# Modeling Palletized Products: The Case of Semi-Filled Bottles under Top-Load Conditions
## 1 Introduction
Figure 1: A multilevel system: (**a**) a single bottle; (**b**) bottles grouped in bundles; (**c**) pallet and palletizing plant. The understanding of what happens during the crushing of the bottle involves a number of complications, and, to detail this complexity, it must be considered that plastic bottles are usually:  
* Made of Polyethylene terephthalate (PET), a material with a restricted elastic zone: by increasing the load the plastic regime is easily reached;
* Designed in complex shapes, including reinforcing and functional elements;
* Manufactured by highly efficient and standardized processes;
* Designed with minimum thicknesses to reduce weight and cost. The bottles are then filled, not completely, with the liquid to be contained and transported (natural water in the present case). By its turn, the partial filling represents one of the most significant reasons able to add complexity to the problem [12]. The air cushion inside the bottle creates a gas-liquid-structure interaction in response to any external mechanical (or thermal) stresses. This is the main reason why assembling a numerical model becomes challenging. Several studies are available where experimental tests were conducted with methods and results similar to those presented in this study. For instance, in [13; 14], the bottle was characterized by a monoaxial compression test, extrapolating the trend of the force as a function of the lowering. In [15], a system was specifically developed for the evaluation of the contact pressures between the bottom of the container and the fixed base, using pressure transducers in order to be experimentally correlated with a drop test. In [16], a probe for monitoring the internal pressure of the bottle was considered, stating the decrease in the volume of air during the deformation phase. In [17], a system of circumferential bands driven by a traction system was described to determine the radial resistance of the component, being able to characterize the body also orthotopically. It must be also considered that there are widespread standard tests, for example, top-load tests (according to ASTM D 2659-95 [18], as in [17; 19; 20]), that lead this kind of test as a common practice, also in the case of industrial applications. The difficulty is not related to the implementation of these tests or in collecting pertinent experimental data, but in the possibility of generating a complete model (theoretical or numerical) able to interpret the information. There are several studies that attempt to tackle this theoretical or numerical modeling through two different approaches. The first type of method, closely linked to the bottle-producing companies, consists of modeling the body according to the geometry, type, and quantity of material necessary to guarantee a predetermined resistance. In numerous studies (e.g., [21; 22]) the simulations were totally focused on the minimization of the thickness to reduce the cost or, as analyzed in [23], to introduce new types of bottles to the market. Once implemented, the characteristics of the material and the discretization of the profiles with a small variable thickness represent a computational challenge, which can be addressed with the models presented in [24; 25; 26]. The second method consists of evaluating the interaction between the liquid, gas, and wall, described in [27]. This type of approach, based on both the Eulerian and Lagrangian models, is used to define the volume of liquid and air present inside the bottle, losing precision in the definition of the container itself. An example of this approach can be found in [28]. Figure 2: (**a**) Preform in PET (polyethylene terephthalate), from which the bottle is obtained through a thermoforming process; (**b**) example of the final geometric complexity of a bottle. The various papers presented in [19; 23] on static and dynamic simulations describe the numerical methods used for the simulation of the bottles.