## Abstract

Lattice structure metamaterials offer a variety of unique and tailorable properties, yet industrial adoption is slowed by manufacturability and inspection-related difficulties. Despite recent advances in laser powder bed fusion additive manufacturing, the sub-millimeter features of lattices are at the edge of process capabilities and suffer from low geometric quality. To better understand their complex process-structure-property (PSP) relationships, octahedron structures were manufactured across a power spectrum, inspected, and mechanically tested. X-ray computed tomography was used to characterize lattice geometry, and demonstrated that lattice strut geometry measures, increased significantly as a function of laser power. Furthermore, lattices are shown to exhibit a direct correlation between laser power and mechanical performance metrics. Performance variations up to 60% are shown as a function of process parameters despite nominally identical geometry. Significant geometry variations are found to be the cause of performance variation, while material properties as measured by microindentation hardness are constant across the studied parameter range. PSP relationships are modeled, and the limitations of these models are explored. It was found that resulting models can predict mechanical performance based on geometric characteristics with R^{2} values of up to 0.86. Finally, mechanistic causes of observed performance changes are discussed.

## 1 Introduction

Additive manufacturing (AM) has revolutionized the existing manufacturing paradigm due to its design freedom and ability to manufacture complex geometries with relative ease and low cost [1]. Laser powder bed fusion (LPBF) is a specific metal AM technique which utilizes a layer-by-layer process to create three-dimensional components from digital design files. This process has recently begun to come into maturity for industrial production and use in advanced engineering applications, as evidenced by its increasing use in industry and significant market growth in recent years [2,3]. Despite these advances, full process adoption for end-use components is inhibited by lack of understanding of process-structure-property (PSP) relationships, particularly for AM structures containing sub-millimeter features.

Lattice structures are a type of metamaterial created by tessellating a unit cell design in three orthogonal directions [4]. The manufacture of these complex structures, which are often made of thousands to millions of sub-millimeter sized struts [5], is enabled by the layer-by-layer approach and microscale manufacturing process of LPBF AM. It is well-demonstrated that lattices possess desirable properties such as high specific stiffness and strength [6,7], high energy absorption capacity [8], heat dissipation capabilities [9], and functionally gradable properties [10]. Despite significant research into these structures the subtle effects of LPBF process variables on structural geometry and subsequent performance, i.e., process-structure-property relationships, is not yet entirely understood nor quantified.

Recent advances in LPBF technology and a heavy research interest in its fundamental processes and applications have led to significant capability increases, specifically regarding dimensional accuracy and repeatability [11–14]. Furthermore, interest in lattice metamaterials has increased in recent years, resulting in improved modeling and manufacturing capabilities [15–20]. Despite these technical improvements, the well-known advantages of LPBF cannot yet be fully realized due to a variety of challenges relating to manufacturing, inspection, qualification, and modeling. Specifically, this study seeks to understand how process variation impacts structure-level performance as well as the sensitivity of these relationships. In the present study, process-structure-property relationships are established for octahedron lattice structures with the goal of elucidating the effects of laser parameter variation on lattice geometry, material properties, and structural performance.

Important to part manufacturing and performance outcomes are measures of structural build quality related to strut geometry, form, and surface topography, each of which is linked to the manufacturing process. While over 50 different parameters exist for LPBF [21], processing spaces are typically created through control of one or more of laser power, scan speed, and hatch spacing, which are common for macroscale parts and for material process development or optimization [22–25]. Variation in effective energy input due to variations in these parameters, occurs in LPBF processes, and will be studied in the present work through laser power variation. It is well-established that these differences can significantly impact the manufactured geometric accuracy in small-scale components and that the magnitude of effects on performance inversely scales with minimum feature size [26]. Consequently, it is of significant interest to understand the variation of part outcomes at the structure and property levels as a function of manufacturing process parameters.

Past work from the author [27] as well as others [28,29] support the general trend that higher laser energy input will result in larger melt pools and oversized features of size. Full-volume metrology using computed tomography (CT) data of lattice samples has routinely demonstrated this phenomenon [27,30–32]. In this regard, microscale features are consistently oversized in their mean value by as much as 50% [30]. While the oversizing effect and poor surface topographies are generally observed, the impact of print surface heterogeneities on mechanical properties of microscale components is often under-addressed and under-studied. Specifically, the precise PSP relationships are not well-quantified at the sub-millimeter scale relevant to lattice struts.

Beyond localized effects, manufacturing defects have been demonstrated to impact bulk lattice behavior [33–35]. Liu et al. [33] addressed lattice performance variation by performing detailed inspections of 316L octet truss lattices. Subsequent compression simulations were developed based on the acquired statistical characterization data and found that strut diameter variation had more significant impacts on lattice elastic modulus and ultimate tensile strength (UTS) than did strut waviness. Importantly, this study demonstrated that geometric defects not only impact lattice properties but also the failure mode. Similarly, Cao et al. [34] used CT-acquired statistical geometry data for a wide geometric quality space and found that, in extreme conditions, lattice performance may decrease by as much as 70%. However, this study did not explore in detail why the observed defects impacted performance or how manufacturing parameters relate to this performance.

Despite the extent of prior work studying lattice performance, a thorough understanding of the relationship between manufacturing process parameters and lattice performance is not yet fully developed. To address this knowledge gap, the present study aims to quantify process-structure-property relationships in lattice structures through use of CT-based geometry inspection, mechanical testing, hardness testing, and subsequent statistical modeling.

## 2 Methods

In the present study, octahedron lattices were manufactured across a range of laser powers and mechanically tested to assess performance. Representative samples were then inspected using X-ray CT and hardness testing with Vickers microindentation to further understand the geometry and material factors contributing to performance variations. Variations seen in geometric characterization and performance data were then analyzed to better understand the impacts of manufacturing parameters on lattice properties. Statistical models were created to quantitatively describe the PSP relationships observed.

The sample geometry utilized in the present study was designed for compressive testing and consisted of octahedron unit cells arranged in a 3 × 3 × 2 pattern as shown in Fig. 1. The octahedron geometry was chosen due to its lightweight design and high energy absorbing capacity. These characteristics, in addition to its relative ease of manufacture compared to more complex unit cell designs, make it a commonly utilized design in industrial applications such as in aerospace and defense. Unit cells were designed using solidworks software with a nominal unit cell geometry bounding box of 4.7 × 4.7 × 2.5 mm (*X*, *Y*, *Z*) and 0.508 mm (0.2 in.) strut diameters. The parts were built in the +*Z*-direction, recoat was done in the −*X*-direction, and argon gas flow in the −*Y* direction, according to the coordinate system indicated in Fig. 1.

Samples were manufactured using an EOS M280 LPBF machine at five different laser power settings ranging from 80% to 120% of the nominal power input of 195 W in equal increments of 10%. The entirety of the sample, i.e., lattice and baseplate, was printed at the modified parameters. Four replicates were produced per condition, for a total of 20 samples. Nominal parameter settings used correspond to those recommended by the machine manufacturer and are summarized in Table 1. These parameters are suggested as a general use parameter set and are not necessarily the optimal set for all geometries, as will be seen in the ensuing. A brush recoater was used to minimize the risk of damage to the delicate struts, a common occurrence when manufacturing microscale features with a solid recoater. Laser power was only modified for the infill parameters and was kept constant at nominal settings for all downskin, upskin, and contour parameters to avoid convolution of many parameters. Following manufacture, samples were separated from the buildplate using wire electrical discharge machining (EDM). The EDM-cut surface was then hand-sanded to remove any remaining burrs on the bottom of the sample to ensure appropriate conditions for compressive mechanical testing.

Print parameter | Value |
---|---|

Laser power | 195 W |

Laser speed | 1083 mm/s |

Hatch spacing | 0.09 mm |

Layer thickness | 20 μm |

Recoater | Brush |

Print parameter | Value |
---|---|

Laser power | 195 W |

Laser speed | 1083 mm/s |

Hatch spacing | 0.09 mm |

Layer thickness | 20 μm |

Recoater | Brush |

Virgin 316L stainless steel powder sourced from Kennametal Additive Manufacturing and conforming to ASTM F3184-16 was used for the manufacture of these samples. Measurements of powder characteristics were performed using a Malvern Mastersizer laser diffraction particle size analyzer. Using this instrument, powder was measured to have Dv10, Dv50, and Dv90 values of 18.7 *µ*m, 29.0 *µ*m, and 44.5 *µ*m, respectively. These values indicate the diameter of the particle at the 10th, 50th, and 90th percentile of the volumetric distribution of the powder.

Assessment of print geometry variation was performed using X-ray CT. One representative sample from each sample condition was inspected using a North Star Imaging 225 kV CT system at a voxel-side length resolution of 15.1 *µ*m. Acceleration voltage and current were tuned to produce high-contrast image for quality surface determination. A combination prefilter stack consisting of 0.5 mm of Cu and 0.4 mm of Al was used to mitigate beam hardening imaging artifacts. Details of the CT inspections are presented in Table 2.

CT parameter | Value |
---|---|

Acceleration voltage | 220 kV |

Current | 23 µA |

Prefilter | 0.5 mm Cu & 0.4 mm Al |

Exposure time | 200 ms |

Projection count | 2100 |

Scan time | 56 min |

Images averaged per projection | 6 |

Resolution/Voxel-side length | 15.1 µm |

CT parameter | Value |
---|---|

Acceleration voltage | 220 kV |

Current | 23 µA |

Prefilter | 0.5 mm Cu & 0.4 mm Al |

Exposure time | 200 ms |

Projection count | 2100 |

Scan time | 56 min |

Images averaged per projection | 6 |

Resolution/Voxel-side length | 15.1 µm |

Using the three-dimensional image information gathered using CT, strut size distributions were assessed quantitatively using a custom struct cross-sectional analysis procedure [36]. This process is summarized graphically in Fig. 2. The geometry inspection was performed as follows. First, a surface determination of the whole-volume scan was made using a global threshold value, the result of which is shown in Fig. 2(a) for the 175.5 W sample. This threshold was determined using an ISO-65 threshold, which was visually determined to produce a “clean” surface free of noise that accurately represents the surface as seen in slice-by-slice scans. This method identifies peaks in the image histogram corresponding to air and material and uses the grayscale value 65% of the way between these in the direction of the material peak as the threshold value. The histogram describing only the lattice section of the scan, excluding the baseplate, was used for this determination. The sample was then registered to the nominal computer-aided design (CAD) model using an iterative closest point best-fit registration. The use of a consistent method for determining the global threshold value ensured a fair and consistent assessment of each lattice’s geometry. Then, the image volume was resampled at the voxel resolution to create image stacks parallel to the axial direction of the struts within the lattice as defined by the CAD model. An example image plane is shown in Fig. 2(b). Interpolation was performed to ensure a gridded image volume in the new coordinate system. For each lattice sample, six image stacks were generated corresponding to the six strut directions contained within the octahedron lattice unit cell.

*x*–

*y*coordinates this produced were used to calculate size, shape, and form deviation metrics including major and minor axes of a best-fit ellipse, elliptical eccentricity, area moment of inertia, and cross-sectional area. The best-fit ellipse was calculated using a least squares approach, which minimizes the sum of the squared error between the fit function and the input strut cross section boundary data points. Eccentricity was used to evaluate shape and was calculated using Eq. (1), where

*a*is the minor axis length and

*b*is the major axis length of the ellipse. Using this definition, a circle would have an eccentricity value of zero. Area moment of inertia was used as a geometric indicator of the strut’s resistance to bending, which is the primary deformation mode of the octahedron lattice. This metric was calculated using Eq. (2), where

*y*is the distance from the

*x*-axis. This axis is defined by the cross section centroid location and is parallel to the horizontal axis of the image.

*A*is the total area of the cross section. After repeating this process at voxel-level increments, statistical models can be created describing the geometry of the lattices.

An example strut cross section with identified and measured features is shown in Fig. 2(c). An exemplar strut is shown as both raw CT volume surface data and the simplified, ellipse representation in Fig. 2(d). From this subfigure, it can be seen that (1) the strut is highly noncylindrical and has significant surface topography features deviating from the nominal, cylindrical geometry and (2) the ellipse representation captures the majority of these features in a highly simplified format. From the gathered cross section information, statistical methods can be employed to model the lattice geometry characteristics, as shown in Fig. 2(e), where a normal distribution is used to model cross-sectional area.

Porosity inspections were additionally performed on the representative lattices used for geometry inspection to verify that lattices were near-fully dense. The Volume Graphics VGDefX porosity inspection algorithm was used, with a minimum pore volume of eight voxels. Porosity inspection was digitally constrained to analyze only the lattice part of the sample as opposed to inclusion of the baseplate.

As is evident from strut shown in Fig. 2(d), conditions for tensile testing cannot be adequately met in the as-printed condition at the size scale of lattice struts due to the tortuous surface topography [37]. While tensile testing is typically used to assess basic material properties, the small-scale nature of lattice struts and the accompanying surface defects can obscure tensile test assessment, which relies upon an assumption of acceptably nominal geometry and ideal test conditions [38]. Therefore, to efficiently characterize hardness of the lattice struts, representative samples were designed and manufactured for testing. Fundamentally, hardness testing allows for the inspection of local mechanical properties. This property can be correlated to tensile strength and is also an indicator of machinability, wear resistance, toughness, and ductility [39].

Hardness samples consisted of columns of lattice cells printed across the explored parameter space. Baseplates were printed using nominal manufacturing parameters to ensure good build quality at the bulk scale, with cell columns made at the appropriate laser powers. Samples were first manufactured and separated from the buildplate using wire EDM. Then, samples were cut to approximate size using wire EDM to the geometry shown in Fig. 3. Following this initial sectioning, samples were mechanically polished to 1200 grit using a Buehler EcoMet 6 grinder-polisher to create a surface appropriate for hardness testing. Hardness testing was performed using Vickers microindentation hardness at 300 gf in accordance with ASTM E384 [39]. Several indentation forces were tested, and it was found that this force level produced indents with diagonal lengths of ∼40 *µ*m, the approximate size suggested by the standard. Three indentations, one in the upper, middle, and lower sections were made for each strut at the approximate axial midpoint between nodes. Six struts were tested for each laser power for a total of 30 struts tested and 90 total indentations. An example of this indentation pattern is shown in Fig. 3(b) captured using a Leica DVM-6 digital microscope.

Following geometric characterization, full-scale samples were compression tested using an Instron 5982 equipped with a 22-kip (98 kN) load cell. Compression tests were performed at a constant displacement rate of 0.021 mm/s, equivalent to a quasi-static strain rate of 0.005 s^{−1}, in accordance with ASTM E9 [40]. It should be noted that all stress-strain measurements utilize the nominal bounding box of the lattice area structure such that the initial length used in strain calculations is 5 mm and the area used in force calculations is 196 mm^{2} (14.1 mm × 14.1 mm). Force and displacement data were gathered at 50 Hz to ensure all relevant response details were captured. Crosshead position was used as displacement data.

Once compression testing was performed, raw data were imported into matlab for further analysis. Using force-displacement data and sample dimensions, stress-strain data were calculated and subsequently several properties. Evaluated properties of interest included elastic modulus (E), yield stress (YS), ultimate tensile strength, maximum compressive stress (MCS), plateau stress (PS), energy absorption (EA), and energy absorption rate. Elastic modulus was calculated by fitting a first-order polynomial to the linear section of the load curve and extracting the slope value. This section was defined as 0.015–0.03 mm/mm strain. Yield stress was calculated by using this modulus value and a 0.02% offset from the fit line used for modulus calculation. Maximum compressive stress was calculated as the first extremum of the initial peak of the stress-strain curve. Plateau stress was defined as the average stress value in the 20–40% compressive strain range and energy absorption was defined as the integral of the stress-strain data up to the plateau stress end strain, as per ISO 13314 [41]. These methods were used to determine properties to be compared across each power setting to assess any discernible trends relating these properties to the manufacturing parameter variation.

## 3 Results

Geometric analysis, performed using the method described above, resulted in a significant amount of data, with ∼450,000 data points representing major and minor best-fit elliptical axis lengths, eccentricity, area moment of inertia, and cross-sectional area gathered from CT inspections of five representative samples. Due to this large amount of data, only select raw results are shown in Fig. 4 to provide a visual of the shape of the data distributions. This figure contains histograms and fitted distributions of strut cross-sectional area presented as probability density functions (PDF). Summary results in the form of a box-and-whisker plots are shown in Fig. 5.

To demonstrate the geometric inspection method results, Fig. 4 shows an exemplar strut characteristic, cross-sectional area, for all struts in each CT inspected sample that was analyzed. As can be noted from this chart, the varying laser powers resulted in significant variation between samples with regard to strut cross-sectional area as well as wide statistical variation within samples. When compared to the nominal value indicated by the green vertical line, it can be seen that significant variation in median cross-sectional area, indicated by the dashed line, exists between samples manufactured at different laser powers. The median value was lowest in the 156.0 W samples and generally increased as the laser power increased, except for the 234.0 W sample. The spread of the data, described by the variance (*σ*^{2}) displayed in the title of each chart, was lowest in the nominal and increased as power deviated from nominal, with variation widest in the lower power samples.

Geometric data were found to not be normally distributed through use of the Kolmogorov–Smirnov (K–S) statistical test for normality. Upon further inspection by way of probability plots, it was found that the best-fit normal distributions modeled the data well within approximately ±1.5 standard deviations, depending upon the geometry characteristic being modeled. However, data at the tails of the measured distributions deviated significantly from the fit normal distributions. To address this shortcoming of the normal distribution models, kernel density estimation (KDE) functions were used to model the data with a normal distribution as the kernel. These nonparametric functions modeled the data well throughout the range of cross-sectional areas observed. This was visually confirmed through use of a quantile-quantile plot.

Geometry results further indicated that horizontal struts had lower average quality than did the diagonally printed struts. This can be seen in Fig. 4 by comparing the fit KDE distributions of the diagonal and horizontal strut cross-sectional area data indicated by the solid and dashed line, respectively. These charts indicate that the horizontal struts deviate significantly from that of the overall distribution, as evidenced by the wider spread of these distributions in this figure.

Figure 5 presents summary data of the CT-based geometry inspection results presented as box-and-whisker plots, where the centerline of the box indicates the median value, box extents indicate the interquartile range (25th and 75th percentiles), and whisker ends indicate the 5th and 95th percentile of the distribution data. For simplicity of visualization, points beyond these whisker extents are not shown. Raw data are not plotted due to the large amount of gathered information, with greater than ∼11,000 data points for each individual distribution. As can be seen in Fig. 5, as laser power increased, the geometry-related metrics generally increased in magnitude as well. This trend was most evident in the major axis, where median values range from 546 *µ*m in the lowest power (175.5 W) sample to 619 *µ*m in the 214.5 W sample, a cumulative increase of 13.4%, before dropping to 595 *µ*m in the highest power (234 W) sample. Similarly, median area moment of inertia, *I _{x}*, ranges between 0.0035 mm

^{4}and 0.0057 mm

^{4}, a percent increase of 63%. Similar trends can be observed in minor axis, eccentricity, and cross-sectional area.

To test for variation between geometric characteristics at power levels, the Kruskal–Wallis test was used. The Kruskal–Wallis test is a nonparametric version of the one-way analysis of variance (ANOVA) test, but does not require the sample populations to be normal distributions as in classic ANOVA. This test was performed to compare the five tested power levels for each geometric characteristic. For all five geometric characteristics shown in Fig. 5, the null hypothesis was rejected with *p*-values less than 0.001, indicating that the samples did not come from the same distribution. Furthermore, a pairwise comparison of each power level population within each geometric characteristic was performed using the Tukey–Kramer test. The results of this test further indicated that differences in geometric characteristics at each tested power level were statistically significant, with all pairwise *p*-values below 0.001.

Beyond variation in strut geometry characteristic sizes, notable deviation was seen from the nominal values. Major axis length, cross-sectional area, and area moment of inertia each had all or most of the median values of these characteristics above the nominal value. This can be seen by comparison to the nominal value indicated by the horizontal, dashed line. In particular, major axis length was far above the nominal value of 508 *µ*m, with the extreme case being the 214.5 W sample in which the median value was 22% above nominal at 619 *µ*m. Conversely, minor axis length was less than the nominal value in each of the inspected samples. In the lowest power sample, the median minor axis length value was 12% below nominal at 448 *µ*m. The combination of these trends in major and minor axis lengths resulted in highly elliptical cross sections, as quantified by the eccentricity parameter. Eccentricity median values can be seen to follow the same general trend as these parameters, but to a lesser magnitude, meaning the cross sections become more elliptical as laser power increased.

Porosity inspections of lattices indicated that, at the voxel-side length resolution of 15.1 *µ*m used in this study, all inspected samples were 99.9% dense, suggesting that porosity should not be a major contributor to any observed performance variation. In addition, no statistically significant differences were found between porosity distributions within the samples. Although gas pores can be seen in the micrograph of the polished section in Fig. 3, these pores are near or beyond the limit of detectability with the CT parameters utilized in this study. However, prior work by the authors has shown that pores of this size contribute very little to tensile mechanical performance variation in microscale LPBF features [42] and are thus expected to have a negligible impact on mechanical testing results.

A single, representative compressive stress-strain curve from each set of laser power level is presented in Fig. 6. This figure shows that the samples produced at higher laser power exhibited higher stresses at given strain values. Increasing stiffness can be seen by variation in the slope of the elastic portion of the curve. This region, indicated by a dashed line rectangle, is shown as an inset within Fig. 6 for better viewing. Moduli from the plotted curves vary between 144 MPa in the 156 W sample and 180 MPa in the 234 W sample. Similarly, the initial peak of the data, occurring around 0.07 mm/mm strain, often referred to as the first MCS, is indicative of the initial plastic hinging failure of the lattice struts. As can be seen in this figure, the stress value increases with increasing laser power, with the lowest power sample MCS stress at 5.2 MPa and the highest power at 6.9 MPa, despite little variation in the global strain value of this event. Following initial plastic hinging behavior, the stress level of the sample decreases as the deformation is absorbed by the first cell failure. Upon densification of the initial failed cells, the stress value rises again as the second layer begins to plastically fail. After a period of absorbing more deformation, the sample begins densification at approximately 0.4 mm/mm strain, after which the sample begins to behave as a solid piece of material as opposed to a cellular structure. Finally, although the samples appear to follow a similar pattern of behavior throughout compression up to 0.5 mm/mm strain, the magnitude of stress values change significantly as a function of the laser power used to manufacture the component. Although representative stress-strain curves are presented for clarity, these trends were observed across all samples mechanically tested, as will be presented in the ensuing.

Extracted properties from this data are shown in graphical form in Fig. 7 and tabular form in Table 3. Mechanical data presented in Fig. 7 are overlaid with a dashed line indicating the line of best fit, calculated using linear least squares. The corresponding *R*^{2} values, indicating how well the line fits the provided data, are shown as figure annotations. Dotted lines are similarly plotted representing the 95% confidence interval of the true model of the presented relationships and were calculated using matlab. Data within each group were fairly concentrated with relatively low variation. Modulus and yield stress both exhibited the most variation and the least distinct groups, as seen by the relatively low overlap between sample groups in the plastic properties of maximum compressive stress, energy absorbed, and plateau stress, as compared to modulus and yield stress. As can be seen from the high *R*^{2} values (∼≥0.90), these process-structure relationships are well-modeled by linear fits in all cases within the tested power range, with the exception of modulus, where the *R*^{2} value was relatively low at 0.57. It is important to consider that outside of the tested power range the fidelity of the developed PS models likely drops significantly. For instance, at very low powers the laser would not sufficiently fuse the metal powder feedstock and thus not create a lattice that could follow the modeled relationships. Plastic properties are shown to have the best correlation with laser power, shown by the *R*^{2} values between 0.91 and 0.94. For each measured property, the nominal power, 195.0 W samples performed slightly above expected based on the modeled trend. Table 3 presents the mean value and standard deviation for each calculated property at the specified power settings.

Laser power (µ ± σ) | |||||
---|---|---|---|---|---|

156 W (80%) | 175.5 W (90%) | 195.0 W (100%) | 214.5 W (110%) | 234.0 W (120%) | |

Modulus (MPa) | 145 ± 5 | 151 ± 17 | 177 ± 17 | 177 ± 10 | 179 ± 7 |

Yield stress (MPa) | 4.26 ± 0.29 | 4.57 ± 0.14 | 4.97 ± 0.14 | 5.12 ± 0.09 | 5.43 ± 0.14 |

Maximum compressive stress (MPa) | 5.37 ± 0.17 | 5.71 ± 0.09 | 6.30 ± 0.13 | 6.52 ± 0.11 | 6.83 ± 0.11 |

Energy absorbed (MW) | 1.87 ± 0.08 | 1.97 ± 0.02 | 2.37 ± 0.04 | 2.43 ± 0.03 | 2.61 ± 0.05 |

Plateau stress (MPa) | 4.82 ± 0.17 | 5.08 ± 0.05 | 6.31 ± 0.13 | 6.46 ± 0.06 | 6.99 ± 0.13 |

Laser power (µ ± σ) | |||||
---|---|---|---|---|---|

156 W (80%) | 175.5 W (90%) | 195.0 W (100%) | 214.5 W (110%) | 234.0 W (120%) | |

Modulus (MPa) | 145 ± 5 | 151 ± 17 | 177 ± 17 | 177 ± 10 | 179 ± 7 |

Yield stress (MPa) | 4.26 ± 0.29 | 4.57 ± 0.14 | 4.97 ± 0.14 | 5.12 ± 0.09 | 5.43 ± 0.14 |

Maximum compressive stress (MPa) | 5.37 ± 0.17 | 5.71 ± 0.09 | 6.30 ± 0.13 | 6.52 ± 0.11 | 6.83 ± 0.11 |

Energy absorbed (MW) | 1.87 ± 0.08 | 1.97 ± 0.02 | 2.37 ± 0.04 | 2.43 ± 0.03 | 2.61 ± 0.05 |

Plateau stress (MPa) | 4.82 ± 0.17 | 5.08 ± 0.05 | 6.31 ± 0.13 | 6.46 ± 0.06 | 6.99 ± 0.13 |

Note: Each power value is shown with the corresponding percentage of nominal power. Variation in properties for mean high and low laser powers is shown as percentage of the mean property at nominal power settings.

Hardness data, taken using Vickers microindentation testing, are shown in Fig. 8. As can be seen from this chart, the mean hardness value was similar across the entire power range. As suggested by the overlapping confidence intervals and confirmed using a one-way ANOVA test, which yielded a *p*-value of 0.805, no statistically significant difference existed between the hardness of these specimens.

## 4 Discussion

As seen in the present results, increasing laser power resulted in lattices with increasing elastic modulus, yield strength, maximum compressive strength, energy absorption, and plateau stress. Regarding hardness testing, no significant difference was seen between the hardness of lattices produced at differing powers, suggesting that variation in structural performance was not related to changes in the constitutive properties of the material resulting from parameter variation. This finding is consistent with others who have examined the hardness of 316L SS produced using different manufacturing parameters [43–45]. For example, Eliasu et al. [43] probed 316L SS manufactured at a variety of laser parameters using Vickers hardness testing, as performed in the present study, and found no statistically significant difference among bulk samples produced within the range of energy density values tested here (80–120 J/mm^{3}). It is worth noting, however, that hardness testing generally indicates yield strength according to Tabor’s relationship [46]. It is only one of many properties that impact AM part performance and is not an all-encompassing indicator of structural performance.

Others have noted, particularly in components with microscale features such as lattices, that UTS and elongation to failure can be affected by printing parameters. However, this is primarily due to the presence of surface topography or large amounts of internal microporosity as opposed to constitutive material differences [42]. The samples examined in the present work consisted of negligible porosity in all cases, as verified by inspection of the computed tomography data, which indicated all samples were >99.9% dense. Additionally, porosity has consistently been found to have minimal effects on lattice performance, as demonstrated in Ref. [47], which simulated lattices with porosity distributions up to 1%, well above the porosity content of the samples used in the present study. Furthermore, porosity effects are nearly always a second-order factor, being largely overshadowed by geometry changes and surface topography effects [37,47].

Geometry inspection results show a correlation with lattice structural performance, unlike hardness values. This can be seen by comparison of geometry measures in Fig. 5 and performance metrics in Fig. 7, which both increase as laser power increases. A direct comparison of the median geometry characteristics and mechanical performance metrics is shown in Fig. 9. This figure models the many structure-property relationships for octahedron lattices using a linear regression first-order modeling approach. In this figure, each row of scatter plots corresponds to a mechanical property and each column to median geometry characteristics.

As can be seen in this figure, significant variation exists among the correlation of structural geometry characteristics and property values, with *R*^{2} goodness of fit values shown on color-coded boxes ranging from 0.46 to 0.86. Elastic modulus is poorly related to all measured geometric characteristics, as evidenced by the *R*^{2} values in the range of 0.52–0.65. All other mechanical properties have varying correlations with geometric characteristics. Notably, eccentricity is the best predictor of all measured mechanical properties. This geometry characteristic yields relatively high *R*^{2} values of 0.86–0.87 for maximum compressive stress, energy absorption, and plateau stress. Eccentricity is still the best predictor for yield strength but has a slightly lower *R*^{2} value of 0.78 in the best case. Similarly, major axis predicts mechanical properties well, being the second-best predictor as measured by the *R*^{2} value for all mechanical properties. Conversely, minor axis is a consistently poor indicator for all mechanical properties, with no *R*^{2} values above 0.58. In the present data, eccentricity had higher correlation with mechanical performance than cross-sectional size parameters.

While the elastic performance of the structures is relatively unaffected by the changes in strut geometry quantified here, the plastic properties of the lattice (MCS, EA, PS) are impacted by subtleties in lattice geometry, as indicated by their well-modeled relationships. Factors contributing to these effects may be stress risers from tortuous surface topography or buckling-initiating surface features, which would not affect a strut’s elastic performance. These phenomena are likely contributing to the low *R*^{2} values of elastic modulus models and relatively high *R*^{2} values of plastic property models.

As shown by the deviation of the 234.0 W datapoints and the plotted line in Fig. 9, the highest power sample mechanical performance is not well-modeled by the relationship derived using all sample performance points. The authors hypothesize that the highest power lattice exhibited the highest performance metrics but did not have correspondingly large geometry due to geometric factors not assessed here. In other words, the laser power effects may have created a maximum strut cross-sectional geometry at 214.5 W, but this may not be true for other aspects of lattice geometry. For example, the increased laser power and correspondingly large melt pools [28,29,48] may have created strut intersections with large fillet radii. As documented in the literature and as observed in other work by the author, increasing fillet radii can substantially increase lattice performance, particularly in bending-dominated structures such as the octahedron unit cell geometry used in the present study [49–52]. Similarly, Portela et al. [52] observed nodal geometry stiffening lattices and impacting bending-dominated lattices more than stretching-dominated ones. Consequently, the authors believe that alternative geometric factors beyond what were measured in the present study, such as fillet size, contributed to the high mechanical performance of the 120% of nominal laser power samples. However, it should be noted that strut geometry measures can be seen to account for the bulk trend of the lattice performance variation observed in the present study. Overall, the above structure-property models have demonstrated the viability of using structure-based inspection procedures for octahedron lattice qualification.

In summary, the above results indicate that material hardness and porosity distributions did not have significant variation across the conditions tested. In contrast, strut size and shape had a marked variation across the tested conditions. The corresponding variation in lattice compressive performance was thus likely attributable to the variation in strut size and shape. The impact of strut size and shape have been well documented in the literature, especially as having more effect on microscale feature performance than porosity. As described above, the variations in laser power give rise to variations in melt pool size and shape, translating to large differences in strut size and shape, which ultimately gave rise to correlated performance variation.

## 5 Conclusions

In the present study, octahedron lattice structures were manufactured at a range of laser powers from 316L stainless steel, inspected for geometry variation using computed tomography, hardness tested, mechanically tested, and the relationships between geometry and mechanical performance modeled. Strut geometry characteristics were quantified through a rigorous image processing methodology, demonstrating wide variation in printed geometry with significant deviation from nominal design. Material property variation was assessed through Vickers microindentation hardness testing, finding no significant difference among manufactured samples. Mechanical compression testing was performed to assess structural performance and found that significant performance variation up to 60% existed between lattices produced at varying laser powers within a ±20% range of nominal. Several sets of regression-based statistical models were created using the gathered process, geometry, material, and performance information. Models included process-structure, process-property, and structure-property models, quantifying with high correlation accuracy these relationships. The major conclusions of this study are summarized as follows:

Octahedron lattice performance metrics change up to 60% as a function of laser power within a ±20% window of nominal laser power.

Hardness differences were not observed between lattices manufactured at various laser powers.

Lattice strut geometry characteristics varied as a function of laser power and were found to contribute to performance variation as a function of power.

Plastic mechanical properties can be modeled with

*R*^{2}values of up to 0.86 using only strut median value geometry characteristic information.Elastic modulus is not well correlated with strut geometry in the range of tested power settings used in the present study.

## Acknowledgment

This work was supported by the National Science Foundation (CMMI-1825640); the U.S. Department of Energy (DE-EE0008303, DE-NA0003525); and the American Society for Nondestructive Testing Graduate Fellowship to EWJ. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

## Author Contribution Statement

**Elliott W. Jost:** Conceptualization, Methodology, Software, Formal Analysis, Investigation, Writing – Original Draft, Writing – Review & Editing, Visualization. **Jonathan Pegues:** Conceptualization, Resources, Writing – Review & Editing. **David G. Moore:** Conceptualization, Resources, Writing – Review & Editing. **Christopher Saldaña:** Conceptualization, Resources, Writing – Review & Editing, Supervision.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.