*2.1. Macroscopic Compression*

The FE software Abaqus was used for the numerical simulation of the RVE [29]. The model generation for macroscopic compression followed [23,24,27,28] and was organized hierarchically along the workflow presented in Figure 1. This workflow was programmed object oriented in Python with classes for the different hierarchy levels, allowing for scripting of the RVE generation and job submission within loops for the variation of input parameters. A postprocessing script handled the simulation analysis and database generation.

The model generation started at the ligament level, where the ligament axis is discretized in *Nelem* FE beam elements with circular cross-section and variable radius *r*. The ligament shape is defined along the axis according to [10,11] by *r* ∗ *sym* = *rmid*/*rend* and *rend*/*l*, where *rmid* and *rend* denote the ligament radius in the middle and at the ends, respectively, and *l* is the ligament length in a diamond unit cell. Together with the topology, the set of ligament geometry parameters *rmid*, *rend*, and *l* define the solid fraction *ϕ*<sup>0</sup> before randomization.

Odermatt et al. [27] developed nodal corrections for 16 ligament shapes that allow for a quantitative prediction of the elastic–plastic response of the RVE up to macroscopic strains of 20%. Details about the ligament geometries, initial solid fractions, and the nodal correction approach can be found in [27]. The extension of the nodal corrected zones is visible in the second column of Figure 1, where nodal corrected elements in the diamond unit cell are displayed in orange. With the nodal correction set "on", their material parameters were modified such that the deformation behavior of the unit cell corresponded to that of an FE solid model of the same ligament shape. Preliminary simulations for decreasing number of elements using a unit cell with periodic boundary conditions confirmed that the nodal correction by [27] performed well in the range from 20 down to 6 FE elements per ligament for both geometries listed in Table 1. Therefore,

6 elements per ligament were chosen in this work for which the relative error in macroscopic stiffness and strength was within 15% error relative to the results of the FE solid model. *Materials* **2021**, *14*, x FOR PEER REVIEW 4 of 24

**Figure 1.** Workflow for generating a representative volume element (RVE) in four steps: (i) ligament, (ii) unit cell, (iii) RAW model after cutting and cleaning, and (iv) computation of the RVE. Varied parameters are highlighted in red color. **Figure 1.** Workflow for generating a representative volume element (RVE) in four steps: (i) ligament, (ii) unit cell, (iii) RAW model after cutting and cleaning, and (iv) computation of the RVE. Varied parameters are highlighted in red color.

tized in ܰ FE beam elements with circular cross-section and variable radius ݎ. The ligament shape is defined along the axis according to [10,11] by ݎ௦௬ <sup>∗</sup> = ݎௗ/ݎௗ and **Table 1.** Ligament shapes used for the generation of two data sets in the low and a high solid fraction regime, respectively.

The model generation started at the ligament level, where the ligament axis is discre-


for a quantitative prediction of the elastic–plastic response of the RVE up to macroscopic strains of 20%. Details about the ligament geometries, initial solid fractions, and the nodal correction approach can be found in [27]. The extension of the nodal corrected zones is visible in the second column of Figure 1, where nodal corrected elements in the diamond unit cell are displayed in orange. With the nodal correction set "on", their material parameters were modified such that the deformation behavior of the unit cell corresponded to that of an FE solid model of the same ligament shape. Preliminary simulations for decreasing number of elements using a unit cell with periodic boundary conditions confirmed For generating an RVE of size *N*, the unit cell is copied *N* + 2 times (origin at [−1, −1, −1]) in each coordinate direction, followed by the randomization of the structure. The degree of randomization is defined by the parameter *A*, which corresponds to a random displacement in space applied to the connecting nodes by an amplitude *A*, which is given as a fraction of the unit cell size *a* [23,24]. Alternatively, one can also choose to displace an FE node in the mid-section of the ligaments by this magnitude normal to the ligament axis [27]. The randomization can be calibrated via the elastic Poisson's ratio and is typically *A* = 0.23 [24].

that the nodal correction by [27] performed well in the range from 20 down to 6 FE elements per ligament for both geometries listed in Table 1. Therefore, 6 elements per ligament were chosen in this work for which the relative error in macroscopic stiffness and strength was within 15% error relative to the results of the FE solid model. **Table 1.** Ligament shapes used for the generation of two data sets in the low and a high solid fraction regime, respectively.  **Shape** ∗ / 0.1232 0.289 0.5 ଶଵܩ Because the coordination of the diamond structure of *z* = 4 is too high in comparison to experimental observations [10,30,31], the connectivity can be reduced by random cutting of a fraction *ζ* of the ligaments [28]. For diamond, the percolation threshold is reached for sufficiently large RVEs at a cut fraction of *ζ* → 0.5, where the average coordination number approaches *z* → 2. For models of smaller size the percolation threshold is reached at lower values and is sensitive to the random realization. In combination with randomly cut ligaments, the randomization *A* can be reduced to values close to 0 to reach the elastic Poisson's ratio measured in experiments [28]. Therefore, we chose these two parameters independent of each other and within comparably large ranges of 0 ≤ *A* ≤ 0.3 and 0 ≤ *ζ* ≤ 0.3.

0.3574 0.346 1.0 ଷଷܩ For generating an RVE of size ܰ, the unit cell is copied ܰ + 2 times (origin at [−1, −1, −1]) in each coordinate direction, followed by the randomization of the structure. The degree of randomization is defined by the parameter ܣ, which corresponds to a random displacement in space applied to the connecting nodes by an amplitude ܣ, which is The resulting RAW model of size 10 × 10 × 10 unit cells is randomly distorted and can contain free floating ligaments due to random cuts. A cleaned RVE is generated by cutting the RAW model to a cubic volume of size *N* = 8 (origin at [0, 0, 0]) by removing all elements outside of this volume. Free floating ligaments are removed by two subsequent cleaning cycles that eliminate dangling ligaments and then re-attach element by element those

displace an FE node in the mid-section of the ligaments by this magnitude normal to the

ligaments that are connected to the residual core of the ligament network. For more details, the reader is referred to the supplementary material that is provided in [28]. The result of the preprocessing is an RVE of dimensions 8 × 8 × 8 unit cells with plane boundaries, consisting of 512 diamond unit cells with a total of 8192 ligaments and 49,152 FE-elements (*A* = 0, *ζ* = 0). Symmetry boundary conditions are applied to FE-nodes in the planes *x* = 0, *y* = 0, and *z* = 0, while macroscopic compression is applied at the top face at the position *z* = *N*.

For simplicity, the model was generated such that the unit cell size corresponds to a unit size of 1 mm. Realistic microstructural dimensions of the ligament and the pore size can be achieved by self-similar scaling of the model to a desired characteristic size, e.g., a ligament diameter of 20–150 nm [5]. Because the material law does not account for size effects, the resulting macroscopic behavior is not affected by such a scaling. However, when the effect of the surface energy is included, the ligament size is important; then also the applied electrode potential must be defined [32]. These two parameters allow for switching of the strength and the plastic Poisson's ratio during macroscopic deformation of the material.

*A* and *ζ* are dimensionless structural parameters describing the random distortion of the connecting nodes as fraction of the unit cell size and the fraction of randomly cut ligaments, respectively. Both parameters modify the solid fraction relative to the initial solid fraction *ϕ*0. According to Roschning et al. [24], we should account for the distortion of the ligament axis by *A* by an increase in solid fraction by using

$$\frac{\varphi\_A}{\varphi\_0} = 1 + 0.15A + 2.91A^2 \,\text{.}\tag{1}$$

whereas the random cutting *ζ* removes a fraction of ligaments and, therefore, mass from the model [28]

$$\frac{\varphi\_{\zeta}}{\varphi\_{0}} = 1 - \zeta. \tag{2}$$

If the RVE is large enough, Equations (1) and (2) can be combined as

$$\frac{\mathcal{Q}}{\mathcal{Q}\_0} = (1 - \zeta) \left( 1 + 0.15A + 2.91A^2 \right). \tag{3}$$

It should be noted that the random cutting *ζ* can lead to a mechanical deactivation of whole regions that are still part of the model. Therefore, *ϕ<sup>ζ</sup>* should not be interpreted as effective solid fraction *ϕ*eff that represents the load bearing mass [33].

In view of the number of parameters that may play a role, we limited the structural variation to the randomization *A* and the cut fraction *ζ* and kept all other structural parameters within each data set constant (ligament aspect ratio *rend*/*l*, ligament shape *r* ∗ *sym*). Two data sets for ligament shapes *G*<sup>21</sup> and *G*<sup>33</sup> (see Table 1) were created, covering a large range from very low ( *ϕ*<sup>0</sup> ∼ 12%) to very high ( *ϕ*<sup>0</sup> ∼ 36%) solid fractions. Because the porosity was computed from 1 − *ϕ*0, the porosity ranged from ∼ 64% to ∼ 88%.

We used nanoporous gold (np-Au) as model material, because in terms of microstructure and mechanical properties this is the best investigated material of a variety of nanoporous metals reported in the literature. The chosen material behavior is plasticity with linear isotropic hardening [23]. This adds two material parameters denoted as yield stress *σy*,*<sup>s</sup>* and work hardening rate *ET*,*<sup>s</sup>* ; the subscript *s* denotes that both parameters are a property of the solid phase, which makes up the 3D network. Both depend on the ligament diameter, which can be manipulated during the sample preparation of the material via the Au/Ag ratio, dealloying conditions, and heat treatment, as demonstrated in [5,32]. The elastic constants for gold are known and were kept constant for all simulations: Young's modulus *E<sup>s</sup>* = 80 GPa, Poisson's ratio *ν<sup>s</sup>* = 0.42. An example of a deformed RVE (*A* = 18%, *ζ* = 26%) is shown in Figure 2a. The stress is evenly distributed over the length of the RVE, which indicates that the overall deformation is homogeneous despite the local structural variations due to the randomization of the ligament network.

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porosity was computed from 1 − ߮

Two data sets for ligament shapes ܩଶଵ and ܩଷଷ (see Table 1) were created, covering a large range from very low (߮~12%) to very high (߮~36%) solid fractions. Because the

the local structural variations due to the randomization of the ligament network.

We used nanoporous gold (np-Au) as model material, because in terms of microstructure and mechanical properties this is the best investigated material of a variety of nanoporous metals reported in the literature. The chosen material behavior is plasticity with linear isotropic hardening [23]. This adds two material parameters denoted as yield stress ߪ௬,௦ and work hardening rate ܧ்,௦; the subscript ݏ denotes that both parameters are a property of the solid phase, which makes up the 3D network. Both depend on the ligament diameter, which can be manipulated during the sample preparation of the material via the Au/Ag ratio, dealloying conditions, and heat treatment, as demonstrated in [5,32]. The elastic constants for gold are known and were kept constant for all simulations: Young's modulus ܧ<sup>௦</sup> = 80 GPa, Poisson's ratio ߥ<sup>௦</sup> = 0.42. An example of a deformed RVE (ܣ = 18%, ߞ = 26% (is shown in Figure 2a. The stress is evenly distributed over the length of the RVE, which indicates that the overall deformation is homogeneous despite

, the porosity ranged from ~64% to ~88%.

**Figure 2.** (**a**) RVE consisting of 8 × 8 × 8 unit cells with ܣ = 18% and ߞ = 26% after compression with 20% strain in the negative z-direction. The purple dashed curves in the magnified image shown on the right side indicate the axis of some cut ligaments. Due to the missing load transmission, the remaining dangling parts show zero stress (blue color). (**b**) Simulation output in the form of stress–strain and stress–plastic strain curves from which the macroscopic yield stress and work hardening rate are determined. **Figure 2.** (**a**) RVE consisting of 8 × 8 × 8 unit cells with *A* = 18% and *ζ* = 26% after compression with 20% strain in the negative z-direction. The purple dashed curves in the magnified image shown on the right side indicate the axis of some cut ligaments. Due to the missing load transmission, the remaining dangling parts show zero stress (blue color). (**b**) Simulation output in the form of stress–strain and stress–plastic strain curves from which the macroscopic yield stress and work hardening rate are determined.

This model makes up a set of variable inputs consisting of 5 independent parameters: This model makes up a set of variable inputs consisting of 5 independent parameters:

$$\begin{array}{c} X = (\varphi\_0, A, \zeta, \sigma\_{y,s}, E\_{T,s}) \\ = (\varphi\_0, A, \zeta, \sigma\_{y,s}, E\_{T,s}) \end{array} \tag{4}$$

lation within the ranges 0 ≤ ܣ ≥ 0.3 , 0 ≥ ߞ ≥ 0.3 , 20 MPa ≤ ߪ௬,௦ ≤ 1000 MPa , and 1 GPa ≤ ܧ்,௦ ≤ 10 GPa, which cover the known range of experimental data. The random distribution of the parameters is uniform for ܣ, ߞ, log ߪ௬,௦, and log ܧ்,௦. Each parameter set is stored together with the job number, which uniquely connects microscopic to macroscopic compression as well as nanoindentation properties in the data processing in Sections 3 and 4. The random choice of the parameter sets has the advantage that the parameter space is evenly filled while no parameter is computed more than once. This avoids patterns that might be unwantedly recognized by the machine learning algorithms. Furthermore, the parameter space can continued to be filled if it turns out that the number of patterns is not sufficient for the analysis. This is particularly useful when the simulations For each initial solid fraction, the remaining parameters are randomly set for each simulation within the ranges 0 ≤ *A* ≤ 0.3, 0 ≤ *ζ* ≤ 0.3, 20 MPa ≤ *σy*,*<sup>s</sup>* ≤ 1000 MPa, and 1 GPa ≤ *ET*,*<sup>s</sup>* ≤ 10 GPa, which cover the known range of experimental data. The random distribution of the parameters is uniform for *A*, *ζ*, log *σy*,*<sup>s</sup>* , and log *ET*,*<sup>s</sup>* . Each parameter set is stored together with the job number, which uniquely connects microscopic to macroscopic compression as well as nanoindentation properties in the data processing in Sections 3 and 4. The random choice of the parameter sets has the advantage that the parameter space is evenly filled while no parameter is computed more than once. This avoids patterns that might be unwantedly recognized by the machine learning algorithms. Furthermore, the parameter space can continued to be filled if it turns out that the number of patterns is not sufficient for the analysis. This is particularly useful when the simulations are computationally expensive. For an example where this strategy is applied in combination with artificial neural networks for solving a complex inverse problem in nanoindentation, the reader is referred to [34].

The resulting compression behavior of each pattern is represented by 5 dependent properties:

$$\mathcal{Y} = \left(\mathbb{E}, \mathbb{v}, \mathbb{v}\_p, \sigma\_{\mathcal{Y}}, \mathbb{E}\_{\mathcal{T}}\right), \tag{5}$$

where *E*, *ν*, *σy*, and *E<sup>T</sup>* denote the macroscopic Young's modulus, elastic Poisson's ratio, yield stress, and work hardening rate, respectively. The computation of the plastic Poisson's ratio *ν<sup>p</sup>* follows [32]

$$\nu\_p = -\frac{\delta \varepsilon\_\perp}{\delta \varepsilon\_\parallel},\tag{6}$$

where *δε*<sup>⊥</sup> and *δε*<sup>k</sup> are increments of true strain normal and parallel to the loading direction, respectively. Because *ν<sup>p</sup>* changes during plastic compression, it is measured at 10% plastic compression strain. As demonstrated in Figure 2b, the predicted stress–plastic strain data is linearly fitted for plastic strains > 1% for obtaining the macroscopic yield stress *σ<sup>y</sup>* and work hardening rate *ET*.

rection, respectively. Because ߥ changes during plastic compression, it is measured at 10% plastic compression strain. As demonstrated in Figure 2b, the predicted stress–plastic strain data is linearly fitted for plastic strains > 1% for obtaining the macroscopic yield

are increments of true strain normal and parallel to the loading di-

where ܧ, ߥ, ߪ௬, and ܧ் denote the macroscopic Young's modulus, elastic Poisson's ratio, yield stress, and work hardening rate, respectively. The computation of the plastic Pois-

are computationally expensive. For an example where this strategy is applied in combination with artificial neural networks for solving a complex inverse problem in

The resulting compression behavior of each pattern is represented by 5 dependent

(5) ,(்ܧ ,௬ߪ ,ߥ ,ߥ ,ܧ) = ܻ

#### *2.2. Nanoindentation 2.2. Nanoindentation*

<sup>∥</sup>ߝߜ and ୄߝߜ where

son's ratio ߥ follows [32]

stress ߪ௬ and work hardening rate ܧ்.

properties:

For the simulation of nanoindentation, the model described in Section 2.1 is extended by adding a conical indenter with an angle of 140.6◦ . For this angle, the volume-to-depth ratio of the conical indenter corresponds to that of a Berkovich tip. Details on the simulation of nanoindentation for solids and thin films can be found, e.g., in [34]. Due to the numerous ligaments that get in contact during the indentation process, an explicit dynamic analysis was required for achieving convergence. A robust load signal was produced by attaching dashpots at the free boundaries (see Figure 3) to damp elastic waves induced by the multiple contact events during the dynamic indentation process. For the simulation of nanoindentation, the model described in Section 2.1 is extended by adding a conical indenter with an angle of 140.6°. For this angle, the volume-to-depth ratio of the conical indenter corresponds to that of a Berkovich tip. Details on the simulation of nanoindentation for solids and thin films can be found, e.g., in [34]. Due to the numerous ligaments that get in contact during the indentation process, an explicit dynamic analysis was required for achieving convergence. A robust load signal was produced by attaching dashpots at the free boundaries (see Figure 3) to damp elastic waves induced by the multiple contact events during the dynamic indentation process.

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nanoindentation, the reader is referred to [34].

**Figure 3.** Nanoindentation model with a conical indenter displaced by 2 unit cells at a speed of 20 mm/s. Dashpots are attached at FE nodes located at free boundaries to stabilize oscillations in the dynamic simulation. Images show the deformation of the RVE at the end of the loading phase, where the color corresponds to (**a**) von Mises stress, and (**b**) displacement magnitude. **Figure 3.** Nanoindentation model with a conical indenter displaced by 2 unit cells at a speed of 20 mm/s. Dashpots are attached at FE nodes located at free boundaries to stabilize oscillations in the dynamic simulation. Images show the deformation of the RVE at the end of the loading phase, where the color corresponds to (**a**) von Mises stress, and (**b**) displacement magnitude.

It can be seen from Figure 3 that the contact of the indenter, modeled as a rigid body, is established with the axis of the beam elements. Therefore, the upper half of the ligaments in contact peek out on the upper side of the indenter surface. Contact among the It can be seen from Figure 3 that the contact of the indenter, modeled as a rigid body, is established with the axis of the beam elements. Therefore, the upper half of the ligaments in contact peek out on the upper side of the indenter surface. Contact among the ligaments is not considered. In principle, this is possible in Abaqus Explicit, but the contact is limited to a pair of a rendered element surface and the axis of a second element. Preliminary studies with this indentation model revealed that such events happen rarely and at a very late stage of the indentation and, therefore, can be neglected in the total force on the indenter. It should be noted that this situation can change once we work with real microstructures and with a contact formulation that accounts for the surface of both contacting ligaments.

The calibration of the indenter velocity and the dashpot parameter is presented in Figure 4 for ligament geometry *G*<sup>21</sup> (*ϕ*<sup>0</sup> = 0.12) with *σy*,*<sup>s</sup>* = 200 MPa, *ET*,*<sup>s</sup>* = 6 GPa, and a randomization *A* = 0.23 [32]. For simplicity, effects of the surface energy are not included, and the cut fraction is set to *ζ* = 0. For uniaxial compression, the predicted stress–strain curve yields the following macroscopic mechanical properties: *E* = 1.9 GPa, *ν* = 0.178, *σ<sup>y</sup>* = 17.8 MPa, and *E<sup>T</sup>* = 108 MPa.

0.178, ߪ<sup>௬</sup> = 17.8 MPa, and ܧ் = 108 MPa.

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tacting ligaments.

Preliminary studies with this indentation model revealed that such events happen rarely and at a very late stage of the indentation and, therefore, can be neglected in the total force on the indenter. It should be noted that this situation can change once we work with real microstructures and with a contact formulation that accounts for the surface of both con-

The calibration of the indenter velocity and the dashpot parameter is presented in Figure 4 for ligament geometry ܩଶଵ (߮ = 0.12) with ߪ௬,௦ = 200 MPa, ܧ்,௦ = 6 GPa, and a randomization ܣ = 0.23] 32[. For simplicity, effects of the surface energy are not included, and the cut fraction is set to ߞ = 0. For uniaxial compression, the predicted stress– strain curve yields the following macroscopic mechanical properties: ܧ = 1.9 GPa, ߥ=

**Figure 4.** Parametric study for adjustment of (**a**) indenter velocity with smooth step function for a dashpot parameter of 10ିଷ Ns/mm and (**b**) dashpot parameter at fixed indenter velocity of 20 mm/s. **Figure 4.** Parametric study for adjustment of (**a**) indenter velocity with smooth step function for a dashpot parameter of 10−<sup>3</sup> Ns/mm and (**b**) dashpot parameter at fixed indenter velocity of 20 mm/s.

For a conical or pyramidal indenter, the load ܲ always increases with the square of the indentation depth ℎ [34], as soon as the indented material acts like a continuum. Therefore, a plot of ܲ/ℎ<sup>ଶ</sup> vs. ℎ should tend towards a constant value. This is reached for ℎ > 1 mm (~1 unit cell), indicating that at this depth value, a sufficient number of ligaments are in contact, and the solution homogenizes over enough microstructural elements. It can be seen from Figure 4 that the results show large scatter for high loading rates and for a low dashpot constant. The load–depth curves converged into a sufficiently steady state solution for an indenter velocity of 20 mm/s, combined with a smooth step function and a dashpot parameter of 10ିଷ Ns/mm. These settings were used for all following simulations, including those shown in Figure 3. The displacement magnitude in Figure 3b shows negligible deformation at the free boundaries, which suggests that the RVE is sufficiently large. With this model and parameter setting, the total CPU time per simulation is ~60 CPUh. Generating a data set with 100 simulations for a selected ligament shape requires ~1 week in real time by parallel computing on 16 CPUs. For a conical or pyramidal indenter, the load *P* always increases with the square of the indentation depth *h* [34], as soon as the indented material acts like a continuum. Therefore, a plot of *P*/*h* <sup>2</sup> vs. *h* should tend towards a constant value. This is reached for *h* > 1 mm (~1 unit cell), indicating that at this depth value, a sufficient number of ligaments are in contact, and the solution homogenizes over enough microstructural elements. It can be seen from Figure 4 that the results show large scatter for high loading rates and for a low dashpot constant. The load–depth curves converged into a sufficiently steady state solution for an indenter velocity of 20 mm/s, combined with a smooth step function and a dashpot parameter of 10−<sup>3</sup> Ns/mm. These settings were used for all following simulations, including those shown in Figure 3. The displacement magnitude in Figure 3b shows negligible deformation at the free boundaries, which suggests that the RVE is sufficiently large. With this model and parameter setting, the total CPU time per simulation is ~60 CPUh. Generating a data set with 100 simulations for a selected ligament shape requires ~1 week in real time by parallel computing on 16 CPUs.

For ℎ > 1 mm, ܲ/ℎ ଶ values are averaged to compute the leading constant ܥ describing the loading curve ܲ = ܥℎ<sup>ଶ</sup> . A robust hardness value can be computed by ܪ= ܣ/௧ܲ , where ܲ<sup>௧</sup> = ܥℎ<sup>௧</sup> ଶ is the load at maximum indentation depth, ܣ = ߨܽ ଶ is the contact area, and ܽ is the contact radius at this depth. For the example shown in Figures 3 and 4, we obtain ܽ = 5.3 mm and a hardness of ܪ = 2.79 MPa. Thus, the hardness value is significantly lower than the macroscopic yield stress, which is ߪ<sup>௬</sup> = 17.8 MPa, whereas the common assumption for foams is that ܪ = ߪ௬ [35–38]. This motivates a detailed investigation of the dependence of ܪ/ߪ௬ with respect to possible effects caused by the network geometry (randomness, connectivity) and elastic–plastic material properties For *h* > 1 mm, *Pi*/*h* 2 *i* values are averaged to compute the leading constant *C* describing the loading curve *P* = *Ch*<sup>2</sup> . A robust hardness value can be computed by *H* = *Pt*/*Ac*, where *P<sup>t</sup>* = *Ch*<sup>2</sup> *t* is the load at maximum indentation depth, *A<sup>c</sup>* = *πa* 2 *c* is the contact area, and *a<sup>c</sup>* is the contact radius at this depth. For the example shown in Figures 3 and 4, we obtain *a<sup>c</sup>* = 5.3 mm and a hardness of *H* = 2.79 MPa. Thus, the hardness value is significantly lower than the macroscopic yield stress, which is *σ<sup>y</sup>* = 17.8 MPa, whereas the common assumption for foams is that *H* = *σ<sup>y</sup>* [35–38]. This motivates a detailed investigation of the dependence of *H*/*σ<sup>y</sup>* with respect to possible effects caused by the network geometry (randomness, connectivity) and elastic–plastic material properties of the ligaments, which is presented in Section 4. The data generation for the nanoindentation simulations uses the same parameter sets as those used for the simulation of macroscopic compression in Section 3, i.e., for each solid fraction, we performed 100 simulations for macroscopic compression and another 100 simulations for nanoindentation. In a few cases, the simulations of the macroscopic compression did not converge. These parameter sets were removed from both databases to avoid confusion in the analysis that combines macroscopic compression with nanoindentation data.

#### **3. Macroscopic Compression**

In the following sections, we reduced the dimensionality of the problem to extract relationships from our data that can be visualized, discussed, and, in the best case, modeled with simple mathematical functions. Our strategy consisted of three steps: (i) dimensional analysis, (ii) principal component analysis, and (iii) visualization and modeling of the relationship with a minimum number of inputs. The dimensional analysis [39] makes use of the physics background and the Buckingham *π* theorem to reduce the problem without loss of accuracy. This turned out to be a useful approach that should always be placed as a first step of feature engineering, because it ensures that the basic physics is incorporated in the input and output data, while at the same time the machine learning algorithms are relieved and their generalization capability is substantially increased [34,40,41].

Principal component analysis (PCA) was applied in conjunction with a multi-layer perceptron (MLP) algorithm using the scikit-learn package [42]. The MLP, also known as artificial neural networks, allows for the analysis of patterns consisting of multiple inputs and outputs with respect to underlying nonlinear dependencies. For details and applications, the reader is referred to [43–45]. After the dimensionality of the problem was reduced, comparably compact MLPs consisting of two hidden layers with 3 and 2 neurons were used for approximation and visualization of the data. This is possible, when the relationship of interest is sufficiently represented by the selected inputs.
