*2.2. Heat Generation Model*

In the process of FSW, welding heat input includes three parts: (1) friction between the shoulder and the workpiece; (2) friction between the pin and the workpiece; and (3) plastic deformation of the material. According to the minimum torque required by the tool to overcome the rotation of friction, the heat produced by friction on the shoulder, the heat produced by friction on the side of the pin, and the heat produced by friction on the bottom of the pin can be calculated as:

$$Q\_{\rm shoulder} = \frac{4}{3}\pi^2 \mu \text{PN} \left(\mathcal{R}\_s^{~3} - \mathcal{R}\_2^{~3}\right) \tag{1}$$

$$Q\_{pin-side} = \frac{4\pi^2 N \mu P}{3\sin\alpha} \left(R\_2^3 - R\_3^3\right) \tag{2}$$

$$Q\_{\rm pin-bottom} = \frac{4}{3}\pi^2 \mu PNR\_3^3 \tag{3}$$

where *Qshoulder*, *Qpin*−*side*, *Qpin*−*bottom* represent heat generation at the shoulder, heat generation at the side of the pin, and heat generation at the bottom of the pin, respectively. μ is the friction coefficient, *P* is the welding pressure, *N* is the rotating speed, *Rs* is the shoulder radius, *R*<sup>2</sup> is the root radius of the pin, *R*<sup>3</sup> is the end radius of the pin, and α is the cone angle of the pin.

In the process of FSW, the material is subjected to severe plastic deformation. Most of the deformation work of the material is transferred to the surrounding area in the form of heat energy. This part of the deformation work accounts for about 90% of the deformation work, while the rest

of the work is used in the plastic deformation of the material. The heat source density generated by plastic deformation is: .

$$q\_p = \overline{\sigma} \alpha\_p \overline{\epsilon} \tag{4}$$

where qp is the heat source density of the plastic deformation of the material, σ is the equivalent stress,

<sup>α</sup><sup>p</sup> is the heat conversion efficiency, which is set as 0.9, and . ε is the equivalent strain rate.

Therefore, the heat produced by plastic deformation of metal materials is:

$$Q\_v = \int\_v \overline{\sigma} a\_p \dot{\overline{\varepsilon}} dV \tag{5}$$

where Qv represents the plastic deformation heat production of the material with volume V.
