## 3 Finite Element Simulations  
Quasi-static finite element simulations are performed using the commercial software LS-Dyna (version 11) [21] and relying on explicit time integration, remembering that very  
\begin{table}
\begin{tabular}{c c c c c} \hline \(a_{1}\) [\(-\)] & \(a_{2}\) (MPa\({}^{-1}\)) & \(a_{3}\) [\(-\)] & \(a_{4}\) (MPa) & \(a_{5}\) [\(-\)] \\ \hline \(-\)0.036 & 0.48 & 0.49 & 0.30 & 0.33 \\ \hline \end{tabular}
\end{table}
Table 1: Parameter values in Equation (2).  
Figure 1: Schematic of the instrumented die (white arrow pointing at the cavity). The eight sensors used to determine the radial (contact) pressure are shown. The inner radius of the die is 5 mm, and the outer radius is 50 mm. The height difference between the sensors is 2.5 mm. More details regarding the experimental setup are found in [19; 20].  
Figure 2: Experimental results pertinent to pressing with the instrumented die. Left graph: Pressing pressure, reaction pressure and radial pressure as function of pressing time. The pressures are recorded at sensors 8 and 4 (see Figure 1). It should be clearly stated that the contact pressure vanishes when the punch has passed the sensor. Right graph: Average frictional coefficient over the whole powder pillar as function of average normal (contact) pressure. The blue curve is pertinent to the experimental test results, while the grey curve is derived based on Equation (2).  
large deformations are present. The quasi-static conditions are ensured, remembering that explicit time integration is at issue, by applying loading at rates.  
The material model adhered to is an extended Drucker-Prager CAP model with density-dependent material functions and parameters [5]. Isotropic hypoelasticity, with a density-dependent elastic bulk modulus, is assumed for obvious, but not simplifying, reasons. It is implemented as standard in LS-Dyna [22], and the material parameters for the WC-Co powder here were defined by [PERSON] et al. in [11]. In [11], the material model and the material parameters and curves are presented in much more detail and subsequently used in finite element simulations. A total of 1248 fully integrated hexagonal S/R solid elements are used to model the WC-Co powder, while the pressing tools, assumed to be rigid, are described using standard shell elements. The geometry, a cylinder with a diameter of 10 mm, is compacted from the top (as shown in Figure 3), and two symmetry planes are used. In Figure 3, the finite element mesh is shown.  
Mortar contacts, described in [21], are used between the pressing tool and powder, and these contacts are implemented for both frictional models. The geometry is obviously not modelled as axisymmetric, but the only reason for this is the limitations in LS-Dyna. Instead, this was accounted for in the model by introducing two symmetry planes in order to ensure geometrically correct conditions.  
The two frictional models in Equations (1) and (2) are accordingly defined in the FE-solver LS-Dyna. When Equation (1), with a constant friction coefficient of 0.2, is concerned, this is a very straightforward procedure. In the case of Equation (2), the frictional behaviour is described using a curve dependent on the contact pressure, as detailed in Equation (2) and Table 1. Any influences from velocity (see [2] for details), time dependence and gravity are neglected. As already stated above, it should be strongly emphasized that it was shown in [2] that stick-slip phenomena can be disregarded.