  In this work, we study the formation and evolution of dark matter halos by
means of the spherical infall model with shell-crossing. We present a framework
to tackle this effect properly based on the numerical follow-up, with time, of
that individual shell of matter that contains always the same fraction of mass
with respect to the total mass. In this first step, we do not include angular
momentum, velocity dispersion or triaxiality. Within this framework - named as
the Spherical Shell Tracker (SST) - we investigate the dependence of the
evolution of the halo with virial mass, with the adopted mass fraction of the
shell, and for different cosmologies. We find that our results are very
sensitive to a variation of the halo virial mass or the mass fraction of the
shell that we consider. However, we obtain a negligible dependence on
cosmology. Furthermore, we show that the effect of shell-crossing plays a
crucial role in the way that the halo reaches the stabilization in radius and
the virial equilibrium. We find that the values currently adopted in the
literature for the actual density contrast at the moment of virialization,
delta_vir, may not be accurate enough. In this context, we stress the problems
related to the definition of a virial mass and a virial radius for the halo.
The question of whether the results found here may be obtained by tracking the
shells with an analytic approximation remains to be explored.
