Issue 49

O. Naimark, Frattura ed Integrità Strutturale, 49 (2019) 272-281; DOI: 10.3221/IGF-ESIS.49.27 275 factor I K there is a difference between the Irwin criteria and the cohesive modulus. The cohesive modulus determines the steady-state character of crack propagation, but not the catastrophic one corresponding to the Griffith-Irwin approach. Non-local formulation of stress criterion was proposed by Novozhilov [3] to introduce the fracture “quantization” scale. According to this formulation the crack propagation condition in Mode I reads: 0 2 d yy C dx d     , (5) where yy  is the normal stress perpendicular to the atomic layer with the space d , C  is the critical stress. The space d is the fracture quantum. The criterion (5) can be used if the complete expression of the stress-field, and not only its asymptotic term, is taken into consideration. The non-local stress criterion can be also used for the interpretation of the cohesion forces ( ) B G s , when the fracture quantum not restricted to be the atomic spacing. The qualitative difference between above mentioned approaches can be shown taking in view the remarks by Fraenkel [18] under the critical analysis of the Griffith approach. Fraenkel wrote that the physically realistic form of the energy U must contain the local minimum ( ) e e U a (Fig. 1, curve 2). The difference in the energy c e U U U    determines the work of the stress field at the crack tip under transition from the steady-state to the unstable regime of crack propagation. This work provides the overcoming of the energy barrier. It is natural to assume that the cohesive modulus, non-local approach, the Finite Fracture Mechanics and the TCD are the force version of this energy barrier. We will show in the following that the metastable energy form, assumed by Fraenkel, has the relationship to the collective behavior of the defect ensemble in the process zone and to the interaction of the defects ensemble with the main crack. O RIGIN OF CHARACTERISTIC LENGTHS AND CRACK VARIETY PATHS tatistical theory of typical mesoscopic defects (microcracks, microshears) allowed us to establish specific type of critical phenomena in solid with defects – the structural-scaling transitions and propose the phenomenology of damage-failure transition [19]. One of the key results of the statistical approach and statistically based phenomenology are the establishment of two “order parameters” responsible for the structure evolution – the defect density tensor ik p ( defect induced deformation ) and the structural scaling parameter   3 0 R r   , which represents the ratio of the spacing between defects R and mean size of structural heterogeneity 0 r , and characterizes the current susceptibility of material to the defects growth [20,21]. Statistically predicted non-equilibrium free energy F represents generalization of the Ginzburg-Landau expansion in terms of mentioned order parameters, defect induced deformation   yy p x p  in uni-axial loading in y -direction and structural scaling parameter  :       2 2 4 6 * 1 1 1 , , 2 4 6 c p F A p Bp C p D p x              , (6) where yy    is the stress,  is the non-locality parameter, , , , A B C D are the material parameters, *  and c  are characteristic values of structural-scaling parameter (bifurcation points) that define the areas of typical nonlinear material responses on the defect growth (quasi-brittle, ductile and fine-grain states) in corresponding ranges of  : * * 1, , 1.3 c c              . Free energy form (6) represents multi-wall potential with qualitative different metastability in the ranges * c      and 1 c     [19]. Free energy release kinetics allows the presentation of damage evolution equation in the form       3 5 * , , p c p p p A p Bp C p D x x                       , (7) S