Received: 23 October 2018 Revised: 20 May 2019 Accepted: 3 July 2019 DOI: 10.1002/pts.2473 RESEARCH ARTICLE Prediction of cushion curves of polymer foams using a nonlinear distributed parameter model Amin Joodaky 1 Gregory S. Batt 2 James M. Gibert 1 1 Advanced Dynamics and Mechanics Lab, Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, IN 47907, USA 2 Department of Food, Nutrition, and Packaging Sciences, Clemson University, SC 29634, USA Correspondence Amin Joodaky, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA. Email: ajoodaky@purdue.edu Funding information Purdue Research Foundation Polymer foams are commonly used in the protective packaging of fragile products. Cushion curves are commonly used within the packaging industry to characterize a foam's impact performance. These curves are two-dimensional representations of the deceleration of an impacting mass versus static stress. Cushion curves are currently generated from exhaustive experimental test data. This study represents the first time that the physics of the mass-cushion impact have been analysed by modelling the foam as nonlinear, continuous rod. Using a single mode of vibration and excluding the effects of damping, the maximum displacement during the impact can be obtained from a polynomial describing the maximum elastic energy in the foam. The displacements can be used to recover the amplitude of the deceleration shock pulse. Numerical and analytical analysis of the model with damping is considered in its ability to predict the shock pulse shape, duration, and amplitude at various static stresses, foam thickness, and drop heights as compared with experimental data. Furthermore, both the analytical and numerical results agree and are primarily within the expected lab-to-lab variability of 18% documented in ASTM D1596 - Standard Test Method for Dynamic Shock Cushioning Characteristics of Packaging Material. KEYWORDS cushion curve, nonlinear dynamics, packaging, polymer foam, shock pulse 1 INTRODUCTION A wide range of packaged products types are susceptible to damage as a result of shock loading. The loading can be due to mishandling during transport, or a transient loading condition caused by the transportation environment, i.e., bumps in the road for truck travel or turbulence for air travel. The shock attenuation properties of a cushion material are experimentally determined using ASTM D1596 Standard Test Method for Shocking Absorbing Characteristics of Package Cushioning Materials. 1 In the test, a mass M is dropped from a height H on a sample cushioning material of area A. The goal of the test is to study the effect of the static stress  s = Mg∕A and drop height on the peak deacceleration of the mass as the foam is compressed. The results are represented as cushion curves that plot peak acceleration during impact versus static stress. 2 Mathematical models of impact response of packaging systems orig- inated in the 1990s. 3-7 These modelling efforts significantly reduced the necessary time to determine cushion curves from the experimen- tal procedures outlined in ASTM D1596. 8 Initially, the mathematical predictions were based on an energy balance during impact 2,3,5-7,9-11 and evolved to both analytical solutions and numerical simulations of the impact process. 12-17 However, with the exception of finite element simulations, 14,18 these efforts were based on a lumped param- eter approximation of the cushioning foam and impacting mass. The key to these modelling efforts is to capture both the dissipative and quasi-static stress strain behaviour of the material. While not directly related to foam 19-21 noted that the initial quasi-static constitutive behaviour in corrugated boxes is similar to foam, and thus, the boxes should offer similar shock absorption properties. This study aims to provide a rigorous analysis beyond the lumped parameter paradigm on the prediction of shock response of cush- ioned materials. The materials used in this study are an expanded polystyrene foam EPS20, 22 Styropor 23 (BASF), and a polyethylene foam, Ethafoam 150 (Nova Chemicals). Like other expanded polymer Packag Technol Sci. 2019;1–12. wileyonlinelibrary.com/journal/pts © 2019 John Wiley & Sons, Ltd. 1