## 2 Materials and Methods
### Numerical Simulation
For material 24, it is necessary to enter, as input data for the plastic zone, the true stress-strain curve from which the elastic component is removed. The hardening slope in the true stress-strain curve must not exceed the initial modulus. Otherwise, it can lead to significant errors in the simulation, especially during repeated loading-unloading. By default, the hardening behaviour is explicitly defined through a user-specified true stress-strain curve, as shown below [22]:  
\[\sigma=f(\varepsilon_{p}) \tag{1}\]  
where \(\varepsilon_{p}\) represents effective plastic strain. In material model 89, hardening is not governed by a stress-strain curve but by a rheological framework that combines elastic-viscoelastic-plastic (with hardening) behaviour. For material 89, the true strain-stress curve is introduced without eliminating the elastic component, as in the case of material 24. It uses the tangent stiffness (slope of the true stress-strain curve) at the point of unloading for reloading. The hardening law is embedded in the flow stress formulation. The flow stress \(\sigma\) for material 89 can be obtained using the following equation [22]:  
\[\sigma=\sigma_{y}\cdot\left(1+C\cdot\varepsilon_{p}^{n}\right) \tag{2}\]where \(\sigma_{y}\) represents yield stress, \(C\) and \(n\) are hardening constants, and \(\epsilon_{p}\) represents effective plastic strain. If material 24 can also be successfully used with metallic materials, material 89 is dedicated to polymeric materials such as PA or HDPE. In order to determine which of the two material models produces results that are closer to the ones obtained experimentally, two analyses were performed for each material, one using model 24 and the other using model 89. Therefore, the longitudinal modulus of elasticity, E = 1839 MPa, density, \(\rho\) = 0.114 \(\times\) 10\({}^{-9}\) t/mm\({}^{3}\), and Poisson's ratio, \(\
u\) = 0.39, were introduced for polyamide, whereas the longitudinal modulus of elasticity, E = 1035.6 MPa, density \(\rho\) = 1.036 \(\times\) 10\({}^{-9}\) t/mm\({}^{3}\), and Poisson's ratio, \(\
u\) = 0.42 were introduced for polyethylene. These data have been entered for both models, with the difference consisting in the introduction of the stress-strain curve. It should be mentioned that for both types of models the strain rate was not taken into account because the incremental forming process is carried out at low strain rates, at which the influence is reduced. Both models started from the conventional stress-strain curve obtained from the uniaxial tensile test. Figures 1 and 2 show the conventional curves for the two materials, namely polyamide and polyethylene. Based on the two curves, the true stress and the true strain were calculated, resulting in the true stress-true strain curve. Subsequently, for material model 24, the elastic component of the strain was removed, retaining only the effective plastic strain. Figure 2 shows the discretized geometric model of the entire assembly that participates in the forming process. A special mention is given to the blank which, following a first discretization, was remeshed in the area subjected to deformation in order to obtain better accuracy in the results. In the simulated incremental forming process, it is only the punch that moves, while the blankholder constantly presses the blank with a force of 2 kN. Since the desired trajectory is a frustum of cone-type trajectory, a spiral trajectory was chosen, presented in Figure 3.  
As can be seen in Figure 3, the trajectory of the punch also includes the areas of \"entry\" into the material and \"exit\" from the material, just as in the real, experimental case. A particularly important issue in dynamic analyses where parts come into contact, as in the present analysis, concerns how the contacts between the parts participating in the forming process are defined. The Ls-Dyna R13.1.0 program provides users with several types of contacts, such as knot-on-knot, knot-on-surface, surface-on-surface, etc.