Method Of Singular Integral Equations Research Articles

Analysis for complex plane cracks in 1D orthorhombic quasicrystals using the singular integral equation method

A singular integral equation method is proposed to analyze the complex plane cracks in one-dimensional (1D) orthorhombic quasicrystals. Using the Somigliana formula, the singular integral equations of the curved crack are derived. Based on the general situation of the curved crack, the singular integral equations of the inclined crack and the arc crack are given. Then the analytical solutions for the singular phonon and phason stresses near the tips of the inclined and the arc crack are obtained. Gauss-Chebyshev quadrature method is introduced to calculate the singular integral equations, and a numerical algorithm for solving the stress intensity factor (SIF) is proposed. Numerical solutions for the phonon and phason SIFs of some examples are solved and discussed.

Singular integral based closed-form solutions for modified EMPS models in semipermeable magneto-electro-elastic materials

The singular integral equation method is applied to present analytical solutions for modified electric-magnetic-polarization saturation (EMPS) models subjected to semipermeable center cracked magneto-electro-elastic (MEE) materials. A generalized methodology is presented to explicitly solve the EMPS model under any arbitrary saturated electric and magnetic conditions using the distributed dislocation method (DDM) and finding the analytical solutions for developed singular integral equations. The explicit expressions are derived for distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), saturated zone lengths, and local stress intensity factor (LSIF) for particular cases of the generalized EMPS model such as constant, linear, quadratic and cubic polynomial varying saturated electric and magnetic conditions. Using an iterative scheme and the explicit solutions of COD, COP, and COI, the exact semipermeable crack-face conditions are evaluated at the center of the crack, and the same applies throughout the crack-surface for the study of semipermeable modified EMPS models. A comparative analysis of the fracture parameters has also been carried out between the results of exact semipermeable crack-face conditions and the results of semipermeable crack-face conditions of the center crack problem to analyze the impact of polynomial varying saturated conditions on the crack-face conditions. Numerical studies performed under different electromagnetic-mechanical loadings, volume fractions, and crack-face conditions demonstrate the increasing effects in the fracture parameters' values with the degree of the polynomial varying saturated conditions.

Quadrature methods for singular integral equations of Mellin type based on the zeros of classical Jacobi polynomials, II

With this paper we continue the investigations started in [6] and concerned with stability conditions for collocation-quadrature methods based on the zeros of classical Jacobi polynomials, not only Chebyshev polynomials. While in [6] we only proved the necessity of certain conditions, here we will show also their sufficiency in particular cases.

An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions

An easy-to-implement recursive fractional spectral-Galerkin method for multi-term weakly singular Volterra integral equations with non-smooth solutions

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ В СТАТИЧНИХ ТРИВИМІРНИХ КРАЙОВИХ ЗАДАЧАХ — КОСОСИМЕТРИЧНА ЗАДАЧА ДЛЯ ШАРУ, ПОСЛАБЛЕНОГО НАСКРІЗНИМ ОТВОРОМ І КОВЗНИМ ЗАЩЕМЛЕННЯМ ТОРЦІВ

Spatial static boundary-value problems of mathematical physics for a layer with a non-circular cylindrical through hole have hardly been solved using the method of singular integral equations (SIE) despite the fact that numerous fundamental theoretical issues have been developed. Methods for the calculation of spectral characteristics are also absent. The paper provides an overview of the methods for solving these problems. A new mathematical model has been constructed, and a new method based on a system of three SIEs has been developed and tested numerically. As a result of a high-precision numerical study, it was found that with an increase in the thickness of the layer, an increase in the relative circumferential stress occurs. In the case of a circular hole, a shift of the maximum relative circumferential stress from the ends to the depth of the layer is observed. In the case of an elliptical hole, with a decrease in one of the radii, an increase in the relative circumferential stress is also observed. Keywords: three-dimensional boundary-value problems, singular integral equations, numerical experiment, static bending, a through hole.

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.

Tunability of Radiation Pattern of the H-Polarized Natural Waves of Dielectric Waveguide with Infinite Graphene Plane and Finite Number of Graphene Strips at THz

We investigate the radiation of the THz natural waves of the dielectric waveguide with graphene plane scattered by finite number of graphene strips. Our mathematically accurate analysis uses the singular integral equations method. The discretization scheme employs the Nystrom-type algorithm. The complex-valued propagation constants of the natural waves and corresponding fields are determined numerically from the equation, which also involves the kernel-function of the singular integral equation. The method we use is meshless and full-wave. The convergence is provided by the mathematical theorems. By varying the chemical potential of graphene and structural geometrical parameters, we examine the elevation angle of the main lobe of the radiation pattern and the radiated power.

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.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

Thermal Fracture of Functionally Graded Coatings with Systems of Cracks: Application of a Model Based on the Rule of Mixtures

This paper is devoted to the problem of the thermal fracture of a functionally graded coating (FGC) on a homogeneous substrate (H), i.e., FGC/H structures. The FGC/H structure was subjected to thermo-mechanical loadings. Systems of interacting cracks were located in the FGC. Typical cracks in such structures include edge cracks, internal cracks, and edge/internal cracks. The material properties and fracture toughness of the FGC were modeled by formulas based on the rule of mixtures. The FGC comprised two constituents, a ceramic on the top and a metal as a homogeneous substrate, with their volume fractions determined by a power law function with the power coefficient λ as the gradation parameter for the FGC. For this study, the method of singular integral equations was used, and the integral equations were solved numerically by the mechanical quadrature method based on the Chebyshev polynomials. Attention was mainly paid to the determination of critical loads and energy release rates for the systems of interacting cracks in the FGCs in order to find ways to increase the fracture resistance of FGC/H structures. As an illustrative example, a system of three edge cracks in the FGC was considered. The crack shielding effect was demonstrated for this system of cracks. Additionally, it was shown that the gradation parameter λ had a great effect on the fracture characteristics. Thus, the proposed model provided a sound basis for the optimization of FGCs in order to improve the fracture resistance of FGC/H structures.

Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials

In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C*-algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials.

МАТЕМАТИЧНА МОДЕЛЬ ВЗАЄМОДІЇ СТАЦІОНАРНИХ SH-ХВИЛЬ З СИСТЕМОЮ КРИВОЛІНІЙНИХ ТРІЩИН У НАПІВПРОСТОРІ

A method for solving mathematical physics problems is proposed for semi-infinite media containing systems of curved crack-slits. Most of the studies known from the literature relate to the problems of diffraction of elastic waves on straight and circular cuts. However, in reality, the crack is usually not straight or circular. Studies have shown that the curvature of a crack significantly affects the value of the dynamic stress intensity coefficients. The value of this parameter also depends on the proximity of the defects to each other, since they always fall within the range of the reflected wave. The stress-strain state of media with complex properties can be effectively modeled by computing complexes in combination with software systems. Most studies are devoted to the development of the finite element method. However, the method of integral equations is very effective for solving anti-planar problems of diffraction theory. The advantage of this method is the reduction in the number of spatial variables, the high speed of convergence, and the possibility of using various efficient numerical solution methods. The method also has the ability to build efficient parallel computing schemes. A unified approach to solving the problem is developed on the basis of singular integral equations (SIEs). The corresponding dynamic boundary value problems for a clamped and force-free half-plane are investigated. The influence of the defect curvature, their interaction, and the proximity of the boundary on the magnitude and nature of the dynamic stress intensity coefficients is studied. Parallel algorithms allow to significantly reduce the computation time and analyse the characteristics of the wave field in more detail. The combination of the SIE method, which reduces the dimensionality of the problem by one, and provides significant savings in computing time due to the parallelization of computational procedures, leads to a significant increase in the efficiency of the proposed algorithm. The method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

Direct methods for singular integral equations and non-homogeneous parabolic PDEs

In this article, the author presented some applications of the Laplace, \(L^2\), and Post-Widder transforms for solving fractional Singular Integral Equations, impulsive differential equation and systems of differential equations. Finally, analytic solution for a non-homogeneous partial differential equation with non-constant coefficients is given. The obtained results reveal that the integral transform method is an effective tool and convenient.

Application of the Deformation Fracture Criterion to Cracking of Disc Specimens with a Central Narrow Slot

Abstract Using the method of singular integral equations, the elastic-plastic problem for cracked Brazilian disk was solved. Based on the Dugdale model and deformation fracture criterion, the relationships between critical load, notch tip opening displacement and length of the plastic strips were established. Also, the comparison between the present solution for the finite domain and the known solution obtained for the semi-infinite notch in the elastic plane was performed.

MODELING THE EFFECT OF STOCHASTIC DEFECTS FORMED IN PRODUCTS DURING MACHINING ON THE LOSS OF THEIR FUNCTIONAL DEPENDENCIES

The article investigates the influence of hereditary defects formed in the surface layer of products from metals of heterogeneous structure on the quality of surfaces treated with finishing methods. The research is based on an integrated approach based on the results of the deterministic theory of defect development and methods of probability theory. The treated layer of the product is considered as a medium weakened by random defects that do not interact with each other, namely: structural changes, cracks, inclusions, the parameters of which are random variables with known laws of their probability distribution. The causes of structural changes, crack formation on the treated surface product depending on different types of probability distribution of dimensions are investigated: length, depth of defects, and their orientation. From these positions, technological possibilities of their elimination by definition of branch of combinations of the technological parameters providing necessary quality of the processed surfaces are considered. Modeling of thermomechanical processes in the treated surface containing hereditary defects is carried out based on thermoelastic equations with discontinuous boundary conditions in the places of their accumulation. The research used the apparatus of boundary value problems of mathematical physics equations, the method of singular integral equations for solving problems of fracture mechanics, Fourier-Laplace integral transformations for obtaining exact solutions, the method of constructing discontinuous functions. The dependences determining the intensity of stresses in the vertices of hereditary defects are obtained. A method for predicting the nature of crack formation depending on the probability distribution of defects, the values of heat flux entering the surface layer of the processed product has been developed. It is established that the increase in the homogeneity of the material leads to an increase in the value of heat flux, which corresponds to a fixed probability of failure.

The axisymmetric contact in couple-stress elasticity taking into account adhesion

In this paper, the spherical indenter contact problem for the couple-stress elasticity by considering the adhesion is investigated. The Maugis–Dugdale adhesive theory is considered to describe the adhesion effect. The mixed boundary problem is solved by using the Hankel transform technical and singular integral equation method. The influences of the adhesion parameter, the characteristic material length parameter and Poisson’s ratio on the pull-off force, the relation between the applied force and indentation depth, the contact stresses and the normal displacement are discussed. The results predicted the importance of size-effect in the adhesive contact. The present work provides a fundamental basis for probing the mechanical properties in the microstructure materials.

An efficient method for singular integral equations of non-normal type with two convolution kernels

This article deals with the solvability and explicit solutions of one class of singular integral equations with two convolution kernels in the non-normal type case. Via using techniques of complex analysis, such equations are transformed into Riemann–Hilbert boundary value problems (R-HPs) with the discontinuous property. We establish a regularity theory and existence of solutions, and obtain the general solutions and the conditions of Noether solvability. Especially, we investigate the asymptotic behaviours of solutions at nodes. Thus, this paper generalizes the theories of integral equations and Riemann–Hilbert boundary value problems.

Stress Concentration in Bounded Compositeplates with Carbon Reinforcement

The research purpose is to develop an approach for determining the stress concentration near the holes in composite structure elements reinforced with carbon fibres. The research is performed on the basis of a numerical-analytic approach using the method of singular integral equations. The paper studies the stress concentration near the holes in composite plate elements of the structures, which are reinforced with carbon fibres. The stresses are determined based on the singular integral equations. The integral equations are solved numerically using the mechanical quadrature method. The stress in the strip is studied at: longitudinal tension; pure bending; three-point bending; with periodically spaced holes. An approach to calculating the stresses in composite strips weakened by holes of different shapes, based on the method of integral equations, has been developed. The equation kernels are formulated on the basis of Green's functions, under which the boundary conditions on straight-line boundaries are satisfied identically. A methodology for calculating the stress concentration near the holes of arbitrary shape in plate elements of the structures has been developed. The results obtained can be used when calculating the strength of composite materials reinforced with carbon fibres.

Singular integral equations in plane wave scattering by infinite graphene strip grating with brake of periodicity

Abstract In this paper, the solution of the H-polarized wave scattering problem by infinite graphene strip grating is obtained. The structure is periodic except two neighboring strips. The distance between these two strips is arbitrary. In particular, such a problem allows to quantify the mutual interaction of graphene strips in the array. The total field is represented as a superposition of the field of currents on the ideally-periodic grating and correction currents induced by the shift of the strips. The analysis is based on the convergent method of singular integral equations. It enables us to study the influence of the correction currents in a wide range from 10 GHz to 6 THz. It is shown that the interaction between graphene strips is strong near plasmon resonances and near the Rayleigh anomaly.

Excitation of guided waves of grounded dielectric slab by a THz plane wave scattered from finite number of embedded graphene strips: Singular integral equation analysis

The scattering and absorption of the H-polarised wave by a finite number of graphene strips embedded into a grounded dielectric slab is considered in the terahertz frequency range. The rigorous and efficient method of singular integral equations is used here. The discretisation is based on the convergent meshless Nystrom-type algorithm. Graphene is modelled as a zero thickness layer with a complex conductivity obtained from the Kubo formalism. Dependences of the total scattering cross section, including the part that is associated with dielectric slab guided waves, and absorption cross section as well as near field patterns are studied. The behaviour of the scattering and absorption characteristics is explained in terms of resonances on the natural modes supported by the studied structure. Special attention is paid to the excitation of the dielectric waveguide eigenwaves near the grating-mode resonances.