stability, nonlinear elastic medium, finite perturbations, elastic potentials, Lyapunov function, stability regions, finite deformations, stability of the solution, zero solution, bifurcation points.

The paper studies aspects of the stability of nonlinear elastic bodies with respect to superimposed finite deformations. The question of studying the basic process of deformation of the initial medium is reduced to solving a nonlinear boundary value problem with variable coefficients for finite perturbations. Solutions for displacement perturbations are selected in the form of series containing eigenfunctions. Using the principle of possible displacements, the question of the stability of the ground state is reduced to the study of the stability of the zero solution of a system of ordinary differential equations with constant coefficients. For this system, a function is constructed, which, with some restrictions on the initial perturbations, will be the Lyapunov function, which is used to find stability regions.

Sumin Alexander Ivanovich, Dr. Sci. Phys. & Math., Professor, Head of the Department of Mathematics of the Military Educational and Scientific Center of the Military – Air Forces ”Military Air Academy named after Professor N E Zhukovsky and Yu A Gagarin”, Voronezh, Russia.

Boger Andrey Alexandrovich, Candidate of Technical Sciences, Associate Professor, Associate Professor of the Department of Mathematics of the Military Educational and Scientific Center of the Military – Air Forces ”Military Air Academy named after Professor N E Zhukovsky and Yu A Gagarin”, Voronezh, Russia.

Sumin Viktor Alexandrovich, Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Mathematics of the Military Educational and Scientific Center of the Military – Air Forces ”Military Air Academy named after Professor N E Zhukovsky and Yu A Gagarin”, Voronezh, Russia.

Ryabov Sergey Vladimirovich, Candidate of Technical Sciences, Associate Professor, Associate Professor of the Department of Mathematics of the Military Educational and Scientific Center of the Military – Air Forces ”Military Air Academy named after Professor N E Zhukovsky and Yu A Gagarin”, Voronezh, Russia.