# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 5 Results
### Identification of Elastoplastic and Fracture Parameters
Additionally, ongoing developments in advanced proxy modeling, particularly in Physics-Informed Neural Networks (PINNs), offer promising prospects for further reducing these errors. However, a robust PINN framework capable of effectively handling embedded discontinuities in fracture mechanics has not yet been established. The identification of elastoplastic and fracture parameters was performed using 200 Markov chains and 5000 computational steps, resulting in a total of 1,000,000 evaluations for different parameter combinations. The entire process took less than 10 min to complete. The results are presented in Figure 10.  
Figure 10: Identification of yield strength \(\sigma_{0}\), hardening modulus \(H\), tensile strength \(\sigma_{n}\) and tensile fracture energy \(G_{fn}\) parameters. The posterior distribution of yield strength \(\sigma_{0}\) exhibits minimal variation, indicating a high level of certainty, with a maximum posterior value of 76.52 N/mm\({}^{2}\). Similarly, the hardening modulus \(H\) shows only slight variation around its maximum posterior value of 1626.01 N/mm\({}^{2}\). The posterior distribution of tensile strength \(\sigma_{n}\) also demonstrates high certainty, with a well-defined peak at 171.11 N/mm\({}^{2}\). Finally, while the posterior distribution of tensile fracture energy \(G_{fn}\) shows slightly greater uncertainty compared to the other parameters, it still features a strong peak around its maximum posterior value of 258.17 N/mm. These maximum posterior values were used in both the proxy model and the full numerical model to compare against the experimental data. The comparison is presented in Figure 11. As shown, the proxy model closely follows the lab measurements throughout the test, with only small deviations (with a maximum error of 2%). The numerical model also aligns very well with the experimental results, particularly in the elastic and elastoplastic regions, which confirms the accuracy of the identification process. A slight discrepancy emerges after the onset of fracture, mainly due to differences between the proxy model and the numerical simulation results (see Figure 9). The maximum error between the laboratory measurement and the numerical model is 8.9%. Nevertheless, the accuracy remains good, especially considering the challenges of modeling nonlinear fracture behavior. Figure 12 shows a comparison of specimen failure observed in the laboratory tests and in the numerical simulations. For the simulation results, cracks are visualized using a color scale ranging from blue to orange, representing the level of energy dissipation, where blue indicates lower values and orange higher ones. For the single\(-\)notched dogbone specimen, both the experimental and numerical results (see Figure 12a,b) confirm the formation of a single dominant crack. This crack initiates at the edge of the notch and propagates toward the end of the specimen perpendicular to the direction of the loading. This confirms that the model accurately captured the failure mechanism. Figure 11: Comparison of the force\(-\)displacement results (single\(-\)notched specimen): Laboratory measurement vs. proxy model result vs. numerical model result.