Variational formulations of the nonlinear equilibrium and stability problems of elastic rods

This article is dedicated to the analysis of the nonlinear plane problems formulated in the special Cosserat-Timoshenko’s theory of elastic rods in Lagrangian description. The problems were solved using conjugate pairs of strain and stress vectors. Equivalence of the differential and variational formulations of the Lagrangian functional was proved. The differential equations of the plane problems of stability were obtained from the second variation of the Lagrangian functional. Deformations of bending, shear and tension-compression were taken into account while finding an exact solution for some stability problems