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Vibrations of an elastic isotropic sphere of incompressible material under uniform initial hydrostatic loading. (English. Russian original) Zbl 0631.73052
Sov. Phys., Dokl. 30, 535-537 (1985); translation from Dokl. Akad. Nauk SSSR 282, 1077-1081 (1985).
In the present article, within the framework of the linearized theory of elasticity, we consider the vibrations of an isotropic elastic sphere of incompressible material with an arbitrary structure of the elastic potential under uniform initial hydrostatic loading. Following the author: Stability of elastic bodies with uniform compression (1979; Zbl 0429.73039), we carry out the investigation in a unified general form for the theory of finite (large) initial strains and two versions of the theory of small initial strains as they apply to the cases in which the initial loading is realized by a “follower” or a “deadweight” load; for the theory of small initial strains, an improved expression is used to determine the “follower” load. The investigation is carried out in Lagrangian coordinates (r,$$\theta$$,$$\phi)$$, which coincide with the spherical coordinates in the natural or initial stress-strain state; by virtue of the incompressibility condition for the initial state in question, the introduction of the indicated Lagrangian coordinates coincides in the natural and initial stress-strain states.
##### MSC:
 74H45 Vibrations in dynamical problems in solid mechanics 74B20 Nonlinear elasticity