PSI - Issue 43

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

256

5

(

)

2 3 4 4 4 16 16 P P P P + − − +

F





( n n   +  + +         ) ( ) 1 2 ( n n ) 1 . 1

(8)

π =   P

(

)

a



0 

2

(

)

Fig. 2. Dependence of the non-dimensional plate loading , ( ) 0 , F a   on the variable, P, i.e. on the plastic zone magnitude around the crack tip, r p , for the different values of strain hardening exponent, n.

2 1 ; ; ; F z    ,

Fig. 3. Dependence of Hypergeometric function,

on the variable, P, for the different values of strain hardening exponent, n.

4. Algorithm for explicit determining of dependence among the plastic zone magnitude and the external loading of the crack It will be shown how, by means of the diagram from the Fig. 2, ( ) ( ) 0 , , F a f P n   = is determined the plastic zone magnitude, p , r explicitly, dependently on the crack loading, F. The algorithm of one curve, for example n = 2, will be described. The diagram is being calculated and constructed with the software Wolfram Mathematica, by filling the Table 1. Firstly, the discreet values, ( ) 0 , F a   from 0.00 to 3.73, are chosen, for the first row in the Table. Afterwards, the belonging value of the force, F , is being calculated, and values are filled in the second row. By means of the function Get Coordinates in the program package Mathematica the belonging variable, P , is read off from the Fig. 2 and the obtained values, P , are written in the third row of the Table. Finally, with so read off values, P , the plastic zone magnitude, p , r is calculated and written in the fourth row of the Table. In the end, by means of the, F , and the, p , r the curve, ( ) p 1 , r f F = is drawn and the diagram as in Fig. 4 is constructed. Table 1. Numerical values of the physical quantities required for a construction of the curves shown diagrammatically in Fig. 4, for n = 2 ( ) 0 , - F a   0 0.5 1 1.5 2 2.5 3 3.5 3.73 F , kN/mm 0 1.550 3.100 4.650 6.200 7.750 9.300 10.850 11.563 P , - 0 0.0406 0.1053 0.1633 0.2071 0.2410 0.2687 0.2915 0.3001 p , r mm 0 0.88376 2.66785 4.85001 7.07067 9.30502 11.61695 13.98082 15.01251 Conclusion The dependence of the plastic zone magnitude around the crack tip, p , r on the monotonously increasing external loading of the crack, F , according to Fig. 1, was investigated. The plate was made from ductile metallic material with a property of non-linear strain hardening. One cohesive model (Dugdale´s model) with the non-linear distribution of the cohesive stresses in the plastic zone, was applied in the analysis. The analytical expression, (4), for the stress intensity factor from the external loading was derived, which is then transformed and presented by means of the newly introduced independent variable, P . The stress intensity factor from the cohesive stresses, ( ) coh p , K a r + was derived in the paper, by Pustaić and Lovrenić - Jugović , (2019, MSMF9), the expression (5). The program package Wolfram