Mechanical molecular scission is the main problem of polymeric drag reducers. The ability to reduce the drag is notably decreased as the molecules break down step by step as time goes on. A number of researchers have given a large part of their time to attempts to further understand the role that some important features play in polymer degradation. Until now, all efforts have been in experimental approaches. This paper is the first attempt to take into account the effect of molecular scission on a drag reducing flow by a direct numerical simulation. We analyse a turbulent plane Couette flow of a FENE-P fluid. Our degradation model is based on the maximum polymer extension length L. Unlike the standard FENE-P model, in which L is a constant, the polymer extension here is a spatio-temporal field L(x, y, z, t). When the molecules are highly stretched, which is measured by the trace of the conformation tensor, their maximum length is locally reduced and, consequently, so is their ability to reduce drag. The degraded L spreads within the domain by means of a transport equation. We show here that with such a simple idea we are able to predict the main aspects of mechanical degradation in the flow, such as the change of the turbulent structures and velocity field, and, consequently, the fall of the drag reduction over time.