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

21

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where, the designation Q ( n ) = Γ[ n /( n +1)] / Γ[(1/2) + n /( n + 1)] was introduced. By further transformation, the expression (9) equals

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As earlier, c = a / 2 is considered. By further arranging it follows

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The non-dimensional largeness R = r p / a and t = F /( σ 0 ∙ a ), ( t – means traction ) are introduced so the expression can simpler be written down

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The cubic equation is obtained from which, numerically, the non-dimensional value of the plastic zone magnitude, R , is determined, for the given non-dimensional crack loading, t . The physical quantities: F , σ 0 , a , and n , need to be assigned. The mathematical software Wolfram Mathematica 7.0 was used for determine the cubic equation roots. 5. Numerical example – the cubic equation solution by means of the mathematical software Wolfram Mathematica 7.0 It will be shown how the plastic zone magnitude, R = r p / a , is determined by means of the derived cubic equation (14), for the given non-dimensional crack loading, t = F /( σ 0 ∙ a ), and how the diagram shown on Fig. 5 is constructed. The program package Wolfram Mathematica 7.0 was used for computing. It is necessary to emphasize that only one curve at the time is computed and constructed, for which the strain hardening exponent , n , is kept constant , while the loading, F , is monotonously increased from 0 to 8 kN/mm. The procedure will be illustrated under an assumption that n = 3. Furthermore, the values a = 10 mm, c = a /2 = 5 mm and σ 0 = 310 MPa will be taken. For so defined values of the parameters, the cubic equation (14) takes the form

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Afterwards, the physical quantity, t , is varied in the quoted limits. For one determined, t , a certain cubic equation is obtained which is necessary to solve by means of the mentioned software and one point is obtained on the curve for, n = 3. For F = 6 kN/mm, ( t = 1.93548), that equation looks like

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