There is a practice to select reinforcing filaments of composite material on basis of their maximal strength and rigidity. On the other hand, the matrices are used, which have lower rigidity and resistance to deformation than those of the reinforcing filaments. That is why, when estimating local destruction in the first approximation, we assume that the inclusions-filaments are absolutely rigid but of a limited tensile strength. Such an assumption simplifies the mathematical description of the problem as well as its solution significantly. Long before a general destruction of such compositions, the processes of local destruction take place, i.e. plastic deformation of the matrix or of the contact matrix-filament layer followed by lamination and tearing of the filaments. The analysis of destruction of composites with short filaments has shown that a plastic flow usually takes place at the edges on the ends of filaments and propagates through width and length of the filament-matrix boundary surface. Similar conclusions were confirmed by application of the optical methods of investigation of stress fields nearby to the reinforcing elements. Using the methods of finite elements and local variations, it is proven that contrary to the crack, the zone of plasticity is shifted towards the inclusion.