Issue 78

A. Gallo et alii, Fracture and Structural Integrity, 78 (2026) 489-503; DOI: 10.3221/IGF-ESIS.78.28

requiring 11 minutes. For ψ = 0.7, the analysis reached a minimum of F = 0.00708 after 5 iterations and 38 minutes. Overall, the F value remained relatively stable across the tested values of ψ , with a percentage difference with respect to optimized value in Tab. 1 ranging from 4% to 6%. Finally, the evaluation was repeated with the initial guesses varied by ±2%. The results corresponding to an increase by 2% resulted in a value of F equal to 0.00706 employing 56 minutes and 6 iterations. On the other hand, decreasing them by the same quantity, a value of F = 0.00736 was obtained in 39 minutes and 8 iterations. For the latter case, the simulation was repeated allowing the automatic variation of the parameter ψ during simulation, resulting F equal to 0.00693 in 117 minutes and 13 iterations. The percentage difference between the obtained objective functions and the one in Tab. 1 ranges from 5% to 10%, although the maximum difference is reduced to 3% by allowing ψ parameter to vary automatically. This result confirms that changing the starting point may require an adaptative reduction in the step size in order to facilitate convergence towards the minimum of the objective function. Calibration of GTN ductile damage model The second case study relates the parameters identification of the Gurson–Tvergaard–Needleman (GTN) ductile damage model [2,5,12,20–22,28]. The GTN model extends classical plasticity formulations by introducing damage variables that account for the progressive degradation of material stiffness and strength leading to fracture in ductile materials. Due to the strong coupling between plasticity and damage mechanisms, and the sensitivity of the response to model parameters, GTN calibration represents a highly nonlinear constrained optimization problem, particularly challenging for conventional trial and-error or semi-empirical procedures. The GTN yield function is defined as: ൌ ൬ ఙ ೐೜ ఙ ೤ ൰ ଶ ൅ 2 ଵ ∗ cosh ቀ ଷ௤ మ ଶ ∙ ௛ ௬ ൗ ቁെ൫ 1 െ ଷ ∗ଶ ൯ൌ 0 (16) where ௘௤ is the von Mises equivalent stress, ௛ is the hydrostatic stress, ௬ is the yield stress, ∗ is the effective void volume fraction, and ଵ , ଶ , and ଷ are material parameters introduced by Tvergaard [5]. The damage evolution is governed by additional parameters controlling void nucleation, growth, and coalescence, whose identification is essential for accurately predicting ductile fracture. In particular, the effective porosity ∗ is expressed as: ∗ ൌ ⎩⎪⎨ ⎪⎧ if ൏ ௖ ௖ ൅ቈ ೜ భ భି ௙ ೎ ௙ ೑ି ௙ ೎ ቉ሺ െ ௖ ሻ ௙ if ൒ ௙ if ௖ ൏ ൏ ௙ (17) where f c is the critical porosity and f f is the fracture porosity. The porosity evolution consists of two terms: ሶ ൌ ሶ ௚௥௢௪௧௛ ൅ ሶ ௡௨௖௟௘௔௧௜௢௡ (18) The growth term, firstly introduced by Gurson, is defined as follows: ሶ ௚௥௢௪௧௛ ൌሺ 1 െ ሻ ∙ ሾ ሺ ሶ ሻሿ | ఌ ೐೗ ≅଴ ≅ሺ 1 െ ሻ ∙ ൫ ሶ ௣௟ ൯ (19) The nucleation term, subsequently introduced by Tvergaard and Needleman, accounts for the formation of new voids due to broken material atomic bonds under growing loading condition. It can be modelled in strain or stress-controlled way. The strain-controlled nucleation porosity ேఌ is obtained from the relationship: ሶ ேఌ ൌ ே ∙ ሶ ௣௟ ൌ ⎩⎨ ⎧ ௙ಿ ഄ ௌಿ ഄ √ଶగ ∙ ି ቎ ቀഄ ೛೗ షഄಿ ቁ మ మೄಿ ഄ మ ቏ ⎭⎬ ⎫ ∙ ሶ ௣௟ (20) where ேఌ is the plastic strain standard deviation and ே is the strain nucleation. The stress-controlled nucleation porosity ேఙ is instead expressed as:

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