# Modeling Palletized Products: The Case of Semi-Filled Bottles under Top-Load Conditions
## 4 Numerical Models
### Solid (3D) Model  
Three-dimensional solid elements were used to evaluate an extreme hypothesis of simplification consisting of a bottle discretized by a homogeneous object, made of a single material. The presence of liquid, air, and the plastic polymers of small thickness were not taken into account separately, but the bottle was analyzed globally, as a single solid body. This body was considered as made by an equivalent material, whose characteristics were set in order to guarantee a mechanical response in line with the experimental tests. From this perspective, the geometry should be also simplified. Starting from the 2D profile, a 3D revolution solid was generated and discretized by solid elements (Figure 12a). Figure 11: Buckling analysis in top-load conditions for a 10 mm vertical displacement, limited to the initial four modes of instability and for a load multiplier (\(k\)) of: (**a**) \(k=0.19\); (**b**) \(k=0.16\); (**c**) \(k=0.12\); (**d**) \(k=0.09\). Figure 12: Finite Element modeling by solid elements: (**a**) 3D bottle model; (**b**) tetrahedral mesh; (**c**) boundary conditions. A critical requirement for efficient and accurate explicit analysis is a high-quality mesh, to be also balanced with the availability of computational resources. Explicit analysis benefitted from the integration of the design phase in the ANSYS Workbench environment, which includes powerful automatic mesh generation. In general, there are two options, one is to create high-quality hex (brick) elements using multi-zone meshing, while the second one automatically uses the swept method to create hex elements. Some geometries, especially those created for manufacturing and imported from CAD data, are too complex to be swept to produce a full hex mesh. Tetrahedral elements can accurately represent small portions of a part's geometry that cannot be swept. On the contrary, tetrahedral elements can be not accurate enough in investigating large deformations and, in addition, mesh size can affect stress results. At least, different mesh density must be tuned to confirm the repeatability of this model. For this reason, it would be advisable to optimize the validity of the mesh, both in terms of element types and dimensions. At the same time, the ultimate goal of the research was to try to represent the behavior of a bottle inside a pallet in the most elementary way possible. For this reason, it was decided to limit the discretization to an automatic meshing by ANSYS tools. In particular, quadrilateral dominant elements were used and the final mesh for explicit analysis consisted of 14,331 nodes and 13,275 elements (Figure 12a). With respect to the first case, similarities and differences existed in the way loads and constraints were applied. The rigid plate on the top (Figure 12b) was set with a 2 mm/s speed (instead of a displacement) while bottle nodes on the bottom were fixed avoiding translations and rotations. In an extreme simplification of this kind, which considers the entire system as homogeneous and discretized by finite elements of the same type, would be possible when a model of 'equivalent material' can offer the same responses as the real system (Figure 13). Ansys Workbench 18.1 also makes it possible to insert the material behavior by additional complex methods, as in the case of the multilinear isotropic hardening or Chaboche kinematic hardening models (Figure 14). These hardening models can be generated by fitting the experimental data and are able to provide rather different responses to the simulation. In Table 2 the accuracy of predictions is evaluated by comparing real and simulated strains in the case of different hardening models. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{2}{c}{**Residual Strain**} & \multicolumn{2}{c}{**Hardening Model**} \\ \hline \hline  & **Measure (\(\epsilon_{\mathbf{e}}\))** & **Bilinear Isotropic** & **Multilinear Isotropic** & **Chaboche Kinematic** \\ \hline mm & 46.5 & 54.9 & 61.9 & 43.2 \\ \hline \(\Delta\) & & +18\% & +33\% & –7\% \\ \hline \end{tabular}
\end{table}