Issue 78

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

൫഑೐೜శ഑೓శ഑ಿ ൯మ మೄಿ ഑ మ ൩ ቑ ∙ ൫ ሶ ௘௤ ൅ ሶ ௛ ൯

ሶ ேఙ ൌ ே ∙ ൫ ሶ ௘௤ ൅ ሶ ௛ ൯ൌቐ ௙ಿ ഑ ௌಿ ഑ √ଶగ ∙ ି ൥

(21) where ேఙ is the standard deviation of ൫ ௘௤ ൅ ௛ ൯ and ே is the stress nucleation. The plasticity mechanism is governed by the equation: ఙ ఙ ೤ ൌ൬ ఙ ఙ ೤ ൅ ଷீ ఙ ೤ ̂ ௣௟ ൰ ௡ (22) where n is the hardening coefficient, ̂ ௣௟ is the accumulated plastic strain, and G is the shear modulus. The same ANSYS MATLAB workflow adopted in the previous JC case was employed. Geometry, boundary conditions, element type, and mesh discretization remained unchanged, while the dynamic analysis was conducted under displacement control with a speed of 1.8 mm/min, according to the standard T228/1 used in [23,24]. Mesh density was validated through convergence analysis. This choice guarantees that variations in the numerical response are primarily driven by changes in material parameters rather than discretization effects. Due to the large number of parameters of GTN formulation and to the sequential nature of the physical mechanisms they describe, the calibration problem cannot be reliably formulated as a purely unconstrained optimization task. Indeed, several GTN parameters have a strong physical interdependence and must satisfy bilateral constraints to ensure a meaningful and stable material response. Inequality constraints were imposed to guarantee the positivity of porosity-related parameters and standard deviations, while bilateral constraints were imposed to maintain physically consistent ordering among parameters describing successive damage stages. For instance, the critical porosity ௖ was constrained to be greater than the initial porosity ଴ and strictly lower than the fracture porosity ௙ , i.e. ଴ ൏ ௖ ൏ ௙ . Additional bounds were assigned to nucleation parameters to ensure realistic strain- and stress-controlled void nucleation behaviour. The constraints were enforced through an interior penalty function approach [7,11,14,18], which transforms original constrained problem into a sequence of unconstrained optimization ones by extending the objective function with penalty terms r p . Considering also the initial porosity f 0 , the GTN parameters to be identified are eight, as shown in Eqn. (23). ൌ൛ ௬ , , ଴ , ே , ே , ேఌ , ௖ , ௙ ൟ் (23) Using the interior penalty method, a modified objective function, referred to as pseudo-objective function, is defined as: ൫ , ௣ , ௣ ᇱ ൯ ൌ ሺ ሻ ൅ ௣ ∑ି ௚ ଵ ೕ ሺ ሻ ே௝ୀଵ ൅ ௣ ᇱ ∑ ሾℎ ௞ ሺ ሻሿ ଶ ெ௞ୀଵ (24) where ௝ ሺ ሻ and ℎ ௞ ሺ ሻ represent inequality and equality constraints, respectively. Since no equality constraints are considered in this case, the Eqn. (14) reduces to: ൫ , ௣ ൯ ൌ ሺ ሻ ൅ ௣ ∑ି ௚ ଵ ೕ ሺ ሻ ே௝ୀଵ (25) increasing the total number of optimization parameters by one, bringing the total to nine. The interior penalty coefficients were progressively increased during the optimization process to gradually tighten the enforcement of the constraints without compromising numerical stability. After convergence of the optimization algorithm, the identified GTN parameters were compared with typical values and admissible ranges reported in the literature for ductile metallic materials [12,21]. The optimized parameter set is reported in Tab. 2. The resulting objective function was about one order of magnitude lower than that obtained using parameters set available in literature [12,21]. The identified parameters exhibit physically consistent values and show a significantly improved agreement with the experimental stress–strain response (Fig. 5), particularly in the post-necking regime where damage evolution governs the material behaviour. Further, from Tab. 2 it is possible to highlight a deviation between the objective function evaluated with reference and MBS optimized values equal to ηൌ ி ೝ೐೑ି ி ೚೛೟ ி ೚೛೟ 100 ൌ 2240% . Such a percentage difference highlights the

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