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

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Fig. 3. Dependence of the Hypergeometric function, 2 F 1 ( α; β; γ; P ), on the variable, P , for the different values of the strain-hardening exponent, n .

Let us examine how the statically equivalent loads, the concentrated force, F = p 0 ∙ a , which acts on a distance, c = (1/2) a , and uniformly distributed, continuous load, p 0 , acting over the crack surface influence on the plastic zone magnitude around the crack tip, r p . If the expression (7) from the article Pustaić et al. (2022, MSMF 10) and the right side of the expression (6), are compared, it can be noticed that: the right sides of that expressions are different, while the left sides are equal. On the right side in the expression (7), the factor (√3+4 P -4 P ²) / (√4 -16 P +16 P ²) = A, appears, while the factor 1 / arc sin (1-2 P ) = B, appears on the right side in the expression (6). These factors were computed for the values of a variable P , 0.00 ≤ P ≤ 0.30, and their numerical values are present ed in the Table 1. Also, their mutual difference is given in the percentage, ∆ = [(A – B) / A] ∙100%, and graphically presented on the Fig. 4.

Table 1. Numerical values of the factors A and B depending on the value of a variable, P . P , - 0.00 0.05 0.10 0.15 0.20 0.25 0.30 A, - 0.86603 0.99225 1.14564 1.33821 1.58990 1.93649 2.44949 B, - 0.63662 0.89304 1.07841 1.28966 1.55400 1.90985 2.43002 ∆,% 26.49 9.9985 5.87 3.63 2.26 1.38 0.79

Fig. 4. Graphical presentation of the difference of the A and B factors, depending on the variable, P . From Table 1 and diagram at Fig. 4 can be concluded that for the equal plastic zone magnitude, r p , different loads will be necessary, according to the expressions (7), Pustaić et al. (2022), and the (6). The loading, F / ( σ 0 ∙ a ) always must