PSI - Issue 28

V. Yu. Filin et al. / Procedia Structural Integrity 28 (2020) 3–10 Filin V.Yu., Ilyin A.V.,Mizetsky A.V. / Procedia Structural Integrity 00 (2020) 000 – 000

7

5

   

   

σ

i m

,

(8)

− 2 exp 3 ε ε α 0 = + cr

σ

where the strain intensity  0 is necessary to created initial pores,  is a constant of material, this constant is proportional to the logarithm of the ratio of critical pore size to its initial value, ( ) 1 2 3 σ σ σ is a hydrostatic stress. Evaluation of the parameters  0 ,  may be based on the following considerations. Hankock and Mackenzie (1976) reported  0 = 0.2 for pure Swedish iron. Modern steel grades have a lot of inclusions increasing their strength so  0 should be less. 0.06 to 0.07 for low-alloyed shipbuilding steel is recommended by Handbook, Gorynin ed. (2009). For the case of uniaxial tension of unnotched specimens, 1/ 3 σ / σ = i m . So for this case α 0.6065 ε ε 0 = + cr . (9) Using a wide known formula where  is a relative reduction of area in the neck that for modern steel is about 0.7, we come to the value  cr = 1.20. Supposing  0 = 0.06, we get  = 1.9. For the case of biaxial bulge of plates, in the middle of the bulge / 2 / 3   = i m . So for this case α 0.3679 ε ε 0 = + cr . (11) At the same time, in this test 2 ε ε radial cr cr = , where the critical radial strain to fracture  cr radial in our experiments reached up to 0.40. This way we come to the values  up to 1.37. The results of Hankock and Mackenzie (1976) led to the values  = 1.1...1.6 that appeared different for different directions in steel of heterogeneous structure. The coefficient  had its minimum for specimens cut in the through thickness direction. Trials of  assessment via the area of experimental load-displacement curves in TKDS test are hindered by uncertain displacement values not only at the expense of the load train compliance. A difference in initial slope of the records was observed within the same series of specimens due to icing. The above considerations did not allow to use for FEM simulation a certain  value in equation (8) as a criterion of ductile fracture. This way, the effect of different  was investigated, the values 1.3, 1.5 and 2.0 were taken for simulation. Calibration of the problem in respect of SIF (fracture toughness) was performed by the solution of a plane strain problem for a wide plate (small scale yield condition is observed) with the same FEMmesh size. Loading of the model was performed until the first attainment of the cleavage fracture criterion (stress equal to  i ). K I for this plate is known from reference literature and is taken for K I a . The found values are as follows:  = 2.0, K I a = 11.5 MPa  m 0.5 ;  = 2.5, K I a = 26.0 MPa  m 0.5 ;  = 2.6, K I a = 29.5 MPa  m 0.5 ;  = 2.7, K I a = 34.1 MPa  m 0.5 ;  = 2.8, K I a = 41.5 MPa  m 0.5 ;  = 3.0, K I a = 71.6 MPa  m 0.5 ;  = 3.3, K I a = 118.4 MPa  m 0.5 . Following the classic approach,  > 3 may be hardly obtained in a real material, however the used simplified model inadequately simulates the crack blunting, so its usefulness had to be checked by a solution of trial problems. It has been shown that for the selected mesh size physically adequate results are obtained up to  = 3.7, so the simulation results up to  = 3.3 may be taken into further consideration. 3. Results of numerical simulation A table below presents the calculation results including the appearance of simulated fracture surfaces (black is cleavage, white is ductile, notch area is omitted). Figure 3 presents real fracture surfaces for comparison. Figure 4 shows an example of  m /  i ratio distribution in fracture during crack propagation. Simulated load-displacement curves are given in Figure 5. 3 1 σ = + + m ψ 1 1 − ε = cr , (10)