PSI - Issue 42

Abdulla Abakarov et al. / Procedia Structural Integrity 42 (2022) 1046–1053

1048

A. Abakarov and Y. Pronina / Structural Integrity Procedia 00 (2019) 000–000

3

The ratio of the initial lengths of the big and small cracks varies from 1 to 20, the distances between them (related to the initial length of the small crack) varies from 0.25 to 5. The initial small crack length is taken constant: l ( t 0 ) = l 0 = 50 µ m. The load, σ ∞ , is set equal to 210 MPa. According to most experimental results, it is assumed that the crack growth rate rises abruptly as the mode I SIF reaches a certain threshold value K SCC I and thereafter increases gradually as stress intensity factor increases. For example, we consider the following model of SCC crack growth rate for type 304 stainless steel in water at 288 °C , discussed by Saito and Kuniya (2001) and Fujii et al. (2015):

3 · 10 − 19 − 1 . 5 · 10 − 20 3 K 7 . 74 · 10 − 21

  

   0 . 443

1 . 1 · 10 − 7  

   −

SCC I

I − K

dl dt =

 2 . 5 · 10 10 · exp

(1)

= 2 MPa √ m. It is required to assess the shielding e ff ect on the growth of cracks of various lengths

where K SCC I

clustered in the considered stack arrays.

3. Methods of solution

To find the length of cracks at any time, the forward Euler method is used. The time step size is chosen so that the relative di ff erence in the results with a considered and a smaller step size was small enough. Static problem of multiple cracks at each time step is solved using the approximate method of Kachanov (1993). This method becomes less accurate at relatively small spacings between the cracks; therefore, the results for relatively large ratios of the big crack length to the small one are not quite accurate at small spacings between the cracks. However, these results agree well with the rest of data and give an idea of the qualitative behavior of the crack system at various geometric parameters. The method of Kachanov is applicable for the problem of finite number of cracks. To approximately estimate the behavior of infinite periodic systems of cracks, we developed a “procedure of periodization”. According to Abakarov and Pronina (2022), the relative di ff erence between the results with 16 and 20 periodic cells in a doubly-periodic crack array becomes less then 1% (however, it doesn’t mean that the global error — compared to infinite stack — is the same). Therefore, we choose the number of periodic cells (containing one big and one small crack) not less than 20. It is clear that the stress state around the central cracks di ff ers from the side ones, thus causing the di ff erence in the cracks rate growth (since it proportional to the mode I SIF). In order to eliminate this secondary e ff ect of the di ff erence in the lengths of the cracks in the central and other cells over time ( t > 0), we set the growth rate of all the cracks of the same initial size to be the same as that calculated for the central cracks. This procedure significantly reduces computational cost.

4. Results and discussion

Since we are interested in the interaction of stacked cracks of di ff erent lengths when the smaller ones lie in the relaxation zone of the bigger ones, every curve is divided by a little circle into two parts: one of them corresponds to the situations when the small cracks are in the relaxation zone around the big ones and vice versa. All the plots for more than 3 cracks are for the cracks in the central cell of a considered array.

4.1. Comparative analysis for di ff erent cracks configurations

Figure 4 shows the relative mode I SIFs at the tips of the big cracks (closest to the center) in two configurations with three cracks, one with five cracks and in the approximately infinite periodic array (“BS44p” — calculated as a stack consisting of 22 periodic cells using the “procedure of periodization”) after 150 hours of the cracks growth. All the values are normalized to the length ( l 0 ) or mode I SIF ( k 0 I ) for a single small crack at initial time ( t 0 = 0) under the same load. Calculations reveal that the length of the big cracks grows almost proportional to their initial length and slightly dependent on the distances between the cracks, regardless of whether the neighboring small cracks fall into the relaxation zone or not. The more the number of big cracks in the array, the less their final length and the