## 2 Methodology
### Numerical Procedure  
The CDFW model was developed using the commercial finite element (FE) modeling software package, DEFORM 11.0. The software is especially capable of solving problems involving a large plastic deformation via remeshing [24]. The geometry and meshing of the FE model are shown in Figure 2.  
A two-dimensional analysis of one-half of the solid element was employed because of the axisymmetric shape of the specimen (Figure 2a). The mesh contained 2125 elements and 2321 nodes (Figure 2b). The dimensions of the modelled parts are consistent with the experiment in terms of width. Three different mesh-sized windows were used. Coarse elements of 0.03 mm were used in the region containing the weld interface. Outside this region, as the distance increases, the element size also increases, and a mesh size of 0.1 mm was used across the model. The effect of the mesh size was also considered in the study. Finally, the mesh size was selected so as not to generate long calculation times. Automatic remeshing was used to maintain a good mesh quality. Many various simulation methods, such as 2D axial-symmetric, 3D full simulation and 2.5D torsion approaches, were initially tested. Finally, a 2.5D torsion simulation including the circumferential effects was accepted for use in this study. A 2.5D model includes all the circumferential velocities needed to account for the torsional friction during welding. In the 2D axial-symmetric approach, the DEFORM code is not able to simulate a rotational speed, and the related shear deformation cannot be computed. The computing time required for a 3D full simulation is very long. In this paper, the torsional friction model has a significant impact not only on heat generation, but also on the deformation of the specimen, as noticed in reference [25]. Figure 2c shows  
Figure 1: Friction welding machine showing placed samples with thermocouple (**a**), and temperature measurement scheme (**b**) where: 1,2,3,4,5,6 are the thermocouples positions. the model with the sticking conditions at the contact point. This axisymmetric model takes into account both the axial force (see lower tooling) and the rotational speed (see upper tooling). The friction coefficient and surface pressures between the weld were taken into account. When the temperature is low, the friction stress is calculated using the Coulomb friction law according to the following formulae:  
\[\tau_{f}=-\mu\sigma_{n}\frac{v_{s}}{|v_{s}|} \tag{1}\]  
where \(\tau_{f}\) is the shear friction, is the friction coefficient, \(\sigma_{n}\) is the normal stress, and \(v_{s}\) is the sliding velocity; the negative sign characterizes the opposition between the friction stress. The friction coefficient \(\mu\) can be estimated as follows:  
\[\mu=aP^{b}T^{c}exp(dV) \tag{2}\]  
where \(P\), \(T\) and \(V\) are the axial pressure, temperature and linear velocity, and \(a\), \(b\), \(c\) and \(d\) are the constants obtained from the experiments. When the temperature is high, the shear friction law is adopted and can be described as follows:  
\[\tau_{f}=-mk\frac{v_{s}}{|v_{s}|} \tag{3}\]  
where \(k\) is the shear yield stress of the material and \(m\) the friction factor [26]. In this study, to determine an appropriate shear friction, preliminary simulations were carried out. It was found that using a shear friction of 0.175 gave the best results for the given temperatures. Data regarding the temperature-dependent thermal conductivity, specific heat and emissivity for AA were used; these were obtained from the DEFORM software library. The thermophysical properties of THA were taken from the ASM Handbook [2]. The properties of both materials for the simulation model are presented in Table 2.  
The viscoplastic behavior of the materials is characterized by the temperature, flow stress and strain rate, and was obtained from the literature [2]. Figure 3 shows the flow stress for the aluminum (Figure 3a) alloy and tungsten composite (Figure 3b) at a strain of \(\varepsilon\) = 0.1 during deformation.