System Of Singular Integral Equations Research Articles

Elastoplastic Limit State of Inhomogeneous Shells of Revolution with Internal Cracks

By using an analog of the δc -model, we reduce the problem of stressed state and limit equilibrium of an inhomogeneous shell of revolution weakened by an internal crack of any configuration with plastic strains developed on the continuation of the crack in the form of a narrow strip to an elastic problem. The indicated elastic problem is then reduced to a system of singular integral equations with unknown limits of integration and discontinuous functions on the right-hand sides. We propose an algorithm for the numerical solution of these systems with regard for the conditions of plasticity of thin shells and the conditions of boundedness for stresses. The effects of loading, geometric parameters, and mechanical characteristics on the crack-opening displacement and the sizes of plastic zones are investigated for cylindrical and spherical shells made of functionally graded materials and containing internal parabolic cracks.

Modeling of deformation of the bimaterial with thin Non-linear interface inclusion

An incremental approach to solving the antiplane problem for bimaterial media with a thin, physically nonlinear inclusion placed on the materials interface is discussed. Using the jump functions method and the coupling problem of boundary values of the analytical functions method we reduce the problem to the system of singular integral equations (SSIE) on jump functions with variable coefficients allowing us to describe any quasi-static loads (monotonous or not) and their influence on the stress-strain state in the bulk. To solve the SSIE problem, an iterative analytical-numerical method is offered for various non-linear deformation models. Numerical calculations are carried out for different values of non-linearity characteristic parameters for the inclusion material. Their parameters are analyzed for a deformed body under a load of a balanced concentrated force system.

The Carleman regularization technique in the modelling of the plane E ‐polarized electromagnetic wave scattering by a flat system of impedance strips

The Carleman method of analytical regularization is used to model the plane E-polarized electromagnetic wave scattering by flat grating of a finite number of impedance strips. It is based on a modification of the singular integral equation technique, the perturbation method and the Carleman inversion formula for the simplest logarithmically singular integral equation. Attention is focussed on the asymptotic model of the plane E-polarized electromagnetic wave scattering by a sparsely filled impedance strip grating. In this case, the corresponding singular integral equation system has an explicit solution. This result is important to verify 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.

Transient response of collinear Griffith cracks in a functionally graded strip bonded between dissimilar elastic strips under shear impact loading

This article analyses the interaction between a central and two symmetrically placed collinear Griffith cracks subjected to transient response under anti-plane shear impact loading. The cracks are situated in a strip constituted by functionally graded material (FGM) bonded between two dissimilar elastic strips of equal thickness. The material properties of FGM are assumed to vary exponentially as a function of thickness. Applying integral transforms, the boundary value problem reduces to a system of singular integral equations in the Laplace transformed domain. These equations are solved numerically using the Lobatto-Chebyshev collocation quadrature approach. The inverse Laplace transform is used to find the approximate expressions of dynamic stress intensity factors (DSIFs). The striking feature of the article is the study of phenomenal changes of shielding and amplification through dynamic stress magnification factors (DSMFs) at the tips of the cracks under the sudden impact loading applied at the upper material surface. The effects of impact load applied at different surfaces, positions of cracks’ axis and the thickness of the strips of the composite material on the possibilities of cracks’ arrest are depicted graphically for different particular cases.

Solvability of a Certain System of Singular Integral Equations with Convex Nonlinearity on the Positive Half-Line

We study a system of nonlinear singular integral equations with a sum-difference kernel on the positive half-line. Such a system occurs (in various representations) in many branches of mathematical physics and applied mathematics. In particular, one encounters a system of equations with a kernel representing a Gaussian distribution and with a power nonlinearity in the dynamic theory of p-adic open-closed strings; in the case when the nonlinearity has a certain exponential structure, such a system occurs in mathematical biology, namely, in the theory of the spatio-temporal distribution of an epidemic. We prove constructive theorems on the existence of nonnegative nontrivial continuous and bounded solutions. We study the uniqueness and the asymptotic behavior of constructed solutions at infinity. In conclusion, we give concrete applied examples of the mentioned equations that satisfy all assumptions of the proved theorems.

Mathematical modeling of thermoelastic state in a tree-component piecewise-homogeneous plate containing a crack

Purpose. A two-dimensional mathematical model of the problem of thermoelasticity for three-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. Graphic dependences of stress intensity factors (SIFs) at the tops of the crack have been built. This would make it possible to obtain the critical values of constant temperature in the two joined dissimilar elastic half-planes containing an inclusion and a crack in order to prevent crack growth, which would not allow the local destruction of the body. Methodology. Based on the method of the function of a complex variable we have studied the two-dimensional thermoelastic state for 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. The two-dimensional mathematical model of the thermoelastic state has been built in order to determine the stress intensity coefficients 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. Numerical solution of the integral equations in the case of constant temperature in the two joined dissimilar elastic half-planes containing the crack and an inclusion was obtained by the mechanical quadrature method. Influence of thermophysical and mechanical properties of an inclusion on the SIF sat the crack types was investigated. Graphic dependences of the stress intensity factors which characterize distribution of the intensity of stresses at the vertices of a crack have been built, as well as on its elastic and thermoelastic characteristics of inclusion. This would make it possible to analyze the intensity of stresses in the neighborhood of a crack vertices depending on the geometrical and mechanical factors. As a result, this allow 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 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 (a two joined dissimilar elastic half-planes containing inclusion and a crack) due to the action of constant temperature is 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 appropriate selection of composite components with appropriate mechanical characteristics.

О разрешимости одной системы сингулярных интегральных уравнений с выпуклой нелинейностью на положительной полупрямой

О разрешимости одной системы сингулярных интегральных уравнений с выпуклой нелинейностью на положительной полупрямой

Micromechanical investigation of thermal stresses of a finite plane containing multiple elliptical inclusions based on the equivalent inclusion method and distributed dislocation method

This paper presents a micromechanical method to analyze the thermal stresses in a finite plane containing multiple elliptical inclusions. Firstly, the Eshelby’s equivalent inclusion method is employed to solve the elastic fields of a two-dimensional infinite plane containing multiple elliptical inclusions under a uniform temperature change. Both the interior Eshelby’s tensor and the exterior Eshelby’s tensor are employed. Then the boundary of the plane is modeled by continuous distributions of dislocation densities. By combining the two steps, a system of singular integral equations is formulated based on the traction-free boundary condition. Then the thermal stresses of the plane can be obtained by the superposition of the stresses obtained by the Eshelby’s equivalent inclusion method and distributed dislocation method. Additionally, some examples are given to show the effects of the presented method. The effects of the material constants, geometric parameters and fiber packing arrangement on the thermal stresses are also studied.

Investigation of interactions among collinear Griffith cracks situated in a functionally graded medium under thermo-mechanical loading

This study deals with the interactions between a central crack and two symmetrically situated collinear Griffith cracks in an infinite functionally graded medium under thermo-mechanical loading. These Griffith cracks are partially insulated. The considered medium is a non-homogeneous isotropic elastic one. The Fourier sine and cosine transforms are used to solve the elasticity and heat conduction equations which are reduced to a system of singular integral equations of the first kind. These equations are solved numerically by using Chebyshev polynomials of the first kind. The approximate expressions of the normalized mode I stress intensity factors and stress magnification factors are found analytically. The main objective of this article is to study the influence of the relative sizes of collinear cracks on mode I stress intensity factors and to find the possibility of crack shielding and amplification. The salient feature of this article is the pictorial presentations of temperature field, thermal crack surface stresses, stress intensity factors, and stress magnification factors under thermal, mechanical, and thermo-mechanical loadings for different particular cases.

Thermal shock fracture associated with a unified fractional heat conduction

Recently, a unified fractional heat conduction model is newly proposed, which is a further improvement of the fractional thermoelasticity theory. It would be interesting to know how the newly developed fractional derivative definitions affects the thermal fracture behavior when some defects such as cracks exist in the medium. This work is aimed at analyzing the thermal fracture problem of an elastic half-space and strip with an insulated Griffith crack based on the unified fractional heat conduction model. Fourier and Laplace transforms are employed to solve the mixed boundary problem associated with a time-fractional partial differential equation. Temperature and stress intensity factors are evaluated by solving a system of singular integral equations. Effects of different fractional derivatives and different parameters on the thermoelastic fields are analyzed. A comparison of the temperature and stress intensity factors between the present model and the Fourier model is made. Numerical results show that different fractional definitions have different memory effects on the transient responses, and the effects of fractional definitions on the responses will vary accordingly with the parameter values.

Fracture Analysis of Griffith Interface Crack in Fine-Grained Piezoelectric Coating/Substrate under Thermal Loading

Coating often plays a role in monitoring and protecting substrates in engineering applications. Interface cracks between the coating and the substrate can lead to crack growth under the action of external loading and will cause device failure. In this paper, the behavior of a fine-grained piezoelectric coating/substrate with a Griffith interface crack under steady-state thermal loading is studied. The temperature field, displacement field, and electric field of the coupling of thermal and electromechanical problems are constructed via integral transformation and the principle of superposition. Thus, problems are transformed into a system of singular integral equations, and the expressions of thermal intensity factor, thermal stress intensity factor, and electric displacement intensity factor are obtained. We used a numerical calculation and a system of singular equations to obtain the relationship of strength factor with material parameters, coating thickness, and crack size.

Liquid vibrations in circular cylindrical tanks with and without baffles under horizontal and vertical excitations

The paper is devoted to issues of numerical simulation of free surface elevations in rigid circular cylindrical fluid-filled tanks under external vertical and horizontal loadings with usage boundary element methods. The liquid in reservoirs is assumed to be ideal and incompressible, and its flow is irrotational. The influence of horizontal and vertical baffle installation is considered and damping effects due baffles are estimated. The influence of elasticity effects is considered. Free, forced and parametrical vibrations are examined. Modes of the free liquid vibrations present basic functions for analysis of forced and parametric vibrations. The modes and frequencies of the free liquid vibrations in baffled and un-baffled circular cylindrical tanks are received by using single- and multi-domain boundary element methods. The new effective procedure is developed for numerical solution of resolving system of singular integral equations. The problems of forced vibrations are reduced to solving the systems of second order ordinary differential equations. For parametric vibrations the system of Mathieu equations with non-zero right-hand parts is obtained. The numerical simulation of different kinds of loadings and baffle configurations is accomplished. The instability conditions of the fluid-filled tank subjected to both horizontal and vertical excitations are considered.

Interaction of multiple straight cracks and elliptical inclusions in a finite plate due to mismatched thermal expansion

The purpose of this paper is to study the interaction between multiple elliptical inclusions and straight cracks in a finite plate due to a uniform temperature change. In this solution, Eshelby’s equivalent inclusion method involving both interior Eshelby’s tensor and exterior Eshelby’s tensor is applied to calculate the thermal stress fields of an infinite plate containing multiple elliptical inclusions under a uniform temperature change first. Then the multiple cracks and the boundary are modeled by continuous distributions of dislocation densities in an infinite plate. Based on the stress boundary conditions of the cracks and boundary, a system of singular integral equations with Cauchy kernels are obtained. After solving the singular integral equations with Gauss–Chebyshev numerical quadrature, the stress intensity factor of each crack can be calculated. Besides, the finite element method is employed to examine the accuracy and efficiency of the presented method. Finally, the effects of the material and geometric parameters on the normalized stress intensity factors of the cracks are studied.

Limit Equilibrium of a Cylindrical Shell with Longitudinal Crack with Regard for the Inertia of the Material

We study the problem of limit equilibrium of a long cylindrical shell with longitudinal crack subjected to the action of a load that varies with time according to the exponential law. For the case of symmetric loading of the crack, we construct a system of singular integral equations. We also study the influence of the rate of changes in the load on the value of the force intensity factor in the vicinity of the crack ends.

Numerical Analysis of Systems of Singular Integral Equations of the First Kind with an Indefinable Index in the Problem of Diffraction of Plane Waves on a Rigid Inclusion

Numerical Analysis of Systems of Singular Integral Equations of the First Kind with an Indefinable Index in the Problem of Diffraction of Plane Waves on a Rigid Inclusion

The Bitsadze–Samarskii Type Problem for Mixed Type Equation in the Domain with the Deviation from the Characteristics

For a mixed elliptic-hyperbolic type equation in a mixed domain, when the elliptic part is a sector of the circle, and the hyperbolic part consists of two characteristic triangles, the Bitsadze–Samarskii type nonlocal problem is investigated. A feature of such problems is that the boundary conditions in the hyperbolic parts of the boundary are determined by a first-order differential operator and pointwise link the values of the partial derivatives of the desired solution on the characteristics with the partial derivatives on arbitrary monotone curves lying inside the characteristic triangles of the equation. To solve the problem, the methods of the theory of partial differential equations and the theory of singular integral equations as well as the methods of energy integrals and complex analysis were used, with the help of which the existence and uniqueness theorem for the solution of the investigated problem is proved. Also, a method of reducing the investigated problem to an equivalent system of singular integral equations is shown, also, a method for solving this system is proposed, which allows to obtain a solution to the problem in explicit form.

Modelling of Impulse Load Influence on the Stress State of Foam Materials with Positive and Negative Poisson’s Ratio

Abstract The influence of impulse load applied for different duration on the distribution of normalised dynamic radial stresses in positive and negative Poisson’s ratio medium was investigated in this study. For solving the non-stationary problem in the case of plane deformation for structurally inhomogeneous materials, the model of Cosserat continuum was applied. This model enables accounting for the influence of shear-rotation deformation of micro-particles of the medium. In the framework of Cosserat elasticity, on applying the Fourier transforms for time variable and developing the boundary integral equation method, solving of the non-stationary problem reduces to the system of singular integral equations, where the components that determine the influence of shear-rotation deformations are selected. The numerical calculations were performed for the foam medium with positive and negative Poisson’s ratio for different values of time duration of impulse. Developed approach can be used to predict the mechanical behaviour of foam auxetic materials obtained at different values of a volumetric compression ratio under the action of time variable load based on analysis of the distribution of radial stresses in foam medium.

The plane H-polarized electromagnetic wave scattering by pre-fractal grating of impedance strips

AbstractThe problem of the plane H-polarized electromagnetic wave scattering by flat pre-fractal impedance strips' gratings is examined. For this purpose, a mathematical model in the form of the first kind singular integral equation system is modified for correct usage. Considerable attention is focused on the asymptotic model of the H-polarized electromagnetic wave scattering by sparsely filled grating, which has an explicit solution. The scattered electromagnetic field in the far-zone is considered in details. Dependences of scattering integral characteristics on values of the strip impedance and angles of the plane H-polarized electromagnetic wave incidence upon the grating are studied.

An axisymmetric crack in a functionally graded thermal barrier coating bonded to a homogeneous elastic substrate under transient thermal loading

In this paper, the fracture problem of an axisymmetric crack in a functionally graded thermal barrier coating (FGTBC) bonded to a homogeneous substrate is considered. The problem is solved for the laminate that is suddenly heated from the upper surface of the FGTBC. The bottom surface of the homogeneous substrate is maintained at the initial temperature. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. By using both the Laplace and Hankel transforms, the thermo-mechanical fracture problem is reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The stress intensity factors of the crack are computed and presented as functions of the normalized time for various values of the nonhomogeneous and geometric parameters.

About the problem of flow past around thin profiles

In this paper some approaches to solving the problems of flow past around thin profiles are presented. We consider the problem of flow past around a contour defined by a set of segments, each of which is close to the segment ( a; a). A formula that gives an approximate velocities distribution in a flow flowing around a system of thin profiles is obtained. A mixed boundary-value problem for an analytic function in the upper half-plane is studied. A system of singular integral equations having a unique solution up to two arbitrary constants in the class of functions that are boundary values of analytic functions in the upper half-plane is obtained. Considered the generalized problem of flow past around thin profiles for infinite numbers of thin profiles.