Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
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IssuesArchive of Issues2008-5pp.726-732

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D. V. Georgievskii, "Structure of polynomial solutions of elasticity equations in stresses," Mech. Solids. 43 (5), 726-732 (2008)
Year 2008 Volume 43 Number 5 Pages 726-732
DOI 10.3103/S0025654408050063
Title Structure of polynomial solutions of elasticity equations in stresses
Author(s) D. V. Georgievskii (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, georgiev@mech.math.msu.su)
Abstract For an isotropic elasticity problem in stresses in three-dimensional space minus the origin, we study solutions that have the singularity 1/r2 and, after the multiplication by r2, polynomially depend on the direction cosines. In this polynomial class, for the equilibrium equation we write out the general solution that is a statically admissible (in the sense of Castigliano) solution of the Kelvin problem. We show that if one or several Beltrami equations are not satisfied, then the classical Kelvin solution becomes nonunique. A method for constructing nonunique solutions of this kind is given. The equivalence of various statements of the elasticity problem in stresses is discussed. For the problem on the action of a lumped force at the vertex of an arbitrary conical elastic body, we write out the exact solution in stresses for the case of an incompressible material. The solution for a compressible material is represented in the form of series in a parameter characterizing the deviation of the Poisson ratio from 1/2. We obtain iterative chains of problems in stresses and conditions for the finiteness of these chains. We also analyze the realizability of a linear-fractional dependence of the solution in stresses on the Poisson ratio.
Received 12 January 2005
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