PSI - Issue 43

Dragan Pustaić et al. / Procedia Structural Integrity 43 (2023) 252 – 257 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

255

4

( ) p , , P P r a = it is obtained,

(

)

(

)

coh p , K a r + [MPa m ]. Inversion of this expression,

p p , , r r P a =

well as one for,

and final dependence assumes the form − The stress intensity factor from the cohesive stresses, ) ( 2 1 2 . Pa P ) p r = (

(

)

coh p , K a r + is determined by means of the Green´s

functions method, as was indicated in the paper Chen et al. (1992). In that paper the author was shown how to use the mentioned method for solving a crack in a plate with finite dimensions. The Green functions, in that case are much more complicated in comparison with those for the cracks in an infinite body. In order to determine the factor, ( ) coh p , K a r + it is necessary to know the distribution of the cohesive stresses, p(x) , within the plastic zone, (3).

b  a

b  a

(

)

(

)

) ( ) 1 1 n + 

1 2

−

( ) ( ) p x m x b x  , d

(

2 b b x  − π

2

 

2  

d

coh K a r

r x a −

x

 = 

+ =



=



p

0 p

(5)



) 1 .   

( n n

)

( ) 1 2

( n n

( ) 1 2; 1 2; 1 2 

( ) 1 ; n n r b

2 .  

2 =  r

1

2 1 F



  +  + +    

+ +



 



p 0

p

The solution (5) was obtained by means of the program package Wolfram Mathematica, http://www.wolfram.com/products/mathematica/. In the expression (5), ( ) x  stands for Gamma function , while ( ) 2 1 ; ; ; z F    denotes the Hypergeometric function. 3. Exact analytical solution for the plastic zone magnitude around the crack tip When the right sides of the expressions (4) and (5) are included in condition (1), the partly arranged expression is obtained which gives the dependence between the external loads of the plate, F , and the plastic zone magnitude, p , r which is comprised in the variable, P

2 1 2 a −









( n n

)

( ) 1 2

( n n

)

( ) 1 2; 1 2; 1 2 

( ) 1 ; n n P

2

π

1

1

.

F

P  

c

2 1 F

(6)

 = 

−   +  + +     

+ +

 



 





0

(

)

2

P

Similarly, it is possible to include, under a square root, the value c = (1/2) a and arrange. As the plate loading, F , is related to the unit of plate thickness, (N/m), in the expression (6) it is necessary to notice, the so-called, non dimensional loading , ( ) 0 , F a   [-]. Finally, the analytical expression (6) assumes the form ( ) ( ) ( ) ( ) ( )   ( ) ( )   2 2 1 2 0 3 4 4 π 1 1 2 1 1 2; 1 2; 1 2 1 ; . 4 16 16 P P F P n n n n F n n P a P P  + − =     +  + +  + +              − + (7) The expression (7) is explicit analytical note, in an inverse form, of the dependence of the non-dimensional loading of the crack and plastic zone magnitude, p , r comprised in the variable, P . If it is assumed that the magnitude, p , r is changed in the interval from 0 to 15 mm, the variable, P , in that case, takes the values from 0.00 to 0.30. For so defined variables, and according to the analytical expression (7), the curves, ( ) ( ) 0 , , F a f P n   = were calculated and constructed, using the program package Wolfram Mathematica. The solutions are presented in Fig. 2. The values of Hypergeometric function , ( ) 2 1 ; ; ; z F    were, also, calculated in this article, using the software Wolfram Mathematica , equally as in the paper Pustaić and Lovrenić - Jugović (2019, MSM F9). As it can be seen from Fig. 3, the Hypergeometric function 2 1 , F in the given interval, is changed between very narrow boundaries from 1.00 to 1.08. If the maximum mistake in the calculation of 8% is accepted, it could be taken that the approximate value of the function, 2 1 , F will be 2 1 1, F  independently of the magnitude of the variable, P , and the strain hardening exponent, n . In that case, the analytical expression (7) becomes much simpler, but it remains enough accurate