# A Comparative Study of Analytical Rosenthal, Finite Element, and Experimental Approaches in Laser Welding of AA5456 Alloy

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Modeling

#### 2.1. Material Properties and Methods

#### 2.2. Thermal Modeling

^{3}, 8 × 8 × 2 mm

^{3}, 6 × 6 × 1.5 mm

^{3}, 4 × 4 × 1 mm

^{3}, and 2 × 2 × 1 mm

^{3}, as domains 1 to 5, respectively. As shown in Figure 2, the temperature variation at point (5, 0, 2.5), as the initial point of laser beam with an average power of 80 W focused on the substrate, was extracted. The difference between these five domains is less than 1%. Therefore, domain 1 was selected for this calculation to reduce the computational cost and time. However, this issue should be taken into account that by increasing the heat input of laser beam, the molten depth can be larger than the height of the substrate determined in the thermal model. Therefore, if the molten depth becomes larger than the domain used in the thermal modeling, the height of the domain needs to be changed with regards to the molten depth. However, in order for the computational cost and time to be decreased with regards to the molten depth and width, which were the main concentration in the present study, a domain with 2.5 mm height was prepared. Additionally, natural convection within a liquid melt pool is neglected in the thermal calculation. By doing so, the molten pool temperature is supposed to be higher than its temperature in real experiments; however, this would affect the solidification process very little from the molten pool boundary in which phase transformation and heat conduction occur [22]. Equation (2) illustrates the distribution mode of laser beam power, which is considered to be Gaussian. In this equation, $\left({x}_{0},{y}_{0}\right)$ is the initial point of the laser beam focused on the substrate, which is equal to (5, 0). Moreover, it should be noted that a pulsed Nd:YAG laser was utilized in the present study; therefore, it has a pulse duration and frequency with which the on-time and off-time of the laser beam can be determined. According to the parameters in Table 2, the pulse duration and frequency are 4 ms and 10 Hz, respectively. This means the laser beam is activated for 4 ms and then deactivated for 96 ms. In this regard, there should be 10 pulses in a second of the laser welding process (the time length from a pulse to the next pulse is around 100 ms). To consider the pulse mode of the laser beam, a step function is defined in which its value is 1 when the laser is active; on the other hand, its value is 0 when the laser beam is off. This function $\phi $, which is dependent on time of the welding process t, is shown below as Equation (3).

## 3. Analytical Rosenthal Equation

- Materials’ properties such as specific heat at a constant pressure, density, and thermal conductivity are not the functions of temperature. In addition, the latent heat of materials is not considered when a phase transformation occurs between the solidus and liquidus temperatures of alloys, especially in the calculation of the specific heat in the mushy zone.
- The heat distribution mode on the substrate is considered to be quasi-stationary due to the consistency of welding velocity and power.
- A point heat source is selected in this analytical approach.
- Heat losses through radiation and convection are not taken into account. Furthermore, the natural convection within the substrate is not involved so that the heat produced in the substrate transfers by the conduction mode.

## 4. Results

#### 4.1. Molten Pool Dimensions

#### 4.2. Predicted Temperature-Dependent Parameters

#### 4.3. Microstructural Evaluation: Primary Dendrite Arm Spacing (PDAS)

## 5. Conclusions

- (1)
- The fusion width and depth obtained from the numerical modeling are consistent with the experimental results up to the heat input of 30 J · mm
^{−1}. However, by increasing the heat input, the discrepancy between the results from the numerical modeling and experimental ones becomes larger. The Rosenthal equation overestimates the melt pool dimensions. The reason may lie in the assumptions performed in the analytical Rosenthal equation in terms of neglecting heat losses through the substrate. - (2)
- With regards to the partially melted zone, the FE model is slightly larger than the experimental results at the heat input of 25 J · mm
^{−1}. However, by increasing the heat input, the discrepancy between the results achieved from the numerical modeling and the results from experiments becomes wider. On the other hand, the Rosenthal equation overestimates the partially melted zone in comparison to the FE model and the experiment; this could be a result of assumptions with which heat losses as well as latent heat of the material are supposed to be neglected. - (3)
- Numerical modeling produces temperature-dependent parameters including temperature gradient, cooling rate, and growth rate, which are higher than those obtained from the analytical method. Moreover, the discrepancy between the predicted results using the numerical modeling and the analytical method becomes larger by enhancing energy density.
- (4)
- The primary dendrite arm spacing (PDAS) is measured using the numerical modeling and the analytical method. As a result, the values obtained from the analytical method are slightly larger than those from the numerical modeling. Furthermore, it is observed that both of the numerical and analytical methods predict the PDAS with somewhat accuracy at the heat input of 30 J · mm
^{−1}regarding the experimental result. However, there should be more experiments to verify these methods as a confident tool for prediction of the microstructure in laser welded materials, especially at different heat inputs. - (5)
- All in all, the Rosenthal equation underestimates thermal results and overestimates microstructural results in comparison to the FE model. It should be taken into account that the discrepancy between the results attained from the analytical method and the numerical modeling becomes larger at higher heat inputs; therefore, there should be a restriction in utilizing the analytical Rosenthal equation at higher heat inputs to investigate the heat transfer behavior of laser welded materials due to the assumptions performed in this particular method. Thus, the numerical modeling can be widely used for investigation of heat transfer behavior of welded components at higher heat inputs, and its accuracy is more than the analytical method because it uses genuine materials’ characteristics.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Variation of thermophysical properties for 5456 aluminum alloy (AA5456 with temperature: (

**a**) specific heat; (

**b**) density; and (

**c**) thermal conductivity.

**Figure 3.**(

**a**) Thermal model: (1) the distribution mode of laser heat source; (2) radiation and convection losses; (3) insulation of side walls; (4) the bottom part of the substrate remained at room temperature; (

**b**) calculation domain used in the simulation.

**Figure 4.**(

**a**) Transverse-section (x-z) of the heat transfer; (

**b**) molten pool achieved from the numerical modeling using an average laser power of 80 W, welding speed of 2 mm.s

^{−1}, and absorptivity of 0.36 at t = 4 ms.

**Figure 5.**Comparison of the fusion width obtained from experiments, the numerical modeling, and the analytical Rosenthal equation.

**Figure 6.**Comparison of the fusion depth obtained from experiments, the numerical modeling, and the analytical Rosenthal equation.

**Figure 7.**(

**a**) Comparison of the partially melted zone thickness obtained from experiments, the numerical modeling, and the analytical method; (

**b**) fitted partially melted zone from the finite-element (FE) model on the cross-section of the melt pool using an average laser power of 50 $W$, welding velocity of 2 $\mathrm{mm}\xb7{\mathrm{s}}^{-1}$, and absorptivity of 0.36.

**Figure 8.**(

**a**) Variation of temperature versus welding time for different points within the molten pool; (

**b**) variation of temperature gradient versus welding time for different points within the molten pool; (

**c**) variation of temperature gradient versus temperature for different points within the molten pool obtained from the numerical modeling using a laser beam with an average power of 80 $\mathrm{W}$, welding velocity of 2 $\mathrm{mm}\xb7{\mathrm{s}}^{-1}$, and absorption coefficient of 0.36.

**Figure 9.**(

**a**) Temperature gradient, (

**b**) cooling rate, and (

**c**) solidification rate achieved from the numerical modeling and the analytical method versus heat input of the laser beam and their comparison.

**Figure 10.**Variation of primary dendrite arm spacing (PDAS) obtained from the experiment, the numerical modeling, and the analytical method versus heat input.

**Figure 11.**Measurement of dendrite arms taken by optical and scanning electron microscopies at the boundary of fusion zone using a laser with an average power of 60 $\mathrm{W}$, welding velocity of 2 $\mathrm{mm}.{\mathrm{s}}^{-1}$, and absorption coefficient of 0.36.

Mg | Mn | Fe | Si | Cr | Cu | Ti | Al |
---|---|---|---|---|---|---|---|

4.7 | 0.66 | 0.22 | 0.09 | 0.09 | 0.01 | 0.03 | Base |

Specimen | Pulse Frequency (Hz) | Pulse Duration (ms) | Peak Power (kW) | Pulse Energy (J) | Average Power (W) | Heat Input (J/mm) |
---|---|---|---|---|---|---|

A1 | 10 | 4 | 1.25 | 5 | 50 | 25 |

A2 | 10 | 4 | 1.5 | 6 | 60 | 30 |

A3 | 10 | 4 | 1.75 | 7 | 70 | 35 |

A4 | 10 | 4 | 2 | 8 | 80 | 40 |

Polynomial Coefficients $\mathit{y}=\mathit{a}+\mathit{b}\mathit{T}+\mathit{c}{\mathit{T}}^{2}$ | |||||||
---|---|---|---|---|---|---|---|

Property $\mathit{y}$ | Unit | $\mathit{a}$ | $\mathit{b}$ | $\mathit{c}$ | Range T (K) | State | Reference |

$D\left(T\right)$ | kg·m^{−3} | 2717.683 | −0.231 | - | $293\le T\le {T}_{m}$ | s | [16,17] |

$D\left(T\right)$ | kg·m^{−3} | 2599.97 | −0.27 | - | ${T}_{m}\le T\le 1680$ | l | [16,17] |

${c}_{p}\left(T\right)$ | J·kg^{−1}·K^{−1} | 787.73 | 0.457306 | - | $293\le T\le {T}_{m}$ | s | [16,18] |

${c}_{p}\left(T\right)$ | J·kg^{−1}·K^{−1} | 1261 | - | - | ${T}_{m}\le T\le 1491$ | l | [16,17] |

$\lambda \left(T\right)$ | W·m^{−1}·K^{−1} | 111.11 | 0.0888 | - | $293\le T\le {T}_{m}$ | s | [16,18] |

$\lambda \left(T\right)$ | W·m^{−1}·K^{−1} | 33.9 | 7.892 × 10^{−2} | −2.099 × 10^{−5} | ${T}_{m}\le T\le 1491$ | l | [16,17] |

Property | Value | Reference |
---|---|---|

Thermal conductivity, $k$ | 117 $\mathrm{W}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1}$ | [27,28] |

Density, $\rho $ | 2670 $\mathrm{kg}.{\mathrm{m}}^{-3}$ | [27,28] |

Specific heat, ${C}_{p}$ | 924 $\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}$ | [27,28] |

Absorptivity, $\lambda $ | 0.36 | [23] |

Solid diffusivity, $\alpha $ | $6.74\times {10}^{-5}{\text{}\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ | [17] |

Feature | Amount | Reference |
---|---|---|

Solidification zone, $\mathsf{\Delta}{T}_{0}$ | 67 $\mathrm{K}$ | [20] |

Gibbs–Thomson coefficient, $\Gamma $ | $1.3\times {10}^{-7}\text{}\mathrm{K}\xb7\mathrm{m}$ | [5] |

Partition coefficient, ${k}_{0}$ | 0.48 | [5] |

Liquid diffusivity, $D$ | ${10}^{-8}{\text{}\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ | [5] |

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**MDPI and ACS Style**

Hekmatjou, H.; Zeng, Z.; Shen, J.; Oliveira, J.P.; Naffakh-Moosavy, H.
A Comparative Study of Analytical Rosenthal, Finite Element, and Experimental Approaches in Laser Welding of AA5456 Alloy. *Metals* **2020**, *10*, 436.
https://doi.org/10.3390/met10040436

**AMA Style**

Hekmatjou H, Zeng Z, Shen J, Oliveira JP, Naffakh-Moosavy H.
A Comparative Study of Analytical Rosenthal, Finite Element, and Experimental Approaches in Laser Welding of AA5456 Alloy. *Metals*. 2020; 10(4):436.
https://doi.org/10.3390/met10040436

**Chicago/Turabian Style**

Hekmatjou, Hamidreza, Zhi Zeng, Jiajia Shen, J. P. Oliveira, and Homam Naffakh-Moosavy.
2020. "A Comparative Study of Analytical Rosenthal, Finite Element, and Experimental Approaches in Laser Welding of AA5456 Alloy" *Metals* 10, no. 4: 436.
https://doi.org/10.3390/met10040436