Additive manufacturing (AM) technologies such as laser-based powder bed fusion (LB-PBF) facilitate the fabrication of complex lattice structures. However, these structures consistently display dimensional variation between the idealised and as-manufactured specimens. This research proposes a method to characterise the impact of common LB-PBF powders (aluminium and titanium alloys) and geometric design parameters (polygon order, effective diameter and inclination angle) on section properties relevant to stiffness and strength of as-manufactured strut elements will be quantified. Micro-computed tomography (µCT) will be applied to algorithmically characterise the as-manufactured variation and identify a scale threshold below which additional geometric resolution does not influence the section properties of as-manufactured parts. This methodology provides a robust and algorithmic, design for additive manufacturing (DFAM) tool to characterise the effects of manufacturing and design parameters on the functional response of AM strut elements, as is required for certification and optimisation.
Research Article
The effect of geometric design and materials on section properties of additively manufactured lattice elements
https://doi.org/10.21203/rs.3.rs-1976866/v1
This work is licensed under a CC BY 4.0 License
Journal Publication
published 03 Apr, 2023
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Laser-based powder bed fusion
Additive manufacturing
Geometric analysis
Micro-computed tomography
Additive manufacturing defects
Design for additive manufacturing
Additive manufacturing (AM) enables the layer-by-layer fabrication of three-dimensional geometry directly from computer-aided design (CAD) data, thereby providing potential advantages over conventional manufacturing. For example, AM can fabricate complex, topology optimised geometry as a single structure, whereas conventional manufacturing may be constrained by the need for tooling access [1]. Despite these advantages, AM processes are subject to a series of technical challenges including relatively low production rates, thermal stresses, potentially high material costs, and geometric uncertainties in the as-manufactured geometry [2]. Metal additive manufacturing (MAM) enables fabrication in a range of fusible metal alloys including light alloys, superalloys, and tool steels. As such, MAM is well suited for the fabrication of high-value lattice applications including medical implants, aerospace components, and custom tooling [3]. While understanding MAM defects within individual lattice strut elements is required to optimise mechanical performance, few fundamental studies have been executed. In response to this shortcoming, a methodology is proposed to algorithmically quantify the section properties of as-manufactured strut elements. This methodology is demonstrated with a design of experiments (DOE) that considers a range of design-relevant control factors for strut elements fabricated by laser-based powder bed fusion (LB-PBF) which includes various cross-section shapes (triangular face up, triangular face down, square, octagonal, and circle), materials (AlSi10Mg and Ti6Al4V), and strut element inclination angles, \({\alpha }_{inclination}\), (35°, 45°, 90°). To enable the algorithmic characterisation of the geometric attributes of these as-manufactured lattice strut elements, micro-computed tomography (µCT) imaging techniques are applied to characterise the three-dimensional geometry of manufactured specimens [4]. This outcome contributes to the fundamental understanding of analytical and as-manufactured quantification methods for AM strut elements, as well as providing a robust DFAM tool for lattice certification and structure optimisation.
1.1 METAL ADDITIVE MANUFACTURING
As a sub-classification of AM, MAM processes sequentially fabricate three-dimensional metal components based on digital CAD data [5, 6]. Fabrication using MAM is typically achieved by the iterative fusion, or adhesion, of metallic input material over a series of layers based on cross-sections taken from digital design data [7]. Prominent examples include sheet lamination (SHL), binder jet technology (BJT), material jetting technology (MJT), material extrusion (MEX), powder bed fusion (PBF), and directed energy deposition (DED). MAM provides an opportunity to fabricate novel, mass optimised, or high-value components such as medical implants and lightweight aerospace components. In this research, LB-PBF is used to fabricate individual strut element specimens due to the compatibility of this method with the fabrication of high-resolution features at small scales with a relatively high degree of dimensional control [8].
1.1.1 MAM DEFECTS ASSOCIATED WITH
As with any manufacturing process, defects occur during the MAM process. In this case it is due to complex physical phenomena and process parameters such as laser power, scan speed, hatch spacing, powder features, powder packing arrangements, density distribution, morphology, as well as thickness, there is high possibility of defect formation [9]. Echeta et al. [10] classified LB-PBF defects into three categories: porosity or incomplete fusion, residual stresses, and surface texture. Zhang et al. [9] defined this porosity as spheroidal voids within the fused powder, where the pore diameter is up to approximately 100 µm. This porosity is formed due to high cooling rates during solidification, leading to dissolved gas that cannot escape from the melt pool which results in voids once the melt pool has solidified. Additionally, insufficient input energy may lead to incomplete fusion and can result in pores in the order of 500 µm. Insufficient energy density or using inappropriate process parameters may cause balling effects on the surface of the fabricated object. The main driving mechanism for a balling defect in LB-PBF is Rayleigh instability and a lack of wetting produces a segmented melt pool with the associated formation of ball shapes [11]. Residual stresses are formed during the PBF processes due to rapid temperature cycling rates that can cause cracking in lattice strut elements [12]. Furthermore, shrinkage occurs during phase transition from a liquid melt pool to a solid structure, potentially leading to surface cracking. Surface texture variation is observed on MAM lattice structures, which is caused by several phenomena. Layer-wise fabrication methods employed during MAM inherently lead to the stair-stepping phenomenon [10]. This is most clearly observed on surfaces inclined to the build direction, with a distinct characteristic surface roughness on both sides of the inclined surface of a lattice structure [4] [13] [14]. The magnitude of roughness on both the upward and downward faces of struts is a function of inclination angle as well as the associated process parameters. This as-manufactured roughness may provide advantages in biomedical applications where it can be beneficial as a biomimetic surface encouraging cellular adhesion [15].
The primary commercial opportunity for AM production is typically high-value applications; consequently, methods to predict and quantify the magnitude and influence of AM defects in as-manufactured components are of critical importance[16] [17]. Echeta et al. [10] comprehensively reviewed the primary measurement methods available for quantifying MAM defects. These methods include µCT, scanning electron microscope (SEM), physical inspection, optical microscopy, and Archimedes’ method. Each of these methods has a specific set of capabilities for quantifying certain attributes of observed manufacturing defects. For this research, µCT provides a robust tool to quantify the geometric defects introduced during MAM and to characterise the associated section properties.
1.2 MECHANICAL RESPONSE OF LATTICE STRUT ELEMENTS
Lattice structures are commonly employed in applications subject to compressive or absorption loading conditions. The mechanical response of individual strut elements depends upon the associated unit cell topology and associated loading conditions. The strut element loading can be categorised as being either bending or stretch dominated [18]; resulting in strut elements subjected predominantly to bending moments or axial loads, as illustrated in Fig. 1a and Fig. 1b respectively. The structural response of the lattice can be determined by Maxwell’s criterion which considers pin-jointed structures and predicts the determinacy of the structure based on the number of struts and nodes [19]. Strut element cross-section properties contribute substantially to structural response, especially for bending and buckling modes. This research provides insight into the structural efficiency of as-manufactured strut elements according to the associated material and geometric design parameters.
1.3 STRUT CROSS-SECTION MANUFACTURABILITY
Alghamdi et al. [21] investigated the effect of polygon order, \(p\), (triangular, square, octagonal, and circular) on the geometry of as-manufactured lattice strut elements fabricated with LB-PBF in aluminium alloy AlSi10Mg, and titanium alloy Ti6Al4V. They observed the tendency for nominally triangular and square cross-sections to become circular upon fabrication, as quantified by the isoperimetric quotient1. This observed effect was particularly strong in the aluminium strut elements and for relatively small cross-section areas. Furthermore, triangular cross-sections were observed to have greater manufacturability when the triangle is oriented with a vertex pointing down (toward the build platen), compared to a vertex pointing upward (away from the build platen). The enhanced manufacturability observed for downward-facing triangular sections appears to be due to a combination of cross-section area and apparent inclination angle; where the downward-facing triangles show both a reduced cross-section area and an increased apparent inclination angle as shown in Table 1. This research proposes to extend the findings of prior investigations to characterise the functional response of as-manufactured strut elements.
Table 1: Effect of apparent angle (blue faces) and edge angle (orange edges) on the manufacturability of triangular sections with vertex pointing upward and downward.
[1] The isoperimetric quotient of a closed contour is the ratio of the contour area to the area of a circle of equal perimeter to the closed curve. It is a measure of ‘circularity’, where a circle yields an isoperimetric quotient of unity [22] W.-Y. Jywe, C.-H. Liu, and C. o.-K. Chen, "The min–max problem for evaluating the form error of a circle," Measurement, vol. 26, no. 4, pp. 273-282, 1999/12/01/ 1999, doi: http://dx.doi.org/10.1016/S0263-2241(99)00052-4.
Cross-section properties can be used to characterise structural design efficiency, including the second moment of area, \(I\); the radius of gyration, \({R}_{g}\); section modulus, \(Z\); elastic shape factor,
\({Ø}_{B}^{e }\) ; and failure shape factor, \({Ø}_{B}^{f }\). These properties will be calculated to analytically compare the efficiency of the specimens assessed in this research, and are briefly defined below.
2.1 SECOND MOMENT OF AREA
The second moment of area is a measure of the capacity of a column to resist buckling or for a beam to resist bending [23]. For the cross-sections of interest, the second moment of area is measured concerning an infinitesimal area, \(dA\), at some distance, \(y\), from the neutral axis (Eq. 1) [24]. For many regular polygonal shapes, such as a circle of arbitrary radius, \(r\) (Eq. 2), the second moment of area can be calculated analytically. A general equation (Eq. 3) is introduced which can be used to obtain the second moment of area for any simple n-sided polygon2 [25], where \({{x}}_{{i}}\)and \({{y}}_{{i}}\) represent the Cartesian coordinates of the \(i\)-th polygon vertex.
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\({I}=\int {{y}}^{2} {d}{A}\) |
Eq. 1 |
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\({{I}}_{{c}{i}{r}{c}{l}{e}}=\frac{{\pi }}{4} {{r}}^{4}\) |
Eq. 2 |
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\({{I}}_{{p}{o}{l}{y}{g}{o}{n}}=\frac{1}{12} \sum _{{i}=1}^{{n}}\left({{x}}_{{i}} {{y}}_{{i}+1}-{{x}}_{{i}+1} {{y}}_{{i}}\right)\left({{x}}_{{i}}^{2}+{{x}}_{{i}} {{x}}_{{i}+1}+{{x}}_{{i}+1}^{2}\right)\) | Eq. 3 |
2.2 RADIUS OF GYRATION
The radius of gyration defines a theoretical distance from the cross-section’s centroid at which the cross-section area can be imagined to be concentrated to achieve the same second moment of area as the actual cross-section (Fig. 2b) [24]. It is a measure of the elastic resistance of the cross-section to buckling or bending [26], and can therefore be useful to compare how different shapes associated with the same cross-section area will behave under loading that induces either buckling or bending (Eq. 4).
\({{R}}_{{g}}=\sqrt{\frac{{I}}{{A}}}\) |
Eq. 4 |
For example, comparing \({R}_{g}\) for a triangular cross-section with that of a circular cross-section shows that triangular sections have larger \({R}_{g}\), indicating they are more efficient when resisting elastic buckling or bending than a circular section with equal area.
2.3 ELASTIC AND FAILURE SHAPE FACTORS FOR BENDING
The shape factor compares the mechanical performance (such as buckling or bending) of a cross-section of interest against a reference circular cross-section of the equivalent area [27]. The shape factor of a solid circular cross-section is unity and as this number increases, the resistance to bending and buckling increases. In addition, the dimensionless nature of shape factors allows for the comparison of shapes independent of scale. The influence of cross-section shape on stiffness can be calculated using the elastic bending shape factor, \({Ø}_{B}^{e }\)(Eq. 5). As strength depends on local stress, the section modulus, \(Z\), must be calculated (Eq. 6), where \({y}_{max}\) is the outer most fibre from the neutral axis subject to compression or tension, to quantify the bending failure shape factor, \({\text{Ø}}_{B}^{f}\). This is the ratio of any given section modulus over that of a circle of equal area (Eq. 7), \({Z}_{circle}\).
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\({{Ø}}_{{B}}^{{e} }= \frac{4 {\pi } {I}}{{{A}}^{2}}\) |
Eq. 5 |
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\({Z}= \frac{{I} }{{{y}}_{{m}{a}{x}}}\) |
Eq. 6 |
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\({{Ø}}_{{B}}^{{f} }=\frac{4 \sqrt{{\pi }} {Z}}{{{A}}^{3/2}}=\frac{{Z}}{{{Z}}_{{c}{i}{r}{c}{l}{e}}}\) |
Eq. 7 |
Structural performance in bending can therefore be characterised by selecting an appropriate combination of material and shape, for both elastic and plastic scenarios [27]. On inspection, the solid triangular cross-section under bending load is stiffer than an equivalent solid circular cross-section by 21%, whereas it has 23% lower strength.
2.4 EFFECTIVE DIAMETER
Effective diameter, \({D}_{eff}\), represents the diameter of a circle that has an equivalent area to the area of a cross-section of interest. In this research, four effective diameters are implemented (3.0 mm, 2.0 mm, 1.0 mm, and 0.5 mm) for four different cross-sections (circular, octagonal, square and triangular) with equivalent areas [1] (Fig. 3).
[2] The polygon points should be ordered in a counter-clockwise direction, voids are included by clockwise ordering.
Recent literature reviews have highlighted a lack of DFAM tools that can quantify the structural integrity of as-manufactured AM strut elements. Cross-section design has a significant impact on strut performance; thus, the shape factor of as-manufactured strut elements should be characterised to assess the strength and stiffness of proposed designs. This research proposes a fundamental methodology for the characterisation of as-manufactured strut elements to quantify the associated geometric and functional properties. This generalisable DFAM tool is implemented specifically on strut element specimens fabricated by PB-LBF. Figure 4 illustrates the proposed methodology's workflow, which can be applied as a guideline to any manufacturing process to ensure optimal production and certification. The proposed method can be classified into the following steps: CAD design, analytical quantification, PB-LBF fabrication, X-ray µCT, as-manufactured quantification, and statistical analysis.
3.1 DESIGN OF EXPERIMENTS
The design of experiments (DOE) contains four control factors including two material types implemented separately as powder feedstock for the LB-PBF process (aluminium alloy AlSi10Mg, and titanium alloy Ti6Al4V); three inclination angles (\({\alpha }_{inclination}\) = 90°, 45°, 35°), four nominal diameters (D = 3.0, 2.0, 1.0, 0.5 mm), and four polygonal cross-sections (circle, octagon, square, triangle vertex up, triangle vertex down). These specimens were arranged in rows dependent upon inclination angle and nested on a plate feature 3 mm thick and then extruded from the plate by a length of 15 mm (Fig. 5), resulting in a total of 120 struts.
3.2 LASER-BASED POWDER BED FUSION
An SLM Solutions 250HL machine was implemented to fabricate the Ti6Al4V struts, and an SLM Solutions 125HL machine manufactured the AlSi10Mg struts. The operational parameters for each machine are displayed in Table 2. Following fabrication, the parts were cooled within the machine to room temperature, and then electrical discharge machining (EDM) was utilized to wire cut the strut specimens from the build plate.
3.3 LASER SCAN STRATEGY
A standard hatch infill scan strategy was implemented to fabricate these struts as displayed in Fig. 5. Initially, a scan path is applied that follows the slice contour of the STL file (this scan is referred to as a border scan). To achieve a high geometrical accuracy to the idealised model, the border scan is positioned inwards from the nominal slice contour by 60 µm (Ti6Al4V) and 120 µm (AlSi10Mg) for melt pool compensation. A secondary border scan is then implemented, within the perimeter of the original border scan. This secondary scan is inset by 90 µm (Ti6Al4V) and 150 µm (AlSi10Mg) from the initial border scan. A hatch pattern inclined at 90\(^\circ\) is then implemented to scan the bulk of the strut's cross-sectional area. This pattern is inset by 90 µm (Ti6Al4V) and 150 µm (AlSi10Mg) from the secondary border scan. To ensure complete melting the hatching is rotated by 66.9\(^\circ\) at each consecutive scan layer. Finally, a single contour scan is implemented, overlapping the hatch pattern perimeter, to smooth the non-uniform edges of hatch pattern. To ensure the contour scan overlaps the edges of the hatch pattern it is inset by 90 µm (Ti6Al4V) and 150 µm (AlSi10Mg) from the secondary border scan.
3.4 MICRO-COMPUTED TOMOGRAPHY
To quantify the strut geometries, a Bruker SKYSCAN X-ray micro µCT machine (Bruker Pty Ltd.) was utilized. The technology used an X-ray tube current of 100 µA, an acceleration voltage of 100 kV, resulting in a pixel size of 8 mm. The AlSi10Mg samples employed an Al 1 mm filter, while the Ti6Al4V samples used a Cu 1 mm filter. To reconstruct the cross-section slices acquired from the µCT, nRECON shadow image reconstruction software (Bruker Pty Ltd.) was employed. These reconstructed grey scale images were then used to identify the strut element boundary and quantify the as-manufactured section properties.
3.5 ALGORITHMIC IMPLEMENTATION
The cross-section geometric properties for both idealised and as-manufactured cases are calculated using a custom-developed MATLAB (R2020b) script that was developed by Almalki et al. [28]. The code starts by obtaining the idealised properties based on the polygon order from the DOE. Then, µCT cross-section images for the as-manufactured case are imported and converted into binary images. From this data, the geometric properties are calculated, as a function of the angular orientation of the cross-section about its centroidal axis as shown inFigure 6 for the second moment of area. To provide statistical distributions of cross-section response, this method is repeated for numerous sequential µCT cross-section images within the image stack
This section presents the cross-section geometric properties of the proposed DOE for both idealised and as-manufactured cases. It is divided into six subsections: The idealised case, which demonstrates how polygon order affects the geometric properties; the comparison between idealised polygon cross-sections and as-manufactured strut element cross-sections for both materials; the second moment of area; the radius of gyration; the elastic shape factor, used to evaluate the stiffness of the as-manufactured geometry; the failure bending shape factor, used to evaluate the strength of the as-manufactured geometry.
4.1 GEOMETRIC PROPERTIES OF IDEALISED STRUT ELEMENTS
The stiffness and strength of strut element specimens associated with the same cross-section area may be characterised by quantifying the idealised geometric properties including the second moment of area, \({I}_{ideal}\), the radius of gyration, \({R}_{g,ideal}\), elastic bending shape factor, \({Ø}_{B,ideal}^{e }\), section modulus, \({Z}_{ideal}\), and bending failure shape factor, \({Ø}_{B,ideal}^{f }\).
These geometric properties are quantified in Fig. 7 for various regular polygonal cross-sections associated with equal cross-section area, for a range of orientations achieved by incremental rotations, \({\theta }_{rotation}\), about the centroid. Figure 7b-d show that the second moment of area, the radius of gyration, and the elastic shape factor remain constant while rotating these idealised polygonal shapes. The elastic shape factor of the octagonal cross-section has a very slight increase of 0.2% over the circular section, whereas square and triangular shapes indicate a 4.7%, and 20.9% increase, respectively. Meanwhile, the section modulus and failure shape factor show a dependence on the polygon’s orientation that increases with lower-order polygons, this is associated with vertex orientation. For example, the failure shape factor of the idealised triangular cross-section ranged from − 23% lower (pointing up, \({\theta }_{rotation}=0^\circ ,120^\circ ,240^\circ\)) to 55% higher (pointing down, \({\theta }_{rotation}=60^\circ ,180^\circ ,300^\circ\)), when compared to that of the circular cross-section, this can clearly be seen in Fig. 7f with the number of peaks corresponding to the number of vertices.
These results are known in structural engineering but have yet to be applied in the design of lattice structure elements enabled by AM. The following subsections investigate how the as-manufactured cross-section varies from the idealised results for these important geometric characteristics.
4.2 IDEALISED VERSUS AS-MANUFACTURED CROSS-SECTION AREA
The as-manufactured polygon cross-section area, \({A}_{CT}\), is affected by manufacturing processes, leading to variation between the idealised and as-manufactured strut element. The effective diameter, \({D}_{eff}\), inclination angle, \({\alpha }_{inclination}\), polygon order, \(p\), and material choice all significantly affect LB-PBF manufacturability.
Figure 8 shows the variation in cross-section area within each of the as-manufactured strut elements. The box plots provide a graphical statistical summary for each cross-section image for the given strut. These include the rectangular box representing the interquartile range (IQR) (25–75% percentiles), the whiskers extending up to 1.5xIQR and the median as the horizontal line within the box. The plots are presented in a graphical array as defined by material (columns), and effective diameter (rows), while cross-section shape and build inclination angle form the horizontal axis labels and the as-manufactured cross-section area forms the vertical axis labels. The expected value based on the idealised shape is presented as horizontal lines spanning the plots. For consistency, each of the geometric characteristics discussed in later subsections is presented using the same graphical array with only the vertical axis changing to reflect the relevant value.
Comparing the aluminium (left column of Fig. 8a-d) to the titanium (right column of Fig. 8e-h), generally, there is greater variation within the individual aluminium specimens than the titanium as indicated by the relative size of the IQR, the exception being at \({D}_{eff}=0.5 mm\). Furthermore, for the aluminium there is a trend that area (median) and variation in the area (box plot size) increases with decreasing inclination angle.
For the titanium (Fig. 8e-h) there is an upward trend in the position of the box plots within each graph, indicating that the cross-section area of the strut elements increases as the shape changes from circular to triangular, i.e., as the polygon order decreases, the area increases. This suggests that more material may be accumulating in the as-manufactured triangular shape than in the circular shape, even though they are intended to be the same area as indicated by the spanning horizontal line.
For the aluminium strut elements (Fig. 8a-d), the same upward trend across the shapes is not visible, however within each shape (clusters of three) there is both a downward trend in the cross-section area and in the variation of the cross-section area with increasing build inclination angle.
These trends indicate that the cross-section area of aluminium strut elements is strongly affected by inclination, accumulating more area and greater variation in the area along a strut element as the build inclination angle is decreased from 90° to 35°. Meanwhile, the titanium strut elements show far less variation in area.
4.3 SECOND MOMENT OF AREA (IDEAL VS CT)
The second moment of area for as-manufactured case, \({I}_{CT}\), is calculated based on extracted data from µCT cross-section images. This extracted data provides the outer boundary, centroid, and as-manufactured area, \({A}_{CT}\), for each image. Each fabricated strut element is imaged many times along its length. Therefore, to quantify \({I}_{CT}\) with image orientation, the extracted boundary is incrementally rotated by a small rotation angle, \({\theta }_{rotation}\), (3.6°) as shown in Fig. 9a. At each rotational angle the \({I}_{CT}\) is calculated as shown in Fig. 9b.
All values of \({I}_{CT}\) are compared with that of the idealised case of a circular cross-section, \({I}_{ideal}\), for each strut, categorized by material (AlSi10Mg and Ti6Al4V), the effective diameter, \({D}_{eff}\), shape, and inclination angle, \({\alpha }_{inclination}\), as shown in Fig. 10. The distribution of \({I}_{CT}\) is presented in the form of box plots, using the same graphical array described in subsection 4.2. \({I}_{ideal}\) is shown as horizontal line segments that increase with decreasing polygon order.
Comparing the aluminium and titanium struts at the same effective diameter, there is significantly more variation in \({I}_{CT}\) across the aluminium struts than across the titanium struts. Suggesting that titanium provides a more consistent stiffness.
The aluminium strut elements seen in Fig. 10a-d, show both \({I}_{CT}\) and \({A}_{CT}\) experience a similar trend. The magnitude and variation of \({I}_{CT}\) within the as-manufactured strut elements show a decreasing trend with increasing build inclination. This is seen by longer IQR boxes for 35° strut elements compared to the 90° struts.
Considering the titanium strut elements in Fig. 10e-h, the increase in \({I}_{CT}\) across the shapes is greater than expected, in comparison to the idealised cross-section. This corresponds with the previous observation that the as-manufactured area, \({A}_{CT}\), increased with decreasing polygon order at a given \({D}_{eff}\).
4.4 RADIUS OF GYRATION (IDEAL VS CT)
The efficiency of a cross-section shape of interest for elastic stability under compression can be evaluated using the radius of gyration, \({R}_{g}\), associated with the cross-section area of interest, as discussed in Section 2.2. Figure 11 shows box plots for the radius of gyration of the as-manufactured strut elements, \({R}_{g,CT}\), compared with the idealised case, \({R}_{g,ideal}\), represented as a green horizontal line segment for both aluminium and titanium. It does not consider the material, which is useful to evaluate the efficiency of the actual shape versus the idealised shape. Figure 11 also evaluates the quality of fabrication and the effectiveness of controlling factors such as \({{\alpha }}_{{i}{n}{c}{l}{i}{n}{a}{t}{i}{o}{n}}\).
Overall, it can be observed that the variation in \({R}_{g,CT}\) is larger in the aluminium (Fig. 11a-d) than the titanium (Fig. 11a-d) strut elements. The variation within individual aluminium strut elements is largest for the 35° and 45° build inclination angle, while the 90° cases are similar to the titanium.
The dependence on shape observed for \({A}_{CT}\) and \({I}_{CT}\) in titanium strut elements\({A}_{CT}\) and \({I}_{CT}\) is not as obvious for \({R}_{g,CT}\), with the relative increase\({\alpha }_{inclination}{R}_{g,ideal}\)
4.5 ELASTIC SHAPE FACTOR (IDEAL VS CT)
The elastic shape factor, \({Ø}_{B}^{e }\), provides the stiffness efficiency of the cross-section shape as discussed previously in Section 2.3. The as-manufactured elastic shape factor, \({Ø}_{B,CT}^{e }\) , is compared with the idealised case, \({Ø}_{B,ideal}^{e }\), in Fig. 12. With the idealised shape factor for the circular cross-section being 1.0, the octagonal, square, and triangular cross-sections are 1.002, 1.047 and 1.209 respectively. Comparing \({Ø}_{B,CT}^{e }\) to \({Ø}_{B,ideal}^{e }\) shows the effect of manufacturing defects and control factors. The orientational dependence observed in \(I\) is again observed in \({Ø}_{B}^{e }\). The shape factor removes size dependence so comparisons can be made purely on the achieved shape and not be confounded with whether more or less material is contributing to the change.
Comparing \({Ø}_{B}^{e }\) between aluminium (Fig. 12a-d) and titanium (Fig. 12e-h), generally, there is less variability within individual titanium strut elements than within individual aluminium strut elements, as can be seen by the size of the IQR for individual box-plots. This is most notable in the 35° and 45° cases. The exception appears to be in strut elements with smaller \({D}_{eff}\) of 0.5 mm, where the aluminium and titanium both show relatively large variation in \({Ø}_{B}^{e }\).
Also noticed at lower effective diameters, the elastic shape factors of the as-manufactured triangular cross-sections are producing results more in line with a circular cross-section. While at higher effective diameters the elastic shape factor of the as-manufactured strut elements better matches the idealised result. The transition for this behaviour occurs at \({D}_{eff}\) of 1.0 mm for the titanium, and between an \({D}_{eff}\) of 2.0 to 3.0 mm for the aluminium.
Another observation is that the median \({Ø}_{B}^{e }\) for each titanium strut element sits at or below the ideal value, however in the aluminium, particularly at an \({D}_{eff}\) of 1.0 and 2.0 mm, the median \({Ø}_{B}^{e }\) for the inclined circular, octagonal and square cross-sections sit above their ideal values. This appears to be an indication of defects introduced during MAM processes, suggesting that as-manufactured defects could increase local stiffness and failure response if they align favourably with loading conditions as illustrated in Fig. 7f, i.e.; unintended additional material is deposited in such a way as to increase \({y}_{max}\), aligning with the optimal orientation.
4.6 FAILURE SHAPE FACTOR (IDEAL VS CT)
The failure shape factor, \({Ø}_{B}^{f}\), can be used to evaluate the manufacturability of a strut element cross-section. With the idealised failure shape factor, \({Ø}_{B,ideal}^{f}\), for the circular cross-section being 1.0, the octagonal, square, and triangular cross-sections experience an orientation dependence and range from 0.95–1.029, 0.83–1.18, and 0.77–1.55 respectively. The failure shape factors for the as-manufactured strut elements, \({Ø}_{B,CT}^{f}\), are compared to the ideal ranges in Fig. 13.
Comparing the aluminium (Fig. 13a-d) and titanium (Fig. 13e-h) cases, the titanium strut elements show greater consistency for a given shape across the three inclination angles, and the square and triangular shapes tend to remain bound by the ideal range, with the distributions better matching the ideal range with increasing effective diameter. This trend is not observed in the aluminium strut elements which show greater variation and is particularly apparent for circular and octagonal shapes.
The implication of the large variation in the failure shape factor is that it shows both an opportunity for improved or reduced strength, based on geometric orientation concerning load conditions. An important observation from these graphs is that the median result typically sits below 1.0, meaning that orientations typically fall into the scenario of reduced strength rather than improved strength should the cross-section orientation be randomly aligned relative to the load direction.
Structural mechanics theory suggests that of the regular polygons, square and triangular cross-sections provide the greatest structural efficiency over the circular cross-section, with elastic shape factors of 1.05 and 1.21 respectively, when correctly aligned to the load direction. This suggests an opportunity for improved stiffness simply through the choice of more efficient cross-section shapes and aligning them to the load direction. For polygons with equivalent cross-section area, \(I\) is the key parameter when considering stiffness and buckling resistance. For irregular cross-sections both \(I\) and \({\text{Ø}}_{B}^{f}\) have orientation dependencies. This orientation dependence is well known and highly exploited in the fields of civil and structural engineering, through the use of shaped sections such as I-beams, though to date has yet to be applied to lattice structures. Given the failure shape factor’s dependence on orientation, it has been shown that strut elements are particularly sensitive to shape defects introduced during the AM process.
Titanium strut elements possessed more consistent geometric characteristics along their length of build for all sizes, though exhibited an unexpected increase in cross-section area with decreasing polygon order. For square and triangular cross-sections, \({\text{Ø}}_{B,CT}^{e}\) and \({\text{Ø}}_{B,CT}^{f}\) approached the predicted values of \({\text{Ø}}_{B,ideal}^{e}\) and \({\text{Ø}}_{B,ideal}^{f}\) with increasing \({D}_{eff}\).
Aluminium strut elements were subject to significant variations in geometric characteristics along their length of build, with the shape being indistinguishable for results with an \({D}_{eff}\) under 2 mm. A strong dependency on apparent angle was observed, with increasing cross-section area and variation in said area, a trend carried into all other characteristics. For the square and triangular cross-sections, \({\text{Ø}}_{B,CT}^{e}\) and \({\text{Ø}}_{B,CT}^{f}\) approached the predicted values of \({\text{Ø}}_{B,ideal}^{e}\) and \({\text{Ø}}_{B,ideal}^{f}\) with increasing \({D}_{eff}\), though results fell short when the \({D}_{eff}\) fell below 2 mm. A manufacturability limit of 2 mm is observed for all cross-sections in aluminium, with results below this being indistinguishable from the circular case.
To determine alternative cross-section shape suitability, the desired performance and the number of manufacturing defects that can be tolerated must be considered. For example, while a traditional I-beam section would provide exceptional stiffness, due to the relatively thin sections involved they would be particularly susceptible to defects at small scales and it is only appropriate for two diametrically opposed loading conditions, in complex loading conditions this section is less than optimal. The second moment of area of regular polygons is independent of the loading direction (unlike irregular polygons). Thus, resistance to elastic bending could be improved by switching to the triangular or square cross-section, regardless of load direction.
There are future opportunities to extend the methodology and analysis to other loading cases, cross-sections, and materials. Both \({\text{Ø}}_{B}^{e}\), characterising elastic bending and buckling resistance, and \({\text{Ø}}_{B}^{f}\), characterising bending failure, have been considered as these are often the dominant behaviours in lattice structures under compression. Other shape factors, such as torsion, must be considered for more complex loading cases. This could lead to an investigation of alternative cross-sections that provide structural efficiencies, such as channels and hollow sections [25]. Furthermore, this methodology can be applied to investigate the manufacturability and as-manufactured structural efficiency of other AM systems [24, 26] and materials.
Laser-based powder bed fusion processes observe a high frequency of geometrical variation between the idealised models and the as-manufactured specimens. This research has developed a design tool to algorithmically quantify the structural impact of as-manufactured geometrical defects.
The as-manufactured strut elements were imaged using µCT and then reconstructed into a stack of as-manufactured strut element cross-section images. The geometric parameters including, \({A}_{CT}\), \({I}_{CT}\), \({\text{Ø}}_{B,CT}^{e}\), and \({\text{Ø}}_{B,CT}^{f}\) have been quantified using the proposed method and verified against idealised cases.
The methodology was applied to two sets of strut elements additively manufactured, one in aluminium alloy AlSi10Mg and another in titanium alloy Ti6Al4V, with varying effective diameters, polygon orders, and inclination angles. Therefore, the proposed methodology can be applied as a DFAM tool for any AM systems and materials. The following key observations were made:
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\({I}_{ideal}\) for the square and triangular cross-sections have a 5% and 21% improvement, respectively, compared to the idealised circular cross-section. This indicates an increased elastic buckling and bending resistance of fabricated strut elements.
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\({I}_{ideal}\) of circular and octagonal cross-sections differ by only 0.2%. Displaying that polygon orders greater than eight do not significantly affect \(I\). This is most beneficial as larger polygons typically have larger file sizes, which can be reduced without losing as-manufactured shape accuracy. Fewer faces also benefits FEA, with lower computational costs.
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It was found that the variation between \({I}_{CT}\) and \({I}_{ideal}\) in aluminium is generally greater than that in titanium, indicating titanium has produces less manufacturing defects. These variations are increased with lower apparent angles due to increased rates of manufacturing defects.
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The effective diameter has a major impact on the manufacturability of the strut elements, particularly on aluminium at 0.5 mm, where it is observed that the polygon order is not contributing any differences on \({I}_{CT}\). Titanium maintains the polygon shape with fewer manufacturing defects, with a flattening of the results starting to appear at 0.5 mm.
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\({\text{Ø}}_{B}^{e}\) and \({\text{Ø}}_{B}^{f}\) are useful to determine the manufacturability of different regular polygon cross-sections. As the effective diameter increased, \({\text{Ø}}_{B,CT}^{e}\) and \({\text{Ø}}_{B,CT}^{f}\) distributions were more closely aligned with the range of their idealised values, \({\text{Ø}}_{B,ideal}^{e}\) and \({\text{Ø}}_{B,ideal}^{f}\), indicating greater accuracy in achieving idealised geometry. Titanium strut elements show this improvement in both \({\text{Ø}}_{B,CT}^{e}\) and \({\text{Ø}}_{B,CT}^{f}\), for each incremental increase in \({D}_{eff}\). Aluminium strut elements show poor replication of both shape factors at all but the largest effective diameter.
Future work will be dedicated to exploring the production of parts using multi jet fusion (MJF), which can fabricate parts significantly faster than LB-PBF, using the developed methodology to determine the manufacturability and mechanical responses of parts fabricated using the MJF process. The comparison of this work and producing parts using MJF opens up the possibility of better understanding the advantage and disadvantages of each process relative to the required application.
ACKNOWLEDGMENTS
The authors acknowledge the use of facilities within RMIT Advanced Manufacturing Precinct and the RMIT Microscopy and Microanalysis Facility. (https://www.rmit.edu.au/about/our-locations-and-acilities/facilities/researchfacilities/advanced-manufacturing-precinct).
FUNDING
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
AVAILABILITY OF DATA AND MATERIAL
The data is available and only can be provided by request of the journal.
CODE AVAILABILITY
The code is available and only can be provided by request of the journal.
DECLARATIONS
Ethics approval: Not applicable.
Consent to participate: Not applicable.
Consent for publication: Not applicable.
Competing interests: The authors declare no competing interests.
- M. M. Martin Leary, Hugh Williams, Eric Yang, Ahmad Alghamdi, , X. Z. Bill Lozanovski, Darpan Shidid, Lena Farahbod-Sternahl, , and I. K. Gerd Witt, Peter Choong, Ma Qian, Milan Brandt, "Inconel 625 Lattice Structures Manufactured By Selective Laser Melting (SLM): Mechanical Properties, Deformation and Failure Modes," (in English), Mater. Des., vol. 157, pp. 179-199, Nov 5 2018, doi: 10.1016/j.matdes.2018.06.010.
- J. Mun, J. Ju, and J. Thurman, "Indirect additive manufacturing based casting (I AM casting) of a lattice structure," in ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 2014, vol. 2A, doi: 10.1115/IMECE2014-38055. [Online]. Available: https://www.scopus.com/inward/record.uri?eid=2-s2.0-84926290434&partnerID=40&md5=7e404aefc05f39f193163e327aa272b3
- W. E. Frazier, "Metal Additive Manufacturing: A Review," Journal of Materials Engineering and Performance, journal article vol. 23, no. 6, pp. 1917-1928, June 01 2014, doi: 10.1007/s11665-014-0958-z.
- A. Alghamdi, "Effect of polygon order on additively manufactured lattice structures: a method for defining the threshold resolution for lattice geometry," The International Journal of Advanced Manufacturing Technology, 2019 2019.
- M. Leary et al., "Inconel 625 lattice structures manufactured by selective laser melting (SLM): Mechanical properties, deformation and failure modes," Materials & Design, vol. 157, pp. 179-199, 2018.
- B. Lozanovski et al., "Computational modelling of strut defects in SLM manufactured lattice structures," (in English), Materials & Design, vol. 171, Jun 5 2019, doi: ARTN 107671 10.1016/j.matdes.2019.107671.
- B. Lozanovski et al., "Computational modelling of strut defects in SLM manufactured lattice structures," Materials & Design, p. 107671, 2019.
- D. Downing, M. Leary, M. McMillan, A. Alghamdi, and M. Brandt, "Heat transfer in lattice structures during metal additive manufacturing: numerical exploration of temperature field evolution," Rapid Prototyping Journal, vol. 25, no. 5, pp. 911-928, 2020.
- B. Zhang, Y. Li, and Q. Bai, "Defect Formation Mechanisms in Selective Laser Melting: A Review," Chinese Journal of Mechanical Engineering, vol. 30, no. 3, pp. 515-527, 2017, doi: 10.1007/s10033-017-0121-5.
- I. Echeta, X. Feng, B. Dutton, R. Leach, and S. Piano, "Review of defects in lattice structures manufactured by powder bed fusion," The International Journal of Advanced Manufacturing Technology, pp. 1-20, 2019.
- R. Li, J. Liu, Y. Shi, L. Wang, and W. Jiang, "Balling behavior of stainless steel and nickel powder during selective laser melting process," The International Journal of Advanced Manufacturing Technology, vol. 59, no. 9, pp. 1025-1035, 2012.
- A. Alghamdi, T. Maconachie, D. Downing, M. Brandt, M. Qian, and M. Leary, "Effect of additive manufactured lattice defects on mechanical properties: an automated method for the enhancement of lattice geometry," The International Journal of Advanced Manufacturing Technology, pp. 1-15, 2020 2020.
- J. Noronha, M. Qian, M. Leary, E. Kyriakou, and M. Brandt, "Hollow-walled lattice materials by additive manufacturing: Design, manufacture, properties, applications and challenges," Current Opinion in Solid State and Materials Science, vol. 25, no. 5, p. 100940, 2021.
- J. Noronha, M. Qian, M. Leary, E. Kyriakou, S. Brudler, and M. Brandt, "Manufacturability of Ti-Al-4V Hollow-Walled Lattice Struts by Laser Powder Bed Fusion," JOM, vol. 73, no. 12, pp. 4199-4208, 2021.
- M. Bächle and R. J. Kohal, "A systematic review of the influence of different titanium surfaces on proliferation, differentiation and protein synthesis of osteoblast‐like MG63 cells," Clinical oral implants research, vol. 15, no. 6, pp. 683-692 %@ 0905-7161, 2004.
- X. Z. Zhang, H. P. Tang, M. Leary, T. Song, L. Jia, and M. Qian, "Toward Manufacturing Quality Ti-6Al-4V Lattice Struts by Selective Electron Beam Melting (SEBM) for Lattice Design," (in English), Jom, vol. 70, no. 9, pp. 1870-1876, Sep 2018, doi: 10.1007/s11837-018-3030-x.
- D. Shidid, M. Leary, P. Choong, and M. Brandt, "Just-in-time Design and Additive Manufacture of Patient-specific Medical Implants," Physics Procedia, vol. 83, pp. 4-14, // 2016, doi: http://dx.doi.org/10.1016/j.phpro.2016.08.002.
- N. A. Fleck, New Approaches to Structural Mechanics, Shells and Biological Structures kluwer (New Approaches to Structural Mechanics, Shells and Biological Structures). Springer Netherlands : Imprint: Springer (in English), 2002.
- V. S. Deshpande, N. A. Fleck, and M. F. Ashby, "Effective properties of the octet-truss lattice material," Journal of the Mechanics and Physics of Solids, vol. 49, no. 8, pp. 1747-1769, 2001.
- D. Kang, S. Park, Y. Son, S. Yeon, S. H. Kim, and I. Kim, "Multi-lattice inner structures for high-strength and light-weight in metal selective laser melting process," Materials & Design, vol. 175, p. 107786, 2019.
- A. Alghamdi et al., "Effect of polygon order on additively manufactured lattice structures: a method for defining the threshold resolution for lattice geometry," Int J Adv Manuf Technol, vol. 105, pp. 2501-2511, October 2019 2019, doi: 10.1007/s00170-019-04168-1.
- W.-Y. Jywe, C.-H. Liu, and C. o.-K. Chen, "The min–max problem for evaluating the form error of a circle," Measurement, vol. 26, no. 4, pp. 273-282, 1999/12/01/ 1999, doi: http://dx.doi.org/10.1016/S0263-2241(99)00052-4.
- M. Clifford, K. Simmons, and P. Shipway, An introduction to mechanical engineering: Part 1. CRC Press, 2009.
- R. Gentle, P. Edwards, and W. Bolton, "Mechanical engineering systems," Elsevier, 2001, ch. 5, pp. 261-265.
- D. Hally, "Calculation of the Moments of Polygons," DEFENCE RESEARCH ESTABLISHMENT SUFFIELD RALSTON (ALBERTA), 1987.
- J. Ochshorn, Structural elements for architects and builders. Elsevier, 2009.
- M. F. Ashby, & Cebon, D., "Materials selection in mechanical design," Le Journal de Physique IV, vol. 3, no. C7, pp. C7-1, 1993.
- A. Almalki et al., "A digital-twin methodology for the non-destructive certification of lattice structures," JOM, vol. 74, no. 4, pp. 1784-1797, 2022.
|
Term |
Definition |
|
AM |
Additive manufacturing |
|
BJT |
Binder jetting technology |
|
CAD |
Computer-aided design |
|
CSP |
Cold spray |
|
DED |
Directed energy deposition |
|
DFAM |
Design for additive manufacturing |
|
DOE |
Design of experiments |
|
IQR |
Interquartile range |
|
LB-PBF |
Laser-based powder bed fusion |
|
MAM |
Metal additive manufacturing |
|
MEX |
Material extrusion |
|
MJF |
Multi jet fusion |
|
MJT |
Material jetting technology |
|
PBF |
Powder bed fusion |
|
PBS |
Powder bed system |
|
PFS |
Powder feed system |
|
SEM |
Scanning electron microscope |
|
SHL |
Sheet lamination |
|
WFS |
Wire feed system |
|
µCT |
Micro-computed tomography |
NOTATION
