## 3 Results and Discussion
### Thermal Cycle during the AFSD Process  
Depending on the friction stir processing parameters [33; 42; 43; 44], the rotating consumable rod during the deposition process and under the machine downward force suffers from both normal and torsional friction forces at its tip during the friction stirring with the fixed substrate on the machine table in the deposition region. To derive an equation for recognizing the amount of heat generated during this process, an annulus with a width dR, an area dA, and an inner radius R at the friction interface between the used rod and the substrate was selected as a representative element for estimation, as illustrated in Figure 6a. This annulus is then subject to a torque of dT. The annulus is the primary heat source in the introductory stage of the deposition process. The generated heat input (Q in watt) to build DPs by the AFSD process is generally described as follows:  
\[\mathrm{dQ}=\omega\times\mathrm{dT} \tag{1}\]  
Figure 5: Axial force cycle during the AFSD process of the AA7075-T6 deposited at various deposition feed rates of (a) 1, (b) 2, (c) 3 and (d) 4 mm/min and 400 rpm. From Equations (1)-(3)  
\[\mathrm{dQ}=\omega\times\mathrm{R}\times\tau_{\mathrm{Friction}}\times\mathrm{dA} \tag{4}\]  
T is the applied torque (in N.m), \(\omega\) is the angular rotation speed (in rad/s). \[\tau_{\mathrm{Friction}}=\tau_{\mathrm{contact}}=\mu\times\mathrm{P}=\mu\times \mathrm{F/A} \tag{5}\]  
\[\mathrm{dA}=\mathrm{R}\times\mathrm{dR}\times\mathrm{d\theta} \tag{6}\]  
\(\tau_{\mathrm{friction}}\) (in Pa) is the friction shear stress at the interface, (F in N) is the axial downward force, (A in m\({}^{2}\)) is the projected areas, and (P in Pa) is the contact pressure. Substitute in Equation (4) using Equations (5) and (6)  
\[\mathrm{dQ}=\ \omega\ \times\ \mathrm{R}\ \times\tau_{\mathrm{contact}}\times \mathrm{R}\ \times\ \mathrm{dR}\ \times\ \mathrm{d\theta} \tag{7}\]  
\[\mathrm{dQ}=\ \omega\ \times\ \mathrm{R}\ \times\mu\ \times\mathrm{P}\ \times\ \mathrm{R}\ \times\ \mathrm{dR}\ \times\ \mathrm{d\theta} \tag{8}\]  
\[\mathrm{dQ}=\ \omega\ \times\ \mathrm{R}^{2}\times\mu\ \times\mathrm{P}\times\mathrm{dR}\ \times\ \mathrm{d\theta} \tag{9}\]  
\[\mathrm{Q}=\int_{0}^{2\pi}\int_{0}^{\mathrm{R}}\omega\ \times\ \mathrm{R}^{2}\times\ \mathrm{P}\ \times\mathrm{dR}\ \times\mathrm{d\theta} \tag{10}\]  
\[\mathrm{Q}=\frac{2}{3}\times\pi\ \times\ \omega\ \times\ \mu\ \times\ \mathrm{R}^{3}\times\frac{ \mathrm{F}}{\mathrm{A}} \tag{11}\]  
Figure 5:0  
Figure 5:1 (**a**) The scheme of an element at the friction interface and (**b**) the temperature measuring position in the substrate. Figure 5:2  
where N is the CRRS (in rpm), R is the consumable rod radius (in m), \(\mu\) is the friction coefficient between the two rubbing materials (equals 1.35) [45] and F is the deposition axial force. By substitution in Equation (11) using Equations (12) and (13), then:  
Figure 5:3  
For an AA7075-T6 rod rotating speed of 400 rpm and different deposition rates (1-4 mm/min), using the measured downward forces at each deposition condition, Table 2 was plotted. It also presents the measured temperature of the AA2024-T4 substrate (T\({}_{1}\)) via a thermocouple at the position shown in Figure 6b and the generated heat input based on Equation (14). The temperature of the deposited layer (T\({}_{2}\)) is calculated depending on the Fourier law of heat conduction by Equation (16)  
Figure 5:4  
By substitution in Equation (15) using Equation (14):  
Figure 5:5  
In this equation, K is the material's conductivity and T\({}_{1}\) and T\({}_{2}\) are the change in the temperatures across a short distance (\(\Delta\)x).