scholarly journals About a new approach to calculating spiral clamps

2019 ◽  
pp. 59-67
Author(s):  
A N Danilin ◽  
S I Zhavoronok ◽  
L N Rabinsky
Keyword(s):  
Boundary Value Problems ◽  
Mutual Influence ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Design Parameters ◽  
Elastic Cylindrical Shell ◽  
Force Transfer ◽  
The Matrix ◽  
Generalized Displacements ◽  
Anisotropic Elastic

The bearing capacity of spiral clamps, which are mounted on wires (cables) for their tension, connection, repair, etc., is studied. The design of spiral clamps is formed from stretched spirals that are wound onto conductors with an interference fit, which makes it possible to obtain tensile connections practically inseparable. The general problem of the interaction of spiral clamps and overhead line conductor layers is formulated. Different asymptotic solutions are given for initial and boundary value problems, and the design parameters of spiral clamps are determined to provide their carrying capacity. A wire layer is represented by the energy approach as an equivalent anisotropic elastic cylindrical shell, and wire construction as a whole is considered as a system of cylindrical shells inserted each other and interacting by forces of pressure and friction. The equivalence of the elastic properties of the shell to the properties of the wire layer is established using energy averaging. The constitutive relations obtained using the Castigliano theorem relate the generalized displacements and the corresponding forces. The matrix in these ratios is a stiffness matrix or flexibility matrix of a spiral wire structure. Such approach allows variety of interaction problems for spiral clamps with conductor layers to be solved, and the force transfer mechanism to be investigated from common positions. Static equations are written from the equilibrium of the elementary shell ring. It is considered that the length of the clamp is so great that the mutual influence of its ends can be neglected; the clamp is modeled as semi-infinite shell. This model allows the different initial and boundary value problems to be formulated, depending on the boundary conditions and clamp mounting methods on a conductor.

scholarly journals Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

2019 ◽  
Vol 24 (1) ◽  
pp. 33 ◽  
Author(s):  
Mikhail Nikabadze ◽  
Armine Ulukhanyan
Keyword(s):  
Boundary Value Problems ◽  
Anisotropic Medium ◽  
Eigenvalue Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Theory Of Elasticity ◽  
Initial Boundary Value Problems ◽  
Special Cases ◽  
Initial Boundary Value

The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators(tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transverselyisotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators(tensors–tensors) of the initial-boundary value problems are constructed that allow decomposinginitial-boundary value problems. We also find the determinant and the tensor of cofactors to the sumof six tensors used for decomposition of initial-boundary value problems. From three-dimensionaldecomposed initial-boundary value problems, the corresponding decomposed initial-boundary valueproblems for the theories of thin bodies are obtained.

scholarly journals ON INCOMPLETE FACTORIZATION IMPLICIT TECHNIQUE FOR 2D ELLIPTIC FD EQUATIONS

2020 ◽  
Vol 25 (1) ◽  
pp. 37-52
Author(s):  
Vladimir Sabinin
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Incomplete Factorization ◽  
Iteration Parameter ◽  
New Variant ◽  
The Matrix ◽  
Point Scheme ◽  
The One

A new variant of Incomplete Factorization Implicit (IFI) iterative technique for 2D elliptic finite-difference (FD) equations is suggested which is differed by applying the matrix tridiagonal algorithm. Its iteration parameter is shown be linked with the one for Alternating Direction Implicit method. An effective set of values for the parameter is suggested. A procedure for enhancing the set of iteration parameters for IFI is proposed. The technique is applied to a 5-point FD scheme, and to a 9-point FD scheme. It is suggested applying the solver for 5-point scheme to solving boundary-value problems for the 9-point scheme, too. The results of numerical experiment with Dirichlet and Neumann boundary-value problems for Poisson equation in a rectangle, and in a quasi-circle are presented. Mixed boundary-value problems in square are considered, too. The effectiveness of IFI is high, and weakly depends on the type of boundary conditions.

Verification of Plasticity Theory with Isotropic Hardening and Additive Decomposition of Left Deformation Tensor Using Digital Image Correlation System

2015 ◽  
Vol 240 ◽  
pp. 61-66 ◽  
Author(s):  
Marcin Gajewski ◽  
Cezary Ajdukiewicz ◽  
Andrzej Piotrowski
Keyword(s):  
Digital Image Correlation ◽  
Boundary Value Problems ◽  
Digital Image ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Experimental Tests ◽  
Image Correlation ◽  
Deformation Tensor ◽  
Isotropic Hardening ◽  
Complex Boundary

The development of measurement methods, and in particular digital image correlation (DIC) systems, which are designed to measure of entire displacements and deformations fields, opens up new areas of research. In general, the materials constitutive relations are formulated in such a way that material parameters could be determined with relatively simple experimental tests carried out on samples with uniform (approximately) stress and strain fields. Then it is possible to apply them to complex boundary value problems formulated e.g. in the small or large deformation theories. The application of DIC allows to verify the accuracy of their predictions by comparing the results of the experiment with solutions to boundary value problems obtained using the finite element method (FEM).

Kinetic Theory of Molecular Gases I: Models of the Linear Waldmann–Snider Collision Operator

1975 ◽  
Vol 53 (13) ◽  
pp. 1266-1278 ◽  
Author(s):  
G. Tenti ◽  
Rashmi C. Desai
Keyword(s):  
Kinetic Theory ◽  
Boundary Value Problems ◽  
Transport Properties ◽  
Boundary Value ◽  
Kinetic Equations ◽  
Collision Operator ◽  
Matrix Elements ◽  
Molecular Gases ◽  
The Matrix ◽  
Modeling Theory

Using a method closely akin to the Gross–Jackson–Sirovich procedure, we present a modeling theory of the linear Waldmann–Snider collision operator. The resulting model kinetic equations are applicable to all regions of wavelength and frequency consistent with the original equation itself. The theory is made parameter free by relating the matrix elements of the collision operator to measured transport properties. It is sophisticated enough to afford a study of both scalar and tensorial phenomena and can be applied to the analysis of a variety of initial and boundary value problems.

scholarly journals Implicit constitutive relations for nonlinear magnetoelastic bodies

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140959 ◽  
Author(s):  
R. Bustamante ◽  
K. R. Rajagopal
Keyword(s):  
Boundary Value Problems ◽  
Large Deformations ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Small Displacement ◽  
Elastic Bodies ◽  
Classical Models ◽  
Linearized Model ◽  
Implicit Constitutive Relations

Implicit constitutive relations that characterize the response of elastic bodies have greatly enhanced the arsenal available at the disposal of the analyst working in the field of elasticity. This class of models were recently extended to describe electroelastic bodies by the present authors. In this paper, we extend the development of implicit constitutive relations to describe the behaviour of elastic bodies that respond to magnetic stimuli. The models that are developed provide a rational way to describe phenomena that have hitherto not been adequately described by the classical models that are in place. After developing implicit constitutive relations for magnetoelastic bodies undergoing large deformations, we consider the linearization of the models within the context of small displacement gradients. We then use the linearized model to describe experimentally observed phenomena which the classical linearized magnetoelastic models are incapable of doing. We also solve several boundary value problems within the context of the models that are developed: extension and shear of a slab, and radial inflation and extension of a cylinder.

scholarly journals On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

2021 ◽  
Vol 31 (4) ◽  
pp. 651-667
Author(s):  
B.Kh. Turmetov ◽  
V.V. Karachik
Keyword(s):  
Integral Representation ◽  
Boundary Value Problems ◽  
Explicit Form ◽  
Poisson Equation ◽  
Green's Function ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Inverse Matrix ◽  
The Matrix ◽  
Neumann Boundary Value Problems

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

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