## 3 Generation of Cross-Correlated Random Fields
### Implementation in COMSOL Multiphysics(r)
#### 4.2.2 Validation of the Algorithm  
For the validation process, a two-dimensional numerical model under the plane strain assumption representing the cross-section of a tensile test specimen was used. The size of the numerical model was \(50\,\mathrm{mm}\,\times\,3\,\mathrm{mm}\). In Figure 8, the numerical model and the corresponding boundary conditions are depicted. To reduce convergence issues during the plastic deformation, the simulation was displacement driven. Besides that, the numerical model was discretized by four-node elements with a size of \(0.25\,\mathrm{mm}\,\times\,0.25\,\mathrm{mm}\). Hence, over the thickness, the cross-section was discretized with 12 and along the cross-section with 200 elements. Due to the fact that COMSOL Multiphysics(r) experiences some issues when combining a user-defined hyperelastic strain-energy density function with plasticity, the provided algorithm was validated in two steps. In the first step, the external material algorithm was validated based on the well-known hyperelastic neo-Hookean material model [75]. Therefore, the strain-energy density function within the presented algorithm was adapted to the one implemented in COMSOL Multiphysics(r) for the neo-Hookean material model given by  
\[\Psi_{\mathrm{Neo}}=\frac{1}{2}\mu(I_{1}-3)-\mu\mathrm{ln}(J)+\frac{1}{2} \Lambda[\mathrm{ln}(J)]^{2}. \tag{60}\]  
Subsequently, two numerical simulations were performed, one with the implemented neo-Hookean hyperelastic material model and the second one with the provided external material algorithm. In both cases, the ideal plasticity and a plain-strain state were  
Figure 8: Two-dimensional numerical model. Figure 7: Structure of the implemented algorithm. assumed. The corresponding material properties are provided in Table 6. Furthermore, a total displacement of \(u_{x}=1\) mm corresponding to a strain of 2% was applied to ensure plastic deformation. In a second step, the strain-energy density function provided by [PERSON] et al. [57] was passed to COMSOL Multiphysics(r) as a user-defined potential. Since this cannot be combined with the ideal plastic behavior due to convergence issues, this was only simulated assuming elastic deformation. The results were compared again to the results provided by Algorithm 1. The corresponding material parameters are also provided in Table 6.  
The results are depicted in Figure 9. The first row gives the result for the isotropic neo-Hookean simulation, and the second row holds the results for the transversely isotropic simulations. The diagrams on the left-hand side contain the stress-strain curves. The von Mises stress is plotted in black, and the horizontal component of the second Piola-Kirchhoff stress is plotted in gray, respectively. Following the theoretical framework provided in Section 4.1, it is important to note that the von Mises stress is given in the intermediate configuration, whereas the second Piola-Kirchhoff stress tensor stands in the reference configuration. The solid lines give the results of the implemented material model in COMSOL Multiphysics(r), and the small circles belong to the results obtained by the external material routine. As indicated by the diagrams on the right-hand side, which provide the deviation of the external material routine to the implemented material model in percent, both validation steps showed good agreement. Both comparisons experienced an increasing deviation at large displacements, which was still very small. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \(\mathbf{\Lambda}\) & \(\mathbf{\mu}\) & \(\mathbf{\kappa}\) & \(\mathbf{\beta}\) & \(\mathbf{\gamma}\) & \(\mathbf{\sigma_{Y}}\) \\  & **GPa** & **GPa** & **GPa** & **GPa** & **GPa** & **MPa** \\ \hline Neo-Hookean & 5.38 & 1.20 & - & - & - & 50 \\ Transversely isotropic & 5.38 & 1.20 & 1.10 & \(-\)0.13 & 1.13 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material parameters validation with neo-Hookean material model.