Article. Published in Computational Mechanics of Solid Mediums. 2009. T. 2, No2. C. 22-39. A model of the destruction of composites and adhesive compounds is proposed, based on the consideration of the zone of the destruction process near the top of the crack. It is believed that the zone of the destruction process is a layer of finite length containing material with partially broken bonds between its individual structural elements (end region), considered as part of a crack. The presence of bonds between the edges of the crack in the end region is modeled by applying to the surfaces of the crack the forces of adhesion caused by the presence of bonds. The analysis of the maximum crack equilibrium within the end region model is performed on the basis of a non-local fracture criterion with an energy condition for advancing the crack top and a kinematic condition for determining the edge advance of the crack end region and includes: 1) determination of the dependence of adhesion forces on crack opening ; 2) evaluation of the stress state near the crack taking into account external loads and adhesion forces ; 3) determination of the dependence of critical external loads on the crack length . This paper discusses all of the above steps. The results of model calculations.