# Numerical Investigation into the Effect of Different Parameters on the Geometrical Precision in the Laser-Based Powder Bed Fusion Process Chain

^{*}

## Abstract

**:**

## 1. Introduction

^{2}. The number of elements required for this study was high, due to the large expected cooling rate and laser spot size. Denlinger [21] used a dynamic mesh coarsening algorithm. The mesh is fine where the laser surface heat flux interacts with the part and coarser in previously built layers, which effectively smears out the temperature and stress field in these layers. This strategy results in a high resolution simulation of a single cuboid, both thermally and mechanically. Similarly, Parry et al. [22] investigated the effect of scanning strategy and geometry of the part on the final deformations. Since they need to investigate the effect of the scanning direction, the laser itself needs to be resolved. This necessitates the choice of a fine mesh and therefore a small computational domain. Bayat et al. [23] simplified the laser by bunching the real layers together in so-called meta-layers and applying flash heating—the top of the meta-layer was exposed to an aggregated heat flux. Yakout et al. [24] investigated the effect of the thermal expansion coefficient and the thermal diffusivity on the residual stress after the LPBF. To achieve this, they simulated different materials and validated the model for one of them. They show that a low thermal coefficient and high thermal diffusivity leads to a reduction of the residual stress. An experimental study by Yadroitsev et al. [25] investigated the mechanisms behind the instability in single tracks analytically.

## 2. Materials and Methods

#### 2.1. Modelling Approach

#### 2.1.1. Thermal Model

#### 2.1.2. Mechanical Model

#### 2.2. Material Properties

^{3}. The parameters for the mechanical model are presented as functions of temperature in Figure 3A,B.

## 3. Results and Discussion

#### 3.1. Overview of the Simulations

#### 3.2. Benchmark Case: Simulation 1

#### 3.3. Mesh Sensitivity Analysis: Simulations 1–5

#### 3.4. Effect of the Duration of the Heat Treatment: Simulations 6 and 7

#### 3.5. Effect of the Heat Treatment Temperature: Simulations 8, 9 and 10

#### 3.6. Correcting the Energy Input of the Primary Process: Simulation 11

^{3}cube, this energy equates to 6375 kJ. On the other hand, it is possible to calculate the total transferred energy for the actual LPBF process. The total energy is equal to

## 4. Conclusions

- The model is capable of modelling the entire process in a limited amount of time and using a limited amount of computational resources. This allows a large number of simulations to be run for estimating the effect of varying certain parameters.
- The effect of changing the process chain sequence, from first heat treating to first removing from the base plate, leads to an increase of the deformation of the used part. This is most likely due to the stress relaxation, which causes deformation without build-up of stresses when heating up, while causing a build-up of stress and limited deformation when cooling down.
- The model also illustrates the capabilities of a generic FE solver to show the effect of the different process chain steps in the additive manufacturing process chain on the part quality.
- The model is not capable of capturing the effect of the duration of the heat treatment or the used temperature accurately due to its insensitivity to these parameters. However, when heating below the relaxation temperature a significant difference is observed in sequence B, since the stresses are no longer relaxed when heating up the beam.
- The stress relaxation does not decrease the stresses in a cantilever beam significantly but does lead to a homogenisation of this stress.
- Correcting the energy input does lead to an improved estimate for the residual stress in the part before post-processing, but since the post-processing changes the stress, the final deflection of a cantilever beam-type part does not differ significantly.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the process chain model including the laser-based powder bed fusion (LPBF) process, stress relief heat treatment and removal of the base plate. Both the thermal and mechanical part of the model are depicted.

**Figure 2.**Parameters for the thermal part of the finite element (FE) model; (

**A**) thermal conductivity as function of temperature; (

**B**) specific heat capacity. Data from [46].

**Figure 3.**Parameters for the mechanical part of the FE model; (

**A)**Young’s modulus and Poisson coefficient of 17-4 precipitation hardening (PH) stainless steel; (

**B)**yield strength and tangent modulus as function of temperature for the bilinear hardening law used. Data from [46].

**Figure 4.**Thermal expansion coefficient as functions of temperature. Data from [46].

**Figure 5.**Part geometry and process sequence; (

**a**) the cantilever part that is used throughout this study; (

**b**) the two different process sequences analysed in this work. In sequence A (top), the heat treatment immediately follows the primary LPBF process, before the part is removed from the base plate. Sequence B (bottom) reverses the order of the heat treatment and removal from the base plate. In the simulation, the beam simulated according to sequence A is in front, while the cantilever following sequence B is in the back.

**Figure 8.**Stress in the cantilever after the primary LPBF process; (

**A**) the normal stress in the x-direction (longitudinal) at the surface of the part; (

**B**) the stresses on the left-most support of the cantilever, indicated in Figure 8D; (

**C**) the same normal stress in longitudinal direction. The stress field is identical; (

**D**) the same stress component as Figure 8A, but on the central cross-section of the part.

**Figure 9.**Vertical deformation in the two cantilevers. The cantilever which underwent sequence A is displayed on the bottom, while the one that followed sequence B is shown on the top.

**Figure 10.**The normal stress in x-direction after the entire process chain is completed. The cantilever which underwent sequence A is displayed on the bottom, while the one that followed sequence B is shown on the top.

**Figure 11.**Stress strain curves at 200 °C for bilinear hardening and power law hardening plastic behaviour.

**Figure 13.**Normal stress in the x-direction in the centre of the top of the cantilever subjected to sequence A. Both the results for simulation 1 and simulation 7, with a long heat treatment, are shown.

**Figure 14.**The results from the simulation with corrected energy by adjusting the deposition temperature; (

**A**) shows the normal stress in the x-direction on the deformed cantilever. The cantilever which underwent sequence A is displayed on the bottom, while the one that followed sequence B is shown on the top; (

**B**) displays the normal stress component across the cantilever going from the bottom of the cantilever to the top, for simulation 1 and simulation 11.

**Table 1.**Overview of all the simulations performed for this study. Note that study 1 is the simulation benchmark.

Number | Parameters | Intentions | |||
---|---|---|---|---|---|

Time | Peak Temperature | Deposition Temperature | Element Size | ||

1 | 7200 s | 350 °C | 1405 °C | 5.00 × 10^{−3} m | Simulation benchmark |

2 | 7200 s | 350 °C | 1405 °C | 6.00 × 10^{−4} m | Investigation of mesh convergence |

3 | 7200 s | 350 °C | 1405 °C | 7.50 × 10^{−4} m | |

4 | 7200 s | 350 °C | 1405 °C | 8.00 × 10^{−4} m | |

5 | 7200 s | 350 °C | 1405 °C | 1.00 × 10^{−3} m | |

6 | 3600 s | 350 °C | 1405 °C | 5.00 × 10^{−4} m | Investigation of effect of dwell time during heat treatment |

7 | 10800 s | 350 °C | 1405 °C | 5.00 × 10^{−4} m | |

8 | 7200 s | 400 °C | 1405 °C | 5.00 × 10^{−4} m | Investigation of effect of heat treatment temperature |

9 | 7200 s | 300 °C | 1405 °C | 5.00 × 10^{−4} m | |

10 | 7200 s | 280°C | 1405 °C | 5.00 × 10^{−4} m | |

11 | 7200 s | 350 °C | 3000 °C | 5.00 × 10^{−4} m | Energy correction approach |

**Table 2.**The values of the maximal longitudinal normal stress after the entire process chain (sequence A).

Simulation Number | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Element size (m) | 5 × 10^{−4} | 6 × 10^{−4} | 7.5 × 10^{−4} | 8 × 10^{−4} | 1 × 10^{−3} |

Longitudinal normal stress (MPa) | 841.6 | 840.5 | 835.1 | 832.8 | 825.7 |

Simulation 8 | Simulation 9 | Simulation 10 | |||||||
---|---|---|---|---|---|---|---|---|---|

Temperature | 400 °C | 300 °C | 280 °C | ||||||

Sequence A | Sequence B | Sequence A | Sequence B | Sequence A | Sequence B | ||||

Displacement in z (m) | 0.0102 | 0.0363 | 0.0102 | 0.0364 | 0.0102 | 0.0103 | |||

End stress (Pa) | 1.00 × 10^{9} | 1.28 × 10^{9} | 1.39 × 10^{9} | 1.28 × 10^{9} | 1.00 × 10^{9} | 1.10 × 10^{9} |

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**MDPI and ACS Style**

De Baere, D.; Moshiri, M.; Mohanty, S.; Tosello, G.; Hattel, J.H. Numerical Investigation into the Effect of Different Parameters on the Geometrical Precision in the Laser-Based Powder Bed Fusion Process Chain. *Appl. Sci.* **2020**, *10*, 3414.
https://doi.org/10.3390/app10103414

**AMA Style**

De Baere D, Moshiri M, Mohanty S, Tosello G, Hattel JH. Numerical Investigation into the Effect of Different Parameters on the Geometrical Precision in the Laser-Based Powder Bed Fusion Process Chain. *Applied Sciences*. 2020; 10(10):3414.
https://doi.org/10.3390/app10103414

**Chicago/Turabian Style**

De Baere, David, Mandanà Moshiri, Sankhya Mohanty, Guido Tosello, and Jesper Henri Hattel. 2020. "Numerical Investigation into the Effect of Different Parameters on the Geometrical Precision in the Laser-Based Powder Bed Fusion Process Chain" *Applied Sciences* 10, no. 10: 3414.
https://doi.org/10.3390/app10103414