A composite package cushioning system consists of multiple layers of cushioning materials in series. Such composite cushioning systems will enhance the protection capability to the packaged products. However, the dynamic analysis of the composite cushioning system becomes more difficult because of the strong nonlinear characteristics of cushioning materials. To overcome the difficulty in calculation, the method of virtual mass is proposed to form the system equations of motion as a set of multidimensional ordinary differential equations so that the Runge-Kutta method can be applied to obtain the numerical solution of the system response. The rationale of the method is shown both analytically and numerically by assessing the reliability of the virtual mass method. For a specific composite cushioning system consisting of two layers of expanded polystyrene (EPS) and expanded polyethylene (EPE) materials, experiments were carried out first for each material to obtain the stress-strain behaviours under both static and dynamic loadings. On the basis of these data, the stress-strain constitutive models were developed for EPS and EPE. Then, experiments were conducted for the composite cushioning system under different system parameters, and the measured data are compared with the analytical results using the method of virtual mass. The agreement between the analytical and the experimental results substantiates the accuracy of the proposed virtual mass method. Copyright © 2012 John Wiley & Sons, Ltd. One of the important functions of transport package is to shield products from damage using cushion elements. To measure the product fragility and determine protection capability of the package, the method of damage boundary curve, proposed originally by Newton 1 for linear packaging systems, has been applied to various nonlinear packaging systems. Because the cushioning materials exhibit stronger and complex nonlinear mechanical characteristics, simplified mathematical models were proposed for the nonlinearity in order to predict damage boundary curves of packaging systems. Wang 2 obtained displacement damage boundary curves for a bilinear packaging system. Hyperbolic tangent type of nonlinear packaging systems were considered 3,4 for investigating the drop damage boundary curve. On the basis of the rotation effect occurring in asymmetric packaging systems, Gao 5,6 studied shock response of nonlinear tangent packaging system from the damage boundary curve point of view.In addition to theoretical analyses, experiments have been conducted to investigate the dynamic performance of typical cushioning materials. Polymeric foam or structural cushioning materials have

Quasi-static compressive and impact behaviours of multi-layered corrugated paperboard (MLCP) cushioning structure were analysed by a recently proposed virtual mass method. First, virtual mass method was applied and verified analytically to solve quasi-static compressive responses for representative two-layer corrugated paperboard cushioning structure. The results show that the two layers in the cushioning structure reach the buckling state in chronological order because of the existence of the small perturbations triggered by inertial force related to virtual mass, which leads to the two typical stress peaks in stress-strain curves. Second, the quasi-static compressive behaviours of MLCP cushioning structure were further studied numerically, showing that the buckling order of multi-layer cushioning structure depends on virtual mass, but the stress-strain curves remain unchanged when the virtual mass is smaller than some certain value. Finally, quasi-static and dynamic impact tests of MLCP cushioning structure composed of C-flute corrugated paperboard were carried out to further validate the capacity of the virtual mass method to describe layer-wise collapse mechanism given the constitutive relationship of the monolayer corrugated paperboard. NUMERICAL EXAMPLESThe rationale of the virtual mass method used for solving the compressive responses of the multi-layered cushioning structure can also be illustrated numerically. Example 1The virtual mass method was applied to obtain the stress-strain curves and deformation-time curves with the following parameters: a 1 = 0.8 MPa, a 2 = 0.2 MPa, a 3 = 2.5 MPa, ε 1 = 0.2, ε 2 = 0.6, h = 0.01 m and v = 0.01 m/s. The stress-strain curves of the two-layered structure are depicted shown in Figure 4a, Figure 4. Quasi-static compressive responses of two-layered cushioning structure with different m 1 (a) stress-strain curves, (b) deformation-time curves and (c) velocity-time curves of the interface.874

Based on the constitutive models of expanded polyethylene (EPE) and C-flute corrugated paperboard (CCP), the drop impact model for expanded polyethylene in series with CCP was established to consider the effect of cushioning action for CCP box. A numerical procedure was adopted for the optimization of the product packaging system by considering the action of the corrugated paperboard box. Then the optimal results were obtained and compared without considering the effect of CCP. Finally, the calculation reliability was proved by the comparison between calculated results and experimental data.

This paper extends the knowledge into the mechanical behaviour characterizations and constitutive modelling of polyethylene (PE) foam under multiple loading and unloading. The mechanical properties of PE foam subjected to single loading cases can be obtained by uniaxial compressive tests at quasi‐static and dynamic states. And the multiple loading and unloading behaviours of the foam can be revealed by consecutive drop tests. The major objective of this research is to propose a phenomenological model consists of shape function and modulus function, which can be predicted compressive response of PE foam for single loading cases. The constitutive models of foamed PE under multiple loading and unloading conditions are established by both using hyperbolic function, where the relations between coefficients and residual strain are introduced. And then, experiments are conducted to validate the proposed model by comparing the constitutive models proposed in this paper and those predicting by finite element software ABAQUS with those by experiments, showing that the proposed models are more accurate for predicting acceleration‐times curves of multiple drop scenarios.