System Of Dual Integral Equations Research Articles

The problem of axisymmetric torsion of a rigid disk parallel to an internally penny crack and interface in a bilayer medium

The aim of this work was to investigate an axisymmetric problem involving a penny-shaped crack embedded in a bi-material with finite thickness under torsion. This torsion is caused by a circular rigid disk attached to the upper layer and fixed by an undeformable support at the lower layer. The location of the crack can be found in the upper layer (case 1) or the lower layer (case 2), both cases are considered. The problem was solved by applying the Hankel transform to reduce it to a system of dual integral equations, which further simplified into Fredholm integral equations of the second-kind, These equations were then solved numerically using the Gaussian quadrature rule. Dimensionless plots were presented to demonstrate the influence of layers thickness, the distance between the support and the crack, and the shear modulus ratio between layers on the stress intensity factor of the crack. The obtained results illustrated the agreement between the analytical solution and the numerical results.

Hot indentation of a FGM-coated thermoelastic half-space by a conical punch: Approximated analytical solution of the contact problem

Hot indentation of a thermoelastic coated half-space by a rigid punch is considered. The coating is assumed to be functionally-graded with all group of properties arbitrary varying with depth. The deformation of the half-space caused by the simultaneous mechanical and thermal loading is considered. The punch is subjected to a normal centrally applied force and preliminary heated to a uniformly distributed temperature. The contact problem is reduced to the solution of a system of dual integral equations over two unknown functions: Hankel transform of normal stresses and heat flux in the contact area. Heat flux is obtained using the bilateral asymptotic method. This method was sufficiently generalized to take into account additional function describing the displacements of the surface caused by thermal heating and obtain contact stresses. The solution is obtained in an analytical form which clearly demonstrates the dependence of the contact characteristics from the initial physical parameters of the problem. Numerical analysis is provided to illustrate the effect of the presence of the coating and its thickness on the results. Differences between the homogeneous and functionally-graded coatings is illustrated.

Axisymmetric frictionless contact between an elastic layer thickness and a circular base under a rigid punch

The study presented in this work deals with analytical methods for an axisymmetric problem of an elastic layer partially reposing on a rigid circular base, and is indented along the upper surface with a rigid punch. The contact between the medium and the base is smooth. This boundary value problem is transformed into a system of dual integral equations. In contrast to the classical approach consisting in resolving the corresponding Fredholm equation of the second kind, the latter equations are obtained from an infinite algebraic system of simultaneous equations, where the particular case [Formula: see text] is verified. The results of this system are also obtained numerically. The normal displacement, normal stress, and the stress singularity factor are given analytically and shown graphically with discussion. By comparison with those predicted by the finite element method, the accuracy of the numerical method is approved.

Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface

Breaking spatial symmetry is an essential requirement for phoretic active particles to swim at low Reynolds number. This fundamental prerequisite for swimming at the micro scale is fulfilled either by chemical patterning of the surface of active particles or alternatively by exploiting geometrical asymmetries to induce chemical gradients and achieve self-propulsion. In the present paper, a far-field analytical model is employed to quantify the leading-order contribution to the induced phoretic velocity of a chemically homogeneous isotropic active colloid near a finite-sized disk of circular shape resting on an interface separating two immiscible viscous incompressible Newtonian fluids. To this aim, the solution of the phoretic problem is formulated as a mixed-boundary-value problem that is subsequently transformed into a system of dual integral equations on the inner and outer domains. Depending on the ratio of different involved viscosities and solute solubilities, the sign of phoretic mobility and chemical activity, as well as the ratio of particle–interface distance to the radius of the disk, the isotropic active particle is found to be repelled from the interface, be attracted to it, or reach a stable hovering state and remain immobile near the interface. Our results may prove useful in controlling and guiding the motion of self-propelled phoretic active particles near aqueous interfaces.

Solving Mixed Boundary Value Problems using Fourier Transform and Dual Integral Equations Method

The research considered the solution of the system of dual integral equations involving Fourier transform occurring in mixed boundary value problems for Laplace’s equation with mixed Dirichlet-Neumann boundary conditions. The dual integral equations involving trigonometric function kernel have been solved using Fourier sine transforms. Graphical solutions with the help of Mathematica were used to demonstrate the effectiveness of this method.
 Keywords: Boundary value problem; Laplace’s equation; Mixed boundary value problems; Fourier transform; Dual integral equations.

Mechanics of multilayered or functionally graded material pressed onto an imperfect rigid base with a circular opening by localized surface loading

Loaded materials constrained by imperfect boundary can be found in many engineering scenarios, including blanking process, contact pressure sensor and soil/rock foundation subjected to underground mining. This paper examines the surface loading problem of a multilayered or functionally graded material (FGM) resting on an imperfect rigid smooth base weakened by a circular opening. The surface loading is axisymmetric and coaxial with the circular opening. This mixed boundary value problem is analogous to a Mode-I penny-shaped crack in multilayered elastic medium. With the aid of the GKS-based method for N-layered elasticity, the problem is formulated as a system of dual integral equations and then reduced to a Fredholm integral equation of the second kind. Solutions of the SIF, COD and full elastic field are expressed in terms of the solution of the integral equation. Numerical results obtained from the present solutions are compared with the those from finite element method. Using the new solutions, numerical studies are also conducted for four examples, including a homogenous layer, a multilayered material, a material with FGM coating and an asphalt pavement structure with FGM properties. These numerical results demonstrate that (1) a small size of surface loading can result in high SIF value at the crack tip; (2) placing the hard layer as the top surface is most effective for reducing ground settlement; (3) the gradation in Poisson’s ratio can have non-negligible effect on the SIF for thin FGM coating; (4) effect of the circular opening to the ground settlement can be significant if its radius is comparable to the total thickness of the upper layers.

Steady azimuthal flow field induced by a rotating sphere near a rigid disk or inside a gap between two coaxially positioned rigid disks

Geometric confinements play an important role in many physical and biological processes and significantly affect the rheology and behavior of colloidal suspensions at low Reynolds numbers. On the basis of the linear Stokes equations, we investigate theoretically and computationally the viscous azimuthal flow induced by the slow rotation of a small spherical particle located in the vicinity of a rigid no-slip disk or inside a gap between two coaxially positioned rigid no-slip disks of the same radius. We formulate the solution of the hydrodynamic problem as a mixed-boundary-value problem in the whole fluid domain, which we subsequently transform into a system of dual integral equations. Near a stationary disk, we show that the resulting integral equation can be reduced into an elementary Abel integral equation that admits a unique analytical solution. Between two coaxially positioned stationary disks, we demonstrate that the flow problem can be transformed into a system of two Fredholm integral equations of the first kind. The latter are solved by means of numerical approaches. Using our solution, we further investigate the effect of the disks on the slow rotational motion of a colloidal particle and provide expressions of the hydrodynamic mobility as a function of the system geometry. We compare our results with corresponding finite-element simulations and observe very good agreement.

A mixed boundary value problem of a cracked elastic medium under torsion

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

The Reissner-Sagoci problem for an interfacial crack in an elastic bilayer medium under torsion of an embedded rigid circular disc

This paper studies an axisymmetric problem of a penny-shaped crack at the interface of a bi-material under a static torsion by an embedded circular rigid disc. By using the Hankel integral transformation method, the angular displacements and the shear stresses are formulated. The faces of the crack are supposed stress free, and the displacement is continuous outside in the crack plane while the rigid circular disc inclusion rotates about the axis passing through their centers and the stress is continuous outside in the rigid disc plane. Considering these mixed conditions associated with the embedded rigid disc and the interfacial crack, the mixed boundary values are taken into account, that are transformed, to a system of dual integral equations. By appropriate transform, the dual integral equations are converted into a regular system of Fredholm integral equations of the second kind with two unknown functions which is then solved by quadrature rule. Numerical results for displacements and stresses in the interaction zone, the stress intensity factor at the edges of the crack and the rigid disc and the applied torque are obtained and discussed according to certain relevant parameters. The torsional effects of the disc on the elastic bilayer are evaluated by the analysis of the Mode III stress intensity factor in the crack vicinity depending on the degree of nonhomogeneity at the interfacial plane, crack size and the depth of embedment of the rigid disc. The efficiency of the method and of the mathematical formulation are checked by comparison with the results available for a relevant analysis in homogeneous solids.

An Axisymmetric Torsion Problem of an Elastic Layer on a Rigid Circular Base

A solution is presented to a doubly mixed boundary value problem of the torsion of an elastic layer, partially resting on a rigid circular base by a circular rigid punch attached to its surface. This problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer expansion formula of the Bessel function we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. Both the two contact stresses under the punch and on the lower face of the layer are expressed as appropriate Chebyshev series. The effects of the radius of the disc with the rigid base and the layer thickness on the displacements, contact stresses as well as the shear stress and the stress singularity factor are discussed. A numerical application is also considered with some concluding results.

Torsion of a cracked elastic body by an embedded semi-infinite rigid cylinder

Torsion of a cracked elastic body by an embedded semi-infinite rigid cylinder is studied. A coaxial penny-shaped crack is situated in the plane of the end of the cylinder. The problem is reduced to a system of dual integral equations including Hankel and Weber–Orr transforms and then, by using ansatzs, to an integral equation of the second kind with Hankel integral operator given on a semi-infinite interval or to an equivalent infinite system of linear algebraic equations with a Hankel matrix. A detailed investigation allowed us to suggest efficient methods for solving equations for any size of the crack. In particular, accurate approximate formulas for the stress intensity factor and the contact stresses are derived, as well as an asymptotic formula for the stress intensity factor when a crack is large. Analytical estimations and calculations manifest a strong increase in the stress intensity factor and the contact stress at the end of the cylinder as the crack tip is very close to the cylinder surface.

An Axisymmetric Contact Problem of a Thermoelastic Layer on a Rigid Circular Base

We study the thermoelastic deformation of an elastic layer. The upper surface of the medium is subjected to a uniform thermal field along a circular area while the layer is resting on a rigid smooth circular base. The doubly mixed boundary value problem is reduced to a pair of systems of dual integral equations. The both system of the heat conduction and the mechanical problems are calculated by solving a dual integral equation systems which are reduced to an infinite algebraic one using a Gegenbauer’s formulas. The stresses and displacements are then obtained as Bessel function series. To get the unknown coefficients, the infinite systems are solved by the truncation method. A closed form solution is given for the displacements, stresses and the stress singularity factors. The effects of the radius of the punch with the rigid base and the layer thickness on the stress field are discussed. A numerical application is also considered with some concluding results.

Axisymmetric Flow due to a Stokeslet Near a Finite-Sized Elastic Membrane

Elastic confinements play an important role in many soft matter systems and affect the transport properties of suspended particles in viscous flow. On the basis of low-Reynolds-number hydrodynamics, we present an analytical theory of the axisymmetric flow induced by a point-force singularity (Stokeslet) directed along the symmetry axis of a finite-sized circular elastic membrane endowed with resistance toward shear and bending. The solution for the viscous incompressible flow surrounding the membrane is formulated as a mixed boundary value problem, which is then reduced into a system of dual integral equations on the inner and outer sides of the domain boundary. We show that the solution of the elastohydrodynamic problem can conveniently be expressed in terms of a set of inhomogeneous Fredholm integral equations of the second kind with logarithmic kernel. Basing on the hydrodynamic flow field, we obtain semi-analytical expressions of the hydrodynamic mobility function for the translational motion perpendicular to a circular membrane. The results are valid to leading-order in the ratio of particle radius to the distance separating the particle from the membrane. In the quasi-steady limit, we find that the particle mobility near a finite-sized membrane is always larger than that predicted near a no-slip disk of the same size. We further show that the bending-related contribution to the hydrodynamic mobility increases monotonically upon decreasing the membrane size, whereas the shear-related contribution displays a minimum value when the particle-membrane distance is equal to the membrane radius. Accordingly, the system behavior may be shear or bending dominated, depending on the geometric and elastic properties of the system. Our results may find applications in the field of nanoparticle-based sensing and drug delivery systems near elastic cell membranes.

Функции податливости электромагнитупругой пьезоэлектрической пьезомагнитной полуплоскости и полупространства с функционально-градиентным или слоистым покрытием

Статья посвящена построению функций податливости осесимметричных и плоских контактных задач электромагнитоупругости для полубесконечных пьезоэлектрических пьезомагнитных тел с функционально градиентными или кусочно-однородными покрытиями. Материалы покрытия и подложки считаются трансверсально-изотропными. Вычисление функций податливости сведено к решению краевых двухточечных задач для систем обыкновенных дифференциальных уравнений с переменными коэффициентами с использованием техники интегральных преобразований. Граничные условия этих краевых задач соответствуют распределенной в некоторой области единичной механической нормальной или касательной, электрической или магнитной нагрузке. Получены парные интегральные уравнения и их системы, описывающие контактные задачи о вдавливании изолированного или проводящего штампов в полупространство (или полуплоскость) с покрытием и задачу об электроде, расположенном на поверхности покрытия. Трансформанты ядер этих интегральных уравнений совпадают с функциями податливости. Предложены специальные аппроксимации трансформант ядер, с использованием которых можно построить замкнутые аналитические решения приближенных парных интегральных уравнений и их систем. Представлены численные результаты, иллюстрирующие свойства всех десяти независимых функций податливости для различных сочетаний свойств покрытия и подложки и законов изменения свойств по глубине. Показано, что при отсутствии касательной нагрузки все функции податливости являются строго положительными. Исследованы условия, при которых функции податливости, соответствующие касательной нагрузке, являются знакопеременными. Проиллюстрированы различия между свойствами функций податливости, соответствующих однородным и функционально градиентным покрытиям.

An Axisymmetric Contact Problem of an Elastic Layer on a Rigid Circular Base

Abstract An analytical solution is presented to a doubly mixed boundary value problem of an elastic layer partially resting on a rigid smooth base. A circular rigid punch is applied to the upper surface of the medium where the contact is supposed to be smooth. The case of the layer with a cylindrical hole was considered by Toshiaki and all [5]. The studied problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer formula we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. The truncation method is used for getting the system coefficients. A closed form solution is given for the displacements, stresses and the stress singularity factors. The stresses and displacements are then obtained as Bessel function series. For the numerical application we give some conclusions on the effects of the radius of the punch with the rigid base and the layer thickness on the displacements, stresses, the load and the stress singularity factors are discussed.

On a Mixed Boundary Value Problem for the Biharmonic Equation in a Strip

The aim of the present work is to consider a mixed boundary value problem for the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges y=0,y = h having clamped conditions on intervals |x|≥a and hinged support conditions for |x|<a. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of solution of system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the dual integral equation to infinite system of linear algebraic equations is also proposed.

Approximate solution of dual integral equations using Chebyshev polynomials

ABSTRACTThe aim of the present work is to introduce solution of special dual integral equations by the orthogonal polynomials. We consider a system of dual integral equations with trigonometric kernels which appear in formulation of the potential distribution of an electrified plate with mixed boundary conditions and convert them to Cauchy-type singular integral equations. We use the Chebyshev orthogonal polynomials to construct approximate solution for Cauchy-type singular integral equations which will solve the main dual integral equations. Numerical results demonstrate effectiveness of this method.

Effect of heat conduction of penny-shaped crack interior on thermal stress intensity factors

This paper analyzes the thermal stress induced by a penny-shaped crack in an elastic solid with uniform steady heat flux. Air inside an opening crack is taken as a thermally conducting medium and the crack is partially insulated. The Hankel transform technique is applied to convert the problem to a system of dual integral equations. Heat flux in the opening crack or temperature gradient across the crack depends on the crack opening displacement. Explicit expressions for the whole temperature change field and heat flux at any position in the cracked medium are given in terms of elementary functions. Thermal stresses and displacements are presented for a solid with a partially insulated crack under remote tensile loading and uniform heat flux. Stress intensity factors (SIFs) are determined. The mode-I SIFs depend only on external tensile loading, and are free of the material properties. The mode-II SIFs are related to both mechanical and thermal loading, in addition to the material properties. Numerical results for a cracked thermoelastic material are presented to show the influence of the thermal conductivity of air of the crack interior on the mode-II SIFs, and indicate that heat conduction of crack affects thermal SIFs. Insulated and isothermal cracks are two limiting cases of a partially insulated crack.

Influence of Initial Stresses on the Fracture of Composite Material Weakened by a Subsurface Mode III Crack

By using the relations of linearized mechanics of deformable solids, we study the space problem of fracture of prestressed semibounded composites weakened by subsurface disk-shaped torsion (mode-III) cracks. With the help of representations of the general solutions of linearized equilibrium equations in terms of harmonic potential functions and Hankel integral transforms, we reduce the problem to a system of dual integral equations and then to the resolving Fredholm integral equation of the second kind. From the analysis of the stress distribution in the vicinity of the crack, we obtain the values of the stress intensity factors and study their dependences on the initial stresses, mechanical characteristics of the components of the composite, and geometric parameters of the problem.

Solvability of Some Classes of Systems of Dual Integral Equations Involving Fourier Transforms

The aim of the present work is to consider the existence and uniqueness problems for some classes of general systems of dual integral equations involving Fourier transforms of generalized functions, which are a generalization of some systems of dual equations encountered in the mixed boundary value problems of mathematical physics and contact problems of elasticity.