# Development of a Numerical Model of the Hot Air Staking Process Based on Experimental Data
## 3 Results and Discussion
### Experimental Study to the Heating and Forming Behavior by Using the Hot Air Staking Process
In contrast to a short heating time, Figure 5b shows the melt formation and the joint for a heating time t\({}_{\rm E}\) of 8 s. Due to the longer heating time, the melt layer is clearly thicker. Consequently, the forming of the rivet pin leads to a predominantly uniform material structure in the rivet head. Especially in the connection area of the rivet head and the rivet pin, entirely plasticized and subsequently recrystallized material could be detected. Furthermore, the materials structure is quite homogeneous and the dominance of the shear zone between melted and solid material areas is much less significant. In comparison to the heating time t\({}_{\rm E}\) of 3 s, there is no detectable weld line in the rivet head. Numerical Analysis of the Heating Behavior in Comparison to the Experimental Data for Hot Air Staking  
In a first step, the heating process was simulated with the software simufact.forming by a 2D-rotationally symmetric thermomechanical process simulation. In order to calculate the temperature distribution within the workpiece, the Fourier heat conduction equation was used to calculate the heat conduction processes in the workpiece. The necessary thermophysical parameters heat capacity and heat conduction were determined experimentally as a function of temperature and implemented in the simulation system in tabular form. Figure 6 shows the heat capacity and thermal conductivity as a function of temperature. Figure 5: Heating and forming behavior characterized by the materials structure in the rivet pin and the rivet head, illustrated in (**a**) by the comparison of a short heating time (t\({}_{\rm E}\) = 3 s) and in (**b**) by a long heating time (t\({}_{\rm E}\) = 8 s)—**(**I**) melted and recrystallized structural zone, (**II**) cold formed structural zone, (**III**) shear zone during forming between melted and non-melted material, (**IV**) weld line. Process parameters: material PP, diameter rivet pin d\({}_{\rm P}\) = 3.0 mm, air temperature T\({}_{\rm A}\) = 300 \({}^{\circ}\)C, air flow rate Q\({}_{\rm A}\) = 75 L/min, heating time t\({}_{\rm E}\) = 3 s/8 s, joining force F\({}_{\rm J}\) = 80 N, cooling time t\({}_{\rm C}\) = 12 s.  
Figure 6: Measured heat capacity and conductivity depending on the temperature. The heating of the rivet pin by the hot air is achieved by a heat transfer at the contact surfaces. In the simulation, a heat transfer coefficient must be defined for this purpose, with which the heating behavior can be realistically represented. The heat transfer coefficient was determined iteratively by comparing the heating behavior on the basis of the thin sections. Regarding the heat flow of the hot streaming air, the heat transfer coefficient to the environment was set to 2000 W/(m\({}^{2}\) K). Thus the component heats up homogeneously from the outside to the inside. On the basis of the thin sections (cf. Figure 3, however, it could be shown that the heating is stronger in the area of the front surface. The reason for this is assumed to be the accumulation of hot air under the forming punch. In order to be able to take this into account numerically, a heating plate was inserted into the model, which simulated the heating caused by the hot air at rest under the punch. Figure 7a shows the simulation setup and Figure 7b a comparison between experiment and simulation. It can be seen that the numerical assumptions made allow the melt formation (red areas) to be simulated realistically in terms of quality and quantity. The calculated temperature distribution is the initial condition at the beginning of the riveting process. Numerical Analysis of the Forming Behavior in Comparison to the Experimental Data for Hot Air Riveting  
The numerical simulation of hot air riveting is very challenging because the plastic material is present in the workpiece in both the solid aggregate state and the molten state. In order to be able to reproduce this numerically, a subroutine was developed with which a distinction can be made between the viscous and elastic-plastic material behavior.