PSI - Issue 68

F.J. Gómez et al. / Procedia Structural Integrity 68 (2025) 734–740 F.J. Gómez, T. Gómez-del-Rio, J. Rodríguez/ Structural Integrity Procedia 00 (2025) 000–000

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failure criteria, and determining material properties such as resilience from the Charpy testing (ASTM 2017) or fracture toughness (Anderson, T.L., 2017)

Nomenclature H

Mean strain energy density auxiliary function

Fracture toughness

K IC K UC

Critical notch stress intensity factor

Characteristic length Normal distribution

l ch N

R Notch radius WAIC Watanabe-Akaike information criterion µ Mean value of the normal distribution Standard deviation of the normal distribution !"# Elastic stress at the tip of the notch s c Maximum tensile strength

The standardized procedure for determining fracture toughness, K IC , typically involves the use of cracked or pre cracked specimens. However, when cracking is costly, or pre-cracking fatigue is not feasible, U-notch specimens are a suitable alternative. Gómez-del-Rio et al. investigated the fracture toughness of polyamide 12, a thermoplastic polymer known for its high strength and durability, using U-notch specimens (Crespo et al 2017, 2019). These researchers proposed the theory of critical distance (Seweryn 1994, Taylor et al, 2005) to describe the fracture process and determined by fitting the criterion parameter and the fracture toughness. It is worth noting that the value obtained through this procedure relies on the adopted failure criteria, introducing additional uncertainty into the process. Gómez et al (2006a, 2006b) conducted a comprehensive analysis on the applicability of failure criteria to U notched specimens in linear elastic materials. Their study encompassed 18 ceramic materials and one linear elastic polymer, where they proposed a non-dimensional formulation to summarize the experimental results into a single band. The researchers compared and evaluated seven different failure criteria, including the Cohesive Zone Model (Bazant and Planas 1998, Elices et al 2001) with linear and rectangular softening curves, the Theory of Critical Distances considering mean stress and maximum stress (Seweryn 1994, Taylor et al, 2005), the Strain Energy Density Criterion (Lazzarin and Zambardi 2001, Berto and Lazzarin 2014), and the Finite Fracture Mechanics (Leguillon 2002). Remarkably, all the criteria demonstrated relatively similar accuracy in explaining the experimental data and were reformulated into comparable mathematical expressions. Additionally, the authors introduced a novel phenomenological failure criterion that effectively fit the experimental results (Gómez et al, 2006a). To address the uncertainties inherent in previous studies, the present document proposes the application of a Bayesian methodology (Gelman et al 2014). Bayesian analysis enables the quantification of uncertainty and enhances prediction accuracy. The method relies on Bayes' Theorem, which updates the initial probability distribution, known as the prior, based on experimental data to obtain an improved probability distribution, known as the posterior. Bayesian methodology has been widely employed in the field of inverse problems, providing a probabilistic framework for estimating unknown parameters from observed data. The next sections will evaluate the uncertainty in the fracture toughness determined from U-notch specimens taking into account experimental errors. The epistemic uncertainty, arising from the unknown failure criterion, will also be analyzed. Finally, a Bayesian methodology for selecting the best failure criterion based on available data will be proposed. 2. Experimental program The material under study is polyamide, PA12. Further comprehensive information about the material and its manufacturing process can be found in (Crespo et al 2017, 2019). Tensile notched specimens were fabricated using