# Measuring and Predicting the Effects of Residual Stresses from Full-Field Data in Laser-Directed Energy Deposition

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## Abstract

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## 1. Introduction

## 2. Aim and Structure of the Approach

## 3. Materials and Methods

#### 3.1. Thermal Expansion Coefficient

#### 3.2. Incremental Hole Drilling

## 4. Stochastic Modeling and Optimization

#### 4.1. Evaluation of the Thermal Expansion Coefficient

#### 4.1.1. Development of the Metamodel

#### 4.1.2. Numerical Modeling of Thin-Wall Specimens

#### 4.1.3. Results

#### 4.2. Stochastic Modeling of the Displacement Field for Incremental Hole Drilling

#### 4.2.1. Numerical Modeling of the Incremental Hole Drilling

#### 4.2.2. Results of the Displacement Field

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**DIC images of the thin-wall samples. (

**a**) Pre-wall geometry where the boundaries are indicated in blue, (

**b**) post-wall geometry after the addition of 20 extra layers where the boundaries are indicated in yellow, (

**c**) ROI used for the DIC for the reference state, (

**d**) surface deformation W(Y,Z) of the post-wall geometry due to the expansion, where W is the in plane deformation in Z-direction. The values are indicative and are not explicitly used in the analysis.

**Figure 3.**DIC results of the displacements W (

**a**,

**c**) and U (

**b**,

**d**) for the four points C0, C1, C2, C3. The plots (

**a**,

**b**) presents the average value for various drilling depths. The contours (

**c**,

**d**), as seen in VIC-3D, illustrate the displacement field for a drilling depth of 1.2 mm.

**Figure 4.**FE model of the thin-wall specimens. (

**a**) Creating and partitioning of the full model due to symmetry, (

**b**) thickness of printed layers, (

**c**) addition of steel material (in white) and stiff base material (in green), (

**d**) meshed structure, (

**e**) nodes (in red) from which the displacements are extracted.

**Figure 5.**FE model of the IHD test. (

**a**) Creating and partitioning of the full model due to symmetry, (

**b**) thickness of printed layers, (

**c**) steel material used in the modeling (in white), (

**d**) meshed structure, (

**e**) nodes (in red) from which the displacements are extracted.

**Figure 6.**Y-Z plane view of the IHD test model illustrating the steel material (in white) and the void section (in red) for different drilling depths. (

**a**) Drilling depth 0.0 mm, (

**b**) drilling depth 0.4 mm, (

**c**) drilling depth 2.0 mm.

**Figure 7.**Comparison of DIC measurements and PC results for the W-displacement field and a drilling depth 1.2 mm. (

**a**) Expected experimental value minus one standard deviation, (

**b**) expected experimental value, (

**c**) expected experimental value plus one standard deviation, (

**d**) expected PC value minus one standard deviation, (

**e**) expected PC value, (

**f**) expected PC value plus one standard deviation.

**Figure 8.**Comparison of DIC measurements and PC results for the U-displacement field and a drilling depth 1.2 mm. (

**a**) Expected experimental value minus one standard deviation, (

**b**) expected experimental value, (

**c**) expected experimental value plus one standard deviation, (

**d**) expected PC value minus one standard deviation, (

**e**) expected PC value, (

**f**) expected PC value plus one standard deviation.

**Figure 9.**Comparison of DIC measurements and PC results for the W−displacement field and a drilling depth 2.0 mm. (

**a**) Expected experimental value minus one standard deviation, (

**b**) expected experimental value, (

**c**) expected experimental value plus one standard deviation, (

**d**) expected PC value minus one standard deviation, (

**e**) expected PC value, (

**f**) expected PC value plus one standard deviation.

**Figure 10.**Comparison of DIC measurements and PC results for the U-displacement field and a drilling depth 2.0 mm. (

**a**) Expected experimental value minus one standard deviation, (

**b**) expected experimental value, (

**c**) expected experimental value plus one standard deviation, (

**d**) expected PC value minus one standard deviation, (

**e**) expected PC value, (

**f**) expected PC value plus one standard deviation.

**Table 1.**Thermal expansion coefficient estimated by the minimization of the L1 metamodels (ML and PC) along the direction Z (${a}_{zz}$) and Y (${a}_{yy}$).

L1 Metamodel | ${\mathit{a}}_{\mathit{zz}}$ (10${}^{-6}{/}^{\circ}$C) | ${\mathit{a}}_{\mathit{yy}}$ (10${}^{-6}{/}^{\circ}$C) |
---|---|---|

ML | 13.91 ± 2.06 | 17.04 ± 0.30 |

PC | 13.76 ± 2.97 | 15.01 ± 1.20 |

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**MDPI and ACS Style**

Polyzos, E.; Pulju, H.; Mäckel, P.; Hinderdael, M.; Ertveldt, J.; Van Hemelrijck, D.; Pyl, L. Measuring and Predicting the Effects of Residual Stresses from Full-Field Data in Laser-Directed Energy Deposition. *Materials* **2023**, *16*, 1444.
https://doi.org/10.3390/ma16041444

**AMA Style**

Polyzos E, Pulju H, Mäckel P, Hinderdael M, Ertveldt J, Van Hemelrijck D, Pyl L. Measuring and Predicting the Effects of Residual Stresses from Full-Field Data in Laser-Directed Energy Deposition. *Materials*. 2023; 16(4):1444.
https://doi.org/10.3390/ma16041444

**Chicago/Turabian Style**

Polyzos, Efstratios, Hendrik Pulju, Peter Mäckel, Michael Hinderdael, Julien Ertveldt, Danny Van Hemelrijck, and Lincy Pyl. 2023. "Measuring and Predicting the Effects of Residual Stresses from Full-Field Data in Laser-Directed Energy Deposition" *Materials* 16, no. 4: 1444.
https://doi.org/10.3390/ma16041444