Virtual Mass Method for Solution of Dynamic Response of Composite Cushion Packaging System

Abstract

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

Impact Responses of Composite Cushioning System considering Critical Component with Simply Supported Beam Type

“…The results with 0.01 and 0.005 kg are almost overlapped in Figure 2(a), indicating the results remain unchanged when virtual mass is smaller than 0.01 kg, which is similar to the phenomenon reported in [9]. The relative velocity of center of the beam to main body is depicted in Figure 2(b).…”

“…The constitutive relationships for the two kinds of polymeric foam are determined from static and impact tests on EPS and EPE specimens, respectively. The constitutive relationships for EPS and EPE are depicted as [9] …”

“…Then cushioning structure such as corrugated [5,6] or honeycomb paperboard [7,8] is utilized to store products during transportation and undergo permanent deformation if the big impact level occurs; hence a portion of kinetic energy is absorbed by outer packaging box, which is neglected in practical design. Lu et al [9,10] proposed virtual mass method to investigate impact responses of multilayered cushioning materials based on each single-layer constitutive relationship. Gao and Lu [11] used Newton-Raphson iteration method to explore the mechanical behaviors of composite cushioning system consisting of polymeric foam and corrugated paperboard.…”

A phenomenological constitutive modelling of polyethylene foam under multiple impact conditions

“…Sherwood utilized tenth‐order polynomial fitting to develop a uniaxial constitutive model for polyurethane foam. In our previous research, combination of seventh‐order polynomial and tangent function was used to obtain the model for expanded PS foam. However, as discussed in the study of Ozturk and Anlas, the results change a lot with small variations of some coefficients in third‐polynomial forms besides an exponential term.…”

Prediction of cushion curves of polymer foams using a nonlinear distributed parameter model

“…These modelling efforts significantly reduced the necessary time to determine cushion curves from the experimental procedures outlined in ASTM D1596 . Initially, the mathematical predictions were based on an energy balance during impact and evolved to both analytical solutions and numerical simulations of the impact process . However, with the exception of finite element simulations, these efforts were based on a lumped parameter approximation of the cushioning foam and impacting mass.…”