## 2 Materials and Methods
### Finite Element Numerical Modeling  
FEM simulations were carried out considering the interference fit of a hard press, type 50H7s6, according to the ISO standard for fits and tolerances [12]. This fit is recommended by such a standard for transmitting both movement and a given torque with a fixed placement of the shaft into the hub in a rotation movement. The dimensions of the interference fit were selected according to the following ratios obtained in previous studies [23]: length ratio \(L/l=4\) and the hub length must be \(l/d=1\), where \(d\) is the shaft diameter. Thus, the reference case has, on the one hand, a solid shaft of radius \(R=25\) mm and a length of \(L=200\) mm and, on the other hand, the hub is a hollow cylinder with a length of \(l=50\) mm  
Figure 1: Scheme of a shrink fit with (**a**) a conventional hub; (**b**) a partial chamfered hub; and (**c**) a full–chamfler hub. and an outer radius of \(r_{o}=50\) mm. Thus, taking into account the tolerances of the selected interference fit 50H7/s6, the resulting maximum radial interference is \(\delta=29.5\) um. Full-chamfer hubs constitute a simple modification of the conventional hub geometry. Thus, only one parameter is needed for defining the hub geometry, i.e., the chamfer angle, \(\alpha\), defined as shown in Figure 1c. However, for partial-chamfer hubs, an additional parameter is required, i.e., the chamfer height, \(l_{r}\), according to Figure 1b. In both cases, full and partial-chamfer hubs, the hub thickness, \(t\), has an influence on stress concentrations, so in this study this parameter was also taken into account. The contact hub length, \(l\), was fixed for all the cases of study. In this way, the maximum transmitted torque is only dependent on contact pressure according to Equation (4). In order to include the influence of the hub thickness, \(t\), in the study, a dimensionless parameter \(\gamma\) was defined as the ratio between the hub thickness and the shaft diameter as follows: \(\gamma=t/d\). Thus, in this study, five different values of \(\gamma\) were considered: (i) an eighth of the shaft diameter, \(\gamma=1/8\), (ii) a quarter of \(d\), \(\gamma=1/4\), (iii) a half, \(\gamma=1/2\) (iv), three quarters, \(\gamma=3/4\), and (v) a hub thickness equal to the shaft diameter, \(\gamma=1\). In addition, a dimensionless parameter \(\beta\), defined as the ratio between the chamfer height and hub thickness, \(\beta=l_{r}/t\), is useful for the analysis of partial-chamfer hubs. Thus, when \(\beta=0\), a conventional hub without a chamfer hub is considered; \(0<\beta<1\) represents partial-chamfer hubs with diverse chamfer heights; and, finally, \(\beta=1\) represents full-chamfer hubs as shown in Figure 2. For each of the values of \(\gamma\), six different values of \(\beta\) are considered as follows: (i) \(\beta=0\) is a conventional case without a chamfer hub; four partial-chamfer hubs progressively increasing the hub thickness are as follows: (ii) \(\beta=1/8\), (iii) \(\beta=1/4\), (iv) \(\beta=1/2\), (v) \(\beta=3/4\), and, finally, (vi) \(\beta=1\), where the geometry of the full-chamfer hubs is characterized by a chamfer height equal to the hub thickness. Finally, for a given hub thickness \(\gamma\) and a chamfer height \(\beta\), the chamfer geometry can be characterized by the chamfer angle \(\alpha\) defined as shown in Figure 3. In this way, the chamfer angle can be obtained from the expression \(\alpha=\text{atan}(l_{z}/l_{r})\) for partial-chamfer hubs and the same expression can be applied to full-chamfer hubs considering the previous condition, \(\beta=1\), i.e., \(l_{r}=t\) and then \(\alpha=\text{atan}(l_{z}/t)\). In this study, the chamfer angle was varied for each one of values of \(\gamma\) and \(\beta\), from \(\alpha=0^{\circ}\) (conventional hub) up to \(\alpha=30^{\circ}\)  
Figure 2: Scheme of the geometric modifications of the hub in terms of the dimensionless parameters used in this study: the chamfer height \(\beta\) and the hub thickness \(\gamma\).