Cauchy Singular Integral Equations Research Articles

SOME CONSIDERATIONS ON NUMERICAL METHODS FOR CAUCHY SINGULAR INTEGRAL EQUATIONS ON THE REAL LINE

Two different direct methods are proposed to solve Cauchy singular integral equations on the real line. The aforementioned methods differ in order to be able to prove their convergence which depends on the smoothness of the known term function in the integral equation.

The size-dependent frictionless contact of piezoelectric materials

According to the couple-stress theory, the size-dependent frictionless contact problem between a transversely isotropic piezoelectric half-plane and a rigid conducting or insulating punch is examined in this paper. With the purpose of describing size effect appearing in material microstructures, the couple-stress theory introduces the characteristic length parameter in the constitutive equations. Using Fourier transform method, frictionless contact problems of flat and cylindrical punches are simplified to Cauchy singular integral equations. In order to determine the electric charge density and normal pressure, these equations are numerically solved by transforming into algebraic ones. The influences of size parameter and total electric charge on the electric charge density, in-plane electric displacement, in-plane stress, and contact pressure are predicted. Results for the surface electromechanical fields forecasted by the couple-stress theory depart significantly from the classical results.

On solving some Cauchy singular integral equations by de la Vallée Poussin filtered approximation

The paper deals with the numerical solution of Cauchy Singular Integral Equations based on some non standard polynomial quasi–projection of de la Vallée Poussin type. Such kind of approximation presents several advantages over classical Lagrange interpolation such as the uniform boundedness of the Lebesgue constants, the near–best order of uniform convergence to any continuous function, and a strong reduction of Gibbs phenomenon. These features will be inherited by the proposed numerical method which is stable and convergent, and provides a near-best polynomial approximation of the sought solution by solving a well conditioned linear system. The numerical tests confirm the theoretical error estimates and, in case of functions subject to Gibbs phenomenon, they show a better local approximation compared with analogous Lagrange projection methods.

On the imperfect interface of a functionally graded thermoelectric layered structure

This paper investigates the frictionless contact problem of a functionally graded thermoelectric (FGTE) layered half-plane under the action of a rigid flat punch. The punch is an electrical and thermal conductor. The electro-thermo-elastic properties of the FGTE materials vary exponentially with thickness. Especially, the imperfect interface conditions are taken into account. The problem is reduced to a pair of Cauchy singular integral equations using the Fourier integral transform technique, which are numerically solved to yield the surface normal electric current density, the surface normal energy flux, and the surface normal contact pressure. The asymptotic analysis of the kernel function is supplied. The intensity factors are provided to characterize the singularity at the punch ends. The contact behavior of the FGTE coating-TE substrate systems is significantly influenced by the imperfect interface condition and the material property gradient according to numerical results. The design of thermoelectric coating structures may benefit from the theoretical research done here.

Error analysis of a residual-based Galerkin’s method for a system of Cauchy singular integral equations with vanishing endpoint conditions

In this paper, we develop Galerkin’s residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind, where N is the total number of Cauchy-type singular integral equations in the system. Without theoretical analysis, a numerical scheme is not justified. Therefore, first, we prove the well-posedness of the system of Cauchy-type singular integral equations with the help of the compactness of an operator. Further, we derive a theoretical error bound and the order of convergence. Also, we show that the resulting system of equations obtained by applying the algorithm is well-posed together with an explicit representation of the solution in matrix form. Finally, we give some illustrative examples to validate the theoretical error bounds numerically.

Scattering of Love wave from an interface crack in a viscoelastic waveguide layer bonded to a piezoelectric substrate: an analytical estimate and numerical validation

This study uses an analytical method to examine the scattering of a Love wave from an interfacial crack in a thin lossy media (viscoelastic material) that serves as a waveguide layer attached to an unbounded loss-less media (piezoelectric material). The viscoelastic material is modeled using the Kelvin–Voigt equation. Phase velocity and attenuation have been calculated by equating the real and imaginary parts of the dispersion relation to zero, respectively, due to the complex wave number of viscoelastic material. Using the integral transform approach and the dislocation density function, the scattering field close to the crack tip is determined, resulting in a Cauchy singular integral equation of the first type. It has been transformed into a system of linear algebraic equations using Gauss-Chebyshev numerical integration. It has been investigated how frequency, waveguide layer thickness, and viscoelastic material loss factor affect phase velocity and Love wave attenuation. The mode-III dynamic stress intensity factor at the left and right crack-tip is investigated for various thicknesses of the waveguide layer and loss factor of viscoelastic material. The outcomes were verified using numerical data from the COMSOL Multiphysics 5.6 FEA software. The findings are fundamental and could be used in the design of particular sensors.

Mechanical modelling the slider‐substrate system with account of sliding

AbstractSliding onset is critical to interface failure or earthquake prediction, yet the physical mechanism is still unclear. In the present work, a model for the slender slider‐substrate system has been proposed. To account for sliding, nucleation of an interfacial dislocation and its subsequent mobility have been analyzed which may help clarify the sliding onset. The solution of this model can be simplified to that of Cauchy singular integral equations. Before the interfacial dislocation nucleation, numerical results thus obtained for interfacial tractions have been demonstrated to be in good consistency with experiments. By introducing a critical criterion for static dislocation nucleation, it has been found that the calculated critical force agrees well with that for sliding precursors and the mobility of the dislocation capture some essential features for the experimentally observed behaviours for sliding precursors. These results may cast new light to the occurrence and understanding of sliding onset.

Several cracks in an orthotropic hollow cylinder coated by an MEE layer under torsion

This paper performs such fracture analysis for an orthotropic hollow cylinder coated by a magneto-electro-elastic (MEE) layer under torsion. The orthotropic hollow cylinder contains several arbitrarily shaped cracks. A uniform electric displacement and magnetic induction are applied on the outer boundary of the MEE coating. At first, the displacement and stress components of the coated hollow cylinder weakened by a Volterra-type screw dislocation are found as a function of the dislocation density through Fourier transform. Next, distributed dislocation technique (DDT) is used to simplify the stress components corresponding to the screw dislocation to a set of Cauchy singular integral equations. The singular integral equations are subsequently solved through a numerical procedure to obtain the unknown dislocation density. Next, the stress intensity factors (SIFs) and the torsional rigidity in the cracked hollow cylinder with its MEE coating are found in terms of the dislocation density function. Several numerical results of the SIFs and torsional rigidity are provided to reveal the effects of the geometrical and physical parameters on fracture behavior. It is concluded that the SIFs can be minimized using suitable values of the electric displacement and magnetic induction.

Multi-physics electrical contact analysis considering the electrical resistance and Joule heating

A multi-physics electrical contact model is developed to predict the contact pressure, temperature, and electric current density (ECD) distributions on the contact surface. The theoretical model takes the interfacial electrical resistance and its resultant Joule heating into account. The contact pressure and contact radius are calculated by solving the first kind of Cauchy singular integral equation. The temperature and ECD are obtained based on the obtained contact pressure. Then, a three-dimensional finite element model is established to simulate the mechanical, thermal, and electrical behaviors of electrical contact. The finite element model can consider both the interfacial electrical resistance and the constriction electrical resistance. The theoretical results of the electrical contact behaviors are compared with the numerical results. The influences of the electrical resistance, normal force, electric potential, and distribution form of the heat flux on the contact pressure, temperature, and ECD distributions are investigated. The results indicate that the proposed multi-physics contact model can provide an accurate prediction of electrical contact behaviors.

Numerical solution of Cauchy singular integral equations of the first kind with index V=1

A method based on Gauss-Chebyshev quadrature and barycentric interpolation is used to obtain the numerical solution of Cauchy singular integral equations of the first kind with index equal to 1 at non-Chebyshev nodes. The unknown function in the equation is first expressed as a product of an appropriate weight function and a truncated weighted series of Chebyshev polynomial of the first kind. Some properties of Chebyshev polynomials are then used to reduce the equation to a system of linear equations. On solving the linear system, the numerical solution of the Cauchy singular integral equation is obtained at Chebyshev nodes, after which barycentric interpolation is used to obtain the numerical solution at non-Chebyshev nodes. When the numerical solution obtained is compared with the analytical solution and the absolute error computed, the results are found to be satisfactory.

The dynamic contact of a viscoelastic coated half-plane under a rigid flat punch

This paper investigates the dynamic contact response of a viscoelastic coated half-plane under a forced rigid punch. This punch, acting on the coating surface, is subjected to a concentrated vertical harmonic force. The coating and half-plane are assumed as viscoelastic materials which can be described by the linear hysteretic damping model. Using the Fourier integral transform technique and the asymptotic method, the dynamic contact problem is reduced to a Cauchy singular integral equation. By using appropriate numerical method, the dynamic contact stress and vertical impedance for the flat punch are calculated. It is found that the dynamic contact behaviors are not only affected by the external excitation frequency but also dependent on the material properties of the coated half-plane.

A finite Zener-Stroh crack interacting with a blunt crack under in-plane deformations

A Zener-Stroh crack is the name given to a microcrack resulting from the coalescence of edge dislocations along a slip plane. The process by which the crack is initiated was proposed in the first instance by Zener and later examined in detail by Stroh. The Zener-Stroh crack can be thought of as being equivalent, in some sense, to the Griffith crack in classical linear elastic fracture mechanics. We study a finite mode I and II Zener-Stroh crack interacting with a nearby mode I and II blunt crack represented by a parabolic cavity. In this respect, the Zener-Stroh crack can be considered as a coalescence of edge dislocations emitted from the blunt crack. A conformal mapping function is introduced which maps the parabolic boundary onto a straight line in the image plane. By treating the Zener-Stroh crack as a pileup of edge dislocations, two Cauchy singular integral equations are constructed in the image plane. The Gauss-Chebyshev integration formula is applied to numerically solve the resulting singular integral equations. The local mode I and II stress intensity factors at the two tips of the Zener-Stroh crack are determined. We establish conditions on the propagation direction of the Zener-Stroh crack away from or toward the blunt crack. Also established are the conditions for the emission of an edge dislocation from the blunt crack by treating the edge dislocation as a Zener-Stroh crack with infinitesimal length. The dislocation emission criteria from the blunt crack are established here in elementary closed form and will thus be particularly convenient for application in engineering practice.

Numerical solution of Cauchy singular integral equations of the first kind with index V=1

A method based on Gauss-Chebyshev quadrature and barycentric interpolation is used to obtain the numerical solution of Cauchy singular integral equations of the first kind with index equal to 1 at non-Chebyshev nodes. The unknown function in the equation is first expressed as a product of an appropriate weight function and a truncated weighted series of Chebyshev polynomial of the first kind. Some properties of Chebyshev polynomials are then used to reduce the equation to a system of linear equations. On solving the linear system, the numerical solution of the Cauchy singular integral equation is obtained at Chebyshev nodes, after which barycentric interpolation is used to obtain the numerical solution at non-Chebyshev nodes. When the numerical solution obtained is compared with the analytical solution and the absolute error computed, the results are found to be satisfactory.

A finite crack in a half-plane under uniform heat flux or surface heat source

A novel and effective method is proposed to determine the temperature and thermal stresses in the case of a finite Griffith crack lying perpendicular to the surface of an isotropic half-plane under uniform remote heat flux or a surface heat source. The surface of the half-plane and the two faces of the crack are otherwise thermally insulating and traction-free. For the heat conduction problem, the thermally insulating crack is simulated by a continuous distribution of heat dipoles, and the resulting Cauchy singular integral equation (SIE) is solved numerically to arrive at the associated density function. The original thermoelastic problem can be treated equivalently as a mode II Zener-Stroh crack (ZSC) under isothermal conditions. The net dislocation of ZSC is determined by the aforementioned density function and the crack is modeled by a pileup of edge dislocations. The resulting SIE is solved numerically leading to the mode II stress intensity factors at the two crack tips induced by uniform remote heat flux or surface heat source. The net dislocation of an actual Zener-Stroh crack can be designed in such a way that the crack will become neutral to the uniform heat flux or surface heat source.

In-plane stress analysis in a cracked functionally graded piezoelectric plane

The fracture behavior of a cracked functionally graded piezoelectric plane was investigated under in-plane mechanical and electrical loading. The electro-elastic properties of the plane exponentially varied along the y-axis. Employing the Fourier transform, stress analysis containing a Volterra edge dislocation in a functionally graded piezoelectric plane was carried out. These solutions were used to derive a system of Cauchy singular integral equations for analyzing the straight and curved cracks. Linear, arc, parabolic and sine wave-shaped cracks were evaluated. The integral equations were numerically solved using the Lobatto–Chebyshev integration formula for dislocation density functions on the crack face which were employed to calculate field intensity factors. The effects of the shaped crack, loading parameters, and non-homogeneity parameter on the stress and electric intensity factors were considered.

The Cauchy method of analytical regularisation in the modelling of plane wave scattering by a flat pre‐fractal system of impedance strips

The Cauchy method of analytical regularisation, first in full explanation, is used for the modelling of the plane wave scattering by a flat pre-fractal system of impedance strips. It is based on a modification of the singular integral equation technique and the correct type of inversion formula of the simplest Cauchy singular integral equation. The approach is effective for the plane H-polarised electromagnetic (or sonic) wave scattering problems. The principle of strict mathematical orderings of the pre-fractal strip grating deals with a five-parts class of perfect fractal sets, which is characterised by a variable positive fractal dimension. Considerable attention is devoted to the asymptotic model of the plane wave scattering by a sparsely filled grating, which has an explicit solution. This result is very important for verification of the mathematical models presented and testing of direct numerical methods. The scattered electromagnetic field in the far zone is successfully analysed with the aid of the asymptotic solution derived for electrically narrow strips.

Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

Elastic cloaking for a periodic distribution of parallel finite cracks

We achieve elastic cloaking for a periodic distribution of an infinite number of parallel finite mode III cracks by means of the complex variable method and the theory of Cauchy singular integral equations. The cloaking bimaterial structure is composed of an undisturbed uniformly stressed left half-plane perfectly bonded via a wavy interface to the right half-plane which contains periodic cracks. The original design of the wavy interface and the positions of the periodic cracks are ultimately reduced to the solution of a Cauchy singular integral equation which can be solved numerically.

Axisymmetric thermoelastic contact of an FGM-coated half-space under a rotating punch

This paper investigates the axisymmetric thermoelastic contact of a functionally graded material (FGM) coated half-space. A rigid insulated punch rotates on the surface of the FGM coating with a constant angular velocity. The frictional heating is generated within the contact region. The thermoelastic properties of the coating vary exponentially along the thickness direction. By using the Hankel integral transform, the problem is reduced to Cauchy singular integral equations, which are solved numerically to obtain the normal contact stress, radial stress and surface temperature. The effects of the friction coefficient, angular velocity, gradient index, coating thickness and punch geometry on the surface thermoelastic contact characteristics are investigated and discussed in detail.

In-plane stress analysis of multiple parallel cracks in an orthotropic FGM medium under time-harmonic loading

This study aimed to investigate the fracture behavior of a cracked orthotropic FGM medium when exposed to an in-plane harmonic load. The mechanical feature of the medium exponentially varied along the y-axis. Using the Fourier transform method, stress was analyzed by assessing Volterra edge dislocation in orthotropic FGM media. The obtained solution was then applied to derive Cauchy singular integral equations to analyze multiple cracks. The mentioned equations were numerically solved to determine dislocation density functions on the crack face which were then used to evaluate dynamic mixed stress intensity factors. The influence of crack interaction, non-homogeneity, and load frequency on the dynamic stress intensity was numerically illustrated through examples.