# Modeling Palletized Products: The Case of Semi-Filled Bottles under Top-Load Conditions
## 1 Introduction  
The irreversible damage of goods during transportation, caused mainly by sloshing, vibration, and shock mechanical solicitations [1], is a common scenario nowadays in the logistics sector, resulting in significant economic losses to both the retailer and manufacturer [2]. The environmental aspect is particularly affected by the greenhouse gas (GHG) emissions to produce and ship goods that, being damaged, will not satisfy the customer and will end up in a landfill. On the other hand, the economic losses can be ultimately managed in some way. For instance, an increase in the price of goods, as estimated with proper methods, as in [3], can be added to ensure the economic sustainability of the industrial investment. On the contrary, the environmental cost inferred by deteriorated goods, as claimed in [4], is often definitive and articulated on multifaceted aspects, such as materials no longer usable, energy and water used for product processing, and fuel and carbon dioxide used during the transport phases. The loss in functionality and usability of goods during transportation can be considered a non-negligible fraction of the whole number of products moved around the world. For instance, a study commissioned by Walmart [5] estimated the phenomenon in 1,176,260 pallets damaged each year, equivalent to 58,813 truckloads of product going into landfill that, at the financial level, equates to about $2400M and 18% earnings. Many leading companies are trying to identify and stem the causes of the deterioration of goods due to transport [6]. The evaluation of the economic repercussions of thesephenomena through in-depth analyses [7] can lead to the identification of solutions to be adopted by the company to minimize losses. Several recent studies [8; 9] have been directed to optimize transport conditions, developing solutions for what is commonly referred to as'secondary packaging', which does not take into consideration packaging materials that are not in direct contact with the product (e.g., blisters, sachets, boxes). In particular, in the case of bottles, the secondary packaging refers to the action of arranging the bottles together into bundles covered with stretch wrap films, positioned side by side and overlapped to form the pallets (Figure 1a). The pallets, which, in the case of bottles, can also contain tens of bundles arranged on several levels, are protected by the deposition of several layers of stretch film. This plastic and thin film is deposited with special implants, combining both the wrapping and pre-tensioning actions of the film itself (Figure 1b). In this way, a stabilization effect is created with respect to external dynamic stresses, which is added to the general protection offered by the presence of the plastic layer outside. Through a suitable choice of the type of film and the wrapping parameters it is possible to guarantee suitable levels of protection with respect to the most common stress conditions [10; 11] that are generated during the transport phases. This is possible as long as the bottles retain their structural integrity. The external packaging can do very little if the stressors on the individual bottles are such that can damage them. This effect is not so rare, considering the attempt to use progressively lighter and slimmer bottles close to the limit conditions, in the face of always large and heavy pallets. The result is that bottles are subjected to various types of loads during transport and with various possibilities of damage, with one of the most dangerous being that which leads to buckling. Plastic bottles have been widely used since 1989 as containers for water, milk, and other beverages. The body is usually made of PET (polyethylene terephthalate) from pre-printed plastics (Figure 2a) and then blown, while the bottle cap is made of PE (polyethylene). The form is not only conceived for an aesthetic purpose, but it also aims to guarantee adequate stiffness during transport (being stacked on top of each other) and specific features during use (Figure 2b).