# Numerical Model for Predicting Bead Geometry and Microstructure in Laser Beam Welding of Inconel 718 Sheets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Developed Model

#### 2.1. Model Basis

_{m}) includes the buoyancy force (S

_{b}), generated as a consequence of the density difference, and the velocity reduction term (S

_{d}), introduced in those elements where the material is in solid state. Material is considered completely rigid and incompressible when it is in solid state, therefore, the velocity of the material in the solid region is zero. This is modeled by the second term in Equation (4), where the parameter ${f}_{l}$, which represents the liquid fraction, has a zero value in the solid and a unit value in the liquid. In order to avoid numerical instabilities due to a zero in the denominator at the solid region, the coefficient e

_{0}must have a small value (10

^{−3}in the present model). Moreover, the coefficient C must have a high value in order to make zero the velocity of the fluid flow in the solid region C = 10

^{6}kg·m

^{−3}·s

^{−1}[31].

_{e}), Equation (5) includes the latent heat (S

_{L}) and the heat exchange at the substrate surface (S

_{C}). Inside this second term, the energy radiated by the laser beam (q

_{laser}) and the heat losses due to radiation and convection (q

_{losses}) are included. As no material vaporization is expected, the model includes only the fusion latent heat, which is defined in Equation (6).

^{−3}value between two subsequent iterations. The same criteria are used for mass, momentum, energy conservation, and VOF equations.

#### 2.2. Initial and Boundary Conditions

#### 2.3. Surface Forces

#### 2.4. Microstructure

_{L}), columnar dendritic microstructure is formed until the solidus temperature (T

_{S}) is reached. This temperature phase-change range is named as the mushy zone [32].

_{L}and the $\gamma $/laves eutectic temperature (T

_{e}). At this juncture, dendritic columns grow mainly in the energetically favorable crystallographic directions, forming the principal axis and, to a lesser extent, in the other transverse secondary directions [9]. The secondary dendrite arm spacing (SDAS) is measured in this research test for subsequent thermal model validation by means of Equation (13). To this end, the mean values are calculated based on ten different measurements for each analyzed welding bead. SDAS is measured in μm, and C is a constant that depends on the material. For the specific case of the Inconel 718, this constant takes a value of 10 [33].

_{L}) and the $\gamma $/laves eutectic temperature (T

_{e}), respectively (Table 2).

## 3. Proposed Methodology for the Model Validation

^{®}25 from SCANLAB (SCANLAB GmbH, Munich, Germany

`)`with a focus length of 265 mm, maximum workspace of 120 × 120 mm

^{2}, and maximum feed rate of 10,000 mm·s

^{−1}. Scan head allows fast movements of the laser beam because of the low inertia of the moving mirrors, giving, as a result, high velocities and accelerations without losing positioning accuracy. Therefore, the laser beam motion is fast enough to consider as a ring-type spot of 1 mm diameter that moves at a ${v}_{f}$ feed rate speed. In this case, a wobble strategy is used for the welding process, see Figure 4. This method allows one to fill an area by describing rings, so a suitable relation between the feed rate (${v}_{f}$) and the peripheral speed (${v}_{p})$ is implemented for achieving minimum overlap and no space among consecutive rings. Therefore, the laser spot must spend the same time for tracing a loop (orbital motion) and for advancing a spot diameter distance (linear movement).

^{−1}and 5 mm·s

^{−1}. The seam length is of 30 mm, enough to ensure steady state is achieved during welding track. All process parameters are detailed in Table 3. Afterwards, all the samples are cut at a 20 mm distance from the beginning of the weld, encapsulated and polished for Marble solution etching, Figure 5. The geometry of the weld beads is revealed by this chemical attack, in order to analyze their cross shape and compare them with the results provided by the model. Moreover, secondary dendrite arm spacing (SDAS) in the samples is measured for the cases where the minimum and maximum powers are applied (350 W and 500 W, respectively). Finally, the measured SDAS is compared with the values predicted by the numerical model.

#### 3.1. Model Parameters

^{−1}and 5 mm·s

^{−1}feed rates are used, respectively.

#### 3.2. Materials

#### 3.3. Experimental Setup

^{−1}(6 l·min

^{−1}through each 80 mm × 2 mm rectangular slot).

## 4. Results

_{f}= 3 mm·s

^{−1}), and molten material starts to drop due to gravity forces. After the laser beam passes by the modeled cross section and there is no external heat input, the material solidifies, resulting in the final shape of the generated weld bead. This final shape, together with the area melted during the whole process, is compared with the experimental results when validating the model.

#### 4.1. Analysis of the Geometry of the Weld Beads

#### 4.2. Microstructure Validation

## 5. Conclusions

- (1)
- The developed model represents accurately the weld beads generated under different process parameters. However, the model performance depends on the analyzed bead feature. For instance, an error lower than the 10% is obtained regarding the weld width, whereas a higher accuracy is reached in the weld depth (an error below 4%).
- (2)
- The developed tool is valid for modeling not only the melt pool dynamics, but also the drop of the molten material once the laser beam melts the whole thickness of the Inconel 718 sheets. The error between the model and the experimental results when modeling the crown and root height is below 0.2 and 0.1 mm, respectively. Therefore, it is concluded that the model has a bigger error when dimensioning the weld crown. This is due to the fact that the model is incapable of predicting accurately the height variation of the weld if the penetration is not complete.
- (3)
- After comparing the internal structure measured in the experimental tests and the values given by the model, it is concluded that the model gives the SDAS with an error below 1.5 microns. The two different areas that are analyzed (M1 and M2) show that the SDAS in the test tubes is slightly higher than the value given by the model. Hence, it is concluded that the predicted cooling rate is also somewhat higher than the real one. This can be originated by the symmetry assumption or the two-dimensional solving of the melt pool dynamics, whereas the physical problem is three-dimensional.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Liu, S.; Mi, G.; Yan, F.; Wang, C.; Jiang, P. Correlation of high power laser welding parameters with real weld geometry and microstructure. Opt. Laser Technol.
**2017**, 94, 59–67. [Google Scholar] [CrossRef] - D’Ostuni, S.; Leo, P.; Casalino, G. FEM Simulation of Dissimilar Aluminum Titanium Fiber Laser Welding Using 2D and 3D Gaussian Heat Sources. Metals
**2017**, 7, 307. [Google Scholar] [CrossRef] - Li, X.; Wang, L.; Yang, L.; Wang, J.; Li, K. Modeling of temperature field and pool formation during linear laser welding of DP1000 steel. J. Mater. Process. Technol.
**2014**, 214, 1844–1851. [Google Scholar] [CrossRef] - Kubiak, M.; Piekarska, W. Comprehensive model of thermal phenomena and phase transformations in laser welding process. Comput. Struct.
**2016**, 172, 29–39. [Google Scholar] [CrossRef] - Tsirkas, S.A. Numerical simulation of the laser welding process for the prediction of temperature distribution on welded aluminium aircraft components. Opt. Laser Technol.
**2018**, 100, 45–56. [Google Scholar] [CrossRef] - Kazemi, K.; Goldak, J.A. Numerical simulation of laser full penetration welding. Comput. Mater. Sci.
**2009**, 44, 841–849. [Google Scholar] [CrossRef] - Venkatesan, K.; Ramanujam, R.; Kuppan, P. Parametric modeling and optimization of laser scanning parameters during laser assisted machining of Inconel 718. Opt. Laser Technol.
**2016**, 78, 10–18. [Google Scholar] [CrossRef] - Anderson, M.; Patwa, R.; Shin, Y.C. Laser-assisted machining of Inconel 718 with an economic analysis. Int. J. Mach. Tools Manuf.
**2006**, 46, 1879–1891. [Google Scholar] [CrossRef] - Ram, G.D.J.; Reddy, A.V.; Rao, K.P.; Reddy, G.M.; Sundar, J.K.S. Microstructure and tensile properties of Inconel 718 pulsed Nd-YAG laser welds. J. Mater. Process. Technol.
**2005**, 167, 73–82. [Google Scholar] - Steen, W.M.; Mazumder, J. Laser Material Processing, 4th ed.; Springer: London, UK, 2010. [Google Scholar]
- Sun, Z.; Karppi, R. The application of electron beam welding for the joining of dissimilar metals: An overview. J. Mater. Process. Technol.
**1996**, 59, 257–267. [Google Scholar] [CrossRef] - Chen, H.; Pinkerton, A.J.; Li, L. Fibre laser welding of dissimilar alloys of Ti-6Al-4V and Inconel 718 for aerospace applications. Int. J. Adv. Manufact. Technol.
**2011**, 52, 977–987. [Google Scholar] [CrossRef] - Davis, J.R. ASM Specialty Handbook: Nickel, Cobalt, and Their Alloys, 1st ed.; ASM International: Materials Park, OH, USA, 2000; 442p. [Google Scholar]
- Dowden, J.M. The Mathematics of Thermal Modeling: An Introduction to the Theory of Laser Material Processing, 1st ed.; CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Mazumder, J. Laser welding. In Laser Materials Processing, 1st ed.; Bass, M., Ed.; Elsevier: New York, NY, USA, 1983; Volume 3, pp. 120–200. [Google Scholar]
- Brown, M.S.; Arnold, C.B. Fundamentals of Laser-Material Interaction and Application to Multiscale Surface Modification. In Laser Precision Microfabrication, 1st ed.; Sugioka, K., Meunier, M., Piqué, A., Eds.; Springer: London, UK, 2010. [Google Scholar]
- Mills, K.C.; Keene, B.J.; Brooks, R.F.; Shirali, A. Marangoni effects in welding. Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**1998**, 356, 911–926. [Google Scholar] [CrossRef] - Swift-Hook, D.T.; Gick, A.E.F. Penetration welding with lasers. Weld. J.
**1973**, 52, 492–499. [Google Scholar] - Klemens, P.G. Heat balance and flow conditions for electron beam and laser welding. J. Appl. Phys.
**1976**, 47, 2165–2174. [Google Scholar] [CrossRef] - Goldak, J.; Chakravarti, A.; Bibby, M. A new finite element model for welding heat sources. Metall. Trans. B
**1984**, 15, 299–305. [Google Scholar] [CrossRef] - Bonollo, F.; Tiziani, A.; Zambon, A. Model for CO
_{2}laser welding of stainless steel, titanium, and nickel: parametric study. Mater. Sci. Technol.**1993**, 9, 1137–1144. [Google Scholar] [CrossRef] - Kaplan, A. A model of deep penetration laser welding based on calculation of the keyhole profile. J. Phys. D Appl. Phys.
**1994**, 27, 1805–1814. [Google Scholar] [CrossRef] - Ducharme, R.; Williams, K.; Kapadia, P.; Dowden, J.; Steen, B.; Glowacki, M. The laser welding of thin metal sheets: An integrated keyhole and weld pool model with supporting experiments. J. Phys. D Appl. Phys.
**1994**, 27, 1619–1627. [Google Scholar] [CrossRef] - Sudnik, W.; Radaj, D.; Breitschwerdt, S.; Erofeew, W. Numerical simulation of weld pool geometry in laser beam welding. J. Phys. D Appl. Phys.
**2000**, 33, 662–671. [Google Scholar] [CrossRef] - Tsirkas, S.A.; Papanikos, P.; Kermanidis, Th. Numerical simulation of the laser welding process in butt-joint specimens. J. Mater. Process. Technol.
**2003**, 134, 59–69. [Google Scholar] [CrossRef] - Gery, D.; Long, H.; Maropoulos, P. Effects of welding speed, energy input and heat source distribution on temperature variations in butt joint welding. J. Mater. Process. Technol.
**2005**, 167, 393–401. [Google Scholar] [CrossRef] - Zhao, H.; Niu, W.; Zhang, B.; Lei, Y.; Kodama, M.; Ishide, T. Modelling of keyhole dynamics and porosity formation considering the adaptive keyhole shape and three-phase coupling during deep-penetration laser welding. J. Phys. D Appl. Phys.
**2011**, 44, 485302. [Google Scholar] [CrossRef] - Kubiak, M.; Piekarska, W.; Saternus, Z.; Domanski, T. Numerical prediction of fusion zone and heat affected zone in hybrid Yb: YAG laser + GMAW welding process with experimental verification. Procedia Eng.
**2016**, 136, 88–94. [Google Scholar] [CrossRef] - Zhang, L.J.; Zhang, J.X.; Gumenyuk, A.; Rethmeier, M.; Na, S.J. Numerical simulation of full penetration laser welding of thick steel plate with high power high brightness laser. J. Mater. Process. Technol.
**2014**, 214, 1710–1720. [Google Scholar] [CrossRef] - Patankar, S.V. Numerical Heat Transfer and Fluid Flow, 1st ed.; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Saldi, Z.S.; Kidess, A.; Kenjeres, S.; Zhao, C.; Richardson, I.M.; Kleijin, C.R. Effect of enhanced heat and mass transport and flow reversal during cool down on weld pool shapes in laser spot welding of steel. Int. J. Heat Mass Transf.
**2013**, 66, 879–888. [Google Scholar] [CrossRef] - Voller, V.R.; Prakash, C. A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems. Int. J. Heat Mass Transf.
**1987**, 30, 1709–1719. [Google Scholar] [CrossRef] - Patel, A.D.; Murty, Y.V. The Effect of Cooling Rate on Microstructural Development in Alloy 718. Superalloys 718, 625, 706 and Various Derivatives. In Proceedings of the International Symposium on Superalloys and Various Derivatives, Pittsburgh, PA, USA, 17–20 June 2001; Loria, E.A., Ed.; TMS (Minerals, Metals, and Materials Society): Warrendale, PA, USA, 2001; pp. 123–132. [Google Scholar]
- Antonsson, T.; Fredriksson, H. The effect of cooling rate on the solidification of Inconel 718. Metall. Mater. Trans. B
**2005**, 36, 85–96. [Google Scholar] [CrossRef] - Zhang, Y.N.; Cao, X.; Wanjara, P. Microstructure and hardness of fiber laser deposited Inconel 718 using filler wire. Int. J. Adv. Manufact. Technol.
**2013**, 69, 2569–2581. [Google Scholar] [CrossRef] - Haynes International. Available online: www.haynesintl.com/alloys/alloy-portfolio_/High-temperature-Alloys/haynes718-alloy/nominal-composition (accessed on 9 July 2018).
- Mills, K.C. Recommended Values of Thermophysical Properties for Selected Commercial Alloys, 1st ed.; Woodhead Publishing: Cambridge, UK, 2002; pp. 181–190. [Google Scholar]

**Figure 5.**Upper view (

**left**), cross section (

**center**) and detail of the microstructure (

**right**) of the Test 6.

**Figure 6.**Variation of the elapsed time required for running the simulation and the obtained error compared with the experimentally measured depth of the weld bead as the element size varies for the case of the Test 4.

**Figure 8.**(

**a**) Experimental setup for ensuring the protective atmosphere during the laser beam welding (LBW) tests; (

**b**) Frontal and lateral schematic views.

**Figure 9.**Evolution of the welding section and material velocity for different time steps as the laser beam passes in Test 5.

**Figure 10.**Scheme of the different cross sections of the weld bead (W: crown width; H: crown height; R: root height; D: penetration depth).

**Figure 11.**Comparison between the modeled and the analyzed cross sections (tests parameters described in Table 3).

**Figure 12.**Simulated secondary dendrite arm spacing (SDAS) for cross section and experimental microstructure details in regions M1 and M2 of Test 8 (500 W and 5 mm·s

^{−1}).

Symbol | Description | Unit |
---|---|---|

u | Fluid velocity in the X axis direction. | m·s^{−1} |

v | Fluid velocity in the Y axis direction. | m·s^{−1} |

U | Absolute fluid velocity. | m·s^{−1} |

$\Delta x$ | Element size in the X axis direction. | m |

$\Delta y$ | Element size in the Y axis direction. | m |

$\rho $ | Material density. | kg·m^{−3} |

$p$ | Pressure value. | N·m^{−2} |

$\mu $ | Material viscosity. | kg·m^{−1}·s^{−1} |

$g$ | Gravitational acceleration constant. | m·s^{−2} |

$\stackrel{\rightharpoonup}{e}$ | Y+ direction unitary vector. | - |

$\gamma $ | Volume fraction (solid/liquid). | - |

${f}_{l}$ | Liquid fraction | - |

${f}_{s}$ | Surface forces. | N |

$\sigma $ | Surface tension. | N·m^{−1} |

$\frac{d\sigma}{dT}$ | Surface tension variation regarding the temperature. | N·m^{−1}·K^{−1} |

$\kappa $ | Surface curvature. | m^{−1} |

$\overrightarrow{n}$ | Vector normal to the surface (solid/liquid–gas interface). | - |

$\beta $ | Coefficient of liquid thermal expansion. | K^{−1} |

$c$ | Specific energy. | J·kg^{−1}·c |

$k$ | Heat conductivity. | W·m^{−1}⋅K^{−1} |

${S}_{L}$ | Fusion latent heat. | J·kg^{−1} |

$T$ | Temperature. | K |

${T}_{S}$ | Solidus temperature. | K |

${T}_{L}$ | Liquidus temperature. | K |

${T}_{\infty}$ | Room temperature. | K |

t | Time variable. | S |

Δt | Time step. | S |

$P$ | Laser power. | W |

${q}_{laser}$ | Laser beam intensity. | W·m^{−2} |

${q}_{losses}$ | Energy losses due to radiation and convection. | W·m^{−2} |

${r}_{out}$ | Outer radius of the laser beam in the wobble strategy. | M |

${r}_{in}$ | Inner radius of the laser beam in the wobble strategy. | M |

$\alpha $ | Absorptivity. | - |

$h$ | Convection coefficient. | W·m^{−2} K^{−1} |

$\epsilon $ | Emissivity. | - |

${\sigma}_{b}$ | Stefan–Boltzmann coefficient. | W·m^{−2} K^{−4} |

$\delta $ | Angle between the laser beam and the normal vector to the surface | rad |

${v}_{f}$ | Welding feed rate | Mm·s^{−1} |

${v}_{p}$ | Peripheral speed in the wobble operation | mm·s^{−1} |

Reaction Stage | Value (°C) |
---|---|

Liquidus on cooling | 1260 |

Solidus on cooling | 1227 |

$\gamma $/laves eutectic on cooling | 1177 |

Test Number | Laser Power (W) | Feed Rate (mm·s^{−1}) | Peripheral Speed (mm·s^{−1}) | Argon Feed (l·min^{−1}) | Seam Length (mm) | Wobble Diameter (mm) |
---|---|---|---|---|---|---|

1 | 350 | 3 | 84.8 | 24 | 30 | 0.9 |

2 | 350 | 5 | 141.4 | 24 | 30 | 0.9 |

3 | 400 | 3 | 84.8 | 24 | 30 | 0.9 |

4 | 400 | 5 | 141.4 | 24 | 30 | 0.9 |

5 | 450 | 3 | 84.8 | 24 | 30 | 0.9 |

6 | 450 | 5 | 141.4 | 24 | 30 | 0.9 |

7 | 500 | 3 | 84.8 | 24 | 30 | 0.9 |

8 | 500 | 5 | 141.4 | 24 | 30 | 0.9 |

Al | B | C | Co | Cr | Cu | Fe | Mn | Mo | Ni |

0.55 | 0.004 | 0.054 | 0.28 | 18.60 | 0.05 | 18.60 | 0.24 | 3.03 | 52.40 |

P | S | Si | Ti | Nb | Ta | Bi | Pb | Ag | |

<0.005 | <0.002 | 0.06 | 0.98 | 4.89 | <0.05 | <0.00003 | <0.0005 | <0.0002 |

**Table 5.**Properties of Inconel 718 (average thermophysical properties of Inconel 718, Copyright © 2002 Woodhead [37]).

Definition | Unit | Value |
---|---|---|

Melting range (T_{m}) | K | 1533–1609 |

Density (ρ) | Kg·m^{−3} | 8190 |

Specific heat (c) | J·kg^{−1}·K^{−1} | 435 |

Conductivity (k) | W·m^{−1}·K^{−1} | 8.9 |

Latent heat fusion (S_{L}) | J·kg^{−1} | 210 × 10^{3} |

Density (${\rho}_{L}$) (liquid phase) | Kg·m^{−3} | 7400 |

Specific heat (${c}_{L}$) (liquid phase) | J·kg^{−1}·K^{−1} | 720 |

Conductivity (k_{L}) (liquid phase) | W·m^{−1}·K^{−1} | 29.6 |

Test Number | Crown Width (W) | Depth (D) | ||||
---|---|---|---|---|---|---|

Experimental (mm) | Model (mm) | Error (%) | Experimental (mm) | Model (mm) | Error (%) | |

1 | 2.16 | 2.30 | 6.38 | 2.00 | 2.00 | 0.00 |

2 | 1.98 | 1.80 | 9.09 | 1.09 | 1.05 | 3.93 |

3 | 2.42 | 2.50 | 3.52 | 2.00 | 2.00 | 0.00 |

4 | 2.07 | 2.00 | 3.19 | 1.30 | 1.33 | 2.47 |

5 | 2.56 | 2.60 | 1.76 | 2.00 | 2.00 | 0.00 |

6 | 2.40 | 2.20 | 8.37 | 1.72 | 1.75 | 1.74 |

7 | 2.82 | 2.62 | 7.13 | 2.00 | 2.00 | 0.00 |

8 | 2.61 | 2.35 | 9.82 | 2.00 | 2.00 | 0.00 |

Test Number | Crown Height (H) | Root Height (R) | ||||
---|---|---|---|---|---|---|

Experimental (mm) | Model (mm) | Error (mm) | Experimental (mm) | Model (mm) | Error (mm) | |

1 | −0.11 | 0.00 | 0.11 | 0.07 | 0.00 | 0.07 |

2 | 0.09 | 0.00 | 0.09 | 0.00 | 0.00 | 0.00 |

3 | −0.16 | −0.10 | 0.06 | 0.31 | 0.25 | 0.06 |

4 | 0.04 | 0.00 | 0.04 | 0.00 | 0.00 | 0.00 |

5 | −0.20 | −0.20 | 0.00 | 0.43 | 0.45 | 0.02 |

6 | 0.16 | 0.00 | 0.16 | 0.00 | 0.00 | 0.00 |

7 | −0.16 | −0.25 | 0.09 | 0.58 | 0.50 | 0.08 |

8 | 0.10 | 0.00 | 0.10 | 0.07 | 0.00 | 0.07 |

Test Number | Area M1 | Area M2 | ||||
---|---|---|---|---|---|---|

Model (μm) | Experimental (μm) | Error (μm) | Model (μm) | Experimental (μm) | Error (μm) | |

1 | 2.71 | 3.19 | −0.48 | 2.77 | 3.70 | −0.93 |

2 | 2.21 | 3.05 | −0.84 | 2.04 | 2.58 | −0.54 |

7 | 2.31 | 3.78 | −1.47 | 2.82 | 4.25 | −1.43 |

8 | 2.08 | 3.54 | −1.46 | 1.83 | 2.63 | −0.80 |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hernando, I.; Arrizubieta, J.I.; Lamikiz, A.; Ukar, E.
Numerical Model for Predicting Bead Geometry and Microstructure in Laser Beam Welding of Inconel 718 Sheets. *Metals* **2018**, *8*, 536.
https://doi.org/10.3390/met8070536

**AMA Style**

Hernando I, Arrizubieta JI, Lamikiz A, Ukar E.
Numerical Model for Predicting Bead Geometry and Microstructure in Laser Beam Welding of Inconel 718 Sheets. *Metals*. 2018; 8(7):536.
https://doi.org/10.3390/met8070536

**Chicago/Turabian Style**

Hernando, Iñigo, Jon Iñaki Arrizubieta, Aitzol Lamikiz, and Eneko Ukar.
2018. "Numerical Model for Predicting Bead Geometry and Microstructure in Laser Beam Welding of Inconel 718 Sheets" *Metals* 8, no. 7: 536.
https://doi.org/10.3390/met8070536