*2.2. Thermal Modeling*

The substrate with the width of 10 mm, the length of 10 mm, and the height of 2.5 mm is involved in the thermal modeling. In general, it should be mentioned that the dimension of actual substrate that is used for the laser welding process is larger than the domain considered for simulation. Nevertheless, the thermal model simply plays a role as a heat container in the welding process; the serious thermal involvement happens adjacent to the upper part of the substrate and decreases abruptly with shifting to the bottom [21]. It was verified with a domain size independence test that the calculation domain is appropriate to not influence the temperature regime during the welding process, while restricting the computational load. To do so, five kinds of domains were used to validate the domain size independence. These were 10 <sup>×</sup> <sup>10</sup> <sup>×</sup> 2.5 mm3, 8 <sup>×</sup> <sup>8</sup> <sup>×</sup> 2 mm3, 6 <sup>×</sup> <sup>6</sup> <sup>×</sup> 1.5 mm3, 4 <sup>×</sup> <sup>4</sup> <sup>×</sup> 1 mm3, and 2 <sup>×</sup> 2 <sup>×</sup> 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, (*x*0, *y*0) 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 ϕ, which is dependent on time of the welding process t, is shown below as Equation (3).

$$q(\mathbf{x}, y, t) = \frac{2\lambda P \varphi(t)}{\pi r\_0^2} \exp\left\{ \frac{-2\left[\left(\mathbf{x} - \mathbf{x}\_0 - Vt\right)^2 + \left(y - y\_0 - Vt\right)^2\right]}{r\_0^2} \right\} \tag{2}$$

$$\varphi(t) = \begin{cases} \begin{array}{c} 1 \ t = D \\ 0 \ t = f - D \end{array} \tag{3} \end{cases} \tag{3}$$

**Figure 2.** Domain size independence test.

In this equation, *P* is the laser power, λ is the absorption coefficient of the material, *r*<sup>0</sup> is the laser beam radius, *V* is the welding velocity, and ϕ(*T*) is the function calculated via Equation (3), in which *D* and *f* are the pulse duration and frequency of the laser equipment, respectively. The absorptivity of aluminum alloy 5456 is reported to be 0.36 [23].

To calculate the amount of radiation losses during laser welding, Equation (4) is performed using a coefficient for radiative heat transfer through the top surface of the substrate:

$$h\_{\text{radiation}} = \varepsilon \sigma (T^2 + T\_{\text{ambient}}^2)(T + T\_{\text{ambient}}) \tag{4}$$

In this equation, *hradiation* is supposed to be the coefficient for radiative heat transfer, σ is a constant coefficient known as the Stefan–Boltzmann constant, ε is the coefficient for emission of the substrate, and *Tambient* is considered to be the ambient temperature; these parameters are reported to be 5.67 J/K2·m4, 0.022, and 293 K, respectively [24,25].

In the thermal model, the temperature of the bottom part of the substrate is kept at room temperature. Based on previous studies, it has been shown that the substrate temperature is not significantly affected by the temperature variation at the bottom part of the substrate [26]. Furthermore, there is an assumption in the thermal model in which the insulation aspect of side walls is taken into account. Figure 3a exhibits the thermal model, which is considered for simulation in this research. Moreover, the calculation domain used in this simulation is demonstrated in Figure 3b. It should be taken into account that the mesh used in the present study is extremely fine (530714 elements) to calculate the temperature-dependent parameters accurately. A tetrahedral-shaped mesh with the minimum element size of 0.002 mm and the element volume ratio of 0.06968 was used to investigate the thermal behavior of joints.

**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.
