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

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.     

智慧课堂在《食品添加剂》课程教学中的实践探索

目的 研究发泡聚乙烯(EPE)缓冲曲线的简便预测方法,利用一条已知缓冲曲线作为预测工具曲线,基于能量密度对已知缓冲曲线进行处理,从而得到EPE缓冲曲线族。方法 根据能量守恒定律,通过得到该缓冲材料的能量密度-冲击应力曲线,绘制不同条件下的缓冲曲线。推导缓冲材料静应力、冲击应力、单位体积吸收能量之间的关系,提出预测缓冲材料冲击最大加速度,以及生成缓冲曲线族的方法。利用密度为24 kg/m³,材料厚度分别为25.4,38.1,50.8,76.2 mm的EPE,跌落高度分别为30.5,45.7,61.0 cm的出版缓冲曲线,以及密度为35 kg/m³,材料厚度为55 mm的EPE,跌落高度分别为46,66,76 cm的试验缓冲曲线,预测不同条件下的EPE缓冲曲线,并与出版缓冲曲线和试验缓冲曲线进行对比和误差分析。结果 选用适当条件范围(跌落高度及材料厚度)的缓冲曲线作为预测工具曲线预测缓冲曲线族,预测结果与已出版或试验缓冲曲线较为接近。结论 此方法可以有效减少试验次数,为快速获得缓冲曲线族提供了新思路。     

Theory and practice of cushion curve: A supplementary discussion

The practice of developing cushion curves has been applied for many years. Known for its tedious development, the process does not correlate to cushion configurations too. This study reviewed conventional theories, practice of cushion curves, and simplified methods, including the conversion from quasi‐static compression chart to cushion curves, and testing‐extrapolated methods based on several applicable tests. In addition, the study examined the effect of configuration on cushion curves. Discussion indicated that the theories of cushion curves are based on two fundamental properties in physics: energy conversion and Newton's second law. Energy density‐based test‐extrapolation approaches reduced the development time significantly without compromising the accuracy of the cushion curves. The edge and corner cushion structure showed a similar pattern to the conventional flat cushion curve, but were shifted horizontally to the right. This paper recommends packaging engineers to apply C‐e curves as the basis of the cushion curves to design cushioning. Finally, energy dissipated during platen drop test is discussed to explain the deformation process in relation to dynamic stress‐strain curve and cushion curve.     

Cushioning properties of moulded pulp

Cushion curves for a moulded pulp material were constructed and compared with cushion curves for an expanded polystyrene (EPS) foam. The ability of the moulded pulp material to provide some shock absorption resulted from a number of ‘dimples’ moulded into the pulp sheet. As an impact was applied to the moulded pulp, these dimples crushed and thereby mitigated the shock. Results showed that the moulded-pulp test samples had good cushioning characteristics for low static loadings, low drop heights, and single impacts. However, the cushioning characteristics of the material were inferior to those of EPS foam at static loadings above about 5kPa, and where higher drop heights or multiple impacts occurred. These properties were improved by using two samples with interlocking dimples. It was concluded that the material had the potential to replace EPS foam in a limited number of situations, and several recommendations for future research were given.     

Evaluation of Static and Dynamic Cushioning Properties of Polyethylene Foam for Determining Its Cushion Curves

A comparison of static and dynamic cushioning properties of polyethylene foam is discussed in this paper. A dynamic factor function is defined and obtained from the analysis on the experimental data from compression and cushion tests. The deformation energy per unit volume is taken into consideration to make an estimate of the maximum strain and dynamic stress data in an impact, and the peak acceleration can be predicted from the compression data corrected for the dynamic effects by the dynamic factor function. Then, the cushion curves at almost every test condition, such as different drop heights or thickness of cushioning material, can be approximated by a specially developed computer code using Matlab. This research can assist in determining the accuracy of cushion curves for all variants of a particular cushioning material from the dynamic factor function and compression test data. This new procedure will greatly simplify the experimental process for determining cushion curves. Copyright © 2014 John Wiley & Sons, Ltd.     

A preliminary study of cushion properties of a 3D printed thermoplastic polyurethane Kelvin foam

The cells in conventional packaging foams have random size and orientation, and the energy‐absorbing behaviour of these foams is determined by the collective contribution of different sizes of cells. In contrast to the random nature of stochastic foams, 3D printing technologies allow engineers to design and produce foams having engineered cellular structures. In this study, engineered cellular structures based on the classic Kelvin 1887 model were 3D printed in 30 × 30 × 30 mm thermoplastic polyurethane cubes with a repeating size of 216 unit cells. One hundred consecutive cyclic compression tests were performed to assess the 3D printed foam's resilience and energy absorption characteristics. The stress‐strain curve of the 3D printed thermoplastic polyurethane foam indicated viscoelastic behaviour and a Mullins effect indicative of resilient rubber. A long wave buckling mode was observed during cyclic compression cycles due to the Kelvin structure. The cushion factor computed from the stress‐strain curve was close to that of a metal spring with linear elasticity. The combination of the 3D printed foam's resilience, its much lower density than rubber, and the complete geometric freedom of the engineered cellular structures offer designers the potential to create high‐performance cushion materials tailored for packaging applications.     

麦秆发泡包装衬垫非线性粘弹性模型及参数识别

由非线性粘弹性材料的一维Volterra-Frecher积分型本构方程推导得出了非线性粘弹性材料的一维微分型本构方程,建立了麦秆发泡包装衬垫的非线性粘弹性模型。根据在一种跌落高度、一种衬垫厚度和静态应力条件下的麦秆发泡包装衬垫的动态压缩试验数据,识别出了其缓冲性能模型参数。     

Evaluation of Two Simplified Methods for Determining Cushion Curves of Closed Cell Foams

Because of the negative environmental impact of using traditional polymeric materials for cushioning in protective packaging, the interest in optimization of cushion design resulting in replacement or reduction of the amount of cushioning materials has increased in recent years. It has been generally accepted that the shock absorbing characteristics for cushioning materials are presented as the so‐called cushion curves, i.e. graphs of peak acceleration of shocks versus the corresponding static stress, presented over a range of static stress conditions for a specific material thickness and at a specific drop height. The cushion curves are mainly used to design and optimize cushion packaging. This information is specific to each packaging manufacturer, particular chemical formulation and manufacturing process. The traditional method to obtain cushion curves of a specific material is described in ASTM D1596, but this method requires enormous amounts of test data, is time consuming and expensive. For this reason, cushion curves provided by manufacturers are rarely updated and often are obsolete. In recent years, alternative methods have been developed to obtain cushion curves in a significantly shorter time required for testing. To achieve a wide use of these methods, it is necessary to know their accuracy in comparison with the ASTM D1596 method. The objective of this paper is to compare two of those alternative methods with the standard ASTM D1596 method, by applying them to obtain the cushion curves of two typical cushioning materials. More specifically, the single compression data method developed by Sek et al. and the stress–energy method (dynamic stress versus dynamic energy) developed by Burgess were applied to obtain cushion curves for expanded polystyrene with density 15 kg/m³ and for polyethylene Ethafoam® (Modisprem S.A. Zaragoza, Spain) with density 29 kg/m³ and are presented in this paper. The effectiveness of these alternative methods for developing cushion curves and their advantages, disadvantages and limitations are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Detection and Modelling of Nonlinear Elastic Response in Damaged Composite Structures

In this paper the nonlinear material response of damaged composite structures under periodic excitation is experimentally and numerically investigated. In particular, the nonlinear wave propagation problem was numerically analysed through a finite element model able to predict the nonlinear interaction of acoustic/ultrasonic waves with damage precursors and micro-cracks. Such a constitutive model is based on the Landau’s semi-analytical approach to account for anharmonic effects of the medium, and is able to provide an understanding of nonlinear elastic phenomena such as the second harmonic generation. Moreover, Kelvin tensorial formulation was used to extend the wave propagation problem in orthotropic materials to the 3D Cartesian space. In this manner, the interaction of the stress waves with the 3D crack could be analysed. This numerical model was then experimentally validated on a composite plate undergone to impact loading. Good agreement between the experimental and numerical second harmonic response was found, showing that this material model can be used as a simple and useful tool for future structural diagnostic applications.     

电脑主机缓冲包装结构缓冲效果的高速摄像评价

为了评价电脑主机缓冲结构的缓冲效果,利用高速摄像测量方法,对电脑主机运输包装件冲击实验进行了冲击响应分析,研究了电脑主机包装件中EPE缓冲垫对电脑主机的防护作用。实验中分别取5,10,15 cm 3个跌落高度,利用跌落冲击瞬间缓冲垫、主机、水平滑台三者冲击作用的瞬态图像,通过采集图像进行处理和分析,得到了电脑主机的位移-时间、加速度-时间曲线以及水平滑台的加速度峰值和衬垫冲击传递率。结果表明,随着冲击高度的增加,电脑主机加速度响应随之增大,缓冲垫的冲击传递率达到50%左右,显示出EPE缓冲结构具有显著的缓冲效果。     

泡沫塑料包装衬垫缓冲性能建模

本文从非线性粘弹性物质的多重积分型本构方程出发,引入塑性应变,推导了粘弹塑性物质的微分型本构方程.进一步考虑材料的损伤特性,建立了泡沫塑料包装衬垫的缓冲性能模型.根据在一种跌落高度、衬垫厚度和静态应力组合条件下的国产可发聚苯乙烯沫泡塑料的连续多次跌落冲击试验数据,识别了其缓冲性能模型参数.动力学计算与试验结果的比较表明,该模型能较好地反映可发聚苯乙烯泡沫塑料包装衬垫在连续多次跌落冲击过程中的加速度、速度、位移变化,冲击能量的吸收性和塑性变形等基本特性.并且仅根据在一种组合条件下的试验数据所识别的这一模型,在其它组合条件下其缓冲性能与试验结果也吻合较好.     

简支曲梁结构的大变形及吸能分析

目的 通过对简支曲梁缓冲器的非线性大变形及能量吸收特性的理论研究,为其缓冲设计与应用提供理论参考。方法 基于Euler–Bernoulli梁理论,以曲梁的曲率半径及截面角为基本参数推导简支圆形曲梁大变形控制方程,考虑压板作用下曲梁的多种变形情况给出曲梁的大变形及变形能的解析表达,进而计算不同外力及初始安装角下缓冲器的变形情况及变形能,并与数值计算结果进行对比。结果 理论计算结果与数值解高度吻合,表明计算方法的可靠性,缓冲器的缓冲系数取决于曲梁材料、初始曲率半径及安装角度,与其数量无关;当初始安装角为 时,缓冲器的最小缓冲系数可取到6.12。结论 所讨论曲梁缓冲器具有明显的非线性大变形特性和良好的缓冲吸能特性,能够替代传统缓冲材料,方便地用于运输系统的缓冲设计中,给出了简支曲梁缓冲器的基本设计方法。     

EPE缓冲系统跌落冲击响应分析的变分迭代法

目的以EPE缓冲包装系统为研究对象,探讨非线性系统跌落冲击响应分析的近似解析方法。方法基于EPE系统跌落冲击试验,建立系统动力学模型,应用变分迭代法求解动力学方程,获得系统响应的一阶近似解,预测系统跌落冲击时间、最大位移及加速度等参数。结果与实测结果比较,系统跌落冲击时间、位移及加速度峰值等相对误差小于5%。结论非线性系统跌落冲击响应分析的变分迭代法具有普遍意义。     

A new method for the determination of cushion curves

A new method for the measurement of shock‐absorbing characteristics of cushioning materials and determination of ‘cushion curves’ is discussed in this paper. The method not only significantly reduces testing time but also improves the accuracy of the estimate of a cushion curve. Cushion curves are determined from the material's static compression characteristics and the impact data (static load/peak acceleration) obtained from a small number of impacts on a cushion tester. However, the method is capable of producing a cushion curve from the measurement of just a single impact. The process involves an iterative least mean squares (ILMS) minimisation of the discrepancy between peak acceleration values predicted from a theoretical model and measured in the impact tests. The algorithm of the ILMS method, examples demonstrating its application and the dynamic effect in impacts of various materials such as the EPU, the EPS and corrugated fibreboard are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

跌落工况下斜支承系统易损件的冲击特性

目的基于斜支承系统双自由度模型,研究系统易损件的跌落冲击特性。方法针对系统无量纲跌落冲击动力学方程,用龙格-库塔数值分析法获得易损件跌落冲击动力学响应,探讨系统支承角、频率比、跌落冲击初始速度、阻尼比等对易损件位移及加速度响应的影响规律。结果通过对易损件位移、加速度响应最值影响因素的分析表明,减小支承角可增加易损件位移响应最值,降低其加速度响应最值,延长响应周期;随着频率比的增加,易损件位移和加速度响应的最值减小;随着初始速度的增加,易损件位移、加速度响应最值上升明显;对于加速度响应最值,系统阻尼比存在最佳值。结论为使斜支承系统获得理想的减振和抗跌落冲击性能,需综合考虑各相关参数。     

Predicting the Effect of Temperature on the Shock Absorption Properties of Polyethylene Foam

Polyethylene (PE) foam is a material used commonly in protective packaging for its shock absorption properties. When developing a package design intended to mitigate shock to the product, decisions are typically made based on established cushion evaluation procedures performed at standard laboratory conditions. Distribution environment temperatures, however, can vary greatly from the condition at which these materials are assessed. The research presented in this paper utilizes the stress–energy method of cushion evaluation and highlights trends in the stress–energy equations of PE foam tested at 12 different temperatures, ranging from ?20°C to 50°C. A quadratic polynomial is used to describe the variation in the stress–energy equation coefficients over the temperature range evaluated. The model developed enables cushion curve prediction for any static stress, drop height, material thickness and temperature expected over the intended range of use of the material. This model is validated by performing additional impact testing of samples at various temperatures and comparing experimentally obtained acceleration values to those predicted by the model. Further model analysis is performed to estimate the optimal static stress for the material at any temperature within the range tested and to study the variation with temperature of this optimal point. Results reveal that the model developed is capable of predicting the shock absorption properties of the material within the range of parameters tested and that the optimal static stress of the material decreases as temperature increases from ?20°C to 50°C. Application to cushion design is made to recommend an approach to designing a PE cushion system for use over a range of temperatures. Copyright © 2016 John Wiley & Sons, Ltd.     

C楞瓦楞纸板动态缓冲模型及应用

利用万能试验机和跌落试验台分别得到C楞瓦楞纸板静态、动态应力-应变数据,建立相应的理论模型,并用最小二乘法识别模型中的参数。给出了瓦楞纸板在缓冲动力学中应用的例子,结果表明所建立的缓冲模型,可以直接用于缓冲包装设计,克服了用最大加速度-静应力曲线来设计缓冲包装需要大量实验和较多的数据的弊端。     

后峰锯齿脉冲激励下双曲正切包装系统的非线性动力学特性研究

建立了二自由度的双曲正切包装系统模型,并应用四阶龙格-库塔法对得到的冲击动力学方程进行数值求解,研究了其冲击响应特性。采用后峰锯齿脉冲作为激励,得到了关键部件的三维冲击谱,进一步讨论了频率比、脉冲激励幅值、包装材料阻尼和脉冲周期对关键部件冲击谱的影响规律。结果表明,频率比、脉冲激励幅值、包装材料阻尼和脉冲周期均对关键部件冲击响应峰值有显著影响。     

Generation of cushion curves from one shock pulse

Three methods for generating performance information relative to the shock absorbing ability of cushioning materials are discussed. The conventional method, which follows the test standard ASTM D1596, requires enormous amounts of test data and can be quite expensive in material costs, laboratory time and equipment. Two methods based on generating the dynamic stress vs. energy density curve for the material are discussed. This single curve replaces all of the published cushion curves. One method requires only 10-20 drops and predicts peak G to within ± 5% of the published values. A new method proposed here produces the same stress vs. energy curve from only one drop.     

Effect of specimen size on test results to determine cushioning characteristics of corrugated fibreboard

This paper investigates the shock absorption properties of corrugated fibre-board when impacts occur on test specimens of different sizes. In particular, the effect of two types of specimen is studied: (1) a circular specimen of the same size as the flat bottom surface of the drop hammer and (2) a rectangular specimen larger than the contact area of the impacting hammer. This study was conducted as a preliminary investigation of the significance of the cushion areas adjacent to the area in contact with the drop hammer, since this occurs in most cases in practise when packaged products rely on the shock absorbing characteristics of the corrugated fibreboard. The results show that the effect is significant and suggests that the use of traditional cushion curves for corrugated fibreboard be re-evaluated.