Simultaneous Integral Equations Research Articles

Axisymmetric torsion of rigid disc parallel to penny shaped crack at the interface of layer bonded by two orthotropic half spaces

Two half-spaces and a layer of finite height of linear elastic orthotropic medium containing a penny-shaped crack and a circular rigid disk are considered such that the crack surface is parallel to the rigid disk, where both are situated at two different interfaces of layer and half-space and perpendicular to the axis of symmetry. The rigid disk is agitated by an axisymmetric torsion. The equations of equilibrium are solved with the use of Hankel’s integral transform method. Then, the tangential displacement components and shear stress components are determined. With the help of mixed-type boundary conditions, a system of integral equations is derived. After that, by utilizing a trial solution, these integral equations are transformed into a pair of simultaneous Fredholm integral equations of the second kind. The numerical results of the Fredholm integral equations are evaluated by converting it into a system of algebraic equations to interpret the physical quantity, like the stress intensity factor (SIF), which is demonstrated graphically. The obtained results are compared to the previously published article to ensure its accuracy.

Singular integral method based closed form solution for moving EMPS model in semipermeable two‐equal‐collinear cracked MEE material

AbstractThis paper presents the analytical closed‐form solutions of moving two equal collinear semipermeable cracks in magneto‐electro‐elastic material considering the strip electric‐magnetic polarization saturation (EMPS) model. To simulate the dynamic/moving cracks problem, two equal collinear Yoffe type cracks moving with a constant subsonic velocity along the plane of the cracks is considered. The problem is solved by considering semipermeable crack face conditions and under the in‐plane electro‐magnetic‐mechanical loading. Applying distributed dislocation technique and symmetry of the collinear cracks, the problem is reduced into simultaneous singular integral equations which have been solved in closed form expressions using finite Hilbert approach. The explicit expressions for dynamic distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), and local stress intensity factors (LSIFs) are evaluated for both the cases of the EMPS model (magnetic zone is smaller or greater than electric zone). For evaluation of inner and outer electric and magnetic zone lengths (MZLs), an iterative approach is applied to solve the developed two non‐linear equations. Also, to implement the semipermeable crack‐face conditions, a bisection iterative method has been applied. Based on the developed closed form solutions, numerical studies are presented for standard fracture parameters with respect to crack propagation velocity, electrical loading and inter‐crack space distance. This study shows that electric displacement and magnetic induction defined over the crack‐surface for semipermeable conditions decrease whereas the numerical values of saturated zone lengths, COD, COP, COI, and LSIFs increase with increasing crack propagation velocity. Moreover, for both static and dynamic cracks problems, the significant crack‐interaction effects have been observed on all the studied fracture parameters except the crack‐face conditions.

A restricted subset selection procedure for selecting the largest normal mean under heteroscedasticity

This article considers the goal of selecting the population with the largest mean among k normal populations when variances are not known. We propose a Stein-type two-sample procedure, denoted by for selecting a nonempty random-size subset of size at most m () that contains the population associated with the largest mean, with a guaranteed minimum probability whenever the distance between the largest mean and the second largest mean is at least where m, and are specified in advance of the experiment. The probability of a correct selection and the expected subset size of are derived. Critical values/procedure parameters that are required for certain k, m, and are obtained by solving simultaneous integral equations and are presented in tables.

Mutual Coupling and Impedance of a VLF Curtain Array in an Anisotropic Magnetoplasma

In this letter, we investigate and propose an analytical method for calculating the mutual impedance of a very low frequency (VLF: 3–30 kHz) curtain array immersed in an anisotropic magnetoplasma. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> simultaneous integral equations satisfied by the array are first built with full consideration of the coupling effect between different elements. Under the assumption that the currents of adjacent elements differ from a fixed phase, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -coupled integral equations are reduced to an independent one, which is solved to determine the complete distribution of current on each element. The self- and mutual impedances of the array are then evaluated through the induced electromotive force method. Computations show that the current is more concentrated at the middle part of the antenna as its electrical length increases, and the error that appeared on both the amplitude and phase of the required voltages verifies the existence of the mutual coupling effect. It is also found that both the self- and mutual impedances will experience certain fluctuations when the electrical length of the elements changes, with maxima and minima occurring at several particular lengths. The approach may provide theoretical support to the prediction of mutual impedance in practical VLF array applications.

Analysis of a mode-I crack in a one-dimensional orthorhombic quasicrystal strip

A mode-I crack in a one-dimensional (1D) orthorhombic quasicrystal (QC) strip under in-plane phonon and phason stress loading is considered. Fourier transforms are applied to reduce the mixed boundary value problem of the mode-I crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and phason stresses and displacement fields near the crack tips have been obtained in an explicit form. The crack-tip singularities of the mode-I crack have been investigated and the intensity factors of the stresses in the phonon and phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack kinking phenomenon. The effect of the thickness ratio of the cracked strip on the SIFs and energy release rates has been investigated. When the thickness of the cracked strip becomes infinite large, the results obtained for the crack problem can be reduced to the analytic solution for a mode-I crack in an infinite 1D orthorhombic QC media.

Analysis of an anti-plane crack in a one-dimensional orthorhombic quasicrystal strip

In this paper, a central crack in a one-dimensional (1D) orthorhombic quasicrystal strip under anti-plane loadings has been studied. Fourier transform is applied to reduce the mixed boundary value problem of the crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and the phason fields near the crack tips have been obtained. The crack-tip singularities of the anti-plane crack have been investigated, and the intensity factors of the stresses in the phonon and the phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack skewing phenomenon. This crack skewing phenomenon is different from the mode-III crack problems in isotropic elastic materials and transversely isotropic elastic materials where the crack propagates along the original crack plane.

A generalized solution to the combo-crack problem—I. Pressure load on crack surface

The axisymmetric elasticity problem of crack combo containing an externally circular crack (ECC) and a coplanar concentric penny-shaped crack (PSC) is mathematically equivalent to the annular contact problem. This problem has been attempted by using Love's strain potential approach, which eventually comes down to solving a pair of simultaneous Fredholm integral equations. Finding the closed-form solutions to the integral equations is difficult, if not impossible. Approximate solutions have been proposed in power series representations, which suffer from two major deficiencies. First, the solutions apply only to a special loading case in which uniform pressure is applied to the whole surface of the interior PSC. Secondly, the accuracy of the solution becomes unsatisfactory when the interior PSC tip is close to the ECC tip. To address these issues, in this paper we revisit this problem by considering a more general loading case in which uniform pressure is applied to a circular region of any size at the center of the PSC's surface. To overcome the lower accuracy caused by power series with limited terms, we numerically solve the pair of simultaneous Fredholm integral equations based on the Gauss-Lobatto quadrature. The high accuracy of our solution in the whole size spectra of the PSC and ECC is verified by finite element simulations. Our paper provides a generalized and more accurate solution to the annular contact problem or the combo crack problem, which deserves to be included in the updated library of the solutions to basic crack problems.

A Study on the Applications of Laplace Transformation

Mathematics plays an important role in our everyday life. Laplace transform is one of the important tools which is used by researchers to find the solutions of various real life problems modeled into differential equations or simultaneous differential equations or Integral equations. In this paper, we are going to study the details on lapace transform, its properties and “Applications of Laplace Transform in Various Fields”. Various uses of Laplace Transforms in the research problems have been highlighted. Detailed applications of Laplace Transform have been discussed.

Analysis on the propagation of Griffith crack in a magnetoelastic self-reinforced strip subjected to moving punch of constant load

The present study analysed the characteristics of a moving Griffith crack in a self-reinforced strip of finite thickness and infinite extent with the moving parallel punches of constant load acting on the boundaries of the strip at both sides due to the propagation of magnetoelastic plane waves under mechanical point loading. With the aid of integral transform technique, problem has been reduced to the pair of simultaneous singular integral equations with Cauchy-type singularities. The point load at the edge of the moving crack is considered in terms of Dirac delta function, and the expression of stress intensity factor (SIF) at the crack tip with constant point loading has been established in closed form by using the well-known properties of Hilbert transformation. Moreover, some of the special cases have been deduced from the obtained expression of SIF for the force of constant intensity, without punch pressure and anisotropy in the considered strip. Numerical computations and graphical demonstrations have been carried out to observe the profound effect of magnetoelastic coupling parameter, punch pressure, crack length, distinct positions of point load and the velocity of crack associated with magnetoelastic plane wave on SIF for self-reinforced materials and isotropic material strip. A comparative study of SIF at the tip of moving crack has been made for the self-reinforced and isotropic materials to highlight some of the important peculiarities of the problem.

Application of distributed dislocation method to curved crack moving near a press-fitted inclusion in a two-dimensional infinite plate

This paper describes the application of the distributed dislocation method to the problem of a curved crack moving near a press-fitted inclusion in a two-dimensional infinite plate which is subjected to a tensile loading. The formulation is based upon the superposition of the following three fundamental problems: a continuous distribution of edge dislocations along the crack line in an infinite plate with an inclusion, residual stress generated by the press-fitted inclusion without the crack, and stress field in the infinite elastic plate containing the circular inclusion, but without the crack, which is subjected to a uniform tensile loading. The superposition leads to simultaneous singular integral equations, which are introduced by relating the traction-free condition along the curved crack line to the dislocation density functions as unknown functions in the integral equations. The stress intensity factors can be derived directly from those dislocation density functions. In this study, the crack-propagation program, which is considered that the crack tip is automatically moved in a direction satisfying the restriction that the stress intensity factor KII is zero, was developed. The developed program is applied to the elastic problem including the press-fitted inclusion and crack, and the influence of the initial crack locations and residual stress generated by inserting the inclusion into the hole on the geometry of the crack path is discussed.

Water wave scattering by three thin vertical barriers arranged asymmetrically in deep water

The paper involves oblique scattering of water waves by three non-identical thin vertical barriers arranged asymmetrically in deep water. The asymmetry in the configuration of the barriers is in the sense that the outer barriers are situated at two different distances from the inner (middle) barrier. Two geometries of the barriers have been considered. In the first one, the two outer barriers are partially immersed and the inner barrier is completely submerged and extends infinitely downwards while the second one is its complementary. Each problem is reduced to a system of three simultaneous first kind integral equations involving differences of potential functions across the barriers. These integral equations are solved approximately by employing single-term Galerkin’s approximations where the single term is the exact solution of the corresponding integral equation arising in the problem of scattering of normally incident water waves by a single thin vertical barrier in deep water. Fairly accurate numerical estimates for the reflection and transmission coefficients are obtained which satisfy the energy identity. For both the configurations of the barriers, the behavior of reflection coefficient is observed by depicting it graphically against wavenumber for different values of various parameters. It is observed that the reflection coefficient vanishes for a sequence of discrete wavenumbers only when the outer two barriers are identical and equidistant from the inner barrier. However, this phenomenon may differ sometimes for the second configuration of the barriers. Numerical results available in the literature are recovered as special cases thereby validating the accuracy of the numerical estimates obtained here.

Symmetric Solutions of Nonlinear Fractional Integral Equations via a New Fixed Point Theorem under FG-Contractive Condition

We set up the existence of a symmetric outcome of a system of simultaneous nonlinear fractional integral equations, that arises in motion of water wave on smooth surface, with the help of a common fixed point theorem satisfying a generalized FG-contractive condition. To accomplish this, we introduce first the concept of generalized FG-contractive condition for two pairs of self-mappings in a complete metric space and then we establish requisites for common fixed point results for weakly compatible mappings followed by a suitable example.

A study of the double edge cracks and a central crack in an orthotropic strip under the tensile loads

In this article, An elastostatic problem with two symmetrically located edge cracks and a centre crack in an orthotropic elastic material under normal loadings is considered. Two harmonic functions are considered which are expressed in terms of displacement of stress and component of axes. The problem is reduced into a pair of simultaneous singular integral equations of the first kind with Cauchy-type singularities, which are solved using Chebyshev polynomials. The normalized stress intensity factors are found against ratios of cracks’ length are computed for different particular cases for the orthotropic material Steel-Mylar and the results are depicted through graphs.

Water wave scattering by three thin vertical barriers with middle one partially immersed and outer two submerged

The problem of oblique water wave scattering by three thin vertical barriers in deep water is investigated here assuming linear theory. The geometrical configurations of the three barriers are such the inner(middle) barrier is partially immersed and the two outer barriers are completely submerged and extend infinitely downwards. A system of three simultaneous integral equations of first kind involving differences of velocity potentials across the barriers has been obtained in the mathematical analysis by employing Havelock’s expansion of water wave potential alongwith Havelock’s inversion formulae. Approximate numerical solutions of these integral equations are obtained by using single-term Galerkin approximations where the single term is chosen to be the exact solution of the integral equation obtained for a single vertical barrier in deep water for normal incidence of surface waves. Fairely accurate numerical estimates for the reflection and transmission coefficients are obtained by solving this system of linear equations. These estimates satisfy the energy identity, thereby justifying the validity of the present method. Due to energy identity, the behaviour of reflection coefficient is discussed by depicting it graphically against wavenumber. Existence of zeros of reflection coefficient is observed only when the two outer barriers are identical i.e. they are submerged from the same depth below the mean free surface. Some earlier results for two completely submerged barriers and a single barrier are recovered as special cases for a particular set of values of the parameters involved in the problem. This provides another check on the correctness of the present method.

Biaxial Loading of a Plate Containing a Hole and Two Co-Axial Through Cracks

Abstract The paper presents the solution linear elasticity problem for an isotropic plate weakened by a hole and two co-axial cracks. The plate is exerted by uniform traction at infinity. The corresponding 2D problem is solved by the method of Kolosova-Muskhelishvili complex potentials. The method implies reduction of the problem to simultaneous singular integral equations (SIE) for the functions defined the region of the cracks and hole. For particular case the solution the SIE is obtained analytically in a closed form. A thorough analysis of the stress intensity factors (SIF) is carried out for various cases of the hole shape: penny-shaped, elliptical and rectangular.

HYDRO2GEN: Non-thermal hydrogen Balmer and Paschen emission in solar flares generated by electron beams

Aim.Sharp rises of hard X-ray (HXR) emission accompanied by Hαline profiles with strong red-shifts up to 4 Å from the central wavelength, often observed at the onset of flares with the Specola Solare Ticinese Telescope (STT) and the Swedish Solar Telescope (SST), are not fully explained by existing radiative models. Moreover, observations of white light (WL) and Balmer continuum emission with the Interface Region Imaging Spectrograph (IRISH) reveal strong co-temporal enhancements and are often nearly co-spatial with HXR emission. These effects indicate a fast effective source of excitation and ionisation of hydrogen atoms in flaring atmospheres associated with HXR emission. In this paper, we investigate electron beams as the agents accounting for the observed hydrogen line and continuum emission.Methods.Flaring atmospheres are considered to be produced by a 1D hydrodynamic response to the injection of an electron beam defining their kinetic temperatures, densities, and macro velocities. We simulated a radiative response in these atmospheres using a fully non-local thermodynamic equilibrium (NLTE) approach for a 5-level plus continuum hydrogen atom model, considering its excitation and ionisation by spontaneous, external, and internal diffusive radiation and by inelastic collisions with thermal and beam electrons. Simultaneous steady-state and integral radiative transfer equations in all optically thick transitions (Lyman and Balmer series) were solved iteratively for all the transitions to define their source functions with the relative accuracy of 10−5. The solutions of the radiative transfer equations were found using the L2 approximation. Resulting intensities of hydrogen line and continuum emission were also calculated for Balmer and Paschen series.Results.We find that inelastic collisions with beam electrons strongly increase excitation and ionisation of hydrogen atoms from the chromosphere to photosphere. This leads to an increase in Lyman continuum radiation, which has high optical thickness, and after the beam is off it governs hydrogen ionisation and leads to the long lasting orders of magnitude enhancement of emission in Balmer and Paschen continua. The ratio of Balmer-to-other-continuum head intensities are found to be correlated with the initial flux of the beam. The height distribution of contribution functions for Paschen continuum emission indicate a close correlation with the observations of heights of WL and HXR emission reported for limb flares. This process also leads to a strong increase of wing emission (Stark’s wings) in Balmer and Paschen lines, which is superimposed on large red-shifted enhancements of Hα-Hγline emission resulting from a downward motion by hydrodynamic shocks. The simulated line profiles are shown to fit closely the observations for various flaring events.

Application of singular integral equation to crack moving near an inclusion in two-dimensional infinite plate

This paper describes application of singular integral equation method to crack moving near an inclusion in two-dimensional infinite plate which is subjected to a tensile loading at infinity. The theoretical formulation is based upon the superposition of the problem of a continuous distribution of edge dislocations spread along the crack locus in an infinite plate with single inclusion and the problem for the same geometry without crack which is subjected to a tensile loading. The superposition leads to simultaneous singular integral equation, and they relate the surface zero-tractions to the dislocation densities along the curved crack locus. The stress intensity factors are derived directly from the crack-tip stress field. Then, the crack tip is automatically moved to a direction as satisfying the restriction that the stress intensity factor KII is zero, which was developed in this study. The direction search is conducted by rotating the tip of virtually incremented crack around the origin crack tip. The search and extension processes are repeated in sequence. In this numerical calculation, the influence of the initial crack locations on crack moving path is discussed.

Interaction of Three Interfacial Cracks between an Orthotropic Half-Plane Bonded to a Dissimilar Orthotropic Layer with Punch

Abstract This article deals with the interactions between a central crack and a pair of outer cracks situated at the interface of an orthotropic elastic half-plane bonded to a dissimilar orthotropic layer with a punch. The problem is reduced to the solution of three simultaneous singular integral equations that are finally solved using Jacobi polynomials. The phenomena of crack shielding and crack amplification have been depicted through graphs for different particular cases.

General model of optical frequency conversion in homogeneous media: Application to second-harmonic generation in an ε -near-zero waveguide

Traditional optical frequency conversion model is well improved in this work. In terms of the dyadic Green's function method, a set of coupled-amplitude equations is reduced under a proposed transition layer assumption, accompanying the simultaneous integral equations. The model, as a generalization of the current frequency conversion theory, is aimed at any one-dimensional thin film or bulk nonlinear structure, allowing for arbitrary optical anisotropy and absorption without pumping and propagating limitations. The assumption reasonably simplifies the strict nonlinear boundary conditions and enables the equations to yield exact radiative field solutions. A field-enhanced phase-matching configuration is designed for second harmonic generation in a lossy $\ensuremath{\epsilon}$-near-zero material. The high contrast of refractive indices between a substrate (silicon) and the material traps the harmonic wave inside and constructs a natural mirror reflection waveguide. A simulation in the lowest guided mode predicts an efficiency enhancement proportional to the relative wave impedance to the fifth power under a resonant condition.

Penny-shaped crack in elastic medium with surface energy effects

In this paper, a penny-shaped crack in an infinite elastic medium subjected to vertical pressure loading at the crack surface under the influence of surface stress is considered. The Gurtin–Murdoch continuum theory of elastic material surfaces is adopted, and the Hankel integral transform is employed to solve this axisymmetric boundary value problem. A set of simultaneous dual integral equations is solved by employing an appropriate numerical solution scheme. Selected numerical results are presented to portray the influence of the surface stress on the elastic field. Numerical results reveal that the surface stress has a significant influence on both stress and displacement fields. It is also found that the material becomes tougher with the presence of surface stress. In addition, the elastic field also shows size-dependent behavior deviating from the classical crack solution. The solutions presented in this study provide fundamental understanding of the influence of surface stress on fracture mechanics problems. It can also be used as a benchmark solution for the development of numerical techniques, such as FEM and BEM, for analysis of more complicated problems associated with cracks under the influence of surface stress.