Asymptotic Distribution of Eigenfunctions and Eigenvalues of the Basic Boundary-Contact Oscillation Problems of the Classical Theory of Elasticity

1999 ◽  
Vol 6 (2) ◽  
pp. 107-126
Author(s):  
T. Burchuladze ◽  
R. Rukhadze
Keyword(s):  
Elastic Medium ◽  
Asymptotic Distribution ◽  
Classical Theory ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Asymptotic Formulas ◽  
Closed Surfaces ◽  
Isotropic Elastic Medium ◽  
Boundary Contact ◽  
Basic Boundary

Abstract The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman's method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained.

Asymptotic Distribution of Eigenelements of the Basic Two-Dimensional Boundary-Contact Problems of Oscillation in Classical and Couple-Stress Theories of Elasticity

2000 ◽  
Vol 7 (1) ◽  
pp. 11-32
Author(s):  
T. Burchuladze ◽  
R. Rukhadze
Keyword(s):  
Couple Stress ◽  
Correlation Method ◽  
Contact Problems ◽  
Asymptotic Formulas ◽  
Two Dimensional ◽  
Closed Curves ◽  
Dimensional Boundary ◽  
Isotropic Elastic Medium ◽  
Boundary Contact ◽  
Basic Boundary

Abstract The basic boundary-contact problems of oscillation are considered for a two-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed curves. Asymptotic formulas for the distribution of eigenfunctions and eigenvalues of the considered problems are derived using the correlation method.

GS02 Quasi-Three-Dimensional Problems for Isotropic Elastic Medium Which Has Many Circular Inclusions

2011 ◽  
Vol 2011 (0) ◽  
pp. _GS02-1_-_GS02-2_
Author(s):  
Mutsumi MIYAGAWA ◽  
Takuo SUZUKI ◽  
Takanobu TAMIYA ◽  
Jyo SHIMURA
Keyword(s):  
Elastic Medium ◽  
Three Dimensional ◽  
Isotropic Elastic Medium

scholarly journals STRESS-STRAIN STATE OF THE CONICAL SHELL OF VARIABLE THICKNESS BASED ON THREE-DIMENSIONAL EQUATIONS OF ELASTICITY THEORY

2020 ◽  
Vol 82 (2) ◽  
pp. 189-200
Author(s):  
Val.V. Firsanov ◽  
V.T. Pham
Keyword(s):  
Differential Equations ◽  
Classical Theory ◽  
Variable Thickness ◽  
Conical Shell ◽  
Strain State ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Median Surface ◽  
Stress Strain ◽  
Stress Strain State

The results of a study of the stress-strain state of a conical shell of variable thickness based on a non-classical theory are presented. The sought-for displacements of the shell are approximated by polynomials in the normal coordinate to the median surface two degrees higher in relation to the classical theory of the Kirchhoff-Love type. When developing the theory, the three-dimensional equations of the theory of elasticity, as well as Lagrange variational principle are used as the equation of the shell state. As the result of minimizing the specified value of the total energy of the shell, a mathematical model is constructed, which is a system of differential equations of equilibrium in the displacements with variable coefficients and the corresponding boundary conditions. Two cases are considered: the shell is under the action of symmetric and asymmetric loads. Two-dimensional equations are transformed to the system of ordinary differential equations by means of trigonometric sequences as per circumferential coordinate. To solve the formulated boundary value problem, finite difference and matrix sweep methods are applied. The calculations have been made by means of a computer program. After having determined the displacements, shell deformations and tangential stresses are found from geometric and physical equations, transverse stresses - from the equilibrium equations of the three-dimensional theory of elasticity. As an example, a conical shell rigidly restrained at the two edges, with asymmetrically varying thickness is considered. Compared are the results of the VAT calculations obtained as per the improved and classical theories. The significant contribution of additional stresses in the boundary zone to the total stress state of the shell is shown. The received results can be used in the strength and durability calculations and tests of machine-building facilities of various purposes.

Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. II

1995 ◽  
Vol 2 (3) ◽  
pp. 259-276
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Arbitrary Configuration ◽  
Anisotropic Bodies ◽  
Present Part ◽  
Classical Potential Theory ◽  
Basic Boundary

Abstract In the first part [Duduchava, Natroshvili and Shargorodsky, Georgian Math. J. 2: 123–140, 1985] of the paper the basic boundary value problems of the mathematical theory of elasticity for three-dimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov () and Bessel-potential () spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

2015 ◽  
Vol 8 (1) ◽  
pp. 82-103
Author(s):  
Palaniyandi Ponnusamy
Keyword(s):  
Classical Theory ◽  
Cross Sections ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Coupled System ◽  
Theory Of Elasticity ◽  
Inviscid Fluid ◽  
Displacement Potential ◽  
Modes Of Vibration

AbstractIn this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.

scholarly journals STRESS-STRAIN STATE OF THE SPHERICAL SHELL EXPOSED TO AN ARBITRARY LOAD BASED ON A NON-CLASSICAL THEORY

2019 ◽  
Vol 81 (3) ◽  
pp. 359-368
Author(s):  
V.V. Firsanov ◽  
V.T. Pham
Keyword(s):  
Classical Theory ◽  
Three Dimensional ◽  
Middle Surface ◽  
Energy Functional ◽  
Theory Of Elasticity ◽  
Natural Boundary ◽  
Equilibrium Equations ◽  
Tangential Stresses ◽  
Natural Boundary Conditions ◽  
External Loads

Considered is the stress state of an isotropic spherical shell exposed to an arbitrary load based on a non-classical theory. When building a mathematical model of the shell, three-dimensional equations of the theory of elasticity are applied. Displacements are represented in the form of polynomials along the coordinate normal to the middle surface two degrees higher relative to the classical theory of the Kirchhoff-Love type. As a result of minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are obtained. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of displacements and external loads into trigonometric series in the circumferential coordinate. Displacements are represented in form of polynomials along the coordinate normal to the middle surface by two degrees higher relative to the classical theory of the Kirchhoff-Love type. Resulting from minimization of the refined value of the Lagrange energy functional, a system of differential equilibrium equations in displacements and natural boundary conditions are received. The task of reducing two-dimensional equations to ordinary differential equations is carried out by decomposing the components of the displacements and the external loads into trigonometric series as per the circumferential coordinate. The formulated boundary problem is solved by the methods of finite differences and matrix sweep. As a result, displacements are obtained in the grid nodes, for approximation of which splines are used. The shell deformations are found using geometric relationship; tangential stresses are received from the correlations of Hooke's law. One of the features of this paper lies in the fact that the transverse stresses are determined by the direct integration of the equilibrium equations of the three-dimensional theory of elasticity. An example of the calculation of a hemispherical shell rigidly restrained along the lower base contour is brought. The shell is exposed to the wind load. Comparison of the results received by the refined theory with the data of the classical theory has shown that in the zone of distortion of the stressed state, the normal tangential stresses are substantially revised and the transverse normal stresses, which are neglected in the classical theory, are of the same magnitude with the maximum values of the main bending stress. Considered is the influence of the relative thickness on the stress state of the shell. It was discovered that the shell thickness significantly increases the error of the classical theory, while determining the stresses and assessing the strength of the elements of the aircraft structures.

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