1 Introduction

Particle shape is of fundamental importance when handling granular materials and may serve as the decisive factor determining the success or failure of a given process. In this context, comprehending the role of particle shape in the flow and packing properties of granular materials represents an active area of research across various sectors, from the pharmaceutical industry [1, 2] to the investigation of granular phenomena [3, 4].

Additive manufacturing (AM) is an important tool to support the investigations in this area. Owing to the freedom it provides in shaping and its rapid prototyping solutions, AM constitutes an integral component of numerous investigations that require particles of non-spherical shapes [5,6,7,8]. In complement to experiments, numerical simulations using the discrete element method (DEM) prove to be a powerful tool for investigating the behavior of granulates of different shapes [9], including applications in the additive manufacturing process [10].

In a recent study, particles of different shapes but possessing the same volume equivalent diameter were manufactured through selective laser sintering (SLS). The study successfully demonstrated the influence of particle shape on flow and packing properties [11] in a systematic way. Furthermore, it was established that a single set of calibration parameter for DEM simulations in combination with idealized particle shape was not capable of describing the behavior of all the particles investigated. Despite sharing the same material, polyamide 12 (PA 12), and being produced on the same SLS machine, DEM simulations using a unique set of frictional parameters could not replicate the experimental behavior of the 3D manufactured parts, even though the idealized shapes of the particles were used in the numerical simulations.

The process of manufacturing parts through AM is known to introduce undesired effects, such as surface roughness and inner porosity, in both metallic [12] and polymeric printed components [13]. In polymeric parts produced through fused filament fabrication (FFF), inner porosity in the range of 4–6% has been identified, depending on the manufacturing strategy [13]. Using SLS technology and polyamide 11 (PA11), reported values for surface roughness \(R_a\) fall within the range of 9–24 \({\upmu }\)m [14]. In their study, a quarter of a hemisphere with a radius of 20 mm was printed. The influence of the blade geometry in the final part roughness was investigated in [15]. The authors found surface roughness \(R_a\) values ranging from an average of 23 to 31 \({\upmu }\)m, depending on the blade adopted.

The work of Zhang et. al [16] is likely the first to incorporate surface effects into the examination of the particle shape’s impact on the bulk behavior of 3D printed particles with well-defined shapes. In that study spheres, briquettes, tetrahedron, rock crystal and CT scanned rock were 3D printed using the Binder Jetting (BJ) technique. Surface roughness was measured both before and after infiltration with a Colour Bond resin, employed in the study to mitigate surface roughness. The particles were subsequently used for mixing and segregation studies. However, there is a gap in the literature regarding a systematic investigation into the distinct shapes to verify differences in roughness among them. Additionally, the true density was computed based on the ideal shape from Computer-Aided Design (CAD) rather than the printed (real) shape, and the mentioned study did not include comparisons with numerical simulations.

In our previous study [11], we primarily emphasized the significance of particle shape in influencing the packing and flow properties of granular matter, conducting a systematic investigation using 3D printed particles. Recognizing the crucial role of DEM simulations in granular matter research, we conducted numerical simulations to validate the experimental findings. Despite employing a calibration procedure to determine the frictional parameters, the simulation outcomes exhibited poor correlation with the experimental results, except for the calibrated (cubic) shape. Considering that all particle shapes were manufactured using the same technique, we anticipated better agreement with experimental results. Our hypothesis is that the manufacturing process was responsible for these discrepancies.

This study aims to explore the impact of the SLS manufacturing process on the final particle surface roughness and density, specifically focusing on how these effects vary among different manufactured shapes. Rather than focusing on specific manufacturing strategies and parameters for SLS, our analysis centers on understanding the influences of AM defects on the packing properties of particles. This investigation incorporates both experimental and numerical perspectives. Micro-Computed Tomography (\(\mu\)-CT) was employed to image 3D-manufactured particles, yielding the real particle shapes. Experiments with these real particle shapes were then simulated by Discrete Element Method (DEM) simulations, using varying surface mesh resolutions. Additionally, the inner porosity of the manufactured parts was calculated based on the \(\mu\)-CT images, and necessary corrections were applied to the particle density. The study reveals a correlation between the resulting surface roughness of the manufacturing process, inner porosity, and the specific shape of the manufactured particles.

2 Methodology

This section covers the production process, characterization, and simulation of particles with different shapes. A brief summary of the methods is given here, and details are provided in the following sections. The particles were produced by selective laser sintering (SLS). The printed particles were imaged using \(\mu\)-CT, yielding volumetric data from which triangulated surfaces were calculated. For each shape, simplifications of the original triangulation containing 100, 500, 1000 and 2000 faces were used. We will refer to those as meshes of different resolutions. For each distinct shape (e.g., cube, octahedron), three different particles were imaged. Subsequently, particles of every shape and mesh resolution underwent characterization based on solidity, convexity and surface roughness. These characteristics are calculated for each of the three particles, average and standard deviation values are shown. Solidity is a measure for the inner porosity of the real manufactured particles, which is crucial when calculating the density for particles used in DEM simulation, as they do not have inner pores. The characterized particles were then employed in DEM simulations of cylinder filling experiments for further comparisons with experimental measurements and the correlation of surface defects introduced by the AM technique and the particle shapes is finally evaluated.

2.1 3D printed particles

Selective laser sintering (FORMIGA P110, EOS, Germany) using Polyamide 12 powder was chosen as the 3D printing manufacturing process. Six different particle shapes were manufactured (cube, octahedron, quatropod, stellated octahedron (stellOcta), tetrahedron and tetrapod), with all particles having the same volume equivalent sphere diameter of 3 mm - see Fig. 1 top. These are the same particles manufactured and employed in our previous study [11].

2.2 \(\mu\)-CT

To acquire comprehensive 3D images of the particle structures, \(\mu\)-CT imaging was employed. This involved placing one particle of each shape onto a sample holder, totaling six samples per record. The samples were fixed to the sample holder by double-sided tape (Tesa, Norderstedt, Germany). The sample holder then was placed in a custom-build X-ray micro-tomography system (XCT-1600HR; MatrixTechnologies, Feldkirchen, Germany). Two-dimensional projections were obtained and reconstructed using custom software from Matrix Technologies, which is based on CERA (Siemens, Munich, Germany). This procedure is repeated three times using different particles of each shape. The final reconstructed volumes have a voxel resolution of 8 µm. For each shape, three different particles have been scanned leading to 18 different particles in total. Images of the \(\mu\)-CT-scanned particles (one representative for each shape) are shown in Fig. 1.

Fig. 1
figure 1

Top: Images of the real 3D printed particles. Bottom: Images of the \(\mu\)-CT scanned particles. From left to right: tetrapod, octahedron, quatropod, cube, tetrahedron and stellated octahedron

Full size image

2.3 Image processing

To isolate and evaluate the obtained particle images the following image analysis steps were carried out (Matlab R2023a):

  1. 1.

    Manual threshold selection for binarization.

  2. 2.

    Removal of small components through opening with a cuboid structuring element with an edge length of 24 \(\upmu m.\)

  3. 3.

    Extraction of the particles, showing internal holes arising from the sintering process.

  4. 4.

    To extract the outer surface of the structure, closing was performed with a spherical structuring element of radius 80 \(\upmu m\), followed by filling of isolated holes.

  5. 5.

    The triangulated outer surface was obtained by calculating the isosurface of the binary volume.

The triangulated outer surface of each particle then was simplified to 100, 500, 1000 and 2000 faces using the Quadric Error based Mesh Simplification algorithm proposed by Garland and Heckbert [17] implemented in MeshLab 2021.07 [18]. The particles with their respective mesh resolutions can be seen in Fig. 2. Besides, an idealized shape mesh is built from the CAD file and investigated for comparisons with the real shape meshes. The idealized shape is characterized by a perfectly smooth surface without sintering artifacts.

Fig. 2
figure 2

All printed particles are depicted with their respective resolutions. From left to right: 100, 500, 1000, and 2000 faces and idealized shape mesh. From top to bottom: cube, octahedron, quatropod, stellated octahedron, tetrahedron and tetrapod

Full size image

2.4 Shape characterization

The real particles scanned in \(\mu\)-CT were characterized in terms of solidity. For this evaluation, the real volumetric shape data obtained from \(\mu\)-CT are used.

The particles of each shape were evaluated in terms of convexity and surface roughness. For convexity, the idealized shape meshes and all resolutions of the real shape meshes were characterized. Since surface roughness would not be meaningful for idealized shapes, only the real shape mesh representations were analyzed. Furthermore, an additional mesh resolution of 10000 faces was used to assess whether increasing the number of surface elements beyond 2000 faces would further alter particle surface roughness and convexity.

2.4.1 Solidity

To characterize the volume of void inclusions in the particles, the solidity \(\phi\) of the particles was determined by

$$\begin{aligned} \phi = \frac{V_{binary}}{V_{closed}} \end{aligned}$$
(1)

where \(V_{binary}\) and \(V_{closed}\) denote the volume of the binarized volume and the outer surface volume respectively. The void spaces (inner porosity) of the particles can be observed in Fig. 3. Values of solidity are shown in Table 1.

Table 1 Table displaying the average and standard deviation values of solidity for the various analyzed shapes
Full size table
Fig. 3
figure 3

The images showcase the particles scanned using micro-CT (\(\mu\)-CT). In the top row, particles are displayed, revealing their three-dimensional inner porous structure. The pores are color-coded based on their surface distance for enhanced visualization. In the bottom row, a slice was taken through the respective particles, exposing the internal voids (depicted in black). The particles featured from left to right are cube, octahedron, quatropod, stellated octahedron, tetrahedron, and tetrapod

Full size image

2.4.2 Convexity

For determining the convexity \(\xi\) of the particles at each resolution level, the volume \(V_{convex}\) of the associated convex hull of the surface triangulation was computed. The convexity is then determined by calculating

$$\begin{aligned} \xi = \frac{V_{triangulated}}{V_{convex}} \end{aligned}$$
(2)

with \(V_{triangulated}\) denoting the volume of the surface triangulation.

2.4.3 Roughness

According to ISO 25178 [19] the surface roughness \(S_a\) is calculated as follows:

$$\begin{aligned} S_a = \frac{1}{A} \iint \limits _A |z(x,y)| dxdy \end{aligned}$$
(3)

Here, A denotes the surface area, while z(x, y) describes the topographic height relative to the locally averaged surface and dxdy is the area associated with the measuring point.

ISO 25178 does not describe a standard procedure for calculating z(x, y) from triangulated volumetric data derived from \(\mu\)-CT measurements. To obtain a related measure, the local deviations are calculated from a smoothed version of the surface triangulation. To estimate this smoothed surface, the curvature-weighted diffusive smoothing filter developed by Desbrun et al. [20] is adopted. This filter involves selecting a smoothing parameter \(\lambda \cdot dt\), with \(\lambda\) denoting the smoothing rate and dt denoting the smoothing time. We set this parameter to \(10^4\), since the resulting roughness is in range of the particle size of the polyamide powder used [21].

As the smoothing filter moves the vertices of the surface mesh along their normal vector, the Euclidean distance \(d_i\) between the unsmoothed and smoothed vertices can be directly evaluated. Furthermore, the area belonging to the \(i-th\) vertex can be calculated as:

$$\begin{aligned} A_{i} = \frac{1}{3} \sum _{j} A_{j} \end{aligned}$$
(4)

In this equation the index j denotes the faces adjacent to the \(i\)th$ vertex.

By replacing |z(x, y)| with \(d_i\) and discretizing Eq. 3, the following area-weighted average roughness \(S^*_a\) can be calculated:

$$\begin{aligned} S^*_{a} = \frac{1}{\sum _{i} A_{i}} \sum _{i} d_{i} \cdot A_{i} \end{aligned}$$
(5)

An example of the smoothing process and subsequent calculation of surface roughness is illustrated in Fig. 4. In this image, tetrapod particles are depicted in their original mesh with various levels of resolution, along with the smoothed surface obtained through the method described in this section.

Fig. 4
figure 4

Tetrapod with a mesh resolution of 1000 faces (top) and 10000 faces (bottom). From left to right: Original mesh, smoothed mesh and the difference of both mesh, colored in terms of deviations \(d_i\) in proportions of average \(S^*_{a}\) values

Full size image

2.5 Numerical simulations

For the numerical analysis, DEM simulations were conducted using the software Ansys Rocky-DEM (version: 23R2, supplier: Ansys Inc.). Particle shapes were modeled through polyhedrons, employing triangulated meshes with varying numbers of faces (resolution) to represent the particles. In DEM, the equations of motion for every individual particle are numerically integrated over time. Rocky uses an Euler integration scheme for velocity and a modified first order scheme for displacement. In our simulations, a time step \(\Delta t\) of \(10^{-6}\) s is used.

When two particles are in contact, a contact force model, consisting of an elastic and a dissipative term, calculates the resulting force in normal and tangential directions. The contact model used is the hysteretic linear-spring model proposed by Walton and Braun [22] in the normal direction. The same model was adopted in the previous work [11]. No rolling friction model is used in the simulations. The normal force model is implemented in Ansys-Rocky in an incremental way, in the form:

$$\begin{aligned} \varvec{F}_{n}^{t}=\left\{ \begin{matrix} min(K_{nl} \varvec{s}_{n}^{t}, \varvec{F}_{n}^{t-\Delta t}+K_{nu}\Delta \varvec{s}_{n})&{} if &{} \Delta \varvec{s}_{n}\ge 0\\ max(\varvec{F}_{n}^{t-\Delta t}+K_{nu} \Delta \varvec{s}_{n}, \lambda K_{nl}\varvec{s}_{n}^{t})&{} if &{} \Delta \varvec{s}_{n}< 0 \end{matrix}\right. \end{aligned}$$
(6)

with \(\varvec{F}_{n}^{t}\) and \(\varvec{F}_{n}^{t-\Delta t}\) the normal elastic–plastic contact forces at the current time t and previous time \(t-\Delta t\), respectively. \(\Delta \varvec{s}_{n}\) is the change in the contact normal overlap during the current time, being positive as particles approach and negative as particles separate from each other. It is given by \(\Delta \varvec{s}_{n}=\varvec{s}_{n}^{t}-\varvec{s}_{n}^{t-\Delta t}\), with \(\varvec{s}_{n}^{t}\) and \(\varvec{s}_{n}^{t-\Delta t}\) the normal overlap values at the current and at the previous time, respectively. \(K_{nl}\) and \(K_{nu}\) are the values of loading and unloading contact stiffness, respectively. \(\lambda\) is a dimensionless small constant whose value in Ansys-Rocky is 0.001. The part of the expression in which this constant is active ensures that, during the unloading, the normal force will return to zero when the overlap decreases to zero, avoiding artificial attraction between particles.

The loading stiffness for particle-particle and particle boundary is obtained from:

$$\begin{aligned} \frac{1}{K_{nl}}= & {} \left\{ \begin{array}{ll} \frac{1}{K_{nl,p_1}}+\frac{1}{K_{nl,p_2}} &{} particle-particle \\ \frac{1}{K_{nl,p}}+\frac{1}{K_{nl,b}} &{} particle-boundary \end{array}\right. \end{aligned}$$
(7)
$$\begin{aligned} {K_{nu}}= & {} \frac{K_{nl}}{\epsilon ^2} \end{aligned}$$
(8)

where subscripts 1 and 2 identify the two contacting particles. The stiffness parameter for particles is obtained from \(K_{nl}=E_pL\) and for boundaries \(K_{nl}=E_bL\). Here \(E_p\) is the particle material’s Young’s modulus, \(E_b\) is the boundary material’s Young’s modulus and L is the particle’s characteristic size (diameter if particle is spherical). More details on the Rocky implementation are found in Rocky User Manual [23]. Further discussions on the derivation and validation of the model are available in [22]. Additional comparisons of this model with other contact models (e.g. Hertz contact model) can be accessed in [24].

In the tangential direction, the linear-spring Coulomb limit force model is employed. The tangential force is obtained from:

$$\begin{aligned} {\varvec{F}_{\tau , e}^t}= \varvec{F}_{\tau , e}^{t-\Delta t}-K_\tau \Delta \varvec{s}_\tau \end{aligned}$$
(9)

where \(\varvec{F}_{\tau , e}^{t-\Delta t}\) is the value of the tangential force in the previous time step, \(\Delta \varvec{s}_\tau\) is the particles’ tangential relative displacement during the current time step and \(K_\tau\) is the tangential stiffness calculated as \(K_\tau =r_K K_{nl}\). Here \(r_K\) is the user-defined tangential stiffness ratio. In our simulation we adopted the Rocky standard value of 1. The tangential force is limited by coulomb’s friction in the form:

$$\begin{aligned} {\varvec{F}_{\tau , e}^t}= min\left(\left| \varvec{F}_{\tau , e}^t \right| ,\mu F_n^t\right)\frac{\varvec{F}_{\tau , e}^t}{\left| \varvec{F}_{\tau , e}^t \right| } \end{aligned}$$
(10)

with \(F_n^t\) the contact normal force at the current time t and \(\upmu\) being the static coefficient of friction \(\upmu _s\) if no sliding takes place at the contact, or the dynamic coefficient of friction \(\mu _d\) is sliding takes place at the contact. In our simulations, both \(\upmu _s\) and \(\upmu _d\) assume the same value. More details are found in Rocky User Manual [23], and additional comparisons with other tangential force models can be accessed in [24].

2.6 Packing characterization

The packing properties of the granular material were characterized by assessing the granular bed porosity. It is important to distinguish between the porosity measured experimentally and the porosity that can be simulated using DEM (Discrete Element Method). Experimentally, porosity measurements include both inner (intra-particle) and outer (inter-particle) porosity. The inner porosity is due to the particle manufacturing technique, which introduces inner pores. However, DEM simulations capture only the outer porosity, as particles in DEM do not have inner pores.

To ensure accuracy in this study, two corrections are necessary. Firstly, the density used in DEM simulations must differ from the true density derived from pycnometer measurements. The true density, determined using a gas pycnometer, only considers the solid volume of the particle. For experimental granular bed porosity, which includes both inner and outer pores, particles are placed in a constant volume cylinder and the mass is measured to determine the true density. Given that DEM particles lack inner pores, there are two scenarios to consider:

  • When simulating with particles of the same volume as obtained from the pycnometer results, this leads to smaller effective contact areas, significantly impacting outer porosity.

  • Using particles with a fixed equivalent sphere diameter of 3 mm and the absolute density from pycnometer measurements results in a larger mass, impacting outer porosity values to a lesser extent.

Therefore, adjusting the absolute density and using a fixed equivalent sphere diameter of 3 mm is essential for DEM simulations. While simulations only capture outer porosity, experiments measure both inner and outer porosity. To extract the values of outer granular bed porosity from experiments, the absolute density needs to be corrected. This is achieved by multiplying the absolute density measured with the pycnometer by the particle solidity. The outer porosity is obtained using the following equation:

$$\begin{aligned} \epsilon _{b,o}=\frac{\left( 1-V_{p,o} \right) }{V_c} \end{aligned}$$
(11)

where \(\epsilon _{b,o}\) is the outer porosity, \(V_c\) is the volume of the container (0.1845 l) and \(V_{p,o}\) is the volume of all particles in the container, considering the particles as if they had no inner pores. This can be understood as the fraction of particle volume relevant for the outer porosity only, and is obtained using the following formula:

$$\begin{aligned} V_{p,o}=\frac{M_p}{\rho _a \cdot s} \end{aligned}$$
(12)

where \(\rho _a\) is the absolute density measured with the pycnometer, \(M_p\) is the mass of all particles in the container and s is the particle solidity.

2.6.1 Granular bed porosity

Granular bed porosity was measured using a lockable funnel and an underlying cylindrical container with a volume of 0.1845 l and a diameter of 0.05 m. Initially, particles were poured into the closed funnel. The shutter was then opened, and particles flowed into the cylindrical container. No more particles were poured into the funnel after the shutter was opened. Afterwards, the volume was adjusted by removing the tip with a spatula and the granular material mass, and granular bed porosity, were determined. This procedure was repeated three times, and the resulting average values, along with their respective standard deviations, were calculated. The experimental results were reported in [11], and further details regarding the experimental procedure can be found in the aforementioned reference. To calculate the granular bed porosity, the true density of the particles was determined using a gas pycnometer [11]. The results are shown in Table 2. The density corrected based solidity values, as discussed in the previous section, is referred to as adjusted density in Table 2.

Table 2 The particle mass, volume and true density were determined experimentally using a pycnometer
Full size table

Numerical simulations replicated the same procedure, although a single simulation sufficed, and granular bed porosity values were averaged across three sub-regions of the particle-filled cylinder. The objective of calculating the porosity through three different regions is to avoid errors due to possible local compaction, as well as avoid the bottom and top of the cylinder, where porosity is expected to be higher. The numerical setup, as well as the upper and lower regions used to calculate porosity, are marked in Fig. 5. The coefficient of friction was systematically varied within the interval ranging from 0.15 to 0.95 for all the shapes and mesh resolutions. In total, 150 simulations were performed. The simulations varied in calculation time, ranging from 1.5 to 30 h, depending on the mesh resolution. Overall, approximately 750 h of GPU computation time were necessary, utilizing an Nvidia RTX 3090 graphics card. An illustrative example of the numerical simulation setup is presented in Fig. 5. Other parameters used in DEM simulations are shown in Table 3.

Fig. 5
figure 5

Images of the numerical setup for granular bed porosity measurement are depicted. On the left, particles are poured in the cylindrical contained through the funnel. Particles are allowed to settle (middle) and granular bed porosity is obtained through three measurements of local granular bed porosity using three regions of equal volume. Upper and lower regions as detailed in red (right image)

Full size image

Furthermore, for a proper comparison of the experimental and numerical results, it is necessary to recalculate the experimental values of granular bed porosity using the corrected density, obtaining experimental values equivalent to particles without inner pores. We will refer to those values as “experimental corrected” in further sections for comparison purposes with experimental values without corrections (based on the true density measured by the pycnometer). These are important corrections, and their impact will be discussed in the results section.

Table 3 Numerical values used in the contact model for the DEM simulations
Full size table

3 Results and discussion

3.1 Volumetric defects-inner porosity

To investigate inner porosity defects and their relation to particle shape, it is crucial to examine the values of solidity presented in Table 1. Notably, solidity varies among different particle shapes. Specifically, when solidity is considered alongside the values of particle convexity, as illustrated in Fig. 6, a direct correlation becomes evident.

The tetrapod and quatropod shapes exhibit the lowest values of solidity, coinciding with their lower convexity values. Conversely, cube and octahedron shapes display the highest solidity values, aligning with their high convexity values. This observation can be attributed to the presence of smaller features, such as "tips", in the tetrapod and quatropod particles (see Fig. 1). These features likely contribute to relatively larger manufacturing errors compared to cube and octahedron shapes, which lack such small features. Further support for this interpretation comes from examining the tetrahedron, which also has small tips but demonstrates an intermediate value of solidity due to its larger convexity.

Another contributing factor to these observations could be the local thermal properties, influencing inner porosity [25]. It is anticipated that smaller features, like the tips of the quatropod, would experience a more pronounced thermal impact during heating and cooling compared to particles with cubic shapes.

Fig. 6
figure 6

Graph illustrating convexity values and their standard deviations for different particle shapes across various mesh resolutions

Full size image

3.2 Surface defects

Now we investigate the surface defects introduced by the AM technique and their relation to particle shape. The results for surface roughness, illustrated in Fig. 7, indicate that different particle shapes exhibit varying surface roughness. This suggests a correlation between particle shape and surface defects. However, it is also observed that increasing the mesh resolution leads to higher surface roughness. To ensure the accuracy of the observed shape-dependent surface roughness and to mitigate potential artifacts introduced by varying mesh resolutions, which could erroneously influence the interpretation of the results, it’s crucial to assess the numerical results using these different mesh resolutions.

Fig. 7
figure 7

Average roughness (\(S^*_{a}\)) values and their standard deviations are depicted for various particle shapes across different mesh resolutions

Full size image

Numerical results for granular bed porosity with a coefficient of friction ranging from 0.35 to 0.75 are presented in Fig. 8. These numerical results are compared with experimental data both before and after the correction of density based on the solidity values - refer to the earlier discussion in Sect. 2.6.1.

An interesting observation is the consistent increase in granular bed porosity as the number of faces grows. This rise is associated with the increasing roughness linked to a higher number of faces. However, there’s a notable exception for meshes with 100 faces for quatropod and tetrapod shapes, which also happen to be the shapes with the lowest convexity values. This discrepancy suggests that 100 faces might be inadequate to capture the macroscopic shape features of the particle, resulting in a shape representation that deviates from the actual shape - refer to, for example, [26] for a detailed study of the shape factors used to characterize particle form. Remarkably, these are the shapes that require the largest number of triangular faces to describe the ideal particle shape representation, approximately two orders of magnitude more than the other ideal shapes.

Fig. 8
figure 8

Average granular bed porosity and standard deviation for various particle shapes using different mesh resolutions, along with simulations employing ideal shapes. A coefficient of static friction ranging from 0.35 to 0.75 was applied. Experimental measurements are included for comparison

Full size image

It’s evident from Fig. 8 that the mesh resolution impacts the granular bed porosity differently across various shapes. Stellated octahedron and tetrahedron exhibit more dispersed values of bed porosity for different mesh resolutions compared to the other shapes studied. This highlights a correlation between the number of faces employed to depict a particle and its shape. Ultimately, after correcting particle density and granular bed porosity, utilizing high-resolution surface meshes, and applying the appropriate coefficient of friction, the numerical results should ideally align with experimental measurements. The numerical values of granular bed porosity notably fall below the experimental measurements before correction but are within the range of experimental results after correction.

3.3 Fit model for friction parameter

To identify frictional parameters replicating the experimental results, granular bed porosity values were plotted (see Fig. 9) against the coefficient of friction for the entire range of simulations (0.15 to 0.95) across various particles and mesh resolutions. To establish a functional relationship, the friction and resulting porosity were fitted with a power law relation, as proposed by Silbert for mono-disperse frictional spheres [27] and for superballs particles varying from spherical to cubic shape [28]. Additionally, corrected experimental results were plotted and the estimated friction values replicating the experimental result were evaluated. In Fig. 9, the intersection between the solid and dashed lines give the estimated coefficient of friction in dependency of the mesh resolution. For certain mesh resolutions, no intersection in the range of 0 to 1 can be observed. The estimated friction values for the 2000 faces resolution are presented in Table 4. However, caution is advised in interpreting these results as the calibrated coefficient of friction, given potential wall effects. While wall effects are expected to be negligible for spherical particles confined in a cylindrical container whose diameter is 16 times larger than the average particle diameter [29], which is the ratio in the current study, their extent for the complex shapes studied here remains uncertain.

It is interesting to note that tetrapod and stellated octahedron exhibit the highest roughness \(S^*_{a}\) values and also the highest coefficients of friction. In contrast, cube, octahedron, and tetrahedron display low frictional values, as well as low surface roughness. The quatropod stands out as an exception, showing high surface roughness but low frictional values. Notably, the quatropod demonstrates a mixed tendency when examining Fig. 9. The ideal mesh shows the highest granular bed porosity values, followed by the 100 and 2000 faces resolutions, which have nearly identical values. For other particles, the 2000 faces mesh resolution exhibits the highest bed porosities, except for the stellated octahedron, where it ranks as the second-highest. This implies that even higher mesh resolutions may be necessary for quatropods, or wall effects could significantly influence the results for that particle shape. Moreover, tetrapod, quatropod, and stellated octahedron consistently possess the highest surface roughness and the lowest convexity values.

Fig. 9
figure 9

The graph illustrates granular bed porosity values (average and standard deviation) plotted against the coefficient of friction for various particles and mesh resolutions. Each color represents a distinct mesh resolution, as specified in the legend. Solid lines denote the least-squares fit of a power law relationship. All coefficients of determination \(R^2\) are above 0.9, except for the tetrahedron with a 1000 faces resolution, for which this value is 0.83

Full size image
Table 4 The estimated coefficients of friction reproducing the corrected experimental porosity for the 2000 faces resolution are presented
Full size table

Using the highest mesh resolution (2000 faces), the coefficient of friction matching the experimentally corrected granular bed porosity was determined for all particles. For other mesh resolutions, including the ideal mesh, the coefficient of friction could only be found for certain particles. Despite all particles being manufactured with the same technique and material, no single mesh resolution can adequately represent the packing and flow properties of 3D printed granular materials across different friction coefficients. This highlights that surface defects introduced by additive manufacturing are not independent of the particle shape.

4 Conclusion

In this study, we have illustrated the significant impact of the Selective Laser Sintering (SLS) manufacturing technique on the packing properties of 3D printed particles, introducing both surface and volumetric defects. The intensity of inner pore defects varies with particle shapes and is directly linked to particle convexity, as shown in Table 1. Particles with lower convexity values, like tetrapod, quatropod, and stellated octahedron, display higher surface roughness and greater inner porosity (solidity).

Surface defects also correlate with particle shape. Tetrapod and stellated octahedron particles, having the highest surface roughness and lowest convexity values, exhibit the highest coefficient of friction to match experimental results. On the other hand, cube, octahedron, and tetrahedron, with lower surface roughness and high convexity, exhibit the lowest coefficient of friction to match experimental results. The tetrapod stands as an exception to this trend.

In general, simulated granular bed porosity tends to increase with the number of faces considered for shape representation, attributed to increased roughness. However, no clear tendency was observed for mesh representations using 100 faces or the idealized mesh. Specifically, for quatropod and tetrapod shapes, the highest granular bed porosity values were found with the 100 faces mesh, suggesting that low-resolution meshes may introduce macroscopic form factor modifications.

Ideal shapes displayed higher granular bed porosities compared to real particle representations for quatropod and stellated octahedron, lower values for octahedron, and mixed values for the other particles. No clear tendency could be identified, despite density corrections, indicating that density correction alone is insufficient, and a surface correction is also necessary.

For efficient DEM simulations, it is advisable to use the lowest number of faces possible. However, there is no single mesh resolution that can adequately represent the packing and flow properties of 3D printed granular materials with a single friction coefficient. A combination of both factors is necessary, as initially explored in [30]. Ultimately, a calibration procedure for each particle shape is required, even when using the same manufacturing technique and material. As illustrated in Fig. 7, increasing the mesh resolutions to 10000 faces resulted in elevated surface roughness compared to 2000 faces, indicating the necessity for additional adjustment of the coefficient of friction for even higher mesh resolutions. Increasing the number of faces in DEM representation might lead to a universal frictional parameter valid for all shapes, requiring only a density correction. Nevertheless, simulations with even more refined meshes may become computationally prohibitive, as the current resolution of 2000 faces already demands significant computational resources for meaningful simulations.

Despite the particles in this study being manufactured using the same AM technique and material, there is no universal adjustment of the coefficient of friction and particle density that can compensate for surface and volumetric defects introduced by the manufacturing technique across all shapes. It is evident from this study that AM defects are shape-dependent, emphasizing the need for specific density correction and calibration of frictional parameters for each shape in numerical simulations.