System Of Singular Integral Equations Research Articles

Обобщённая математическая модель основных задач теории упругости для анизотропных тел

Introduction. The assessment of the strength of underground structures is a complex task that requires a comprehensive approach. In many cases empirical methods based on experience are not sufficient. For effective application of analytical methods, it is necessary to use mathematical modelling. This allows to simplify the problem considerably, to consider different variants, to identify cases to obtain a solution in a closed form. Materials and methods. The model is based on a system of singular integral equations with a shift and complex conjugate terms. The solution of the system is the complex elastic potential. Systems of singular equations with a Carleman shift are considered on the space of random functions converging in the mean square to expand the possibility of practical use of the work results. Results. The results of the study are: a mathematical model formulation, the development of an algorithm for reducing a system of singular equations to a system of known Fredholm integral equations of the second kind, a method for determining the solvability conditions and the number of solutions. Discussion. As a result of the study: (1) a general algorithm for finding a solution to a mathematical model is obtained; (2) the methodology for determining the number of solutions and the type of solvability conditions is given; (3) The solution algorithm is extended for the stochastic case; (4) The possibility of wider application of the mathematical apparatus of functions of complex variable for research of strength properties of rocks is proved. Conclusion. The proposed comprehensive approach to assessing the strength characteristics of underground structures, combining a set of analytical methods in conjunction with mathematical modelling, expands the variability of calculation and allows to simplify the problem significantly and obtain a solution in closed form. Resume. The paper develops a mathematical model of the stress state of an elastic body, which allows solving the basic problems of elasticity theory for a variety of media. Suggestions for practical applications and directions for future research. The results of the study make it possible to improve the efficiency of analytical methods in the study of the strength of underground structures. The obtained results are applicable for predicting the strength and safety of underground structures.

Impulsive impact of a twin hull

An impulsively starting motion of two cylindrical bodies floating on a free liquid surface is considered. The shape of the cross-section of each body and the distance between them are arbitrary. The integral hodograph method is advanced to derive the complex velocity potential defined in a rectangular parameter region in terms of the elliptic quasi-doubly periodic Jacobi theta functions. A system of singular integral equations in the velocity magnitude on the free surface and in the slope of the wetted part of each body is derived using the kinematic boundary condition, which is then solved numerically. The velocity field, the pressure impulse on the bodies and the added mass coefficients of each body immediately after the impact are determined in a wide range of distances between the bodies and for cross-sectional shapes such as the flat plate and half-circle.

Aeolian Liquid Vibrations in Conical Tanks with Baffles under Wind Loading with Fuzzy Parameters

The Aeolian liquid vibrations in conical reservoirs caused by low-velocity, steady winds have been under consideration. Both amplitudes and dominant frequencies of wind loadings have been constantly changed, so to adequately describe the vibration process, fuzzy logic methods have been applied. At the first stage, the crisp initial value problem for conical shells with and without baffles has been considered. The liquid inside the reservoirs has been supposed to be an incompressible and ideal one, and its flow induced by the forced harmonic excitation, has been considered as potential. So, there exists a potential to satisfy the Laplace equation. The impermeable condition has been used at wetted surface boundaries of the shell, whereas the dynamic and kinematic boundary conditions have been set on the free liquid surface. A system of singular integral equations has been obtained for values of the velocity potential and the function describing the free surface rise. Its solution has been gained by boundary element methods. The crisp boundary value problem has been reduced to the second-order system of differential equations. After receiving the crisp solution of this system, the initial data have been fuzzified, involving triangular fuzzy numbers, and the fuzzy initial value problem has been formulated. The numerical solution to this problem with uncertain intervals involved has been obtained and analyzed.

An application of Lobatto–Chebyshev method to a nonhomogeneous plane problem with two cracks

In this manuscript, Lobatto–Chebyshev method, which is an effective collocation method, is applied to a system of singular integral equations, which leads from a nonhomogeneous plane problem. It is assumed that there are two cracks in a nonhomogeneous medium. The problem is formulated in the view of the basics of the elasticity theory and boundary conditions of the problem. By using the method of singular integral equation, the problem is converted to a system of first kind Cauchy type singular integral equations. It is aimed to determine the stress intensity factors (SIFs) of the crack problem. It is seen that Lobatto–Chebyshev quadrature has many advantages in determining SIFs. To verify the validity of the method, the obtained results corresponding to the one crack case are compared with the results in literature.

A numerical investigation of crack behavior near a fixed boundary using singular integral equation and finite element methods

In this paper, we present the method of integral transforms and the Gauss-Chebyshev quadrature methods to solve the problem of a crack parallel to a fixed boundary under remote tension. We derive a system of singular integral equations of the first kind, specific to the problem at hand, which we numerically solve using Gauss-Chebyshev integration. We specialize our results to the problem of a crack in an infinite plate under remote tension, and show that the relative error in our numerically derived solutions is within machine precision of the closed-form analytical solutions. Stress intensity factors are calculated that are in excellent agreement with those derived by others using different methods. We also demonstrate that both the stress intensity factors and normal σyy(x,y) and shear σxy(x,y) stress fields derived via numerical solution of the singular integral equations, compare well with those determined using the commercially available Abaqus finite element code where the crack is modeled using the eXtended Finite Element Method.

NUMERICAL MODELLING OF ELECTROMAGNETIC WAVE SCATTERING ON GRATINGS WITH ONE-DIMENSIONAL QUASI-FRACTAL PERIOD STRUCTURE

Numerical modelling of the properties of E-polarised and H-polarised waves scattered on periodic screened quasi-fractal gratings is carried out. The location of the band system at each period is determined by the principle of constructing a generalized symmetric Cantor set at a certain step of the algorithm. A mathematical model of the problems based on systems of boundary singular integral equations of the first kind was used in the study. These systems of equations were obtained using the method of parametric representations of singular and hypersingular integral operators. The systems of singular integral equations were solved numerically using the computational schemes of the method of discrete singularities. The solutions of these equations are used to obtain the main characteristics of the electric and magnetic fields. This experiment proved the possibility of using the MDS computational scheme to analyse systems containing 8 – 16 bands at different distances from each other. Graphs of the dependence of harmonic amplitudes on the wavenumber, point plots of absolute values of all non-zero harmonics at resonant wavenumber values, and maps of electric and magnetic field components in the region above the grating were obtained. It is confirmed that the overall field structure in the case of normal incidence is significantly influenced by all harmonics with absolute numbers from 0 to 50. The harmonics had a large number of resonances that were observed at different values of the wavenumber. This led to a complex structure of the isolines of absolute values of the scattered electric and magnetic field amplitudes in the region above the structure, and a significant difference in amplitude values with small changes in coordinates. In the future, it is planned to carry out computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case considered in this paper. The proposed structure may be of interest for the design of multimode broadband antennas.

A problem for a material surface attached to the boundary of an elastic semi-plane

A problem for a nano-sized material surface attached to the boundary of an elastic isotropic semi-plane is considered. The semi-plane and the material surface are assumed to be in a plane strain state under the action of a normal external tractions applied to the material surface. The material surface is modeled using the Steigmann–Ogden form of surface energy. The problem is solved using integral representations of stresses and displacements. With the help of these representations, the problem can be reduced to either a system of two singular integral equations or a single singular integral equation. Two types of material surface tip conditions are considered: tip conditions with an uncompensated surface prestress (free tip conditions) and tip conditions with a compensated surface prestress term. The numerical solution of the system of singular integral equations is obtained by expanding each unknown function into a series based on the Chebyshev polynomials. Then, the approximations of the unknown functions can be obtained from a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. It is shown that both the surface parameters and the type of tip conditions have significant influence on the behavior of the material system.

Bending of a plate with circular rigid inclusion and system of cracks on the assumption of strip crack face contact

The article presents a solution to the bending problem for an infinite isotropic plate with a circular rigid inclusion and arbitrarily oriented straight through cracks. It is assumed that under the action of an external load at infinity, the faces of all cracks are in smooth contact along a region of constant width near the upper surface of the plate. The solution is obtained using methods of the theory of functions of complex variable and complex potentials and is reduced to a system of singular integral equations, which is numerically solved by the mechanical quadrature method. On the basis of numerical analysis, the graphical dependences of the contact force, forces and moments intensity factors are constructed at various parameters of the problem.

A BRIEF OVERVIEW OF STATIONARY TWO-DIMENSIONAL THERMOELASTIC STATE MODELS IN HOMOGENEOUS AND PIECEWISE-HOMOGENEOUS BODIES WITH CRACKS

Purpose. A two-dimensional mathematical model of the problem of thermo-elasticity for piecewise-homogeneous component plate containing a crack has been built. The stress intensity coefficients in the vertices of the crack increase affecting strength of the body significantly. This leads to the growth of a crack and, as a result, to further local destruction of a material. Therefore, such a model reflects, to some extent, the destruction mechanism of the elements of engineering structures with cracks. Methodology. Based on the method of the function of a complex variable we have studied the two-dimensional thermoelastic state for the body with crack as stress concentrators. As result, the problem of thermoelasticity was reduced to a system of two singular integral equations (SIE) of the first and second kind, a numerical solution of which was found by the method of mechanical quadratures. Findings. The two-dimensional mathematical model of the thermoelastic state has been built in order to determine the stress intensity factors at the top of the crack and inclusion. The systems of singular integral equations of the first and second kinds of the specified problem on closed (contour of inclusion) and open (crack) contours are constructed. The influence of thermophysical and mechanical properties of inclusion on the SIF sat the crack types was investigated. The dependences of the stress intensity factor which characterizes the distribution of the intensity of stresses at the vertices of a crack have been built, as well as its elastic and thermoelastic characteristics of inclusion. This would make it possible to analyze the intensity of stresses in the neighborhoods of crack vertices depending on the geometrical and mechanical factors. As a result, this allows to determine the critical values of temperature in the three-component plate containing a crack in order to prevent the growth of the crack, as well as to prevent the local destruction of the body. It was found that the appropriate selection of mechanical and thermophysical characteristics of the components of a three-component plate containing a crack can be useful to achieve an improvement in body strength in terms of the mechanics of destruction by reducing stress intensity factors at the crack’s vertices. Originality. The solutions of the new two-dimensional problem of thermoelasticity for a specified region due to the action of constant temperature as well as due to local heating by a heat flux were obtained. The studied model is the generalization of the previous models to determine the two-dimensional thermoelastic state in a piecewise homogeneous plate weakened by internal cracks. Practical value. The practical application of this model is a more complete description of the stress-strain state in piecewise homogeneous structural elements with cracks operating under temperature loads. The results of numerical calculations obtained from the solution of systems of equations and presented in the form of graphs can be used in the design of rational modes of operation of structural elements. This takes into account the possibility of preventing the growth of cracks by the appropriate selection of composite components with appropriate mechanical characteristics.

КОМП’ЮТЕРНЕ МОДЕЛЮВАННЯ РОЗСІЮВАННЯ Е-ПОЛЯРИЗОВАНИХ ХВИЛЬ НА ЕКРАНОВАНІЙ ПРЕДКАНТОРОВІЙ СИСТЕМІ РЕФЛЕКТОРІВ

Computer simulation of electromagnetic wave scattering on a shielded periodic reflector system, with a special period structure is considered. The electrodynamic reflective structure consists of a screen and a periodic system of strips. The cross-section of the strips is a system of segments formed by the principle of constructing a generalized Cantor set at each of the periods. Іt was used a mathematical model based on the reduction of the initial boundary value problems for the Helmholtz equation to systems of singular integral equations. The choice of this approach did not require analytical calculations when changing the number of lattice elements. Using this approach, it was not necessary to make changes to the computer implementation of the algorithm. The computer experiment has proved the possibility of applying the discrete singularities method computational scheme for the considered structure. It is proved that the structure constructed according to the proposed principle has the properties of broadband and multimode for the studied values of the structure parameters. In particular, for the studied structures, the resonance occurred when the wavenumber was changed from 0 to 50, with a significant increase in amplitude occurring at harmonics with numbers up to 40. This phenomenon led to a complex structure of the lines of the absolute values of the amplitude of the scattered electric field above the grating. The presence of small areas with an increase in the amplitude of the scattered field by a factor of 5-6 compared to the amplitude the incident electromagnetic wave was observed. These properties show practical interest in the use of this type of antennas to improve electronic camouflage and tactical radio communications. In the future, it is planned to conduct computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case discussed in this paper.

On non-trivial solvability of one system of non-linear integral equations on the real axis

A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann-Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $\pm\infty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.

Complex Potentials and Integral Equations for Thermoelasticity Problem of Orthotropic Cracked Plate

In the present article, the plane thermoelasticity problem of an infinite orthotropic cracked plate, is studied. The interaction between thermal and mechanical field, in a plate containing a crack and a heat source is investigated. The plate is submitted to normal stresses at infinity, where a heat flow is also activated. Applying the complex potentials method and the technique of Cauchy integrals, a system of singular integral equations is obtained. These equations, allow the determination of the temperature’s distribution, but also of the stress state at any point of the medium.

Flexural gravity wave scattering due to a pair of asymmetric rectangular trenches

The research presented here investigates the interaction of oblique flexural gravity waves with a pair of asymmetric trenches. The fluid considered here is bounded above by an infinite thin uniform ice sheet of particular flexural rigidity. Velocity potential describing the motion in the fluid is obtained explicitly in terms of non-orthogonal eigenfunctions. The mode-coupling relations satisfied by the eigenfunction are given. Using matching conditions along the vertical boundaries of the trench walls the problem is transformed into a system of singular integral equations. The unknown functions involved in the integral equations are calculated numerically using multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials. The reflection and transmission coefficients are obtained numerically and depicted in the number of figures for both symmetric and asymmetric cases. The energy balance relation satisfied by the reflection and transmission coefficients is obtained. The results are analyzed for various non-dimensional parameters describing the model. The results using the present method are compared with previously published works available in the literature. It is found that a thin ice sheet of uniform flexural rigidity has a significant effect on the scattering behavior of the wave in presence of wide trenches.

Frictionless discontinuous contact problem of the thermoelectric layer resting on a rigid base

Neglecting the friction between the indenter and the thermoelectric layer (TEL), the discontinuous contact problem between the TEL and the rigid base is studied by using elastic theory and integral transformation technology. The external load is applied to the TEL through the rigid base. When the value of the external load exceeds the critical load (CL) separation occurs along the interface between the TEL and the rigid base. The discontinuous contact problem is transformed into a system of singular integral equations (SIES) by boundary conditions. Numerical results of the contact stress and the separation distance between the TEL and the rigid base are considered by the external load, the specific weight of the TEL and the width of the indenter. The results show that the larger external load and the specific weight of the TEL will cause the separation distance between the TEL and the rigid base to increase. However, the increase in the width of the rigid indenter has the opposite effect on the contact stress between the TEL and the rigid base. The model developed in this article provides guidance for the design and safety and reliability judgment of thermoelectric components in practical applications.

Scattering by PEC half-plane and disk placed on parallel planes

AbstractThe scattering by the perfectly electric conducting (PEC) half-plane and PEC zero thickness disk placed on parallel planes is considered. The fields are represented in the spectral domain, i.e. in the domain of Fourier transform. The operator equations with respect to the Fourier amplitudes of the scattered field are obtained. The kernel functions of these equations contain poles. After regularization procedure, which is connected with the elimination of the poles, operator equations are converted to the system of singular integral equations. The convergence of the solution is based on the corresponding theorems. The scattered field consists of the plane wave, reflected by the infinite part of the half-plane, cylindrical waves, which appear as a result of scattering by the edge of the half-plane, and spherical waves, which appear as a result of scattering by the disk and multiple re-scattering by the disk-half-plane. The total near-field distribution and far-field patterns of cylindrical waves are presented.

SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR ANISOTROPIC PLATES AND SHELLS BY BOUNDARY ELEMENTS METHOD

Modern mechanical engineering sets the tasks of calculating thin-walled structures that combine lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of anisotropic materials and plastics seems justified. The problems of the theory of plates and shells belong to the class of boundary value problems, the analytical solution of which, due to various circumstances (nonlinearity of differential equations, complexity of geometry and boundary conditions, etc.), cannot bedetermined. Numerical methods help to solve this problem. Among numerical methods, undeservedly little attention is paid to the boundary element method. In this regard, the further development of indirect method of compensating loads for solving problems of the anisotropic plates and shells theory based on the applicationof exact fundamental solutions is relevant.The paper considers the application of the indirect boundary element method for solving of an anisotropic plates and shells nonlinear deformation problem. Since the kernels of the system of singular integral equations to which the solution of the problem is reduced are expressed in terms of the fundamental solution and itsderivatives, first of all, the article provides a method for determining the fundamental solutions to the problem of bending and the plane stress state of an anisotropic plate. The displacement vector is determined from the solution of linear equations system describing the bending and plane stress state of an anisotropic plate. The solution of the system is performed by the method of compensating loads, according to which the area representing the plan of the shallow shell is supplemented to an infinite plane, and on the contour that limits the area, compensating loads are applied to the infinite plate. Integral equations of indirect BEM are given. In this paper, the study of nonlinear deformation of anisotropic plates and shallow shells is carried out using the deflection load dependencies. The deflection at a given point on the median surface of the shell was taken as the leading parameter.

Existence of solution of infinite systems of singular integral equations of two variables in C(I × I, ℓp) with I = [0, T], T > 0 and 1 < p < ∞ using Hausdorff measure of noncompactness

In this article, we discuss the solvability of infinite systems of singular integral equations of two variables in the Banach sequence spaces C(I ? I, ?p) with I = [0, T], T > 0 and 1 < p < ? with the help of Meir-Keeler condensing operators and Hausdorff measure of noncompactness. With an example, we illustrate our findings.

Thermoelastic contact problem of a rigid punch sliding on a functionally graded piezoelectric layered half plane with heat generation

In this study, the plane steady state thermoelastic frictional contact problem between a rigid cylindrical punch and a functionally graded piezoelectric material (FGPM) layer bonded to the homogeneous half plane is considered. The rigid punch is assumed to be a perfect electric conductor and thermal insulator and slides over the FGPM layer with a small constant speed and heat flux is generated due to the friction. Applying the Fourier integral transform technique and thermoelasticity, general stress, displacement, and electrical expressions for the thermoelastic piezoelectric contact problem are derived. Using the boundary conditions, the contact problem is reduced to Cauchy-type singular integral equations of the second kind in which the contact stress, the electrical displacement and the contact widths are unknown. The system of singular integral equation is solved numerically using Gauss-Jacobi and Gauss-Chebyshev integration formula. The effect of material inhomogeneity parameter, moving velocity, friction coefficient, external load, electric charge, punch radius on the contact stress, in-plane stress, electric displacement, and generated temperature are discussed in detail.

Effect of T-stress on the fracture in an infinite two-dimensional decagonal quasicrystals of a cruciform crack with unequal arms

The T-stress near the tips of a cruciform crack with unequal arms in the two-dimensional (2D) decagonal quasicrystals (QCs) are analyzed. The associated boundary value problems can transform to a system of singular integral equations by using the integral transformation technique. According to the stress difference method, T-stresses of phonon field and phason field can be expressed as the sum of an integral involving crack opening displacements (CODs) and applied loading at infinity. CODs play a leading role in determining T-stresses of phonon field and phason field. The role of the T-stresses in brittle fracture for 2D decagonal QCs is emphasized. Moreover, the influence of the ratio of two crack-arm lengths b/a and the CODs on the T-stresses of a cruciform crack is examined.

A new system of singular integral equations for a curvilinear crack in bonded materials

The modified complex potentials (MCPs) functions are used to develop a new system of singular integral equations (SIEs) for a curvilinear crack in the upper part of bonded materials subjected to shear mode stress with the help of continuity conditions for resultant force and displacement functions. The unknown dislocation distribution function is mapped into a square root singularity function by using curved length coordinate method and the traction along the crack as the right hand term. The Gaussian quadrature rules were used to obtain the numerical solution for a new system of SIEs in order to compute the nondimensional stress intensity factors (SIFs) for these problems. Our results agree with those of the previous works. The findings have revealed that the nondimensional SIFs depend on the elastic constant ratio, crack geometries and the position of the cracks.