2.2.1. Geometric Model and Boundary Conditions

The FSW process is a dynamic nonlinear analysis based on the Lagrange method. The finite element software DEFORM-3D (V10.2, Scientific Forming Technologies Corporation, Columbus, OH, USA) was used to simulate the whole process of friction stir welding. The material of the tool was W6, and the workpiece size was 150 mm × 100 mm × 6 mm for a 2A14-T6 lightweight aluminum alloy. The tool shoulder diameter of the tool is 16.3 mm, the maximum diameter of the tool pin is 8.15 mm, and the length of the tool pin is 5.65 mm.

The geometric model of the friction stir welding is shown in Figure 3. To make the simulation results more accurate, three sets of meshes are added to the workpiece weld area, and the minimum mesh is divided into 0.6 mm. Because the pin participates in the material flow of the welding process, the tool plays an important role in the flow of the material. However, considering the solution duration, the mesh size of the tool shoulder and the welding influence area are gradually increased, which ensures the calculation accuracy and improves the calculation efficiency [31]. A set of mesh window is also set on the tool, and the mesh window is set to move synchronously with the tool during the welding process.

**Figure 3.** Geometric model in finite element model (FEM) simulation.

Due to the large plastic deformation of the material that is to be welded during the FSW process, the mesh adaptive re-division technique can be used to control the mesh distortion caused by the rotation and translation of the tool. The mesh distortion is controlled by the absolute minimum mesh size. The absolute mesh size can ensure the accuracy of the solution but also increase the solution time [32]. The total number of workpiece tetrahedral elements is divided into 68,762. The total number of mixing head tetrahedral meshes is 35,104. In the simulation process, since the tool material strength is much higher than the workpiece material, the tool is set as a rigid body, and the workpiece is set as a rigid plastic body. In order to simplify the model, the fixture and the backing plate are not built in the model, and the bottom and side degrees of freedom of the workpiece are fully constrained.

## 2.2.2. Finite Element Formula

A rigid viscoplastic model with a von-Miss yield criterion is used [33]. The finite element formula for rigid viscoplastic materials is based on the variational principle, where the allowable velocity should satisfy the conditions of compatibility and incompressibility, as shown in Equation (14):

$$\eta = \bigcup\_{V} E(\dot{\varepsilon}\_{ij})dV - \bigcup\_{\widetilde{S}\_F} F\_i u\_i dS\_\prime \tag{14}$$

where *Fi*, *V*, *SF* and *E*( . ε*ij*) are respectively the surface traction force, the workpiece volume, the force surface and the plastic deformation power function. A penalty function for incompressibility is added to eliminate the incompressibility constraint on the allowable velocity field. The actual velocity field can now be determined by the steady value of the variation equation [34], which is expressed as:

$$
\delta\eta = \int\_{V} \overline{\sigma} \delta\overline{\varepsilon} dV + \lambda \int\_{V} \dot{\varepsilon}\_{V} \delta\dot{\varepsilon}\_{V} dV - \int\_{S\_{F}} F\_{i} \delta u\_{i} dS\_{i} \tag{15}
$$

where . ε, . ε*ij*, and σ *ij* are the effective strain rate, strain rate component and deviatoric stress component. λ, and δ*ui* are, respectively, the large penalty factors and arbitrary variables. δ . ε*<sup>V</sup>* and δ . ε represent the change in the strain rate derived from δ*ui*.

#### 2.2.3. Material Model

A proper selection of material models is critical to an accurate solution in the simulation process. The material changes from a solid state to a viscous state, so it is necessary to define a wide range of

strain, strain rate and flow stress values at temperatures. The flow stress is defined as a function of the strain, strain rate, and temperature, and Equation (16) is used to define the flow stress:

$$
\overline{\sigma} = \overline{\sigma}(\varepsilon, \dot{\overline{\varepsilon}}, T),
\tag{16}
$$

Among them, σ is the flow stress, ε is the strain, and *T* is the temperature.

The flow stress is temperature-dependent and strain rate sensitive. Figure 4 shows the flow stress curve of the 2A14 aluminum alloy [35]. The flow stress data is imported into the numerical model. As can be seen from Figure 4, the material is sensitive to the temperature at high strain rates. As the temperature rises, the flow stress decreases. The main reason for this is that the thermal vibration energy of the crystal lattice becomes larger at a high temperature and the external force required for the dislocation movement decreases. The thermal properties of the 2A14-T6 workpiece and W6 tool steel are summarized in Table 2.

**Figure 4.** Flow stress curve of the 2A14 aluminum alloy.


