Method Of Singular Integral Equations Research Articles

Method of singular integral equations in scattering by double-layer infinite strip grating with several strips removed in every layer

The H-polarized plane wave scattering by the infinite double-layer strip grating with several strips removed in every layer is considered. The scattered field is represented as a sum of the field of currents on the ideally periodic double-layer grating and field of correction currents excited due to the absence of strips. The Fourier amplitudes of these fields are obtained from the singular integral equation with additional conditions. The solution is rigorous. Numerical results for the near and far fields are presented.

Investigation of Bending of Anisotropic Plates with Inclusions with the Help of Singular Integral Equations

We consider the problem of bending of an anisotropic plate of constant thickness with rigid inclusions. The algorithm of its solution is constructed on the basis of reduction of the problem of bending of a multiply connected anisotropic plate with inclusions to the evaluation of the Lekhnitskii potentials for an auxiliary plane problem of the theory of elasticity with properly chosen elastic constants. To solve this auxiliary problem, we use the method of singular integral equations in the complex form. We consider the problems of bending for plates with inclusions subjected either to the action of moments at infinity (for infinitely large plates) or to transverse loading (for finite plates). We also present some examples of evaluation of stresses acting in the plates containing rigid inclusions of different shapes or systems of inclusions of this kind.

Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

INTERFERENCE OF CLOSABLE CRACKS AND NARROW SLITS IN AN ELASTIC PLATE UNDER BENDING

The problem of bending of an infinite plate containing an array of trough closable cracks and narrow slits is considered in a two-dimensional statement. A crack is treated as a mathematical cut, the edges of which are able to contact along the line on the plate outside. A slit is referred to as a cut with contact stress-free surfaces and the negative jump of normal displacement can occur on this cut. The crack closure caused by bending deformation was studied based on the classical hypothesis of direct normal and previously developed model of the contact of edges along the line. A new boundary problem for a couple of biharmonic equations of plane stress and plate bending with interconnected boundary conditions in the form of inequalities on the cuts is formulated. The method of singular integral equations was applied in order to develop approximate analytical and numerical solutions to the problem. The forces and moments intensity factors near the peaks of defects and contact reaction on the closed edges of the cracks are calculated. A detailed analysis was carried out for parallel rectilinear crack and slit, depending on their relative location. Presented results demonstrate qualitative differences in the stress concentration near the defects of different nature.

Analysis of Cracked Body Strengthened by Adhesively Bonded Patches by BEM-FEM Coupling

This paper presents an efficient numerical technique capable of handling the stress analysis of three-dimensional cracked bodies strengthened by adhesively bonded patches. The proposed technique is implemented within the framework of the coupling of the weakly singular boundary integral equation method and the standard finite element procedure. The former is applied to efficiently treat the elastic body containing cracks, whereas the latter is adopted to handle both the adhesive layers and patches. The approximation of the near-front relative crack-face displacement is enhanced by using local interpolation functions that can capture the right asymptotic behavior. This also offers the direct calculation of the stress intensity factors along the crack front. A selected set of results is reported to demonstrate the capability of the proposed technique and the influence of various parameters on the performance of the strengthening.

Deformation of an Isotropic Plate with Periodic System of Curvilinear Holes and Plasticity Bands

By the method of singular integral equations, we solve a plane elastoplastic problem of fracture mechanics for an isotropic plane with an infinite series of curvilinear holes and plasticity bands at their tips. We study the influence of the shapes of smooth holes and the radii of rounding of the contours at their tips on the opening displacements and lengths of the plasticity bands. The solutions of the corresponding problems for semiinfinite bilateral rounded notches whose tips serve as the origins of plasticity bands are obtained by the limit transition in the case where the relative distance between the holes tends to zero. On the basis of the deformation criterion of fracture and solutions of the periodic elastoplastic problem, we approximately determine the strength of rectangular specimens with bilateral U-shaped notches.

Stressed State of an Orthotropic Plane with Two-Section Kinked Crack Under Antiplane Deformation

The method of singular integral equations is used to obtain the solution of the antiplane problem of fracture mechanics for an orthotropic plane with two-section kinked crack with regard for the specific features of stresses at the tips of angular cracks. For this purpose, we apply the unified approach developed earlier to the solution of the problems of stress concentration in isotropic bodies with pointed and rounded notches.

Heat Concentration around a Cylindrical Interface Crack in a Composite Tube

Cracks always form at the interface of discrepant materials in composite structures, which influence thermal performances of the structures under transient thermal loadings remarkably. The heat concentration around a cylindrical interface crack in a bilayered composite tube has not been resolved in literature and thus is investigated in this paper based on the singular integral equation method. The time variable in the two-dimensional temperature governing equation, derived from the non-Fourier theory, is eliminated using the Laplace transformation technique and then solved exactly in the Laplacian domain by the employment of a superposition method. The heat concentration degree caused by the interface crack is judged quantitatively with the employment of heat flux intensity factor. After restoring the results in the time domain using a numerical Laplace inversion technique, the effects of thermal resistance of crack, liner material, and crack length on the results are analyzed with a numerical case study. It is found that heat flux intensity factor is material-dependent, and steel is the best liner material among the three potential materials used for sustaining transiently high temperature loadings.

Stressed State of a Plane with Periodic System of Closely Located Curvilinear Holes with Edge Cracks

A plane periodic problem of the theory of elasticity for an isotropic plane with infinite row of closely located curvilinear holes with edge cracks is solved by the method of singular integral equations. The stress intensity factors (SIF) at the tips of edge cracks propagating from symmetric holes of various shapes (elliptic holes, rhombic holes with rounded vertices, and narrow slots) are computed for arbitrary distances between the holes under the conditions of tension of the plane at infinity (mode I). The SIF for the edge cracks at the rounded vertices of the corresponding bilateral notches in the elastic plane are found as a result of the limit transition as the distances between the holes approach zero.

Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions

This paper investigates the influence of crack geometry, crack-face and loading conditions, and the permittivity of a medium inside the crack gap on intensity factors of planar and non-planar cracks in linear piezoelectric media. A weakly singular boundary integral equation method together with the near-front approximation is adopted to accurately determine the intensity factors. Obtained results indicate that the non-flat crack surface, the electric field, and the permittivity of a medium inside the crack gap play a crucial role on the behavior of intensity factors. The mode-I stress intensity factors (KI) for two representative non-planar cracks under different crack-face conditions are found significantly different and they possess both upper and lower bounds. In addition, KI for impermeable and semi-permeable non-planar cracks treated depends strongly on the electric field whereas those of impermeable, permeable, and semi-permeable penny-shaped cracks are identical and independent of the electric field. The stress/electric intensity factors predicted by permeable and energetically consistent models are, respectively, independent of and dependent on the electric field for the penny-shaped crack and the two representative non-planar cracks. Also, the permittivity of a medium inside the crack gap strongly affects the intensity factors for all crack configurations considered except for KI of the semi-permeable penny-shaped crack.

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).

Influence of Flexible Coating on the Limit Equilibrium of a Spherical Shell with Meridional Crack

We study the problem of tension of a spherical shell reinforced by a flexible coating and weakened by a meridional crack. The coating is modeled by a hinge connecting the faces of the cut on one of the face surfaces of the shell. The stressed state near the crack tips and the distribution of the hinge reaction over the coating are determined by the method of singular integral equations. The limit state of the shell is analyzed with regard for the bounded strength of the coating.

Specimen for the Determination of Fatigue Thresholds Under Cyclic Transverse Shear

We develop a basic element of the procedure of construction of the near-threshold part of the diagram of fatigue fracture for metallic materials under the conditions of transverse shear. It is proposed to use a square sample with edge cut and a force scheme of its cyclic loading. We determine the strain-stress state of a square plate with edge crack (regarded as a prototype of the proposed specimen) under the conditions of transverse shear by two independent methods (finite-element method and the method of singular integral equations). The stress intensity factor KII is determined for a broad interval of relative crack lengths with regard for the smooth contact of the crack faces and the presence of friction between the faces.

Scattering of the H-polarized plane wave by finite and semi-infinite multilayer systems of infinite graphene strip gratings in the THz range

AbstractWe consider the scattering and absorption of the H-polarized plane wave by finite and semi-infinite systems of layers in the THz range. Every layer consists of an infinite graphene strip grating embedded into a dielectric slab. The solution of the problem we obtain in several steps. First, with the use of the method of singular integral equations we obtain scattering matrices of a single layer. Then, we present equations for the finite and semi-infinite systems of layers relatively the Fourier amplitudes of the scattered field. The frequency dependences of the reflection, transmission, and absorption coefficients demonstrate the variety of resonances: plasmon, slab-mode, grating-mode resonances, and resonances of the multilayer structure.

Distribution of Stresses Near Angular Notches on an Orthotropic Elastic Plane Under Conditions of Antiplane Deformation

By using a solution of the antiplane eigenvalue problem of the theory of elasticity for an elastic orthotropic wedge, we construct the distribution of singular stresses and displacements in a neighborhood of the tip of the corresponding angular notch under the conditions of antiplane deformation. By the method of singular integral equations, we solve the corresponding problem for an orthotropic plane with semiinfinite angular rounded notch and establish the relationship between the stress intensity and stress concentration factors at the sharp and rounded tips of angular notches.

Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading

The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and singular integral equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.

Stressed State of a Quasiorthotropic Half Plane with Curvilinear Edge

By the method of singular integral equations, we consider a plane periodic problem of the theory of elasticity for a quasiorthotropic half plane containing an infinite row of edge angular rounded notches. By using the unified approach to the solution of the problems of stress concentration near notches with sharp and rounded vertices, we determine the stress intensity factors at the sharp tips of the corresponding edge notches. The same method is used for the solution of the problem of stress distribution in a quasiorthotropic half plane with sinusoidal edge.

Diffraction of H-polarized wave by planar venetian-blind type grating

Diffraction of the plane H- polarized electromagnetic wave by finite graphene venetian-blind type grating is considered. The operator equations with respect to the spectral functions of the scattered field are presented. The reflection operator of a single graphene strip is obtained with the use of the method of singular integral equations. The dependencies of the total scattering and absorption cross section vs. frequency are represented.

Revisit of antiplane shear problems for an interface crack: Does the stress intensity factor for the interface Mode III crack depend on the bimaterial modulus?

An overview of the results for Mode III interface crack problems available in the literature, as well as those obtained by the authors, is presented in the paper. The analytical and semi-analytical solutions for the main fracture characteristics (i.e. stress intensity factors and energy release rates) are discussed with respect to their dependency on bimaterial constants. Detailed formulation for the antiplane shear problem for an interface crack interacting with internal cracks is presented using the methods of complex potentials and singular integral equations. Some particular solutions (e.g. for the problem of a singularity interacting with an interface and for the Mode III interface crack) are obtained, and these can serve as benchmark solutions or as parts of other similar problems. The approximate closed-form solution is derived for a special case of the interaction between the interface crack and internal micro-cracks. The resulting form of the solution for the stress intensity factors is rather simple and its structure is easily analyzed. A theoretical analysis of the explicit form of expressions for the stress intensity factors shows the conditions under which the solution for the interface crack depends on the bimaterial mismatch, and those under which it does not depend. This formula also allows to define the condition, when a small internal crack does not influence the stress intensity factor of the interface crack (i.e. invisible for the interface crack).

Two-Dimensional Periodic Problem of the Theory of Elasticity for an Isotropic Plane with Curvilinear Holes and Edge Cracks

By the method of singular integral equations, we solve a two-dimensional periodic problem of the theory of elasticity for an isotropic plane with infinitely many curvilinear holes whose contours serve as origins of edge curvilinear cracks. By the method of quadratures, we reduce the obtained system of integral equations to a complex system of linear algebraic equations. We also determine the stress intensity factors at the tips of edge cracks when the plane is uniformly loaded at infinity (tension or transverse shear). Finally, the influence of the shapes and relative sizes of holes and cracks on the stress intensity factors is investigated for various external loads.