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 behaviour. These schemes have shown to lead to excellent convergence properties that make them cost efficient alternative to explicit schemes generally advocated as the best choice for these problems. To qualify the theory, examples are presented, whereby performance of the crushed shock-absorber devices, as well as their resolution based on the dynamic buckling of thin steel cylinder phenomenon are given. In these examples, it is shown that implicit schemes are more efficient (less CPU-cost for a given precision of the contact situation and capable of reaching higher precision) than the explicit scheme, despite that both use the Penalty method for the treatment of contact problems.