  This research report is devoted to the study of the conservation and the
dissipation properties of the mechanical energy of several time--integration
methods dedicated to the elasto--dynamics with unilateral contact. Given that
the direct application of the standard schemes as the Newmark schemes or the
generalized--$\alpha$ schemes leads to energy blow-up, we study two schemes
dedicated to the time--integration of nonsmooth systems with contact: the
Moreau--Jean scheme and the nonsmooth generalized--$\alpha$ scheme. The energy
conservation and dissipation properties of the Moreau--Jean is firstly shown.
In a second step, the nonsmooth generalized--$\alpha$ scheme is studied by
adapting the previous works of Krenk and H{\o}gsberg in the context of
unilateral contact. Finally, the known properties of the Newmark and the
Hilber--Hughes--Taylor (HHT) scheme in the unconstrained case are extended
without any further assumptions to the case with contact.
