Method Of Singular Integral Equations Research Articles

Periodic System of Closely Located Holes in a Quasiorthotropic Plane

We study a plane periodic problem of the theory of elasticity for a quasiorthotropic plane with an infinite series of closely located curvilinear holes by the method of singular integral equations. By using a unified approach to the solution of the problems of stress concentration factors near holes with sharp and rounded vertices, we obtain the stress intensity factors at sharp vertices of the holes. The stress concentration factors at rounded and sharp tips of two-sided parabolic and U-shaped notches are found as a result of the limit transition.

Mathematical Modelling of Stationary Thermoelastic State for a Plate with Periodic System of Inclusions and Cracks

Abstract Two-dimensional stationary problem of heat conduction and thermoelasticity for infinite elastic body containing periodic system of inclusions and cracks is considered. Solution of the problem is constructed using the method of singular integral equations (SIEs). The numerical solution of the system integral equations are obtained by the method of mechanical quadrature for a plate heated by a heat flow, containing periodic system elliptic inclusions and thermally insulated cracks. There are obtained graphic dependences of stress intensity factors (SIFs), which characterise the distribution of intensity of stresses at the tops of a crack, depending on the length of crack, elastic and thermoelastic characteristics inclusion, relative position of crack and inclusion.

Mathematical Details on Singular Integral Equation Method for Solving Crack Problems

This article provides a detail derivation of a singular Fredholm integral equation for the solution of a mixed mode crack problem in a nonhomogeneous medium. The integral equation derived here has already been addressed by F. Delale and F. Erdogan (Delale & Erdogan 1983), one of the most cited and pioneer papers in fracture mechanics that uses singulalr integral equation method (SIEM) to solve crack problems. However, probably due to its limit of paper length, some mathematical details are not provided to bring this powerful method, SIEM, to its full strength. In this paper we fill in the mathematical gaps, and both analytical and numerical parts are addressed in details. Some discussions from the view point of differential equations are given, and new numerical outcomes under different loading functions are provided.

The crack propagation path under multiple moving contact loads in rolling contact fatigue

This paper proposes a model which considers multiple moving loads to calculate the remaining life of a surface under the rolling contact fatigue. This model is based on the linear fracture mechanics for determining the characteristics of cyclic crack growth. The body under the rolling contact is modeled by a half-plane with an edge crack. Rolling contact is simulated by a translational motion of concentrated forces along the half-plane boundary. The fracture mechanics problem is formulated based on the singular integral equations method and these equations are solved at each step of crack growth, resulting in the stress intensity factors at each step. Then, the crack growth is modeled using the mixed mode fracture criterion for finding the crack path during the growth and the Paris equation for determining the amount of fatigue crack propagation. The effect of the number of moving loads with similar and different speeds applied on the edge of the half-plane, the value of coefficient of friction and the distance between the moving loads with the similar speeds on the crack propagation path and the remaining life have been investigated. It is seen that by increasing the number of the moving loads with the similar speeds on the boundary of the half-plane, increasing the distance between the two and three concentrated moving loads and when two concentrated moving loads move with different speeds, the crack propagation angle compared to the initial crack angle increases and the crack is distracting more inside the surface. As a result, the surface cracks join together and grow towards the surface, causing pitting be reduced.

Analysis for T-stress of cracks in 3D anisotropic elastic media by weakly singular integral equation method

This paper proposes an efficient numerical procedure for computing T-stresses of cracks in three-dimensional, linearly elastic, infinite media. The technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated. A pair of weakly singular, weak-form, displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack-face fields. Besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data, use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack-face displacement fields. A weakly singular symmetric Galerkin boundary element method together with the special near-front approximation is utilized to solve for the unknown relative crack-face displacement whereas the sum of the crack-face displacement is obtained from the displacement boundary integral equation for cracks via the Galerkin technique. The latter step is one of the essential aspects of the present study that provides the direct means for determining the T-stress data in terms of the sum of the crack-face displacement in the neighborhood of the crack-front. An extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique.

КРИТЕРИЙ ОДНОЗНАЧНОЙ РАЗРЕШИМОСТИ СПЕКТРАЛЬНОЙ ЗАДАЧИ ПУАНКАРЕ В ЦИЛИНДРИЧЕСКОЙ ОБЛАСТИ ДЛЯ ОДНОГО КЛАССА МНОГОМЕРНЫХ ЭЛЛИПТИЧЕСКИХ УРАВНЕНИЙ

Two-dimensional spectral problems for elliptic equations are well studied, and their multidimensional analogs, as far as the author knows, are little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of a fundamental nature, since the method of singular integral equations, which is very attractive and convenient, used for two-dimensional problems, cannot be used here because of the lack of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. In the cylindrical domain of Euclidean space, for a single class of multidimensional elliptic equations, the spectral Poincare problem. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.

Method of singular integral equations in liquid vibration problems for coaxial shells

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

Згин зосередженою силою тонкої пластини, розташованої на пружній основі та послабленої контактною тріщиною

The paper considers the two-dimensional formulation of the problem of the contact interaction of the crack edges in a plate bent by the concentrated force on the elastic Winkler foundation. The crack closure is described using the model of contact along a line in one of the plate surfaces. Within the framework of this model, the boundary value problem is formulated for the equations of the classical theories of plate bending on the elastic foundation and a plane stress state with interrelated tension and bending conditions on the crack line. The obtained boundary value problem has been solved using singular integral equations method. Based on numerical solutions of the integral equation the dependences of forces and moments intensity factors in the vicinity of the defect tips and distribution of contact forces along the crack line on the parameters of elastic foundation stiffness and the coordinate of the application point of the load have been investigated. The effect of crack closure and influence of the elastic foundation stiffness on the limit equilibrium of the plate, depending on the coordinate of the point of application of the concentrated force, has been evaluated. The area of the correctness of the problem statement when the crack closure occurs throughout its length has been established. It was found that the crack closure leads to the appearance of nonzero forces intensity factor, reduction of the moments intensity factor and increase of the limit load. The dependences of the forces and moments intensity factors and the limit load on the dimensionless coordinate of the point of application of the concentrated force are nonmonotonic. Numerical analysis showed that increasing the elastic foundation stiffness, as well as the displacement of the point of application of the force from the center of the cut, increase the limit load and weaken the contact reaction.

Investigation of the interaction of two parallel shifted cracks in plate bending adjusted for their closure

The problem of interaction of two parallel shifted cracks in plate bending is considered. The cracks closure has been investigated in the classical two-dimensional statement, using the model of smooth contact along a line. The influence of the relative position of cracks and of the contact of their edges on the forces and moment intensity factors has been studied by the singular integral equations method.

On the special angle of surface cracks propagation in the railway rail heads

Investigation of crack-like surface defects on running surfaces of railway rails was carried out using the Keer-Bryant model scheme. For this purpose by singular integral equation method, the contact problem for an elastic half-plane with an edge crack was solved, taking into account the general conditions of crack faces interaction (slip, stick, opening). Dependence of the range of stress intensity factors KII in contact cycle on crack orientation angle for different service parameters, namely friction between the wheel and the rail, length of the crack and friction between its faces was investigated in detail. As a result, it has been shown that surface cracks on running surfaces of rails grow in the direction of the rolling stock traffic at shallow characteristic angle by transverse shear mechanism. This angle is a basis for formation of typical surface contact fatigue damages, such as pitting, squat, checks. The conditions for finding it have been put down. Characteristic angle values found theoretically agree with engineering data well.

Method of singular integral equations in diffraction by semiinfinite grating: E -polarization case

Diffraction of the E-polarized electromagnetic wave by a semiinfinite strip grating is considered. The scattered field is represented as a superposition of the field induced by the currents on the strips of an infinite periodic grating and the field induced by the correction current excited due to end of the grating. A singular integral equation with additional conditions for correction current density is obtained. A solution for the infinite periodic grating in the E-polarization case is obtained from the corresponding solution for the H-polarization case using the duality principle. Numerical results for the current density and far fields distribution are represented.

Discretization Methods for Three-Dimensional Singular Integral Equations of Electromagnetism

Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.

Simulation of infinite periodic graphene planar grating in the THz range by the method of singular integral equations

Simulation of infinite periodic graphene planar grating in the THz range by the method of singular integral equations

Calculation of trajectories and the rate of growth of curvilinear fatigue cracks in isotropic and composite plates

The methods of calculating the trajectories and the rate of growth of curvilinear fatigue cracks in isotropic and composite plate structure elements during cyclic loading along straight or curvilinear trajectories are developed. For isotropic and anisotropic materials, the methodes are developed on the basis of the force criterion of destruction with the additional application of the fatigue fracture diagrams. To find the change in the shape of the cracks in the loading process, the step-by-step method was used. At each stage, the direction of the growth of all vertices of cracks and the lengths of their arcs was found on the basis of determining the intensity coefficients of stresses by the method of singular integral equations. The results of calculations of the cracks system growth process are presented.

Singular integral equations in diffraction by multilayer grating of graphene strips in the THz range

Diffraction of the H-polarized electromagnetic wave by multilayer graphene grating in the THz range is considered. The scattered field is represented in the spectral domain via unknown spectral functions. The mathematical model is based on the graphene surface impedance and the method of singular integral equations. The numerical solution is obtained by the Nystrom-type method of discrete singularities. The scattering and absorption characteristics as functions of frequency and distance between layers are studied.

Calculation of the Input Impedance of a Strip Oscillator by the Singular Integral Equation Method

A method is developed for calculation of the input impedance of a strip oscillator (SO) located in the free space. The singular integral equation with the Cauchy singularity is obtained for the unknown function of the current density distribution on the SO surface. Plots of the input impedance depending on the SO length are given.

Modeling of Graphene Planar Grating in the THz Range by the Method of Singular Integral Equations

AbstractDiffraction of the$H$- polarized electromagnetic wave by the planar graphene grating in the THz range is considered. The scattering and absorption characteristics are studied. The scattered field is represented in the spectral domain via unknown spectral function. The mathematical model is based on the graphene surface impedance and the method of singular integral equations. The numerical solution is obtained by the Nystrom-type method of discrete singularities.

Review and evaluation of weight functions and stress intensity factors for edge-cracked finite-width plate

Various analytical and numerical weight functions (WFs) for the edge-cracked finite-width plate (ECFWP) configuration have been developed in the literature for efficient fracture mechanics analyses involving complicated/arbitrary load conditions. The objective of this paper is to make a comprehensive review and critical evaluation of the existing WFs determined by using various approaches, e.g. singular integral equation methods; one/two reference load case(s) based analytical approaches; fitting and interpolation to handbook solutions; numerical methods such as the complex Taylor series expansion and finite element analysis. The accuracy levels of existing WFs are assessed in rigorous manner by benchmarking the corresponding Green’s functions (GFs) against well-recognized accurate solutions. A highly accurate and wide-ranging analytical ECFWP WF is presented in closed form. Stress intensity factor and crack opening displacement solutions for various load cases of practical interest are easily determined with high accuracy by using the well-verified analytical ECFWP WF for this crack configuration to demonstrate the methods.

Stressed State of a Plane Wedge-Shaped Specimen with Edge Crack Under Uniaxial Tension

We propose a computational model of plane compact wedge-shaped specimens with stability regions of the stress intensity factor at the tip of an edge crack whose length increases. This model is used for the determination of the crack resistance of materials with regard for the influence of various operating factors. On the basis of the method of singular integral equations, we obtain the numerical solutions of model problems for wedge-shaped specimens subjected to uniaxial eccentric tension. We reveal the influence of the modes of loading of these specimens and their geometric parameters on the behavior of the stress intensity factor depending on the crack length and establish the ranges of the geometric parameters guaranteeing the presence of maximum stability regions for the stress intensity factor in the process of quasistatic crack growth.

Управління лінійними неоднорідними системами методами сингулярних інтегральних рівнянь

Управління лінійними неоднорідними системами методами сингулярних інтегральних рівнянь