## 1 Introduction
The thickness of cushioning materials and the load-bearing area should offer enough protection against the different dynamic inputs to  
Figure 1: The conceptual structure of a packaging system using cushioning material. minimize the cost of product damage. However, these factors should be also minimized to keep the package costs and volumes down. The cushion characteristics of a polymer foam are affected by their mechanical properties. Based on a study by [PERSON] and [PERSON] [8], these are related to the properties of the polymer's cell wall and cell geometry and can be understood in terms of bending, elastic buckling, and plastic collapse. New simple formulas and description of a large data for mechanical properties of polymeric foams were developed. [PERSON] et al. [9] constructed mechanism-mode maps and used them to create energy-absorption diagrams for foams. The energy of the impact that must be absorbed by the foam can be determined by Equation (1):  
\[W=\int_{0}^{\epsilon_{0}}\sigma\,d\epsilon\ +\ \int_{\epsilon_{0}}^{\epsilon^{*}} \sigma(\epsilon)\;d\epsilon, \tag{1}\]  
where \(W\) is the energy absorbed per unit volume (J/m\({}^{3}\)), \(\epsilon_{0}\) is the strain at the linear-elastic limit, \(\epsilon\) is the nominal compressive strain, \(\epsilon^{*}\) is the strain, and \(\sigma\) is the stress. Figure 2 shows a general stress-strain curve for a foam [9]. The optimum foam material and density can be determined using energy-absorption diagrams. Two examples of the application of this method can be seen in the study by [PERSON] [10]. The modelization of stress-strain relation and the calculation of energy absorption and its efficiency for different cushion materials have been studied [11; 12; 13; 14; 15; 16; 17]. Another method used in cushion design is the Janssen factor, which is the ratio of the actual peak acceleration of the actual foam divided by the acceleration of an ideal foam [18]. [PERSON] and [PERSON] [19] developed packaging-selection diagrams with greater generality and simplicity than those developed using the Janssen factor or energy-absorption diagrams. Their diagrams can be used to narrow down the range of densities and choices of foams to be tested during the determining process for the cushion curves. Cushion curves are mainly applied during the design of protective packaging both in the packaging industry and in the Six-Step method [7; 19; 20; 21; 22]. The performance properties of cushioning materials against shock or vibration inputs are represented in these curves. A large amount of experimental data is required to determine these, as new curves must be developed for each drop height, and every curve contains values for different thicknesses and static stresses of the loading [19]. Many researchers have developed methods for designing cushion curves that are less time-consuming and more cost-effective [23]. These studies were mostly based on the use of static compressive stress-strain data [24; 25; 26], dynamic stress vs. energy curves [27; 28; 29; 30; 31; 32; 33] and strain and strain rate (DF, or dynamic factor) [34; 35]. [PERSON] and [PERSON] developed an impact oscillator model that is able to predict a shock pulse's shape, duration, and amplitude at various static stresses and drop heights [36]. FEA (finite element analysis) also can be applied to evaluate the performance of cushioning materials [37; 38]. Figure 2: A stress–strain curve for a foam. [PERSON] et al. [39] and [PERSON] et al. [40] studied the dropping shock response of a tangent nonlinear packaging system and developed the homotopy perturbation method using an auxiliary term and [PERSON]'s modified homotopy perturbation method coupled with the energy method. [PERSON] and [PERSON] [41] and [PERSON] et al. [42] investigated the damping characteristics of foams, which are relatively unfamiliar to the packaging industry compared to the cushion curve. [PERSON]. [41] proposed a contact force law based on a nonlinear viscoelastic model for modeling impacts of a cushion foam.