*3.1. Dimensional Analysis*

The mechanical behavior of the RVE can written in form of dependencies for the elastic and plastic macroscopic properties

$$f\_e(E, \nu) = f\_e\left(\frac{r\_{mid}}{r\_{end}}, \frac{r\_{end}}{l}, \ E\_{\text{s}}, \ \nu\_{\text{s}}, A, \zeta\right) \tag{7}$$

and

$$\left(\sigma\_{y\prime}E\_{T\prime}\upsilon\_p\right) = f\_p\left(\frac{r\_{\rm mid}}{r\_{\rm end}}, \frac{r\_{\rm end}}{l}, \ E\_{\rm s\prime}\ \upsilon\_{\rm s\prime}\sigma\_{y\prime s\prime}, E\_{T\prime\prime}A\_{\prime}\zeta\right),\tag{8}$$

respectively. Assuming that the ligament shape is sufficiently represented by the initial solid fraction, Equations (7) and (8) simplify to

$$(E, \nu) = \operatorname{g}\_{\varepsilon}(\varphi\_0, E\_{\rm s}, \upsilon\_{\rm s}, A\_{\prime}\zeta\_{\rm s}),\tag{9}$$

$$g\left(\sigma\_{\mathcal{Y}}, E\_{\mathcal{T}}, \nu\_{\mathcal{P}}\right) = \mathcal{g}\_{\mathcal{P}}\left(\mathcal{q}\_{0\prime}, E\_{\text{s}\prime}, \nu\_{\text{s}\prime}\sigma\_{\mathcal{Y},\text{s}\prime}, E\_{\text{T},\text{s}\prime}A\_{\text{t}}\zeta\right). \tag{10}$$

First, we used a priori knowledge in form of the Gibson–Ashby scaling law *E*/*E<sup>s</sup>* = *C<sup>E</sup> ϕ* 2 [35]. The leading constant *C<sup>E</sup>* depends on the unit cell geometry, which in our case was defined by the diamond structure and the chosen ligament shape. To simplify Equation (9) with respect to the Young's modulus, we can assume that the Poisson's ratio of the ligaments has no effect on the macroscopic deformation of the RVE, which results mainly from bending of the ligaments [23]. Combining both aspects and include Equation (3) for computing the solid fraction, we can reduce Equation (9) to a dependence of only two microstructural descriptors,

$$\frac{E}{E\_\text{s}\varrho^2} = \mathcal{g}\_E^\*(A, \zeta)\_\prime \tag{11}$$

which can be evaluated easily by visualization of the data in a 3D plot. If such a plot confirms Equation (11), the varying ligament shape is sufficiently represented in the solid fraction *ϕ*. Furthermore, *g* ∗ *E* represents a generalized Gibson–Ashby law that considers the dependence from the degree of randomization and cuts of the 3D network, which is not captured simply by the solid fraction. It also extends the master curve proposed in [28], which was produced using perfectly ordered RVEs, a single solid fraction, and constant material behavior.

Along the same line of thinking, it follows for the simplification of Equation (9) with respect to Poisson's ratio that a dimensionless macroscopic property can only depend on dimensionless microscopic quantities, i.e., the Young's modulus *E<sup>s</sup>* plays no role. In the same way as before, we can remove a dependence of *ν<sup>s</sup>* . The macroscopic Poission's ratio can be understood as the result of the translation of the vertical compression deformation into a lateral expansion by the architecture of the deforming 3D network, defined by *A* and *ζ*. This argument is in line with Gibson and Ashby, who stated that the Poisson's ratio is expected to be independent of the relative density [46]. Thus, we get

$$\mathbf{v} = \mathbf{g}\_{\nu}^{\*}(A, \mathbb{Q}). \tag{12}$$

Concerning the increased number of independent parameters in Equation (10), dimensionality reduction would support both their understanding and modeling of their relationships responsible for the plastic response. Before going into the analysis of the data, it is useful to rewrite this equation in dimensionless form. Again, we can assume that *E<sup>s</sup>* and *ν<sup>s</sup>* have no effect. For plasticity, this can only be assumed as long as *σy*,*<sup>s</sup>* ≪ *E<sup>s</sup>* and *ET*,*<sup>s</sup>* ≪ *E<sup>s</sup>* . Otherwise, we would combine comparable contributions of elastic and plastic deformation in the macroscopic response of the RVE, which requires the consideration of two dimensionless parameters for describing the elastic plastic behavior, namely *σy*,*s*/*E<sup>s</sup>* and *ET*,*s*/*E<sup>s</sup>* . Using the Buckingham *π* theorem [39], we can eliminate one more argument without loss of generality. One way is to normalize the macroscopic properties on the left side by their respective solid properties in the form

$$\left(\frac{\sigma\_y}{\sigma\_{y,s}}, \frac{E\_T}{E\_{T,s}}, \nu\_p\right) = \mathfrak{F}\_p\left(\varphi, \frac{E\_{T,s}}{\sigma\_{y,s}}, A, \zeta\right). \tag{13}$$

Again, we can incorporate the Gibson–Ashby scaling law for the yield stress *σy*/*σy*,*<sup>s</sup>* = *Cσ<sup>y</sup> ϕ* 3/2 [35], which yields for the first output

$$\frac{\sigma\_y}{\sigma\_{ys}\,\phi^{3/2}} = \mathfrak{E}\_{\sigma\_y}^\* \left( \frac{E\_{T,s}}{\sigma\_{y,s}} \, ^\prime A \, \zeta \right) . \tag{14}$$

Concerning the second output of Equation (13), it is unknown which scaling is appropriate, because the work hardening rate is a slope in the stress–plastic strain diagram. Intuitively, one would follow Equation (14) in favor of an exponent of 3/2. We can answer this question together with PCA and keep the exponent *β* in the scaling flexible, such that

$$\frac{E\_T}{E\_{T,s}q^{\beta}} = \mathfrak{f}\_{E\_T}^\* \left( \frac{E\_{T,s}}{\sigma\_{y,s}}, A, \zeta \right). \tag{15}$$

Alternatively to Equation (13), only dependent variables are used for normalization of the output

$$\left(\frac{E\_T}{\sigma\_y}, \nu\_p\right) = \widetilde{\g}\_p \left(\varphi, \frac{E\_{T,s}}{\sigma\_{ys}}, A, \zeta\right). \tag{16}$$

The choice between the two methods of normalization depends on the potential application. Equation (16) has the advantage that all quantities on the left side are experimentally accessible, such that it could be possible to invert *<sup>g</sup>*e*<sup>p</sup>* and to obtain some insight into material or structural properties of the nanoporous metal based on macroscopic compression testing.
