**3. Analytical Rosenthal Equation**

Rosenthal [6] has proposed an analytical method to estimate the thermal characteristics of materials during fusion welding in conduction mode. Therefore, this equation can be used in laser welding processes undergoing conduction mode welding and not key hole, in order to understand the temperature-dependent behavior of materials during welding. The Rosenthal equation is a simple method to predict the thermal behavior of materials, which makes it applicable to various manufacturing processes such that time-dependent parameters like temperature gradient, cooling rate, and solidification rate are able to be calculated using this equation. For this purpose, there are some assumptions that have been taken into account to develop this particular equation, which are as follows [6]:


In order for the analytical Rosenthal equation to be applicable in the pulsed mode of a laser beam as a heat generator in the welding process, an additional assumption is made that the laser beam is active in the time range of its pulse duration 4 ms. To do so, Equation (3) should be performed directly in the Rosenthal equation. Therefore, the modified Rosenthal equation is achieved and shown in Equation (5):

$$T = T\_0 + \frac{\lambda P \varphi(t)}{2\pi kr} \exp\left[-\frac{V(r+\xi)}{2\alpha}\right] \tag{5}$$

According to Equation (5), *T*<sup>0</sup> is the base material starting temperature, *k* is the thermal conductivity, *V* is the welding velocity, α is the thermal diffusivity, and ϕ(*t*) is the function that was previously described in Equation (3). It should be taken into account that in Equation (5), the welding is performed in the direction of the *y*-axis; so, to perform welding velocity and time in the equation, *y* − *Vt* is substituted by ξ; r is the distance of any point on the substrate from the laser beam, which can be computed by *x*<sup>2</sup> + ξ<sup>2</sup> + *z*2. As discussed before, materials' properties are not the functions of temperature while using in the analytical solution. As a result, the properties mentioned in Table 4 are used in the analytical Rosenthal equation to predict the thermal behavior of the material in laser welding.

**Table 4.** Room-temperature thermal properties of aluminum alloy 5456 used in the analytical method.

