermoelastic stress field

ABSTRACTTemperature changes near the end of a curvilinear crack in the presence of end areas with cohesive forces of the material are considered. It is assumed that these areas adjoin to the crack tip and their sizes are comparable with the crack size. The goal of local changes of temperature is to delay or inhibit the growth of a curvilinear crack. A boundary value problem on equilibrium of a bridged curvilinear crack under the action of external tensile loads, induced thermoelastic temperature field, and tractions in the bonds preventing its disclosure is reduced to the system of nonlinear singular differential equations with a Cauchy-type kernel. Normal and tangential tractions in the bonds are found from the solution of this system. The stress intensity factors are calculated. The condition of limit equilibrium of a curvilinear crack with an end area is formed based on two-parameter fracture criterion.

Applications of the Clifford algebra valued boundary element method to electromagnetic scattering problems

Electromagnetic problems governed by Maxwell's equations are solved by using a Clifford algebra valued boundary element method (BEM). The well-known Maxwell's equations consist of eight pieces of scalar partial differential equations of the first order. They can be rewritten in terms of the language of Clifford analysis as a nonhomogeneous k-Dirac equation with a Clifford algebra valued function. It includes three-component electric fields and three-component magnetic fields. Furthermore, we derive Clifford algebra valued boundary integral equations (BIEs) with Cauchy-type kernels and then develop a Clifford algebra valued BEM to solve electromagnetic scattering problems. To deal with the problem of the Cauchy principal value, we use a simple Clifford algebra valued k-monogenic function to exactly evaluate the Cauchy principal value. Free of calculating the solid angle for the boundary point is gained. The remaining boundary integral is easily calculated by using a numerical quadrature except the part of Cauchy principal value. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. In the numerical implementation, we introduce an oriented surface element instead of the unit outward normal vector and the ordinary surface element. In addition, we adopt the Dirac matrices to express the bases of Clifford algebra Cl3(ℂ). We also use an orthogonal matrix to transform global boundary densities into local boundary densities for satisfying boundary condition straightforward. Finally, two electromagnetic scattering problems with a perfect spherical conductor and a prolate spheroidal conductor are both considered to examine the validity of the Clifford algebra valued BEM with Cauchy-type kernels.

A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel

The main objective of this paper is to give an efficient numerical method for the solution of the second kind Fredholm integral equation with Cauchy type kernel. Although, numerical treatment of Singular integral equations (SIEs) has been considered by many researchers and many numerical methods have been proposed for the solution of this equation, strong singularity of the Cauchy singular integral operator still remains as a challenge to numerical methods. Most of the previous methods rely on specific quadrature rule or suitable base functions for capturing the singularity. Here we focus on operator transformation and graded mesh. In addition, we study error analysis. In this regard, efficiency of reproducing kernel Hilbert space (RKHS) method using smooth transformation on the graded mesh is improved and compared with some other numerical methods.

Elastic Equilibrium of a Space Containing a Thin Curved Elastic Inclusion in Longitudinal Shear

We study the problem of antiplane deformation of an isotropic medium containing a thin elastic curved inhomogeneity. The methods used for the solution of this problem are based on the application of the method of jump functions and the conditions of interaction of the matrix containing a thin curvilinear inclusion and the solution of the resultant system of singular integral equations with Cauchy-type kernels by the collocation method. Numerous examples are considered. The results of evaluation of the stress intensity factors for a crack and an absolutely rigid inclusion along a circular arc are compared with the corresponding analytic results. For a crack along a symmetric parabolic arc, the stressed state is thoroughly investigated. We also study the influence of the modulus of elasticity and the shape of the curvature of inhomogeneity (circular arc, parabola, or a half of a cosine curve) on the generalized stress intensity factors. It is shown that, for the stress intensity factors near the tip of the inhomogeneity, the inclination of the tangent at the tip to the plane of application of the shear forces is of determining importance.

Solution of the Periodic Problem of Propagation of Cohesive Cracks Under Longitudinal Shear

UDC 539.375 We consider a punched body weakened by surface cohesive cracks under the conditions of longitudinal shear. The solution of the problem of equilibrium of this body is reduced to the solution of an infinite algebraic system of equations and one singular integral equation with Cauchy-type kernel.

Numerical Solution of the Singular Integral Equations of the First Kind on the Curve

In this work we present a numerical solution for singular integral equations of the first kind on the oriented smooth contour with Cauchy type kernel. For this one we use an adapted quadratic approximation constructed by the author for this goal, based on the Simpson rule. Many examples are treated in order to prove the efficiency of this approximation. AMS Subject Classification (2000). 45D05, 45E05, 45L05, 45L10 and 65R20

Solution of second kind integral equation with Cauchy type kernel using Daubechies scale function

A method based on the Daubechies scale function has been devised to solve numerically a class of second kind integral equations with a Cauchy type kernel. The convergence associated to the method has been established. The method is illustrated with integral equations in this class whose exact solutions are known. The numerical results obtained by the present method are seen to differ negligibly from the exact results.

Optimal and Approximate Solutions of Singular Integral Equations by Means of Reproducing Kernels

A reproducing kernel method is proposed to obtain the optimal and approximate solutions of Carleman singular integral equations. Therefore, we will be mostly interested in singular integral equations with a Cauchy type kernel and whose coefficients are real or complex valued functions. The new method and corresponding concepts allow the analysis of associated discrete singular integral equations and corresponding inverse source problems in appropriate frameworks.

Nucleation of cracks in a perforated heat-releasing material with temperature-dependent elastic properties

A mathematical model of crack nucleation in a perforated heat-releasing material attenuated by a biperiodic system of cooling cylindrical channels with a circular cross section is constructed. Solving the problem of equilibrium of an isotropic perforated heat-releasing material with temperature-dependent properties containing nucleating cracks is reduced to solving systems of algebraic and nonlinear singular integral equations with a Cauchy-type kernel. The forces in crack nucleation regions are found by using the solution of these equations. The condition of crack emergence is formulated with allowance for the criterion of ultimate stretching of bonds in the material.

Cracks with interfacial bonds 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, fracture of the materials of the elements of the contact pair occurs. The friction surface of the bushing contains rectilinear cracks with the end zones characterized by the presence of interaction between the cracks faces. The boundary-value problem of equilibrium of the hub of a friction pair with a crack reduced to solving system of nonlinear singular integro-differential equations with a Cauchy type kernel.

Adapted linear approximation for singular integrals

Purpose The purpose of this work is to present an approximation for singular integrals of Cauchy type kernel on a smooth-oriented contour.

Crack nucleation in a strip of varying thickness under force loading

A mathematical model of crack nucleation in a strip of varying thickness under force loading is constructed. It is assumed that the loading of the strip by external forces gives rise to prefracture zones, which are modeled as regions of weakened interparticle bonds in the material. Solution of the problem of the equilibrium of an isotropic strip of varying thickness, with a crack nucleus reduces to solving a system of nonlinear singular integral equations with a Cauchy type kernel to determine the forces in the region of crack nucleation. The condition of crack nucleation in a strip of varying thickness is formulated using the criterion of the limiting stretching of material bonds.