Classical and non-classical implicit time integration schemes are presented for the transient response of highly nonlinear problems of large deformation analysis exhibiting important material energy dissipation. Surprisingly these schemes lead to excellent convergence properties that make them cost efficient alternative to explicit schemes generally advocated as the best choice for these problems. Numerical illustrations presented herein show the crushing of a shock-absorber device, as well as the resolution of the dynamical buckling of a thin-walled steel cylinder. In these examples, it is shown that implicit schemes are more efficient and require less CPU-cost for a given precision of the contact situation and higher precision reachable than the explicit scheme. Both of which are using the Penalty method for the treatment of the contact problem.