*3.2. Principal Component Analysis*

At first glance, principal component analysis (PCA) [47,48] appears to be meaningless for our case, because there are no linear dependencies among the inputs that could be easily eliminated. PCA of linearly independent inputs simply translates the original inputs into a smaller number of components by linear combination. In case that each original input carries important information, this leads to a loss of information and to an increase in the predicted error in a subsequent MLP regression. In contrast, a successful reduction to a fewer number of components without a substantial increase in the prediction uncertainty shows that there is a potential for the reduction of the dimensionality of the problem and, furthermore, it delivers a feeling for the number of inputs that can be removed. The advantage of PCA is that the data can be quickly analyzed, and it becomes clear which elements of a relationship are the promising candidates for a deeper analysis. the problem and, furthermore, it delivers a feeling for the number of inputs that can be removed. The advantage of PCA is that the data can be quickly analyzed, and it becomes clear which elements of a relationship are the promising candidates for a deeper analysis.

At first glance, principal component analysis (PCA) [47,48] appears to be meaningless for our case, because there are no linear dependencies among the inputs that could be easily eliminated. PCA of linearly independent inputs simply translates the original inputs into a smaller number of components by linear combination. In case that each original input carries important information, this leads to a loss of information and to an increase in the predicted error in a subsequent MLP regression. In contrast, a successful reduction to a fewer number of components without a substantial increase in the prediction uncertainty shows that there is a potential for the reduction of the dimensionality of

Because the equations for elasticity can be easily visualized, Equations (11) and (12) are omitted here. The mapping of PCA with an MLP regression of Equation (14) is shown in Figure 5a. For these regressions, consistently 10 neurons in a single hidden layer were used. The results for 3 components corresponded to the dimensionality of the raw input data and reproduced the accuracy of the MLP prediction without PCA, validating that no information was lost by the transformation. With reduction of the components, computed mean values of the absolute prediction error were 0.121, 0.221, and 0.342 for 3, 2, and 1 components, respectively. As can be seen from the inserted plot (orange), the error doubled with each component that was reduced. The scatter plot in Figure 5a suggests to visualize the data in form of a 3D plot, where a parametrization with one of the three inputs is required, which is presented in Section 3.4. Because the equations for elasticity can be easily visualized, Equations (11) and (12) are omitted here. The mapping of PCA with an MLP regression of Equation (14) is shown in Figure 5a. For these regressions, consistently 10 neurons in a single hidden layer were used. The results for 3 components corresponded to the dimensionality of the raw input data and reproduced the accuracy of the MLP prediction without PCA, validating that no information was lost by the transformation. With reduction of the components, computed mean values of the absolute prediction error were 0.121, 0.221, and 0.342 for 3, 2, and 1 components, respectively. As can be seen from the inserted plot (orange), the error doubled with each component that was reduced. The scatter plot in Figure 5a suggests to visualize the data in form of a 3D plot, where a parametrization with one of the three inputs is required, which is presented in Section 3.4.

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*3.2. Principal Component Analysis* 

**Figure 5.** Result of PCA followed by MLP regression with a single hidden layer of 10 neurons for a decreasing number of components: (**a**) scaled yield stress, Equation (14); (**b**) scaled work hardening rate, Equation (15). **Figure 5.** Result of PCA followed by MLP regression with a single hidden layer of 10 neurons for a decreasing number of components: (**a**) scaled yield stress, Equation (14); (**b**) scaled work hardening rate, Equation (15).

A first investigation of Equation (15) with 3 components and an exponent ߚ = 3/2 similar to Equation (14) led to two main groups in the scatter plot (not shown), which could be combined in a narrow scatter band by changing the exponent to ߚ = 2) black open boxes in Figure 5b. Thus, the data suggested that the work hardening rate should be scaled in the same way as the Young's modulus. Using this exponent, we obtained mean values of the absolute error of 0.070, 0.177, and 0.317 for 3, 2, and 1 components, respec-A first investigation of Equation (15) with 3 components and an exponent *β* = 3/2 similar to Equation (14) led to two main groups in the scatter plot (not shown), which could be combined in a narrow scatter band by changing the exponent to *β* = 2 (black open boxes in Figure 5b. Thus, the data suggested that the work hardening rate should be scaled in the same way as the Young's modulus. Using this exponent, we obtained mean values of the absolute error of 0.070, 0.177, and 0.317 for 3, 2, and 1 components, respectively.

tively. Next, it was of interest to quantify the highest possible reduction of arguments of ݃ in Equation (16). The more significant the outcome is, the better are the chances for deriving a relationship that can potentially also be solved with respect to one of the arguments. This would be a valuable aid in accessing local structural or mechanical properties from Next, it was of interest to quantify the highest possible reduction of arguments of *<sup>g</sup>*e*<sup>p</sup>* in Equation (16). The more significant the outcome is, the better are the chances for deriving a relationship that can potentially also be solved with respect to one of the arguments. This would be a valuable aid in accessing local structural or mechanical properties from comparably simple macroscopic tests. Figure 6 presents the outcome of a PCA of *<sup>g</sup>*e*<sup>p</sup>* followed by MLP. The PCA was first applied simultaneously to both ligament shapes *G*<sup>21</sup> and *G*33. Each dimensionless parameter on the left side of Equation (16) is individually evaluated.

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**Figure 6.** Quality of MLP regression after PCA with decreasing number of components. (**a**) ܧ்/ߪ௬, Equation (16) with ≤ 4 components and (**b**) ܧ்/)ߪ௬߮ ଵ/ଶ) scaled according to Equation (18) with ≤ 3 components. **Figure 6.** Quality of MLP regression after PCA with decreasing number of components. (**a**) *ET*/*σy*, Equation (16) with ≤ 4 components and (**b**) *ET*/ *σyϕ* 1/2 scaled according to Equation (18) with ≤ 3 components.

The importance of the correct scaling was demonstrated for the analysis of Equation (16). The results for an output in the form of the ratio ܧ்/ߪ௬, shown in Figure 6a, revealed that the argument of Equation (16) could not be easily reduced without adding considerable error. The reduction from 3 to 2 components led to a separation into two branches (reddish colors) that resulted from the two solid fractions. Further reduction did not change the result much. The importance of the correct scaling was demonstrated for the analysis of Equation (16). The results for an output in the form of the ratio *ET*/*σy*, shown in Figure 6a, revealed that the argument of Equation (16) could not be easily reduced without adding considerable error. The reduction from 3 to 2 components led to a separation into two branches (reddish colors) that resulted from the two solid fractions. Further reduction did not change the result much.

comparably simple macroscopic tests. Figure 6 presents the outcome of a PCA of ݃ followed by MLP. The PCA was first applied simultaneously to both ligament shapes ܩଶଵ and ܩଷଷ. Each dimensionless parameter on the left side of Equation (16) is individually

However, dividing Equation (15) by Equation (14) for ߚ = 2 yields However, dividing Equation (15) by Equation (14) for *β* = 2 yields

$$\frac{E\_T \rho^{\frac{E}{2} \dagger} \mathcal{I}^{\rho^{3/2} \sigma\_{y,\mathbf{e}}}\_{\sigma\_y \rho^2} \frac{\sigma\_{y,\mathbf{e}}}{E\_{T,s}} = \widetilde{\mathcal{S}}\_p^\* \left( \begin{matrix} \widetilde{\mathbf{E}}\_{\widetilde{\mathbf{e}} \mathbf{a} \mathbf{s}}^{T,s} \\ \sigma\_{y,s} \end{matrix} , A \begin{matrix} \zeta \\ \sigma\_{y,s} \end{matrix} \right) . \tag{17}$$

which can be rewritten as which can be rewritten as

evaluated.

$$\text{Wurke ɛn ɔe ɛwweritɛn as }\\\frac{E\_{\overline{E}\_{\overline{E}},\mu^{1/2}}}{\sigma\_{y}\rho^{1/2}} \stackrel{E\_T}{=} \mathcal{S}\_p^\mu \left(\frac{E\_{\overline{E},s}}{\sigma\_{y,s}}, A, \zeta\right). \tag{18}$$

pendent of the number of components, suggesting that the argument of Equation (18) can be reduced to a single component. Because the output of Equation (18) can be expected to mainly depend on the corresponding ratio ܧ்,௦/ߪ௬,௦ , the visualization can right away move to a 2D scatter plot of ܧ்/)ߪ௬߮ ଵ/ଶ) versus this quantity. In contrast to the output ܧ்/ߪ௬, the plastic Poisson's ratio ߥ showed a large scatter For this type of scaling, shown in Figure 6b, PCA delivered almost a perfect match, independent of the number of components, suggesting that the argument of Equation (18) can be reduced to a single component. Because the output of Equation (18) can be expected to mainly depend on the corresponding ratio *ET*,*s*/*σy*,*<sup>s</sup>* , the visualization can right away move to a 2D scatter plot of *ET*/(*σyϕ* 1/2) versus this quantity.

that is almost invariant to the number of components (not shown). Therefore, no further reduction of the dimensionality is possible for this parameter. This is further discussed along with the visualization of the data in Section 3.4. In contrast to the output *ET*/*σy*, the plastic Poisson's ratio *ν<sup>p</sup>* showed a large scatter that is almost invariant to the number of components (not shown). Therefore, no further reduction of the dimensionality is possible for this parameter. This is further discussed along with the visualization of the data in Section 3.4.

#### *3.3. Macroscopic Elastic Properties 3.3. Macroscopic Elastic Properties*

The dependencies for the elastic properties according to Equations (11) and (12) are visualized in Figures 7a and 8. In these Figures, the randomly distributed simulation data are shown as spheres, and the predictions of the MLP regressions are shown as 3D contour plots. As can be seen from Figure 7, the scaling of Young's modulus removes most of the effect stemming from the solid fraction, such that ݃ா ∗ can be written as dependence of The dependencies for the elastic properties according to Equations (11) and (12) are visualized in Figures 7a and 8. In these Figures, the randomly distributed simulation data are shown as spheres, and the predictions of the MLP regressions are shown as 3D contour plots. As can be seen from Figure 7, the scaling of Young's modulus removes most of the effect stemming from the solid fraction, such that *g* ∗ *E* can be written as dependence of only two structural parameters *A* and *ζ*. The surfaces approximating the individual solid fractions are slightly shifted in the lower regions and intersect at *E*/(*Esϕ* 2 ) ∼ 1.5.

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fractions are slightly shifted in the lower regions and intersect at ܧ/)ܧ௦߮

only two structural parameters ܣ and ߞ. The surfaces approximating the individual solid

ଶ )~1.5.

**Figure 7.** Approximation of the simulation results for the scaled Young's modulus ܧ/)ܧ௦߮ ଶ ). Black and red spheres represent data from ligament shapes ܩଶଵ and ܩଷଷ, respectively, and MLP regressions are shown as contour plot as function of randomization ܣ and cut fraction ߞ:) **a**) 3D plot of Equation (11), (**b**) quality of approximation by Equation (19). **Figure 7.** Approximation of the simulation results for the scaled Young's modulus *E*/(*Esϕ* 2 ). Black and red spheres represent data from ligament shapes *G*<sup>21</sup> and *G*33, respectively, and MLP regressions are shown as contour plot as function of randomization *A* and cut fraction *ζ*: (**a**) 3D plot of Equation (11), (**b**) quality of approximation by Equation (19). *Materials* **2021**, *14*, x FOR PEER REVIEW 14 of 24

networks. In summary, as suggested by the PCA, the visualization in Figure 8 confirms that the dimensionality of this relationship cannot be further reduced. **Figure 8.** Visualization of the simulation results for the macroscopic Poisson's ratio. Black and red spheres represent data for ligament shapes ܩଶଵ and ܩଷଷ, respectively. MLP regressions of both data sets are shown as contour plots. **Figure 8.** Visualization of the simulation results for the macroscopic Poisson's ratio. Black and red spheres represent data for ligament shapes *G*<sup>21</sup> and *G*33, respectively. MLP regressions of both data sets are shown as contour plots.

*3.4. Macroscopic Plastic Properties*  With the outcome of the PCA in mind, Equation (14) is visualized in Figure 9a. Each pair of contour plots correspond to the two solid fractions, the effect of which is captured by the scaling with ߮ ଷ/ଶ. In addition to uncertainties and numerical errors, the remaining gap within each pair could be a result, e.g., of torsion that scales with ߮ and can have some 10% contribution to the deformation as soon as the ligaments are randomized [25]. The effect of log (ܧ்,௦/ߪ௬,௦) is remarkable in all regions of the plot and is around a factor of 2, but also the dependencies of ܣ and ߞ are significant. This explains why all three Overall, the contour plot in Figure 7a confirms the existing understanding about the effect of *A* and *ζ*, which both lower the macroscopic Young's modulus of the ligament network [28]. Additionally, with *ζ* approaching the percolation threshold, one observes a smooth transition into a horizontal tangent with the *x*-axis. Interestingly, after considering both parameters in the computation of the solid fraction, the remaining effect is almost identical, as can be seen from the horizontal isolines in Figure 7a. Combining them in the *x*-axis in Figure 7b reveals where the two data sets start to separate. The dashed line indicates that up to a value of *A* + *ζ* = 0.3 the data can be fitted by

$$\frac{E}{E\_s \rho^2} \approx 2.5 - 5(\mathbf{A} + \mathbf{\zeta}).\tag{19}$$

Young's modulus shown in Figure 7a. The macroscopic Poisson's ratio shown in Figure 8 behaves differently. Again, the effect of the randomization *A* is at least as strong as the effect of the cut fraction *ζ*. However, in agreement to the findings in [28], the cut fraction of *ζ* has no effect for *A* ∼ 0.25, while

For obtaining a first impression on the dependence of ܧ்/ߪ௬ according to Equation (16), the data is visualized in Figure 10. The dependence of log (ܧ்/ߪ௬) is clearly the strongest and nicely correlated with log (ܧ்,௦/ߪ௬,௦), whereas ܣ and ߞ have no or only a

(**a**) (**b**) **Figure 9.** Visualization using MLP regression (shown as contour plot) along with black and red spheres corresponding to the data from ligament shapes ܩଶଵ and ܩଷଷ, respectively. (**a**) Scaled yield stress after Equation (14). Each pair of contour plots represents the two solid fractions. The parametrization of log (ܧ்,௦/ߪ௬,௦) corresponds to the values 0.5 and 2.5. (**b**) Scaled work hardening rate after Equation (15) with an exponent ߚ = 2, where the inputs for the MLP regression are

reduced to ܣ and ߞ. The pair of planes represent the two solid fractions.

it is large for lower values of *A* and of opposite sign for *A* = 0.3. The Poisson's ratio is only slightly sensitive to *ϕ* in the regime of low values of *ζ*, i.e., for fully connected networks. In summary, as suggested by the PCA, the visualization in Figure 8 confirms that the dimensionality of this relationship cannot be further reduced. **Figure 8.** Visualization of the simulation results for the macroscopic Poisson's ratio. Black and red spheres represent data for ligament shapes ܩଶଵ and ܩଷଷ, respectively. MLP regressions of both data sets are shown as contour plots.

#### *3.4. Macroscopic Plastic Properties 3.4. Macroscopic Plastic Properties*

With the outcome of the PCA in mind, Equation (14) is visualized in Figure 9a. Each pair of contour plots correspond to the two solid fractions, the effect of which is captured by the scaling with *ϕ* 3/2. In addition to uncertainties and numerical errors, the remaining gap within each pair could be a result, e.g., of torsion that scales with *ϕ* and can have some 10% contribution to the deformation as soon as the ligaments are randomized [25]. The effect of log(*ET*,*s*/*σy*,*s*) is remarkable in all regions of the plot and is around a factor of 2, but also the dependencies of *A* and *ζ* are significant. This explains why all three parameters need to be kept for a good representation of Equation (14), as indicated by the PCA. The proper scaling of the work hardening rate with an exponent of *β* = 2 is confirmed with Figure 9b, which is in appearance and range of values very close to that of the Young's modulus shown in Figure 7a. With the outcome of the PCA in mind, Equation (14) is visualized in Figure 9a. Each pair of contour plots correspond to the two solid fractions, the effect of which is captured by the scaling with ߮ ଷ/ଶ. In addition to uncertainties and numerical errors, the remaining gap within each pair could be a result, e.g., of torsion that scales with ߮ and can have some 10% contribution to the deformation as soon as the ligaments are randomized [25]. The effect of log (ܧ்,௦/ߪ௬,௦) is remarkable in all regions of the plot and is around a factor of 2, but also the dependencies of ܣ and ߞ are significant. This explains why all three parameters need to be kept for a good representation of Equation (14), as indicated by the PCA. The proper scaling of the work hardening rate with an exponent of ߚ = 2 is confirmed with Figure 9b, which is in appearance and range of values very close to that of the Young's modulus shown in Figure 7a.

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**Figure 9.** Visualization using MLP regression (shown as contour plot) along with black and red spheres corresponding to the data from ligament shapes ܩଶଵ and ܩଷଷ, respectively. (**a**) Scaled yield stress after Equation (14). Each pair of contour plots represents the two solid fractions. The parametrization of log (ܧ்,௦/ߪ௬,௦) corresponds to the values 0.5 and 2.5. (**b**) Scaled work hardening rate after Equation (15) with an exponent ߚ = 2, where the inputs for the MLP regression are reduced to ܣ and ߞ. The pair of planes represent the two solid fractions. **Figure 9.** Visualization using MLP regression (shown as contour plot) along with black and red spheres corresponding to the data from ligament shapes *G*<sup>21</sup> and *G*33, respectively. (**a**) Scaled yield stress after Equation (14). Each pair of contour plots represents the two solid fractions. The parametrization of log(*ET*,*s*/*σy*,*s*) corresponds to the values 0.5 and 2.5. (**b**) Scaled work hardening rate after Equation (15) with an exponent *β* = 2, where the inputs for the MLP regression are reduced to *A* and *ζ*. The pair of planes represent the two solid fractions.

For obtaining a first impression on the dependence of ܧ்/ߪ௬ according to Equation (16), the data is visualized in Figure 10. The dependence of log (ܧ்/ߪ௬) is clearly the strongest and nicely correlated with log (ܧ்,௦/ߪ௬,௦), whereas ܣ and ߞ have no or only a For obtaining a first impression on the dependence of *ET*/*σ<sup>y</sup>* according to Equation (16), the data is visualized in Figure 10. The dependence of log(*ET*/*σy*) is clearly the strongest and nicely correlated with log(*ET*,*s*/*σy*,*s*), whereas *A* and *ζ* have no or only a small effect, respectively. Therefore, the data can be plotted as log(*ET*/*σy*) versus log(*ET*,*s*/*σy*,*s*) in a 2D scatter plot. The correlation shown in Figure 11 is linear over the whole range from 0.5 ≤ log(*ET*,*s*/*σy*,*s*) ≤ 2.5, i.e., from almost perfectly plastic to strongly work hardening materials. The slope is positive, i.e., an increase in the ratio *ET*,*s*/*σy*,*<sup>s</sup>* increases the corresponding ratio *ET*/*σ<sup>y</sup>* in the macroscopic behavior, which is expected. The scatter around the linear fit is, with few exceptions, ±0.1, which corresponds to 25% in a linear scaling. This scatter results in part from the additional dependence of *ζ*, which is visible in a tilt of the contour plots in Figure 10.

is visible in a tilt of the contour plots in Figure 10.

is visible in a tilt of the contour plots in Figure 10.

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small effect, respectively. Therefore, the data can be plotted as log (ܧ்/ߪ௬) versus log (ܧ்,௦/ߪ௬,௦) in a 2D scatter plot. The correlation shown in Figure 11 is linear over the whole range from 0.5 ≤ log (ܧ்,௦/ߪ௬,௦) ≤ 2.5, i.e., from almost perfectly plastic to strongly work hardening materials. The slope is positive, i.e., an increase in the ratio ܧ்,௦/ߪ௬,௦ increases the corresponding ratio ܧ்/ߪ௬ in the macroscopic behavior, which is expected. The scatter around the linear fit is, with few exceptions, ±0.1, which corresponds to 25% in a linear scaling. This scatter results in part from the additional dependence of ߞ, which

small effect, respectively. Therefore, the data can be plotted as log (ܧ்/ߪ௬) versus log (ܧ்,௦/ߪ௬,௦) in a 2D scatter plot. The correlation shown in Figure 11 is linear over the whole range from 0.5 ≤ log (ܧ்,௦/ߪ௬,௦) ≤ 2.5, i.e., from almost perfectly plastic to strongly work hardening materials. The slope is positive, i.e., an increase in the ratio ܧ்,௦/ߪ௬,௦ increases the corresponding ratio ܧ்/ߪ௬ in the macroscopic behavior, which is expected. The scatter around the linear fit is, with few exceptions, ±0.1, which corresponds to 25% in a linear scaling. This scatter results in part from the additional dependence of ߞ, which

**Figure 10.** Approximation of the simulation results for the macroscopic plastic properties (spheres) by MLP regression (shown as contour plot) as functions of randomization ܣ and cut fraction ߞ:) **a**) ܩଶଵ: ߮ = 0.12, (**b**) ܩଷଷ: ߮ = 0.35. **Figure 10.** Approximation of the simulation results for the macroscopic plastic properties (spheres) by MLP regression (shown as contour plot) as functions of randomization *A* and cut fraction *ζ*: (**a**) *G*21: *ϕ* = 0.12, (**b**) *G*33: *ϕ* = 0.35. **Figure 10.** Approximation of the simulation results for the macroscopic plastic properties (spheres) by MLP regression (shown as contour plot) as functions of randomization ܣ and cut fraction ߞ:) **a**) ܩଶଵ: ߮ = 0.12, (**b**) ܩଷଷ: ߮ = 0.35.

Equation (18) for the two data sets ܩଶଵ (black), ߮ = 0.12 and ܩଷଷ (red), ߮ = 0.35. Linear fits given as ݕ = ݉ݔ + ܾ) ;**b**) plot of ܧ்/ߪ௬ for one and two levels of hierarchy according to Equations (21)–(23), respectively, showing the decreasing influence of ܧ்,௦/ߪ௬,௦ with increasing hierarchy level. The parameters of the linear fits for log (ܧ்/ߪ௬) depend on the solid fraction, which **Figure 11.** (**a**) Scaled work hardening rate to yield stress reduced to 2D plots for ܧ்/ߪ௬—Equation (16) and ܧ்/)ߪ௬߮ ଵ/ଶ)— Equation (18) for the two data sets ܩଶଵ (black), ߮ = 0.12 and ܩଷଷ (red), ߮ = 0.35. Linear fits given as ݕ = ݉ݔ + ܾ) ;**b**) plot of ܧ்/ߪ௬ for one and two levels of hierarchy according to Equations (21)–(23), respectively, showing the decreasing influence of ܧ்,௦/ߪ௬,௦ with increasing hierarchy level. **Figure 11.** (**a**) Scaled work hardening rate to yield stress reduced to 2D plots for *ET*/*σy*—Equation (16) and *ET*/(*σyϕ* 1/2) —Equation (18) for the two data sets *G*<sup>21</sup> (black), *ϕ*<sup>0</sup> = 0.12 and *G*<sup>33</sup> (red), *ϕ*<sup>0</sup> = 0.35. Linear fits given as *y* = *mx* + *b*; (**b**) plot of *ET*/*σ<sup>y</sup>* for one and two levels of hierarchy according to Equations (21)–(23), respectively, showing the decreasing influence of *ET*,*s*/*σy*,*<sup>s</sup>* with increasing hierarchy level.

could be incorporated, e.g., by linear interpolation between them, because this effect is small compared to the range of log (ܧ்/ߪ௬). Alternatively, we can make use of the scaling according to Equation (18), which removes the effect of the solid fraction, such that the two data sets are merged in the scatter plot in Figure 11a for log (ܧ்/)ߪ௬߮ ଵ/ଶ)). The correlation is again linear in the log–log plot and can be fitted with The parameters of the linear fits for log (ܧ்/ߪ௬) depend on the solid fraction, which could be incorporated, e.g., by linear interpolation between them, because this effect is small compared to the range of log (ܧ்/ߪ௬). Alternatively, we can make use of the scaling according to Equation (18), which removes the effect of the solid fraction, such that the two data sets are merged in the scatter plot in Figure 11a for log (ܧ்/)ߪ௬߮ ଵ/ଶ)). The correlation is again linear in the log–log plot and can be fitted with The parameters of the linear fits for log(*ET*/*σy*) depend on the solid fraction, which could be incorporated, e.g., by linear interpolation between them, because this effect is small compared to the range of log(*ET*/*σy*). Alternatively, we can make use of the scaling according to Equation (18), which removes the effect of the solid fraction, such that the two data sets are merged in the scatter plot in Figure 11a for log *ET*/(*σyϕ* 1/2 ). The correlation is again linear in the log–log plot and can be fitted with

$$\log(E\_T/(\sigma\_y \rho^{\frac{1}{2}})) = 0.18 + 0.7 \log(E\_{T,s}/\sigma\_{y,s}).\tag{20}$$

Equation (20) can be rewritten as

$$\frac{E\_T}{\sigma\_y} = b\sqrt{\rho} \left(\frac{E\_{T,s}}{\sigma\_{y,s}}\right)^\gamma \,\,\,\,\tag{21}$$

with *b* = 1.514 and *γ* = 0.7.

Shi et al. [6] developed scaling laws for the macroscopic Young's modulus and yield stress (in general denoted as property *P*) for a hierarchically nested network of *n* levels of the form *P*net = *b <sup>n</sup>Psϕ*<sup>e</sup> *nβ* . This results from the recursive application of the Gibson– Ashby scaling law *P*eff = *bP*s*ϕ <sup>β</sup>* under the assumption of a strong self-similarity *<sup>ϕ</sup>*net <sup>=</sup> *<sup>ϕ</sup>*<sup>e</sup> *n* . Here, *P*<sup>s</sup> is the mechanical property of the solid phase, *P*eff is the effective (homogenized) value, and *P*net is the result of the net value of *P*. For the work hardening to yield stress ratio as given by Equation (21), the property itself scales with an exponent *P γ* , such that the effective properties on the next hierarchy level are *<sup>P</sup>*eff,*<sup>j</sup>* <sup>=</sup> *<sup>b</sup>ϕ*<sup>e</sup> *βP γ* eff,*j*−<sup>1</sup> with *<sup>β</sup>* <sup>=</sup> 0.5. Therefore, *ET*/*σ<sup>y</sup>* for a material with two and three levels of hierarchy is given by

$$\frac{E\_T}{\sigma\_y} = b^{1+\gamma} \sqrt{\tilde{\varphi}^{1+\gamma}} \left(\frac{E\_{T,s}}{\sigma\_{y,s}}\right)^{\gamma^2} \tag{22}$$

and

$$\frac{E\_T}{\sigma\_y} = b^{1 + (1+\gamma)\gamma} \sqrt{\tilde{\rho}^{1+(1+\gamma)\gamma}} \left(\frac{E\_{T,s}}{\sigma\_{y,s}}\right)^{\gamma^3},\tag{23}$$

respectively. For two levels, the total solid fraction is *<sup>ϕ</sup>* <sup>=</sup> *<sup>ϕ</sup>*<sup>e</sup> 2 , which ranges from 0.119 to 0.165 [6]. In this case, *<sup>ϕ</sup>*<sup>e</sup> ranges from 0.345 to 0.406, which changes the leading term in Equation (22) by 14%. Because of the exponent *γ* <sup>2</sup> = 0.49, a similar effect would require a variation in the material properties *ET*,*s*/*σy*,*<sup>s</sup>* by a factor of 1.33. This trend is shown in Figure 11b, for a variation of *ET*,*s*/*σy*,*<sup>s</sup>* over two orders of magnitude. If we add a third level of hierarchy, the effect of *ET*,*s*/*σy*,*<sup>s</sup>* becomes even smaller *γ* <sup>3</sup> = 0.343 . We can therefore speculate that *ET*/*σ<sup>y</sup>* → 1 with increasing number of hierarchy levels and *ET*/*σ<sup>y</sup>* reduces to a function of *<sup>ϕ</sup>*e. Section <sup>4</sup> shows that *<sup>E</sup>T*/*σ<sup>y</sup>* is important in the interpretation of the measured hardness.

Finally, the dependency of the plastic Poisson's ratio *ν<sup>p</sup>* in Equation (13) on *A*, *ζ*, and log(*ET*,*s*/*σy*,*s*) is visualized in Figure 12. The MLP regressions shown in Figure 12a reveal that the dependence of *ν<sup>p</sup>* on log(*ET*,*s*/*σy*,*s*) is significant for low solid fractions, while the effect of the cut fraction *ζ* is rather small. This changes for high solid fractions shown in Figure 12b, where the effect of log(*ET*,*s*/*σy*,*s*) is small but the effect of the cut fraction *ζ* has the same importance as the randomization *A*. In combination, both parameters can be used to tune the plastic Poisson's ratio over a large range from ∼ 0.3 to ∼ 0.1. The complex dependency indicates that the multiaxial plastic deformation behavior of nanoporous metals can strongly vary and needs to be determined individually for each microstructure. Additionally, the ligament diameter and surface energy have an important effect on the plastic Poisson's ratio, as shown in [32,49]. These experiments show a comparably large range of values from 0 to 0.2 for increasing ligament size. This range is included in the simulation data shown in Figure 12. *Materials* **2021**, *14*, x FOR PEER REVIEW 17 of 24

**Figure 12.** Visualization of the plastic Poisson's ratio ߥ in Equation (13) for (**a**) ܩଶଵ: ߮ = 0.12 and (**b**) ܩଷଷ: ߮ = 0.35. **Figure 12.** Visualization of the plastic Poisson's ratio *ν<sup>p</sup>* in Equation (13) for (**a**) *G*21: *ϕ* = 0.12 and (**b**) *G*33: *ϕ* = 0.35.

eling of the relationship with a minimum number of inputs.

soft solid phase can be neglected, i.e., ߪ௬,௦ ≪ ܧ<sup>௦</sup>

*4.2. Principal Component Analysis* 

<sup>௦</sup>ܧ ݈, ,ௗݎ ,ௗݎഥ൫ܪ = ܪ

ு ఙ <sup>∗</sup>ܪ =

fluence of the underlying structural and mechanical properties of the nanofoam. To this end, we performed a dimensionality reduction along the same line as in Section 3 with (i) dimensional analysis, (ii) principal component analysis, and (iii) visualization and mod-

The major output of a nanoindentation experiment was the hardness ܪ, which can

The hardness mainly scales with the macroscopic yield stress, as this is the case for bulk materials [50]. Hence, writing Equation (25) in dimensionless form and considering that ܧ்,௦/ߪ௬,௦ can be replaced by a dependence of ܧ்/ߪ௬ and ߮ using Equation (21), this leads to a relationship that includes only macroscopic properties and structural parame-

> ൬߮, ா ఙ

For further reduction of Equation (26), we used the hardness results from the indentation simulations described in Section 2.2 that were carried out with the same parameter sets as the simulations for uniaxial compression in Section 2.1. A PCA of Equation (26), shown in Figure 13, suggested that the four arguments could be reduced to one, when an uncertainty in the predicted ܪ/ߪ௬ from ±0.1 to ±0.3 is acceptable. A reduction of the uncertainty to ±0.2 would already require at least three components. This potential for simplification is also reflected in the factor by which the absolute mean error is increased due to the reduction of the number of components, shown in the insert (orange) in Figure

As in Section 3.1, we represented the ligament shape defined by ݎௗ, ݎௗ, and ݈ by the solid fraction ߮ and, furthermore, assumed that the hardness is governed by plastic and structural parameters, while the effect of the elastic material parameters of the comparably

<sup>௦</sup>ߥ ,

and ܧ்,௦ ≪ ܧ<sup>௦</sup>

(25) .൯ߞ ,ܣ ,௦்,ܧ ,௦,௬ߪ ߮,൫ܪ = ܪ

(24) .൯ߞ ,ܣ ,௦்,ܧ ,௦,௬ߪ ,

(26) .൰ߞ ,ܣ ,

. This reduces Equation (24)

**4. Nanoindentation** 

*4.1. Dimensional Analysis* 

be written as

to

ters:

#### **4. Nanoindentation**

In this section, the dependence of the hardness was analyzed with respect to the influence of the underlying structural and mechanical properties of the nanofoam. To this end, we performed a dimensionality reduction along the same line as in Section 3 with (i) dimensional analysis, (ii) principal component analysis, and (iii) visualization and modeling of the relationship with a minimum number of inputs.
