In this paper, the loss of stability of materials under the influence of mechanical and thermal loads is investigated. The mathematical model is based on the general theory of stability of nonlinear viscoelastic bodies with respect to finite perturbations. A nonlinear boundary value problem with variable coefficients is written out concerning finite perturbations arising from the deformation of a viscoelastic medium. An analytical solution is obtained in the form of a BubnovGalerkin expansion in terms of eigenfunctions. The systems of equations for determining the corresponding coefficients are solved numerically. The number of terms of the series is limited based on the theory of bifurcations and methods of studying dynamic dissipative systems. The area of stability relative to the permissible initial disturbances is determined when changing the values of the load parameters. It is shown that in the region of initial disturbances as a result of a finite number of bifurcations, the deformation process remains stable and a hierarchy of stable equilibrium states is observed.

Alexander I. Sumin, Doctor of Physical and Mathematical Sciences, Professor; e-mail: sumin_ai@mail.ru;

Rita S. Sumina, Candidate of Technical Sciences, Docent; e-mail: rsumina@mail.ru Alexander L. Frolov, Candidate of Physical and Mathematical Sciences, Professor; e-mail: al-frol@yandex.ru

Oksana          А.        Frolova,        Candidate      of      Physical      and      Mathematical      Sciences;              e-mail: oksanafrola@yandex.ru