## 6 Summary and Conclusions  
In this manuscript, a probabilistic modeling approach for the elasto-plastic material behavior at finite strain based on second-order Gaussian random fields was introduced. The description of the material behavior was based on the Kroner-Lee decomposition. Following this, the strain-energy density function introduced by [PERSON] et al. [57] was combined with a classical \(J_{2}\)-plasticity formulation assuming ideal plastic behavior. Since this material model was not available in commercially available finite element codes, first, an algorithm covering the transversely isotropic elastic-ideal plastic material behavior  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Strain Level**} & \multicolumn{2}{c}{\(\mathbf{l_{x}=15\) mm}} & \multicolumn{2}{c}{\(\mathbf{l_{x}=10\) mm}} & \multicolumn{2}{c}{\(\mathbf{l_{x}=5\) mm}} & \multicolumn{2}{c}{**Experimental Data**} \\ \cline{2-9}  & **mean** & **std** & **mean** & **std** & **mean** & **std** & **mean** & **std** \\  & **MPa** & **MPa** & **MPa** & **MPa** & **MPa** & **MPa** & **MPa** & **MPa** \\ \hline
0.2 & 16.6 & 0.1 & 16.6 & 0.1 & 16.6 & 0.1 & 16.1 & 1.8 \\
0.5 & 41.6 & 0.3 & 41.4 & 0.3 & 41.3 & 0.2 & 38.7 & 3.7 \\
1.0 & 80.0 & 1.2 & 78.6 & 1.6 & 76.9 & 1.7 & 70.0 & 5.8 \\
1.5 & 99.2 & 5.2 & 97.7 & 2.6 & 92 was developed and implemented in COMSOL Multiphysics(r). The validation based on the well-known neo-Hookean material model in the elastic and plastic domain, as well as utilizing a user-defined strain-energy density function in the elastic domain showed an excellent conformity. Therefore, in the second step, based on the correlation analysis performed in a previous study [24], cross-correlated second-order Gaussian random fields were generated to represent the spatial distribution, the strain-energy density function parameter \(\Lambda\), and the yield strength induced by the probabilistic characteristics of the microstructure. The remaining parameters were assumed to be homogeneous at this state of the study. With the spatially distributed material properties, tensile tests for specimens made of PBT-GF-30 were simulated. Even though the initial modeling approach was based on the elastic-ideal plastic material behavior, the spatial distribution of the two material parameters led to localized plastic deformation covering the whole measurement length. This caused a strain-dependent reduction of the material stiffness. To validate the numerical results and the presented modeling approach, uniaxial tensile tests were conducted to characterize the elastic and plastic deformation of PBT-GF-30. The obtained stress-strain curves were compared to the numerical results. It was shown that, with an decreasing correlation length, the numerical data matched the experimental results quite well. The deviation at large strain can be assigned to the assumption of ideal plasticity. Furthermore, this initial investigation did not show a significant deviation between the results of different realizations at a low strain level. The reasons were the reduction of the complexity by representing only one parameter of the strain-energy density function by a random field and the limitation of Gaussian random fields. To ensure only positive values for the yield strength, low values were underestimated. Summarizing, it can be concluded that the presented initial probabilistic approach is suitable for the numerical simulation of the elasto-plastic material behavior of SFRCs at finite strain including the spatial distribution on the component level without the need for an explicit microstructural modeling. However, the approach is not limited to SFRCs. It can be used to represent any material system with a probabilistic microstructure, which is the key advantage of the presented approach. Future research will focus on the implementation of specimen unloading, hardening, non-Gaussian random fields, and the representation of additional parameters by stochastic methods.