## 2 Materials and Methods  
### Material Statistics  
In order to develop a computational framework to numerically generate material volumes with identical microstructural properties, a reference material with a simple and very well-specified microstructure was needed. Additionally, for each distinct phase, there should exist available data regarding their mechanical properties, e.g., elastic constants and stress-strain curves of the constituents. In this work, the AA7075 aluminum alloy was selected, as its microstructure has been thoroughly studied in previous works [34, 39, 40, 41, 42]. As the AA7075 alloy contains various alloying elements, mainly Zn, Mg, Fe, and Cu, the phases occurring after the precipitation hardening process may also vary [41, 42]. Nonetheless, in its simplest form, the alloy can be considered to contain three main phases: one matrix \(\alpha\)-phase, consisting predominantly of Al (-94% at. ), in which \(\eta\)- and \(\eta^{\prime}\)-phase (MgZn\({}_{2}\) or of similar chemical type) nanosized particles are dispersed; two precipitate phases, one of which is Fe-bearing (Al\(\gamma\)Cu\({}_{2}\)Fe) and the other is Si-bearing (Mg\({}_{2}\)Si). The volume fractions of the two precipitate phases are 1.29% and 0.34% for Al\(\gamma\)Cu\({}_{2}\)Fe and Mg\({}_{2}\)Si, respectively [39, 43]. For the proposed framework, the objective was to employ image data of the material's microstructure, obtained by imaging techniques, such as SEM, TEM, or tomography. With proper processing, it is possible to calculate volume fractions as well as size distributions of the different phases. The latter may be used to statistically describe the equivalent sphere diameter (ESD) of each precipitate phase, which denotes the diameter of a sphere with a volume equivalent to those of the randomly shaped precipitates. The first step was to utilize an SEM image of AA7075 alloy as a starting point, shown in Figure 1a. This image was segmented into three parts, as can be seen in Figure 1b, so that each grayscale color may represent the three different phases of the alloy. Next, by utilizing ImageJ software, each of the two segmented parts referring to the precipitates with a circularity between 0.20 and 1 was selected, and the particle areas were collected (Figure 1c,d). From the ImageJ analysis, results were obtained for both the surface area fraction of each phase and the distribution of their sizes. The as-measured surface area fractions of the precipitate phases (\(f_{Al_{Y}Cu_{2}Fe}\) and \(f_{Mg_{2}Si}\), respectively) are presented in Table 1 and compared with the volume fractions from the literature, to determine whether the image satisfactorily reflects the microstructure statistics. Relatively small differences are observed, probably due to the assumption that the surface fractions also correspond to the actual volume fractions of the bulk material. A significant error of 59% appears for the volume fraction of the Mg\({}_{2}\)Si phase, due to the small number of available particles to be measured, compared to the Al\({}_{Y}\)Cu\({}_{2}\)Fe phase. This error is non-negligible considering the phase fraction per se; however, it is negligible considering the material in its entirety, as the volume fraction of this phase is about 0.3%. For the purposes of this work, this difference is not expected to seriously affect the results. The analysis also provided the areas (\(A\)) of all particles, in units of \(\upmu\)m\({}^{2}\), which were converted into diameter values (\(d_{A}=\sqrt{4A/\pi}\)) assuming that the particles were spherical in shape. Next, the particles' diameter distribution was fitted with a lognormal distribution,