PSI - Issue 74

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

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The plate is made of ductile metallic material, which has the property of non-linear strain-hardening. The strain hardening of a material can be good described by Ramberg-Osgood analytical law. A cohesive model (Dugdale´s model), with non-linear distribution of the cohesive stresses within the plastic zone, was used in the analysis, in the same way as the authors Hoffman and Seeger suggested in their paper, (1985). The expression for the stress intensity factor, SIF, from external, continuously distributed loading, was derived, the analytical expressions, (2) and (3). Equally, the stress intensity factor from the cohesive stresses was derived, the expressions (4) and (5). The program package Wolfram Mathematica was used for their determining. The solution for, K coh ( b ), is given through the ratio of the special functions – the Gamma functions and the Hypergeometric function, 2 F 1 [ α; β; γ; P ]. The exact analytical solution for the plastic zone magnitude, r p , was obtained from the non-singularity stress condition in the point, ( a + r p , 0). That solution was written down in an inverse form, (6). The non-dimensional loading, p 0 / σ 0 , was presented through the variable, P , and the strain-hardening exponent, n . By using the software package Wolfram Mathematica, the family of curves was computed and constructed, as it is shown on the Fig. 2. By applying the algorithm, described in a section 4, for n = 2, the explicit record in form r p / a = f 1 ( p 0 / σ 0 ) is derived. The family of curves, presented on the Fig. 5, was obtained for the discrete values of the strain-hardening exponent, n = 2, 3, 5, 10, 50 and 1000. Generally, it can be concluded that, the plastic zone magnitude, r p , will be greater as the strain-hardening of the material is smaller , for the same level of external loading. It means, greater , r p , for greater , n . The biggest plastic zone magnitude, r p , will appear at the elastic-perfectly plastic material. 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, Engineering Fracture 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., 2024. Analyzing Plastic Zone Magnitude around a Crack Tip: exact vs. approximate analytical Solution, In: Book of Abstracts of the 24 th European Conference on Fracture, (ECF 24), pp. 132 and poster presentation, Božić, Ž., Domazet, Ž., Basan, R., Vrdoljak, M., Andrić, M., eds., Zagreb, Croatia. Pustaić, D., Lovrenić - Jugović, M., 2025. A novel Approach for Determining and Analyzing the Magnitude of the Plastic Zone around a Crack Tip: an Algorithm for an approximate analytical Solution. Structural Integrity Procedia 68, 16-23. Pustaić, D., Lovrenić - Jugović, M., 2023. Cohesive Model Appl ication 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, pp. 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 Croat ian 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: Proceedings of the 5th International Congress 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. Ph.D. 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/ .