Аннотация: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.