Solution Of Singular Integral Equations Research Articles

Аналіз граничної рівноваги зігнутої сферичної оболонки з колінеарними тріщинами за локальним та інтегральним критеріями руйнування

The stressed-strain state and limit equilibrium of shallow spherical shell weakened by two cross-cutting meridional collinear cracks is studied in the two-dimensional formulation. The crack closure caused by bending deformation was taken into account based on the model of the crack edges contact along a line in one of the face surfaces of the shell. The boundary problem for equations of classical shell theory with interrelated conditions along the line of the cracks is formulated within the framework of such model. Singular integral equation for the unknown jump of normal rotation angle on the cracks edges has been elaborated. Based on numerical solutions of singular integral equation the stressed-strain state and limit equilibrium of the spherical shell depending on the parameters of shell curvature and distance between cracks are investigated. Using the local and integral through-the-thickness energy failure criteria of linear mechanics of fracture, the upper and lower values of limit load were established. It was found that the upper estimate of the limit load according to the integral criterion is approximately twice the magnitude of the lower estimate according to the local criterion.

"Optimal quadrature formulas for approximate solution of the rst kind singular integral equation with Cauchy kernel"

"In the present paper in $L_2^{(m)}(-1,1)$ space the optimal quadrature formulas with derivatives are constructed for approximate solution of a singular integral equation of the first kind with Cauchy kernel. Approximate solution of the singular integral equation is obtained applying the optimal quadrature formulas. Explicit forms of coefficients for the of optimal quadrature formulas are obtained. Some numerical results are presented."

SOLUTION OF A TWO-DIMENSIONAL BOUNDARY VALUE PROBLEM OF HEAT CONDUCTION IN A DEGENERATING DOMAIN

In the paper we consider the boundary value problem of heat conduction outside the cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary condition contain a derivative with respect to the time variable. The peculiarity of the problem under consideration consists precisely in the presence of a moving boundary and the degeneration of the solution domain into a point at the initial moment of time. The well-known classical methods are generally not applicable to this type of problems. By the method of heat potentials, such boundary value problems of heat conduction are reduced to the solution of singular Volterra type integral equations of the second kind A singular Volterra type equation is understood as an equation whose kernel has the following property: the integral of the kernel of the equation does not tend to zero as the upper limit tends to the lower one. Such integral equations cannot be solved by the method of successive approximations, and in most cases the corresponding homogeneous integral equations have nonzero solutions. We prove a theorem on the solvability of the considered boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We found a nonzero solution of this singular integral equation.

An Approximate Solution of Singular Integral Equations Using Chebyshev Series for a Class of Functions That are Zero at One End and Infinite at the Other End of the Integration Interval

An Approximate Solution of Singular Integral Equations Using Chebyshev Series for a Class of Functions That are Zero at One End and Infinite at the Other End of the Integration Interval

On the Stress Transfer and Mode Mixity for a Film of Finite Length Bonded to an Elastic Substrate

The configuration of film-substrate structure is widely involved in many aspects of technologies. To ensure high reliability and sound performance, it is of importance to understand the stress distribution and the adhesion at the interface. In the present work, within mind the film of finite width, a mechanical model has been proposed to analyze the traction at the interface between the film and an elastic substrate that may be subjected to either uniaxial stretching or bending. For incompressible substrates, the problem may be reduced to the solution of singular integral equations of Cauchy type and the tractions at the film’s edges have been revealed to exhibit the square-root singularity. Mode mixity has been calculated. By comparing the strain in the film when the system is subjected to uniaxial stretching with that from experiments on the adhesion between graphene and substrates, an agreement has been found and the implications of the present model in the measurement of adhesion energy have been discussed.

An approximate solution of singular integral equations using the Chebyshev series on the class of functions vanishing at one end and turning into infinity at the other end of the integration interval

An approximate solution of singular integral equations using the Chebyshev series on the class of functions vanishing at one end and turning into infinity at the other end of the integration interval

Partial Range Completeness of Case Eigenfunctions and Numerical Solution of Singular Integral Equations of Particle Transport Problems

We study numerical solution of Singular Integral Equations (SIE) of particle transport theory. We convert them into matrix equations by standard discretization process. It is found that the matrices are highly ill-conditioned and can be solved by Singular Value Decomposition (SVD) method. One expects that matrices resulting from expansions over Partial Range will not be ill-conditioned. We find this is not true though their ill-conditioning is an order of magnitude less than those of full or half range. Reasons for this phenomenon are explained.

Equilibrium internal fractures in elastic bodies supported by thin flexible coatings

Objective. In this paper, the authors study problems of a plane strain of elastic bodies containing internal rectilinear fractures. In each case, the margins of the considered areas are supported by thin flexible coatings. The first part of the paper is devoted to the problem of an infinite elastic wedge, the faces of which are free from the outside and reinforced with a thin flexible material, and the bisector contains a rectilinear fracture with regular forces applied to the margins, and to the study of the stress concentration at the fracture vertices. In the second part of the paper, the authors consider the problem of an equilibrium radial internal fracture in the cross-section of a round pipe. The inner surface of the pipe experiences hydrostatic pressure; the outer surface is reinforced with a thin flexible coating. The purpose of the study in each of the presented tasks is to determine the values of the influence factor.Methods. Both problems are united by a single approach, in which the presence of a coating is modeled mathematically, using special marginal conditions obtained based on an asymptotic analysis of the exact solution for a strip or ring flexible coating of small relative thickness. In the first issue, the singular integral equation is derived using the Mellin transform, which allows proceeding to the solution of a system of ordinary differential equations and obtaining a singular integral equation relative to the derivative of the discontinuity function of the first kind with a Cauchy kernel. In the second issue, discontinuous solutions are constructed using the Fourier series, resulting in a singular integral equation of a similar structure. Previously, similar ideas were successfully implemented by the authors in the study of the problem of the equilibrium state of a strip with a coating weakened by an internal transverse fracture under arbitrary conditions on the lower edge of the strip.Conclusion. Singular integral equations for the considered problems are obtained. The collocation method is used to construct solutions of singular integral equations for various combinations of geometric and physical characteristics of issues. In all the considered cases, the values of the influence factor were calculated. The analysis of changes in the influence factor depending on various combinations of geometric parameters and mechanical characteristics of problems is carried out. It is noted that with increasing rigidity of the coating and increasing its thickness, the values of the influence factor decrease; the increase in the value of the influence factor is provided by approaching the fracture to the body margin and increasing its relative length.

Exact Solutions for Interaction of Parallel Screw Dislocations with a Wedge Crack in One-Dimensional Hexagonal Quasicrystal with Piezoelectric Effects

The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.

Линейные функциональные уравнения в гельдеровых классах функций на простой гладкой кривой

The article describes linear functional equations on simple smooth curves with a shift function having a non-zero derivative satisfying the Hölder condition, and fixed points only at the ends of the curve. The objective of the article is to find the conditions of the existence and uniqueness of the solution of such equations in the Hölder class functions with the coefficient and the right-hand side satisfying the Hölder conditions. These conditions are obtained depending on the values of the equation coefficient at the ends of the curve. Various specifics at the ends of the curve are considered. The indicators of the Hölder solutions are determined. The possibilities of applying linear functional equations to the study and solution of singular integral equations with logarithmic singularities are shown.

Solving Finite Part Singular Integral Equations Using Orthogonal Polynomials

In this paper, we use the Gauss–Jacobi quadrature formula to get an approximate solution for a finite part singular integral equation. The zeros of Jacobi polynomials and Jacobi functions of the second kind are used to construct a square system of algebraic equations which yield an interpolating function for the regular part of the solution of the singular integral equation. In a special case, another approach consists in converting the finite part singular integral equation to a Fredholm integral equation of the second kind using the analytical solution of a simple finite part singular integral equation. To simplify the calculation of kernel and right-hand side of the obtained Fredholm equation, we use the new polynomials $$J_n(x)$$, which admit a recurrence relation. The equivalent form of the integral equation in the last case is used to investigate the error bound of the approximated solution.

Interface crack between dissimilar one-dimensional hexagonal quasicrystals with piezoelectric effect

In this paper, an interface crack between dissimilar one-dimensional (1D) hexagonal quasicrystals with piezoelectric effect under anti-plane shear and in-plane electric loadings is studied. By using integral transform techniques, the mixed boundary value problem for the interface crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert problems with an exact solution. An analytical full-field solution for phonon and phason stresses, electric fields and electric displacement in the cracked bi-materials is given, and of particular interest, the analytical expression of the phonon and phason stresses and electric displacements along the interface is obtained. The crack sliding displacements of the interface crack are provided, and it is found that the phonon and phason stress distributions inside the dissimilar quasicrystal material are independent of the material properties under the anti-plane shear and in-plane electric loadings. The results of the stress intensity factors energy release rate indicate that the crack propagation can either be enhanced or retarded depending on the magnitude and direction of the electric loadings.

Solution of Singular Integral Equations of the First Kind with Cauchy Kernel

In this paper an analytic method is developed for solving Cauchy type singular integral equations of the first kind, over a finite interval. Chebyshev polynomials of the first kind, $T_n(x)$, second kind, $U_n(x)$, third kind, $V_n(x)$, and fourth kind, $W_n(x)$, corresponding to respective weight functions $W^{(1)}(x)=\frac{1}{\sqrt{1-x^2}},W^{(2)}(x)=\sqrt{1-x^2},W^{(3)}(x)=\sqrt{\frac{1+x}{1-x}},$ and $~ W^{(3)}(x)=\sqrt{\frac{1-x}{1+x}}, $ have been used to obtain the complete analytical solutions for four different cases.

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.

Size effect on the fracture resistance of rough and frictional cracks

Elastic fracture mechanics commonly defines the fracture resistance of brittle materials within an idealised picture of planar and traction free cracks. An efficient approach to describe the interface conditions in realcracks, such as those occurring in concrete, ceramics or stones, is to include the effect of both roughness and friction by means of a constitutive relationship between opposite points on the interface. In the present paper,we use a numerical technique, based on the solution of singular integral equations, to derive the near-tip stress field with various interface conditions. Then, the technique is applied to investigate the size effect of the interface roughness, where such an effect is related to the ratio between the characteristic length of the roughness and the nominal length of the crack. It is found that the resulting near-tip stresses can be profoundly influenced by the crack path, particularly for short cracks.

Interface crack between magnetoelectroelastic and orthotropic half-spaces under anti-plane loading

In this paper, an interface crack between magnetoelectroelastic and orthotropic half-spaces under anti-plane shear and in-plane electric and magnetic loadings has been studied. By using integral transform techniques the mixed boundary value problem for interface crack was reduced to the solution of singular integral equations, which can be further reduced to solving a Riemann-Hilbert problem with closed form solution. An analytic full-field solution for stresses, electric fields, magnetic fields, electric displacement and magnetic induction in the cracked bi-materials is given, and of particular interests, the analytical expression of the stresses, electric displacements and magnetic inductions along the interface has been obtained. The crack sliding displacements (CSDs) of the interface crack are provided, and it is found that the stress distributions inside the orthotropic material are dependent on the elastic stiffness of the orthotropic material. The results of the stress intensity factors indicate that the crack propagation can either be enhanced or retarded depending on the magnitude and direction of the electric and magnetic loadings.

Crack interaction with nanoscale twinning near a second-phase particle in fine-grained magnesium alloy

A theoretical model is established to address the effect of nanoscale twinning near a second-phase particle on crack growth in fine-grained magnesium alloys. The numerical solutions of singular integral equations are obtained by the considering complex variable method of Muskhelishvili, the superposition principle of elasticity, and the distributed dislocation technique. The expressions of stress intensity factors near the left crack tip are derived, and the energy release rate (ERR) characterizing the condition for crack propagation is also calculated. The influences of relevant parameters such as the location of nanoscale twin, the size of particle, and the relative distance between the inhomogeneity and the left crack tip on the ERR are examined in detail. The results indicate that the ERR is strongly influenced by the nanoscale twinning and the second-phase particle. The hard inhomogeneity decreases the ERR while the soft inhomogeneity increases the ERR when nanoscale twinning is not taken into account. At the same time, there is competition between the effects of the hard particle and nanoscale twinning on the ERR, and the nanoscale twin band with an optimum size displays the best toughening effect.

Interface crack between magnetoelectroelastic and orthotropic half-spaces under in-plane loading

In this paper, an interface crack between magnetoelectroelastic and orthotropic half-spaces has been studied in detail. By using integral transform techniques the present mixed boundary value problem was reduced to the solution of singular integral equations, which can be further reduced to solving a Riemann-Hilbert problem with closed form solution. The crack-tip singularities of the interface crack have been investigated for possible combinations of the magnetoelectroelastic and orthotropic materials, a criterion based on the coefficient of the Riemann-Hilbert problem is introduced to study the possible singularity behavior of the interface crack, and it is shown that there can be either oscillatory or non-oscillatory singularity for the interface crack depending on the particular combination of the bi-materials. A closed form solution for stresses, electric fields, magnetic fields, electric displacement and magnetic induction in the cracked biomaterials is given, and of particular interests, the analytical expression of the stresses, electric displacements and magnetic inductions along the interface has been obtained.

On the contact interaction of two identical stringers with an elastic semi-infinite continuous or vertically cracked plate

This paper considers two connected contact problems on the interaction of stringers with an elastic semi-infinite plate. In the first problem, an elastic half-infinite continuous plate is reinforced on its boundary by two identical stringers exposed to a tensile external force. In the second problem, in the presence of the same stringers, the plate contains a collinear system of cracks on its vertical axis. The solution of both problems is reduced to the solution of singular integral equations (SIE) that are solved by a known numerical-analytical method.

Combined effects of cooperative grain boundary sliding and migration and reinforced particles on crack growth in fine-grained Mg alloys

A theoretical model is developed to describe the combined influences of cooperative grain boundary (GB) sliding and migration and reinforced particles on crack propagation in fine-grained Mg alloys. The numerical solutions of singular integral equations are derived on the basis of complex variable method of Muskhelishvili, the superposition principle of elasticity and distributed dislocation technique. The expressions of stress intensity factors (SIFs) near the right tip of basal crack are obtained and the energy release rate (ERR) characterizing the condition for crack growth is calculated. The influences of important parameters such as the location of cooperative GB sliding and migration, the size of reinforced particle and the misorientation angle of higher angle grain boundaries on the ERR are discussed in detail. The results show that the fracture toughness of fine-grained Mg alloys can be improved by cooperative GB sliding and migration and particle hardening and refining. There exists an optimum particle size that makes the fracture toughness of fine-grained Mg alloys best. Besides, the GB migration remarkably contributes to the fracture toughness of fine-grained Mg alloys and there is an optimum migration distance making the fracture toughness of fine-grained Mg alloys best.