In the present work the problem of dynamic crack initiation in the brittle solid body was studied using the non-classical approaches of continuum mechanics: peridynamics and incubation time fracture model. For this purpose, structural and incubation time criteria were formulated as criteria of the bond failure. These approaches consider the stress fields in the vicinity of the crack tip, therefore the stress-based failure criteria of the bonds were suggested. Despite the fact that both approaches consider the fracture process as a sequential failure of the structural blocks of a small certain size, which is a material constant, an element size independent approach was proposed. Damage models were implemented in the open-source peridynamic numerical package Peridigm for the state based ordinary material model. As a result, numerically obtained stress intensity factor (SIF) - fracture initiation time dependence fits well the experimental data found in works by Ravi-Chandar K., Knauss W.G., (1984a).