Abstract
Mechanical properties of metallic foams are highly dependent on the characteristics of porosity. In the case of foams manufactured using space holder particles (SHPs), parameters such as percentage porosity, pore size and cell wall thickness are directly contingent on the SHP used. In this work, different fractal distributions of SHPs were introduced in order to modify the resulting porosity. An Al–Si–Cu alloy was used as raw material for producing metallic foams by infiltration, while NaCl particles of 2 and 4 mm were used as SHPs, modifying the smalltolarge particle ratio and fractal dimensions. Cylindrical foams of 10 cm in height and 5 cm in diameter were obtained by infiltrating the Al alloy into the SHPs. Results showed that the use of fractal distribution allowed to slightly increase the percentage porosity, whereas the most important effect was the decrease in cell wall thickness when fractal dimension increased. Mechanical properties were directly affected by these modifications, decreasing when fractality was induced. Finite element analysis models of the foams were obtained in a combination with discrete element method, in order to simulate their compressive elastic behaviors. Young’s modulus estimations were in excellent agreement with experimental results, validating the effect of fractal inclusion and the importance of the correct model selection.
Introduction
Among the new applications of Al and other alloys, metallic foams are becoming increasingly important. These materials have an excellent combination of properties originating from their porosity and the intrinsic properties of the alloys, of which the most critical are low density and highenergy absorption [1, 2]. In the case of Al foams and due to these properties, the most important applications are related to structural, automotive and aeronautical applications [3, 4]. Depending on the desired application, porosity of these materials can be modified, in particular percentage porosity, pore size, cell wall thickness and interconnectivity. Processes involving the incorporation of removable space holder particles (SHPs) are most often used when a higher porosity control is needed [5], because it has been demonstrated that porosity can be changed by varying the SHP characteristics and arrangement prior to the final manufacturing process [6]. In the case of infiltration, SHPs are used as a preform, infiltrating the molten metal in a second stage and finally dissolving the SHP, or even including a reinforcement for manufacturing hybrid composites [7]. Thus, the morphology and size of the SHPs and the mixing process determine the final porosity. Different particles have been used, with wide size distributions and shapes, e.g., NaCl and KBr [6, 8]. It has been demonstrated that within certain physical limitations using spherical particles of homogeneous size a wide range of porosities can be obtained (approximately 62% for real random arrangements). The use of particles with irregular shapes slightly modifies this value due to a better accommodation after the mixing process [9]. Nevertheless, if a significant increase in the percentage porosity is desired, particles of different sizes may be used and the relative quantities of small and big particles modified. Maximum packing appears for different relative quantities of particles, which depends on the particle sizes [10, 11]. A previous study [6] showed that the distribution of the particles used as space holder phase could be considered as fractal, with a high quantity of small diameter particles compared to a lower quantity of bigger particles. The concept of fractal was proposed by Mandelbrot et al. [12] and is based on nonEuclidean geometry theory to describe irregularities in nature. One way to define fractals is by distribution, as in the case of particles, obtaining a straight line from a log–log frequency histogram of particle sizes [6]. Although the increase in the relative sizes of particles used as SHP is the most significant way to maximize porosity (due to the accommodation of the small particles in the interstitial holes between the bigger ones), a limitation for this is the mixing process, where small particles tend to segregate by gravity. Consequently, the increase in the relative quantities between small and big particles could be a better way to increase fractal distribution and obtain a higher porosity. Taking these premises into account, the first objective of this work is to obtain Al foams by infiltration using different distributions of SHPs and to analyze the effect of these modifications on the porosity characteristics and on mechanical properties of the foams obtained. Furthermore, it is also very important to have tools for estimating the possible mechanical behavior of the resulting foams. Finite element analysis (FEA) is one of the most effective methods to predict foam properties due to its ability for obtaining pore arrangements close to real foams [6, 13]. Further refinement can include the combination of FEA with discrete element method (DEM) [14]. In the case of foams in the present work, using random arrangements and pores of different sizes, DEM–FEA will allow replication of models that better estimate the elastic behavior of these materials. Then, the second objective of this work is to generate porous networks using DEM–FEA by attempting to reproduce the porosity created by the interaction between SHPs of different sizes and relative quantities, used as a preform in the infiltration process. It is expected that the results obtained demonstrate the importance of the fractal distribution of the SHPs on the characteristics of porosity and the mechanical properties of the foams as well as their modeling for the design and controlled fabrication of metallic foams.
Experimental
In order to obtain fractal distributions of porosity, the first step was to sieve and separate commercial NaCl particles into two average sizes: 2 mm and 4 mm. In Fig. 1a, examples of these particles can be observed, in which their irregular morphologies can be noted, with shape factor near 1.0. These particles were mixed following three different criteria: (1) ratio 1:1 (only particles of 4 mm), (2) ratio 2:1 (2 particles of 2 mm per one particle of 4 mm) and ratio 4:1 (4 particles of 2 mm per one particle of 4 mm). This leads to fractal dimensions (D_{f}) of 1 and 2, respectively, for cases 2 and 3. D_{f} was obtained from log (relative quantities)/log (size ratio). Figure 1b–d represents the expected unit cells for each case, which characterizes the distribution after mixing, being each particle or unit cell in contact with neighbors. It is important to state that after mixing of SHPs, accommodation between particles leads to the superposition of the unit cells observed in Fig. 1b–d, filling much of the empty spaces. The use of spheres for models has been accepted in different studies [6, 14] which use SHPs, being important for minimizing the model complexity and the computer requirements compared to the use of prisms. Angular pores are under higher stresses than spherical pores, for which stress is quite uniform. Nevertheless, FEA studies using rounded or angular shapes have demonstrated that the differences in the resulting Young’s moduli are not significant, being then possible approximations to simple shapes without sacrificing the accuracy of the model [15].
Each one of the three NaCl particle distributions obtained were introduced (as SHP preform) in a 5cmdiameter and 30cmhigh AISI 314 stainless steel cylinder with a sealed base, filled to a height of 10 cm in order to obtain foams of these dimensions. Enough pieces of a 332 Al–Si–Cu alloy (from a recycled automotive monoblock, composition in Table 1) were introduced into the cylinder over the NaCl preform. The cylinder was then introduced into the top of a Prefinsa HRC4 electric resistance furnace, and the NaCl preform was infiltrated with the molten Al alloy by gravity when the temperature of the furnace was raised to 700 °C. No gas pressure was needed for infiltration thanks to the high fluidity of this alloy, provided by the high Si content, so an Ar atmosphere was only used to avoid oxidation.
The cylinder was extracted from the furnace and air cooled. The cylindrical Al–NaCl composites formed were removed from the steel cylinder and immersed in water to dissolve the SHPs. The density of the foams was obtained by calculating the volume of the cylindrical samples and measuring their mass, while the porosity was measured by means of the relative density and characterized by optical (OM) and scanning electron microscopies (SEM). These characterizations were carried out using a Labomed Med 400 OM and a Jeol JSM IT300LV SEM operated at 20 kV, respectively. Mechanical behavior of the foams was determined by compression tests on machined cylinders 20 mm in diameter and 16 mm in height samples, using a universal testing machine Instron 5500R at a constant crosshead speed of 0.5 mm/s, according to ASTM E909. The averages of three different specimens of each foam were presented for determining the most important mechanical parameters, i.e., Young’s modulus, yield point and stress for densification (end of plateau). Measured characteristics of the porosity were percentage and cell wall thickness, using three foams per each SHP distribution and averaging them for obtaining reproducible results.
Modeling and simulation
For the estimation of the Young’s modulus of the foams, and in order to have predictions before manufacturing according to the different SHP arrays, a combination of DEM and FEA was used. Due to the complexity of the inclusion of particles with different sizes and relative quantities, we decided that DEM could accomplish the required conditions for the generation of the coordinates for SHP location. It was necessary to recreate the mixture of particles with these characteristics, and DEM is a useful tool for modeling the behavior of granular systems; in fact, it can be used for generating the final position of the SHPs in the mixing process [16]. Consequently, DEM was used to model the interaction of spheres placed in the cylindrical mold. As two kinds of particles were introduced into the models, the distance between the centers of the spheres and their interactions were modified accordingly. The quantity of spheres was selected in order to fill all the cylindrical space and was slightly larger than the final geometry used in FEA models. This point is relevant in order to obtain pores on exterior surfaces as occurs in real foam structures. LIGGGHTS^{®} DEM simulation software was used [17], and the coordinates obtained were postprocessed as input in ANSYS 18.1 Design Modeler script in order to create the CAD models of the foams.
FEA models consisted of cylindrical specimens, also of 20 mm in diameter and 16 mm in height (diameter/height ratio equal 0.8), with resulting porosities depending on the aforementioned characteristics, and will be compared to the real foams obtained by infiltration. The pores were modeled as spheres of 4 and 2 mm in diameter according to the fractal conditions. Figure 2a–c shows models of cylindrical foams with different porosities engendered through the DEM–FEA combination, with pores of two sizes and different relative quantities. As observed, pores are randomly distributed as in real foams, where their distribution is typically aperiodic, nonuniform and disordered [18].
For Young’s modulus determination, the foams were uniaxially compressed. Tennode 3D tetrahedral SOLID 187 structural elements were employed for meshing. In order to ensure convergence of the numerical solution, convergence analysis was carried out gradually increasing the number of elements and verifying the local stress behavior. The nodes were kept in the same plane for the upper face of the cylinder applying the couplednode boundary condition. Young’s modulus (E_{z}) was obtained from the response to the compression along the zaxis using strain (ε_{z}), determined using the displacement in zaxis \( u_{z} \) Young’s modulus of 64 GPa and Poisson’s ratio of 0.33 were taken from the compressive tests of the compact aluminum alloy.
Results and discussion
Figure 3a shows a representative cylindrical sample of actual Al foams obtained by infiltration using NaCl as SHP, and Fig. 3b–d shows cross sections of foams with different contents of SHPs. It can be observed that for all foams the distribution of the pores is homogeneous. Notable differences can be observed by comparing Fig. 3b (SHPs of single size) with Fig. 3c, d (fractal SHP distributions). These differences will be discussed below.
Figure 4a–c depicts a combination of sections of real and modeled cross sections showing the distribution of the SHPs in the resulting NaCl–metal composite and pores obtained from the FEA models. Their distributions and the similarity can be compared between their unit cells and those expected according to Fig. 1b–d. The use of composites instead of foams was preferred as it was better for observation purposes. As can be seen, for the case of the foams with singlesize SHPs (Fig. 4a) the characteristic unit cell consists of only one size, while in the case of fractal distributions unit cells contain features of several sizes, as would be expected. In Table 2, the characteristics of the porosity for experimental and FEA modeled foams may be observed. With the insertion of particles of different sizes, the percentage porosity increased (leading to a decrease in densities), a fact that could be explained by insertion of small particles in the interstitial positions between the larger ones. There is no significant change when the relative quantities increased from 2:1 to 4:1. A plausible explanation is that although the insertion of particles with different sizes increases packing of particles, the increase in the relative quantities between small and large particles does not always increase porosity because it also depends on their relative sizes, as discussed above [9,10,11]. Nevertheless, cell wall thickness decreased markedly, even for the foams with different pore sizes (2:1 and 4:1). This decrease for these two foams is more relevant because their porosities are similar. The effect of this parameter on mechanical properties will be analyzed below. It can be noted that these behaviors are similar for both experimental and FEA modeled foams, denoting the effectiveness of a DEM–FEA combination for creating models closer to real foam topologies. Comparing real and model foams, in the case of wall thickness it can be observed that for models these values are higher, and is explained by some agglomeration occurring in the models.
The mechanical behavior of the foams was studied experimentally with compression testing. Stress–strain curves for foams obtained using the three different conditions of SHPs are shown in Fig. 5. As can be observed, all curves exhibit the characteristic behavior of foams, showing three regions: (1) an initial linear elastic region at very low strain, (2) an extended plateau region at a relative constant stress level where the stress increases slowly as the cells deform plastically and collapse, and (3) a densification region where the collapsed cells are compacted together, increasing again the stress [19]. Mechanical properties obtained from these curves are summarized in Table 3. Young’s moduli were calculated using a correction in order to determine the zero point applying a tangent to the point of the maximum slope of the elastic region. Stress for densification (end of plateau region) was determined according to the change in the inflection point of the curve. This means that the second derivate of the function that fits this curve vanishes and changes sign at that point, being the end of the effective deformation [19]. It is observed that all mechanical parameters, i.e., yield stress, Young’s modulus and stress for densification decreased with the increase in particle size ratio from 1:1 to 2:1 and 4:1. This behavior is mainly originated by the lower porosity (62.59%) for the foams with relative pore quantities 1:1, compared to foams with relative quantities of 2:1 (65.92%) and 4:1 (64.81%). The consequent decrease in cell wall thickness contributes to the decrease in mechanical properties. Now, for the foams with pore quantities of 2:1 and 4:1 porosities were similar (~ 1% different), being even higher for foams with quantities 2:1. Then, it could be expected that mechanical properties were better for foams 4:1. Nevertheless, for this case cell wall thickness was the parameter that more impacted mechanical properties, leading to better mechanical properties for foams 2:1, which presented higher wall thickness than foams 4:1.
For the analysis of the simulations, Fig. 6a–c presents graphical distributions of the displacements resulting from the applied stresses, for foams modeled with the three different particle ratios. As can be seen, maximum displacements were obtained for the foams with ratio 4:1 (1.376 mm), and 0.9323 mm and 0.30269 mm for ratios 2:1 and 1:1, respectively. Besides, Fig. 6d–f shows the distribution of the von Mises stresses, with maxima values corresponding to zones with lower cell wall thicknesses. It is clearly observed that the extension of these zones is smaller for foams with ratios 4:1. These results show that the presence of pores of different sizes led to a higher stress concentration and then to higher deformation than the models with uniform pore sizes, and were used for Young’s moduli determinations.
In order to compare experimental values of Young’s modulus and FEA estimations, Fig. 7a presents these values for the foams with different relative quantities of pores. For comparative purposes, various models found in the literature are also included and are as follows:
Model of Zhu et al. [20]:
Model of Warren and Kraynik [21]:
Model of Gan et al. [22]:
where E_{s} is the elastic modulus of the material of the foam and ρ its relative density. As can be clearly observed, FEA estimations are very close to the experimental results, decreasing with the increase in the relative quantities of particles with different sizes. As was mentioned above, elastic moduli are highly dependent not only on the percentage porosity, but also on the cell wall thickness. Consequently, Eqs. 1–3 overpredict the Young’s modulus, because they only use the relative densities (directly dependent on porosity percentage). Moreover, there are significant differences between the cases of fractal arrangements. For literature models, the values are almost constant because they only account for the percentage porosity and no other characteristics of porosity such as cell wall thickness. These results are compared in Fig. 7b, where relative differences are depicted, showing that overpredictions as high as 8 times are reached. This demonstrates the important effect of the model selection, depending on the foam manufacturing process and conditions, as was in our case modifying SHPs. On the other hand, FEA modeling accounts for other parameters and not only the percentage porosity, leading to closer predictions to the experimental values.
Conclusions
After the analysis of the effect of SHPs with different quantities of small and large particles, for producing foams by infiltration, the following conclusions can be drawn:

1.
The inclusion of particles of different sizes (fractals) increased the porosity of the foams. The modification in the relative quantities of small and big particles created an important reduction in cell wall thickness, without significant variations in the percentage porosity.

2.
It was possible to generate different models using the DEM–FEA model combination, varying the relative quantities between pores of different sizes.

3.
Mechanical behavior of the foams significantly decreased with the insertion of particles of different sizes, for both experimental and FEA modeled foams.

4.
The increase in the porosity percentage was the parameter that most decreased the mechanical behavior of the foams. For similar porosities, the decrease in cell wall thickness was the main cause for decreasing mechanical properties.

5.
Experimental and FEA estimates of Young’s modulus were in excellent agreement, confirming the importance of using models which are more representative of the random distribution of pores in real foams.

6.
Comparative models showed that if only the percentage porosity is taken into account for Young’s modulus prediction, the differences between model estimations and the experimental values can reach very high values.
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Acknowledgements
I. Alfonso would like to acknowledge the financial support from SEP CONACYT 285215 and UNAM PAPIIT IN117316. R. Drew acknowledges the financial support from PREI DGAPA UNAM. L. Pérez acknowledges the financial support from the Advanced Center for Electrical and Electronic Engineering, AC3E, Basal Project FB0008, CONICYT. R. Ganesan acknowledges the financial support from the Chilean Agency CONICYT (FONDECYT Project 3150411).
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Carranza, J.C., Pérez, L., Ganesan, R. et al. Effect of fractal distribution of the porosity on mechanical properties of Al foams manufactured by infiltration. J Braz. Soc. Mech. Sci. Eng. 41, 379 (2019). https://doi.org/10.1007/s4043001918767
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Keywords
 Foam
 FEA
 DEM
 Fractal
 Infiltration