Mechanical Properties of Additively Manufactured Periodic Cellular Structures and Design Variations

Due to advancements made in computer-aided design and additive manufacturing, cellular structures are emerging as a significant engineering and materials interest. One reason for this growing interest is the tradeoff in material properties that can be achieved through the use of cellular designs. For example, a highly porous metal periodic cellular structure can provide an advantageous strength-to-weight ratio with improved energy absorption and dispersion characteristics, while offering a significant mass reduction to its solid counterpart. There are numerous potential applications within the structures discipline where light-weighting components would provide immense benefits, such as decreased fuel consumption and reduced vibrations in the automotive and aerospace industries [1,2]. At the same time, the energy absorbing capabilities of metal foams and lattices provide the significant benefit for vehicle impact protection [3] or ballistic protection [4].

Cellular materials are defined as a combination of solid material and void space, typically filled with air. There are three principal factors that influence the mechanical properties of a cellular structure. They are the mechanical properties of the base material from which the structure is made, the relative density of the cellular structure, and the topology, or design, used [5].

As the methods for experimentation and analysis of cellular materials are quite different than those of continuum methods, a brief introduction of specific features and relevant terminology is presented.

Cell Size. The average measured cross-sectional distance or diameter of the cell. Sometimes referred to as the cell length or mean cell diameter.

Cell Wall Thickness. The average measured thickness of the cell edges or structural members of the cell. Often referred to simply as the thickness.

While there are varying classes of cellular structures, often defined by cellular distribution or design, this research is primarily concerned with a specific subset of cellular architectures that are defined by periodic replication of individual cellular units with well-defined structural characteristics incorporating an open cell design, which are often referred to as lattices. There are two common variations of lattice structures: strut-based and surface-based networks [6]. Strut-based networks are defined by a joint and frame design, similar to a truss, and are considered stretch-dominated structures due to the fact that the structural members are loaded through tension or compression, which leads to the members stretching to carry the load [7]. Surface-based networks can be thought of being similar to traditional foams and are considered bending-dominated structures due to the surface failure modes of bending, buckling, or crushing [7]. These differences in loading and failure behavior indicate that each lattice type will be better suited for different applications. The nature of loading and failure of the strut-based networks indicate that these lattice types will perform better under uniaxial loading than surface-based networks and will be characterized by a higher strength-to-weight ratio. However, the longer plateau region of the surface-based lattice response, along with the bending deformation behavior, indicates that surfaced-based designs will perform better in energy-absorbing applications [5]. As some of the more unique characteristics provided by lattice designs, especially considering survivability and impact applications, involve their energy absorption and dispersion characteristics, the primary focus of this research will be on surface-based lattice designs based upon triply periodic minimal surfaces (TPMS).

Figure 1 depicts a typical engineering stress–strain response curve of a surface-based lattice design from uniaxial compression testing. For lattice structures, the stress is determined as the load over the bounding cross-sectional area, and the strain is calculated normally as displacement over the original specimen length. These terms will be mathematically defined in Sec. “Mechanical Testing.” Three distinct response phases are present within this curve. First, the curve displays a linear-elastic response up to its elastic yield strength. In this region, the modulus of elasticity (E) or Young’s modulus of the design can be determined, along with the Yield Strength (σ_y). The second phase of the curve is the plastic response of the cells, noted by cell failure and collapse, and is characterized by the plateau stress (σ_pl). The third and final stage present in the response curve is called densification, which is noted by a sharp rise in stress. This is where all the cells within the structure have collapsed onto one another and the material is near solid again.

While the base TPMS architectures have been tested using a variety of materials [6,8–10,11], previous research has focused on the basic designs of the surfaces. This research effort expands into testing derivative designs based up on the TPMS analytical expressions, characterizing their mechanical performance and deformation behavior, along with comparing the results with the source design. Characterizing these new designs will broaden the scope of potential lattices for use in a wide variety of applications. As mentioned, these designed cellular structures represent an emergent class of engineering material, with enormous implications across the engineering and survivability disciplines.

TPMS are uniquely structured surfaces that are described by three distinct criteria [12]. First, the surfaces are symmetric and periodic in all three axes. Second, the surfaces are area-minimizing, i.e., the surface comprises the smallest possible area that bounds the region. Finally, the surfaces have zero mean curvature over a single cell or symmetric structure. Mean curvature can be defined as a measure of the surface curvature in relation to differential geometry and can be thought of the average principal curvature of the surface [13]. In the natural world, the first criterion, being symmetric and periodic in all three axes, guarantees that the surface mean curvature is zero. The periodic nature of these designs, allows them to be patterned to fill specific dimensional spaces.

There are several different TPMS designs that have been found and are well described in Schoen’s seminal work on periodic surfaces [12]. Three of these designs were chosen for analysis and will be described in more detail. These lattice structures are the Primitive (Schwarz P), Diamond (Schwarz D), and Schoen’s I-WP. In the TPMS surface equations that follow, x, y, and z are the surface’s Cartesian coordinates in three-dimensional space, and m represents a periodicity scaling factor, which is the ratio of the desired cell size to π. As the equations used to describe these surfaces are based on trigonometric functions, the periodicity scaling factor resizes the cells within the prescribed volume, which in turn sets the cell density within the structure.

Cubes were utilized for determining the mechanical properties of the TPMS designs. The choice to utilize cubes in quasi-static uniaxial compression testing was based on previous testing that used cylindrical test specimens [15,16], as specified by American Society for Testing and Materials (ASTM) E9-19 [17], in which all of the TPMS cylindrical specimens exhibited buckling over uniaxial compression beyond the elastic regime.

The process used to develop the necessary test specimens consisted of three steps prior to fabrication. The first step involved creating individual lattice cells, based on the previously shown equations (Eqs. (3)–(5)) within the Matrix Laboratory (matlab) software suite. In addition to the TPMS cells, two variational designs were developed for each base TPMS cell through manipulation of their respective trigonometric functions. The primary means for developing the variational designs was achieved by making the design expressions equal to a shape variable, c, versus being equal to zero as is the case for the true minimal surfaces. The value of c controls the shape of the surface, as well as the overall surface area, which means that the variational designs are no longer necessarily minimal surfaces although they remain triply periodic designs. The values for c were chosen based on the influence the variable had on each design. The first value chosen was zero, as that corresponds to the minimal surface representation, then the value was increased to determine when the surface could no longer be maintained as an open network for the desired surface thicknesses. From this point, the c value was reduced slightly to add a buffer that would ensure there were no closed cells within the specimen during the manufacturing process. For the Primitive and I-WP designs, this value of c was then mirrored across the origin to determine the third point. While the Diamond design produced the same structure for the same positive and negative c value; therefore, a mid-point between zero and the upper c value was used as the third variation. The two Primitive variations are shown in Figs. 2(h) and 2(i), the Diamond variations are shown in Figs. 2(h) and 2(i), and the I-WP variations are shown in Figs. 2(e) and 2(f).

Once the single cells were verified, the cell was replicated within matlab to create a rectangular lattice structure to match the desired build dimensions. The build dimensions were based on the test specimen width and height, with the build height equal to the specimen width plus one cell size. The addition of the single cell size to the specimen length allowed for material loss when the specimen was removed from the build plate. The individual cell size was set to 5 mm for the compression cubes to mitigate some of the intrinsic size and cell density effects [6]. In the second step of the specimen development process, the structure file was ported into the rhinoceros 6 (rhino 6) computer-aided design software [18]. This step entailed setting the desired surface thickness to achieve a range of relative densities based on the specimen design. The final step was completed in Materialise Magics, which is used to prepare files for additive manufacturing on the General Electric (GE) Concept Laser M2 Cusing metal 3D printer at the Air Force Institute of Technology (AFIT). In this step, the specimen was first checked for errors, and then the build strategy was determined based on printer settings and user input. Finally, the part file was transferred to the 3D printing machine for manufacturing.

The material manufactured in this study was produced by selective laser melting (SLM) of gas atomized spherical Inconel 718 (IN718) powder produced by Powder Alloy Corporation of Loveland, OH, shown in Fig. 3. The chemical composition of the IN718 powder is shown in Table 1. Several images of the powder were collected with a Tescan MIRA-3 field emission scanning electron microscope (FE-SEM). The zeiss zen image analysis software was used to analyze the SEM images to determine the powder size distribution shown in Fig. 4. The powder had a mean particle size of 19.18 μm with other relevant size data shown in Table 2.

A GE Concept Laser M2 SLM machine equipped with a 400-W continuous-wave Ytterbium fiber laser was utilized to manufacture all of the samples tested in this research. A high purity nitrogen shield gas was used to inert the build chamber prior to and during manufacturing. In SLM fabrication, a thin layer of powder is spread across the build surface, then the laser melts and fuses the powder based on a computed scan strategy that determines the build pattern and timing information. Once a layer is completed, another layer of powder is added, and the process is repeated until the build is complete. There are six primary parameters that can be controlled during the fabrication process: laser power, scan speed, laser spot size, hatch or scan spacing, contour offset, and powder layer thickness. Changes in these parameters cause significant variation in densification and microstructure, both of which can affect the resultant material properties of the manufactured parts [19]. Commercial SLM machines segment areas of a design into categories based upon physical locations in a design. Areas near the surface of a part will typically use lower power in an attempt to achieve lower surface roughness, while thick interior areas use high laser power and higher scanning speeds to increase productivity. The thin-wall nature of the designs utilized in this research led to the Concept Laser machine software primarily applying the Contour laser parameters that are designed to produce the best possible surface finish. The Contour laser parameters utilized a 120-W laser power, 280 mm/s scan speed, 50 μm spot size, and a 90 μm contour offset. The contour offset is the offset distance from the computer model’s external surface to the center of the laser scan track, which is based on the material melt pool width. Additionally, the layer thickness was set to 40 μm in order to be greater than the D90 value of the IN718 powder. Hatch spacing is often set for laser scanning in bulk sections but the lack of such features in the designs used in this research made selection of this parameter unnecessary.

Uniaxial compression testing is a relatively well-known technique for determining material properties in the static and quasi-static regimes, with established procedures [17,20]. Generally speaking, the test process consists of subjecting the specimen of interest to an increasing axial compression force, utilizing either displacement or load control, where both force and displacement data are collected for determining the compression properties of the material or design. Outlined here are the specific compression testing equipment and data reduction methods that were used as part of this research.

All of the static and quasi-static compression testing for this research effort was performed on an MTS Systems Corporation (MTS) Model 810 Material Test System. The Model 810 is a servo-hydraulic Universal Testing Machine (UTM) that incorporates the Model 318.50 load unit, which is capable of producing forces up to 500 kN, or 110 kip, in compression. The system in use also incorporates an MTS 609 Alignment Fixture. The alignment fixture allows for alignment adjustments while the system is under a preload, which reduces the misalignment error due to applying a load to the system. An MTS 661.23C-01 axial force transducer was also used, which is force rated to 500 kN, 110 kip. The grip assembly used was an MTS 647.50 Vee-Wedge Grip Assemblies. These grip assemblies are side-loaded hydraulic grips that aid in platen alignment through use of a sting attachment. The sting attachment screws into the bottom of the platens and when clamped into the vee-wedge automatically aligns along the test area centerline. The MTS 643.10B Fixed Compression Platens that were used are made of a case-hardened steel alloy and incorporate etched rings on the test platform to further aid in test specimen alignment. The machine is calibrated annually by MTS service professionals following ASTM E4-16 calibration procedures. The precision of the UTM as configured is 0.001 mm for displacement and 13 N for force.

Following fabrication, the specimens were measured, weighed, and the actual relative densities were determined. Each of the TPMS designs and their variations were plotted to compare the actual relative density against the designed relative density. The designed relative density range for each specimen was determined by the use of three common surface thicknesses. This technique led to different relative density ranges for each specimen, but was utilized to ensure consistent fabrication quality across the designs. A comparison between the as designed relative density and the as fabricated relative density can be found in Fig. 5, with the Diamond design variations depicted in Fig. 5(a), the I-WP design variations in Fig. 5(b), and the Primitive design variations in Fig. 5(c). For all of the designs, there was a greater difference between the actual and designed relative densities at the lower density values.

This difference can be attributed to several factors, namely the intricacy of the design, variation in surface thickness, and bonding of loose powder into the melt surface. The effects of the design intricacy can be seen when comparing the trends of the Diamond and Primitive designs against the results of the I-WP design. The I-WP design has closer surface features, which are regions that are likely to have greater variation in surface thickness, as well as higher inadvertent fusing of loose powder. Some additional variation in surface thickness is due to the fabrication parameters, such as laser beam size and power. The laser parameters directly influence the size of the melt pool, which sets the fineness of the scan pattern and thereby the precision of the surface features. The melt pool parameters also have a direct impact on the amount of loose powder that will inadvertently bond to the structure.

Scanning electron microscope (SEM) images were taken of manufactured parts for analysis of the print quality, which directly impacts the actual relative density of the specimens. A representative sampling of images for each of the three base cell designs can be seen in Fig. 6. As seen in these images, there is a significant amount of inadvertent fusing of powder to the cell surface. This additional material will increase the relative density of the printed part above that of the designed specimen. Another area of analysis is a comparison of the printed surface thickness to the designed surface thickness. For all three of these images, the designed surface thickness was 500 μm, the SEM images were used with digital image correlation techniques to determine the actual surface thickness of the specimens. For the Diamond cell design, Fig. 6(a), the actual surface thickness calculated was less than the designed value, with an average surface thickness of 476 μm. The combination of additional powder with a smaller surface thickness lead to the actual relative density of the Diamond cellular designs being remarkably close to the designed relative density. The I-WP cell design, Fig. 6(b), had an increased surface thickness compared to the designed value, having an average surface thickness of 524 μm. With the increased surface thickness and addition of partially fused powder, the I-WP actual relative density was greater than the designed relative density for all of the manufactured specimens. Lastly, the Primitive cell design, Fig. 6(c), also had an increased surface thickness when compared to the designed thickness, although not as much as the I-WP, which had an average surface thickness of 507 μm. As with the I-WP design, the Primitive design displayed an increased surface thickness and inadvertent powder fusing, which caused the actual relative density of the design to be greater than the designed relative density.

The compressive mechanical properties of each of the lattices were derived from the engineering stress-strain response curve. Two specimens were tested for each lattice design, with the stress-strain responses being averaged to eliminate some of the variability due to additive manufacturing. The response curves are depicted in Figs. 7–9. In Figs. 7 and 9, abrupt changes in the stress-strain curve can be seen in the plateau region of the response, which is indicative of cellular failure. On the individual cell level, failure for both the Diamond and Primitive designs was typified by through-surface fracture, which can easily be seen in the 35% and 50% strain images of Figs. 13–15 and 19–21. The surface fractures led to the stress drops in the plateau response, which was then followed by an increase in specimen stiffness that resulted in the subsequent rise in the stress response. In Fig. 8, there is significantly less variation within the plateau region, which is consistent with the exhibited deformation behavior of the I-WP lattices. With the more collective failure of cells being prevalent in the I-WP designs, individual cell failure was predominantly seen through bending and buckling of the lattice surface. All of the stress-strain curves show a high level of consistency between the specific design’s response. However, there are some noticeable differences between the different structural designs.

While the overall value of the plateau stress is predominantly dependent on the relative density of the design, the slope of the plateau region appears highly dependent on architecture. The three I-WP designs had the greatest plateau slope, which is likely due to early interaction between the cellular structures. The Primitive designs displayed a nearly flat plateau region, indicating that there is little interaction between the cells prior to reaching densification strain. The Diamond designs showed an intermediate response when compare to the others.

The densification strain was highly dependent on the relative density of the specimen, in that densification occurred earlier in higher density specimens, but there were appreciable differences between the architectures again. The Primitive cellular designs presented the highest densification strains of the three base architectures, again likely due to the lack of early interaction between the cell structures. The I-WP designs registering higher densification strains than the Diamond designs. This may be due to the nature of deformation between the designs, with the I-WP being through collective collapse and the Diamond being through shear bands.

Additionally, there was a difference in the slope of the response within the densification region. The Primitive responses also displayed the highest densification slope. With the delayed interaction between cellular structure, the rate of densification was considerably higher for the Primitive design leading to the higher slope. The I-WP designs had the next highest densification slope of the designs. This may be due to the collective nature of deformation within these structures. Finally, the Diamond designs had the lowest densification slope, which may be attributed to the combination of shear deformation and plateau region interaction.

The modulus of elasticity was determined through the analysis of the linear elastic portion of the response curve through a comparison of the tangent modulus and secant modulus, see Fig. 10, as well as the more common least squares linear regression curve fit. The tangent modulus is determined as the slope of a line tangent to the linear elastic stress-strain response, and the secant modulus is determined as the slope of a line from the origin that intersects the stress-strain response curve within the linear elastic region. The values obtained through both methods were within one percent of each other.

The summarized mechanical property power law fit coefficients for these lattices are presented in Table 3. Across the range of mechanical property curve fits, the average R² value was 0.9937.

The exponential fit coefficient, n, for the modulus of elasticity provides some additional insight into the nature of the designs deformation behavior [21]. If the exponent value of the curve fit is approximately equal to one, then the lattice will, in general, carry the loading through tension or compression, which is considered stretching-dominated deformation. This indicates that the stiffness of stretch-dominated materials will change linearly with its relative density. However, if the exponent value is approximately equal to two, then the lattice will carry the loading through bending, buckling, or crushing, which is considered bending-dominated deformation. A common trend seen with bending-dominated deformation is that structural failure happens through the thickness of the surface. This means that the stiffness of bending-dominated materials will change quadratically with a change in its relative density. In general, the nature of loading and failure of the stretching-dominated lattices suggest that these architectures would perform better under uniaxial loading than the bending-dominated designs, yielding a higher strength-to-weight ratio. However, as mentioned, the plateau region of the bending-dominated structure response tended to be elongated when compared to that of an equivalent stretching-dominated design, signifying that bending-dominated designs would perform better within energy-absorbing applications [5]. With these exponent values being between the stretching-dominated and bending-dominated values, there is a mixed-mode of deformation, which was seen in Figs. 13–21.

For comparison of the lattice designs, the mechanical properties of interest were plotted against the actual relative density for each specimen, shown in Figs. 11 and 12. An overall trend seen across the four plots is that there is a greater effect of cellular design at lower relative densities, noted by the larger spread of points between designs. This is likely due to the underlying material properties becoming dominant within the mechanical response as the relative density increases.

For the elastic modulus, Fig. 11(a), the base Diamond TPMS and its variations showed the best performance across the range of evaluated relative densities. At the upper end of the relative density range, the base I-WP and its first variation indicated similar results to the Diamond, but again the tendency is for all of the results to align with increasing relative density. All three of the Primitive designs performed relatively consistent within the range as well, with the first Primitive variation slightly outperforming the other two designs. The second variation of the I-WP TPMS exhibited significantly lower performance than the other lattice designs for elastic modulus, which may be attributed to difficulty manufacturing fine geometric features arising in this variational design.

Similar results were seen for the yield strength, Fig. 11(b). All three of the Diamond designs performed the best over the entire range, with nearly identical results. The I-WP, its first variation, and the three Primitive designs all performed relatively equal to each other, but with slightly lower values than the Diamond lattices. Again, I-WP variation 2 exhibited considerably worse performance than the other eight lattices. This is consistent with the results from the elastic modulus, since the modulus is used in determining the yield strength with the 0.2% Offset method.

Even with the less stable plateau response, seen in Fig. 7, the Diamond design and its variations displayed higher plateau stress values than the I-WP and Primitive designs, Fig. 12(a). Unlike Fig. 11, for the plateau stress, the second I-WP variational design did not show markedly different results than the other two I-WP designs. Similar to the elastic modulus results, the first Primitive variational design performed better than the base Primitive TPMS and variation 2 design.

The results for the toughness, Fig. 12(b), follow the same trend as for the plateau stress. This was not unexpected, as the toughness is determined by finding the area beneath the stress-strain response curve, and the plateau stress directly influences the design toughness. The only differences between variations of a specific design were found in the Primitive design. Its two variations had larger toughness values than its base TPMS design. The toughness is of unique importance, as it is an indication of the ability for the structure to absorb energy, which can be balanced with a more productive strength-to-weight ratio.

Images representative of the structure responses under uniaxial compression are presented in Figs. 13–21. The strain values depicted in these figures range from load initiation, 0% strain, to shortly prior to reaching the densification strain, 50% strain. This allows for analysis through the mechanical failure region of the lattice structure and not the material. It is worth noting that the relative density between the different lattices is not consistent. However, the failure patterns observed were consistent across relative densities for each of the lattice types. This indicates that the deformation behavior of lattice structures is remarkably independent of relative density, but is decidedly dependent on the topology of the lattice. While the relative density of the structure does not play a significant role in the deformation behavior, it will impact the degree to which failure artifacts are present in the stress-strain response. At higher relative densities, the stress fluctuations that are indicative of cell failure will be less pronounced or even masked, see Fig. 7(a).

As depicted in Figs. 13–15, the deformation pattern was slightly different between the Diamond TPMS lattice and its variations. While all three designs indicated a shear failure pattern, the base Diamond lattice showed dual shear bands forming throughout the strain range, or failure along two diagonal lines of cells. Both variational designs showed failure along a single shear band or failure along a single diagonal line of cells. Due to the topology of the Diamond cell, single shear deformation can lead to cell layer collapse along that diagonal axis, which will reduce the amount of stress fluctuations within the stress-strain response when compared to other designs that fail through shear deformation, such as the Primitive design.

As presented in Figs. 16–18, the deformation pattern was relatively consistent between the I-WP TPMS lattice and its variations. In all three lattice designs, the overarching failure mechanism is uniform failure across a horizontal row of cells. For the base I-WP and its first variational design, there is near failure of full rows starting at 35% strain, see Figs. 16(c) and 17(c). This deformation pattern holds through 50% strain. However, up to 20% strain for all three designs, and for the second variational design, there is a collective structure deformation, where all of the horizontal rows are deforming in the same manner. This uniform failure leads to the smooth hardening like response, with minimal stress fluctuations, as seen in the stress-strain response curves of the I-WP specimens, Fig. 8.

As detailed in Figs. 19–21, the deformation pattern was again relatively consistent between the base Primitive TPMS lattice and its variations. For all three lattice designs dual shear bands can be seen forming as early as 20% strain. For the base Primitive lattice, one of the shear bands was more dominant than the other, see Fig. 19(c), however as the strain accumulated it maintained its dual shear failure, see Fig. 19(d). In both of the Primitive variational designs, the dual shear band failure progressed comparatively along both failure lines throughout the tested strain range. The shear deformation in conjunction with the Primitive topology leads to increased stress concentration points, which in turn leads to more pronounced cell failure. These failure points can be seen in the stress-strain response curves through the stress fluctuations within the plateau region, see Fig. 9.

In all cases, surface fracture began around 30% strain and can be seen in the majority of the 35% strain images. Additionally, it is interesting to note that the cellular failure is more prevalent near the loading surface in all three of the Primitive designs, where cellular failure within the I-WP designs was more extensive near the stationary surface. The Diamond designs failed near uniformly along the height of the specimen. These differences may be due to surface boundary loading along the platens, which could lead to constrained deformation due to friction on the contact surface or potentially inconsistencies in the fabrication process that could lead to early failure in a region. Additionally, the cell topology likely plays a role, as the cell design dictates the amount of structural surface contact at the loading platen, as well as impacting the print quality.

In this study, nine different periodic cellular structures were designed based on three base TPMS cell topologies. These designs were then fabricated from IN718 utilizing additive manufacturing techniques. Scanning electron microscopy was used to assess the print quality, namely accuracy of the print and any notable defects. The specimens were then tested under quasi-static uniaxial compression in order for their mechanical properties to be determined from the stress-strain response. The relationship between the design relative density and mechanical properties was fit with a power law curve, which provided insight into the deformation behavior of the structures. All of the specimen designs indicated a mixed mode of deformation between stretching-dominated and bending-dominated deformation. Imagery taken during the compression testing was also used to show the deformation patterns of the different lattices. The Diamond and Primitive designs displayed strong indications of shear failure, where the I-WP designs had a uniform failure along the horizontal rows of cells.

In comparing the mechanical response of the designs, there were three notable results. First, the Diamond design, including its variations, showed the best performance in all four of the evaluated mechanical properties. This indicates that the Diamond design provides the best stiffness-to-weight ratio and superior energy absorption capability, which was seen in the calculated toughness of the design specimens. Second, in general, the novel derivative designs did not perform as well as their source design. However, the results do open up some trade space that could be used in the engineering design process. Within the lattice design groupings, the Primitive variational designs provided better performance than the base design for a couple mechanical properties, most notably in toughness. Superior performance of variational designs opens the range of possibilities for future studies in cellular optimization techniques. Finally, the cellular topology has a larger impact on the mechanical response of the structure at lower relative densities, whereas the mechanical response is more heavily influenced by the underlying material properties at higher relative densities.

The authors would like to thank Dr. Martin Schmidt of the Air Force Office of Scientific Research (AFOSR) for his support of this research.

There are no conflicts of interest.