scholarly journals Investigation of the second main problem of elasticity for a layer with n cylindrical inclusions

2021 ◽  
pp. 156-166
Author(s):  
Vitaly Miroshnikov ◽  
Tetiana Denisova
Keyword(s):  
Boundary Conditions ◽  
Reduction Method ◽  
Fourier Method ◽  
High Accuracy ◽  
Theory Of Elasticity ◽  
System Of Equations ◽  
Spatial Problem ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Cylindrical Inclusions

When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy. In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed. The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10-2 for values from 0 to 1. Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions. In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared. The result has shown that close placement of reinforcement bars increases stresses  on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained. The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.

Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion

2019 ◽  
Vol 968 ◽  
pp. 413-420
Author(s):  
Vitaly Yu. Miroshnikov ◽  
Alla V. Medvedeva ◽  
Sergei V. Oleshkevich
Keyword(s):  
Stress State ◽  
Reduction Method ◽  
Elastic Inclusion ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Geometrical Parameters ◽  
Spatial Problem ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

A spatial problem of the theory of elasticity for the layer with an infinite round cylindrical inclusion is investigated. At the boundaries of the layer, displacements are given. The cylindrical elastic inclusion is rigidly coupled with the layer and their boundary surfaces do not intersect. The solution to the spatial problem is obtained by the generalized Fourier method, with regard to the Lamé system of equations. The obtained infinite systems of linear algebraic equations are solved by a reduction method. As a result, the values ​​of displacements and stresses in the elastic body are determined. A comparative analysis of the stress state for different geometrical parameters is carried out, and a comparison is made with the stress state in the layer with a cylindrical cavity.

LEAST-SQUARES MESHFREE COLLOCATION METHOD

2012 ◽  
Vol 09 (02) ◽  
pp. 1240031 ◽  
Author(s):  
BO-NAN JIANG
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Least Squares ◽  
Collocation Method ◽  
Present Method ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Dual Variables

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

scholarly journals Particular properties of estimation of partial capacitances of insulation of three core power cables by applying aggregate measurements

2021 ◽  
pp. 15-20
Author(s):  
Ivan Kostiukov
Keyword(s):  
Root Mean Square Error ◽  
Mean Square Error ◽  
Root Mean Square ◽  
Least Squares Method ◽  
Power Cable ◽  
System Of Equations ◽  
Power Cables ◽  
Mean Square ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

This paper presents a description of specific properties of determining the values of partial capacitances of insulation gaps in power cables with paper insulation for various ways of forming and solving the system of linear algebraic equations. Possible ways of inspection the insulation of three core power cables for the estimation of values of partial capacitances by applying aggregate measurements which are based on various ways of connection of emittance meter to tested sample of power cable are given. Estimation of partial capacitances by the direct solution of a system of linear algebraic equations, by minimizing the root mean square error of solving an overdetermined system of equations by the least squares method, as well as by finding a normal solution of an indefinite system of equations by the pseudo-inverse matrix, is also considered. It is shown that minimization of the root mean square error by the least squares method and the direct solution of system of equations show quite similar results for the case of estimation of partial capacitances by means of aggregate measurements, at the same time the solution of an indefinite system of equations by the method of a pseudo-inverted matrix allows to reproduce rather accurately only 3 out of 6 values of partial capacitances. The uneven effect of frequency on the electrical capacitance of the insulation gaps between the cores of the power cable and between its cores and the sheath is shown. It was proposed to use the frequency dependence of the electrical capacitance of insulation gaps as an informative parameter about the technical state of insulating gaps between the cores of the power cable and between its cores and its sheath.   Keywords: root mean square error; least squares method; system of linear algebraic equations; dielectric losses; dielectric permittivity.

Influence Functions in the Numerical Analysis of Bending of Thin Elastic Plates

1984 ◽  
Vol 8 (2) ◽  
pp. 103-114 ◽  
Author(s):  
Mohammed F.N. Mohsen ◽  
Ali A. Al-Gadhib ◽  
Mohammed H. Baluch
Keyword(s):  
Boundary Conditions ◽  
Influence Function ◽  
Infinite Plate ◽  
Thin Plates ◽  
Influence Functions ◽  
Algebraic Equations ◽  
Elastic Plates ◽  
The Real ◽  
Linear Algebraic Equations ◽  
The Moment

A numerical method for the linear analysis of thin plates of arbitrary plan form and subjected to arbitrary loading and boundary conditions is presented in this paper. This method is an extension of the Wu-Altiero method [1] where use has been made of the force influence function for an infinite plate, whereas the work contained in this paper is based on the use of the moment influence function of an infinite plate. The technique basically involves embedding the real plate into a fictitious infinite plate for which the moment influence function is known. N points are prescribed at the plate boundary at which the boundary conditions for the original problem are collocated by means of 2N fictitious moments placed around contours outside the domain of the real plate. A system of 2N linear algebraic equations in the unknown moments is obtained. The solution of the system yields the unknown moments. These may in turn be used to compute deflection, moments or shear at any point in the thin plate. Finally, the method is extended to include influence functions of both concentrated forces and concentrated moments. This is obtained by applying concentrated moments and forces simultaneously on the contours located outside the domain of the plate.

scholarly journals Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

2021 ◽  
Vol 20 ◽  
pp. 717-728
Author(s):  
Boris M. Shumilov
Keyword(s):  
Band System ◽  
Cubic Spline ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Diagonal Dominance ◽  
Sweep Method ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Initial Scale ◽  
Strict Diagonal Dominance

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

scholarly journals Construction Fragile Digital Watermark Used by a Team of Authors in Executable Files

2020 ◽  
Vol 18 (1) ◽  
pp. 65-73
Author(s):  
I. V. Nechta
Keyword(s):  
Unique Solution ◽  
Digital Watermark ◽  
The Other ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Statistical Research ◽  
Other Hand ◽  
Linear Algebraic Equations ◽  
Software Product ◽  
The One

According to statistical research, a violation of license agreements annually causes huge losses to software companies. On the one hand, illegal copies of the software product are created, on the other hand, some fragments of the programs are used by third parties unauthorized. Another important problem is the violation of the program integrity, for example, in terms of blocking functions of the license key checking. In this regard, the task of construction methods for protecting intellectual property in software applications is highly relevant. Previously known methods solve this problem by means of fragile digital watermarks. Below is presented a method for constructing a fragile digital watermark used in executable files. A model of a developers team creating software product protected by DWM is considered. The application of this method will allow to reveal the fact of the container integrity violation, on the one hand, and, on the other hand, will allow the author, if it is necessary, to confirm his participation in the development and embedding of the DWM. In this method we use mathematical properties of systems of linear algebraic equations, digital signature and cryptographic hash functions. The scheme is based on the Kronecker – Capelli theorem. To find the group password the co-author who is in the group finds one root of the consistent system of linear algebraic equations. The indicated system consists of n equations and contains n + 1 variables. For an outsider who is not in the group, the system of equations does not have a unique solution. The co-author of the group is able to calculate one variable in system based on their passport data. Therefore, the system of equations for such co-author has a unique solution. Various attacks on a protected by the new method are analyzed, and it is shown its efficiency. The constructed method can be applied in companies with a large team of developers.

scholarly journals ГРАНИЧНЫЕ УСЛОВИЯ КОНТАКТНОГО ТИПА В ЗАДАЧЕ О КРУГОВОЙ ЦИЛИНДРИЧЕСКОЙ ПОЛОСТИ В УПРУГОМ ПРОСТРАНСТВЕ

2018 ◽  
pp. 121-128
Author(s):  
В. Ю. Мирошников ◽  
Т. В. Денисова ◽  
В. С. Проценко
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Numerical Comparison ◽  
Elastic Space ◽  
Dimensional Problem ◽  
Cylindrical Coordinates ◽  
Contact Type ◽  
Tangential Stresses

A three-dimensional problem of the theory of elasticity is considered, when contact-type conditions (normal displacements and tangential stresses) are given on a cylindrical cavity in elastic space. The solution is obtained on the basis of the Fourier method with respect to the Lame equations in cylindrical coordinates. The solvability and uniqueness of the problem for these boundary conditions is proved. Normal and tangential stresses are found in the elastic body. A numerical comparison is made of the influence of the boundary conditions in the form of displacements and boundary conditions of the contact type on the stressed state of the elastic space.

Asymptotic Behavior of the Solution to the Axisymmetric Dynamical Problem of the Elasticity Theory for the Transversally Isotropic Spherical Layer of Small Thickness

2020 ◽  
pp. 61-71
Author(s):  
M.F. Mehdiyev ◽  
N.K. Akhmedov ◽  
S.M. Yusubova
Keyword(s):  
Dynamic Problem ◽  
Transversely Isotropic ◽  
Spherical Layer ◽  
Theory Of Elasticity ◽  
Dynamical Problem ◽  
Algebraic Equations ◽  
Elastic Band ◽  
Small Thickness ◽  
Homogeneous Solutions ◽  
Linear Algebraic Equations

In this paper, we study the axisymmetric dynamic problem of the theory of elasticity for the transversely isotropic spherical layer of small thickness that does not contain any of the poles 0 and π. It is assumed that the lateral surface of the sphere is free of stresses, and boundary conditions are set on conical sections. Using the method of asymptotic integration of equations of the theory of elasticity, the dynamic problem of this theory is analyzed for the transversely isotropic spherical layer as the thin-walled parameter tends to zero. A possible form of wave formation in the transversely isotropic spherical layer has been studied depending on the frequency of the influencing forces. Homogeneous solutions are constructed and their classification is given. Asymptotic expansions of the homogeneous solutions are obtained, which make possible to calculate the stress-strain state for various values of the frequency of the influencing forces. It is shown that for the high-frequency oscillations in the first term of the asymptotics, the dispersion equation coincides with the well-known Rayleigh-Lamb equation for the elastic band. In the general case of loading on the sphere using the Hamilton variational principle, the boundary-value problem is reduced to the solving infinite systems of linear algebraic equations.

scholarly journals Periodic solutions of high accuracy to the forced Duffing equation: Perturbation series in the forcing amplitude

1987 ◽  
Vol 29 (1) ◽  
pp. 21-38 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Periodic Solutions ◽  
Phase Plane ◽  
High Accuracy ◽  
Perturbation Series ◽  
Duffing Equation ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Linear Resonance ◽  
Limiting Values

Abstract“Steady state” periodic solutions are sought to the forced Duffing equation. The solutions are expressed as formal Fourier series, giving rise to an infinite system of non-linear algebraic equations for the Fourier coefficients. This system is then solved using perturbation series in the amplitude of the forcing term. Solution profiles of high accuracy and phase-plane orbits are presented. The existence of limiting values of the forcing amplitude is discussed, and points of non-linear resonance are identified.

New Analytical Solutions of Buckling Problem of Rotationally-Restrained Rectangular Thin Plates

2019 ◽  
Vol 11 (10) ◽  
pp. 1950101 ◽  
Author(s):  
Salamat Ullah ◽  
Jinghui Zhang ◽  
Yang Zhong
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
Integral Transformation ◽  
Solution Procedure ◽  
Thin Plates ◽  
Algebraic Equations ◽  
Transform Method ◽  
Linear Algebraic Equations ◽  
Title Problem ◽  
Finite Integral

A double finite sine integral transform method is employed to analyze the buckling problem of rectangular thin plate with rotationally-restrained boundary condition. The method provides more reasonable and theoretical procedure than conventional inverse/semi-inverse methods through eliminating the need to preselect the deflection function. Unlike the methods based on Fourier series, the finite integral transform directly solves the governing equation, which automatically involves the boundary conditions. In the solution procedure, after performing integral transformation the title problem is converted to solve a fully regular infinite system of linear algebraic equations with the unknowns determined by satisfying associated boundary conditions. Then, through some mathematical manipulation the analytical buckling solution is elegantly achieved in a straightforward procedure. Various edge flexibilities are investigated through selecting the rotational fixity factor, including simply supported and clamped edges as limiting situations. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparing with the existing literature as well as the finite element method using (ABAQUS) software.

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