# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 2 Modeling of Fracture Propagation
### Constitutive Law
Taking this into account, an obtained measurement \(\mathbf{d}_{m}\) can be expressed as a realization of a random variable [23; 50]:  
\[\mathbf{d}_{m}(\omega_{e})=\mathbf{u}_{\text{true}}+\mathbf{\epsilon}(\omega_{e}) =\mathcal{G}(\mathbf{z}_{\text{true}})+\epsilon(\omega_{e}). \tag{15}\]  
Here, \(\mathbf{u}_{\text{true}}\) represents the measurable variable predicted by using the \"true\" parameter values \(\mathbf{z}_{\text{true}}\), while \(\mathbf{\epsilon}\) accounts for the the combined errors in the measurement and the modeling, depending on the specific realization \(\omega_{e}\). These errors are described by their probability density function \(\pi_{e}\). It is further assumed that the realizations of the errors \(\mathbf{\epsilon}\) are independent of the realizations of the uncertain input parameters \(\mathbf{z}\), ensuring that the random variables \(\mathbf{z}\) and \(\mathbf{\epsilon}\) are independent. Accurately defining the error distribution is a crucial step in the identification process. This work considers three types of additive errors: measurement errors, numerical model errors, and proxy modeling errors  
\[\mathbf{\epsilon}=\mathbf{\epsilon}_{\text{meas}}+\mathbf{\epsilon}_{\text{model}}+\mathbf{ \epsilon}_{\text{proxy}}, \tag{16}\]  
where the last term, \(\mathbf{\epsilon}_{\text{proxy}}\), arises when the computationally expensive numerical simulation is replaced by a proxy model. These three error components are assumed to be mutually independent. The measurement error \(\mathbf{\epsilon}_{\text{meas}}\) is often assumed to follow a normal distribution, \(\mathbf{\epsilon}_{\text{meas}}\sim\mathcal{N}(\mathbf{0},\mathbf{C}_{\text{meas}})\), centered around the experimental measurements. This assumption is justified by the use of standardized calibration and testing procedures, which ensure consistent and reliable measurement performance. The covariance matrix \(\mathbf{C}_{\text{meas}}\) is taken as diagonal, assuming independent error components. The diagonal entries \(\sigma_{i}^{2}\) are defined based on the manufacturer's specifications of the measurement device, which may be expressed as a percentage of the measured values. This percentage typically reflects the maximum possible error, which is here interpreted as a \(3\sigma_{i}\) deviation under the normal distribution assumption. Consequently, the standard deviation \(\sigma_{i}\) of the measurement error is calculated by dividing the maximum error by three. In contrast, modeling and proxy errors are inherently more complex. They may exhibit correlation, systematic bias, or non-Gaussian behavior, especially near crack initiation. If not properly addressed, such simplifications can lead to underestimated uncertainties and biased parameter estimates, even if the model fits the experimental data. Various strategies exist to handle numerical model errors. One approach is to treat the associated covariance matrix as an additional unknown within the identification process [25]. Alternatively, it can be estimated separately as a deterministic, input-dependent function, based on the constitutive laws employed and comparisons between model predictions and experimental data. To avoid the risk of incorporating incorrectly estimated bias and correlations, a simple analysis was conducted comparing model predictions with experimental results for non-linear data. This revealed that modeling error exhibited noticeable bias only in the later stages of testing, where increased uncertainty was already anticipated. In contrast, proxy model errors were generally small and only weakly correlated, with a negligible impact on the later parts of the response. Based on these findings, we adopt the simplifying assumption that these error components are also zero-mean, mutually independent, and Gaussian. This is considered a pragmatic compromise in the absence of reliable bias quantification.