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

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4

The assumption about a small plastic zone around the crack tip, SSY, is introduced, where r p / 2 b ≈ 0 can be taken. The solution of the integral in the expression (4) amounts to

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The solution (5) is given as the ratio of the special functions – the Gamma functions , found by means of the program package Wolfram Mathematica , http://www.wolfram.com/products/mathematica/. If this solution is compared with the exact analytical solution for that problem, Pustaić et al. (2022, MSMF 10), it can be seen that the difference between those solutions is only in a fact that the Hypergeometric function , 2 F 1 , does not appear in the expression (5). In other words, 2 F 1 is now equal to one, 2 F 1 = 1, independently on the magnitude of a variable, P , and strain hardening exponent , n. If the variable, P , is varied in the limits from 0 to 0.30 and the strain hardening exponent, n , changes between 2 ≤ n ≤ 1000, the Hypergeometric function, 2 F 1 , will be changed in a very narrow interval from 1.00 to 1.08, as seen on Fig. 3. Consequently, if the maximum possible mistake in the calculation of 8% is accepted, the approximate solution (5) can be considered enough accurate and reliable. 3. Approximate analytical solution for the plastic zone magnitude around the crack tip When the right sides of the expressions (2) and (5) are equalized, congruently to the stress non-singularity condition in the tip of the fictitious elastic crack, K ext = │ K coh │, the dependence of the external loading of the crack, F , and the plastic zone magnitude, r p , is obtained, included in the variable, P . After partial arrangement that expression equals

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The variable, c , as already mentioned, can obtain any arbitrary value in the interval 0 ≤ c ≤ a . If one discrete value from that interval is taken, c = (1/2) a , and the expression (6) is rearranged in a way that the non-dimensional loading , F /( σ 0 ∙ a ), is included, it can be written

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From the structure of the expression (7) can be seen that the non-dimensional loading depends on the variable, P , and the strain hardening exponent, n . The plastic zone magnitude, r p , is included in the variable, P . In order to construct the diagram F /( σ 0 ∙ a ) = f ( P , n ), one numerical example is considered, in which value of a = 10 mm is taken and the plastic zone magnitude, r p , is changed within the limits from 0 to15 mm. The variable, P , is changed, in that case, within the interval from 0 to 0.3. The strain hardening exponent, n , varies within the interval from 2 (material with high hardening) to 1000 (almost elastic-perfectly plastic material). For the so defined parameters, using the program package Wolfram Mathematica , the diagram on Fig. 2 was computed and constructed. The concentrated forces, F , [N/m], act on the crack surface and open it. The distance of those forces in relation to the axis, y , [m], which is, at the same time, the axis of symmetry, c , [m], according to Fig. 1a, Pustaić et al. (2023, Structural Integrity Procedia). The distance of the force, F , can be varied within the interval 0 ≤ c ≤ a . By varying the distance, c , within that interval, it has been investigated how the position of the forces on the crack surface influences on the plastic zone magnitude around the crack tip and on crack opening, as well. It is started from the expression (6), published in the paper Pustaić et al. (2023, Structural Integrity Procedia). The assumption about small plastic zone around the crack tip (SSY condition) is introduced and can be taken that the Hypergeometric function , 2 F 1 , is nearly equal one, 2 F 1 ≈ 1. After that assumption, the expression (6) is strongly simplified. Also, it was taken that the strain hardening exponent , n , is equal 2, ( n = 2). The expression (6), in non-dimensional form , now becomes