PSI - Issue 74

Dragan Pustaić et al. / Procedia Structural Integrity 74 (2025) 70 – 76 Dragan Pustaić / St ructural Integrity Procedia 00 (20 2 5) 000 – 000

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arc sin ( a / b ), appears. The expressions (3) and (5) we will rearrange by introducing new independent variable, P = r p / [2 b ] = r p / [2( a + r p )]. Formally, it can be written, P = P ( r p , a ). By inversion of that expression, r p = r p ( P , a ) is obtained, what finally assumes the form r p = (2 Pa ) / (1 – 2 P ). When the right sides of the expressions (3) and (5) are put in the non-singularity condition of the stresses, in the tip of fictitious elastic crack, the dependence between the non-dimensional, external loading of a crack, p 0 / σ 0 , and the new independent variable, P , is obtained, expression (6). After separation the variables in that equation, the solution was found by means of software Wolfram Mathematica. It is necessary to point out that, in these step, only the dependence between the non-dimensional loading and the new independent variable, P , is possible to obtain, just it was plotted in Fig. 2. From the expression (6), it is impossible get the inverse solution in which the plastic zone magnitude, r p , will be expressed as a function of non-dimensional crack loading. It was a reason, while we established the new algorithm for an explicit determining the dependence of the plastic zone magnitude on the external loading, like in the section 4.

3. Exact analytical solution for the plastic zone magnitude around the crack tip The final expression, in an arranged form, looks like ( ) ( ) ( ) ( ) ( ) 0 2 1 1 π 12; 12; 12 1 ; 1 2 1 n n p t P F n n P n n σ   Γ +      = = ⋅ ⋅ ⋅ + +       Γ + + 

(

)

arc sin 1 2 . P −

(6)

 

 





0

From the structure of the expression (6), it can be seen that the non-dimensional loading of the crack, t = p 0 / σ 0 , depends on the variable, P , and the strain-hardening exponent, n . The plastic zone magnitude, r p , is included in the variable, P . One numerical example will be examined in order that to construct the diagram of interdependency, p 0 / σ 0 = f ( P , n ). In that example, we will assume that the half of the crack length amounts, a = 10 mm, and the plastic zone magnitude, r p , will be changed in the limits from 0 to 15 mm. Then the variable, P , will be changed within the interval from 0 to 0.3. The strain-hardening exponent, n , will vary within the interval from 2, (the material with big hardening) to 1000 (almost elastic-perfectly plastic material). With so defined parameters, by using the program package Wolfram Mathematica , the diagrams on Figs. 2 and 3 were computed and constructed. The results will be commented in the Conclusion of the paper.

Fig. 2. Dependence of non-dimensional loading of the crack, p 0 / σ 0 , on the variable, P , in which the plastic zone magnitude, r p , is included, for the different values of strain-hardening exponent, n .