## 2 Finite Element Model
### Finite Element Modeling  
This paper provides a simulation study of the rotary ultrasonic machining (RUM)-assisted cutting process of the NHC core. This simulation is based on a numerical FE model developed with the Abaqus/Explicit software. The cell walls of the Nomex honeycomb structure are composed of two-thirds single-cell walls and one-third double-cell walls. The corresponding thicknesses of single- and double-cell walls are set to 0.13 mm and 0.26 mm, respectively, in the numerical model. In this approach, the meshing of the NHC core walls is achieved using reduced-integration four-node S4R shell elements, as shown in Figure 5a. Throughout the numerical modeling, the UCK tool is supposed to be rigid, which means that it is non-deformable during the milling process. Therefore, the UCK milling cutter is meshed using rigid quadrangular elements with four nodes (R3D4), as presented in Figure 5b. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mechanical Properties** & \\ \hline Density [g/cm\({}^{3}\)] & 1.4 \\ E [MPa] & 3400 \\ Poisson’s ratio & 0.3 \\ Yield strengths for simple wall thickness (MPa) & 29 \\ Yield strengths for double wall thickness (MPa) & 61 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The mechanical properties of Nomex paper [22; 23]. Figure 4: Evolution of the stress and deformation of the Nomex material [22; 23]. Before starting the implementation of the numerical code, it is essential to conduct a preliminary phase aimed at examining the nonlinear parameters that can impact the convergence of the numerical results, in particular the type and size of the mesh. It is important to note that using a fine mesh size can result in a substantial increase in calculation time, particularly in the context of 3D configurations. It is, therefore, essential to find an optimal balance between the size of the mesh and the exactness of the results, while maintaining a reasonable calculation time. As part of the simulation of the machining process, a general contact was used to describe the interaction between the UCK milling cutter and the Nomex honeycomb core. Since the walls of the Nomex honeycomb structure are thin, the contact between it and the cutting tool is considered punctual. For this purpose, the friction coefficient adopted to carry out the numerical simulations is set at 0.1. In order to ensure permanent contact between the milling cutter and the workpiece, an initial integration between the part and the tool was set up, taking into account the geometric particularities of the NHC core and the UCK milling cutter (see Figure 5a). The boundary conditions were adopted according to the experimental process [20] so that the lower surface of the NHC structure remained stationary and incapable of undergoing displacements or rotations. For this purpose, in the numerical modeling, complete fixation is imposed on the bottom surface of the NHC core, preventing any translational movement (\(\mathrm{U_{x}=U_{y}=U_{z}=0}\)) along the X, Y, and Z axes as well as any rotation around these axes (\(\mathrm{U_{Rx}=U_{Ry}=U_{Rz}=0}\)), as shown in Figure 6b. The difference between conventional milling and ultrasonic vibration-assisted axial milling is the introduction of ultrasonic vibrations along the Z axis at the end of the milling cutter. The ultrasonic rotary machining (RUM) process represents an approach in mechanical manufacturing that involves the synchronization of three types of motions. These motions include translation of the tool along the axis OY, characterized by the feed rate \(V_{f}\), rotation of the cutting tool around the axis OZ, expressed by the spindle speed n, and the vibration of the tool along the OZ axis, generating a sinusoidal ultrasonic wave (see Figure 6). To guarantee precise monitoring of the milling simulation process, a reference point denoted RP, was assigned along the axis of revolution of the cutting tool in accordance with the representation in Figure 5b.