PSI - Issue 68

Dragan Pustaić et al. / Procedia Structural Integrity 68 (2025) 16 – 23 Dragan Pustaić, Martina Lovrenić-Jugović / Structural Integrity Procedia 00 (2025) 000–000

22

7

The roots of that cubic equation are: R 1 = - 1.93909, R 2 = - 0.851309 and R 3 = 0.790397. Three real roots are always obtained. Two of them are always negative and only one is positive. The negative solutions are not physically acceptable and the true solution is only the positive one (the plastic zone magnitude, r p , is positive quantity). Now, one point on a curve for n = 3 was obtained. Afterwards, the loading, t , is increased / decreased and the second cubic equation is obtained. After solving it and obtaining the second point, the computing procedure is so continued. The loading quantity, F , respectively, t , and the computed values, R , and r p , are presented in the Table 1 , for n = 3.

Table 1. Numerical values of the loading and the plastic zone magnitude, necessary for construction of the curve, diagrammatically presented on Fig. 5, for n = 3

F, kN/mm

0

1

2

3

4

5

6

7

8

t, -

0 0 0 0

0.32258 0.10406 0.044982

0.64516 0.41623 0.154552

0.96774 0.93652 0.295772

1.29032 1.66493 0.45279 4.52790

1.61290 2.60146 0.618769

1.93548 3.74610 0.790397

2.25806 5.09886 0.965862

2.58064 6.65973 1.14407 11.4407

t 2 , -

R = r p / a, -

r p , mm

0.44982

1.54552

2.95772

6.18769

7.90397

9.65862

The note of the cubic equation in the program package Wolfram Mathematica 7.0 , is quoted for n = 3 and F = 6 kN/mm, along with the result:

[ ] [ ] !" # $ %%%%%%%%&'()* + I -. + - /0112314. #0I/2314 /5 % 5 ! ! ! ! = + ! ! == [ ] { } { } { } { } !"# $ % &&&&& $'()(*( + & *',-$)*( + & *'.(*)(. ' ! ! ! = !" !" !

(17)

(18)

Fig. 5. Dependence of plastic zone magnitude, r p / a , around the crack tip on a monotonously increasing external loading of the crack, F /( σ 0 ∙ a ), for the different values of strain hardening exponent, n .

In the same way, the remaining values of the plastic zone magnitude, r p , are computed, point by point, for the other values of strain hardening exponent, n . It is assumed, in this paper, that strain hardening exponent, n , takes the values, n = 2, 3, 5, 10 and 1000. The obtained results are presented in the form of a diagram on Fig. 5. The retrospect on the results and comments will be summarized in the conclusion of the paper.