System Of Singular Integral Equations Research Articles

A Circular Inclusion and Two Radial Coaxial Cracks with Contacting Faces in a Piecewise Homogeneous Isotropic Plate Under Bending

Abstract The bending problem of an infinite, piecewise homogeneous, isotropic plate with circular interfacial zone and two coaxial radial cracks is solved on the assumption of crack closure along a line on the plate surface. Using the theory of functions of a complex variable, complex potentials and a superposition of plane problem of the elasticity theory and plate bending problem, the solution is obtained in the form of a system of singular integral equations, which is numerically solved after reducing to a system of linear algebraic equations by the mechanical quadrature method. Numerical results are presented for the forces and moments intensity factors, contact forces between crack faces and critical load for various geometrical and mechanical task parameters.

NUMERICAL SOLUTION OF A SYSTEM OF SINGULAR INTEGRAL EQUATION WITH HILBERT AND CAUCHY KERNELS

In the paper a specific system of first kind singular integral equations with the Hilbert and Cauchy kernels arising when solving some problems of electrostatics and electrodynamics is studied. The method of discrete singularities is applied for constructing its discrete mathematical model, which is a system of n linear algebraic equations. Under the additional smoothness assumptions on the right-hand parts of the equations of the initial system and regularity assumptions on the kernels of the integrals in them the obtained system of linear algebraic equations is proved to admit a unique solution for n sufficiently large. The rate of convergence of the solution of the discrete problem to the exact solution of the system of singular integral equations is estimated.

Transient interaction of edge interfacial cracks in bonded dissimilar strips with a functionally graded interlayer: Antiplane deformation

Transient stress intensification due to the interaction of two parallel, edge interfacial cracks in bonded media with a functionally graded interlayer is investigated under the condition of antiplane shear impact. The graded interlayer is assumed to exhibit power-law variations in shear modulus and mass density between dissimilar, homogeneous semi-infinite strips. Laplace and Fourier integral transforms are employed to derive a system of singular integral equations with generalized Cauchy kernels, which is solved in the Laplace domain. Mode III stress intensity factors are defined and evaluated as a function of time. Numerical results include variations of dynamic stress intensity factors, addressing the impact-induced interaction between two nearby edge interfacial cracks affected by geometric and material parameters of the bonded system. More specifically, this study elaborates on the effects that layer thickness, crack length, shear modulus, and mass density have on the dynamic overshoot characteristics of transient behavior in the near-tip regions.

Thermally active interphase inclusion in a smooth contact conditions with transversely isotropic half-spaces

An exact solution of the stationary thermoelasticity problem about interfacial circular absolutely rigid inclusion, which is in the smooth contact conditions in a piecewise homogeneous transversely homogeneous space, is constructed. The task with the help of the constructed discontinuous solution, by the method of singular integral relations, is reduced to a system of singular integral equations (SIE). An exact solution has been built for the specified SIE, as a result, dependences of translational displacement of the inclusion on temperature, the resultant load, main moment and thermomechanical characteristics of transversely isotropic materials have been obtained.

Investigation of transient thermal-mechanical behavior of a cracked viscoelastic material using time-fractional dual-phase-lag theory

In the past decade, viscoelastic materials (like polymer composites, biological tissues, rubber, etc.) have been increasingly used in a variety of industries due to their excellent multifunctional properties. In this work, by extending the fractional calculus to dual-phase-lag (DPL) heat conduction theory, the transient thermal–mechanical response in cracked viscoelastic materials under thermal shock is analyzed. With the aids of Fourier and Laplace transform, the thermal-viscoelastic problem is converted into a system of singular integral equations with Cauchy kernels of the first kind, which are then solved numerically. The parametric study is performed and the numerical results of temperatures, intensity factors of temperature gradient (IFTG) and stress intensity factors (SIFs) around the crack tips, are shown graphically to illustrate the effects of time fractional order, thermal lags and viscoelastic relaxation on the thermoelastic response.

Detailed Solution of a System of Singular Integral Equations for Mixed Mode Fracture in Functionally Graded Materials

A mixed mode crack problem in functionally graded materials is formulated to a system of Cauchy singular Fredholm integral equations, then the system is solved by the singular integral equation method (SIEM). This specific crack problem has already been solved by N. Konda and F. Erdogan (Konda & Erdogan 1994). However, many mathematical details have been left out. In this paper we provide a detailed derivation, both analytical and numerical, on the formulation as well as the solution to the system of singular Fredholm integral equations. The research results include crack displacement profiles and stress intensity factors for both mode I and mode II, and the outcomes are consistent with the paper by Konda & Erdogan (Konda & Erdogan 1994).

A quadrature approach to the generalized frictionless shearing contact problem

In this study, the generalization of a frictionless contact problem in case of shearing deformation for an elastic inhomogeneous half space is presented. The basic equations of the elasticity theory and Fourier transform technique are applied to the problem to derive the system of singular integral equations. The obtained system of singular integral equations is solved by a quadrature approach. The numerical results are presented for the case of $N=1$, $N=2$, $N=3$, where $N$ denotes the number of the punches whose base are flat.

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact The problem of the interaction of harmonic waves with a thin elastic circular inclusion, which is located in an elastic isotropic body (matrix), is solved. On both sides of the inclusion between it and the body (matrix), the conditions of smooth contact are realized. The solution method is based on representing the displacements in the matrix through discontinuous solutions of the Lamé equations for harmonic vibrations. This made it possible to reduce the problem to Fredholm integral equations of the second kind with respect to functions associated with jumps in normal stress and radial displacement to included ones. After the realization of the boundary conditions on the sides of the inclusion, a system of singular integral equations is obtained to determine these jumps. Keywords: elastic inclusions, cylindrical waves, matrix, stress intensity factor.

Methods for solving incorrect tasks of electrodynamic analysis of radiating structures based on chiral metamaterials

This article is devoted to the development of correct methods for electrodynamic analysis of single and multi-element radiating structures with substrates of chiral metamaterials. An approach to the electrodynamic analysis of such antennas based on the method of singular integral representations of the field and the method of surface impedances is proposed. A system of singular integral equations with respect to unknown distribution functions of current density over emitters is obtained. The proposed method was tested using the example of solving a test problem using the Feko software package. The prospects of using such ICs in MIMO systems are shown, which make it possible to increase their efficiency, in particular, to increase throughput. The effect of azimuthal scattering of electromagnetic waves is also presented, which can be used to create low-profile antennas of subscriber radio communication stations installed on mobile objects.

Physically nonlinear deformation of a thin interphase inclusion under antiplane problem conditions

The longitudinal shear problem of the bimaterial with thin physically nonlinear inclusion at the interface matrix materials is considered. The solution of the formulated problem is constructed by the method of the conjugation of limit values of analytical functions with the use of the jump function method. A model of thin inclusion with arbitrary nonlinear strain characteristics is constructed. The solution of the problem is reduced to a system of singular integral equations with variable coefficients. A convergent iteration method for solving such a system for different types of physically nonlinear deformation is proposed. An incremental calculation method for calculating stress-strain state under multistep (including cyclic) quasi-static loading is developed. Numerical calculations of the body stress-strain state for various values of the parameters of the nonlinearity of the inclusion material are carried out. Their influence on the mode of deformation of the matrix under loading by a balanced system of concentrated forces is investigated.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

Mixed-mode computation of the transient dynamic stress intensity factor for multiple interface cracks

In this paper, the transient dynamic stress intensity factors are estimated for multiple cracks which are located at the interface between a nonhomogeneous elastic half-plane and an elastic half-plane. The material properties of the nonhomogeneous half-plane vary continuously in the y-direction. First, Laplace and Fourier transforms are applied to reduce the mixed boundary value problem to a system of singular integral equations with Cauchy kernels which are solved numerically. By numerical Laplace inversion technique, the dislocation density on the crack faces is obtained and then transient dynamic stress intensity factors at crack tips have been determined. For validity and accuracy of the method, the results are compared with other references and very good agreement is shown. The influences of nonhomogeneity parameters, crack length, the variation in time and the interaction between of cracks on the transient dynamic stress intensity factors are studied. It is observed that the mode I stress intensity factors decrease regularly with increasing the FG constant and the results are briefly discussed.

Solvability of infinite system of nonlinear singular integral equations in the C(Itimes I, c) space and modified semi-analytic method to find a closed-form of solution

‎In this article‎, ‎we discuss about solvability of infinite systems of singular integral equations with two variables in the Banach sequence space $C(I times I‎, ‎c)$ by applying measure of noncompactness‎ and Meir-Keeler condensing operators‎. By presenting an example, we have illustrated our results‎. ‎For validity of the results we introduce a modified semi-analytic method in the case of two variables to make an iteration algorithm to find a closed-form of solution for the above problem‎. ‎The numerical results show that the produced sequence for approximating the solution of example is in the $c$ space with a high accuracy‎.

Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches

The problem of propagation of obliquely incident surface water waves over a pair of asymmetrical rectangular trenches in a channel of finite depth is examined for its solution. The mathematical problem is handled for its numerical solution with the aid of a system of singular integral equations of first kind. The resulting integral equations are solved numerically by using suitably designed polynomial approximations of the unknown functions. System of linear algebraic equations is obtained by utilizing the zeros of Chebyshev polynomial of first kind as collocation points and hence the occurrence of ill-conditioned matrix is avoided. The effectiveness of the pair of trenches is studied by analyzing the physical quantities namely the reflection and transmission coefficients. As a special limiting case, the results for a single trench problem are derived and are found to be in excellent agreement with the results available in the literature. The effect of various parameters are analyzed through different graphs for a pair of asymmetrical trenches. The energy balance relation is derived and used to check the accuracy of numerical results.

Contact Problem for Layered Medium Supported by a Wedge

In this paper the plane contact problem for a layered medium supported by an elastic wedge is considered. A compressive load is applied to the top of the layered medium through a frictionless rigid punch. It is assumed that all contact areas are frictionless and only compressive normal tractions can be transmitted through the interfaces. Hence, the length of the contact region along the layers and layer-wedge interface is finite and are also unknown. The problem is formulated in terms of a system of singular integral equations in which the unknown functions are the pressure between layers in layered medium and between layered medium and the wedge. The analytical formulation of layered medium supported by wedge is given. As a numerical example, contact length and stress intensity factor at the tip of rails are calculated and the result are given by diagrams.

Formulation of the Algorithm to Calculate Constants Characterizing Rock Mass with a Stope

A method is proposed for solving inverse problems characterized by a set of parameters, which uses a system of singular integral equations connecting the boundary values of stress and displacement components and excluding regularization. The calculation involves specifying the static, kinematic and dynamic aspects and adapting them to the conditions of mining a specific seam. The static aspect is understood as classical calculation of the stress-strain state in the vicinity of a stope, the kinematic one takes into account the dead weight of rocks, and the dynamic one considers mining the seam and damage accumulation in enclosing rocks.

Strength of inhomogeneous rib pillars

The author analyzes stress state of a piecewise-homogeneous pillar with a curved interface. Based on the system of singular integral equations, relations are derived for stresses and displacements at the boundary of the pillar and at the interfaces of the pillar components. Examples of numerical implementation are given.

On Solution of Singular Integral Equations by Operator Method

In this paper, we study the exact solution of singular integral equations using two methods, including Adomian Decomposition Method and Elzaki Transform Method. We propose an analytical method for solving singular integral equations and system of singular integral equations, and have some goals in our paper related to suggested technique for solving singular integral equations. The primary goal is for giving analytical solutions of such equations with simple steps, another goal is to compare the suggested method with other methods used in this study.

Mixed mode transient analysis of functionally graded piezoelectric plane weakened by multiple cracks

In this paper, a functionally graded piezoelectric (FGP) plane weakened by multiple parallel cracks is investigated under mixed mode mechanical and electrical impacts. The system contains Volterra type glide and climb edge dislocations. The material properties of the system are assumed to vary exponentially along the x-axis. At first, the stress field and electric displacement in FGP plane are derived by using a single dislocation. Then, a system of singular integral equations with a simple Cauchy kernel of the transient dynamic problem is obtained by determining a distributed dislocation density on the crack surface and using the Fourier and Laplace integral transforms, which are solved numerically using the Lobatto-Chebyshev integration formula. The dynamic field intensity factors are determined through numerical Laplace inversion and the distributed dislocation technique (DDT). Finally, various examples are provided to investigate the influence of the material properties, electromechanical coupling factor, loading conditions and cracks arrangement on the dynamic fracture behavior of the interacting cracks.

Automated numerical simulation of the propagation of multiple cracks in a finite plane using the distributed dislocation method

In this paper, an automated numerical simulation of the propagation of multiple cracks in a finite elastic plane by the distributed dislocation method is developed. Firstly, a solution to the problem of a two-dimensional finite elastic plane containing multiple straight cracks and kinked cracks is presented. A serial of distributed dislocations in an infinite plane are used to model all the cracks and the boundary of the finite plane. The mixed-mode stress intensity factors of all the cracks can be calculated by solving a system of singular integral equations with the Gauss–Chebyshev quadrature method. Based on the solution, the propagation of multiple cracks is modeled according to the maximum circumferential stress criterion and Paris' law. Several numerical examples are presented to show the accuracy and efficiency of this method for the simulation of multiple cracks in a 2D finite plane.