PSI - Issue 28

Vedernikova Alena et al. / Procedia Structural Integrity 28 (2020) 1160–1166 Author name / Structural Integrity Procedia 00 (2019) 000–000

1163

4

where t K is the stress concentration factor,   is the nominal stress, n r is the notch root radius. The eqs. (4), (7) with initial condition   y t 0 p x, t 0   and boundary conditions   y x 0 p x, t 0   ,   yy p x, t  were solved numerically for different values of nominal stress (model material). 0  . One of the possible explanations of this proposal linked with existence of two singularities related to the stress field at the crack tip and blow-up kinetics of damage localization is presented in the work Naimark (2019). It was concluded from analysis of numerical results that the initiation of fracture process requests a fulfillment of synchronous conditions: the stress should be bigger than critical value 0  (inherent material strength) in some area near the stress concentrator and the length of this area should be bigger than some critical spatial scale c l     c : l l        x 0, l   ,    y 0 x    (Fig. 2b). When the stress is bigger than 0  but the spatial length of this fluctuation is too small to initiate the dissipative structure     0 cσ l , l < l    we can consider the stable situation (Fig. 2a). For this simulation were used following material parameters: s 2  , 3   , a 256  , q 16  , k 1  m 2 . x 0 x +     The extremum of the function p qp     lead to the value of the stress at which the unlimited growth of the defects begins. This stress value can be associated with the inherent material strength

(a)

(b)

Fig. 2. Spatial-time defect evolution near stress concentrator for different initial stress distribution: (a) without blow-up regime; (b) blow-up regime. Line (1) – inherent material strength, (2) - stress distribution (Eq. 7), (3) - spatial defect distribution (Eq. 4). Fig. 3 presents a relations between the analytical estimation of spatial scale of dissipative structure by Eq. 6 ( c L ), numerically obtained value of fundamental length ( TN L ) and scale c l  for different parameters  and s .

Fig. 3. Relations between analytically (L c ) and numerically (L TN ) estimated spatial scale of dissipative structure and spatial scale of initial stress fluctuation (l cσ ).