PSI - Issue 23

Dragan Pustaić et al. / Procedia Structural Integrity 23 (2019) 27 – 32 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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specified values of the parameters a, p r and n. If the mistake in calculating from maximum 7% is accepted, it could be taken with the enough accuracy that it is 2 1 1, F  Pustaić and Lovrenić - Jugović (2018). In that case, the analytical expression (8) is much simplified, but it remains enough accurate and reliable for determining the plastic zone magnitude. So, the expression (8) now becomes         2 π 1 1 2 1 . t P n n n n                 (9) There are, really, only little differences in comparison with the expression (8) and the results obtained from the expression (9) can be considered enough accurate and reliable. This solution is the same as that obtained under the assumption of small plastic zone around crack tip, (12), only, what it is written in the inverse form. 4. Assumption about small plastic zone around crack tip If there is a limitation on forming the small plastic zone around crack tip ( small scale yielding ), then the ratio p 2 , r b in the equation (5), can be taken nearly equal zero, i.e. p 2 0. r b  Expression (5), in that case, takes the form This integral can be solved fully exact, analytically. The solution can be found in the mathematical handbook and can be expressed through the special Gamma functions. The stress intensity coefficient is obtained for the case of small plastic zone           coh p 0 2 1 1 2 1 K b r n n n n                   (11) If the solution (11) is compared with the one obtained by program package Wolfram Mathematica 7.0, (2017), (6), it can be seen that the solution (11) will be identically equal to the solution (6), if in the solution (6) the value of hypergeometric function 2 1 F is taken to be equal to 1.   K b r  coh   1    0   1 1 n 1  0 1 2     d . p 2 π      (10)

Fig. 4. Dependence of plastic zone magnitude around crack tip p r a on monotonous increasing external loading of a plate 0 ,    for different values of strain hardening exponent n.