Static and Dynamics of a Rod at the Longitudinal Loading

A short review of works about static and dynamical stability of a thin rod under axial compression is given. By using linear static approach a critical compression has been found by L. Euler. In the paper of M. A. Lavrentiev and A. Y. Ishlinsky it has been established that at intensive loading which essentially exceeds the Eulerian one, the maximum growth of the lateral deflection corresponds to the mode with a large number of waves in the longitudinal direction. The following researches are connected with the longitudinal waves influence. The conditions of parametric resonances appearing and also the cases of stability loss under load less than the Eulerian one are found. Under quasi-linear approach the beating effect with energy transition from longitudinal vibrations into transversal ones and vice versa is established. At a long-time action of the load exceeding the Eulerian one both linear and quasi-linear approaches do not lead to finite values of transversal amplitude. That is why the non-linear approach is used and the growth of the post-critical deformations of the rod is studied. The connection of the deformation picture with the effect discovered by M. A. Lavrentiev and A. Y. Ishlinsky with the Eulerian elastics is marked.