## 5 Application to Tensile Test Specim
### Numerical Model  
The numerical model used to validate the novel modeling approach on the component level based on tensile test specimens was identical to the one used in Section 4.2.2. As already discussed in Section 3.2, only one parameter of the strain-energy density function and the yield strength were represented by the cross-correlated second-order Gaussian random fields. This reduced the complexity at this state of the investigation. Since, in this case, the standard deviation of the material parameter \(\Lambda\) did not cover the total standard deviation of elasticity tensor element \(C_{11}\), which is significantly related to the structural behavior under uniaxial tension, the corresponding value was set to 1.74 GPa. Furthermore, the experimental data clearly revealed that local plastic deformation occurred even at low stress levels. This was related to the yield strength of the matrix material and to the plastic deformation induced by the stress peaks at the edges of the reinforcing elements. In addition, the maximal stress level of the data sheet and, hence, the results of the homogenization were not reached. Therefore, the mean value and the standard deviation were adjusted and set to 100 MPa and 31 MPa, respectively. The remaining parameters were assumed to be homogeneous and were set to the mean values obtained by a previous correlation analysis [24]. Furthermore, a plain strain state was employed. This was eligible because the microstructure of the SFRC consisted of layers with different main fiber orientations [77; 78]. The core layer, which was located in the center of the cross-section, was characterized by a fiber orientation perpendicular to the main flow direction of the injection mold process. With respect to the specimens used in this study, the fibers within the small core layer were perpendicular to the load direction and, hence, inhibited the out-of-plane deformation.  
Further crucial parameters for the numerical simulation were the correlation lengths of the random fields. Since the correlation length significantly influences the local distribution of the material properties, it was strongly linked to the standard deviation of the numerical results. Therefore, in this study, different values were used to analyze the influence more in detail. In Figure 10, realizations for the strain-energy density function parameter \(\Lambda\) are depicted for a correlation length in the horizontal direction of \(l_{x}=5\,\)mm, \(l_{x}=10\,\)mm, and \(l_{x}=20\,\)mm. In the vertical direction, the correlation length was set to \(l_{y}=3\,\)mm. The effect of the correlation length on the random field is clearly observable. In analogy to the description of the oscillations, the correlation length can be interpreted as the wavelength of the dominant eigenfunctions.  
In this paper, the numerical simulations were performed for horizontal correlation lengths of \(l_{x}=5\,\mathrm{mm}\), \(l_{x}=10\,\mathrm{mm}\), and \(l_{x}=15\,\mathrm{mm}\). The correlation length in the vertical direction was set to \(0.3\,\mathrm{mm}\) for all simulations. This value was chosen to allow plastic deformation at different locations, as was observed during the experimental investigations. In combination with the ideal plastic modeling approach, the resulting stress-strain relation approximated the experimental data for the reinforced material. Were the correlation length be identical to the specimen's thickness of \(3\,\mathrm{mm}\), the plastic deformation was limited to the region where the yield strength was reached first, because the surrounding material cannot take higher stress levels. The obtained results would then be characterized purely by the ideal plastic behavior of the matrix material.  
The numerical simulations were carried out for eight different realizations per correlation length. Hence, in total, 24 simulations were conducted. The results are presented below.