# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 6 Conclusions  
This work demonstrates the potential of the Bayesian approach for solving the inverse problem of material parameter identification, specifically applied to the failure behavior of aluminum 6060 alloy specimens. Within the applied methodology, a numerical model based on the embedded discontinuity method (Q6 ED) is used, in which cracks are modeled locally with their own location and orientation. This allows for realistic and mesh-independent fracture results. To achieve a computationally efficient approach, a generalized polynomial chaos expansion (gPCE) was used as a proxy model. This approach has already proven to be efficient and accurate for simpler fracture cases, while also supporting straightforward sensitivity analysis, which is valuable for both test design and understanding parameter influence. However, gPCE introduces approximation errors that must be carefully controlled across the entire response surface. If not adequately managed, these errors can bias the identification process by overfitting regions with lower proxy error, thereby compromising the accuracy of the identified parameters. The experimental setup involved tensile testing using a universal testing machine (UTM), with deformation monitored through simple yet effective measurements, including internal UTM sensors (force, crossbar position, and extensometers), LVDT sensors with an additional data acquisition unit, and an optical measurement system. Unlike standard laboratory specimens, the aluminum samples in this study were intentionally designed with notches to induce stress concentrations, which makes them non-standard and typically unsuitable for direct parameter extraction from test data alone. A key aspect of the Bayesian method is the definition of the error model. In this study, measurement error, modeling error, and proxy model error were assumed to be mutually independent and normally distributed with zero mean. While this assumption is widely used due to its simplicity and analytical convenience, it may not always hold, particularly in the context of nonlinear fracture mechanics. In this work, the assumption of unbiased and uncorrelated measurement errors was considered reasonable due to reliable calibration and stable testing conditions. In contrast, modeling and proxy errors are inherently more complex. They may exhibit correlation, systematic bias, or non-Gaussian behavior, especially near crack initiation. Nevertheless, the error analysis for non-linear data showed that the impact of such limitations was limited. Consequently, the adopted simplifications were deemed justified and did not significantly compromise the validity of the results within the scope of this study. The Bayesian inversion process successfully provided credible posterior values for key parameters across all behavioral phases: elastic ([PERSON]'s modulus \(E\), Poisson's ratio \(\
u\)), elastoplastic (yield strength \(\sigma_{0}\), hardening modulus \(H\)), and fracture parameters (tensile strength \(\sigma_{n}\), tensile fracture energy \(G_{fn}\)). The resulting posterior distributions show limited variability, highlighting the robustness of the Bayesian methodology and the effectiveness of the integrated numerical and experimental framework. However, this level of certainty may not be achieved with different numerical models or experimental conditions. This highlights the importance of applying probabilistic methods for nonlinear and non-standard measurements over deterministic approaches, as they provide a means to quantify uncertainty, which is essential for the reliability and interpretability of the results. The identified elastic parameters from the rectangular specimen were shown to be applicable to all the subsequently tested specimens. The maximum a posteriori (MAP) values of the elastoplastic and fracture parameters, obtained from the single\(-\)notched specimen, were validated through comparisons with independent measurement data and additional results from the double\(-\)notched specimen.