## 2 Materials and Methods
### Temperature-Dependent Material Properties  
In the generation of an RVE, each grain is assigned to a phase. In Figure 2b an example is shown for a sample with three phases. From the literature, a series of single crystal properties can be obtained, as shown in Table 1 in Voigt notation. In general, these elastic constants are computed from measurements to wave velocities, or strains and stresses. The values of ferrite are very close (differences less than 1.5%) and those of austenite differ up to 5%. This may indicate that the assumed temperatures might be off. Figure 3: An example of replication and randomization. On the left (**a**), the phase is shown, replicated three times in \(y\)-direction. The middle image (**b**) shows an ordinary replication of the first Bunge Euler angle. Note the repetitive patterns in the zooms. The image on the right (**c**) shows the results of Euler angle randomization per phase after replication, which eliminates the repetitive pattern in the texture. After randomization, the phases corresponding with (**c**) still remaining the same as in (**a**). The colors are corresponding with those used in Figure 2.  
A dimensionless metric that describes the anisotropy of a crystal, known the Zener ratio or Zener's anisotropy, has been defined for cubic crystals [44; 48] as  
\[\mathcal{Z}=\frac{2C_{44}}{C_{11}-C_{12}}. \tag{1}\]  
The ratio is (close to) 1 for isotropic media. The isotropy of the crystalline materials is also listed in Table 1.  
The elastic properties of steel depend on the temperature, as has been shown e.g., by [PERSON] [41]. Based on GLUS(r) measurements (method described in [13; 49]) and data inversion [50], the temperature dependencies of the aforementioned material types have been obtained. The elastic constants \(C_{ij}(T)\) were experimentally determined for each material in Table 1 at a discrete number of different temperatures. A polynomial function was fitted to this data to obtain the temperature dependency. The temperature dependency was modelled as  
\[C_{ij}(T)=\Big{(}a_{1,ij}T^{2}+a_{2,ij}T+1\Big{)}C_{ij}(T_{\text{REF}}), \tag{2}\]  
where \(C_{ij}(T_{\text{REF}})\) is the elastic constant \(C_{ij}\) at the reference temperature \(T_{\text{REF}}\) (also listed in Table 1), \(T\) is the temperature in the simulation, and \(\big{[}a_{1,ij}\quad a_{2,ij}\big{]}\) are the experimentally determined polynomial coefficients. In the simulations, the required elastic constants of a material at a specific temperature were obtained by entering the constants at the reference temperature (from Table 1) and the polynomial coefficients into Equation (2). An example is shown in Figure 4a, where the temperature dependency of the elastic coefficients \(C_{11}\), \(C_{12}\), and \(C_{44}\) is plotted. The curve is determined based on the aforementioned measurements in the range between 100 \({}^{\circ}\)C and 500 \({}^{\circ}\)C, but extrapolated values down to \(-273\)\({}^{\circ}\)C are also plotted as a dotted line. The experimental values which [PERSON] lists in his article [41] (originally measured by [PERSON] [42]) are shown as well. Although there are deviations between [PERSON]'s values and the curve, the trends are comparable. Temperature dependencies of other phases have not been found in the literature but based on the match between the extrapolated experimental results and [PERSON]'s results, the approach is considered to be valid. Note that the temperature dependency of the elasticity matrix is different for each component, both in absolute and in relative sense. For ferrite, the \(C_{12}\) shows a much smaller relative decrease than \(C_{11}\), which has in turn a smaller relative decrease than \(C_{44}\). The