A method for the construction op the approximate solution of the mixed axisymmetric problem in the theory of elasticity

Many applied problems in the theory of elasticity can be reduced to the solution of singular integral equations either linear or nonlinear. In this paper we shall study a nonlinear system of singular integral equations which appear on the closed Lipanouv surface in an ideal medium [4]. We shall find a cubic mechanical method which corresponds to the system and prove its convergence; we obtained a discrete operator which corresponds to this system and study its properties and then a solution to the resulting system of the nonlinear equations which leads to an approximate solution for the original system and its convergence.

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Analysis of axisymmetric problem from the theory of elasticity for an isotropic cylinder of small thickness with alternating elasticity modules

Eastern-European Journal of Enterprise Technologies ◽

10.15587/1729-4061.2019.162153 ◽

2019 ◽

Vol 2 (7 (98)) ◽

pp. 13-19

Author(s):

Natik Akhmedov ◽

Sevda Akbarova ◽

Jalala Ismayilova

Keyword(s):

Axisymmetric Problem ◽

Theory Of Elasticity ◽

Isotropic Cylinder ◽

Small Thickness

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Approximate Elasticity Solution for Orthotopic Cylinder Under Hydrostatic Pressure and Band Loads

Journal of Applied Mechanics ◽

10.1115/1.3408416 ◽

1970 ◽

Vol 37 (1) ◽

pp. 101-108 ◽

Cited By ~ 10

Author(s):

A. P. Misovec ◽

J. Kempner

Keyword(s):

Approximate Solution ◽

Shell Theory ◽

Good Accuracy ◽

Three Dimensional ◽

Transversely Isotropic ◽

External Pressure ◽

Theory Of Elasticity ◽

Numerical Comparison ◽

Axial Loads ◽

Special Case

An approximate solution to the Navier equations of the three-dimensional theory of elasticity for an axisymmetric orthotopic circular cylinder subjected to internal and external pressure, axial loads, and closely spaced periodic radial loads is developed. Numerical comparison with the exact solution for the special case of a transversely isotropic cylinder subjected to periodic band loads shows that very good accuracy is obtainable. When the results of the approximate solution are compared with previously obtained results of a Flu¨gge-type shell solution of a ring-reinforced orthotropic cylinder, it is found that the shell theory gives fairly accurate representations of the deformations and stresses except in the neighborhood of discontinuous loads. The addition of transverse shear deformations does not improve the accuracy of the shell solution.

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On the Axisymmetric Problem of the Theory of Elasticity for an Infinite Region Containing Two Spherical Cavities

Journal of Applied Mechanics ◽

10.1115/1.4010401 ◽

1952 ◽

Vol 19 (1) ◽

pp. 19-27

Author(s):

E. Sternberg ◽

M. A. Sadowsky

Keyword(s):

Axisymmetric Problem ◽

Three Dimensional ◽

Theory Of Elasticity ◽

Uniform Field ◽

Dimensional Problem ◽

Spherical Cavities ◽

Infinite Region ◽

In Series ◽

The Common ◽

Axis Of Symmetry

Abstract This paper contains a solution in series form for the stress distribution in an infinite elastic medium which possesses two spherical cavities of the same size. The loading consists of tractions applied to the cavities, as well as of a uniform field of tractions at infinity, and both are assumed to be symmetric with respect to the common axis of symmetry of the cavities and with respect to the plane of geometric symmetry perpendicular to this axis. The loading is otherwise unrestricted. The solution is based upon the Boussinesq stress-function approach and apparently constitutes the first application of spherical dipolar co-ordinates in the theory of elasticity. Numerical evaluations are given for the case in which the surfaces of the cavities are free from tractions and the stress field at infinity is hydrostatic. The results illustrate the interference of two sources of stress concentration in a three-dimensional problem. The approach used here may be extended to cope with the general equilibrium problem for a region bounded by two nonconcentric spheres.

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