# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 5 Results
### Identification of Elastoplastic and Fracture Parameters  
This phase of the identification process involves the elastoplastic hardening of the material, the development of microcracks, the formation of the macrocrack, and finally, the material failure. These processes result in complex datasets that are much more challenging to interpret compared to elastic parameters, resulting in significantly higher computational effort needed. To train the gPCE model, 20,000 samples were generated from the prior distributions using QMC sampling, with an additional 5000 samples used for testing. These computations performed on a cluster computer required approximately 70 h to complete. After validation and testing, a ninth-degree polynomial gPCE with 715 coefficients per timestep was found to provide the most accurate results. Regarding the definition of the error distributions, the standard deviation of the measurement error \(\sigma_{i}\) was set to 0.167% of the measured value. The modeling error is defined by comparing the laboratory measurements with the results from the numerical model, based on knowledge of the constitutive laws used. Here, we divide the modeling error into two parts. The first part relates to the linear elastic and elastoplastic hardening behavior, as well as the onset of microcrack formation. For this part, a standard deviation of \(\sigma_{i}=25\) N is chosen. This value is selected because the numerical model produces identical outputs for the same input parameters, whereas real materials exhibit inherent variations. Since the elastic parameters were fixed in the previous phase and by considering the small influence of the nonlinear behavior of microcracks, this error distribution is selected to cover potential variations. The second part of the modeling error relates to the merging of microcracks, the formation of macrocracks, and the final material failure. In the numerical model, energy dissipation is represented by an exponential softening law. However, this is only an approximation of a highly nonlinear real-world phenomenon. Given the potential for larger variations when sudden material failure occurs, the error distribution for this part is set to be significantly larger, with \(\sigma_{i}=100\) N. Finally, the proxy model error is determined by comparing its predictions for the testing samples with the results from the numerical model. These error distributions, along with laboratory measurements, are presented in Figure 9.  
The biggest error comes from the proxy modeling. This is primarily due to the complex and nonlinear nature of the problem, for which we have a complex, nonsmooth response surface that is difficult to approximate. Because a wide range of prior distributions was  
Figure 9: Identification of elastoplastic and fracture parameters (single\(-\)notched specimen)\(-\)95% CI error distributions. considered, the samples in the testing set exhibit substantial variation. At a given timestep, some samples remain in the linear elastic regime, others undergo elastoplastic hardening, some initiate cracking, and others may approach complete failure. Capturing such a diverse dataset is challenging for different proxy modeling techniques [49]. Therefore, when analyzing fracture propagation problems, it is inevitable that relatively larger errors will be encountered in proxy modeling. That is why it is important to include these errors inside the identification process through the error model to ensure that the uncertainty introduced by the proxy is properly reflected in the posterior distributions of the identified parameters. One suggested strategy to reduce proxy modeling errors is to perform multiple iterations of the identification process for the same test. After the initial identification, the resulting posterior distributions can be broadened using a chosen uncertainty coefficient and then used as new prior distributions. These updated priors typically exhibit less variability, which simplifies the response surface and improves the quality of the proxy model in subsequent iterations.