# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 5 Results
### Result Verification  
The verification of the identified elastic, elastoplastic, and fracture parameters was carried out for two cases by comparing the numerical model results using the MAP estimate of the parameters with two distinct sets of experimental data. The first case involved a comparison of displacements and strain field values on a single\(-\)notched dogbone specimen, with data obtained by DIC camera. The second verification used force\(-\)displacement data from a double -notched dogbone specimen (see Figure 3c). This approach allowed for validation not only against different types of measurements from the specimen used for identification but also against specimens with different geometries (see Table 3). For each verification case, 20 samples were generated from the posterior parameter distributions using QMC sampling. These parameter sets were then used in simulations, and the resulting outputs were compared with experimental data to evaluate the accuracy and robustness of the identification methodology. Figure 12: Comparison of the failure mechanism for the notched dogbone specimens: (**a**) Failure of the single\(-\)notched specimen in laboratory (**b**) Failure of the single\(-\)notched specimen in numerical simulation (**c**) Failure of the double\(-\)notched specimen in laboratory (**d**) Failure of the double\(-\)notched specimen in numerical simulation. Colored arrows indicate crack propagation directions. For the first verification case, two points on the specimen were selected: (12 mm, 100 mm) and (12 mm, 120 mm). These points were located on opposite sides of the crack. The vertical displacement difference between these points was extracted from DIC measurements and compared with numerical results, as shown in Figure 13. The results reveal some differences between the numerical and DIC data. The largest discrepancy appears at the beginning of the test, mainly due to the difficulty of DIC systems in capturing very small displacement values accurately. As the test progresses and displacements become larger, the measurements from DIC become more reliable. In the middle of the test, where the measured values are higher, the DIC measurements become more reliable. At this stage, the observed differences are largely attributed to the same sources of error noted in the force\(-\)displacement comparison, particularly those arising from the proxy model. The maximum error between the DIC measurements and the numerical model is 11.9%. Toward the end of the test, these differences continue to decrease, resulting in an increasingly close match between the numerical predictions and the DIC data. Additionally, a comparison of strain field values was performed for a total displacement of 3.899 mm. The strain components were computed from the displacement gradients using standard DIC post-processing methods. This comparison also showed very good agreement between the two results, both in the shape of the strain fields and the magnitude of the strain values (see Figure 14). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Specimen** & **Measurement** \\ \(\sigma_{0},H,\sigma_{n},G_{fn}\) & single\(-\)notched & Vertical displacements and strain fields (DIC) \\ \(\sigma_{0},H,\sigma_{n},G_{fn}\) & double\(-\)notched & Force\(-\)displacement (UTM) \\ \hline \hline \end{tabular} \({}^{1}\)Accuracy of the identified elastic parameter values (\(E\), \(\
u\)) was confirmed by matching of numerical model results with experimental measurements in the linear elastic region of force\(-\)displacement measurements for the single-notch and double-notch specimens.  
\end{table} 
Table 3: Verification process of elastic, elastoplastic, and fracture parameters through comparison of laboratory measurements and numerical model predictions (two verification cases) \({}^{1}\). Figure 13: Comparison of the vertical displacement difference between two points (single\(-\)notched specimen): DIC measurements vs. numerical model results using 20 samples of parameter values drawn from the posterior distribution.