Cushioning properties and application of expanded polystyrene for a dynamic nonlinear system

To explore the cushioning properties of expanded polystyrene (EPS), a novel two‐degree‐of‐freedom mathematical model is proposed with a dynamic nonlinear system and successfully implemented it. The maximum response acceleration of the product was first determined in the novel dynamic mathematical model and used as a parameter for evaluating product damage. Furthermore, a drop test was carried out to verify the theoretical model and showed good consistency. Based on the correct theoretical model, a number of simulations with different drop heights and thicknesses of the cushion were done to investigate cushioning properties of EPS. The results show that EPS can reduce the maximum response acceleration of the product by more than 20% of the excitation acceleration. Thus, EPS has good cushioning ability and can be used as packaging material to protect products. Finally, the theoretical model was applied to an example application. A new method for constructing the cushioning curve is recommended, which could replace the traditional ASTM D 1596 drop test method.


INTRODUCTION
When products are transported from the warehouse to consumers, they may be damaged by various hazards such as shock, vibration and transportation environment. [1][2][3] Therefore, many types of cushioning materials and models have been developed and applied to protect the products from these hazards, especially to minimize the effect of shock.
One way of reducing the damage to products is to put adequate cushioning foam in the package to dissipate input vibration and impact energy to which it is subjected during transport. Polymer foams are generally considered to have good cushioning performance due to ignoring the effect of strain rate. [4][5][6][7] Many systems have used this performance for cushioning package design to protect products from damage. The Transport Packaging Laboratory of Kobe University has used various drop tests, including the free-fall test, 8 the shock test 9 and the hybrid drop test, 10 to analyze the cushioning properties of expanded polyethylene (EPE) subjected to shock and vibration during transport by considering F I G U R E 1 A package includes a product box and cushioning material. the package as a mechanical friction-viscous damping model (FVD). Most recently, Fu et al. 11 put forward a constitutive model for polyethylene foam, which is very helpful in the shock response analysis of the foams under multiple loadings. In addition, Ge et al. 3,12 developed a new cushion foam that could be considered a high performance cushioning material for packaging applications. However, when studying the cushioning characteristics of foam materials, it is also crucial to correctly establish a mathematical model of the cushioning package system based on the practical situation.
In order to study the cushioning properties of the packaging system, a mathematical impact model needs to be developed. In most previous studies, the packaged product was considered as a single mass block or two degree of freedom mass-spring system with a critical component. [13][14][15] However, most products, such as smartphones, iPads, and so forth, are usually supplied in a box, which is placed on the cushion material, and together put into a corrugated cardboard box to form a package, as shown in Figure 1. Almost all previous studies did not pay attention to studying the influence of drop shock on boxed products to propose a reasonable cushioning packaging design. The present research is to demonstrate a new packaging model using expanded polystyrene (EPS) as the cushioning material. The product's box has a small mass, which has little influence on the system and can be ignored, while the product has greater mass.
Another way of reducing damage to products is according to the cushion curves of special materials, which is determined by ASTM D 1596, 16 to provide a reasonable cushioning design. Cushion curves are graphical representations of a cushion material's ability to limit the transmission of shock (called G level) to a product. G level is plotted on the vertical axis versus static loading (weight divided by contact area) on the horizontal axis. 17 It represents relationships among the drop heights, the fragility of the product, the cushion foam's thickness, and static stress. However, it takes a lot of laboratory time and experimental cost to generate all the curve information, although its application can protect the products from shock and vibration to a certain extent. For the sake of simplifying the process of determining cushion curves, many alternative methods have been investigated by taking advantage of a technique that relies on the dynamic stress-strain curve of the cushioning material. 7,18-20 However, the dynamic stress-strain curve only has a good cushioning performance for closed-cell cushioning materials. 19,21 Therefore, the concept of C-e (C is the static cushion factor and e is expressed as impact energy per unit volume) curves is proposed, which is used instead of traditional cushion curves to design edge and corner cushion structures. 22 Sek 21 carried out a new procedure to reduce the experimental time for determining cushion curves by studying the static and dynamic cushioning performance of polyethylene foam. Among all those methods that have been presented to generate the cushion curves, simplifying the process of determining the cushion curves is still one of the significant challenges in package engineering.
In this paper, a novel packaging model with a dynamic nonlinear system is proposed and verified by drop tests. Also, a new simplified method of determining the cushion curves by studying the cushioning characteristics of EPS for a packing box is developed. The theoretical analysis for the proposed model is introduced in Section 2, followed by the drop test in Section 3. In Section 4, an application is discussed to illustrate how to use the proposed theory to generate cushion curves and design cushioning for the packing box. The conclusions are summarized in Section 5.

Mathematical model
In this section, a novel model of the packaging system including the product box and the cushioning material is presented, as depicted in Figure 2. As observed in Figure 2, the cushioning material is regarded as the outer cushion and is considered to be a nonlinear spring with the dynamic stiffness coefficient of k 1 . The product box contains a product and an inner cushion and is located on the outer cushion. The role of the inner cushion is to connect the box (m 1 ) and the product (m 2 ) and is considered to be a linear spring with the stiffness coefficient k 2 . Therefore, the packaging system could be treated as a two degree of freedom mathematical model system with no damping and ignoring the outer corrugated carton, as shown in Figure 3. Here, x 1 (t) and x 2 (t) are the displacement response of the boxed product and the product, respectively. Symbols 1 and 1 represent the stress and strain of the outer cushion with a dynamic nonlinear spring, while symbols 2 and 2 denote the stress and strain of the inner cushion with a linear spring.
In this study, expanded polystyrene (EPS) is used as the outer cushion with a density of 55 kg/m 3 while hard plastic is the inner cushion with a density of 1.5 g/cm 3 . The relationship between stress and strain is determined by a polynomial function using Matlab and the data from an Instron 5566 universal test machine, as shown in Figures 4-6, where the F I G U R E 2 The packaged product. Colors: purple--product, gray-inner cushion material, green--box cased, white--outer cushion material.

F I G U R E 3
The mathematical model of the packaged product. Purple represents the product's mass of m 2 , while green represents the box's mass of m 1 . k 1 represents the outer cushion material and has a nonlinear constitutive relationship. 1 denotes stress while 1 states strain. k 2 is the inner cushion material, which has a linear constitutive relationship. 2 and 2 represent its stress and strain, respectively. x 2 is the response displacement of m 2 . x 2 is the response displacement of m 1 + m 2 .

F I G U R E 4
Instron 5566. The white ellipse represents the pressure plate, which is used to compress the cushioning material.

F I G U R E 5
The inner cushion material with stiffness of k 2 . cushion material is compressed between two parallel platens. Thus, the stress-strain relationship of the outer cushion is obtained as: The stress-strain curve of the inner cushion is fitted as: The outer cushion material with the stiffness of k 1 .The cushion material is placed between the upper and bottom platen.

Energy analysis during the drop process
In a drop, when the package collides with the ground, the cushioning material is compressed, which is a dynamic nonlinear process, as illustrated in Figure 7. Unfortunately, the dynamic nonlinear stress-strain curve of cushioning materials is not known in advance and is very difficult to obtain experimentally. However, energy is conserved during the dynamic impact process. It is assumed that the impact energy is completely absorbed by the outer cushion, resulting in maximum deformation of the outer cushion and not transmitted to the product box. Taking m 1 + m 2 as the research objective, the energy balance is formulated as: 23,24 (m 1 + m 2 ) gh where is the maximum strain of the cushion material and g is the acceleration due to gravity. h denotes the drop height of the packaged product. t 1 and A 1 represent the thickness and the contact area of the outer cushion, respectively. Moreover, (m 1 + m 2 ) gh∕A 1 t 1 is the energy absorbed per unit volume of the outer cushion and is equal to ∫ 0 1 ( 1 ) d , which means the area under the dynamic stress-strain curve up to the maximum strain value ( ), as shown in Figure 8. The maximum impact accelerationẍ C of the boxed product is computed by:

The governing equation of the packaging system
The product box and the product are regarded as an uncoupled two-degree of freedom system. Generally, the mass of the product box is small, which has little influence on the packaging system. Therefore, the boxed product is considered to be subjected to free vibration. According to Newton's second law, the equation of motion of the boxed product can be formulated as: where x 1 (t) is the displacement response of the boxed product. The maximum response acceleration of the boxed product can be expressed as:ẍ where w C is the natural frequency of the boxed product with w C = √ k 1 ∕ (m 1 + m 2 ). Substituting Equation (4) in Equation (7), Equation (7) is rewritten as:ẍ In terms of the product, the input acceleration comes from the response acceleration of the box, which acts as the excitation for the product and causes the product to undergo forced vibration. Thus, the governing equation of the product can be formulated as: Applying Duhamel's integral, the response acceleration of the product can be easily obtained as: where w p is the natural frequency of the product with w p = √ k 2 ∕m 2 , and the shock duration is 0 ≤ t ≤ ∕w C . The maximum response acceleration is compared with the fragility value of the product to assess the possibility of damage to the product.

Experimental equipment
Experiments were carried out using dropping equipment to verify the physical model described above, as shown in Figure 9. The experimental model considers three main parts: the outer cushion, the inner cushion, and the product. The product was clamped on the equipment and the experimental parameters are given in Table 1.

Validation
The peak response acceleration was recorded for each drop test using an accelerometer attached to the product, and the results are detailed in Table 2. The measured peak response accelerations were compared with the simulation, showing good agreement. The results indicated that the proposed mathematical model and method are correct and could be applied to an actual packaging system.

The cushioning properties of EPS
The drop experiments reported in Sections 3.1 and 3.2 showed that the proposed model and theory of the packaging system are correct and reasonable. Thus, the cushioning properties of EPS will be presented as follows: Several sets of data for different drop heights and the different thicknesses of the outer cushion were simulated. For case 1, when the boxed product with a mass of m 1 + m 2 = 2.5 kg dropped in free-fall from different heights (h) and the thickness of the outer cushion (t 1 ) was 0.07 m, the maximum input acceleration, the maximum response acceleration of the product, the maximum strain of the outer cushion and the stiffness coefficient (k 1 ) of the outer cushion are summarized in Table 3.
For case 2, when the boxed product with a mass of 2.5 kg fell freely from a given drop height (h) of 0.61 m while the outer cushion (t 1 ) had different thicknesses, the simulation results are given in Table 4.
It can be clearly seen from Table 3 that, when the drop height (h) varies, the outcomes are different, that is, the input acceleration and response acceleration of the product, the stiffness values (k 1 ), and the maximum strain (%) of the outer cushion. These simulation results show an upward trend with the increase in drop height, which is consistent with the objective facts. It was found that, as the input acceleration increases, the response acceleration of the product accordingly increased up to the maximum input and response accelerations of the product. However, the response acceleration is much lower than the input acceleration, as shown in Figure 10A.

Test data
Simulation data    Table 4 summarizes the peak response and input acceleration of the product, the maximum strain information, and the stiffness coefficient of the outer cushion with different thicknesses at the same drop height. These results revealed a downward trend with the increase in the thickness of the outer cushion. In terms of the peak input and response acceleration of the product, it was noted that the response acceleration of the product decreases with decrease of the input acceleration, but it is lower than the excitation acceleration of the product, as seen in Figure 10B.

Order (#) Product acceleration (m/s 2 ) Product acceleration (m/s 2 )
In conclusion, the outer cushion has a good cushioning ability that can reduce the response acceleration of the product below the input acceleration. From the above theoretical simulation results, the author believes that the results are reasonable and correct, because the simulation results are in line with objective laws.

APPLICATION
In this section, an application is presented based on the above theory. A novel method is proposed to simplify the process of determining cushion curves.

Development of cushion curves
Based on the above simulation analysis and experimental verification, a family of G-values versus static loading (s) cushion curves can be obtained, as depicted in Figure 11. According to Figure 11, several cushion curves with different stiffness values are graphed on a set of the G versus s axes for varying drop height and thickness. Here, k 1 and k 2 are from Table 4 while k 3 and k 4 come from Table 3. k 5 is generated from random input values. The proposed theory is able to generate any cushion curve for that specific material instead of undertaking a large number of sample drops.

Cushioning packaging design
In other words, if the stiffness values for the cushion material are known beforehand, there will be a corresponding cushion curve to find a practical design. As an example, the G-value of the product was taken as 60 g and the initial cushion thickness as 0.04 m, with the stiffness value of the cushion's material being k 1 .
According to the cushion curve, in the range of static stress, the three points a, b and c can be determined with the corresponding static stresses being s a = mg∕A a , s b = mg∕A b and s c = mg∕A c , respectively, as shown in Figure 11. Then, the range of bearing areas can be calculated as, A a = mg∕s a , A b = mg∕s b and A c = mg∕s c . It can be seen that A b uses the least material, followed by A c and last A a .
In this study, the three static stresses were used to produce a cushioning package design for the bottom area of the product box. A method for determining cushion curves was developed to design cushions for the packing box based on the stiffness of the cushion material. Hence, package designers do not need to perform any tests to establish cushion curves.

CONCLUSIONS
This paper proposed a new packaging mechanics model for a boxed product with a dynamic nonlinear two-degree of freedom system to study the cushioning properties of EPS. This new analysis model has an inner cushion with a constant stiffness coefficient k 2 and an outer cushion with a dynamic stiffness coefficient k 1 . The proposed mathematical model and theory were verified by drop tests and showed good agreement. Furthermore, extensive theoretical simulations were implemented to test the cushioning performance of the EPS, which indicated good cushioning properties. Finally, on the basis of the theoretical simulation, a new method for simplifying the cushion curves was presented. It can provide cushion curves for varying drop height and any cushion thickness, as well as maximum strain information for the cushion material. The proposed method is relatively quick and inexpensive and could be regarded as an alternative approach to using the traditional ASTM D 1596 method.