System Of Dual Integral Equations Research Articles

Dynamic Response of a Piezoelectric Material with a Conducting Rigid Inclusion

The problem of a conducting rigid inclusion embedded in an infinite piezoelectric matrix is considered under the action of combined electromechanical impact loads. By using integral transform techniques, the mixed initial-boundary value problem for the case of anti-plane shear load and in-plane electric field is transformed into two systems of dual integral equations, the solutions of which give the singularity coefficients of electroelastic field near the inclusion tips in closed-form in the Laplace transform domain. Numerical results for the stress singularity coefficient in the physical space are presented graphically by numerically solving the resulting Fredholm integral equation and carrying out the numerical inversion of Laplace transform for a PZT-5H material with a conducting rigid line inclusion.

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.

Flowing partially penetrating well: solution to a mixed-type boundary value problem

A new semi-analytic solution to the mixed-type boundary value problem for a flowing partially penetrating well with infinitesimal skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-infinite vertical extent or to packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res. 1978; 14:929–34], our DE solution does not suffer from numerical oscillations. The new solution is validated by matching the corresponding finite-difference solution and is computationally much more efficient. An automated (Newton–Raphson) parameter identification algorithm is proposed for field test inversion, utilizing the DE solution for the forward model. The procedure is computationally efficient and converges to correct parameter values. A solution for the partially penetrating flowing well with no skin and a drawdown–drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993; 30:600–6], is compared to the DE solution. The D–D solution leads to physically inconsistent infinite total flow rate to the well, when no skin effect is considered. The DE solution, on the other hand, produces accurate results.

On crack problem in an elastic ponderable layer

The paper deals with the plane problem of stress distribution in an elastic ponderable layer with a stationary edge crack normal to the boundary plane. The layer is situated and fixed on a rigid foundation. The stresses are caused by action of body forces. By using the method of Fourier transforms the problem is reduced to a system of dual integral equations and next, to a Fredholm integral equation of the second kind. The numerical analysis of the Fredholm equation permitted to determine the stress intensity factor and the crack opening displacement.

Anti-plane shear crack growth rate of piezoelectric ceramic body with finite width

The theory of linear piezoelectricity is applied to develop an anti-plane crack growth rate equation of a finite crack in a piezoelectric ceramic body with finite width. Plastic zone is assumed to be confined to a sheet ahead of both crack edges similar to the strip model for in-plane loading. The procedure consists of reducing a system of dual integral equations to a Fredholm integral equation of the second kind. The accumulated plastic displacement criterion is used for developing a solution for the crack growth rate. Numerical values of crack growth rate are obtained and the results are displayed graphically to exhibit the electroelastic interactions.

Dynamic response of a crack in a functionally graded material between two dissimilar half planes under anti-plane shear impact load

This article provides a theoretical and numerical treatment of a finite crack subjected to an anti-plane shear impact load in a media with spatially varying elastic properties. This variation is in a direction perpendicular to the crack surfaces. The elastic properties of the interfacial layer are assumed to vary continuously between that of two dissimilar homogeneous bonding layers. Laplace and Fourier transforms are applied to reduce this mixed boundary value problem to a system of dual integral equations which in turn will be reduced to a standard Fredholm integral equation of the second kind. The Fredholm integral equation is solved in the Laplace transform plane numerically. The time inversion is accomplished by a numerical scheme. The dynamic stress intensity factor is found to either increase or decrease with the crack length to the layer thickness depending on the relative magnitudes of the material properties of the adjoining layer.

Fracture in Functionally Gradient Materials Subjected to the Time-Dependent Anti-Plane Shear Load

In this article, a finite crack subjected to a general time-dependent anti-plane shear load in functionally gradient materials is studied. The elastic properties of the interfacial layer are assumed to vary continuously between those of two dissimilar homogeneous bonding layers. Laplace and Fourier transforms are applied to reduce this mixed boundary value problem to a system of dual integral equations which in turn will be reduced to a standard Fredholm integral equation of the second kind. The dynamic crack tip stress fields are determined analytically while the dynamic stress intensity factors are evaluated numerically from the standard Fredholm integral equation. The Laplace inversion is performed numerically by using Gauss-quadrature and Jacobi polynomials. In general, the dynamic stress intensity factor is found to be a function of the crack length, location of the crack in interfacial layer, and material properties of the surrounding layers as well as interfacial layer.

Solution of a static mixed boundary value problem

A wide class of electromagnetic problems can be expressed as a system of dual integral equations. These kinds of integral equations occur in boundary value problems wherein there is one integral equation for a certain region, and another for the rest of the region. In this paper it is shown that the equation for the potential distribution due to an insulating cylinder column with a band electrode in a conducting space, representing a well logging problem, can be put into a standard form of dual integral equations, which can then be transformed into a system of linear equation by means of a Neumann series. A general method to compute the coefficients of the linear system is discussed in order to obtain some approximate formulas and the plot of the conductance as a function of the ratio a/ Δ (radius/half-length of the cylinder column).

High-frequency surface currents induced on a spherical reflector by the edge-diffraction of obliquely incident waves

An explicit expression of the high-frequency surface current excited on a perfectly conducting spherical cap by the edge-diffraction of an obliquely incident wave is derived. The result is given in the GTD terminology and shows the influence of the incidence angles on the transfer functions. To this end the real (physical) space is embedded into a twofold extended abstract space in which an equivalent canonical problem is established. This latter yields a system of dual integral equations whose kernels were not previously treated. By inverting these new kernels, the system of dual equations is reduced to two independent Wiener-Hopf problems. From the asymptotic solutions of these problems, one derives the expression of the edge-excited current along with the known geometrical optics term. The resutlt can be used directly in practical applications. If the incident ray becomes normal to the edge, then the resulting expressions are reduced to the already known expressions. The basic idea and general formulas used here can also be used to solve other diffraction problems with similar geometry.

A new method to compute the capacitance of the circular patch resonator

Discusses an analytical solution for the capacitance of the circular microstrip patch resonator. Shows that the electrostatic problem can be formulated as a system of dual integral equations, which is reduced to a single Fredholm integral equation of the second kind with a continuous kernel. Discusses the solution of this new integral equation in a certain range of the parameters, and derives some useful formulas for the capacitance. Finally, gives plots of the capacitance in a wide range of the relative permittivity.

Capacitance of a cylindrical system

A wide class of electromagnetic problems can be expressed as a system of dual integral equations. These kinds of integral equations occur in boundary value problems wherein there is an integral equation for a certain region and another for the rest of the region. In this paper it is shown that the integral equation for the charge density on a hollow metallic cylinder of finite length enclosed in another cylinder of infinite length can be put into «a standard form» of dual integral equations, which can be transformed into a numerically well-posed system of linear equation by means of a Neumann series. A general method to compute the coefficients of the linear system is discussed and some plots of the charge density distributions, and of the capacitance as a function of the ratioh/r1 (half-length/radius of the cylinder) are given. The range of validity of the classical asymptotic expansion is finally discussed.

A new method of solution of Hallén’s problem

A wide class of electromagnetic scattering problems can be expressed as a system of dual integral equations. This kind of integral equation, occurring in boundary value problems wherein there is one equation for a certain region and another for the dual domain, is usual in diffraction problems. Considerable attention has been drawn by many researchers in the field of optics, acoustics, scattering of elastic waves, accelerator physics, and antenna theory. Wiener–Hopf techniques enable us to solve such kinds of integral equations when the two regions are contiguous and semi-infinite. Unfortunately Wiener–Hopf techniques do not apply if one of the regions is finite; this is the case for realistic scatterers, such as irises, antennas, and drift tubes in accelerators. In this article a general method for solving such dual integral equations is discussed and applied to a particular case: Hallén’s equation of cylindrical antennas. This method consists of a transformation of the dual integral equations into a Fredholm integral equation of the second kind and then into a linear system of algebraic equations. Comparisons with results obtained by other methods for a wide range of frequencies show the accuracy and the robustness of the proposed one.

Capacitance of a hollow cylinder

A wide class of electromagnetic problems can be expressed as a system of dual integral equations. These kinds of integral equations occur in boundary value problems wherein there is an integral equation for a certain region and another for the rest of the region. In this paper it is shown that the equation for the charge density on a hollow metallic cylinder can be put into ‘a standard form’ of dual integral equations, which can be transformed into a system of linear equation by means of a Neumann series. A general method to compute the coefficients of the linear system is discussed and the plot of the capacitance as a function of the ratioh/a (half-length/radius of the cylinder) is given. The range of validity of some classical approximate formulas is finally discussed.

The motion generated by a slowly rising disk in an unbounded rotating fluid for arbitrary Taylor number

The motion of a disk rising steadily parallel to the axis of rotation in a uniformly rotating unbounded liquid is considered. In the limit of zero Rossby number the linear viscous equations of motion are reduced to a system of dual integral equations which renders an ‘exact’ solution for arbitrary values of the Taylor number, Ta. The investigation is focused on the drag and the flow field. In the limits of small and large Ta the asymptotic results of the present formulation agree with – and extend – previous investigations by different approaches.A particular novel feature, for large Ta, is the contribution of the Ekman-layer flux to the outer motion. New insight into the structure of the Taylor column is gained; in particular, it is shown that the main part of the column is a ‘bubble’ of recirculating fluid, detached from the body and not communicating with the Ekman layer. However, it turns out that the essential discrepancy in drag between experiments (Maxworthy 1970) and previous theories cannot be attributed to the Ekman-layer suction effect.

The effect of internal pressure on a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half-spaces

In the linear theory of micropolar elasticity, the problem of a penny-shaped crack at the interface of two bonded dissimilar micropolar elastic half spaces is studied. The problem is first reduced to a system of dual integral equations which are further reduced to the solution of Riemann-Hilbert problem. Further stresses at the rims of cracks and in the vicinity have been evaluated.

Debonding of thin external layer exerted by thermodiffusive fluxes

The stress distributions in a matrix, generated by thermodiffusive fluxes, are determined for the case of a thin, inextensible membrane bonded to the bounding plane over the outer regionr ∈ (b>0, ∞). The problem has been reduced to solving a corresponding system of dual integral equations. There exist singularities in the distribution of contact shear stresses, namely one over square root type atr=b while a logarithmic one at the lines of jumps of temperature and/or mass diffusion.

Effect of Crack on Ridge of Plate Structure

The present study is concerned with the membrane and bending stresses induced in a folded plate in the presence of a crack at the ridge of the folded plate. The classical method of solving dual integral equations by converting them into singular integral equations is applied for determination of stresses in a folded plate structure which is weakened by a crack. The crack is located along the ridge of the folded plate structure whose behavior is governed by the coupled set of equations pertaining to bending and stretching deformations. The presence of the crack leads to the formation of a system of dual integral equations with Cauchy kernels. These are solved numerically by applying the Chebyshev integration formula in conjunction with certain collocation techniques.

Diffraction of shear waves by a rigid strip in a medium of monoclinic type

The paper discusses the two dimensional problem of diffraction of shear waves by a rigid strip in an infinite medium of monoclinic type. This problem is reduced to a system of dual integral equations of which the solution provides the diffracted field. The method of steepest descent has been used in the determination of the diffracted fields at a large distance from the strip. Diffraction pattern for displacement and stress field have been computed and the effect of anisotropy is distinctly marked.

Dynamic response of elliptical footings

Dynamic response of an elliptical footing in frictionless contact with a homogeneous elastic half-space is considered. Both vertical and horizontal vibrations are treated. In the case of the vertical vibration, the mixed boundary value problem gives rise to a set of dual integral equations. For the horizontal vibration, we have a system of dual integral equations. The dual integral equations which are two dimensional in nature are reduced to two-dimensional Fredholm integral equations of the first kind. They are then recast in a suitable form after separating out the static solution. Successive low-frequency terms are then obtained by utilising the static solution. The series solutions up to ω2, ω being the frequency, are obtained, and analytical results for the dynamic compliances are obtained. In the limiting case of a circular footing, our results are in agreement with those of previous authors.

Diffraction of magnetoelastic shear waves by a rigid strip

This paper discusses the two-dimensional problem of the diffraction of magnetoelastic shear waves by a rigid strip in an infinite elastic perfect conductor. This problem is reduced to a system of dual integral equations of which the solution provides the diffracted field. The method of steepest descent has been used in the determination of the diffracted fields at a large distance from the strip. Diffraction patterns for displacement and stress fields have been computed and the effect of the magnetic field is distinctly marked.