# 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
The element temperature serves as trigger criterion. If this is above the temperature of the melting temperature range of the plastic, the viscous temperature-dependent material behavior is assigned to the corresponding element. Otherwise the elastic-plastic material behavior is used. The conceptual approach of the subroutine for case distinction and material assignment is shown in Figure 8.  
Figure 8: Basic concept of the subroutine for case distinction and material assignment. Figure 7: **(a)** Simulation setup for the calculation of the heating process and **(b)** comparison of the experimental and simulation data for different heating times t\({}_{\text{E}}\). In order to describe the material behavior for the elastic-plastic state, the yield stress is described according to the Hensel-Spitlet approach see Equation (1). \[k_{f}=A.e^{m_{1}\cdot T}.\varphi^{m_{2}}.e^{\frac{m_{4}}{\Psi}}\cdot\dot{\varphi} ^{m_{3}} \tag{1}\]  
Using this phenomenological approach, the yield stress is described as a function of the plastic strain, the strain rate, and the forming temperature. The flow curves were determined by cylinder upsetting test and on the basis of these the parameters of the Hensel-Spitlet approach could be determined. Figure 9 shows the calculated yield stress \(k_{f}\) according to Equation (1) as a function of the plastic strain and the temperature at a strain rate of 0.1 s\({}^{-1}\) and the determined parameters. A detailed description of how to carry out the cylinder upsetting tests and determine the flow curves is given in [11]. Furthermore, the flow behavior of the molten plastic must be described within the process simulation for the Lagrange approach. In this paper this is done with the so called Ostwald-de Waele power law model in the form  
\[\tau=k\big{(}\dot{\gamma}\big{)}^{n} \tag{2}\]  
(with shear stress \(\tau\) and shear rate \(\dot{\gamma}\)). In this form, however, the equation cannot be used with the subroutine because, similar to the solid state, the yield stress has to be defined as a function of the plastic strain or strain rate. By applying the shape change energy hypothesis or the comparative forming rate according to von Mises Equation (2) can be reformulated according to [6]:  
Figure 3.0  
By means of rheometer experiments, the complex viscosity \(\eta^{*}\) as a function of shear rate \(\dot{\gamma}\) can be determined experimentally for different temperatures. The forming speed can be calculated directly from the shear rate \(\dot{\gamma}\) according to Equation (4):  
\[\dot{\varphi}=\frac{1}{\sqrt{3}}\cdot  
This equation is defined within the FE-tool simufact.forming, and by applying the method of least squares, the coefficients \(C_{1}\) and \(C_{2}\) for Equation (5) can be determined for each test temperature from Equation (3). In [12] the approach for determining the coefficients is explained. Table 1 shows the determined coefficients for the corresponding testing temperature. Figure 10 summarizes the experiments for material data determination, the analytical approaches used, and the resulting yield stresses for both solid (a) and the molten state (b). The suitability of material characterization by means of cylindrical upsetting tests for plastics in the solid state for compressive stresses was demonstrated in [11]. Furthermore, the numerical simulation and verification of the behavior of molten adhesive by means of a rate-power-law model was carried out in [12]. The approaches used in this thesis are therefore suitable. The necessary thermal material parameters for thermal capacity and the temperature-dependent thermal conductivity shown in Figure 6 were also implemented. The process simulation was carried out with a 2D rotationally symmetrical model using the real tool geometry. The punch speed was assumed to be constant (v\({}_{\text{Punch}}\) = 21.25 mm/s) and was determined experimentally by measuring the process duration.