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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In the expression (6),   x  denotes the Gamma function, while 



2 1 ; ; ; z F    denotes the Hypergeometric function.

3. Exact analytical solution for the magnitude of cohesive zone around crack tip

,   is equal

The stress intensity coefficient at the tip of fictitious elastic crack, from the external loading

yy   





ext K a r b       By inserting the right sides of this expression and that in (6) into the condition (1), the next equation is obtained, derived in the paper Pustaić and Lovrenić - Jugović (2018) , which gives the dependence among the plastic zone magnitude around crack tip p r and the external loading   , for the different values of strain hardening exponent n                     2 2 0 p p 2 2 1 p p π 2 1 1 2 1 1 2; 1 2; 1 2 1 ; 2 . a r r n n n n F n n r a r                                 (7) This equation (7) is possible to write in the inverse form, i.e. in the form in which the plate loading   will be presented in depending of plastic zone magnitude p r . The new mark, 0 t     is introduced and a new independent variable, as well, as:   p p 2 , P r a r    as it was derived in the paper Pustaić and Lovrenić - Jugović (2018) . From the equation (7), it will be obtained               2 1 2 π 1 1 2 1 1 2; 1 2; 1 2 1 ; . t P n n n n F n n P                        (8) p π .

Fig. 3. Dependence of Hypergeometric function  2 1 ; ; ; F z    on the new variable P, for the different values of strain hardening exponent n. 

Fig. 2. Dependence of plate loading 0 t     on the new variable P, in which is the plastic zone magnitude r p incorporated and for the different values of strain hardening exponent n.

If it is taken that the quantity a = 10 mm and if it is assumed that the magnitude p r is changed from 0 to 15 mm, in that case, the variable P is changed into the limits from 0.00 to 0.30. For, so defined the variables, and on the basis of analytical expression (8), the curves, ( , ), t t P n  are calculated and constructed using the program package Wolfram Mathematica 7.0, (2017), and they are shown at the Fig. 2. Also, the values of Hypergeometric function   2 1 ; ; ; F z    are calculated and plotted, using the software package Wolfram Mathematica 7.0, (2017). As it can be seen from the Fig. 3, the Hypergeometric function 2 1 , F is changed in a very narrow interval from 1.00 to 1.07, for the