scholarly journals Two-level mathematical models for determining the stress state and life plate with a hole

2021 ◽  
pp. 55-59
Author(s):  
Mariya Ihorivna Shapovalova ◽  
Oleksii Oleksandrovich Vodka
Keyword(s):  
Stress State ◽  
Yield Surface ◽  
Structural Damage ◽  
Theory Of Elasticity ◽  
Practical Significance ◽  
Material Structure ◽  
Algebraic Equations ◽  
Mathematical Methods ◽  
Plastic Deformations ◽  
Linear Algebraic Equations

Modern trends in the development of mechanical engineering and other industries related to the production of materials and structures with a given set of physical, mechanical, and technological properties are aimed at reducing material consumption, energy consumption, increasing accuracy, reliability, and competitiveness of the manufactured product. Therefore, the creation of mathematical methods for assessing the stress state of structural elements based on the analysis of the elastic characteristics of a material, taking into account the peculiarities of its internal microstructure, is an actual task. The considered algorithm includes the following stages: identification of strength parameters using data obtained from images of the material microstructure; study of the stress-strain state of the model based on the variational-difference finite element method; formation of a system of linear algebraic equations for solving the problem of analyzing the elastic properties of a material using the plane problem of the theory of elasticity; construction of the material yield surface for a series of tests based on the strength criteria of composite materials, taking into account the different resistance of the material under tensile and compressive loads. Based on the developed mathematical model, the SSS and the yield surface of the plate with a hole are estimated. Structural analysis is performed at the macro and micro levels. The occurrence of plastic deformations at the micro-level can lead to the development of cracks and structural damage at the macro level. As a result of the study, the probability of plastic deformation in the plate is determined, and the critical zones of the model are established. The practical significance of the results obtained is to create an approach to assessing the mechanical properties of a material, such as elastic modulus, shear modulus, Poisson's ratio, and their probabilistic characteristics following the internal material structure. The proposed approach contributes to the expansion of knowledge about the material and allows to increase the valuable information obtained by modeling. To assess the probability of plastic deformations, the generated method uses the entire set of probabilistic characteristics of the yield surface.

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.

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.

Methods of mathematical modeling for predicting university entry into TOP-100 world university rankings

2020 ◽  
Vol 19 (7) ◽  
pp. 1360-1384
Author(s):  
V.M. Moskovkin ◽  
Zhang He
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Volterra Equations ◽  
University Rankings ◽  
State University ◽  
Algebraic Equations ◽  
Mathematical Methods ◽  
World University Rankings ◽  
Linear Algebraic Equations ◽  
The World

Subject. The article focuses on mathematical methods for predicting university entry into TOP-100 World University Rankings. Objectives. The purpose of the study is to develop mathematical modeling techniques for predicting university entry into TOP-100 World University Rankings. Methods. We consider two approaches in mathematical simulation for the said purpose. The first approach employs population dynamics equations, including the Verhulst and the Lotka-Volterra equations. The second approach uses linear algebraic equation in three variables together with the variables constraints. Results. Population dynamics equations enable to model improvement in university performance expressed by Total Score or Overall Score indicator, to define intra-university competition and competition between universities of the World University Rankings. Linear algebraic equations provide projections of a university's specified position in the world rankings. We made clear mathematical statement of the problems; solved the problem that relates to entering TOP-100 of three global rankings (ARWU, THE, QS) by the Moscow State University and St. Petersburg University. Conclusions. We assume that the offered approaches will be useful for university managers who monitor and guide their university positioning.

scholarly journals Method of integro-differential equations for interpreting the results of vertical electrical sounding of the soil

2021 ◽  
pp. 67-70
Author(s):  
D. G. Koliushko ◽  
S. S. Rudenko ◽  
A. N. Saliba
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Power Plants ◽  
Current Source ◽  
Soil Surface ◽  
Observation Point ◽  
Practical Significance ◽  
Maximum Relative Error ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper is devoted to the problem of determining the geoelectric structure of the soil within the procedure of testing the grounding arrangements of existing power plants and substations to the required depth in conditions of dense development. To solve the problem, it was proposed to use the Schlumbergers method , which has a greater sounding depth compared to the Wenner electrode array. The purpose of the work is to develop a mathematical model for interpreting the results of soil sounding by the Schlumberger method in the form of a four-layer geoelectric structure. Methodology. To construct a mathematical model, it is proposed to use the solution of a particular problem about the field of a point current source, which, like the observation point, is located in the first layer of a four-layer soil. Based on this expressions, a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients ai and bi was compiled. On the basis of its analytical solution, an expression for the potential of the electric field was obtained for conducting VES (the point current source and the observation point are located only on the soil surface). Results. Comparison of the results of soil sounding by the Schlumberger installation and the interpretation of its results for the same points shows a sufficient degree of approximation: the maximum relative error does not exceed 9.7 % (for the second point), and the average relative error is 3.6 %. Originality. Based on the obtained expression, a test version of the program was implemented in Visual Basic for Applications to interpret the results of VES by the Schlumberger method. To check the obtained expressions, the interpretation of the VES results was carried out on the territory of a 150 kV substation of one of the mining and processing plants in the city of Kriviy Rih. Practical significance. The developed mathematical model will make it possible to increase the sounding depth, and, consequently, the accuracy of determining the standardized parameters of the grounding arrangements of power stations and substations.

MODELING OF DEFORMATION OF OBJECTS OPERATING UNDER CONDITIONS OF TEMPERATURE INFLUENCES

2019 ◽  
pp. 38-44
Author(s):  
G. V. Grygorchuk
Keyword(s):  
Stress State ◽  
Stress Tensor ◽  
Sugar Industry ◽  
Parametric Representation ◽  
Algebraic Equations ◽  
Linear Elasticity Theory ◽  
Metric Tensor ◽  
Ambient Temperatures ◽  
State Process ◽  
Linear Algebraic Equations

A mathematical model of the deformation and stress state process for long rotating objects complex geometric configuration is proposed. The model of the process of deformation of plots, which are combinations of sections of rectilinear, spherical and conical forms, is proposed, parametric representation of the law of motion of the specified plots in the initial and control moments of time is constructed, on the basis of which the components of vectors of local bases and components of the metric tensor for the specified types of plots are determined. What are the components of the stress tensor. The theory of systems of systems of linear algebraic equations was used to determine the geometric characteristics of a deformed spherical section, as well as the apparatus of the Hermite interpolation polynomials for modeling the axis deformation process. Within the framework of linear elasticity theory, the components of the stress tensor arising from non-isothermal processes characteristic of sugar industry technological objects operation have been calculated. The components of the stress tensor, which allows to take into account the features of the deformed object at different ambient temperatures, have been refined. The model of the stress state of objects is offered, the test calculations for the model sites are carried out, the directions of further researches are revealed.

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.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

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