## 3 Generation of Cross-Correlated Random Fields
### Application to the Elastic-Ideal Plastic Material Behavior of SFRCs  
In this section, the presented framework for the generation of cross-correlated random fields is used to represent the material properties of the elastic-ideal plastic material behavior of SFRCs. The information of the correlation structure was derived and discussed in detail by [PERSON] et al. [24]. It is shown that the correlation structure of the material parameters is approximated best by a combination of the triangle correlation function, defined as  
\[\rho(\xi_{1},\xi_{2})= \begin{cases}a\Big{(}1-\frac{|\xi_{1}|}{b_{1}}\Big{)} \Big{(}1-\frac{|\xi_{2}|}{b_{2}}\Big{)}&\text{if }|\xi_{1}|\leq b_{1},\;|\xi_{2}|\leq b_{2}\\ 0&\text{else}\end{cases} , \tag{17}\]  
and an exponential correlation function [32; 48; 53]:  
\[\rho(\xi_{1},\xi_{2})=a\exp^{-\frac{|\xi_{1}|}{b_{1}}-\frac{|\xi_{2}|}{b_{2}}}. \tag{18}\]  
The parameter \(a\) indicates the strength of the correlation between two parameters; \(\xi_{1}\) and \(\xi_{2}\) hold the spatial coordinates, and \(b_{1}\) and \(b_{2}\) are the corresponding correlation lengths, respectively. In the case of auto-correlation, \(a\) is equal to one; in the case of cross-correlation, \(a\) can take values between \(-1\) and \(1\). The values of \(a\) were set in accordance with the results obtained by [PERSON] et al. [24]. In this initial study, the numerical simulation of the elastic-ideal plastic material behavior of SFRCs, only two parameters were described by spatially distributed material properties to reduce the complexity of the nonlinear numerical simulations. To ensure that both the elastic and plastic domain are captured by the stochastic multiscale approach, one parameter of the strain-energy density function, which is related to the elastic domain, and the yield strength are represented by random fields. Based on this, the routine can be reduced to two parameters capturing the cross-correlation of \(\Lambda\) and \(\sigma_{y}\). The matrix holding the factor \(a\) for each material parameter combination reads  
\[\mathbf{a}= \begin{bmatrix}1.00&0.53\\ 0.53&1.00\end{bmatrix} . \tag{19}\]  
In addition to the parameter \(a\), the required mean values, standard deviations, and correlation lengths are provided in Table 2. Since the correlation length cannot be determined solely on numerical data, the values were normalized to the correlation length of the parameter \(\Lambda\). For more details about the material model and the provided data, the reader is kindly referred to Section 4.1 and the correlation analysis presented by the authors [24], respectively. With this information at hand, realizations of random fields for these two material parameters were generated using the numerical methods for two-dimensional cross-correlated homogeneous second-order random fields presented in Section 3.1. In accordance with the numerical simulation of the tensile test in Section 4 and the experimental procedure in Section 2, the domain has a size of \(50\,\mathrm{mm}\times 3\,\mathrm{mm}\). One realization of the parameter set is presented in Figure 3. The correlation length of \(\Lambda\) was set to \(l_{x}=20\,\mathrm{mm}\) and \(l_{y}=3\,\mathrm{mm}\) in the horizontal and vertical direction, respectively. In addition, the quality of the procedure was analyzed. Therefore, in total, 1000 realizations for each parameter were generated. Based on the obtained data, the correlation structure was derived and compared to the results presented in the former study by the authors [24]. Figure 4 holds the results for \(\Lambda\) and \(\sigma_{y}\). The first two diagrams give the analytic auto-correlation functions of \(\Lambda\) and \(\sigma_{y}\), based on the results obtained in a previous correlation analysis [24] and the correlation function based on the 1000 realizations, respectively. In the last diagram, the cross-correlation function is plotted. For a better overview, the correlation function plots are limited to the horizontal direction.