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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6. Conclusion The goal of this article was to develop the appropriate algorithm for computing the plastic zone magnitude, r p , around the crack tip, depending on the crack loading, F , but at the assumption of a small plastic zone around the crack tip, SSY, ( r p /2 b ≈ 0). In that way, the approximate analytical solution for, r p , is obtained. The aim was to investigate how big mistake is taken in the calculation by introducing an assumption about the small plastic zone. The question is, are the obtained results accurate and how much, in comparison to those obtained by the exact analytical procedure described in the paper published at the international conference MSMF10, Brno (Pustaić et al., 2022), respectively, in the paper published in the international journal Structural Integrity Procedia (Pustaić et al., 2023). It was expected that the appropriate analytical expressions will be simpler under the assumption about the small plastic zone, which proved to be correct. If the reliability of those approximate results is acceptable the proposed algorithm is efficient and useful for engineering application. By introducing an assumption about small plastic zone around the crack tip, r p , in this paper, the cubic equation (12) is obtained, describing the dependence of the plastic zone magnitude, r p , on the external crack loading, F . The equation (12), respectively, (13) or (14) is fully exact under introduced assumption. From that equation, the non dimensional value of the plastic zone magnitude, R , is determined numerically in dependence on assigned non dimensional crack loading, t . The whole procedure of computing is described in the numerical example at the chapter 5. The computation was performed by using the mathematical software Wolfram Mathematica 7.0 , http://www.wolfram.com/products/mathematica/. If the solution (5) is compared with the exact analytical solution found in the paper Pustaić et al., (2022, MSMF 10), it can be clearly seen that the Hypergeometric function , 2 F 1 , is no longer present, or in other words, the function, 2 F 1 , is now equal to one, i.e., 2 F 1 = 1, independently on the magnitude of a variable, P , and the strain hardening exponent , n . The assumption of the small plastic zone around the crack tip, r p , is equivalent to the assumption that the Hypergeometric function is equal to one, 2 F 1 = 1, in the interval 0.00 ≤ P ≤ 0.30 and 2 ≤ n ≤ 1000, Fig. 3. The function, 2 F 1 , is varied in that interval in very narrow limits, 1.00 ≤ 2 F 1 ≤ 1.077, Fig. 3. The biggest possible mistake in the calculation will be smaller than 8%. The maximum approximate non-dimensional loading, F / ( σ 0 ∙ a ), (for: n = 2), will be smaller for approximately 7% in comparison to the appropriate exact loading, as seen from Fig. 2. By comparison of normalized plastic zone magnitude, r p / a , (for: n = 1000 and F / ( σ 0 ∙ a ) = 2.58) it can be seen that the approximate solution, (1.4), will be approximately 4.5% greater from an exact , (1.34). References Chen, X. G., Wu, X. R., Yan, M. G., 1992. Dugdale Model for Strain-Hardening Materials. Engineering Fracture Mechanics 41 (6), 843-871. Guo, W., 1993. Elastic-Plastic Three Dimensional Crack Border Field - I. Singular Structure of the Field, Engin. Fract. Mechanics 46 (1), 93-104. Guo, W., 1995. Elastic-Plastic Three Dimensional Crack Border Field - III. Fracture Parameters. Engineering Fracture Mechanics 51 (1), 51-71. Hoffman, M., Seeger, T., 1985. Dugdale Solutions for Strain-Hardening Materials. The Crack Tip Opening Displacement in Elastic-Plastic Fracture Mechanics. In: Proceedings of the Workshop on the CTOD Methodology, Geesthacht, 57-77. Neimitz, A., 2000. Dugdale Model Modification due to the Geometry Induced Plastic Constraints. Engineering Fracture Mechanics 67, 251-261. Neimitz, A., 2004. Modification of Dugdale Model to Include the Work Hardening and in- and out-of-Plane Constraints. Engineering Fracture Mechanics 71 (11), 1585-1600. Neimitz, A., Graba, M., 2008. Analytical-Numerical Hybrid Method to Determine the Stress Field in Front of the Crack in 3D Elastic-Plastic Structural Element. In: Proceedings of the 17th European Conference of Fracture (ECF 17), Brno, Czech Republic, 514-521. Pustaić, D., Lovrenić-Jugović, M., 2023. Cohesive Model Application in the Assessment of Plastic Zone Magnitude for one Particular Case of Crack Loading. Structural Integrity Procedia 43, 252-257. Pustaić, D., Lovrenić-Jugović, M., 2019. Mathematical Approach of Crack Tip Plasticity, In: Abstract Booklet of the 9th International Conference on Materials Structure & Micromechanics of Fracture, p. 162 and poster presentation, Šandera, P., ed., Brno, Czech Republic. Pustaić, D., Lovrenić-Jugović, M., 2019. More accurate Mathematical Description in the Assessment of Plastic Zone Magnitude around the Crack Tip. Structural Integrity Procedia 23, 27-32. Pustaić, D., Lovrenić-Jugović, M., 2018. Mathematical Modeling of Cohesive Zone in the Ductile Metallic Materials. In: Proceedings of the 9th International Congress of Croatian Society of Mechanics. Marović, P. et al. (eds.). Croatian Society of Mechanics. USB, Split, Croatia. Pustaić, D., Lovrenić, M., 2006. Analytical and Numerical Investigation of Crack Opening in Strain-Hardening Material. In: Proc. of the 5th International Congres of Croatian Society of Mechanics. Matejiček, F. et al. (eds.). Croatian Society of Mechanics. CD-ROM. Trogir, Croatia. Pustaić, D., 1990. Contribution to the Stress Analysis in the thin Plates in a non-linear Range. PhD Thesis, University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture. Zagreb, Croatia. Wolfram Mathematica, Version 7.0, 2017. Wolfram Research Inc., Champaign II, http://www.wolfram.com/products/mathematica/.