An axisymmetric problem on the interaction of a periodic system of coaxial cracks in an infinite material is investigated by the use of approaches of the linearized mechanics of solids. Two non-classical mechanisms of fracture are considered, namely, fracture of solids with initial stresses acting parallel to the cracks' planes and fracture of materials under compression along cracks. Numerical results are obtained for a highly elastic material with the Bartenev - Khazanovich elastic potential. The dependence of the parameters of fracture on the loading conditions, physico-mechanical characteristics of materials, and geometric parameters of the problem is analyzed.