2.2.4. Dislocation Motion Model

The plastic deformation of the metal results from the dislocation movement of the material, so the dislocation density changes constantly in the process of friction stir welding, and the material flow changes based on this model. The dislocation motion is expressed as follows:

$$d\rho\_i = (h - \dot{r}\rho\_i)d\varepsilon\_\prime \tag{17}$$

$$h = h\_0 \left(\frac{\dot{\overline{\varepsilon}}}{\dot{\varepsilon}\_0}\right)^m \cdot \exp\left(\frac{mQ}{RT}\right). \tag{18}$$

$$\dot{r} = \dot{r}\_0 \left(\frac{\dot{\overline{\varepsilon}}}{\dot{\varepsilon}\_0}\right)^{-m} \cdot \exp\left(\frac{-m\underline{Q}}{RT}\right) \tag{19}$$

where <sup>ρ</sup>*<sup>i</sup>* is the dislocation density, *<sup>h</sup>* is the height of the action range of the dislocation stress field, . *r* is the radius of the action range of the dislocation stress field, *<sup>m</sup>* is the rate sensitivity, and . ε<sup>0</sup> is the non-dynamic strain rate.
