Variational functionals for two-dimensional equilibrium and stability problems of Cosserat-Timoshenko elastic rods

This article deals with nonlinear two-dimensional problem of the theory of elastic Cosserat-Timoshenko rods in the material (Lagrangian) description with energy conjugate stress and deformation vectors. Equivalence of the differential and variational formulations of the problem was proved for smooth solutions. The expression for the second variation of the Lagrangian functional was derived. The differential equations for the stability problem were obtained from the second variation of the Lagrangian functional. Two types of equation of plane problems of stability of equilibrium are obtained: variational equations for initial system of differential equations and Euler equations for the second variation of the Lagrangian functional.Exact solution of the stability problem accounting for the deformations of bending, shear and tension-compression was obtained for the pivotally supported rod.