Incorporation of Defects in Finite Elements to Model Effective Mechanical Properties of Metamaterial Cells Printed by Selective Laser Melting
Abstract
:1. Introduction
- Three-dimensional printing equipment. In a typical Selective Laser Melting (SLM) or powder bed fusion (PBF) procedure, the main elements of the equipment are the laser, the base plate, the powder deposition mechanism and the printing chamber. Defects can be associated with any of them. Regarding the laser, poor calibration can induce several types of defects in the piece such as, for example, unmelted parts, porosity, or low precision [66]. Regarding the relevance of the base plate, defects usually appear due to insufficient thickness in that plate, causing poor heat dissipation between the component and the base plate. In this case, apart from geometric defects, the defects can also consist of cracks of delaminations [67]. The SLM process operates in an inert gas atmosphere, such as argon and nitrogen, needing a chamber. Factors such as the rate of gas filling or even its trajectory may contribute to the appearance of defects that also affect the geometric precision of the piece or the porosity of the printed material [68]. The powder deposition mechanism also contributes to the defects. For example, an uneven deposition of the powder layers causes a bad physical interaction between the laser beam and the material, resulting again in defects, such as porosity [69].
- Manufacturing sequence interactions. The interaction between the newly deposited powder, the molten powder and the laser beam affects the creation of defects that influence the finish of the piece. These interactions determine the energy density transferred to the material, defined as the applied energy per unit volume to a material during a PBF process. For an SLM process, the energy density E is defined by the following equation [70]:
- Deposition materials. The quality of the powders during the manufacturing process is an important factor for the final result of the obtained solid part, affecting both its precision and its quality. The flowability of powder and its apparent density (volumetric density) are two of the characteristics that have a major influence in the final quality. Fewer defects are obtained when spherical and smooth particles are used in the process, with an approximate size of 10–45 m in SLM [72].
- Orientation and preparation of specimen. Two aspects have a crucial impact on the quality and performance of the resulting pieces [73,74]: the printing direction, the specimen orientation, and the auxiliary supports. Orientation has importance in the final properties and in the anisotropy of these properties. Orientation also affects how heat flows towards the base plate. Supports have also this function, along with reducing residual stresses that are created in some parts of the piece.
2. Types of Defects Modeled
2.1. Undulations
2.2. Conicity in the Edges of the Lattice
2.3. Porosity
3. Methodology to Model Defects in the Lattice
Cell (See Figure 3) | Subelement Section Type | Section (mm2) | Cell Height (mm) | E (GPa) | |
---|---|---|---|---|---|
Face center cubic (FCC) | Square | 7 | 70 | 0.33 | |
Octet truss (OT) | Square | 7 | 70 | 0.33 | |
Truncated octahedron (TO) | Square | 7 | 70 | 0.33 |
3.1. Defects Considered and Their Simulation
3.1.1. Lattice Bar Undulations
3.1.2. Conicity in the Edges of the Lattice
3.1.3. Porosity Gradient
3.1.4. Computational Modeling Sequence
4. Sensitivity Analysis
4.1. Position of the Nodes
4.2. Porosity
4.3. Conicity
5. Experimental Validation
5.1. Methods: Experimental Determination of Defects
- Diameter and node deviations: Digital scans of the latticed structures [116] are used to measure the diameter distribution of the beams. Diameters are measured along the different beams randomly at 50 different locations to extract the statistical distributions. The same optical images are used to identify the centroids of different sections and calculate the geometrical deviation from the ideal central beam line. and Z direction deviations are calculated using both and plane photographs. In total, 483 sections are analyzed for the solid fraction case, 496 sections for , and 535 sections for along 30 different beams for each case as shown in Table 3 (Nodal deviation) and Table 4 (Conicity).
- Porosity: X-ray tomographies of the lattice structures are performed in [116] to identify internal defects. Thirty random beams are reconstructed and analyzed to extract the porosity fraction of each beam. The homogenized results of each beam are used to produce the statistical distribution of the percentage of porosity within the beams, see Table 5.
5.2. Methods: Finite Element Modeling
- Error in the position of the nodes: For all the ideal solid fractions s, the parameter is set to because the experimental data indicate that it can be considered that all the nodes have changed with respect to their original position in the CAD model, see Table 3. Therefore, all mesh nodes have been altered.
- Conicity in the edges of the lattice: For each ideal solid fraction s, the distribution of measured diameters is translated to edges of the finite element mesh, see Table 4. This distribution results in a corresponding distribution of the Young modulus for the 12 elements of each edge, given in Table 6.
- Porosity gradient: This defect results in a variation of Young’s modulus E with respect to the reference value . Given the large variety of pore types, the physical relationship that exists between porosity and moduli is not evident. However, the following expression is used in this work because of the good correlation obtained [124]:
5.3. Results
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Cell | Section Type | ND (mm) | E (GPa) | Height (mm) | |
---|---|---|---|---|---|
Cubic edge (s = 15) | Circular | 0.33 | 114 | 0.33 | 1.36 |
Cubic edge (s = 25) | Circular | 0.42 | 114 | 0.33 | 1.36 |
Cubic edge (s = 35) | Circular | 0.53 | 114 | 0.33 | 1.36 |
Density Distribution per Range (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|
Deviation Range (m) | = 0.15 | = 0.25 | = 0.35 | ||||||
x | y | z | x | y | z | x | y | z | |
0–15 | 81 | 86 | 76 | 86 | 88 | 79 | 91 | 89 | 80 |
15–30 | 7 | 5 | 9 | 4 | 4 | 4 | 2 | 2 | 6 |
30–45 | 4 | 3 | 5 | 3 | 2 | 6 | 2 | 1 | 3 |
45–60 | 3 | 2 | 4 | 2 | 2 | 4 | 2 | 1 | 4 |
60–75 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 3 | 3 |
75–90 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
90–105 | 1 | 0 | 2 | 1 | 0 | 3 | 0 | 2 | 2 |
Diameter Range (μm) | Density Distribution per Range (%) | ||
---|---|---|---|
= 0.15 | = 0.25 | = 0.35 | |
200–225 | 3 | 0 | 0 |
225–250 | 6 | 0 | 0 |
250–275 | 11 | 0 | 0 |
275–300 | 18 | 0 | 0 |
300–325 | 30 | 1.4 | 0 |
325–350 | 18 | 2.6 | 0 |
350–375 | 7 | 4.2 | 0 |
375–400 | 4 | 9.2 | 0 |
400–425 | 2 | 23.4 | 0.2 |
425–450 | 1 | 28.3 | 2 |
450–475 | 0 | 18.2 | 4 |
475–500 | 0 | 6.3 | 6.4 |
500–525 | 0 | 4.3 | 22 |
525–550 | 0 | 1.8 | 29 |
550–575 | 0 | 0.3 | 24 |
575–600 | 0 | 0 | 8 |
600–625 | 0 | 0 | 2.3 |
625–650 | 0 | 0 | 2.1 |
Porosity Range (%) | Density Distribution per Range (%) | ||
---|---|---|---|
= 0.15 | = 0.25 | = 0.35 | |
0–1 | 5.4 | 12.3 | 7.9 |
1–2 | 8.9 | 18.6 | 14.2 |
2–3 | 22.9 | 25.6 | 29.6 |
3–4 | 16.5 | 18.6 | 18.6 |
4–5 | 14.6 | 14.6 | 10.4 |
5–6 | 8.2 | 3.2 | 3.2 |
6–7 | 9.2 | 1.9 | 5.8 |
7–8 | 5.1 | 1.5 | 4.5 |
8–9 | 2.1 | 0.8 | 2.1 |
9–10 | 0.9 | 0.9 | 0.9 |
10–11 | 1.2 | 0.8 | 1.3 |
11–12 | 2.1 | 1.2 | 1.4 |
12–13 | 2.9 | 0.0 | 0.1 |
13–14 | 0.0 | 0.0 | 0.0 |
14–15 | 0.0 | 0.0 | 0.0 |
15–16 | 0.0 | 0.0 | 0.0 |
Cell | Experiment | Simulation (CAD) | Simulation (X-ray) |
---|---|---|---|
Cubic edge (s = 15) | 3471 ± 57 | 7193 ± 31 | 8095 ± 36 |
Cubic edge (s = 25) | 7665 ± 691 | 15,894 ± 82 | 16,488 ± 80 |
Cubic edge (s = 35) | 12,546 ± 912 | 23,798 ± 152 | 25,471 ± 137 |
Cell | Experimental Modulus (MPa) | Simulation (CAD) with Defects (Maximum) | Simulation (CAD) with Defects (Minimum) |
---|---|---|---|
Cubic edge (s = 15) | 3471 ± 57 | 3863.70 | 3785.29 |
Cubic edge (s = 25) | 7665 ± 691 | 7483.22 | 7305.15 |
Cubic edge (s = 35) | 12,546 ± 912 | 12,152.50 | 11,873.99 |
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Vera-Rodríguez, G.; Moreno-Corrales, L.; Marín-González, I.; Barba, D.; Montáns, F.J.; Sanz-Gómez, M.Á. Incorporation of Defects in Finite Elements to Model Effective Mechanical Properties of Metamaterial Cells Printed by Selective Laser Melting. Sustainability 2024, 16, 1167. https://doi.org/10.3390/su16031167
Vera-Rodríguez G, Moreno-Corrales L, Marín-González I, Barba D, Montáns FJ, Sanz-Gómez MÁ. Incorporation of Defects in Finite Elements to Model Effective Mechanical Properties of Metamaterial Cells Printed by Selective Laser Melting. Sustainability. 2024; 16(3):1167. https://doi.org/10.3390/su16031167
Chicago/Turabian StyleVera-Rodríguez, Gonzalo, Laura Moreno-Corrales, Iván Marín-González, Daniel Barba, Francisco J. Montáns, and Miguel Ángel Sanz-Gómez. 2024. "Incorporation of Defects in Finite Elements to Model Effective Mechanical Properties of Metamaterial Cells Printed by Selective Laser Melting" Sustainability 16, no. 3: 1167. https://doi.org/10.3390/su16031167
APA StyleVera-Rodríguez, G., Moreno-Corrales, L., Marín-González, I., Barba, D., Montáns, F. J., & Sanz-Gómez, M. Á. (2024). Incorporation of Defects in Finite Elements to Model Effective Mechanical Properties of Metamaterial Cells Printed by Selective Laser Melting. Sustainability, 16(3), 1167. https://doi.org/10.3390/su16031167