# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 4 Test Design and Experimental Setup
### Test Design  
The uncertain parameters of the aluminum alloy are treated as random variables, initially characterized by a uniform prior distribution \(z_{i}\sim\mathcal{U}(a,b)\). This choice reflects a non-informative prior defined between the bounds given based on the best engineering judgment. These bounds are given in Table 1. The parameter identification process is conducted in two phases. In the first phase, the rectangular specimen (see Figure 3a) is used to identify the elastic parameters (\(E\) and \(\
u\)). Although [PERSON]'s modulus \(E\) and [PERSON]'s  
Figure 3: Aluminum alloy specimens: (**a**) Rectangular (**b**) Dogbone single\(-\)notched (**c**) Dogbone double\(-\)notched.  
ratio \(\
u\) are standard parameters in engineering, for stochastic computations, it is often more advantageous to work with the bulk modulus \(K\) and shear modulus \(G\), as they directly correspond to the eigenvalues of the constitutive matrix. Ensuring the positivity of \(K\) and \(G\) automatically guarantees the positive definiteness of the constitutive matrix. Nevertheless, in this work, we use the parameter pair \((E,\
u)\), and enforce the positivity of \(K\) and \(G\) through appropriately bounded distributions of these parameters.  
The proposed laboratory setup for parameter identification includes both force\(-\)displacement measurements and deformation measurements taken from the specimen's lateral surfaces. In the middle of the specimen, two points were selected on opposite sides of the lateral boundary, and the sum of their horizontal displacements is used as a measurement. To validate the feasibility of this approach prior to experimental testing, a Sobol sensitivity analysis was performed. A set of 200 parameter combinations from the prior distribution was generated using a Quasi-Monte Carlo (QMC) sampling technique, ensuring a well-distributed sample set. For each generated sample, virtual measurements were collected at different time steps using a linearly incrementing displacement control. Then a gPCE model was trained using 150 random samples, while 50 were used for accuracy testing. The degree of the polynomial basis was optimized to achieve the best accuracy. The results of the sensitivity analysis, presented in Figure 4, confirm that the selected measurements effectively capture the influence of both elastic parameters.  
For the identification of the elastoplastic hardening and fracture parameters, the single\(-\)notched dog bone specimen is used (see Figure 3b). For this case, only a  
\begin{table}
\begin{tabular}{l c c c} \hline
**Parameter** & **Distribution** & **Left Bound (_a_)** & **Right Bound (_b_)** \\ \hline \(E\) & \(U(a,b)\) & \(50,000\) N/mm\({}^{2}\) & \(100,000\) N/mm\({}^{2}\) \\ \(\
u\) & \(U(a,b)\) & 0.2 & 0.4 \\ \(\sigma_{0}\) & \(U(a,b)\) & \(50\) N/mm\({}^{2}\) & \(150\) N/mm\({}^{2}\) \\ \(H\) & \(U(a,b)\) & \(2000\) N/mm\({}^{2}\) & \(6000\) N/mm\({}^{2}\) \\ \(\sigma_{n}\) & \(U(a,b)\) & \(150\) N/mm\({}^{2}\) & \(300\) N/mm\({}^{2}\) \\ \(G_{fn}\) & \(U(a,b)\) & \(100\) N/mm & \(300\) N/mm \\ \hline \end{tabular}
\end{table}
Table 1: Prior distributions for material parameters.  
Figure 4: Sobol sensitivity analysis (rectangular specimen): Partial variances of (**a**) The force\(-\)displacement measurement (**b**) Lateral deformation measurement.  
force\(-\)displacement measurement was selected for the identification process. Following the same procedure as the elastic parameter identification, a set of 2000 combinations of parameters was generated using QMC sampling and a corresponding gPCE model was trained. The results, presented in Figure 5, show that force\(-\)displacement measurement is sufficient to capture the influences of the elastoplastic and fracture model parameters.