PSI - Issue 36

Oleksandr Andreykiv et al. / Procedia Structural Integrity 36 (2022) 36–42 Oleksandr Andreykiv, Andri і Babii, Iryna Dolinska et al. / Structural Integrity Procedia 00 (2021) 000 – 000

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4

4. Determination of subcritical fatigue crack growth period Further, it is assumed that a rectilinear surface crack of depth 0 0.001m l = has been initiated in the tube (Fig. 2b), and obtains a more complex form during the propagation (Fig. 2c). Then, the lifetime of a beam element of rectangular profile can be determined by the following formula:

(1) s s N N N N  = + + , (2) i

(4)

where (1) s N is the propagation period of the rectilinear surface crack of initial depth 0 0.001m l = to the complex configuration that entirely occupies one tube wall and partially the two others (Fig. 2c); (2) s N is the period necessary for this complex crack to reach the critical size causing failure of the considered element. In order to determine the lifetime of the boom element by the formula (4), a computational model of crack propagation from its initial to the final size is built. For this reason, the proposed earlier energy approach (Andreikiv et al. (2011), Andreikiv et al. (2017a), Malezhyk et al. (2019)) is applied. The boom is assumed to be loaded by the moments , i M when the crack reaches the depth i l . Thereby, the crack growth rate equation V dl dN = is written in the form:

(1)

(2)

  

   

dW

W



dl

p

p

.

(5)

  − − 

=

f

t

dN

l



dN

Here (1) ( ) p W N is the work of plastic deformations during the tube unloading from the action of the moment M and compression of the fracture process zone; it depends only on the number of loading cycles and is generated by the body itself; ( ) 2 ( ) p W l is the work of plastic deformations in the fracture process zone near the crack tip during the tube unloading from the moments i M , that depends only on the crack length l ; 0 t t f    = is the specific work of plastic deformations in the fracture process zone near the crack tip; 0 f C f    = is its critical value; t  is the crack tip opening displacement (CTOD); c  is the critical CTOD; 0 f  is the average loading in the fracture process zone. The functions (1) ( ) p W N and ( ) 2 ( ) p W l in the equation (5) are determined similarly to the studies by Andreikiv et al. (2011), Andreikiv et al. (2017a) as follows: ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 0 max 2 2 2 2 0 0 1 0 (1 ) ( ), 0, 25(1 ) , p t t th l n p f i Mt th i W N N R W l R x l x dx            = = − −   = − − −     (6) where 0  is the fatigue material property determined experimentally; th  is the threshold CTOD t  that does not lead to crack propagation; min max ; t t R    = ( ) x  is the delta-function; Mt  is the crack tip opening displacement under the action of the moments i M ; i l is the length of the fatigue crack, when the boom is loaded by the moment i M . After substitution of (6) into (5) with respect to the findings by Andreikiv et al. (2016), Andreikiv et al. (2018b), an equation is obtained for determination of the subcritical crack growth period s N N = in the cracked tube wall loaded by the moments M and i M : ( ) ( ) ( ) ( ) ( ) 2 2 2 0 2 2 2 0 1 1 0, 25 1 t th n c t i Mt th i R l dl dN R l l l            =   −  −  =   − − − −  −   . (7)

For completeness of the model, the following initial and final conditions are added to the equation (7):