Vibration of a Timoshenko beam supporting arbitrary large pre-deformation

We present an induced geometrically exact theory for the three-dimensional vibration of a beam undergoing finite transformation. The beam model coincides with a curvilinear Cosserat body and may be seen as an extension of the Timoshenko beam model. No particular hypothesis is used for the constitutive laws (in the framework of hyperelasticity), the geometry at rest or boundary conditions. The method leads to a weak formulation of the equations of vibration. The obtained internal energy is symmetric and leads to a self-adjoint operator that casts into a geometrical and material parts. Both may be written explicitly in terms of the finite transformation. The results are applied on the vibration of a beam supporting a finite longitudinal strain. The nonlinear effects according to the pre-stress are explicitly detailed for this example: instability, buckling.