# Thermal Modeling of Polyamide 12 Powder in the Selective Laser Sintering Process Using the Discrete Element Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Discrete Element Method

#### 2.1. Preliminary Steps

- The precision of the simulations: a small-time step is essential for precise and detailed solutions.
- The duration of the simulation: if the simulation is too long, a larger time step is required.
- The influence of the chosen integration scheme: explicit schemes are most commonly used in the literature, and the critical time step limits the selected time step.

#### 2.2. Contact Search

#### 2.3. Contact Laws

#### 2.3.1. Mechanical Contact Law

#### 2.3.2. Thermal Contact Law

#### 2.4. Time Step Integration

## 3. Methodology of the Model

#### 3.1. Particle System Definition

#### 3.2. Contact Search

#### 3.3. Contact Modeling

#### 3.3.1. Inter-Particle Contact Law

#### 3.3.2. Contact Law with the Boundary Planes

#### 3.4. Energy Deposition

#### 3.5. Time Step Integration

#### 3.6. Implementation

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yan, C.; Shi, Y.; Li, Z.; Wen, S.; Wei, Q. Selective Laser Sintering Additive Manufacturing Technology. In 3D Printing Technology Series; Academic Press: Cambridge, MA, USA, 2021. [Google Scholar] [CrossRef]
- Wilms, M.B.; Rittinghaus, S.-K.; Goßling, M.; Gökce, B. Additive manufacturing of oxide-dispersion strengthened alloys: Materials, synthesis and manufacturing. Prog. Mater. Sci.
**2023**, 133, 101049, ISSN 0079-6425. [Google Scholar] [CrossRef] - Gusarov, A.; Kruth, J.-P. Modelling of radiation transfer in metallic powders at laser treatment. Int. J. Heat Mass Transf.
**2005**, 48, 3423–3434. [Google Scholar] [CrossRef] - Dong, L.; Makradi, A.; Ahzi, S.; Remond, Y. Finite Element Analysis of Temperature and Density Distributions in Selective Laser Sintering Process. Mater. Sci. Forum
**2007**, 553, 75–80. [Google Scholar] [CrossRef] - Dong, L.; Makradi, A.; Ahzi, S.; Remond, Y. Three-dimensional transient finite element analysis of the selective laser sintering process. J. Mater. Process. Technol.
**2009**, 209, 700–706. [Google Scholar] [CrossRef] - Fischer, P.; Karapatis, N.; Romano, V.; Glardon, R.; Weber, H. A model for the interaction of near-infrared laser pulses with metal powders in selective laser sintering. Appl. Phys. A
**2002**, 74, 467–474. [Google Scholar] [CrossRef] - Steuben, J.C.; Iliopoulos, A.P.; Michopoulos, J.G. Discrete element modeling of particle-based additive manufacturing processes. Comput. Methods Appl. Mech. Eng.
**2016**, 305, 537–561. [Google Scholar] [CrossRef] [Green Version] - Lanzl, L.; Wudy, K.; Drexler, M.; Drummer, D. Laser-high-speed-DSC: Process-oriented Thermal Analysis of PA 12 in Selective Laser Sintering. Phys. Procedia
**2016**, 83, 981–990. [Google Scholar] [CrossRef] [Green Version] - Yaagoubi, H.; Abouchadi, H.; Janan, M.T. Numerical simulation of heat transfer in the selective laser sintering process of Polyamide12. Energy Rep.
**2021**, 7, 189–199. [Google Scholar] [CrossRef] - Cundall, P.A.; Strack, O.D.L. A discrete numerical model for granular assemblies. Géotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Oñate, E.; Owen, R. Particle-Based Methods: Fundamentals and Applications; Springer Science & Business Media: Berlin, Germany, 2011. [Google Scholar]
- Haeri, S.; Wang, Y.; Ghita, O.; Sun, J. Discrete element simulation and experimental study of powder spreading process in additive manufacturing. Powder Technol.
**2017**, 306, 45–54. [Google Scholar] [CrossRef] [Green Version] - Haeri, S. Optimisation of blade type spreaders for powder bed preparation in Additive Manufacturing using DEM simulations. Powder Technol.
**2017**, 321, 94–104. [Google Scholar] [CrossRef] [Green Version] - Nezami, E.G.; Hashash, Y.M.; Zhao, D.; Ghaboussi, J. A fast contact detection algorithm for 3-D discrete element method. Comput. Geotech.
**2004**, 31, 575–587. [Google Scholar] [CrossRef] - Nezami, E.G.; Hashash, Y.M.A.; Zhao, D.; Ghaboussi, J. Shortest link method for contact detection in discrete element method. Int. J. Numer. Anal. Methods Géoméch.
**2006**, 30, 783–801. [Google Scholar] [CrossRef] - Raschdorf, S.; Kolonko, M. A comparison of data structures for the simulation of polydisperse particle packings. Int. J. Numer. Methods Eng.
**2011**, 85, 625–639. [Google Scholar] [CrossRef] - MArmstrong, M.; Mehrabi, H.; Naveed, N. An overview of modern metal additive manufacturing technology. J. Manuf. Process.
**2022**, 84, 1001–1029, ISSN 1526-6125. [Google Scholar] [CrossRef] - Xin, L.; Boutaous, M.; Xin, S.; Siginer, D.A. Multiphysical modeling of the heating phase in the polymer powder bed fusion process. Addit. Manuf.
**2017**, 18, 121–135. [Google Scholar] [CrossRef] - Rougier, E.; Munjiza, A.; John, N.W.M. Numerical comparison of some explicit time integration schemes used in DEM, FEM/DEM and molecular dynamics. Int. J. Numer. Methods Eng.
**2004**, 61, 856–879. [Google Scholar] [CrossRef] - Drummer, D.; Drexler, M.; Wudy, K. Density of Laser Molten Polymer Parts as Function of Powder Coating Process during Additive Manufacturing. Procedia Eng.
**2015**, 102, 1908–1917. [Google Scholar] [CrossRef] [Green Version] - Osmanlic, F. Modeling of Selective Laser Sintering of Viscoelastic Polymers. Ph.D. Thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany, 2019. [Google Scholar]
- Franco, A.; Lanzetta, M.; Romoli, L. Experimental analysis of selective laser sintering of polyamide powders: An energy perspective. J. Clean. Prod.
**2010**, 18, 1722–1730. [Google Scholar] [CrossRef] - Stichel, T.; Frick, T.; Laumer, T.; Tenner, F.; Hausotte, T.; Merklein, M.; Schmidt, M. A Round Robin study for Selective Laser Sintering of polyamide 12: Microstructural origin of the mechanical properties. Opt. Laser Technol.
**2017**, 89, 31–40. [Google Scholar] [CrossRef] - Soldner, D.; Steinmann, P.; Mergheim, J. Modeling crystallization kinetics for selective laser sintering of polyamide 12. GAMM-Mitteilungen
**2021**, 44, e202100011. [Google Scholar] [CrossRef] - Yao, B.; Li, Z.; Zhu, F. Effect of powder recycling on anisotropic tensile properties of selective laser sintered PA2200 polyamide. Eur. Polym. J.
**2020**, 141, 110093. [Google Scholar] [CrossRef] - Ai, Y.; Liu, X.; Huang, Y.; Yu, L. Numerical analysis of the influence of molten pool instability on the weld formation during the high speed fiber laser welding. Int. J. Heat Mass Transf.
**2020**, 160, 120103. [Google Scholar] [CrossRef]

**Figure 2.**Schematization of the parameters associated with the contact mechanics between the particles.

**Figure 6.**Volume particle diameter distribution density for PA12 powders used in additive manufacturing [20].

**Figure 10.**The temperature evolution at the center of the laser beam in our DEM simulations with the parameters from Table 1.

**Figure 12.**Visualization of the temperature in the powder bed under the effect of a stationary laser at three different times (a = 0.001 s, b = 0.003 s, c = 0.006 s).

Parameter | Notation | Value |
---|---|---|

Laser Power | $P$ | 1.7 W |

Preheating Temperature of the Powder | ${T}_{i}$ | 170 °C |

Sintering Temperature | ${T}_{s}$ | 180 °C |

The Temperature of the Chamber | ${T}_{a}$ | 170 °C |

Laser Beam Radius | ${r}_{l}$ | 200 $\mathsf{\mu}\mathrm{m}$ |

Particle Radius | ${r}_{i}$ | 25 $\mathsf{\mu}\mathrm{m}$ |

Simulation Time | $\tau $ | 0.006 s |

Time Step | $\u2206\tau $ | 0.001 s |

The Density of The Powder | $\rho $ | 1000 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ |

Thermal Conductivity | $\lambda $ | 0.28 $\mathrm{W}/\mathrm{m}\xb7\mathrm{K}$ |

Domain Porosity | $\epsilon $ | 0.8 (FEM), 0.78 (DEM) |

Stefan–Boltzmann Coefficient | ${\sigma}_{SB}$ | 5.67 × ${10}^{-8}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lakraimi, R.; Abouchadi, H.; Janan, M.T.; Chehri, A.; Saadane, R.
Thermal Modeling of Polyamide 12 Powder in the Selective Laser Sintering Process Using the Discrete Element Method. *Materials* **2023**, *16*, 753.
https://doi.org/10.3390/ma16020753

**AMA Style**

Lakraimi R, Abouchadi H, Janan MT, Chehri A, Saadane R.
Thermal Modeling of Polyamide 12 Powder in the Selective Laser Sintering Process Using the Discrete Element Method. *Materials*. 2023; 16(2):753.
https://doi.org/10.3390/ma16020753

**Chicago/Turabian Style**

Lakraimi, Reda, Hamid Abouchadi, Mourad Taha Janan, Abdellah Chehri, and Rachid Saadane.
2023. "Thermal Modeling of Polyamide 12 Powder in the Selective Laser Sintering Process Using the Discrete Element Method" *Materials* 16, no. 2: 753.
https://doi.org/10.3390/ma16020753