PSI - Issue 74

Dragan Pustaić et al. / Procedia Structural Integrity 74 (2025) 70 – 76 Dragan Pustaić / Stru ctural Integrity Procedia 00 (20 2 5) 000 – 000

72

3

Fig. 1. ( a) The uniformly distributed, continuous loading, p 0 , is acting over the crack surface, which opens the crack. The small plastic zones of the length, r p , are formed around the crack tips; (b) the non-linearly distributed, cohesive stresses are acting on a part, r p , of a fictitious elastic crack.

2. Stress intensity factors from the external loading and from the cohesive stresses The stress intensity factor, (SIF), from the external loading of a crack by the concentrate forces, F , [N/m], on a distance, c , [m], within the tip of a fictitious elastic crack, was derived in the PhD thesis, Pustaić (1990 ). Now, according to the Fig. 1.a, F = p 0 ∙ d x and the distance, c = x , it can be written

0 ∫ a

0 ∫ a

0 ∫ a

p

p

d

b x b x + −

b x b x − +

π b

x

( ) ( ) K b K b K b = + − = ( )

d

d 2 =

.

(2)

x

x p

0

0

⋅

+

⋅

⋅

ext

I

I

0

π

π

b

b

⋅

⋅

2 b x −

2

By integrating the expression (2), it is obtained

π b

a b

( ) ext K b p

0 2 = ⋅

arc sin .

(3)

⋅

In a same way, it is possible to determine the stress intensity factor from the cohesive stresses, K coh ( b ), [MPa√m]. Now, according to the Fig. 1.b, it is necessary to include, F = - p (x)∙d x , and we obtain

b ∫ a

d

π b

x

( )

( )

2 =− ⋅

.

(4)

coh K b

p x

⋅

2 b x −

2

The right side of the expression (1) it is necessary to include before integrating. According to its structure, the expression (4) is analogous to the expression (2) and it was derived on the same way. The solution of the integral can be found by means of the mathematical software Wolfram Mathematica, and equals

{

}

{

}

( )

( n n

)

( ) 1 2

( n n

)

( ) 12; 12; 12 

( n n + +

) 1 ;

2 . 

(5)

- = ⋅ σ

2

1

1

coh K b

r

2 1 F

r b

⋅ Γ + Γ + + ⋅    



 







0

p

p

The solution for the K coh ( b ) is given through the ratio of the special functions – the gamma functions and the hypergeometric functions, 2 F 1 [ α ; β ; γ ; P ]. Within the expression (3), for the K ext ( b ), the inverse trigonometric function,