Dynamics of plane motion of an elastic rod

Plane motion of an elastic rod, subjected to a compressive force is analyzed. Equations of motions are derived for the case when deformations are not small. It is assumed that the compressive force has a periodic component, so that parametric instability is possible. Stability boundary is estimated analytically and determined numerically. In the special case of static loading a new formula for the slenderness ratio, below which there is no buckling, is obtained. In deriving the differential equations of motion a generalized constitutive equations taking into account compressibility and shear stresses are used.