The experimental studies of the fracture of composite materials under uniaxial loading along the reinforcing elements (in unidirectional composites, mainly) showed that a specific fracture manifesting itself as a partition of material into separate parts along the line of action of external loading takes place. The above phenomenon has not been observed for homogeneous materials and is characteristic only for composite materials and their full fracture occurs in planes and surfaces along the reinforcing elements. It is known that one of the main reasons of occurrence of such a type of fracture may be a slight curving of the reinforcing elements. In papers by Akbarov, on the basis of a model of the piecewise-homogeneous body using exact three-dimensional (3-D) linear equations of elasticity theory, the method by means of which the stress distribution in laminated composites with the curved structures is determined has been developed, and the interpretation of the above effect is proposed. However, as a criterion of a material fracture, a macroscopic criterion of fracture has been used. As is well-known, any macroscopic criterion of a fracture can give only a rough estimate of fracture and cannot describe sufficiently a rigid mechanism of a fracture. The mentioned mechanism Of a fracture can be established only within the framework of the crack mechanics in composites considered. For this purpose, in the present paper, based on the piecewise-homogeneous body model using exact equations of the elasticity theory, the method for studying the crack problems in composites with periodically curved layers has been suggested with concrete problems as an example. In this case, according to Desai-Mac-Harry hypothesis, the cracks are assumed to be located in the most dangerous parts of matrix layers and their edges are parallel to the direction of external forces.