Vol. 14, No. 1, 2019

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Analytical investigation of free vibrations of a bounded nonlinear bulk-elastic medium in a field of mass forces

Eugene I. Ryzhak and Svetlana V. Sinyukhina

Vol. 14 (2019), No. 1, 61–95
Abstract

Free vibrations of a homogeneous nonlinear bulk-elastic medium, namely a solid with negligible shear stiffness, occupying a bounded domain and nonuniformly deformed under the action of a field of mass forces are investigated through variational methods. The nonlinear constitutive law of bulk elasticity is assumed to be arbitrary, the applied field of mass forces is assumed to be an arbitrary potential field, and conditions of free sliding are prescribed on the whole boundary. The associated problem of free vibrations corresponds to significantly nonuniform distribution of mechanical parameters of the medium, which results in significantly varying coefficients of governing equations — a case where standard methods are inapplicable and results of analysis are almost absent. A crucial element of presented variational analysis is the use of derived by the authors earlier the canonical form for the second variation of the total potential energy. This canonical form enables to state and prove a modified spectral theorem, and additionally a comparison theorem for the free vibration frequencies of different media in different fields of mass forces, provided the media occupy domains possessing the same or similar shapes. For some special shapes, the bilateral bounds for all the free vibration frequencies are obtained. The results are illustrated by clarifying examples.

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Keywords
free vibrations, bulk-elastic medium, field of mass forces, variational methods, bilateral bounds for frequencies
Milestones
Received: 5 August 2018
Revised: 13 November 2018
Accepted: 24 December 2018
Published: 7 April 2019
Authors
Eugene I. Ryzhak
O. Schmidt Institute of Physics of the Earth
Russian Academy of Sciences
10, B. Gruzinskaya St.
Moscow, 123242
Russia
Svetlana V. Sinyukhina
O. Schmidt Institute of Physics of the Earth
Russian Academy of Sciences
10, B. Gruzinskaya St.
Moscow, 123242
Russia