On the Stress‐Function Approaches of Boussinesq and Timpe to the Axisymmetric Problem of Elasticity Theory

1951 ◽  
Vol 22 (9) ◽  
pp. 1121-1124 ◽  
Author(s):  
E. Sternberg ◽  
R. A. Eubanks ◽  
M. A. Sadowsky
Keyword(s):  
Elasticity Theory ◽  
Stress Function ◽  
Axisymmetric Problem

scholarly journals Stress Function Approach in Anisotropic Elasticity Theory.

1992 ◽  
Vol 58 (554) ◽  
pp. 1839-1844
Author(s):  
Hideaki KASANO ◽  
Hiroyuki MATSUMOTO ◽  
Ichiro NAKAHARA
Keyword(s):  
Elasticity Theory ◽  
Stress Function ◽  
Anisotropic Elasticity ◽  
Function Approach

Asymptotic behavior of the solution of an axisymmetric problem of elasticity theory for a sphere with variable elasticity modules

2020 ◽  
Vol 25 (12) ◽  
pp. 2231-2251 ◽  
Author(s):  
Natiq K Akhmedov ◽  
Natavan S Gasanova
Keyword(s):  
Elasticity Theory ◽  
Edge Effect ◽  
Characteristic Equation ◽  
Axisymmetric Problem ◽  
Applied Theory ◽  
Lagrange Principle ◽  
Homogeneous Solutions ◽  
Inhomogeneous Sphere ◽  
Radially Inhomogeneous ◽  
Theory Of Shells

In the paper the axisymmetric problem of elasticity theory is studied for the radially inhomogeneous sphere of small thickness that does not contain any of the poles 0 and [Formula: see text]. Here the case is considered when the elasticity modules vary linearly with respect to the radius. It is assumed that the lateral surface of the sphere is free of stresses, and at the ends of the sphere (at the conical sections) the stresses are set, leaving it in equilibrium. A characteristic equation is obtained and, based on its asymptotic analysis, the existence of three groups of roots is established with respect to the small parameter characterizing the thickness of the sphere. The corresponding homogeneous solutions are constructed, depending on the roots of the characteristic equation. It is shown that the penetrating solution corresponds to the first group of roots. The second group of roots corresponds to the solution of the edge effect type, similar to the edge effect in the applied theory of shells. The third group of roots corresponds to the boundary layer type solution localized in the conical sections. The solution corresponding to the first and second groups of roots determines the internal stress–strain state of the sphere. In the first term of the asymptotic, they can be considered as a solution in the applied theory of shells. The question of satisfying the boundary conditions at the ends (on the conical sections) of the sphere is considered using the variational Lagrange principle.

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