Characterization and modeling of the mechanical behavior of polymer foam

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(1)Daniel Thor Morton. Doctoral thesis. Doctoral theses at NTNU, 2021:173. Doctoral theses at NTNU, 2021:173. NTNU Norwegian University of Science and Technology Thesis for the Degree of Philosophiae Doctor Faculty of Engineering Department of Structural Engineering. ISBN 978-82-326-6245-6 (printed ver.) ISBN 978-82-326-5699-8 (electronic ver.) ISSN 1503-8181 (printed ver.) ISSN 2703-8084 (online ver.). Daniel Thor Morton. Characterization and modeling of the mechanical behavior of polymer foam.

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(3) Daniel Thor Morton. Characterization and modeling of the mechanical behavior of polymer foam. Thesis for the Degree of Philosophiae Doctor Trondheim, June 2021 Norwegian University of Science and Technology Faculty of Engineering Department of Structural Engineering.

(4) NTNU Norwegian University of Science and Technology Thesis for the Degree of Philosophiae Doctor Faculty of Engineering Department of Structural Engineering © Daniel Thor Morton ISBN 978-82-326-6245-6 (printed ver.) ISBN 978-82-326-5699-8 (electronic ver.) ISSN 1503-8181 (printed ver.) ISSN 2703-8084 (online ver.) Doctoral theses at NTNU, 2021:173 Printed by NTNU Grafisk senter.

(5) Preface This thesis is submitted in partial fulfilment of the requirement for the degree of Philosophiae Doctor (PhD) in Structural Engineering at the Norwegian University of Science and Technology (NTNU). The work has been conducted at the Structural Impact Laboratory (SIMLab) at the Department of Structural Engineering, NTNU. The work was supervised by Professor Aase Reyes, Professor Arild Holm Clausen, and Professor Odd Sture Hopperstad. The thesis is written as a monograph and comprises two main parts, Part I and Part II. The author has been responsible for the drafting of this thesis, the experimental work, and the numerical work. The experiments were prepared and conducted in collaboration with NTNU lab technicians, a Master’s student, and summer interns. The scanning electron microscopy (SEM) images of the foam were taken at the Department of Materials Science and Engineering, NTNU with assistance from Mr. Sergey Khromov. The differential scanning calorimetry (DSC) and Fourier-transform infrared spectroscopy (FTIR) analyses were organized by Dr. Frode Grytten and conducted at SINTEF Materials and Nanotechnology.. Daniel Thor Morton Trondheim, Norway March 10, 2021. i.

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(7) Abstract As the emphasis on pedestrian safety in the automotive industry is increasing, polymer foams and their application in energy absorption and impact mitigation are growing in relevance. As the automotive industry moves towards virtual testing and design of new products, the accuracy of the numerical tools employed are a critical part of improving the safety of products and increasing the efficiency of the design process. The work in this thesis is twofold and covers characterization of the mechanical properties and micromechanical modeling of polymer foams. Firstly, we aim to better understand how polymer foams, and expanded polypropylene (EPP) in particular, respond to different loading scenarios and environmental conditions. This will help engineers to better utilize the material and ensure that a foam component will function as intended, within the design parameters. From extensive mechanical testing of EPP foam, we observe that the material has a strong temperature and rate dependence. The difference in transverse strain response during uniaxial tensile and compressive loading is also captured. Furthermore, combined compression and shear loading tests illustrate the complexity of the mechanical response when the material is subjected to non-proportional loading. Secondly, a micromechanical model has been developed to better understand the mechanical response of foam structures subjected to complex loading conditions. This facilitates a better understanding of the limitations in existing constitutive models and will help aid the development of future models that more accurately represent the mechanical behavior of the material. The suggested modeling framework captures the key features of the mechanical response observed in the experimental testing. This includes the difference between the uniaxial compression and tension stress-strain response, and the difference in transverse contraction. The influence of multiaxial and non-proportional loading is also captured by simulations. . iii.

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(9) Acknowledgements I would like to start by expressing gratitude to my supervisors, Professor Aase Reyes, Professor Arild Holm Clausen, and Professor Odd Sture Hopperstad. Thank you for the invaluable guidance, help, and motivation you have provided throughout the process of completing this research. Your dedication and contribution to the research group and University community is admirable and greatly appreciated. I would also like to recognize the support provided by the Center for Advanced Structural Analysis (CASA), a Center for Research-based Innovation at the Norwegian University of Science and Technology (NTNU), and the Department of Structural Engineering, NTNU, who have funded the project. The community at the SIMLab research group has been invaluable these past four years. I could not have asked for a better group of colleagues, both professionally and socially. Thank you. To the people with whom I have shared an office, thank you for the time we have spent together. I cherish the discussions we have had, and the help you have provided. I would also like to thank Mr. Trond Auestad, Mr. Tore Wisth, and others who have helped with producing specimens, instrumentation, and execution of the experimental program. Your assistance has been greatly appreciated. To my family, friends, and partner Senta, thank you for your support and encouragement. It means the world to me, and I am lucky to have you in my life.. v.

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(11) Contents Preface. i. Abstract. iii. Acknowledgements. v. Nomenclature. xi. Acronyms. xv. 1 Introduction 1.1 Background and motivation 1.2 Objectives . . . . . . . . . . 1.3 Scope . . . . . . . . . . . . 1.4 Public contributions . . . .. I. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. Characterization of EPP foam. 2 Literature review 2.1 Polymer foams . . . 2.2 Compression testing 2.3 Tension testing . . . 2.4 Shear testing . . . . 2.5 Triaxial testing . . .. 1 1 2 3 4. 5 . . . . .. 7 7 14 28 31 38. 3 Experimental studies 3.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Experimental program . . . . . . . . . . . . . . . . . . . . . . . .. 43 43 46. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. vii. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . ..

(12) Contents 3.3 3.4 3.5 3.6 3.7 3.8 3.9. Compression tests: Quasi-static behavior . . . . . . . Compression tests: Strain rate dependence . . . . . . Compression tests: Temperature dependence . . . . . Compression tests: Unloading and reloading behavior Tensile tests . . . . . . . . . . . . . . . . . . . . . . . Shear tests . . . . . . . . . . . . . . . . . . . . . . . . Three-point bending tests . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . 49 . 71 . 81 . 87 . 91 . 104 . 115. 4 Summary of Part I 123 4.1 Response overview . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2 Application of strain rate and temperature dependence . . . . . . 130. II. Modeling of foams. 137. 5 Constitutive modeling 5.1 Abaqus material models . . . . . . . . . . 5.2 LS-DYNA material models . . . . . . . . . 5.3 Comparison of LS-DYNA material models 5.4 Other foam models . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 139 140 147 151 159 162. 6 Micromechanical model 6.1 Background . . . . . . . . . . . . . . 6.2 Foam analysis and morphology . . . 6.3 Bulk material properties . . . . . . . 6.4 Micromechanical modeling framework. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 163 164 175 179 183. settings . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 201 201 208 215 228. . . . .. 7 Micromechanical model verification 7.1 Overview of simulations and simulation 7.2 Non-periodic model . . . . . . . . . . . 7.3 Periodic model . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . .. . . . .. . . . .. 8 Validation of micromechanical model with experimental data 229 8.1 Modeling of EPP . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.2 Modeling of PVC . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 viii.

(13) Contents 9 Virtual experiments 9.1 Density and material distribution 9.2 Internal friction . . . . . . . . . . 9.3 Transverse strain . . . . . . . . . 9.4 Multiaxial yielding . . . . . . . . 9.5 Non-proportional loading . . . . . 9.6 Summary . . . . . . . . . . . . . 10 Summary of Part II. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 261 263 268 270 271 277 279 281. 11 Conclusions 283 11.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 283 11.2 Suggestions for further work . . . . . . . . . . . . . . . . . . . . . 285. ix.

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(15) Nomenclature α. Shape parameter for Deshpande-Fleck yield surface. αiso. Shape parameter for Deshpande-Fleck yield surface with isotropic hardening (Abaqus). αvol. Shape parameter for Deshpande-Fleck yield surface with volumetric hardening (Abaqus). σ. Stress tensor. σ0. Deviatoric stress tensor. σ sk. Skeletal stress tensor. ε. Strain tensor. ε0. Deviatoric strain tensor. ∆L. Change in length. ε̇, ė. Strain rate. ε̇0 , ėe Reference strain rate γ. Engineering shear strain. σ̂sd. Normalized standard deviation. µ. coefficient of friction (COF). ν. Poisson’s ratio. ρ. Relative density xi.

(16) Nomenclature Avoid Average void area Deq. Average equivalent diameter (sphere). t. Average thickness. φ. Material distribution. ψ. Sphericity. ρ. Density. ρs. Density of solid. σ. True stress. σc,∗ el. Estimated elastic collapse stress. σc,∗ pl. Estimated plastic collapse stress. σe. von Mises equivalent stress. σH , sH Hydrostatic stress σy. Yield stress of foam. σc , sc Compressive collapse stress σD, df Deviatoric stress, Deshpande-Fleck σeq. Desphande-Fleck equivalent stress. σg , sg Gas stress contribution σH, df Hydrostatic stress, Deshpande-Fleck σS, df Axial stress, Deshpande-Fleck σsd. Standard deviation. σsk , ssk Skeletal stress σT, df Radial stress, Deshpande-Fleck σuss xii. Ultimate shear strength.

(17) Nomenclature σuts. Ultimate tensile strength. σys. Yield stress of solid. τ. Engineering shear stress. θ. Lode angle. ε. Logarithmic strain. εl. Longitudinal logarithmic strain. εt. Transverse logarithmic strain. εe, p. von Mises effective plastic strain. εeq, df Equivalent plastic strain, MAT154 (LS-DYNA) εeq, pl Equivalent plastic strain measure εe. von Mises effective strain. εv, p. Volumetric plastic strain. εv , ev Volumetric strain ξ. Normalized coordinate. A. Current cross-sectional area. A0. Undeformed cross-sectional area. Aobject Area of object Asphere Area of sphere Aeq. Equivalent cross-sectional area (strut). Cv. Cell volume. Deq. Equivalent diameter (sphere). E. Young’s modulus. e. Engineering strain xiii.

(18) Nomenclature E∗. Estimated Young’s modulus. EH. Hardening modulus. Es. Young’s modulus of solid. F. Force. k. Machine compliance. L. Length. L0. Undeformed length. n. number. p0. Original internal pressure. patm. Atmospheric pressure. pg. Gauge pressure. s. Engineering stress. T. Temperature. t. Thickness. Tg. Glass transition temperature. Tm. Melting temperature. u. Displacement. V. Volume. V0. Original volume. Ve. Volume of edge bulk material in foam. Vg. Gas volume. Vg0. Original gas volume. Vs. Volume of solid. Vt. Total volume of bulk material in foam. w. Shear displacement. xiv.

(19) Acronyms aPP. Atactic-polypropylene. ASTM American Society for Testing and Materials COF. Coefficient of friction. DIC. Digital image correlation. DSC. Differential scanning calorimetry. EPP. Expanded polypropylene. EPS. Expanded polystyrene. ESEM. Environmental scanning electron microscope. FE. Finite element. FEA. Finite element analysis. FTIR. Fourier-transform infrared spectroscopy. GUI. Graphical user interface. HDPE. High-density polyethylene. ISO. International Organization for Standardization. LDPE. Low-density polyethylene. LVDT. Linear voltage displacement transducer. xv.

(20) Acronyms MID. Material identifier. OEM. Original equipment manufacturer. PDF. Probability density function. PE. Polyethylene. PMI. Polymethacrylimide. PP. Polypropylene. PS. Polystyrene. PU. Polyurethane. PUR. Rigid polyurethane. PVC. Polyvinyl chloride. QS. Quasi-static. RVE. Representative volume element. SEM. Scanning electron microscopy. SHPB. Split-Hopkinson pressure bar. SID. Simulation identifier. TID. Tessellation identifier. UTM. Universal testing machine. XCMT X-ray computed microtomography XCT. X-ray computed tomography. XPS. Extruded polystyrene. xvi.

(21) 1. Introduction 1.1. Background and motivation. Understanding the mechanical properties of foams is an important component of efficiently using the materials for engineering purposes. The automotive industry, for example, utilizes foam in bumpers for mitigating pedestrian impact and this requires a thorough understanding of the mechanical behavior of the material. Polymer foams are considered good impact mitigators due to their unique stress-strain response in compression. This type of material permits large deformations at relatively low stresses, ideal for energy absorption. Interest in the mechanical response and numerical representation has risen due to the prevalence of digital product development, for example in crash simulations, where the material response needs to be accurately described. The use of numerical tools like finite element analysis (FEA) software is an increasingly important part of the product development cycle, for example in the design of car bumpers or bicycle helmets. Accurate application of such numerical tools not only depends on the mechanical behavior of the material, but also on the material model, the algorithms, and equations used to describe the response. For optimal results, the mechanical behavior must be understood and the appropriate material models must be used. To evaluate the efficacy of currently available material models for foams, and promote the development of improved models, we have characterized the mechanical response of expanded polypropylene (EPP) foam used by the automotive industry. Supplementing the mechanical testing, we have also developed a micromechanical modeling framework that allows us to better understand the behavior of the foams in a wider range of deformation states. This also allows us to investigate the different mechanisms defining the complex response. Part I, Characterization of EPP foam, covers the mechanical characterization of the material at hand. We conducted a comprehensive test series on EPP 1.

(22) 1.2. Objectives. Chapter 1. foam, in an attempt to characterize EPP behavior during different loading and environmental conditions. Components in, for example, a car, will experience a range of impact velocities and ambient temperatures. Samples from different foam bumpers have been tested in both compression and tension at different temperatures and with different strain rates, along with quasistatic bending and shear testing at room temperature. A review of different methods for testing foams in different ways is also presented in Chapter 2. Part II, Modeling of foams presents the modeling and simulation component of the thesis. The representation of the mechanical response of polymer foams is challenging due to its complex internal geometry, causing a highly non-linear response. It is difficult to conduct sufficient experimental tests to fully define the behavior for an arbitrary stress or strain state. The constitutive material models available today also ingest a limited amount of experimental data and infer the general response based on the assumptions of the individual models. Based on the characterized material from Part I, some of the existing constitutive material models available in the FEA software ANSYS®LS-DYNA® [1] have been evaluated, identifying pros and cons associated with specific loading conditions. Most of the evaluated models fall short when describing the combined compression and shear response. As research on constitutive modeling of foams is continuously ongoing, a brief overview of other efforts is also presented. By better understanding the response of foam subjected to complex loading cases, we can more accurately choose between the existing constitutive models, or evaluate the behavior of models under development. A micromechanical representation of foam has been developed by emulating a foam morphology and assigning suitable bulk material properties to the structure. This model can be used to expand the understanding of the mechanical behavior of calibrated models, or be used with generic parameters to improve the understanding of the fundamental deformation mechanisms of polymer foams. The modeling framework is presented in Chapter 6, along with background information on polymer foam morphology and other micromechanical modeling efforts.. 1.2. Objectives. The objective of this thesis is to improve the understanding of the mechanical response and modeling challenges of polymer foams. By combining extensive mechanical characterization and micromechanical modeling, we aim to broaden the understanding of the deformation mechanisms and response of polymer foams 2.

(23) Chapter 1. 1.3. Scope. and more effectively evaluate the efficacy of current and future constitutive models. Key research objectives for this thesis are as follows: • Review state-of-the-art methods for mechanical characterization of polymer foams. • Characterize the mechanical response of polymer foam for different loading conditions and environmental conditions like strain rate and temperature. • Evaluate the applicability of currently available constitutive material models for representing polymer foams. • Establish a micromechanical modeling framework for polymer foams, and use this to better understand the mechanical behavior. • Evaluate the multiaxial stress-strain response of the micromechanical model and the implications for constitutive modeling of foam.. 1.3. Scope. As is evident in the literature review of mechanical testing of polymer foams, Chapter 2, there are a wide variety of polymer foams, both in terms of bulk materials and manufacturing processes. In the interest of limiting the scope of the thesis, we imposed some restrictions: • The primary material emphasised in this thesis is expanded polypropylene (EPP) foam, a foam of primary interest to several of the research partners of the research center, CASA. EPP is characterized and modeled. • Modeling of polyvinyl chloride (PVC) is a secondary goal, as the multiaxial yield behavior is the foundation for the Deshpande and Fleck yield surface, a popular phenomenological yield surface for foams. • The mechanical tests are carried out at the Department of Structural Engineering, NTNU, using the available uniaxial test machines. • The numerical simulation work is conducted using the FEA software LSDYNA, and is limited to the quasi-static behavior. 3.

(24) 1.4. Public contributions. 1.4. Chapter 1. Public contributions. Part of the work in this thesis has been presented publicly, either in a peer-reviewed article or as a conference contribution. The code used for some of the analysis and processing has also been made publicly available on GitHub.. 1.4.1. Journal publications. 1. D. T. Morton, A. Reyes, A. H. Clausen, O. S. Hopperstad. Mechanical response of low density expanded polypropylene foams in compression and tension at different loading rates and temperatures. Materials Today Communications, 23:100917, 2020, DOI:10.1016/j.mtcomm.2020.10091. 1.4.2. Conference contributions. 1. D. T. Morton, A. Reyes, A. H. Clausen, O. S. Hopperstad. Experimental investigation of expanded polypropylene foam (Poster). In 17th International conference on Deformation, Yield and Fracture of Polymers - DYFP 2018, Kerkrade, Netherlands 25th-29th March 2018. 2. D. T. Morton, A. Reyes, A. H. Clausen, O. S. Hopperstad. Mechanical properties of expanded polypropylene foam at different temperatures. In 5th Cellular Materials - CellMAT 2018, Bad Staffelstein, Germany 24th-26th October 2018.. 1.4.3. Code. The Python code used for some of the numerical work is available through a public GitHub repository: github.com/DanielThorM. This includes • tessToPy: Python representation of Neper tessellations with regularization of periodic structures. • foamModeling: Python representation of the discretized foam structure with templates for LS-DYNA model generation and processing. • keywordGenerator: Python script for creating LS-DYNA .key files.. 4.

(25) Part I Characterization of EPP foam. 5.

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(27) 2. Literature review The accurate characterization of the mechanical response under different loading conditions is an important step in calibrating and successfully using numerical tools to predict the behavior of a component. Today, there exists a variety of different test standards which facilitate the comparison of different foams, but not necessarily comparison across different test setups. The tests’ main focus is typically on establishing some basic qualities, but not always describing the full stress-strain relationship. Relevant test standards can be found in both American Society for Testing and Materials (ASTM) and International Organization for Standardization (ISO) standards. Examples include ASTM D1621 [2] and ISO 844 [3] for finding the Compressive Properties of Rigid Cellular Plastics, among others. A review of different approaches used to test foams is presented in this chapter, where methods for characterizing the mechanical response of foams subjected to different loading conditions like compression, tension, shear, and more, are outlined. In each section, some of the applicable test methods and available standards are mentioned. An overview of the response and important consequences for applications using foams is also included for each loading case.. 2.1. Polymer foams. Polymer foams are a subset of cellular solids, a term, as stated by Gibson and Ashby [4], encompassing materials comprising an assembly of cells which are defined by solid edges or walls. Gibson and Ashby defined true cellular solids as materials with a relative density below 0.3. The relative density ρ̄ is defined as ρ=. ρ ρs. (2.1). where ρ and ρs are the density of foam and solid(bulk) material, respectively. Relative densities above 0.3 are better represented as a solid with isolated voids 7.

(28) 2.1. Polymer foams. Chapter 2. and are omitted from the scope of this report. Within cellular solids you find a wide specter of base materials including metal, wood, glass, and our material of interest; polymers. Within the realm of polymer foams, there is a variety of base materials such as polypropylene (PP), polystyrene (PS), and more. Depending on the properties of the base material and the manufacturing process, the internal cells of the foam are typically predominantly either open or closed, i.e., the internal surface of the foam is open to the surroundings or fully isolated. The different foams can also be rigid, or highly flexible, again depending on the base material and the manufacturing process. This section aims to give a broad overview of what polymer foams are, and can be used for. Different applications of foam are illustrated with examples which highlight the beneficial properties for each case. Expanded polypropylene (EPP) is emphasised as an example, with a brief introduction to the chemistry and manufacturing process.. 2.1.1. Applications. Polymer foams can today be found in a myriad of products and the scope is seemingly ever growing. Polymer foams are an integral part of people’s every day life. You will find them in bicycle helmets, running shoes, mattresses, car bumpers, cushions, building insulation, airplane wings, coffee cups, life jackets, and many other applications. In the following section, we will highlight a few of these applications and elaborate on why applications of polymer foams are of great importance. The common examples illustrated below fall into four major categories of applications as suggested by Gibson and Ashby [4], namely, thermal insulation, packaging, structural, and buoyancy. Here, we will consider packaging to encompass protective applications which include energy absorption. All applications mentioned depend on the mechanical properties. Energy absorption and strength of foam structures are directly dependent on the mechanical properties, while insulation and life jackets must withstand operational loads despite load bearing not being the primary function. Closed cell foams have the potential to be great insulators. Air itself is an efficient insulator and the closed cell structure inhibits air circulation and therefore limits heat transfer through the foam. To have a functional material, the foam must have the desired mechanical properties, e.g., be stiff and strong enough to be mounted to a structure. With a continuously expanding range of mechanical properties, it is also possible to consider creating structural components which 8.

(29) Chapter 2. 2.1. Polymer foams. are inherently good insulators. In this case, an accurate representation of the mechanical properties is crucial. A helmet’s primary purpose is absorbing energy from an impact and reducing the acceleration experienced by the brain [5]. A well-tailored foam material has a plateau region where the compressive stress is close to constant for a large amount of compressive strain, allowing a lower peak force and a lower, less harmful, acceleration. The role of polymer foam as a load-bearing material is primarily exemplified through the use of sandwich structures. A layer of denser and stronger material on the surface of a foam core results in a larger second area moment, allowing the stress to be distributed more efficiently across the cross section. This reduces the maximum stress and inclination to buckle while the weight is kept low. Life jackets provide flotation by having a lower relative density than water and being attached to a person. To provide this extra buoyancy, the foam needs to withstand the hydrostatic pressure imposed by submersion in addition to maintaining its shape during normal use.. 2.1.2. Overview of polymer foams. Polymer foams were first introduced in the 1920s with latex foams. Since then, methods for producing foams from many different polymers have emerged, among them phenolic, polyurethane (PU) and polystyrene (PS) [6] and later polyolefin foams including polyethylene (PE) and polypropylene (PP) [7]. Two types of internal foam structures are illustrated in Figure 2.2. The primary interest in the mechanical properties of polymeric foams is related to compressive stresses. Compression of polymeric foams is in general characterized by three stages, as seen in Figure 2.1. The small strain response is considered elastic and the stress is typically steadily increasing with strain. The following plateau region is where there are large strains but no significant increase in stress, and is often the result of a successive collapse of the cell structure as the strain increases. At large compressive strains, the stress will increase rapidly due to internal contact of cell walls caused by the large deformation. This is also referred to as densification. As the void volume of the material goes towards zero, the response will more closely resemble the bulk material properties. Beyond the bulk material, foams are often labeled as open or closed cell foams. The open cell foams are air permeable, and it will be the cell edges which govern the mechanical response. For closed cell foams, gas, typically air, is trapped within 9.

(30) Chapter 2. Engineering stress. 2.1. Polymer foams. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Engineering strain[mm/mm] Figure 2.1: Stress-strain response of foam in compression.. (a). (b). Figure 2.2: Examples of (a) PS and (b) EPP foam structures, both with nominal density ρ = 44 kg/m3 [8]1 .. the cell. When deformed, this trapped gas will contribute to the mechanical response of the foam if there is any volumetric deformation, until the pressure inside a given cell increases so much that the cell wall ruptures. Typical bulk materials are polypropylene (PP), polystyrene (PS), polyurethane (PU), polyethylene (PE), polymethacrylimide (PMI), and polyvinyl chloride (PVC). 1 Reprinted from Polymer Testing, Vol. 53, Cronin, D.S. and Ouellet, S., Low density polyethylene, expanded polystyrene and expanded polypropylene: Strain rate and size effects on mechanical properties, p 40-50, Copyright 2015, with permission from Elsevier.. 10.

(31) Chapter 2. 2.1. Polymer foams. Both rigid foams like PMI and PVC, and more flexible foams like PP and PU, can be tailored to suite a wide range of applications depending on the composition of the bulk material and the manufacturing method of the foam.. 2.1.3. Expanded polypropylene (EPP). As foams are seeing increased use across the board, the use of expanded polypropylene (EPP) is also expanding. Today you will find EPP in, for example, high end head protection and in the bumpers of cars. The main characteristics of the bulk material and the manufacturing process are covered in this section. The production of EPP starts with PP resin particles being mixed with a number of other chemicals and a volatile blowing agent dissolved in a liquid. The pressure of the mixture is higher than the vapor pressure of the blowing agent and the mixture is heated until the particles soften and the blowing agent is impregnated in the particles. A following reduction in pressure creates an internal overpressure, expanding the particles into pre-foamed PP beads [9]. The subsequent manufacturing process consists of filling the pre-foamed beads into a mold of the desired shape and then heating the particles, causing the beads to expand and bond together, filling the mold with a rigid part [10]. The heating is done by blowing steam into the mold [7]. The steam acts both as a heater and a blowing agent due to the water vapor penetrating the foam before condensing. The condensed water will then vaporize and expand upon subsequent cooling and reduction in pressure. The parts then have to be aged to dry out the water trapped within the beads [11]. The process is illustrated in Figure 2.3. The process of interbead bonding of PP beads has not been definitely established, but recent papers by Zhai et al. [12, 13] suggest a viable mechanism based on the crystal melting behavior. During the heating of the mold, parts of the crystalline structure are dissolved such that diffusion of amorphous polymer occurs at the interface of adjacent expanding beads. During cooling, part of the amorphous region recrystallizes across the bead boundaries. Zhai et al. [12, 13] also investigated how the molding temperature and pressure affect the strength of the interbead bonding. Cuboid samples were tested in tension to establish the ultimate tensile strength. A higher steam pressure makes the beads better conform and increases the intersecting area, exemplified by the bead boundaries becoming fuzzy and interbead voids not being observed. Low steam pressure molding of 2.0 bar gave primarily interbead fracture, whereas a higher steam pressure upwards of 3.0 bar shifted the fracture mode towards intrabead fracture. Interbead fracture refers to 11.

(32) 2.1. Polymer foams. Chapter 2. Figure 2.3: Process for producing an EPP part [14]2 .. failure between the beads, while intrabead fracture refers to failure through the beads. Another production effect worth noting is the higher density closer to the surface of a mold. Bouix et al. [15] have shown that the molded blocks they used had a drastically higher density close to the molding surface, but variations within the block were low.. 2.1.4. Polypropylene (PP). PP, the main component of EPP foam, is a semi-crystalline thermoplastic polymer. Thermoplastics are characterized by being remoldable as opposed to a thermoset which cross links with irreversible chemical bonds. A semi-crystalline polymer is composed of two phases, amorphous and crystalline. The amorphous phase can be described as an unorganized tangle of polymer chains which behaves like a glassy solid at temperatures below the glass transition (Tg ) and as either a viscous liquid or a rubbery solid above Tg , depending on the molecular weight. The crystalline phase, contrary to the amorphous phase, is a highly ordered structure where the polymer chains fold and orient themselves parallel to each other in lamellae. The crystalline structure remains at temperatures above Tg and up to the melting temperature Tm where the crystalline phase is dissolved. The combination of the two phases produces a ”leathery state” at temperatures between Tg and Tm [16]. The glass transition temperature for polypropylene is typically around −10 ◦C but varies with the quantity of comonomers (additives) as their presence prevents the polymer from fully crystallizing [17]. 2 Used with permission of John Wiley & Sons - Books, from Foamcore Blow-Molded Structural Components for Transportation Applications, Steven R. Sopher, Plastics engineering, Volume 71, Issue 9, Copyright 2015; permission conveyed through Copyright Clearance Center, Inc.. 12.

(33) Chapter 2. 2.1. Polymer foams. The production of PP can be accomplished by using either stereospecific ZieglerNata catalysts or metallocene catalysts which allow the controlled polymerization of the propylene monomer [17]. The details of the process are outside the scope of this thesis. The polymerization of propylene yields a chain −CH2 − CH(CH3 )− where the CH has a methylene(CH3 ) group attached. The orientation of these groups can be isotactic, syndiodactic, or atactic, where the methylene groups are attached on the same side, alternating sides, or on random sides of the main chain, respectively. In a semi-crystalline PP, the crystalline phase contains isotactic PP and the amorphous phase contains both isotactic and atactic PP. The isotactic portion of the amorphous phase can crystallize slowly over time, slightly altering the material’s properties [18]. A homopolymer PP contains only propylene, whereas copolymer PP contains a fraction of ethylene (or higher alkenes). The ethylene fraction reduces the tendency to crystallize, creating a less rigid polymer [17]. Depending on the grade of the PP (Homo-/random-/block copolymer), unmodified PP has a density of 898 - 908 kg/m3 . The Young’s modulus is between 900 and 1550 MPa with random and block copolymers being the softest. The yield stress ranges between 24 and 35 MPa, with random and block copolymers being the weakest. A typical PP composition has a maximum continuous use temperature of around 100 ◦C. The continuous use temperature is the ambient temperature which causes the tensile strength to decrease by half after 100,000 hours at the elevated temperature and without stresses [17]. Another relevant aspect is the brittle temperature of the polymer. The PP grade affects the brittle temperature, with homopolymers ranging from 5 ◦C to 15 ◦C, random copolymers −10 ◦C to 15 ◦C, and block copolymer −40 ◦C to 10 ◦C. Further information about PP properties can be found in ”Practical Guide to Polypropylene” by Tripathi [17].. 13.

(34) 2.2. Compression testing. 2.2. Chapter 2. Compression testing. Compression testing is a widely used method to characterize polymer foam response. This test is of particular interest as it best describes the response in the most common mode of deformation, i.e., impact mitigation and energy absorption. By successfully determining the stress-strain response of the foam it is possible to calibrate the material models used in finite element (FE) simulations. The following section outlines some of the currently available information about compression testing of polymer foams. The basic method of testing is outlined, along with alternative approaches for characterizing the compressive behavior. Some qualities like rate dependence and transverse expansion of different foams are also presented to give an overview of what response to expect. An overview of references for compression testing of different types of foams is included in Table 2.1. Table 2.1: References on compressive testing of different foams.. Material. Compression testing references. PU/PUR. [19–32]. PE/LDPE/HDPE. [8, 19, 33–36]. PP/EPP. [8, 15, 20, 21, 28, 33–35, 37–43]. PS/EPS/XPS. [8, 20, 21, 28, 34, 40, 41, 43–46]. PVC. [47–52]. PMI. [49, 53–55]. Before presenting the background literature on mechanical characterisation, some common definitions used throughout this thesis are introduced. Some literature might use slightly different nomenclature or definitions, but any referenced results have been adapted to the presented definitions (e.g. sign of compressive collapse stress, σc ), if possible. The engineering strain, e, is defined as e=. ∆L , L0. (2.2). where ∆L is the change in sample length and L0 is the original length. The change 14.

(35) Chapter 2. 2.2. Compression testing. in sample length is defined to be negative in compression. The logarithmic strain is given by L = ln(1 + e) (2.3) ε = ln L0 Both strain measures assume homogeneous deformation over the initial distance encompassed by L0 , resulting in av averaged strain measure. Foams tested in compression typically exhibit some localization, as will be discussed later. The initial strain rate of a test is defined as ε̇ = ė =. L̇ , L0. ∆L 1. (2.4). where the dot denotes the time derivative of a variable. For small strains, the true strain rate and engineering strain rate are equal. The engineering stress, s is defined as s=. F , A0. (2.5). where F is the force on the sample and A0 is the initial cross-sectional area. The force on the sample is defined to be negative in compression. The true stress or Cauchy stress, σ, is given by F (2.6) σ= , A where A is the current cross-sectional area corresponding to each load increment. The stress-strain response in compression is the most commonly presented data from a compression test. This will typically be plotted in the first quadrant, using the absolute values of both the stress and strain. A representative stress-strain curve is seen in Figure 2.4. An initially linear elastic region is followed by a plateau where the slope of the stress-strain curve is smaller than in the initial region. The plateau is followed by a rapid increase in stiffness, also referred to as densification. These three stages correspond to elastic deformation of the foam structure, local collapse and buckling, and the onset of internal self-contact. Notably, foams exhibit low transverse expansion during compression, and the difference between the true and engineering stress is often assumed negligible. From the stress-strain response it is possible to extract information describing the response in general terms. Common parameters for quantifying the compressive response are shown in Figure 2.5. The compressive Young’s modulus, or elastic 15.

(36) 2.2. Compression testing. Chapter 2. Engineering stress, |s|, [MPa]. 0.4. 0.3. 0.2. 0.1. 0.0 0.00. 0.25 0.50 0.75 1.00 Logarithmic strain, |ε|. 1.25. Figure 2.4: Example of compressive response for EPP foam.. modulus E, is the slope of the graph in the initial linear region, defined by E=. dσ , dε. |ε| 1 and/or. d2 σ =0 dε2. (2.7). The hardening modulus, EH , is the slope of the stress-strain curve in the inflection point of the plateau region and is defined by EH =. dσ , dε. d2 σ = 0, dε2. |ε| 0. (2.8). The hardening modulus is primarily used when determining the collapse stress of the foam. The collapse stress, σc is the true stress is used or sc if the engineering stress is used, is typically considered the largest stress of either the local maximum following the linear region [4] for rigid/brittle foams, or the intersection point of the tangent to the linear region and the tangent of the plateau region [15], for flexible foams, as seen in Figure 2.5. This value is also referred to as collapse initiation stress [56], yield stress [57], plastic collapse stress, or elastic buckling stress [4]. One important factor controlling the mechanical properties of foams is the density, ρ, most commonly given in kg/m3 . The density of the foams referenced in 16.

(37) Chapter 2. 2.2. Compression testing. Engineering stress, |s|, [MPa]. 0.2. EH. σc 0.1. 0.0 0.0. E Flexible Rigid 0.2 0.4 Logarithmic strain, |ε|. 0.6. Figure 2.5: Common parameters describing the compressive response of foam.. specific papers might be included in parentheses following the material description, e.g., EPP (35 kg/m3 ), in this thesis. This property provides important information about the foam, as it highly influences the material response.. 2.2.1. Quasi-static testing methods. The methods employed for quasi-static testing are principally identical across the board, with a testing machine comprising one fixed surface and one moving, parallel surface which compresses the foam sample. The compressing surfaces are also called compression plates, and the moving element is commonly referred to as the crosshead. An illustration of this setup is seen in Figure 2.6. These tests are often conducted with a constant cross-head velocity, meaning that the true strain rate changes throughout the test. The testing speed is therefore commonly described by the initial strain rate, described by Equation (2.4) The compressive response is typically tested at low, quasi-static strain rates at room temperature. Higher strain rates are also tested, but may require more complex machines and instrumentation. The initial strain rate, defined in Equation (2.4), is typically 10−5 s−1 to 10−1 s−1 for quasistatic testing. At quasi-static strain rates the effects of inertia 17.

(38) 2.2. Compression testing. Chapter 2. F. L0. Cross head platen. ∆L Foam Rigid base. Figure 2.6: Illustration of compression of a foam sample.. and self-heating are negligible The main differences between the methods are found in the specimen geometry, with samples typically being either cuboids or cylinders, and the method of determining the compressive strain. The three most common methods for determining the compressive strain are cross-head displacement, extensometers, and digital image correlation (DIC). The majority of tests in the literature report the engineering stress-strain curve resulting from the cross-head displacement and load cell. The cross-head displacement yields an accurate global strain measure for the particular sample. The force from the load cell combined with the original cross-sectional area gives a reasonably accurate stress measure due to foam’s low transverse expansion during compression. More involved methods of deformation measuring, like DIC, have been employed to characterize the full strain field of a sample. For example, the deformation of a typical low-density foam tends to localize as the compressive strain increases [22], and the low transverse expansion at large strains can be quantified [26]. To facilitate the comparison of results from different labs when testing foams, several test standards have been established. The two most common sources for standard testing methods are American Society for Testing and Materials (ASTM) and International Organization for Standardization (ISO). For compression testing, both ASTM and ISO provide several different standards applicable to different subsets of foam, for example, ASTM D1621 [2] and ISO 844 [3] for finding the compressive properties of rigid cellular plastics, or ASTM D3574 [58] and ISO 18.

(39) Chapter 2. 2.2. Compression testing. 3386 [59] for compressive properties of cellular plastics. The sample dimensions required in these standards vary, but the cross-sectional area is typically allowed to be between 25 cm2 and 230 cm2 and the sample must be shorter than the smallest cross-sectional dimension. None of the standards, at the time of review, refers to full-field measuring techniques like DIC. Testing not strictly following a standard is also common. The most typical deviation is related to sample size, often restricted by the size or quantity of the supplied material. The method of determining the strain might also be different. Calibration of FE material models, for example, often requires complete stressstrain curves, whereas the standards often only define key parameters like Young’s modulus and yield stress. In addition to using the cross-head displacement to establish the global strain, DIC has also been used to determine the local strain field in some studies [22, 26, 40, 42, 52]. This data can be used to determine the deformation behavior like localization or transverse strain from which the true stress can be established. An example of the local strain field during a compressive test is seen in Figure 2.7. Here, one can see how the strain localizes at several different places on the surface of the specimen. Dai et al. [22] found that in uniaxial compression, a higher density foam (PU 53 kg/m3 ) did not exhibit any localization of strain, which is reflected in a shorter plateau region than the lower density foam (PU 14 kg/m3 ). The lower density foam developed a local band of higher strain in the middle of the sample, which resulted in a clear stress plateau. Another interesting approach is X-ray computed microtomography (XCMT) which was used by Roux et al. [60] to determine the 3D strain field of an EPP foam sample. They do not report a general foam density but suggest that the method is suitable for determining the strain heterogeneity in a sample. In situ environmental scanning electron microscope (ESEM) imaging of quasistatic compression of PMI foam has also been used to find the micromechanical mode of deformation [61]. The surface of the sample was imaged while compressed to a given strain. Results from this approach indicate that low density PMI foam deforms by elastic buckling of cell walls, while denser samples deform by plastic bending of the cell walls.. 2.2.2. Dynamic testing methods. Dynamic testing of polymer foams is used to determine the rate dependence at intermediate to high strain rates. This rate dependence is relevant when modeling 19.

(40) 2.2. Compression testing. Chapter 2. Figure 2.7: Example of strain localization during compressive deformation of EPP foam [42].. high speed impacts, and testing is typically done in uniaxial compression. A comprehensive review of the dynamic compressive behavior of cellular materials is presented by Sun and Li [56]. Intermediate strain rate (10−1 s−1 to 102 s−1 ) compression testing can be accomplished using, e.g., a drop tower, a flywheel, or a pendulum apparatus. At these strain rates, inertia and self-heating could start affecting the response, depending on the material. The three test methods are illustrated in Figure 2.8 Drop tower testing is a principally simple method of dynamic testing. An elevated mass on vertical guides is released from a predetermined height above the sample. The sample is then initially overwhelmed by the falling mass, allowing a near constant compression velocity for a portion of the compression. The force can be calculated from an accelerometer on the falling mass, or a load cell on either 20.

(41) Chapter 2. 2.2. Compression testing. (a). (b). (c). Figure 2.8: Machines for intermediate strain rate testing; (a) drop tower, (b) flywheel, and (c) pendulum. S indicates the sample location and F indicates potential force transducer or accelerometer location.. the impactor or the base. The displacement can be measured using, for example, a laser displacement meter. Ouellet et al. [34] describe a setup used to characterize EPS (60 kg/m3 to 112 kg/m3 ), PE (80 kg/m3 to 110 kg/m3 ), and PU (128 kg/m3 to 192 kg/m3 ) foams at strain rates between 101 s−1 to 102 s−1 . The force was determined using an accelerometer attached to an impactor with a mass of 7 kg. The displacement of the falling plate was measured using an optical displacement system. They include the entire compressive strain region in their measurement and due to decreasing strain rate towards the final strain, they report each curve with an averaged strain rate. A flywheel is another method of compressing samples at intermediate strain rates. The setup comprises a flywheel and a mechanism allowing instantaneous transfer of energy from a rotating flywheel to a translating punch or platen used to compress the sample. The large rotational energy of the flywheel allows a constant compression velocity until the desired strain is reached. An example of this setup is reported by Bouix et al. [15]. With this setup it is possible to load the samples at strain rates between 50 s−1 to 800 s−1 . The force exerted on the sample is measured by a piezo-electric force sensor located at the stationary side of the sample, and the displacement of the punch is measured using a laser displacement meter. A pendulum setup utilizes the potential energy of an elevated mass swinging in an arc. Both the force on the sample and displacement of the pendulum head can be measured in the same manner as with the drop tower. Cronin and Ouellet [8] used a pendulum setup to investigate sample size dependence of PE (45 kg/m3 to 70 kg/m3 ), EPS (56 kg/m3 to 70 kg/m3 ) and EPP (56 kg/m3 to 70 kg/m3 ) foams 21.

(42) 2.2. Compression testing. Chapter 2. at different strain rates. Here, the load was measured using a Quartz Load Washer and the displacement of the pendulum using a laser displacement system. The setup was used to load the samples at a strain rate of approximately 102 s−1 . To achieve higher strain rates (102 s−1 to 104 s−1 ), it is possible to use, e.g., a launched projectile or a Split-Hopkinson bar. A launched projectile functions much like the drop tower test but allows for a larger rate of deformation, and can be accelerated using, for example, compressed gas or gunpowder. The launched projectile with or without the sample strikes a stationary plate, with or without a sample, supported by a load cell. The deformation can be tracked by a high speed camera, which also could allow for detailed strain measurements if used together with DIC. An example of this setup is presented by Koohbor et al. [62]. They used a shock tube to accelerate a projectile with a mass of 70 g to impact and characterize the high strain rate response of dense PU foam (560 kg/m3 ; this material falls outside the definition of a cellular material suggested by Gibson and Ashby). The force on the sample was logged by three sensors placed in a triangle behind the sample, such that any off-axis loading would be captured. The sample was coated by a speckle pattern and a high speed camera imaged the deformation. The displacements and strains were determined using DIC. The split-Hopkinson pressure bar (SHPB) is another method of achieving high strain rates. A long bar is impacted by a striker, causing a stress wave. The incident stress wave travels along the bar before reaching the interface with the sample. Here, the wave splits into a reflected and a transmitted wave as there is an impedance mismatch between the bar and the sample. The stress that propagates through the sample is transferred to the transmitted bar behind the sample. By measuring the strain in the bar throughout the propagation of the shock wave, the magnitude of the three different waves can be determined and the stress in the sample calculated. An example of a SHPB is seen in Ouellet et al. [34] where they subject EPS, PU, and PE foams to strain rates between 500 s−1 to 2500 s−1 . Cylindrical samples with a thickness of 3 mm to 5 mm were selected as this allowed for uniform deformation. Diameters between 10 mm to 20 mm were tested but 12.7 mm was used for most tests. The setup comprised a 2.44 m long acrylic incident and transmitted bars, both with a 24.5 mm diameter. The striker was 0.71 m long and propelled by compressed gas. The specimen deformation was determined from images captured by high-speed camera. A more applicable bar material for the SHPB setup is suggested by Bouix et al. [15] who conducted high strain rate tests on EPP using bars of Nylon PA6. This combination has a lower 22.

(43) Chapter 2. 2.2. Compression testing. impedance mismatch allowing for a better signal. It is noted that it is difficult to interpret the stress in the initial compression region (ε < 0.1) for an SHPB test due to the large variation in strain rate and lack of equilibrium in the sample. Song and Chen [63] and Zhao and Gary [64] have reviewed these aspects in further detail. The problem of equilibrium in samples under high strain rates has also been addressed by Koohbor et al. [62] who tested rigid closed cell foam using a projectile and shock tube. The full strain field during rapid deformation was found to be heterogeneous, hence the inertia forces need to be included in the analysis. This was, however, a very dense foam and the influence of inertia is presumably less for the lower density foams most commonly found in the literature.. 2.2.3. Foam behavior. Typical qualities of foam subjected to compression are presented below. This is intended to provide an overview of qualities to keep in mind when working with or evaluating different foams. Depending on the bulk material, the compressive behavior past the collapse stress can be closer to a rigid, or brittle, behavior on one side, and flexible on the other. Two stress-strain curves of XPS and EPP are seen in Figure 2.9. The XPS response exhibits sharp peaks in the stress-strain response, indicating a brittle failure. The EPP response is smoother, indicating a more elastic or ductile response in the loading phase. EPP also generally seems to have a higher degree of recovery after unloading compared to EPS, a type of PS foam [8]. Within foams of similar bulk materials, one of the main parameters defining the response, e.g., collapse stress and Young’s modulus, is the density of the foam ρ. The properties of different densities of EPP are seen in Figure 2.10. Similar trends are seen for EPS [43] and the density dependence is relevant for most types of foams [4]. Testing of different foams at higher strain rates emphasises that the high strain rate dependency seen in bulk polymer is also seen in polymer foams. At strain rates between 10−2 s−1 to 102 s−1 , the collapse stress of EPS, PE, and PU foams is found to have a linear dependence on the logarithm of the strain rate [57]. Here, EPS foam had a 50% increase in collapse stress over the tested range, PE foam had a 167% increase, and PU foam had a 60% increase. Testing at lower strain rates, 4 × 10−3 s−1 to 5 s−1 , indicates a low strain rate dependency for PS foam, medium dependency for PP foam, and a very high dependency for 23.

(44) 2.2. Compression testing. Chapter 2. 3.0 XPS, 45 kg/m3 EPP, 50 kg/m3. True stress, |σ|, [MPa]. 2.5 2.0 1.5 1.0 0.5 0.0 0.0. 0.5 1.0 1.5 2.0 Logarithmic strain, |ε|. 2.5. Figure 2.9: Example of stress-strain curve XPS and EPP in compression [40].. PU foam, which at low strain rates only retains a fraction of its compressive strength [20, 21]. Higher strain rates show a non-linear dependency on the logarithm of the strain rate for low-density polyethylene (LDPE) and EPS foams. There appears to be a lower dependency in the range between 10−2 s−1 to 102 s−1 before a sharp increase in collapse stress for strain rates above 102 s−1 [34, 65]. Further testing on EPS and EPP foam by Cronin and Ouellet [8] shows a significant increase in stress at strain rates of 102 s−1 , which means that this study finds a hardening effect for strain rates a decade lower than the findings presented by Ouellet et al. [34, 65]. Some of the results from the findings of Cronin and Ouellet [8] and Ouellet et al. [34, 65] are presented in Figure 2.11, where some of the data has been aggregated for comparison. The strain rate dependence of EPS foam is corroborated by Chen et al. [44] where there is a significant increase in strain rate dependence above 102 s−1 . This effect is not seen for LDPE which has a low dependence on the strain rate even for a strain rate of 102 s−1 . Similar to the results mentioned above, EPP foam also exhibits a significant increase in collapse stress for strain rates above 2 × 102 s−1 , as investigated by Bouix et al. [15]. Here, the plateau stress modulus was found to be strain rate dependent for larger densities and it is hypothesized that micro-inertia effects of 24.

(45) Chapter 2. 2.2. Compression testing 25 Collapse stress Young’s modulus. 0.8. 20. 0.6. 15. 0.4. 10. 0.2. 5. 0.0 0. 50 100 Density, ρ, [kg/m3 ]. Young’s modulus, E, [MPa]. Collapse stress, σc , [MPa]. 1.0. 0 150. Figure 2.10: Collapse stress and Young’s modulus of EPP as a function of density [43].. thicker cell walls cause the increased strength at higher strains, as the micro-inertia counters buckling. Similar strain rate dependencies are often seen in bulk polymers used in foams, such as PP [66], PE [67] and others [68], and are frequently referred to as bilinear with regard to the logarithm of the strain rate. This will be further elaborated on in Part II, Section 6.3. Poisson’s ratio ν relates to the expansion or contraction of a material transversely to a longitudinal deformation and is defined as ν=−. εt εl. (2.9). where εt is the transverse strain and εl is the longitudinal logarithmic strain [69]. This definition applies to infinitesimal strains, where the engineering and logarithmic strain are approximately equal. For transverse contraction at larger strains, particularly for non-linear behavior, a more detailed description is suggested by Smith et al. [70] as the Poisson’s function, ν(ε). For small increments in strain, ν(ε) is given by dεt ν(εl ) = − (2.10) dεl 25.

(46) 2.2. Compression testing. Chapter 2. 3.5 EPS [8] (ρ = 35 kg/m3 ) Eng. stress, |s|, [MPa]. 3.0 2.5. EPS [8] (ρ = 44 kg/m3 ) EPS [34] (ρ = 61 kg/m3 ). 2.0 1.5 1.0 0.5 0.0 10−3. 10−1 101 103 Average strain rate, |ė|, [s−1 ]. Figure 2.11: Rate dependence of EPS, reported stress at compressive engineering strain |e| = 0.5, as investigated by Cronin et al. [8] and Ouellet et al. [34].. Generally, it is seen that Poisson’s ratio during elastic compression of foams is large (0.2-0.3) while it approaches zero when loaded beyond the initial elastic region. Poisson’s ratio in the plateau region of the compression response is often referred to as plastic Poisson’s ratio [25, 71]. Using DIC it is possible to get a detailed view of the evolution of the transverse expansion. Pierron [26] evaluated several approaches for determining Poisson’s function, Equation (2.10), for PU foam (30 kg/m3 ) in compression. The averaged transverse expansion at each increment, adapted from Pierron, is seen in Figure 2.12. Poisson’s ratio of other foams includes ν = 0.225 for EPS (60 kg/m3 ) and ν = 0.175 for EPS (120 kg/m3 ) [45], ν ≈ 0.33 for PU (50 kg/m3 ) [71], and ν = 0.35 for PVC (100 kg/m3 ) [35]. Isotropy and homogeneity are important features when trying to represent foam behavior accurately. Anisotropy is typically arising in extruded foams as they tend to expand primarily in the vertical direction relative to a horizontal extrusion plane. Inhomogeneity typically manifests itself through either differences in morphology or local variations in density. Generally, extruded foams are anisotropic because they are allowed to rise freely in one direction. Expanded bead foams, on the other hand, are close to 26.

(47) Chapter 2. 2.2. Compression testing 0.7 PU, 30 kg/m3. 0.6 0.5. t − dε dεl. 0.4 0.3 0.2 0.1 0.0 −0.1. 0.0. 0.5 1.0 Logarithmic strain, |εl |. 1.5. Figure 2.12: Poisson’s function for PU foam (30 kg/m3 ) [26].. isotropic due to molding with hydrostatic pressure. PVC foam is found to be anisotropic with regard to the stiffness, collapse stress, and tensile failure strength, which are typically higher parallel to the rise direction than the rise direction [35, 47, 48]. Extruded PU, PE [22, 72] and XPS [33, 40] foams have also been found to be anisotropic, particularly if the component surface is included. Expanded bead foams typically have less anisotropy as they are molded under hydrostatic pressure. EPP and one example of LDPE are close to isotropic if the samples are extracted at a reasonable distance from the mold surface [15, 33]. This is also the case for EPS [33, 72], and PMI [61]. Testing of foams might be influenced by size effects arising from, e.g., inhomogenity or large internal cell size relative to the sample size. Ouellet et al. [65] studied the size effect of LDPE and EPS samples under quasi-static loading and in the Split-Hopkinson bar, and found the size effect of EPS to be significant. It is noted that the density scatter for the EPS samples was larger than for LDPE. The scatter in the quasi-static compression response was, however, not linked to the variation in density, and they suggest that the response also might be affected by foam uniformity and cell integrity at the cut surface. Cronin et al. [8] investigated the size effect of EPP, EPS, and LDPE foams in uniaxial compression. EPS appears to be highly dependent on the sample size, 27.

(48) 2.3. Tension testing. Chapter 2. with a significant increase (2×) in strength across the range of sample sizes of 10 mm, 17 mm, and 35 mm. Sample size had a limited impact on the EPP samples and there was almost no effect on the LDPE samples. This is consistent with the findings by Ouellet et al. [65].. 2.3. Tension testing. Compared to compression testing, tension testing of polymer foams has not received much attention. This is expected, as tensile failure is not a commonly desired failure mode, and compressive testing is easier to perform. In sandwich structures, it is usually shear failure or core-sheet delamination that causes failure, while the other common application of foam is energy absorption where compressive failure is desired. Nevertheless, the tensile response is important for accurate modeling, especially if fracture is relevant for the application. An overview of references for tension testing of different types of foams is included in Table 2.2. Table 2.2: References on tension testing of different foams.. Material. Tension testing references. PU/PUR. [20–23, 32, 73]. PE/LDPE/HDPE. [74]. PP/EPP. [5, 42, 75]. PS/EPS/XPS. [44, 76, 77]. PVC. [35, 47, 48, 50, 52, 78–81]. PMI. [55, 61, 82]. The tensile response of foam typically differs significantly from the compressive response. A typical tensile stress-strain curve is seen in Figure 2.13. There is no pronounced collapse as one would see in compression, e.g., Figure 2.4. The typical quantities used to describe the tensile response of foam are Young’s modulus E and the ultimate failure strength, σuts . Young’s modulus in tension is given by Equation (2.7), presented in Section 2.2 about compression testing. The ultimate tensile strength σuts is defined as the maximum true stress seen in the 28.

(49) Chapter 2. 2.3. Tension testing 0.8. True stress, |σ|, [MPa]. σuts. 0.4. 0.2 E 0.0 0.0. 0.1 0.2 Logarithmic strain, ε. 0.3. Figure 2.13: Example of tensile response and common parameters describing the tensile response for EPP.. test. An illustration of Young’s modulus E and the ultimate tensile strength σuts for a tension test is given in Figure 2.13.. 2.3.1. Quasi-static testing methods. Tension testing has a larger variety in boundary conditions and test protocols compared to compression testing. In addition to different specimen geometries and methods of determining the strain, there are several types of fixtures, e.g., bonding of a sample to fixtures, illustrated in Figure 2.14(a), or mechanically restraining the sample, illustrated in Figure 2.14(b). The tensile fixture of foam specimens, both bonding and mechanical, is challenging due to a wide variety of base materials and the compressible nature of cellular materials. Most tensile testing is conducted at quasi-static strain rates. There is a variety of ASTM and ISO standards, along with other original testing procedures found in articles. Most of the reviewed literature utilizes one of the standard methods as a basis, although some articles lack a definitive description of how the stresses and strains are determined. Typically, the tensile strain appears to be calculated from either the machine displacement or an attached extensometer. These are commonly only used in the longitudinal direction. The stress is again found by 29.

(50) 2.3. Tension testing F. Chapter 2 F. Cross head Cross head Foam clamps Foam Base clamps Adhesive Rigid base (a). (b). Figure 2.14: Examples of tension testing fixtures; (a) Bonding of sample used by, for example, ASTM D1623 [83] and ASTM C297 [84], and (b) Mechanically restraining the sample, used by ASTM D3574 [58], D3575 [85] and ISO 1798 [86].. dividing the load on the sample by a reference cross-sectional area. The different standards employed for tension testing of foams can be roughly grouped into flexible cellular solids (ASTM D3574 [58], ASTM D3575 [85] and ISO 1798 [86]), rigid cellular solids (ISO 1926 [87], ASTM D1623 [83] and ASTM C297 [84]) and plastics (ASTM D638 [88] and ISO 527 [89, 90]). These tests typically either mechanically attach samples, for example, by clamping, or bond fixtures to the samples. The sample shape varies, but is often cuboid or dumbbellshaped. A method differing from the ASTM and ISO standards was presented by Deshpande and Fleck [47]. Dumbbell-shaped samples with flanges on the ends were bolted to the rig on either end. The global displacement in the axial direction was used to estimate the strain. It is not further specified how the strain was calculated. One example of using DIC to calculate the strains in a tensile test is found in Kidane [80]. PVC foam (150 kg/m3 to 176 kg/m3 ) was tested in an electromechanical testing machine. A dog-bone specimen with a gauge length of 134 mm was painted white with a random black speckle pattern.. 2.3.2. Dynamic testing methods. There appears to be limited work done at intermediate or higher strain rates. Testing of EPS by Chen et al. [44] at strain rates 10−3 s−1 to 3 × 101 s−1 is one 30.

(51) Chapter 2. 2.4. Shear testing. example. The intermediate strain rate was achieved by accelerating the cross head to the desired speed before clamping onto the previously free end of the sample. This allowed for a near instantaneous application of the desired strain rate. The sample dumbbell shape adhered to ASTM D638. There is little additional information on how the strain and strain rates are calculated and only limited results are provided.. 2.3.3. Foam behavior. The main difference from the compressive response is the lack of collapse stress, and a larger transverse contraction. PU foam, for example, has been found to have an initial apparent Poisson’s ratio, defined by Equation (2.9) in Section 2.2.3, between 0.3 and 0.4, before increasing to 0.6-0.8 for engineering strains approaching 0.4 [73]. Investigating the multiaxial yield behavior of PVC foam (100 kg/m3 ), Shafiq et al. [35] found the Poisson’s ratio to be around 0.35 in compression and around 0.4 in tension. It is assumed that these values are extracted in the initial phase of loading.. 2.4. Shear testing. In addition to the compressive and tensile response of foam, the shear response is also important for and accurate representation of the response. In helmets, for example, the rotational acceleration from an impact has a high damage potential [91]. The shear or combined compression and shear response of a helmet foam is therefore important to understand. This section outlines some of the methods for determining the shear response of foam and some of the resulting findings. An overview of references for shear testing of different types of foams is included in Table 2.3. Several methods and standards for determining the shear properties of cellular foams can be used. The methods can typically be grouped into either simple shear or bending tests. Rigid foams like PMI are typically tested in bending, while less rigid foams like EPP and PU are more commonly tested in simple shear. Some examples of combined compression and shear testing are also included in this section, as they are also highly relevant in capturing the response to a likely loading mode. 31.

(52) 2.4. Shear testing. Chapter 2. Table 2.3: References on shear testing of different foams.. Material. Shear testing references. PU. [22, 32]. PE/LDPE/HDPE. [74]. PP/EPP. [5, 92, 93]. EPS/XPS/PS. [20, 21, 77]. PVC. [47, 50, 52, 79, 94, 95]. PMI. [54, 61, 82, 96, 97]. The magnitude of shear deformation is given by the shear strain. The engineering shear strain γ is given by γ=. w L0. (2.11). where w is the deformation normal to the height L0 of the sample. The corresponding engineering shear stress τ is given by τ=. F A0. (2.12). where F is the applied force and A is the cross-sectional area normal to the height. An illustration of simple shear deformation is seen in Figure 2.15. A representative stress-strain curve for a double lap shear test is seen in Figure 2.16. From this response, the shear modulus, G and the ultimate shear strength τuss can be estimated. The shear modulus G is defined as G=. dτ dγ. (2.13). and is the slope of the initial part of the stress-strain curve, as seen in Figure 2.16. The ultimate shear strength τuss is the maximal shear stress before the sample can withstand, but could also be defined at the first sign of failure. 32.

(53) Chapter 2. 2.4. Shear testing w. τ. L0. Figure 2.15: Illustration of generic simple shear response.. 2.4.1. Shear testing methods. Examples of test setups used to determine the shear properties of foam, for example by approximating simple shear, are seen in Figure 2.17. The test methods and applicable standards are described below. Three of the standards covering the shear strength of cellular materials are ASTM C273 [98], ASTM C393 [99] and ISO 1922 [100]. ASTM C273 and ISO 1922, usually referred to as block shear tests, are similar in concept. ASTM C273 obtains the shear stress-strain relationship by measuring the force and relative displacement of two plates bonded to the sample. The loading direction is diagonal across the specimen as seen in Figure 2.17(b). The ISO 1922 standard transfers the load through the centerline of the sample, parallel to the loading plates, as seen in Figure 2.17(a). ASTM C393 is a three- or four-point bending test of a sandwich structure, illustrated in Figure 2.17(f). The two block shear test methods were evaluated by O’Connor [101], who conducted finite element analysis (FEA) and comparative experimental work using ASTM C273 and BS4370 (similar to ISO1922). Given stiff shear fixtures, and measuring the relative displacement of the plates rather than the cross-head displacement, the two methods provided indistinguishable results. The FEA showed significant stress concentrations, but the effect of this is not further elaborated. It is mentioned that these methods mainly are used for comparative evaluation for quality control purposes and hence their accuracy is less important than their precision. These tests do not prevent transverse strain, which results in a change in thickness during testing. The four-point bending test described in ASTM C393 was evaluated by Fathi et al. [94], who used DIC to quantify the shear strain distribution. They 33.

(54) 2.4. Shear testing. Chapter 2 0.3. Shear stress, |τ | [Mpa]. τuss 0.2. 0.1. G 0.0 0.0. 0.1 0.2 0.3 Engineering shear strain, |γ|. 0.4. Figure 2.16: Example of shear response and common parameters for EPP.. found that the four-point bending test slightly underestimated the shear modulus and slightly overestimated the shear strength relative to the ASTM C273 block shear test. They note that the block shear test might not be able to capture the critical strains at failure, as the strain measure is based on the global displacement. The double lap shear test, seen in Figure 2.17(e), is another method for determining the shear response. This configuration will prevent compressive or tensile strains normal to the loading direction which the block shear tests are susceptible to. This setup has been used in several studies [5, 77, 102, 103]. The Arcan test [104], often used to determine the shear properties of metals and composites, may also be used for foams. A specimen can either be made completely from the material, or the material can be bonded between a two-piece fixture, as illustrated in Figure 2.17(h). The stress is determined by the smallest cross section and the load on the fixture. The shear strain can be determined using either a strain gauge or DIC [95]. This method has been used on PVC foam by Deshpande et al. [47] and was found to be in reasonable agreement with the double lap shear test. A picture frame shear test is another alternative for replicating pure shear. This fixture is illustrated in Figure 2.17(g). The two diagonal corners of a hinged system resembling a picture frame pulled apart. The extended picture-frame shear 34.

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