*4.1. Dimensional Analysis*

The major output of a nanoindentation experiment was the hardness *H*, which can be written as

$$H = \overline{H} \begin{pmatrix} r\_{\text{mid}} \ r\_{\text{end}} \ l\_{\text{\\_}} \ \mathcal{E}\_{\text{s}} \ \upsilon\_{\text{s}} \ \sigma\_{\text{y},\text{s}} \ \mathcal{E}\_{\text{T},\text{s}} \ \mathcal{A}\_{\text{\\_}} \mathcal{J}\_{\text{\\_}} \end{pmatrix} . \tag{24}$$

As in Section 3.1, we represented the ligament shape defined by *rmid*, *rend*, and *l* by the solid fraction *ϕ* and, furthermore, assumed that the hardness is governed by plastic and structural parameters, while the effect of the elastic material parameters of the comparably soft solid phase can be neglected, i.e., *σy*,*<sup>s</sup>* ≪ *E<sup>s</sup>* and *ET*,*<sup>s</sup>* ≪ *E<sup>s</sup>* . This reduces Equation (24) to

$$H = \hat{H}(\varphi\_{\prime}, \sigma\_{\prime \sharp \prime}, E\_{T, \sharp \prime}, A\_{\prime} \zeta). \tag{25}$$

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 *ET*,*s*/*σy*,*<sup>s</sup>* can be replaced by a dependence of *ET*/*σ<sup>y</sup>* and *ϕ* using Equation (21), this leads to a relationship that includes only macroscopic properties and structural parameters:

$$\frac{H}{\sigma\_{\!y}} = \hat{H}^\* \left( \varphi, \frac{E\_T}{\sigma\_{\!y}}, A, \zeta \right). \tag{26}$$

#### *4.2. Principal Component Analysis*

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 *H*/*σ<sup>y</sup>* 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 13. By a reduction to a single component, this error measure is only increased by a factor of 1.3, which is a very low value compared to the results in Section 3.2. *Materials* **2021**, *14*, x FOR PEER REVIEW 18 of 24 13. By a reduction to a single component, this error measure is only increased by a factor of 1.3, which is a very low value compared to the results in Section 3.2.

*4.3. Hardness* 

<sup>௬</sup>หߪ/ܪ ≈ <sup>௬</sup>ߪ/ܪ

ୀ

**Figure 13.** Principal component analysis of the dependence of ܪ/ߪ௬ with respect to structural and macroscopic mechanical properties following Equation (26), combining data sets ܩଶଵ and ܩଷଷ. **Figure 13.** Principal component analysis of the dependence of *H*/*σ<sup>y</sup>* with respect to structural and macroscopic mechanical properties following Equation (26), combining data sets *G*<sup>21</sup> and *G*33.

The quantitative dependence of the normalized hardness ܪ/ߪ௬ as function of the structural parameters (ܣ, ߞ (and macroscopic material properties log (ܧ்/ߪ௬) is shown in Figure 14. The MLP regressions are shown as contour plots in Figure 14a,b, confirming that log (ܧ்/ߪ௬) is the most important parameter, followed by the randomization ܣ, which has a moderate effect, whereas the effect of the cut fraction ߞ can be neglected. In Figure 14c,d, the axis of the cut fraction ߞ is replaced by log (ܧ்/ߪ௬). A small effect of the randomization ܣ with a negative slope in the low solid fraction data can be expressed by

the uncertainty when structural effects are not taken into account. For solid fractions of typical samples with ߮ ≈ 0.32, the effect caused by structural disorder is negligible.

− 0.16ܣ. For ܣ = 0.3, this effect is ∆ܪ/ߪ<sup>௬</sup> ≤ 0.05, which corresponds to

(**a**) (**b**)

#### *4.3. Hardness 4.3. Hardness*

The quantitative dependence of the normalized hardness *H*/*σ<sup>y</sup>* as function of the structural parameters (*A*, *ζ*) and macroscopic material properties log *ET*/*σ<sup>y</sup>* is shown in Figure 14. The MLP regressions are shown as contour plots in Figure 14a,b, confirming that log *ET*/*σ<sup>y</sup>* is the most important parameter, followed by the randomization *A*, which has a moderate effect, whereas the effect of the cut fraction *ζ* can be neglected. In Figure 14c,d, the axis of the cut fraction *ζ* is replaced by log *ET*/*σ<sup>y</sup>* . A small effect of the randomization *A* with a negative slope in the low solid fraction data can be expressed by *H*/*σ<sup>y</sup>* ≈ *H*/*σ<sup>y</sup> <sup>A</sup>*=<sup>0</sup> − 0.16*A*. For *<sup>A</sup>* = 0.3, this effect is <sup>∆</sup>*H*/*σ<sup>y</sup>* ≤ 0.05, which corresponds to the uncertainty when structural effects are not taken into account. For solid fractions of typical samples with *ϕ* ≈ 0.32, the effect caused by structural disorder is negligible. The quantitative dependence of the normalized hardness ܪ/ߪ௬ as function of the structural parameters (ܣ, ߞ (and macroscopic material properties log (ܧ்/ߪ௬) is shown in Figure 14. The MLP regressions are shown as contour plots in Figure 14a,b, confirming that log (ܧ்/ߪ௬) is the most important parameter, followed by the randomization ܣ, which has a moderate effect, whereas the effect of the cut fraction ߞ can be neglected. In Figure 14c,d, the axis of the cut fraction ߞ is replaced by log (ܧ்/ߪ௬). A small effect of the randomization ܣ with a negative slope in the low solid fraction data can be expressed by <sup>௬</sup>หߪ/ܪ ≈ <sup>௬</sup>ߪ/ܪ ୀ − 0.16ܣ. For ܣ = 0.3, this effect is ∆ܪ/ߪ<sup>௬</sup> ≤ 0.05, which corresponds to the uncertainty when structural effects are not taken into account. For solid fractions of typical samples with ߮ ≈ 0.32, the effect caused by structural disorder is negligible.

**Figure 13.** Principal component analysis of the dependence of ܪ/ߪ௬ with respect to structural and macroscopic mechanical properties following Equation (26), combining data sets ܩଶଵ and ܩଷଷ.

13. By a reduction to a single component, this error measure is only increased by a factor

*Materials* **2021**, *14*, x FOR PEER REVIEW 18 of 24

of 1.3, which is a very low value compared to the results in Section 3.2.

**Figure 14.** (**a**,**b**) Dependence of the normalized hardness ܪ/ߪ௬ as function of structural properties (ܣ, ߞ (and macroscopic material properties log (ܧ்/ߪ௬); (**c**,**d**) reduction of dimensionality by elimination of the cut fraction ߞ, confirming that the simulation data can be represented by a simple dependence ܪ/ߪ௬(ܧ்/ߪ௬). Ligament geometries are (**a**,**c**) ܩଶଵ, ߮ = 0.12; (**b**,**d**) ܩଷଷ, ߮ = 0.35. **Figure 14.** (**a**,**b**) Dependence of the normalized hardness *H*/*σ<sup>y</sup>* as function of structural properties (*A*, *ζ* ) and macroscopic material properties log *ET*/*σ<sup>y</sup>* ; (**c**,**d**) reduction of dimensionality by elimination of the cut fraction *ζ*, confirming that the simulation data can be represented by a simple dependence *H*/*σ<sup>y</sup> ET*/*σ<sup>y</sup>* . Ligament geometries are (**a**,**c**) *G*21, *ϕ* = 0.12; (**b**,**d**) *G*33, *ϕ* = 0.35.

We could further reduce the relationship to a 2D scatter plot, shown in Figure 15, which is fitted with a linear relation We could further reduce the relationship to a 2D scatter plot, shown in Figure 15, which is fitted with a linear relation

$$\frac{H}{\sigma\_y} \stackrel{H}{\underset{\text{\textquotedblleft}}{\text{\textquotedblleft}}} \overline{H}\_0^\* \stackrel{H\_0^\*}{\rightarrow} m\_H \frac{m\_{\overline{\text{\textquotedblleft}}}^{E\_T}}{\sigma\_y} \tag{27}$$

correspond to the standard deviation of ±0.11. Both ligament geometries are combined in this plot, which indicates that the dependence Equation (27) is applicable for a broad where for our data, we obtained *H*∗ <sup>0</sup> = 0.41 and *m<sup>H</sup>* = 0.035. In this Figure, the error bars correspond to the standard deviation of ±0.11. Both ligament geometries are combined in

[11,27] were added. These data points, entered as star symbols, represent the combinations of ݎௗ/݈ ∈ {0.231,0.289, 0.346,0.404} and ݎௗ/ݎௗ ∈ {0.5, 0.75, 1.0, 1.25} for two values ܧ்,௦/ߪ௬,௦ ∈ {3.16, 50.0}. This adds 32 simulations that provide an insight into possible dependencies of the ligament shape and solid fraction. The results are added in Figure 15 as blue stars, where the blue curves connect simulation results of constant ݎௗ/݈ and the line thickness increases with the value of ݎௗ/݈. All results are within the scatter of the random simulations for geometries ܩଶଵ and ܩଷଷ, confirming that Equation (27) holds for all ligament shapes and solid fractions within the given scatter band. While for ܧ்,௦/ߪ௬,௦ = 3.16 the data scatter around a spot in the lower left area of the plot, the results for ܧ்,௦/ߪ௬,௦ = 50 show that with increasing ݎௗ/ݎௗ and solid fraction ߮ the data points systematically move towards larger ratios ܪ/ߪ௬. The same applies to the random data, when we compare the range of values for the geometries ܩଶଵ(߮ = 0.12) and ܩଷଷ

(߮ = 0.35) in black and red, respectively.

range of structures and is insensitive to microstructural parameters.

this plot, which indicates that the dependence Equation (27) is applicable for a broad range of structures and is insensitive to microstructural parameters. *Materials* **2021**, *14*, x FOR PEER REVIEW 20 of 24

**Figure 15.** (**a**) Correlation of ܪ/ߪ௬ with ܧ்/ߪ௬ for ligament shapes ܩଶଵ (߮ = 0.12) and ܩଷଷ (߮ = 0.36). (**b**) Deformation shown for a slice of one unit cell of the indented RVE with ܣ = 0.18 and ߞ = 0.26, for which the stress–strain curve is shown in Figure 2b. **Figure 15.** (**a**) Correlation of *H*/*σ<sup>y</sup>* with *ET*/*σ<sup>y</sup>* for ligament shapes *G*<sup>21</sup> (*ϕ*<sup>0</sup> = 0.12 ) and *G*<sup>33</sup> (*ϕ*<sup>0</sup> = 0.36). (**b**) Deformation shown for a slice of one unit cell of the indented RVE with *A* = 0.18 and *ζ* = 0.26, for which the stress–strain curve is shown in Figure 2b.

Despite the common assumption for foams ܪ/ߪ<sup>௬</sup> = 1, it seems reasonable that a porous material tends towards a bulk solid for a high solid fraction. However, it is difficult to understand that the hardness can fall below the macroscopic yield stress. This could be caused by the way the macroscopic stress–strain behavior has been translated into the material parameters ܧ் and ߪ௬, as shown in Figure 2b. The macroscopic yield stress is read from the linear fit of the stress–strain curve for plastic strains > 1%. This procedure removes initial nonlinearities that could be interpreted as microplasticity. However, microplasticity does not exist in our continuum model; therefore, the true yield stress is usually lower than that determined from the linear fit. In the example shown in Figure 2b, the measured yield stress determined from the linear hardening model was 7.3 MPa, whereas at 0.2% plastic strain, the stress reached a value of only 5 MPa. For ܪ = 3.2 MPa, the ratio ܪ/ߪ௬ then changed from 0.43 to 0.64, if the yield stress at 0.2% plastic strain was used. This explains in part why the hardness can be lower than the yield stress. It could be speculated that another contribution might stem from the reduction of the For confirmation, additional simulations for all 16 ligament geometries *G*<sup>11</sup> to *G*<sup>44</sup> [11,27] were added. These data points, entered as star symbols, represent the combinations of *rend*/*l* ∈ {0.231, 0.289, 0.346, 0.404} and *rmid*/*rend* ∈ {0.5, 0.75, 1.0, 1.25} for two values *ET*,*s*/*σy*,*<sup>s</sup>* ∈ {3.16, 50.0}. This adds 32 simulations that provide an insight into possible dependencies of the ligament shape and solid fraction. The results are added in Figure 15 as blue stars, where the blue curves connect simulation results of constant *rend*/*l* and the line thickness increases with the value of *rend*/*l*. All results are within the scatter of the random simulations for geometries *G*<sup>21</sup> and *G*33, confirming that Equation (27) holds for all ligament shapes and solid fractions within the given scatter band. While for *ET*,*s*/*σy*,*<sup>s</sup>* = 3.16 the data scatter around a spot in the lower left area of the plot, the results for *ET*,*s*/*σy*,*<sup>s</sup>* = 50 show that with increasing *rmid*/*rend* and solid fraction *ϕ* the data points systematically move towards larger ratios *H*/*σy*. The same applies to the random data, when we compare the range of values for the geometries *G*21(*ϕ*<sup>0</sup> = 0.12) and *G*<sup>33</sup> (*ϕ*<sup>0</sup> = 0.35) in black and red, respectively.

connectivity of the ligament network, as shown in Figure 15b. The RVE is characterized by a cut fraction of ߞ = 0.26, i.e., almost a third of the ligaments in the RVE are broken. This leads to large pores, which become comparable to the indentation depth and the contact radius. When a microstructural length and the indentation depth are of the same order, the simulation shows a size effect. For 3D networks, this problem becomes relevant when approaching the percolation threshold and is difficult to solve [51]. An increase in the normalized indentation load ܲ/ℎ<sup>ଶ</sup> , shown as insert in Figure 15b, apparently confirms this effect. However, one would then also expect a systematic bias in the data in the form of a dependence of the cut fraction, i.e., ܪ/ߪ<sup>௬</sup> → 1 for ߞ → 0, but such a trend is not present in Figure 14a,b. It is therefore possible that values ܪ/ߪ<sup>௬</sup> < 1 exist. Because this has important implications on the interpretation of hardness data of foams in general, an indepth investigation should be the scope of future work. Despite the common assumption for foams *H*/*σ<sup>y</sup>* = 1, it seems reasonable that a porous material tends towards a bulk solid for a high solid fraction. However, it is difficult to understand that the hardness can fall below the macroscopic yield stress. This could be caused by the way the macroscopic stress–strain behavior has been translated into the material parameters *E<sup>T</sup>* and *σy*, as shown in Figure 2b. The macroscopic yield stress is read from the linear fit of the stress–strain curve for plastic strains > 1%. This procedure removes initial nonlinearities that could be interpreted as microplasticity. However, microplasticity does not exist in our continuum model; therefore, the true yield stress is usually lower than that determined from the linear fit. In the example shown in Figure 2b, the measured yield stress determined from the linear hardening model was 7.3 MPa, whereas at 0.2% plastic strain, the stress reached a value of only 5 MPa. For *H* = 3.2 MPa, the ratio *H*/*σ<sup>y</sup>* then changed from 0.43 to 0.64, if the yield stress at 0.2% plastic strain was used. This explains in part why the hardness can be lower than the yield stress.

**5. Summary and Conclusions**  Nanoporous metals with their complex microstructure represent an ideal candidate It could be speculated that another contribution might stem from the reduction of the connectivity of the ligament network, as shown in Figure 15b. The RVE is characterized by

the sample preparation, it is possible to tune the microstructure and macroscopic mechanical properties within a large design space. This includes, among others, the solid fraction,

a cut fraction of *ζ* = 0.26, i.e., almost a third of the ligaments in the RVE are broken. This leads to large pores, which become comparable to the indentation depth and the contact radius. When a microstructural length and the indentation depth are of the same order, the simulation shows a size effect. For 3D networks, this problem becomes relevant when approaching the percolation threshold and is difficult to solve [51]. An increase in the normalized indentation load *P*/*h* 2 , shown as insert in Figure 15b, apparently confirms this effect. However, one would then also expect a systematic bias in the data in the form of a dependence of the cut fraction, i.e., *H*/*σ<sup>y</sup>* → 1 for *ζ* → 0, but such a trend is not present in Figure 14a,b. It is therefore possible that values *H*/*σ<sup>y</sup>* < 1 exist. Because this has important implications on the interpretation of hardness data of foams in general, an in-depth investigation should be the scope of future work.

#### **5. Summary and Conclusions**

Nanoporous metals with their complex microstructure represent an ideal candidate for method developments that combine data and AI. With a few parameters controlling the sample preparation, it is possible to tune the microstructure and macroscopic mechanical properties within a large design space. This includes, among others, the solid fraction, ligament size, and the connectivity density. It has been recently demonstrated that the versatile dealloying process allows hierarchically organized nanoporous metals with superior macroscopic properties to be produced compared to those with only one hierarchy level [6]. Via the microstructure, it is possible to tune the macroscopic properties, such as Young's modulus, yield strength, elastic and plastic Poisson's ratio, and hardness in wide ranges. This makes this class of materials not only attractive for various applications, such as sensing or actuation in combination with light weighting, but it is also an ideal science case for the demonstration of the capabilities of dimensionality reduction methods.

To this end, the generation of ~200 data sets for macroscopic compression and nanoindentation was realized with the help of an efficient FE-beam modeling technique. The parameter space consists of five independent inputs (microstructure, material parameters) and six dependent outputs (macroscopic compression behavior and hardness). It was systematically analyzed in three steps by means of a dimensional analysis including a priori knowledge about the problem at hand, principal component analysis, and visualization. In the latter two steps, machine learning served as key for analyzing the existence and quality of approximations on the presented data sets.

From the outcome, we conclude that, independent of the size of the data set, it is always recommendable to start with a dimensional analysis. This ensures that the analyzed dependency is formulated in a physically reasonable manner and it allows the dimensionality of the problem to be reduced by usually two quantities in quasi static mechanics or by three for dynamic problems. At this stage, it is advisable to incorporate *a priori* knowledge from the literature or by reasoning, which can further simplify the problem considerably. How well this has been done and by how many components the dependency can be further reduced can be easily tested by machine learning in combination with principal component analysis. If no deeper understanding is needed, the outcome in the form of a black box would already be a sufficient computer model of the relationship hidden in the presented data.

Deeper insight can be gained by visualization, which is also supported by machine learning. Here, the multilayer perceptron first approximates the design space from the randomly distributed data and then is applied for continuous mapping along selected inputs in the form of contour plots. This serves the validation of the previous steps as well as for a better understanding of the quantitative dependence of a specific output, e.g., the hardness to yield stress ratio, of inputs, e.g., the randomization or mechanical properties. In this way, the major dependences can be identified from a limited number of data and unimportant inputs can be eliminated. Furthermore, one obtains a measure for the uncertainty due to ignored inputs that have a non-negligible effect.

For the scientific case at hand, which is the microstructure–property relationship of nanoporous metals, there are several important findings, which are applicable not only to Au but to any metal, as long as it can be described with the chosen elastic–plastic material behavior and microstructure. Our analysis showed that the ratio of the work hardening rate to the yield stress *ET*,*s*/*σy*,*<sup>s</sup>* represents a key property that can mapped to the corresponding macroscopic ratio *ET*/*σ<sup>y</sup>* in a log–log scaling. It is therefore possible to invert this relationship for measured macroscopic behavior, which allows one to gain an important insight in the amount of work hardening present in the solid phase, relative to its yield stress. Work hardening implies storage of defects in the nanoscaled ligaments, and its existence has been a matter of debate. The derived relationship can help to quantitatively underpin speculations that are in favor [52] or contradict [13,23] the mechanistic model of dislocation starvation in nanosized metallic objects simply by translating the macroscopic test data into those of the solid phase.

In addition to the known Gibson–Ashby scaling laws for Young's modulus and yield strength, one for the work hardening rate is added, which uses the same exponent of 2 as the Young's modulus. This is unexpected, because the work hardening rate is a slope defined by two flow stresses at different plastic strains and the yield stress, which is one of them, scales with an exponent of 1.5. Additionally, the appearance and range of the relationship as functions of randomization and cut fraction are very similar to that of the Young's modulus.

Another important finding is the linear relation between *H*/*σ<sup>y</sup>* and *ET*/*σy*. The common assumption that for hardness testing of foams *H* = *σ<sup>y</sup>* [35], which is also used in the interpretation of nanoindentation of np-Au [37,38], turned out to be a special case for *ET*/*σ<sup>y</sup>* ∼ 17. The range of the *H*/*σ<sup>y</sup>* data is surprisingly large and exceeds the common values for porous and bulk solids of 1 and 3, respectively, towards lower values: A large number of data are within the range 0.5 ≤ *H*/*σ<sup>y</sup>* ≤ 1. This can in part be explained by how the macroscopic stress–plastic strain curve is modeled. Another reason could be a size effect that results from large pores for samples with very low connectivity, but it appears that still values of *H*/*σ<sup>y</sup>* < 1 exist. Because this has important implications on the interpretation of hardness data, an in-depth investigation will be the scope of future work.

Finally, for hierarchic materials with a nested network [6], our results suggest that the effect of *ET*,*s*/*σy*,*<sup>s</sup>* becomes small or even negligible with respect to *ET*/*σy*. With increasing levels of hierarchy, it can be expected that the normalized hardness *H*/*σ<sup>y</sup>* changes from a dependence of *ET*/*σy*, which holds for a common nanoporous metal, towards a dependence of the solid fraction *ϕ*.

**Funding:** Support was provided by Deutsche Forschungsgemeinschaft—Project Number 192346071— SFB 986 "Tailor-Made Multi-Scale Materials Systems: M3," project B4.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data sets analyzed in this work are available via the repository TORE (https://tore.tuhh.de/ accessed on 3 April 2021) published under https://doi.org/10.15480/3 36.3411.

**Acknowledgments:** N.H. acknowledges Ilona Ryl, who carried out preliminary FE simulations with a hybrid FE-beam and FE-solid modeling technique.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


*Article*
