# Warpage Analysis and Control of Thin-Walled Structures Manufactured by Laser Powder Bed Fusion

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^{2}

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## Abstract

**:**

## 1. Introduction

^{7}°C/m), particularly when printing thin-walled structures [9,10,11,12,13].

## 2. Experimental Campaign

^{3}) with an annealed Ti-6Al-4V base-plate (250 × 250 × 25 mm

^{3}) and a controlled atmosphere of pure argon (the oxygen content is restricted to 100 ppm).

- Single-wall structures of different thicknesses and heights;
- Cylindrical structures of different diameters and heights;
- Square-section structures of several thicknesses;
- Open-section structures (e.g., semi-cylinder, L-shape, etc.).

#### Scale Effect on the Part Warpage

## 3. Numerical Modeling

- Recoating: the baseplate is lowered to accommodate a new powder layer;
- Laser-scanning: the laser heat source generates a molten pool which follows a user-defined trajectory to selectively melt the powder bed consolidating the new layer comprising the AM build.

#### 3.1. Mechanical Problem

**s**+ ▽p +

**b**= 0

**u**− e

^{T}) − p/K = 0

**b**are the body forces per unit of volume and K(T) is the temperature-dependent bulk modulus and p and

**s**, are the hydrostatic (pressure) and the deviatoric components of the Cauchy stress tensor,

**σ**. Thus:

**σ**= p

**I**+

**s**

**ε**(

**u**) = ▽

^{s}

**u**, can also be split in its volumetric and deviatoric parts, e

_{vol}= ▽·

**u**and

**e**, respectively, so that:

**ε**= e

_{vol}

**I**/3 +

**e**

**s**(

**u**) = 2G(

**e**(

**u**) −

**e**

^{inh})

**e**

^{inh}is the inherent strain tensor, as:

**e**

^{inh}= e

^{T}

**I**+

**e**

^{vp}

^{T}(T), and the visco-plastic strains,

**e**

^{vp}.

**e**

^{vp}= diag(e

_{l}, e

_{t}, e

_{z}) =

**I**(e

_{l}, e

_{t}, e

_{z})

^{T}

_{l}, e

_{t}and e

_{z}are the longitudinal (aligned with the scanning direction), transversal and vertical (building direction) visco-plastic components, respectively [33,34,35]. In LPBF, the thickness of the deposited layer is very small, so e

_{z}can be neglected. Additionally, thin-walled parts can be assumed to be in a quasi-plane-stress condition, so that the only inherent strain component to be considered is the one along the mid-line of the cross-section, regardless of the scanning pattern.

#### 3.2. Thermal Problem

_{scan}= A/H

_{HAZ}= l

_{scan}× d × s = (d/H) × (s/t) × V

_{scan}computed as:

_{scan}= l

_{scan}/V

_{scan}= A/(H × V

_{scan})

_{scan}is the scanning velocity. Observe that the LPBF process also includes the recoating operation, that is a cooling phase of duration Δt

_{cool}.

_{eq}= n·(Δt

_{scan}+ Δt

_{cool})

_{lump}, so that:

_{eq}= ηP/V

_{lump}

#### 3.3. Geometrical Models and FE Meshes

^{2}within the horizontal XY plane, and 0.3 mm in the building direction (a lumping layer-height). All these thin-walled work-pieces have the same height of 50 mm. The LPBF process is simulated with a sequence of 167 time-steps through a layer-by-layer activation strategy. Each layer used for the simulation has a thickness of 0.3 mm, thus including 10 physical layers (30 µm of loose powder is spread at each recoating step).

^{2}) and an emissivity of 0.35 [39]. The ambient temperature is set to 26°C during the whole LPBF process. The efficiency of the LPBF process, when Ti-6Al-4V loose powder is used, is defined in terms of the heat power absorption coefficient fixed to η = 0.4 [39].

## 4. Calibration and Results

#### 4.1. Calibration of the Numerical Model

#### 4.2. Results and Discussion on the Warpage Mechanism of the Thin-Walled Structures

_{xx}and σ

_{zz}, respectively, at two stages of the manufacturing. It can be noted in Figure 12b that the internal surface sustains compressive stresses while tensile stresses are induced in the external surface. As a consequence, the wall deforms outwards. The vertical tensile stresses induced by the high cooling rate and the small bending stiffness of these structures also promote the bulging as shown in Figure 12c.

_{xx}) and the vertical stresses (σ

_{zz}). During the building process, large tensile stresses develop at the top and two side edges while compressive stresses are produced in the central area of the built. As a result, warpage is visible along the lateral edges (see Figure 11). Note that a similar stress distribution and bulging phenomenon also take place when fabricating the single-wall and the square thin-walled section (see Figure 4 and Figure 5).

## 5. Structural Optimization for Warpage Control

## 6. Conclusions

- The wall thickness plays a significant role on the warpage of the final part. Thicker walled structures present reduced warpage.
- Increasing the build height as well as reducing the wall curvature causes larger warpage, as shown for the cylindrical structures.
- Open sections (e.g., semi-cylinder, L-shape, etc.) are more prone to warpage than closed ones (e.g., cylinder and square section) because of their reduced structural stiffness.
- The use of vertical stiffeners enables locally enhancing the structural stiffness of thin-walled structures, minimizing the residual warpage induced by the LPBF process.
- FE analysis of LPBF processes is a useful tool to analyse different thin-walled structures in order to predict the actual warpage. The developed FE model has been calibrated with experimental 3D-scanning images.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Residual warpage (out-of-plane displacement) for the single-wall structures for different wall-thicknesses: (

**a**) 1 mm; (

**b**) 2 mm; (

**c**) 5 mm, respectively. (

**d**) Maximum warpage recorded in the middle section.

**Figure 4.**Residual warpage contour-fills of the thin-walled structures with different heights: (

**a**) 37 mm; (

**b**) 43 mm; (

**c**) 50 mm, respectively. (

**d**) Maximum warpages recorded in the middle section.

**Figure 5.**Contour-fills of the final warpage of the cylindrical thin-wall structures with different building heights: (

**a**) 30 mm; (

**b**) 50 mm; (

**c**) 70mm, respectively. (

**d**) Maximum warpages recorded in the middle section.

**Figure 6.**Contour-fills of the final warpage of the cylindrical thin-wall structures with different diameters of: (

**a**) 50 mm; (

**b**) 70 mm; (

**c**) 90 mm; (

**d**) Maximum warpages recorded in the middle section.

**Figure 7.**Contour-fills of the final warpage of the square thin-wall section with different wall-thicknesses: (

**a**) 1 mm; (

**b**) 2 mm; (

**c**) 5 mm, respectively. (

**d**) Maximum warpages recorded in the middle section.

**Figure 10.**Comparison of the out-of-plane displacement fields obtained through 3D scanning and numerical simulation: (

**a**) semi-cylinder; (

**b**) cylinder.

**Figure 11.**Comparison of the out-of-plane displacement fields obtained through (

**a**) 3D scanning and (

**b**) numerical simulation for the L-shaped structure.

**Figure 12.**Semi-cylindrical part: the evolutions of (

**a**) total displacements and (

**b**) longitudinal stresses (σ

_{xx}) as well as (

**c**) vertical stresses (σ

_{zz}).

**Figure 15.**Warpage of the open (

**a**) semi-cylindrical and (

**b**) L-shaped thin-walled parts with and without vertical stiffeners.

**Figure 16.**Residual von Mises stresses of the open (

**a**) semi-cylindrical and (

**b**) L-shaped parts with and without vertical stiffeners.

Al | V | O | H | N | C | Fe | Si | Ti |
---|---|---|---|---|---|---|---|---|

6.28 | 3.90 | 0.098 | 0.002 | 0.020 | 0.008 | 0.022 | 0.026 | Balance |

Laser Power (W) | Layer Thickness (µm) | Scan Speed (mm/s) | Hatch Spacing (µm) | Laser BeamDiameter (µm) |
---|---|---|---|---|

200 | 30 | 1000 | 100 | 100 |

**Table 3.**Dimensions of the proposed thin-walled structures and the numbers of FE elements and nodes.

Thin-Walled Parts | Dimensions | Number of FE Elements | Number of Nodes |
---|---|---|---|

Semi-cylindrical part | Φ50 mm × H50 mm | 21,606 | 37,618 |

Cylindrical part | Φ50 mm × H50 mm | 34,632 | 63,336 |

L-shaped part | L50 mm × W50 mm × H50 mm | 34,366 | 55,392 |

**Table 4.**Temperature-dependent material properties of Ti-6Al-4V alloy [5].

Temperature (°C) | Density (kg/m^{3}) | Specific Heat (J/(kg·°C)) | Thermal Conductivity (W/(m·°C)) | Poisson’s Ratio | Young’s Modulus (GPa) | Thermal Dilatancy (μm/m/ °C) | Yield Stress (MPa) |
---|---|---|---|---|---|---|---|

20 | 4420 | 546 | 7 | 0.345 | 110 | 8.78 | 850 |

205 | 4395 | 584 | 8.75 | 0.35 | 100 | 10 | 630 |

500 | 4350 | 651 | 12.6 | 0.37 | 76 | 11.2 | 470 |

995 | 4282 | 753 | 22.7 | 0.43 | 15 | 12.3 | 13 |

1100 | 4267 | 641 | 19.3 | 0.43 | 5 | 12.4 | 5 |

1200 | 4252 | 660 | 21 | 0.43 | 4 | 12.42 | 1 |

1600 | 4198 | 732 | 25.8 | 0.43 | 1 | 12.5 | 0.5 |

1650 | 3886 | 831 | 35 | 0.43 | 0.1 | 12.5 | 0.1 |

2000 | 3818 | 831 | 35 | 0.43 | 0.01 | 12.5 | 0.01 |

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**MDPI and ACS Style**

Lu, X.; Chiumenti, M.; Cervera, M.; Tan, H.; Lin, X.; Wang, S. Warpage Analysis and Control of Thin-Walled Structures Manufactured by Laser Powder Bed Fusion. *Metals* **2021**, *11*, 686.
https://doi.org/10.3390/met11050686

**AMA Style**

Lu X, Chiumenti M, Cervera M, Tan H, Lin X, Wang S. Warpage Analysis and Control of Thin-Walled Structures Manufactured by Laser Powder Bed Fusion. *Metals*. 2021; 11(5):686.
https://doi.org/10.3390/met11050686

**Chicago/Turabian Style**

Lu, Xufei, Michele Chiumenti, Miguel Cervera, Hua Tan, Xin Lin, and Song Wang. 2021. "Warpage Analysis and Control of Thin-Walled Structures Manufactured by Laser Powder Bed Fusion" *Metals* 11, no. 5: 686.
https://doi.org/10.3390/met11050686