scholarly journals ELASTOPLASTIC PROBLEM FOR PERFORATED PLATE IN TRANSVERSE SHEAR

2021 ◽  
Vol 08 (04) ◽  
pp. 29-40
Author(s):  
Rafail Mehdiyev Rafail Mehdiyev ◽  
Alekber Mehdiyev Alekber Mehdiyev ◽  
Rustam Mammadov Rustam Mammadov
Keyword(s):  
Thin Plate ◽  
Singular Integral ◽  
Transverse Shear ◽  
Perforated Plate ◽  
Singular Integral Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Circular Holes ◽  
Periodic Stress ◽  
Unequal Length

A solution is given to the problem of transverse shear of a thin plate clamped along the edges of the holes and weakened by a doubly periodic system of rectilinear through cracks with plastic end zones collinear to the abscissa and ordinate axes of unequal length. General representations of solutions are constructed that describe the class of problems with a doubly periodic stress distribution outside circular holes and rectilinear cracks with end zones of plastic deformations. Satisfying the boundary conditions, the solution of the problem of the theory of shear plates is reduced to two infinite systems of algebraic equations and two singular integral equations. Then each singular integral equation is reduced to a finite system of linear algebraic equations. Keywords: perforated thin plate, straight cracks with end zones, transverse bending, plastic deformation zones.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Finite-part singular integral approximations in Hilbert spaces

2004 ◽  
Vol 2004 (52) ◽  
pp. 2787-2793
Author(s):  
E. G. Ladopoulos ◽  
G. Tsamasphyros ◽  
V. A. Zisis
Keyword(s):  
Unit Circle ◽  
Integral Equations ◽  
Singular Integral ◽  
Hilbert Spaces ◽  
Existence Theorems ◽  
Singular Integral Equations ◽  
Finite Part ◽  
Algebraic Equations ◽  
Integration Interval ◽  
Linear Algebraic Equations

Some new approximation methods are proposed for the numerical evaluation of the finite-part singular integral equations defined on Hilbert spaces when their singularity consists of a homeomorphism of the integration interval, which is a unit circle, on itself. Therefore, some existence theorems are proved for the solutions of the finite-part singular integral equations, approximated by several systems of linear algebraic equations. The method is further extended for the proof of the existence of solutions for systems of finite-part singular integral equations defined on Hilbert spaces, when their singularity consists of a system of diffeomorphisms of the integration interval, which is a unit circle, on itself.

scholarly journals Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Pingrun Li
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Complex Analysis ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Convolution Type ◽  
Algebraic Equations ◽  
Periodic Coefficients ◽  
Linear Algebraic Equations ◽  
Classical Boundary

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class L2[-π,π]. Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.

scholarly journals Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

2014 ◽  
Vol 13 (01) ◽  
pp. 1-21 ◽  
Author(s):  
L. P. Castro ◽  
E. M. Rojas ◽  
S. Saitoh ◽  
N. M. Tuan ◽  
P. D. Tuan
Keyword(s):  
Boundary Value Problems ◽  
Unit Circle ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Degenerate Kernel ◽  
Algebraic Equations ◽  
Explicit Formulas ◽  
Riemann Boundary ◽  
Linear Algebraic Equations

By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section.

scholarly journals Solution of Cauchy-Type Singular Integral Equations of the First Kind with Zeros of Jacobi Polynomials as Interpolation Nodes

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
G. E. Okecha
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Singular Integral Equations ◽  
Quadrature Rule ◽  
Convergence Result ◽  
Cauchy Type ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Lagrangian Interpolation

Of concern in this paper is the numerical solution of Cauchy-type singular integral equations of the first kind at a discrete set of points. A quadrature rule based on Lagrangian interpolation, with the zeros of Jacobi polynomials as nodes, is developed to solve these equations. The problem is reduced to a system of linear algebraic equations. A theoretical convergence result for the approximation is provided. A few numerical results are given to illustrate and validate the power of the method developed. Our method is more accurate than some earlier methods developed to tackle this problem.

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

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

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

A Boundary Element Method for Plane Anisotropic Elastic Media

1990 ◽  
Vol 57 (3) ◽  
pp. 600-606 ◽  
Author(s):  
Kyu J. Lee ◽  
A. K. Mal
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Singular Integral Equations ◽  
Boundary Curve ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Numerical Formulation ◽  
Complex Linear ◽  
Anisotropic Elastic ◽  
Improved Accuracy

The general problem of plane anisotropic elastostatics is formulated in terms of a system of singular integral equations with Cauchy kernels by means of the classical stress function approach. The integral equations are represented over the image of the boundary in the complex plane and a numerical scheme is developed for their solution. The boundary curve is discretized and suitable polynomial approximations of the unknown functions in terms of the complex variable are introduced. This reduces the equations to a set of complex linear algebraic equations which can be inverted to yield the stresses in a straightforward manner. The major difference between the present technique and the previous ones is in the numerical formulation. The integral equations are discretized in the complex plane and not in terms of real variables which depend on arc length, resulting in improved accuracy in presence of strong boundary curvature.

scholarly journals Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

Stress Concentration in a Stretched Cylindrical Shell With Two Equal Circular Holes

1975 ◽  
Vol 42 (1) ◽  
pp. 105-109 ◽  
Author(s):  
P. Seide ◽  
A. S. Hafiz
Keyword(s):  
Cylindrical Shell ◽  
Stress Concentration ◽  
Stress Distribution ◽  
Circular Cylindrical Shell ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Wide Range ◽  
Circular Holes ◽  
Limiting Case ◽  
Infinite Set

In this investigation, the stress distribution due to uniaxial tension of an infinitely long, thin, circular cylindrical shell with two equal small circular holes located along a generator is obtained. The problem is solved by the superposition of solutions previously obtained for a cylinder with a single circular hole. The satisfaction of boundary conditions on the free surfaces of the holes, together with uniqueness and overall equilibrium conditions, yields an infinite set of linear algebraic equations involving Hankel and Bessel functions of complex argument. The stress distribution along the boundaries of the holes and the interior of the shell is investigated. In particular, the value of the maximum stress is calculated for a wide range of parameters, including the limiting case in which the holes almost touch and the limiting case in which the radius of the cylinder becomes very large. As is the case for a flat plate, the stress-concentration factor is reduced by the presence of another hole.

Plane Analysis of Finite Multilayered Media With Multiple Aligned Cracks—Part I: Theory

2006 ◽  
Vol 74 (1) ◽  
pp. 128-143 ◽  
Author(s):  
Linfeng Chen ◽  
Marek-Jerzy Pindera
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Mixed Boundary ◽  
Plane Deformation ◽  
Singular Integral Equations ◽  
Cauchy Type ◽  
Algebraic Equations ◽  
Energy Release Rates ◽  
Global Stiffness Matrix ◽  
Bounding Surfaces

Elasticity solutions are developed for finite multilayered domains weakened by aligned cracks that are in a state of generalized plane deformation under two types of end constraints. Multilayered domains consist of an arbitrary number of finite-length and finite-height isotropic, orthotropic or monoclinic layers typical of differently oriented, unidirectionally reinforced laminas arranged in any sequence in the plane in which the analysis is conducted. The solution methodology admits any number of arbitrarily distributed interacting or noninteracting cracks parallel to the horizontal bounding surfaces at specified elevations or interfaces. Based on half-range Fourier series and the local/global stiffness matrix approach, the mixed boundary-value problem is reduced to a system of coupled singular integral equations of the Cauchy type with kernels formulated in terms of the unknown displacement discontinuities. Solutions to these integral equations are obtained by representing the unknown interfacial displacement discontinuities in terms of Jacobi or Chebyshev polynomials with unknown coefficients. The application of orthogonality properties of these polynomials produces a system of algebraic equations that determines the unknown coefficients. Stress intensity factors and energy release rates are derived from dominant parts of the singular integral equations. In Part I of this paper we outline the analytical development of this technique. In Part II we verify this solution and present new fundamental results relevant to the existing and emerging technologies.

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